615 results
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2. A Note on a Paper of Aivazidis, Safonova and Skiba
- Author
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M. M. Al-Shomrani, Adolfo Ballester-Bolinches, and A. A. Heliel
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Subnormal subgroup ,Combinatorics ,Mathematics::Group Theory ,Finite group ,General Mathematics ,Mathematics - Abstract
The main result of this paper states that if $${\mathcal {F}}$$ is a subgroup-closed saturated formation of full characteristic, then the $${\mathcal {F}}$$ -residual of a K- $${\mathcal {F}}$$ -subnormal subgroup S of a finite group G is a large subgroup of G provided that the $${\mathcal {F}}$$ -hypercentre of every subgroup X of G containing S is contained in the $${\mathcal {F}}$$ -residual of X. This extends a recent result of Aivazidis, Safonova and Skiba.
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- 2021
3. Hill representations for ∗-linear matrix maps
- Author
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A. van der Merwe and S. ter Horst
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Combinatorics ,Linear map ,Matrix (mathematics) ,General Mathematics ,Nonnegative matrix ,Linear matrix ,Hermitian matrix ,Mathematics - Abstract
In the paper (Hill, 1973) from 1973 R.D. Hill studied linear matrix maps L : ℂ q × q → ℂ n × n which map Hermitian matrices to Hermitian matrices, or equivalently, preserve adjoints, i.e., L ( V ∗ ) = L ( V ) ∗ , via representations of the form L ( V ) = ∑ k , l = 1 m H k l A l V A k ∗ , V ∈ ℂ q × q , for matrices A 1 , … , A m ∈ ℂ n × q and continued his study of such representations in later work, sometimes with co-authors, to completely positive matrix maps and associated matrix reorderings. In this paper we expand the study of such representations, referred to as Hill representations here, in various directions. In particular, we describe which matrices A 1 , … , A m can appear in Hill representations (provided the number m is minimal) and determine the associated Hill matrix H = H k l explicitly. Also, we describe how different Hill representations of L (again with m minimal) are related and investigate further the implication of ∗ -linearity on the linear map L .
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- 2022
4. Post-quantum Simpson's type inequalities for coordinated convex functions
- Author
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Xuexiao You, Saowaluck Chasreechai, Muhammad Ali, Ghulam Murtaza, Thanin Sitthiwirattham, and Sotiris K. Ntouyas
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Inequality ,General Mathematics ,media_common.quotation_subject ,simpson's inequalities ,co-ordinated convexity ,Combinatorics ,(p ,QA1-939 ,Convex function ,Quantum ,post quantum calculus ,q)-integrals ,Mathematics ,media_common - Abstract
In this paper, we prove some new Simpson's type inequalities for partial $ (p, q) $-differentiable convex functions of two variables in the context of $ (p, q) $-calculus. We also show that the findings in this paper are generalizations of comparable findings in the literature.
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- 2022
5. Analyzing the Weyl Construction for Dynamical Cartan Subalgebras
- Author
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Elizabeth Gillaspy, Anna Duwenig, and Rachael Norton
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General Mathematics ,01 natural sciences ,Section (fiber bundle) ,Combinatorics ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,46L05, 22D25, 22A22 ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Twist ,Operator Algebras (math.OA) ,Mathematics::Representation Theory ,Quotient ,Mathematics ,Science & Technology ,Mathematics::Operator Algebras ,010102 general mathematics ,Spectrum (functional analysis) ,Mathematics - Operator Algebras ,Cartan subalgebra ,C-ASTERISK-ALGEBRAS ,Physical Sciences ,010307 mathematical physics ,EQUIVALENCE - Abstract
When the reduced twisted $C^*$-algebra $C^*_r(\mathcal{G}, c)$ of a non-principal groupoid $\mathcal{G}$ admits a Cartan subalgebra, Renault's work on Cartan subalgebras implies the existence of another groupoid description of $C^*_r(\mathcal{G}, c)$. In an earlier paper, joint with Reznikoff and Wright, we identified situations where such a Cartan subalgebra arises from a subgroupoid $\mathcal{S}$ of $\mathcal{G}$. In this paper, we study the relationship between the original groupoids $\mathcal{S}, \mathcal{G}$ and the Weyl groupoid and twist associated to the Cartan pair. We first identify the spectrum $\mathfrak{B}$ of the Cartan subalgebra $C^*_r(\mathcal{S}, c)$. We then show that the quotient groupoid $\mathcal{G}/\mathcal{S}$ acts on $\mathfrak{B}$, and that the corresponding action groupoid is exactly the Weyl groupoid of the Cartan pair. Lastly we show that, if the quotient map $\mathcal{G}\to\mathcal{G}/\mathcal{S}$ admits a continuous section, then the Weyl twist is also given by an explicit continuous $2$-cocycle on $\mathcal{G}/\mathcal{S} \ltimes \mathfrak{B}$., 32 pages
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- 2022
6. Real subset sums and posets with an involution
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Cinzia Bisi, Tommaso Gentile, and Giampiero Chiaselotti
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Computer Science::Information Retrieval ,General Mathematics ,Carry (arithmetic) ,Astrophysics::Instrumentation and Methods for Astrophysics ,Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ,Context (language use) ,Combinatorics ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Computer Science::General Literature ,Order (group theory) ,Involution (philosophy) ,Partially ordered set ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
In this paper, we carry out in an abstract order context some real subset combinatorial problems. Specifically, let [Formula: see text] be a finite poset, where [Formula: see text] is an order-reversing and involutive map such that [Formula: see text] for each [Formula: see text]. Let [Formula: see text] be the Boolean lattice with two elements and [Formula: see text] the family of all the order-preserving 2-valued maps [Formula: see text] such that [Formula: see text] if [Formula: see text] for all [Formula: see text]. In this paper, we build a family [Formula: see text] of particular subsets of [Formula: see text], that we call [Formula: see text]-bases on [Formula: see text], and we determine a bijection between the family [Formula: see text] and the family [Formula: see text]. In such a bijection, a [Formula: see text]-basis [Formula: see text] on [Formula: see text] corresponds to a map [Formula: see text] whose restriction of [Formula: see text] to [Formula: see text] is the smallest 2-valued partial map on [Formula: see text] which has [Formula: see text] as its unique extension in [Formula: see text]. Next we show how each [Formula: see text]-basis on [Formula: see text] becomes, in a particular context, a sub-system of a larger system of linear inequalities, whose compatibility implies the compatibility of the whole system.
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- 2021
7. Metric properties of Cayley graphs of alternating groups
- Author
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M.S. Olshevskyi
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Combinatorics ,Cayley graph ,General Mathematics ,Metric (mathematics) ,Mathematics - Abstract
A well known diameter search problem for finite groups with respect to its systems of generators is considered. The problem can be formulated as follows: find the diameter of a group over its system of generators. The diameter of a group over a specific system of generators is the diameter of the corresponding Cayley graph. It is considered alternating groups with classic irreducible system of generators consisting of cycles with length three of the form $(1,2,k)$. The main part of the paper concentrates on analysis how even permutations decompose with respect to this system of generators. The rules for moving generators from permutation's decomposition from left to right and from right to left are introduced. These rules give rise for transformations of decompositions, that do not increase their lengths. They are applied for removing fixed points of a permutation, that were included in its decomposition. Based on this rule the stability of system of generators is proved. The strict growing property of the system of generators is also proved, as the corollary of transformation rules and the stability property. It is considered homogeneous theory, that was introduced in the previous author's paper. For the series of alternating groups with systems of generators mentioned above it is shown that this series is uniform and homogeneous. It makes possible to apply the homogeneous down search algorithm to compute the diameter. This algorithm is applied and exact values of diameters for alternating groups of degree up to 43 are computed.
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- 2021
8. The Optimal Graph Whose Least Eigenvalue is Minimal among All Graphs via 1-2 Adjacency Matrix
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Nudrat Aamir, Lubna Gul, Gohar Ali, and Usama Waheed
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Combinatorics ,Article Subject ,General Mathematics ,QA1-939 ,Adjacency matrix ,Mathematics ,Graph ,Eigenvalues and eigenvectors - Abstract
All graphs under consideration are finite, simple, connected, and undirected. Adjacency matrix of a graph G is 0,1 matrix A = a i j = 0 , i f v i = v j o r d v i , v j ≥ 2 1 , i f d v i , v j = 1. . Here in this paper, we discussed new type of adjacency matrix known by 1-2 adjacency matrix defined as A 1,2 G = a i j = 0 , i f v i = v j o r d v i , v j ≥ 3 1 , i f d v i , v j = 2 , from eigenvalues of the graph, we mean eigenvalues of the 1-2 adjacency matrix. Let T n c be the set of the complement of trees of order n . In this paper, we characterized a unique graph whose least eigenvalue is minimal among all the graphs in T n c .
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- 2021
9. Covering by homothets and illuminating convex bodies
- Author
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Alexey Glazyrin
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Conjecture ,Applied Mathematics ,General Mathematics ,Discrete geometry ,Boundary (topology) ,Metric Geometry (math.MG) ,Upper and lower bounds ,Infimum and supremum ,Homothetic transformation ,Combinatorics ,Mathematics - Metric Geometry ,Hausdorff dimension ,FOS: Mathematics ,Mathematics::Metric Geometry ,Convex body ,Mathematics - Abstract
The paper is devoted to coverings by translative homothets and illuminations of convex bodies. For a given positive number $\alpha$ and a convex body $B$, $g_{\alpha}(B)$ is the infimum of $\alpha$-powers of finitely many homothety coefficients less than 1 such that there is a covering of $B$ by translative homothets with these coefficients. $h_{\alpha}(B)$ is the minimal number of directions such that the boundary of $B$ can be illuminated by this number of directions except for a subset whose Hausdorff dimension is less than $\alpha$. In this paper, we prove that $g_{\alpha}(B)\leq h_{\alpha}(B)$, find upper and lower bounds for both numbers, and discuss several general conjectures. In particular, we show that $h_{\alpha} (B) > 2^{d-\alpha}$ for almost all $\alpha$ and $d$ when $B$ is the $d$-dimensional cube, thus disproving the conjecture from Research Problems in Discrete Geometry by Brass, Moser, and Pach.
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- 2021
10. Additive subgroups generated by noncommutative polynomials
- Author
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Tsiu-Kwen Lee
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Combinatorics ,Ring (mathematics) ,Polynomial ,General Mathematics ,Unital ,Image (category theory) ,Structure (category theory) ,Ideal (ring theory) ,Algebra over a field ,Noncommutative geometry ,Mathematics - Abstract
Let R be an algebra. Given a noncommutative polynomial f, let f(R) stand for the additive subgroup of R generated by the image of f. For a unital or an affine algebra R, $$S_k(R)$$ is completely determined for any standard polynomial $$S_k$$ when R is generated by $$S_k(R)$$ as an ideal. Motivated by Bresar’s paper [Adv. Math. 374 (2020), 107346, 21 pp] and Robert’s paper [J. Oper. Theory 75 (2016), 387–408], under certain conditions we also prove that f(R) is equal to either [R, R] or the whole ring R. We obtain these results by studying the structure of Lie ideals L of a ring R whenever R is generated by [R, L] as an ideal.
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- 2021
11. Navier-Stokes equations under slip boundary conditions: Lower bounds to the minimal amplitude of possible time-discontinuities of solutions with two components in L∞(L3)
- Author
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Hugo Beirão da Veiga and Jiaqi Yang
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Combinatorics ,Amplitude ,General Mathematics ,Boundary (topology) ,Slip (materials science) ,Boundary value problem ,Classification of discontinuities ,Navier–Stokes equations ,Omega ,Mathematics ,Bar (unit) - Abstract
The main purpose of this paper is to extend the result obtained by Beirao da Veiga (2000) from the whole-space case to slip boundary cases. Denote by u two components of the velocity u. To fix ideas set ū = (u1,u2, 0) (the half-space) or $${\boldsymbol{\bar u}} = {\hat u_1}{\hat e_1} + {\hat u_2}{\hat e_2}$$ (the general boundary case (see (7.1))). We show that there exists a constant K, which enjoys very simple and significant expressions such that if at some time τ ∈ (0,T) one has $$\lim {\sup _{t \to \tau - 0}}\left\| {{\boldsymbol{\bar u}}(t)} \right\|_{{L^3}(\Omega )}^3 < \left\| {{\boldsymbol{\bar u}}(\tau )} \right\|_{{L^3}(\Omega )}^3 + K$$ , then u is continuous at τ with values in L3(Ω). Roughly speaking, the above norm-discontinuity of merely two components of the velocity cannot occur for steps’ amplitudes smaller than K. In particular, if the above condition holds at each τ ∈ (0,T), the solution is smooth in (0,T) × Ω. Note that here there is no limitation on the width of the norms $$\left\| {{\boldsymbol{\bar u}}(t)} \right\|_{{L^3}(\Omega )}^3$$ . So K is independent of these quantities. Many other related results are discussed and compared among them. This is a second main aim of this paper. New results are proved in Sections 5–7.
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- 2021
12. Gaussian Asymptotics of Jack Measures on Partitions From Weighted Enumeration of Ribbon Paths
- Author
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Alexander Moll
- Subjects
Spectral theory ,Generalization ,General Mathematics ,Gaussian ,Probability (math.PR) ,Mathematical proof ,Combinatorics ,symbols.namesake ,Mathematics::Quantum Algebra ,Ribbon ,FOS: Mathematics ,symbols ,Enumeration ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,Limit (mathematics) ,Mathematics::Representation Theory ,Cumulant ,Mathematics - Probability ,Mathematics - Abstract
In this paper we determine two asymptotic results for Jack measures on partitions, a model defined by two specializations of Jack polynomials proposed by Borodin-Olshanski in [European J. Combin. 26.6 (2005): 795-834]. Assuming these two specializations are the same, we derive limit shapes and Gaussian fluctuations for the anisotropic profiles of these random partitions in three asymptotic regimes associated to diverging, fixed, and vanishing values of the Jack parameter. To do so, we introduce a generalization of Motzkin paths we call "ribbon paths", show for general Jack measures that certain joint cumulants are weighted sums of connected ribbon paths on $n$ sites with $n-1+g$ pairings, and derive our two results from the contributions of $(n,g)=(1,0)$ and $(2,0)$, respectively. Our analysis makes use of Nazarov-Sklyanin's spectral theory for Jack polynomials. As a consequence, we give new proofs of several results for Schur measures, Plancherel measures, and Jack-Plancherel measures. In addition, we relate our weighted sums of ribbon paths to the weighted sums of ribbon graphs of maps on non-oriented real surfaces recently introduced by Chapuy-Dol\k{e}ga., Comment: Several results in this paper first appeared in the author's unpublished monograph arXiv:1508.03063. Version 2: revised and accepted for publication in International Mathematics Research Notices (IMRN)
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- 2021
13. Approximation of functions of H$$\ddot{o}$$lder class and solution of ODE and PDE by extended Haar wavelet operational matrix
- Author
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Priya Kumari and Shyam Lal
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Combinatorics ,Approximation theory ,Wavelet ,Partial differential equation ,Exact solutions in general relativity ,General Mathematics ,Ode ,Estimator ,Interval (mathematics) ,Haar wavelet ,Mathematics - Abstract
In this paper, extended H $$\ddot{o}$$ lder class $$H_\alpha ^{(w)}[0,\mu )$$ is considered. This class is the generalization of H $$\ddot{o}$$ lder class $$H_\alpha [0,\mu )$$ . Three new estimators $$E_N^{(1)}(f), E_N^{(2)}(f)$$ and $$E_N^{(3)}(f)$$ of functions of classes $$H_\alpha [0,\mu )$$ and $$H_\alpha ^{(w)}[0,\mu )$$ have been obtained. These estimators are best in approximation of functions by wavelet methods. The estimators obtained in this paper and the solution of ordinary and partial differential equations by extended Haar wavelet operational matrix method in the interval $$[0,\mu )$$ and its comparison with exact solution for different values of $$\mu$$ are the significant achievement of this research paper in approximation theory as well as Wavelet Analysis.
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- 2021
14. Approximation by a new sequence of operators involving Apostol-Genocchi polynomials
- Author
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D. K. Verma, Naokant Deo, and Chandra Prakash
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Combinatorics ,General Mathematics ,Mathematics ,Sequence (medicine) - Abstract
The main objective of this paper is to construct a new sequence of operators involving Apostol-Genocchi polynomials based on certain parameters. We investigate the rate of convergence of the operators given in this paper using second-order modulus of continuity and Voronovskaja type approximation theorem. Moreover, we find weighted approximation result of the given operators. Finally, we derive the Kantorovich variant of the given operators and discussed the approximation results.
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- 2021
15. Unitary representations of type B rational Cherednik algebras and crystal combinatorics
- Author
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Emily Norton
- Subjects
Functor ,Unitarity ,General Mathematics ,Type (model theory) ,Unitary state ,Fock space ,Combinatorics ,Irreducible representation ,FOS: Mathematics ,Mathematics - Combinatorics ,Partition (number theory) ,Component (group theory) ,Combinatorics (math.CO) ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Mathematics - Representation Theory ,Mathematics - Abstract
We compare crystal combinatorics of the level 2 Fock space with the classification of unitary irreducible representations of type B rational Cherednik algebras to study how unitarity behaves under parabolic restriction. First, we show that any finite-dimensional unitary irreducible representation of such an algebra is labeled by a bipartition consisting of a rectangular partition in one component and the empty partition in the other component. This is a new proof of a result that can be deduced from theorems of Montarani and Etingof-Stoica. Second, we show that the crystal operators that remove boxes preserve the combinatorial conditions for unitarity, and that the parabolic restriction functors categorifying the crystals send irreducible unitary representations to unitary representations. Third, we find the supports of the unitary representations., This paper supersedes arXiv:1907.00919 and contains that paper as a subsection. 35 pages, some color figures
- Published
- 2021
16. On the minimum value of the condition number of polynomials
- Author
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Carlos Beltrán, Fátima Lizarte, and Universidad de Cantabria
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Sequence ,Degree (graph theory) ,Mathematics - Complex Variables ,Applied Mathematics ,General Mathematics ,Univariate ,Term (logic) ,Combinatorics ,Computational Mathematics ,Integer ,Simple (abstract algebra) ,FOS: Mathematics ,30E10, 30C15, 31A15 ,Complex Variables (math.CV) ,Constant (mathematics) ,Condition number ,Mathematics - Abstract
In 1993, Shub and Smale posed the problem of finding a sequence of univariate polynomials of degree $N$ with condition number bounded above by $N$. In a previous paper by C. Belt\'an, U. Etayo, J. Marzo and J. Ortega-Cerd\`a, it was proved that the optimal value of the condition number is of the form $O(\sqrt{N})$, and the sequence demanded by Shub and Smale was described by a closed formula (for large enough $N\geqslant N_0$ with $N_0$ unknown) and by a search algorithm for the rest of the cases. In this paper we find concrete estimates for the constant hidden in the $O(\sqrt{N})$ term and we describe a simple formula for a sequence of polynomials whose condition number is at most $N$, valid for all $N=4M^2$, with $M$ a positive integer., Comment: 21 pages
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- 2021
17. New fixed point theorems for orthogonal $F_m$-contractions in incomplete $m$-metric spaces
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A. Shoaib, F. Uddin, M. Mehmood, and H. Isik
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Combinatorics ,Metric space ,General Mathematics ,Fixed-point theorem ,Mathematics - Abstract
In this paper, we introduce the concept of orthogonal $m$-metric spaces and by using $F_m$-contraction in orthogonal $m$-metric spaces, we give the concept of orthogonal $F_m$-contraction (briefly, $\bot_{F_m}$-contraction) and investigate fixed point results for such mappings. Many existing results in the literature appear to be special case of results proved in this paper. An example to support our main results is also mentioned.
- Published
- 2021
18. Fekete-Szegö problem for starlike functions connected withk-Fibonacci numbers
- Author
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Serap Bulut
- Subjects
Combinatorics ,Subordination (linguistics) ,Fibonacci number ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Analytic function ,Mathematics - Abstract
In a recent paper, Sokół et al. [Applications of k-Fibonacci numbers for the starlike analytic functions, Hacet. J. Math. Stat. 44(1) (2015), 121{127] obtained an upper bound for the Fekete-Szegö functionalϕλwhenλ 2R of functions belong to the classSLkconnected withk-Fibonacci numbers. The main purpose of this paper is to obtain sharp bounds forϕλbothλ 2R andλ 2C.
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- 2021
19. Maximal families of nodal varieties with defect
- Author
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REMKE NANNE KLOOSTERMAN
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Surface (mathematics) ,Double cover ,Degree (graph theory) ,Plane (geometry) ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Combinatorics ,Mathematics - Algebraic Geometry ,Hypersurface ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,NODAL ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
In this paper we prove that a nodal hypersurface in P^4 with defect has at least (d-1)^2 nodes, and if it has at most 2(d-2)(d-1) nodes and d>6 then it contains either a plane or a quadric surface. Furthermore, we prove that a nodal double cover of P^3 ramified along a surface of degree 2d with defect has at least d(2d-1) nodes. We construct the largest dimensional family of nodal degree d hypersurfaces in P^(2n+2) with defect for d sufficiently large., v2: A proof for the Ciliberto-Di Gennaro conjecture is added (Section 5); Some minor corrections in the other sections. v3: some minor corrections in the abstract v4: The proof for the Ciliberto-Di Gennaro conjecture has been modified; The paper is split into two parts, the complete intersection case will be discussed in a different paper
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- 2021
20. Fragility of nonconvergence in preferential attachment graphs with three types
- Author
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Ben Andrews and Jonathan Jordan
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Random graph ,Vertex (graph theory) ,05C82 ,General Mathematics ,Probability (math.PR) ,Type (model theory) ,Preferential attachment ,Graph ,Combinatorics ,Fragility ,FOS: Mathematics ,Tournament ,Node (circuits) ,Mathematics - Probability ,Mathematics - Abstract
Preferential attachment networks are a type of random network where new nodes are connected to existing ones at random, and are more likely to connect to those that already have many connections. We investigate further a family of models introduced by Antunovi\'{c}, Mossel and R\'{a}cz where each vertex in a preferential attachment graph is assigned a type, based on the types of its neighbours. Instances of this type of process where the proportions of each type present do not converge over time seem to be rare. Previous work found that a "rock-paper-scissors" setup where each new node's type was determined by a rock-paper-scissors contest between its two neighbours does not converge. Here, two cases similar to that are considered, one which is like the above but with an arbitrarily small chance of picking a random type and one where there are four neighbours which perform a knockout tournament to determine the new type. These two new setups, despite seeming very similar to the rock-paper-scissors model, do in fact converge, perhaps surprisingly., Comment: 7 pages, 2 figures
- Published
- 2021
21. Inverse problems of the Erdős-Ko-Rado type theorems for families of vector spaces and permutations
- Author
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Xiangliang Kong, Bingchen Qian, Yuanxiao Xi, and Gennian Ge
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Combinatorics ,Matrix (mathematics) ,Intersection ,General Mathematics ,Structure (category theory) ,Intersection number ,Inverse problem ,Type (model theory) ,Linear subspace ,Mathematics ,Vector space - Abstract
Ever since the famous Erdős-Ko-Rado theorem initiated the study of intersecting families of subsets, extremal problems regarding intersecting properties of families of various combinatorial objects have been extensively investigated. Among them, studies about families of subsets, vector spaces and permutations are of particular concerns. Recently, we proposed a new quantitative intersection problem for families of subsets: For $${\cal F} \subseteq \left({\matrix{{[n]} \cr k \cr}} \right)$$ , define its total intersection number as $${\cal I}({\cal F}) = \sum\nolimits_{{F_1},{F_2} \in {\cal F}} {\left| {{F_1} \cap {F_2}} \right|} $$ . Then, what is the structure of $${\cal F}$$ when it has the maximal total intersection number among all the families in $$\left({\matrix{{[n]} \cr k \cr}} \right)$$ with the same family size? In a recent paper, Kong and Ge (2020) studied this problem and characterized extremal structures of families maximizing the total intersection number of given sizes. In this paper, we consider the analogues of this problem for families of vector spaces and permutations. For certain ranges of family size, we provide structural characterizations for both families of subspaces and families of permutations having maximal total intersection numbers. To some extent, these results determine the unique structure of the optimal family for some certain values of $$\left| {\cal F} \right|$$ and characterize the relationship between having maximal total intersection number and being intersecting. Besides, we also show several upper bounds on the total intersection numbers for both families of subspaces and families of permutations of given sizes.
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- 2021
22. Sumsets of Wythoff sequences, Fibonacci representation, and beyond
- Author
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Jeffrey Shallit
- Subjects
FOS: Computer and information sciences ,Fibonacci number ,Mathematics - Number Theory ,Discrete Mathematics (cs.DM) ,Formal Languages and Automata Theory (cs.FL) ,General Mathematics ,Computer Science - Formal Languages and Automata Theory ,Of the form ,Combinatorics ,Alpha (programming language) ,Simple (abstract algebra) ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,Number Theory (math.NT) ,Representation (mathematics) ,Computer Science - Discrete Mathematics ,Mathematics - Abstract
Let $$\alpha = (1+\sqrt{5})/2$$ and define the lower and upper Wythoff sequences by $$a_i = \lfloor i \alpha \rfloor $$ , $$b_i = \lfloor i \alpha ^2 \rfloor $$ for $$i \ge 1$$ . In a recent interesting paper, Kawsumarng et al. proved a number of results about numbers representable as sums of the form $$a_i + a_j$$ , $$b_i + b_j$$ , $$a_i + b_j$$ , and so forth. In this paper I show how to derive all of their results, using one simple idea and existing free software called Walnut. The key idea is that for each of their sumsets, there is a relatively small automaton accepting the Fibonacci representation of the numbers represented. I also show how the automaton approach can easily prove other results.
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- 2021
23. Finite Homogeneous Subspaces of Euclidean Spaces
- Author
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V. N. Berestovskiĭ and Yu. G. Nikonorov
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Convex hull ,General Mathematics ,Archimedean solid ,Combinatorics ,symbols.namesake ,Polyhedron ,Metric space ,symbols ,Tetrahedron ,Mathematics::Metric Geometry ,Cube ,Isometry group ,Mathematics ,Regular polytope - Abstract
The paper is devoted to the study of the metric properties of regular and semiregular polyhedra in Euclidean spaces. In the first part, we prove that every regular polytope of dimension greater or equal than 4, and different from 120-cell in $$\mathbb {E}^4 $$ is such that the set of its vertices is a Clifford–Wolf homogeneous finite metric space. The second part of the paper is devoted to the study of special properties of Archimedean solids. In particular, for each Archimedean solid, its description is given as the convex hull of the orbit of a suitable point of a regular tetrahedron, cube or dodecahedron under the action of the corresponding isometry group.
- Published
- 2021
24. A fractional $$p(x,\cdot )$$-Laplacian problem involving a singular term
- Author
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K. Saoudi, A. Mokhtari, and N. T. Chung
- Subjects
Symmetric function ,Sobolev space ,Combinatorics ,Continuous function (set theory) ,Applied Mathematics ,General Mathematics ,Bounded function ,Domain (ring theory) ,Lambda ,Laplace operator ,Omega ,Mathematics - Abstract
This paper deals with a class of singular problems involving the fractional $$p(x,\cdot )$$ -Laplace operator of the form $$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}_{p(x,\cdot )}u(x)= \frac{\lambda }{u^{\gamma (x)}}+u^{q(x)-1} &{} \hbox {in }\Omega , \\ u>0, \;\;\text {in}\;\; \Omega &{} \hbox {} \\ u=0 \;\;\text {on}\;\;{\mathbb {R}}^N\setminus \Omega , &{} \hbox {} \end{array} \right. \end{aligned}$$ where $$\Omega $$ is a smooth bounded domain in $${\mathbb {R}}^N$$ ( $$N\ge 3$$ ), $$00$$ small enough. To our best knowledge, this paper is one of the first attempts in the study of singular problems involving fractional $$p(x,\cdot )$$ -Laplace operators.
- Published
- 2021
25. Limit theorems for linear random fields with tapered innovations. II: The stable case
- Author
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Vygantas Paulauskas and Julius Damarackas
- Subjects
Combinatorics ,010104 statistics & probability ,Number theory ,Random field ,General Mathematics ,010102 general mathematics ,Limit (mathematics) ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In the paper, we consider the limit behavior of partial-sum random field (r.f.) $$ \left.{S}_n\left({t}_1,{t}_2;\right)X\left(b\left(\mathbf{n}\right)\right)\right)={\sum}_{k=1}^{\left[{n}_1{t}_1\right]}{\sum}_{l=1}^{\left[{n}_2{t}_2\right]}{X}_{k,l}\left(b\left(\mathbf{n}\right)\right), $$ where $$ \left\{{X}_{k,l}\left(b\left(\mathbf{n}\right)\right)={\sum}_{i=0}^{\infty }{\sum}_{j=0}^{\infty }{c}_{i,j}{\upxi}_{k-i,l-j}\left(b\left(\mathbf{n}\right)\right),k,l\in \mathrm{\mathbb{Z}}\right\},n\ge 1, $$ is a family (indexed by n = (n1, n2), ni ≥ 1) of linear r.f.s with filter ci,j = aibj and innovations ξk,l(b(n)) having heavy-tailed tapered distributions with tapering parameter b(n) growing to infinity as n → ∞. In [V. Paulauskas, Limit theorems for linear random fields with tapered innovations. I: The Gaussian case, Lith. Math. J., 61(2):261–273, 2021], we considered the so-called hard tapering as b(n) grows relatively slowly and the limit r.f.s for appropriately normalized Sn(t1, t2;X(b(n))) are Gaussian. In this paper, we consider the case of soft tapering where b(n) grows more rapidly in comparison with the case of hard tapering and stable limit r.f.s.We consider cases where the sequences {ai} and {bj} are long-range, short-range, and negatively dependent.
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- 2021
26. A group of Pythagorean triples using the inradius
- Author
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Howard Sporn
- Subjects
Combinatorics ,Coprime integers ,Group (mathematics) ,General Mathematics ,Pythagorean triple ,Right triangle ,Mathematics ,Incircle and excircles of a triangle - Abstract
Pythagorean triples are triples of integers (a, b, c) satisfying the equation a2 + b2 = c2. For the purpose of this paper, we will take a, b and c to be positive, unless otherwise stated. Then, of course, it follows that a triple represents the lengths of sides of a right triangle. Also, for the purpose of this paper, we will consider the triples (a, b, c) and (b, a, c) to be distinct, even though they represent the same right triangle. A primitive Pythagorean triple is one for which a, b and c are relatively prime.
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- 2021
27. An improvement on Furstenberg’s intersection problem
- Author
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Han Yu
- Subjects
Combinatorics ,Intersection ,Applied Mathematics ,General Mathematics ,Bounded function ,010102 general mathematics ,Dimension (graph theory) ,Zero (complex analysis) ,0101 mathematics ,Invariant (mathematics) ,Dynamical system (definition) ,01 natural sciences ,Mathematics - Abstract
In this paper, we study a problem posed by Furstenberg on intersections between × 2 , × 3 \times 2, \times 3 invariant sets. We present here a direct geometrical counting argument to revisit a theorem of Wu and Shmerkin. This argument can be used to obtain further improvements. For example, we show that if A 2 , A 3 ⊂ [ 0 , 1 ] A_2,A_3\subset [0,1] are closed and × 2 , × 3 \times 2, \times 3 invariant respectively, assuming that dim A 2 + dim A 3 > 1 \dim A_2+\dim A_3>1 then A 2 ∩ ( u A 3 + v ) A_2\cap (uA_3+v) is sparse (defined in this paper) and has box dimension zero uniformly with respect to the real parameters u , v u,v such that u u and u − 1 u^{-1} are both bounded away from 0 0 .
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- 2021
28. Generalization of some fractional versions of Hadamard inequalities via exponentially (α,h−m)-convex functions
- Author
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Ghulam Farid, Waqas Nazeer, Hafsa Yasmeen, Yu-Pei Lv, and Chahn Yong Jung
- Subjects
Generalization ,General Mathematics ,Regular polygon ,Function (mathematics) ,Hadamard inequality ,h−m)-convex function ,hadamard inequality ,exponentionally (α ,Combinatorics ,Alpha (programming language) ,Exponential growth ,Hadamard transform ,riemann-liouville fractional integrals ,(α ,QA1-939 ,Convex function ,Mathematics - Abstract
In this paper we give Hadamard inequalities for exponentially $ (\alpha, h-m) $-convex functions using Riemann-Liouville fractional integrals for strictly increasing function. Results for Riemann-Liouville fractional integrals of convex, $ m $-convex, $ s $-convex, $ (\alpha, m) $-convex, $ (s, m) $-convex, $ (h-m) $-convex, $ (\alpha, h-m) $-convex, exponentially convex, exponentially $ m $-convex, exponentially $ s $-convex, exponentially $ (s, m) $-convex, exponentially $ (h-m) $-convex, exponentially $ (\alpha, h-m) $-convex functions are particular cases of the results of this paper. The error estimations of these inequalities by using two fractional integral identities are also given.
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- 2021
29. Degrees of Enumerations of Countable Wehner-Like Families
- Author
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I. Sh. Kalimullin and M. Kh. Faizrahmanov
- Subjects
Statistics and Probability ,Class (set theory) ,Degree (graph theory) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Spectrum (topology) ,010305 fluids & plasmas ,Combinatorics ,0103 physical sciences ,Enumeration ,Countable set ,Family of sets ,0101 mathematics ,Turing ,computer ,Finite set ,computer.programming_language ,Mathematics - Abstract
This paper is a survey of results on countable families with natural degree spectra. These results were obtained by a modification of the methodology proposed by Wechner, who first found a family of sets with the spectrum consisting precisely of nonzero Turing degrees. Based on this method, many researchers obtained examples of families with other natural spectra. In addition, in this paper we extend these results and present new examples of natural spectra. In particular, we construct a family of finite sets with the spectrum consisting of exactly non-K-trivial degrees and also we find new sufficient conditions on $$ {\Delta}_2^0 $$ -degree a, which guarantees that the class {x : x ≰ a} is the degree spectrum of some family. Finally, we give a survey of our recent results on the degree spectra of α-families, where α is an arbitrary computable ordinal.
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- 2021
30. On a class number formula of Hurwitz
- Author
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William Duke, Árpád Tóth, and Özlem Imamoglu
- Subjects
Binary quadratic forms ,Combinatorics ,class numbers ,Hurwitz ,Applied Mathematics ,General Mathematics ,Binary quadratic form ,Class number formula ,Mathematics - Abstract
In a little-known paper Hurwitz gave an infinite series representation of the class number for positive definite binary quadratic forms. In this paper we give a similar formula in the indefinite case. We also give a simple proof of Hurwitz's formula and indicate some extensions., Journal of the European Mathematical Society, 23 (12), ISSN:1435-9855, ISSN:1435-9863
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- 2021
31. Generation of colored graphs with isomorphism rejection
- Author
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P. V. Razumovsky and M. B. Abrosimov
- Subjects
General Computer Science ,generator ,Mechanical Engineering ,General Mathematics ,Colored graph ,Computational Mechanics ,graph labeling ,color graph ,graph ,Combinatorics ,Mechanics of Materials ,graph coloring ,QA1-939 ,Isomorphism ,Mathematics ,MathematicsofComputing_DISCRETEMATHEMATICS - Abstract
In the article we consider graphs whose vertices or edges are colored in a given number of colors — vertex and edge colorings. The study of colorings of graphs began in the middle of the 19th century, but the main attention is paid to proper colorings, in which the restriction applies that the colors of adjacent vertices or edges must be different. This paper considers colorings of graphs without any restrictions. We study the problem of generating all non-isomorphic vertex and edge $k$-colorings of a given graph without direct checking for isomorphism. The problem of generating non-isomorphic edge $k$-colorings is reduced to the problem of constructing all vertex $k$-colorings of a graph. Methods for generating graphs without direct checking for isomorphism or isomorphism rejection are based on the method of canonical representatives. The idea of the method is that a method for encoding graphs is proposed and a certain rule is chosen according to which one of all isomorphic graphs is declared canonical. All codes are built and only the canonical ones are accepted. Often, the representative with the largest or smallest code is chosen as the canonical one. In practice, generating all codes requires large computational resources; therefore, various methods of enumeration optimization are used. The paper proposes two algorithms for solving the problem of generating vertex $k$-colorings with isomorphism rejection by McKay and Reed – Faradzhev methods. A comparison of the proposed algorithms for generating colorings on two classes of graphs — paths and cycles is made. Computational experiments show that the Reed – Faradzhev method is faster for paths and cycles.
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- 2021
32. Variations of Weyl Type Theorems for Upper Triangular Operator Matrices
- Author
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M. H. M. Rashid
- Subjects
Set (abstract data type) ,Combinatorics ,Operator matrix ,General Mathematics ,Triangular matrix ,Banach space ,Extension (predicate logic) ,Type (model theory) ,Lambda ,Mathematics ,Bounded operator - Abstract
Let $\mathcal X$ be a Banach space and let T be a bounded linear operator on $\mathcal {X}$ . We denote by S(T) the set of all complex $\lambda \in \mathcal {C}$ such that T does not have the single-valued extension property. In this paper it is shown that if MC is a 2 × 2 upper triangular operator matrix acting on the Banach space $\mathcal {X} \oplus \mathcal {Y}$ , then the passage from σLD(A) ∪ σLD(B) to σLD(MC) is accomplished by removing certain open subsets of σd(A) ∩ σLD(B) from the former, that is, there is the equality σLD(A) ∪ σLD(B) = σLD(MC) ∪ℵ, where ℵ is the union of certain of the holes in σLD(MC) which happen to be subsets of σd(A) ∩ σLD(B). Generalized Weyl’s theorem and generalized Browder’s theorem are liable to fail for 2 × 2 operator matrices. In this paper, we also explore how generalized Weyl’ theorem, generalized Browder’s theorem, generalized a-Weyl’s theorem and generalized a-Browder’s theorem survive for 2 × 2 upper triangular operator matrices on the Banach space.
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- 2021
33. Noncommutative Counting Invariants and Curve Complexes
- Author
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Ludmil Katzarkov and George Dimitrov
- Subjects
Intersection theory ,medicine.medical_specialty ,Functor ,Conjecture ,Group (mathematics) ,General Mathematics ,010102 general mathematics ,Quiver ,Type (model theory) ,01 natural sciences ,Combinatorics ,0103 physical sciences ,medicine ,010307 mathematical physics ,0101 mathematics ,Partially ordered set ,Commutative property ,Mathematics - Abstract
In our previous paper, viewing $D^b(K(l))$ as a noncommutative curve, where $K(l)$ is the Kronecker quiver with $l$-arrows, we introduced categorical invariants via counting of noncommutative curves. Roughly, these invariants are sets of subcategories in a given category and their quotients. The noncommutative curve-counting invariants are obtained by restricting the subcategories to be equivalent to $D^b(K(l))$. The general definition, however, defines a larger class of invariants and many of them behave properly with respect to fully faithful functors. Here, after recalling the definition, we focus on the examples and extend our studies beyond counting. We enrich our invariants with the following structures: the inclusion of subcategories makes them partially ordered sets and considering semi-orthogonal pairs of subcategories as edges amounts to directed graphs. It turns out that the problem for counting $D^b(A_k)$ in $D^b(A_n)$ has a geometric combinatorial parallel - counting of maps between polygons. Estimating the numbers counting noncommutative curves in $D^b({\mathbb P}^2)$ modulo the group of autoequivalences, we prove finiteness and that the exact determining of these numbers leads to a solution of Markov problem. Via homological mirror symmetry, this gives a new approach to this problem. Regarding the structure of a partially ordered set mentioned above, we initiate intersection theory of noncommutative curves focusing on the case of noncommutative genus zero. The above-mentioned structure of a directed graph (and related simplicial complex) is a categorical analogue of the classical curve complex, introduced by Harvey and Harrer. The paper contains pictures of the graphs in many examples and also presents an approach to Markov conjecture via counting of subgraphs in a graph associated with $D^b({{\mathbb{P}}}^2)$. Some of the results proved here were announced in a previous work.
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- 2021
34. On the size of subsets of $$\mathbb{F}_p^n$$ without p distinct elements summing to zero
- Author
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Lisa Sauermann
- Subjects
Mathematics - Number Theory ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Zero (complex analysis) ,Lattice (group) ,0102 computer and information sciences ,Infinity ,01 natural sciences ,Upper and lower bounds ,Prime (order theory) ,Combinatorics ,Integer ,010201 computation theory & mathematics ,FOS: Mathematics ,Mathematics - Combinatorics ,Maximum size ,Combinatorics (math.CO) ,Number Theory (math.NT) ,0101 mathematics ,Constant (mathematics) ,media_common ,Mathematics - Abstract
Let us fix a prime $p$. The Erd\H{o}s-Ginzburg-Ziv problem asks for the minimum integer $s$ such that any collection of $s$ points in the lattice $\mathbb{Z}^n$ contains $p$ points whose centroid is also a lattice point in $\mathbb{Z}^n$. For large $n$, this is essentially equivalent to asking for the maximum size of a subset of $\mathbb{F}_p^n$ without $p$ distinct elements summing to zero. In this paper, we give a new upper bound for this problem for any fixed prime $p\geq 5$ and large $n$. In particular, we prove that any subset of $\mathbb{F}_p^n$ without $p$ distinct elements summing to zero has size at most $C_p\cdot \left(2\sqrt{p}\right)^n$, where $C_p$ is a constant only depending on $p$. For $p$ and $n$ going to infinity, our bound is of the form $p^{(1/2)\cdot (1+o(1))n}$, whereas all previously known upper bounds were of the form $p^{(1-o(1))n}$ (with $p^n$ being a trivial bound). Our proof uses the so-called multi-colored sum-free theorem which is a consequence of the Croot-Lev-Pach polynomial method. This method and its consequences were already applied by Naslund as well as by Fox and the author to prove bounds for the problem studied in this paper. However, using some key new ideas, we significantly improve their bounds., Comment: 11 pages
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- 2021
35. A new obstruction for normal spanning trees
- Author
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Max Pitz
- Subjects
Aleph ,Spanning tree ,General Mathematics ,010102 general mathematics ,Minor (linear algebra) ,Type (model theory) ,01 natural sciences ,Graph ,Combinatorics ,Mathematics::Logic ,Arbitrarily large ,Cardinality ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,Connectivity ,05C83, 05C05, 05C63 ,Mathematics - Abstract
In a paper from 2001 (Journal of the LMS), Diestel and Leader offered a proof that a connected graph has a normal spanning tree if and only if it does not contain a minor from two specific forbidden classes of graphs, all of cardinality $\aleph_1$. Unfortunately, their proof contains a gap, and their result is incorrect. In this paper, we construct a third type of obstruction: an $\aleph_1$-sized graph without a normal spanning tree that contains neither of the two types described by Diestel and Leader as a minor. Further, we show that any list of forbidden minors characterising the graphs with normal spanning trees must contain graphs of arbitrarily large cardinality., Comment: 9 pages. arXiv admin note: text overlap with arXiv:2005.02833
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- 2021
36. On Classes of Subcompact Spaces
- Author
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Alexander V. Osipov, E. G. Pytkeev, and V. I. Belugin
- Subjects
Condensed Matter::Quantum Gases ,Combinatorics ,Compact space ,High Energy Physics::Lattice ,General Mathematics ,Cardinal number ,Hausdorff space ,Space (mathematics) ,Mathematics - Abstract
This paper continues the study of P. S. Alexandroff’s problem: When can a Hausdorff space $$X$$ be one-to-one continuously mapped onto a compact Hausdorff space? For a cardinal number $$\tau$$ , the classes of $$a_\tau$$ -spaces and strict $$a_\tau$$ -spaces are defined. A compact space $$X$$ is called an $$a_\tau$$ -space if, for any $$C\in[X]^{\le\tau}$$ , there exists a one-to-one continuous mapping of $$X\setminus C$$ onto a compact space. A compact space $$X$$ is called a strict $$a_\tau$$ -space if, for any $$C\in[X]^{\le\tau}$$ , there exits a one-to-one continuous mapping of $$X\setminus C$$ onto a compact space $$Y$$ , and this mapping can be continuously extended to the whole space $$X$$ . In this paper, we study properties of the classes of $$a_\tau$$ - and strict $$a_\tau$$ -spaces by using Raukhvarger’s method of special continuous paritions.
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- 2021
37. Ideal, non-extended formulations for disjunctive constraints admitting a network representation
- Author
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Markó Horváth and Tamás Kis
- Subjects
Combinatorics ,Cardinality ,Series (mathematics) ,Unit vector ,General Mathematics ,Embedding ,Polytope ,QA75 Electronic computers. Computer science / számítástechnika, számítógéptudomány ,Ideal (ring theory) ,Characterization (mathematics) ,Type (model theory) ,Software ,Mathematics - Abstract
In this paper we reconsider a known technique for constructing strong MIP formulations for disjunctive constraints of the form $$x \in \bigcup _{i=1}^m P_i$$ x ∈ ⋃ i = 1 m P i , where the $$P_i$$ P i are polytopes. The formulation is based on the Cayley Embedding of the union of polytopes, namely, $$Q := \mathrm {conv}(\bigcup _{i=1}^m P_i\times \{\epsilon ^i\})$$ Q : = conv ( ⋃ i = 1 m P i × { ϵ i } ) , where $$\epsilon ^i$$ ϵ i is the ith unit vector in $${\mathbb {R}}^m$$ R m . Our main contribution is a full characterization of the facets of Q, provided it has a certain network representation. In the second half of the paper, we work-out a number of applications from the literature, e.g., special ordered sets of type 2, logical constraints, the cardinality indicating polytope, union of simplicies, etc., along with a more complex recent example. Furthermore, we describe a new formulation for piecewise linear functions defined on a grid triangulation of a rectangular region $$D \subset {\mathbb {R}}^d$$ D ⊂ R d using a logarithmic number of auxilirary variables in the number of gridpoints in D for any fixed d. The series of applications demonstrates the richness of the class of disjunctive constraints for which our method can be applied.
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- 2022
38. Corrigenda to 'Cohen-Macaulay bipartite graphs in arbitrary codimension'
- Author
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Rahim Zaare-Nahandi, Hassan Haghighi, and Siamak Yassemi
- Subjects
Combinatorics ,Applied Mathematics ,General Mathematics ,Bipartite graph ,Codimension ,Mathematics - Abstract
A misuse of terminology has occurred in the statement and proof of Theorem 4.1 in our paper [Proc. Amer. Math. Soc. 143 (2015), pp. 1981–1989] which caused some justifiable misinterpretation of this result. To recover this result we provide a new definition and give the statement of our result in terms of this definition. The proof of the new version is an improvement of the old proof. The effect of the new definition on further relevant results in our paper has been adopted in a remark.
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- 2021
39. The annihilators comaximal graph
- Author
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Saeed Rajaee
- Subjects
Combinatorics ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Computer Science::Information Retrieval ,General Mathematics ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Astrophysics::Instrumentation and Methods for Astrophysics ,Computer Science::General Literature ,Graph (abstract data type) ,Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
In this paper, we introduce and study a new kind of graph related to a unitary module [Formula: see text] on a commutative ring [Formula: see text] with identity, namely the annihilators comaximal graph of submodules of [Formula: see text], denoted by [Formula: see text]. The (undirected) graph [Formula: see text] is with vertices of all non-trivial submodules of [Formula: see text] and two vertices [Formula: see text] of [Formula: see text] are adjacent if and only if their annihilators are comaximal ideals of [Formula: see text], i.e. [Formula: see text]. The main purpose of this paper is to investigate the interplay between the graph-theoretic properties of [Formula: see text] and the module-theoretic properties of [Formula: see text]. We study the annihilators comaximal graph [Formula: see text] in terms of the powers of the decomposition of [Formula: see text] to product distinct prime numbers in some special cases.
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- 2021
40. Polynomial and Pseudopolynomial Procedures for Solving Interval Two-Sided (Max, Plus)-Linear Systems
- Author
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Helena Myšková and Ján Plavka
- Subjects
Polynomial ,Similarity (geometry) ,max-plus matrix ,interval solution ,General Mathematics ,Linear system ,Solution set ,Combinatorics ,Transformation (function) ,Binary operation ,Linear algebra ,QA1-939 ,Computer Science (miscellaneous) ,Interval (graph theory) ,Engineering (miscellaneous) ,Mathematics ,solvability - Abstract
Max-plus algebra is the similarity of the classical linear algebra with two binary operations, maximum and addition. The notation Ax = Bx, where A, B are given (interval) matrices, represents (interval) two-sided (max, plus)-linear system. For the solvability of Ax = Bx, there are some pseudopolynomial algorithms, but a polynomial algorithm is still waiting for an appearance. The paper deals with the analysis of solvability of two-sided (max, plus)-linear equations with inexact (interval) data. The purpose of the paper is to get efficient necessary and sufficient conditions for solvability of the interval systems using the property of the solution set of the non-interval system Ax = Bx. The main contribution of the paper is a transformation of weak versions of solvability to either subeigenvector problems or to non-interval two-sided (max, plus)-linear systems and obtaining the equivalent polynomially checked conditions for the strong versions of solvability.
- Published
- 2021
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41. On commuting automorphisms and central automorphisms of finite 2-groups of almost maximal class
- Author
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Mehri Akhavan Malayeri and Nazila Azimi Shahrabi
- Subjects
Combinatorics ,Class (set theory) ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,Structure (category theory) ,Algebra over a field ,Automorphism ,Mathematics - Abstract
Let G be a finite 2-group. In our recent papers, we proved that in a finite 2-group of almost maximal class, the set of all commuting automorphisms, $$\mathcal {A}(G)=\lbrace \alpha \in Aut(G) :x\alpha (x)=\alpha (x)x~~for~ all~ x\in G\rbrace $$ is equal to the group of all central automorphisms, $$Aut_{c}(G)$$ , except only for five ones. Also, we determined the structure of $$Aut_{c}(G)$$ and $$\mathcal {A}(G)$$ for these five groups. Using these results, in this paper, we find the structure of $$\mathcal {A}(G)=Aut_{c}(G)$$ for the remaining 2-groups of almost maximal class. Also, we prove the following results
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- 2021
42. D-Magic Oriented Graphs
- Author
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Alison Marr and Rinovia Simanjuntak
- Subjects
oriented graphs ,digraph labeling ,distance magic labeling ,D-magic labeling ,Physics and Astronomy (miscellaneous) ,General Mathematics ,Astrophysics::High Energy Astrophysical Phenomena ,Magic (programming) ,Construct (python library) ,Graph ,Vertex (geometry) ,Combinatorics ,Set (abstract data type) ,Magic constant ,Multipartite ,Chemistry (miscellaneous) ,Homogeneous space ,Computer Science (miscellaneous) ,Physics::Atomic and Molecular Clusters ,QA1-939 ,Mathematics ,MathematicsofComputing_DISCRETEMATHEMATICS - Abstract
In this paper, we define D-magic labelings for oriented graphs where D is a distance set. In particular, we label the vertices of the graph with distinct integers {1,2,…,|V(G)|} in such a way that the sum of all the vertex labels that are a distance in D away from a given vertex is the same across all vertices. We give some results related to the magic constant, construct a few infinite families of D-magic graphs, and examine trees, cycles, and multipartite graphs. This definition grew out of the definition of D-magic (undirected) graphs. This paper explores some of the symmetries we see between the undirected and directed version of D-magic labelings.
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- 2021
43. The moduli space of matroids
- Author
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Oliver Lorscheid, Matthew Baker, and Dynamical Systems, Geometry & Mathematical Physics
- Subjects
Mathematics::Combinatorics ,Functor ,F-geometry ,Group (mathematics) ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,Tracts ,01 natural sciences ,Matroid ,Moduli space ,Combinatorics ,Matroids ,Mathematics - Algebraic Geometry ,Morphism ,010201 computation theory & mathematics ,Scheme (mathematics) ,Blueprints ,FOS: Mathematics ,Isomorphism ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics ,Initial and terminal objects - Abstract
In the first part of the paper, we clarify the connections between several algebraic objects appearing in matroid theory: both partial fields and hyperfields are fuzzy rings, fuzzy rings are tracts, and these relations are compatible with the respective matroid theories. Moreover, fuzzy rings are ordered blueprints and lie in the intersection of tracts with ordered blueprints; we call the objects of this intersection pastures. In the second part, we construct moduli spaces for matroids over pastures. We show that, for any non-empty finite set $E$, the functor taking a pasture $F$ to the set of isomorphism classes of rank-$r$ $F$-matroids on $E$ is representable by an ordered blue scheme $Mat(r,E)$, the moduli space of rank-$r$ matroids on $E$. In the third part, we draw conclusions on matroid theory. A classical rank-$r$ matroid $M$ on $E$ corresponds to a $\mathbb{K}$-valued point of $Mat(r,E)$ where $\mathbb{K}$ is the Krasner hyperfield. Such a point defines a residue pasture $k_M$, which we call the universal pasture of $M$. We show that for every pasture $F$, morphisms $k_M\to F$ are canonically in bijection with $F$-matroid structures on $M$. An analogous weak universal pasture $k_M^w$ classifies weak $F$-matroid structures on $M$. The unit group of $k_M^w$ can be canonically identified with the Tutte group of $M$. We call the sub-pasture $k_M^f$ of $k_M^w$ generated by ``cross-ratios' the foundation of $M$,. It parametrizes rescaling classes of weak $F$-matroid structures on $M$, and its unit group is coincides with the inner Tutte group of $M$. We show that a matroid $M$ is regular if and only if its foundation is the regular partial field, and a non-regular matroid $M$ is binary if and only if its foundation is the field with two elements. This yields a new proof of the fact that a matroid is regular if and only if it is both binary and orientable., 85 pages; some additional material, e.g. a new section 5.6; the terminology has been adapted to the usage in follow-up papers
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- 2021
44. McKay Quivers and Lusztig Algebras of Some Finite Groups
- Author
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Ragnar-Olaf Buchweitz, Matthew Lewis, Colin Ingalls, and Eleonore Faber
- Subjects
General Mathematics ,Field (mathematics) ,Group Theory (math.GR) ,01 natural sciences ,Combinatorics ,Elementary algebra ,Symmetric group ,FOS: Mathematics ,Mathematics - Combinatorics ,0101 mathematics ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Mathematics ,Finite group ,05E10 16G20 16S35 16S37 20F55 20C30 ,010102 general mathematics ,Quiver ,Mathematics - Rings and Algebras ,010101 applied mathematics ,Clifford theory ,Rings and Algebras (math.RA) ,Combinatorics (math.CO) ,Mathematics - Group Theory ,Mathematics - Representation Theory ,Vector space ,Group ring - Abstract
We are interested in the McKay quiver $\Gamma(G)$ and skew group rings $A*G$, where $G$ is a finite subgroup of $\mathrm{GL}(V)$, where $V$ is a finite dimensional vector space over a field $K$, and $A$ is a $K-G$-algebra. These skew group rings appear in Auslander's version of the McKay correspondence. In the first part of this paper we consider complex reflection groups $G \subseteq \mathrm{GL}(V)$ and find a combinatorial method, making use of Young diagrams, to construct the McKay quivers for the groups $G(r,p,n)$. We first look at the case $G(1,1,n)$, which is isomorphic to the symmetric group $S_n$, followed by $G(r,1,n)$ for $r >1$. Then, using Clifford theory, we can determine the McKay quiver for any $G(r,p,n)$ and thus for all finite irreducible complex reflection groups up to finitely many exceptions. In the second part of the paper we consider a more conceptual approach to McKay quivers of arbitrary finite groups: we define the Lusztig algebra $\widetilde A(G)$ of a finite group $G \subseteq \mathrm{GL}(V)$, which is Morita equivalent to the skew group ring $A*G$. This description gives us an embedding of the basic algebra Morita equivalent to $A*G$ into a matrix algebra over $A$., Comment: v2: minor revision, final version to appear in Algebr. Represent. Theory
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- 2021
45. Blocking sets of tangent and external lines to an elliptic quadric in PG(3, q)
- Author
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Puspendu Pradhan, Bart De Bruyn, and Binod Kumar Sahoo
- Subjects
Quadric ,Q) ,irreducible conic ,General Mathematics ,Tangent ,Characterization (mathematics) ,Blocking (statistics) ,Combinatorics ,Mathematics and Statistics ,elliptic quadric ,blocking set ,ovoid ,PG(2 ,Projective space ,Mathematics - Abstract
Consider an elliptic quadric $$Q^-(3,q)$$ in $$\mathrm{PG}(3,q)$$ . Let $$\mathcal {E}$$ and $$\mathcal {T}$$ denote the set of all lines of $$\mathrm{PG}(3,q)$$ which meet $$Q^-(3,q)$$ in 0 and 1 point, respectively. In this paper, we characterize the minimum size $$(\mathcal {T}\cup \mathcal {E})$$ -blocking sets and give a different proof for the characterization of minimum size $$\mathcal {E}$$ -blocking sets in $$\mathrm{PG}(3,q)$$ which works for all q. We also discuss whether the main results of this paper (Theorems 1.6 and 1.7) can be extended to ovoids in $$\mathrm{PG}(3,q)$$ .
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- 2021
46. Khovanov Homology for Links in #r(S2×S1)
- Author
-
Michael Willis
- Subjects
Combinatorics ,Khovanov homology ,Path (topology) ,symbols.namesake ,Root of unity ,General Mathematics ,Homotopy ,Euler characteristic ,symbols ,Homology (mathematics) ,Invariant (mathematics) ,Link (knot theory) ,Mathematics - Abstract
We revisit Rozansky’s construction of Khovanov homology for links in S 2 × S 1 , extending it to define the Khovanov homology Kh ( L ) for links L in M r = # r ( S 2 × S 1 ) for any r . The graded Euler characteristic of Kh ( L ) can be used to recover WRT invariants at certain roots of unity and also recovers the evaluation of L in the skein module S ( M r ) of Hoste and Przytycki when L is null-homologous in M r . The construction also allows for a clear path toward defining a Lee’s homology Kh ' ( L ) and associated s -invariant for such L , which we will explore in an upcoming paper. We also give an equivalent construction for the Khovanov homology of the knotification of a link in S 3 and show directly that this is invariant under handle-slides, in the hope of lifting this version to give a stable homotopy type for such knotifications in a future paper.
- Published
- 2021
47. Cosets of normal subgroups and powers of conjugacy classes
- Author
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María José Felipe and Antonio Beltrán
- Subjects
Normal subgroup ,cosets of normal subgroups ,characters ,General Mathematics ,Powers of conjugacy classes ,powers of conjugacy classes ,Combinatorics ,Conjugacy classes ,Mathematics::Group Theory ,Conjugacy class ,Cosets of normal subgroups ,Coset ,Characters ,MATEMATICA APLICADA ,conjugacy classes ,Mathematics - Abstract
[EN] Let G be a finite group and let K=xG be the conjugacy class of an element x of G. In this paper, it is proved that if N is a normal subgroup of G such that the coset xN is the union of K and K-1 (the conjugacy class of the inverse of x), then N and the subgroup ¿K¿ are solvable. As an application, we prove that if there exists a natural number n >= 2 such that Kn=K?K-1, then ¿K¿ is solvable., The authors are grateful to the referee for careful reading and many helpful comments and improvements on the paper. This research is partially supported by the Spanish Government, Proyecto PGC2018-096872-B-I00 and by Generalitat Valenciana, Proyecto AICO-2020-298. The first named author is also supported by Proyecto UJI-B2019-03.
- Published
- 2021
48. The Inequalities of Merris and Foregger for Permanents
- Author
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Divya K. Udayan and Kanagasabapathi Somasundaram
- Subjects
Doubly stochastic matrix ,Conjecture ,Physics and Astronomy (miscellaneous) ,General Mathematics ,MathematicsofComputing_GENERAL ,permanent ,Foregger’s inequality ,Combinatorics ,Matrix (mathematics) ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Integer ,Chemistry (miscellaneous) ,Symmetric group ,Linear algebra ,Merris conjecture ,Computer Science (miscellaneous) ,QA1-939 ,Order (group theory) ,Elementary symmetric polynomial ,doubly stochastic matrices ,Mathematics - Abstract
Conjectures on permanents are well-known unsettled conjectures in linear algebra. Let A be an n×n matrix and Sn be the symmetric group on n element set. The permanent of A is defined as perA=∑σ∈Sn∏i=1naiσ(i). The Merris conjectured that for all n×n doubly stochastic matrices (denoted by Ωn), nperA≥min1≤i≤n∑j=1nperA(j|i), where A(j|i) denotes the matrix obtained from A by deleting the jth row and ith column. Foregger raised a question whether per(tJn+(1−t)A)≤perA for 0≤t≤nn−1 and for all A∈Ωn, where Jn is a doubly stochastic matrix with each entry 1n. The Merris conjecture is one of the well-known conjectures on permanents. This conjecture is still open for n≥4. In this paper, we prove the Merris inequality for some classes of matrices. We use the sub permanent inequalities to prove our results. Foregger’s inequality is also one of the well-known inequalities on permanents, and it is not yet proved for n≥5. Using the concepts of elementary symmetric function and subpermanents, we prove the Foregger’s inequality for n=5 in [0.25, 0.6248]. Let σk(A) be the sum of all subpermanents of order k. Holens and Dokovic proposed a conjecture (Holen–Dokovic conjecture), which states that if A∈Ωn,A≠Jn and k is an integer, 1≤k≤n, then σk(A)≥(n−k+1)2nkσk−1(A). In this paper, we disprove the conjecture for n=k=4.
- Published
- 2021
49. On generalized zero-divisor graphs of a non-commutative ring with respect to an ideal
- Author
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Priyanka Pratim Baruah and Kuntala Patra
- Subjects
Ring (mathematics) ,Computer Science::Information Retrieval ,General Mathematics ,Astrophysics::Instrumentation and Methods for Astrophysics ,Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ,Commutative ring ,Girth (graph theory) ,Graph ,Combinatorics ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Computer Science::General Literature ,Ideal (ring theory) ,ComputingMilieux_MISCELLANEOUS ,Zero divisor ,MathematicsofComputing_DISCRETEMATHEMATICS ,Mathematics - Abstract
Let R be a non-commutative ring, and I be an ideal of R. In this paper, we generalize the definition of the zero-divisor graph of R with respect to I, and define several generalized zero-divisor graphs of R with respect to I. In this paper, we investigate the ring-theoretic properties of R and the graph-theoretic properties of all the generalized zero-divisor graphs. We study some basic properties of these generalized zero-divisor graphs related to the connectedness, the diameter and the girth. We also investigate some properties of these generalized zero-divisor graphs with respect to primal ideals.
- Published
- 2021
50. The universality of Hughes-free division rings
- Author
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Andrei Jaikin-Zapirain and UAM. Departamento de Matemáticas
- Subjects
Group (mathematics) ,Matemáticas ,General Mathematics ,Existential quantification ,010102 general mathematics ,Universality (philosophy) ,General Physics and Astronomy ,Universal division ring of fractions ,Division (mathematics) ,01 natural sciences ,Combinatorics ,Crossed product ,0103 physical sciences ,Hughes-free division ring ,Division ring ,010307 mathematical physics ,0101 mathematics ,Locally indicable groups ,Mathematics - Abstract
Let E∗ G be a crossed product of a division ring E and a locally indicable group G. Hughes showed that up to E∗ G-isomorphism, there exists at most one Hughes-free division E∗G-ring. However, the existence of a Hughes-free division E∗ G-ring DE∗G for an arbitrary locally indicable group G is still an open question. Nevertheless, DE∗G exists, for example, if G is amenable or G is bi-orderable. In this paper we study, whether DE∗G is the universal division ring of fractions in some of these cases. In particular, we show that if G is a residually-(locally indicable and amenable) group, then there exists DE[G] and it is universal. In Appendix we give a description of DE[G] when G is a RFRS group, This paper is partially supported by the Spanish Ministry of Science and Innovation through the grant MTM2017-82690-P and the “Severo Ochoa Programme for Centres of Excellence in R&D” (CEX2019-000904-S4). I would like to thank Dawid Kielak and an anonymous referee for useful suggestions and comments
- Published
- 2021
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