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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 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]

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]

We give counterexamples for some statements of Borwein and sufficient conditions for the validity of these. [ABSTRACT FROM AUTHOR]

In this paper, we study the Onsager-Machlup function and its relationship to the Freidlin-Wentzell function for measures equivalent to arbitrary infinite dimensional Gaussian measures. The Onsager-Machlup function can serve as a density on infinite dimensional spaces, where a uniform measure does not exist, and has been seen as the Lagrangian for the "most likely element". The Freidlin-Wentzell rate function is the large deviations rate function for small-noise limits and has also been identified as a Lagrangian for the "most likely element". This leads to a conundrum - what is the relationship between these two functions? We show both pointwise and G-convergence (which is essentially the convergence of minimizers) of the Onsager-Machlup function under the small-noise limit to the Freidlin-Wentzell function - and give an expression for both. That is, we show that the small-noise limit of the most likely element is the most likely element in the small noise limit for infinite dimensional measures that are equivalent to a Gaussian. Examples of measures include the law of solutions to path-dependent stochastic differential equations and the law of an infinite system of random algebraic equations. [ABSTRACT FROM AUTHOR]

A comparison of the performance on the final exam for subpopulations of students in College Algebra was used to assess the effectiveness of online homework. Data was collected for two small groups of students, one with traditional paper homework and the other with online homework. The groups of students were further classified by incoming skill level measured by a departmental placement test. The data suggests that online homework was beneficial to the incoming higher-skilled group, whereas, the lower-skilled students did not benefit from online homework. [ABSTRACT FROM PUBLISHER]

The article presents abstracts related to mathematics including the equivalent of P to NP-intersect, the application of default logic variants in default logic, and basics of specification.

The present paper continues a cycle of papers, in which the Yoneda algebras were calculated for several families of algebras of dihedral and semidihedral type in the classification by K. Erdmann. Using the technique of a previous paper, a description of the Yoneda algebras for both families of local algebras occurring in this classification is given. Namely, a conjecture about the structure of the minimal free resolution of a (unique) simple module is stated, which is based on some empirical observations, and after establishing this conjecture, “cohomology information" is derived from the resolution discovered, and, as a result, this allows us to describe the Yoneda algebras of the algebras under consideration, It is noted that a similar technique was applied in computation of the Hochschild cohomology algebra for some finite-dimensional algebras. Bibliography: 23 titles. [ABSTRACT FROM AUTHOR]

In this paper, finite element methods for the problems of wave propagation in a fluid-saturated porous medium are discussed. The medium is composed of a porous elastic solid (soil, rock, etc.) saturated by a compressible viscous fluid (oil, water, etc.), and the fluid may flow relatively to the solid. Biot's lowfrequency dynamic equations are chosen to describe the problems mentioned above, with stress-given boundary conditions, ABGs (Absorbing Boundary Conditions) on artificial boundaries and conditions on interfaces between the fluid-saturated porous medium and elastic solids. In the paper, a new kind of discrete ABCs is presented, and a discrete-time Galerkin method are utilized for obtaining approximate solutions. The numerical results show that they both are effective. Two dilatational waves (fast wave P1 and slow wave P2) and one rotational wave (S wave) are clearly visible in the figures of computational results, which coincide with theoretical analysis very well. [ABSTRACT FROM AUTHOR]

The article discusses about developing expressive mathematical language in young children. Topics discussed include that teachers can help develop children's expressive mathematical language by giving them opportunities to speak and write as they use hands-on materials alongside, discusses types of mathematical writing including exploratory writing, argumentative writing and mathematically creative writing; and it also includes samples in which teachers notice the math in what a child is doing.

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]

The theory of nonnegative matrices is an example of a theory motivated in its origins and development by purely mathematical concerns that later proved to have a remarkably broad spectrum of applications to such diverse fields as probability theory, numerical analysis, economics, dynamical programming, and demography. At the heart of the theory is what is usually known as the Perron–Frobenius Theorem. It was inspired by a theorem of Oskar Perron on positive matrices, usually called Perron’s Theorem. This paper is primarily concerned with the origins of Perron’s Theorem in his masterful work on ordinary and generalized continued fractions (1907) and its role in inspiring the remarkable work of Frobenius on nonnegative matrices (1912) that produced, inter alia, the Perron–Frobenius Theorem. The paper is not at all intended exclusively for readers with expertise in the theory of nonnegative matrices. Anyone with a basic grounding in linear algebra should be able to read this article and come away with a good understanding of the Perron–Frobenius Theorem as well as its historical origins. The final section of the paper considers the first major application of the Perron–Frobenius Theorem, namely, to the theory of Markov chains. When he introduced the eponymous chains in 1908, Markov adumbrated several key notions and results of the Perron–Frobenius theory albeit within the much simpler context of stochastic matrices; but it was by means of Frobenius’ 1912 paper that the linear algebraic foundations of Markov’s theory for nonpositive stochastic matrices were first established by R. Von Mises and V.I. Romanovsky. [ABSTRACT FROM AUTHOR]

The paper reports some new solutions for heat transfer through extended surfaces or fins using the symbolic algebra package Maple 8, which is widely available. The four specific problems chosen for the present study are: (a) a rectangular convection fin with the heat transfer coefficient varying either linearly or exponentially with the distance from the base, (b) a truncated conical spine with convection at both ends, (c) a heat-generating annular fin with a constant base heat flux and an adiabatic tip, and (d) a convection fin array made of a rectangular fin and two triangular fins. Each problem is formulated in a manner that makes its solution novel and distinct from what is available in the literature. Solutions are provided in symbolic forms. Using the numerical and graphical capabilities of Maple, the results are presented in the form of numerical data as well as graphical displays. The paper demonstrates that Maple provides an effective and convenient tool for the analysis of extended surface heat transfer problems that otherwise demand tedious algebraic manipulations. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Cauchy problem for an elliptic equation in a plane domain. The problem is ill-posed. Using the method of quasi-reversibility, an approximation to the exact solution is given. Using Carleman's inequatily, we derive a sharp error estimate. [ABSTRACT FROM AUTHOR]

In this paper we introduce a membrane system (named λP systems) in which the computation is performed through certain operations on the tree structure of the membranes. The objects within the membranes play the role of catalysts for the operations. The result of the computation is the final configuration of the system. We show that λP systems can simulate pure λ-calculus and so they have universal computational power. We also show that NP-complete problems can be solved in polynomial time in this way by showing that 3SAT is solvable in linear time with linear input. [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]

A one-dimensional deformed Heisenberg algebra [X, P] = i f (P) is studied. We answer the question: for what function of deformation f (P) does there exist a nonzero minimal uncertainty in position (minimal length)? We also find an explicit expression for the minimal length in the case of an arbitrary function of deformation. [ABSTRACT FROM AUTHOR]

In this paper, some methods are given to construct dipolarizations in Kac–Moody algebras as well as their some subalgebras, and an illustrative example is provided. The dipolarizations found in this paper include symmetric and non-symmetric ones. [ABSTRACT FROM AUTHOR]

Some preconditioners for accelerating the classical iterative methods are given in Zhang et al. [Y. Zhang and T.Z. Huang, A class of optimal preconditioners and their applications, Proceedings of the Seventh International Conference on Matrix Theory and Its Applications in China, 2006. Y. Zhang, T.Z. Huang, and X.P. Liu, Modified iterative methods for nonnegative matrices and M-matrices linear systems, Comput. Math. Appl. 50 (2005), pp. 1587-1602. Y. Zhang, T.Z. Huang, X.P. Liu, A class of preconditioners based on the (I+S(α))-type preconditioning matrices for solving linear systems, Appl. Math. Comp. 189 (2007), pp. 1737-1748]. Another kind of preconditioners approximating the inverse of a symmetric positive definite matrix was given in Simons and Yao [G. Simons, Y. Yao, Approximating the inverse of a symmetric positive definite matrix, Linear Algebra Appl. 281 (1998), pp. 97-103]. Zhang et al. 's preconditioners and Simons and Yao's are generalized in this paper. These preconditioners are all of low construction cost, which all could be taken as approximate inverse of M-matrices. Numerical experiments of these preconditioners applied with Krylov subspace methods show the effectiveness and performance, which also show that the preconditioners proposed in this paper are better approximate inverse for M-matrices than Simons'. [ABSTRACT FROM AUTHOR]

A common thread in all of the recent results concerning the testing of dense graphs is the use of Szemerédi's regularity lemma. In this paper we show that in some sense this is not a coincidence. Our first result is that the property defined by having any given Szemerédi-partition is testable with a constant number of queries. Our second and main result is a purely combinatorial characterization of the graph properties that are testable with a constant number of queries. This characterization (roughly) says that a graph property P can be tested with a constant number of queries if and only if testing P can be reduced to testing the property of satisfying one of finitely many Szemerédi-partitions. This means that in some sense, testing for Szemerédi-partitions is as hard as testing any testable graph property. We thus resolve one of the main open problems in the area of property-testing, which was first raised by Goldreich, Goldwasser, and Ron [J. ACM, 45 (1998), pp. 653-750] in the paper that initiated the study of graph property-testing. This characterization also gives an intuitive explanation as to what makes a graph property testable. [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]

This paper deals with a parametric family of convex semi-infinite optimization problems for which linear perturbations of the objective function and continuous perturbations of the right-hand side of the constraint system are allowed. In this context, Cánovas et al. (SIAM J. Optim. 18:717–732, []) introduced a sufficient condition (called ENC in the present paper) for the strong Lipschitz stability of the optimal set mapping. Now, we show that ENC also entails high stability for the minimal subsets of indices involved in the KKT conditions, yielding a nice behavior not only for the optimal set mapping, but also for its inverse. Roughly speaking, points near optimal solutions are optimal for proximal parameters. In particular, this fact leads us to a remarkable simplification of a certain expression for the (metric) regularity modulus given in Cánovas et al. (J. Glob. Optim. 41:1–13, []) (and based on Ioffe (Usp. Mat. Nauk 55(3):103–162, []; Control Cybern. 32:543–554, [])), which provides a key step in further research oriented to find more computable expressions of this regularity modulus. [ABSTRACT FROM AUTHOR]

This paper is a continuation of the study begun in the previous paper with the same title. We analyze ER(2)16*+8(RP2n) and compute ER(2)*(RP16K+1), and use these to prove more nonimmersion theorems for RPn, including many in fairly low dimensions. In particular, we get 12 new nonimmersion results for RPn where n < 192, the range included in the tables Don Davis keeps. These complement the 10 already found in the first paper. [ABSTRACT FROM AUTHOR]

Abstract  This paper is a sequel to the studies on classification properties of involutive divisions reported in [1]. An example is given in which the minimal involutive basis of a particular monomial ideal for the “Janet antipode” n! orderings of variables. This example disproves the hypothesis that the minimal involutive basis for continuous and constructive divisions always coincides with the Janet basis for some ordering of variables. [ABSTRACT FROM AUTHOR]

This paper considers the Steiner Minimal Tree (SMT) problem in the rectilinear and octilinear planes. The study is motivated by the physical design of VLSI: The rectilinear case corresponds to the currently used M-architecture, which uses either horizontal or vertical routing, while the octilinear case corresponds to a new routing technique, X-architecture, that is based on the pervasive use of diagonal directions. The experimental studies show that the X-architecture demonstrates a length reduction of more than 10–20%. In this paper, we make a theoretical study on the lengths of SMTs in these two planes. Our mathematical analysis confirms that the length reduction is significant as the previous experimental studies claimed, but the reduction for three points is not as significant as for two points. We also obtain the lower and upper bounds on the expected lengths of SMTs in these two planes for arbitrary number of points. [ABSTRACT FROM AUTHOR]

In this paper, we provide theoretical analysis for a cubic regularization of Newton method as applied to unconstrained minimization problem. For this scheme, we prove general local convergence results. However, the main contribution of the paper is related to global worst-case complexity bounds for different problem classes including some nonconvex cases. It is shown that the search direction can be computed by standard linear algebra technique. [ABSTRACT FROM AUTHOR]

Let R be an arbitrary finite commutative local ring. In this paper, we obtain a necessary and sufficient condition for a function over R to be a polynomial function. Before this paper, necessary and sufficient conditions for a function to be a polynomial function over some special finite commutative local rings were obtained. [ABSTRACT FROM AUTHOR]

Assume that m ⩾ 2, p is a prime number, (m,p(p - 1)) = 1, -1 ∉ (p) ⊂ (∤/m∤)* and [(∤/m∤)* : (p)] = 4. In this paper, we calculate the value of Gauss sum G(X) = Σx∈F*qX(x)ζT(x)P over ...q, where q = pf, f = φ(m)/4, X is a multiplicative character of ...q and T is the trace map from ...q to ...p. Under our assumptions, G(X) belongs to the decomposition field K of p in ℚ(ζm) and K is an imaginary quartic abelian number field. When the Galois group Gal(K/ℚ) is cyclic, we have studied this cyclic case in another paper: "Gauss sums of index four: (1) cyclic case" (accepted by Acta Mathematica Sinica, 2003). In this paper we deal with the non-cyclic case. [ABSTRACT FROM AUTHOR]

Time delay due to maturation time, capturing time or other reasons widely exists in biological systems. In this paper, a predator-prey system of Leslie type with diffusion and time delay is studied based on mathematical analysis and numerical simulations. Conditions for both delay induced and diffusion induced Turing instability are obtained by using bifurcation theory. Furthermore, a series of numerical simulations are performed to illustrate the spatial patterns, which reveal the information of density changes of both prey and predator populations. The obtained results show that the interaction between diffusion and time delay may give rise to rich dynamics in ecosystems. [ABSTRACT FROM AUTHOR]

Fix , let be the loop Lie algebra over with basis and relations , where . In this paper, a formal distribution Lie algebra of is constructed. Then the associated conformal algebra is studied, where has a -basis with -brackets and . In particular, we determine the conformal derivations and rank one conformal modules of this conformal algebra. Finally, we study the central extensions and extensions of conformal modules. [ABSTRACT FROM AUTHOR]

Waveform design is a pivotal component of the fully adaptive radar construct. In this paper, we consider waveform design for radar space time adaptive processing (STAP), accounting for the waveform dependence of the clutter correlation matrix. Due to this dependence, in general, the joint problem of receiver filter optimization and radar waveform design becomes an intractable, nonconvex optimization problem, Nevertheless, it is, however, shown to be individually convex either in the filter or in the waveform variables. We derive constrained versions of a) the alternating minimization algorithm, b) proximal alternating minimization, and c) the constant modulus alternating minimization, which, at each step, iteratively optimizes either the STAP filter or the waveform independently. A fast and slow time model permits waveform design in radar STAP, but the primary bottleneck is the computational complexity of the algorithms. [ABSTRACT FROM PUBLISHER]

In previous work it was shown that the lower triangular generalized Hausdorff matrix Hαa with nonzero entries hnk= (n+α+1)-1, for a ≥ 0, is subnormal on Ι² if and only if α=0,1,2, …. For 0h : ..., is subnormal when .... In this paper we show that, when dj = Γ( j+1)Γ(h)/Γ( j+h), the square of the weights chosen above, then the corresponding operator Ch is bounded on Ι² for 0 < h < 3/2, that Hα is bounded on Ι² for all non-integer α < 0, and that Ch is closely related to Hh-1. This relationship leads to our main result that Ch is only subnormal when h = 1, when it corresponds to the original Cesaro operator with α = 0 and each dj = 1. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some new Lyapunov-type inequalities for a class of even-order linear differential equations, the results are new and generalize and improve some early results in this field. [ABSTRACT FROM AUTHOR]

This paper introduces a generalization of the frequently discussed problem of finding the optimum redundancy that maximizes the reliability of a network of components. Past work has restricted consideration to arrangements of redundant components called series-parallel networks. This paper allows a much broader class of arrangements called series-parallelseries networks. It is important to consider such arrangements for realistic situations in which components have more than one failure mode, or the combination of parallel paths introduces a failure probability. A dynamic programming algorithm is used to solve the more general problem for the case in which there are no constraints on the optimum solution. The algorithm is extended to handle multiple constraints using dominance and a variety of elimination methods to reduce the storage required in a compurer implementation of the algorithm. Problems with as many as 15 serial components and three constraints have been solved with reasonable digital computer computation times. [ABSTRACT FROM AUTHOR]

This paper gives general conditions for the convergence of a class of cutting-plane algorithms without requiring that the constraint sets for the subproblems be sequentially nested. Conditions are given under which inactive constraints may be dropped after each subproblem. Procedures for generating cutting-planes include those of KELLEY, CHENEY AND GOLDSTEIN, and a generalization of the one used by both ZOUTENDIJK and VEXNoTT. For algorithms with nested constraint sets, these conditions reduce to a special case of those of ZANGWILL for such problems and include as special cases the algorithms of Kelley, Cheney and Goldstein, and Veinott. Finally, the paper gives an arithmetic convergence rate. [ABSTRACT FROM AUTHOR]

The article is a comment on an article "Matrix Theory and Cost Allocation," by researchers T.H. Williams and C.H. Griffin, published in the July 1964 issue of the journal "The Accounting Review." According to the author, Williams and Griffin were the first to select this topic as an illustration of the application of matrix algebra to accounting problems. The model presented by Williams and Griffin is a matrix formulation of the popular simultaneous equations method of past years. It was pointed out that the aggregate cost of service departments after allocation in the Williams and Griffin model is more than the combined direct cost before allocation. The scope of this paper has two dimensions: first, to attempt to clarify the matrix algebra approaches that have already been posed and place them in perspective; second, to present a new matrix model of reciprocal service cost allocation and relate it to those previously presented. It would be assumed that each service department will have some of its cost allocated to some of the producing departments.

This paper presents new simple approaches for evaluating determinant and inverse of a matrix. The choice of pivot selection has been kept arbitrary thus they reduce the error while solving an ill conditioned system. Computation of determinant of a matrix has been made more efficient by saving unnecessary data storage and also by reducing the order of the matrix at each iteration, while dictionary notation [1] has been incorporated for computing the matrix inverse thereby saving unnecessary calculations. These algorithms are highly class room oriented, easy to use and implemented by students. By taking the advantage of flexibility in pivot selection, one may easily avoid development of the fractions by most. Unlike the matrix inversion method [2] and [3], the presented algorithms obviate the use of permutations and inverse permutations. [ABSTRACT FROM AUTHOR]

In this paper, a unique and more distinctive encryption algorithm is proposed. This is based on the complexity of highly nonlinear S box in Flesnelet domain. The nonlinear pattern is transformed further to enhance the confusion in the dummy data using Fresnelet technique. The security level of the encrypted image boosts using the algebra of Galois field in Fresnelet domain. At first level, the Fresnelet transform is used to propagate the given information with desired wavelength at specified distance. It decomposes given secret data into four complex subbands. These complex sub-bands are separated into two components of real subband data and imaginary subband data. At second level, the net subband data, produced at the first level, is deteriorated to non-linear diffused pattern using the unique S-box defined on the Galois field . In the diffusion process, the permuted image is substituted via dynamic algebraic S-box substitution. We prove through various analysis techniques that the proposed scheme enhances the cipher security level, extensively. [ABSTRACT FROM AUTHOR]

In this paper, we describe the BIBasis package designed for REDUCE and Macaulay2 computer algebra systems, which allows one to compute Boolean involutive bases and Gröbner bases. The implementations and user interfaces of the package for both systems are described in the respective sections of the paper. Also, we present results of comparisons of BIBasis with other packages and algorithms for constructing Boolean Gröbner bases available in the computer algebra systems. [ABSTRACT FROM AUTHOR]

The paper reports on a computer algebra program LSSS (Linear Selective Systems Solver) for solving linear algebraic systems with rational coefficients. The program is especially efficient for very large sparse systems that have a solution in which many variables take the value zero. The program is applied to the symmetry investigation of a non-abelian Laurent ODE introduced recently by M. Kontsevich. The computed symmetries confirmed that a Lax pair found for this system earlier generates all first integrals of degree at least up to 14. [ABSTRACT FROM AUTHOR]

The aim of this paper is to characterize the formal power series which have purely periodic β-expansions in Pisot or Salem unit base under some condition. Furthermore, we will prove that if β is a quadratic Pisot unit base, then every rational f in the unit disk has a purely periodic β-expansion and discuss their periods. [ABSTRACT FROM AUTHOR]

In a 1966 paper by C.A.B. Smith, the short and long conjunctive sums of games are defined and methods are described for determining the theoretical winner of a game constructed using one type of these sums. In this paper, we develop a method for determining the winner of a game constructed using arbitrary combinations of these sums. [ABSTRACT FROM AUTHOR]

This paper deals with good fuzzy preorders on fuzzy power structures. It is shown that a fuzzy preorder R on an algebra $${(X,\mathbb{F})}$$ is compatible if and only if it is Hoare good, if and only if it is Smyth good. [ABSTRACT FROM AUTHOR]

In this paper, it is proved that every variety of commutative unary algebras of a finite signature has a finite basis of identities. [ABSTRACT FROM AUTHOR]

In this paper, we study the structure of finite groups G under the assumption that certain abelian subgroups of largest possible exponent of prime power order lie in the 픉-hypercenter of G. We improve and extend some results of Ezzat, Shaalan, Elashiry, Asaad, Ramadan, and Buckley. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the multiple-objective optimization problems, considering an improved definition of generalized type I univex function. Some optimality conditions as well as some duality relations are established. [ABSTRACT FROM AUTHOR]