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Abstract: Abstract Let (X, d, μ) be a metric measure space with non-negative Ricci curvature. This paper is concerned with the boundary behavior of harmonic function on the (open) upper half-space X × ℝ+. We derive that a function f of bounded mean oscillation (BMO) is the trace of harmonic function u(x, t) on X × ℝ+, u(x, 0) = f(x), whenever u satisfies the following Carleson measure condition $$\mathop {{\rm{sup}}}\limits_{{x_B},{r_B}} \int_0^{{r_B}} {{{\rlap{--} \smallint }_{B\left( {{x_B},{r_B}} \right)}}} {\left {t\nabla u\left( {x,t} \right)} \right ^2}{\rm{d}}\mu \left( x \right){{{\rm{d}}t} \over t} \le C < \infty ,$$ where ∇ = (∇x,∂t) denotes the total gradient and B(xB, rB) denotes the (open) ball centered at xB with radius rB. Conversely, the above condition characterizes all the harmonic functions whose traces are in BMO space. PubDate: 2022-06-01

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Abstract: Abstract In this survey, we give a neat summary of the applications of the multi-resolution analysis to the studies of Besov-Q type spaces \(\dot B_{p,q}^{{\gamma _1},{\gamma _2}}\) (ℝn)and Triebel-Lizorkin-Q type spaces \(\dot B_{p,q}^{{\gamma _1},{\gamma _2}}\) (ℝn). We will state briefly the recent progress on the wavelet characterizations, the boundedness of Calderón-Zygmund operators, the boundary value problem of \(\dot B_{p,q}^{{\gamma _1},{\gamma _2}}\) (ℝn) and \(\dot F_{p,q}^{{\gamma _1},{\gamma _2}}\) (ℝn). We also present the recent developments on the well-posedness of fluid equations with small data in \(\dot B_{p,q}^{{\gamma _1},{\gamma _2}}\) (ℝn) and \(\dot F_{p,q}^{{\gamma _1},{\gamma _2}}\) (ℝn). PubDate: 2022-06-01

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Abstract: Abstract In this paper, we study the category of corepresentations of a monoidal comonad. We show that it is a semisimple category if and only if the monoidal comonad is a cosemisipmle (coseparable) comonad, and it is a braided category if and only if the monoidal comonad admit a cobraided structure. At last, as an application, the braided structure and the semisimplicity of the Hom-comodule category of a monoidal Hom-bialgebra are discussed. PubDate: 2022-06-01

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Abstract: Abstract A coloring of a graph G is injective if its restriction to the neighbour of any vertex is injective. The injective chromatic number χi(G) of a graph G is the least k such that there is an injective k-coloring. In this paper, we prove that for each planar graph with g ≥ 5 and Δ (G) ≥ 20, χi(G) ≤ Δ(G)+ 3. PubDate: 2022-06-01

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Abstract: Abstract In this paper we introduce the history and present situation of the computation of the cohomology rings of Kac-Moody groups, their flag manifolds and classifying spaces, and give some problems and conjectures that deserve further study. PubDate: 2022-06-01

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Abstract: Abstract Positive-instantaneous frequency representation for transient signals has always been a great concern due to its theoretical and practical importance, although the involved concept itself is paradoxical. The desire and practice of uniqueness of such frequency representation (decomposition) raise the related topics in approximation. During approximately the last two decades there has formulated a signal decomposition and reconstruction method rooted in harmonic and complex analysis giving rise to the desired signal representations. The method decomposes any signal into a few basic signals that possess positive instantaneous frequencies. The theory has profound relations to classical mathematics and can be generalized to signals defined in higher dimensional manifolds with vector and matrix values, and in particular, promotes kernel approximation for multi-variate functions. This article mainly serves as a survey. It also gives two important technical proofs of which one for a general convergence result (Theorem 3.4), and the other for necessity of multiple kernel (Lemma 3.7). Expositorily, for a given real-valued signal f one can associate it with a Hardy space function F whose real part coincides with f. Such function F has the form F = f + iH f, where H stands for the Hilbert transformation of the context. We develop fast converging expansions of F in orthogonal terms of the form $$F = \sum\limits_{k = 1}^\infty {{c_k}{B_k},} $$ where Bk’s are also Hardy space functions but with the additional properties $${B_k}(t) = {\rho _k}(t){e^{i{\theta _k}(t)}},\,\,\,\,\,\,{\rho _k} \geqslant 0,\,\,\,\,\,\,\,\theta _k^\prime (t) \geqslant 0,\,\,\,\,\,\,{\rm{a}}.{\rm{e}}.$$ The original real-valued function f is accordingly expanded $$f = \sum\limits_{k = 1}^\infty {{\rho _k}(t)\cos {\theta _k}(t)} $$ which, besides the properties of ρk and θk given above, also satisfies $$H({\rho _k}\cos {\theta _k})(t) = {\rho _k}(t)\sin {\theta _k}(t).$$ Real-valued functions f (t)= ρ(t)cos θ(t) that satisfy the condition $$\rho \geqslant 0,\,\,\,\,\,\,\,{\theta ^\prime }(t) \geqslant 0,\,\,\,\,\,\,\,\,H(\rho \cos \theta )(t) = \rho (t)\sin \theta (t)$$ are called mono-components. If f is a mono-component, then the phase derivative θ′(t) is defined to be instantaneous frequency of f. The above described positive-instantaneous frequency expansion is a generalization of the Fourier series expansion. Mono-components are crucial to understand the concept instantaneous frequency. We will present several most important mono-component function classes. Decompositions of signals into mono-components are called adaptive Fourier decompositions (AFDs). We note that some scopes of the studies on the 1D mono-components and AFDs can be extended to vector-valued or even matrix-valued signals defined on higher dimensional manifolds. We finally provide an account of related studies in pure and applied mathematics. PubDate: 2022-06-01

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Abstract: Abstract Let A ⊂ ℤN, and $${f_A}(s) = \left\{ {\matrix{{1 - {{\left A \right } \over N},} & {{\rm{for}}\,\,s \in A,} \cr { - {{\left A \right } \over N},} & {{\rm{for}}\,\,s \notin A.} \cr } } \right.$$ We define the pseudorandom measure of order k of the subset A as follows $${P_k}(A,N) = \mathop {\max }\limits_D \left {\sum\limits_{n \in {\mathbb{Z}_N}} {{f_A}(n + {c_1}){f_A}(n + {c_2}) \cdots {f_A}(n + {c_k})} } \right ,$$ where the maximum is taken over all D = (c1,c2,⋯,ck) ∈ ℤk with 0 ≼ c1< c2< ⋯ <ck ≼ N − 1. The subset A ⊂ ℤN is considered as a pseudorandom subset of degree k if Pk (A, N) is “small” in terms of N. We establish a link between the Gowers norm and our pseudorandom measure, and show that “good” pseudorandom subsets must have “small” Gowers norm. We give an example to suggest that subsets with “small” Gowers norm may have large pseudorandom measure. Finally, we prove that the pseudorandom subset of degree L(k) contains an arithmetic progression of length k, where $$L(k) = 2 \cdot {\rm{lcm}}\left( {2,4, \ldots ,2\left\lfloor {{k \over 2}} \right\rfloor } \right),\,\,\,\,\,\,{\rm{for}}\,k \geqslant 4,$$ and lcm(α1, a2,…,aι) denotes the least common multiple of a1,a2,…,aι. PubDate: 2022-04-01

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Abstract: Abstract We discuss the role of differential equations in Lie group representation theory. We use Kashiwara’s pentagon as a reference frame for the real representation theory and then report on some work arising from its p-adic analogue by Emerton, Kisin, Patel, Huyghe, Schmidt, Strauch using Berthelot’s theory of arithmetic \({\cal D}\) -modules and Schneider-Stuhler theory of sheaves on buildings. PubDate: 2022-04-01

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Abstract: Abstract The book-embedding problem arises in several area, such as very large scale integration (VLSI) design and routing multilayer printed circuit boards (PCBs). It can be used into various practical application fields. A book embedding of a graph G is an embedding of its vertices along the spine of a book, and an embedding of its edges to the pages such that edges embedded on the same page do not intersect. The minimum number of pages in which a graph G can be embedded is called the pagenumber or book-thickness of the graph G. It is an important measure of the quality for book-embedding. It is NP-hard to research the pagenumber of book-embedding for a graph G. This paper summarizes the studies on the book-embedding of planar graphs in recent years. PubDate: 2022-04-01

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Abstract: Abstract In this paper, a tumor immune model with time delay is studied. First, the stability of nonnegative equilibria is analyzed. Then the time delay τ is selected as a bifurcation parameter and the existence of Hopf bifurcation is proved. Finally, by using the canonical method and the central manifold theory, the criteria for judging the direction and stability of Hopf bifurcation are given. PubDate: 2022-04-01

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Abstract: Abstract The Firefighter Problem on a graph can be viewed as a simplified model of the spread of contagion, fire, rumor, computer virus, etc. The fire breaks out at one or more vertices in a graph at the first round, and the firefighter chooses some vertices to protect. The fire spreads to all non-protected neighbors at the beginning of each time-step. The process stops when the fire can no longer spread. The Firefighter Problem has attracted considerable attention since it was introduced in 1995. In this paper we provide a survey on recent research progress of this field, including algorithms and complexity, Firefighter Problem for special graphs (finite and infinite) and digraphs, surviving rate and burning number of graphs. We also collect some open problems and possible research subjects. PubDate: 2022-04-01

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Abstract: Abstract In this paper, we firstly construct several new kinds of Sidon spaces and Sidon sets by investigating some known results. Secondly, using these Sidon spaces, we will present a construction of cyclic subspace codes with cardinality \(\tau \cdot {{{q^n} - 1} \over {q - 1}}\) and minimum distance 2k−2, where τ is a positive integer. We furthermore give some cyclic subspace codes with size \(2\tau \cdot {{{q^n} - 1} \over {q - 1}}\) and without changing the minimum distance 2k−2. PubDate: 2022-04-01

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Abstract: Abstract The random weighting method is an emerging computing method in statistics. In this paper, we propose a novel estimation of the survival function for right censored data based on the random weighting method. Under some regularity conditions, we prove the strong consistency of this estimation. PubDate: 2022-02-01

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Abstract: Abstract In this paper, we consider the weighted local polynomial calibration estimation and imputation estimation of a non-parametric function when the data are right censored and the censoring indicators are missing at random, and establish the asymptotic normality of these estimators. As their applications, we derive the weighted local linear calibration estimators and imputation estimations of the conditional distribution function, the conditional density function and the conditional quantile function, and investigate the asymptotic normality of these estimators. Finally, the simulation studies are conducted to illustrate the finite sample performance of the estimators. PubDate: 2022-02-01

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Abstract: Abstract This paper is a survey for development of linear distributed parameter system. At first we point out some questions existing in current study of control theory for the Lp linear system with an unbounded control operator and an unbounded observation operator, such as stabilization problem and observer theory that are closely relevant to state feedback operator. After then we survey briefly some results on relevant problems that are related to solvability of linear differential equations in general Banach space and semigroup perturbations. As a principle, we propose a concept of admissible state feedback operator for system (A, B). Finally we give an existence result of admissible state feedback operators, including semigroup generation and the equivalent conditions of admissibility of state feedback operators, for an Lp well-posed system. PubDate: 2022-02-01

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Abstract: Abstract We survey some unsolvable conjectures in finite p-groups and their research progress. PubDate: 2022-02-01

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Abstract: Abstract The core of the nonparametric/semiparametric Bayesian analysis is to relax the particular parametric assumptions on the distributions of interest to be unknown and random, and assign them a prior. Selecting a suitable prior therefore is especially critical in the nonparametric Bayesian fitting. As the distribution of distribution, Dirichlet process (DP) is the most appreciated nonparametric prior due to its nice theoretical proprieties, modeling flexibility and computational feasibility. In this paper, we review and summarize some developments of DP during the past decades. Our focus is mainly concentrated upon its theoretical properties, various extensions, statistical modeling and applications to the latent variable models. PubDate: 2022-02-01

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Abstract: Abstract An augmented Lagrange algorithm for nonlinear optimizations with second-order cone constraints is proposed based on a Löwner operator associated with a potential function for the optimization problems with inequality constraints. The favorable properties of both the Löwner operator and the corresponding augmented Lagrangian are discussed. And under some mild assumptions, the rate of convergence of the augmented Lagrange algorithm is studied in detail. PubDate: 2022-02-01

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Abstract: Abstract Suppose that H is a subgroup of a finite group G. We call H is semipermutable in G if HK = KH for any subgroup K of G such that (∣H∣, ∣K∣) = 1; H is s-semipermutable in G if HGp = GpH, for any Sylow p-subgroup Gp of G such that (∣H∣, p) = 1. These two concepts have been received the attention of many scholars in group theory since they were introduced by Professor Zhongmu Chen in 1987. In recent decades, there are a lot of papers published via the application of these concepts. Here we summarize the results in this area and gives some thoughts in the research process. PubDate: 2022-02-01

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Abstract: Abstract The Hermitian tensor is an extension of Hermitian matrices and plays an important role in quantum information research. It is known that every symmetric tensor has a symmetric CP-decomposition. However, symmetric Hermitian tensor is not the case. In this paper, we obtain a necessary and sufficient condition for symmetric Hermitian decomposability of symmetric Hermitian tensors. When a symmetric Hermitian decomposable tensor space is regarded as a linear space over the real number field, we also obtain its dimension formula and basis. Moreover, if the tensor is symmetric Hermitian decomposable, then the symmetric Hermitian decomposition can be obtained by using the symmetric Hermitian basis. In the application of quantum information, the symmetric Hermitian decomposability condition can be used to determine the symmetry separability of symmetric quantum mixed states. PubDate: 2021-11-06 DOI: 10.1007/s11464-021-0927-4