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Abstract: Abstract Let p be an odd prime. By using the elementary methods we prove that: (1) if 2 ∤ x, p = ±3 (mod 8), the Diophantine equation (2x − 1)(py − 1) = 2z2 has no positive integer solution except when p = 3 or p is of the form \(p = 2a_0^2 + 1\) , where a0 > 1 is an odd positive integer. (2) if 2 ∤ x, 2 ∣; y, y ≠ 2, 4, then the Diophantine equation (2x − 1)(py − 1) = 2z2 has no positive integer solution. PubDate: 2021-10-01

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Abstract: Abstract It is well known that people can derive the radiation MHD model from an MHD-P1 approximate model. As pointed out by F. Xie and C. Klingenberg (2018), the uniform regularity estimates play an important role in the convergence from an MHD-P1 approximate model to the radiation MHD model. The aim of this paper is to prove the uniform regularity of strong solutions to an isentropic compressible MHD-P1 approximate model arising in radiation hydrodynamics. Here we use the bilinear commutator and product estimates to obtain our result. PubDate: 2021-10-01

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Abstract: Abstract We prove that for normal operators N1, \({N_2} \in {\cal L}({\cal H})\) , the generalized commutator [N1, N2; X] approaches zero when [N1, N2; [N1, N2; X]] tends to zero in the norm of the Schatten-von Neumann class \({{\cal C}_p}\) with p > 1 and X varies in a bounded set of such a class. PubDate: 2021-10-01

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Abstract: Abstract Let G be a finite group. If G has two rows which differ in only one entry in the character table, we call G an RD1-group. We investigate the character tables of RD1-groups and get some necessary and sufficient conditions about RD1-groups. PubDate: 2021-10-01

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Abstract: Abstract We study Morse-Bott functions with two critical values (equivalently, non-constant without saddles) on closed surfaces. We show that only four surfaces admit such functions (though in higher dimensions, we construct many such manifolds, e.g. as fiber bundles over already constructed manifolds with the same property). We study properties of such functions. Namely, their Reeb graphs are path or cycle graphs; any path graph, and any cycle graph with an even number of vertices, is isomorphic to the Reeb graph of such a function. They have a specific number of center singularities and singular circles with nonorientable normal bundle, and an unlimited number (with some conditions) of singular circles with orientable normal bundle. They can, or cannot, be chosen as the height function associated with an immersion of the surface in the three-dimensional space, depending on the surface and the Reeb graph. In addition, for an arbitrary Morse-Bott function on a closed surface, we show that the Euler characteristic of the surface is determined by the isolated singularities and does not depend on the singular circles. PubDate: 2021-10-01

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Abstract: Abstract Let T be a tree. Then a vertex of T with degree one is a leaf of T and a vertex of degree at least three is a branch vertex of T. The set of leaves of T is denoted by L(T) and the set of branch vertices of T is denoted by B(T). For two distinct vertices u, v of T, let PT[u, v] denote the unique path in T connecting u and v. Let T be a tree with B(T) ≠ ∅. For each leaf x of T, let yx denote the nearest branch vertex to x. We delete V(PT[x, yx]) {yx} from T for all x ∈ L(T). The resulting subtree of T is called the reducible stem of T and denoted by R_Stem(T). We give sharp sufficient conditions on the degree sum for a graph to have a spanning tree whose reducible stem has a few branch vertices. PubDate: 2021-10-01

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Abstract: Abstract We study algebraic properties of two Toeplitz operators on the weighted Bergman space on the unit disk with harmonic symbols. In particular the product property and commutative property are discussed. Further we apply our results to solve a compactness problem of the product of two Hankel operators on the weighted Bergman space on the unit bidisk. PubDate: 2021-10-01

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Abstract: Abstract A perturbation of the de Rham complex was introduced by Witten for an exact 1-form Θ and later extended by Novikov for a closed 1-form on a Riemannian manifold M. We use invariance theory to show that the perturbed index density is independent of Θ; this result was established previously by J. A. Álvarez López, Y. A. Kordyukov and E. Leichtnam (2020) using other methods. We also show the higher order heat trace asymptotics of the perturbed de Rham complex exhibit nontrivial dependence on Θ. We establish similar results for manifolds with boundary imposing suitable boundary conditions and give an equivariant version for the local Lefschetz trace density. In the setting of the Dolbeault complex, one requires Θ to be a \(\overline \partial \) closed 1-form to define a local index density; we show in contrast to the de Rham complex, that this exhibits a nontrivial dependence on Θ even in the setting of Riemann surfaces. PubDate: 2021-10-01

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Abstract: Abstract We introduce a higher dimensional analogue of the Engel structure, motivated by the Cartan prolongation of contact manifolds. We study the stability of such structure, generalizing the Gray-type stability results for Engel manifolds. We also derive local normal forms defining such a distribution. PubDate: 2021-10-01

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Abstract: Abstract We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation with the initial data u0 ∈ X, where \(X \in \{M_{2,q}^s(\mathbb{R}),\,{H^\sigma}(\mathbb{T}),\,{H^{{s_1}}}(\mathbb{R}) + {H^{{s_2}}}(\mathbb{T})\}\) and q ∈ [1, 2], s ⩾ 0, or σ ⩾ 0, or s2 ⩾ s1 ⩾ 0. Moreover, if M 2,q s (ℝ) ↪ L3(ℝ), or if \(\sigma \geqslant {1 \over 6}\) , or if \({s_1} \geqslant {1 \over 6}\) and \({s_2} > {1 \over 2}\) we how that the Cauchy problem is unconditionally wellposed in X. Similar results hold true for all higher order nonlinear Schrödinger equations and mixed order NLS due to a factorization property of the corresponding phase factors. For the proof we employ the normal form reduction via the differentiation by parts technique and build upon our previous work. PubDate: 2021-10-01

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Abstract: Abstract Let K be a field, and let G be a group. In the present paper, we investigate when the group ring K[G] has finite weak dimension and finite Gorenstein weak dimension. We give some analogous versions of Serre’s theorem for the weak dimension and the Gorenstein weak dimension. PubDate: 2021-10-01

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Abstract: Abstract We explore the connection between atomicity in Prüfer domains and their corresponding class groups. We observe that a class group of infinite order is necessary for non-Noetherian almost Dedekind and Prüfer domains of finite character to be atomic. We construct a non-Noetherian almost Dedekind domain and exhibit a generating set for the ideal class semigroup. PubDate: 2021-10-01

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Abstract: Abstract Let R be a Noetherian ring, and I and J be two ideals of R. Let S be a Serre subcategory of the category of R-modules satisfying the condition CI and M be a ZD-module. As a generalization of the S-depth(I, M) and depth(I, J, M), the S-depth of (I, J) on M is defined as \(S{\rm{ - depth}}\left( {I,J,M} \right) = \inf \left\{ {S{\rm{ - depth}}\left( {\mathfrak{a},M} \right): \mathfrak{a}\in \widetilde{W}\left( {I,J} \right)} \right\}\) , and some properties of this concept are investigated. The relations between S-depth(I, J, M) and H I,J i (M) are studied, and it is proved that S-depth(I, J, M) = inf{i: H I,J i (M) ∉ S}, where S is a Serre subcategory closed under taking injective hulls. Some conditions are provided that local cohomology modules with respect to a pair of ideals coincide with ordinary local cohomology modules under these conditions. Let SuppRH I,J i (M) be a finite subset of Max(R) for all i < t, where M is an arbitrary R-module and t is an integer. It is shown that there are distinct maximal ideals \(\mathfrak{m}_1,\mathfrak{m}_2,...\mathfrak{m}_k\in{W(I,J)}\) such that \(H_{I,J}^i\left( M \right) \cong H_{\mathfrak{m}{_1}}^i\left( M \right) \oplus H_{\mathfrak{m}{_2}}^i\left( M \right) \oplus \ldots \oplus H_{\mathfrak{m}_{_k}}^i\left( M \right)\) for all i < t. PubDate: 2021-10-01

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Abstract: Abstract Tilting theory plays an important role in the representation theory of coalgebras. This paper seeks how to apply the theory of localization and colocalization to tilting torsion theory in the category of comodules. In order to better understand the process, we give the (co)localization for morphisms, (pre)covers and special precovers. For that reason, we investigate the (co)localization in tilting torsion theory for coalgebras. PubDate: 2021-10-01

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Abstract: Abstract for each squarefree monomial ideal I ⊂ S = k[x1, …, xn], we associate a simple finite graph GI by using the first linear syzygies of I. The nodes of GI are the generators of I, and two vertices ui and uj are adjacent if there exist variables x, y such that xui = yuj. In the cases, where GI is a cycle or a tree, we show that I has a linear resolution if and only if I has linear quotients and if and only if I is variable-decomposable. In addition, with the same assumption on GI, we characterize all squarefree monomial ideals with a linear resolution. Using our results, we characterize all Cohen-Macaulay codimension 2 monomial ideals with a linear resolution. As another application of our results, we also characterize all Cohen-Macaulay simplicial complexes in the case, where \({G_{\rm{\Delta }}} \cong {G_{{I_{\rm{\Delta }}} \vee }}\) is a cycle or a tree. PubDate: 2021-10-01

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Abstract: Abstract We provide a construction of monomial ideals in R = K[x, y] such that μ(I2) < μ(I), where μ denotes the least number of generators. This construction generalizes the main result of S. Eliahou, J. Herzog, M. Mohammadi Saem (2018). Working in the ring R, we generalize the definition of a Freiman ideal which was introduced in J. Herzog, G. Zhu (2019) and then we give a complete characterization of such ideals. A particular case of this characterization leads to some further investigations on μ(Ik) that generalize some results of S. Eliahou, J. Herzog, M. Mohammadi Saem (2018), J. Herzog, M. Mohammadi Saem, N. Zamani (2019), and J. Herzog, A. Asloob Qureshi, M. Mohammadi Saem (2019). PubDate: 2021-10-01

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Abstract: Abstract An m × n matrix R with nonnegative entries is called row stochastic if the sum of entries on every row of R is 1. Let Mm,n be the set of all m × n real matrices. For A, B ∈ Mm,n, we say that A is row Hadamard majorized by B (denoted by A ≺ RHB) if there exists an m × n row stochastic matrix R such that A = R ο B, where X ο Y is the Hadamard product (entrywise product) of matrices X, Y ∈ Mm,n. In this paper, we consider the concept of row Hadamard majorization as a relation on Mm,n and characterize the structure of all linear operators T: Mm,n → Mm,n preserving (or strongly preserving) row Hadamard majorization. Also, we find a theoretic graph connection with linear preservers (or strong linear preservers) of row Hadamard majorization, and we give some equivalent conditions for these linear operators on Mn. PubDate: 2021-10-01

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Abstract: Abstract We study the generalized k-connectivity κk(G) as introduced by Hager in 1985, as well as the more recently introduced generalized k-edge-connectivity λk(G). We determine the exact value of κk (G) and λk (G) for the line graphs and total graphs of trees, unicyclic graphs, and also for complete graphs for the case k = 3. PubDate: 2021-10-01

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Abstract: Abstract We give some characterizations for radial Minkowski additive operators and prove a new characterization of balls. Finally, we show the property of radial Minkowski homomorphism. PubDate: 2021-10-01

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Abstract: Abstract A subgroup H of a finite group G is said to be SS-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K is S-quasinormal in K. We analyze how certain properties of SS-supplemented subgroups influence the structure of finite groups. Our results improve and generalize several recent results. PubDate: 2021-10-01