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In 2010, Gábor Czédli and E. Tamás Schmidt mentioned that the best cover-preserving embedding of a given semimodular lattice is not known yet [A cover-preserving embedding of semimodular lattices into geometric lattices. Advances in Mathematics, 225, 2455–2463 (2010)]. That is to say: What are the geometric lattices G such that a given finite semimodular lattice L has a cover-preserving embedding into G with the smallest ∣G∣? In this paper, we propose an algorithm to calculate all the best extending cover-preserving geometric lattices G of a given semimodular lattice L and prove that the length and the number of atoms of every best extending cover-preserving geometric lattice G equal the length of L and the number of non-zero join-irreducible elements of L, respectively. Therefore, we solve the problem on the best cover-preserving embedding of a given semimodular lattice raised by Gábor Czédli and E. Tamás Schmidt. [ABSTRACT FROM AUTHOR]

In this paper, we characterize that the boundedness, compactness and spectral structure of dual Toeplitz operators acting on the orthogonal complement of the harmonic Bergman space. This generalizes the corresponding results for dual Toeplitz operators on the orthogonal complement of the Bergman space due to Stroethoff and Zheng's paper [Trans. Amer. Math. Soc., 354, 2495–2520 (2002)]. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the idea of Song, Zhao and Huang [Czechoslov. Math. J., 70, 483–504 (2020)] and introduce the notion of right (left) Gorenstein subcategory r G (C , D) (l G (C , D)) , relative to two additive full subcategories C and D of an abelian category A . Under the assumption that C ⊆ D , we prove that the right Gorenstein subcategory r G (C , D) possesses many nice properties that it is closed under extensions, kernels of epimorphisms and direct summands. When C ⊆ D and C ⊥ D , we show that the right Gorenstein subcategory r G (C , D) admits some kind of stability. Then we discuss a resolution dimension for an object in A , called r G (C , D) -projective dimension. Finally, we prove that if (U , V) is a hereditary cotorsion pair with kernel C has enough injectives, such that U ⊆ D and U ⊥ D , then (r G (C , D) , r e s C ^ , C) is a weak Auslander—Buchweitz context, where res C ^ is the subcategory of A consisting of objects with finite C -projective dimension. [ABSTRACT FROM AUTHOR]

Zhao and Ho asked in a recent paper that for each T0 space X, whether KB(X) (the set of all irreducible closed sets of X whose suprema exist) is the canonical k-bounded sobrification of X in the sense of Keimel and Lawson. In this paper, we construct a counterexample to give a negative answer. We also consider the subcategory Topκ of the category Top0 of T0 spaces, and prove that the category KBSob of k-bounded sober spaces is a full reflective subcategory of the category KBSob of k-bounded sober spaces is a full reflective subcategory of the category Topκ. [ABSTRACT FROM AUTHOR]

This paper contains two parts toward studying abelian varieties from the classification point of view. In a series of papers [Doc. Math., 21, 1607–1643 (2016)], [Taiwanese J. Math., 20(4), 723–741 (2016)], etc., the current authors and T. C. Yang obtain explicit formulas for the numbers of superspecial abelian surfaces over finite fields. In this paper, we give an explicit formula for the size of the isogeny class of simple abelian surfaces with real Weil number √q. This establishes a key step that extends our previous explicit calculation of superspecial abelian surfaces to those of supersingular abelian surfaces. The second part is to introduce the notion of genera and idealcomplexes of abelian varieties with additional structures in a general setting. The purpose is to generalize the previous work by the second named author [Forum Math., 22(3), 565–582 (2010)] on abelian varieties with additional structures to similitude classes, which establishes more results on the connection between geometrically defined and arithmetically defined masses for further investigations. [ABSTRACT FROM AUTHOR]

Let ( T, d) be a dendrite with finite branch points and f be a continuous map from T to T. Denote by ω( x, f) and P( f) the ω-limit set of x under f and the set of periodic points of f, respectively. Write Ω( x, f) = { y| there exist a sequence of points x ∈ T and a sequence of positive integers n < n < ··· such that lim x = x and lim $$f^{n_{k}}$$ ( x ) = y}. In this paper, we show that the following statements are equivalent: (1) f is equicontinuous. (2) ω( x, f) = Ω( x, f) for any x ∈ T. (3) ∩ f ( T) = P( f), and ω( x, f) is a periodic orbit for every x ∈ T and map h: x → ω( x, f) ( x ∈ T) is continuous. (4) Ω( x, f) is a periodic orbit for any x ∈ T. [ABSTRACT FROM AUTHOR]

A simple graph G is a 2-tree if G = K3, or G has a vertex v of degree 2, whose neighbors are adjacent, and G — v is a 2-tree. Clearly, if G is a 2-tree on n vertices, then E(G) = 2n — 3. A non-increasing sequence π = (d1,...,dn) of nonnegative integers is a graphic sequence if it is realizable by a simple graph G on n vertices. [Acta Math. Sin. Engl. Ser., 25, 795–802 (2009)] proved that if k ≥ 2, n ≥ and π = (d1,...,dn) is a graphic sequence with , then p has a realization containing every 1-tree (the usual tree) on k vertices. Moreover, the lower bound (k — 2)n is the best possible. This is a variation of a conjecture due to Erdos and Sós. In this paper, we investigate an analogue problem for 2-trees and prove that if k ≥ 3 is an integer with k = i (mod3), π = (d1,...,dn is a graphic sequence with , then π has a realization containing every 2-tree on k vertices. Moreover, the lower bound max is the best possible. This result implies a conjecture due to [Discrete Math. Theor. Comput. Sci., 17(3), 315–326 (2016)]. [ABSTRACT FROM AUTHOR]

Let R be a ring, M be a R-bimodule and m, n be two fixed nonnegative integers with m + n ≠ 0. An additive mapping δ from R into M is called an (m, n)-Jordan derivation if (m + n)δ(A2) = 2mAδ(A) + 2nδ(A)A for every A in R. In this paper, we prove that every (m, n)-Jordan derivation with m ≠ n from a C* -algebra into its Banach bimodule is zero. An additive mapping δ from R into M is called a (m, n)-Jordan derivable mapping at W in R if (m + n)δ(AB + BA) = 2mδ(A)B + 2mδ(B)A + 2nAδ(B) + 2nBδ(A) for each A and B in R with AB = BA = W. We prove that if M is a unital A-bimodule with a left (right) separating set generated algebraically by all idempotents in A, then every (m, n)-Jordan derivable mapping at zero from A into M is identical with zero. We also show that if A and B are two unital algebras, M is a faithful unital (A, B)-bimodule and U=[AMNB] is a generalized matrix algebra, then every (m, n)-Jordan derivable mapping at zero from U into itself is equal to zero. [ABSTRACT FROM AUTHOR]

In this paper, we consider the weakly damped wave equations with hereditary effects, and the nonlinearity f satisfies critical growth. The delay term g(t, ut) may be driven by a function with very weak assumptions, namely, just measurability. We analyze the well-posedness of solutions and verify the existence of the pullback D-attractor in CH01(Ω)×CL2(Ω) by constructing the energy functional and combining with the idea of the contractive function. [ABSTRACT FROM AUTHOR]

In this paper, we present a local Csörgö-Révész type functional limit theorem for increments of Brownian motion and give its convergence rate. The results also extend the functional forms of Lévy’s modulus of continuity for Brownian motion. [ABSTRACT FROM AUTHOR]

Let S = ( a , ..., a ; b , ..., b ), where a , ..., a and b , ..., b are two nonincreasing sequences of nonnegative integers. The pair S = ( a , ..., a ; b , ..., b ) is said to be a bigraphic pair if there is a simple bipartite graph G = ( X ∪ Y, E) such that a , ..., a and b , ..., b are the degrees of the vertices in X and Y, respectively. Let Z be the cyclic group of order 3. Define σ( Z , m, n) to be the minimum integer k such that every bigraphic pair S = ( a , ..., a ; b , ..., b ) with a , b ≥ 2 and σ( S) = a + ⋯ + a ≥ k has a Z -connected realization. For n = m, Yin [ Discrete Math., 339, 2018-2026 (2016)] recently determined the values of σ( Z , m, m) for m ≥ 4. In this paper, we completely determine the values of σ( Z , m, n) for m ≥ n ≥ 4. [ABSTRACT FROM AUTHOR]

In this paper, we present some basic notions of simple ordered semihypergroups and regular ordered Krasner hyperrings and prove some results in this respect. In addition, we describe pure hyperideals of ordered Krasner hyperrings and investigate some properties of them. Finally, some results concerning purely prime hyperideals are proved. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the intersection numbers of nearly Kirkman triple systems. J [ v] is the set of all integers k such that there is a pair of NKTS( v)s with a common uncovered collection of 2-subset intersecting in k triples. It has been established that $${J_N}\left[ v \right] = \left\{ {0,1,...,\frac{{v\left( {v - 2} \right)}}{6} - 6,\frac{{v\left( {v - 2} \right)}}{6} - 4,\frac{{v\left( {v - 2} \right)}}{6}} \right\}$$ for any integers v = 0 (mod 6) and v ≥ 66. For v ≤ 60, there are 8 cases left undecided. [ABSTRACT FROM AUTHOR]

In this paper, we prove that a product F × F of saturated fusion systems is exotic if and only if at least one of the factors is exotic. This result provides a method to construct new exotic fusion systems by known exotic fusion systems. We also investigate central products of saturated fusion systems. [ABSTRACT FROM AUTHOR]

In this paper, by using the parametric representation of spirallike mappings, we construct to obtain the growth theorem of spirallike mappings on Reinhardt domain B. Moreover, the distortion theorem of spirallike mappings is obtained. [ABSTRACT FROM AUTHOR]

By Petersen's Theorem, a bridgeless cubic graph has a 2-factor. Fleischner ( Discrete Math., 101, 33-37 (1992)) has extended this result to bridgeless graphs of minimum degree at least three by showing that every such graph has an even factor without isolated vertices. Let m > 0 be even and m > 0 be odd. In this paper, we prove that every m-edge-connected graph with minimum degree at least m + 1 contains an even factor with minimum degree at least m and every ( m + 1)-edge-connected graph contains an odd factor with minimum degree at least m, which further extends Fleischner's result. Moreover, we show that our results are best possible. [ABSTRACT FROM AUTHOR]

For 0 < p < 1, Haberl and Ludwig defined the notions of symmetric L-intersection body and nonsymmetric L-intersection body. In this paper, we introduce the general L-intersection bodies. Furthermore, the Busemann-Petty problems for the general L-intersection bodies are shown. [ABSTRACT FROM AUTHOR]

We show that, up to isomorphism, there is a unique non-CI connected cubic Cayley graph on the dihedral group of order 2n for each even number n ≥ 4. This answers in the negative the question of Li whether all connected cubic Cayley graphs are CI-graphs (Discrete Math., 256, 301–334 (2002)). As an application, a formula is derived for the number of isomorphism classes of connected cubic Cayley graphs on dihedral groups, which generalises the earlier formula of Huang et al. dealing with the particular case when n is a prime (Acta Math. Sin., Engl. Ser., 33, 996–1011 (2017)). As another application, a short proof is also given for a result on sparse circulant matrices obtained by Wiedemann and Zieve (arXiv preprint, (2007)). [ABSTRACT FROM AUTHOR]

This is a set of lecture notes for the first author's lectures on the difference equations in 2019 at the Institute of Advanced Study for Mathematics at Zhejiang University. We focus on explicit computations and examples. The convergence of local solutions is discussed. [ABSTRACT FROM AUTHOR]

In this article, we showed that positive relative entropy implies mean Li-Yorke chaos along some good sequences in the fiber. Our result extended Li and Qiao's result in [Monatsh. Math., 186 (2018), 153–173]. [ABSTRACT FROM AUTHOR]

Let B be the unit disc in ℝ2, ℋ be the completion of C 0 ∞ (B) under the norm ∥ u ∥ H = (∫ B | ▽ u | 2 d x − ∫ B u 2 (1 − | x | 2) 2 d x) 1 2 , ∀ u ∈ C 0 ∞ (B). By the method of blow-up analysis and an argument of rearrangement with respect to the standard hyperbolic metric d v H = d x (1 − | x | 2 ) 2 , we prove that, for any fixed α , 0 ≤ α < λ p (B) = inf u ∈ H , u ≠ 0 ∥ u ∥ H 2 ∥ u ∥ p 2 , the supremum sup u ∈ H , ∥ u ∥ H = 1 ∫ B e 4 π (1 + α ∥ u ∥ p 2) u 2 d x < + ∞ , ∀ p > 1. This is an analog of early results of Lu—Yang (Discrete Contin. Dyn. Syst., 2009) and Yang (Trans. Amer. Math. Soc., 2007), and extends those of Wang—Ye (Adv. Math., 2012) and Yang—Zhu (Ann. Global Anal. Geom., 2016). [ABSTRACT FROM AUTHOR]

In this article, we investigate the arithmetic behavior of the function D3(n) which counts the number of 3-regular tripartitions of n. For example, we show that for α ≥ 1 and n ≥ 0, D3(32αn+11⋅32α−1−14)≡0(mod32α+3) and D3(32αn+7⋅32α−1−14)≡0(mod32α+2). [ABSTRACT FROM AUTHOR]

Let X be a Peano continuum having a free arc. If X admits a sensitive open map, then X either is homeomorphic to the closed interval [0, 1], or is homeomorphic to the unit circle S. [ABSTRACT FROM AUTHOR]

Let G = ( V ( G), E( G)) be a graph with vertex set V ( G) and edge set E( G). For two distinct vertices x and y of a graph G, let R{ x, y} denote the set of vertices z such that the distance from x to z is not equal to the distance from y to z in G. For a function g defined on V ( G) and for U ⊆ V ( G), let g( U) = Σ g( s). A real-valued function g: V ( G) → [0, 1] is a resolving function of G if g( R{ x, y}) ≥ 1 for any two distinct vertices x, y ∈ V ( G). The fractional metric dimension dim ( G) of a graph G is min{ g( V ( G)): g is a resolving function of G}. Let G and G be disjoint copies of a graph G, and let σ: V ( G) → V ( G) be a bijection. Then, a permutation graph G = ( V, E) has the vertex set V = V ( G) ∪ V ( G) and the edge set E = E( G) ∪ E( G) ∪ { uv | v = σ( u)}. First, we determine dim f ( T) for any tree T. We show that $1 < \dim _f (G_\sigma ) \leqslant \tfrac{1} {2}(|V(G)| + |S(G)|) $ for any connected graph G of order at least 3, where S( G) denotes the set of support vertices of G. We also show that, for any ɛ > 0, there exists a permutation graph G such that dim( G) - 1 < ε. We give examples showing that neither is there a function h such that dim( G) < h(dim ( G)) for all pairs ( G, σ), nor is there a function h such that h(dim ( G)) > dim( G)) for all pairs ( G, σ). Furthermore, we investigate dim( G)) when G is a complete k-partite graph or a cycle. [ABSTRACT FROM AUTHOR]

We consider context-free grammars of the form $G = \{ f \to f^{b_1 + b_2 + 1} g^{a_1 + a_2 } ,g \to f^{b_1 } g^{a_1 + 1} \} $, where a and b are integers subject to certain positivity conditions. Such a grammar G gives rise to triangular arrays { T( n, k)} satisfying a three-term recurrence relation. Many combinatorial sequences can be generated in this way. Let T( x) = Σ T( n, k) x. Based on the differential operator with respect to G, we define a sequence of linear operators P such that T( x) = P( T( x)). Applying the characterization of real stability preserving linear operators on the multivariate polynomials due to Borcea and Brändén, we obtain a necessary and sufficient condition for the operator P to be real stability preserving for any n. As a consequence, we are led to a sufficient condition for the real-rootedness of the polynomials defined by certain triangular arrays, obtained by Wang and Yeh. Moreover, as special cases we obtain grammars that lead to identities involving the Whitney numbers and the Bessel numbers. [ABSTRACT FROM AUTHOR]

We obtain the explicit formulae for the harmonic Bergman kernels of B\{0} and $\mathbb{R}^n \backslash \overline {B_n } $ and study the connection between harmonic Bergman kernel and weighted harmonic Bergman kernel. We also get the explicit formula for the weighted harmonic Bergman kernel of B\{0} with the weight $\tfrac{1} {{|x|^4 }} $. [ABSTRACT FROM AUTHOR]

Let $\mathbb{F}_p $ be the finite field of p elements with p prime. If A is a subset of $\mathbb{F}_p $ and g is an element of $\mathbb{F}_p^* $ with order ν, then . [ABSTRACT FROM AUTHOR]