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2. On a paper of Hasse concerning the Eisenstein reciprocity law.
- Author
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Vostokov, S., Ivanov, M., and Pak, G.
- Subjects
- *
RECIPROCITY theorems , *MATHEMATICS , *ALGEBRA , *MATHEMATICAL analysis , *MATHEMATICAL combinations - Abstract
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]
- Published
- 2009
- Full Text
- View/download PDF
3. Some remarks on Hilbertian fields (An appendix to the paper “Galois averages” by R. Massy)
- Author
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Jensen, C.U. and Massy, Richard
- Subjects
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ALGEBRA , *NUMBER theory , *MATHEMATICAL analysis , *ARITHMETIC functions - Abstract
Abstract: The paper gives proofs of some results just claimed in [R. Massy, Galois averages, J. Number Theory 113 (2005) 244–275]. For instance, it is proved that for a finite non-trivial separable extension , , of Hilbertian fields finitely generated over their prime field, the quotient group , for the corresponding multiplicative groups of non-zero elements, cannot be a torsion group of finite exponent. [Copyright &y& Elsevier]
- Published
- 2006
- Full Text
- View/download PDF
4. Some remarks on a paper by L. Carlitz
- Author
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Dominici, Diego
- Subjects
- *
POLYNOMIALS , *ALGEBRA , *MATHEMATICS , *MATHEMATICAL analysis - Abstract
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]
- Published
- 2007
- Full Text
- View/download PDF
5. The Small-Noise Limit of the Most Likely Element is the Most Likely Element in the Small-Noise Limit.
- Author
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Selk, Zachary and Honnappa, Harsha
- Subjects
- *
STOCHASTIC differential equations , *GAUSSIAN distribution , *ALGEBRA , *MATHEMATICS , *MATHEMATICAL analysis - Abstract
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]
- Published
- 2024
- Full Text
- View/download PDF
6. A decidable theory involving addition of differentiable real functions.
- Author
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Buriola, Gabriele, Cantone, Domenico, Cincotti, Gianluca, Omodeo, Eugenio G., and Spartà, Gaetano T.
- Subjects
- *
DIFFERENTIABLE functions , *MATHEMATICAL analysis , *REAL numbers , *DERIVATIVES (Mathematics) , *ALGEBRA , *REAL variables - Abstract
This paper enriches a pre-existing decision algorithm, which in turn augmented a fragment of Tarski's elementary algebra with one-argument real functions endowed with a continuous first derivative. In its present (still quantifier-free) version, our decidable language embodies the addition of functions and multiplication of functions by scalars; the issue we address is the one of satisfiability. As regards real numbers, individual variables and constructs designating the basic arithmetic operations are available, along with comparison relators. As regards functions, we have variables of another sort, out of which compound terms are formed by means of constructs designating addition and differentiation. An array of predicates designates various relationships between functions, as well as function properties, that may hold over intervals of the real line; those are: function comparisons, strict and non-strict monotonicity / convexity / concavity, comparisons between the derivative of a function and a real-valued term. Our decision method consists in preprocessing the given formula into an equi-satisfiable quantifier-free formula of the elementary algebra of real numbers, whose satisfiability can then be checked by means of Tarski's decision method. No direct reference to functions will appear in the target formula, each function variable having been superseded by a collection of stub real variables; hence, in order to prove that the proposed translation is satisfiability-preserving, we must figure out a flexible-enough family of interpolating C 1 functions that can accommodate a model for the source formula whenever the target formula turns out to be satisfiable. With respect to the results announced in earlier papers of the same stream, a significant effort went into designing the family of interpolating functions so that it could meet the new constraints stemming from the presence of function addition (along with differentiation) among the constructs of our fragment of mathematical analysis. • A formal language RDF* concerned with differentiable real functions is proposed. • The class of differentiable functions treated is closed under addition. • The expressive power of RDF* is illustrated through a gallery of examples. • A satisfiability-preserving algorithm reducing RDF* formulas into an existential sentence of Tarskian algebra is presented. • The correctness of the reduction algorithm is reported. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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7. ROUGHNESS OF FILTERS IN EQUALITY ALGEBRAS.
- Author
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Rezaei, Gholam Reza, Borzooei, Rajab Ali, Kologhani, Mona Aaly, and Young Bae Jun
- Subjects
- *
ROUGH sets , *ALGEBRA , *MATHEMATICAL analysis , *STATISTICAL decision making - Abstract
Rough set theory is an excellent mathematical tool for the analysis of a vague description of actions in decision problems. Now, in this paper by considering the notion of an equality algebra, the notion of the lower and the upper approximations are introduced and some properties of them are given. Moreover, it is proved that the lower and the upper approximations define an interior operator and a closure operator, respectively. Also, using D-lower and D-upper approximation, conditions for a nonempty subset to be definable are provided and investigated that under which condition D-lower and D-upper approximation can be filter. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
8. Neural network architectures using min-plus algebra for solving certain high-dimensional optimal control problems and Hamilton–Jacobi PDEs.
- Author
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Darbon, Jérôme, Dower, Peter M., and Meng, Tingwei
- Subjects
- *
AUTOMATIC control systems , *ALGEBRA , *MATHEMATICAL analysis , *PARTIAL differential equations - Abstract
Solving high-dimensional optimal control problems and corresponding Hamilton–Jacobi PDEs are important but challenging problems in control engineering. In this paper, we propose two abstract neural network architectures which are, respectively, used to compute the value function and the optimal control for certain class of high-dimensional optimal control problems. We provide the mathematical analysis for the two abstract architectures. We also show several numerical results computed using the deep neural network implementations of these abstract architectures. A preliminary implementation of our proposed neural network architecture on FPGAs shows promising speedup compared to CPUs. This work paves the way to leverage efficient dedicated hardware designed for neural networks to solve high-dimensional optimal control problems and Hamilton–Jacobi PDEs. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
9. On quasi-monoidal comonads and their corepresentations.
- Author
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Wang, Dingguo and Zhang, Xiaohui
- Subjects
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ALGEBRA , *MATHEMATICAL equivalence , *GENERALIZATION , *MATHEMATICAL models , *MATHEMATICAL analysis - Abstract
In this paper, we define and study quasi-monoidal comonads on a monoidal category. It generalize the (Hom type) coquasi-bialgebras to a non-braided setting. We investigate their corepresentations and their coquasitriangular structures. We also discuss their gauge equivalence relations. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
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10. Translation and modulation invariant Banach spaces of tempered distributions satisfy the metric approximation property.
- Subjects
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BANACH spaces , *MATHEMATICAL analysis , *FUNCTION spaces , *ALGEBRA , *MATHEMATICAL convolutions - Abstract
In this paper, we establish the validity of the so-called Bounded Approximation Property (BAP) for a comprehensive class of translation and modulation invariant Banach spaces (B, ∥ · ∥B) of tempered distributions on the Euclidean space ℝ d . In fact, such spaces have a double module structure, over some Beurling algebra with respect to convolution, and with respect to pointwise multiplication over some Fourier Beurling algebra. Combining this double module structure with functional analytic arguments which describe the approximation of convolution operators by discrete convolutions we are able to verify the BAP, in fact, for most cases even the Metric Approximation Property (MAP) for such Banach space. The family of spaces under consideration is very rich and contains virtually all the classical function spaces relevant for mathematical analysis, as long as the Schwartz space (ℝ d) is dense in (B, ∥ · ∥B). In particular, all the reflexive spaces in this family are included. Moreover, this family of Banach spaces is closed with respect to intersections, sums and various interpolation methods. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
11. Comments on a paper “A Hermitian Morita theorem for algebras with anti-structure”
- Author
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Dasgupta, Bhanumati
- Subjects
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ALGEBRA , *MATHEMATICS , *MATHEMATICAL analysis , *ALGORITHMS - Abstract
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]
- Published
- 2007
- Full Text
- View/download PDF
12. Type 3 Multi Fuzzy Sets.
- Author
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Hemabala, K. and Kumar, B. Srinivasa
- Subjects
- *
FUZZY sets , *ALGEBRA , *INTERSECTION graph theory , *MATHEMATICAL formulas , *MATHEMATICAL analysis - Abstract
In this paper, we present the initiative of type 3 multi fuzzy sets which is an extension of type3 fuzzy sets. we have attempted to propose an extension of type2 multi fuzzy sets into a type3 multi fuzzy sets. In the wake of characterizing type 3 multi fuzzy sets, we examine the algebraic properties of these sets including set-theoretic operation such as complement, inclusion, union, intersection with example. Also we illustrate some algebraic properties. [ABSTRACT FROM AUTHOR]
- Published
- 2022
13. On $ n $-slice algebras and related algebras.
- Author
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Guo, Jin-Yun, Xiao, Cong, and Lu, Xiaojian
- Subjects
- *
ALGEBRA , *FINITE groups , *QUADRATIC equations , *ABELIAN groups , *MATHEMATICAL analysis - Abstract
The -slice algebra is introduced as a generalization of path algebra in higher dimensional representation theory. In this paper, we give a classification of -slice algebras via their -preprojective algebras and the trivial extensions of their quadratic duals. One can always relate tame -slice algebras to the McKay quiver of a finite subgroup of . In the case of , we describe the relations for the -slice algebras related to the McKay quiver of finite Abelian subgroups of and of the finite subgroups obtained from embedding into . [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
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14. On $\sigma$ -LCD Codes.
- Author
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Carlet, Claude, Mesnager, Sihem, Tang, Chunming, and Qi, Yanfeng
- Subjects
- *
EUCLIDEAN geometry , *LINEAR statistical models , *BINARY codes , *ERROR-correcting codes , *MATHEMATICAL analysis - Abstract
Linear complementary pairs (LCPs) of codes play an important role in armoring implementations against side-channel attacks and fault injection attacks. One of the most common ways to construct LCP of codes is to use Euclidean linear complementary dual (LCD) codes. In this paper, we first introduce the concept of linear codes with $\sigma $ complementary dual ($\sigma $ -LCD), which includes known Euclidean LCD codes, Hermitian LCD codes, and Galois LCD codes. Like Euclidean LCD codes, $\sigma $ -LCD codes can also be used to construct LCP of codes. We show that for $q > 2$ , all $q$ -ary linear codes are $\sigma $ -LCD, and for every binary linear code $\mathcal C$ , the code $\{0\}\times \mathcal C$ is $\sigma $ -LCD. Furthermore, we study deeply $\sigma $ -LCD generalized quasi-cyclic (GQC) codes. In particular, we provide the characterizations of $\sigma $ -LCD GQC codes, self-orthogonal GQC codes, and self-dual GQC codes, respectively. Moreover, we provide the constructions of asymptotically good $\sigma $ -LCD GQC codes. Finally, we focus on $\sigma $ -LCD abelian codes and prove that all abelian codes in a semi-simple group algebra are $\sigma $ -LCD. The results derived in this paper extend those on the classical LCD codes and show that $\sigma $ -LCD codes allow the construction of LCP of codes more easily and with more flexibility. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
15. Polynomial control on stability, inversion and powers of matrices on simple graphs.
- Author
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Shin, Chang Eon and Sun, Qiyu
- Subjects
- *
POLYNOMIALS , *MATRICES (Mathematics) , *NUMERICAL analysis , *MATHEMATICAL analysis , *ALGEBRA - Abstract
Abstract Spatially distributed networks of large size arise in a variety of science and engineering problems, such as wireless sensor networks and smart power grids. Most of their features can be described by properties of their state-space matrices whose entries have indices in the vertex set of a graph. In this paper, we introduce novel algebras of Beurling type that contain matrices on a connected simple graph having polynomial off-diagonal decay, and we show that they are Banach subalgebras of B (ℓ p) , 1 ≤ p ≤ ∞ , the space of all bounded operators on the space ℓ p of all p -summable sequences. The ℓ p -stability of state-space matrices is an essential hypothesis for the robustness of spatially distributed networks. In this paper, we establish the equivalence among ℓ p -stabilities of matrices in Beurling algebras for different exponents 1 ≤ p ≤ ∞ , with quantitative analysis for the lower stability bounds. Admission of norm-control inversion plays a crucial role in some engineering practice. In this paper, we prove that matrices in Beurling subalgebras of B (ℓ 2) have norm-controlled inversion and we find a norm-controlled polynomial with close to optimal degree. Polynomial estimate to powers of matrices is important for numerical implementation of spatially distributed networks. In this paper, we apply our results on norm-controlled inversion to obtain a polynomial estimate to powers of matrices in Beurling algebras. The polynomial estimate is a noncommutative extension about convolution powers of a complex function and is applicable to estimate the probability of hopping from one agent to another agent in a stationary Markov chain on a spatially distributed network. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
16. Reply to “Comments on the paper: On the properties of equidifferent OWA operator”
- Author
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Liu, Xinwang
- Subjects
- *
ALGEBRA , *MATHEMATICAL analysis , *MATHEMATICAL models of decision making , *ANALYSIS of variance - Abstract
Abstract: In reply to Péter Majlender, the connection between the (maximum spread) equidifferent OWA operator weights and the analytical method for the minimum variance OWA operator problem [R. Fullér, P. Majlender, On obtaining minimal variability OWA operator weights, Fuzzy Sets and Systems 136 (2003) 203–215] is pointed out and the differences between them are clarified. [Copyright &y& Elsevier]
- Published
- 2006
- Full Text
- View/download PDF
17. On Simple Iterative Fractional Order Differential Equations.
- Author
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Damag, Faten H. and Adem Kiliçman
- Subjects
- *
FRACTIONAL differential equations , *MATHEMATICAL analysis , *NUMERICAL analysis , *ALGEBRA - Abstract
In this paper, the simple fractional iterative differential equation will be the focus of study Dβv(s) = vn(s); v(s0) = a: where s0; v0 ∈ I = [0, b], and 0 < β < 1. One class of the iterative fractional differential equations is the simple iterative fractional differential equation. In this paper, the local existence and uniqueness results for the fractional order of degree n of simple iterative fractional differential equation are proven and the solutions found by using power series. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
18. On an Algorithmic Algebra over Simple-Named Complex-Valued Nominative Data.
- Author
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Ivanov, Ievgen, Korniłowicz, Artur, and Nikitchenko, Mykola
- Subjects
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ALGEBRA , *ALGORITHMS , *SEMANTICS , *MATHEMATICAL models , *MATHEMATICAL analysis - Abstract
This paper continues formalization in the Mizar system [2, 1] of basic notions of the composition-nominative approach to program semantics [14] which was started in [8, 12, 10]. The composition-nominative approach studies mathematical models of computer programs and data on various levels of abstraction and generality and provides tools for reasoning about their properties. In particular, data in computer systems are modeled as nominative data [15]. Besides formalization of semantics of programs, certain elements of the composition-nominative approach were applied to abstract systems in a mathematical systems theory [4, 6, 7, 5, 3]. In the paper we give a formal definition of the notions of a binominative function over given sets of names and values (i.e. a partial function which maps simple-named complex-valued nominative data to such data) and a nominative predicate (a partial predicate on simple-named complex-valued nominative data). The sets of such binominative functions and nominative predicates form the carrier of the generalized Glushkov algorithmic algebra for simple-named complex-valued nominative data [15]. This algebra can be used to formalize algorithms which operate on various data structures (such as multidimensional arrays, lists, etc.) and reason about their properties. In particular, we formalize the operations of this algebra which require a specification of a data domain and which include the existential quantifier, the assignment composition, the composition of superposition into a predicate, the composition of superposition into a binominative function, the name checking predicate. The details on formalization of nominative data and the operations of the algorithmic algebra over them are described in [11, 13, 9]. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
19. On Algebras of Algorithms and Specifications over Uninterpreted Data.
- Author
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Ivanov, Ievgen, Korniłowicz, Artur, and Nikitchenko, Mykola
- Subjects
- *
ALGEBRA , *ALGORITHMS , *SEMANTICS , *MATHEMATICAL models , *MATHEMATICAL analysis - Abstract
This paper continues formalization in Mizar [2, 1] of basic notions of the composition-nominative approach to program semantics [13] which was started in [8, 11]. The composition-nominative approach studies mathematical models of computer programs and data on various levels of abstraction and generality and provides tools for reasoning about their properties. Besides formalization of semantics of programs, certain elements of the composition-nominative approach were applied to abstract systems in a mathematical systems theory [4, 6, 7, 5, 3]. In the paper we introduce a definition of the notion of a binominative function over a set D understood as a partial function which maps elements of D to D. The sets of binominative functions and nominative predicates [11] over given sets form the carrier of the generalized Glushkov algorithmic algebra [14]. This algebra can be used to formalize algorithms which operate on various data structures (such as multidimensional arrays, lists, etc.) and reason about their properties. We formalize the operations of this algebra (also called compositions) which are valid over uninterpretated data and which include among others the sequential composition, the prediction composition, the branching composition, the monotone Floyd-Hoare composition, and the cycle composition. The details on formalization of nominative data and the operations of the algorithmic algebra over them are described in [10, 12, 9]. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
20. Number of solutions to kax + lby = cz.
- Author
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Deng, Naijuan, Yuan, Pingzhi, and Luo, Wenyu
- Subjects
- *
INTEGERS , *EQUATIONS , *ALGEBRA , *NUMERICAL analysis , *MATHEMATICAL analysis - Abstract
Text Let k , l , a , b , c be positive integers such that gcd ( k a , l b ) = 1 , min { a , b , c } > 1 , a ≠ 3 , b ≠ 3 and 2 ∤ c . In this paper, we prove that there are at most four solutions in positive integers ( x , y , z ) to the equation k a x + l b y = c z and at most two solutions when 2 ∤ ( u ( l / k ) ) , where u ( m ) is the least positive integer t with m t ≡ 1 ( mod c ) . Video For a video summary of this paper, please visit https://youtu.be/Dt3Y7TDMxlg . [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
21. Some variations of dual Euler–Rodrigues formula with an application to point–line geometry.
- Author
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Kahveci, Derya, Gök, İsmail, and Yaylı, Yusuf
- Subjects
- *
QUATERNIONS , *EUCLIDEAN geometry , *ALGEBRA , *MATHEMATICAL analysis , *DUAL space - Abstract
This paper examines the Euler–Rodrigues formula in dual 3-space D 3 by analyzing its variations such as vectorial form, exponential map, point–line theory and quaternions which have some intrinsic relations. Contrary to the Euclidean case, dual rotation in dual 3-space corresponds to a screw motion in Euclidean 3-space. This paper begins by explaining dual motion in terms of the given dual axis and angle. It will then go on to express dual Euler–Rodrigues formula with algebraic methods. Furthermore, an application of dual Euler–Rodrigues formula to point–line geometry is accomplished and point–line displacement operator is obtained by dual Euler–Rodrigues formula. Finally, dual Euler–Rodrigues formula is presented with the help of dual Euler–Rodrigues parameters that is expressed as a dual quaternion. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
22. Excision theory in the dihedral and reflexive (co)homology of algebras.
- Author
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Noreldeen, Alaa Hassan and Liu, Lishan
- Subjects
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HOMOLOGY theory , *COHOMOLOGY theory , *ALGEBRA , *MATHEMATICAL analysis , *MATHEMATICAL models - Abstract
In this paper, we study an excision theorem of the dihedral and reflexive (co)homology theory of associative algebras. That is, for such an extension, we obtain a six-term exact sequence in the dihedral cohomology. Also, we present and prove the relation between cyclic and dihedral cohomology of algebras and some examples. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
23. Hyper B -Ideals in Hyper B -Algebra.
- Author
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VICEDO, ANN LESLIE O. and VILELA, JOCELYN P.
- Subjects
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ALGEBRA , *HYPERGROUPS , *GROUP theory , *MATHEMATICAL analysis , *GENERALIZATION - Abstract
Hyperstructure theory has many applications to several areas of pure and applied sciences. This paper investigates some properties of hyper B -algebras which is a generalization of B -algebras. This paper also introduces the notion of hyper B -ideals, weak hyper B -ideals and strong hyper B -ideals in hyper B - algebras and gives some relations among these hyper B -ideals. Relations between hyper B -ideals and subhyper B -algebras of H is also discussed. Moreover, homomorphism on hyper B -algebra is defined and some related properties are given. Finally, proof of the relationship between the hyper B -algebras and hypergroups is provided. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
24. Imprimitive permutations in primitive groups.
- Author
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Araújo, J., Araújo, J.P., Cameron, P.J., Dobson, T., Hulpke, A., and Lopes, P.
- Subjects
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PERMUTATIONS , *GROUP theory , *ALGORITHMS , *ALGEBRA , *MATHEMATICAL analysis - Abstract
The goal of this paper is to study primitive groups that are contained in the union of maximal (in the symmetric group) imprimitive groups. The study of types of permutations that appear inside primitive groups goes back to the origins of the theory of permutation groups. However, this is another instance of a situation common in mathematics in which a very natural problem turns out to be extremely difficult. Fortunately, the enormous progresses of the last few decades seem to allow a new momentum on the attack to this problem. In this paper we prove that there are infinite families of primitive groups contained in the union of imprimitive groups and propose a new hierarchy for primitive groups based on that fact. In addition we introduce some algorithms to handle permutations, provide the corresponding GAP implementation, solve some open problems, and propose a large list of open problems. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
25. Contradiction-Tolerant Process Algebra with Propositional Signals.
- Author
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Bergstra, J. A. and Middelburg, C. A.
- Subjects
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PROPOSITION (Logic) , *COMPUTER multitasking , *MATHEMATICAL analysis , *ALGEBRA , *TIME-sharing computer systems - Abstract
In a previous paper, an ACP-style process algebra was proposed in which propositions are used as the visible part of the state of processes and as state conditions under which processes may proceed. This process algebra, called ACPps, is built on classical propositional logic. In this paper, we present a version of ACPps built on a paraconsistent propositional logic which is essentially the same as CLuNs. There are many systems that would have to deal with selfcontradictory states if no special measures were taken. For a number of these systems, it is conceivable that accepting self-contradictory states and dealing with them in a way based on a paraconsistent logic is an alternative to taking special measures. The presented version of ACPps can be suited for the description and analysis of systems that deal with self-contradictory states in a way based on the above-mentioned paraconsistent logic. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
26. UMBRAL CALCULUS APPROACH TO r-STIRLING NUMBERS OF THE SECOND KIND AND r-BELL POLYNOMIALS.
- Author
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TAEKYUN KIM, DAE SAN KIM, HYUCK-IN KWON, and JONGKYUM KWON
- Subjects
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CALCULUS , *POLYNOMIALS , *MATHEMATICAL analysis , *MATHEMATICAL functions , *ALGEBRA - Abstract
In this paper, we would like to use umbral calculus in order to derive some properties, recurrence relations and identities related to r-Stirling numbers of second kind and r-Bell polynomials. In particular, we will express the r-Bell polynomials as linear combinations of many well-known families of special polynomials. [ABSTRACT FROM AUTHOR]
- Published
- 2019
27. POLY-GENOCCHI POLYNOMIALS WITH UMBRAL CALCULUS VIEWPOINT.
- Author
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TAEKYUN KIM, DAE SAN KIM, GWAN-WOO JANG, and JONGKYUM KWON
- Subjects
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CALCULUS , *POLYNOMIALS , *ALGEBRA , *MATHEMATICS , *MATHEMATICAL analysis - Abstract
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]
- Published
- 2019
28. q-ANALOGUE OF MODIFIED DEGENERATE CHANGHEE POLYNOMIALS AND NUMBERS.
- Author
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KWON, JONGKYUM and PARK, JIN-WOO
- Subjects
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POLYNOMIALS , *MATHEMATICAL analysis , *MATHEMATICS , *ALGEBRA , *CONTINUOUS functions - Abstract
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]
- Published
- 2019
29. A GENERALIZATION OF SOME RESULTS FOR APPELL POLYNOMIALS TO SHEFFER POLYNOMIALS.
- Author
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TAEKYUN KIM, DAE SAN KIM, GWAN-WOO JANG, and LEE-CHAE JANG
- Subjects
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POLYNOMIALS , *CALCULUS , *ALGEBRA , *MATHEMATICAL functions , *MATHEMATICAL analysis - Abstract
Recently, Mihoubi and Taharbouchet gave some interesting method of obtain-ing certain identities for Appell polynomials of arbitrary orders starting from the given identities for Appell polynomials of fixed orders. In addition, they illustrated their method with several examples. The purpose of this paper is to note that their method can be gen-eralized so as to include any Sheffer polynomials. Also, we will provide many examples that illustrate our results. [ABSTRACT FROM AUTHOR]
- Published
- 2019
30. On Diagonalization of Matrices in an Arbitrary Field.
- Author
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Shmatkov, V. D.
- Subjects
- *
MATRICES (Mathematics) , *MATHEMATICAL functions , *ARBITRARY constants , *MATHEMATICAL analysis , *ALGEBRA - Abstract
This paper presents a simple way for diagonalization of matrices in an arbitrary field, with which one can calculate functions of matrices. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
31. Derivations of Leavitt path algebras.
- Author
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Lopatkin, Viktor
- Subjects
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ALGEBRA , *TOEPLITZ matrices , *CIRCULANT matrices , *MATRICES (Mathematics) , *MATHEMATICAL analysis - Abstract
Abstract In this paper, we describe the K -module H H 1 (L K (Γ)) of outer derivations of the Leavitt path algebra L K (Γ) of a row-finite graph Γ with coefficients in an associative commutative ring K with unit. We explicitly describe a set of generators of H H 1 (L K (Γ)) and relations among them. We also describe a Lie algebra structure of outer derivation algebra of the Toeplitz algebra. We prove that every derivation of a Leavitt path algebra can be extended to a derivation of the corresponding C ⁎ -algebra. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
32. A generalization of strongly monomial groups.
- Author
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Bakshi, Gurmeet K. and Kaur, Gurleen
- Subjects
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Q-groups , *ALGEBRA , *SUBGROUP growth , *GROUP theory , *MATHEMATICAL analysis - Abstract
Abstract Olivieri, del Río and Simón defined strongly monomial groups and a significant result proved by them is the explicit description of the simple components of the rational group algebra Q G of a strongly monomial group G. In this paper, generalized strongly monomial groups are defined, which is a generalization of strongly monomial groups. Beside strongly monomial groups, an extensive list of important families of groups which are generalized strongly monomial is produced. A description of the simple components of Q G , when G is a generalized strongly monomial group, is also provided. Furthermore, the work by Jespers, Olteanu, del Río and Van Gelder on the construction of a subgroup of finite index in the group of central units of integral group ring of a strongly monomial group has also been generalized. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
33. On representations of Fuss–Catalan algebras.
- Author
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Hussein, Ahmed B.
- Subjects
- *
ALGEBRA , *MATHEMATICS , *COMPLEX numbers , *NUMBER theory , *MATHEMATICAL analysis - Abstract
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]
- Published
- 2019
- Full Text
- View/download PDF
34. Kac determinant and singular vector of the level N representation of Ding-Iohara-Miki algebra.
- Author
-
Ohkubo, Yusuke
- Subjects
- *
MATHEMATICAL singularities , *DEFORMATIONS (Mechanics) , *ALGEBRA , *VECTOR algebra , *MATHEMATICAL analysis - Abstract
In this paper, we obtain the formula for the Kac determinant of the algebra arising from the level N representation of the Ding-Iohara-Miki algebra. It is also discovered that its singular vectors correspond to generalized Macdonald functions (the q-deformed version of the AFLT basis). [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
35. Triangulated Structures Induced by Triangle Functors.
- Author
-
Zhao, Zhibing, Du, Xianneng, and Bao, Yanhong
- Subjects
- *
TRIANGULATED categories , *FUNCTOR theory , *MATHEMATICAL analysis , *ALGEBRA , *MATHEMATICS theorems - Abstract
Given a triangle functor F: A→B, the authors introduce the half image hImF, which is an additive category closely related to F. If F is full or faithful, then hImF admits a natural triangulated structure. However, in general, one can not expect that hImF has a natural triangulated structure. The aim of this paper is to prove that hImF admits a natural triangulated structure if and only if F satisfies the condition (SM). If this is the case, hImF is triangle-equivalent to the Verdier quotient A/KerF. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
36. Generalized Power UP-Algebras.
- Author
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Akarachai Satirad, Phakawat Mosrijai, and Aiyared Iampan
- Subjects
- *
PROOF theory , *ALGEBRA , *MATHEMATICAL analysis - Abstract
The power UP-algebras of types 1 and 2 were proved by Iampan [1]. In this paper, we prove the generalized power UP-algebras of types 1 and 2, and find its cardinality. [ABSTRACT FROM AUTHOR]
- Published
- 2019
37. Ground states of two-component attractive Bose–Einstein condensates I: Existence and uniqueness.
- Author
-
Guo, Yujin, Li, Shuai, Wei, Juncheng, and Zeng, Xiaoyu
- Subjects
- *
EQUATIONS , *NUMERICAL analysis , *ALGEBRA , *MATRICES (Mathematics) , *MATHEMATICAL analysis - Abstract
Abstract We study ground states of two-component Bose–Einstein condensates (BEC) with trapping potentials in R 2 , where the intraspecies interaction (− a 1 , − a 2) and the interspecies interaction − β are both attractive, i. e , a 1 , a 2 and β are all positive. The existence and non-existence of ground states are classified completely by investigating equivalently the associated L 2 -critical constraint variational problem. The uniqueness and symmetry-breaking of ground states are also analyzed under different types of trapping potentials as β ↗ β ⁎ = a ⁎ + (a ⁎ − a 1) (a ⁎ − a 2) , where 0 < a i < a ⁎ : = ‖ w ‖ 2 2 (i = 1 , 2) is fixed and w is the unique positive solution of Δ w − w + w 3 = 0 in R 2. The semi-trivial limit behavior of ground states is tackled in the companion paper [12]. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
38. Convolution operators on measure algebras of KPC-hypergroups.
- Author
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Székelyhidi, László, Tabatabaie, Seyyed Mohammad, and Sadathoseyni, Bentol Hoda
- Subjects
- *
HYPERGROUPS , *GROUP theory , *MATHEMATICAL analysis , *ALGEBRA , *MATHEMATICS - Abstract
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]
- Published
- 2019
- Full Text
- View/download PDF
39. ON BOREL MAPS, CALIBRATED s-IDEALS, AND HOMOGENEITY.
- Author
-
POL, R. and ZAKRZEWSKI, P.
- Subjects
- *
BOREL subgroups , *MATHEMATICAL analysis , *LATTICE theory , *MATHEMATICS theorems , *ALGEBRA - Abstract
Let μ be a Borel measure on a compactum X. The main objects in this paper are s-ideals Ipdimq, J0pμq, Jf pμq of Borel sets in X that can be covered by countably many compacta which are finite-dimensional, or of μ-measure null, or of finite μ-measure, respectively. Answering a question of J. Zapletal, we shall show that for the Hilbert cube, the s-ideal Ipdimq is not homogeneous in a strong way. We shall also show that in some natural instances of measures μ with nonhomogeneous s-ideals J0pμq or Jf pμq, the completions of the quotient Boolean algebras BorelpXq{J0pμq or BorelpXq{Jf pμq may be homogeneous. We discuss the topic in a more general setting, involving calibrated s-ideals. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
40. Maximizing monotone submodular functions over the integer lattice.
- Author
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Soma, Tasuku and Yoshida, Yuichi
- Subjects
- *
MATHEMATICAL functions , *ALGORITHMS , *VECTORS (Calculus) , *ALGEBRA , *MATHEMATICAL analysis - Abstract
The problem of maximizing non-negative monotone submodular functions under a certain constraint has been intensively studied in the last decade. In this paper, we address the problem for functions defined over the integer lattice. Suppose that a non-negative monotone submodular function f:Z+n→R+ is given via an evaluation oracle. Assume further that f satisfies the diminishing return property, which is not an immediate consequence of submodularity when the domain is the integer lattice. Given this, we design polynomial-time (1-1/e-ϵ)-approximation algorithms for a cardinality constraint, a polymatroid constraint, and a knapsack constraint. For a cardinality constraint, we also provide a (1-1/e-ϵ)-approximation algorithm with slightly worse time complexity that does not rely on the diminishing return property. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
41. Polyhedral approximation in mixed-integer convex optimization.
- Author
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Lubin, Miles, Yamangil, Emre, Bent, Russell, and Vielma, Juan Pablo
- Subjects
- *
ALGORITHMS , *MATHEMATICAL optimization , *ALGEBRA , *MATHEMATICAL analysis , *MACHINE learning - Abstract
Generalizing both mixed-integer linear optimization and convex optimization, mixed-integer convex optimization possesses broad modeling power but has seen relatively few advances in general-purpose solvers in recent years. In this paper, we intend to provide a broadly accessible introduction to our recent work in developing algorithms and software for this problem class. Our approach is based on constructing polyhedral outer approximations of the convex constraints, resulting in a global solution by solving a finite number of mixed-integer linear and continuous convex subproblems. The key advance we present is to strengthen the polyhedral approximations by constructing them in a higher-dimensional space. In order to automate this extended formulation we rely on the algebraic modeling technique of disciplined convex programming (DCP), and for generality and ease of implementation we use conic representations of the convex constraints. Although our framework requires a manual translation of existing models into DCP form, after performing this transformation on the MINLPLIB2 benchmark library we were able to solve a number of unsolved instances and on many other instances achieve superior performance compared with state-of-the-art solvers like Bonmin, SCIP, and Artelys Knitro. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
42. ON THE ANDERSON-BADAWI ωR[x](I[X]) = ωR(I) CONJECTURE.
- Author
-
NASEHPOUR, PEYMAN
- Subjects
- *
MATHEMATICAL analysis , *ALGEBRA , *GAUSSIAN beams , *GAUSSIAN distribution , *TORSION - Abstract
Let R be a commutative ring with an identity different from zero and n be a positive integer. Anderson and Badawi, in their paper on n-absorbing ideals, define a proper ideal I of a commutative ring R to be an n-absorbing ideal of R, if whenever x1 . . . xn+1 ∊ I for x1 , . . . , xn+1 ∊ ωR , then there are n of the Xi's whose product is in I and conjecture that ωR[x](I[X]) = ωR(I) for any ideal I of an arbitrary ring R, where ωR(I) = min{n: I is an n-absorbing ideal of R}. In th e present paper, we use content formula techniques to prove th a that their conjecture is true, if one of the following conditions hold: (1) The ring R is a Prüfer domain. (2) The ring R is a Gaussian ring such th a that its additive group is torsion-free. (3) The additive group of the ring R is torsion-free and I is a radical ideal of R. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
43. Differential graded categories and Deligne conjecture.
- Author
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Shoikhet, Boris
- Subjects
- *
MATHEMATICAL analysis , *ABELIAN categories , *CATEGORIES (Mathematics) , *ALGEBRA , *LOGICAL prediction - Abstract
We prove a version of the Deligne conjecture for n -fold monoidal abelian categories A over a field k of characteristic 0, assuming some compatibility and non-degeneracy conditions for A . The output of our construction is a weak Leinster ( n , 1 ) -algebra over k , a relaxed version of the concept of Leinster n -algebra in A lg ( k ) . The difference between the Leinster original definition and our relaxed one is apparent when n > 1 , for n = 1 both concepts coincide. We believe that there exists a functor from weak Leinster ( n , 1 ) -algebras over k to C • ( E n + 1 , k ) -algebras, well-defined when k = Q , and preserving weak equivalences. For the case n = 1 such a functor is constructed in [31] by elementary simplicial methods, providing (together with this paper) a complete solution for 1-monoidal abelian categories. Our approach to Deligne conjecture is divided into two parts. The first part, completed in the present paper, provides a construction of a weak Leinster ( n , 1 ) -algebra over k , out of an n -fold monoidal k -linear abelian category (provided the compatibility and non-degeneracy condition are fulfilled). The second part (still open for n > 1 ) is a passage from weak Leinster ( n , 1 ) -algebras to C • ( E n + 1 , k ) -algebras. As an application, we prove in Theorem 8.1 that the Gerstenhaber–Schack complex of a Hopf algebra over a field k of characteristic 0 admits a structure of a weak Leinster ( 2 , 1 ) -algebra over k extending the Yoneda structure. It relies on our earlier construction [30] of a 2-fold monoidal structure on the abelian category of tetramodules over a bialgebra. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
44. Linear matrix inequalities for globally monotonic tracking control.
- Author
-
Garone, Emanuele and Ntogramatzidis, Lorenzo
- Subjects
- *
MATRIX inequalities , *MATHEMATICAL analysis , *ALGEBRA , *LINEAR differential equations , *ELECTRONIC controllers - Abstract
This paper addresses the problem of achieving monotonic tracking control for any initial condition (also referred to as global monotonic tracking control ). This property is shown to be equivalent to global non-overshooting as well as to global non-undershooting (i.e., non-overshooting and non-undershooting for any initial condition, respectively). The main objective of this paper is to prove that a stable system is globally monotonic if and only if all the rows of the output matrix are left eigenvectors of the space transition matrix. This property allows one to formulate the design of a controller which ensures global monotonic tracking as a convex optimization problem described by a set of Linear Matrix Inequalities (LMIs). [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
45. Isomorphism Classes of 10-Dimensional Filiform Leibniz Algebras.
- Author
-
Kasim, Suzila Mohd., Rakhimov, Isamiddin S., and Said Husain, Sharifah Kartini
- Subjects
- *
ISOMORPHISM (Mathematics) , *SET theory , *DIMENSIONAL analysis , *ALGEBRA , *MATHEMATICAL analysis , *INVARIANTS (Mathematics) - Abstract
This paper implements Rakhimov-Bekbaev approach to present a complete list of isomorphism classes of a subclass of complex filiform Leibniz algebras obtained from naturally graded non-Lie filiform Leibniz algebra. This class is split into two subclasses. In this paper we shall consider the second class which is denoted by SLbn in dimension n. The isomorphism criteria in terms of invariant functions for SLb10 are presented. We represent SLb10as a union of subsets and show that some of these subsets are represented as union of infinitely many orbits (a set of isomorphic to each other algebras) while others are represented as just a single orbit. In former case we give invariant functions to distinguish the orbits, while for the latter case the representatives of the single orbits are provided. As a result, we give the list of isomorphism classes with the table of multiplications. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
46. Undirected replicas of directional binary algebras.
- Author
-
Matczak, K. and Smith, J. D. H.
- Subjects
- *
BINARY number system , *ALGEBRA , *MONOIDS , *SEMIGROUPS (Algebra) , *MULTIPLICATION , *MATHEMATICAL analysis - Abstract
Following the prototype of dimonoids, directional algebras are obtained from universal algebras by splitting each fundamental operation into a number of distinct fundamental operations corresponding to directions or selected arguments in the original fundamental operation. Thus dimonoids are directional semigroups, with left- and right-directed multiplications. Directional quasigroups appear in a number of versions, depending on the axiomatization chosen for quasigroups, but this paper concentrates on 4-diquasigroups, which incorporate a left and right quasigroup structure. While introducing several new instances of 4-diquasigroups, including dicores and group-representable diquasigroups, the paper is primarily devoted to the study of undirected replicas of directional binary algebras, dimonoids, digroups, and diquasigroups, where the two directed multiplications are identified. Undirected replicas of diquasigroups are two-sided quasigroups, and thus offer a new approach to the construction of quasigroups of various kinds. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
47. A C algebra of pseudodifferential operators on the half line.
- Author
-
Arsenović, Miloš
- Subjects
- *
PSEUDODIFFERENTIAL operators , *OPERATOR theory , *FREDHOLM operators , *COMMUTATORS (Operator theory) , *ALGEBRA , *MATHEMATICAL analysis , *NUMERICAL analysis - Abstract
In this paper we employ a C-algebra approach to the study of Fredholm properties of differential and pseudodifferential operators on the half line. The algebra investigated in this paper has compact commutators, so the Gelfand theory applies to the quotient algebra, and we obtain an explicit description of the corresponding maximal ideal space and necessary and sufficient conditions for Fredholmness of operators in the algebra. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
48. Relative weak injectivity of operator system pairs.
- Author
-
Bhattacharya, Angshuman
- Subjects
- *
INJECTIVE functions , *OPERATOR theory , *ALGEBRA , *MATHEMATICAL analysis , *GROUP theory , *NUMERICAL analysis - Abstract
The concept of a relatively weakly injective pair of operator systems is introduced and studied in this paper, motivated by relative weak injectivity in the C*-algebra category. E. Kirchberg [11] proved that the C\*-algebra C\*(F∞) of the free group F∞ on countably many generators characterises relative weak injectivity for pairs of C\*-algebras by means of the maximal tensor product. One of the main results of this paper shows that C\*(F∞) also characterises relative weak injectivity in the operator system category. A key tool is the theory of operator system tensor products [9,10]. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
49. On the Center of Kumjian-Pask Algebras Associated to Finitely Aligned k-Graph.
- Author
-
Gozali, Sumanang Muhtar, Rosjanuardi, Rizky, and Yusnitha, Isnie
- Subjects
- *
ALGEBRA , *GRAPH theory , *FINITE element method , *MATHEMATICAL analysis , *MATHEMATICAL models - Abstract
Pino, Huef, Clark, and Raeburn introduced the Kumjian-Pask algebra associated to row finite k-graph Λ without sources. Clark and Pangalela generalized the notion of Kumjian-Pask algebra to finitely aligned k-graph. In this paper, we survey the center of Kumjian-Pask algebras for the case finitely aligned k-graph. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
50. On Genocchi Operational Matrix of Fractional Integration for Solving Fractional Differential Equations.
- Author
-
Abdulnasir Isah and Chang Phang
- Subjects
- *
FRACTIONAL integrals , *MATHEMATICS , *POLYNOMIALS , *MATHEMATICAL analysis , *NUMERICAL analysis , *EQUATIONS , *ALGEBRA - Abstract
In this paper we present a new numerical method for solving fractional differential equations (FDEs) based on Genocchi polynomials operational matrix through collocation method. The operational matrix of fractional integration in Riemann-Liouville sense is derived. The upper bound for the error of the operational matrix of fractional integration is also shown. The properties of Genocchi polynomials are utilized to reduce the given problems to a system of algebraic equations. Illustrative examples are finally given to show the simplicity, accuracy and applicability of the method. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
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