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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]

It is proved that the set of those natural numbers which cannot be written as n-Ω( n) is of positive lower density. Here Ω( n) is the number of the prime power divisors of n. This is a refinement of a theorem of F. Luca. [ABSTRACT FROM AUTHOR]

There is no single canonical polynomial-time version of the Axiom of Choice (AC); several statements of AC that are equivalent in Zermelo-Fraenkel (ZF) set theory are already inequivalent from a constructive point of view, and are similarly inequivalent from a complexity-theoretic point of view. In this paper we show that many classical formulations of AC, when restricted to polynomial time in natural ways, are equivalent to standard complexity-theoretic hypotheses, including several that were of interest to Selman. This provides a unified view of these hypotheses, and we hope provides additional motivation for studying some of the lesser-known hypotheses that appear here. Additionally, because several classical forms of AC are formulated in terms of cardinals, we develop a theory of polynomial-time cardinality. Nerode & Remmel (Contemp. Math. 106, 1990 and Springer Lec. Notes Math. 1432, 1990) developed a related theory, but restricted to unary sets. Downey (Math. Reviews MR1071525) suggested that such a theory over larger alphabets could have interesting connections to more standard complexity questions, and we illustrate some of those connections here. The connections between AC, cardinality, and complexity questions also allow us to highlight some of Selman's work. We hope this paper is more of a beginning than an end, introducing new concepts and raising many new questions, ripe for further research. [ABSTRACT FROM AUTHOR]

Growing use of the internet in educational contexts has been prominent in recent years. In this survey paper, we describe how the internet is transforming the mathematics classroom and mathematics teacher education. We use as references several reviews of use of the internet in mathematics education settings made in recent years to determine how the field has evolved. We identify three domains in which new approaches are being generated by mathematic educators: principles of design of new settings; social interaction and construction knowledge; and tools and resources. The papers in this issue reflect different perspectives developed in the last decade in these three domains, providing evidence of the advances in theoretical frameworks and support in the generation of new meanings for old constructs such as 'tool', 'resources' or 'learning setting'. We firstly highlight the different ways in which the use of digital technologies generates new ways of thinking about mathematics and the settings in which it is learnt, and how mathematics teacher educators frame the new initiatives of initial training and professional development. In this survey paper, we identify trends for future research regarding theoretical and methodological aspects, and recognise new opportunities requiring further engagement. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the Cauchy problem of inhomogeneous convolution equations of a fairly general nature. To solve the problems posed here, we apply the operator method proposed in some earlier papers by the author. The solutions of the problems under consideration are found using an effective method in the form of well-convergent vector-valued power series. The proposed method ensures the continuity of the obtained solutions with respect to the initial data and the inhomogeneous term of the equation. [ABSTRACT FROM AUTHOR]

In this paper, I present what I call the symmetry conception of God within 1st order, extensional, non-well-founded set theory. The symmetry conception comes in two versions. According to the first, God is that unique being that is universally symmetrical with respect to set membership. According to the second, God is the universally symmetrical set of all sets that are universally symmetrical with respect to set membership. I present a number of theorems, most importantly that any universally symmetrical set is identical to its essence, that show that the two symmetry conceptions intersect with some dominant theological conceptions of God. The theorems also show that both of the symmetry conceptions of God entail that God has a fractal like structure. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new concept of sets and a new class of functions called E-univex sets and E-univex functions, respectively. For an E-differentiable function, the concept of E-univexity is introduced by generalizing several concepts of generalized convexity earlier defined into optimization theory. In addition, some properties of E-differentiable E-univex functions are investigated. Further, also concepts of E-differentiable generalized E-univexity are introduced. Then, the sufficiency of the so-called E-Karush–Kuhn–Tucker necessary optimality conditions are proved for an E-differentiable nonlinear optimization problem in which the involved functions are E-univex and/or generalized E-univex. [ABSTRACT FROM AUTHOR]

Zhao and Ho asked in a recent paper that for each T0 space X, whether KB(X) (the set of all irreducible closed sets of X whose suprema exist) is the canonical k-bounded sobrification of X in the sense of Keimel and Lawson. In this paper, we construct a counterexample to give a negative answer. We also consider the subcategory Topκ of the category Top0 of T0 spaces, and prove that the category KBSob of k-bounded sober spaces is a full reflective subcategory of the category KBSob of k-bounded sober spaces is a full reflective subcategory of the category Topκ. [ABSTRACT FROM AUTHOR]

The problem of ranking of elements from some finite set on the basis of nearest adjoining order method for pairwise comparisons is investigated in this paper. It is assumed that in the set under consideration there exists a weak preference relation, which is to be identified estimated) on the basis of pairwise comparisons in the form of difference of ranks. Moreover, the results of comparisons may be disturbed with random errors; the assumptions made about error distributions are not restrictive. The paper comprises: the problem formulation definitions, assumptions, and optimisation problem, which provides the NAO solution) and the theoretical background - the form of distributions of random variables which make it possible to determine the properties of NAO solution, in particular, evaluation of the probability, that the NAO solution is equivalent to the errorless one. The approach presented in the paper can be extended to the case of more than one comparison for each pair of elements, i.e., completely formalised multi-experts ranking procedure. [ABSTRACT FROM AUTHOR]

Research question: This paper examines the overarching question of to what extent different analytic choices may influence the inference about country-specific cross-sectional and trend estimates in international large-scale assessments. We take data from the assessment of PISA mathematics proficiency from the four rounds from 2003 to 2012 as a case study. Methods: In particular, four key methodological factors are considered as analytical choices in the rescaling and analysis of the data: (1) The selection of country sub-samples for item calibration differing at three factor levels. (2) The item sample refering to two sets of mathematics items used within PISA. (3) The estimation method used for item calibration: marginal maximum likelihood estimation method as implemented in R package TAM or an pairwise row averaging approach as implemented in the R package pairwise. (4) The type of linking method: concurrent calibration or separate calibration with successive chain linking. Findings: It turned out that analytical decisions for scaling did affect the PISA outcomes. The factors of choosing different calibration samples, estimation method and linking method tend to show only small effects on the country-specific cross-sectional and trend estimates. However, the selection of different link items seems to have a decisive influence on country ranking and development trends between and within countries. [ABSTRACT FROM AUTHOR]

In the Univalent Foundations of mathematics spatial notions like "point" and "path" are primitive, rather than derived, and all of mathematics is encoded in terms of them. A Homotopy Type Theory is any formal system which realizes this idea. In this paper I will focus on the question of whether a Homotopy Type Theory (as a formalism for the Univalent Foundations) can be justified intuitively as a theory of shapes in the same way that ZFC (as a formalism for set-theoretic foundations) can be justified intuitively as a theory of collections. I first clarify what such an "intuitive justification" should be by distinguishing between formal and pre-formal "meaning explanations" in the vein of Martin-Löf. I then go on to develop a pre-formal meaning explanation for HoTT in terms of primitive spatial notions like "shape", "path" etc. [ABSTRACT FROM AUTHOR]

The theorem proved in this paper establishes conditions under which a Banach space X is an Asplund space (i.e., its dual space is a space with the Radon-Nikodym (RN) property). The theorem is formulated in terms of the existence of a supersequentially compact set in ( B( X **), ω *), where B( X **) stands for the unit ball of the second dual of X and ω* for the weak topology on the ball. The example presented in the paper shows that one cannot get rid of some restrictive conditions in the theorem in general. [ABSTRACT FROM AUTHOR]

Given a convex disk K and a positive integer k, let $$\vartheta _T^k(K)$$ and $$\vartheta _L^k(K)$$ denote the k-fold translative covering density and the k-fold lattice covering density of K, respectively. Let T be a triangle. In a very recent paper, Sriamorn (, ) proved that $$\vartheta _L^k(T)=\frac{2k+1}{2}$$ . In this paper, we will show that $$\vartheta _T^k(T)=\vartheta _L^k(T)$$ . [ABSTRACT FROM AUTHOR]

Codes having higher information rates are desirable, since a higher rate code implies a more efficient use of redundancy than a lower rate code. However, when choosing a code for a particular application, we must also consider the error-correcting capabilities of the code. There is a basic trade-off between code rate and minimum distance. The smaller the code rate, the larger is the minimum distance and vice-versa. This paper proposes a simple coding scheme that can construct a code with higher information rate from an existing code. First, the paper derives a low-rate $$\mathcal {C}'(n', k', d')$$ -code from an existing $$\mathcal {C}(n, k, d)$$ -code, where $$\frac{k}{n} \ge \frac{k'}{n'}$$ . An associated decoding procedure for the newly derived class of low-rate codes is also described. Finally, the proposed coding scheme combines $$\mathcal {C}$$ and a set of $$\mathcal {C}'$$ s to obtain a $$\mathcal {C}''(n'', k'', d'')$$ -code with $$\frac{k''}{n''} \ge \frac{k}{n} \ge \frac{k'}{n'}$$ . Kronecker product is used as a basic tool in the coding procedure. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel approach to the construction of new finite algebras and describe the congruence lattices of these algebras. Given a finite algebra $${\langle B_0, \ldots \rangle}$$, let $${B_1,B_2, \ldots , B_K}$$ be sets that either intersect B or intersect each other at certain points. We construct an overalgebra $${\langle A, FA \rangle}$$, by which we mean an expansion of $${\langle B_0, \ldots \rangle}$$ with universe $${A = B_0 \cup B_1 \cup \ldots \cup B_K}$$, and a certain set F of unary operations that includes mappings e satisfying $${e^2_i = e_i}$$ and e( A) = B, for $${0 \leq i \leq K}$$. We explore two such constructions and prove results about the shape of the new congruence lattices Con $${\langle A, F_A \rangle}$$ that result. Thus, descriptions of some new classes of finitely representable lattices is one contribution of this paper. Another, perhaps more significant, contribution is the announcement of a novel approach to the discovery of new classes of representable lattices, the full potential of which we have only begun to explore. [ABSTRACT FROM AUTHOR]

In this paper, we study the unit-neighbourhood of the tree bisection and reconnection operation on unrooted binary phylogenetic trees. Specifically, we provide a recursive method to calculate the size of the unit-neighbourhood for any tree in the space $${\fancyscript{T}_n}$$ of unrooted binary phylogenetic trees with n-leaves. We also give both upper and lower bounds on this size for all trees in $${\fancyscript{T}_n}$$, and characterize those trees for which the stated upper bound is sharp. [ABSTRACT FROM AUTHOR]

In this paper a method to construct Kripke models for subtheories of constructive set theory is introduced that uses constructions from classical model theory such as constructible sets and generic extensions. Under the main construction all axioms except the collection axioms can be shown to hold in the constructed Kripke model. It is shown that by carefully choosing the classical models various instances of the collection axioms, such as exponentiation, can be forced to hold as well. The paper does not contain any deep results. It consists of first observations on the subject, and is meant to introduce some notions that could serve as a foundation for further research. [ABSTRACT FROM AUTHOR]

In this paper I consider some logical and mathematical aspects of the discussion of the identity and individuality of quantum entities. I shall point out that for some aspects of the discussion, the logical basis cannot be put aside; on the contrary, it leads us to unavoidable conclusions which may have consequences in how we articulate certain concepts related to quantum theory. Behind the discussion, there is a general argument which suggests the possibility of a metaphysics of non-individuals, based on a reasonable interpretation of quantum basic entities. I close the paper with a suggestion that consists in emphasizing that quanta should be referred to by the cardinalities of the collections to which they belong, for which an adequate mathematical framework seems to be possible. [ABSTRACT FROM AUTHOR]

In the recent paper [1] the claim is made that a probabilistic version of a common fixed point theorem of Pant holds. We provide some examples to demonstrate that this claim is false unless some additional conditions are imposed. Our note is desired to complete the interesting results in the quoted paper. [ABSTRACT FROM AUTHOR]

In this paper, we prove the Hyers-Ulam-Rassias stability of isometric homomorphisms in proper CQ*-algebras for the following Cauchy-Jensen additive mapping: The concept of Hyers-Ulam-Rassias stability originated from the Th.M. Rassias’ stability theorem that appeared in the paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300. This is applied to investigate isometric isomorphisms between proper CQ*-algebras. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate how the fixed point property and its negation behave when a covering relation is added to the order. We prove that every finite ordered set which is not totally ordered and which is dismantlable by retractables, respectively by irreducibles, has an upper cover (in its extension lattice) which is also dismantlable by retractables, respectively by irreducibles. We also provide examples of finite ordered sets having the fixed point property so that none of their upper covers has the fixed point property. [ABSTRACT FROM AUTHOR]

This paper pursues two goals. Its first goal is to clear up the "identity problem" faced by the structuralist interpretation of mathematics. Its second goal, through the consideration of examples coming in particular from the theory of permutations, is to examine cases of misunderstandings in mathematics fit to cast some light on mathematical understanding in general. The common thread shared by these two goals is the notion of setting. The study of a mathematical object almost always goes together with the choice of a particular setting, and the understanding of the workings of mathematical settings is an essential component of mathematical knowledge. It is claimed that the recognition of mathematical settings, as features distinct from both mathematical structures and the systems which instantiate those structures, allows one to classify most of understandable misunderstandings in mathematics, and also to solve the identity problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of computing the real dimension of a given semi-algebraic subset of R k, where R is a real closed field. We prove that the dimension k′ of a semi-algebraic set described by s polynomials of degree d in k variables can be computed in time . This result slightly improves the result by Vorobjov, who described an algorithm with complexity bound (sd)O(k′(k−k′)) for the same problem. The complexity bound of the algorithm described in this paper has a better dependence on the number s of polynomials in the input. Bibliography: 22 titles. [ABSTRACT FROM AUTHOR]

The splitting method is one of the most powerful and well-studied approaches to solving various NP-hard problems. The main idea of this method is to split the input instance of a problem into several simpler instances (further simplified by certain simplification rules) such that when the solution for each of them is found, one can construct the solution for the initial instance in polynomial time. There exists a huge number of papers describing algorithms of this type, and usually a considerable part of such a paper is devoted to case analysis. In this paper, we present a program that, given a set of simplification rules, automatically generates a proof of an upper bound on the running time of a splitting algorithm using these rules. As an example, we report the results of experiments with such a program for the SAT, MAXSAT, and (n, 3)- MAXSAT (the MAXSAT problem for the case where every variable in the formula appears at most three times) problems. Bibliography: 13 titles. [ABSTRACT FROM AUTHOR]

Quasi-set theory is a first-order theory without identity, which allows us to cope with non-individuals in a sense. A weaker equivalence relation called “indistinguishability” is an extension of identity in the sense that if x is identical to y then x and y are indistinguishable, although the reciprocal is not always valid. The interesting point is that quasi-set theory provides us with a useful mathematical background for dealing with collections of indistinguishable elementary quantum particles. In the present paper, however, we show that even in quasi-set theory it is possible to label objects that are considered as non-individuals. This is the first paper of a series that will be dedicated to the philosophical and physical implications of our main mathematical result presented here. [ABSTRACT FROM AUTHOR]

Ann-category is called almost strict if its natural equivalences, except a natural equivalence of commutativity and of the distributivity, are identities. The purpose of this paper is to prove that every Ann-category is Ann-equivalent to an almost strict Ann-category of the type (R, M) and to give new interpretations of the cohomology groups H3 (R, M) of the rings R. The present paper consists, in a certain sense, of an extension of our results in [4-6]. Reading the present paper requires certain knowledge of the main results, which were announced in [3]. For completeness, we briefly recall some of the material that will be indispensable for the understanding of this paper. Throughout this paper, for the tensorial product of two objects A and B, we write AB instead of A ⊗ B, but for the morphisms we still write f ⊗ g to avoid confusion with composition. [ABSTRACT FROM AUTHOR]

The present paper is a continuation of ref. 4, where the truncated two-point correlation function for a class of lattice spin systems was proved to have exponential decay at low temperature, under a weak coupling assumption. In this paper we compute the asymptotics of the correlation function as the temperature goes to zero. This paper thus extends ref. 3 in two directions: The Hamiltonian function is allowed to have several local minima other than a unique global minimum, and we do not require translation invariance of the Hamiltonian function. We are in particular able to handle spin systems on a general lattice. [ABSTRACT FROM AUTHOR]

Index sets are an integral and fundamental part of every mathematical modeling language. They assist the modeler in grouping various objects and entities. Index sets are also used extensively in the mathematical notation to write an expression in a concise way. An example is the sigma notation for formulating the summation of an unknown number n of terms. In this paper, the concept of index set is introduced in the context of modeling languages. The main objective is to propose an extension and generalization of the concept of index sets, which is the concept of hierarchical index sets. The paper concludes with an application, which clearly shows the usefulness of this concept. [ABSTRACT FROM AUTHOR]

The notion of coupled fixed point was introduced in 2006 by Bhaskar and Lakshmikantham. On the other hand, Filipovićet al. [M. Filipovićet al., 'Remarks on 'Cone metric spaces and fixed-point theorems of T-Kannan and T-Chatterjea contractive mappings',' Math. Comput. Modelling 54, 1467-1472 (2011)] proved several fixed and periodic point theorems for solid cones on cone metric spaces. In this paper we prove some coupled fixed-point theorems for certain T-contractions and study the existence of solutions of a system of nonlinear integral equations using the results of our work. The results of this paper extend and generalize well-known comparable results in the literature. [ABSTRACT FROM AUTHOR]

A generalized strong external difference family (briefly (v,m;k1,⋯,km;λ1,⋯,λm)-GSEDF) was introduced by Paterson and Stinson in 2016. In this paper, we give some nonexistence results for GSEDFs. In particular, we prove that a (v,3;k1,k2,k3;λ1,λ2,λ3)-GSEDF does not exist when k1+k2+k31, t>1 and v≡t≡1(mod2). Then we use it to obtain some new GSEDFs for m=2. In particular, for any prime power q with q≡1(mod4), we show that there exists a (qt,2;(q-1)2n-1,qt-12;(q-1)2n-2,(q-1)2n-2)-GSEDF, where t=p1p2⋯pn, pi>1, 1≤i≤n, p1,p2,⋯,pn are odd integers. [ABSTRACT FROM AUTHOR]

Given a bipartite graph G=(A∪B,E) with strict preference lists and given an edge e∗∈E, we ask if there exists a popular matching in G that contains e∗. We call this the popular edge problem. A matching M is popular if there is no matching M′ such that the vertices that prefer M′ to M outnumber those that prefer M to M′. It is known that every stable matching is popular; however G may have no stable matching with the edge e∗. In this paper we identify another natural subclass of popular matchings called “dominant matchings” and show that if there is a popular matching that contains the edge e∗, then there is either a stable matching that contains e∗ or a dominant matching that contains e∗. This allows us to design a linear time algorithm for identifying the set of popular edges. When preference lists are complete, we show an O(n3) algorithm to find a popular matching containing a given set of edges or report that none exists, where n=|A|+|B|. [ABSTRACT FROM AUTHOR]

In this paper, the authors find the best numbers α and β such that for all a, b>0 with a≠ b, where $ \bar{C}(a,b)=\frac{2(a^{2}+ab+b^{2})}{3(a+b)}$ and $T(a,b)=\frac {2}{\pi }{\int }_{0}^{{\pi }/{2}}\sqrt {a^{2}{\cos ^{2}{\theta }}+b^{2}{\sin ^{2}{\theta }}}\, \mathrm {d} \theta $ denote respectively the centroidal mean and Toader mean of two positive numbers a and b. [ABSTRACT FROM AUTHOR]

In this paper, the concept of dual normal relations on sets is introduced and generalized. Intrinsic characterizations of them are obtained. [ABSTRACT FROM AUTHOR]

In the model of Perfectly Secure Message Transmission Schemes (PSMTs), there are n channels between a sender and a receiver, and they share no key. An infinitely powerful adversary A can corrupt (observe and forge) the messages sent through some subset of n channels. For non-threshold adversaries called Q, Kumar et al. showed a many round PSMT (Ashwin Kumar et al. On perfectly secure communication over arbitrary networks. PODC 2002, pp. 193-202, ). In this paper, we show round efficient PSMTs against Q-adevrsaries. We first give a 3-round PSMT which runs in polynomial time in the size of the underlying linear secret sharing scheme. We next present a 2-round PSMT which is inefficient in general. (However, it is efficient for some special case.) [ABSTRACT FROM AUTHOR]

In this paper, we discuss the stability of the sets of efficient points of vector-valued optimization problems when the data of the approximate problems converges to the data of the original problem in the sense of Painlevé-Kuratowski. Our results improve the corresponding results obtained by Lucchetti and Miglierina (Optimization 53(5-6):517-528, , Section 3). [ABSTRACT FROM AUTHOR]

In this paper, the existence and multiplicity of a class of double resonant semilinear elliptic equations with the Dirichlet boundary value are studied. [ABSTRACT FROM AUTHOR]

This paper is concerned with the L- L estimates of the solutions for a class of pseudodifferential equations under some suitable degenerate assumptions. As applications, these estimates can be used to show that a generalized Schrödinger operator with some integrable potential generates a fractionally integrated group in L(ℝ). [ABSTRACT FROM AUTHOR]

In this paper all cocyclic Hadamard matrices of order less than 40 are classified. That is, all such Hadamard matrices are explicitly constructed, up to Hadamard equivalence. This represents a significant extension and completion of work by de Launey and Ito. The theory of cocyclic development is discussed, and an algorithm for determining whether a given Hadamard matrix is cocyclic is described. Since all Hadamard matrices of order at most 28 have been classified, this algorithm suffices to classify cocyclic Hadamard matrices of order at most 28. Not even the total numbers of Hadamard matrices of orders 32 and 36 are known. Thus we use a different method to construct all cocyclic Hadamard matrices at these orders. A result of de Launey, Flannery and Horadam on the relationship between cocyclic Hadamard matrices and relative difference sets is used in the classification of cocyclic Hadamard matrices of orders 32 and 36. This is achieved through a complete enumeration and construction of (4 t, 2, 4 t, 2 t)-relative difference sets in the groups of orders 64 and 72. [ABSTRACT FROM AUTHOR]

Decision making is the process of finding the best option among the feasible alternatives. In classical multiple-criteria decision-making (MCDM) methods, the ratings and the weights of the criteria are known precisely. However, if decision makers cannot reach an agreement on the method of defining linguistic variables based on the fuzzy sets, the interval-valued fuzzy set theory can provide a more accurate modeling. In this paper, the interval-valued fuzzy VIKOR method is presented, aiming at solving MCDM problems in which the weights of criteria are unequal, using interval-valued fuzzy set concepts. For application and verification, this study presents a numerical example and builds a practical maintenance strategy selection problem to verify our proposed method. Moreover, a comparison is made between the interval-valued fuzzy VIKOR and other adapted MCDM interval-valued fuzzy number-based. [ABSTRACT FROM AUTHOR]

Let G be a finite abelian group and A be a subset of G. We say that A is complete if every element of G can be represented as a sum of different elements of A. In this paper, we study the following question What is the structure of a large incomplete set? We show that such a set is essentially contained in a maximal subgroup. As a co-product, we obtain a new proof for several earlier results, including a new proof for Diderrich's conjecture in large groups. [ABSTRACT FROM AUTHOR]

In this paper, we prove that if two generalized incidence rings I( P1 ,R1) and I( P2 ,R2) are elementarily equivalent, then the corresponding ordered sets ( P1 ,R1) and ( P2 ,R2) are elementarily equivalent. [ABSTRACT FROM AUTHOR]

A subset U of vertices of a graph G is called a determining set if every automorphism of G is uniquely determined by its action on the vertices of U. A subset W is called a resolving set if every vertex in G is uniquely determined by its distances to the vertices of W. Determining (resolving) sets are said to have the exchange property in G if whenever S and R are minimal determining (resolving) sets for G and $${r\in R}$$ , then there exists $${s\in S}$$ so that $${S-\{s\} \cup \{r\}}$$ is a minimal determining (resolving) set. This work examines graph families in which these sets do, or do not, have the exchange property. This paper shows that neither determining sets nor resolving sets have the exchange property in all graphs, but that both have the exchange property in trees. It also gives an infinite graph family ( n-wheels where n ≥ 8) in which determining sets have the exchange property but resolving sets do not. Further, this paper provides necessary and sufficient conditions for determining sets to have the exchange property in an outerplanar graph. [ABSTRACT FROM AUTHOR]

The paper is concerned with the bad company problem as an instance of a more general difficulty in the philosophy of mathematics. The paper focuses on the prospects of stability as a necessary condition on acceptability. However, the conclusion of the paper is largely negative. As a solution to the bad company problem, stability would undermine the prospects of a neo-Fregean foundation for set theory, and, as a solution to the more general difficulty, it would impose an unreasonable constraint on mathematical practice. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the following implicit iteration scheme recently introduced by Xu and Ori [ Numer. Funct. Anal. Optim., 22, (2001) 767–773]: and to prove several strongly and weakly convergent theorems of the iteration for a finite family of pseudocontractive mappings under condition α n ∈ (0, b] ⊂ (0, 1). [ABSTRACT FROM AUTHOR]

Abstract  This paper discusses the problem of modeling on triangulated surfaces with geodesic curves. In the first part of the paper we define a new class of curves, called geodesic Bézier curves, that are suitable for modeling on manifold triangulations. As a natural generalization of Bézier curves, the new curves are as smooth as possible. In the second part we discuss the construction of C 0 and C 1 piecewise Bézier splines. We also describe how to perform editing operations, such as trimming, using these curves. Special care is taken to achieve interactive rates for modeling tasks. The third part is devoted to the definition and study of convex sets on triangulated surfaces. We derive the convex hull property of geodesic Bézier curves. [ABSTRACT FROM AUTHOR]

In this paper I discuss possible ways of measuring the power of arithmetical theories, and the possiblity of making an explication in Carnap’s sense of this concept. Chaitin formulates several suggestions how to construct measures, and these suggestions are reviewed together with some new and old critical arguments. I also briefly review a measure I have designed together with some shortcomings of this measure. The conclusion of the paper is that it is not possible to formulate an explication of the concept. [ABSTRACT FROM AUTHOR]

This paper is about diameters of compact sets. These are chords of maximal length. On one hand we see that the sum of the angles under which we see a diameter from two points of the set separated by the diameter is never smaller than 5 π/6. On the other hand we describe cases in which the diameter of a point-symmetric set must join symmetric points. [ABSTRACT FROM AUTHOR]

Abstract.  In this paper we consider the category Q-Mod of modules over a given quantale Q. The paper is motivated by constructions and results from the category of modules over a ring. We show that the category Q-Mod is monadic, consider its relation to the category Q-Top of Q-topological spaces and generalize a method of completion of partially ordered sets. [ABSTRACT FROM AUTHOR]

In [10], the notion of homogeneous perfect sets as a generalization of Cantor type sets is introduced and their Hausdorff and lower box-counting dimensions are studied. In this paper, we determine their exact packing and upper box-counting dimensions based on the length of their fundamental intervals and the gaps between them. Some known results concerning the dimensions of Cantor type sets are generalized. [ABSTRACT FROM AUTHOR]

In earlier papers, for “large” (but otherwise unspecified) subsets A, B of Z p and for h( x) ∈ Z p [ x], Gyarmati studied the solvability of the equations a + b = h( x), resp. ab = h( x) with a ∈ A, b ∈ B, x ∈ Z p , and for large subsets A, B, C, D of Z p Sárközy showed the solvability of the equations a + b = cd, resp. ab + 1 = cd with a ∈ A, b ∈ B, c ∈ C, d ∈ D. In this series of papers equations of this type will be studied in finite fields. In particular, in Part I of the series we will prove the necessary character sum estimates of independent interest some of which generalize earlier results. [ABSTRACT FROM AUTHOR]