A Scalar Arching Model

In this paper we propose an extension of the model of Liu, Coppersmith et al., in order to allow for arch formation. This extended model qualitatively captures interesting properties of granular materials due to fluctuations of stress paths, such as stress fluctuations and stick-slip motion.

Introduction
Granular media are materials where fluctuations are very large. When filling a silo with grains, a part of the weight of the grains is supported by the walls of the silo, meaning that the bottom plate of the silo only carries an apparent weight W. This effect is well known and was studied by Janssen at the end of the last century. When repeating this procedure with the very same amount of grain, one observe large fluctuations of W, typically of order 10 − 25%! On a given silo, W can also vary drastically because of very small perturbations, such as variations of temperature or density [3,4].
These effects can be understood in terms of arching. As a matter of fact, stress propagation in granular media is strongly inhomogeneous: clear stress paths are present and carry an important part of the total weight of the grains. Those paths, or arches, are completely different from a realisation to another, leading (potentially) to very different values of W. Furthermore, the geometry of these paths can easily rearrange under small perturbations, inducing strong fluctuations in W.
Another phenomenon closely connected to the presence of these arches is that of the stick-slip motion. Imagine trying to push some granular material through a tube with a piston. Moments where the system jams (stick) and moments where the system slides (slip) will irregulary alternate. This is explained by the fact that the stress paths network is sometimes able to resist to the applied force, and sometimes is not, depending on its structure (which fluctuates).
In this paper, we propose a very simple numerical model which qualitatively captures some features of the above effects. This model will be called the Scalar Arching Model, or sam in the following.

The sam
The model we are going to present is an extension of that proposed by Liu, Coppersmith et al. [1,2], which allows for arch formation. This model only deals with the vertical normal component of the stress tensor, σ zz = w, the 'weight'. In that sense, the model is scalar. This is obviously a simplifying choice which may not be justified. Work is in progress for extending this model to a fully tensorial description (see e.g. [5]). Again for simplicity, we will keep to a two dimensional situation. Note that arching effects are expected to be more pronounced than in three dimensions. The system   Figure (a) represents the system we study. An apparent weight W is measured at the bottom plate of a silo. Figure (b) shows the underlying lattice with which we modelize the granular medium. Indexes i and n are such that −L ≤ i ≤ L and 0 ≤ n ≤ H, where 2L and H are respectively the width and the height of the silo. On the particular configuration of figure (b), the force w1 is much larger than w2, meaning that the shear force acting on the grain (i,n) is strong enough to remove the contacts between the grains (i, n) and (i − 1, n + 1).
we would like to describe is shown in figure 1-(a). Following Liu et al.'s approach, the granular packing is represented by a regular square lattice. Each site is a 'grain' labelled by two integers (i, n) giving its horizontal and vertical coordinate. All the randomness of the local packing, the friction, the size and shape of the grains, is assumed to be encoded into random transmission coefficients q ± . The rule for the propagation of the weight is the following: where w g is the weight of a grain. This rule simply means that each grain supports the force of its two upwards neighbours and share its own load randomly between its two downwards neighbours. (The rôle of correlations has been recently discussed in [6]). The mass conservation constraint imposes q + (i, n) + q − (i, n) = 1. At this stage, the model is the one considered in [1,2]. We now include a local slip condition: when the shear on a given grain is too strong, the grain can slip and lose its contact with its neighbours opposite to the direction of the shear. More precisely, one defines a threshold R c such that . These rules lead to an avalanchelike process: suppose, as shown on figure 1-(b), that the link between the grains (i, n) and (i−1, n+1) is removed because the force w 1 is much larger than w 2 . The force from (i, n) to (i + 1, n + 1) is then very likely to be much larger than the force from (i + 2, n) to (i + 1, n + 1), and the link between the grains (i + 1, n + 1) and (i, n + 2) is very likely to be removed as well. This process (that was called static avalanche in [7]) can be interpreted as an arch formation.
A particular value of W is associated to a particular configuration of the stress paths. At the walls, forces are absorbed with the following rule: if i = ±L, the fraction q ± (±L, n) of the load w(±L, n) is absorbed. Depending on the configuration of the stress paths, a larger or smaller part of the weight of the grains is then supported by the walls, leading to differents values of W. We studied the probability distribution function (p.d.f.) of the values of W from many different configurations of the stress paths (i.e. from many different silos). The relative standard deviation from the mean value is as large as 20%, as found experimentally. We also looked at the p.d.f. of the local weight w(i, H) and found it to be a power-law, meaning that extremely large values are expected.
The most striking feature of this model is that the stress paths network generated by the sam (as shown in figure 2) is very likely to rearrange under small perturbations. Our control parameter for generating perturbations is the threshold R c . This coefficient does not actually represent a specific physical quantity but can be seen as representing the temperature and/or the compacity. As a matter of fact, contacts between grains ar very sensible to small changes of those two quantities. What we observed in the sam is that when R c is changed by a very small amount, the stress paths network sometimes rearranges and sometimes does not. When it does, W changes by a relative amount ∆ which can be either small or large, meaning that the actual value of ∆ is not correlated to the amount by which R c has been changed. Furthermore, ∆ is found to be power-law distributed: P (∆) ∼ 1/∆, which means that large rearrangements of the stress paths network are as probable as small ones. In that sense, the sam captures the feature that granular matter is extremely 'fragile', i.e., sensitive to small perturbations. Actually, our model is close in spirit to the large class of 'SOC' (Self-Organized Critical) models [8], initially proposed to describe true 'dynamical' avalanches in granular media.

Stick-slip motion
When the bottom plate of the silo is used as an upwards pushing piston (see figure 3), an irregular stick-slip motion of the system is observed [9]. This behaviour can be understood by the fact that the stress paths network rearranges, generating configurations where it can resist to F (sticking or jamming situations) and configurations where it cannot (slipping or sliding situations). In this section, we will explain how one can slightly modify the original version of the sam to describe this situation. The idea is to look at the sam picture 'upside down'. We neglect the weight of the grains and focus on the transmission of F through the grains, from the bottom of the cell to the walls and the free surface. All propagation rules are kept the same. The only noticeable change from last section is what happens at the walls. Because strong arching is expected at the walls when F is applied, we introduce a new parameter α, the 'jamming ability' of the walls. With probability 1− α absorbing rules of the last section apply. With probability α however, the load w(±, L) is completely absorbed by the wall, meaning that a local arch is strongly 'anchored' on the wall. Such situations are essential to get the system jammed. In addition to the q ± (i, n), we now then have new random numbers α ± (n) which, compared to α, determines which absorbing rule applied at site (±L, n).
We then propose the following algorithm. For a given force F , and a given set of random numbers q ± (i, n) and α ± (n), • we calculate F w and F s , respectively the total forces on the walls and on the top surface of the silo. Obviously, F = F w + F s . • if F s = 0, the grains do not move. It is a stick situation. We then increase the applied force F of ∆F and the time t of ∆t. In order to mimick the mechanical noise, we also change all random numbers with probability p, and go back to the first point.
• if F s > 0, the static equilibrium conditions for the system are not satisfied, which means that grains are moving. It is a slip situation. We then decrease the applied force F by ∆F , change all random numbers (because the flow motion completely rearranges the packing), and go back to the first point. The simulation starts at t = 0 with F = 0 and let F (t) evolve. It is important to note that our model is a pure static model: no dynamics is included. Therefore, the motion of the grains during the slipping moment is assumed to be infinitely quick. We then actually describe only sticking situations, separated by slipping moments which have two effects: untighten the spring governing the external force F , and reinitialize the structure of the packing (i.e. the random numbers). Such an algorithm leads indeed to an irregular stick-slip motion.  Depending on the values of the parameters controlling the system, two different behaviours are observed. For small values of the aspect ratio b or the 'jamming ability' α, situations where the system never jams for ever are typically observed, see figure 4-(a). We called these situations slipping stick-slip phases. On the contrary, sticking stick-slip phases are seen for large values of b and α, see figure 4-(b). The critical curves α c (p) or α c (R c ) can be plotted for a fixed b, which separate the two regimes. We caracterized the first phase by the first return time τ , i.e. the interval of time between two consecutive times where F vanishes, and the second one by the slope s = F (t)/t. Just below the critical point, τ is be power-law distributed with an exponent 3/2, meaning that F simply behaves like a random walk. When α → α c we find τ ∼ 1/(α c − α) and s ∼ α − α c for α > α c . More details can be found in [10].


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The author purpofes to publifh by degrees, the more remarkable cafes of his Hofpital practice; and in order to enable the reader to compare more properly the courfe and fymptoms of the difeafe with the eftedis of the remedies that have been employed, he com-, municates a feries of cafes of each principal difeafe. The remedies that are made'ufe of are the moil fimple and the lead expenftve.
When emetics are indicated, he ufes, inftead of ipecacuanha, 15 grains of gratiola, fometimes mixed with 5 grains of rhubarb. In the gangrene, he applies, inftead of the bark, the cortex Hippocaftani and falic ; inftead of farfaparilla he gives the carex arenaria. In intermittent fevers he has fuccefsfully employed the cortex Soimydae ; in the fluor albus he found the faba pichurim, half a drachm pro dofe for three times a day, very efficacious. For blifters he recommends a mixture of wax, turpentine, and the powder of Spanifh fiies. In contufions, equal parts of vinegar and water are Serviceable. After confiderable wounds or operations, he gives a fimple tin dure of opium every two or three hours, confifting of op. crud. dr. j.fpir. vin. reftific. unc.j. He employed the cortex caribous in intermittent fevers without fuccefs, but with fome ufe in the lumbago ; he never remarked any emetic quality of that remedy during its ufe. Such, and fimilar remarks, are contained in this work, which deferves the attention of hofpital phyficians, and of thofe who are obliged to have recourfe to cheap and fimple remedies. The greateft part of this volume, however, is devoted to the lefions of the head, of which 217 cafes fell under the author's care, from the ift. of November 1795, to the lft. of March 1800, 127 of which ended fatally. The method of treatment confifts of cold fomentations on the fhaved head, of venefettion, leeches, and the internal ufe of camphor; but he recommends, above every thing, the timely application of cold fomentations, of the ufe of which lie is fo much perfuaded, fhat in moft cafes the fatal event might be attributed to their being neglefted; but .they ought always to be kept properly cold, by the addition of ice, fnovv, &c. Thefe are the contents of this ufeful publicafon, the continuation of which we heartiiy approve. The Author of the work before us has been favourably known to the public as an acute reafoner, an able controverfialift, and a very zealous defender of thofe theoretical {peculations on various difeafes, which proclaimed the genius of the late John Hunter.
In his " Obfervations on Morbid Poifons," (publilhed in 1795>) Dr. Adams has taken great pains to eftablifli a very lingular opinion concerning the origin of carcinoma, which had been merely hinted at by Dr. Hunter, and which is again urfued at confiderable length in the treatife before us.
There are few difeafes in which the utmoft length of conje&tire, and the greateft latitude of experiment, are more to be allowed and encouraged than in cancer.
A melancholy conviftion of the extent of its ravages, and of the inability of artto^check it progrefc by any certain rules, hat been lately evinced by an inftitution in this metropolis, for the exprefs purpofe of inveftigating the nature of this dreadful calamity 5 and we feel difpofed to give a confiderable fliare of attention to the conje&ures advanced by the ingenious author of the prefent publication. As this is to be confidered as a fequcl to the chapter in the Observations on Morbid Poisons, juft alluded to, we fliall beg to prefent our Readers with the fum and fubftance of it in as few words as pofiible.
Schirrus, Dr. A. obferves, or the early ftage of carcinoma, is always compofed of certain cysts, which, he thinks, constitute its true charafter, and which are filled with fluids of different completions. Mr. Hunter has fpoken of the cylls as being cancerous hydatids, and this is the opinion which Dr. A. follows up in his fubfequent obfervations, and which conftitutes their originality. Thefe cyfts have fome peculiarites in their appearance and progrefs; the one, that their increafe is not towards the furface, like matter in a common abfcefs, but in every direttion ; another peculiarity in the difeafe is a difpofition to fungate even before the fkin is broken, and in a very high degree after it has become an open fore. After this period the progrefs of the difeafe is uncertain, fometimes with a rapid ulceration of the fungus, which, in floughing away, expofes feveral roundifli fovcoli, which are the cyfts with their contents, changed from their original appearance to that of an opake pale, refembling half diflolved leather. This alternate progrefs, namely, of the extenfion of thefe cyfts, of the formation of fungus around them, of the floughing of this fungus and the confequent difcharge of the contents of the cyfts, and of the attempts at granulating Dr. Adams, on the Cancerous Breast. milating which are made by the furrounding parts, forms the principal phenomena of this difeafe.
We fliall not further purfne the arguments of the Author in his Obfervations on Morbid Poifons, in order to eftabliih his opinion on the origin of cancer ; but he concludes the fubjeft by propofin^ a fet of queries, wherein are the following : Is the fimple hydatid the firft form of carcinoma? Have thefe hydatids a life independent of the fiibjett in which they grow, excepting as parafites? &c. &c. The anfwer to feveral of thefe queries is given by the Author in the work which we are now to confider, and to which it is high time to revert. This volume confifts of correfpondence between the Author and Dr. Baillie, Mr. Cline, Dr. Babington, Mr. Abernethy, and Dr. Stokes; in which Dr. A. ftates pretty fully his peculiar opinions, and his correfpondents fome of the obje&ions which may be urged againft them, and other very valuable illuftrative fa&s. Perhaps it would have affifted the reader in forming a clearer idea of th? Author's theory, if the fads on which it is founded had not been prefented in fo defultory a form ; but the authenticity which is attached to original correfpondence may make amends for this defeft.
Almoft the whole of our knowledge on that curious -parafitical animal, the human Hydatid, is contained in an excellant paper by globules being found of various fizes floating in the liquor, feems to prove.that they aie originally formed there, and not in the coats of the hydatid, upon which they.are afterwards depofited. The number of thole that had young ones in them, was few in proportion to the others.. : ; " The hy datids "in their growth and decay appear to pnfs through various ftages"; the'y are firft found floating in the fluid that fills the hydatid, and afterw ards attached to its coats. The hydatid thus pregnant with ycvn g, if the expreflion may be allowed, adheres to the neighbouring parts, increafes in fize, and becomes itfelf a fac, containing numerous fmall hydatids. Thefc, after a certain time, decay, and the fkins or empty bags are fqueezed together into a fubftance like ifinglafs. It is probable they ftill undergo a further change; two fmall bodies of the fize of the common bean, of a cheefe-like confiftence, and covered with a {kin, were taken notice of adhering to the bladder, near its neck; it may be a queftion whether thofe were not the remains of hydatids ? but that mull be determined by future obfervations." The hydatids of fheep differ from the human hydatid, in poffefiing befides a more mufcuiar bag filled with a tranfparent liquor, a mouth or longitudinal aperture, and a neck compofed of rings.
Thefe hydatids are contained in the abdomen of the animal; others are fometimes found in the brain, which again want the mouth and rietk above defcribed, and give the difeafe called the Jtaggers. Each'kind'of hydatid, when put in warm water, moves about brifkly, with a kind of periilaltic motion of the whole body, though it has'been taken out of the fheep for feveral hours.
??'The firft letter contained in this volume is addreffed to James Pitcairn, Efq. in .which Dr. A. makes fome obfervations on the ftrufture of the hydatid, and obferves, very juftly, that " the moil *f fimple idea of animal life we can well form, is that of the hytf datid, confifting only of a membranous bag containing a tranf-** parent -fluid. Thofe  ferved, I fhould be inclined to believe, that cyfts are only fometimes formed in a fcirrhous ftrudture, but are not cffential to itln this, however, I may be miftaken ; and it may be found by a more minute observation, that trie formation of" cyfts always conftitutes a part of a fcirrhous ftrudture. If you fhould be able to eftablifn this or any other general obfervation about the nature of fcirrhus, it will give me real fatisfa&ion. " I have known a fubftance which pofiefled the common characters ?f fcirrhous ftrufture to be converted into a kind of bony matter.
In this, I believe, that the earthy part will be generally found to be in a larger proportion to the animal part than in common bone. Mufcular and membranous parts I have known to be affetted with fcirrhus, as well as thofe which are ftridtly glandular. A fatty membrane I have feep affefted with the Hjme difeafe. The fat was almoft as hard as a piece of griftle." In the fourth letter, (from the author to Dr. Baillie) he explains more at large his opinion on the nature of hydatid, and the proofs of its exiftence in carcinoma. In this part of the fubjedt the author is fo clear and precife, that our readers cannot form a better idea of it than in his own words. " By the term hydatid," he obferves, " I mean an animal, confifting only of a cyft and its contents, incapable of exifting but in living animal matter, having the powers of fecreting or abforbing from its nidus the food which fills its whole cavity, and of producing an offspring fimilar to itfelf by no generative organs that can be traced. If  Dr. Adams, on the Cancerous Breast fhould be objefted that the etymology of the word would confine it to thofe cy{Is which contain a watery fluid,^ I would remark, that fuch is not the cafe with the common hydatid, the contents of which are coagulable, and fometimes tinged with red particles of blood. For this and other reafons, I fhould divide thefe into hydads lymphatica, and hydatis cruenta." *' That which is the fubjeft of our prefent enquiry, I fhould call hydatis carcinomatofa, which, befides the difference of its contents, has alfo the property of ftimulating the part in which it lives to form a kind of fungus, for purpofes I fhall endeavour hereafter to point out: my firft bufinefs is to prove, if poffible, the animalcular exiftence of carcinoma ; for this fungus, though in the cancerous breaft it is ufually confidered as the whole of the fcirrhus, appears to me only an appendage to the carcinomatous hydatid." " As you have been lefs converfant with the cancer in the breaft: than the vifcera, you may, perhaps, not have examined with your Hfual accuracy a quantity of apparently difeafed fat; which, when the part'is very much enlarged, is found inclofed in different portions of the fcirrhus, and of which the whole pofterior part of the tumour, as well as the fpa.ce from that to the axilla is ufually made up. This difeafed fat, as it is fometimes called from its appearing thinner, more tranfparent, and of a greeniih yellow hue, I have found to have all the properties afcribed to the hydatid in the human fubjedt." Dr. A. then reverts to the proofs of the feparate exiftence of the common hydatid given in Dr. Bailiie's Morbid Anatomy. The chief is, that the hydatid of the fneep, when taken out of the body and put into warm water, lias been feen to move. This, indeed, has not been obferved with the hydatid of the human liver; but as Dr. B. obferves, the reafon may be that the body is feldom examined till many hours after death. In the cafe given by Dr. Hunter, which we have juft alluded to, he found likewife that the hydatids when opened, retained a ftrong ccntratfils farce fo as to roll themfelves up in part. This power of contraction, Dr. Adams obferves, if not dependant on elafticity, is a fufficient proof of the life of thefe hydatids. If, therefore, the fame property, unconnected with elafticity, be found in the cancerous breaft, it will be fufficient (Dr. A. imagines) to eftablifh the point of the hydatid-life of this difeafed part. This he de-mon# rates in the following manner. exhibits this appearance, nor is it poHible in this ftage to dif~ tinguifh at firft light what I call carcinomatous hydatids frofti common fat. By this it appears that the yellow-green, tranfparency of the fat, and the contractile power of the inclofing tunics could only arife from life, and the degree of heat with which that life was attended." The next point on which Dr: A., exercifes his conje&ures, is oa the ufe of the formation of cartilage before ulceration, afid of the fungus after it, which fpring up round the ulcer; calling in aid of his explanation that principle fo well laid down by Hunter, that living animals lodged within another living body, do not whilfl alive ftimulate to fuppuration the parts which form their nidus; but that when dead they a?t as mere extraneous matter, aYid caufe this procefs to take place: Dr. A. thence explains the ufe of the carcinomatous fungus in the following terms: " Now, if carcinomata pafs through the fame ftages as Dr. J. Hunter has remarked of the common or lympahtic hydatid, is it not probable that on the death of any of them fuppuration will follow, and that this fuppuration may expofe the living hydatids in fuch a manner that many of them may die from not being furrounded by living animal matter? To prevent this, I conceive 3. fungus is formed, which inclofes individuals or clufters of them in feparate compartments, fo that the death of one fet produces no effeCt on the reft." " As far as my obfervation extends,' this fungus grows in every dire&ion where it is neceffary to preferve the hydatids. When z clufter of hydatids dies, the fungus between it and the furface ulcerates, or floughs flowly, till the compartment containing them is expofed. By this time, if the progrefs has been very flow, ali the tunics of the hydatids are detached, and the furface being clean will make an attempt at healing. If no dead hydatids are in the neighbourhood, it will often, for a time, fcab, or even fkin over.
But if, when the cavity is expofed, fome of the tunics of the hydatids retain their attachments, the attempt at healing will only produce an exuberance of fungus with retorted edges. This will continue till all the tunics or fragments of them are detached; after which, if no new impediment arifes, the edges will take a different direction, and the part heal for a time." The fifth letter, (from Mr. Cline to the author). The writer obferves, that he has often met with the cells in cancerous tumours, containing a greenifh yellow liquor, as mentioned by Dr. A. but they did not give Mr. C. the idea of being living hydatids. The writer alfo gives the following valuable practical directions concerning the removal of encyfted tumours.
" When an encyfted tumour is fmall, it may be cured by an opening that will difcharge its contents; and then by applying lapis fepticus to the internal furface, fo as to deftroy the cyft. But when they are very large, this praCtice muft be dangerous, becaufe the fuppurating furface might be greater than the conftitution is able to fupport. The removal or dcftruCtion of the cyft is generally Dr. Alams, on the Cancerous Breast? generally neceffary, for it rarely has any difpofition toheal. Evc? the leavinoa very fmall part of the cyft will be fufficicnt to prevent the complete healing of the wound." In the fucceeding letter, (in anfwer to the former,) Dr. Adams endeavours to explain the nature of the various cavities or cylts conne&ed with carcinoma. He fuppofes three kinds. The firlt is the common hydatid, in which, after the contents have been walhed away, on throwing the amputated part in water, the cyft Hill preferves its figure, and remains an empty cavity, or only filled with ferum. The fecond kind of cyft is a cavity filled with a gelatinous fubftance, which the author fuppofes to be carcinomatous hydatids, which have gone through their various ftages of birth, growth, and decay, and are retained in the enclofed fungus.
A third fpecies is a cavity compofed of cells, filled with a dark bloody fluid, to which the author gives the appellation of Hydatis Cruenia.
In the eighth letter, (which is from the author to Mr. Aberr nethy,) Pr. A. explains more at large the final caufe of the generation of fungus. This is in a great meafure a repetition of what we have'given our readers from the fourth letter; but the following paffages will include all the moll material fubftance of the author's hypothefis, which therefore we fhall quote. " In examining a carcinomatous breaft amputated in an early period, we meet with little or no fungus. By an early ftage I mean, before the difeafe, how long foever it may have exifted, has made any confiderable progrefs. If the progrefs has been fufficient to exhibit any fuperficial marks, by a circumfcribed puckering of the fkin, we find the fungus ufually confined to the fpace between the carcinomatous hydatids and the furface; but if the difeafe has made confiderable progrefs, fo that the whole breaft is much enlarged, it is then that we find various compartments in the fungus filled with hydatids in different Hates of their progrefs towards maturity and death. <c Hence it feems as if the hydatids'had a period of exiflence lhort in proportion as, their powers of multiplication are greater. Till they multiply (fuppofing them in a fituation that affords them a nidus for it) they appear more or lefs in a torpid ftate, occafion-. ally growing, and at other times ftationary. But the death, or perhaps even the approach towards death, of any individual or number of carcinomatous hydatids, inftautly becomes a fiimulus to the furrounding parts to generate this fungus, which, by feparating the dead from the living, produces in different parts of the fame breaft two different adtions at the fame time. One is a kind of ulceration, or more commonly continual iloughing of the fungus which inclofes the dead hydatids; the other is the formation of new fungus to protect the living hydatid, and in many inftarices, if not in all, the fungus becomes itfelf a nidas for the -generation as well as proteftion of future hydatids. That it does io for hydatis cruenta, when fuch are the contents of a cancerous breaft, we have every proof that our fenfes can furnilh. For in thefe cafes the gg ttie fungus is always much fofter and fpreads fafler, if the integuments are removed by the knife, cauftic, or ulceration, and the whole appearance when removed is fimilar to the defcription of thofe hydatids which have efcaped from the uterus, adhering to a fpongy fubltance refembling or fcrving as a placenta." In a private letter to the Editors, Mr. C. complains that his Statement, which appeared in our laft Number " was not intended to tie obtruded upon the public in fo crude and incomplete a form, nor with the verbal errors it contains. He does not think his opponeiit juftifiable in fending what was inteiided for a fmall circle, before the public at large, without the knowledge and conferit of the writer." In the prefent pamphlet, Mr. C. has given that Statement of the Cafe which he wiflies to be eonfidered the only one intended by him for general infpe&ion. It does not differ materially in fafts or opinions from that given in the Journal. 'A fecond Cafe is added, very (imilar to the firft, which was attended by Mr. C's friend, Mr. Davey, who gives the following i-elation of-it. " When I firff faw her, which was riot lefs than two-hours and a half from the conimendement of the attack, fh? was juft recovered from a fit, which, from its fymptoms,' (defcribed to me by the perfons prefent) I judged to be apopledtic, arid of that kind which is termed by writers of the moil refpeftable authority, the nervous apoplexy.

"
She was now perfeS^ly colle&ed in her fenfes, but hiid a difficulty in articulation, with a want of recolleftiori'. To explain which I muft obferve, her ideas of things were perfectly accurate, though Ihe had forgotten .the words which exprelTed them; I fay collected, becaufe when {he was at a lofs for a proper word,' if you mentioned.it, fhe direftly ufed it. " She complained of head-ach, itcknefs, naufea, nuffibrfgfs ^ all over her right fide/ with want of power to riiove that arm, thigh and leg, without the greateft exertion, and then could merely drag them after her.
For fome days previous to this attack flie had , been troubled with vertigo, naufea, general uneafinefs, pain in the head, and drowfinefs. .The bowels had been regular.


s2fieldsofstudy:
Computer Science
Physics

text:
Violation of S-matrix factorization in massive Thirring model

We present a counter example which shows the violation of the S-matrix factorization in the massive Thirring model. This is done by solving the PBC equations of the massive Thirring model exactly but numerically. The violation of the S-matrix factorization is related to the fact that the crossing symmetry and the factorization do not commute with each other. This confirms that the soliton antisoliton S-matrix factorization picture of the sine-Gordon model is semiclassical and does not lead to a full quantization procedure of the massive Thirring model.

Introduction
Integrable field theories in 1+1 dimensions have been studied in various fields of research. The Heisenberg xxx model is solved by the Bethe ansatz technique [1]. The solution is confirmed most elegantly by the Yang-Baxter equation [2][3][4]. Also, the Heisenberg xxx model with spin 1 2 is equivalent to the Thirring model which is solved exactly in the continuum field theory.
Since the Yang-Baxter equation corresponds to the S-matrix factorization, there is, therefore, no question about the S-matrix factorization in this case.
Also, the Heisenberg xyz model with spin 1 2 has been studied quite extensively [5][6][7][8][9]. Since this model is proved to be equivalent to the massive Thirring model in the continuum limit [10], it should be interesting to study these models from different points of view.
The S-matrix factorization is also assumed for the massive Thirring model or sine-Gordon field theory [11]. With this ansatz of S-matrix factorization, the spectrum of the sine-Gordon model has been obtained and is found to be consistent with the spectrum obtained by the WKB method [12][13][14].
Therefore, it has been believed that the factorization of the S-matrix for the massive Thirring model holds exactly since the bound state spectrum obtained by the WKB method was considered to be exact. However, the recent investigations of the bound state problem of the massive Thirring model from the light cone method as well as from the Bethe ansatz technique have shown that there is only one bound state, on the contrary to the WKB result [15][16][17][18]. Indeed, it is proved that the WKB result of the bound state spectrum for the massive Thirring model is not exact [15][16][17]. This suggests that we should reexamine the S-matrix factorization for the massive Thirring model since the S-matrix factorization has also been believed to hold in an exact fashion.
Until now, this factorization ansatz of the S-matrix for the massive Thirring model has never been questioned seriously. In fact, the factorization of the S-matrix for the particle particle scattering holds exactly. This can be easily seen by using the Bethe ansatz wave functions. However, it is nontrivial to obtain the S-matrix factorization for the particle-antiparticle scattering case, as we will see it later. Now, the point is that the factorization of the S-matrix has been obtained for the sine-Gordon field theory with soliton anti-soliton pictures. Therefore, in this case, one does not have to worry about the antiparticle property.
However, the soliton and antisoliton are only semiclassical objects and there is no clear way to quantize it. This is related to the fact that the soliton is a solution of the field equation, but not the field itself.
Therefore, even though one can easily write down the particle antiparticle scattering S-matrix with the factorization ansatz starting from the particle particle S-matrix factorization property, the identification of the soliton and antisoliton as the fermion and antifermion cannot be justified quantum mechanically.
In this paper, we present a counter example which shows the violation of the S-matrix factorization for particle hole scattering case in the massive Thirring model. This is based on the Bethe ansatz solutions which we obtain exactly by numerically solving the PBC equations. The structure of the Bethe ansatz solutions is quite similar to the free fermion field theory.
The ultraviolet cutoff Λ is determined by the fermion number N when we put the theory into the box with the length of L. In free field theory, the Λ is given as while, in the massive Thirring model, Λ can be obtained as the function of the N/L after we solve the PBC equations properly. Any physical observables can be obtained by letting the N and L infinity. Here, we note that this is the simplest but the best way to define the field theory.
In order to discuss the S-matrix factorization, we have to determine the vacuum and then create particle hole states as long as we are interested in the charge zero sector. Since the number operator commutes with the hamiltonian of the massive Thirring model, we are only concerned with the charge zero sector.
Here, we show that the S-matrix factorization does not commute with the crossing symmetry. Since the lagrangian of the massive Thirring model is invariant under the charge conjugation, one tends to believe that the crossing symmetry should always hold. Indeed, the crossing symmetry itself holds. However, one must be careful for the order of operations in quantum mechanics. There is no guarantee that one can make use of the crossing symmetry for the S-matrix of the particle hole scattering with the factorization properties together.
In order to check the validity of the S-matrix factorization, we have to solve the equations of the periodic boundary conditions (PBC) and construct the vacuum. However, this is not sufficient if we want to deal with the particle hole scattering. We have to solve the PBC equations for n−particle n−hole states.
The important point is that the PBC equations for n−particle n−hole states are different not only from those for the vacuum states but also from each other. Therefore, we have to first solve these PBC equations to determine the rapidities of the vacuum as well as those of the n−particle n−hole states.
It is always an interesting question how one can construct the particle hole state under the situation where the rapidity distribution of the particle hole states are different from those of the vacuum state. In other words, how can we find or identify the rapidity of the hole state if the rapidities of the negative energy particles for the n−particle n−hole states are different from those of the vacuum state ? Later, we will see that we can find the rapidity of the hole state in the limit of infinitely large N (the number of particle) and L (the box size). In this limit, we can maintain the particle hole picture.
Here, a question arises. Can we also prove the factorization of the S-matrix even for the particle hole scattering case ? As can be seen from the above statement, the hole states carry the information of all the negative energy particles since the rapidities of the particle hole state are different from the vacuum. Therefore, it is highly nontrivial that the S-matrix for the particle hole scattering can be factorized.
Indeed, we will show below that the S-matrix for the particle hole scattering is not factorizable since the factorization and the limit of N and L infinity be justified only when one neglects the commutability of the operators. This means that the soliton antisoliton picture is indeed semiclassical. Therefore, in order to make a correct correspondence between the soliton and fermion in a fully quantized sense, we should not rely on the S-matrix factorizations.
This paper is organized as follows. In the next section, we will briefly describe Zamolodchikov's S-matrix factorization method. Then, in section 3, we discuss the Bethe ansatz solution for the massive Thirring model. In section 4, we present the numerical solutions of the Bethe ansatz PBC equations in order to define the vacuum as well as the n−partcile n−hole states.
Then, section 5 will treat the S-matrix for the particle hole scattering. In particular, we will present the numerical evidences that the S-matrix of the particle hole scattering is not factorizable. Section 6 will summarize what we have understood from this work.

Zamolodchikov's S-matrix factorization
In integrable field theories, there are infinite conservation laws which induce strong selection rules [11]. For example, there is no particle creation or annihilation in the scattering process. Also, there is no change of the momentum before and after the scattering process. This means that the scattering is always elastic.
The S-matrix for the two body scattering case can be defined as where A i (α)'s describe the particle states with the rapidity α.
In the same way, we can consider the N body S-matrix The basic assumption of the S-matrix factorization is that this S-matrix can be written by the product of the two body S-matrices.
For simplicity, we consider the three body case. One assumes that the S-matrix of the three body scattering can be written as This assumption can be checked by making use of the Bethe ansatz solutions to the massive Thirring model for the particle-particle scattering case.
In this three body scattering case, we can consider the following two different processes. In the first case, the particle 1 scatters with the particle 2, and then the particle 1 scatters with the particle 3, and finally the particle 3 scatters with the particle 2. On the other hand, in the second case, the particle 2 first scatters with the particle 3, and then the particle 3 scatters with the particle 1, and finally the particle 1 scatters with the particle 2.
Since the final state is the same between the above two processes, the two scattering events should give the same scattering process. Therefore, we This is the Yang-Baxter equation.
Now, Zamolodchikov further assumes that the S-matrix should satisfy the unitarity, the analyticity and the crossing symmetry. The unitarity reads The analyticity is written as Finally, the crossing symmetry can be stated as whereī andl denote the anti-particles of i and l states. Here, the antisoliton of the sine-Gordon model is identified as the antifermion of the massive Thirring model. Together with the above assumptions, Zamolodchikov obtained the functional equation with the particles and antiparticles included.
However, for the massive Thirring model, it is nontrivial to define the antiparticle state in the nonperturbative sense. This is because one has to first determine the vacuum and then construct the particle hole states.
The problem here is that the rapidity distributions for the vacuum and the particle hole states are different from each other since the periodic boundary condition equations are different. Therefore, it is nontrivial to make hole states referring to the vacuum rapidity distribution.
The question we want to address here is whether this crossing symmetry (eq. (2.7)) can be satisfied together with the factorization of the S-matrix or not.
It is obvious that the crossing symmetry itself should hold since the Lagrangian of the massive Thirring model possesses this symmetry. Indeed, this can also be proved by the Bethe ansatz solutions.
However, it is highly nontrivial whether the factorization for the particle hole scattering S-matrix can also hold or not. This is what we want to check in this paper and we prove numerically that the particle hole S-matrices do not satisfy the factorization.

Bethe ansatz solution of massive Thirring model
The massive Thirring model is a 1+1 dimensional field theory with current current interactions [19]. Its lagrangian density can be written as where the fermion current j µ is written as Choosing a basis where γ 5 is diagonal, the hamiltonian is written as Now, we define the number operator N as This number operator N commutes with H. Therefore, when we construct physical states, we must always consider physical quantities with the same particle number N as the vacuum. For different particle number state, the vacuum is different and thus the model space itself is different.
where β i is related to the momentum k i and the energy E i of the i-th particle as Here, β i 's are complex variables.
ǫ(x) is a step function and is defined as The phase shift function φ(β i − β j ) can be explicitly written as In this case, the eigenvalue equation becomes Also, ψ(x, β) can be written in terms of ψ 1 (x) and ψ 2 (x) as, From the definition of the rapidity variable β i 's, one sees that for positive energy particles, β i 's are real while for negative energy particles, β i takes the form iπ − α i where α i 's are real. Therefore, in what follows, we denote the positive energy particle rapidity by β i and the negative energy particle rapidity by α i . Therefore, we demand that Ψ(x 1 , .., x N ) be periodic in each argument x i . This gives the boundary condition This leads to the following PBC equations, Taking the logarithm of eq.(3.14), we obtain where n i 's are integers. These are equations which we should now solve.

Numerical Solutions
The parameters we have in the PBC equations are the box length L and the particle number N. In field theory, we introduce the ultraviolet cutoff parameter. In the PBC equations, the cutoff parameter is the particle number N. This is quite similar to the free field theory where the cutoff parameter is defined by the particle number N. Once we determine the particle number N and the box length L, then we determine all the rapidity values necessary to obtain any physical quantity. In this sense, the Bethe ansatz solution can be well defined as a field theory. Any physical observables can be obtained by letting the N and L infinity.
There is also an important quantity which is the density of the system ρ.

It becomes
Here, the system is fully characterized by the density ρ. For later convenience, we define the effective density ρ 0 as where L 0 and N 0 are defined as L 0 = m 0 L and N 0 = 1 2 (N − 1), respectively.
It is important to note that the beta function of the massive Thirring model vanishes to all orders [20], and thus there is no need to worry about the renormalization of the coupling constant. However, one has to be careful for the coupling constant normalization arising from the way one makes the current regularization. Here, we employ the Schwinger's normalization throughout this paper.
In what follows, we solve the PBC equations numerically in the same manner as in ref.
[15]. The numerical method to solve the PBC equations is explained in detail in ref. [15]. We note that the errors involved in the numerical calculations are very small. For example, the numerical uncertainty for the rapidity values will appear at the 10 −6 level, and therefore, this does not generate any problem in the present discussion.
(a) Vacuum state The PBC equations for the vacuum which is filled with negative energy where n i runs as There is no ambiguity to determine the vacuum. The vacuum is indeed constructed uniquely [15].
(b) 1p − 1h configuration One particle-one hole (1p − 1h) state can be made by taking out one negative energy particle (the n i 0 particle) and putting it into a positive energy state. In this case, the PBC equations become where β i 0 can be a complex variable as long as it can satisfy eqs. (c) 2p − 2h configurations In the same way as above, we can make two particle-two hole (2p −2h) states. Here, we take out the n i 1 and n i 2 particles and put them into positive energy states. The PBC equations for the two particle-two hole states become All the configurations one can construct for the two particle two hole states turn out to be the scattering states in this model [15].

Factorization of S-matrix
Since the vacuum state is denoted by the rapidities α 1 , .., α N , |vac >= |α 1 , ..., α N > we can define the S-matrix of the vacuum to vacuum transition. In this case, we can write it as where S 0 (α i , α j ) denotes the two body S-matrix between i and j particles to make a transition from x i < x j to x i > x j states. It can be written as where φ(α i − α j ) can be written explicitly as Note that these particles here are negative energy ones. Obviously, the vacuum to vacuum transition must be unity, Therefore, we have always the constraint We also give a correspondence between S 0 (α i , α j ) as defined here and S pp pp (α i , α j ) as defined in section 2.
where pp denotes the particle particle scattering. If it is for particle hole scattering case, the S-matrix can be written as S hp ph (α i , β h j ) as will be soon explained below. It should be noted that, in the massive Thirring model, the states are completely specified by the rapidity variables α with the index of particle or hole. These are the quantum numbers that can specify the states.
(a) S-matrix Now, we first define the one particle one hole state. This can be denoted as where β h 1 is the rapidity of the hole state when N and L become infinity. But for the moment, it is only symbolically written since α † i 's differ from α i 's of the vacuum state. Therefore, the S-matrix for the 1p − 1h state can be written as Here, we note that, in the large N and L limits, we find that as will be shown later numerically in Table 1.
In what follows, we only discuss those particle hole scattering processes where the particle states are always involved. We do not consider here particle particle scattering or hole hole scattering processes.
We note that this particular example is sufficient since we are only interested in finding an example of the violation of the factorization.
Therefore, we define the S-matrix for the particle-hole scattering in the following way, Next, we want to define the two particle two hole state. This can be denoted as Again, β h 1 and β h 2 are written symbolically.
Therefore, the S-matrix for the 2p − 2p state can be written as Here again, we note that, in the large N and L limits, we find that In the same way as S hp ph , we define the 2p − 2h scattering process where the particle states are involved.
Now that the α † i and α ‡ i approach to the α i for the large N and L limits, one may think that one can replace the α † i and α ‡ i by the α i . We denote these S-matrices by S hp ph (β 1 , β h 1 ) (V ) and S hphp phph (β 1 , β h 1 , β 2 , β h 2 ) (V ) . Therefore, if we take the large N and L limits at this stage without care for the difference between α i , α † i and α ‡ i , then we obtain that the S-matrix S hphp phph (β 1 , β h 1 , β 2 , β h 2 ) (V ) for the 2p − 2h state can be factorized into 1p − 1h S-matrices. That is, Eq.(5.12) shows that the S-matrix of the two−particle two−hole scattering seems indeed factorized. This is the main reason why people believe that the factorization ansatz for the particle hole scattering should hold in the same way as the massless case or in other words as the particle particle scattering case.

Based on the factorization ansatz of the S-matrix, Zamolodchikov and
Zamolodchikov obtained the Yang-Baxter type functional equation for the S-matrix which can determine the shape of the S-matrix itself by the simple algebraic manner [11]. However, the derivation of the functional equation implicitly assumes the factorization of the particle hole scattering S-matrix. Even though the factorization of the particle hole S-matrix is valid in terms of their rapidity variables, it is nontrivial to show that this factorization can hold even for the common rapidity variables with the vacuum.
Therefore, one must be careful for the treatment of the rapidity variables when one calculates the S-matrix. In particular, one should carefully treat the order of the operations when one makes the large N and L limit in quantum mechanics. The large limits of N and L should be taken as late as possible.

(b) Large L and N limits
In what follows, we show that, in the large N and L limits, eq.(5.12) does not hold. In order to see it, we first rewrite the S-matrix of the 1p − 1h state in terms of the vacuum rapidities.
where δ i is defined as the difference between α i and α † i , that is, δ i = α † i − α i . Since δ i is quite small for the large values of N and L, we can expand S 0 (β 1 , α j + δ j ) in terms of δ i . Here, we expand the phase shift function 14) The first terms of eqs.(5.14) correspond to the S-matrices which are factorizable. Now, the important point is that the δ i itself becomes zero as the values of N and L become infinity but the summation of δ i stays finite in this limit. We will discuss it in detail below. Therefore, it is important to keep the second term for the S-matrix evaluation.
In the same way, we rewrite the S-matrix of the 2p − 2h state.
where ǫ i is defined as ǫ i = α ‡ i − α i . Since ǫ i is very small for the large values of N and L, we can expand S 0 (β 1 , α k + ǫ k ) and S 0 (β 2 , α l + ǫ l ) in terms of ǫ i in the same way as eqs.(5.14) We can also check later that the ǫ i itself becomes zero when N and L become infinity. However, the summation of ǫ i is finite and therefore, we have to keep them when we want to show the S-matrix factorization. Now, we rewrite eqs.(5.13) and (5.15) in terms of eqs.(5.14) and (5.16).
As mentioned above, we are only concerned with the particle hole scattering S matrix factorization.
Therefore, eqs.(5.13) and (5.15) can be written as The important point is that δ k contains all the information of the 1p − 1h state and therefore depends on β 1 and β h 1 which are the quantum numbers that characterize the 1p − 1h state. In the same manner, the summation ǫ k depends on β 1 , β h 1 , β 2 and β h 2 which characterize the 2p − 2h state.  Table 1, we plot several values of the rapidity as the function of N with the fixed value of the density ρ 0 . This clearly shows that the rapidities of the 1p − 1h and 2p − 2h states approach to those of the vacuum state when N becomes very large. Therefore, eq.(5.12) indeed holds. However, we must be careful when we treat any physical quantities which depend on the sum of the rapidity difference between the vacuum state and the 1p − 1h or 2p − 2h configurations.
The important point is that, even though each rapidity of the 1p − 1h and 2p − 2h states approaches to that of the vacuum, the sum of the rapidity differences stays finite.
In order to see the effect mentioned above, we define the following quantities where β 1 and β 2 correspond to the rapidity values for the n 1 and n 2 states.
In Table 2, we plot the values of D 1p1h , E 1p1h , D 2p2h and E 2p2h as the function of N and L 0 for the lowest scattering state. As can be seen, the calculated values of D 1p1h and D 2p2h do not depend very much on the N and L 0 as long as we keep the same density ρ 0 .
However, as can be seen from Table 2, the E 1p1h and the E 2p2h decrease as N increases. They behave as Therefore, these quantities do not survive at the large N limit.
This clearly shows that the sum of the rapidities times the derivative of the S-matrix is finite even though each rapidity value becomes zero when N and L become infinity. Therefore, we prove that these second terms do not vanish.
Now, it is obvious that the D 1p1h (n 1 ) and D 2p2h (n 1 , n 2 ) are quite different from each other since they carry the information of the many body nature of the particle hole states. Indeed, the D 2p2h (n 1 , n 2 ) depends on β 1 , β 2 , β h 1 and β h 2 . Therefore, the factorization of the particle hole S-matrix cannot be justified. In fact, as we see below, the numerical results confirm the difference between the D 1p1h (n 1 ) and D 2p2h (n 1 , n 2 ) . (d) Field theory limit Now, we should take the field theory limit since this is the only way to compare our results with Zamolodchikov's factorization ansatz. For the field theory limit, we have to let ρ → ∞ [15]. In the case of the bound state problem, when we let ρ → ∞, then we should take the limit of m 0 → 0, keeping the excited state energy finite. In order to see it more concretely, we repeat here how it is done for the bound state problem. The excited state energy ∆E is written as [15] where A and B are some constants which depend on the coupling constant. Here, α is a constant with positive value which is smaller than unity. In this case, when we make ρ → ∞, keeping ∆E finite, we can make a fine tuning of m 0 such that This means that we should let m 0 → 0.
However, in the present case, the situation is a bit different and even simpler since the correction factor itself does not have any dimensions. Therefore, the field theory limit is just that we let the ρ 0 infinity. In fig.1, we show the calculated absolute values of D 1p1h and D 2p2h as the function of ρ −1 0 . In this case, they can be well parameterized by the following functions as where C (i) , D (i) and κ 2 are some constants which depend on the coupling constant and the rapidity values of the particle hole states.
As can be seen, when the ρ 0 goes to infinity, then the D 1p1h and D 2p2h approach to some finite numbers. Namely, they become This means that we have made the field theory limit in a correct way.
The values of C (i) , D (i) and κ 2 for several cases of the particle hole excited states are shown in Table 3.
From the fig.1, we know that the breaking of the factorization of the S-matrix is the order of |D 1p1h − D 2p2h | which is smaller than unity.
We do not know whether this value of |D 1p1h − D 2p2h | can be said to be large or small. In any case, the S-matrix factorization is violated at the elementary level.
Here, we again note that the S-matrix factorization for the particle particle scattering must hold. This is essentially the same as the vacuum case. It is interesting to observe that the energy of the vacuum state can be obtained analytically, though the vacuum energy alone is not physically very meaningful.

Conclusions
We have presented a numerical proof that the S-matrix factorization for the particle hole scattering in the massive Thirring model is not satisfied.
This is mainly because the factorization of the S-matrix and the crossing symmetry do not commute with each other.
It is always simpler to find a counter example of the violation of the Smatrix factorization than to prove the validity of the factorization. This is clear since we only have to find out any type of example which shows the violation of the factorization. This is indeed the point we have worked out in this paper. The present calculations present a convincing evidence that the S-matrix factorization for the particle hole scattering does not hold exactly.
Here, we want to examine the implication and consequence of this result.
As stressed in this paper, it is important to realize that the S-matrix factorization should hold for the particle particle scattering even for the massive Thirring model. This can be easily seen from the Bethe ansatz solution which is eq.(3.5). Indeed, if one looks at the vacuum to vacuum transition eq.(5.1), then one sees that the S-matrix is factorized into the product of the two body S-matrices. This is the consequence of the particle particle scattering. Clearly, the massless case must have this nice feature of the S-matrix factorization since, in this case, one can treat all the scattering processes as two species of fermion scattering without going to the particle hole scattering. Therefore, those models which are equivalent to the six vertex models should receive no effects from the present investigation.
However, those models which are equivalent to the eight vertex models should be carefully treated if one is concerned with the bound state spectrum with S-matrix factorization ansatz. At least, we now know that the  Table 3.   Fig.1a. This is the excited state with n i 0 = 2, n i 1 = 1 and n i 2 = −2. We plot the rapidity values for several cases of n i for the vacuum, 1p −1h state with n i 0 = 0 and 2p − 2h state with n i 1 = 1, n i 2 = −1. The density ρ 0 is fixed to ρ 0 = 20. Table 2 (g 0 = 6.28) We plot the values of D 1p1h , E 1p1h , D 2p2h and E 2p2h for several cases of the number of the particles N for the lowest scattering states ( n i 0 = 1, n i 1 = 1, n i 2 = −1. ). The density ρ 0 is fixed to ρ 0 = 20. D 1p1h (n i 0 ) (g 0 = 6.28) We plot the values of the parameters C (1) , D (1) . The numbers in parenthesis indicate error bars in the fitting. D 2p2h (n i 1 , n i 2 ) (g 0 = 6.28) We plot the values of the parameters C (2) , D (2) and κ 2 . The numbers in parenthesis indicate error bars in the fitting.


A Logical Relational Approach for Information Retrieval Indexing

In a relational indexing approach (see e.g. Farradane's work), information is carried by a fixed set of relationship types over an underlying set of terms. The idea is that the essence of the meaning of information is encapsulated in the relationships between terms. The importance of relationships is now widely recognized within many fields such as relational databases and knowledge representation formalisms. These fields have substantially improved our understanding of relationships and the problems involved in trying to formalize them. However, although those relationships can be correctly represented by almost all the well-known formalisms in such fields, they are not exploited as much as the objects by concrete operations. In information retrieval, previous attempts at managing relationships have mainly addressed structural aspects, and exclude the manipulation of index expressions by relational operations. This paper suggests a prime use of the relation properties through a logical framework, in a way that it can improve the effectiveness of the matching operation.

retrieval system is still viewed as a system that selects documents on the basis of a matching operation between a document representation d and a query representation q. If the matching operation deems a document as being sufficiently similar to the query, then the document is assumed likelytoberelevantand returned to the user. According to the logical model of information retrieval, the task of the system can be described as the extraction, form the document base, of those documents d that, given a query q, make the formula d ! q valid, where d and q are formulae of the underlying logic, and "!" denotes the logical consequence decision formalized by the logic in question. Such a logic-based approach originates in the work of Cooper [5] who provided a formal definition of relevance in terms of logical consequences. However, this approach to information retrieval and to relevance, as well as the classical approaches [6,7], is commonly called the topical approach [8,9] in which only the document's and the query's representations matter. As mentioned by Saracevic [10], it has been known for a long time that topicality is not the only criterion of a user's relevance judgment. A number of others factors also affect the relevance result such as the user's knowledge, the expected use of information, the application domain and so on. Van Rijsbergen [1] pointed out this problem when he proposed the use of a non-classical logic for information retrieval.
A general part of information retrieval systems is the index language, which is the language used to represent the documents in the collection, and the request of the seeker. The creation of the internal representation of a document is a prominent function of any information retrieval system. It is often referred to as the indexing process. The outcome of this process is a set of index expressions that supposedly summarize the information content of a document. The index expressions can be keywords, parse trees, semantic structures and so on. Central to our approach in this paper is the assumption that the output of the internal representation can influence considerably the effectiveness of the information retrieval system, especially when this output consists of some semantic structures (conceptual graphs, parse trees, infons, etc.). Indeed, as mentioned by [11], the most intricate or carefully designed retrieval algorithm can not compensate for inappropriately represented documents. It is then evident that the index language must be much more expressive and richly structured than the keywords-based languages usually adopted by the classical information retrieval systems. As a consequence, the index expressions should have a complex internal structure that we must handle with the utmost care: the more complex the index expressions are in their structures and contents, the more the underlying semantics of this complexity has to be made explicit. We claim here that the knowledge we have about the resulting indexing expressions can improve the matching process.
Recently, some authors make use of the knowledge representation formalisms to analysis document's information content and information need, and to evaluate the relevance of a document to a query. Good examples of such formalisms, experienced within the RIME [12] and MIRTL [13] projects, are based on a formalism derived from the notion of Conceptual Dependency and on Terminological Logics, respectively. Usually, these formalisms allow for a number of term-forming operators by means of which one may build semantic structures starting from a basic repertory of simple terms and relationships (see figure 1) 1 . Hence, a semantic structure is an expression of the form E i Relation E j ,w h e r eE i and E j are simple or complex terms, and Relation is a relationship holding between the terms. As a consequence, one can explore in addition to the thesaurus of terms (synonymy, related terms, etc.), the relationships properties, like symmetry, transitivity and so on, to derive more knowledge. This can be done through a logical framework following the ideas of Nie [14]. He showed that the logical model suggested by van Rijsbergen [1] seems to be a very promising approach to represent the impact of the system's knowledge on the matching operation. 1 In this example, bears-on and has-for-value are relations, while OPACITY,LUNG and TISSULAR are terms.
Hence, various kinds of inference rules can then be determined, and they form the derivation system which drives the plausible inference process between the documents and the queries. This derivation system will be defined over the language of index expressions. has-for-value

Keywords:
Opacity, Lung, Tissular The contributions made by this paper are twofold. Firstly, some useful relational operations in information retrieval are given. These operations allow us to handle the relationships that may occur in the documents and the queries representations. Secondly, we can generate more, hopefully precise, derivations and as a consequence more relevant answers to the user. The remainder of the paper is organized as follows. In Section 2 we recall the classical indexing approaches used in information retrieval. In Section 3 we will show how the semantical properties of index expressions can be used to generate more derivations rules. We define rules that take into account the relationships involved in these index expressions. In Section 4 we will show how the relationships will be organized in case of real information retrieval systems, some particular problems concerning the application of such an approach to information retrieval are also mentioned, while further directions of investigations are described in Section 5.

Current logical approaches
One can define a relation as a set of tuples that represents a relationship among objects in the universe of discourse.
Each tuple is a finite, ordered sequence of objects [15]. Tuples are in the universe of discourse, and can be represented as individual objects. Usually, in the literature the words relation and relationship refer to the same thing. In CycL [15], relations are called relationships and in LOOM [16], relations are called relations. Usually, relations are denoted by predicates. A fact that a particular tuple is an element of a relation is denoted by relation-name (arg 1 , arg 2 ...arg n ), where the arg i are the objects in the tuple. In the case of binary relations, the fact can be read as "arg 1 is relationname arg 2 "or"arelation-name of arg 1 is arg 2 ".
One of the first indexing strategies used in information retrieval was the extraction of keywords or topics from documents and the valuation of their importance according to their frequency of appearance. It has often been noted BCS IR Colloquium, 1997 that this approach neglects the important relations between keywords [9]. Sometimes, those relationships are even removed from the indexing process by an initial selection of words that removes the common words in the document (like "is", "on", "of", etc.); This is mainly done out of practical considerations. Such a classical approach can benefit from the availability of efficient algorithms that automatically extract the keywords from the document. As a result the relationships in which the keywords were involved are no longer taken into account. However, these relationships could be used to provide contextual information for driving the matching process. In fact, in the real word, relationships between objects play an important role. In the field of relational databases [17] or information systems engineering (e.g. Merise [18]), the importance of relations has already been recognized.
In the well-known information retrieval CACM test collection, we have noted a high occurrence frequency for some relations. After analysis, we have converted this collection into an homogeneous representation including only the title and the abstract fields. The number N of words in the collection was then computed. We has developed a tool that allows to search for sentences containing a sequence of words. It is possible to include parameters in these sentences. Hence, by the query "the #parameter on #parameter" we mean that we are looking for all sentences in the collection beginning with the word "the" followed by any string, followed by the relation "on" and ending with any string. Some example of these sentences are shown in the figure 2, with their respective probability of occurrence in the collection, according to the number N.  In information retrieval the representation of the relationships between keywords takes its root in the work of Farradane described in [19,20]. Farradane introduced the idea that much of the meaning in information objects is denoted in the relationships between terms. For example, "John drinks a cup" exhibits a functional dependence relationship type between John and cup. Hence, whereas in classical information retrieval approaches such a sentence will be indexed by the two keywords John and cup, Farradane projected the idea that the relationship drink must also be represented in the final index of the sentence. In fact, in this special example, the indexing of the sentence by the keywords John and cup can even be called erroneous, as obviously John drinks the content of the cup (which can be wine, coke, water, etc.) and not the cup itself. The representation of the relationship drink will avoid such an ambiguity as we all know that we don't drink cups but only the liquid they contain.
A parallel was drawn with the conceptual model from the database world where relationship types between entities play an important role; the index description of a document consisting solely of keywords would be like an BCS IR Colloquium, 1997 entity relationship model without relationship types. In Farradane's work, information was carried by a fixed set of relationship types over an underlying set of terms. This conception bears a close resemblance to a large class of knowledge formalisms such as terminological logics [21], conceptual graphs [22] and situation theory [3]. A terminological logic is a subset of first order logic with equality that contains only unary relations, representing sets of objects in the domain (referred to as concepts) and binary relations (called roles) linking together the objects of the domain. A conceptual graph is a bipartite, connected, finite and oriented graph of concepts and conceptual relations.
In the graphs, concept nodes represent entities, attributes, states and events, and relation nodes show how the concepts are interconnected [22]. Finally, the situation theory formalism introduces the important notion of infon [23,3]. An infon is a structure that represents the information that a relation R holds or does not hold between a particular set of objects. The use of those three formalisms in information retrieval was partially motivated by the fact that they all allow to represent more complex index terms and they offer enough capabilities to represent objects and relationships between objects [13,24,25,26,27]. Recent studies [28,9] about the impact of structured documents on both indexing and retrieving have shown the need to represent some aggregative relationships that satisfy a set of properties and constraints.
However, if the relationships between terms can be represented by almost all the actual knowledge representation formalisms, they are not exploited as much as the objects by the primitive operations given by those knowledge formalisms. Moreover, no behavior can explicitly be associated to relationship types, and neither does relation-based reasoning allow one to explore the implicit knowledge that can be of beneficial use in the matching process. For example, it has been shown in [29], that although the terminological logics are able to represent the relationships needed to index complex and structured documents, it is far from clear how to describe their characteristics. For instance, it is not possible to specify mathematical properties about roles (symmetry, transitivity, ...) without considering all possible derivable facts as a part of the knowledge base. Let us take a simple example: assume that Aggregateos i ;o s j and Aggregateos j ;o s k are two assertions (facts) in the terminological knowledge base . Also, assume that the Aggregate role is transitive. In such case, there is no possibility in terminological logics to infer Aggregateos i ;o s k from the knowledge base . Hence, a document indexed by Aggregateos i ;o s j and Aggregateos j ;o s k will not be retrieved by a query containing Aggregateos i ;o s k . In order to avoid this problem, we must add explicitly this derivable fact to the knowledge-base . However doing this would be very expensive, especially in cases where the document base contains several thousands of structural objects. In order to avoid practical problems like this, the introduction of some relational operations can help to describe the relationships behavior. Hence, we aim in the following sections to operate directly on the relationship types that can occur in semantical index expressions. We will propose a technique that can infer new information by analyzing the properties of the relations such as symmetry, transitivity, semantical behavior, arrangement with others relationships, and so on. This will be done through a general logical framework for studying relationships and their semantical properties. This framework captures the semantical information of the relations for information retrieval purposes by specifying their properties by way of inference rules.

A logical relational approach
Suppose one wants to build a logical system for a given application. First of all, one must specify which language L is to be used for defining the notion of well formed formulae in this system. This will allow us to construct the axioms and rules capturing the system. In other words, it is not enough to know the behavior of this logical system, as one must also know how it is presented. Here, we are interested in a logical framework for information retrieval.
Hence, if one considers the deduction as the retrieval operation, this means that the relevance relationship between document and query can be established in terms of the axioms and rules belonging to the logical system. As a result, the index language we use for the description of documents and queries constitutes the one specifying the syntax of a well formed formula in the above system. Following Farradane's idea, the index expressions must be able to represent the following form : E i Relation E j where E i and E j are simple or complex descriptors, and Relation is a relation holding between the descriptors. We choose the situation theory [30,23] as the derivation system's language: The semantical content of each document depends greatly on the relationships between objects and the index expressions it contains, and these relationships are the means of describing how the objects are combined. It seems interesting then to capture the behavior and the properties of these relationships through a set of inference rules in order to generate more knowledge about the content of each document. Such relationship properties will be defined by the set of axioms and rules contained in the derivation system.  By definition we assume that the aboutness inference 2; is reflexive, this seems to be an inherent property of aboutness in many IR models [3]; i.e., 8e 2LO;R;e2; e. Using this property, one may infer that the operator is idempotent: The empty index expression " is a neutral element for : 8e 2L;e "2;e A query can been seen as a request for information. It can therefore also been represented as an aggregation of expressions from the language LO; R. What remains now is the question of how to derive the query from the document description in a way that will improve the effectiveness of the matching operation.
Given the definitions above, we offer to operate directly on relations, that means we propose to exploit their semantical properties and the information they bear implicitly. Hence, in addition to the semantic networks that establish the relationships that objects have to other objects (i.e., the thesaurus), usually used in classical information retrieval approaches, for each relation we give its properties, its behavior and its arrangement with others relations.
All these information will be captured through a set of derivation rules.

The Mathematical Properties
General mathematical properties can be used to augment relations implicitly contained in the knowledge's system.
Such a relations can be used to improve the matching process between the document and the query. For instance suppose that in an image, we have three objects o 1 ;o 2 and o 3 and that they are connected by the "cover" spatial relationships, then we can add a new "cover" relationship between o 1 and o 3 such as:: cover cover cover o1 o2 o3 Indeed, in the raster mode, we say that an object o 1 covers another object o 2 if, and only if, all the pixels of the object o 1 are included in the pixels of the object o 2 , i.e. pixel(o 1 ) pixel(o 2 ). This relation is transitive as mentioned by [32].
It should be noted that these mathematical properties are more easy to formalize when they are associated to binary relations. However, although it is theoretically possible to split a relation into binary relations [22], we choose here to handle both binary and N-ary relations. Indeed, it is perhaps the case that due to some practical considerations, one may need some N-ary relations. For instance in [29], it has been shown that in order to cope with the complexity of structured multimedia documents, some tertiary relations are needed. To represent such relations, we need two primitive binary relations which are more difficult to manage: we must check if all the properties of the N-ary relation are faithfully represented. In fact one can easily see that the only use of binary relations requires the creation of a large number of relations which makes dealing with them hard and rather complex their treatments. Using the N-ary relations, the number of relations will decrease and a better computational behavior of the underlying operational system will be expected. On the other hard, one can remark that only the N-ary relations having an interest in information retrieval will be considered in the framework. As their semantical properties are known, it is not hard to include them such as derivation rules.
The mathematical properties of relations can be defined as follows: Sequentiality hhR;e1;e2;:::;e k ;1ii hhR;e k ;e k+1 :::;e k+m ;1ii hhR;:::e N,1 :::;e k+m ;1ii In case R is an N-ary relation, e N,1 denotes a combination of N , 1 expressions chosen from the set e 1 , ..., e k+m,1 . We can have N , 1 possible derivations, according to which combination of expressions we take.
For example in case of tertiary relations, the following derivations are valid if the relation R satisfy the above For instance, the relation "Between" that is a tertiary relation satisfy the two above derivation rules. This property is a general case of the transitivity relation.
The permutation rule expresses the claim that the order of linking some objects in the relation does not have any influence on the derivation process.

Permutation
hhR;e1;e2;:::;ei;:::;ej;:::;e k ;1ii hhR;e1;e2;:::;ej;:::;ei;:::;e k ;1ii Consider for instance the relation Between. If we assume that hhBetween;X;Y;Z;1ii means "X is between Y and Z", then the document "Reading is between London and Bristol" is about the query "Reading is between Bristol and London", as the Between relation is symmetric. We have the following instantiation of the "Permutation"rule: Example hhBetween;Reading;London;Bristol;1ii hhBetween;Reading;Bristol;London;1ii In case of binary relations, this property corresponds to the classical notion of Symmetry. For instance, the relation Married-To is Symmetric. The symmetry property is primarily intended to increase the number of the aboutness theorems [3].
Relations could have the behavior of a function. In the work of Chiaramella and al. [28], it has been shown that in order to allow a solid formal basis to express the retrieval of structured documents, some specific functions must be introduced. They can satisfy one of the following logical constraints: there is no i 1 ;:::;i n ;e 1 ;e 2 such that Ri 1 ;:::;i n ;e 1 ^Ri 1 ;:::;i n ;e 2 there is no e; i 1 ;:::;i n ;j 1 ;:::;j m such that Re; i 1 ;:::;i n ^Re; j 1 ;:::;j m In our logical framework, the first constraint can be expressed as follows:  We can have then the following example: Example hhPa rt-Of;o1;o2;1ii hhContained-In;o2;o3;1ii hhContained-In;o1;o3;1ii

Link with others relations
The definition of links between relations enables one to deduce new information about the indexing expressions, using relation-based reasoning. Hence, relations can be handled as well as concepts.
Like the synonymy in case of keywords or concepts (considered as unary relations), one may express that two relations are the same. The following "Alias" rule is a way to specify that two relations have the same extension and then that they are logically equivalent. Like for the terms (keywords, concepts, etc.), in information retrieval, this rule may be useful to avoid the omission of some relevant answers when the name of the relation mentioned by the user is not the same as the one used in the indexing process. A link between relationships that can be interesting for information retrieval is introduced by the following Inversion rule. This rule signifies the fact that the relations R 1 and R 2 are equivalent when their arguments are swapped is mentioned. Relationships of this kind are often used in some knowledge representation formalisms, especially in the case of binary relations. For instance in many terminological logics, a special operator inv is introduced. This operator applied to a given role (binary relation) produces an inversion of its arguments. This derivation rule allows for instance to derive that an image in which A tree is on the left of a house is relevant to a query looking for all images representing A relation R 1 can be a sub-relation of another relation R 2 . Intuitively, if these relations are viewed as sets of tuples, R 1 is a subset of R 2 . In other words, every tuple of R 1 isalsoatupleofR 2 , i.e., if R 1 holds for some objects o 1 ;o 2 ;:::;o n , then the relation R 2 holds for the same arguments. The following Subrelation rule expresses this fact. Note that a relation and its sub-relation must have the same arity N. The next rule denotes the Simultaneity property. Let for instance n and m be the respective arities of R 1 and R 2 . Let i 1 ;i 2 ;:::;i n,1 and j 1 ;j 2 ;:::;j m,1 be a set of indexing expressions. The Simultaneity property states the fact that each indexing expression e taking part in the relation R 1 by R 1 e; i 1 ;i 2 ;:::;i n,1 , necessarily takes part in the relation R 2 such that R 2 e; j 1 ;j 2 ;:::;j m,1 and vice versa. In our framework, this property can be expressed as follows: Simultaneity hhR1;e;i1;:::;in,1;1ii hhR2;e;j1;:::;jm,1;1ii For instance, if we assume that hhPa rents;x;y;z;1ii means "the parents of x are y and z, with y being the mother and with z being the father", then a document about hhPa rents;x;y;z;1ii is also about hhHave-As-Father,x ,z;1ii.
Another rule claims that for some pairs of relations R 1 and R 2 having the same arity N, there is no set of objects o 1 ;o 2 ;:::;o n that can be linked together at the same time by R 1 and R 2 . In terms of our framework, we write: Preclusion hhR1;e1;e2;:::;en;1ii hhR2;e1;e2;:::;en;0ii In case of binary relations, if a document is about an expression such as hhW ith; TRAIN; RESERVATION;1ii,then this document is not about a query looking for Trains without Reservation,i.e.hhW ithout; TRAIN; RESERVATION;1ii.
Here, the relations "With"and"Without" preclude each other. This property is interesting for information retrieval as it may be used to determine the non-aboutness [3].

A theoretical study of relations
In the previous section, we have presented a general logical mechanism of relation-based reasoning. It is based on a selection of rules that possess some important properties, according to real cases we have encountered in today's information retrieval systems [32,29]. Detailed investigations of what an interesting relation for information retrieval consists of are needed in order to develop an operational system. Indeed, it remains clear that only a specific set of relations are useful or need to be characterized in case of information retrieval. For instance, as mentioned by Palmer [33], the verbs do not convey useful properties. On the other side, we already know that one needs a specific class of aggregative relations in order to handle the structured multimedia documents [28]. For example, one can represent the information that a book b 1 is composed of two chapters c 1 and c 2 by the relations Aggregate(b 1 , c 1 )and Aggregate(b 1 , c 2 ), with Aggregate an aggregative relation. Another class of useful relations in information retrieval is exposed in the work of Mechkour described in [32]. This work proposes a model for images retrieval and describes a class of spatial relationships that may occur in an image. Twelve spatial relations were proposed in this model, like far, near, touch, in, etc..
Our logical mechanism can be integrated to all the knowledge formalisms that don't allow for relation-based reasoning. Hence, this will resolve their limitations in case where the query refer to some common properties about the relations. The result is an hybrid model that combines the original characteristics of the chosen formalism and our relation-based framework, in a way that it can handle relations, as well as concepts. For instance, assume that a document d is represented by the full nodes in the figure 4. Hence, this document will be retrieved by the query q, only if the properties of the relations Aggregate and Cover are applied. As introduced by Sowa [ Figure 4: A relation-based reasoning example Some specific classes of relations can be defined. These classes will serve as a basis to express the properties of relations, their relation relationships and their links with others relations, in order to make explicit their behavior.
Indeed, in addition to the fact that it is more easy to determine those properties at the level of the same class, it is usually the case that only relations of a common class can be agenced or linked together. A study of the different kind of relationships used in the information retrieval literature shows that mainly prepositions and verbs are used as relations in the indexing expressions. For example, in Palmer's work [33], an index expression consists of a noun phrase, called a conceptual group, where the relationships are essentially French prepositions. Bruza's approach adopts the same principle with English prepositions [34].
As a consequence, a classification of this particular kind of relations can have an interest for our approach. In the The relation-based mechanism will rely on such a classification to determine for each relation its properties and behavior. This will be done within each specified class on the relations it contains. Depending on the information contained in the corpus and the document's nature (textual, multimedia, structured, etc.), other classes may be added and analyzed, such as the class of aggregative relations in case of structured documents.
It should be noted that it is difficult to use prepositions correctly. Most of them have several different functions; for instance the dictionary lists eighteen main uses of the preposition at [35], though probably only few of them are of interest in information retrieval. In order to differentiate those kind of relations and to specify exactly what kind of the relation at we use 3 , one may specify a signature for each relation. The signature of an n-ary relation R is a sequence of n objects type ht 1 ;t 2 ;:::;t n i that specifies the types of the objects that can be linked by this relation.
Here a hierarchy (lattice) of objects is needed. To be valid each argument (object) of the tuple must be a specialization (a sub-class) of the corresponding type in the signature, according to the hierarchy of objects. For example the relation Loc 4 linking an Object to a place may have h ; PLACEi as a signature, where denotes the universal object denoting all the individuals of the domain. We write t 1 t 2 to mention the fact that the type t 1 is a specialization of the type Using the signature, one is able to express the similarity between the sentences of the type "the adventures of Alice" and "Alice's adventure". This can not be deduced by Palmer's [33] and Bruza's [34] approaches. Hence, for each relation, one may specify the following parts: The label of the corresponding relation, such as Loc for Locality relation.
The class of the relation, such as EXACT for relation loc, considered as a special case of the at preposition in the figure 4.
The signature of the relation, like h ; PLACEi for the relation Loc.
An informal comment about the use of the relation, such as "A locality is a relation linking an Object to a Place".
The Mathematical properties of the relation, for example the Loc relation may satisfy the Left Exclusivity rule.

Its arrangement with others relations
Their links with others relations, for instance: the Loc relation is a Subrelation of the relation Place (see figure   4),hereweusetheSubrelation rule.  Table 1: Prepositions information is not added but rejected. Depending on the context reasoning approach and the established arrangement between relations, this problem can be resolved. For instance, as the relation hhFather-Of;a;b;1ii precludes the relation hhChild-Of;a;b;1ii, it will be impossible in case of monotonic reasoning to add an information like hhChild-Of; John; Jack;1ii if hhFather-Of; John; Jack;1ii already exists in the system's knowledge about the document. Some other questions can be discussed such as the influence of the application order of the rules on the matching process, or whether all the derivable information from the indexing expression should be added to the system's knowledge about the document. For instance, an automatic application of the inversion or Alias rules may produce redundant information.

Conclusion
In information retrieval some indexing processes combine information items by bringing them into a relationship, in order to describe the information content of a document more precisely. In this paper we have proposed a logical framework for studying such relationships and their impact on the matching process. The framework captures some relationship features and properties. Within this framework, rules have been outlined that enables the generation of new implicit information contained in the document. The work presented here does not refer to a particular indexing language: our relational indexing approach does not mention in which way the index terms are achieved.
The indexing terms could be any element of any complex language, such as conceptual graphs, terminological logics and so on. The construction of the framework is work in progress. It constitutes a first step in the understanding of relations and their influence on the matching process in information retrieval. Our interest is focused on the theoretical aspects, refining the accuracy of the framework and evaluating the effectiveness of the relation approach on both recall an precision. Although the expressive power of our framework has been demonstrated, we still have paid no attention to uncertainty aspects. This is an important focus for future work. For example, the valuation of the fact that the index expression hhTa lk in g -To; John; Mary; ;1ii is about hhListening-To; Mary; John; ;1ii is very often true but not always. It depends on the context, for instance whether they are in the classroom or in a meeting. From an implementation point of view, we are investigating this framework within a conceptual graph environment [32].