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This paper proposes a framework for constructing high-order linearly implicit exponential integrators that conserve a quadratic invariant. This is then applied to the scalar auxiliary variable (SAV) approach. Quadratic invariants are significant objects that are present in various physical equations and also in computationally efficient conservative schemes for general invariants. For instance, the SAV approach converts the invariant into a quadratic form by introducing scalar auxiliary variables, which have been intensively studied in recent years. In this vein, Sato et al. (Appl. Numer. Math. 187, 71-88 2023) proposed high-order linearly implicit schemes that conserve a quadratic invariant. In this study, it is shown that their method can be effectively merged with the Lawson transformation, a technique commonly utilized in the construction of exponential integrators. It is also demonstrated that combining the constructed exponential integrators and the SAV approach yields schemes that are computationally less expensive. Specifically, the main part of the computational cost is the product of several matrix exponentials and vectors, which are parallelizable. Moreover, we conduct some mathematical analyses on the proposed schemes. [ABSTRACT FROM AUTHOR]

This special issue of the Journal Mathematical Methods in the Applied Sciences on the topic of Modelling, Analysis, and Applications is devoted to the development of mathematical modelling, analysis, and applications, from theoretical and numerical perspectives, involving different applied sciences and engineering. It collects invited contributions, from selected quality full papers submitted to the International Conference on Mathematical Analysis and Applications in Science and Engineering (ICMA2SC2022). The conference took place in Porto, Portugal in 27–29 June 2022 (https://www.isep.ipp.pt/Page/ViewPage/ICMASC). [ABSTRACT FROM AUTHOR]

In the present paper, necessary and sufficient conditions are given for the equality of the power rezidue symbols $$ {\left( {\frac{\alpha }{a}} \right)_n} $$ and $$ {\left( {\frac{\alpha }{a}} \right)_n} $$ in the cyclotomic field ℚ(ζ n), 2 ∤ n, for a ∈ ℤ, ( a, n) = 1. This result is a generalization of the classical Eisenstein reciprocity law and its continuation in a Hasse’s paper. Bibliography: 3 titles. [ABSTRACT FROM AUTHOR]

Abstract  We generalize a result of F. Luca and A. Sankaranarayanan by proving that the set of n for which ϕ(1) ⃛ ϕ(n) is squareful is of zero density. A similar statement holds for σ (n) instead of ϕ(n) and for some other multiplicative functions. [ABSTRACT FROM AUTHOR]

Abstract: We study a family of orthogonal polynomials which generalizes a sequence of polynomials considered by L. Carlitz. We show that they are a special case of the Sheffer polynomials and point out some interesting connections with certain Sobolev orthogonal polynomials. [Copyright &y& Elsevier]

In this note, we slightly generalize Theorem 2 in the paper by M. Tunç and point out that the assumption of Theorem 3 is not sufficient. A misuse of the term 'mean' is also discussed. [ABSTRACT FROM AUTHOR]

This article provides instructions in preparing a paper for publication in the "Journal of Optimization Theory and Applications." Some of these guidelines are the following: 1)submission of manuscripts in triplicate; 2) reference to English as the official language of the journal; 3) inclusion of an abstract of at least 50 to 100 words in each contribution; and, 4) the abstract should be followed by a list of four to five key words identifying the subject.

Among several number triangles that exist in mathematics, Pascal's triangle is the most fascinating and prominent structure possessing numerous properties. In this paper, we will introduce a number triangle consisting of positive integers arranged in ascending to descending order so that the entries in each row would be mirror images with respect to its middle number. Using this simple number triangle, we have proved six amusing results which interestingly depend only as functions of its centred numbers. Moreover some of the concepts and results discussed in this paper have connection with Tamil literary work and triangular numbers. [ABSTRACT FROM AUTHOR]

Abstract: In 1.9 of the paper [A. Hahn, A Hermitian Morita theorem for algebras with anti-structure, J. Algebra 93 (1985) 215–235], should be replaced by . This leads to minor changes in the rest of the paper where the ring should be replaced by its opposite and vice versa. [Copyright &y& Elsevier]

Grey wolf optimization (GWO) algorithm is a new emerging algorithm that is based on the social hierarchy of grey wolves as well as their hunting and cooperation strategies. Introduced in 2014, this algorithm has been used by a large number of researchers and designers, such that the number of citations to the original paper exceeded many other algorithms. In a recent study by Niu et al., one of the main drawbacks of this algorithm for optimizing real‐world problems was introduced. In summary, they showed that GWO's performance degrades as the optimal solution of the problem diverges from 0. In this paper, by introducing a straightforward modification to the original GWO algorithm, that is, neglecting its social hierarchy, the authors were able to largely eliminate this defect and open a new perspective for future use of this algorithm. The efficiency of the proposed method was validated by applying it to benchmark and real‐world engineering problems. [ABSTRACT FROM AUTHOR]

Š. Schwarz in his paper [SCHWARZ, Š.: Zur Theorie der Halbgruppen, Sborník prác Prírodovedeckej fakulty Slovenskej univerzity v Bratislave, Vol. VI, Bratislava, 1943, 64 pp.] proved the existence of maximal subgroups in periodic semigroups and a decade later he brought into play the maximal subsemigroups and thus he embodied the idempotents in the structural description of semigroups [SCHWARZ, Š.: Contribution to the theory of torsion semigroups, Czechoslovak Math. J. 3(1) (1953), 7–21]. Later in his papers he showed that a proper description of these structural elements can be used to (re)prove many useful and important results in algebra and number theory. The present paper gives a survey of selected results scattered throughout the literature where an semigroup approach based on tools like idempotent, maximal subgroup or maximal subsemigroup either led to a new insight into the substance of the known results or helped to discover new approach to solve problems. Special attention will be given to some disregarded historical connections between semigroup and ring theory. [ABSTRACT FROM AUTHOR]

The Laguerre calculus is widely used for the inversion of differential operators on the Heisenberg group. Applying the Laguerre calculus established on nilpotent Lie groups of step two in Chang et al. [The Laguerre calculus on the nilpotent Lie group of step two. Preprint; 2019. Available from: ], we find the explicit formulas for the heat kernel of sub-Laplace operator and the fundamental solution of power of sub-Laplace operator on nilpotent Lie groups of step two. Calin, Chang and Markina [Generalized Hamilton–Jacobi equation and heat kernel on step two nilpotent Lie groups. In: Gustafsson B, Vasil'ev A, editors. Analysis and mathematical physics. Basel: Birkhüser; 2009 (Trends in mathematics)] also get the formulas for the heat kernel of sub-Laplace operator on nilpotent Lie groups of step two by using the Hamiltonian and Lagrangian formalisms that are related to geometric mechanics. In this paper, we use a totally different method to prove our main results by using the Laguerre calculus, which is more direct from the point of view of Fourier analysis. [ABSTRACT FROM AUTHOR]

Hausman and Shapiro gave an estimate for the number of practical numbers to be 0(x ⁄(logx)β ) for every positive β<1/2 ( 1/log2 -1)². In this paper, we generalize Hausman and Shapiro bound by proving the number of t-practical numbers n (n ≤x ), ( t ≥1 ) to be 0(x ⁄(logx)β ) for every positive β<1/2 ( 1/log2 -1)², and 1≤t≤exp.(( logx ) δ1 ) for any δ1 satisfying 0< δ1 <1-1(1 +√2 β) log2 . We mean by the t-practical number n, the number in which every integer 1≤m≤tn is of the form m= ∑ d/n cd d, 0 ≤ cd≤ t [ABSTRACT FROM AUTHOR]

Let r ≥ 2 and r be even. An r -hypergraph G on n vertices is called odd-colorable if there exists a map φ : [ n ] → [ r ] such that for any edge { j 1 , j 2 , … , j r } of G , we have φ ( j 1 ) + φ ( j 2 ) + ⋅ ⋯ ⋅ + φ ( j r ) ≡ r ∕ 2 ( mod r ) . In this paper, we first determine that, if r = 2 q ( 2 t + 1 ) and n ≥ 2 q ( 2 q − 1 ) r , then the maximum chromatic number in the class of the odd-colorable r -hypergraphs on n vertices is 2 q , which answers a question raised by V. Nikiforov recently in Nikiforov (2017). We also study some applications of the spectral symmetry of the odd-colorable r -hypergraphs given in the same paper by V. Nikiforov. We show that the Laplacian spectrum S p e c ( L ( G ) ) and the signless Laplacian spectrum S p e c ( Q ( G ) ) of an r -hypergraph G are equal if and only if r is even and G is odd-colorable. As an application of this result, we give an affirmative answer for the remaining unsolved case r ⁄ ≡ 0 ( m o d 4 ) of a question raised in Shao et al. (2015) about whether S p e c ( L ( G ) ) = S p e c ( Q ( G ) ) implies that L ( G ) and Q ( G ) have the same H-spectrum. [ABSTRACT FROM AUTHOR]

In this paper, we will investigate the effect of diffusion, which is ubiquitous in nature, on the immune system using a reaction-diffusion model in order to understand the dynamical behavior of complex patterns and control the dynamics of different patterns. Through control theory and linear stability analysis of local equilibrium, we obtain the optimal condition under which the system loses stability and a Turing pattern occurs. By combining mathematical analysis and numerical simulation, we show the possible patterns and how these patterns evolve. In addition, we establish a bridge between the complex patterns and the biological mechanism using the results from a previous study in Nature Cell Biology. The results in this paper can help us better understand the biological significance of the immune system. [ABSTRACT FROM AUTHOR]

In the paper James et al. (1995), the authors established a compact framework for general n × n system of chromatography (1.1) by using the kinetic formulation coupled with the compensated compactness method. However, how to construct suitable approximated solutions { u i l } of system (1.1) and then to prove the compactness of η ( u i l ) t + q ( u i l ) x in H l o c − 1 , for the entropy–entropy flux pairs ( η , q ) constructed by the kinetic formulation, with respect to the sequence { u i l } , is an open problem. In this paper, we construct the approximated solutions { u i ε } by using the parabolic viscosity method. By carefully calculating the Riemann invariants of system (1.1) , we obtained all necessary estimates in the compact framework of James et al. (1995), and gave a complete proof of the global existence of weak solutions for the Cauchy problem (1.1) with the bounded, nonnegative initial data (1.2) . As a direct by-product, when the total variation of the initial data is bounded, we obtained a simple proof of the existence of global weak solutions by applying the Div-Curl lemma in the compensated compactness theorem to some pairs of functions ( c , f ( u i ) ) , where c is a constant. [ABSTRACT FROM AUTHOR]

The notions of "mathematicalness" and "mathematicability" of nature appear in the context of attempts at explaining the effectiveness of mathematics in the description of the world. Mathematicalness of nature means that structures of the world are mathematical. But is this true? Is nature mathematical? In the paper some reasons for mathematicalness of nature are considered. Mathematical analysis is widely used in physics. Its application requires continuity of time and space. There are also different kinds of infinity in the mathematical theories used in physics. This raises the issue: whether the material world is continuous or we "impose" on nature certain properties in order to use convenient mathematical tools. Is mathematics a useful tool, or does it reflect nature? So, is nature mathematical or only mathematicable? The article shows that mathematicalness of nature is only a metaphysical hypothesis. [ABSTRACT FROM AUTHOR]

The paper presents some interesting aspects related to the biography and works of Romanian mathematician Caius Iacob (1912--1992). He was famous for his works in the fields of mathematical analysis, fluid mechanics, classical hydrodynamics and compressible-flow theory. At the age of 19, he graduated from the Mathematics Faculty in Bucharest, and then he went to Paris to continue his studies at the Faculty of Sciences, where he worked on a PhD thesis under the advice of famous French mathematician Henri Villat. On 24 June 1935, Caius Iacob successfully presented to the Sorbonne committee his PhD thesis about "Determination of conjugated harmonic functions with some limit conditions, and their applications in hydrodynamics". Returning to Romania, Caius Iacob had a long and successful career teaching mathematics and mechanics at the universities of Timişoara, Cluj and Bucharest. His most important work is considered the "Mathematical introduction to the mechanics of fluids". This book, providing original ways to work with classical hydrodynamics and compressibleflow theory, was published in Romanian in 1952 and in French in 1959. In 1955, he was elected a Corresponding Member of the Romanian Academy, becoming a titular Member in 1963. He was also President of the Mathematics Section of the Romanian Academy from 1980 until the end of his life, in 1992. In 1991, he initiated the foundation of the "Romanian Academy Institute of Applied Mathematics". In 2001 the institute merged with the "Centre for Mathematical Statistics", which had been created in 1964 by mathematician Gheorghe Mihoc, thus creating the "Gheorghe Mihoc -- Caius Iacob Institute of Mathematical Statistics and Applied Mathematics" of the Romanian Academy. [ABSTRACT FROM AUTHOR]

In this paper, we prove the generalized Hyers-Ulam stability for the following cubic functional equation with involution f(2x + y) + f(2x + σ(y)) - 2f(x + y) - 2f(x + σ(y)) - 12f(x) = 0 in modular spaces by using a fixed point theorem. [ABSTRACT FROM AUTHOR]

The global existence issue in critical spaces for compressible Navier–Stokes equations, was addressed by Danchin (Invent Math 141:579–614, 2000) and then developed by Charve and Danchin (Arch Rational Mech Anal 198:233–271, 2010), Chen et al. (Commun Pure Appl Math 63:1173–1224, 2010) and Haspot (Arch Rational Mech Anal 202:427–460, 2011) in more general Lp setting. The main aim of this paper is to exhibit (more precisely) time-decay estimates of solutions constructed in the critical regularity framework. To the best of our knowledge, the low-frequency assumption usually plays a key role in the large-time asymptotics of solutions, which was firstly observed by Matsumura and Nishida (J Math Kyoto Univ 20:67–104, 1980) in the L 1 (R d) space. We now claim a new low-frequency assumption for barotropic compressible Navier–Stokes equations, which may be of interest in the mathematical analysis of viscous fluids. Precisely, if the initial density and velocity additionally belong to some Besov space B ˙ 2 , ∞ - σ 1 (R d) with the regularity σ 1 ∈ (1 - d / 2 , 2 d / p - d / 2 ] , then a sharp time-weighted inequality including enough time-decay information can be available, where optimal decay exponents for the high frequencies are exhibited. The proof mainly depends on some non standard Besov product estimates. As a by-product, those optimal time-decay rates of Lq–Lr type are also captured in the critical framework. [ABSTRACT FROM AUTHOR]

This paper presents a new method to prove the nonuniform dichotomy spectrum theorem. Chu et al. [Bull. Sci. Math. 139 (2015), pp. 538-557] and Zhang [J. Funct. Anal. 267 (2014), pp. 1889-1916] generalized the dichotomy spectrum in Siegmund [J. Dynam. Differential Equations 14 (2002), pp. 243-258] to the nonuniform dichotomy spectrum and the authors in these works employed linear integral manifolds (stable and unstable) to establish the spectral theorem. They then used the spectrum theorem to study reducibility. We prove the nonuniform dichotomy spectrum by way of contradiction. In particular, we employ the nonuniform kinematically similarity (nonuniform reducibility) to reduce the shift system into two blocks and then we get a contradiction based on a technique in mathematical analysis. The method in the proof is completely different from previous works. [ABSTRACT FROM AUTHOR]

In this paper, for any smooth function on [0, 1]², we give an asymptotic representation of hyperbolic cross approximations of its Fourier series whose principal part is determined by the values of the function at vertexes of [0, 1]² and present a novel approach to estimates of the upper bounds of approximation errors. At the same time, we also give an asymptotic formula of partial sum approximations whose principal part is determined by not only partial derivatives at vertexes of [0, 1]², but also mean values on each side. Comparing asymptotic representations of these two kinds of approximation, we find that although in general the hyperbolic cross approximation is better than the partial sum ap- proximation, the partial sum approximation possibly work better under some cases, and we also give the corresponding necessary and sufficient condition to characterize these cases. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of analytic functions associated with Mittag-Leffler fuction in the open unit disk. Several properties of functions belonging to this class are derived. [ABSTRACT FROM AUTHOR]

In this paper, for 𝒞 a strict braided monoidal category with tensor product ⊗, we improve the definition of Hom-associative algebra by removing the multiplicativity condition of the automorphismα. After that we state the close connection between the classical notions of (co)algebra, (co)module and Hopf algebra and the corresponding ones in the Hom-world. As a consequence we can study the questions about the Hom-world by passing to the classical one, and the main contribution of this paper is to illustrate the suitable procedure to move from one side to the other. Finally, we apply our techniques to get in the Hom setting some classical results about cleft and Galois extensions. [ABSTRACT FROM PUBLISHER]

Creativity is a complex, multi-faceted concept encompassing a variety of related aspects, abilities, properties and behaviours. If we wish to study creativity scientifically, then a tractable and well-articulated model of creativity is required. Such a model would be of great value to researchers investigating the nature of creativity and in particular, those concerned with the evaluation of creative practice. This paper describes a unique approach to developing a suitable model of how creative behaviour emerges that is based on the words people use to describe the concept. Using techniques from the field of statistical natural language processing, we identify a collection of fourteen key components of creativity through an analysis of a corpus of academic papers on the topic. Words are identified which appear significantly often in connection with discussions of the concept. Using a measure of lexical similarity to help cluster these words, a number of distinct themes emerge, which collectively contribute to a comprehensive and multi-perspective model of creativity. The components provide an ontology of creativity: a set of building blocks which can be used to model creative practice in a variety of domains. The components have been employed in two case studies to evaluate the creativity of computational systems and have proven useful in articulating achievements of this work and directions for further research. [ABSTRACT FROM AUTHOR]

In this paper, we consider three types of functions given by the sums of products of poly-Genocchi and poly-Bernoulli functions and derive their Fourier series expansions. Moreover, we will express each of them in terms of Bernoulli functions. [ABSTRACT FROM AUTHOR]

In this paper, we, motivated by Mihet [2], give the concept of two nonlinear contractions ((φ, ε-λ)-contraction and (f, bn)-contraction) in KM-fuzzy metric spaces, and obtained some fixed point theorems. We answer the open question posed by Mihet in [2, open question 2]. Finally, an example can be used to be exemplify our results. [ABSTRACT FROM AUTHOR]

In this paper, we study differential equations arising from the generating functions of generalized degenerate tangent polynomials. We give explicit identities for the generalized degenerate tangent polynomials arising from differential equations. [ABSTRACT FROM AUTHOR]

The objective of this paper is to investigate and extend some Pachpatte type dynamic inequalities on time scales in three independent variables which provide explicit bounds on unknown functions and their derivatives. Some applications are also discussed here in order to illustrate the usefulness of our results. [ABSTRACT FROM AUTHOR]

In this paper, we would like to exploit umbral calculus in order to derive explicit expressions, some properties, recurrence relations and identities for poly-Genocchi polynomials. [ABSTRACT FROM AUTHOR]

In this paper we introduce a generalized form of the well known ToDD's difference equation and give the closed form expressions for this generalized form . In other words , we have the following nonlinear rational partial difference equation T ‹X1,X2,X3, …,Xn› = 1 + T ‹X1 - 1,X2 - 1, …,Xn - 1› + T ‹X1 - 2,X2 - 2, …,Xn - 2› T ‹X1 - 3,X2 - 3,X3 - 3, …,Xn - 3› where X1,X2, …:,Xn ∊ ℕ,and the initial values T ‹p1, p2, …:, pn› ,T ‹p2, p1, p3, p4, …, pn› ,T ‹p2, p3, p1, p4, …, pn›,… …,T ‹p2, p3, p4, …p1, pn›,T ‹p2 - 3, p3 - 3, p4 - 3, …pn - 3, p1› are real numbers with p1 ∊ {0,-1,-2} and p2, p3, …, pn ∊ ℕ such that T ‹p1, p2, …:, pn› ≠ 0 ,T ‹p2, p1, p3, p4, …, pn› ≠ 0 , T ‹p2, p3, p1, p4, …, pn› ≠ 0,…,T ‹p2 - 3, p3 - 3, p4 - 3, …pn - 3, p1› ≠ 0. We will use a novel technique to prove the results by using what we call 'piecewise n-dimensional mathematical induction' which we intro- duce here for the first time . We will obvious that this new concept represents generalized form for many types of mathematical induction . As a direct consequences , we investigate and drive the explicit solutions for the well known ordinary ToDD's difference Equation . [ABSTRACT FROM AUTHOR]

In this paper, we develop some fixed point theorems by using auxiliary functions for maps satisfying a rational type contractive condition in partially ordered S-metric spaces. Conditions for uniqueness of fixed point are also discussed. Our results generalize some existing results in the literature of S-metric spaces. [ABSTRACT FROM AUTHOR]

The Changhee polynomials and numbers are introduced in [3], and some interesting identities and properties of these polynomials are found by many researcher. In this paper, we consider the q-analogue of modified degen-erated Changhee polynomials and derive some new and interesting identities and properties of those polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of positive solutions of the fully third-order boundary value problem... is continuous. Some inequality conditions on f to guarantee the existence of positive solution are presented. These inequality conditions allow that ƒ(t; x; y; z) may be superlinear or sublinear growth on x, y and z as |(x; y; z)| → 0 and |(x; y; z)| → ∞. [ABSTRACT FROM AUTHOR]

Abstract Given a finite group G , we say that G has property P n if for every prime integer p , G has at most n − 1 irreducible characters whose degrees are multiples of p. In this paper, we classify all finite groups that have property P 4. We show that the groups satisfying property P 4 are exactly the finite groups with at most three nonlinear irreducible characters, one solvable group of order 168, SL 2 (3) , A 5 , S 5 , PSL 2 (7) and A 6. [ABSTRACT FROM AUTHOR]

Abstract Let g E H be the Eguchi–Hanson metric on the blow-up of ℂ 2 at the origin. In this paper we show that m g E H is not balanced for any positive integer m. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the representation theory of the Fuss–Catalan algebras, FC n (a , b). We prove that this algebra is cellular with a cellular basis and forms a tower of recollement, as defined by Cox, Martin, Parker, and Xi [7] , and hence, it is quasi-hereditary algebra if a , b are non-zero complex numbers. [ABSTRACT FROM AUTHOR]

Let T and S be two self-adjoint positive invertible operators of a complex Hilbert space H. In this paper, we investigate some inequalities involving the quadratic forms of the weighted arithmetic and harmonic means of T and S. Application for operator entropies is also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we use Schauder and Banach fixed point theorems to study the existence, uniqueness and stability of bounded nonhomogeneous iterative functional differential equations of the form x' (t) = λ1x(t) + λ2x[2](t)+...+λnx[n](t)+f(t). [ABSTRACT FROM AUTHOR]

In this paper we study the probability that the commutator of a randomly chosen pair of elements, one from a subring of a finite ring and other from the ring itself is equal to a given element of the ring. [ABSTRACT FROM AUTHOR]

In this paper, we study varieties and characterize convolution operators on the algebras related to the new structures of KPC-hypergroups, which are a generalization of the classical ones. [ABSTRACT FROM AUTHOR]

Necessary conditions for the existence of a super‐simple, decomposable, near‐resolvable (v,4,6)‐balanced incomplete block design (BIBD) whose 2‐component subdesigns are both near‐resolvable (v,4,3)‐BIBDs are v≡1 (mod 4) and v≥17. In this paper, we show that these necessary conditions are sufficient. Using these designs, we also establish that the necessary conditions for the existence of a super‐simple near‐resolvable (v,4,3)‐RBIBD, namely v≡1 (mod 4) and v≥9, are sufficient. A few new pairwise balanced designs are also given. [ABSTRACT FROM AUTHOR]

In this paper, we prove the Hyers-Ulam stability and the Hyers-Ulam-Rassias stability of the following two functional equations φ(x)=xφ((1-α)x+α)+(1-x)φ((1-β)x), x∊[0,1],0<α≤β<1, and φ(x)=xφ(f(x))+1(1-x)φ(g(x)),x∊[0,1] which is an open problem raised by Berinde and Khan [Berinde, V. and Khan, A. R., On a functional equation arising in mathematical biology and theory of learning, Creat. Math. Inform., 24 (2015), No. 1, 9-16] [ABSTRACT FROM AUTHOR]

The augmented Zagreb index (AZI) of a graph G is defined as AZI (G) =∑uv∊E(G) (dudv/du+dv-2)3 where E(G) and du denote set of edges of G and degree of the vertex u in G respectively. In this paper we establish some general results and bounds of AZI for certain unicyclic graphs and their corresponding chemical representation. We also obtain some results pertaining to AZI of certain trees. [ABSTRACT FROM AUTHOR]

In this paper, we study the further improvements of the reverse Young and Heinz inequalities for the wider range of v, namely v∈R. These modified inequalities are used to establish corresponding operator inequalities on a Hilbert space. [ABSTRACT FROM AUTHOR]

Algebraic topology studies topological spaces with the help of tools from abstract algebra. The main focus of this paper is to show that many concepts from algebraic topology can be conveniently expressed in terms of (normal) factor graphs. As an application, we give an alternative proof of a classical duality result of Kramers and Wannier, which expresses the partition function of the 2-D Ising model at a low temperature in terms of the partition function of the 2-D Ising model at a high temperature. Moreover, we discuss analogous results for the 3-D Ising model and the Potts model. [ABSTRACT FROM AUTHOR]

J. Jakubík noted in [JAKUBÍK, J.: Modular Lattice of Locally Finite Length, Acta Sci. Math. 37 (1975), 79–82] that F. Šik in the unpublished manuscript proved that in the class of upper semimodular lattices of locally finite length, modularity is equivalent to the lack of cover-preserving sublattices isomorphic to S7. In the present paper we extend the scope of Šik's theorem to the class of upper semimodular, upper continuous and strongly atomic lattices. Moreover, we show that corresponding result of Jakubík from [JAKUBÍK, J.: Modular Lattice of Locally Finite Length, Acta Sci. Math. 37 (1975), 79–82] cannot be strengthened is analogous way. [ABSTRACT FROM AUTHOR]

A chance constrained optimization problem over a finite distribution involves a set of scenario constraints from which a small subset can be violated. We consider the setting where all scenario constraints are mixed-integer convex. Existing works typically consider a mixed integer nonlinear programming (MINLP) formulation of this problem by introducing binary variables to indicate which constraint systems are to be satisfied or violated. A variety of cutting plane approaches for this MINLP formulation have been developed. In this paper we consider a family of cuts in the original space rather than those in the extended space of the MINLP reformulation. These cuts, known as quantile cuts, can be viewed as a projection of the well known family of mixing inequalities for the MINLP reformulation onto the original problem space. We show that the closure of the infinite family of all quantile cuts has a finite description. An important corollary of this result is that for linear chance constrained problems the quantile closure is polyhedral. We further show that a recursive application of quantile closure operations recovers the convex hull of the nonconvex chance constrained set in the limit, and in the pure integer setting the convergence is finite. We show that separation of quantile cuts is in general NP-hard, develop a heuristic separation method, and demonstrate its effectiveness through a computational study. We also study an approximation of the quantile closure and propose a generalization by grouping scenarios. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the family of quotients of internally quasicontinuous functions. Moreover, we study cardinal invariants related to quotients in the case of internally quasicontinuous functions and the complement of this family. [ABSTRACT FROM AUTHOR]

Self-reference has played a prominent role in the development of metamathematics in the past century, starting with Gödel's first incompleteness theorem. Given the nature of this and other results in the area, the informal understanding of self-reference in arithmetic has sufficed so far. Recently, however, it has been argued that for other related issues in metamathematics and philosophical logic a precise notion of self-reference and, more generally, reference is actually required. These notions have been so far elusive and are surrounded by an aura of scepticism that has kept most philosophers away. In this paper I suggest we shouldn't give up all hope. First, I introduce the reader to these issues. Second, I discuss the conditions a good notion of reference in arithmetic must satisfy. Accordingly, I then introduce adequate notions of reference for the language of first-order arithmetic, which I show to be fruitful for addressing the aforementioned issues in metamathematics. [ABSTRACT FROM AUTHOR]