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Chapter 11: - Page 3 of 12
Los Baños
(English version of "El Filibusterismo")
He explained the situation angrily to all the occupants of the sala, addressing himself especially to the three walking about, as if he had selected them for judges. The general played thus, he replied with such a card, Padre Irene had a certain card; he led, and then that fool of a Padre Irene didn't play his card! Padre Irene was giving the game away! It was a devil of a way to play! His mother's son had not come here to rack his brains for nothing and lose his money!
Then he added, turning very red,
If the booby thinks my money grows on every bush!... On top of the fact that my Indians are beginning to haggle over payments! Fuming, and disregarding the excuses of Padre Irene, who tried to explain while he rubbed the tip of his beak in order to conceal his sly smile, he went into the billiardroom.
Padre Fernandez, would you like to take a hand? asked Fray Sibyla.
I'm a very poor player, replied the friar with a grimace.
Then get Simoun, said the General.
Eh, Simoun! Eh, Mister, won't you try a hand?
What is your disposition concerning the arms for sporting purposes? asked the secretary, taking advantage of the pause.
Simoun thrust his head through the doorway.
Don't you want to take Padre Camorra's place, Señor Sindbad? inquired Padre Irene.
You can bet diamonds instead of chips.
I don't care if I do, replied Simoun, advancing while he brushed the chalk from his hands.
What will you bet?
What should we bet? returned Padre Sibyla.
The General can bet what he likes, but we priests, clerics—
Bah! interrupted Simoun ironically.
You and Padre Irene can pay with deeds of charity, prayers, and virtues, eh?
You know that the virtues a person may possess, gravely argued Padre Sibyla,
are not like the diamonds that may pass from hand to hand, to be sold and resold. They are inherent in the being, they are essential attributes of the subject—
I'll be satisfied then if you pay me with promises, replied Simoun jestingly.
You, Padre Sibyla, instead of paying me five something or other in money, will say, for example: for five days I renounce poverty, humility, and obedience. You, Padre Irene: I renounce chastity, liberality, and so on. Those are small matters, and I'm putting up my diamonds.

subset: commoncrawl


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Euclidean Geometry • Equitable Distributing • Linear Programming • Set Theory • Nonstandard Analysis • Advice • Topology • Number Theory • Computation of Time (Previous | Next)
Preliminary remarks: In the following section, the definitions established in the chapters on Set Theory and Topology are used, and usually take \(m, n \in {}^{\omega}\mathbb{N}^{*}\). Integration and differentiation are studied on an arbitrary non-empty subset \(A\) from \({}^{(\omega)}\mathbb{K}^{n}\). Every element not in the image set is replaced by the neighbouring element in the target set, and if multiple choices are possible one single choice is selected. Otherwise, the mapping is not meaningfully defined.
Definition: The function \(||\cdot||: \mathbb{V} \rightarrow {}^{(\omega)}\mathbb{R}_{\ge 0}\) where \(\mathbb{V}\) is a vector space over \({}^{(\omega)}\mathbb{K}\) is called a norm, if for all \(x, y \in \mathbb{V}\) and \(\lambda \in {}^{(\omega)}\mathbb{K}\), it holds that: \(||x|| = 0 \Rightarrow x = 0\) (definiteness), \(||\lambda x|| = |\lambda| \; ||x||\) (homogeneity), and \(||x + y|| \le ||x|| + ||y||\) (triangle inequality). The dimension of \(\mathbb{V}\) is defined as the maximal number of linearly independent vectors, and is denoted by dim \(\mathbb{V}\). The norms \({||\cdot||}_{a}\) and \({||\cdot||}_{b}\) are said to be equivalent if there exist non-infinitesimal \(\sigma, \tau \in {}^{c}\mathbb{R}_{>0}\) such that, for all \(x \in \mathbb{V}\), it holds that:\[\sigma||x||{}_{b} \le ||x||{}_{a} \le \tau||x||{}_{b}.\]Theorem: Let \(N\) be the set of all norms in \(\mathbb{V}\). Every norm on \(V\) is equivalent if and only if \({||x||}_{a}/{||x||}_{b}\) is finite but not infinitesimal for all \({||\cdot||}_{a}, {||\cdot||}_{b} \in N\) and all \(x \in \mathbb{V}^{*}\).\(\triangle\)
Proof: Set \(\sigma := \text{min }\{{||x||}_{a}/{||x||}_{b}: x \in \mathbb{V}^{*}\}\) and \(\tau := \text{max }\{{||x||}_{a}/{||x||}_{b}: x \in \mathbb{V}^{*}\}.\square\)
Definition: The area or half of the circumference of the unit circle defines pi \(\pi\). Let \(\iota := \pi/2\). Euler's number \(e\) as the solution of the equation \({x}^{i\pi} = -1\). Then the logarithm function \(\ln\) is defined by \({e}^{\ln \, z} = z\) and the corresponding power function by \({z}^{s} = {e}^{s \, \ln \, z}\) for complex \(s\) and \(z\). This allows us to give a formal definition of exponentiation. The logarithm of \(a \in {}^{\omega}\mathbb{C} \setminus {}^{\omega}\mathbb{R}_{\le 0}\) and base \(b \in {}^{c}\mathbb{R}_{>0}\) is denoted by \({_b}{a}\). In the following, let \(||\cdot||\) be the Euclidean norm.\(\triangle\)
Remark: The preceding definition of \(e\) is \(\mathcal{O}(\hat{c})\) larger than that by \({(1 + \hat{c})}^{c}\). The exponential series being exactly differentiated with as many terms as possible justifies the former. This deviation can have negative consequences when attempting to calculate as precisely as possible: Typically, resorting to approximations will be necessary. Adjoining the symbol \(\infty \gg \varsigma^2\) to the real numbers allows calculations like having a constant. If \(\pm0\) is replaced by \(\pm\hat{\infty}\), the calculations become unique and consistent. A generalisation to other sets and norms is easy if the latter are equivalent.
Definition: The function \({\mu}_{h}: A \rightarrow \mathbb{R}_{\ge 0}\) where \(A \subseteq {}^{(\omega)}\mathbb{C}^{n}\) is an \(m\)-dimensional set with \(h \in \mathbb{R}_{>0}\) less than or equal the minimal distance of the points in \(A, m \in {}^{\omega}\mathbb{N}_{\le 2n}\), \({\mu}_{h}(A) := |A| {h}^{m}\) and \({\mu}_{h}(\emptyset) = |\emptyset| = 0\) is called the exact h-measure of \(A\) and \(A\) is said to be h-measurable. Let the exact standard measure be \({\mu}_{\text{d0}}\) (d0 may be omitted).\(\triangle\)
Remark: This answers the measure problem positively: \({\mu}_{h}(A)\) is clearly additive and uniquely determined, i.e. if \(A\) is the union of pairwise disjoint \(h\)-homogeneous sets \({A}_{j}\) for \(j \in J \subseteq \mathbb{N}\), then\[{{\mu }_{h}}(A)=\sum\limits_{j \in J}{{{\mu }_{h}}\left( {{A}_{j}} \right)}.\]It is also strictly monotone, i.e. if \(h\)-homogeneous sets \({A}_{1}, {A}_{2} \subseteq {}^{(\omega)}\mathbb{K}^{n}\) satisfy \({A}_{1} \subset {A}_{2}\), then \({\mu}_{h}({A}_{1}) < {\mu}_{h}({A}_{2})\). If \(h\) is not equal on all considered sets \({A}_{j}\), the minimum of all \(h\) is chosen and the homogenisation follows as described in Set Theory.
Remark: The exact \(h\)-measure is more precise than other measures and is optimal. It simply considers the neighbourhood relations of points and its value is neither smaller nor greater than the distances of points parallel to the coordinate axes. Concepts such as \(\sigma\)-algebras and null sets are dispensable, since the only null set in this context is the empty set \(\emptyset\). Nonstandard mathematics also does without the concept of compactness in any form.
Examples: Consider the set \(A \subset [0, 1[\) of points, whose least significant bit is 1 (0) in their (conventionally) real binary representation. Then \({\mu}_{\text{d0}}(A) = \hat{2}\). Real numbers represent a further refinement of the conventionally real numbers, by dividing the conventionally real intervals into considerably finer sub-intervals. Since A is an infinite and conventionally uncountable union of individual points (without the neighbouring points of [0, 1[ in \(A\)) and these points are Lebesgue null sets, \(A\) is not Lebesgue measurable, however it is exactly measurable. Similarly, consider the subset \(Q\) of [0, 1[ \(\times\) [0, 1[ of all points with least significant bit 1 (0) in both coordinates. This set has exact measure \({\mu}_{\text{d0}}(Q) = \hat{4}\).
Definition: Neighbouring points in \(A\) are described by means of the irreflexive symmetric neighbourhood relation \(B \subseteq {A}^{2}\). The function \(\gamma: C \rightarrow A \subseteq \mathbb{C}{}^{n}\), where \(C \subseteq \mathbb{R}\) is \(h\)-homogeneous and h is infinitesimal, is called a path if \(||\gamma(x) - \gamma(y)||\) is infinitesimal for all neighbouring points \(x, y \in C\) and (\(\gamma(x), \gamma(y)) \in B\). The neighbourhood relations of \(B\) in \(A\) are systematically written as (predecessor, successor) with the notation \(({z}_{0}, \curvearrowright {z}_{0})\) or \((\curvearrowleft {z}_{0}, {z}_{0})\), where \(\curvearrowright\) is pronounced "post" and \(\curvearrowleft\) is pronounced "pre". This applies analogously to the neighbourhood relation \(D \subseteq C{}^{2}\).\(\triangle\)
Definition: Let \({z}_{0} \in A \subseteq \mathbb{K}^{n}\) and \(f: A \rightarrow {}^{(c)}\mathbb{K}^{m}\). In the following, proofs for predecessors will be omitted, since they are analogous to the proofs for successors. If \(||f(\curvearrowright B {z}_{0}) - f({z}_{0})|| < \alpha\) for infinitesimal \(\alpha \in {}^{(\omega)}\mathbb{R}{}_{>0}\), \(f\) is defined \(\alpha B\)-successor-continuous in \({z}_{0}\) in the direction \(\curvearrowright B {z}_{0}\). If the exact modulus of \(\alpha\) does not matter, \(\alpha\) may be omitted in the notation. If \(f\) is \(\alpha B\)-successor-continuous for all \({z}_{0}\) and \(\curvearrowright B {z}_{0}\), it simply is defined \(\alpha B\)-continuous. It holds that \(\alpha\) is the degree of continuity. If the inequality only holds for \(\alpha = \hat{c}\), \(f\) simply is defined (\(B\)-successor-)continuous. The property of \(\alpha B\)-predecessor-continuity is defined analogously.\(\triangle\)
Remark: In practice, choose \(\alpha\) by estimating \(f\) (for example after considering any jump discontinuities). If \(B\) is obvious or irrelevant, it may be omitted - as below, when \(B = {}^{(\omega)}\mathbb{K}{}^{2n}\).
Example: The function \(f: \mathbb{R} \rightarrow \{\pm 1\}\) with \(f(x) = {(-1)}^{x/\text{d0}}\) is nowhere successor-continuous on \(\mathbb{R}\), but its absolute value is (cf. Number Theory). Here, \(x/\)d0 is an integer since \(\mathbb{R}\) is d0-homogeneous. Setting \(f(x) = 1\) for rational \(x\) and \(= -1\) otherwise, then \(f(x)\) is partially d0-successor-continuous on non-rational numbers, unlike the conventional notion of continuity.
Example of a Peano curve (from [739], p. 188): "Consider the even, periodic function \(g: \mathbb{R} \rightarrow \mathbb{R}\) with period 2 and image [0, 1] defined by\[{g}(t)=\left\{ \begin{array}{cl} 0 & \text{for }0\le t<\tfrac{1}{3} \\ 3t-1 & \text{for }\tfrac{1}{3}\le t<\tfrac{2}{3} \\ 1 & \text{for }\tfrac{2}{3}\le t\le 1. \\ \end{array} \right.\,\] Clearly, g is fully specified by this definition, and continuous. Now let the function \(\phi: I = [0, 1] \rightarrow \mathbb{R}^{2}\) be defined by\[\phi(t) = \left( {\sum\limits_{k = 0}^{\infty} {\frac{{g({4^{2k}}t)}}{{{2^{k + 1}}}},} \sum\limits_{k = 0}^{\infty} {\frac{{g({4^{2k + 1}}t)}}{{{2^{k + 1}}}}} } \right)."\]The function \(\phi\) is at least continuous since the sums are ultimately locally linear functions in \(t\), when \(\infty\) is replaced by \(\omega\). It would however be an error to believe that [0, 1] can be bijectively mapped onto \({[0, 1]}^{2}\) in this way: the powers of four in \(g\), and the values 0 and 1 taken by \(g\) in two sub-intervals thin out \({[0, 1]}^{2}\) so much that a bijection is clearly impossible. Restricting the proof to rational points only is simply insufficient.
Definition: For \(f: A \rightarrow {}^{(\omega)}\mathbb{K}{}^{m}\),\[{d}_{\curvearrowright B z}f(z) := f(\curvearrowright B z) - f(z)\]is called \(B\)-successor-differential of \(f\) in the direction \(\curvearrowright B z\) for \(z \in A\). If dim \(A = n\), then \({d}_{\curvearrowright B z}f(z)\) can be specified by \(d((\curvearrowright B){z}_{1}, ... , (\curvearrowright B){z}_{n})f(z\)). If \(f\) is the identity, i.e. \(f(z) = z\), then \({d}_{\curvearrowright B z}Bz\) can be written instead of \({d}_{\curvearrowright B z}f(z)\). If \(A\) or \(\curvearrowright B z\) is obvious or irrelevant, it can be omitted. The conventional real case can be defined analogously to the above.\(\triangle\)
Remark: If the modulus of the \(B\)-successor-differential of \(f\) in the direction \(\curvearrowright B z\) at \(z \in A\) is smaller than \(\alpha\) and infinitesimal, then \(f\) is also \(\alpha B\)-successor-continuous at that point.
Definition: An (infinitely) real-valued function with arguments \(\in {}^{(\omega)}\mathbb{K}{}^{n}\) is said to be convex (concave) if the line segment between any two points on the graph of the function lies above (below) or on the graph. It is said to be strictly convex (concave) if "or on" can be omitted.\(\triangle\)
Definition: The \(m\) arithmetic means of all \({f}_{k}(\curvearrowright B z)\) of \(f(z)\) give the \(m\) averaged normed tangential normal vectors of \(m\) (uniquely determined) hyperplanes, defining the \(mn\) continuous partial derivatives of the Jacobian matrix of \(f\), which is not necessarily continuous. The hyperplanes are taken to pass through \({f}_{k}(\curvearrowright B z)\) and \(f(z)\) translated towards 0. The moduli of their coefficients are minimised by a very simple linear programme (cf. Linear Programming).\(\triangle\)
Theorem: Every in \(A \subseteq {}^{(\omega)}\mathbb{K}{}^{n}\) convex resp. concave function \(f: A \rightarrow {}^{(\omega)}\mathbb{R}\) is \(\alpha B\)-successor-continuous and \(B\)-successor-differentiable.\(\square\)
Definition: A point \(x (\curvearrowright x)\) of a function \(f: A \subseteq {}^{\omega}\mathbb{R} \rightarrow {}^{\omega}\mathbb{R}\) is said to be a right (left) jump discontinuity of \(s := |f(\curvearrowright x) - f(x)|\) upwards (downwards), or vice versa if \(s > \hat{\omega}\).\(\triangle\)
Theorem: A monotone function \(f: [a, b] \rightarrow {}^{\omega}\mathbb{R}\) has at most \(2\omega^2 - 1\) jump discontinuities.
Proof: Between \(-\omega\) and \(\omega\), at most \(2\omega^2\) jump discontinuities with a jump of \(\hat{\omega}\) are possible. If the function does not decrease at non-discontinuities, like a step function, the claim follows.\(\square\)
Remark: This theorem corrects Froda's theorem and makes it more precise. If each set is preceded by the superscript \({}^{c}\), the statement for conventional sets is obtained.
Definition: The partial derivative in the direction \(\curvearrowright B {z}_{k}\) of \(F: A \rightarrow {}^{(\omega)}\mathbb{K}\) at \(z = ({z}_{1}, ..., {z}_{n}) \in A \subseteq {}^{(\omega)}\mathbb{K}^{n}\) with \(k \in \mathbb{N}_{\le n}^*\) is defined as\[\frac{\partial B\,F(z)}{\partial B\,{{z}_{k}}}:=\frac{F({{z}_{1}},\,...,\,\curvearrowright B\,{{z}_{k}},\,...,\,{{z}_{n}})-F(z)}{\curvearrowright B\,{{z}_{k}}-{{z}_{k}}}.\]With this notation, if the function \(f\) satisfies \(f = ({f}_{1}, ..., {f}_{n}): A \rightarrow {}^{(\omega)}\mathbb{K}^{n}\) with \(z \in A \subseteq {}^{(\omega)}\mathbb{K}^{n}\)\[f(z)=\left( \frac{F(\curvearrowright B{{z}_{1}},{{z}_{2}},...,{{z}_{n}})-F({{z}_{1}},...,{{z}_{n}})}{(\curvearrowright B{{z}_{1}}-{{z}_{1}})},...,\frac{F({{z}_{1}},...,{{z}_{n-1}},\curvearrowright B{{z}_{n}})-F({{z}_{1}},...,{{z}_{n}})}{(\curvearrowright B{{z}_{n}}-{{z}_{n}})} \right)=\left( \frac{\partial B{{F}_{1}}(z)}{\partial B{{z}_{1}}},\,\,...\,\,,\,\,\frac{\partial B{{F}_{n}}(z)}{\partial B{{z}_{n}}} \right)=\text{grad }{{B}_{\curvearrowright Bz}}\,F(z)\,=\,\nabla {{B}_{\curvearrowright Bz}}\,F(z),\]then \(f(z)\) is said to be exact \(B\)-successor-derivative \({F'}_{\curvearrowright B z} B(z)\) or the exact \(B\)-successor-gradient \(\text{grad }_{\curvearrowright B z} F(z)\) of the function \(F\) at \(z\), which is said to be exactly \(B\)-differentiable at \(z\) in the direction \(\curvearrowright B z\), provided that each quotient exists in \({}^{(\omega)}\mathbb{K}\). \(\nabla\) is the Nabla operator. If this definition is satisfied for every \(z \in A\), then \(F\) is said to be an exactly \(B\)-differentiable \(B\)-antiderivative of \(f\). For \(x \in {}^{(\omega)}\mathbb{R}\), the left and right \(B\)-antiderivatives \({F}_{l}(x)\) and \({F}_{r}(x)\) distinguish between the cases of the corresponding \(B\)-derivatives.
If \(A\) or \(\curvearrowright B z\) are obvious from context or irrelevant, they can be omitted. The conventional case may be obtained analogously to the above. In the case \(n = 1\), \({F'}_{r}B(x)\) is defined the right exact \(B\)-derivative for \(\curvearrowright B x > x \in {}^{(\omega)}\mathbb{R}\) and that \({F'}_{l}B(x)\) is the left exact \(B\)-derivative for \(\curvearrowright B x < x\). If both directions have the same value, \(F'B(z)\) is called the exact derivative. When \(A ={}^{c}\mathbb{C}\) and \(n = 1\), this reduces to the conventional case where \(F\) is holomorphic.\(\triangle\)
Remark: Clearly, the \(B\)-antiderivatives of a given function only differ by an additive constant. The \(B\)-antiderivatives of discontinuous functions can typically only be derived by adding and appropriately recombining easier \(\alpha B\)-continuous functions (e.g. by reversing the rules of differentiation).
Remark: The rules stated below can be extended to (infinite) complex sets and left exact derivatives. The sets and neighbourhood relations will be subsequently omitted. Let \(f\) and \(g\) be right exactly differentiable (infinite) real functions at \(x \in A \subseteq {}^{(\omega)}\mathbb{R}\).
Chain rule: For \(x \in A \subseteq {}^{(\omega)}\mathbb{R}, B \subseteq {A}^{2}, f: A \rightarrow C \subseteq {}^{(\omega)}\mathbb{R}, D \subseteq {C}^{2}, g: C \rightarrow {}^{(\omega)}\mathbb{R}\), choosing \(f(\curvearrowright B x) = \curvearrowright D f(x)\)), it holds that:\[{g'}_{r}B(f(x)) = {g'}_{r}D(f(x)) {f'}_{r}B(x).\]Proof:\[{{{g}'}_{r}}B(f(x))=\frac{g(f(\curvearrowright Bx))-g(f(x))}{f(\curvearrowright Bx)-f(x)}\frac{f(\curvearrowright Bx)-f(x)}{\curvearrowright Bx-x}=\frac{g(\curvearrowright Df(x))-g(f(x))}{\curvearrowright Df(x)-f(x)}{{{f}'}_{r}}B(x)={{{g}'}_{r}}D(f(x)){{{f}'}_{r}}B(x).\square\]Product rule: \((fg){'}_{r}(x) = {f'}_{r}(x) g(x) + f(\curvearrowright x) {g'}_{r}(x)= {f'}_{r}(x) g(\curvearrowright x) + f(x) {g'}_{r}(x).\)
Proof: Add and subtract \(f(\curvearrowright x) g(x)\) resp. \(f(x) g(\curvearrowright x)\) in the numerator.\(\square\)
Quotient rule: Suppose that the denominators of the following quotients are non-zero. Then:\[\left( \frac{f}{g} \right)_{r}^{\prime }(x)=\frac{{{{{f}'}}_{r}}(x)\,g(x)-f(x)\,{{{{g}'}}_{r}}(x)}{g(x)\,g(\curvearrowright x)}=\frac{{{{{f}'}}_{r}}(x)\,g(\curvearrowright x)-f(\curvearrowright x)\,{{{{g}'}}_{r}}(x)}{g(x)\,g(\curvearrowright x)}.\]Proof: Add and subtract \(f(x) g(x)\) resp. \(f(\curvearrowright x) g(\curvearrowright x)\) in the numerator.\(\square\)
Remark: For the product and quotient rule to be as precise as the conventional one, the arguments and function values must belong to a smaller level of infinity than \(1/\)d0, and \(f\) and \(g\) must be sufficiently (\(\alpha\)-) continuous at \(x \in A\). I.e. \(\alpha\) must be sufficiently small) to allow \(\curvearrowright x\) to be replaced by \(x\). An analogous principle holds for infinitesimal arguments.
Remark: The right exact derivative of the inverse function\[{f}^{-1}{'}_{r}(y) = 1/{f'}_{r}(x)\]Can be derived from \(y = f(x)\) and the identity \(x = {f}^{-1}(f(x))\) using the chain rule with an equal level of precision. L'Hôpital's rule also makes sense for (\(\alpha\)-) continuous functions \(f\) and \(g\) such that \(f(v) = g(v) = 0\) with \(v \in A\) and \(g(\curvearrowright v) \ne 0\), and may be stated as:\[\frac{f(\curvearrowright v)}{g(\curvearrowright v)}=\frac{f(\curvearrowright v)-f(v)}{g(\curvearrowright v)-g(v)}=\frac{{{{{f}'}}_{r}}(v)}{{{{{g}'}}_{r}}(v)}.\]
Remark: Differentiability is thus easy to establish. In the (conventional) (infinite) real case, set\[{{{F}'}_{b}}B(v)\,:=\,\frac{F(\curvearrowright B\,v)-F(\curvearrowleft B\,v)}{\curvearrowright B\,v-\curvearrowleft B\,v}\] wherever this quotient is defined. This is especially useful when \(\curvearrowright B v - v = v - \curvearrowleft B v\), and the combined derivatives both have the same sign. This definition has the advantage of allowing us to view \({F'}_{b} \; B(v)\) as the "tangent slope" at the point \(v\), especially when \(F\) is \(\alpha B\)-continuous at \(v\). It also results in simpler rules of differentiation, in particular since a derivative value of 0 is most suitable for cases with opposite signs (see below). In other cases, simply calculate the arithmetic mean of both exact derivatives. This can be extended to the (conventional) complex numbers analogously.
Definition: Given \(z \in A \subseteq {}^{(\omega)}\mathbb{K}^{n}\), \[\int\limits_{z\in A}{f(z)dBz:=\sum\limits_{z\in A}{f(z)(\curvearrowright B\,z-z)}}\]is called the exact \(B\)-integral of the vector field \(f = ({f}_{1}, ..., {f}_{n}): A \rightarrow {}^{(\omega)}\mathbb{K}^{n}\) on \(A\) and \(f(z)\) is said to be\(B\)-integrable. If this requires removing at least one point from \(A\), then the exact \(B\)-integral is called improper.
For \(\gamma: [a, b[ \, \cap \, C \rightarrow A \subseteq {}^{(\omega)}\mathbb{K}^{n}, C \subseteq \mathbb{R}\), and \(f = ({f}_{1}, ..., {f}_{n}): A \rightarrow {}^{(\omega)}\mathbb{K}^{n}\)\[\int\limits_{\gamma }{f(\zeta )dB\zeta =}\int\limits_{t\in [a,b[ \, \cap \, C}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}D(t)dDt}\]where \(dDt > 0, \curvearrowright D t \in ]a, b] \, \cap \, C\), choosing \(\curvearrowright B \gamma(t) = \gamma(\curvearrowright D t)\), since \(\zeta = \gamma(t)\) and \(dB\zeta = \gamma(\curvearrowright D t) - \gamma(t) = {\gamma'}_{\curvearrowright }D(t) dDt\) (i.e. in particular for \(C = \mathbb{R}, B\) maximal in \(\mathbb{C}^{2}\), and \(D\) maximal in \(\mathbb{R}^{2})\), is called the exact \(B\)-line integral of the vector field \(f\) along the path \(\gamma\). Improper exact \(B\)-line integrals are defined analogously to exact \(B\)-integrals, except that only interval end points may be removed from \([a, b[ \, \cap \, C\).\(\triangle\)
Remark: The value of the exact line integral on \({}^{(c)}\mathbb{K}\) is usually consistent with the conventional line integral; however, \(f\) does not need to be continuous and the proper \(B\)-line integral exists always. It can easily be seen that the exact \(B\)-line integral is linear and monotone in the (conventional) (infinite) real case. The art of integration lies in correctly combining the summands of a sum.
Definition: For all \(x \in V\) of an \(h\)-homogeneous \(n\)-volume \(V \subseteq [{a}_{1}, {b}_{1}] \times...\times [{a}_{n}, {b}_{n}] \subseteq {}^{(\omega)}\mathbb{R}^{n}\) with \(B = {B}_{1}\times...\times{B}_{n}, {B}_{k} \subseteq {[{a}_{k}, {b}_{k}]}^{2}\) and \(|{dB}_{k}{x}_{k}| = h\) for all \(k \in \mathbb{N}_{\le n}^*\)\[\int\limits_{x\in V}{f(x){dBx}}:=\int\limits_{x\in V}{f(x)dB({{x}_{1}},\,...,{{x}_{n}})}:=\int\limits_{{{a}_{n}}}^{{{b}_{n}}}{...\int\limits_{{{a}_{1}}}^{{{b}_{1}}}{f(x)d{{B}_{1}}{{x}_{1}}\,...\,d{{B}_{n}}{{x}_{n}}}}\]is called the exact \(B\)-volume integral of the \(B\)-volume integrable function \(f: {}^{(\omega)}\mathbb{R}^{n} \rightarrow {}^{(\omega)}\mathbb{R}\) with \(f(x) := 0\) for all \(x \in {}^{(\omega)}\mathbb{R}^{n} \setminus V\). Improper exact \(B\)-volume integrals are defined analogously to exact \(B\)-integrals.\(\triangle\)
Remark: Because \(\mathbb{C}\) and \(\mathbb{R}^{2}\) are isomorphic, something similar exists in the complex case and\[\int\limits_{x\in V}{dBx={{\mu }_{h}}(V)}.\]Example: Using the exact \(B\)-volume integral in contrast to the Lebesgue integral,\[||f|{{|}_{p}}:={{\left( \int\limits_{x\in V}{||f(x)|{{|}^{p}}dBx} \right)}^{\hat{p}}}\]satisfies for arbitrary \(f: {}^{(\omega)}\mathbb{R}^{n} \rightarrow {}^{(\omega)}\mathbb{R}^{m}\) and \(p \in [1, \omega]\) all the properties of a norm, also definiteness.
Example: Let \([a, b[ \, \cap \, h{}^{\omega}\mathbb{Z}\) be a non-empty \(h\)-homogeneous subset of \([a, b[{}^{\omega}\mathbb{R}\), and write \(B \subseteq [a, b[ \, \cap \, h{}^{\omega}\mathbb{Z} \times ]a, b]h{}^{\omega}\mathbb{Z}\). Now let \({T}_{r}\) be a right \(B\)-antiderivative of a not necessarily convergent Taylor series \(t\) on \([a, b[ \, \cap \, h{}^{\omega}\mathbb{Z}\) and define \(f(x) := t(x) + {\varepsilon(-1)}^{x/h}\) for conventionally real \(x\) and \(\varepsilon \ge \hat{c}\). For \(h = \hat{c}\), \(f\) is nowhere continuous, and thus is conventionally nowhere differentiable or integrable on \([a, b[ \, \cap \, h{}^{\omega}\mathbb{Z}\), but for all \(h\) it gives the following exact result\[ f_{r}^{\prime }B(x)=t_{r}^{\prime }B(x)-2\widehat{dBx}\varepsilon {{(-1)}^{x/h}}\]und\[\int\limits_{x\in [a,b[ \, \cap \, h{}^{\omega }\mathbb{Z}}{f(x)dBx={{T}_{r}}(b)-{{T}_{r}}(a)+\,}\hat{2}\varepsilon \left( {{(-1)}^{a/h}}-{{(-1)}^{b/h}} \right).\]Example: The conventionally non-measurable middle-thirds Cantor set \({C}_{\hat{3}}\) has measure \({\mu}_{\text{d0}}({C}_{\hat{3}}) = {\delta}^{-\omega}\) for \(\delta := \frac{2}{3}\). Consider the function \(c: [0, 1] \rightarrow \{0, {\delta}^{\omega}\}\) defined by \(c(x) = {\delta}^{\omega}\) for \(x \in {C}_{\hat{3}}\) and \(c(x) = 0\) for \(x \in [0, 1] \setminus {C}_{\hat{3}}\). Then\[\int\limits_{x \in {{C}_{\hat{3}}}}{c(x)dx=\sum\limits_{x=0}^{1}{c(x)dx}}={{\delta}^{\omega}}{{\mu }_{\text{d0}}}\left( {{C}_{\hat{3}}} \right)=1.\]Definition: A sequence \(({a}_{k})\) with members \({a}_{k}\) is a mapping from \({}^{(\omega)}\mathbb{Z}\) to \({}^{(\omega)}\mathbb{C}^{m}: k \mapsto {a}_{k}\). A series is a equence \(({s}_{k})\) with \(m \in {}^{(\omega)}\mathbb{Z}\) and partial sums\[{{s}_{k}}=\sum\limits_{j=m}^{k}{{{a}_{j}}}.\triangle\]
Remark: Sums may be arbitrarily rearranged according to the associative, commutative, and distributive laws if care is taken to calculate them correctly (using Landau symbols).
Fubini's theorem: For \(X, Y \subseteq {}^{(\omega)}\mathbb{K}\), \(f: X\times Y \rightarrow {}^{(\omega)}\mathbb{K}\) satisfies\[\int\limits_{Y}{\int\limits_{X}{f(x,\,y)dBx\,}dBy}=\int\limits_{X\times Y}{f(x,\,y)dB(x,\,y)}=\int\limits_{X}{\int\limits_{Y}{f(x,\,y)dBy\,}dBx}.\]Proof: Reorder the sums corresponding to these integrals.\(\square\)
Example: Since\[\int\limits_{[a,\,b[\times [r,\,s[}{\frac{\left( {{x}^{2}}-{{y}^{2}} \right)}{{{\left( {{x}^{2}}+{{y}^{2}} \right)}^{2}}}{{d}^{2}}(x,\,y)}=\int\limits_{a}^{b}{\left. \frac{ydx}{{{x}^{2}}+{{y}^{2}}} \right|_{r}^{s}}=-\int\limits_{r}^{s}{\left. \frac{xdy}{{{x}^{2}}+{{y}^{2}}} \right|_{a}^{b}}=\arctan \frac{s}{b}-\arctan \frac{r}{b}+\arctan \frac{s}{a}-\arctan \frac{r}{a}\]by the principle of latest substitution (see below), the (improper) integral\[I(a,b):=\int\limits_{[a,\,b{{[}^{2}}}{\frac{\left( {{x}^{2}}-{{y}^{2}} \right)}{{{\left( {{x}^{2}}+{{y}^{2}} \right)}^{2}}}{{d}^{2}}(x,\,y)}=\arctan \frac{b}{b}-\arctan \frac{a}{b}+\arctan \frac{b}{a}-\arctan \frac{a}{a}= \iota - \iota = 0\]is obtained and not\[I(0,1)=\int\limits_{0}^{1}{\int\limits_{0}^{1}{\frac{\left( {{x}^{2}}-{{y}^{2}} \right)}{{{\left( {{x}^{2}}+{{y}^{2}} \right)}^{2}}}dy\,dx}}=\int\limits_{0}^{1}{\frac{dx}{1+{{x}^{2}}}}=\frac{\iota}{2}\ne -\frac{\iota}{2}=-\int\limits_{0}^{1}{\frac{dy}{1+{{y}^{2}}}}=\int\limits_{0}^{1}{\int\limits_{0}^{1}{\frac{\left( {{x}^{2}}-{{y}^{2}} \right)}{{{\left( {{x}^{2}}+{{y}^{2}} \right)}^{2}}}dx\,dy}}=I(0,1).\]Definition: A sequence \(({a}_{k})\) with \(k \in {}^{(\omega)}\mathbb{N}^{*}, {a}_{k} \in {}^{(\omega)}\mathbb{C}\) and \(\alpha \in ]0, \hat{c}]\) is said to be \(\alpha\)-convergent to \(a \in {}^{(\omega)}\mathbb{C}\) if there exists \(m \in {}^{(\omega)}\mathbb{N}^{*}\) satisfying \(|{a}_{k} - a| < \alpha\) for all \({a}_{k}\) with \(k \ge m\) such that the difference max \(k - m\) is not too large. The set \(\alpha\)-\(A\) of all such \(a\) is called set of \(\alpha\)-limit values of \(({a}_{k})\). An appropriately and uniquely determined representative of this set (e.g. the final value or mean value) is called the \(\alpha\)-limit value \(\alpha\)-\(a\). In the special case \(a = 0\), the sequence is called a zero sequence. If the inequality only holds for \(\alpha = \hat{c}\), the \(\alpha\)- is omitted from the notation.
Remark: The choice of \(k\) will usually be maximal and of \(\alpha\) minimal. Conventional limit values are often only precise to less than \(\mathcal{O}(\hat{\omega})\) and in general are too imprecise, since they are often e.g. (arbitrarily) algebraic (of a certain degree) or transcendental. The conventional formulation of the definition of conventional convergence, which always requires infinitely many or almost all members of the sequence to have an arbitrarily small distance from the limit value and only allows finitely many to have larger distance, needs to be extended, since otherwise only the largest index of each sequence is taken into account and considered to be relevant (cf. \cite{813}, p. 144). Only then is the monotone convergence valid (cf. [813], p. 155).
Remark: The statement that each positive number may be represented by a determined, unique, infinite decimal fraction is baseless because of the fundamental theorem of set theory (cf. p. 27 f.). Furthermore, any proof claiming that, for \(\varepsilon \in {}^{(\omega)}\mathbb{R}_{>0}\) - in particular whenever the phrase "for all conventionally reals \(\varepsilon > 0\)" is used - there exists a real number \(\varepsilon\hat{r}\) with real \(r \in {}^{(\omega)}\mathbb{R}_{>1}\), is false, because of \(\varepsilon := \; \curvearrowright 0\) or an infinite regression occurs. Therefore, in the \(\varepsilon\delta\)-definition of the limit value (it is questionable that \(\delta\) exists, p. 235 f.) and in the \(\varepsilon\delta\)-definition of continuity (see p. 215) \(\varepsilon\) must be restricted to specific multiples of \(\curvearrowright 0\): Consider for example the real function that doubles every real value but is not even uniformly continuous.
Remark: Uniform continuity need not be considered, since in general \(\delta := \; \curvearrowright 0\) and \(\varepsilon\) accordingly larger. If two function values do not satisfy the conditions, then the function is not continuous at that point. Thus, continuity is equivalent to uniform continuity, by choosing the largest \(\varepsilon\) from all admissible infinitesimal values. It is also easy to show that continuity is equivalent to Hölder continuity if infinite real constants are allowed. The same is true for uniform convergence, since simply the maximum of the indices may be chosen satisfying each argument as the index that satisfies everything, and \(\acute{\omega}\) is sufficient in every case. If this is not true for a given argument, then pointwise convergence also fails. Thus, uniform convergence is equivalent to pointwise convergence, by choosing the largest of all admissible infinitesimal values.
Intermediate value theorem: Let \(f: [a, b] \rightarrow {}^{(\omega)}\mathbb{R}\) be \(\alpha\)-continuous on \([a, b]\). Then \(f(x)\) takes every value between min \(f(x)\) and max \(f(x)\) to a precision of \(< \alpha\) as \(x\) ranges over \(\in [a, b]\). If \(f\) is continuous on \({}^{\omega}\mathbb{R}\), then it takes every value of \({}^{c}\mathbb{R}\) between min \(f(x)\) and max \(f(x)\).
Proof: Between min \(f(x)\) and max \(f(x)\), there is a gapless chain of overlapping \(\alpha\)-neighbourhoods centred around each \(f(x)\), by \(\alpha\)-continuity of \(f\). The second part of the claim follows from the fact that a deviation \(|f(\curvearrowright x) - f(x)| < \hat{c}\) or \(|f(x) - f(\curvearrowleft x)| < \hat{c}\) in \({}^{c}\mathbb{R}\) is smaller than the conventionally maximal admissible resolution.\(\square\)
Definition: The derivative of a function \(f: A \rightarrow {}^{(\omega)}\mathbb{R}\) where \(A \subseteq {}^{(\omega)}\mathbb{R}\), is defined to be zero if and only if 0 lies in the interval defined by the boundaries of the left and right exact derivatives.\(\triangle\)
Example: The (2d0)-continuous function \(f: {}^{(\omega)}\mathbb{R} \rightarrow \{0, \text{d0}\}\) defined by \(f(x):=\hat{2}\text{d0}\left( {(-1)}^{x/\text{d0}}+1 \right)\) consists of only the local minima 0 and the local maxima d0, and only has the left and right exact derivatives \(\pm 1\).
Example: The function \(f: [0, 1] \rightarrow [-\varsigma/\grave{\varsigma}, \varsigma/\grave{\varsigma}]\) for \(f(x) := (-1)^q q/\grave{q}\), if \(x\) is rational and has the denominator \(q \in \mathbb{Q}_{> 0}\), and \(f(x) := 0\) else, has the two relative extrema \(\pm \varsigma/\grave{\varsigma}\) (cf. [763], p. 24).
Example: For \(s \in {}^{(\omega)}\mathbb{C}\) where Re\((s) \le 1\), \(\zeta(s) = \sum\limits_{n=1}^{\omega}{n^{-s}}\) has definitely no analytic continuation (cf. [949], p. 4) and no zeros. This disproves the Riemann hypothesis:\[\sum_{n=1}^{\mathrm{\omega}}\frac{{(-1)}^n}{n^s}=2^{-\acute{s}}\sum_{n=1}^{\mathrm{\omega/2}}n^{-s}-\sum_{n=1}^{\mathrm{\omega}}n^{-s}\neq\left(2^{-\acute{s}}-1\right)\sum_{n=1}^{\mathrm{\omega(/2)}}n^{-s}\]First fundamental theorem of exact differential and integral calculus for line integrals: The function\[F(z)=\int\limits_{\gamma }{f(\zeta )dB\zeta }\]where \(\gamma: [d, x[ \, \cap \, C \rightarrow A \subseteq {}^{(\omega)}\mathbb{K}, C \subseteq \mathbb{R}, f: A \rightarrow {}^{(\omega)}\mathbb{K}, d \in [a, b[ \, \cap \, C\), and choosing \(\curvearrowright B \gamma(x) = \gamma(\curvearrowright D x)\) is exactly \(B\)-differentiable, and for all \(x \in [a, b[ \, \cap \, C\) and \(z = \gamma(x)\)\[F' \curvearrowright B(z) = f(z).\]Proof:\[dB(F(z))=\int\limits_{t\in [d,x] \, \cap \, C}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}D(t)dDt}-\int\limits_{t\in [d,x[ \, \cap \, C}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}D(t)dDt}=\int\limits_{x}{f(\gamma (t))\frac{\gamma (\curvearrowright Dt)-\gamma (t)}{\curvearrowright Dt-t}dDt}=f(\gamma (x)){{{\gamma }'}_{\curvearrowright }}D(x)dDx=\,f(\gamma (x))(\curvearrowright B\gamma (x)-\gamma (x))=f(z)dBz.\square\]Second fundamental theorem of exact differential and integral calculus for line integrals: According to the conditions from above, it holds with \(\gamma: [a, b[ \, \cap \, C \rightarrow {}^{(\omega)}\mathbb{K}\) that\[ F(\gamma (b))-F(\gamma (a))=\int\limits_{\gamma }{{{{{F}'}}_{\curvearrowright }}B(\zeta )dB\zeta }.\]Proof:\[F(\gamma (b))-F(\gamma (a))=\sum\limits_{t\in [a,b[ \, \cap \, C}{F(\curvearrowright B\,\gamma (t))}-F(\gamma (t))=\sum\limits_{t\in [a,b[ \, \cap \, C}{{{{{F}'}}_{\curvearrowright }}B(\gamma (t))(\curvearrowright B\,\gamma (t)-\gamma (t))}=\int\limits_{t\in [a,b[ \, \cap \, C}{{{{{F}'}}_{\curvearrowright }}B(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}D(t)dDt}=\int\limits_{\gamma }{{{{{F}'}}_{\curvearrowright }}B(\zeta )dB\zeta }.\square\]Corollary: If \(f\) has an antiderivative \(F\) on a closed path \(\gamma\), it holds with the conditions above that\[\oint\limits_{\gamma }{f(\zeta )dB\zeta :=}\int\limits_{\gamma }{f(\zeta )dB\zeta }=0.\square\]Remark: The conventionally real case of both fundamental theorems may be established analogously. Given \(u, v \in [a, b[ \, \cap \, C, u \ne v\) and \(\gamma(u) = \gamma(v)\), it may be the case that \(\curvearrowright B \gamma(u) \ne \; \curvearrowright B \gamma(v)\).
Definition: Let according to the trapezoidal rule\[\int\limits_{z\in A}^{=}{f(z)dBz:=\sum\limits_{z\in A}{\frac{(f(z)+f(\curvearrowright B\,z))}{2}(\curvearrowright B\,z-z)}}.\]Let according to the midpoint rule - assuming that \((z + \curvearrowright B z)/2\) exists -\[\int\limits_{z\in A}^{\doteq }{f(z)dBz:=\sum\limits_{z\in A}{f\left( \frac{z\,+\curvearrowright Bz}{2} \right)(\curvearrowright B\,z-z)}}.\triangle\]Remark: Since this tightened exact \(B\)-integral is clearly independent of the direction, it justifies (implicitly) theorems that cancel integral values in opposite directions, such as Green's theorem (see below). In the first fundamental theorem, the derivative \(dB(F(z))/dBz\) can be tightened to the arithmetic mean \((f(z) + f(\curvearrowright B z))/2\) resp. \((f(z + \curvearrowright B z)/2)\), and similarly, in the second fundamental theorem, \(F(\gamma(b)) - F(\gamma(a))\) can be tightened to \((F(\gamma(b)) + F(\curvearrowleft B \gamma(b)))/2 - (F(\gamma(a)) + F(\curvearrowright B \gamma(a)))/2\) resp. \(F((\gamma(b) + \curvearrowleft B \gamma(b))/2) - F((\gamma(a) + \curvearrowright B \gamma(a))/2)\), which yields approximately the original results when \(f\) and \(F\) are sufficiently \(\alpha\)-continuous at the boundary.
Leibniz integral rule: For \(f: {}^{(\omega)}\mathbb{K}^{n+1} \rightarrow {}^{(\omega)}\mathbb{K}, a, b: {}^{(\omega)}\mathbb{K}^{n} \rightarrow {}^{(\omega)}\mathbb{K}, \curvearrowright B x := {(s, {x}_{2}, ..., {x}_{n})}^{T}\), and \(s \in {}^{(\omega)}\mathbb{K} \setminus \{{x}_{1}\}\), choosing \(\curvearrowright D a(x) = a(\curvearrowright B x)\) and \(\curvearrowright D b(x) = b(\curvearrowright B x)\), it holds that\[\frac{\partial }{\partial {{x}_{1}}}\left( \int\limits_{a(x)}^{b(x)}{f(x,t)dDt} \right)=\int\limits_{a(x)}^{b(x)}{\frac{\partial f(x,t)}{\partial {{x}_{1}}}dDt}+\frac{\partial b(x)}{\partial {{x}_{1}}}f(\curvearrowright Bx,b(x))-\frac{\partial a(x)}{\partial {{x}_{1}}}f(\curvearrowright Bx,a(x)).\]Proof:\[\begin{aligned}\frac{\partial }{\partial {{x}_{1}}}\left( \int\limits_{a(x)}^{b(x)}{f(x,t)dDt} \right) &={\left( \int\limits_{a(\curvearrowright Bx)}^{b(\curvearrowright Bx)}{f(\curvearrowright Bx,t)dDt}-\int\limits_{a(x)}^{b(x)}{f(x,t)dDt} \right)}/{\partial {{x}_{1}}}\;={\left( \int\limits_{a(x)}^{b(x)}{(f(\curvearrowright Bx,t)-f(x,t))dDt}+\int\limits_{b(x)}^{b(\curvearrowright Bx)}{f(\curvearrowright Bx,t)dDt}-\int\limits_{a(x)}^{a(\curvearrowright Bx)}{f(\curvearrowright Bx,t)dDt} \right)}/{\partial {{x}_{1}}}\; \\ &=\int\limits_{a(x)}^{b(x)}{\frac{\partial f(x,t)}{\partial {{x}_{1}}}dDt}+\frac{\partial b(x)}{\partial {{x}_{1}}}f(\curvearrowright Bx,b(x))-\frac{\partial a(x)}{\partial {{x}_{1}}}f(\curvearrowright Bx,a(x)).\square\end{aligned}\]Remark: Integrating happens in the complex plane over a path whose start and end points are the limits of integration. If \(\curvearrowright D a(x) \ne a(\curvearrowright B x)\), then the final summand must be multiplied by \((\curvearrowright D a(x) - a(x))/(a(\curvearrowright B x) - a(x))\), and if \(\curvearrowright D b(x) \ne b(\curvearrowright B x)\), then the penultimate summand must be multiplied by \((\curvearrowright D b(x) - b(x))/(b(\curvearrowright B x) - b(x))\).
Example: Sine and cosine must be defined for period \(2\pi\), since their power series only converge for finite arguments.
Examples (cf. [813], p. 540 - 543 with \(n \in {}^{\omega}\mathbb{N}^{*}\) and \(x \in [0, 1]\) in each case):
1. The sequence \({f}_{n}(x) = \sin(nx)/\sqrt{n}\) does not tend to \(f(x) = 0\) as \(n \rightarrow \omega\), but instead to \(f(x) = \sin(\omega x)/\sqrt{\omega}\) with (continuous) derivative \(f'(x) = \cos(\omega x) \sqrt{\omega}\) instead of \(f'(x) = 0\).
2. The sequence \({f}_{n}(x) = x - \hat{n}x^{n}\) tends to \(f(x) = x - \hat{\omega}{x}^{\omega}\) as \(n \rightarrow \omega\) instead of \(f(x) = x\) with (continuous) derivative \(f'(x) = 1 - {x}^{\acute{\omega}}\) instead of \(f'(x) = 1\). Conventionally, \({f}_{n}(x) = 1 - {x}^{\acute{n}}\) is discontinuous at the point \(x = 1\).
3. The sequence \({f}_{n}(x) = (\hat{2}{n}^{2} - |{n}^{3}(x - \widehat{2n})|)(1 - \text{sgn}(x - \hat{n}))\) (or alternatively, expressed in terms of continuously differentiable functions\[{{f}_{n}}(x)=\left\{ \begin{array}{cl} 2{{n}^{3}}x & \text{for }x\in \left[ 0,\widehat{2n} \right] \\ 2{{n}^{2}}-2{{n}^{3}}x & \text{for }x\in \left] \widehat{2n},\hat{n} \right] \\ 0 & \text{for }x\in \left] \hat{n},1 \right] \\ \end{array} \right.\,\]does always not tend to 0 as \(n \rightarrow \omega\), but instead tends to different values depending on the value of \(x\) (replace \(n\) by \(\omega\) in \({f}_{n}(x))\). Furthermore, it holds that\[\int\limits_{x \in [0,1[} {{f_n}dx} = \hat{2}{n}\]and\[\int\limits_{x\in [0,1[}{f dx}=\hat{2}{\omega}\]instead of\[\int\limits_{x \in [0,1[} {fdx} = 0\]supposedly because \(f(x) = 0\).
4. The sequence \({f}_{n}(x) = (\hat{2}n - |{n}^{2}(x - \widehat{2n})|)(1 - \text{sgn}(x - \hat{n}))\) (or alternatively, expressed in terms of continuously differentiable functions)\[{{f}_{n}}(x)=\left\{ \begin{array}{cl} 2{{n}^{2}}x & \text{for }x\in \left[ 0,\widehat{2n} \right] \\ 2n-2{{n}^{2}}x & \text{for }x\in \left] \widehat{2n},\hat{n} \right] \\ 0 & \text{for }x\in \left] \hat{n},1 \right] \\ \end{array} \right.\,\]does not always tend to 0 as \(n \rightarrow \omega\), but instead tends to different values depending on the value of \(x\) (replace \(n\) by \(\omega\) in \({f}_{n}(x)\)). Furthermore, it holds that\[\int\limits_{x \in [0,1[} {{f_n}dx} = \int\limits_{x \in [0,1[} {fdx} = {\hat{2}}\]instead of\[\int\limits_{x \in [0,1[} {fdx} = 0\]supposedly because \(f(x) = 0\).
5. The sequence \({f}_{n}(x) = nx(-\acute{x})^{n}\) does not tend to \(f(x) = 0\) as \(n \rightarrow \omega\), but instead to the continuous function \(f(x) = {\omega x(-\acute{x})}^{\omega}\), and takes the value \(\hat{e}\) when \(x = \hat{\omega}\).
Finiteness criterion for series: Let \(j, k, m, n \in \mathbb{N}\). The modulus \(S_n := \left| \sum\limits_{k=0}^{n}{s_k} \right|\) for \(s_k \in {}^{(\omega)}\mathbb{C}\) is finite, if and only if for a monotonically decreasing sequence \(({d}_{j})\) such that \( d_j \in {}^{c}\mathbb{R}_{\ge 0}\) holds: \(S_n = \sum\limits_{j=0}^{m}{{{\left( -1 \right)}^{j}}{{d}_{j}}}.\)
Proof: The sum is bounded below by 0 and bounded above by \({d}_{0}\). The claim follows directly from the ability to arbitrarily rearrange summands, sort them according to their signs and sizes, and recombine them or split them into separate sums.\(\square\)
Example: From the alternating harmonic series, it follows that\[\sum\limits_{n=1}^{\omega }{{{\left( -1 \right)}^{n}}}\left( \omega -\hat{n} \right)={_e}2.\]Definition: The following rearrangement for \({a}_{m}, {b}_{n} \in {}^{(\omega)}\mathbb{K}\) gives the series product and corrects the Cauchy product:\[\sum\limits_{m=1}^{\omega }{{{a}_{m}}}\sum\limits_{n=1}^{\omega }{{{b}_{n}}}=\sum\limits_{m=1}^{\omega }{\left( \sum\limits_{n=1}^{m}{\left( {{a}_{n}}{{b}_{m-\acute{n}}}+{{a}_{\omega -\acute{n}}}{{b}_{\omega -m+n}} \right)}-{{a}_{m}}{{b}_{\omega -\acute{m}}} \right)}.\triangle\]Example: For following series product (cf. [763], p. 61 f.), it holds that:\[\left(\sum_{m=1}^{\mathrm{\omega}}\frac{{(-1)}^m}{\sqrt m}\right)^2=\sum_{m=1}^{\mathrm{\omega}}{\left(\sqrt{\frac{\hat{m}}{\mathrm{\omega}-\acute{m}}}-\sum_{n=1}^{m}{{(-1)}^m\left(\sqrt{\frac{\hat{n}}{m-\acute{n}}}+\sqrt{\frac{\widehat{\mathrm{\omega}-\acute{n}}}{\mathrm{\omega}-m\ \mathrm{+\ }n}}\right)}\right)=0,36590...\ }\ \ \ll\frac{{\zeta\left(\hat{2}\right)}^2}{3+2\sqrt2}.\]Example: The signum function sgn yields the following series product (cf. [763], p. 62): \[\sum\limits_{m=0}^{\omega }{{2}^{{{m}^{\text{sgn}(m)}}}}\sum\limits_{n=0}^{\omega}{\text{sgn}(n-\gamma)} = \acute{\omega}{2}^{\grave{\omega}}\gg -2.\]Definition: Let \(f: A \rightarrow {}^{(\omega)}\mathbb{K}\) for \(A \subseteq {}^{(\omega)}\mathbb{K}\). The left-hand side of\[\frac{d_{\curvearrowright B\,z}^{2}Bf(z)}{{{(d\curvearrowright B\,z)}^{2}}}:=\frac{f(\curvearrowright B(\curvearrowright B\,z))-2f(\curvearrowright B\,z)+f(z)}{{{(d\curvearrowright B\,z)}^{2}}}\]is called the second derivative of \(f\) at \(z \in A\) in the direction \(\curvearrowright B z.\triangle\)
Remark: Higher derivatives are defined analogously. Every number \({m}_{n} \in {}^{\omega}\mathbb{N}\) for \(n \in {}^{\omega}\mathbb{N}^{*}\) of derivatives is written as an exponent after the \(n\)-th variable to be differentiated. If \(n \ge 2\), the derivatives are called partial and \(d\) is replacedby \(\partial\). The exponent to be specified in the numerator is the sum of all \({m}_{n}\). Taylor series only make sense for \(\omega\)-times \(\alpha\)-continuously differentiable functions, due to approximating and convergence-related behaviour.
Exchange theorem: The result of multiple partial derivatives of a function \(f: A \rightarrow {}^{(\omega)}\mathbb{K}\) is independent of the order of differentiation, provided that variables are only evaluated and limits are only computed at the end, if applicable (principle of latest substitution).
Proof: The derivative is uniquely determined: This is clear up to the second derivative, and the result follows by (transfinite) induction for higher-order derivatives.\(\square\)
Example: Let \(f: {}^{\omega}\mathbb{R}^{2} \rightarrow {}^{\omega}\mathbb{R}\) be defined as \(f(0, 0) = 0\) and \(f(x, y) = {xy}^{3}/({x}^{2} + {y}^{2})\) otherwise. Then:\[\frac{{{\partial ^2}f}}{{\partial x\partial y}} = \frac{{{y^6} + 6{x^2}{y^4} - 3{x^4}{y^2}}}{{{{({x^2} + {y^2})}^3}}} = \frac{{{\partial ^2}f}}{{\partial y\partial x}}\]with value \(\hat{2}\) at the point (0, 0), even though the equation\[\frac{{\partial f}}{{\partial x}} = \frac{{{y^5} - {x^2}{y^3}}}{{{{({x^2} + {y^2})}^2}}} \ne \frac{{x{y^4} + 3{x^3}{y^2}}}{{{{({x^2} + {y^2})}^2}}} = \frac{{\partial f}}{{\partial y}}\]is equal to \(y\) on the left for \(x = 0\) and 0 on the right for \(y = 0\). Partially differentiating the left-hand side with respect to \(y\) gives \(1 \ne 0\), which is the partial derivative of the right-hand side with respect to \(x\).
Theorem: Splitting \(F: A \rightarrow {}^{(\omega)}\mathbb{C}\) into real and imaginary parts \(F(z) := U(z) + i V(z) := f(x, y) := u(x, y) + i v(x, y)\), and given infinitesimal \(h = |dBx| = |dBy|, h\)-homogeneous \(A \subseteq {}^{(\omega)}\mathbb{C}\), with the neighbourhood relation \(B \subseteq {A}^{2}\) for every \(z = x + i y \in A\) is holomorphic and\[r(h):=\frac{{\partial{}^{2}}Bf(x,y)}{\partial Bx\partial By\,}h\]is infinitesimal if and only if the Cauchy-Riemann partial differential equations\[\frac{{\partial Bu}}{{\partial Bx}} = \frac{{\partial Bv}}{{\partial By}},\,\,\frac{{\partial Bv}}{{\partial Bx}} = - \frac{{\partial Bu}}{{\partial By}},\]are satisfied by \(B\) in both the \(\curvearrowright\) direction and the \(\curvearrowleft\) direction.
Proof: Since\[\begin{aligned}F'B(z) &= \frac{{F(z \pm \partial Bx) - F(z)}}{{\pm \partial Bx}} = \frac{{F(z \pm i\partial By) - F(z)}}{{\pm i\partial By}} = \frac{{F(z + dBz) - F(z)}}{{dBz}} = \frac{{\partial Bu}}{{\partial Bx}} + i\frac{{\partial Bv}}{{\partial Bx}} = \frac{{\partial Bv}}{{\partial By}} - i\frac{{\partial Bu}}{{\partial By}} = \frac{{u(x \pm \partial Bx,y) + i\,v(x \pm \partial Bx,y) - u(x,y) - i\,v(x,y)}}{{\pm \partial Bx}} \\ &= \frac{{u(x,y \pm \partial By) + i\,v(x,y \pm \partial By) - u(x,y) - i\,v(x,y)}}{{\pm i\partial By}} = \frac{{\partial Bf}}{{\partial Bx}} = - i\frac{{\partial Bf}}{{\partial By}} = \hat{2}\left( {\frac{{\partial Bf}}{{\partial Bx}} - i\frac{{\partial Bf}}{{\partial By}}} \right) = \frac{{\partial BF}}{{\partial Bz}}\end{aligned}\]and \(dBz = dBx + i dBy\) for every derivative defined on \(A\), and since\[\begin{aligned}&u(\curvearrowright Bx,y)-u(x,y)+u(x,\curvearrowright By)-u(x,y)+u(\curvearrowright Bx,\curvearrowright By)-u(\curvearrowright Bx,y)-u(x,\curvearrowright By)+u(x,y) =u(\curvearrowright Bx,\curvearrowright By)-u(x,y) \\ &=\frac{\partial Bu(x,y)}{\partial Bx}dBx+\frac{\partial Bu(x,y)}{\partial By}dBy+\frac{\partial Bu(\curvearrowright Bx,y)}{\partial By}dBy-\frac{\partial Bu(x,y)}{\partial By}dBy =\frac{\partial Bu(x,y)}{\partial Bx}dBx+\frac{\partial Bu(x,y)}{\partial By}dBy+\frac{{{\partial}^{2}}Bu(x,y)}{\partial Bx\partial By}dBxdBy=dBU(z)\end{aligned}\]
as well as the analogous formulas for \(v\) and in the \(\curvearrowleft\) direction, it holds that\[F'B(z)\,dBz = dBF(z) = dBU(z) + i\,dBV(z) = \,\left( {\begin{array}{*{20}{c}}{\frac{{\partial Bu}}{{\partial Bx}}} & {\frac{{\partial Bu}}{{\partial By}}}\\{i\frac{{\partial Bv}}{{\partial Bx}}} & {i\frac{{\partial Bv}}{{\partial By}}}\end{array}} \right)\left( {\begin{array}{*{20}{c}}{dBx}\\{dBy}\end{array}} \right) + \frac{{{\partial ^2}Bf(x,y)}}{{\partial Bx\partial By}}dBxdBy.\]The assumptions allow to neglect the final summand, and so the claim follows.\(\square\)
Remark: In particular, the final summand may be neglected whenever \(f\) is continuous. The following necessary and sufficient condition is valid for \(F\) to be holomorphic:\[F'B(\bar z) = \frac{{\partial Bf}}{{\partial Bx}} = i\frac{{\partial Bf}}{{\partial By}} = \hat{2}\left( {\frac{{\partial Bf}}{{\partial Bx}} + i\frac{{\partial Bf}}{{\partial By}}} \right) = \frac{{\partial BF}}{{\partial B\bar z}} = 0.\]Definition: When integrating identical paths in opposite positive and negative directions, the counter-directional rule for integrals is adopted, stating that when following the path in the negative direction, the function value of the successor of the argument must be chosen if the function is too discontinuous, implying that the integral sums to 0 over both directions.\(\triangle\)
Remark: This convention is required in order to ensure that integrals that are expected to sum to zero do in fact do so. Without it, they could potentially have a significantly different value.
Counter-directional theorem: If the path \(\gamma: [a, b[ \, \cap \, C \rightarrow V\) with \(C \subseteq \mathbb{R}\) passes the edges of every \(n\)-cube of side length d0 in the \(n\)-volume \(V \subseteq {}^{(\omega)}\mathbb{R}^{n}\) with \(n \in \mathbb{N}_{\ge 2}\) exactly once, where the opposite edges in all two-dimensional faces of every \(n\)-cube are traversed in reverse direction, but uniformly, then, for \(D \subseteq \mathbb{R}^{2}, B \subseteq {V}^{2}, f = ({f}_{1}, ..., {f}_{n}): V \rightarrow {}^{(\omega)}\mathbb{R}^{n}, \gamma(t) = x, \gamma(\curvearrowright D t) = \curvearrowright B x\) and \({V}_{\curvearrowright } := \{\curvearrowright B x \in V: x \in V, \curvearrowright B x \ne \curvearrowleft B x\}\), it holds that\[\int\limits_{t \in [a,b[ \, \cap \, C}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}(t)dDt}=\int\limits_{\begin{smallmatrix} (x,\curvearrowright B\,x) \\ \in V\times {{V}_{\curvearrowright}} \end{smallmatrix}}{f(x)dBx}=\int\limits_{\begin{smallmatrix} t \in [a,b[ \, \cap \, C, \\ \gamma | {\partial{}^{\acute{n}}} V \end{smallmatrix}}{f(\gamma (t)){{{{\gamma }'}}_{\curvearrowright }}(t)dDt}.\]Proof: If two arbitrary squares are considered with common edge of length d0 included in one plane, then only the edges of \(V\times{V}_{\curvearrowright}\) are not passed in both directions for the same function value. They all, and thus the path to be passed, are exactly contained in \({\partial}^{\acute{n}}V.\square\)
Remark: It is not difficult to transfer both definition and theorem to the complex numbers.
Green's theorem: Given neighbourhood relations \(B \subseteq {A}^{2}\) for some simply connected \(h\)-set \(A \subseteq {}^{(\omega)}\mathbb{R}^{2}\), infinitesimal \(h = |dBx|= |dBy| = |\curvearrowright B \gamma(t) - \gamma(t)| = \mathcal{O}({\hat{\omega}}^{m})\), sufficiently large \(m \in \mathbb{N}^{*}, (x, y) \in A, {A}^{-} := \{(x, y) \in A : (x + h, y + h) \in A\}\), and a simply closed path \(\gamma: [a, b[\rightarrow \partial A\) followed anticlockwise, choosing \(\curvearrowright B \gamma(t) = \gamma(\curvearrowright D t)\) for \(t \in [a, b[, D \subseteq {[a, b]}^{2}\), the following equation holds for sufficiently \(\alpha\)-continuous functions \(u, v: A \rightarrow \mathbb{R}\) with not necessarily continuous partial derivatives \(\partial Bu/\partial Bx, \partial Bu/\partial By, \partial Bv/\partial Bx\) and \(\partial Bv/\partial By\):\[\int\limits_{\gamma }{(u\,dBx+v\,dBy)}=\int\limits_{(x,y)\in {{A}^{-}}}{\left( \frac{\partial Bv}{\partial Bx}-\frac{\partial Bu}{\partial By} \right)dB(x,y)}.\]Proof: Wlog the case \(A := \{(x, y) : r \le x \le s, f(x) \le y \le g(x)\}, r, s \in {}^{(\omega)}\mathbb{R}, f, g : \partial A \rightarrow {}^{(\omega)}\mathbb{R}\) is proved, since the proof is analogous for each case rotated by \(\iota\), and every simply connected \(h\)-set is a union of such sets. It is simply shown that\[\int\limits_{\gamma }{u\,dBx}=-\int\limits_{(x,y)\in {{A}^{-}}}{\frac{\partial Bu}{\partial By}dB(x,y)}\]since the other relation may is given analogously. Since the regions of \(\gamma\) where \(dBx = 0\) do not contribute to the integral, for negligibly small \(t := h(u(s, g(s)) - u(r, g(r)))\), it holds that\[-\int\limits_{\gamma }{u\,dBx}-t=\int\limits_{r}^{s}{u(x,g(x))dBx}-\int\limits_{r}^{s}{u(x,f(x))dBx}=\int\limits_{r}^{s}{\int\limits_{f(x)}^{g(x)}{\frac{\partial Bu}{\partial By}}dBydBx}=\int\limits_{(x,y)\in {{A}^{-}}}{\frac{\partial Bu}{\partial By}dB(x,y)}.\square\]Remark: The choice of \(m\) depends on the required number of sets of the type specified in the above proof, the union of which yields the simply connected \(h\)-set.
Goursat's integral lemma: If \(f\) is holomorphic on a triangle \(\Delta \subseteq {}^{(\omega)}\mathbb{C}\) but does not have an antiderivative on \(\Delta\), then\[I:=\int\limits_{\partial \Delta }{f(\zeta )dB\zeta }=0.\]Refutation of conventional proofs based on estimation by means of a complete triangulation: The direction in which \(\partial\Delta\) is traversed is irrelevant. If \(\Delta\) is fully triangulated, then wlog every minimal triangle \({\Delta}_{s} \subseteq \Delta\) must either satisfy\[{I_s}: = \int\limits_{\partial {\Delta _s}} {f(\zeta )dB\zeta } = f({z_1})({z_2} - {z_1}) + f({z_2})({z_3} - {z_2}) + f({z_1})({z_1} - {z_3}) = (f({z_1}) - f({z_2}))({z_2} - {z_3}) = 0\]or\[\begin{aligned}\int\limits_{\partial {\Delta _s}} {f(\zeta )dB\zeta } &= f({z_1})({z_2} - {z_1}) + f({z_2})({z_3} - {z_2}) + f({z_3})({z_1} - {z_3}) = (f({z_1}) - f({z_2})){z_2} + (f({z_2}) - f({z_3})){z_3} + (f({z_3}) - f({z_1})){z_1}\\ &= f'({z_2})\left( {({z_1} - {z_2}){z_2} - ({z_3} - {z_2}){z_3} + ({z_3} - {z_2}){z_1} - ({z_1} - {z_2}){z_1}} \right) = f'({z_2})\left( {({z_3} - {z_2})({z_1} - {z_3}) - {{({z_1} - {z_2})}^2}} \right) = 0\end{aligned}\]where \({z}_{1}, {z}_{2}\), and \({z}_{3}\) represent the vertices of \({\Delta}_{s}\). By holomorphicity and cyclic permutations, this can only happen for \(f({z}_{1}) = f({z}_{2}) = f({z}_{3})\). If every adjacent triangle to \(\Delta\) is considered, deduce that \(f\) must be constant, which contradicts the assumptions. This is because the term in large brackets is translation-invariant, since otherwise set \({z}_{3} := 0\) wlog, making this term 0, in which case \({z}_{1} = {z}_{2}(1 \pm i\sqrt{3})/2\) and \(|{z}_{1}| = |{z}_{2}| = |{z}_{1} - {z}_{2}|\). However, since every horizontal and vertical line is homogeneous on \({}^{(\omega)}\mathbb{C}\), this cannot happen, otherwise the corresponding sub-triangle would be equilateral and not isosceles and right-angled. Therefore, in both cases, \(|{I}_{s}|\) must be at least \(|f'({z}_{2}) \mathcal{O}({\text{d0}}^{2})|\), by selecting the vertices 0, |d0| and \(i|\text{d0}|\) wlog. Denoting the perimeter of a triangle by \(L\), then it holds that \(|I| \le {4}^{m} |{I}_{s}|\) for an infinite natural number \(m\), and also \({2}^{m} = L(\partial\Delta)/|\mathcal{O}({\text{d0}}^{2})|\) since \(L(\partial\Delta) = {2}^{m} L(\partial{\Delta}_{s})\) and \(L(\partial{\Delta}_{s}) = |\mathcal{O}({\text{d0}}^{2})|\). Therefore, it holds that \(|I| \le |f'({z}_{2}) {L(\partial\Delta)}^{2}/\mathcal{O}({\text{d0}}^{2})|\), causing the desired estimate \(|I| \le |\mathcal{O}(dB\zeta)|\) to fail, for example if \(|f'({z}_{2}) {L(\partial\Delta)}^{2}|\) is larger than \(|\mathcal{O}({\text{d0}}^{2})|.\square\)
Cauchy's integral theorem: Given the neighbourhood relations \(B \subseteq {A}^{2}\) and \(D \subseteq [a, b]\) for some simply connected \(h\)-set \(A \subseteq {}^{\omega}\mathbb{C}\), infinitesimal \(h\), a holomorphic function \(f: A \rightarrow {}^{\omega}\mathbb{C}\) and a closed path \(\gamma: [a, b[\rightarrow \partial A\), choosing \(\curvearrowright B \gamma(t) = \gamma(\curvearrowright D t)\) for \(t \in [a, b[\), it holds that\[\int\limits_{\gamma }{f(z)dBz}=0.\]Proof: By the Cauchy-Riemann partial differential equations and Green's theorem, with \(x := \text{Re} \, z, y := \text{Im} \, z, u := \text{Re} \, f, v := \text{Im} \, f\) and \({A}^{-} := \{z \in A : z + h + ih \in A\}\), it holds that\[\int\limits_{\gamma }{f(z)dBz}=\int\limits_{\gamma }{\left( u+iv \right)\left( dBx+idBy \right)}=\int\limits_{z\in {{A}^{-}}}{\left( i\left( \frac{\partial Bu}{\partial Bx}-\frac{\partial Bv}{\partial By} \right)-\left( \frac{\partial Bv}{\partial Bx}+\frac{\partial Bu}{\partial By} \right) \right)dB(x,y)}=0.\square\]Fundamental theorem of algebra: For every non-constant polynomial \(p \in {}^{(\omega)}\mathbb{C}\), there exists some \(z \in {}^{(\omega)}\mathbb{C}\) such that \(p(z) = 0\).
Indirect proof: By performing an affine substitution of variables, reduce to the case \(1/p(0) \ne \mathcal{O}(\text{d0})\). Suppose that \(p(z) \ne 0\) for all \(z \in {}^{(\omega)}\mathbb{C}\). Since \(f(z) := 1/p(z)\) is holomorphic, it holds that \(f(1/\text{d0}) = \mathcal{O}(\text{d0})\). By the mean value inequality \(|f(0)| \le {|f|}_{\gamma}\) (see [473], p. 160) for \(\gamma = \partial\mathbb{B}_{r}(0)\) and arbitrary \(r \in {}^{(\omega)}\mathbb{R}_{>0}\), and hence \(f(0) = \mathcal{O}(\text{d0})\), which is a contradiction.\(\square\)
Remark: The in \({\mathbb{B}}_{\omega}(0) \subset {}^{\omega}\mathbb{C}\) (entire) functions \(f(z) = \sum\limits_{k=1}^{\omega }{{{z}^{k}}{{{\hat{\omega }}}^{k+1}}}\) and \(g(z) = \hat{\omega }z\) give counterexamples to Liouville's (generalised) theorem and Picard's little theorem because of \(|f(z)| < 1\) and \(|g(z)| \le 1\). The function \(f(\hat{z})\) for \(z \in {\mathbb{B}}_{\omega}(0)^{*}\) discounts Picard's great theorem. The function \(b(z) := \hat{c}z\) for \(z \in {\mathbb{B}}_{c}(0) \subset {}^{c}\mathbb{C}\) maps the simply connected \({\mathbb{B}}_{c}(0)\) holomorphicly, but not necessarily injectively or surjectively to \(\mathbb{D}\). It disproves the Riemann mapping theorem and the (generalised) Poincaré conjecture.
Remark: If \(\hat{\omega}\) is identified with 0, the main theorem of Cauchy's theory of functions is true and can be proven according to Dixon (as in [473], p. 228 f.), especially as the limit mentioned there shall be 0 resp. \(\hat{r}\) tends to 0 for \(r \in {}^{\omega}\mathbb{R}_{>0}\) tending to \(\omega\).
Definition: A point \({z}_{0} \in M \subseteq {}^{(\omega)}\mathbb{C}^{n}\) or belonging to a sequence \(({a}_{k})\) for \({a}_{k} \in {}^{(\omega)}\mathbb{C}^{n}\) and an (infinite) natural number \(k\) is called a (proper) \(\alpha\)-accumulation point of \(M\) or of the sequence, if the ball \(\mathbb{B}_{\alpha}({z}_{0}) \subseteq {}^{(\omega)}\mathbb{C}^{n}\) with centre \({z}_{0}\) and infinitesimal radius \(\alpha\) contains infinitely many points from \(M\) or infinitely many pairwise distinct members of the sequence. If \(\alpha = \hat{\omega}a\) holds here, the \(\alpha\)-accumulation point is simply called an accumulation point.\(\triangle\)
Remark: Let \(p(z) = \prod\limits_{k=0}^{\omega}{\left( z-{{d}_{k}} \right)}\) with \(z \in {}^{\omega}\mathbb{C}\) be an infinite product with pairwise distinct zeros \({d}_{k} \in \mathbb{B}_{\hat{\omega}}(0) \subset \mathbb{D}\), chosen in such a way that \(|f({d}_{k})| < \hat{\omega}\) for a function \(f\) holomorphic on a region \(G \subseteq \mathbb{C}\) and that \(f(0) = 0\). Suppose that \(G\) contains \(\mathbb{B}_{\hat{\omega}}(0)\) completely. This can always be achieved by means of coordinate transformations provided that \(G\) is sufficiently "large".
Then the coincidence set \(\{\zeta \in G : f(\zeta) = g(\zeta)\}\) of the on \(G\) holomorphic function \(g(z) := f(z) + p(z)\) contains an accumulation point at 0, and \(f \ne g\), contradicting the statement of the identity theorem. Examples of such \(f\) include functions with a zero at 0 that are restricted to \(\mathbb{B}_{\hat{\omega}}(0)\) and holomorphic on \(G\). Since \(p(z)\) can take every conventional complex number, the deviation between \(f\) and \(g\) is non-negligible.
The identity theorem is also contradicted by the (local) fact that all derivatives \({u}^{(n)}({z}_{0}) = {v}^{(n)}({z}_{0})\) of two functions \(u\) and \(v\) can be equal at \({z}_{0} \in G\) for all \(n\), but that \(u\) and \(v\) may significantly differ further away maintaining to be holomorphic, since some holomorphic function has to be developed into a Taylor series with approximated powers.
Extending to \(\prod\limits_{k=0}^{\left| \mathbb{N}^{*} \right|}{\left( z-{{d}_{k}} \right)}\) allows entire functions with an infinite number of zeros to be constructed. The set of zeros is not necessarily discrete. Thus, the set of all functions that are holomorphic on a region \(G\) may contain zero divisors. Functions such as polynomials with \(n > 2\) pairwise distinct zeros once again give counterexamples to Picard's little theorem, since they omit at least \(\acute{n}\) values in \(\mathbb{C}\).
Remark: From the identity\[{{s}^{(0)}}(x):=\sum\limits_{m=0}^{n}{{{(-x)}^{m}}}=\frac{1-{{(-x)}^{\grave{n}}}}{\grave{x}}\]for real or complex \(x\), by differentiating, it can be deduced that:\[{{s}^{(1)}}(x)=-\sum\limits_{m=1}^{n}{m{{(-x)}^{\acute{m}}}}=\frac{\grave{n}{{(-x)}^{n}}-n{{(-x)}^{\grave{n}}}-1}{{{\grave{x}}^{2}}},\]when the modulus of \(x\), \(dx\) or \(\widehat{dx}\) have different orders of magnitude.
For sufficiently small \(x\), and sufficiently, but not excessively large \(n\), this formula can be further simplified to \(-1/{\grave{x}}^{2}\), and also remains valid when \(x \ge 1\) is not excessively large. By successively multiplying \({s}^{(j)}(x)\) by \(x\) for \(j \in {}^{\omega}\mathbb{N}^{*}\) and subsequently differentiating, other formulas can be derived for \({s}^{(j+1)}(x)\), providing an example of divergent series that has previously not always been correctly calculated.
For all complex \(z, \; {s}^{(0)}(z)\) holds. In the problematic special case of \(z = -1\), this follows from L'Hôpital's rule. However, if \({s}^{(0)}(-x)\) is integrated from 0 to 1 and set \(n := \omega\), an integral expression for \({_e}\omega + \gamma\) is obtained in terms of Euler's constant \(\gamma\). Substituting \(y := -\acute{x}\), by the binomial series a series is obtained with almost exclusively infinite coefficients; if \({_e}\omega\) is also expressed as a series, even an expression for \(\gamma\) is obtained.
If the numerator of \({s}^{(0)}(z)\) is illegitimately simplified, finding incorrect results is risked, especially when \(|z| \ge 1\). So \({s}^{(0)}(-{e}^{i\pi})\) is e.g. 0 for odd \(n\), and 1 for even \(n\), but not \(\hat{2}\).
Theorem: Using the digamma function \(\psi\), it holds for \(n \in {}^{\omega}2\mathbb{N}^{*}\), small \(\varepsilon \in ]0, 1]\) and \({{d}_{\varepsilon k n}}:={{\varepsilon}^{{\hat{n}}}}{e}^{\hat{n}2k\pi i}\) that\[\zeta(\grave{n}) = \underset{\varepsilon \to 0}{\mathop{\lim }}\,\widehat{-\varepsilon n}\sum\limits_{k=1}^{n}{\left( \gamma +\psi ({{d}_{\varepsilon k n}}) \right)}+\mathcal{O}(\varepsilon)\]and\[\zeta(\grave{n}) = \underset{\varepsilon \to 0}{\mathop{\lim }}\,\widehat{2\varepsilon n}\sum\limits_{k=1}^{n}{\left( \psi ({{d}_{\varepsilon k n}}{{i}^{\hat{n}2}})-\psi ({{d}_{\varepsilon k n}}) \right)}+\mathcal{O}({{\varepsilon }^{2}}).\]Proof: The claim results easily via the geometric series from ([474], p. 37 - 42): \[\psi (z)+\gamma +\hat{z}=\sum\limits_{m=1}^{\omega }{\left( \hat{m}-\widehat{m+z} \right)}=-\sum\limits_{m=1}^{\omega }{\zeta(\grave{m}){{(-z)}^{k}}}=z\sum\limits_{m=1}^{\omega}{\hat{m}\widehat{m+z}}.\square\]Remark: The slowly convergent series on the right-hand side may be well accelerated by Euler summation. Odd roots of unity lead to analogous representations.
Corollary: For \(z \in \mathbb{B}_{1-\hat{c}}(0) \subset \mathbb{D}, s \in {}^{\omega}\mathbb{C}\) and \(\hat{c} \le \text{Re} \; s < 1 + \hat{c}\), it holds with the convergent series\[v(s,z):=\sum\limits_{m=1}^{\omega }{\zeta (m+s){{z}^{m}}}=z\sum\limits_{m=1}^{\omega }{{{{\hat{m}}}^{s}}\widehat{m-z}}\]and \(u(s, z) := z \, v(s, z)\):\[\zeta (\acute{n}+s)=\underset{\varepsilon \to 0}{\mathop{\lim }}\,\widehat{\varepsilon n}\sum\limits_{k=1}^{n}{u(s,{{d}_{\varepsilon k n}})}+\mathcal{O}(\varepsilon)\]and\[\zeta (n+s)=\underset{\varepsilon \to 0}{\mathop{\lim }}\,\widehat{\varepsilon n}\sum\limits_{k=1}^{n}{v(s,{{d}_{\varepsilon k n}})}+\mathcal{O}(\varepsilon).\square\]
Remark: Adding \(p\hat{z} + qz + r\) to \(u(s, z)\) resp. \(v(s, z)\) with \(p, q, r \in {}^{\omega}\mathbb{C}\) barely matters.
Corollary: With \(a \in D \subseteq \mathbb{C}\), it holds by Taylor's theorem (cf. [473], p. 165 f.) for a function\[f(z)=\sum\limits_{m=0}^{\omega }{\widehat{m!}\,{{f}^{(m)}}(a){{(z-a)}^{m}}}\]holomorphic in the domain \(D\) and \(g(a, z) := zf(z + a)\) resp. \(h(a, z) := f(z + a) - f(a)\) that\[{{f}^{(\acute{n})}}(a)=\underset{\varepsilon \to 0}{\mathop{\lim }}\,\widehat{\varepsilon n}\,\acute{n}!\sum\limits_{k=1}^{n}{g(a, {{d}_{\varepsilon k n}})}+\mathcal{O}(\varepsilon )\]and\[{{f}^{(n)}}(a)=\underset{\varepsilon \to 0}{\mathop{\lim }}\,\hat{\varepsilon}\,\acute{n}!\sum\limits_{k=1}^{n}{h(a, {{d}_{\varepsilon k n}})}+\mathcal{O}(\varepsilon).\square\]
Remark: The precision may be arbitrarily increased by successively inserting the terms.
© 2010-2018 by Boris Haase
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TORONTO – Another batch of September call-ups joined the Toronto Blue Jays on Tuesday and pushed the number of active players inside a jammed clubhouse to 32, and counting. That's approaching a spring-training-esque allotment of bodies to close out the 2018 season and manager John Gibbons suddenly finds himself with the needs of several more people to juggle.
What is he going to do with all those players?
"That's a good question," Gibbons said before a solid Ryan Borucki outing went to waste in a 4-0 loss to the Tampa Bay Rays on Tuesday night. "A lot of the guys that come up won't necessarily play a lot, but I'll make sure to try and get them in. I think that's important."
Joining the crowd earlier in the day were infielder Richard Urena, outfielders Jonathan Davis and Dwight Smith Jr., and first baseman Rowdy Tellez, while Jake Petricka was activated to replace Joe Biagini, who was placed on the disabled list with a left oblique strain.
Outfielder Anthony Alford and catcher Reese McGuire are expected to come up later this month, Justin Shafer will be up once his 10-day option period expires while right-hander David Paulino, part of the return from Houston for Roberto Osuna, could be activated in the days ahead after recovering from a shoulder injury.
The advice from Gibbons to the kids?
"Just watch the game, basically enjoy it," he said. "Don't expect to take a ton of BP because we're killing our coaches anyway. Don't overdo it, put it that way. Just enjoy the experience, take advantage of it, watch and learn. They'll all go to nice stadiums in our division, they'll see all the primary places we play.
"I don't know what else I'd tell them. Don't eat too much of the food. You laugh, that happens a lot and they swell up."
For now, the only thing swollen is the roster and there are a few reasons for that.
• Adding bodies, particularly on the pitching side, is necessary in part to survive the last leg of the six-month marathon that is the regular season, especially with Marcus Stroman's season still uncertain due to his ongoing blister issues. Gibbons thinks Stroman should shut it down for the rest of the year but the right-hander hopes to keep pitching and they'll re-evaluate in a couple of days to determine what he might be able to offer.
• In some cases, the Blue Jays are looking to reward players who have had good minor-league seasons with a touch of the big-leagues. Assistant GM Ben Cherington made the suggestion and certain players were promoted knowing their window of opportunity would be limited because the club already has several youngsters they're promoting around the diamond.
• Finally, even though they won't necessarily be playing much, the Blue Jays have a looming 40-man roster crunch coming and bubble players will be watched closely in how they go about their work, take in information and act as teammates. Right now, the 40 is full, Marco Estrada and Tyler Clippard are the only pending free agents left and Troy Tulowitzki will need to be re-instated from the 60-day DL. Players who must be added to the 40 or be exposed in the Rule 5 draft include right-handers Hector Perez, Jordan Romano, Jacob Waguespack and Corey Copping and outfielders Forrest Wall and Harold Ramirez, so the Blue Jays have lots of decisions to make.
The clubhouse is now a very different place.
"Definitely a change – for the last few years it's been nothing but vets," said Justin Smoak, suddenly among the few elder statesman left. "It's weird, but I remember my first day like it was yesterday, I'll never forget that memory. All these guys, it's something they've worked for their entire baseball life, to try to get to this point, so it's exciting for the young guys to be here and just get a taste of what it's like and hopefully have some success."
Of the position players, Urena (up the middle) and Davis (as a pinch-runner) are likely to get the most use, while the arms should be fairly busy.
Petricka, who is arbitation eligible, Thomas Pannone and Taylor Guerrieri logged the final three frames behind Borucki, who allowed two runs on three hits and two walks over six frames with five strikeouts.
The start was the 13th this year for the 24-year-old, who is almost like a veteran on a roster now dotted with rookies.
"Coming here in the beginning, it was cool having all these veteran guys," said Borucki. "It's also a very cool opportunity seeing all these guys I've played with in the minor-leagues. Rowdy, J.D. (Davis), I've been playing with those guys since 2014. It's cool to see them move up the ranks and get their opportunity here."
For the fans who have demanded a rebuild of the Blue Jays, well, this is it.
"I do think it's important, especially in our situation, that these guys come up and they get a taste of it, they get a feel of it," said Gibbons, adding later: "These are the guys that are part of the future. A lot of first-timers. Get that out of the way, they'll know what to expect next year when that time comes – if it comes."


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Advanced Lessons
Once you've taught basic obedience and a few tricks, you may want
to move on to advanced training behaviors. One example of an advanced
behavior that is easily taught using clicker training is increasing
the threshold of a behavior before reinforcement.
Consider heeling, for example. Your dog may have learned to heel
on command, but you find his attention wanders after about five
seconds and he begins to look away or start to wander. To increase
the time he will stay at your heel, you start delaying the time
at which you click and treat. For example, if he begins looking
away at five seconds, click and treat at four, which will buy you
another five seconds for a total of nine. Then move it up to clicking
and treating at seven seconds, and so forth, until your dog is happily
heeling for however long you want him to.
The main point in increasing thresholds is to take it slow and
make sure to vary the schedule of reinforcement, so your dog doesn't
learn to anticipate his treat. The way to do that is to occasionally
treat at the very beginning of the behavior, so your dog never knows
when the treat may be forthcoming. This will help keep him focused
on the behavior and prevent his attention from wandering.
Back : Clicker
Home : Pet
Supplies Review


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Young Animal Lover is Well-Rounded Student
Patch Whiz Kid of the Week: McKenzie Lamica
Whiz Kid's Age: 9 and a half: Birthday Sept. 18
Whiz Kid's School: Spencer Elementary
Whiz Kid's Accomplishment: McKenzie is not your typical 9-year-old girl, she is extremely well-rounded. She is a wonderful big sister to Emma, 6. She loves animals, she cares for her two dogs (Esther and Tina), four pet rats (Cody, Bean, Stalk and Oreo) and a toad that she found on the street and named it Lizzy.
Well, as we can see McKenzie sure does love animals, she also enjoys helping out her community, she is an active member in the 4H Program which works to prepare young people to make a positive impact in their communities and the world.
McKenzie also excels in school; she is a great student that is very empathic and ambitious. She plays the saxophone, plays basketball — winning many trophies for her skills and leadership. As if this is not enough, McKenzie also has great skills in both math and reading. She takes part in an after-school program at Spencer, the Young Scholars Academy.
McKenzie loves to read Harry Potter books — she is now reading the last book in the series. Her favorite food is hot wings.
Whiz Kid's Key to Awesomeness: "I'm awesome because I try hard," said McKenzie.
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To begin, first have students massage their jaw line and temples. Second, practice deep breathing and holding breath. Third, practice ragdoll forward bends while exhaling loudly. Next, have students walk around room while producing short vocalizations from the diaphragm. Then, split class into two or three groups and have each group stand in a circle. One person in each group will then produce a noise that will then be produced by everyone in the circle single file and in order until it comes back around to the person who originally produced the noise. The original noise producer will then produce that noise one final time and then the next person in the circle will produce a new noise and the cycle will repeat. Continue until every person has produced their own noise and go as fast as possible.
After each circle has completed their sound production activity, have all the groups stand in a circle together but in a way so that each group is still visually separate from each other. Have students participate in a singing round.
*Ensure that you have just enough groups per line of the chosen song.
- Gets voices ready for singing
- Teamwork
- Students may not be comfortable with singing
- Preparing students for Christmas recitals
- Getting students comfortable with singing and/or singing in front of classmates and teachers
- Can be used by teachers to strengthen their vocals for all the talking they will do as teachers
- Can be used to help memorize a song- repetitively singing and hearing the same song
- Can be used as a concentration tool


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National Research Council Releases New Framework for K-12 Science Education
by Joe Savrock (August 2011)
A recently released National Research Council (NRC) report presents a new framework to guide K-12 science education. The report, titled A Framework for K-12 Science Education: Practices, Crosscutting Concepts, and Core Ideas, calls for a shift in the way science is learned in the United States.
Penn State faculty member Deborah C. Smith is an appointed member of the NRC committee that contributed to the report. The 18-member committee, which consists of experts in science education and scientists and engineers from a variety of disciplines, sees the need for significant improvements in how science is taught and learned in the U.S.
The framework builds on the Benchmarks for Science Literacy (American Association for the Advancement of Science, 1993), the 1996 NRC National Science Education Standards, the Framework for the 2009 National Assessment of Educational Progress, and the Science College Board Standards for College Success (2009), as well as on years of research on teaching and learning science.
In an interview published last fall, Smith said the earlier standards "were a good start on identifying important scientific ideas and inquiry practices. Now that we have some perspective on those standards and how they've been used, and lots more research about students' science learning, it was time to rethink a more up-to-date framework."
In a recent webinar, offered jointly by the NRC and the National Science Teachers Association, presenters discussed the new framework's three core dimensions—scientific and engineering practices, crosscutting concepts, disciplinary core ideas. The Framework will serve as the foundation for new K-12 science education standards, developed and coordinated by the nonprofit educational organization Achieve Inc., in collaboration with a small group of states who will adopt them for state science standards.
"The new framework is designed to help students gradually deepen their knowledge of core ideas in four disciplinary areas as a coherent developmental progression over multiple years of school, rather than the acquisition of shallow disconnected knowledge of many topics," said Smith. The framework specifies core ideas in life sciences; physical sciences; earth and space sciences; and engineering, technology and the applications of science that the committee recommends all students should understand by the time they finish high school. In each of these areas, the report provides progressions of initial to more sophisticated understandings of core ideas across the K-2, 3-5, 6-8, and 9-12 grade bands, given what is known from research.
The committee also strongly emphasizes seeing science and engineering as a set of practices—for example, engaging students in planning and carrying out investigations, in developing and using models, and in using argumentation from evidence to make claims—rather than a single, linear "method." The authors describe developmental progressions in students' uses of these practices and in their understandings of how they contribute to making scientific and engineering knowledge.
In addition, the framework identifies seven crosscutting concepts, such as "cause and effect" and "stability and change," that have explanatory value across much of science and engineering. According to the report, these concepts should be taught in the context of core ideas from the disciplines of science, with teachers and students using a common language for them across disciplines, so that students understand how the same concept is relevant in many fields. Crosscutting concepts, says the report, should be woven into science and engineering as students progress from kindergarten through 12th grade.
"Given the years of research on science teaching and learning, both in classrooms and elsewhere, we have a much clearer vision of children's amazing capacities for authentic participation and understanding in science and engineering, when given appropriate environments that encourage and support them," said Smith, assistant professor of science education. Her own research has revealed how kindergartners—in a classroom in which these practices, core ideas, and crosscutting concepts were embedded—can develop scientific discourses and practices, use evidence from their investigations to make claims, evaluate competing explanations, and design tests to resolve conflicts.
The impact of the framework is likely to be far-reaching, and not only in developing new national and state science standards. The committee also strongly states that issues of equity and diversity in access to and support within excellent opportunities to learn science and engineering are paramount in making sure that all K-12 students can learn. In addition, the design of K-12 science and engineering curricula that engage students in the three dimensions will be an immediate challenge. Similarly, the committee emphasizes that the development of authentic performance assessments (some examples of which are given in the report), perhaps shared in consortia across states, will be a critical factor in implementing the framework's vision. The report reviews the status of current research on such curricula and assessments. It also provides recommendations for needed research in several areas, including teacher education as well as professional development with boards of education, administrators, and teachers.
Smith says, "For example, our Penn State science teacher education programs already embed much of what the framework recommends, because our faculty are deeply involved in policy, research, and work with teachers on many of these issues. There are great opportunities for collaborative research and development here in Pennsylvania."
Smith is collaborating with Tyrone (Pa.) Elementary School to explore how teachers and children can create the kinds of science and engineering learning opportunities recommended in the report.


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ORGANIZER: Joey
Valentin Dubuc
Joey Parker wrote -As many of you know I traveled to Kenya, Africa last summer. This trip put life into perspective and truly made me grow up as an individual. The organization I worked with in Kenya was, "The Kenyan Children's Foundation." This organization works with various schools in Kenya and orphanages, as well as various other projects helping create a more sustainable life for many.
When I got back from Africa, I really wanted to do all I could to get people involved here in the states, so now I have the opportunity to launch a project and hope that people will join me in making a difference. If this project goes as planned I will do another one. I truly feel with the "Internet revolution" we can use this to our advantage and help make a difference and save lives.
When the projects are completed in Kenya, I will have photos for everyone to see. It is always exciting when you donate money and see where it is going and the final product actually completed.
The Project: The Greenhouse
You may be asking, "Why a greenhouse?" This greenhouse will be going at the farm to help produce fresh produce to help make the farm self-sustainable. The farm was established to one day produce enough profit to permanently sustain the children's home at Meru. But before this goal is met we must create these greenhouses to make the farm self-sustainable, then we can move forward. It is a process, and I hope you all will join on this adventure with me.
Why should you support this?
The food raised at the farm will be sold into the community for the people to live off of. The Kenyan Children's Foundation makes no money on any of this. It is all a local endeavor, hiring local people, to support local families, schools and orphanages.
Our goal: $4,500
This includes putting it up, establishing water supply and drip irrigation, planting the crops, fertilizers and chemicals. The best thing is that the company that puts in the greenhouse also buys the produce.


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Cannabinoid Church: Baked by the Dozen by Ranger Apothecary – April 2 2017
Cannabinoid Church: Baked by the Dozen
A&C, DoD, and our growers will be dropping knowledge and breaking nugs as always.
We will be teaching a gluten free, sugar free cookie recipe that's guaranteed to open your eyes to the world of edibles. Rather than mixing all the ingredients at the event, we'll have them present, as well as the end product.
Attendees will learn about decarbing cannabis oil, infusing phospholipids, and how to weigh dough into estimated dosage sized servings.
Best of all, we'll be giving free cookies away to those in attendance this week.
As a reminder, this is a private residence in proper behavior is required. No sales or donations for cannabis or its derivatives will take place on the premesis.


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744 Bedok Reservoir Road
Singapore 470744
11:00am - 10:00pm
This cafe has one of the worst customer service!!
I've visited this cafe at Bedok Reservoir a few times as it is conveniently located near my work place and home. Food is OK but the service has always been substandard. Employees there don't really give a damn about customer service so don't expect them to smile and greet you warmly. Manager has the "blackest" face. She's like having a bad day every time and will never put on a smile even when eye contact. So, I have known their service has always been poor.
But I am extremely appalled by their service today!! My mother and I arrived on a regular weekday, Wednesday at 12.30pm. There was a long standing queue outside the cafe and the cafe was packed. I looked inside the cafe and saw the service staff noticed me so I proceeded to stand behind the line.
The weather was hot and there are no fans installed outside. The queue was moving very slowly. After waiting for a long hour, it's finally our turn to be seated next. I saw the servers clearing away the table and we were waiting for them to signal us into the cafe. Right at this moment, a group of people queuing behind us waltz into the cafe and got seated at the table that was just cleared. My mother and I were shocked because they came 45 mins after us and how can they just cut our queue??!! I quickly went into the cafe and informed the staff (her name was Amy) that we were supposed to be seated next. To prove myself I even pointed out the couple who was waiting in front of us earlier to Amy, saying we were behind them. Amy then asked the couple if their names were on the waiting list. The couple said no. She looked at me speechless and did not resolve the issue. Left me and my mum hanging. Left the other couple lost for words but they were lucky to be seated.
What was even more frustrating was that, there was a f**king waiting list all along and none of the staff inside actually came out to let us queuing customers know that we can actually write down our names and get called when it's our table is ready??!! All of us could have avoided the heat and went somewhere else first!!
I questioned Amy why we were not informed of the waiting list. She looked at me dumbfounded and awkwardly asked me to join a table that clearly wasn't meant to be shared with. It was a table meant for 4 pax already seated by a family of three. She wants my mum and I to be seated at the corner of the table, not even seated opposite of each other. It clearly was going to be very uncomfortable arrangement for them and the family we were sharing with.
I couldn't believe that besides messing up the waiting list system, she couldn't even resolve the issue by acknowledging her mistake and politely asked those behind me to give up the table for my mother and I since we have been waiting in the line for the last hour.
My mother and I were terribly upset to be put in such awkward and bad situation. We walked away and never going to come back.
To the owner of this cafe, your staff is not well trained, especially Amy. I believe many of us can agree that customer service is extremely important for running a cafe. Of course, I am not expecting to be pampered with a hai di lao sort of service, but at the very least, your customers should be treated fairly, respected and served with a smile. Your staff has failed to deliver even the most basic customer service. You have lost yourself a customer today, I suggest you do something about it asap.
decent cafe food if you are near the hood. the toast was nice though.
(full tank $15.9 / meaty mac & cheese $14.9)
3 pancakes with earl grey ice cream and fruits, drizzled with maple syrup. Preferred the ice cream to the pancakes but overall it's quite good as it's not too sweet.
The strands of spaghetti was drenched in thick and creamy chilli crab sauce. Served with real crab meat, the sauce was tongue-tingling as the spiciness kicked in. I would have preferred the less spicy sauce and bigger chunks of crab meat then the shredded pieces. Overall it was tasty and retained the chilli crab flavour, but not one of my favourites.
A good cup of joe that was medium-bodied, nutty and bittersweet
Pulled pork was yummy 😋 but the waffles was soggy😑. The honey BBQ ribs (not in picture) was dry. Nice cafe in the east but the food need some quality control


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Does mental diversion (Penal Code 1001.36) apply retroactively?
We're starting to get case law decisions from Appellate Courts about the Mental Health Diversion Law (California Penal Code 1001.36, from AB 1810) Governor Brown inked this bill late June 2018 and it became effective June 27, 2018. You can see my blog about that law, although there seem to be some amendments being made to it.
One question frequently asked is whether this law will apply only going forward? Or will it apply to older cases.
From this September 28, 2018 decision People v. Frahs*, the rule is this mental health diversion law applies retroactively.
[update: August 26, 2019: we now have several cases holding whether Mental Health Diversion is retroactive…and the Courts are in disagreement with each other. I blogged about the updated cases here]
In Frahs, the People charged him in 2016 for robbery. At trial, "Frahs put on evidence that he suffers from a form of schizophrenia" but the jurors convicted him anyway. The Court also found a prior strike to be true and sentenced him to 9 years prison.
Let's step through this:
First, how does retroactively work?
Taking language straight from Frahs, "[i]n general, statutes are presumed to apply prospectively unless they state otherwise. However, the presumption against retroactivity does not apply when the Legislature reduces the punishment for criminal conduct."
The legal reasoning from the Supreme Court "is that when a statute reduces or ameliorates the punishment, it is presumed that the Legislature has determined the offense no longer merits the greater punishment, and this rationale applies even if the defendant was convicted and sentenced before the statute became effective."
How does it apply to the mental health diversion laws in California?
Frahs holds this diversion "does not lessen the punishment for a particular crime. However, for a defendant with a diagnosed mental disorder, it is unquestionably an 'ameliorating benefit' to have the opportunity for diversion—and ultimately a possible dismissal—under section 1001.36."
This is also important language in this opinion about how broadly the Court interprets applying this new diversion law:
"Further, it appears that the Legislature intended the mental health diversion program to apply as broadly as possible: 'The purpose of this chapter is to promote (a) Increased diversion of individuals with mental disorders to mitigate the individuals' entry and reentry into the criminal justice system while protecting public safety.'"
So what happens for Frahs?
The Court reverses the conviction and sentence, and remands the case for 1001.36 diversion evaluation. Although the Court says "he appears to meet at least one of the threshold requirements (a diagnosed mental disorder)."
*Cited as 27 Cal.App.5th 784 (2018)


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2 results for tag: pleasure
A Visual Guide To The Inner Seasons
This is a visual guide to the inner seasons of menstruality, you'll find the more you develop menstrual cycle awareness, the greater the gifts you'll receive. If you want a window into your own, unique inner seasons, charting your cycle or if you don't have one charting by the moon will get you started. A Menstrual Medicine Circle will deepen your understanding of what getting in the way of your well-being and what your next steps are.
Beppina Wildfire – She Comes Fully As Herself
This is part of the Pants of Empowerment project, where women share the voice of their feminine nature and in return, receive a fantastic pair of pants to empower them. All names have been changed. What are you aware of in your pelvic bowl? I felt a sense of resilience and safety when I was walking through my pelvic bowl. Connecting with the Womb She's a crone, an old lady. But she has a twinkle in her eye, she represents the lineage of all the women who've gone before me, the women who support me. She connects to all the other aspects; my sexuality in my clitoris for example. My sexuality is ...


text: Supra dropped a part of Shake Junt rider Spencer Hamilton. With a deep bag of tricks and an eye for spots, Spenny's been on a long path of destruction around the world. From Vancouver to Venice Beach, from the Big Apple to Barcelona, Spencer's always down to roll and always puts in the work to get the clip. Check the full edit and be prepared for more from him in the near future.