Structure.

Cardiolipin (CL) is a kind of diphosphatidylglycerol lipid. Two phosphatidic acid moieties connect with a glycerol backbone in the center to form a dimeric structure. So it has four alkyl groups and potentially carries two negative charges. As there are four distinct alkyl chains in cardiolipin, the potential for complexity of this molecule species is enormous. However, in most animal tissues, cardiolipin contains 18-carbon fatty alkyl chains with 2 unsaturated bonds on each of them. It has been proposed that the (18:2)4 acyl chain configuration is an important structural requirement for the high affinity of CL to inner membrane proteins in mammalian mitochondria. However, studies with isolated enzyme preparations indicate that its importance may vary depending on the protein examined.  Since there are two phosphates in the molecule, each of them can catch one proton. Although it has a symmetric structure, ionizing one phosphate happens at a very different levels of acidity than ionizing both: pK = 3 and pK > 7.5. So under normal physiological conditions (wherein pH is around 7), the molecule may carry only one negative charge. The hydroxyl groups (–OH and –O) on phosphate would form a stable intramolecular hydrogen bond with the centered glycerol's hydroxyl group, thus forming a bicyclic resonance structure. This structure traps one proton, which is quite helpful for oxidative phosphorylation.  As the head group forms such compact bicycle structure, the head group area is quite small relative to the big tail region consisting of 4 acyl chains. Based on this special structure, the fluorescent mitochondrial indicator, nonyl acridine orange (NAO) was introduced in 1982, and was later found to target mitochondria by binding to CL. NAO has a very large head and small tail structure which can compensate with cardiolipin's small head large tail structure, and arrange in a highly ordered way. Several studies were published utilizing NAO both as a quantitative mitochondrial indicator and an indicator of CL content in mitochondria. However, NAO is influenced by membrane potential and/or the spatial arrangement of CL, so it's not proper to use NAO for CL or mitochondria quantitative studies of intact respiring mitochondria. But NAO still represents a simple method of assessing CL content.

Metabolism and catabolism.

Metabolism.

Eukaryotic pathway.

In eukaryotes such as yeasts, plants and animals, the synthesis processes are believed to happen in mitochondria. The first step is the acylation of glycerol-3-phosphate by a glycerol-3-phosphate acyltransferase. Then acylglycerol-3-phosphate can be once more acylated to form a phosphatidic acid (PA). With the help of the enzyme CDP-DAG synthase (CDS) (phosphatidate cytidylyltransferase), PA is converted into cytidinediphosphate-diacylglycerol (CDP-DAG). The following step is conversion of CDP-DAG to phosphatidylglycerol phosphate (PGP) by the enzyme PGP synthase, followed by dephosphorylation by PTPMT1 to form PG. Finally, a molecule of CDP-DAG is bound to PG to form one molecule of cardiolipin, catalyzed by the mitochondria-localized enzyme cardiolipin synthase (CLS).

Prokaryotic pathway.

In prokaryotes such as bacteria, diphosphatidylglycerol synthase catalyses a transfer of the phosphatidyl moiety of one phosphatidylglycerol to the free 3'-hydroxyl group of another, with the elimination of one molecule of glycerol, via the action of an enzyme related to phospholipase D. The enzyme can operate in reverse under some physiological conditions to remove cardiolipin.

Catabolism.

Catabolism of cardiolipin may happen by the catalysis of phospholipase A2 (PLA) to remove fatty acyl groups. Phospholipase D (PLD) in the mitochondrion hydrolyses cardiolipin to phosphatidic acid.

Functions.

Regulates aggregate structures.

Because of cardiolipin's unique structure, a change in pH and the presence of divalent cations can induce a structural change. CL shows a great variety of forms of aggregates. It is found that in the presence of Ca or other divalent cations, CL can be induced to have a lamellar-to-hexagonal (L-H) phase transition. And it is believed to have a close connection with membrane fusion.

Facilitates the quaternary structure.

The enzyme cytochrome c oxidase or Complex IV is a large transmembrane protein complex found in bacteria and the mitochondrion. It is the last enzyme in the respiratory electron transport chain of mitochondria (or bacteria) located in the mitochondrial (or bacterial) membrane. It receives an electron from each of four cytochrome c molecules, and transfers them to one oxygen molecule, converting molecular oxygen to two molecules of water. Complex IV has been shown to require two associated CL molecules in order to maintain its full enzymatic function. Cytochrome bc1(Complex III) also needs cardiolipin to maintain its quaternary structure and functional role. Complex V of the oxidative phosphorylation machinery also displays high binding affinity for CL, binding four molecules of CL per molecule of complex V.

Triggers apoptosis.

Cardiolipin distribution to the outer mitochondrial membrane would lead to apoptosis of the cells, as evidenced by cytochrome c (cyt c) release, Caspase-8 activation, MOMP induction and NLRP3 inflammasome activation. During apoptosis, cyt c is released from the intermembrane spaces of mitochondria into the cytosol. Cyt c can then bind to the IP3 receptor on endoplasmic reticulum, stimulating calcium release, which then reacts back to cause the release of cyt c. When the calcium concentration reaches a toxic level, this causes cell death. Cytochrome c is thought to play a role in apoptosis via the release of apoptotic factors from the mitochondria. A cardiolipin-specific oxygenase produces CL hydroperoxides which can result in the conformation change of the lipid. The oxidized CL transfers from the inner membrane to the outer membrane, and then helps to form a permeable pore which releases cyt c.

Serves as proton trap for oxidative phosphorylation.

During the oxidative phosphorylation process catalyzed by Complex IV, large quantities of protons are transferred from one side of the membrane to another side causing a large pH change. CL is suggested to function as a proton trap within the mitochondrial membranes, thereby strictly localizing the proton pool and minimizing the changes in pH in the mitochondrial intermembrane space.  This function is due to CL's unique structure. As stated above, CL can trap a proton within the bicyclic structure while carrying a negative charge. Thus, this bicyclic structure can serve as an electron buffer pool to release or absorb protons to maintain the pH near the membranes.

Clinical significance.

Barth syndrome.

Barth syndrome is a rare genetic disorder that was recognised in the 1970s to cause infantile death. It has a mutation in the gene coding for tafazzin, an enzyme involved in the biosynthesis of cardiolipin. Tafazzin is an indispensable enzyme to synthesize cardiolipin in eukaryotes involved in the remodeling of CL acyl chains by transferring linoleic acid from PC to monolysocardiolipin. Mutation of tafazzin would cause insufficient cardiolipin remodeling. However, it appears that cells compensate and ATP production is similar or higher than normal cells. Females heterozygous for the trait are unaffected. Sufferers of this condition have mitochondria that are abnormal. Cardiomyopathy and general weakness is common to these patients.

Parkinson's disease and Alzheimer's disease.

Oxidative stress and lipid peroxidation are believed to be contributing factors leading to neuronal loss and mitochondrial dysfunction in the substantia nigra in Parkinson's disease, and may play an early role in the pathogenesis of Alzheimer's disease. It is reported that CL content in the brain decreases with aging, and a recent study on rat brain shows it results from lipid peroxidation in mitochondria exposed to free radical stress. Another study shows that the CL biosynthesis pathway may be selectively impaired, causing 20% reduction and composition change of the CL content. It's also associated with a 15% reduction in linked complex I/III activity of the electron transport chain, which is thought to be a critical factor in the development of Parkinson's disease.

Nonalcoholic fatty liver disease and heart failure.

Recently, it is reported that in non-alcoholic fatty liver disease and heart failure, decreased CL levels and change in acyl chain composition are also observed in the mitochondrial dysfunction. However, the role of CL in aging and ischemia/reperfusion is still controversial.

Tangier disease.

Tangier disease is also linked to CL abnormalities. Tangier disease is characterized by very low blood plasma levels of High-Density Lipoprotein (HDL) cholesterol ("good cholesterol"), accumulation of cholesteryl esters in tissues, and an increased risk for developing cardiovascular disease. Unlike Barth syndrome, Tangier disease is mainly caused by abnormal enhanced production of CL. Studies show that there are three to fivefold increase of CL level in Tangier disease. Because increased CL levels would enhance cholesterol oxidation, and then the formation of oxysterols would consequently increase cholesterol efflux. This process could function as an escape mechanism to remove excess cholesterol from the cell.

Diabetes.

Heart disease is twice as common in people with diabetes. In diabetics, cardiovascular complications occur at an earlier age and often result in premature death, making heart disease the major killer of diabetic people. Cardiolipin has been found to be deficient in the heart at the earliest stages of diabetes, possibly due to a lipid-digesting enzyme that becomes more active in diabetic heart muscle.

Syphilis.

Cardiolipin from a cow heart is used as an antigen in the Wassermann test for syphilis. Anti-cardiolipin antibodies can also be increased in numerous other conditions, including systemic lupus erythematosus, malaria and tuberculosis, so this test is not specific.

HIV-1.

Human immunodeficiency virus-1 (HIV-1) has infected more than 60 million people worldwide. HIV-1 envelope glycoprotein contains at least four sites for neutralizing antibodies. Among these sites, the membrane-proximal region (MPR) is particularly attractive as an antibody target because it facilitates viral entry into T cells and is highly conserved among viral strains. However, it is found that two antibodies directed against 2F5, 4E10 in MPR react with self-antigens, including cardiolipin. Thus, it's difficult for such antibodies to be elicited by vaccination.

Cancer.

It was first proposed by Otto Heinrich Warburg that cancer originated from irreversible injury to mitochondrial respiration, but the structural basis for this injury has remained elusive. Since cardiolipin is an important phospholipid found almost exclusively in the inner mitochondrial membrane and very essential in maintaining mitochondrial function, it is suggested that abnormalities in CL can impair mitochondrial function and bioenergetics. A study published in 2008 on mouse brain tumors supporting Warburg's cancer theory shows major abnormalities in CL content or composition in all tumors.

Antiphospholipid syndrome.

Patients with anti-cardiolipin antibodies (Antiphospholipid syndrome) can have recurrent thrombotic events even early in their mid- to late-teen years. These events can occur in vessels in which thrombosis may be relatively uncommon, such as the hepatic or renal veins. These antibodies are usually picked up in young women with recurrent spontaneous abortions. In anti-cardiolipin-mediated autoimmune disease, there is a dependency on the apolipoprotein H for recognition.

Additional anti-cardiolipin diseases.

Bartonella infection.

Bartonellosis is a serious chronic bacterial infection shared by both cats and humans. Spinella found that one patient with bartonella henselae also had anti-cardiolipin antibodies, suggesting that bartonella may trigger their production.

Chronic fatigue syndrome.

Chronic fatigue syndrome is debilitating illness of unknown cause that often follows an acute viral infection. According to one research study, 95% of CFS patients have anti-cardiolipin antibodies.

Cardiolipin (IUPAC name 1,3-bis("sn"-3’-phosphatidyl)-"sn"-glycerol, also known as Calcutta antigen) is an important component of the inner mitochondrial membrane, where it constitutes about 20% of the total lipid composition. It can also be found in the membranes of most bacteria. The name "cardiolipin" is derived from the fact that it was first found in animal hearts. It was first isolated from beef heart in the early 1940s. In mammalian cells, but also in plant cells, cardiolipin (CL) is found almost exclusively in the inner mitochondrial membrane, where it is essential for the optimal function of numerous enzymes that are involved in mitochondrial energy metabolism.

Kardiolipin je glycerolfosfolipid sestávající ze dvou molekul fosfatidátů napojených kovalentně na molekulu glycerolu. Najdeme ho zejména ve vnitřní membráně mitochondrií a v plazmatické membráně bakterií.


Biological function.

DHT is biologically important for sexual differentiation of the male genitalia during embryogenesis, maturation of the penis and scrotum at puberty, growth of facial, body, and pubic hair, and development and maintenance of the prostate gland and seminal vesicles. It is produced from the less potent testosterone by the enzyme 5α-reductase in select tissues, and is the primary androgen in the genitals, prostate gland, seminal vesicles, skin, and hair follicles.  DHT signals mainly in an intracrine and paracrine manner in the tissues in which it is produced, playing only a minor role, if any, as a circulating endocrine hormone. Circulating levels of DHT are 1/10th and 1/20th those of testosterone in terms of total and free concentrations, respectively, whereas local DHT levels may be up to 10 times those of testosterone in tissues with high 5α-reductase expression such as the prostate gland. In addition, unlike testosterone, DHT is inactivated by 3α-hydroxysteroid dehydrogenase (3α-HSD) into the very weak androgen 3α-androstanediol in various tissues such as muscle, adipose, and liver among others, and in relation to this, DHT has been reported to be a very poor anabolic agent when administered exogenously as a medication.  In addition to normal biological functions, DHT also plays an important causative role in a number of androgen-dependent conditions including hair conditions like hirsutism (excessive facial/body hair growth) and pattern hair loss (androgenic alopecia or pattern baldness) and prostate diseases such as benign prostatic hyperplasia (BPH) and prostate cancer. 5α-Reductase inhibitors, which prevent DHT synthesis, are effective in the prevention and treatment of these conditions. Additionally, DHT may play a function in skeletal muscle amino acid transporter recruitment and function.  Metabolites of DHT have been found to act as neurosteroids with their own AR-independent biological activity. 3α-Androstanediol is a potent positive allosteric modulator of the GABA receptor, while 3β-androstanediol is a potent and selective agonist of the estrogen receptor (ER) subtype ERβ. These metabolites may play important roles in the central effects of DHT and by extension testosterone, including their antidepressant, anxiolytic, rewarding/hedonic, anti-stress, and pro-cognitive effects.

5α-Reductase deficiency.

Much of the biological role of DHT has been elucidated in studies of individuals with congenital 5α-reductase type II deficiency, an intersex condition caused by a loss-of-function mutation in the gene encoding 5α-reductase type II, the major enzyme responsible for the production of DHT in the body. It is characterized by a defective and non-functional 5α-reductase type II enzyme and a partial but majority loss of DHT production in the body. In the condition, circulating testosterone levels are within or slightly above the normal male range, but DHT levels are low (around 30% of normal), and the ratio of circulating testosterone to DHT is greatly elevated (at about 3.5 to 5 times higher than normal).  Genetic males (46,XY) with 5α-reductase type II deficiency are born with undervirilization including pseudohermaphroditism (ambiguous genitalia), pseudovaginal perineoscrotal hypospadias, and usually undescended testes. Their external genitalia are female-like, with micropenis (a small, clitoris-like phallus), a partially unfused, labia-like scrotum, and a blind-ending, shallow vaginal pouch. Due to their lack of conspicuous male genitalia, genetic males with the condition are typically raised as girls. At the time of puberty however, they develop striking phenotypically masculine secondary sexual characteristics including partial virilization of the genitals (enlargement of the phallus into a near-functional penis and descent of the testes), voice deepening, typical male musculoskeletal development, and no menstruation, breast development, or other signs of feminization that occur during female puberty. In addition, normal libido and spontaneous erections develop, they usually show a sexual preference for females, and almost all develop a male gender identity.  Nonetheless, males with 5α-reductase type II deficiency exhibit signs of continued undervirilization in a number of domains. Facial hair was absent or sparse in a relatively large group of Dominican males with the condition, known as the Güevedoces. However, more facial hair has been observed in patients with the disorder from other parts of the world, although facial hair was still reduced relative to that of other men in the same communities. The divergent findings may reflect racial differences in androgen-dependent hair growth. A female pattern of androgenic hair growth, with terminal hair largely restricted to the axillae and lower pubic triangle, is observed in males with the condition. No temporal recession of the hairline or androgenic alopecia (pattern hair loss or baldness) has been observed in any of the cases of 5α-reductase type II deficiency that have been reported, whereas this is normally seen to some degree in almost all Caucasian males. Individuals with 5α-reductase type II deficiency were initially reported to have no incidence of acne, but subsequent research indicated normal sebum secretion and acne incidence.  In genetic males with 5α-reductase type II deficiency, the prostate gland is rudimentary or absent, and if present, remains small, underdeveloped, and unpalpable throughout life. In addition, neither BPH nor prostate cancer have been reported in these individuals. Genetic males with the condition generally show oligozoospermia due to undescended testes, but spermatogenesis is reported to be normal in those with testes that have descended, and there are case instances of men with the condition successfully fathering children.  Unlike males, genetic females with 5α-reductase type II deficiency are phenotypically normal. However, similarly to genetic males with the condition, they show reduced body hair growth, including an absence of hair on the arms and legs, slightly decreased axillary hair, and moderately decreased pubic hair. On the other hand, sebum production is normal. This is in accordance with the fact that sebum secretion appears to be entirely under the control of 5α-reductase type I.

5α-Reductase inhibitors.

5α-Reductase inhibitors like finasteride and dutasteride inhibit 5α-reductase type II and/or other isoforms and are able to decrease circulating DHT levels by 65 to 98% depending on the 5α-reductase inhibitor in question. As such, similarly to the case of 5α-reductase type II deficiency, they provide useful insights in the elucidation of the biological functions of DHT. 5α-Reductase inhibitors were developed and are used primarily for the treatment of BPH. The drugs are able to significantly reduce the size of the prostate gland and to alleviate symptoms of the condition. Long-term treatment with 5α-reductase inhibitors is also able to significantly reduce the overall risk of prostate cancer, although a simultaneous small increase in the risk of certain high-grade tumors has been observed. In addition to prostate diseases, 5α-reductase inhibitors have subsequently been developed and introduced for the treatment of pattern hair loss in men. They are able to prevent further progression of hair loss in most men with the condition and to produce some recovery of hair in about two-thirds of men. 5α-Reductase inhibitors seem to be less effective for pattern hair loss in women on the other hand, although they do still show some effectiveness. Aside from pattern hair loss, the drugs are also useful in the treatment of hirsutism and can greatly reduce facial and body hair growth in women with the condition.  5α-Reductase inhibitors are overall well tolerated and show a low incidence of adverse effects. Sexual dysfunction, including erectile dysfunction, loss of libido, and reduced ejaculate volume, may occur in 3.4 to 15.8% of men treated with finasteride or dutasteride. A small increase in the risk of affective symptoms including depression, anxiety, and self-harm may be seen. Both the sexual dysfunction and affective symptoms may be due partially or fully to prevention of the synthesis of neurosteroids like allopregnanolone rather necessarily than due to inhibition of DHT production. A very small risk of gynecomastia has been associated with 5α-reductase inhibitors (1.2 to 3.5%). Based on reports of 5α-reductase type II deficiency in males and the effectiveness of 5α-reductase inhibitors for hirsutism in women, reduced body and/or facial hair growth is a likely potential side effect of these drugs in men. There are very few studies evaluating the side effects of 5α-reductase inhibitors in women. However, due to the known role of DHT in male sexual differentiation, 5α-reductase inhibitors may cause birth defects such as ambiguous genitalia in the male fetuses of pregnant women. As such, they are not used in women during pregnancy.  MK-386 is a selective 5α-reductase type I inhibitor which was never marketed. Whereas 5α-reductase type II inhibitors achieve much higher reductions in circulating DHT production, MK-386 decreases circulating DHT levels by 20 to 30%. Conversely, it was found to decrease sebum DHT levels by 55% in men versus a modest reduction of only 15% for finasteride. However, MK-386 failed to show significant effectiveness in a subsequent clinical study for the treatment of acne.

Biological activity.

DHT is a potent agonist of the AR, and is in fact the most potent known endogenous ligand of the receptor. It has an affinity (K) of 0.25 to 0.5 nM for the human AR, which is about 2- to 3-fold higher than that of testosterone (K = 0.4 to 1.0 nM) and 15–30 times higher than that of adrenal androgens. In addition, the dissociation rate of DHT from the AR is 5-fold slower than that of testosterone. The EC of DHT for activation of the AR is 0.13 nM, which is about 5-fold stronger than that of testosterone (EC = 0.66 nM). In bioassays, DHT has been found to be 2.5- to 10-fold more potent than testosterone.  The elimination half-life of DHT in the body (53 minutes) is longer than that of testosterone (34 minutes), and this may account for some of the difference in their potency. A study of transdermal DHT and testosterone treatment reported terminal half-lives of 2.83 hours and 1.29 hours, respectively.  Unlike other androgens such as testosterone, DHT cannot be converted by the enzyme aromatase into an estrogen like estradiol. Therefore, it is frequently used in research settings to distinguish between the effects of testosterone caused by binding to the AR and those caused by testosterone's conversion to estradiol and subsequent binding to and activation of ERs. Although DHT cannot be aromatized, it is still transformed into metabolites with significant ER affinity and activity. These are 3α-androstanediol and 3β-androstanediol, which are predominant agonists of the ERβ.

Biochemistry.

Biosynthesis.

DHT is synthesized irreversibly from testosterone by the enzyme 5α-reductase. This occurs in various tissues including the genitals (penis, scrotum, clitoris, labia majora), prostate gland, skin, hair follicles, liver, and brain. Around 5 to 7% of testosterone undergoes 5α-reduction into DHT, and approximately 200 to 300 μg of DHT is synthesized in the body per day. Most DHT is produced in peripheral tissues like the skin and liver, whereas most "circulating" DHT originates specifically from the liver. The testes and prostate gland contribute relatively little to concentrations of DHT in circulation.  There are two major isoforms of 5α-reductase, SRD5A1 (type I) and SRD5A2 (type II), with the latter being the most biologically important isoenzyme. There is also third 5α-reductase: SRD5A3. SRD5A2 is most highly expressed in the genitals, prostate gland, epididymides, seminal vesicles, genital skin, facial and chest hair follicles, and liver, while lower expression is observed in certain brain areas, non-genital skin/hair follicles, testes, and kidneys. SRD5A1 is most highly expressed in non-genital skin/hair follicles, the liver, and certain brain areas, while lower levels are present in the prostate, epididymides, seminal vesicles, genital skin, testes, adrenal glands, and kidneys. In the skin, 5α-reductase is expressed in sebaceous glands, sweat glands, epidermal cells, and hair follicles. Both isoenzymes are expressed in scalp hair follicles, although SRD5A2 predominates in these cells. The SRD5A2 subtype is the almost exclusive isoform expressed in the prostate gland.

Distribution.

The plasma protein binding of DHT is more than 99%. In men, approximately 0.88% of DHT is unbound and hence free, while in premenopausal women, about 0.47–0.48% is unbound. In men, DHT is bound 49.7% to sex hormone-binding globulin (SHBG), 39.2% to albumin, and 0.22% to corticosteroid-binding globulin (CBG), while in premenopausal women, DHT is bound 78.1–78.4% to SHBG, 21.0–21.3% to albumin, and 0.12% to CBG. In late pregnancy, only 0.07% of DHT is unbound in women; 97.8% is bound to SHBG while 2.15% is bound to albumin and 0.04% is bound to CBG. DHT has higher affinity for SHBG than does testosterone, estradiol, or any other steroid hormone.

Metabolism.

DHT is inactivated in the liver and extrahepatic tissues like the skin into 3α-androstanediol and 3β-androstanediol by the enzymes 3α-hydroxysteroid dehydrogenase and 3β-hydroxysteroid dehydrogenase, respectively. These metabolites are in turn converted, respectively, into androsterone and epiandrosterone, then conjugated (via glucuronidation and/or sulfation), released into circulation, and excreted in urine.  Unlike testosterone, DHT cannot be aromatized into an estrogen like estradiol, and for this reason, has no propensity for estrogenic effects.

Excretion.

DHT is excreted in the urine as metabolites, such as conjugates of 3α-androstanediol and androsterone.

Levels.

Serum DHT levels are about 10% of those of testosterone, but levels in the prostate gland are 5- to 10-fold higher than those of testosterone due to a more than 90% conversion of testosterone into DHT by locally expressed 5α-reductase. For this reason, and in addition to the fact that DHT is much more potent as an AR agonist than is testosterone, DHT is considered to be the major androgen of the prostate gland.

Medical use.

DHT is available in pharmaceutical formulations for medical use as an androgen or anabolic–androgenic steroid (AAS). It is used mainly in the treatment of male hypogonadism. When used as a medication, dihydrotestosterone is referred to as "androstanolone" () or as "stanolone" (), and is sold under brand names such as "Andractim" among others. The availability of pharmaceutical DHT is limited; it is not available in the United States or Canada, but is available in certain European countries. The available formulations of DHT include buccal or sublingual tablets, topical gels, and, as esters in oil, injectables like androstanolone propionate and androstanolone valerate.

Chemistry.

DHT, also known as 5α-androstan-17β-ol-3-one, is a naturally occurring androstane steroid with a ketone group at the C3 position and a hydroxyl group at the C17β position. It is the derivative of testosterone in which the double bond between the C4 and C5 positions has been reduced or hydrogenated.

History.

DHT was first synthesized by Adolf Butenandt and his colleagues in 1935. It was prepared via hydrogenation of testosterone, which had been discovered earlier that year. DHT was introduced for medical use as an AAS in 1953, and was noted to be more potent than testosterone but with reduced androgenicity. It was not elucidated to be an endogenous substance until 1956, when it was shown to be formed from testosterone in rat liver homogenates. In addition, the biological importance of DHT was not realized until the early 1960s, when it was found to be produced by 5α-reductase from circulating testosterone in target tissues like the prostate gland and seminal vesicles and was found to be more potent than testosterone in bioassays. The biological functions of DHT in humans became much more clearly defined upon the discovery and characterization of 5α-reductase type II deficiency in 1974. DHT was the last major sex hormone, the others being testosterone, estradiol, and progesterone, to be discovered, and is unique in that it is the only major sex hormone that functions principally as an intracrine and paracrine hormone rather than as an endocrine hormone.

Dihydrotestosterone (DHT, 5α-dihydrotestosterone, 5α-DHT, androstanolone or stanolone) is an endogenous androgen sex steroid and hormone. The enzyme 5α-reductase catalyzes the formation of DHT from testosterone in certain tissues including the prostate gland, seminal vesicles, epididymides, skin, hair follicles, liver, and brain. This enzyme mediates reduction of the C4-5 double bond of testosterone. Relative to testosterone, DHT is considerably more potent as an agonist of the androgen receptor (AR).

Dihydrotestosteron (DHT) nebo přesněji 5α-dihydrotestosteron je vlastní účinná forma testosteronu. Vzniká redukcí testosteronu na uhlíku č. 5 v cílových orgánech (zejména semenné váčky, prostata, zevní genitálie a části kůže a ochlupení). Enzymem, jenž katabolizuje tuto redukci testosteronu na DHT, je 5α-reduktáza.


Von Neumannova–Bernaysova–Gödelova teorie množin

Classes in set theory.

The uses of classes.

Classes have several uses in NBG:

Axiom schema versus class existence theorem.

Once classes are added to the language of ZFC, it is easy to transform ZFC into a set theory with classes. First, the axiom schema of class comprehension is added. This axiom schema states: For every formula formula_18 that quantifies only over sets, there exists a class formula_19 consisting of the satisfying the formula—that is, formula_20 Then the axiom schema of replacement is replaced by a single axiom that uses a class. Finally, ZFC's axiom of extensionality is modified to handle classes: If two classes have the same elements, then they are identical. The other axioms of ZFC are not modified.  This theory is not finitely axiomatized. ZFC's replacement schema has been replaced by a single axiom, but the axiom schema of class comprehension has been introduced.  To produce a theory with finitely many axioms, the axiom schema of class comprehension is first replaced with finitely many class existence axioms. Then these axioms are used to prove the class existence theorem which implies every instance of the axiom schema. The proof of this theorem requires only seven class existence axioms, which are used to convert the construction of a formula into the construction of a class satisfying the formula.

Axiomatization of NBG.

Classes and sets.

NBG has two types of objects: classes and sets. Intuitively, every set is also a class. There are two ways to axiomatize this. Bernays used many-sorted logic with two sorts: classes and sets. Gödel avoided sorts by introducing primitive predicates: formula_21 for "formula_19 is a class" and formula_23 for "formula_19 is a set" (in German, "set" is "Menge"). He also introduced axioms stating that every set is a class and that if class formula_19 is a member of a class, then formula_19 is a set. Using predicates is the standard way to eliminate sorts. Elliott Mendelson modified Gödel's approach by having everything be a class and defining the set predicate formula_27 as formula_28 This modification eliminates Gödel's class predicate and his two axioms.  Bernays' two-sorted approach may appear more natural at first, but it creates a more complex theory. In Bernays' theory, every set has two representations: one as a set and the other as a class. Also, there are two membership relations: the first, denoted by "∈", is between two sets; the second, denoted by "η", is between a set and a class. This redundancy is required by many-sorted logic because variables of different sorts range over disjoint subdomains of the domain of discourse.  The differences between these two approaches does not affect what can be proved, but it does affect how statements are written. In Gödel's approach, formula_29 where formula_19 and formula_31 are classes is a valid statement. In Bernays' approach this statement has no meaning. However, if formula_19 is a set, there is an equivalent statement: Define "set formula_33 represents class formula_19" if they have the same sets as members—that is, formula_35 The statement formula_36 where set formula_33 represents class formula_19 is equivalent to Gödel's formula_39  The approach adopted in this article is that of Gödel with Mendelson's modification. This means that NBG is an axiomatic system in first-order predicate logic with equality, and its only primitive notions are class and the membership relation.

Definitions and axioms of extensionality and pairing.

A set is a class that belongs to at least one class: formula_19 is a set if and only if formula_41. A class that is not a set is called a proper class: formula_19 is a proper class if and only if formula_43. Therefore, every class is either a set or a proper class, and no class is both (if the theory is consistent).  Gödel introduced the convention that uppercase variables range over classes, while lowercase variables range over sets. Gödel also used names that begin with an uppercase letter to denote particular classes, including functions and relations defined on the class of all sets. Gödel's convention is used in this article. It allows us to write:  The following axioms and definitions are needed for the proof of the class existence theorem.  Axiom of extensionality. If two classes have the same elements, then they are identical.  This axiom generalizes ZFC's axiom of extensionality to classes.  Axiom of pairing. If formula_49 and formula_50 are sets, then there exists a set formula_51 whose only members are formula_49 and formula_50.  As in ZFC, the axiom of extensionality implies the uniqueness of the set formula_51, which allows us to introduce the notation formula_56  Ordered pairs are defined by: Tuples are defined inductively using ordered pairs:

Class existence axioms and axiom of regularity.

Class existence axioms will be used to prove the class existence theorem: For every formula in formula_60 free set variables that quantifies only over sets, there exists a class of that satisfy it. The following example starts with two classes that are functions and builds a composite function. This example illustrates the techniques that are needed to prove the class existence theorem, which lead to the class existence axioms that are needed.  The class existence axioms are divided into two groups: axioms handling language primitives and axioms handling tuples. There are four axioms in the first group and three axioms in the second group.  Axioms for handling language primitives:  Membership. There exists a class formula_61 containing all the ordered pairs whose first component is a member of the second component.  Intersection (conjunction). For any two classes formula_19 and formula_64, there is a class formula_31 consisting precisely of the sets that belong to both formula_19 and formula_64.  Complement (negation). For any class formula_19, there is a class formula_64 consisting precisely of the sets not belonging to formula_19.  Domain (existential quantifier). For any class formula_19, there is a class formula_64 consisting precisely of the first components of the ordered pairs of formula_19.  By the axiom of extensionality, class formula_31 in the intersection axiom and class formula_64 in the complement and domain axioms are unique. They will be denoted by: formula_79 formula_80 and formula_81 respectively. On the other hand, extensionality is not applicable to formula_61 in the membership axiom since it specifies only those sets in formula_61 that are ordered pairs.  The first three axioms imply the existence of the empty class and the class of all sets: The membership axiom implies the existence of a class formula_84 The intersection and complement axioms imply the existence of formula_85, which is empty. By the axiom of extensionality, this class is unique; it is denoted by formula_86 The complement of formula_87 is the class formula_88 of all sets, which is also unique by extensionality. The set predicate formula_27, which was defined as formula_90, is now redefined as formula_91 to avoid quantifying over classes.  Axioms for handling tuples:  Product by formula_88. For any class formula_19, there is a class formula_64 consisting of the ordered pairs whose first component belongs to formula_19.  Circular permutation. For any class formula_19, there is a class formula_64 whose 3tuples are obtained by applying the circular permutation formula_99 to the 3tuples of formula_19.  Transposition. For any class formula_19, there is a class formula_64 whose 3tuples are obtained by transposing the last two components of the 3tuples of formula_19.  By extensionality, the product by formula_88 axiom implies the existence of a unique class, which is denoted by formula_107 This axiom is used to define the class formula_108 of all : formula_109 and formula_110 If formula_19 is a class, extensionality implies that formula_112 is the unique class consisting of the of formula_113 For example, the membership axiom produces a class formula_61 that may contain elements that are not ordered pairs, while the intersection formula_115 contains only the ordered pairs of formula_61.  The circular permutation and transposition axioms do not imply the existence of unique classes because they specify only the 3tuples of class formula_117 By specifying the 3tuples, these axioms also specify the for formula_118 since: formula_119 The axioms for handling tuples and the domain axiom imply the following lemma, which is used in the proof of the class existence theorem.  Tuple lemma.  Proof:Class formula_124: Apply product by formula_88 to formula_19 to produce formula_127
 Class formula_128: Apply transposition to formula_124 to produce formula_130
 Class formula_131: Apply circular permutation to formula_124 to produce formula_133
 Class formula_134: Apply circular permutation to formula_128, then apply domain to produce formula_136  One more axiom is needed to prove the class existence theorem: the axiom of regularity. Since the existence of the empty class has been proved, the usual statement of this axiom is given.  Axiom of regularity. Every nonempty set has at least one element with which it has no element in common.  This axiom implies that a set cannot belong to itself: Assume that formula_138 and let formula_139 Then formula_140 since formula_141 This contradicts the axiom of regularity because formula_49 is the only element in formula_143 Therefore, formula_144 The axiom of regularity also prohibits infinite descending membership sequences of sets: formula_145  Gödel stated regularity for classes rather than for sets in his 1940 monograph, which was based on lectures given in 1938. In 1939, he proved that regularity for sets implies regularity for classes.

Class existence theorem.

Class existence theorem. Let formula_146 be a formula that quantifies only over sets and contains no free variables other than formula_147 (not necessarily all of these). Then for all formula_148, there exists a unique class formula_19 of such that: formula_150 The class formula_19 is denoted by formula_152  The theorem's proof will be done in two steps:  Transformation rules. In rules 1 and 2 below, formula_157 and formula_158 denote set or class variables. These two rules eliminate all occurrences of class variables before an formula_11 and all occurrences of equality. Each time rule 1 or 2 is applied to a subformula, formula_160 is chosen so that formula_161 differs from the other variables in the current formula. The three rules are repeated until there are no subformulas to which they can be applied. This produces a formula that is built only with formula_154, formula_155, formula_156, formula_11, set variables, and class variables formula_166 where formula_166 does not appear before an formula_11.  Transformation rules: bound variables. Consider the composite function formula of example 1 with its free set variables replaced by formula_177 and formula_178: formula_179 The inductive proof will remove formula_180, which produces the formula formula_181 However, since the class existence theorem is stated for subscripted variables, this formula does not have the form expected by the induction hypothesis. This problem is solved by replacing the variable formula_182 with formula_183 Bound variables within nested quantifiers are handled by increasing the subscript by one for each successive quantifier. This leads to rule 4, which must be applied after the other rules since rules 1 and 2 produce quantified variables.  Proof of the class existence theorem. The proof starts by applying the transformation rules to the given formula to produce a transformed formula. Since this formula is equivalent to the given formula, the proof is completed by proving the class existence theorem for transformed formulas.  Gödel pointed out that the class existence theorem "is a metatheorem, that is, a theorem about the system [NBG], not in the system..." It is a theorem about NBG because it is proved in the metatheory by induction on NBG formulas. Also, its proof—instead of invoking finitely many NBG axioms—inductively describes how to use NBG axioms to construct a class satisfying a given formula. For every formula, this description can be turned into a constructive existence proof that is in NBG. Therefore, this metatheorem can generate the NBG proofs that replace uses of NBG's class existence theorem.  A recursive computer program succinctly captures the construction of a class from a given formula. The definition of this program does not depend on the proof of the class existence theorem. However, this proof is needed to prove that the class constructed by the program satisfies the given formula and is built using the axioms. This program is written in pseudocode that uses a Pascal-style case statement.  formula_184
 formula_185  formula_186
 formula_189  Let formula_153 be the formula of example 2. The function call formula_192 generates the class formula_193 which is compared below with formula_194 This shows that the construction of the class formula_19 mirrors the construction of its defining formula formula_194

Extending the class existence theorem.

Gödel extended the class existence theorem to formulas formula_153 containing relations over classes (such as formula_199 and the unary relation formula_200), special classes (such as formula_9), and operations (such as formula_202 and formula_203). To extend the class existence theorem, the formulas defining relations, special classes, and operations must quantify only over sets. Then formula_153 can be transformed into an equivalent formula satisfying the hypothesis of the class existence theorem.  The following definitions specify how formulas define relations, special classes, and operations:  A is defined by:  The following transformation rules eliminate relations, special classes, and operations. Each time rule 2b, 3b, or 4 is applied to a subformula, formula_160 is chosen so that formula_161 differs from the other variables in the current formula. The rules are repeated until there are no subformulas to which they can be applied. formula_217 and formula_157 denote terms.  Class existence theorem (extended version). Let formula_146 be a formula that quantifies only over sets, contains no free variables other than formula_220, and may contain relations, special classes, and operations defined by formulas that quantify only over sets. Then for all formula_221 there exists a unique class formula_19 of such that formula_150  Proof: Apply the transformation rules to formula_153 to produce an equivalent formula containing no relations, special classes, or operations. This formula satisfies the hypothesis of the class existence theorem. Therefore, for all formula_221 there is a unique class formula_19 of satisfying formula_227

Set axioms.

The axioms of pairing and regularity, which were needed for the proof of the class existence theorem, have been given above. NBG contains four other set axioms. Three of these axioms deal with class operations being applied to sets.  Definition. formula_228 is a function if formula_229  In set theory, the definition of a function does not require specifying the domain or codomain of the function (see Function (set theory)). NBG's definition of function generalizes ZFC's definition from a set of ordered pairs to a class of ordered pairs.  ZFC's definitions of the set operations of image, union, and power set are also generalized to class operations. The image of class formula_19 under the function formula_228 is formula_232 This definition does not require that formula_233 The union of class formula_19 is formula_235 The power class of formula_19 is formula_237 The extended version of the class existence theorem implies the existence of these classes. The axioms of replacement, union, and power set imply that when these operations are applied to sets, they produce sets.  Axiom of replacement. If formula_228 is a function and formula_33 is a set, then formula_240, the image of formula_33 under formula_228, is a set.  Not having the requirement formula_244 in the definition of formula_245 produces a stronger axiom of replacement, which is used in the following proof. Theorem (NBG's axiom of separation). If formula_19 is a set and formula_64 is a subclass of formula_193 then formula_64 is a set.
 Proof: The class existence theorem constructs the restriction of the identity function to formula_64: formula_251 Since the image of formula_19 under formula_253 is formula_64, the axiom of replacement implies that formula_64 is a set. This proof depends on the definition of image not having the requirement formula_244 since formula_257 rather than formula_258  Axiom of union. If formula_33 is a set, then there is a set containing formula_260  Axiom of power set. If formula_33 is a set, then there is a set containing formula_263  Theorem. If formula_33 is a set, then formula_266 and formula_267 are sets.
 Proof: The axiom of union states that formula_266 is a subclass of a set formula_269, so the axiom of separation implies formula_266 is a set. Likewise, the axiom of power set states that formula_267 is a subclass of a set formula_269, so the axiom of separation implies that formula_267 is a set.  Axiom of infinity. There exists a nonempty set formula_33 such that for all formula_49 in formula_33, there exists a formula_50 in formula_33 such that formula_49 is a proper subset of formula_50.  The axioms of infinity and replacement prove the existence of the empty set. In the discussion of the class existence axioms, the existence of the empty class formula_87 was proved. We now prove that formula_87 is a set. Let function formula_284 and let formula_33 be the set given by the axiom of infinity. By replacement, the image of formula_33 under formula_228, which equals formula_87, is a set.  NBG's axiom of infinity is implied by ZFC's axiom of infinity: formula_289 The first conjunct of ZFC's axiom, formula_290, implies the first conjunct of NBG's axiom. The second conjunct of ZFC's axiom, formula_291, implies the second conjunct of NBG's axiom since formula_292 To prove ZFC's axiom of infinity from NBG's axiom of infinity requires some of the other NBG axioms (see Weak axiom of infinity).

Axiom of global choice.

The class concept allows NBG to have a stronger axiom of choice than ZFC. A choice function is a function formula_5 defined on a set formula_4 of nonempty sets such that formula_7 for all formula_8 ZFC's axiom of choice states that there exists a choice function for every set of nonempty sets. A global choice function is a function formula_1 defined on the class of all nonempty sets such that formula_2 for every nonempty set formula_3 The axiom of global choice states that there exists a global choice function. This axiom implies ZFC's axiom of choice since for every set formula_4 of nonempty sets, formula_301 (the restriction of formula_1 to formula_4) is a choice function for formula_304 In 1964, William B. Easton proved that global choice is stronger than the axiom of choice by using forcing to construct a model that satisfies the axiom of choice and all the axioms of NBG except the axiom of global choice. The axiom of global choice is equivalent to every class having a well-ordering, while ZFC's axiom of choice is equivalent to every set having a well-ordering.  Axiom of global choice. There exists a function that chooses an element from every nonempty set.

History.

Von Neumann's 1925 axiom system.

Von Neumann published an introductory article on his axiom system in 1925. In 1928, he provided a detailed treatment of his system. Von Neumann based his axiom system on two domains of primitive objects: functions and arguments. These domains overlap—objects that are in both domains are called argument-functions. Functions correspond to classes in NBG, and argument-functions correspond to sets. Von Neumann's primitive operation is function application, denoted by ["a", "x"] rather than "a"("x") where "a" is a function and "x" is an argument. This operation produces an argument. Von Neumann defined classes and sets using functions and argument-functions that take only two values, "A" and "B". He defined "x" ∈ "a" if ["a", "x"] ≠ "A".  Von Neumann's work in set theory was influenced by Georg Cantor's articles, Ernst Zermelo's 1908 axioms for set theory, and the 1922 critiques of Zermelo's set theory that were given independently by Abraham Fraenkel and Thoralf Skolem. Both Fraenkel and Skolem pointed out that Zermelo's axioms cannot prove the existence of the set {"Z", "Z", "Z",...} where "Z" is the set of natural numbers and "Z" is the power set of "Z". They then introduced the axiom of replacement, which would guarantee the existence of such sets. However, they were reluctant to adopt this axiom: Fraenkel stated "that Replacement was too strong an axiom for 'general set theory'", while "Skolem only wrote that 'we could introduce' Replacement".  Von Neumann worked on the problems of Zermelo set theory and provided solutions for some of them:

Von Neumann's 1929 axiom system.

In 1929, von Neumann published an article containing the axioms that would lead to NBG. This article was motivated by his concern about the consistency of the axiom of limitation of size. He stated that this axiom "does a lot, actually too much." Besides implying the axioms of separation and replacement, and the well-ordering theorem, it also implies that any class whose cardinality is less than that of "V" is a set. Von Neumann thought that this last implication went beyond Cantorian set theory and concluded: "We must therefore discuss whether its [the axiom's] consistency is not even more problematic than an axiomatization of set theory that does not go beyond the necessary Cantorian framework."  Von Neumann started his consistency investigation by introducing his 1929 axiom system, which contains all the axioms of his 1925 axiom system except the axiom of limitation of size. He replaced this axiom with two of its consequences, the axiom of replacement and a choice axiom. Von Neumann's choice axiom states: "Every relation "R" has a subclass that is a function with the same domain as "R"."  Let "S" be von Neumann's 1929 axiom system. Von Neumann introduced the axiom system "S" + Regularity (which consists of "S" and the axiom of regularity) to demonstrate that his 1925 system is consistent relative to "S". He proved: These results imply: If "S" is consistent, then von Neumann's 1925 axiom system is consistent. Proof: If "S" is consistent, then "S" + Regularity is consistent (result 1). Using proof by contradiction, assume that the 1925 axiom system is inconsistent, or equivalently: the 1925 axiom system implies a contradiction. Since "S" + Regularity implies the axioms of the 1925 system (result 2), "S" + Regularity also implies a contradiction. However, this contradicts the consistency of "S" + Regularity. Therefore, if "S" is consistent, then von Neumann's 1925 axiom system is consistent.  Since "S" is his 1929 axiom system, von Neumann's 1925 axiom system is consistent relative to his 1929 axiom system, which is closer to Cantorian set theory. The major differences between Cantorian set theory and the 1929 axiom system are classes and von Neumann's choice axiom. The axiom system "S" + Regularity was modified by Bernays and Gödel to produce the equivalent NBG axiom system.

Bernays' axiom system.

In 1929, Paul Bernays started modifying von Neumann's new axiom system by taking classes and sets as primitives. He published his work in a series of articles appearing from 1937 to 1954. Bernays stated that:  Bernays handled sets and classes in a two-sorted logic and introduced two membership primitives: one for membership in sets and one for membership in classes. With these primitives, he rewrote and simplified von Neumann's 1929 axioms. Bernays also included the axiom of regularity in his axiom system.

Gödel's axiom system (NBG).

In 1931, Bernays sent a letter containing his set theory to Kurt Gödel. Gödel simplified Bernays' theory by making every set a class, which allowed him to use just one sort and one membership primitive. He also weakened some of Bernays' axioms and replaced von Neumann's choice axiom with the equivalent axiom of global choice. Gödel used his axioms in his 1940 monograph on the relative consistency of global choice and the generalized continuum hypothesis.  Several reasons have been given for Gödel choosing NBG for his monograph:  Gödel's achievement together with the details of his presentation led to the prominence that NBG would enjoy for the next two decades. In 1963, Paul Cohen proved his independence proofs for ZF with the help of some tools that Gödel had developed for his relative consistency proofs for NBG. Later, ZFC became more popular than NBG. This was caused by several factors, including the extra work required to handle forcing in NBG, Cohen's 1966 presentation of forcing, which used ZF, and the proof that NBG is a conservative extension of ZFC.

NBG, ZFC, and MK.

NBG is not logically equivalent to ZFC because its language is more expressive: it can make statements about classes, which cannot be made in ZFC. However, NBG and ZFC imply the same statements about sets. Therefore, NBG is a conservative extension of ZFC. NBG implies theorems that ZFC does not imply, but since NBG is a conservative extension, these theorems must involve proper classes. For example, NBG implies that the proper class "V" can be well-ordered and that every proper class can be put into one-to-one correspondence with "V".  One consequence of conservative extension is that ZFC and NBG are equiconsistent. Proving this uses the principle of explosion: from a contradiction, everything is provable. Assume that either ZFC or NBG is inconsistent. Then the inconsistent theory implies the contradictory statements ∅ = ∅ and ∅ ≠ ∅, which are statements about sets. By the conservative extension property, the other theory also implies these statements. Therefore, it is also inconsistent. So although NBG is more expressive, it is equiconsistent with ZFC. This result together with von Neumann's 1929 relative consistency proof implies that his 1925 axiom system with the axiom of limitation of size is equiconsistent with ZFC. This completely resolves von Neumann's concern about the relative consistency of this powerful axiom since ZFC is within the Cantorian framework.  Even though NBG is a conservative extension of ZFC, a theorem may have a shorter and more elegant proof in NBG than in ZFC (or vice versa). For a survey of known results of this nature, see.  Morse–Kelley set theory has an axiom schema of class comprehension that includes formulas whose quantifiers range over classes. MK is a stronger theory than NBG because MK proves the consistency of NBG, while Gödel's second incompleteness theorem implies that NBG cannot prove the consistency of NBG.  For a discussion of some ontological and other philosophical issues posed by NBG, especially when contrasted with ZFC and MK, see Appendix C of.

Models.

ZFC, NBG, and MK have models describable in terms of the cumulative hierarchy "V" and the constructible hierarchy "L". Let "V" include an inaccessible cardinal κ, let "X" ⊆ "V", and let Def("X") denote the class of first-order definable subsets of "X" with parameters. In symbols where "formula_306" denotes the model with domain formula_307 and relation formula_11, and "formula_309" denotes the satisfaction relation:  Then:

Category theory.

The ontology of NBG provides scaffolding for speaking about "large objects" without risking paradox. For instance, in some developments of category theory, a "large category" is defined as one whose objects and morphisms make up a proper class. On the other hand, a "small category" is one whose objects and morphisms are members of a set. Thus, we can speak of the "category of all sets" or "category of all small categories" without risking paradox since NBG supports large categories.  However, NBG does not support a "category of all categories" since large categories would be members of it and NBG does not allow proper classes to be members of anything. An ontological extension that enables us to talk formally about such a "category" is the conglomerate, which is a collection of classes. Then the "category of all categories" is defined by its objects: the conglomerate of all categories; and its morphisms: the conglomerate of all morphisms from "A" to "B" where "A" and "B" are objects. On whether an ontology including classes as well as sets is adequate for category theory, see.

In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel-Choice set theory (ZFC). NBG introduces the notion of class, which is a collection of sets defined by a formula whose quantifiers range only over sets. NBG can define classes that are larger than sets, such as the class of all sets and the class of all ordinals. Morse–Kelley set theory (MK) allows classes to be defined by formulas whose quantifiers range over classes. NBG is finitely axiomatizable, while ZFC and MK are not.

Von Neumannova-Bernaysova-Gödelova teorie množin (někdy také označovaná jako Gödelova-Bernaysova teorie množin nebo NBG či GB) je jedním z nejšířeji přijatých a používaných axiomatických systémů teorie množin.


Background.

"Share My World" marked several personal and professional changes in Blige’s life and career. Following the departure of label head Andre Harrell the year before, Blige defected from Uptown Records in favor of its MCA parent. Meanwhile, she severed professional ties with long-time producer, manager and mentor Sean "Puffy" Combs shortly before the production of "Share My World" began. His absence was filled with a bevy of high-profile producers, such as: Rodney Jerkins, Jimmy Jam and Terry Lewis, Babyface, Bryce Wilson and R. Kelly. The end result produced an album that was less entrenched in the hip hop soul of her first two albums, and replaced with a style that was more aligned with R&B.  During the making and run of her second album "My Life" (1994), Blige had reportedly experienced clinical depression, while also battling drug and alcohol addiction, and enduring an often turbulent relationship with K-Ci Hailey — all of which heavily influenced the dark mood of that album. In late 1996, however, Blige reportedly made a concerted effort to clean up her life and subsequently found herself in more positive frame of mind while recording "Share My World", which influenced the albums noticeably lighter mood.

Release and reception.

Commercial performance.

For the "Billboard" issue dated May 10, 1997, "Share My World" debuted at number one on both the US "Billboard" 200 and R&B Albums chart with sales of 240,000 copies, marking Blige's first official number-one album on the "Billboard" 200 chart. The album had ended the four-week stint of "Life After Death" by The Notorious B.I.G., which was released posthumously five weeks earlier.  In addition to debuting number one in the US, the album also debuted in the top ten in countries such as Canada, Sweden and the UK. The album also reached the top forty in France, Germany and New Zealand.  "Share My World" was certified Platinum by the Recording Industry Association of America (RIAA), for shipments of one million copies in the US on July 1, 1997. On November 17, 1997, "Share My World" was later certified double platinum by the RIAA and during an episode of "MTV Live", Blige was presented with the plaque by host Ananda Lewis. The album has since sold three million copies domestically by May 1999, garnering Blige's third consecutive album to reach multi-platinum status by the RIAA.

Critical reception.

Alex Henderson of AllMusic wrote in his review, "Her strongest and most confident effort up to that point, "Share" had much more character, personality, and honesty than most of the assembly line fare dominating urban radio in 1997. For all their slickness, emotive cuts like "Get to Know You Better," "Love Is All We Need," and "Keep Your Head" left no doubt that Blige was indeed a singer of depth and substance. Although high tech, the production of everyone from R. Kelly (with whom she duets on the inviting "It's On") and Babyface to Jimmy Jam and Terry Lewis doesn't come across as forced or robotic, but, in fact, is impressively organic." Ernest Hardy of "Rolling Stone" commended Blige's transition from sound to singing, writing that "On "Share My World" even Blige's harshest critics will have to concede that she's moved beyond sound to real singing. Listen to "Seven Days," "Missing You" and the already-classic "Not Gon' Cry" (also on the "Waiting to Exhale" soundtrack), and you hear Blige's signature ache married to newfound technique. There's shading, depth and control in her vocals now." Steve Jones of "USA Today" said the songs "run the usual gamut of love themes, but it's Blige's powerful, emotional deliveries and street sensibility that separate her from the competition." "Village Voice" critic Robert Christgau said "Blige is a diva for her own time. As befits her hip hop ethos, she's never soft if often vulnerable, and as befits her hip hop aesthetic, she plays her natural vocal cadences for melodic signature and sometimes hook. She redefines the New York accent for the '90s. And she's taken two straight follow-ups to the next level."  Jonathan Bernstein from "Entertainment Weekly" was more critical, finding "Share My World" "uneven" and the songwriting "meandering and half finished." Tom Moon later wrote in "The Rolling Stone Album Guide" (2004) that the record displayed "Blige's hit-song savvy but fewer memorable performances" than previous albums.

Track listing.

Sample credits

Personnel.

Credits for "Share My World" adapted from AllMusic.

Share My World is the third studio album by American R&B singer-songwriter Mary J. Blige, released by MCA Records on Earth Day, April 22, 1997. The album became Blige’s first to open at number one on the US "Billboard" 200 album chart. Moreover, it is her first album where she serves as an executive producer, alongside Steve Stoute, who also shared executive producer credits on the album.

Share My World je třetí řadové album americké soulové zpěvačky Mary J. Blige, je to zároveň její první číslo jedna na americkém albovém žebříčku.


History.

Origins - 1929.

František Janeček started the company after working for several years in the armament industry. Although he didn't have any experience with motorcycles, he was familiar with production techniques. Janeček chose between the Austrian double piston two-stroke motor of Puch, the Berlin two-stroke of Schliha, and the new Wanderer 500 cc. Janeček chose the Wanderer. Because of the collapse of the German motor industry, Wanderer had already decided to stop the production. This is because they didn't think it would be profitable to directly compete with BMW. The first model was introduced on October 23, 1929. This was a 500 cc 4-cycle engine with 12 kW of power (18 hp) and fuel consumption of 6 liters per 100 km. Although priced highly, through the first years (and several constructional fixes) this motorcycle was successful and was considered reliable.

1930s.

Due to the economic recession of the early 1930, a cheaper and simpler motorcycle was needed. Year 1932 marked the introduction of JAWA 175, with its 3.6 kW engine. This light (70 kg) machine was capable of speeds up to 80 km/h and fuel consumption of 3.5 liters per 100 km. The first year of production was an immediate success, selling over 3000 of the JAWA 175, almost three times the number of the 500 cc model over three years of production. The production of this model was finally stopped 1946. In 1938 JAWA was the first to offer test rides during exhibition show. This test motorcycle was a custom JAWA 175 equipped with dual handlebars, second pair being in the back for the instructor. Over the years total of 27,535 units of the JAWA 175 were built. It had a high demand.

1940s.

The company founder František Janeček died. Post-war production of JAWA 250/350 motorcycles restarted, and production of the JAWA 350 Ogar motorcycle began.

1950s.

Motorcycles produced in the 1950s include the 250, 350/353-kývačka, and 350/354-Kývačka, 500 OHC and the 50 Pionýr and Jawetta mopeds. A powerful 2-stroke motorcycle known as the JAWA 250/350 with the compact engine, rear wheel suspension and many other innovations was exported to more than 120 countries worldwide. It was one of the most successful models of JAWA.

1960s.

In the early 1960s, Jawa invented as the first in the world an automatic clutch for motorcycles. The automatic centrifugal clutch was designed for Jawa 250/559 and 350/360. Because the clutch has been patented worldwide by Jawa and it was copied in the Honda 50 Cub motorcycle, Honda had to retreat, pay a fine and pay a license fees from each motorcycle sold.

Other activities.

JAWA automobiles.

In 1934 Jawa introduced its first car, the Jawa 700, which was based on the DKW F2. Produced was overall 1,002 vehicles. The sports car special Jawa 750 was built only in six pieces for the 1000-mile Czechoslovakian race in 1933-35. It is a legend in the history of domestic motor sports. In 1937, the modified Jawa 600 Minor was introduced, which replaced JAWA 700. Production continued in limited numbers throughout WWII, and some were assembled in the immediate postwar period. Over 14,000 vehicles were produced and more than half were exported abroad. The successor was Aero Minor, that was however manufactured by Aero and not by Jawa, from 1946 to 1952.

JAWA mopeds.

Under the name Jawa were in foreign markets sold a variety of mopeds with two-stroke engines. In Czechoslovakia (and foreign markets after 1989), these mopeds were known as Babetta. The mopeds were produced since 1970 in Slovakian Považská Bystrica and Kolárovo. It was the first company to include electronic ignition on mopeds. In the first half of the 1990s, sales and exports gradually began to decline and the production of all Babetta types was definitely discontinued in 1999.

JAWA in India.

Ideal Jawa India Ltd based in Mysore was an Indian motorcycle company, which produced licensed Jawa motorcycles beginning in 1960 under the brand name "Jawa" and from 1973 as "Yezdi". The company stopped production in 1996. Jawa motorcycles have a cult following to this present day. Earlier models were manufactured in Czechoslovakia and bore the CZ Jawa emblem on the side of the fuel tank. The locally manufactured models always had O within the Jawa emblem. The Jawa 250 Type 353/04 designated as 'A' Type, Yezdi 250 'B' Type, Jawa 350 type 634 Twin and Yezdi 250 Monarch models are driven to this day in various parts of the country. Jawa and Yezdi bikes, especially the ones with fuel tank paddings and ignition systems on the fuel tank are now collectors items.  In October 2016, Mahindra & Mahindra, through its subsidiary Classic Legends Private Limited (CLPL), signed a licensing deal to launch motorbikes under the JAWA brand name in India and some other East Asian countries. In November 15, 2018, the Classic Legends launched three motorcycles in India: Jawa 300, Jawa 42 and Jawa Perak. They are manufactured in Pithampur.  The Jawa 300 is inspired by the company’s popular classic Jawa 250, while the Jawa 42 is a more modern and urban offering. The Jawa Perak, however, is a completely different model, a custom bobber motorcycle. Jawa 300 and Jawa 42 share the same technical specifications - 293cc (20,4 kW), front disc, rear drum, single channel ABS, fuel injection system, liquid cooled engine with radiator and kerb weight of 170 kg. In terms of appearance, classic Jawa has more of chrome while the Jawa 42 has a modern feel to attract younger riders. It is a competitor to the Royal Enfield bikes in Indian markets. Jawa Perak has 334cc (22,3 kW) engine.  Indian customers' interest in new machines exceeded all expectations. The originally planned production capacity was not enough, the production line was therefore doubled at about 30,000 motorcycles a year and the start of serial production had to move on to later. JAWA Motorcycles (Classic Legends) inaugurated the first Jawa dealership in December 2018 in the Pune region of Maharashtra.  They launched an initiative #ForeverHeroes to lend a helping hand to the children of martyrs, by auctioning the first 13 motorcycles from its production batch. This fundraiser was a first-of-its-kind event which helped Jawa raise a staggering Rs 1,49,25,000 from just 13 bikes. The entire amount received in the fundraising auction will be utilized towards education for the children of martyrs of the Indian Armed Forces.  Indian Jawa 300 will be sold in the Czech Republic from autumn 2020, under the name Jawa 300 CL. Jawa 42 should also be imported here over time.

Racing.

Historically JAWA had been very active in racing, and by far the most active East-European manufacturer in mechanical sports. In track racing JAWA sustained a presence in the World Championship until the mid-sixties with respectable performances considering their limited budget. In Motocross the firm built an impressive record before its four-stroke engines became superseded by two-stroke engines. In speedway racing, dirt track racing and ice racing where four-stroke engines were still at an advantage, the firm remained a dominant force. The separate factory where these competition motorcycles were built became a separate company following the privatizations of the 1990s with the fall of Communism.  Speedway World Champions who have ridden successfully on Jawa bikes include Ivan Mauger, Barry Briggs, Ove Fundin, Ole Olsen, Michael Lee, Tony Rickardsson and Kelvin Tatum.

Recent information.

Following the dissolution of Czechoslovakia and the dismantling of the communist bloc, the Jawa consumer motorcycles and the Speedway competition bikes were divided into separate companies. In the nineties, JAWA Speedway bikes were successful, while the Jawa company has struggled. As of 2006 the JAWA mostly produced bikes similar to Honda 250 cc, 125 cc and 50 cc motorcycles and large bike with a 650 cc Rotax engine (this engine can be also found in BMW F650 series).  Most sold is the iconic 350 cc two-stroke twin, which is almost mechanically unchanged since the 1970s but very reliable, currently are successfully sold mainly to the countries of Latin America.  The large bike with 650 cc Rotax engine was coming in three different variations. JAWA 650 classic was a retro bike with classic proportions. JAWA 650 Style was aimed for city streets and finally JAWA 650 Dakar was a full-scale enduro bike similar to Honda XL Transalp. The Jawa 660 was the successor to the Jawa 650, it was sold between 2011-2018.  Currently models (2020) sold by the JAWA company are: Jawa 350/640 - two-stroke engine Style/Military/Retro and four-stroke engine Jawa 350 OHC (meets the EURO IV standards).

JAWA () is a motorcycle and moped manufacturer founded in Prague, Czechoslovakia in 1929 by František Janeček, who bought the motorcycle division of Wanderer. The name JAWA was established by concatenating the first letters of Janeček and Wanderer. In the past, especially in the 1950s, JAWA was one of the top motorcycle manufacturers and exported its 350 into over 120 countries. The best known model was the "350" Pérák and in the 1970s the "350" Californian. It appeared in the typical black and red coloring from Californian beaches to New Zealand. After 1990, there was a significant loss of production. A successor company was formed in 1997 in Týnec nad Sázavou, continuing the name as JAWA Moto.

Jawa je nejstarší fungující československá firma zabývající se výrobou motocyklů už od roku 1929. Jméno firmy vzniklo složením jména zakladatele Františka Janečka, který zakoupil licenci na výrobu motocyklů Wanderer. V současnosti se firma oficiálně nazývá Jawa Moto spol. s r.o. a sídlí v Týnci nad Sázavou, jejím vlastníkem je podnikatel ing. Jiří Gerle. Dále existuje firma JAWA Divišov a.s., která se zabývá výrobou plochodrážních speciálů a sídlí v Divišově.


Biography.

Origin and early career.

Isaac was the son of Manuel Erotikos Komnenos, who reportedly served as "strategos autokrator" of the East under Emperor Basil II and defended Nicaea against the rebel Bardas Skleros in 978. His mother's name was Maria, about whom nothing else is known. Manuel's native language was Greek and modern scholarship considers the family to have been of Greek origin. It is said that the family name was derived from the city of Komne, near Philippopolis in Thrace. Isaac was born.  As Maria had died early, on his deathbed in 1020, Isaac's father commended his two surviving sons Isaac and John to the care of Emperor Basil II. According to Nikephoros Bryennios the Younger, the two children were raised with the outmost solicitude: they were raised at the Stoudion monastery with the best tutors, while care was taken to teach them how to hunt and military exercises. As soon as they came of age, Isaac and his brother joined the imperial bodyguard, the "Hetaireia".  At a young age, perhaps as early as 1025, Isaac married Catherine of Bulgaria (born ), a daughter of Ivan Vladislav (), the last Tsar of the First Bulgarian Empire. From he held the post of "stratopedarches" of the East—likely denoting that he was "domestikos ton scholon", commander-in-chief, of the eastern field army, but this title is not explicitly attested—and the ranks of "magistros" and "vestes". He was dismissed by Empress Theodora in 1054, and replaced by her eunuch confidant, the "proedros" Theodore.

Revolt of the eastern generals.

Michael VI and the military leadership.

When Michael VI Bringas came to the throne in 1056, Isaac was chosen to lead a deputation of eastern generals to the new emperor. Michael VI engaged in mass promotions of individuals—in the eyes of the contemporary courtier Michael Psellos, to an excessive degree—and the military sought to partake in the emperor's bounty. This was not a trivial matter: the debasement of the Byzantine currency under Constantine IX Monomachos () had affected military pay—not coincidentally presided over by none other than Michael Bringas, who was then military logothete—and while civil officials were compensated by being raised to higher dignities (which commanded higher salaries, "rogai"), the army was not. This exacerbated the already simmering dislike of the military aristocracy for the "regime of eunuchs and civilian politicians" that had dominated the empire during the last decades of the Macedonian dynasty. At Easter 1057, the traditional time when the emperor paid title holders their stipends, the delegation presented itself before the emperor. Along with Isaac, the delegation included the "magistros" Katakalon Kekaumenos, who had just been dismissed as "doux" of Antioch; the "vestarches" Michael Bourtzes, whose namesake grandfather had captured Antioch for Byzantium a century earlier; Constantine Doukas, married to a niece of the Patriarch of Constantinople Michael Keroularios; his brother John Doukas, a friend of Psellos; and others not explicitly named. As the historian Anthony Kaldellis comments, this was a formidable assemblage, as the families represented in it, all of them descended from military men promoted by the warrior-emperor Basil II, would define "the future of the empire for the next thirty years, indeed for the next century and more".  Psellos himself was an eyewitness at the reception of the generals' delegation, and claims that the emperor began abusing them at once; he then made Isaac, as the leader of the deputation, and his second, Kekaumenos, stand forth, and proceeded to denounce him, claiming that he was responsible for "all but losing Antioch" and "corrupting his army", being a coward and incompetent, and of having misappropriated army funds for his own use. John Skylitzes, who wrote later in the century, reports that the emperor treated the generals courteously, but agrees that he refused outright to consider the honours they claimed for themselves, notably the promotion of Isaac and Kekaumenos to the rank of "proedros".  The effect of the emperor's attitude on the army leadership was profound, and turned them against Michael. A second delegation to the chief minister, the "protosynkellos" Leo Paraspondylos, was received in similar manner, and according to Psellos Isaac could barely restrain his colleagues from attacking the emperor then and there, in his own throne room. At length, a plot was formed against the emperor, and despite Isaac's own reluctance, according to Psellos, he was nominated as its leader.  The conspirators contacted the veteran general Nikephoros Bryennios—who had unsuccessfully tried to usurp the throne from Theodora—but had recently been recalled by Michael VI as commander of the Macedonian army, and he apparently agreed to support them. Soon after, however, Bryennios left with his troops for Asia Minor, to campaign against the Turks. Once in the Anatolic Theme he quarreled with the army treasurer, threw him in prison, and appropriated the funds to pay his soldiers as he saw fit. This was seen by another local commander as a sign of rebellion; Bryennios was arrested and blinded.

Proclamation of Isaac as emperor and the Battle of Hades.

Fearing that their plot was about to be discovered, the eastern generals hastened to act: the conspirators resident in the Anatolic Theme, Romanos Skleros, Michael Bourtzes, Nikephoros Botaneiates and the sons of Basil Argyros, hastened to find Isaac Komnenos at his estates near Kastamon in Paphlagonia, and on 8 June 1057, at a place called Gounaria, proclaimed him emperor. It is unclear whether any of the rebels held command of troops; rather, according to Kaldellis, "they had to canvass for support among the officers and soldiers and forge orders of imperial appointment for themselves". Thus Skylitzes reports that Kekaumenos had to forge imperial letters to mobilize the regiments of the Armeniac Theme. With this force he went to join Komnenos. Leaving his family for safety with his brother at the fortress of Pemolissa on the banks of the Halys River, Komnenos advanced west towards Constantinople.  At the same time, the western regiments, and some of the eastern ones too, remained loyal to Michael VI. The emperor placed them under the command of Theodora's eunuch favourite, the "proedros" Theodore, and Aaron, Isaac's brother-in-law; unlike previously, he now showered his commanders with honours to secure their allegiance. The loyalist army assembled at Nicomedia, controlling the direct route to the capital. Therefore Komnenos turned south and seized Nicaea as his base of operations.  The two armies met at the Battle of Hades, near Nicaea. In a hard-fought battle with many casualties, the loyalist left defeated the right wing of the rebel army. On the other flank, Kekaumenos broke through the loyalists to capture their camp and decide the battle, while Isaac held the centre.

Negotiations and downfall of Michael VI.

Michael VI then attempted to negotiate with the rebels, sending Psellos, Leo Alopos, and the former "mesazon" of Constantine IX, Constantine Leichoudes, to Isaac's camp. Michael offered to adopt Isaac as his son and to grant him the title of "Caesar", making him effectively his successor, but this was rejected in a public audience. Psellos claims that Isaac was inclined to accept, the pressure of the assembled troops, who vocally refused it, forced him to agree with his supporters. At a private meeting afterwards Isaac insisted that he had accepted the title of emperor only under the pressure of his followers, and secretly accepted the offer, provided that Michael would also share "some, at any rate, of his imperial power", so that he could make appointments and reward his followers, especially in the military.  The envoys returned to Constantinople, and rapidly secured Michael's consent to the proposal; the Emperor explicitly agreed to pardon Isaac's followers, and to accord Isaac additional honours above those of "Caesar", setting him up almost as a co-emperor ("symbasileus"). As a sign of good faith, furthermore, Paraspondylos was dismissed from office. When the envoys returned to Isaac with these news, he publicly accepted the proposal and prepared to enter the capital.  Back in Constantinople, however, a crowd of officials assembled in the Hagia Sophia and began protesting that by making a deal, the Emperor was forcing them to renounce their oaths to oppose the rebels. Going a step further, they began themselves acclaiming Isaac as emperor. After a short while, on 30 August, Michael Keroularios and the clergy joined their cause, raising suspicions that this "spontaneous" assembly had been planned by the ambitious and wily Patriarch all along. Pressured by Keroularios and wanting to avoid bloodshed in the city, Michael agreed to abdicate. He was quickly tonsured and retired to a monastery. On the next day, 31 August, Isaac and his entourage crossed the Bosporus into Constantinople and entered the palace; on 1 September, he was crowned emperor by the Patriarch in the Hagia Sophia.

Reign.

The first act of the new emperor was to reward his partisans: his fellow conspirators were named to high offices—his own brother John was named "domestikos ton scholon" of the West and received the high title of "kouropalates", which was also awarded to Kekaumenos and Bryennios. The troops that had followed him received a donative and were quickly sent back east, to avoid any trouble with the populace of Constantinople. Patriarch Michael Keroularios was also rewarded for his support, by receiving sole authority for all personnel and financial matters of the Church, which were previously under the purview of the emperor, while the Patriarch's nephews received high court titles.

Fiscal reforms.

Isaac's rise to power was a turning point in Byzantine history, marking the definitive end of the long-lived Macedonian dynasty. Although powerful generals had previously suborned power, they had ruled alongside the Macedonian emperors; Isaac was the first military strongman to usurp power outright since the 9th century. This was reflected in the coinage struck in his name, which uniquely showed him holding a drawn sword; while it may have simply indicated his intention to restore "capable military rule" (Kaldellis), it came to be understood as a claim to rule by right of conquest, and even as expressing an impious belief "that his accomplishments came not from God but from his own prowess". Certainly it highlighted Isaac's determination to make reforms and restore the effectiveness of the army.  The task he faced was truly herculean, as the politically weak emperors of the previous thirty years had fostered corruption and inefficiency, handing out titles and their attendant state salaries ("rogai") in exchange for support. The devaluation of the coinage under Constantine IX had been a first reaction to the brewing crisis, but Isaac was the first emperor who certainly faced a budget deficit. To fund his cherished army, Isaac was therefore obliged to begin strict economies: he reduced or abolished the "rogai" of those who had been awarded titles, enforced a stricter and more efficient collection of taxes, reclaimed misappropriated imperial estates, and cancelled grants of such lands and tax exemptions made under Constantine IX and Michael VI, particularly those that had been granted to monasteries and churches, using a law of Nikephoros II Phokas. Even though salaries of officials, especially members of the Senate, were cut, Isaac's efforts were enthusiastically received even among some senior members of the civil bureaucracy, judging by the comments of Psellos and Michael Attaleiates.

Downfall of Keroularios.

The only point of criticism raised by Psellos is his haste and severity, judging that by a more gradual and judicious, step by step approach, he would have reaped greater success with far less opposition. Thus his appropriation of Church lands provoked the reaction of Michael Keroularios, with whom Isaac's relations had been steadily deteriorating. The Patriarch's role in Isaac's accession and his extensive new powers over the Church quickly went to his head. He is said to have admonished and berated the emperor, and even going as far as threatening to destroy him "like an oven he had made". He is also alleged to have worn imperial purple boots, a privilege restricted to the emperor, and which may indicate, according to Kaldellis, that Keroularios was influenced by Papal theories and conceived of the secular and clerical powers as co-equal, a traditional Byzantine approach known as a symphonia.  Finally, on 8 November 1058, while Keroularios was visiting a church outside the city walls, and hence was away from his supporters in the urban mob, Isaac sent the Varangian Guard to arrest him and take him to Prokonnesos, where he was placed under house arrest. Isaac applied considerable pressure on Keroularios to resign, but the latter steadfastly refused. In the end, the emperor decided to convene a synod against the Patriarch. This too was to take place away from the capital, somewhere in Thrace, with Psellos, who had himself been earlier persecuted by Keroularios, as the chief accuser. In the event, Keroularios died on 21 January 1059, before the synod could take place. Isaac appointed the bureaucrat Constantine Leichoudes as the new patriarch.

Military situation.

The rebellion and civil war that brought Isaac to the throne had concentrated Byzantium's military might away from its borders. The contemporary Armenian historian Aristakes Lastivertsi reports that the Georgian lord Ivane took advantage of this opportunity to capture two Byzantine frontier forts as well as an imperial tax collector, and lay siege to Theodosiopolis. The Byzantine "doux" at Ani drove him off, but Ivane then called upon some Turks for assistance. About a month after Isaac's coronation, these raiders reached Melitene, whose inhabitants were allowed to depart before the city was plundered by the Turks. Local Byzantine troops managed to blockade the mountain passes, forcing the raiders to winter in the region of Chorzane. In spring 1058, the Turks were ambushed and defeated while they attacked the fort of Mormrans, leaving most of their captives behind. At about the same time or shortly after, another Turkish raid into Taron was heavily defeated by the Byzantine defenders. Melitene was repaired and refortified, and made the seat of a "doux".  Constantine IX had famously abolished the military obligations of the Armenian thematic troops in exchange for cash payments, a step widely regarded, both by contemporaries and modern historians, as having catastrophic consequences for Byzantium's eastern defences, especially against the mounting Turkish threat. While Isaac does not appear to have acted to restore the thematic armies, according to Kaldellis, the reaction of the local forces to these events does not appear to indicate a degradation of Byzantium's defensive abilities in the East, but rather the continued and successful application of old-established counter-raiding principles as codified in Nikephoros Phokas' "De velitatione bellica" a century earlier.  Isaac led only one military expedition, in late summer of 1059, into the Balkan provinces that had been suffering raids by the Hungarians and the Pechenegs. The details of the campaign are obscure, but the two had possibly entered into an alliance. At Serdica, the emperor made a treaty with the Hungarians—who appear to have kept the fortress town of Sirmium—before moving against the Pechenegs in the area of Moesia. Most of the Pechenegs submitted again to imperial authority. The only major combat was against the fortified stronghold of a certain Selte, a recalcitrant Pecheneg leader.  On its return march the army was caught in a sudden storm on 24 September. Many men and supplies were lost, while Isaac barely escaped death when a tree struck by lightning fell next to him. This was followed by the false rumour that a tax assessor in the eastern provinces was plotting rebellion, and Isaac hastened back to the capital. Despite these events, Psellos claims that at this point Isaac's character changed markedly, and that he became "more haughty to such an extent that he held everyone else in contempt", including his own brother.

Illness, abdication, and death.

Isaac was a passionate hunter with both the horse and the falcon, spending much time at a hunting lodge outside Constantinople. On a hunt he fell ill. As the fever lasted for several days, Isaac, fearing he would die soon, named Constantine Doukas as his successor on 22 November 1059, and agreed to resign and retire to a monastery. Psellos claims that he was the main author of this nomination, even against the initial opposition of Empress Catherine.  According to Psellos, Isaac began to recover soon after Doukas' nomination, and started reconsidering his decision. Psellos again took the decisive step of having Doukas publicly acclaimed as emperor on 23 December, with Psellos putting the purple sandals on his feet. Isaac then resigned to his fate, and was tonsured as a monk, retiring to the Stoudion monastery. Psellos' prominent role in these events may simply be exaggeration and self-promotion, especially as he was writing this part of his history during the reign of Constantine Doukas and his son Michael VII Doukas (). No contemporary or later source, not even during the Komnenian dynasty (1081–1180), described or implied a coup by Doukas and his supporters, and the legality of the transition was never questioned.  Empress Catherine remained at the palace, and was even allowed to be mentioned first in the imperial acclamations, with Doukas coming second. This joint reign lasted for a brief while, before she too retired to the Myrelaion monastery under the monastic name of Xene. Isaac lived the remainder of his life as a simple monk in Stoudion, readily performing menial tasks until he died in late 1060.

Family.

Isaac was married to Catherine of Bulgaria, a daughter of Ivan Vladislav, the last ruler of the First Bulgarian Empire. Isaac raised her to the position of "Augusta". The couple had at least two children:

Isaac I Komnenos or Comnenus (, "Isaakios ho Komnēnos"; – 1060) was Byzantine Emperor from 1057 to 1059, the first reigning member of the Komnenian dynasty.

Izák I. Komnenos (1005–1061 v Konstantinopoli) byl byzantským císařem v letech 1057 až 1059 a prvním vládnoucím členem pocházejícím z dynastie Komnenovců.


History.

Arab studies talk about the history of the Middle East and North Africa, before the rise of Islam to the present time. Covering a wide range of topics, such as methods, approaches, colonial history, gender, environmental and legal dimensions. It depends on the political, economic, social and cultural history of the region.

Linguistic.

Arabic is a language spoken by more than 422 million people from the ocean to the Gulf, as the Arabs say. This includes Morocco, Mauritania and Western Sahara in the west, and extends to Iraq, the Gulf states and Somalia in the east. The official language of 26 countries, one of the six official languages of the United Nations. It is also the sacred language of over 1.7 billion Muslims around the world, and the language written by some of the greatest works of literature, science and history in the world. According to the teachings of Islam, classical Arabic is the language in which God chose to speak to mankind through Muhammad in the seventh century of the Christian era. It is the language of the Quran, the holy book of Islam. This is the language of Islamic and classical texts. Modern Arabic is the language of books, news broadcasts, poetry and political speeches throughout the Arab world, a language that every child in primary school learns to read and write, a diverse language of Arabic poetic traditions, the precise language of theologians and theologians of the Internet. Knowledge of Arabic provides an opportunity to connect with people throughout the Middle East, providing access to the richness and passion of the contemporary Arab world. Arabic is a way to explore nearly 14 centuries of one of the most sophisticated, diverse, and rich intellectual traditions in the world.

Culture and society.

Culture and society in the Arab world, from structures, institutions, art, poetry, letters, practices, and definitions of identity, based on anthropology, sociology, literature and religious studies.

Development.

Development and political economy in the Arab world, focusing on economic and social development, education, humanitarian aid, and gender and environmental dimensions of development. This concentration is based on economic history, political economy, sociology and politics.

Politics.

Contemporary political developments in the Arab world and the Middle East. The program covers the study of domineering, nationalism, local institutions, politics, war, peacemaking, identity, security policies and environmental security. It relies on comparative policies, international relations, history, science, political economy and development.

Themes.

History of Arabs.

To understand the history of Arabs provides the indispensable basis to understand all aspects of Arabs and its culture. Themes of special interest are:

Theology.

"Kalam" () is one of the "religious sciences" of Islam. In Arabic, the word means "discussion" and refers to the Arabic tradition of seeking theological principles through dialectic. A scholar of "kalam" is referred to as a "mutakallim".

Philosophy.

Arabic philosophy is a part of Arab studies. It is a longstanding attempt to create harmony between faith, reason or philosophy, and the religious teachings of Arabs. A Muslim engaged in this field is called a Arab philosopher. It is divided in fields like:

Sciences.

Arabic science is science in the context of traditional ideas of Arabs, including its ethics and prohibitions. An Arab engaged in this field is called a Muslim scientist This is not the same as science as conducted by any Muslim in a secular context.

Architecture.

Arabic architecture is the entire range of architecture that has evolved within Arab culture in the course of the history of Arabs. Hence the term encompasses religious buildings as well as secular ones, historic as well as modern expressions and the production of all places that have come under the varying levels of Islamic influence.

Arab studies or Arabic studies is an academic discipline centered on the study of Arabs and Arab World. It consists of several disciplines such as anthropology, sociology, linguistics, historiography, archaeology, Anthropology, Cultural studies, Economics, Geography, History, International relations, Law, Literature, Philosophy, Psychology, Political science, Public administration and Sociology. The field draws from old Arabic chronicles, records and oral literature, in addition to written accounts and traditions about Arabs from explorers and geographers in the Arab World (Middle East-North Africa).

Arabistika je filologický vědní obor, který se zabývá arabským jazykem, arabskou literaturou, kulturou a dějinami Blízkého východu a Afriky. V ČR lze studovat arabistiku – jako hlavní studijní obor – pouze na Filozofické fakultě Univerzity Karlovy v Praze.