# Logarithmic scaling of planar random walk’s local times

Authors: Péter Nándori and Zeyu Shen

We prove that the local time process of a planar simple random walk, when time is scaled logarithmically, converges to a non-degenerate pure jump process. The convergence takes place in the Skorokhod space with respect to the M1 topology and fails to hold in the J1 topology.

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# On the construction of a joint distribution given two discrete conditionals

Authors: Indranil Ghosh and Saralees Nadarajah

Consider a two-dimensional discrete random variable (X, Y) with possible values 1, 2, . . . , I for X and 1, 2, . . . , J for Y. For specifying the distribution of (X, Y), suppose both conditional distributions of X given Y and Y given X are specified. In this paper, we address the problem of determining whether a given set of constraints involving marginal and conditional probabilities and expectations of functions are compatible or most nearly compatible. To this end, we incorporate all those information with the Kullback-Leibler (K-L) divergence and power divergence statistics to obtain the most nearly compatible probability distribution when the two conditionals are not compatible, under the discrete set up. Finally, a comparative study is carried out between the K-L divergence and power divergence statistics for some illustrative examples.

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# Reliability characteristics of bivariate Rayleigh distribution and concomitants of its order statistics and record values

Authors: Saeid Tahmasebi, Ali Akbar Jafari and Mohammad Ahsanullah

The Rayleigh distribution is an important model in applications such as noise theory, height of the sea waves and wave length. In this paper, we first study the reliability characteristics of Morgenstern type bivariate Rayleigh distribution (MTBRD). Then, we investigate some properties on concomitants of order statistics and record values in MTBRD. Finally, we propose the best linear unbiased estimator for a parameter of MTBRD using both complete and censored samples based on concomitants of order statistics.

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# Riesz fractional derivatives of solutions of differential equation y (4) + xy = 0

Author: Alireza Ansari

In this paper, we get integral representations for the quintic Airy functions as the four linearly independent solutions of differential equation y(4) + xy = 0. Also, new integral representations for the products of these functions are obtained in terms of the Bessel functions and the Riesz fractional derivatives of these products are given.

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# Simulation based comparison of stochastic claims reserving models in general insurance

Authors: Miklós Arató, László Martinek and Miklós Mályusz

The appropriate estimation of incurred but not reported (IBNR) reserves is traditionally one of the most important task for property and casualty actuaries. As certain claims are reported considerably later after their occurrence, the amount and appropriateness of the reserves have a substantial effect on the financial results of institutions. In recent years, stochastic reserving methods have become increasingly widespread, supported by broad actuarial literature, describing development models and evaluation techniques.

The cardinal aim of the present paper is to compare the appropriateness of several stochastic estimation methods, supposing different distributional underlying development models. For lack of analytical formulae in most of the model settings relevant from a practical perspective, due to the complex behavior of summed variables, simulations are performed to approximate distributions and results. Considering that the number of runoff triangles is generally limited, stochastically simulated scenarios contribute to feasible solutions. Stochastic reserving is taken into account as a stochastic forecast, thus comparison techniques developed for stochastic forecasts can be applied, opening up new informative perspectives beyond classical prediction measures, such as the mean square error of prediction.

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# Convergence theorems for common fixed points of a sequence of nonexpansive nonself mappings in CAT(0) spaces

Authors: Shanguang Qian and Wei-Qi

Based on an original idea, i.e., a special way of choosing the indexes of the involved mappings, we propose a new iterative algorithm for approximating common fixed points of a sequence of nonexpansive nonself mappings, and some convergence theorems are established in the framework of CAT(0) spaces. Our results extend the previous results restricted to the situation of a single mapping.

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# The countable type properties in free paratopological groups

Authors: Fucai Lin, Chuan Liu and Kexiu Zhang

A space X is of countable type (resp. subcountable type) if every compact subspace F of X is contained in a compact subspace K that is of countable character (resp. countable pseudocharacter) in X. In this paper, we mainly show that: (1) For a functionally Hausdorff space X, the free paratopological group FP(X)and the free abelian paratopological group AP(X) are of countable type if and only if X is discrete; (2) For a functionally Hausdorff space X, if the free abelian paratopological group AP(X) is of subcountable type then X has countable pseudocharacter. Moreover, we also show that, for an arbitrary Hausdorff μ-space X, if AP 2(X) or FP 2(X) is locally compact, then X is a topological sum of a compact space and a discrete space.

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# Existence of three solutions for impulsive perturbed elastic beam fourth-order equations of Kirchhoff-type

Authors: Shapour Heidarkhani and Amjad Salari

In this paper, we study the existence of multiple solutions for a class of impulsive perturbed elastic beam equations of Kirchhoff-type. We give a new criteria for guaranteeing that the impulsive perturbed elastic beam equations of Kirchhoff-type have at least three generalized solutions by using a variational method and a critical points theorem of B. Ricceri.

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# Homoclinic orbits of 2nth-order difference equations involving p-Laplacian

Authors: Haiping Shi, Xia Liu and Yuanbiao Zhang

By making use of the critical point theory, we establish some new existence criteria to guarantee that a 2nth-order nonlinear difference equation containing both advance and retardation with p-Laplacian has a nontrivial homoclinic orbit. Our conditions on the potential are rather relaxed, and some existing results in the literature are improved.

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# Linear independence results for the reciprocal sums of Fibonacci numbers associated with Dirichlet characters

Authors: Hiromi Ei, Florian Luca and Yohei Tachiya

Let {F n}n≥0 be the sequence of Fibonacci numbers. The aim of this paper is to give linear independence results over $ℚ(5)$ for the infinite series $∑n=1∞χj(n)/Fn$ with certain nonprincipal real Dirichlet characters χ j. We also deduce the irrationality results for the special principal Dirichlet characters and for other multiplicative functions.

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# On multiple Borsuk numbers in normed spaces

Authors: Zsolt Lángi and Márton Naszódi

Hujter and Lángi defined the k-fold Borsuk number of a set S in Euclidean n-space of diameter d > 0 as the smallest cardinality of a family F of subsets of S, of diameters strictly less than d, such that every point of S belongs to at least k members of F.

We investigate whether a k-fold Borsuk covering of a set S in a finite dimensional real normed space can be extended to a completion of S. Furthermore, we determine the k-fold Borsuk number of sets in not angled normed planes, and give a partial characterization for sets in angled planes.

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# On rings with annihilator condition

In this paper we study rings R with the property that every finitely generated ideal of R consisting entirely of zero divisors has a nonzero annihilator. The class of commutative rings with this property is quite large; for example, noetherian rings, rings whose prime ideals are maximal, the polynomial ring R[x] and rings whose classical ring of quotients are von Neumann regular. We continue to study conditions under which right mininjective rings, right FP-injective rings, right weakly continuous rings, right extending rings, one sided duo rings, semiregular rings and semiperfect rings have this property.

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# On two conjectures on b-coloring of graph products

Authors: S. Francis Raj and T. Kavaskar

A b-coloring of a graph G with k colors is a proper coloring of G using k colors in which each color class contains a color dominating vertex, that is, a vertex which has a neighbor in each of the other color classes. The largest positive integer k for which G has a b-coloring using k colors is the b-chromatic number b(G) of G. It was conjectured in [10], that for any two graphs G and H, b(G[H]) ≦ b(G) − 1|V (H)| + Δ(H) + 1 and b(GH) ≦ max {b(G)(Δ(H) + 1), b(H) Δ(G) + 1)}, where G[H] and GH denotes the lexicographic product and the strong product of G and H, respectively. In this paper, we disprove both conjectures.

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# Walsh-Lebesgue points and restricted convergence of multi-dimensional Walsh-Fourier series

Author: Ferenc Weisz

A new concept of Walsh-Lebesgue points is introduced for higher dimensions and it is proved that almost every point is a modified Walsh-Lebesgue point of an integrable function. It is shown that the Walsh-Fejér means σ n f of a function fL 1[0, 1)d converge to f at each modified Walsh-Lebesgue point, whenever n→∞ and n is in a cone. The same is proved for other summability means, such as for the Weierstrass, Abel, Picard, Bessel, Cesàro, de La Vallée-Poussin, Rogosinski and Riesz summations.

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# Extended arcsine distribution to proportional data: Properties and applications

Authors: Gauss M. Cordeiro, Artur J. Lemonte and Ana K. Campelo

We propose a new two-parameter continuous model called the extended arcsine distribution restricted to the unit interval. It is a very competitive model to the beta and Kumaraswamy distributions for modeling percentages, rates, fractions and proportions. We provide a mathematical treatment of the new distribution including explicit expressions for the ordinary and incomplete moments, mean deviations, Bonferroni and Lorenz curves, generating and quantile functions, Shannon entropy and order statistics. Maximum likelihood is used to estimate the model parameters and the expected information matrix is determined. We demonstrate by means of two applications to proportional data that it can give consistently a better fit than other important statistical models.

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# Livšic’s theorem for q-Sturm—Liouville operators

Authors: Hüseyin Tuna and Aytekin Eryilmaz

In this paper, we study dissipative q-Sturm—Liouville operators in Weyl’s limit circle case. We describe all maximal dissipative, maximal accretive, self adjoint extensions of q-Sturm—Liouville operators. Using Livšic’s theorems, we prove a theorem on completeness of the system of eigenvectors and associated vectors of the dissipative q-Sturm—Liouville operators.

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# A note on the maximal operators of Vilenkin—Nörlund means with non-increasing coefficients

In [14] we investigated some Vilenkin—Nörlund means with non-increasing coefficients. In particular, it was proved that under some special conditions the maximal operators of such summabily methods are bounded from the Hardy space H 1/(1+α) to the space weak-L 1/(1+α), (0 < α ≦ 1). In this paper we construct a martingale in the space H 1/(1+α), which satisfies the conditions considered in [14], and so that the maximal operators of these Vilenkin—Nörlund means with non-increasing coefficients are not bounded from the Hardy space H 1/(1+α) to the space L 1/(1+α). In particular, this shows that the conditions under which the result in [14] is proved are in a sense sharp. Moreover, as further applications, some well-known and new results are pointed out.

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# On characterizations and infinite divisibility of recently introduced distributions

Author: G. G. Hamedani

We present here characterizations of the most recently introduced continuous univariate distributions based on: (i) a simple relationship between two truncated moments; (ii) truncated moments of certain functions of the 1th order statistic; (iii) truncated moments of certain functions of the n th order statistic; (iv) truncated moment of certain function of the random variable. We like to mention that the characterization (i) which is expressed in terms of the ratio of truncated moments is stable in the sense of weak convergence. We will also point out that some of these distributions are infinitely divisible via Bondesson’s 1979 classifications.

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# On M p -embedded primary subgroups of finite groups

Authors: Jia Zhang and Long Miao

A subgroup H of G is called M p-embedded in G, if there exists a p-nilpotent subgroup B of G such that H p ∈ Sylp(B) and B is M p-supplemented in G. In this paper, we use M p-embedded subgroups to study the structure of finite groups.

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# On the approximation properties of Cesàro means of negative order of Vilenkin—Fourier series

In this paper we establish approximation properties of Cesàro (C, −α) means with α ∈ (0, 1) of Vilenkin—Fourier series. This result allows one to obtain a condition which is sufficient for the convergence of the means σ n α(f, x) to f(x) in the L p-metric.

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# The principal ideal theorem for w-Noetherian rings

Authors: Huayu Yin and Fanggui Wang

In this paper, we concern the Principal Ideal Theorem (PIT) for w-Noetherian rings. Let R be a w-Noetherian ring and a be a nonzero nonunit element of R. If p is a prime ideal of R minimal over (a), then ht p ≦ 1.

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# Approximation by k-th order modifications of Szász—Mirakyan operators

Authors: Tuncer Acar, Ali Aral and Ioan Raşa

In this paper, we study the k-th order Kantorovich type modication of Szász—Mirakyan operators. We first establish explicit formulas giving the images of monomials and the moments up to order six. Using this modification, we present a quantitative Voronovskaya theorem for differentiated Szász—Mirakyan operators in weighted spaces. The approximation properties such as rate of convergence and simultaneous approximation by the new constructions are also obtained.

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# Characterization of stably simple curve singularities

Authors: Muhammad Ahsan Binyamin, Junaid Alam Khan, Faira Kanwal Janjua and Naveed Hussain

In this article we characterize the classification of stably simple curve singularities given by V. I. Arnold, in terms of invariants. On the basis of this characterization we describe an implementation of a classifier for stably simple curve singularities in the computer algebra system SINGULAR.

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# On the convergence of double Fourier integrals of functions of bounded variation on ℝ2

Authors: Bhikha Lila Ghodadra and Vanda Fülöp

We investigate the pointwise and uniform convergence of the symmetric rectangular partial (also called Dirichlet) integrals of the double Fourier integral of a function that is Lebesgue integrable and of bounded variation over ℝ2. Our theorem is a two-dimensional extension of a theorem of Móricz (see Theorem 3 in [10]) concerning the single Fourier integrals, which is more general than the two-dimensional extension given by Móricz himself (see Theorem 3 in [11]).

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# On the refinements of Hadamard-type inequalities and their applications for cubature formulas

Author: Kai-Chen Hsu

In this paper, we shall establish some Hadamard-type inequalities for differentiable coordinated convex functions in a rectangle from the plane in two variables. Through these inequalities, more precise estimates could be obtained. Some examples and applications to cubature formulas are also provided.

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# Yet some more non-finite axiomatizability results for algebras of relations and ways to avoid them

Author: Tarek Sayed Ahmed

Let α be an infinite ordinal. Let RCAα denote the variety of representable cylindric algebras of dimension α. Modifying Andréka’s methods of splitting, we show that the variety RQEAα of representable quasi-polyadic equality algebras of dimension α is not axiomatized by a set of universal formulas containing only finitely many variables over the variety RQAα of representable quasi-polyadic algebras of dimension α. This strengthens a seminal result due to Sain and Thompson, answers a question posed by Andréka, and lifts to the transfinite a result of hers proved for finite dimensions > 2. Using the modified method of splitting, we show that all known complexity results on universal axiomatizations of RCAα (proved by Andréka) transfer to universal axiomatizations of RQEAα. From such results it can be inferred that any algebraizable extension of L ω,ω is severely incomplete if we insist on Tarskian square semantics. Ways of circumventing the strong non-negative axiomatizability results hitherto obtained in the first part of the paper, such as guarding semantics, and /or expanding the signature of RQEAω by substitutions indexed by transformations coming from a finitely presented subsemigroup of (ω ω, ○) containing all transpositions and replacements, are surveyed, discussed, and elaborated upon.

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# Asymptotic stability of solutions for a class of nonlinear functional integral equations of fractional order

Authors: Ümit Çakan and İsmet Özdemir

In this paper, using a Darbo type fixed point theorem associated with the measure of noncompactness we prove a theorem on the existence of asymptotically stable solutions of some nonlinear functional integral equations in the space of continuous and bounded functions on R+ = [0,∞). We also give some examples satisfying the conditions our existence theorem.

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# *-Baer property for rings with involution

The consistent way of investigating rings with involution, briefly *-rings, is to study them in the category of *-rings with morphisms preserving also involution. In this paper we continue the study of *-rings and the notion of *-reduced *-rings is introduced and their properties are studied. We introduce also the class of *-Baer *-rings. This class is defined in terms of *-annihilators and principal *-biideals, and it naturally extends the class of Baer *-rings. The use of *-biideals makes this concept more consistent with the involution than the use of right ideals in the notion of Baer *-rings. We prove that each *-Baer *-ring is semiprime. Moreover, we show that the property of *-Baer extends to both the *-corner and the center of the *-ring. Finally, we discuss the relation between *-Baer and quasi-Baer *-rings; the generalization of Baer *-ring.

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# Differential sandwich theorems of p-valent functions defined by a certain linear operator

Authors: Ping He and Defei Zhang

In this paper we introduce differential subordination and superordination properties for certain subclasses of analytic functions involving certain linear operator, and obtain sandwich-type results for the functions belonging to these classes.

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# Large forbidden configurations and design theory

Authors: R. P. Anstee and Attila Sali

Let forb(m, F) denote the maximum number of columns possible in a (0, 1)-matrix A that has no repeated columns and has no submatrix which is a row and column permutation of F. We consider cases where the configuration F has a number of columns that grows with m. For a k × l matrix G, define s · G to be the concatenation of s copies of G. In a number of cases we determine forb(m, m α · G) is Θ(m k). Results of Keevash on the existence of designs provide constructions that can be used to give asymptotic lower bounds. An induction idea of Anstee and Lu is useful in obtaining upper bounds.

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# Mixed-type reverse order laws for the group inverses in rings with involution

Authors: Dijana Mosić and Dragan S. Djordjević

We investigate some equivalent conditions for the reverse order laws (ab)# = b a # and (ab)# = b # a in rings with involution. Similar results for (ab)# = b # a* and (ab)# = b*a # are presented too.

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# A survey of product posterior distributions

In Bayesian statistics, one frequently encounters priors and posteriors that are product of two probability density functions. In this paper, we discuss three such priors/posteriors, provide motivation and derive expressions for their moments, median and mode. Forty seven motivating examples are discussed. We expect that this paper could serve as a useful reference for practitioners of Bayesian statistics. It could also encourage further research in this area.

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# Characterization of power function distribution based on spacings

Authors: G. G. Hamedani and H. W. Volkmer

A characterization of power function distribution based on the distribution of spacings is presented here extending the existing characterizations of the uniform distribution in this direction.

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# Conditional oscillation of Euler type half-linear differential equations with unbounded coefficients

Authors: Jaroslav Jaroš and Michal Veselý

The oscillatory properties of half-linear second order Euler type differential equations are studied, where the coefficients of the considered equations can be unbounded. For these equations, we prove an oscillation criterion and a non-oscillation one. We also mention a corollary which shows how our criteria improve the known results. In the corollary, the criteria give an explicit oscillation constant.

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# Existence of periodic solutions of fourth-order p-Laplacian difference equations

Authors: Haiping Shi, Xia Liu and Yuanbiao Zhang

By making use of the critical point theory, the existence of periodic solutions for fourth-order nonlinear p-Laplacian difference equations is obtained. The main approach used in our paper is a variational technique and the Saddle Point Theorem. The problem is to solve the existence of periodic solutions of fourth-order nonlinear p-Laplacian difference equations. The results obtained successfully generalize and complement the existing one.

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# Existence results for a second order nonlocal boundary value problem at resonance

Author: Katarzyna Szymańska-Dȩbowska

The paper focuses on existence of solutions of a system of nonlocal resonant boundary value problems $x″=f(t,x), x′(0)=0, x(1)=∫01x(s)dg(s)$, where f : [0, 1] × ℝk → ℝk is continuous and g : [0, 1] → ℝk is a function of bounded variation. Imposing on the function f the following condition: the limit limλ→∞ f(t, λ a) exists uniformly in aS k−1, we have shown that the problem has at least one solution.

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# The solvability of some nonlinear functional integral equations

Authors: İsmet Özdemir and Ümit Çakan

In this paper, using a Darbo type fixed point theorem associated with the measure of noncompactness we prove a theorem on the existence of solutions of some nonlinear functional integral equations in the space of continuous functions on interval [0, a]. We give also some examples which show that the obtained results are applicable

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# Strong approximation of Black-Scholes theory based on simple random walks

Authors: Zsolt Nika and Tamás Szabados

A basic model in financial mathematics was introduced by Black, Scholes and Merton in 1973. A classical discrete approximation in distribution is the binomial model given by Cox, Ross and Rubinstein in 1979. In this work we give a strong (almost sure, pathwise) discrete approximation of the BSM model using a suitable nested sequence of simple, symmetric random walks. The approximation extends to the stock price process, the value process, the replicating portfolio, and the greeks. An important tool in the approximation is a discrete version of the Feynman-Kac formula as well.

Our aim is to show that from an elementary discrete approach, by taking simple limits, one may get the continuous versions. We think that such an approach can be advantageous for both research and applications. Moreover, it is hoped that this approach has pedagogical merits as well: gives insight and seems suitable for teaching students whose mathematical background may not contain e.g. measure theory or stochastic analysis.

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# Turán type inequalities for confluent hypergeometric functions of the second kind

Authors: Árpád Baricz, Saminathan Ponnusamy and Sanjeev Singh

In this paper we deduce some tight Turán type inequalities for Tricomi confluent hypergeometric functions of the second kind, which in some cases improve the existing results in the literature. We also give alternative proofs for some already established Turán type inequalities. Moreover, by using these Turán type inequalities, we deduce some new inequalities for Tricomi confluent hypergeometric functions of the second kind. The key tool in the proof of the Turán type inequalities is an integral representation for a quotient of Tricomi confluent hypergeometric functions, which arises in the study of the infinite divisibility of the Fisher-Snedecor F distribution.

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# Certain congruences on eventually regular semigroups I

Author: Roman S. Gigoń

A semigroup is called eventually regular if each of its elements has a regular power. In this paper we study certain fundamental congruences on an eventually regular semigroup. We generalize some results of Howie and Lallement (1966) and LaTorre (1983). In particular, we give a full description of the semilattice of group congruences (together with the least such a congruence) on an arbitrary eventually regular (orthodox) semigroup. Moreover, we investigate UBG-congruences on an eventually regular semigroup. Finally, we study the eventually regular subdirect products of an E-unitary semigroup and a Clifford semigroup.

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# Classes of simplicial complexes which admit nontrivial Cohen-Macaulay modifications

The set of Cohen-Macaulay monomial ideals with a given radical contains the so-called Cohen-Macaulay modifications. Not all Cohen-Macaulay squarefree monomial ideals admit nontrivial Cohen-Macaulay modifications. We present classes of Cohen-Macaulay squarefree monomial ideals with infinitely many nontrivial Cohen-Macaulay modifications.

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# Combinatorics of poly-Bernoulli numbers

Authors: Beáta Bényi and Péter Hajnal

The Bn (k) poly-Bernoulli numbers — a natural generalization of classical Bernoulli numbers (B n = Bn (1)) — were introduced by Kaneko in 1997. When the parameter k is negative then Bn (k) is a nonnegative number. Brewbaker was the first to give combinatorial interpretation of these numbers. He proved that Bn (−k) counts the so called lonesum 0–1 matrices of size n × k. Several other interpretations were pointed out. We survey these and give new ones. Our new interpretation, for example, gives a transparent, combinatorial explanation of Kaneko’s recursive formula for poly-Bernoulli numbers.

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# Differential operators on the weighted densities on the supercircle S 1|1

Authors: Nader Belghith, Mabrouk Ben Ammar and Nizar Ben Fraj

Over the (1, 1)-dimensional real supercircle, we consider the K(1)-modules D λ,μ k of linear differential operators of order k acting on the superspaces of weighted densities, where K(1) is the Lie superalgebra of contact vector fields. We give, in contrast to the classical setting, a classification of these modules. This work is the simplest superization of a result by Gargoubi and Ovsienko.

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# A note on rings with the summand sum property

Author: Shen Liang

A ring R is called right SSP (SIP) if the sum (intersection) of any two direct summands of R R is also a direct summand. Left SSP (SIP) rings are defined similarly. There are several interesting results on rings with SSP. For example, R is right SSP if and only if R is left SSP, and R is a von Neumann regular ring if and only if Mn(R) is SSP for some n > 1. It is shown that R is a semisimple ring if and only if the column finite matrix ring ℂFM(R) is SSP, where ℕ is the set of natural numbers. Some known results are proved in an easy way through idempotents of rings. Moreover, some new results on SSP rings are given.

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# On p-supersolvability of finite groups

Let G be a finite group. A subgroup H of G is said to be s-permutable in G if H permutes with all Sylow subgroups of G. Let H be a subgroup of G and let H sG be the subgroup of H generated by all those subgroups of H which are s-permutable in G. A subgroup H of G is called n-embedded in G if G has a normal subgroup T such that H G = HT and HTH sG, where H G is the normal closure of H in G. We investigate the influence of n-embedded subgroups of the p-nilpotency and p-supersolvability of G.

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# On the average number of normals through points of a convex body

Authors: Gábor Domokos and Zsolt Lángi

In 1944, Santaló asked about the average number of normals through a point of a given convex body. Since then, numerous results appeared in the literature about this problem. The aim of this paper is to add to this list some new, recent developments. We point out connections of the problem to static equilibria of rigid bodies as well as to geometric partial differential equations of surface evolution.

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# Summability of general orthonormal Fourier series

Authors: L. Gogoladze and V. Tsagareishvili

S. Banach in [1] proved that for any function fL 2(0, 1), f ≁ 0, there exists an ONS (orthonormal system) such that the Fourier series of this function is not summable a.e. by the method (C, α), α > 0.

D. Menshov found the conditions which should be satisfied by the Fourier coefficients of the function for the summability a.e. of its Fourier series by the method (C, α), α > 0.

In this paper the necessary and sufficient conditions are found which should be satisfied by the ONS functions (φ n(x)) so that the Fourier coefficients (by this system) of functions from class Lip 1 or A (absolutely continuous) satisfy the conditions of D. Menshov.

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# The support localization property of the strongly embedded subspaces of Banach function spaces

Authors: Pilar Rueda and Enrique A. Sánchez Pérez

Motivated by the well known Kadec-Pełczynski disjointification theorem, we undertake an analysis of the supports of non-zero functions in strongly embedded subspaces of Banach functions spaces. The main aim is to isolate those properties that bring additional information on strongly embedded subspaces. This is the case of the support localization property, which is a necessary condition fulfilled by all strongly embedded subspaces. Several examples that involve Rademacher functions, the Volterra operator, Lorentz spaces or Orlicz spaces are provided.

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Authors: Zoltán Sebestyén and Zsigmond Tarcsay

An extension of von Neumann’s characterization of essentially selfadjoint operators is given among not necessarily densely defined symmetric operators which are assumed to be closable. Our arguments are of algebraic nature and follow the idea of [1].

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# Productly linearly independent sequences

Authors: Jaroslav Hančl, Katarína Korčeková and Lukáš Novotný

We introduce the two new concepts, productly linearly independent sequences and productly irrational sequences. Then we prove a criterion for which certain infinite sequences of rational numbers are productly linearly independent. As a consequence we obtain a criterion for the irrationality of infinite products and a criterion for a sequence to be productly irrational.

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# A remark on generalization of convexity in L 1-convergence of cosine series

Author: S. P. Zhou

The present note will present a different and direct way to generalize the convexity while keep the classical results for L 1-convergence still alive.

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# A separation theorem for totally-sewn 4-polytopes

Authors: T. Bisztriczky and F. Fodor

The Separation Problem, originally posed by K. Bezdek in [1], asks for the minimum number s(O, K) of hyperplanes needed to strictly separate an interior point O in a convex body K from all faces of K. It is conjectured that s(O, K) ≦ 2d in d-dimensional Euclidean space. We prove this conjecture for the class of all totally-sewn neighbourly 4-dimensional polytopes.

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# Some quadratic irrationals with explicit continued fraction and Engel series expansions

Authors: Narakorn Rompurk Kanasri, Vichian Laohakosol and Tawat Changphas

A remarkable class of quadratic irrational elements having both explicit Engel series and continued fraction expansions in the field of Laurent series, mimicking the case of real numbers discovered by Sierpiński and later extended by Tamura, is constructed. Linear integer-valued polynomials which can be applied to construct such class are determined. Corresponding results in the case of real numbers are mentioned.

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# An upper bound theorem concerning lattice polytopes

Author: Gábor Hegedüs

R. P. Stanley proved the Upper Bound Conjecture in 1975. We imitate his proof for the Ehrhart rings.

We give some upper bounds for the volume of integrally closed lattice polytopes. We derive some inequalities for the δ-vector of integrally closed lattice polytopes. Finally we apply our results for reflexive integrally closed and order polytopes.

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# The conditions of provable security of block ciphers against truncated differential attack

Author: Victor Ruzhentsev

The modified method of estimation of the resistance of block ciphers to truncated byte differential attack is proposed. The previously known method estimate the truncated byte differential probability for Rijndael-like ciphers. In this paper we spread the sphere of application of that method on wider class of ciphers. The proposed method based on searching the most probable truncated byte differential characteristics and verification of sufficient conditions of effective byte differentials absence.

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# Constructing S-boxes with low multiplicative complexity

Author: Pavol Zajac

Multiplicative complexity is the minimum number of AND-gates required to implement a given Boolean function in (AND, XOR) algebra. It is a good measure of a hardware complexity of an S-box, but an S-box cannot have too low multiplicative complexity due to security constraints. In this article we focus on generic constructions that can be used to find good n×n S-boxes with low multiplicative complexity. We tested these constructions in the specific case when n = 8. We were able to find 8 × 8 S-boxes with multiplicative complexity at most 16 (which is half of the known bound on multiplicative complexity of the AES S-box), while providing a reasonable resistance against linear and differential cryptanalysis.

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# Cryptanalysis of chosen symmetric homomorphic schemes

Authors: Damian Vizár and Serge Vaudenay

Since Gentry’s breakthrough result was introduced in the year 2009, the homomorphic encryption has become a very popular topic. The main contribution of Gentry’s thesis [5] was, that it has proven, that it actually is possible to design a fully homomorphic encryption scheme. However ground-breaking Gentry’s result was, the designs, that employ the bootstrapping technique suffer from terrible performance both in key generation and homomorphic evaluation of circuits. Some authors tried to design schemes, that could evaluate homomorphic circuits of arbitrarily many inputs without need of bootstrapping. This paper introduces the notion of symmetric homomorphic encryption, and analyses the security of four such proposals, published in three different papers ([2], [7], [10]). Our result is a known plaintext key-recovery attack on every one of these schemes.

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# Cryptanalysis of the HaF family of hash functions

Authors: Mateusz Buczek and Marcin Kontak

HaF is a family of hash functions developed in Poland at Poznán University of Technology, see [1, 2]. It is a classical Merkle-Damgård construction with the output sizes of 256, 512 or 1024 bits. In this paper we present a collision attack with negligible complexity (collisions can be found without using a computer) for all the members of HaF family. We have also shown that the improved function (without the critical transformation) is still insecure. It is possible to find a preimage for a short message with the complexity lower than the exhaustive search. We are also able to create some fixed points with a complexity of single compression function call.

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# Explicit constructions of extremal graphs and new multivariate cryptosystems

Author: Vasyl Ustimenko

New multivariate cryptosystems are introduced. Sequences f(n) of bijective polynomial transformations of bijective multivariate transformations of affine spaces K n, n = 2, 3, ... , where K is a finite commutative ring with special properties, are used for the constructions of cryptosystems. On axiomatic level, the concept of a family of multivariate maps with invertible decomposition is proposed. Such decomposition is used as private key in a public key infrastructure. Requirements of polynomiality of degree and density allow to estimate the complexity of encryption procedure for a public user. The concepts of stable family and family of increasing order are motivated by studies of discrete logarithm problem in Cremona group. Statement on the existence of families of multivariate maps of polynomial degree and polynomial density with the invertible decomposition is formulated. We observe known explicit constructions of special families of multivariate maps. They correspond to explicit constructions of families of nonlinear algebraic graphs of increasing girth which appeared in Extremal Graph Theory. The families are generated by pseudorandom walks on graphs. This fact ensures the existence of invertible decomposition; a certain girth property guarantees the increase of order for the family of multivariate maps, good expansion properties of families of graphs lead to good mixing properties of graph based private key algorithms. We describe the general schemes of cryptographic applications of such families (public key infrastructure, symbolic Diffie—Hellman protocol, functional versions of El Gamal algorithm).

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# Explicit endomorphism of the Jacobian of a hyperelliptic function field of genus 2 using base field operations

Authors: Eduardo Ruiz Duarte and Octavio Páez Osuna

We present an efficient endomorphism for the Jacobian of a curve C of genus 2 for divisors having a Non disjoint support. This extends the work of Costello and Lauter in [12] who calculated explicit formulæ for divisor doubling and addition of divisors with disjoint support in JF(C) using only base field operations. Explicit formulæ is presented for this third case and a different approach for divisor doubling.

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# Fault-injection based backdoors in Pseudo Random Number Generators

Author: Norbert Tihanyi

The main aim of this paper is to present the concept of fault-injection backdoors in Random Number Generators. Backdoors can be activated by fault-injection techniques. Presented algorithms can be used in embedded systems like smart-cards and hardware security modules in order to implement subliminal channels in random number generators.

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# Hierarchy for classes of garbling schemes

Authors: Tommi Meskanen, Valtteri Niemi and Noora Nieminen

The methods for secure outsourcing and secure one-time programs have recently been of great research interest. Garbling schemes are regarded as a promising technique for these applications while Bellare, Hoang and Rogaway introduced the first formal security notions for garbling schemes in [3, 4]. Ever since, even more security notions have been introduced and garbling schemes have been categorized in different security classes according to these notions. In this paper, we introduce new security classes of garbling schemes and build a hierarchy for the security classes including the known classes as well as classes introduced in this paper.