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 \({\log_{d-1} n}\).     

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 \({\lim_{\varepsilon\rightharpoonup 0}\frac{\gamma(p,\varepsilon)}{\varepsilon} < \infty.}\)     

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 \({h(\Omega_{a,b})=1}\).     

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 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 \({0 < p \leq\infty}\).     

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.     

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.     

{{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 pluri-canonical systems of arithmetic surfaces

Let \({S}\) be a Dedekind scheme with perfect residue fields at closed points. Let \({f: X\rightarrow S}\) be a minimal regular arithmetic surface of fibre genus at least 2 and let \({f': X'\rightarrow S}\) be the canonical model of \({f}\). It is well known that \({\omega_{X'/S}}\) is relatively ample. In this paper we prove that \({\omega_{X'/S}^{\otimes n}}\) is relatively very ample for all \({n\geq 3}\).     

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 \({X}\).     

On the Construction of Composite Wannier Functions

We give a constructive proof for the existence of a Bloch basis of rank \({N}\) which is both smooth (real analytic) and periodic with respect to its \({d}\)-dimensional quasi-momenta, when \({1\leq d\leq 2}\) and \({N\geq 1}\). The constructed Bloch basis is conjugation symmetric when the underlying projection has this symmetry, hence the corresponding exponentially localized composite Wannier functions are real. In the second part of the paper, we show that by adding a weak, globally bounded but not necessarily constant magnetic field, the existence of a localized basis is preserved.     

Gelfand type problem for singular quadratic quasilinear equations

In this paper, we study the existence of positive solutions for the quasilinear elliptic singular problem

$$\left\{\begin{array}{ll}-\Delta u + c\,\frac{|\nabla u|^2}{u^\gamma} = \lambda\,f(u), \quad \quad \mbox{in $\Omega$},\\ u=0, \quad \qquad \qquad \qquad \quad \, \, \, \, \, \mbox{on $\partial$$\Omega$},\end{array}\right.$$

where \({c,\lambda >0}\), \({\gamma \in (0,1)}\), f is strictly increasing and derivable in \({[0,\infty)}\) with \({f(0)>0}\). We show that there exists \({\lambda^*>0}\) such that \({(0,\lambda^*]}\) is the maximal set of values such there exists solution. In addition, we prove that for \({\lambda<\lambda^*}\) there exists minimal and bounded solutions. Moreover, we give sufficient conditions for existence and regularity of solutions for \({\lambda=\lambda^*}\).

     

Continued fraction normality is not preserved along arithmetic progressions

It is well known that if \({0.a_1a_2a_3\ldots}\) is the base-\({b}\) expansion of a number normal to base-\({b}\), then the numbers \({0.a_ka_{m+k}a_{2m+k}\ldots}\) for \({m\ge 2}\), \({k\ge 1}\) are all normal to base-\({b}\) as well. In contrast, given a continued fraction expansion \({\langle a_1,a_2,a_3,\ldots\rangle}\) that is normal (now with respect to the continued fraction expansion), we show that for any integers \({m\ge 2}\), \({k\ge 1}\), the continued fraction \({\langle a_k, a_{m+k},a_{2m+k},a_{3m+k},\ldots\rangle}\) will never be normal.     

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 \({n \to \infty}\).     

Applications of strongly convergent sequences to Fourier series by means of modulus functions

The aim of this work is to estimate sums involving P(n), the largest prime factor of an integer \({n \geqq 2}\) under digital constraints \({{f(P(n)) \equiv a}{\rm mod} b}\), for every \({a \in \mathbb{Z}}\) and an integer \({b \geqq 2}\) where f is a strongly q-additive function with integer values (i.e. \({f(aq^j + b) = f(a) + f(b)}\), with \({(a, b, j) \in \mathbb{N}^3}\), \({{0 \leqq b} < q^j}\)). We also estimate the cardinality of the set \({\{{n \leqq x, f(P(n) + c)} \equiv {a {\rm mod} b}, P(n) \equiv l {\rm mod} k\}}\), where \({c \in \mathbb{Z}}\), \({k \geqq 2}\).     

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.     

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.     

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.