A fragmentation model with neighborhood interaction

Several models of fragmentation have been studied that suppose random fracture forces. In this article, we did a numerical study on a dynamic model for fragmentation in which the fracture forces are generated by neighboring fragments and are proportional to the size of the common boundary between two fragments. The following assumptions were also considered: the material defects are represented by a random distribution of point flaws; the total mass is conserved; and the iterative fracture of each fragment is randomly stopped by a condition that considers a constant probability and a minimal fragment size. The motivation for this model was to determine under what circumstances a continuous fragmentation model with fracture forces defined by the neighbors’ interaction produces results that are in agreement with the experimental evidence. The main result of this work establishes that the fragment size distribution follows a power-law for fragments of greater area than the minimal fragment size m_fs. The visualizations present complex fracture patterns that resemble real systems.     

The polytopologies of transfinite provability logic

Provability logics are modal or polymodal systems designed for modeling the behavior of Gödel’s provability predicate and its natural extensions. If Λ is any ordinal, the Gödel-Löb calculus GLP _Λ contains one modality [λ] for each λ < Λ, representing provability predicates of increasing strength. GLP _ω has no non-trivial Kripke frames, but it is sound and complete for its topological semantics, as was shown by Icard for the variable-free fragment and more recently by Beklemishev and Gabelaia for the full logic. In this paper we generalize Beklemishev and Gabelaia’s result to GLP _Λ for countable Λ. We also introduce provability ambiances, which are topological models where valuations of formulas are restricted. With this we show completeness of GLP _Λ for the class of provability ambiances based on Icard polytopologies.     

Herbrand consistency of some finite fragments of bounded arithmetical theories

We formalize the notion of Herbrand Consistency in an appropriate way for bounded arithmetics, and show the existence of a finite fragment of IΔ₀ whose Herbrand Consistency is not provable in IΔ₀. We also show the existence of an IΔ₀-derivable Π ₁-sentence such that IΔ₀ cannot prove its Herbrand Consistency.     

Positive provability logic for uniform reflection principles

We deal with the fragment of modal logic consisting of implications of formulas built up from the variables and the constant ‘true’ by conjunction and diamonds only. The weaker language allows one to interpret the diamonds as the uniform reflection schemata in arithmetic, possibly of unrestricted logical complexity. We formulate an arithmetically complete calculus with modalities labeled by natural numbers and ω, where ω   corresponds to the full uniform reflection schema, whereas n<ω$n<\omega$ corresponds to its restriction to arithmetical _Πn+1${\Pi }_{n+1}$-formulas. This calculus is shown to be complete w.r.t. a suitable class of finite Kripke models and to be decidable in polynomial time.     

Infinitary calculus for a restricted first-order linear temporal logic without contraction on quantified formulas

An infinitary calculus for a restricted fragment of the first-order linear temporal logic is considered. We prove that for this fragment one can construct the infinitary calculusG _Lω ^* without contraction on predicate formulas. The calculusG _Lω ^* possesses the following properties: (1) the succedent rule for the existential quantifier is included into the corresponding axiom; (2) the premise of the antecedent rule for the universal quantifier does not contain a duplicate of the main formula. The soundness and completness ofG _Lω ^* are also proved. Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 3, pp. 378–397, July–September, 1999. Translated by R. Lapinskas     

Monotone operators on Gödel logic

We consider an extension of Gödel logic by a unary operator that enables the addition of non-negative reals to truth-values. Although its propositional fragment has a simple proof system, first-order validity is Π ₂-hard. We explain the close connection to Scarpellini’s result on Π ₂-hardness of ?ukasiewicz’s logic.     

Two interfacial shear strength calculations based on the single fiber composite test

The fragmentation of a single fiber embedded in a polymer matrix upon stretching (SFC test) provides valuable information on the fiber-matrix bond strength (), which determines stress transfer through the interface and, thus, significantly affects the mechanical properties of the composite material. However, the calculated bond strength appears to depend on data interpretation, i.e., on the applied theoretical model, since the direct result of the SFC test is the fiber fragment length distribution rather than the value. Two approaches are used in SFC testing for calculation of the bond strength: 1) the Kelly-Tyson model, in which the matrix is assumed to be totally elastic and 2) the Cox model using the elastic constants of the fiber and the matrix. In this paper, an attempt has been made to compare these two approaches employing theory as well as the experimental data of several authors. The dependence of the tensile stress in the fiber and the interfacial shear stress on various factors has been analyzed. For both models, the mean interfacial shear stress in the fragment of critical length (l_c) was shown to satisfy the same formula () = (_cD)/2l_c, where D is the fiber diameter and _c is the tensile strength of a fiber at gauge length equal to l_c. However, the critical lengths from the Kelly-Tyson approach and Cox model are differently related to the fragment length distribution parameters such as the mean fragment length. This discrepancy results in different () values for the same experimental data set. While the main parameter in the Kelly-Tyson model assumed constant for a given fiber-matrix pair is the interfacial shear strength, the ultimate (local) bond strength _ult may be seen as the corresponding parameter in the Cox model. Various _ult values were obtained for carbon fiber-epoxy matrix systems by analyzing the data of continuously monitored single fiber composite tests. Whereas the mean value of the interfacial shear stress calculated in the Cox approach was comparable to the interfacial shear strength from the Kelly-Tyson model, its ultimate value characterizing the true adhesional bond strength appeared to be three or four times greater.To be presented at the Ninth International Conference on the Mechanics of Composite Materials in Riga, Latvia, October, 1995.Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 4, pp. 446–461, July–August, 1995.     

A semantical proof of De Jongh's theorem

Summary In 1969, De Jongh proved the maximality of a fragment of intuitionistic predicate calculus forHA. Leivant strengthened the theorem in 1975, using proof-theoretical tools (normalisation of infinitary sequent calculi). By a refinement of De Jongh's original method (using Beth models instead of Kripke models and sheafs of partial combinatory algebras), a semantical proof is given of a result that is almost as good as Leivant's. Furthermore, it is shown thatHA can be extended to Higher Order Heyting Arithmetic+all true ₂ ⁰ -sentences + transfinite induction over primitive recursive well-orderings. As a corollary of the proof, maximality of intuitionistic predicate calculus is established wrt. an abstract realisability notion defined over a suitable expansion ofHA.     

The Equilibrium Behavior of Reversible Coagulation-Fragmentation Processes

The coagulation-fragmentation process models the stochastic evolution of a population of N particles distributed into groups of different sizes that coagulate and fragment at given rates. The process arises in a variety of contexts and has been intensively studied for a long time. As a result, different approximations to the model were suggested. Our paper deals with the exact model which is viewed as a time-homogeneous interacting particle system on the state space _N, the set of all partitions of N. We obtain the stationary distribution (invariant measure) on _N for the whole class of reversible coagulation-fragmentation processes, and derive explicit expressions for important functionals of this measure, in particular, the expected numbers of groups of all sizes at the steady state. We also establish a characterization of the transition rates that guarantee the reversibility of the process. Finally, we make a comparative study of our exact solution and the approximation given by the steady-state solution of the coagulation-fragmentation integral equation, which is known in the literature. We show that in some cases the latter approximation can considerably deviate from the exact solution.     

Analytic and numerical analysis of some statistical features of fragmentation processes

In this work, we investigate the following inhomogeneous model of fragmentation with an exponential splitting probability: suppose that an initial particle having a mass x₀>0 (large enough) is broken, with a size-dependent probability , into b>1 pieces. With probability 1−p(x₀), this fragment is left unchanged for ever. The process of fragmentation is repeated recursively for all pieces.Some statistical features are studied, chiefly the fragment size distribution, number of fragments and partition function of the random fragmentation tree. Also, we give details of the numerical analysis of these features.     

Some weak fragments of Martin’s axiom related to the rectangle refining property

We introduce the anti-rectangle refining property for forcing notions and investigate fragments of Martin’s axiom for ℵ₁ dense sets related to the anti-rectangle refining property, which is close to some fragment of Martin’s axiom for ℵ₁ dense sets related to the rectangle refining property, and prove that they are really weaker fragments. T. Yorioka was partially supported by Grant-in-aids for Scientific Research No.16340022 and No.18840022.     

Lambek calculus with one division and one primitive type permitting empty antecedents

The following assertion is proved: an inference rule given by a scheme is admissible in the Lambek calculus with one division L*(\) permitting empty antecedents if and only if it is admissible in the fragment of L*(\) with one primitive type L*(\; p ₁). To do that, a type substitution is used which reduces the derivability in L*(\) to the derivability in L*(\; p ₁).     

Non-standard finite fields overIΔ0+Ω1

We consider residue fields of primes in the well-known fragment of arithmeticIΔ₀+Ω₁. We prove that each such residue field has exactly one extension of each degree. The standard proofs use counting and the Frobenius map. Since little is known about these topics in fragments, we looked for, and found, another proof using permutation groups and the elements of Galois cohomology. This proof fits nicely intoIΔ₀ + Ω₁ using, instead of exponentiation, exponentiation modulo a prime.     

Homogeneous Models and Stable Diagrams

Some fragment is studied of stability theory in the category of D-sets. Conditions are given for existence of D-homogeneous models of however large power. A categoricity theorem is proven for the class of (D,)-homogeneous models.     

Many-Particle Models with Two Constants of Interaction

We discuss integrable models of quantum field theory and statistical mechanics. The dynamics and kinematics of these models are defined by Hamiltonians with symmetries determined by Lie algebras. The paper is devoted to the characterization of models such that the root vectors of their symmetry algebras run two orbits under the action of the Weyl group. Such properties possess root systems of the type B _N , C _N , and G ₂. The main focus is on models with the symmetry of algebras B _N . In this case the main characteristics of the process are obtained from the system of Yang-Baxter equation and the reflection equations.We consider the Calogero-Moser and Calogero-Sutherland models and also the formalisms of the Lax and Dunkl operators. The connection between these formalisms and method of describing these models in terms of the generalized Knizhnik-Zamolodchikov equations with the system of roots of the type B _N by the example of the Gaudin model with reflection are discussed. Examples of many-particle systems that interact with each other with reflections are presented.__________Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 10, Suzdal Conference-4, 2003.     

Phase transitions with semi-diffuse interfaces

In this paper, we examine new “phase-field” models with semi-diffuse interfaces. These models have the property that the −1/+1 planar phase transitions take place over a finite interval. The models also support multiple interface solutions with interfaces centered at arbitrary points L₁<L₂<?<L_N. These solutions correspond to local minima of an entropy functional (see (3.3) and (3.7)) rather than saddle points and are dynamically stable. The classical models have no such exact solutions but they do support solutions with N equally spaced transition points where the order parameter transitions between values p_min(N) and p_max(N) satisfying −1<p_min(N)<0<p_max(N)<1. These solutions of the classical model are saddle points of the entropy functional associated with those models and are not dynamically stable.     

The decision problem for some logics for finite words on infinite alphabets

This paper is a follow-up to a previous paper where the logical characterization of n-ary synchronous relations due to Eilenbeig, Elgot, and Shepherdson was investigated in the case where the alphabet has infinitely many letters. Here we show that modifying one of the predicates leads to a completely different picture for infinite alphabets, though it does not change the expressive power for finite alphabets. Indeed, roughly speaking, being able to express the fact that two words end with the same symbol leads to an undecidable theory, already for the Σ₂ fragment. Finally, we show that the existential fragment is decidable. Bibliography: 19 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 358, 2008, pp. 100–119.     

Existence of many L∞,λ-equivalent,non- isomorphic models of T of power λ

Suppose T is not superstable, T ⊆ T₁ (both first-order theories). If λ>∥T₁∥ is regular or strong limit, we construct 2^λ non-isomorphic, pairwise L_∞,λ-equivalent models of T of power λ, which are reducts of models of T₁. Note, however, that the proof applies to the class of models of T, T (superstable but) with dop or otop and even to appropriate non-elementary classes as well.     

Axiomatization and Polynomial Solvability of Strictly Positive Fragments of Certain Modal Logics

The fragment of the language of modal logic that consists of all implications A → B, where A and B are built from variables, the constant ⊤ (truth), and the connectives ∧ and ◊₁,◊₂,...,◊ _m . For the polymodal logic S5 _m (the logic of m equivalence relations) and the logic K4.3 (the logic of irreflexive linear orders), an axiomatization of such fragments is found and their algorithmic decidability in polynomial time is proved.

     

Regression of directed graphs on independent effects for density and reciprocity

In common models for dyadic network regression, the density and reciprocity parameters are dependent on each other. Here, the j₁ and j₂ models are introduced with a density parameter that represents the log odds of a single tie. Consequently, the density and reciprocity parameters are independent and the interpretation of both parameters more straightforward. Estimation procedures and simulation results for these new models are discussed as well as an illustrative example.