Spaces with normal weights and Hadamard gap series

We give a sufficient and necessary condition for an analytic function f(z) on the unit disc \({\mathbb{D}}\) with Hadamard gaps, that is, for \({f(z)=\sum_{k=1}^{\infty}a_kz^{n_k}}\) where \({n_{k+1}/n_k\geq\lambda >1 }\) for all \({k\in \mathbb{N}}\), to belong to the weighted-type space \({ H_\mu^{\infty}}\), under some condition posed on the weight function μ. We can define the corresponding little weighted-type space \({H_{\mu,0}^{\infty}}\) and give a criterion for functions to belong to it.     

Finite groups with small automizers of {\mathcal {F}}-subgroups

Let \({ \mathcal {F}}\) be a saturated formation and G a finite group such that \({N_{G} (H^{\mathcal {F}})/C_{G} (H^{\mathcal {F}})\cong Inn(H^{\mathcal {F}})}\) for every subgroup H of G. If the minimal non-\({ \mathcal {F}}\)-group is soluble, then \({G \in \mathcal {F}}\).     

Complex homomorphisms on self-conjugate algebras of functions

Let X be a non-void set and A be a subalgebra of \({\mathbb{C}^{X}}\) . We call a \({\mathbb{C}}\) -linear functional \({\varphi}\) on A a 1-evaluation if \({\varphi(f) \in f(X) }\) for all \({f\in A}\) . From the classical Gleason–Kahane–?elazko theorem, it follows that if X in addition is a compact Hausdorff space then a mapping \({\varphi}\) of \({C_{\mathbb{C}}(X) }\) into \({\mathbb{C}}\) is a 1-evaluation if and only if \({\varphi}\) is a \({\mathbb{C}}\) -homomorphism. In this paper, we aim to investigate the extent to which this equivalence between 1-evaluations and \({\mathbb{C}}\) -homomorphisms can be generalized to a wider class of self-conjugate subalgebras of \({\mathbb{C}^{X}}\) . In this regards, we prove that a \({\mathbb{C}}\) -linear functional on a self-conjugate subalgebra A of \({\mathbb{C}^{X}}\) is a positive \({\mathbb{C}}\) -homomorphism if and only if \({\varphi}\) is a \({\overline{1}}\) -evaluation, that is, \({\varphi(f) \in\overline{f\left(X\right)}}\) for all \({f\in A}\) . As consequences of our general study, we prove that 1-evaluations and \({\mathbb{C}}\) -homomorphisms on \({C_{\mathbb{C}}\left( X\right)}\) coincide for any topological space X and we get a new characterization of realcompact topological spaces.     

The Schur functor on tensor powers

Let M be a left module for the Schur algebra S(n, r), and let \({s \in \mathbb{Z}^+}\) . Then \({M^{\otimes s}}\) is a \({(S(n,\,rs), F{\mathfrak{S}_{s}})}\) -bimodule, where the symmetric group \({{\mathfrak{S}_s}}\) on s letters acts on the right by place permutations. We show that the Schur functor f _rs sends \({M^{\otimes s}}\) to the \({(F{\mathfrak{S}_{rs}},F{\mathfrak{S}_s})}\) -bimodule \({F\mathfrak{S}_{rs}\otimes_{F(\mathfrak{S}_{r}\wr{\mathfrak{S}_s})} ((f_rM)^{\otimes s}\otimes_{F} F{\mathfrak{S}_s})}\) . As a corollary, we obtain the image under the Schur functor of the Lie power L ^s (M), exterior power \({\bigwedge^s(M)}\) of M and symmetric power S ^s (M).     

{L^{p}} stability for the Boltzmann equation near vacuum

The paper is concerned with the uniform time stability in the Lebesgue space \({L^{p}(\mathbb{R}^{3} \times \mathbb{R}^{3})}\) of solutions to the Boltzmann equation near vacuum. Precisely, for the soft potential case \({-2 < \gamma < 0}\), there exists \(p_{\gamma} > 1\) such that the nonnegative solution with algebraic decay rate in x, v at infinity is stable with respect to small initial data uniformly in time in \({L^{p}}\) with \({1 \leq p < p_{\gamma}}\).     

Arens regularity of projective tensor products

For completely contractive Banach algebras A and B (respectively operator algebras A and B), the necessary and sufficient conditions for the operator space projective tensor product \({A\widehat{\otimes}B}\) (respectively the Haagerup tensor product \({A\otimes^{h}B}\)) to be Arens regular are obtained. Using the non-commutative Grothendieck inequality, we show that, for C*-algebras A and B, \({A\otimes^{\gamma} B}\) is Arens regular if \({A\widehat{\otimes}B}\) and \({A\widehat{\otimes}B^{op}}\) are Arens regular whereas \({A\widehat{\otimes}B}\) is Arens regular if and only if \({A\otimes^{h}B}\) and \({B\otimes^{h}A}\) are, where \({\otimes^h}\), \({\otimes^{\gamma}}\), and \({\widehat{\otimes}}\) are the Haagerup, the Banach space projective tensor norm, and the operator space projective tensor norm, respectively.     

Riesz Transform Characterizations of Hardy Spaces Associated to Degenerate Elliptic Operators

Let \({L_{w}}{:=-w^{-1}{\rm div}(A\nabla)}\) be the degenerate elliptic operator on the Euclidean space \({{\mathbb{R}^{n}}}\), where w is a Muckenhoupt \({A_{2}({\mathbb{R}^{n}})}\) weight. In this article, the authors establish the Riesz transform characterization of the Hardy space \({H^{p}_{L_{w}}({\mathbb{R}}^{n})}\) associated with L_w, for \({w \in A_{q}({\mathbb{R}}^{n}) \cap RH_{\frac{n}{n-2}}({\mathbb{R}^{n}})}\) with \({n \geq 3}\), \({q \in [1,2]}\) and \({p \in (q(\frac{1}{r}+\frac{q-1}{2}+\frac{1}{n})^{-1},1]}\) if, for some \({r \in (1,\,2]}\), \({{\{tL_w e^{-tL_w}\}}_{t\geq 0}}\) satisfies the weighted \({L^{r}-L^{2}}\) full off-diagonal estimates.     

Maximum Principle for H-Surfaces in the Unit Cone and Dirichlet’s Problem for their Equation in Central Projection

In the unit cone\({\mathcal{C} := \{(x, y, z)} \in {\mathbb R}^{3} : {x}^{2} + {y}^{2} < {z}^{2}, {z} > {0}\}\) we establish a geometric maximum principle for H-surfaces, where its mean curvature \({H = H(x, y, z)}\) is optimally bounded. Consequently, these surfaces cannot touch the conical boundary \({\partial \mathcal{C}}\) at interior points and have to approach \({\partial \mathcal{C}}\) transversally. By a nonlinear continuity method, we then solve the Dirichlet problem of the H-surface equation in central projection for Jordan-domains \({\Omega}\) which are strictly convex in the following sense: On its whole boundary \({\partial \mathcal{C}(\Omega)}\) their associate cone \({\mathcal{C}(\Omega) := \{(rx, ry, r) \in {\mathbb R}^{3} : (x, y) \in \Omega, r \in (0,+\infty)}\}\) admits rotated unit cones \({O \circ \mathcal{C}}\) as solids of support, where \({O \in {\mathbb R}^{3\times3}}\) represents a rotation in the Euclidean space. Thus we construct the unique H-surface with one-to-one central projection onto these domains \({\Omega}\) bounding a given Jordan-contour \({\Gamma \subset \mathcal{C} \backslash \{0\}}\) with one-toone central projection.     

Inequalities between Dirichlet and Neumann eigenvalues of vibrating strings

The Dirichlet eigenvalues \({\{\lambda_{n}\}_{n=1}^{\infty}}\) and Neumann eigenvalues \({\{\mu_{n}\}_{n=1}^{\infty}}\) of the string equation \({\varphi'' (x) +\lambda \rho (x) \varphi(x) =0}\) are considered. It is known that \({ \mu_{n} < \lambda_{n} < \mu_{n+2}}\) for all n. The purpose of this paper is to provide conditions on the mass density \({\rho(x)}\) under which \({\lambda_{n} < \mu_{n+1}}\) or \({\mu_{n+1} < \lambda_{n}.}\)     

Abelian covers of alternating groups

Let \({C={\rm inf} (k/n)\sum_{i=1}^n x_i(x_{i+1}+\cdots+x_{i+k})^{-1}}\), where the infimum is taken over all pairs of integers \({n\geq k\geq 1}\) and all positive \({x_1,\ldots,x_n}\), \({x_{n+i}=x_i}\). We prove that \({\ln 2 \leq C < 0.9305}\). In the definition of the constant C, the operation \({{\rm inf}_{k}\, {\rm inf}_{n}\, {\rm inf}_{x}}\) can be replaced by \({{\rm lim}_{k \to \infty}\, {\rm lim}_{n \to \infty} {\rm inf}_{x}}\).     

Open book structures on semi-algebraic manifolds

Given a C ² semi-algebraic mapping \({F} : {\mathbb{R}^N \rightarrow \mathbb{R}^p}\), we consider its restriction to \({W \hookrightarrow \mathbb{R^{N}}}\) an embedded closed semi-algebraic manifold of dimension \({n-1 \geq p \geq 2}\) and introduce sufficient conditions for the existence of a fibration structure (generalized open book structure) induced by the projection \({\frac{F}{\Vert F \Vert}:W{\setminus} F^{-1}(0) \to S^{p-1}}\). Moreover, we show that the well known local and global Milnor fibrations, in the real and complex settings, follow as a byproduct by considering W as spheres of small and big radii, respectively. Furthermore, we consider the composition mapping of F with the canonical projection \({\pi: \mathbb{R}^{p} \to \mathbb{R}^{p-1}}\) and prove that the fibers of \({\frac{F}{\Vert F \Vert}}\) and \({\frac{\pi \circ F}{\Vert \pi \circ F \Vert}}\) are homotopy equivalent. We also show several formulae relating the Euler characteristics of the fiber of the projection \({\frac{F}{\Vert F \Vert}}\) and \({W \cap F^{-1}(0)}\). Similar formulae are proved for mappings obtained after composition of F with canonical projections.     

The rate of decay of stable periodic solutions for Duffing equation with <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-conditions

For a new class of g(t, x), the existence, uniqueness and stability of \({2\pi}\)-periodic solution of Duffing equation \({x'' + cx' + g(t, x) = h(t)}\) are presented. Moreover, the unique \({2\pi}\)-periodic solution is (exponentially asymptotically stable) and its rate of exponential decay c/2 is sharp. The new criterion characterizes \({g_{x}^{\prime}(t, x) - c^2/4}\) with L ^p -norms \({(p \in [1, \infty])}\), and the classical criterion employs the \({L^{\infty}}\)-norm. The advantage is that we can deal with the case that \({g_{x}^{\prime}(t, x) - c^2/4}\) is beyond the optimal bounds of the \({L^{\infty}}\)-norm, because of the difference between the L ^p -norm and the \({L^{\infty}}\)-norm.     

Intersection properties of radial solutions and global bifurcation diagrams for supercritical quasilinear elliptic equations

We study the positive solution \({u(r,\rho)}\) of the quasilinear elliptic equation

$$\begin{cases}r^{-(\gamma-1)}(r^{\alpha}|u^{\prime}|^{\beta-1}u^{\prime})^{\prime}+|u|^{p-1}u=0, & 0 < r < \infty,\\ u(0) = \rho > 0,\ u^{\prime}(0)=0.\end{cases}$$

This class of differential operators includes the usual Laplace, m-Laplace, and k-Hessian operators in the space of radial functions. The equation has a singular positive solution u ^*(r) under certain conditions on \({\alpha}\), \({\beta}\), \({\gamma}\), and p. A generalized Joseph–Lundgren exponent, which we denote by \({p^*_{JL}}\), is obtained. We study the intersection numbers between \({u(r,\rho)}\) and u ^*(r) and between \({u(r,\rho_0)}\) and \({u(r,\rho_1)}\), and see that \({p^*_{JL}}\) plays an important role. We also determine the bifurcation diagram of the problem

$$\begin{cases}r^{-(\gamma-1)}(r^{\alpha}|u^{\prime}|^{\beta-1}u^{\prime})^{\prime} + \lambda(u+1)^p=0, & 0 < r < 1,\\ u(r) > 0, & 0 \le r < 1,\\ u^{\prime}(0)=0,\ u(1)=0.\end{cases}$$

The main technique used in the proofs is a phase plane analysis.

     

Hecke algebras for {{\rm GL}_n} over local fields

We study the local Hecke algebra \({\mathcal{H}_{G}(K)}\) for \({G = {\rm GL}_{n}}\) and K a non-archimedean local field of characteristic zero. We show that for \({G = {\rm GL}_{2}}\) and any two such fields K and L, there is a Morita equivalence \({\mathcal{H}_{G}(K) \sim_{M} \mathcal{H}_{G}(L)}\), by using the Bernstein decomposition of the Hecke algebra and determining the intertwining algebras that yield the Bernstein blocks up to Morita equivalence. By contrast, we prove that for \({G = {\rm GL}_{n}}\), there is an algebra isomorphism \({\mathcal{H}_{G}(K) \cong \mathcal{H}_{G}(L)}\) which is an isometry for the induced \({L^1}\)-norm if and only if there is a field isomorphism \({K \cong L}\).     

The Leavitt Path Algebras of Generalized Cayley Graphs

Let n be a positive integer. For each \({0 \leq j \leq n-1}\), we let \({C_{n}^{j}}\) denote Cayley graph for the cyclic group \({\mathbb{Z}_n}\) with respect to the subset \({\{1, j\}}\). For any such pair (n, j), we compute the size of the Grothendieck group of the Leavitt path algebra \({L_K(C_{n}^{j})}\); the analysis is related to a collection of integer sequences described by Haselgrove in the 1940s. When j = 0, 1, or 2, we are able to extract enough additional information about the structure of these Grothendieck groups so that we may apply a Kirchberg-Phillips-type result to explicitly realize the algebras \({L_K(C_{n}^{j})}\) as the Leavitt path algebras of graphs having at most three vertices. The analysis in the j = 2 case leads us to some perhaps surprising and apparently nontrivial connections to the classical Fibonacci sequence.     

Heat kernel approach for sup-norm bounds for cusp forms of integral and half-integral weight

In this article, using the heat kernel approach from Bouche (Asymptotic results for Hermitian line bundles over complex manifolds: The heat kernel approach, Higher-dimensional complex varieties, pp 67–81, de Gruyter, Berlin, 1996), we derive sup-norm bounds for cusp forms of integral and half-integral weight. Let \({\Gamma\subset \mathrm{PSL}_{2}(\mathbb{R})}\) be a cocompact Fuchsian subgroup of first kind. For \({k \in \frac{1}{2} \mathbb{Z}}\) (or \({k \in 2\mathbb{Z}}\)), let \({S^{k}_{\nu}(\Gamma)}\) denote the complex vector space of cusp forms of weight-k and nebentypus \({\nu^{2k}}\) (\({\nu^{k\slash 2}}\), if \({k \in 2\mathbb{Z}}\)) with respect to \({\Gamma}\), where \({\nu}\) is a unitary character. Let \({\lbrace f_{1},\ldots,f_{j_{k}} \rbrace}\) denote an orthonormal basis of \({S^{k}_{\nu}(\Gamma)}\). In this article, we show that as \({k \rightarrow \infty,}\) the sup-norm for \({\sum_{i=1}^{j_{k}}y^{k}|f_{i}(z)|^{2}}\) is bounded by O(k), where the implied constant is independent of \({\Gamma}\). Furthermore, using results from Berman (Math. Z. 248:325–344, 2004), we extend these results to the case when \({\Gamma}\) is cofinite.     

Toeplitz Operators from One Fock Space to Another

In this paper, we study Toeplitz operators T _μ from one Fock space \({F^{p}_{\alpha}}\) to another \({F^{q}_{\alpha}}\) for 1 < p, q < ∞ with positive Borel measures μ as symbols. We characterize the boundedness (and compactness) of \({T_\mu: F^{p}_{\alpha} \to F^{q}_{\alpha}}\) in terms of the averaging function \({\widehat{\mu}_r}\) and the t-Berezin transform \({\widetilde{\mu}_t}\) respectively. Quite differently from the Bergman space case, we show that T _μ is bounded (or compact) from \({F^{p}_{\alpha}}\) to \({F^{q}_{\alpha}}\) for some p ≤ q if and only if T _μ is bounded (or compact) from \({F^{p}_{\alpha}}\) to \({F^{q}_{\alpha}}\) for all p ≤ q. In order to prove our main results on T _μ , we introduce and characterize (vanishing) (p, q)-Fock Carleson measures on C ⁿ .     

Characterizations of the Vertex Operator Algebras ${V_{L}^{T}}$ and VLO${V_{L}^{O}}$

In this paper, we give characterizations of the rational vertex operator algebras \({V_{L}^{T}}\) and \({V_{L}^{O}}\), where L is the root lattice of type A ₁, T is the tetrahedral group, and O is the octahedral group. By these two characterizations, the classification of rational VOAs of central charge 1 is reduced to the characterization of \({V_{L}^{I}}\) where I is the icosahedral group.     

Minimal tiled orders of finite global dimension

In this paper, we consider the sequence of balancing and Lucas balancing numbers. The balancing numbers \({B_n}\) are given by the recurrence \({B_n = 6 B_{n-1} - B_{n-2}}\) with initial conditions \({B_0 = 0, B_1 = 1}\) and its associated Lucas balancing numbers \({C_n}\) are given by the recurrence \({C_n = 6 C_{n-1} - C_{n-2}}\) with initial conditions \({C_0 = 1, C_1 = 3}\). First we find the perfect powers in the sequence of balancing and Lucas balancing numbers. We also identify those Lucas balancing numbers which are products of a power of 3 and a perfect power. Using this property of Lucas balancing numbers, we solve a conjecture regarding the non-existence of positive integral solution (x, y) for the Diophantine equation \({2x^2 + 1 = 3^b y^m}\) for any even positive integers b and m with \({m > 2}\), given in (Int J Number Theory 11:1259–1274, 2015). Also we prove that the Diophantine equations \({B_n B_{n+d}\ldots B_{n+(k-1)d} = y^m}\) and \({C_n C_{n+d}\ldots C_{n+(k-1)d} = y^m}\) have no solution for any positive integers n, d, k, y, and m with \({m \geq 2, y \geq 2}\) and gcd\({(n,d) = 1}\).     

Remarks on Complex Symmetric Operators

An operator \({T\in{\mathcal{L}}({\mathcal{H}})}\) is said to be complex symmetric if there exists a conjugation C on \({{\mathcal H}}\) such that \({T= CT^{\ast}C}\). In this paper, we study the spectral radius algebras for complex symmetric operators. In particular, we prove that if A is a complex symmetric operator, then the spectral radius algebra \({{\mathcal B}_{A}}\) associated with A has a nontrivial invariant subspace under some conditions. Finally, we give some relations between \({P_{\tilde{A}}}\) and \({P_{\widetilde{A^{\ast}}}}\) (defined below) when A is complex symmetric.