Journal articles on the topic 'Ultimately periodic infinite'

We study ultimate periodicity properties related to overlaps between the suffixes of a left-infinite word [Formula: see text] and the prefixes of a right-infinite word [Formula: see text]. The main theorem states that the set of minimum lengths of words [Formula: see text] and [Formula: see text] such that [Formula: see text] or [Formula: see text] is finite, where [Formula: see text] runs over positive integers and [Formula: see text] and [Formula: see text] are respectively the suffix of [Formula: see text] and the prefix of [Formula: see text] of length [Formula: see text], if and only if [Formula: see text] and [Formula: see text] are ultimately periodic words of the form [Formula: see text] and [Formula: see text] for some finite words [Formula: see text], [Formula: see text] and [Formula: see text].

Infinite words on a finite non-empty alphabet have been investigated in various respects. We will consider here two important strategies in approaching such words; one of them proceeds from particular to general, while the other proceeds from general to particular. As we shall see, the respective hierarchies don't interfer. There is between them an empty space waiting for investigation. 1.) C. S. Calude, K. Salomaa, S. Yu (eds.). Advances and Trends in Automata and Formal Languages. A Collection of Papers in Honour of the 60th Birthday of Helmut Jürgensen.

Consider a nonstandard numeration system like the one built over the Fibonacci sequence where nonnegative integers are represented by words over {0,1} without two consecutive 1. Given a set X of integers such that the language of their greedy representations in this system is accepted by a finite automaton, we consider the problem of deciding whether or not X is a finite union of arithmetic progressions. We obtain a decision procedure for this problem, under some hypothesis about the considered numeration system. In a second part, we obtain an analogous decision result for a particular class of abstract numeration systems built on an infinite regular language.

In [A. Frid, S. Puzynina and L. Q. Zamboni, On palindromic factorization of words, Adv. in Appl. Math. 50 (2013) 737–748], it was conjectured that any infinite word whose palindromic lengths of factors are bounded is ultimately periodic. We introduce variants of this conjecture and prove this conjecture when the bound is 2. Especially we introduce left and right greedy palindromic lengths. These lengths are always greater than or equals to the initial palindromic length. When the greedy left (or right) palindromic lengths of prefixes of a word are bounded then this word is ultimately periodic.

We achieve elastic cloaking for a periodic distribution of an infinite number of parallel finite mode III cracks by means of the complex variable method and the theory of Cauchy singular integral equations. The cloaking bimaterial structure is composed of an undisturbed uniformly stressed left half-plane perfectly bonded via a wavy interface to the right half-plane which contains periodic cracks. The original design of the wavy interface and the positions of the periodic cracks are ultimately reduced to the solution of a Cauchy singular integral equation which can be solved numerically.

We investigate the orbits of automaton semigroups and groups to obtain algorithmic and structural results, both for general automata but also for some special subclasses. First, we show that a more general version of the finiteness problem for automaton groups is undecidable. This problem is equivalent to the finiteness problem for left principal ideals in automaton semigroups generated by complete and reversible automata. Then, we look at w-word (i.\,e.\ right infinite words) with a finite orbit. We show that every automaton yielding an w-word with a finite orbit already yields an ultimately periodic one, which is not periodic in general, however. On the algorithmic side, we observe that it is not possible to decide whether a given periodic w-word has an infinite orbit and that we cannot check whether a given reversible and complete automaton admits an w-word with a finite orbit, a reciprocal problem to the finiteness problem for automaton semigroups in the reversible case. Finally, we look at automaton groups generated by reversible but not bi-reversible automata and show that many words have infinite orbits under the action of such automata.

AbstractWe introduce a family of infinite nonamenable discrete groups as an interpolation of the Higman–Thompson groups by using the topological full groups of the groupoids defined by $\beta $-expansions of real numbers. They are regarded as full groups of certain interpolated Cuntz algebras, and realized as groups of piecewise-linear functions on the unit interval in the real line if the $\beta $-expansion of $1$ is finite or ultimately periodic. We also classify them by a number-theoretical property of $\beta $.

In this work, we investigate the temporal characteristics of acoustic ray propagation through simulated, weakly turbulent temperature fields. In a first set of experiments, we generate ensembles of random scalar fields from randomly oriented Fourier temperature modes. Then, by integrating the ray trace equations, we estimate the distribution of arrival times for rays propagating a distance R through them. We demonstrate that these arrival time distributions are Gaussian for both axial and 3-D propagation and are primarily determined by the lower wave numbers of the 1-D fluctuation spectrum. In a second set of experiments, we generate random fields comprised of Fourier modes prescribed on a lattice, as in "box model" turbulence. In these simulations, we find that acoustic travel times are significantly affected both by the periodicity of the fields and by the direction of acoustic propagation with respect to the orientation of the box. Both effects can ultimately be attributable to an inadequate representation of the low wave number region of the 1-D spectrum. We suggest that these artifacts of simulated periodic fields may preclude their use for acoustic propagation studies.

With ideas from data compression and combinatorics on words, we introduce a complexity measure for words, called repetition complexity, which quantifies the amount of repetition in a word. The repetition complexity of w, R (w), is defined as the smallest amount of space needed to store w when reduced by repeatedly applying the following procedure: n consecutive occurrences uu…u of the same subword u of w are stored as (u,n). The repetition complexity has interesting relations with well-known complexity measures, such as subword complexity, SUB , and Lempel-Ziv complexity, LZ . We have always R (w)≥ LZ (w) and could even be that the former is linear while the latter is only logarithmic; e.g., this happens for prefixes of certain infinite words obtained by iterated morphisms. An infinite word α being ultimately periodic is equivalent to: (i) [Formula: see text], (ii) [Formula: see text], and (iii) [Formula: see text]. De Bruijn words, well known for their high subword complexity, are shown to have almost highest repetition complexity; the precise complexity remains open. R (w) can be computed in time [Formula: see text] and it is open, and probably very difficult, to find fast algorithms.

The simplest infinite sequences that are not ultimately periodic are pure morphic sequences: fixed points of particular morphisms mapping single symbols to strings of symbols. A basic way to visualize a sequence is by a turtle curve: for every alphabet symbol fix an angle, and then consecutively for all sequence elements draw a unit segment and turn the drawing direction by the corresponding angle. This paper investigates turtle curves of pure morphic sequences. In particular, criteria are given for turtle curves being finite (consisting of finitely many segments), and for being fractal or self-similar: it contains an up-scaled copy of itself. Also space-filling turtle curves are considered, and a turtle curve that is dense in the plane. As a particular result we give an exact relationship between the Koch curve and a turtle curve for the Thue–Morse sequence, where until now for such a result only approximations were known.

The instability of an annular layer coated on the interior side of an outer circular tube and surrounding another annular layer coated on the exterior side of an inner circular tube, is studied in the absence of an imposed flow due to a pressure gradient or boundary motion. As the radius of the inner cylinder tends to vanish and the radius of the outer cylinder tends to infinity, the inner layer reduces to a liquid thread suspended in a quiescent infinite ambient fluid. The fluids are separated by a membrane that exhibits constant surface tension and develops elastic tensions due to deformation from the unstressed cylindrical shape. The surface tension is responsible for the Rayleigh capillary instability, but the elastic tensions resist the deformation and slow down or even prevent the growth of small perturbations. In the first part of this paper, we formulate the linear stability problem for axisymmetric perturbations, and derive a nonlinear eigenvalue system whose solution produces the complex phase velocity of the normal modes. When inertial effects are negligible, there are two normal modes; one is stable under any conditions, and the second may be unstable when the interfacial elasticity is sufficiently small compared to surface tension, and the wavelength of the perturbation is sufficiently long. Stability graphs are presented to illustrate the properties of the normal modes and their dependence on the ratio of the viscosity of the outer to inner fluid, the interfacial elasticity, and the ratios of the cylinders' radii to the interface radius. The results show that as the interfacial elasticity tends to vanish, the unconditionally stable mode becomes physically irrelevant by requiring extremely large ratios of axial to lateral displacement of material points along the trace of the membrane in an azimuthal plane. In the second part of this paper, we investigate the nonlinear instability of an infinite thread in the limit of vanishing Reynolds numbers by dynamical simulation based on a boundary-integral method. In the problem formulation, the elastic tensions derive from a constitutive equation for a thin sheet of an incompressible isotropic elastic solid described by Mooney's constitutive law. The numerical results suggest that the interfacial elasticity ultimately restrains the growth of disturbances and leads to slowly evolving periodic shapes, in agreement with laboratory observations.

Propagation of a curved shock is governed by a system of shock ray equations which is coupled to an infinite system of transport equations along these rays. For a two-dimensional weak shock, it has been suggested that this system can be approximated by a hyperbolic system of four partial differential equations in a ray coordinate system, which consists of two independent variables (ζ, t) where the curves t = constant give successive positions of the shock and ζ = constant give rays. The equations show that shock rays not only stretch longitudinally due to finite amplitude on a shock front but also turn due to a non-uniform distribution of the shock strength on it. These changes finally lead to a modification of the amplitude of the shock strength. Since discontinuities in the form of kinks appear on the shock, it is necessary to study the problem by using the correct conservation form of these equations. We use such a system of equations in conservation form to construct a total-variation-bounded finite difference scheme. The numerical solution captures converging shock fronts with a pair of kinks on them – the shock front emerges without the usual folds in the caustic region. The shock strength, even when the shock passes through the caustic region, remains so small that the small-amplitude theory remains valid. The shock strength ultimately decays with a well-defined geometrical shape of the shock front – a pair of kinks which separate a central disc from a pair of wings on the two sides. We also study the ultimate shape and decay of shocks of initially periodic shapes and plane shocks with a dent and a bulge.

SynopsisIt is the aim of this article to consider some problems arising from the non local-compactness of the phase space for functional differential equations. The compact boundedness, that is, the boundedness depending on each compact set involving the initial values, is proved to be implied from the ultimate boundedness for periodic systems of functional differential equations on Cγ: = {φ ∊ C((–∞,0]) Note that it is known that the compactness cannot be dropped in the above. An example is also given to show that the asymptotic stability is not necessarily uniform even for periodic functional differential equations on Co.

In this paper a one-parameter class of four-dimensional, reversible vector fields is investigated near an equilibrium. We call the parameter μ and place the equilibrium at 0. The differential at 0 is supposed to have ±iq, q > 0, as simple eigenvalues and 0 as a double, nonsemisimple eigenvalue. Our ultimate goal is to construct homoclinic connections of periodic orbits of arbitrary small size, in fact we shall show that the oscillations of the homoclinic orbits at infinity are bounded by a flat function of μ. This result receives its significance from the still unsolved question as to whether solutions exist which are homoclinic to the equilibrium or whether the amplitudes of the oscillations at infinity have a positive infimum. First we construct the periodic solutions. In contrast to previous work, we find these in a full rectangle [0, K0] × ]0,μ0], where K measures the amplitude of the periodic orbits. Then we show that for each n ∈ ℕ there is a μn and a family of periodic solutions X(μ), μ ∈]0,μn[, of Size μn. To each of these solutions, we can find two homoclinic orbits, which are distinguished by their phase shift at infinity. One example of such a vector field occurs when describing the flow of an inviscid irrotational fluid layer under the influence of gravity and small surface tension (Bond number b < ⅓), for a Froude number F close to 1. In this context a homoclinic solution to a periodic orbit is called a generalised solitary wave. Our work shows that there exist solitary waves with oscillations at infinity of order less than |μ|n for every n.

AbstractAn upper bound on the rate of vertical heat transport is established in three dimensions for stress-free velocity boundary conditions on horizontally periodic plates. A variation of the background method is implemented that allows negative values of the quadratic form to yield ‘small’ ($O\left(1/ \mathit{Pr}\right)$) corrections to the subsequent bound. For large (but finite) Prandtl numbers this bound is an improvement over the ‘ultimate’$R{a}^{1/ 2} $scaling and, in the limit of infinite$Pr$, agrees with the bound of$R{a}^{5/ 12} $recently derived in that limit for stress-free boundaries.

The oscillation problem is examined for a rectangular sealing ring composed of a class of transversely isotropic incompressible vulcanized rubber materials about radial direction, where the sealing ring is subjected to a suddenly applied radial load at its inner surface. A nonlinear ordinary differential equation that describes the radial motion of the sealing ring is obtained. It is proved that if the applied load is lower than the critical load, the motion of the rubber ring with time will present a nonlinear periodic oscillation, while if it exceeds the critical load, the motion will increase infinitely with the increasing time and so the rubber ring will be destroyed ultimately.

An elastic medium is considered, weakened by a doubly periodic system of round holes, filled with absolutely rigid inclusions, soldered along the bypass and has a crack initiation. The medium (binder) is weakened by two periodic systems of rectilinear crack initiation directed collinear to the abscissa and ordinate axes, and their sizes are not the same. General representations are constructed that describe a class of problems with a doubly periodic stress distribution outside circular holes and cracks under transverse shear. The analysis of the limiting equilibrium of cracks in the framework of the end zone model is carried out on the basis of a nonlocal fracture criterion with a force condition for the propagation of the crack tip and a deformation condition for determining the advancement of the edge of the end zone of the crack. Basic resolving equations are obtained in the form of infinite algebraic systems and three nonlinear singular integro-differential equations. The equations in each approximation were solved by the Gaussian method with the choice of the principal element for different values of the order of M, depending on the radius of the holes. Calculations were carried out to determine the forces in the connections of the end zones and the ultimate loads causing the growth of cracks.

For ductile solids with periodic microstructures (e.g., honeycombs, fiber-reinforced composites, cellular solids) which are loaded primarily in compression, their ultimate failure is related to the onset of a buckling mode. Consequently, for periodic solids of infinite extent, one can define as the onset of failure the first occurrence of a bifurcation in the fundamental solution, for which all cells deform identically. By following all possible loading paths in strain or stress space, one can construct onset-of-failure surfaces for finitely strained, rate-independent solids with arbitrary microstructures. The calculations required are based on a Bloch wave analysis on the deformed unit cell. The presentation of the general theory is followed by the description of a numerical algorithm which reduces the size of stability matrices by an order of magnitude, thus improving the computational efficiency for the case of continuum unit cells. The theory is subsequently applied to porous and particle-reinforced hyperelastic solids with circular inclusions of variable stiffness. The corresponding failure surfaces in strain-space, the wavelength of the instabilities, and their dependence on micro-geometry and macroscopic loading conditions are presented and discussed.

Background: An elastic medium is considered that is weakened by a doubly periodic system of round holes filled with washers made of a homogeneous elastic material and has cohesive cracks, the surface of which is uniformly covered with a cylindrical non-metallic film. The binder (medium) is weakened by two doubly periodic systems of rectilinear cohesive cracks, directed parallel to the abscissa and ordinate axes, and the length of the crack and their dimensions are not the same (different). General concepts are constructed that describe a class of problems with doubly periodic stress distribution outside circular holes and cohesive cracks under longitudinal shear. The analysis of the limit equilibrium of cracks within the framework of the end zone model is performed on the basis of a nonlocal failure criterion with a force condition for the advancement of the crack tip and a deformation condition for determining the advance of the edge of the end zone of the crack. In solving the problem, the main resolving equations are obtained in the form of infinite algebraic systems and three nonlinear singular integro-differential equations. The equations obtained in each approximation were solved by the Gaussian method with the choice of the main element for different order values depending on the radius of the holes. Calculations were carried out to determine the forces in the connections of the end zones and the ultimate loads causing crack growth. Objective: In the considered problem, the problem of longitudinal slip weakened by cracks parallel to the x axis in the filling medium with two pairs of two periodic cracks weakened by 2 periodic circular holes is considered. In the problem, the boundary problem along the contour of the circular holes is determined, and at the same time, the surface problem is established for the cracks parallel to the x and y axes. The solution of the problem is sought analytically, and taking into account the boundary condition, a system of linear algebraic equations for circular holes and 3 singular integral equations for cracks is obtained. Theoretical Framework: The problem is solved analytically, as the system of linear algebraic equations obtained for circular holes and the system of linear integral equations are jointly solved, and the stress intensity factor at the end of the cracks is determined. Method: The problem was solved using "Kolosova Muskhalashvili" functions and "Collondia's method of solving singular integral equations". Results and Discussion: The main essence of the considered issue is that the stress intensity factor at the end of the cracks is determined according to Fig. 1. Research Implications: For the first time, the stress intensity factor was calculated in composite materials with two pairs of biperiodic cracks of different lengths weakened by biperiodic round holes. Originality/Value: For the first time, the stress intensity coefficient was determined in composite materials with two periodic two pairs of cohesive cracks.

In this condensed paper, by combining the insights from E-Infinity theory, along with Plato‘s initiatory insights into the golden section imbedded in his Principles of the One and Indefinite Dyad, David Bohm‘s ontological framework of the superimplicate, implicate and explicate orders, andthe pervasive presence throughout physics, chemistry, biology and cosmology of the golden ratio (often veiled in Fibonacci and Lucas numbers), a profound golden mean number system emerges underlying the cosmos, nature and consciousness. This ubiquitous presence is evident in quantum mechanics, including quark masses, the chaos border, fine structure constant and entanglement, entropy and thermodynamic equilibrium, the periodic table of elements, nanotechnology, crystallography, computing, digital information, cryptography, genetics, nucleotide arrangement, Homo sapiens and Neanderthal genomes, DNA structure, cardiac anatomy and physiology, biometric measurements of the human and mammalian skulls, weather turbulence, plant phyllotaxis, planetary orbits and sizes, black holes, dark energy, dark matter, and even cosmogenesis – the very origin and structure of the universe. This has been pragmatically extended through the most ingenious biomimicry, from robotics, artificial intelligence, engineering and urban design, to extensions throughout history in architecture, music and the arts. We propose herein a grand unification of the sciences, arts and consciousness, rooted in an ontological superstructure known to the ancients as the One and Indefinite Dyad, that gives rise to a golden mean number system which is the substructure of all existence.

It is well-known that Sturmian sequences are the non ultimately periodic sequences that are balanced over a 2-letter alphabet. They are also characterized by their complexity: they have exactly $(n+1)$ distinct factors of length $n$. A natural generalization of Sturmian sequences is the set of infinite episturmian sequences. These sequences are not necessarily balanced over a $k$-letter alphabet, nor are they necessarily aperiodic. In this paper, we characterize balanced episturmian sequences, periodic or not, and prove Fraenkel's conjecture for the special case of episturmian sequences. It appears that balanced episturmian sequences are all ultimately periodic and they can be classified in 3 families.

Cameron has introduced a natural one-to-one correspondence between infinite binary sequences and sets of positive integers with the property that no two elements add up to a third. He observed that, if a sum-free set is ultimately periodic, so is the corresponding binary sequence, and asked if the converse also holds. We introduce the concept of difference density and show how this can be used to test specific sets. These tests produce further evidence of a positive nature that certain sets are, in fact, not ultimately periodic.

The shifts of an infinite word $W=a_0a_1\cdots$ are the words $W_i=a_ia_{i+1}\cdots$. As a measure of the complexity of a word $W$, we consider the order type of the set of shifts, ordered lexicographically. We consider morphic words (fixed points of a morphism under a coding) that are not ultimately periodic. Our main result in this setting is that if the first letter of $W$ appears at least twice in $W$, then the shifts of the aperiodic image of $W$ under a coding are dense in the sense that there is a shift strictly between any two shifts. In particular, any purely morphic binary word is either ultimately periodic or its shifts are dense. As a concrete example, we give an explicit order-preserving bijection between the shifts of the Thue--Morse word and $(0,1]\cap {\mathbb Q}$. We then give special consideration to morphisms on 3 letters whose shifts do not contain an infinite decreasing sequence.

AbstractWe proceed with the study of ultimate periodicity properties related to overlaps between the suffixes of a left-infinite word $$\lambda $$ λ and the prefixes of a right-infinite word $$\rho $$ ρ . For a positive integer n, let g(n) be n minus the maximum length of overlaps between the suffix of $$\lambda $$ λ and the prefix of $$\rho $$ ρ of length n. In a recent publication we have shown that the function g has finite image if and only if $$\lambda $$ λ and $$\rho $$ ρ are ultimately periodic words with a same root. In this paper we give an asymptotic characterization of words $$\lambda $$ λ and $$\rho $$ ρ for which the function g has finite image. We prove that this condition is true if and only if the sequence $$\big (g(n)/n\big )_n$$ ( g ( n ) / n ) n tends to zero

A morphic word is obtained by iterating a morphism to generate an infinite word, and then applying a coding. We characterize morphic words with polynomial growth in terms of a new type of infinite word called a $\textit{zigzag word}$. A zigzag word is represented by an initial string, followed by a finite list of terms, each of which repeats for each $n \geq 1$ in one of three ways: it grows forward [$t(1)\ t(2)\ \dotsm\ t(n)]$, backward [$t(n)\ \dotsm\ t(2)\ t(1)$], or just occurs once [$t$]. Each term can recursively contain subterms with their own forward and backward repetitions. We show that an infinite word is morphic with growth $\Theta(n^k)$ iff it is a zigzag word of depth $k$. As corollaries, we obtain that the morphic words with growth $O(n)$ are exactly the ultimately periodic words, and the morphic words with growth $O(n^2)$ are exactly the multilinear words.

Once a significant number of LMFRs (liquid-metal cooled fast nuclear reactors) employ wire-wrapped fuel assemblies, considerable attention should be directed to this engineering feature. Given the many contact points and tight gaps, wire-wrapped fuel assemblies are difficult to analyze without numerical tools, out of which CFD (computational fluid dynamic) is the leading tool in the field. This work presents a CFD analysis of a wire-wrapped infinite sub-channel, based on MYRRHA’s fuel assembly geometry and with LBE (lead bismuth eutectic) coolant properties implemented as temperature-dependent polynomial curves. Periodic boundary conditions were applied, and the turbulence model chosen was k-w SST. All calculations were run on ANSYS Fluent R19.3. Following good engineering practices for CFD simulations, a GCI (grid convergence index) analysis was carried out, in order to ensure grid/mesh independence on the results. Ultimately, the main goal of this work was to evaluate if the implementation of temperature-dependent thermal-physical properties for LBE coolant would lead to any different results, when compared to the static properties. However, no significant result change was observed in this case, only an increase in computational time, which leads to the conclusion that for small domains with periodic boundaries, the implementation of temperature-dependent coolant properties is not justifiable.

Abstract Graph neural networks (GNN) are very popular methods in machine learning and have been applied very successfully to the prediction of the properties of molecules and materials. First-order GNNs are well known to be incomplete, i.e., there exist graphs that are distinct but appear identical when seen through the lens of the GNN. More complicated schemes have thus been designed to increase their resolving power. Applications to molecules (and more generally, point clouds), however, add a geometric dimension to the problem. The most straightforward and prevalent approach to construct graph representation for molecules regards atoms as vertices in a graph and draws a bond between each pair of atoms within a chosen cutoff. Bonds can be decorated with the distance between atoms, and the resulting ``distance graph NNs'' (dGNN) have empirically demonstrated excellent resolving power and are widely used in chemical ML, {with all known indistinguishable configurations being resolved in the fully-connected limit, which is equivalent to infinite or sufficiently large cutoff.} Here we {present a counterexample that proves that dGNNs are not complete even for the restricted case of fully-connected graphs induced by 3D atom clouds.} We construct pairs of distinct point clouds whose associated graphs are, for any cutoff radius, equivalent based on a first-order Weisfeiler-Lehman test. This class of degenerate structures includes chemically-plausible configurations, {both for isolated structures and for infinite structures that are periodic in 1, 2, and 3 dimensions.} The existence of indistinguishable configurations sets an ultimate limit to the expressive power of some of the well-established GNN architectures for atomistic machine learning. Models that explicitly use angular or directional information in the description of atomic environments can resolve this class of degeneracies.

Abstract Upon compression, some soft granular crystals undergo pattern transformation. Recent studies have unveiled that the underlying mechanism of this transformation is closely tied to microscopic instability, resulting in symmetry breaking. This intriguing phenomenon gives rise to unconventional mechanical properties in the granular crystals, paving the way for potential metamaterial application. However, no consistent approach has been reported for studying other unexplored transformable granular crystals. In this study, we propose a systematic approach to identify a new set of pattern-transformable diatomic granular crystals having tunable phononic band-gaps. Utilizing diatomic compact packing as a foundation, we first present a catalog of viable particle arrangements, considering the instability arising from kinematic constraints between articles. Subsequently, simple mass-spring models are constructed based on the contact network of the aforementioned particle arrangements. To identify pattern-transformable granular crystals, these mass-spring models are employed for both instability analyses within the linear perturbation framework and quasi-static analyses involving infinitely-periodic configurations. The conclusive pattern transformation in these chosen granular crystals is ultimately validated through detailed finite element models employing continuum elements. Furthermore, the impact of their pattern transformation under compression is highlighted by observing the evolution in their phononic band structure.

One of the most important issues in solid mechanics is the development and application of optimal methods for estimating the resistance of dispersion materials. Currently, the theory of propagation of cracks in solid bodies is widespread due to the wide use of high-strength materials and large-sized structures and devices in various fields of technology. The article discusses the methods of studying the problems of the interaction of two factors - strengthening the panel with stiffening ribs made of unidirectional composite material and weakening it with a number of circular holes and cracks protruding from its contour. An infinite isotropic elastic riveted panel weakened by a periodic system of circular holes and two cracks along the abscissa axis segments near each hole is considered. The contours of the holes and the edges of the cracks are free from external forces. Transverse stiffness ribs made of metal composite material are riveted to the plate. In the calculation scheme, the effect of riveted reinforcing ribs is replaced by the concentrated forces applied at the points of location of the rivets. The values of the concentrated forces are found depending on the geometrical and physical parameters of the problem. Stress intensity factors and ultimate loadings were calculated depending on the geometrical and physical parameters of the problem. It has been shown that there is a stable phase of crack development under certain specified conditions. The development of initial plastic deformations (embryonic cracks) emerging from the contour of circular holes in a plate reinforced with stiffness ribs is studied. The material of the plate is elastic-perfect-plastic satisfying the Tresca-Saint-Venant condition. It is assumed that the plastic deformations are concentrated along some slip lines emanating from the contour of the hole. By satisfying the boundary conditions, the solution of the problem leads to a singular integral equation. Then the singular equation of the problem is brought to a system of algebraic equations without the intermediate step of bringing it to the Fredholm equation. The resulting system was solved by the Gaussian method with the selection of principal elements for different values of n (n is the number of Chebyshev nodes in the division of the interval). The values of concentrated forces are found depending on the geometrical and physical parameters of the problem. The dependences of the length of the plasticity bands on the applied load, the geometrical parameters of the problem and the yield point of the material were found. Using the local dissipation condition (KPT-criteria), a ratio was obtained to determine the destructive edge loading.

Despite the period's fashionable aspiration to a materialist, scientific objectivity, the new wilderness revealed by the microscope in the nineteenth century did not lend itself quickly or easily to sober, observational consensus. Rather, the nature of the microscopic world was, like the cosmos, largely open to interpretation. Since techniques of observation were largely undeveloped, many microscopists were not certain precisely what it was they were to look for, nor of the nature of their subjects. Did monstrosity lurk at the threshold, or was the microscope a window to the divine designs of the creator? Monstrosity and the microscopic may be a familiar relationship today, but prior to Pasteur and Koch's development of a germ theory of disease in the 1870s, the invisible world revealed by the microscope was not especially horrific, nor did it invalidate long-standing notions of the divinity of Nature. It is more than probable that many microorganisms were, prior to their identification as causal agents of disease, looked upon and admired as beautiful natural specimens. Certain microscopists may have suspected early on that all was not well at the microscopic level (suspicion of wilderness is traditional within the Western cartographic project), but by and large nineteenth century microscopy was deeply enmeshed in the extensive romanticism of the period, and most texts on the nature of the microorganism prior to the late nineteenth century tend to emphasise (in retrospect, a little naively), their embodiment of the amazing, wonderful complexity of the natural world. Germany was the center of this modern fusion of romanticism, naturalism, and microscopic visuality, where the prolific microgeologist, Christian Godfried Ehrenberg (1795 - 1876) achieved considerable attention through his discovery of the intricately symmetrical, skeletal remains of unknown microorganisms in the calacerous tertiaries of Sicily and Greece, and Oran in Africa. Documenting these fossils in Microgeologie (1854), he established for them the group Polycystina, in which he also included a series of forms making up nearly the whole of a silicious sandstone prevailing through an extensive district of Barbadoes. These widely admired microscopic sea-dwelling organisms were later discovered and studied in their living state by Johannes Muller, who named them Radiolaria. Ehrenberg's pursuit of natural beauty, rather than monstrosity, was clearly appealing throughout the mid-to-late nineteenth century. Central to the aesthetic evaluation of the natural world inspired by his discoveries was a privileging of symmetrical forms as divine signifiers. Drawing heavily from Ehrenberg's approach to the natural world, it had been the intention of Gideon Algernon Mantell, Vice-President of the Geological Society of London and author of The Invisible World Revealed by the Microscope (1850), to "impart just and comprehensive views of the grandeur and harmony of the Creation, and of the Infinite Wisdom and Beneficence of its Divine Author; and which, in every condition and circumstance of life, will prove a never-failing source of pleasure and instruction" (ix-x). An admirable project indeed, but increasingly problematic in the wake of evidence suggesting the infinite wisdom and beneficence of the divine author included the scripting of destructive, ruthless, mindless, invisible agents of suffering and death against which human beings were granted little, if any, defence. What did such evidence say of our allegedly privileged role in the story of life on Earth? Where might the raw, biological body reside within such an arrangement? Precisely at the vulnerable center of the controversy surrounding the nature of its own existence. Not surprisingly, consensus on what the body actually is has always been fairly frail, since it closed its modern formation in conjunction with the revelation of the body's mysterious, "hidden powers" through the lens of the microscope, which radically expanded, and confused, the cartographic field. Renaissance anatomical representation, thought once to be so authoritative and thorough (maybe too thorough), now seemed superficial. And moreover, as shown by the discovery of electricity and its extensive, shockingly experimental application to the body, we were enigmatic entities indeed, consisting of, and vulnerable to, mysterious, untamed forces of attraction and repulsion. The invention of the "Leyden jar" in the eighteenth century, which allowed the storage and regulation of electrical charge, had been turned almost immediately to the human body, often with all the playful naivete of a child. As Sarah Bakewell (2000) writes: One experimenter, Jean-Antoine Nollet (1700-70), liked to demonstrate the power of the new equipment by lining up 180 of the king's guards with hands clasped and connecting the man on the end to a Leyden jar, so that the whole line leaped involuntarily into the air. (36) The discovery that the biological body was an electrical organism unquestionably inspired the exorbitant interest in the "ether" that underpinned much nineteenth century spiritualism, horror fiction, and the emergence of paranoia as a cultural condition in the modern era. Most notably, it disrupted the notion of an external God in favour of a "divine power" running through, and thus connecting, all life. And as psychiatry has since discovered, the relation of the body to such a deeper, all-pervasive, unmappable power - an ontology in which matter has no empty spaces - is "profoundly schizoid" (Anti-Oedipus 19). But this did not prevent its intrusion into nineteenth century science. Biologist Ernst Haeckel (1834 - 1919), nineteenth century Germany's most vocal advocator of Darwinism, openly subscribed to a mystical, arguably delusional approach to the natural world. Drawn to study of the microscopic by Ehrenberg, Haeckel was likewise attracted to the patterned aesthetic of the natural world, especially its production of symmetrical forms. Although he drew his fair share of critics, it is unlikely he was ever considered "sick", since neither paranoia nor schizophrenia were recognised illnesses at the time. Yet in retrospect his writings clearly indicate a commitment to what would now be regarded as a paranoid/schizophrenic ontology in which "matter has no empty spaces". Haeckel's recourse to monism may be understood, at least in part, as a reaction to the agency panic provoked by the invasion narrative central to the germ theory of disease: if all is One, notions of "invasion" become redundant and transformed into the internalised self-regulation of the whole. Devoted to monism, Haeckel was adamant that "ever more clearly are we compelled by reflection to recognise that God is not to be placed over against the material world as an external being, but must be placed as a "divine power" or "moving spirit" within the cosmos itself" (Monism 15). This conception of God is synonymous with that discussed by Deleuze and Guttari in their exploration of the nervous illness of Judge Daniel Schreber, in which God is defined as the Omnitudo realitatis, from which all secondary realities are derived by a process of division (Anti-Oedipus 13). Like a textbook schizophrenic, Haeckel stressed the oneness of the cosmos, its operation under fundamental conditions of attraction and repulsion, the indissoluble connection between energy and matter, the mind and embodiment, and God and the world. His obsession with the "secret powers" of the Creator led him to adopt the notion of a "cosmic ether", which was itself almost totally dependent on contemporary research into the properties of electricity. Haeckel wrote that "the ether itself is no longer hypothetical; its existence can at any moment be demonstrated by electrical and optical experiment" (Monism 23). Recognising the inherent conflict of nature whilst providing convincing evidence of its divine, harmonious beauty through his hundreds of spectacularly symmetrical, mandala-like representations of Radiolarians and other microscopic forms in Die Radiolarian (1862) and Kunstformen der Natur (1899), Haeckel furthered his views through several popular manifestos such as Monism as Connecting Religion and Science: The Confession of Faith of a Man of Science (1894), The Wonders of Life: A Popular Study of Biological Philosophy (1905), and The Riddle of the Universe at the Close of the Nineteenth Century (1911). For Haeckel, clearly entranced by the hypersignificance of nature, the struggle for biological survival was also a mystical one, and thus divinely inspired. Tying this notion together with the Volkish tradition, and clearly influenced by the emerging germ theory, which emphasised conflict as precondition for (apparently mythic) harmony, Haeckel wrote that: We now know that the whole of organic nature on our planet exists only by a relentless war of all against all. Thousands of animals and plants must daily perish in every part of the earth, in order that a few chosen individuals may continue to subsist and to enjoy life. But even the existence of these favoured few is a continual conflict with threatening dangers of every kind. Thousands of hopeful germs perish uselessly every minute. The raging war of interests in human society is only a feeble picture of the unceasing and terrible war of existence which reigns throughout the whole of the living world. The beautiful dream of God's goodness and wisdom in nature, to which as children we listened so devoutly fifty years ago, no longer finds credit now - at least among educated people who think. It has disappeared before our deeper acquaintance with the mutual relations of organisms, the advancement of ecology and sociology, and our knowledge of parasite life and pathology. (Monism 73-74). The "war of existence", according to Haeckel, was ultimately an expression of the ethereal power of an omnipresent God. Denying real difference between matter and energy, he also implicitly denied the agency of the subject, instead positing the war of existence as a self-regulating flow of divine power. Biological survival was thus synonymous with the triumph of divine embodiment. Since Haeckel was resolutely convinced that nature was hierarchically structured (with the Aryan Volk fairly close to the top), so too were its expressions of God. And since God was not a being external to the Self, but rather the vital spirit or soul running through all being, divinity may be contained by organisms in varying degrees depending on their level of evolution. Domination of others was thus a prerequisite for the pursuit of God. And this was the essence of Haeckel's highly problematic distortion of the Darwinist theory of evolution: At the lowest stage, the rude - we may say animal - phase of prehistoric primitive man, is the "ape-man", who, in the course of the tertiary period, has only to a limited degree raised himself above his immediate pithecoid ancestors, the anthropoid apes. Next come successive stages of the lowest and simplest kind of culture, such as only the rudest of still existing primitive peoples enable us in some measure to conceive. These "savages" are succeeded by peoples of a low civilisation, and from these again, by a long series of intermediate steps, we rise little by little to the more highly civilised nations. To these alone - of the twelve races of mankind only to the Mediterranean and Mongolian - are we indebted for what is usually called "universal history. (Monism 5-6) This fairly crude, very German take on Darwinism, with its emphasis on the transference of biological principles to the social realm, contributed to the establishment of the preconditions for the emergence of National Socialism in that country shortly after Haeckel's death in 1919. In The Scientific Origins of National Socialism (1971), Daniel Gasman reveals the extent of Haeckel's descent into mysticism and its part in the wider development of the Volkish myths that underpinned Nazism in the twentieth century. And although the "sick" ideals of Nazism are undeniably deplorable, upon review of the cultural circumstances in which Haeckel's ideas developed, many of them seem inevitable for a frightened, paranoid culture convinced - based on scientific evidence - that life itself can only ever be a form of war: the very notion that continues to underpin, and indeed sustain, the germ theory of disease in the modern era References Bakewell, Sarah. "It's Alive!" Fortean Times October 2000: 34-39. Carpenter, William B. The Microscope and its Revelations. London: J & A Churchill, 1891. Deleuze, Gilles, and Felix Guattari. Anti-Oedipus: Capitalism and Schizophrenia. 1972. Trans. Robert Hurley, Mark Seem, and Helen R. Lane. Minneapolis: U of Minnesota P, 1983. Gasman, Daniel. The Scientific Origins of National Socialism: Social Darwinism in Ernst Haeckel and the German Monist League. London: McDonald, 1971. Haeckel, Ernst. Die Radiolarien (Rhizopoda Radiaria). Berlin, 1862. ---. Monism as Connecting Religion and Science: The Confession of Faith of a Man of Science. London: Adam and Charles Black, 1894. ---. Kunstformen der Natur. 2 vols. Leipzig and Wien, 1899. ---. The Riddle of the Universe at the Close of the Nineteenth Century. Watts and Co., 1911. ---. The Wonders of Life. London: Watts and Co., 1905. Mantell, Gideon Algernon. The Invisible World Revealed by the Microscope; or, Thoughts on Animalcules. London: John Murray, 1850. Tomes, Nancy. The Gospel of Germs: Men, Women, and the Microbe in American Life. Cambridge: Harvard UP, 1998.

If you cannot understand my argument, and declare “It’s Greek to me,” you are quoting Shakespeare.Bernard LevinPsoriatic arthritis, in its acute or “generalised” stage, is unbearably painful. Exacerbating the crippling of the joints, the entire surface of the skin is covered with lesions only moderately salved by anti-inflammatory ointment, the application of which is as painful as the ailment it seeks to relieve: NURSE MILLS: I’ll be as gentle as I can.Marlow’s face again fills the screen, intense concentration, comical strain, and a whispered urgency in the voice over—MARLOW: (Voice over) Think of something boring—For Christ’s sake think of something very very boring—Speech a speech by Ted Heath a sentence long sentence from Bernard Levin a quiz by Christopher Booker a—oh think think—! Really boring! A Welsh male-voice choir—Everything in Punch—Oh! Oh! — (Potter 17-18)Marlow’s collation of boring things as a frantic liturgy is an attempt to distract himself from a tumescence that is both unwanted and out of place. Although bed-ridden and in constant pain, he is still sensitive to erogenous stimulation, even when it is incidental. The act of recollection, of garnering lists of things that bore him, distracts him from his immediate situation as he struggles with the mental anguish of the prospect of a humiliating orgasm. Literary lists do many things. They provide richness of detail, assemble and corroborate the materiality of the world of which they are a part and provide insight into the psyche and motivation of the collator. The sheer desperation of Dennis Potter’s Marlow attests to the arbitrariness of the list, the simple requirement that discrete and unrelated items can be assembled in linear order, without any obligation for topical concatenation. In its interrogative form, the list can serve a more urgent and distressing purpose than distraction:GOLDBERG: What do you use for pyjamas?STANLEY: Nothing.GOLDBERG: You verminate the sheet of your birth.MCCANN: What about the Albigensenist heresy?GOLDBERG: Who watered the wicket in Melbourne?MCCANN: What about the blessed Oliver Plunkett?(Pinter 51)The interrogative non sequitur is an established feature of the art of intimidation. It is designed to exert maximum stress in the subject through the use of obscure asides and the endowing of trivial detail with profundity. Harold Pinter’s use of it in The Birthday Party reveals how central it was to his “theatre of menace.” The other tactic, which also draws on the logic of the inventory to be both sequential and discontinuous, is to break the subject’s will through a machine-like barrage of rhetorical questions that leave no time for answers.Pinter learned from Samuel Beckett the pitiless, unforgiving logic of trivial detail pushed to extremes. Think of Molloy’s dilemma of the sucking stones. In order for all sixteen stones that he carries with him to be sucked at least once to assuage his hunger, a reliable system has to be hit upon:Taking a stone from the right pocket of my greatcoat, and putting it in my mouth, I replaced it in the right pocket of my greatcoat by a stone from the right pocket of my trousers, which I replaced with a stone from the left pocket of my trousers, which I replaced by a stone from the left pocket of my greatcoat, which I replaced with the stone that was in my mouth, as soon as I had finished sucking it. Thus there were still four stones in each of my four pockets, but not quite the same stones. And when the desire to suck took hold of me again, I drew again on the right pocket of my greatcoat, certain of not taking the same stone as the last time. And while I sucked it I rearranged the other stones in the way I have just described. And so on. (Beckett, Molloy 69)And so on for six pages. Exhaustive permutation within a finite lexical set is common in Beckett. In the novel Watt the eponymous central character is charged with serving his unseen master’s dinner as well as tidying up afterwards. A simple and bucolic enough task it would seem. But Beckett’s characters are not satisfied with conjecture, the simple assumption that someone must be responsible for Mr. Knott’s dining arrangements. Like Molloy’s solution to the sucking stone problem, all possible scenarios must be considered to explain the conundrum of how and why Watt never saw Knott at mealtime. Twelve possibilities are offered, among them that1. Mr. Knott was responsible for the arrangement, and knew that he was responsible for the arrangement, and knew that such an arrangement existed, and was content.2. Mr. Knott was not responsible for the arrangement, but knew who was responsible for the arrangement, and knew that such an arrangement existed, and was content.(Beckett, Watt 86)This stringent adherence to detail, absurd and exasperating as it is, is the work of fiction, the persistence of a viable, believable thing called Watt who exists as long as his thought is made manifest on a page. All writers face this pernicious prospect of having to confront and satisfy “fiction’s gargantuan appetite for fact, for detail, for documentation” (Kenner 70). A writer’s writer (Philip Marlow) Dennis Potter’s singing detective struggles with the acute consciousness that words eventually will fail him. His struggle to overcome verbal entropy is a spectre that haunts the entire literary imagination, for when the words stop the world stops.Beckett made this struggle the very stuff of his work, declaring famously that all he wanted to do as a writer was to leave “a stain upon the silence” (quoted in Bair 681). His characters deteriorate from recognisable people (Hamm in Endgame, Winnie in Happy Days) to mere ciphers of speech acts (the bodiless head Listener in That Time, Mouth in Not I). During this process they provide us with the vocabulary of entropy, a horror most eloquently expressed at the end of The Unnamable: I can’t go on, you must go on, I’ll go on, you must say words, as long as there are any, until they find me, until they say me, strange pain, strange sin, you must go on, perhaps it’s done already, perhaps they have said me already, perhaps they have carried me to the threshold of my story, before the door that opens on my story, that would surprise me, if it opens, it will be I, it will be the silence, where I am, I don’t know, I’ll never know, in the silence you don’t know, you must go on, I can’t go on, I’ll go on. (Beckett, Molloy 418)The importance Beckett accorded to pauses in his writing, from breaks in dialogue to punctuation, stresses the pacing of utterance that is in sync with the rhythm of human breath. This is acutely underlined in Jack MacGowran’s extraordinary gramophone recording of the above passage from The Unnamable. There is exhaustion in his voice, but it is inflected by an urgent push for the next words to forestall the last gasp. And what might appear to be parsimony is in fact the very commerce of writing itself. It is an economy of necessity, when any words will suffice to sustain presence in the face of imminent silence.Hugh Kenner has written eloquently on the relationship between writing and entropy, drawing on field and number theory to demonstrate how the business of fiction is forever in the process of generating variation within a finite set. The “stoic comedian,” as he figures the writer facing the blank page, self-consciously practices their art in the full cognisance that they select “elements from a closed set, and then (arrange) them inside a closed field” (Kenner 94). The nouveau roman (a genre conceived and practiced in Beckett’s lean shadow) is remembered in literary history as a rather austere, po-faced formalism that foregrounded things at the expense of human psychology or social interaction. But it is emblematic of Kenner’s portrait of stoicism as an attitude to writing that confronts the nature of fiction itself, on its own terms, as a practice “which is endlessly arranging things” (13):The bulge of the bank also begins to take effect starting from the fifth row: this row, as a matter of fact, also possesses only twenty-one trees, whereas it should have twenty-two for a true trapezoid and twenty-three for a rectangle (uneven row). (Robbe-Grillet 21)As a matter of fact. The nouveau roman made a fine if myopic art of isolating detail for detail’s sake. However, it shares with both Beckett’s minimalism and Joyce’s maximalism the obligation of fiction to fill its world with stuff (“maximalism” is a term coined by Michel Delville and Andrew Norris in relation to the musical scores of Frank Zappa that opposes the minimalism of John Cage’s work). Kenner asks, in The Stoic Comedians, where do the “thousands on thousands of things come from, that clutter Ulysses?” His answer is simple, from “a convention” and this prosaic response takes us to the heart of the matter with respect to the impact on writing of Isaac Newton’s unforgiving Second Law of Thermodynamics. In the law’s strictest physical sense of the dissipation of heat, of the loss of energy within any closed system that moves, the stipulation of the Second Law predicts that words will, of necessity, stop in any form governed by convention (be it of horror, comedy, tragedy, the Bildungsroman, etc.). Building upon and at the same time refining the early work on motion and mass theorised by Aristotle, Kepler, and Galileo, inter alia, Newton refined both the laws and language of classical mechanics. It was from Wiener’s literary reading of Newton that Kenner segued from the loss of energy within any closed system (entropy) to the running silent out of words within fiction.In the wake of Norbert Wiener’s cybernetic turn in thinking in the 1940s, which was highly influenced by Newton’s Second Law, fiction would never again be considered in the same way (metafiction was a term coined in part to recognise this shift; the nouveau roman another). Far from delivering a reassured and reassuring present-ness, an integrated and ongoing cosmos, fiction is an isometric exercise in the struggle against entropy, of a world in imminent danger of running out of energy, of not-being:“His hand took his hat from the peg over his initialled heavy overcoat…” Four nouns, and the book’s world is heavier by four things. One, the hat, “Plasto’s high grade,” will remain in play to the end. The hand we shall continue to take for granted: it is Bloom’s; it goes with his body, which we are not to stop imagining. The peg and the overcoat will fade. “On the doorstep he felt in his hip pocket for the latchkey. Not there. In the trousers I left off.” Four more things. (Kenner 87)This passage from The Stoic Comedians is a tour de force of the conjuror’s art, slowing down the subliminal process of the illusion for us to see the fragility of fiction’s precarious grip on the verge of silence, heroically “filling four hundred empty pages with combinations of twenty-six different letters” (xiii). Kenner situates Joyce in a comic tradition, preceded by Gustave Flaubert and followed by Beckett, of exhaustive fictive possibility. The stoic, he tells us, “is one who considers, with neither panic nor indifference, that the field of possibilities available to him is large perhaps, or small perhaps, but closed” (he is prompt in reminding us that among novelists, gamblers and ethical theorists, the stoic is also a proponent of the Second Law of Thermodynamics) (xiii). If Joyce is the comedian of the inventory, then it is Flaubert, comedian of the Enlightenment, who is his immediate ancestor. Bouvard and Pécuchet (1881) is an unfinished novel written in the shadow of the Encyclopaedia, an apparatus of the literate mind that sought complete knowledge. But like the Encyclopaedia particularly and the Enlightenment more generally, it is fragmentation that determines its approach to and categorisation of detail as information about the world. Bouvard and Pécuchet ends, appropriately, in a frayed list of details, pronouncements and ephemera.In the face of an unassailable impasse, all that is left Flaubert is the list. For more than thirty years he constructed the Dictionary of Received Ideas in the shadow of the truncated Bouvard and Pécuchet. And in doing so he created for the nineteenth century mind “a handbook for novelists” (Kenner 19), a breakdown of all we know “into little pieces so arranged that they can be found one at a time” (3): ACADEMY, FRENCH: Run it down but try to belong to it if you can.GREEK: Whatever one cannot understand is Greek.KORAN: Book about Mohammed, which is all about women.MACHIAVELLIAN: Word only to be spoken with a shudder.PHILOSOPHY: Always snigger at it.WAGNER: Snigger when you hear his name and joke about the music of the future. (Flaubert, Dictionary 293-330)This is a sample of the exhaustion that issues from the tireless pursuit of categorisation, classification, and the mania for ordered information. The Dictionary manifests the Enlightenment’s insatiable hunger for received ideas, an unwieldy background noise of popular opinion, general knowledge, expertise, and hearsay. In both Bouvard and Pécuchet and the Dictionary, exhaustion was the foundation of a comic art as it was for both Joyce and Beckett after him, for the simple reason that it includes everything and neglects nothing. It is comedy born of overwhelming competence, a sublime impertinence, though not of manners or social etiquette, but rather, with a nod to Oscar Wilde, the impertinence of being definitive (a droll epithet that, not surprisingly, was the title of Kenner’s 1982 Times Literary Supplement review of Richard Ellmann’s revised and augmented biography of Joyce).The inventory, then, is the underlining physio-semiotics of fictional mechanics, an elegiac resistance to the thread of fiction fraying into nothingness. The motif of thermodynamics is no mere literary conceit here. Consider the opening sentence in Borges:Of the many problems which exercised the reckless discernment of Lönnrot, none was so strange—so rigorously strange, shall we say—as the periodic series of bloody events which culminated at the villa of Triste-le-Roy, amid the ceaseless aroma of the eucalypti. (Borges 76)The subordinate clause, as a means of adjectival and adverbial augmentation, implies a potentially infinite sentence through the sheer force of grammatical convention, a machine-like resistance to running out of puff:Under the notable influence of Chesterton (contriver and embellisher of elegant mysteries) and the palace counsellor Leibniz (inventor of the pre-established harmony), in my idle afternoons I have imagined this story plot which I shall perhaps write someday and which already justifies me somehow. (72)In “Pierre Menard, Author of the Quixote,” a single adjective charmed with emphasis will do to imply an unseen network:The visible work left by this novelist is easily and briefly enumerated. (Borges 36)The annotation of this network is the inexorable issue of the inflection: “I have said that Menard’s work can be easily enumerated. Having examined with care his personal files, I find that they contain the following items.” (37) This is a sample selection from nineteen entries:a) A Symbolist sonnet which appeared twice (with variants) in the review La conque (issues of March and October 1899).o) A transposition into alexandrines of Paul Valéry’s Le cimitière marin (N.R.F., January 1928).p) An invective against Paul Valéry, in the Papers for the Suppression of Reality of Jacques Reboul. (37-38)Lists, when we encounter them in Jorge Luis Borges, are always contextual, supplying necessary detail to expand upon character and situation. And they are always intertextual, anchoring this specific fictional world to others (imaginary, real, fabulatory or yet to come). The collation and annotation of the literary works of an imagined author (Pierre Menard) of an invented author (Edmond Teste) of an actual author (Paul Valéry) creates a recursive, yet generative, feedback loop of reference and literary progeny. As long as one of these authors continues to write, or write of the work of at least one of the others, a persistent fictional present tense is ensured.Consider Hillel Schwartz’s use of the list in his Making Noise (2011). It not only lists what can and is inevitably heard, in this instance the European 1700s, but what it, or local aural colour, is heard over:Earthy: criers of artichokes, asparagus, baskets, beans, beer, bells, biscuits, brooms, buttermilk, candles, six-pence-a-pound fair cherries, chickens, clothesline, cockles, combs, coal, crabs, cucumbers, death lists, door mats, eels, fresh eggs, firewood, flowers, garlic, hake, herring, ink, ivy, jokebooks, lace, lanterns, lemons, lettuce, mackeral, matches […]. (Schwartz 143)The extended list and the catalogue, when encountered as formalist set pieces in fiction or, as in Schwartz’s case, non-fiction, are the expansive equivalent of le mot juste, the self-conscious, painstaking selection of the right word, the specific detail. Of Ulysses, Kenner observes that it was perfectly natural that it “should have attracted the attention of a group of scholars who wanted practice in compiling a word-index to some extensive piece of prose (Miles Hanley, Word Index to Ulysses, 1937). More than any other work of fiction, it suggests by its texture, often by the very look of its pages, that it has been painstakingly assembled out of single words…” (31-32). In a book already crammed with detail, with persistent reference to itself, to other texts, other media, such formalist set pieces as the following from the oneiric “Circe” episode self-consciously perform for our scrutiny fiction’s insatiable hunger for more words, for invention, the Latin root of which also gives us the word inventory:The van of the procession appears headed by John Howard Parnell, city marshal, in a chessboard tabard, the Athlone Poursuivant and Ulster King of Arms. They are followed by the Right Honourable Joseph Hutchinson, lord mayor Dublin, the lord mayor of Cork, their worships the mayors of Limerick, Galway, Sligo and Waterford, twentyeight Irish representative peers, sirdars, grandees and maharajahs bearing the cloth of estate, the Dublin Metropolitan Fire Brigade, the chapter of the saints of finance in their plutocratic order of precedence, the bishop of Down and Connor, His Eminence Michael cardinal Logue archbishop of Armagh, primate of all Ireland, His Grace, the most reverend Dr William Alexander, archbishop of Armagh, primate of all Ireland, the chief rabbi, the Presbyterian moderator, the heads of the Baptist, Anabaptist, Methodist and Moravian chapels and the honorary secretary of the society of friends. (Joyce, Ulysses 602-604)Such examples demonstrate how Joycean inventories break from narrative as architectonic, stand-alone assemblages of information. They are Rabelaisian irruptions, like Philip Marlow’s lesions, that erupt in swollen bas-relief. The exaggerated, at times hysterical, quality of such lists, perform the hallucinatory work of displacement and condensation (the Homeric parallel here is the transformation of Odysseus’s men into swine by the witch Circe). Freudian, not to mention Stindberg-ian dream-work brings together and juxtaposes images and details that only make sense as non-sense (realistic but not real), such as the extraordinary explosive gathering of civic, commercial, political, chivalric representatives of Dublin in this foreshortened excerpt of Bloom’s regal campaign for his “new Bloomusalem” (606).The text’s formidable echolalia, whereby motifs recur and recapitulate into leitmotifs, ensures that the act of reading Ulysses is always cross-referential, suggesting the persistence of a conjured world that is always already still coming into being through reading. And it is of course this forestalling of Newton’s Second Law that Joyce brazenly conducts, in both the textual and physical sense, in Finnegans Wake. The Wake is an impossible book in that it infinitely sustains the circulation of words within a closed system, creating a weird feedback loop of cyclical return. It is a text that can run indefinitely through the force of its own momentum without coming to a conclusion. In a text in which the author’s alter ego is described in terms of the technology of inscription (Shem the Penman) and his craft as being a “punsil shapner,” (Joyce, Finnegans 98) Norbert Wiener’s descriptive example of feedback as the forestalling of entropy in the conscious act of picking up a pencil is apt: One we have determined this, our motion proceeds in such a way that we may say roughly that the amount by which the pencil is not yet picked up is decreased at each stage. (Wiener 7) The Wake overcomes the book’s, and indeed writing’s, struggle with entropy through the constant return of energy into its closed system as a cycle of endless return. Its generative algorithm can be represented thus: “… a long the riverrun …” (628-3). The Wake’s sense of unending confounds and contradicts, in advance, Frank Kermode’s averring to Newton’s Second Law in his insistence that the progression of all narrative fiction is defined in terms of the “sense of an ending,” the expectation of a conclusion, whereby the termination of words makes “possible a satisfying consonance with the origins and with the middle” (Kermode 17). It is the realisation of the novel imagined by Silas Flannery, the fictitious author in Italo Calvino’s If on a winter’s night a traveller, an incipit that “maintains for its whole duration the potentiality of the beginning” (Calvino 140). Finnegans Wake is unique in terms of the history of the novel (if that is indeed what it is) in that it is never read, but (as Joseph Frank observed of Joyce generally) “can only be re-read” (Frank 19). With Wiener’s allegory of feedback no doubt in mind, Jacques Derrida’s cybernetic account of the act of reading Joyce comes, like a form of echolalia, on the heels of Calvino’s incipit, his perpetual sustaining of the beginning: you stay on the edge of reading Joyce—for me this has been going on for twenty-five or thirty years—and the endless plunge throws you back onto the river-bank, on the brink of another possible immersion, ad infinitum … In any case, I have the feeling that I haven’t yet begun to read Joyce, and this “not having begun to read” is sometimes the most singular and active relationship I have with his work. (Derrida 148) Derrida wonders if this process of ongoing immersion in the text is typical of all works of literature and not just the Wake. The question is rhetorical and resonates into silence. And it is silence, ultimately, that hovers as a mute herald of the end when words will simply run out.Post(script)It is in the nature of all writing that it is read in the absence of its author. Perhaps the most typical form of writing, then, is the suicide note. In an extraordinary essay, “Goodbye, Cruel Words,” Mark Dery wonders why it has been “so neglected as a literary genre” and promptly sets about reviewing its decisive characteristics. Curiously, the list features amongst its many forms: I’m done with lifeI’m no goodI’m dead. (Dery 262)And references to lists of types of suicide notes are among Dery’s own notes to the essay. With its implicit generic capacity to intransitively add more detail, the list becomes in the light of the terminal letter a condition of writing itself. The irony of this is not lost on Dery as he ponders the impotent stoicism of the scribbler setting about the mordant task of writing for the last time. Writing at the last gasp, as Dery portrays it, is a form of dogged, radical will. But his concluding remarks are reflective of his melancholy attitude to this most desperate act of writing at degree zero: “The awful truth (unthinkable to a writer) is that eloquent suicide notes are rarer than rare because suicide is the moment when language fails—fails to hoist us out of the pit, fails even to express the unbearable weight” (264) of someone on the precipice of the very last word they will ever think, let alone write. Ihab Hassan (1967) and George Steiner (1967), it would seem, were latecomers as proselytisers of the language of silence. But there is a queer, uncanny optimism at work at the terminal moment of writing when, contra Dery, words prevail on the verge of “endless, silent night.” (264) Perhaps when Newton’s Second Law no longer has carriage over mortal life, words take on a weird half-life of their own. Writing, after Socrates, does indeed circulate indiscriminately among its readers. There is a dark irony associated with last words. When life ceases, words continue to have the final say as long as they are read, and in so doing they sustain an unlikely, and in their own way, stoical sense of unending.ReferencesBair, Deirdre. Samuel Beckett: A Biography. London: Jonathan Cape, 1978.Beckett, Samuel. Molloy Malone Dies. The Unnamable. London: John Calder, 1973.---. Watt. London: John Calder, 1976.Borges, Jorge Luis. Labyrinths. Selected Stories & Other Writings. Ed. Donald A. Yates & James E. Irby. New York: New Directions, 1964.Calvino, Italo. If On A Winter’s Night A Traveller. Trans. 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