On the Convergence to Equilibrium of Unbounded Observables Under a Family of Intermittent Interval Maps

We consider a family \({\{T_{r}: [0, 1] \circlearrowleft \}_{r\in[0, 1]}}\) of Markov interval maps interpolating between the tent map \({T_{0}}\) and the Farey map \({T_{1}}\). Letting \({\mathcal{P}_{r}}\) denote the Perron–Frobenius operator of \({T_{r}}\), we show, for \({\beta \in [0, 1]}\) and \({\alpha \in (0, 1)}\), that the asymptotic behaviour of the iterates of \({\mathcal{P}_{r}}\) applied to observables with a singularity at \({\beta}\) of order \({\alpha}\) is dependent on the structure of the \({\omega}\)-limit set of \({\beta}\) with respect to \({T_{r}}\). The results presented here are some of the first to deal with convergence to equilibrium of observables with singularities.     

Mutually prime sequences of inner functions and the ranks of subspaces over the bidisk

Let \({\{\varphi_n(z)\}_{n\ge0}}\) be a sequence of inner functions satisfying that \({\zeta_n(z):=\varphi_n(z)/\varphi_{n+1}(z)\in H^\infty(z)}\) for every n ≥ 0 and \({\{\varphi_n(z)\}_{n\ge0}}\) have no nonconstant common inner divisors. Associated with it, we have a Rudin type invariant subspace \({\mathcal{M}}\) of \({H^2(\mathbb{D}^2)}\) . We write \({\mathcal{N}= H^2(\mathbb{D}^2)\ominus\mathcal{M}}\) . If \({\{\zeta_n(z)\}_{n\ge0}}\) ia a mutually prime sequence, then we shall prove that \({rank_{\{T^\ast_z,T^\ast_w\}} \mathcal{N}=1}\) and \({rank_{\{\mathcal{F}^\ast_z\}}(\mathcal{M}\ominus w\mathcal{M})=1}\) , where \({\mathcal{F}_z}\) is the fringe operator on \({\mathcal{M}\ominus w\mathcal{M}}\) .     

A Beurling-Blecher-Labuschagne Theorem for Noncommutative Hardy Spaces Associated with Semifinite von Neumann Algebras

We prove a Beurling-Blecher-Labuschagne theorem for \({H^\infty}\)-invariant spaces of \({L^p(\mathcal{M},\tau)}\) when \({0 < p \leq\infty}\), using Arveson’s non-commutative Hardy space \({H^\infty}\) in relation to a von Neumann algebra \({\mathcal{M}}\) with a semifinite, faithful, normal tracial weight \({\tau}\). Using the main result, we are able to completely characterize all \({H^\infty}\)-invariant subspaces of \({L^p(\mathcal{M} \rtimes_\alpha \mathbb{Z},\tau)}\), where \({\mathcal{M} \rtimes_\alpha \mathbb{Z} }\) is a crossed product of a semifinite von Neumann algebra \({\mathcal{M}}\) by the integer group \({\mathbb{Z}}\), and \({H^\infty}\) is a non-selfadjoint crossed product of \({\mathcal{M}}\) by \({\mathbb{Z}^+}\). As an example, we characterize all \({H^\infty}\)-invariant subspaces of the Schatten p-class \({S^p(\mathcal{H})}\), where \({H^\infty}\) is the lower triangular subalgebra of \({B(\mathcal{H})}\), for each \({0 < p \leq\infty}\).     

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.     

The Eight Cayley–Dickson Doubling Products

The purpose of this paper is to identify all eight of the basic Cayley–Dickson doubling products. A Cayley–Dickson algebra \({\mathbb{A}_{N+1}}\) of dimension \({2^{N+1}}\) consists of all ordered pairs of elements of a Cayley–Dickson algebra \({\mathbb{A}_{N}}\) of dimension \({2^N}\) where the product \({(a, b)(c, d)}\) of elements of \({\mathbb{A}_{N+1}}\) is defined in terms of a pair of second degree binomials \({(f(a, b, c, d), g(a, b, c,d))}\) satisfying certain properties. The polynomial pair\({(f, g)}\) is called a ‘doubling product.’ While \({\mathbb{A}_{0}}\) may denote any ring, here it is taken to be the set \({\mathbb{R}}\) of real numbers. The binomials \({f}\) and \({g}\) should be devised such that \({\mathbb{A}_{1} = \mathbb{C}}\) the complex numbers, \({\mathbb{A}_{2} = \mathbb{H}}\) the quaternions, and \({\mathbb{A}_{3} = \mathbb{O}}\) the octonions. Historically, various researchers have used different yet equivalent doubling products.     

On stable Baire classes

We introduce and study adhesive spaces. Using this concept we obtain a characterization of stable Baire maps \({f : X\to Y}\) of the class \({\alpha}\) for wide classes of topological spaces. In particular, we prove that for a topological space X and a contractible space Y a map \({f : X \to Y}\) belongs to the nth stable Baire class if and only if there exist a sequence \({(f_k)_{k=1}^\infty}\) of continuous maps \({f_k : {X \to Y}}\) and a sequence \({(F_k)_{k=1}^\infty}\) of functionally ambiguous sets of the nth class in X such that \({f|_{F_k}=f_k|_{F_k}}\) for every k. Moreover, we show that every monotone function \({f : \mathbb{R} \to \mathbb{R}}\) is of the \({\alpha}\) th stable Baire class if and only if it belongs to the first stable Baire class.     

Centralizing traces with automorphisms on triangular algebras

Let \({\mathcal{T}}\) be a triangular algebra over a commutative ring \({\mathcal{R}}\), \({\xi}\) be an automorphism of \({\mathcal{T}}\) and \({\mathcal{Z}_{\xi}(\mathcal{T})}\) be the \({\xi}\)-center of \({\mathcal{T}}\). Suppose that \({\mathfrak{q}\colon \mathcal{T}\times \mathcal{T}\longrightarrow \mathcal{T}}\) is an \({\mathcal{R}}\)-bilinear mapping and that \({\mathfrak{T}_{\mathfrak{q}}\colon \mathcal{T}\longrightarrow \mathcal{T}}\) is a trace of \({\mathfrak{q}}\). The aim of this article is to describe the form of \({\mathfrak{T}_{\mathfrak{q}}}\) satisfying the commuting condition \({[\mathfrak{T}_{\mathfrak{q}}(x), x]_{\xi}=0}\) (resp. the centralizing condition \({[\mathfrak{T}_{\mathfrak{q}}(x), x]_{\xi}\in \mathcal{Z}_\xi(\mathcal{T})}\)) for all \({x\in \mathcal{T}}\). More precisely, we will consider the question of when \({\mathfrak{T}_{\mathfrak{q}}}\) satisfying the previous condition has the so-called proper form.     

Weyl Type Asymptotics and Bounds for the Eigenvalues of Functional-Difference Operators for Mirror Curves

We investigate Weyl type asymptotics of functional-difference operators associated to mirror curves of special del Pezzo Calabi-Yau threefolds. These operators are \({H(\zeta) = U + U^{-1} + V + \zeta V^{-1}}\) and \({H_{m,n} = U + V + q^{-mn}U^{-m}V^{-n}}\), where \({U}\) and \({V}\) are self-adjoint Weyl operators satisfying \({UV = q^{2}VU}\) with \({q = {\rm e}^{{\rm i}\pi b^{2}}}\), \({b > 0}\) and \({\zeta > 0}\), \({m, n \in \mathbb{N}}\). We prove that \({H(\zeta)}\) and \({H_{m,n}}\) are self-adjoint operators with purely discrete spectrum on \({L^{2}(\mathbb{R})}\). Using the coherent state transform we find the asymptotical behaviour for the Riesz mean \({\sum_{j\ge 1}(\lambda - \lambda_{j})_{+}}\) as \({\lambda \to \infty}\) and prove the Weyl law for the eigenvalue counting function \({N(\lambda)}\) for these operators, which imply that their inverses are of trace class.     

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 \({K \cong L}\).     

Asymptotic behavior of the p-torsion functions as p goes to 1

Let \({\Omega}\) be a Lipschitz bounded domain of \({\mathbb{R}^N}\), \({N\geq2}\), and let \({u_p\in W_0^{1,p}(\Omega)}\) denote the p-torsion function of \({\Omega}\), p > 1. It is observed that the value 1 for the Cheeger constant \({h(\Omega)}\) is threshold with respect to the asymptotic behavior of u_p, as \({p\rightarrow 1^+}\), in the following sense: when \({h(\Omega) > 1}\), one has \({\lim_{p\rightarrow 1^+}\left\|u_{p}\right\| _{L^\infty(\Omega)}=0}\), and when \({h(\Omega) < 1}\), one has \({\lim_{p\rightarrow 1^+}\left\|u_p\right\| _{L^\infty(\Omega)}=\infty}\). In the case \({h(\Omega)=1}\), it is proved that \({\limsup_{p\rightarrow1^+}\left\|u_p\right\|_{L^\infty(\Omega)}<\infty}\). For a radial annulus \({\Omega_{a,b}}\), with inner radius a and outer radius b, it is proved that \({\lim_{p\rightarrow 1^+}\left\|u_p\right\| _{L^\infty(\Omega_{a,b})}=0}\) when \({h(\Omega_{a,b})=1}\).     

The Katchalski–Lewis transversal problem for regular polygons

If every k-membered subfamily of a family of plane convex bodies has a line transversal, then we say that this family has property T(k). We say that a family \({\mathcal{F}}\) has property \({T-m}\), if there exists a subfamily \({\mathcal{G} \subset \mathcal{F}}\) with \({|\mathcal{F} - \mathcal{G}| \le m}\) admitting a line transversal. Heppes [7] posed the problem whether there exists a convex body K in the plane such that if \({\mathcal{F}}\) is a finite T(3)-family of disjoint translates of K, then m = 3 is the smallest value for which \({\mathcal{F}}\) has property \({T-m}\). In this paper, we study this open problem in terms of finite T(3)-families of pairwise disjoint translates of a regular 2n-gon \({(n \ge 5)}\). We find out that, for \({5 \le n \le 34}\), the family has property \({T - 3}\) ; for \({n \ge 35}\), the family has property \({T - 2}\).     

A Unified Framework for Linear Dimensionality Reduction in L1

For a family of interpolation norms \({\| \cdot \|_{1,2,s}}\) on \({\mathbb{R}^{n}}\), we provide a distribution over random matrices \({\Phi_s \in \mathbb{R}^{m \times n}}\) parametrized by sparsity level s such that for a fixed set X of K points in \({\mathbb{R}^{n}}\), if \({m \geq C s \log(K)}\) then with high probability, \({\frac{1}{2}\| \varvec{x} \|_{1,2,s} \leq \| \Phi_s (\varvec{x}) \|_1 \leq 2 \| \varvec{x} \|_{1,2,s}}\) for all \({\varvec{x} \in X}\). Several existing results in the literature roughly reduce to special cases of this result at different values of s: For s = n, \({\| \varvec{x} \|_{1,2,n}\equiv \| \varvec{x} \|_{1}}\) and we recover that dimension reducing linear maps can preserve the ?₁-norm up to a distortion proportional to the dimension reduction factor, which is known to be the best possible such result. For s = 1, \({\| \varvec{x} \|_{1,2,1}\equiv \| \varvec{x} \|_{2}}\), and we recover an ?₂/?₁ variant of the Johnson–Lindenstrauss Lemma for Gaussian random matrices. Finally, if \({\varvec{x}}\) is s- sparse, then \({\| \varvec{x} \|_{1,2,s} = \| \varvec{x} \|_1}\) and we recover that s-sparse vectors in \({\ell_1^n}\) embed into \({\ell_1^{\mathcal{O}(s \log(n))}}\) via sparse random matrix constructions.     

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.     

Convergence in p-mean for arrays of row-wise extended negatively dependent random variables

It is known that the maximal operator \({\sigma^{\kappa,*}(f)} := sup_{n \in \mathbf{P}}{|{\sigma}_{n}^{\kappa} (f)|}\) is bounded from the dyadic Hardy space \({H_{p}}\) into the space \({L_{p}}\) for \({p > 2/3}\) [6]. Moreover, Goginava and Nagy showed that \({\sigma^{\kappa,*}}\) is not bounded from the Hardy space \({H_{2/3}}\) to the space \({L_{2/3}}\) [9]. The main aim of this paper is to investigate the case \({0 < p < 2/3}\). We show that the weighted maximal operator \({\tilde{\sigma}^{\kappa,*,p}(f) :=sup_{n\in \mathbf{P}} \frac{|{\sigma}_{n}^\kappa (f)|}{n^{2/p-3}}}\), is bounded from the Hardy space \({H_{p}}\) into the space \({L_{p}}\) for any \({0 < p < 2/3}\). With its aid we provide a necessary and sufficient condition for the convergence of Walsh–Kaczmarz–Marcinkiewicz means in terms of modulus of continuity on the Hardy space \({H_p}\), and prove a strong convergence theorem for this means.     

Dirac–Krein Systems on Star Graphs

Suppose that \({\mathcal {M}}\) is a countably decomposable type II\({_1}\) von Neumann algebra and \({\mathcal {A}}\) is a separable, non-nuclear, unital C\({^*}\)-algebra. We show that, if \({\mathcal {M}}\) has Property \({\Gamma}\), then the similarity degree of \({\mathcal {M}}\) is less than or equal to 5. If \({\mathcal {A}}\) has Property c\({^*}\)-\({\Gamma}\), then the similarity degree of \({\mathcal {A}}\) is equal to 3. In particular, the similarity degree of a \({\mathcal {Z}}\)-stable, separable, non-nuclear, unital C\({^*}\)-algebra is equal to 3.     

A functional equation with involution related to the cosine function

Let G be an abelian group, \({\mathbb{C}}\) be the field of complex numbers, \({\alpha \in G}\) be any fixed element and \({\sigma : G \to G}\) be an involution. In this paper, we determine the general solution \({f, g : G \to \mathbb{C}}\) of the functional equation \({f(x + \sigma y + \alpha) + g(x + y + \alpha) = 2f(x)f(y)}\) for all \({x, y \in G}\).     

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 \({{\rm lim}_{k \to \infty}\, {\rm lim}_{n \to \infty} {\rm inf}_{x}}\).     

Orthogonal Polynomials for Modified Chebyshev Measure of the First Kind

Given numbers \({n,s \in \mathbb{N}}\), \({n \geq 2}\), and the \({n}\)th-degree monic Chebyshev polynomial of the first kind \({\widehat T_n(x)}\), the polynomial system “induced” by \({\widehat T_n(x)}\) is the system of orthogonal polynomials \({\{p_{k}^{n,s} \}}\) corresponding to the modified measure \({d \sigma^{n,s}(x)=\widehat T^{2s}_n(x) d\sigma(x)}\), where \({d\sigma(x)=1/\sqrt{1-x^{2}}dx}\) is the Chebyshev measure of the first kind. Here we are concerned with the problem of determining the coefficients in the three-term recurrence relation for the polynomials \({p^{n,s}_{k}}\). The desired coefficients are obtained analytically in a closed form.     

Sharp weighted bounds for the Hardy–Littlewood maximal operators on Musielak–Orlicz spaces

Let \({\varphi}\) be a Musielak–Orlicz function satisfying that, for any \({(x,\,t)\in{\mathbb R}^n \times [0, \infty)}\), \({\varphi(\cdot,\,t)}\) belongs to the Muckenhoupt weight class \({A_\infty({\mathbb R}^n)}\) with the critical weight exponent \({q(\varphi) \in [1,\,\infty)}\) and \({\varphi(x,\,\cdot)}\) is an Orlicz function with uniformly lower type \({p^{-}_{\varphi}}\) and uniformly upper type \({p^+_\varphi}\) satisfying \({q(\varphi) < p^{-}_{\varphi}\le p^{+}_{\varphi} < \infty}\). In this paper, the author obtains a sharp weighted bound involving \({A_\infty}\) constant for the Hardy–Littlewood maximal operator on the Musielak–Orlicz space \({L^{\varphi}}\). This result recovers the known sharp weighted estimate established by Hytönen et al. in [J. Funct. Anal. 263:3883–3899, 2012].     

Weighted $${{L^p}}$$-Liouville theorems for hypoelliptic partial differential operators on Lie groups

We prove weighted \({L^p}\)-Liouville theorems for a class of second-order hypoelliptic partial differential operators \({\mathcal{L}}\) on Lie groups \({\mathbb{G}}\) whose underlying manifold is \({n}\)-dimensional space. We show that a natural weight is the right-invariant measure \(\check{H}\) of \({\mathbb{G}}\). We also prove Liouville-type theorems for \({C^{2}}\) subsolutions in \({L^{p}(\mathbb{G},\check{H})}\). We provide examples of operators to which our results apply, jointly with an application to the uniqueness for the Cauchy problem for the evolution operator \({\mathcal{L}-\partial_{t}}\).