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 \({0 < p \leq\infty}\).     

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.     

Weak law of large numbers for linear processes

We establish sufficient conditions for the Marcinkiewicz–Zygmund type weak law of large numbers for a linear process \({\{X_k:k\in\mathbb Z\}}\) defined by \({X_k=\sum_{j=0}^\infty\psi_j\varepsilon_{k-j}}\) for \({k\in\mathbb Z}\), where \({\{\psi_j:j\in\mathbb Z\}\subset\mathbb R}\) and \({\{\varepsilon_k:k\in\mathbb Z\}}\) are independent and identically distributed random variables such that \({{x^p\Pr\{|\varepsilon_0| > x\}\to 0}}\) as \({{x\to \infty}}\) with \({1 < p < 2}\) and \({E \varepsilon_0=0}\). We use an abstract norming sequence that does not grow faster than \({n^{1/p}}\) if \({\sum|\psi_j| < \infty}\). If \({\sum|\psi_j|=\infty}\), the abstract norming sequence might grow faster than \({n^{1/p}}\) as we illustrate with an example. Also, we investigate the rate of convergence in the Marcinkiewicz–Zygmund type weak law of large numbers for the linear process.     

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

On geometric properties of the generating function for the Ramanujan sequence

The Ramanujan sequence \(\{\theta _{n}\}_{n \ge 0}\), defined as \(\theta _{0}= {1}/{2}\), \({n^{n}} \theta _{n}/{n !} = {e^{n}}/{2} - \sum _{k=0}^{n-1} {n^{k}}/{k !}\, \), \(n \ge 1\), has been studied on many occasions and in many different contexts. Adell and Jodrá (Ramanujan J 16:1–5, 2008) and Koumandos (Ramanujan J 30:447–459, 2013) showed, respectively, that the sequences \(\{\theta _{n}\}_{n \ge 0}\) and \(\{4/135 - n \cdot (\theta _{n}- 1/3 )\}_{n \ge 0}\) are completely monotone. In the present paper, we establish that the sequence \(\{(n+1) (\theta _{n}- 1/3 )\}_{n \ge 0}\) is also completely monotone. Furthermore, we prove that the analytic function \((\theta _{1}- 1/3 )^{-1}\sum _{n=1}^{\infty } (\theta _{n}- 1/3 ) z^{n} / n^{\alpha }\) is universally starlike for every \(\alpha \ge 1\) in the slit domain \(\mathbb {C}\setminus [1,\infty )\). This seems to be the first result putting the Ramanujan sequence into the context of analytic univalent functions and is a step towards a previous stronger conjecture, proposed by Ruscheweyh et al. (Israel J Math 171:285–304, 2009), namely that the function \((\theta _{1}- 1/3 )^{-1}\sum _{n=1}^{\infty } (\theta _{n}- 1/3 ) z^{n} \) is universally convex.     

About the Uniform Hölder Continuity of Generalized Riemann Function

In this paper, we study the uniform Hölder continuity of the generalized Riemann function \({R_{\alpha,\beta} \,\,{\rm (with}\,\, \alpha > 1 \,\,{\rm and}\,\, \beta > 0}\)) defined by

$$R_{\alpha,\beta}(x) = \sum_{n=1}^{+\infty} \frac{\sin(\pi n^\beta x)}{n^\alpha},\quad x \in \mathbb{R},$$

using its continuous wavelet transform. In particular, we show that the exponent we find is optimal. We also analyse the behaviour of \({R_{\alpha,\beta} \,\,{\rm as}\,\, \beta}\) tends to infinity.

     

Cutoff on all Ramanujan graphs

We show that on every Ramanujan graph \({G}\), the simple random walk exhibits cutoff: when \({G}\) has \({n}\) vertices and degree \({d}\), the total-variation distance of the walk from the uniform distribution at time \({t=\frac{d}{d-2} \log_{d-1} n + s\sqrt{\log n}}\) is asymptotically \({{\mathbb{P}}(Z > c \, s)}\) where \({Z}\) is a standard normal variable and \({c=c(d)}\) is an explicit constant. Furthermore, for all \({1 \leq p \leq \infty}\), \({d}\)-regular Ramanujan graphs minimize the asymptotic \({L^p}\)-mixing time for SRW among all \({d}\)-regular graphs. Our proof also shows that, for every vertex \({x}\) in \({G}\) as above, its distance from \({n-o(n)}\) of the vertices is asymptotically \({\log_{d-1} n}\).     

Rectifiability of harmonic measure

In the present paper we prove that for any open connected set \({\Omega\subset\mathbb{R}^{n+1}}\), \({n\geq 1}\), and any \({E\subset \partial \Omega}\) with \({\mathcal{H}^n(E)<\infty}\), absolute continuity of the harmonic measure \({\omega}\) with respect to the Hausdorff measure on E implies that \({\omega|_E}\) is rectifiable. This solves an open problem on harmonic measure which turns out to be an old conjecture even in the planar case \({n=1}\).     

Closed Convex Hulls of Unitary Orbits in {C^{*}_{r}(\mathbb{F}_{\infty})}

Let \({C^*_r(\mathbb{F}_{\infty})}\) be the reduced C*-algebra of the free group on infinitely many generators. Say that \({a, b \in C^*_r(\mathbb{F}_{\infty})_{SA}}\). Then \({a}\) is majorized by \({b}\) if and only if \({a \in \overline{Conv(U(b))}.}\) In particular, \({\tau(b)1 \in \overline{Conv(U(b))}.}\) Moreover, in the above results, we provide uniform bounds for the number of unitary conjugates needed for a given approximation. In the above, \({Conv(U(b))}\) is the convex hull of the unitary orbit of \({b}\) in \({C^*_r(\mathbb{F}_{\infty})}\).     

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 \({h(\Omega_{a,b})=1}\).     

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.     

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].     

On complete sequences

A sequence A of nonnegative integers is called complete if all sufficiently large integers can be represented as the sum of distinct terms taken form A. For a sequence \({S=\{s_{1}, s_{2}, \dots\}}\) of positive integers and a positive real number α, let S _α denote the sequence \({\{\lfloor\alpha s_{1}\rfloor, \lfloor\alpha s_{2}\rfloor, \dots\}}\), where \({\lfloor x \rfloor}\) denotes the greatest integer not greater than x. Let \({{U_S = \{\alpha \mid S_\alpha} \, is complete\}}\). Hegyvári [6] proved that if \({\lim_{n\to\infty} (s_{n+1}-s_{n})=+ \infty}\), \({s_{n+1} < \gamma s_{n}}\) for all integers \({n \geqq n_{0}}\), where \({1 < \gamma < 2}\), and \({U_{S}\ne\emptyset}\), then \({\mu(U_{S}) > 0}\), where \({\mu(U_{S})}\) is the Lebesgue measure of U _S . Yong-Gao Chen and the first author [4] proved that, if \({s_{n+1} < \gamma s_{n}}\) for all integers \({n \geqq n_{0}}\), where \({1 < \gamma \leqq 7/4=1.75}\), then \({\mu(U_{S}) > 0}\). In this paper, we prove that the conclusion holds for \({1 < \gamma \leqq \sqrt[4]{13}=1.898\dots\;}\).     

A modular-type formula for $$(x;q)_\infty $$

Let \(q=\text {e}^{2\pi i\tau }, \mathfrak {I}\tau >0\), \(x=\text {e}^{2\pi i{z}}\), \({z}\in \mathbb {C}\), and \((x;q)_\infty =\prod _{n\ge 0}(1-xq^n)\). Let \((q,x)\mapsto ({q_1},{x_1})\) be the classical modular substitution given by the relations \({q_1}=\text {e}^{-2\pi i/\tau }\) and \({x_1}=\text {e}^{2\pi i{z}/{\tau }}\). The main goal of this paper is to give a modular-type representation for the infinite product \((x;q)_\infty \), this means, to compare the function defined by \((x;q)_\infty \) with that given by \(({x_1};{q_1})_\infty \). Inspired by the work (Stieltjes in Collected Papers, Springer, New York, 1993) of Stieltjes on semi-convergent series, we are led to a “closed” analytic formula for the ratio \((x;q)_\infty /({x_1};{q_1})_\infty \) by means of the dilogarithm combined with a Laplace type integral, which admits a divergent series as Taylor expansion at \(\log q=0\). Thus, the function \((x;q)_\infty \) is linked with its modular transform \(({x_1};{q_1})_\infty \) in such an explicit manner that one can directly find the modular formulae known for Dedekind’s Eta function, Jacobi Theta function, and also for certain Lambert series. Moreover, one can remark that our results allow Ramanujan’s formula (Berndt in Ramanujan’s notebooks, Springer, New York, 1994, Entry 6’, p. 268) (see also Ramanujan in Notebook 2, Tata Institute of Fundamental Research, Bombay, 1957, p. 284) to be completed as a convergent expression for the infinite product \((x;q)_\infty \).     

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.     

Orthogonal projections of cubes and regular simplices into a plane

We show that for every \({k\ge 2}\) and \({n\ge k}\), there is an \({n}\)-dimensional unit cube in \({\mathbb{R}^n}\) which is mapped to a regular \({2k}\)-gon by an orthogonal projection in \({\mathbb{R}^n}\) onto a \({2}\)-dimensional subspace. Moreover, by increasing dimension \({n}\), arbitrary large regular \({2k}\)-gon can be obtained in such a way. On the other hand, for every \({m\ge 3}\) and \({n\ge m-1}\), there is an \({n}\)-dimensional regular simplex of unit edge in \({\mathbb{R}^n}\) which is mapped to a regular \({m}\)-gon by an orthogonal projection onto a plane. Moreover, contrary to the cube case, arbitrary small regular \({m}\)-gon can be obtained in such a way, by increasing dimension \({n}\).     

Differentiability of arithmetic Fourier series arising from Eisenstein series

Let \(k\in \mathbb {N}^*\) be even. We consider two trigonometric series \( F_k(x)= \sum _{n=1}^\infty \frac{\sigma _{k-1}(n)}{n^{k+1}} \sin (2\pi n x)\) and \(G_k(x)= \sum _{n=1}^\infty \frac{\sigma _{k-1}(n)}{n^{k+1}} \cos (2\pi n x),\) where \(\sigma _{k-1}\) is the divisor function. They converge on \(\mathbb {R}\) to continuous functions. In this paper, we examine the differentiability of \(F_k\) and \(G_k\). These functions are related to Eisenstein series, and their (quasi-)modular properties allow us to apply the method proposed by Itatsu in 1981 in the study of the Riemann series. We focus on the case \(k=2\) and we show that the sine series exhibits a different behaviour with respect to differentiability than the cosine series. We prove that the differentiability of \(F_2\) at an irrational x is related to the continued fraction expansion of x. We estimate the modulus of continuity of \(F_2\). We formulate a conjecture concerning differentiability of \(F_k\) and \(G_k\) for any k even.     

Some remarks on energy inequalities for harmonic maps with potential

We call the \({\delta}\)-vector of an integral convex polytope of dimension d flat if the \({\delta}\)-vector is of the form \({(1,0,\ldots,0,a,\ldots,a,0,\ldots,0)}\), where \({a \geq 1}\). In this paper, we give the complete characterization of possible flat \({\delta}\)-vectors. Moreover, for an integral convex polytope \({\mathcal{P}\subset \mathbb{R}^N}\) of dimension d, we let \({i(\mathcal{P},n)=|n\mathcal{P}\cap \mathbb{Z}^N|}\) and \({i^*(\mathcal{P},n)=|n(\mathcal{P} {\setminus}\partial \mathcal{P})\cap \mathbb{Z}^N|}\). By this characterization, we show that for any \({d \geq 1}\) and for any \({k,\ell \geq 0}\) with \({k+\ell \leq d-1}\), there exist integral convex polytopes \({\mathcal{P}}\) and \({\mathcal{Q}}\) of dimension d such that (i) For \({t=1,\ldots,k}\), we have \({i(\mathcal{P},t)=i(\mathcal{Q},t),}\) (ii) For \({t=1,\ldots,\ell}\), we have \({i^*(\mathcal{P},t)=i^*(\mathcal{Q},t)}\), and (iii) \({i(\mathcal{P},k+1) \neq i(\mathcal{Q},k+1)}\) and \({i^*(\mathcal{P},\ell+1)\neq i^*(\mathcal{Q},\ell+1)}\).     

Generalizations of an Expansion Formula for Top to Random Shuffles

In the top to random shuffle, the first \({a}\) cards are removed from a deck of \({n}\) cards \({12 \cdots n}\) and then inserted back into the deck. This action can be studied by treating the top to random shuffle as an element \({B_a}\), which we define formally in Section 2, of the algebra \({{\mathbb{Q}[S_n]}}\). For \({a = 1}\), Garsia in “On the powers of top to random shuffling” (2002) derived an expansion formula for \({{B^k_1}}\) for \({{k \leq n}}\), though his proof for the formula was non-bijective. We prove, bijectively, an expansion formula for the arbitrary finite product \({B_{a1} B_{a2} \cdots B_{ak}}\) where \({a_{1}, \cdots , a_{k}}\) are positive integers, from which an improved version of Garsia’s aforementioned formula follows. We show some applications of this formula for \({B_{a1} B_{a2} \cdots B_{ak}}\), which include enumeration and calculating probabilities. Then for an arbitrary group \({G}\) we define the group of \({G}\)-permutations \({{S^G_n} := {G \wr S_n}}\) and further generalize the aforementioned expansion formula to the algebra \({{\mathbb{Q} [ S^G_n ]}}\) for the case of finite \({G}\), and we show how other similar expansion formulae in \({{\mathbb{Q} [S_n]}}\) can be generalized to \({{\mathbb{Q} [S^G_n]}}\).     

Generalized commutativity of lattice-ordered groups II

Commutative \({\ell}\)-groups G (in which for all \({x, y \in G, xy = yx}\)) were studied long ago. This was then generalized to the study of \({\ell}\)-groups G in which for a given integer n and for all \({x, y \in G, x^{n}y^{n} = y^{n}x^{n}}\). It was then discovered that if for all \({x, y \in G}\), both \({x^{n}y^{n} = y^{n}x^{n}}\) and \({x^{m}y^{m} = y^{m}x^{m}}\) for two different integers m, n, then also \({x^{d}y^{d} = y^{d}x^{d}}\), where d is the greatest common divisor of m, n.