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 \({\log_{d-1} n}\).     

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 \({0 < p \leq\infty}\).     

Semitotal Domination in Claw-Free Cubic Graphs

In this paper, we continue the study of semitotal domination in graphs in [Discrete Math. 324, 13–18 (2014)]. A set \({S}\) of vertices in \({G}\) is a semitotal dominating set of \({G}\) if it is a dominating set of \({G}\) and every vertex in \({S}\) is within distance 2 of another vertex of \({S}\). The semitotal domination number, \({{\gamma_{t2}}(G)}\), is the minimum cardinality of a semitotal dominating set of \({G}\). This domination parameter is squeezed between arguably the two most important domination parameters; namely, the domination number, \({\gamma (G)}\), and the total domination number, \({{\gamma_{t}}(G)}\). We observe that \({\gamma (G) \leq {\gamma_{t2}}(G) \leq {\gamma_{t}}(G)}\). A claw-free graph is a graph that does not contain \({K_{1, \, 3}}\) as an induced subgraph. We prove that if \({G}\) is a connected, claw-free, cubic graph of order \({n \geq 10}\), then \({{\gamma_{t2}}(G) \leq 4n/11}\).     

A remark on hereditarily nonparadoxical sets

Call a set \({A \subseteq \mathbb {R}}\)paradoxical if there are disjoint \({A_0, A_1 \subseteq A}\) such that both \({A_0}\) and \({A_1}\) are equidecomposable with \({A}\) via countabbly many translations. \({X \subseteq \mathbb {R}}\) is hereditarily nonparadoxical if no uncountable subset of \({X}\) is paradoxical. Penconek raised the question if every hereditarily nonparadoxical set \({X \subseteq \mathbb {R}}\) is the union of countably many sets, each omitting nontrivial solutions of \({x - y = z - t}\). Nowik showed that the answer is ‘yes’, as long as \({|X| \leq \aleph_\omega}\). Here we show that consistently there exists a counterexample of cardinality \({\aleph_{\omega+1}}\) and it is also consistent that the continuum is arbitrarily large and Penconek’s statement holds for any \({X}\).     

The heat semigroup on sectorial domains,highly singular initial values and applications

We show that the heat semigroup is well defined on the Banach space \({\mathcal{X}_{m,\gamma} = \{ \psi:\Omega_m \to \mathbb{R} ;\; |x|^{\gamma +2m}(\prod_{i=1}^m x_i)^{-1}\psi(x) \in L^\infty(\Omega_m)\},}\) \({0 < \gamma < N}\), where \({\Omega_m=\{(x_1,\, x_2,\, \ldots,\, x_N) \in \mathbb{R}^{N};\; x_i > 0,\, 1\leq i\leq m\},}\) \({1\leq m\leq N}\). We then investigate the large time behavior of solutions of the heat equation \({u_{t}-\Delta u=0}\) for t > 0 and \({x \in \Omega_m.}\) Using certain notions from dynamical systems, we show that the large time behavior is related to the spatial asymptotic behavior of its initial value. Since the space \({\mathcal{X}_{m, \gamma}}\) contains highly singular initial data, which can be extended to all of \({\mathbb{R}^{N}}\) by antisymmetry, we also obtain new results on the complexity in the asymptotic behavior of solutions for the heat equation on the whole space.     

On generalized Hyers-Ulam stability of additive mapings on restricted domains of Banach spaces

Let \({(G,\cdot)}\) be a group (not necessarily Abelian) with unit \({e}\) and \({E}\) be a Banach space. In this paper, we show that there exist \({\alpha(p) > 0}\) for any \({0 < p < 1}\) and \({\beta(p,\varepsilon),\gamma(p,\varepsilon) > 0}\) for any \({0 < \varepsilon < \alpha(p)}\), such that for any surjective map \({f: G\rightarrow E}\) satisfying \({\big|\|f(x) + f(y)\|-\|f(xy) \|\big|\leq\varepsilon \|f(x)+f(y)\|^p}\) for all \({x,y\in G}\), there is a unique additive \({T:G\rightarrow E}\) such that \({\|f(x)-T(x)\|\leq\gamma(p,\varepsilon)\|f(x)\|^p}\) for all \({x\in G}\) satisfying \({\|f(x)\|\geq\beta(p,\varepsilon)}\). Moreover, we have \({\lim_{\varepsilon\rightharpoonup 0}\frac{\gamma(p,\varepsilon)}{\varepsilon} < \infty.}\)     

Quasisymmetric dimension distortion of Ahlfors regular subsets of a metric space

We show that if \({f\colon X\to Y}\) is a quasisymmetric mapping between Ahlfors regular spaces, then \({dim_H f(E)\leq dim_H E}\) for “almost every” bounded Ahlfors regular set \({E\subseteq X}\). If additionally, \({X}\) and \({Y}\) are Loewner spaces then \({dim_H f(E)=dim_H E}\) for “almost every" Ahlfors regular set \({E\subset X}\). The precise statements of these results are given in terms of Fuglede’s modulus of measures. As a corollary of these general theorems we show that if \({f}\) is a quasiconformal map of \({\mathbb{R}^N}\), \({N\geq 2}\), then for Lebesgue a.e. \({y\in\mathbb{R}^N}\) we have \({dim_H f(y+E) = dim_H E}\). A similar result holds for Carnot groups as well. For planar quasiconformal maps, our general estimates imply that if \({E \subset {\mathbb{R}}}\) is Ahlfors \({d}\)-regular, \({d < 1}\), then some component of \({f(E \times {\mathbb{R}})}\) has dimension at most \({2/(d+1)}\), and we construct examples to show this bound is sharp. In addition, we show there is a \({1}\)-dimensional set \({S\subseteq \mathbb R}\) and planar quasiconformal map \({f}\) such that \({f({\mathbb{R}} \times S)}\) contains no rectifiable sub-arcs. These results generalize work of Balogh et al. (J Math Pures Appl (2)99:125–149, 2013) and answer questions posed in Balogh et al. (J Math Pures Appl (2)99:125–149, 2013) and Capogna et al. (Mapping theory in metric spaces. http://aimpl.org/mappingmetric, 2016).     

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 \({i^*(\mathcal{P},\ell+1)\neq i^*(\mathcal{Q},\ell+1)}\).     

On the angle sum of lines

What is the maximum of the sum of the pairwise (non-obtuse) angles formed by n lines in the Euclidean 3-space? This question was posed by Fejes Tóth in (Acta Math Acad Sci Hung 10:13–19, 1959). Fejes Tóth solved the problem for \({n \leq 6}\), and proved the asymptotic upper bound \({n^{2} \pi /5}\) as \({n \to \infty}\). He conjectured that the maximum is asymptotically equal to \({n^{2} \pi /6}\) as \({n \to \infty}\). The main result of this paper is an upper bound on the sum of the angles of n lines in the Euclidean 3-space that is asymptotically equal to \({3n^{2} \pi /16}\) as \({n \to \infty}\).     

Integral means of weighted Hardy–Bloch functions

Let \({\mathcal{B}^\omega(p, q, B_d)}\) denote the \({\omega}\)-weighted Hardy–Bloch space on the unit ball B _d of \({\mathbb{C}^d}\), \({d\ge 1}\). For \({2< p,q < \infty}\) and \({f\in \mathcal{B}^\omega(p, q, B_d)}\), we obtain sharp estimates on the growth of the p-integral means M _p (f, r) as \({r\to 1-}\).     

On the k-free values of the polynomial {xy^k + C}

Consider the polynomial \({f(x, y) = xy^k + C}\) for \({k \geq 2}\) and any nonzero integer constant C. We derive an asymptotic formula for the k-free values of \({f(x, y)}\) when \({x, y \leq H}\). We also prove a similar result for the k-free values of \({f(p, q)}\) when \({p, q \leq H}\) are primes, thus extending Erd?s’ conjecture for our specific polynomial. The strongest tool we use is a recent generalization of the determinant method due to Reuss.     

Kleene algebras with implication

Inspired by an old construction due to J. Kalman that relates distributive lattices and centered Kleene algebras, in this paper we study an equivalence for certain categories whose objects are algebras with implication \({(H, \bigwedge, \bigvee, \rightarrow, 0,1)}\) which satisfy the following property for every \({a,b,c\, \in\, H}\): if \({a \leq b \rightarrow c}\), then \({a \bigwedge b \leq c}\).     

{{C^m}} Positive Eigenvectors for Linear Operators Arising in the Computation of Hausdorff Dimension

We consider a broad class of linear Perron–Frobenius operators \({\Lambda:X \rightarrow X}\), where \({X}\) is a real Banach space of \({C^m}\) functions. We prove the existence of a strictly positive \({C^m}\) eigenvector \({v}\) with eigenvalue \({r=r(\Lambda) =}\) the spectral radius of \({\Lambda}\). We prove (see Theorem 6.5 in Sect. 6 of this paper) that \({r(\Lambda)}\) is an algebraically simple eigenvalue and that, if \({\sigma(\Lambda)}\) denotes the spectrum of the complexification of \({\Lambda,\sigma(\Lambda) \backslash \{r(\Lambda)\}\subseteq \{\zeta \in \mathbb{C} \big| |\zeta| \le r_*\}}\), where \({r_* < r(\Lambda)}\). Furthermore, if \({u \in X}\) is any strictly positive function, \({(\frac 1r \Lambda)^k(u) \rightarrow s_u v}\) as \({k \rightarrow \infty}\), where \({s_u > 0}\) and convergence is in the norm topology on \({X}\). In applications to the computation of Hausdorff dimension, one is given a parametrized family \({\Lambda_s,s > s_*}\), of such operators and one wants to determine the (unique) value \({s_0}\) such that \({r(\Lambda_{s_0})=1}\). In another paper (Falk and Nussbaum in C\({^{\rm m}}\) Eigenfunctions of Perron–Frobenius operators and a new approach to numerical computation of Hausdorff dimension, submitted) we prove that explicit estimates on the partial derivatives of the positive eigenvector \({v_s}\) of \({\Lambda_s}\) can be obtained and that this information can be used to give rigorous, sharp upper and lower bounds for \({s_0}\).     

On classification of groups having Schur multiplier of maximum order

Let \({G}\) be a non-abelian finite \({p}\)-group of order \({p^n}\) with \({|G'| = p^k}\). Let \({M(G)}\) denote the Schur multiplier of \({G}\). Niroomand proved that \({|M(G)| \leq p^{\frac{1}{2}(n-k-1)(n+k-2)+1}}\). In this article we classify \({p}\)-groups \({G}\) of nilpotency class 2 for which \({|M(G)|}\) attains this bound.     

Goldbach’s conjecture in arithmetic progressions: number and size of exceptional prime moduli

Set \({T=N^{\frac{1}{3}-\epsilon}}\). It is proved that for all but \({\ll TL^{-H},\,H > 0}\), exceptional prime numbers \({k\leq T}\) and almost all integers b ₁, b ₂ co-prime to k, almost all integers \({n\sim N}\) satisfying \({n\equiv b_{1}+b_{2}(mod\,k)}\) can be written as the sum of two primes p ₁ and p ₂ satisfying \({p_{i}\equiv b_{i}(mod\,k),\,i=1,2}\). For prime numbers \({k\leq N^{\frac{5}{24}-\epsilon}}\), this result is even true for all but \({\ll (\log\,N)^{D}}\) primes k and all integers b ₁, b ₂ co-prime to k.     

Symbol <Emphasis Type="Italic">p</Emphasis>-algebras of prime degree and their <Emphasis Type="Italic">p</Emphasis>-central subspaces

We prove that the maximal dimension of a p-central subspace of the generic symbol p-algebra of prime degree p is \({p+1}\). We do it by proving the following number theoretic fact: let \({\{s_1,\dots,s_{p+1}\}}\) be \({p+1}\) distinct nonzero elements in the additive group \({G=(\mathbb{Z}/p \mathbb{Z}) \times (\mathbb{Z}/p \mathbb{Z})}\), then every nonzero element \({g \in G}\) can be expressed as \({d_1 s_1+\dots+d_{p+1} s_{p+1}}\) for some non-negative integers \({d_1,\dots,d_{p+1}}\) with \({d_1+\dots+d_{p+1}\leq p-1}\).     

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.     

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 \({{\mathbb{Q} [S^G_n]}}\).     

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 \({n}\).     

Cantor’s intersection theorem for <Emphasis Type="Italic">K</Emphasis>-metric spaces with a solid cone and a contraction principle

We establish an extension of Cantor’s intersection theorem for a \({K}\)-metric space (\({X, d}\)), where \({d}\) is a generalized metric taking values in a solid cone \({K}\) in a Banach space \({E}\). This generalizes a recent result of Alnafei, Radenovi? and Shahzad (2011) obtained for a \({K}\)-metric space over a solid strongly minihedral cone. Next we show that our Cantor’s theorem yields a special case of a generalization of Banach’s contraction principle given very recently by Cvetkovi? and Rako?evi? (2014): we assume that a mapping \({T}\) satisfies the condition “\({d(Tx, Ty) \preceq \Lambda (d(x, y))}\)” for \({x, y \in X}\), where \({\preceq}\) is a partial order induced by \({K}\), and \({\Lambda : E \rightarrow E}\) is a linear positive operator with the spectral radius less than one. We also obtain new characterizations of convergence in the sense of Huang and Zhang in a \({K}\)-metric space.