57 results
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2. A CRITERION FOR THE JACOBSON SEMISIMPLICITY OF THE GREEN RING OF A FINITE TENSOR CATEGORY
- Author
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Zhihua Wang, Yinhuo Zhang, Libin Li, WANG, Zhihua, Li, Libin, and ZHANG, Yinhuo
- Subjects
Algebra ,Ring (mathematics) ,Pure mathematics ,General Mathematics ,Tensor (intrinsic definition) ,finite tensor category ,green ring ,Casimir number, Jacobson radical, Frobenius algebra ,010102 general mathematics ,0103 physical sciences ,Foundation (engineering) ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
This paper deals with the Green ring $\mathcal{G}(\mathcal{C})$ of a finite tensor category $\mathcal{C}$ with finitely many isomorphism classes of indecomposable objects over an algebraically closed field. The first part of this paper deals with the question of when the Green ring $\mathcal{G}(\mathcal{C})$, or the Green algebra $\mathcal{G}(\mathcal{C})\otimes_{\mathbb {Z}}$K over a field K, is Jacobson semisimple (namely, has zero Jacobson radical). It turns out that $\mathcal{G}(\mathcal{C})\otimes_{\mathbb {Z}}$K is Jacobson semisimple if and only if the Casimir number of $\mathcal{C}$ is not zero in K. For the Green ring $\mathcal{G}(\mathcal{C})$ itself, $\mathcal{G}(\mathcal{C})$ is Jacobson semisimple if and only if the Casimir number of $\mathcal{C}$ is not zero. The second part of this paper focuses on the case where $\mathcal{C}=\text{Rep}(\mathbb {k}G)$ for a cyclic group G of order p over a field $\mathbb {k}$ of characteristic p. In this case, the Casimir number of $\mathcal{C}$ is computable and is shown to be 2p2. This leads to a complete description of the Jacobson radical of the Green algebra $\mathcal{G}(\mathcal{C})\otimes_{\mathbb {Z}}$K over any field K.
- Published
- 2017
3. ON THE HOMOGENIZED ENVELOPING ALGEBRA OF THE LIE ALGEBRA Sℓ(2,ℂ) II
- Author
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Roberto Martinez-Villa
- Subjects
General Mathematics ,010102 general mathematics ,Universal enveloping algebra ,Witt algebra ,010103 numerical & computational mathematics ,01 natural sciences ,Graded Lie algebra ,Lie conformal algebra ,Filtered algebra ,Algebra ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Lie algebra ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Cellular algebra ,0101 mathematics ,Generalized Kac–Moody algebra ,Mathematics - Abstract
In a previous paper, we studied the homogenized enveloping algebra of the Lie algebrasℓ(2,ℂ) and the homogenized Verma modules. The aim of this paper is to study the homogenization$\mathcal{O}$Bof the Bernstein–Gelfand–Gelfand category$\mathcal{O}$of sℓ(2,ℂ), and to apply the ideas developed jointly with J. Mondragón in our work on Groebner basis algebras, to give the relations between the categories$\mathcal{O}$Band$\mathcal{O}$as well as, between the derived categories$\mathcal{D}$b($\mathcal{O}$B) and$\mathcal{D}$b($\mathcal{O}$).
- Published
- 2016
4. AN INVERSE THEOREM FOR THE GOWERSU4-NORM
- Author
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Terence Tao, Tamar Ziegler, and Ben Green
- Subjects
Conjecture ,Mathematics - Number Theory ,General Mathematics ,010102 general mathematics ,Linear system ,Inverse ,Dynamical Systems (math.DS) ,0102 computer and information sciences ,01 natural sciences ,Combinatorics ,010201 computation theory & mathematics ,Norm (mathematics) ,Bounded function ,FOS: Mathematics ,Number Theory (math.NT) ,Mathematics - Dynamical Systems ,0101 mathematics ,Mathematics - Abstract
We prove the so-called inverse conjecture for the Gowers U^{s+1}-norm in the case s = 3 (the cases s < 3 being established in previous literature). That is, we establish that if f : [N] -> C is a function with |f(n)| = ��then there is a bounded complexity 3-step nilsequence F(g(n)��) which correlates with f. The approach seems to generalise so as to prove the inverse conjecture for s >= 4 as well, and a longer paper will follow concerning this. By combining this with several previous papers of the first two authors one obtains the generalised Hardy-Littlewood prime-tuples conjecture for any linear system of complexity at most 3. In particular, we have an asymptotic for the number of 5-term arithmetic progressions p_1 < p_2 < p_3 < p_4 < p_5, 49 pages, to appear in Glasgow J. Math. Fixed a problem with the file (the paper appeared in duplicate)
- Published
- 2010
5. A NOTE ON THE CLASSIFICATION OF NONCOMPACT QUASI-EINSTEIN MANIFOLDS WITH VANISHING CONDITION ON THE WEYL TENSOR
- Author
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M. Matos Neto and H. Baltazar
- Subjects
Weyl tensor ,symbols.namesake ,010308 nuclear & particles physics ,General Mathematics ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,010102 general mathematics ,0103 physical sciences ,symbols ,0101 mathematics ,Einstein ,01 natural sciences ,Mathematical physics ,Mathematics - Abstract
The aim of this paper is to study complete (noncompact) m-quasi-Einstein manifolds with λ=0 satisfying a fourth-order vanishing condition on the Weyl tensor and zero radial Weyl curvature. In this case, we are able to prove that an m-quasi-Einstein manifold (m>1) with λ=0 on a simply connected n-dimensional manifold(Mn, g), (n ≥ 4), of nonnegative Ricci curvature and zero radial Weyl curvature must be a warped product with (n–1)–dimensional Einstein fiber, provided that M has fourth-order divergence-free Weyl tensor (i.e. div4W =0).
- Published
- 2021
6. THE YONEDA EXT AND ARBITRARY COPRODUCTS IN ABELIAN CATEGORIES
- Author
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Alejandro Argudín-Monroy
- Subjects
Pure mathematics ,18E99, 18G15 ,General Mathematics ,010102 general mathematics ,Coproduct ,Mathematics - Category Theory ,01 natural sciences ,0103 physical sciences ,FOS: Mathematics ,Category Theory (math.CT) ,010307 mathematical physics ,0101 mathematics ,Abelian group ,Mathematics - Abstract
There are well known identities that involve the Ext bifunctor, coproducts, and products in Ab4 and Ab4* abelian categories with enough projectives and enough injectives. Namely, for every such category $\mathcal{A}$, the isomorphisms $\operatorname{Ext}^n (\bigoplus_{i\in I}A_{i},X) \cong \prod_{i\in I} \operatorname{Ext}^n(A_{i},X)$ and $\operatorname{Ext}^n (X,\prod_{i\in I}A_{i}) \cong \prod_{i\in I}\operatorname{Ext}^n (X,A_{i})$ always exist. The goal of this paper is to show similar isomorphisms for the Yoneda Ext in Ab4 and Ab4* abelian categories with not necessarily enough projectives nor injectives. The desired isomorphisms are constructed explicitely by using limits and colimits., 20 pages
- Published
- 2021
7. ON SOLVABILITY OF CERTAIN EQUATIONS OF ARBITRARY LENGTH OVER TORSION-FREE GROUPS
- Author
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Muhammad Akram, Muhammad Fazeel Anwar, and Mairaj Bibi
- Subjects
Pure mathematics ,Conjecture ,Group (mathematics) ,General Mathematics ,010102 general mathematics ,Torsion (algebra) ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Let G be a nontrivial torsion-free group and $s\left( t \right) = {g_1}{t^{{\varepsilon _1}}}{g_2}{t^{{\varepsilon _2}}} \ldots {g_n}{t^{{\varepsilon _n}}} = 1\left( {{g_i} \in G,{\varepsilon_i} = \pm 1} \right)$ be an equation over G containing no blocks of the form ${t^{- 1}}{g_i}{t^{ - 1}},{g_i} \in G$. In this paper, we show that $s\left( t \right) = 1$ has a solution over G provided a single relation on coefficients of s(t) holds. We also generalize our results to equations containing higher powers of t. The later equations are also related to Kaplansky zero-divisor conjecture.
- Published
- 2020
8. D3-MODULES VERSUS D4-MODULES – APPLICATIONS TO QUIVERS
- Author
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Rachid Tribak, Derya Keskin Tütüncü, and Gabriella D′Este
- Subjects
Class (set theory) ,Pure mathematics ,Direct sum ,General Mathematics ,010102 general mathematics ,Dedekind domain ,010103 numerical & computational mathematics ,01 natural sciences ,Prime (order theory) ,Discrete valuation ring ,Homomorphism ,0101 mathematics ,Commutative property ,Mathematics - Abstract
A module M is called a D4-module if, whenever A and B are submodules of M with M = A ⊕ B and f : A → B is a homomorphism with Imf a direct summand of B, then Kerf is a direct summand of A. The class of D4-modules contains the class of D3-modules, and hence the class of semi-projective modules, and so the class of Rickart modules. In this paper we prove that, over a commutative Dedekind domain R, for an R-module M which is a direct sum of cyclic submodules, M is direct projective (equivalently, it is semi-projective) iff M is D3 iff M is D4. Also we prove that, over a prime PI-ring, for a divisible R-module X, X is direct projective (equivalently, it is Rickart) iff X ⊕ X is D4. We determine some D3-modules and D4-modules over a discrete valuation ring, as well. We give some relevant examples. We also provide several examples on D3-modules and D4-modules via quivers.
- Published
- 2020
9. CONGRUENCE PRIMES FOR SIEGEL MODULAR FORMS OF PARAMODULAR LEVEL AND APPLICATIONS TO THE BLOCH–KATO CONJECTURE
- Author
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Huixi Li and Jim Brown
- Subjects
Pure mathematics ,Conjecture ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Modular form ,Automorphic form ,Congruence relation ,01 natural sciences ,0103 physical sciences ,Congruence (manifolds) ,Functional equation (L-function) ,010307 mathematical physics ,0101 mathematics ,Class number ,Mathematics ,Siegel modular form - Abstract
It has been well established that congruences between automorphic forms have far-reaching applications in arithmetic. In this paper, we construct congruences for Siegel–Hilbert modular forms defined over a totally real field of class number 1. As an application of this general congruence, we produce congruences between paramodular Saito–Kurokawa lifts and non-lifted Siegel modular forms. These congruences are used to produce evidence for the Bloch–Kato conjecture for elliptic newforms of square-free level and odd functional equation.
- Published
- 2020
10. THE CASIMIR NUMBER AND THE DETERMINANT OF A FUSION CATEGORY
- Author
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Zhihua Wang, Libin Li, and Gongxiang Liu
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Zero (complex analysis) ,Field (mathematics) ,01 natural sciences ,Casimir effect ,03 medical and health sciences ,0302 clinical medicine ,Prime factor ,Exponent ,030212 general & internal medicine ,0101 mathematics ,Cauchy's integral theorem ,Algebraically closed field ,Complex number ,Mathematics - Abstract
Let $\mathcal{C}$ be a fusion category over an algebraically closed field $\mathbb{k}$ of arbitrary characteristic. Two numerical invariants of $\mathcal{C}$ , that is, the Casimir number and the determinant of $\mathcal{C}$ are considered in this paper. These two numbers are both positive integers and admit the property that the Grothendieck algebra $(\mathcal{C})\otimes_{\mathbb{Z}}K$ over any field K is semisimple if and only if any of these numbers is not zero in K. This shows that these two numbers have the same prime factors. If moreover $\mathcal{C}$ is pivotal, it gives a numerical criterion that $\mathcal{C}$ is nondegenerate if and only if any of these numbers is not zero in $\mathbb{k}$ . For the case that $\mathcal{C}$ is a spherical fusion category over the field $\mathbb{C}$ of complex numbers, these two numbers and the Frobenius–Schur exponent of $\mathcal{C}$ share the same prime factors. This may be thought of as another version of the Cauchy theorem for spherical fusion categories.
- Published
- 2020
11. ON GILP’S GROUP-THEORETIC APPROACH TO FALCONER’S DISTANCE PROBLEM
- Author
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Han Yu
- Subjects
Discrete mathematics ,Dimension (vector space) ,Group (mathematics) ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Distance problem ,010307 mathematical physics ,0101 mathematics ,Borel set ,01 natural sciences ,Mathematics - Abstract
In this paper, we follow and extend a group-theoretic method introduced by Greenleaf–Iosevich–Liu–Palsson (GILP) to study finite points configurations spanned by Borel sets in $\mathbb{R}^n,n\geq 2,n\in\mathbb{N}.$ We remove a technical continuity condition in a GILP’s theorem in [Revista Mat. Iberoamer31 (2015), 799–810]. This allows us to extend the Wolff–Erdogan dimension bound for distance sets to finite points configurations with k points for $k\in\{2,\dots,n+1\}$ forming a $(k-1)$ -simplex.
- Published
- 2020
12. DUALIZING INVOLUTIONS ON THE METAPLECTIC GL(2) à la TUPAN
- Author
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Kumar Balasubramanian and Ekta Tiwari
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Zero (complex analysis) ,01 natural sciences ,Lift (mathematics) ,Admissible representation ,Metaplectic group ,0103 physical sciences ,Elementary proof ,Involution (philosophy) ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
Let F be a non-Archimedean local field of characteristic zero. Let G = GL(2, F) and $3\widetildeG = \widetilde{GL}(2,F)$ be the metaplectic group. Let τ be the standard involution on G. A well-known theorem of Gelfand and Kazhdan says that the standard involution takes any irreducible admissible representation of G to its contragredient. In such a case, we say that τ is a dualizing involution. In this paper, we make some modifications and adapt a topological argument of Tupan to the metaplectic group $\widetildeG$ and give an elementary proof that any lift of the standard involution to $\widetildeG$ ; is also a dualizing involution.
- Published
- 2020
13. NONBINARY DELSARTE–GOETHALS CODES AND FINITE SEMIFIELDS
- Author
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Ignacio F. Rúa
- Subjects
Discrete mathematics ,General Mathematics ,010102 general mathematics ,Galois theory ,0102 computer and information sciences ,Type (model theory) ,01 natural sciences ,Connection (mathematics) ,Nonlinear system ,010201 computation theory & mathematics ,Order (group theory) ,Binary code ,0101 mathematics ,Symplectic geometry ,Mathematics - Abstract
Symplectic finite semifields can be used to construct nonlinear binary codes of Kerdock type (i.e., with the same parameters of the Kerdock codes, a subclass of Delsarte–Goethals codes). In this paper, we introduce nonbinary Delsarte–Goethals codes of parameters $(q^{m+1}\ ,\ q^{m(r+2)+2}\ ,\ {\frac{q-1}{q}(q^{m+1}-q^{\frac{m+1}{2}+r})})$ over a Galois field of order $q=2^l$ , for all $0\le r\le\frac{m-1}{2}$ , with m ≥ 3 odd, and show the connection of this construction to finite semifields.
- Published
- 2020
14. ON SOME QUESTIONS OF PARTITIO NUMERORUM: TRES CUBI
- Author
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Robert C. Vaughan
- Subjects
Combinatorics ,General Mathematics ,010102 general mathematics ,Function (mathematics) ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
This paper is concerned with the function r3(n), the number of representations of n as the sum of at most three positive cubes, $$r_3(n) = {\mathrm{card}}\{\mathbf m\in\mathbb Z^3: m_1^3+m_2^3+m_3^3=n, m_j\ge1\}.$$ , Our understanding of this function is surprisingly poor, and we examine various averages of it. In particular $${\sum_{m=1}^nr_3(m),\,\sum_{m=1}^nr_3(m)^2}$$ and $${\sum_{\substack{ n\le x\\ n\equiv a\,\mathrm{mod}\,q }} r_3(n).\}$$
- Published
- 2020
15. BGG CATEGORY FOR THE QUANTUM SCHRÖDINGER ALGEBRA
- Author
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Yang Li and Genqiang Liu
- Subjects
Subcategory ,Weyl algebra ,Functor ,Quantum group ,General Mathematics ,010102 general mathematics ,Quiver ,Universal enveloping algebra ,Mathematics - Rings and Algebras ,010103 numerical & computational mathematics ,01 natural sciences ,Algebra ,Mathematics::Quantum Algebra ,Mathematics::Category Theory ,Mathematics - Quantum Algebra ,Lie algebra ,0101 mathematics ,Mathematics::Representation Theory ,Central charge ,Mathematics - Representation Theory ,Mathematics - Abstract
In this paper, we study the BGG category $\mathcal{O}$ for the quantum Schr{\"o}dinger algebra $U_q(\mathfrak{s})$, where $q$ is a nonzero complex number which is not a root of unity. If the central charge $\dot z\neq 0$, using the module $B_{\dot z}$ over the quantum Weyl algebra $H_q$, we show that there is an equivalence between the full subcategory $\mathcal{O}[\dot z]$ consisting of modules with the central charge $\dot z$ and the BGG category $\mathcal{O}^{(\mathfrak{sl}_2)}$ for the quantum group $U_q(\mathfrak{sl}_2)$. In the case that $\dot z=0$, we study the subcategory $\mathcal{A}$ consisting of finite dimensional $U_q(\mathfrak{s})$-modules of type $1$ with zero action of $Z$. Motivated by the ideas in \cite{DLMZ, Mak}, we directly construct an equivalent functor from $\mathcal{A}$ to the category of finite dimensional representations of an infinite quiver with some quadratic relations. As a corollary, we show that the category of finite dimensional $U_q(\mathfrak{s})$-modules is wild., Comment: 18 pages
- Published
- 2020
16. ANNIHILATOR-STABILITY AND TWO QUESTIONS OF NICHOLSON
- Author
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Yiqiang Zhou and Guoli Xia
- Subjects
Ring (mathematics) ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Stability (learning theory) ,Triangular matrix ,010103 numerical & computational mathematics ,01 natural sciences ,Negative - answer ,Annihilator ,Multiplicatively closed set ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,0101 mathematics ,Element (category theory) ,Unit (ring theory) ,Mathematics - Abstract
An element a in a ring R is left annihilator-stable (or left AS) if, whenever $Ra+{\rm l}(b)=R$ with $b\in R$ , $a-u\in {\rm l}(b)$ for a unit u in R, and the ring R is a left AS ring if each of its elements is left AS. In this paper, we show that the left AS elements in a ring form a multiplicatively closed set, giving an affirmative answer to a question of Nicholson [J. Pure Appl. Alg.221 (2017), 2557–2572.]. This result is used to obtain a necessary and sufficient condition for a formal triangular matrix ring to be left AS. As an application, we provide examples of left AS rings R over which the triangular matrix rings ${\mathbb T}_n(R)$ are not left AS for all $n\ge 2$ . These examples give a negative answer to another question of Nicholson [J. Pure Appl. Alg.221 (2017), 2557–2572.] whether R/J(R) being left AS implies that R is left AS.
- Published
- 2020
17. FRACTIONAL SCHRÖDINGER–POISSON SYSTEM WITH SINGULARITY: EXISTENCE, UNIQUENESS, AND ASYMPTOTIC BEHAVIOR
- Author
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Shengbin Yu and Jianqing Chen
- Subjects
General Mathematics ,010102 general mathematics ,Monotonic function ,Lambda ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Variational method ,Singularity ,Convergence (routing) ,symbols ,Uniqueness ,0101 mathematics ,Poisson system ,Schrödinger's cat ,Mathematics ,Mathematical physics - Abstract
In this paper, we consider the following fractional Schrödinger–Poisson system with singularity \begin{equation*} \left \{\begin{array}{lcl} ({-}\Delta)^s u+V(x)u+\lambda \phi u = f(x)u^{-\gamma}, &&\quad x\in\mathbb{R}^3,\\ ({-}\Delta)^t \phi = u^2, &&\quad x\in\mathbb{R}^3,\\ u>0,&&\quad x\in\mathbb{R}^3, \end{array}\right. \end{equation*} where 0 < γ < 1, λ > 0 and 0 < s ≤ t < 1 with 4s + 2t > 3. Under certain assumptions on V and f, we show the existence, uniqueness, and monotonicity of positive solution uλ using the variational method. We also give a convergence property of uλ as λ → 0, when λ is regarded as a positive parameter.
- Published
- 2020
18. ON THE MONODROMY AND GALOIS GROUP OF CONICS LYING ON HEISENBERG INVARIANT QUARTIC K3 SURFACES
- Author
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Florian J S C Bouyer
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Galois group ,01 natural sciences ,Moduli space ,K3 surface ,Field of definition ,Monodromy ,Conic section ,Quartic function ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Abstract
In [5], Eklund showed that a general (ℤ/2ℤ)4 -invariant quartic K3 surface contains at least 320 conics. In this paper, we analyse the field of definition of those conics as well as their Monodromy group. As a result, we prove that the moduli space of (ℤ/2ℤ)4-invariant quartic K3 surface with a certain marked conic has 10 irreducible components.
- Published
- 2019
19. QUOTIENT CATEGORIES OF n-ABELIAN CATEGORIES
- Author
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Jiaqun Wei and Qilian Zheng
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Mutation (genetic algorithm) ,0211 other engineering and technologies ,021107 urban & regional planning ,02 engineering and technology ,0101 mathematics ,Abelian group ,01 natural sciences ,Quotient ,Mathematics - Abstract
The notion of mutation pairs of subcategories in an n-abelian category is defined in this paper. Let ${\cal D} \subseteq {\cal Z}$ be subcategories of an n-abelian category ${\cal A}$. Then the quotient category ${\cal Z}/{\cal D}$ carries naturally an (n + 2) -angulated structure whenever $ ({\cal Z},{\cal Z}) $ forms a ${\cal D} \subseteq {\cal Z}$-mutation pair and ${\cal Z}$ is extension-closed. Moreover, we introduce strongly functorially finite subcategories of n-abelian categories and show that the corresponding quotient categories are one-sided (n + 2)-angulated categories. Finally, we study homological finiteness of subcategories in a mutation pair.
- Published
- 2019
20. BIFURCATION PROPERTIES FOR A CLASS OF CHOQUARD EQUATION IN WHOLE ℝ3
- Author
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Alânnio B. Nóbrega, Claudianor O. Alves, and Romildo N. de Lima
- Subjects
010101 applied mathematics ,Combinatorics ,Degree (graph theory) ,General Mathematics ,010102 general mathematics ,Hölder condition ,0101 mathematics ,01 natural sciences ,Bifurcation ,Mathematics - Abstract
This paper concerns the study of some bifurcation properties for the following class of Choquard-type equations:(P)$$\left\{ {\begin{array}{*{20}{l}} { - \Delta u = \lambda f(x)\left[ {u + \left( {{I_\alpha }*f( \cdot )H(u)} \right)h(u)} \right],{\rm{ in }} \ {{\mathbb{R}}^3},}\\ {{{\lim }_{|x| \to \infty }}u(x) = 0,\quad u(x) > 0,\quad x \in {{\mathbb{R}}^3},\quad u \in {D^{1,2}}({{\mathbb{R}}^3}),} \end{array}} \right.$$where${I_\alpha }(x) = 1/|x{|^\alpha },\,\alpha \in (0,3),\,\lambda > 0,\,f:{{\mathbb{R}}^3} \to {\mathbb{R}}$is a positive continuous function andh:${\mathbb{R}} \to {\mathbb{R}}$is a bounded Hölder continuous function. The main tools used are Leray–Schauder degree theory and a global bifurcation result due to Rabinowitz.
- Published
- 2019
21. α-TYPE CHEVALLEY–EILENBERG COHOMOLOGY OF HOM-LIE ALGEBRAS AND BIALGEBRAS
- Author
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Benedikt Hurle and Abdenacer Makhlouf
- Subjects
Pure mathematics ,Computer Science::Information Retrieval ,General Mathematics ,010102 general mathematics ,Deformation theory ,Non-associative algebra ,Structure (category theory) ,Type (model theory) ,Mathematics::Algebraic Topology ,01 natural sciences ,Cohomology ,Bracket (mathematics) ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,0103 physical sciences ,Lie algebra ,010307 mathematical physics ,Lie theory ,0101 mathematics ,Mathematics - Abstract
The purpose of this paper is to define an α-type cohomology, which we call α-type Chevalley–Eilenberg cohomology, for Hom-Lie algebras. We relate it to the known Chevalley–Eilenberg cohomology and provide explicit computations for some examples. Moreover, using this cohomology, we study formal deformations of Hom-Lie algebras, where the bracket as well as the structure map α are deformed. Furthermore, we provide a generalization of the grand crochet and study, in a particular case, the α-type cohomology for Hom-Lie bialgebras.
- Published
- 2019
22. CHARACTERIZATIONS OF BERGER SPHERES FROM THE VIEWPOINT OF SUBMANIFOLD THEORY
- Author
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In-Bae Kim, Sadahiro Maeda, and Byung Hak Kim
- Subjects
Pure mathematics ,Geodesic ,General Mathematics ,Complex projective space ,010102 general mathematics ,0103 physical sciences ,SPHERES ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,Submanifold ,01 natural sciences ,Mathematics - Abstract
In this paper, Berger spheres are regarded as geodesic spheres with sufficiently big radii in a complex projective space. We characterize such real hypersurfaces by investigating their geodesics and contact structures from the viewpoint of submanifold theory.
- Published
- 2019
23. ORIENTED FLIP GRAPHS, NONCROSSING TREE PARTITIONS, AND REPRESENTATION THEORY OF TILING ALGEBRAS
- Author
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Thomas McConville and Alexander Garver
- Subjects
General Mathematics ,010102 general mathematics ,Quiver ,0102 computer and information sciences ,01 natural sciences ,Representation theory ,Combinatorics ,010201 computation theory & mathematics ,Bounded function ,Torsion (algebra) ,0101 mathematics ,Mathematics::Representation Theory ,Indecomposable module ,Mathematics - Abstract
The purpose of this paper is to understand lattices of certain subcategories in module categories of representation-finite gentle algebras called tiling algebras, as introduced by Coelho Simões and Parsons. We present combinatorial models for torsion pairs and wide subcategories in the module category of tiling algebras. Our models use the oriented flip graphs and noncrossing tree partitions, previously introduced by the authors, and a description of the extension spaces between indecomposable modules over tiling algebras. In addition, we classify two-term simple-minded collections in bounded derived categories of tiling algebras. As a consequence, we obtain a characterization of c-matrices for any quiver mutation-equivalent to a type A Dynkin quiver.
- Published
- 2019
24. ON NONLOCAL NONLINEAR ELLIPTIC PROBLEMS WITH THE FRACTIONAL LAPLACIAN
- Author
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Li Ma
- Subjects
010101 applied mathematics ,Nonlinear system ,General Mathematics ,010102 general mathematics ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Applied mathematics ,0101 mathematics ,Fractional Laplacian ,01 natural sciences ,Mathematics - Abstract
In this paper, we study the existence of positive solutions to a semilinear nonlocal elliptic problem with the fractional α-Laplacian on Rn, 0 < α < n. We show that the problem has infinitely many positive solutions in $ {C^\tau}({R^n})\bigcap H_{loc}^{\alpha /2}({R^n}) $. Moreover, each of these solutions tends to some positive constant limit at infinity. We can extend our previous result about sub-elliptic problem to the nonlocal problem on Rn. We also show for α ∊ (0, 2) that in some cases, by the use of Hardy’s inequality, there is a nontrivial non-negative $ H_{loc}^{\alpha /2}({R^n}) $ weak solution to the problem $$ {( - \Delta )^{\alpha /2}}u(x) = K(x){u^p} \quad {\rm{ in}} \ {R^n}, $$ where K(x) = K(|x|) is a non-negative non-increasing continuous radial function in Rn and p > 1.
- Published
- 2019
25. ON IWASAWA THEORY OF RUBIN–STARK UNITS AND NARROW CLASS GROUPS
- Author
-
Youness Mazigh
- Subjects
Pure mathematics ,Class (set theory) ,Degree (graph theory) ,General Mathematics ,Modulo ,010102 general mathematics ,Iwasawa theory ,01 natural sciences ,0103 physical sciences ,010307 mathematical physics ,Ideal (ring theory) ,Inverse limit ,0101 mathematics ,Totally real number field ,Abelian group ,Mathematics - Abstract
Let K be a totally real number field of degree r. Let K∞ denote the cyclotomic -extension of K, and let L∞ be a finite extension of K∞, abelian over K. The goal of this paper is to compare the characteristic ideal of the χ-quotient of the projective limit of the narrow class groups to the χ-quotient of the projective limit of the rth exterior power of totally positive units modulo a subgroup of Rubin–Stark units, for some $\overline{\mathbb{Q}_{2}}$-irreducible characters χ of Gal(L∞/K∞).
- Published
- 2018
26. π-TYPE FERMIONS AND π-TYPE KP HIERARCHY
- Author
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Na Wang and Chuanzhong Li
- Subjects
Pure mathematics ,Hierarchy (mathematics) ,Generalization ,General Mathematics ,010102 general mathematics ,Fermion ,Construct (python library) ,Type (model theory) ,01 natural sciences ,Symmetric function ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In this paper, we first construct π-type Fermions. According to these, we define π-type Boson–Fermion correspondence which is a generalization of the classical Boson–Fermion correspondence. We can obtain π-type symmetric functions Sλπ from the π-type Boson–Fermion correspondence, analogously to the way we get the Schur functions Sλ from the classical Boson–Fermion correspondence (which is the same thing as the Jacobi–Trudi formula). Then as a generalization of KP hierarchy, we construct the π-type KP hierarchy and obtain its tau functions.
- Published
- 2018
27. DIFFERENTIAL GRADED ENDOMORPHISM ALGEBRAS, COHOMOLOGY RINGS AND DERIVED EQUIVALENCES
- Author
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Jie Zhang, Zhen Peng, and Shengyong Pan
- Subjects
Pure mathematics ,Endomorphism ,Homotopy category ,General Mathematics ,Mathematics::Rings and Algebras ,010102 general mathematics ,01 natural sciences ,Cohomology ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Special case ,Equivalence (formal languages) ,Mathematics - Abstract
In this paper, we will consider derived equivalences for differential graded endomorphism algebras by Keller's approaches. First, we construct derived equivalences of differential graded algebras which are endomorphism algebras of the objects from a triangle in the homotopy category of differential graded algebras. We also obtain derived equivalences of differential graded endomorphism algebras from a standard derived equivalence of finite dimensional algebras. Moreover, under some conditions, the cohomology rings of these differential graded endomorphism algebras are also derived equivalent. Then we give an affirmative answer to a problem of Dugas (A construction of derived equivalent pairs of symmetric algebras, Proc. Amer. Math. Soc. 143 (2015), 2281–2300) in some special case.
- Published
- 2018
28. EXISTENCE AND CONCENTRATION OF SOLUTION FOR A NON-LOCAL REGIONAL SCHRÖDINGER EQUATION WITH COMPETING POTENTIALS
- Author
-
Claudianor O. Alves and César E. Torres Ledesma
- Subjects
General Mathematics ,010102 general mathematics ,Non local ,01 natural sciences ,Schrödinger equation ,symbols.namesake ,0103 physical sciences ,symbols ,010307 mathematical physics ,Ball (mathematics) ,0101 mathematics ,Fractional Laplacian ,Mathematical physics ,Mathematics - Abstract
In this paper, we study the existence and concentration phenomena of solutions for the following non-local regional Schrödinger equation$$\begin{equation*} \left\{ \begin{array}{l} \epsilon^{2\alpha}(-\Delta)_\rho^{\alpha} u + Q(x)u = K(x)|u|^{p-1}u,\;\;\mbox{in}\;\; \mathbb{R}^n,\\ u\in H^{\alpha}(\mathbb{R}^n) \end{array} \right. \end{equation*}$$where ϵ is a positive parameter, 0 < α < 1,$1,n> 2α; (−Δ)ραis a variational version of the regional fractional Laplacian, whose range of scope is a ball with radius ρ(x) > 0, ρ,Q,Kare competing functions.
- Published
- 2018
29. COMPLEX PRODUCT STRUCTURES ON HOM-LIE ALGEBRAS
- Author
-
L. Nourmohammadifar and Esmaeil Peyghan
- Subjects
Pure mathematics ,Symmetric product ,General Mathematics ,Mathematics::Rings and Algebras ,010102 general mathematics ,01 natural sciences ,Mathematics::Category Theory ,0103 physical sciences ,Lie algebra ,Physics::Accelerator Physics ,010307 mathematical physics ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Abstract
In this paper, we introduce the notion of complex product structures on hom-Lie algebras and show that a hom-Lie algebra carrying a complex product structure is a double hom-Lie algebra and it is also endowed with a hom-left symmetric product. Moreover, we show that a complex product structure on a hom-Lie algebra determines uniquely a left symmetric product such that the complex and the product structures are invariant with respect to it. Finally, we introduce the notion of hyper-para-Kähler hom-Lie algebras and we present an example of hyper-para-Kähler hom-Lie algebras.
- Published
- 2018
30. FREE ACTION OF FINITE GROUPS ON SPACES OF COHOMOLOGY TYPE (0, b)
- Author
-
Tej Bahadur Singh, Hemant Kumar Singh, and Somorjit K. Singh
- Subjects
Finite group ,Pure mathematics ,Group (mathematics) ,General Mathematics ,010102 general mathematics ,Quaternion group ,Type (model theory) ,Space (mathematics) ,01 natural sciences ,Prime (order theory) ,Cohomology ,Action (physics) ,0103 physical sciences ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
Let G be a finite group acting freely on a finitistic space X having cohomology type (0, b) (for example, $\mathbb S$n × $\mathbb S$2n is a space of type (0, 1) and the one-point union $\mathbb S$n ∨ $\mathbb S$2n ∨ $\mathbb S$3n is a space of type (0, 0)). It is known that a finite group G that contains ℤp ⊕ ℤp ⊕ ℤp, p a prime, cannot act freely on $\mathbb S$n × $\mathbb S$2n. In this paper, we show that if a finite group G acts freely on a space of type (0, 1), where n is odd, then G cannot contain ℤp ⊕ ℤp, p an odd prime. For spaces of cohomology type (0, 0), we show that every p-subgroup of G is either cyclic or a generalized quaternion group. Moreover, for n even, it is shown that ℤ2 is the only group that can act freely on X.
- Published
- 2018
31. POTENTIALS OF A FROBENIUS-LIKE STRUCTURE
- Author
-
Alexander Varchenko and Claus Hertling
- Subjects
Power series ,General Mathematics ,010102 general mathematics ,Coordinate vector ,01 natural sciences ,Matroid ,Combinatorics ,0103 physical sciences ,Partition (number theory) ,010307 mathematical physics ,0101 mathematics ,Tuple ,Mathematics ,Ansatz - Abstract
This paper proves the existence of potentials of the first and second kind of a Frobenius like structure in a frame, which encompasses families of arrangements. The frame uses the notion of matroids. For the proof of the existence of the potentials, a power series ansatz is made. The proof that it works requires that certain decompositions of tuples of coordinate vector fields are related by certain elementary transformations. This is shown with a nontrivial result on matroid partition.
- Published
- 2018
32. THREE-DIMENSIONAL ISOLATED QUOTIENT SINGULARITIES IN EVEN CHARACTERISTIC
- Author
-
D. A. Stepanov and Vladimir Shchigolev
- Subjects
Pure mathematics ,Finite group ,Complement (group theory) ,Group (mathematics) ,General Mathematics ,010102 general mathematics ,Invariant subspace ,Mathematical analysis ,Field (mathematics) ,01 natural sciences ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,SL2(R) ,Quotient ,Vector space ,Mathematics - Abstract
This paper is a complement to the work of the second author on modular quotient singularities in odd characteristic. Here, we prove that if V is a three-dimensional vector space over a field of characteristic 2 and G < GL(V) is a finite subgroup generated by pseudoreflections and possessing a two-dimensional invariant subspace W such that the restriction of G to W is isomorphic to the group SL2(𝔽2n), then the quotient V/G is non-singular. This, together with earlier known results on modular quotient singularities, implies first that a theorem of Kemper and Malle on irreducible groups generated by pseudoreflections generalizes to reducible groups in dimension three, and, second, that the classification of three-dimensional isolated singularities that are quotients of a vector space by a linear finite group reduces to Vincent's classification of non-modular isolated quotient singularities.
- Published
- 2017
33. THE BOUNDED APPROXIMATION PROPERTY FOR THE WEIGHTED SPACES OF HOLOMORPHIC MAPPINGS ON BANACH SPACES
- Author
-
Deepika Baweja and Manjul Gupta
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Banach manifold ,Finite-rank operator ,Hardy space ,Infinite-dimensional holomorphy ,01 natural sciences ,Bounded operator ,010101 applied mathematics ,symbols.namesake ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,symbols ,Interpolation space ,Birnbaum–Orlicz space ,0101 mathematics ,Lp space ,Mathematics - Abstract
In this paper, we study the bounded approximation property for the weighted space$\mathcal{HV}$(U) of holomorphic mappings defined on a balanced open subsetUof a Banach spaceEand its predual$\mathcal{GV}$(U), where$\mathcal{V}$is a countable family of weights. After obtaining an$\mathcal{S}$-absolute decomposition for the space$\mathcal{GV}$(U), we show thatEhas the bounded approximation property if and only if$\mathcal{GV}$(U) has. In case$\mathcal{V}$consists of a single weightv, an analogous characterization for the metric approximation property for a Banach spaceEhas been obtained in terms of the metric approximation property for the space$\mathcal{G}_v$(U).
- Published
- 2017
34. BRAIDED MIXED DATUMS AND THEIR APPLICATIONS ON HOM-QUANTUM GROUPS
- Author
-
Xiaohui Zhang and Lihong Dong
- Subjects
Theoretical physics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Geodetic datum ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Quantum ,Mathematics - Abstract
In this paper, we mainly provide a categorical view on the braided structures appearing in the Hom-quantum groups. Let $\mathcal{C}$ be a monoidal category on which F is a bimonad, G is a bicomonad, and ϕ is a distributive law, we discuss the necessary and sufficient conditions for $\mathcal{C}^G_F(\varphi)$, the category of mixed bimodules to be monoidal and braided. As applications, we discuss the Hom-type (co)quasitriangular structures, the Hom–Yetter–Drinfeld modules, and the Hom–Long dimodules.
- Published
- 2017
35. PERTURBATION OF BANACH SPACE OPERATORS WITH A COMPLEMENTED RANGE
- Author
-
Carlos S. Kubrusly and B.P. Duggal
- Subjects
Pure mathematics ,Approximation property ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Eberlein–Šmulian theorem ,Infinite-dimensional vector function ,Banach space ,Finite-rank operator ,Banach manifold ,Operator theory ,01 natural sciences ,Operator space ,010101 applied mathematics ,0101 mathematics ,Mathematics - Abstract
Let${\mathcal C}[{\mathcal X}]$be any class of operators on a Banach space${\mathcal X}$, and let${Holo}^{-1}({\mathcal C})$denote the class of operatorsAfor which there exists a holomorphic functionfon a neighbourhood${\mathcal N}$of the spectrum σ(A) ofAsuch thatfis non-constant on connected components of${\mathcal N}$andf(A) lies in${\mathcal C}$. Let${{\mathcal R}[{\mathcal X}]}$denote the class of Riesz operators in${{\mathcal B}[{\mathcal X}]}$. This paper considers perturbation of operators$A\in\Phi_{+}({\mathcal X})\Cup\Phi_{-}({\mathcal X})$(the class of all upper or lower [semi] Fredholm operators) by commuting operators in$B\in{Holo}^{-1}({\mathcal R}[{\mathcal X}])$. We prove (amongst other results) that if πB(B) = ∏mi= 1(B− μi) is Riesz, then there exist decompositions${\mathcal X}=\oplus_{i=1}^m{{\mathcal X}_i}$and$B=\oplus_{i=1}^m{B|_{{\mathcal X}_i}}=\oplus_{i=1}^m{B_i}$such that: (i) If λ ≠ 0, then$\pi_B(A,\lambda)=\prod_{i=1}^m{(A-\lambda\mu_i)^{\alpha_i}} \in\Phi_{+}({\mathcal X})$(resp.,$\in\Phi_{-}({\mathcal X})$) if and only if$A-\lambda B_0-\lambda\mu_i\in\Phi_{+}({\mathcal X})$(resp.,$\in\Phi_{-}({\mathcal X})$), and (ii) (case λ = 0)$A\in\Phi_{+}({\mathcal X})$(resp.,$\in\Phi_{-}({\mathcal X})$) if and only if$A-B_0\in\Phi_{+}({\mathcal X})$(resp.,$\in\Phi_{-}({\mathcal X})$), whereB0= ⊕mi= 1(Bi− μi); (iii) if$\pi_B(A,\lambda)\in\Phi_{+}({\mathcal X})$(resp.,$\in\Phi_{-}({\mathcal X})$) for some λ ≠ 0, then$A-\lambda B\in\Phi_{+}({\mathcal X})$(resp.,$\in\Phi_{-}({\mathcal X})$).
- Published
- 2017
36. A NOTE ON THE CONNECTEDNESS OF THE BRANCH LOCUS OF RATIONAL MAPS
- Author
-
Saúl Quispe and Rubén A. Hidalgo
- Subjects
Social connectedness ,General Mathematics ,010102 general mathematics ,Holomorphic function ,010103 numerical & computational mathematics ,Automorphism ,01 natural sciences ,Cubic plane curve ,Moduli space ,Combinatorics ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,0101 mathematics ,Locus (mathematics) ,Orbifold ,Mathematics - Abstract
Milnor proved that the moduli space Md of rational maps of degree d ≥ 2 has a complex orbifold structure of dimension 2(d − 1). Let us denote by ${\mathcal S}$d the singular locus of Md and by ${\mathcal B}$d the branch locus, that is, the equivalence classes of rational maps with non-trivial holomorphic automorphisms. Milnor observed that we may identify M2 with ℂ2 and, within that identification, that ${\mathcal B}$2 is a cubic curve; so ${\mathcal B}$2 is connected and ${\mathcal S}$2 = ∅. If d ≥ 3, then it is well known that ${\mathcal S}$d = ${\mathcal B}$d. In this paper, we use simple arguments to prove the connectivity of ${\mathcal S}$d.
- Published
- 2017
37. AN EFFECTIVE BOUND FOR THE CYCLOTOMIC LOXTON–KEDLAYA RANK
- Author
-
Florin Stan, Alexandru Zaharescu, and Constantin N. Beli
- Subjects
Combinatorics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Rank (graph theory) ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we provide an explicit upper bound for the Loxton–Kedlaya rank of the maximal abelian extension of ℚ.
- Published
- 2017
38. EIGENVALUES OF GEOMETRIC OPERATORS RELATED TO THE WITTEN LAPLACIAN UNDER THE RICCI FLOW
- Author
-
Shouwen Fang, Fei Yang, and Peng Zhu
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Ricci flow ,Riemannian manifold ,Curvature ,01 natural sciences ,Manifold ,Operator (computer programming) ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Laplace operator ,Eigenvalues and eigenvectors ,Scalar curvature ,Mathematics - Abstract
Let (M, g(t)) be a compact Riemannian manifold and the metric g(t) evolve by the Ricci flow. In the paper, we prove that the eigenvalues of geometric operator −Δφ + $\frac{R}{2}$ are non-decreasing under the Ricci flow for manifold M with some curvature conditions, where Δφ is the Witten Laplacian operator, φ ∈ C2(M), and R is the scalar curvature with respect to the metric g(t). We also derive the evolution of eigenvalues under the normalized Ricci flow. As a consequence, we show that compact steady Ricci breather with these curvature conditions must be trivial.
- Published
- 2017
39. ON ∞-COMPLEX SYMMETRIC OPERATORS
- Author
-
Eungil Ko, Ji Eun Lee, and Muneo Cho
- Subjects
Algebra ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we study spectral properties and local spectral properties of ∞-complex symmetric operators T. In particular, we prove that if T is an ∞-complex symmetric operator, then T has the decomposition property (δ) if and only if T is decomposable. Moreover, we show that if T and S are ∞-complex symmetric operators, then so is T ⊗ S.
- Published
- 2017
40. PERTURBATIONS FROM INDEFINITE SYMMETRIC ELLIPTIC BOUNDARY VALUE PROBLEMS
- Author
-
Xianhua Tang and Liang Zhang
- Subjects
General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Mathematical analysis ,Multiplicity (mathematics) ,Infinity ,01 natural sciences ,Omega ,010101 applied mathematics ,Combinatorics ,Bounded function ,Domain (ring theory) ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Boundary value problem ,0101 mathematics ,Mathematics ,media_common - Abstract
In this paper, we study the multiplicity of solutions for the following problem: $$\begin{equation*} \begin{cases} -\Delta u-\Delta(|u|^{\alpha})|u|^{\alpha-2}u=g(x,u)+\theta h(x,u), \ \ x\in \Omega,\\ u=0, \ \ x\in \partial\Omega, \end{cases} \end{equation*}$$ where α ≥ 2, Ω is a smooth bounded domain in ${\mathbb{R}}$N, θ is a parameter and g, h ∈ C($\bar{\Omega}$ × ${\mathbb{R}}$). Under the assumptions that g(x, u) is odd and locally superlinear at infinity in u, we prove that for any j ∈ $\mathbb{N}$ there exists ϵj > 0 such that if |θ| ≤ ϵj, the above problem possesses at least j distinct solutions. Our results generalize some known results in the literature and are new even in the symmetric situation.
- Published
- 2017
41. SOME SPHERE THEOREMS FOR SUBMANIFOLDS WITH POSITIVE BIORTHOGONAL CURVATURE
- Author
-
Elzimar Rufino
- Subjects
Pure mathematics ,010308 nuclear & particles physics ,General Mathematics ,Second fundamental form ,010102 general mathematics ,Mathematical analysis ,Submanifold ,Curvature ,01 natural sciences ,Norm (mathematics) ,Biorthogonal system ,0103 physical sciences ,Mathematics::Differential Geometry ,Diffeomorphism ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
The purpose of this paper is to investigate sphere theorems for submanifolds with positive biorthogonal (sectional) curvature. We provide some upper bounds for the full norm of the second fundamental form under which a compact submanifold must be diffeomorphic to a sphere.
- Published
- 2017
42. EQUIVALENCE OF MODELS FOR EQUIVARIANT (∞, 1)-CATEGORIES
- Author
-
Julia E. Bergner
- Subjects
Discrete mathematics ,Pure mathematics ,Group (mathematics) ,Discrete group ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Equivariant map ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Equivalence (measure theory) ,Mathematics - Abstract
In this paper, we show that the known models for (∞, 1)-categories can all be extended to equivariant versions for any discrete groupG. We show that in two of the models we can also consider actions of any simplicial groupG.
- Published
- 2016
43. A TOWER OF RIEMANN SURFACES WHICH CANNOT BE DEFINED OVER THEIR FIELD OF MODULI
- Author
-
Mariela Carvacho, Michela Artebani, Saúl Quispe, and Rubén A. Hidalgo
- Subjects
Mathematics::Number Theory ,General Mathematics ,Riemann surface ,010102 general mathematics ,Mathematical analysis ,Field (mathematics) ,01 natural sciences ,Tower (mathematics) ,Moduli ,Riemann–Hurwitz formula ,Moduli of algebraic curves ,symbols.namesake ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Mathematics::Algebraic Geometry ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
Explicit examples of both hyperelliptic and non-hyperelliptic curves which cannot be defined over their field of moduli are known in the literature. In this paper, we construct a tower of explicit examples of such kind of curves. In that tower there are both hyperelliptic curves and non-hyperelliptic curves.
- Published
- 2016
44. DISTRIBUTIVE LATTICES OF TILTING MODULES AND SUPPORT τ-TILTING MODULES OVER PATH ALGEBRAS
- Author
-
Yichao Yang
- Subjects
Path (topology) ,Pure mathematics ,Distributive property ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we study the poset of basic tilting kQ-modules when Q is a Dynkin quiver, and the poset of basic support τ-tilting kQ-modules when Q is a connected acyclic quiver respectively. It is shown that the first poset is a distributive lattice if and only if Q is of types $\mathbb{A}_{1}$, $\mathbb{A}_{2}$ or $\mathbb{A}_{3}$ with a non-linear orientation and the second poset is a distributive lattice if and only if Q is of type $\mathbb{A}_{1}$.
- Published
- 2016
45. REAL HYPERSURFACES OF NON-FLAT COMPLEX SPACE FORMS WITH GENERALIZED ξ-PARALLEL JACOBI STRUCTURE OPERATOR
- Author
-
Th. Theofanidis
- Subjects
Pure mathematics ,Jacobi operator ,General Mathematics ,Operator (physics) ,010102 general mathematics ,Mathematical analysis ,0102 computer and information sciences ,01 natural sciences ,symbols.namesake ,Jacobi eigenvalue algorithm ,Complex space ,Jacobi rotation ,Multiplication operator ,010201 computation theory & mathematics ,Tangent space ,symbols ,0101 mathematics ,Subspace topology ,Mathematics - Abstract
The aim of the present paper is the classification of real hypersurfaces M equipped with the condition Al = lA, l = R(., ξ)ξ, restricted in a subspace of the tangent space TpM of M at a point p. This class is large and difficult to classify, therefore a second condition is imposed: (∇ξl)X = ω(X)ξ + ψ(X)lX, where ω(X), ψ(X) are 1-forms. The last condition is studied for the first time and is much weaker than ∇ξl = 0 which has been studied so far. The Jacobi Structure Operator satisfying this weaker condition can be called generalized ξ-parallel Jacobi Structure Operator.
- Published
- 2015
46. CLASSES OF SEQUENTIALLY LIMITED OPERATORS
- Author
-
Jan Fourie and Elroy D. Zeekoei
- Subjects
Pure mathematics ,Class (set theory) ,General Mathematics ,010102 general mathematics ,Banach space ,Ideal norm ,01 natural sciences ,Operator (computer programming) ,0103 physical sciences ,Multiplication ,010307 mathematical physics ,0101 mathematics ,Focus (optics) ,Mathematics ,Vector space ,Normed vector space - Abstract
The purpose of this paper is to present a brief discussion of both the normed space of operator p-summable sequences in a Banach space and the normed space of sequentially p-limited operators. The focus is on proving that the vector space of all operator p-summable sequences in a Banach space is a Banach space itself and that the class of sequentially p-limited operators is a Banach operator ideal with respect to a suitable ideal norm- and to discuss some other properties and multiplication results of related classes of operators. These results are shown to fit into a general discussion of operator [Y,p]-summable sequences and relevant operator ideals.
- Published
- 2015
47. LEBESGUE DECOMPOSITION FOR REPRESENTABLE FUNCTIONALS ON *-ALGEBRAS
- Author
-
Zsigmond Tarcsay
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,General Mathematics ,Unital ,010102 general mathematics ,010103 numerical & computational mathematics ,Absolute continuity ,Primary 46L45, 47A67, Secondary 46K10 ,Lebesgue integration ,01 natural sciences ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,symbols.namesake ,FOS: Mathematics ,Decomposition (computer science) ,symbols ,0101 mathematics ,Mathematics - Abstract
We offer a Lebesgue-type decomposition of a representable functional on a *-algebra into absolutely continuous and singular parts with respect to another. Such a result was proved by Zs. Szűcs due to a general Lebesgue decomposition theorem of S. Hassi, H.S.V. de Snoo, and Z. Sebestyén concerning non-negative Hermitian forms. In this paper, we provide a self-contained proof of Szűcs' result and in addition we prove that the corresponding absolutely continuous parts are absolutely continuous with respect to each other.
- Published
- 2015
48. FACTORIALS AND THE RAMANUJAN FUNCTION
- Author
-
Florian Luca and Jhon J. Bravo
- Subjects
Logarithm ,General Mathematics ,Diophantine equation ,Ramanujan summation ,010102 general mathematics ,Function (mathematics) ,01 natural sciences ,Ramanujan's sum ,Combinatorics ,symbols.namesake ,symbols ,Ramanujan tau function ,0101 mathematics ,Ramanujan prime ,Mathematics - Abstract
In 2006, F. Luca and I. E. Shparlinski (Proc. Indian Acad. Sci. (Math. Sci.)116(1) (2006), 1–8) proved that there are only finitely many pairs (n, m) of positive integers which satisfy the Diophantine equation |τ(n!)|=m!, where τ is the Ramanujan function. In this paper, we follow the same approach of Luca and Shparlinski (Proc. Indian Acad. Sci. (Math. Sci.)116(1) (2006), 1–8) to determine all solutions of the above equation. The proof of our main theorem uses linear forms in two logarithms and arithmetic properties of the Ramanujan function.
- Published
- 2015
49. STABLE PROPERTIES OF HYPERRELEXIVITY
- Author
-
G. K. Eleftherakis
- Subjects
General Mathematics ,010102 general mathematics ,Hilbert space ,Space (mathematics) ,01 natural sciences ,Combinatorics ,symbols.namesake ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,0103 physical sciences ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,symbols ,Equivalence relation ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
Recently, a new equivalence relation between weak* closed operator spaces acting on Hilbert spaces has appeared. Two weak* closed operator spaces ${\mathcal{U}}$, ${\mathcal{V}}$ are called weak TRO equivalent if there exist ternary rings of operators ${\mathcal{M}}$i, i = 1, 2 such that ${\mathcal{U}}=[{\mathcal{M}}_2{\mathcal{V}}{\mathcal{M}}_1^*]^{-w^*}, {\mathcal{V}}=[{\mathcal{M}}_2^*{\mathcal{U}}{\mathcal{M}}_1]^{-w^*}.$ Weak TRO equivalent spaces are stably isomorphic, and conversely, stably isomorphic dual operator spaces have normal completely isometric representations with weak TRO equivalent images. In this paper, we prove that if ${\mathcal{U}}$ and ${\mathcal{V}}$ are weak TRO equivalent operator spaces and the space of I × I matrices with entries in ${\mathcal{U}}$, MIw(${\mathcal{U}}$), is hyperreflexive for suitable infinite I, then so is MIw(${\mathcal{V}}$). We describe situations where if ${\mathcal{L}}$1, ${\mathcal{L}}$2 are isomorphic lattices, then the corresponding algebras Alg($\mathcal{L}$1), Alg($\mathcal{L}$2) have the same complete hyperreflexivity constant.
- Published
- 2015
50. MOTIVIC DONALDSON–THOMAS THEORY AND THE ROLE OF ORIENTATION DATA
- Author
-
Ben Davison
- Subjects
010308 nuclear & particles physics ,General Mathematics ,010102 general mathematics ,Donaldson–Thomas theory ,Orientation (graph theory) ,Topology ,01 natural sciences ,Wall-crossing ,Algebra ,Simple (abstract algebra) ,Component (UML) ,0103 physical sciences ,0101 mathematics ,Mathematics - Abstract
In this paper we introduce and motivate the concept of orientation data, as it appears in the framework for motivic Donaldson–Thomas theory built by Kontsevich and Soibelman. By concentrating on a single simple example we explain the role of orientation data in defining the integration map, a central component of the wall crossing formula.
- Published
- 2015
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