Cartoon Approximation with $$\alpha $$-Curvelets

It is well-known that curvelets provide optimal approximations for so-called cartoon images which are defined as piecewise \(C^2\)-functions, separated by a \(C^2\) singularity curve. In this paper, we consider the more general case of piecewise \(C^\beta \)-functions, separated by a \(C^\beta \) singularity curve for \(\beta \in (1,2]\). We first prove a benchmark result for the possibly achievable best N-term approximation rate for this more general signal model. Then we introduce what we call \(\alpha \)-curvelets, which are systems that interpolate between wavelet systems on the one hand (\(\alpha = 1\)) and curvelet systems on the other hand (\(\alpha = \frac{1}{2}\)). Our main result states that those frames achieve this optimal rate for \(\alpha = \frac{1}{\beta }\), up to \(\log \)-factors.     

Few associative triples,isotopisms and groups

Let Q be a quasigroup. For \(\alpha ,\beta \in S_Q\) let \(Q_{\alpha ,\beta }\) be the principal isotope \(x*y = \alpha (x)\beta (y)\). Put \(\mathbf a(Q)= |\{(x,y,z)\in Q^3;\) \(x(yz)) = (xy)z\}|\) and assume that \(|Q|=n\). Then \(\sum _{\alpha ,\beta }\mathbf a(Q_{\alpha ,\beta })/(n!)^2 = n^2(1+(n-1)^{-1})\), and for every \(\alpha \in S_Q\) there is \(\sum _\beta \mathbf a(Q_{\alpha ,\beta })/n! = n(n-1)^{-1}\sum _x(f_x^2-2f_x+n)\ge n^2\), where \(f_x=|\{y\in Q;\) \( y = \alpha (y)x\}|\). If G is a group and \(\alpha \) is an orthomorphism, then \(\mathbf a(G_{\alpha ,\beta })=n^2\) for every \(\beta \in S_Q\). A detailed case study of \(\mathbf a(G_{\alpha ,\beta })\) is made for the situation when \(G = \mathbb Z_{2d}\), and both \(\alpha \) and \(\beta \) are “natural” near-orthomorphisms. Asymptotically, \(\mathbf a(G_{\alpha ,\beta })>3n\) if G is an abelian group of order n. Computational results: \(\mathbf a(7) = 17\) and \(\mathbf a(8) \le 21\), where \(\mathbf a(n) = \min \{\mathbf a(Q);\) \( |Q|=n\}\). There are also determined minimum values for \(\mathbf a(G_{\alpha ,\beta })\), G a group of order \(\le 8\).     

Some Ricci-flat ($$\alpha ,\beta $$)-metrics

In this paper, we study a special class of Finsler metrics, \((\alpha ,\beta )\)-metrics, defined by \(F=\alpha \phi (\beta /\alpha )\), where \(\alpha \) is a Riemannian metric and \(\beta \) is a 1-form. We find an equation that characterizes Ricci-flat \((\alpha ,\beta )\)-metrics under the condition that the length of \(\beta \) with respect to \(\alpha \) is constant.     

Lengths of words in transformation semigroups generated by digraphs

Given a simple digraph D on n vertices (with \(n\ge 2\)), there is a natural construction of a semigroup of transformations \(\langle D\rangle \). For any edge (a, b) of D, let \(a\rightarrow b\) be the idempotent of rank \(n-1\) mapping a to b and fixing all vertices other than a; then, define \(\langle D\rangle \) to be the semigroup generated by \(a \rightarrow b\) for all \((a,b) \in E(D)\). For \(\alpha \in \langle D\rangle \), let \(\ell (D,\alpha )\) be the minimal length of a word in E(D) expressing \(\alpha \). It is well known that the semigroup \(\mathrm {Sing}_n\) of all transformations of rank at most \(n-1\) is generated by its idempotents of rank \(n-1\). When \(D=K_n\) is the complete undirected graph, Howie and Iwahori, independently, obtained a formula to calculate \(\ell (K_n,\alpha )\), for any \(\alpha \in \langle K_n\rangle = \mathrm {Sing}_n\); however, no analogous non-trivial results are known when \(D \ne K_n\). In this paper, we characterise all simple digraphs D such that either \(\ell (D,\alpha )\) is equal to Howie–Iwahori’s formula for all \(\alpha \in \langle D\rangle \), or \(\ell (D,\alpha ) = n - \mathrm {fix}(\alpha )\) for all \(\alpha \in \langle D\rangle \), or \(\ell (D,\alpha ) = n - \mathrm {rk}(\alpha )\) for all \(\alpha \in \langle D\rangle \). We also obtain bounds for \(\ell (D,\alpha )\) when D is an acyclic digraph or a strong tournament (the latter case corresponds to a smallest generating set of idempotents of rank \(n-1\) of \(\mathrm {Sing}_n\)). We finish the paper with a list of conjectures and open problems.     

Closures of Hardy and Hardy–Sobolev Spaces in the Bloch Type Space on the Unit Ball

For \(0<\alpha <\infty \), \(0<p<\infty \) and \(0<s<\infty \), we characterize the closures in the \(\alpha \)-Bloch norm of \(\alpha \)-Bloch functions that are in a Hardy space \(H^p\) and in a Hardy–Sobolev space \(H^p_s\) on the unit ball of \(\mathbb {C}^n\).     

Transience and Recurrence of Random Walks on Percolation Clusters in an Ultrametric Space

We study transience and recurrence of simple random walks on percolation clusters in the hierarchical group of order N, which is an ultrametric space. The connection probability on the hierarchical group for two points separated by distance k is of the form \(c_k/N^{k(1+\delta )}, \delta >0\), with \(c_k=C_0+C_1\log k+C_2k^\alpha \), non-negative constants \(C_0, C_1, C_2\), and \(\alpha >0\). Percolation occurs for \(\delta <1\), and for the critical case, \(\delta =1\), \(\alpha >0\) and sufficiently large \(C_2\). We show that in the case \(\delta <1\) the walk is transient, and in the case \(\delta =1,C_2>0,\alpha >0\) there exists a critical \(\alpha _\mathrm{c}\in (0,\infty )\) such that the walk is recurrent for \(\alpha <\alpha _\mathrm{c}\) and transient for \(\alpha >\alpha _\mathrm{c}\). The proofs involve ultrametric random graphs, graph diameters, path lengths, and electric circuit theory. Some comparisons are made with behaviours of simple random walks on long-range percolation clusters in the one-dimensional Euclidean lattice.     

Phase transition in loop percolation

We are interested in the clusters formed by a Poisson ensemble of Markovian loops on infinite graphs. This model was introduced and studied in Le Jan (C R Math Acad Sci Paris 350(13–14):643–646, 2012, Ill J Math 57(2):525–558, 2013). It is a model with long range correlations with two parameters \(\alpha \) and \(\kappa \). The non-negative parameter \(\alpha \) measures the amount of loops, and \(\kappa \) plays the role of killing on vertices penalizing (\(\kappa >0\)) or favoring (\(\kappa <0\)) appearance of large loops. It was shown in Le Jan (Ill J Math 57(2):525–558, 2013) that for any fixed \(\kappa \) and large enough \(\alpha \), there exists an infinite cluster in the loop percolation on \({\mathbb {Z}}^d\). In the present article, we show a non-trivial phase transition on the integer lattice \({\mathbb {Z}}^d\) (\(d\ge 3\)) for \(\kappa =0\). More precisely, we show that there is no loop percolation for \(\kappa =0\) and \(\alpha \) small enough. Interestingly, we observe a critical like behavior on the whole sub-critical domain of \(\alpha \), namely, for \(\kappa =0\) and any sub-critical value of \(\alpha \), the probability of one-arm event decays at most polynomially. For \(d\ge 5\), we prove that there exists a non-trivial threshold for the finiteness of the expected cluster size. For \(\alpha \) below this threshold, we calculate, up to a constant factor, the decay of the probability of one-arm event, two point function, and the tail distribution of the cluster size. These rates are comparable with the ones obtained from a single large loop and only depend on the dimension. For \(d=3\) or 4, we give better lower bounds on the decay of the probability of one-arm event, which show importance of small loops for long connections. In addition, we show that the one-arm exponent in dimension 3 depends on the intensity \(\alpha \).     

Heat Content for Stable Processes in Domains of $$\mathbb {R}^d$$

This paper studies the small time behavior of the heat content for rotationally invariant \(\alpha \)-stable processes, \(0<\alpha \le 2\), in domains of \(\mathbb {R}^d\). Unlike the asymptotics for the heat trace, the behavior of the heat content differs depending on the range of \(\alpha \) according to \(0<\alpha <1\), \(\alpha =1\) and \(1<\alpha \le 2\).     

Asymptotic diophantine approximation: the multiplicative case

Let \(\alpha \) and \(\beta \) be irrational real numbers and \(0<\varepsilon <1/30\). We prove a precise estimate for the number of positive integers \(q\le Q\) that satisfy \(\Vert q\alpha \Vert \cdot \Vert q\beta \Vert <\varepsilon \). If we choose \(\varepsilon \) as a function of Q, we get asymptotics as Q gets large, provided \(\varepsilon Q\) grows quickly enough in terms of the (multiplicative) Diophantine type of \((\alpha ,\beta )\), e.g., if \((\alpha ,\beta )\) is a counterexample to Littlewood’s conjecture, then we only need that \(\varepsilon Q\) tends to infinity. Our result yields a new upper bound on sums of reciprocals of products of fractional parts and sheds some light on a recent question of Lê and Vaaler.     

Sharpness of Complex Interpolation on $$\alpha $$-Modulation Spaces

In this paper, we solve a long standing problem on the modulation spaces, \(\alpha \)-modulation spaces and Besov spaces. We establish sharp conditions for the complex interpolation between these function spaces. We show that no \(\alpha \)-modulation space \(M_{p,q}^{s,\alpha }\) can be regarded as the interpolation space between \(M_{p_1,q_1}^{s_1,\alpha _1}\) and \(M_{p_2,q_2}^{s_2,\alpha _2}\), unless \(\alpha _1\) is equal to \(\alpha _2\), essentially. Especially, our results show that the \(\alpha \)-modulation spaces can not be obtained by complex interpolation between modulation spaces and Besov spaces.     

Multiplicity Results for the Fractional Laplacian in Expanding Domains

In this paper, we establish a multiplicity result of nontrivial weak solutions for the problem \((-\Delta )^{\alpha } u +u= h(u)\)    in \(\Omega _{\lambda }\), \(u=0\)    on \(\partial \Omega _{\lambda }\), where \(\Omega _{\lambda }=\lambda \Omega \), \(\Omega \) is a smooth and bounded domain in \({\mathbb {R}}^N, N>2\alpha \), \(\lambda \) is a positive parameter, \(\alpha \in (0,1)\), \((-\Delta )^{\alpha }\) is the fractional Laplacian and the nonlinear term h(u) has subcritical growth. We use minimax methods, the Ljusternick–Schnirelmann and Morse theories to get multiplicity results depending on the topology of \(\Omega \).     

Two-step estimation of ergodic Lévy driven SDE

We consider high frequency samples from ergodic Lévy driven stochastic differential equation with drift coefficient \(a(x,\alpha )\) and scale coefficient \(c(x,\gamma )\) involving unknown parameters \(\alpha \) and \(\gamma \). We suppose that the Lévy measure \(\nu _{0}\), has all order moments but is not fully specified. We will prove the joint asymptotic normality of some estimators of \(\alpha \), \(\gamma \) and a class of functional parameter \(\int \varphi (z)\nu _0(dz)\), which are constructed in a two-step manner: first, we use the Gaussian quasi-likelihood for estimation of \((\alpha ,\gamma )\); and then, for estimating \(\int \varphi (z)\nu _0(dz)\) we make use of the method of moments based on the Euler-type residual with the the previously obtained quasi-likelihood estimator.     

Minimal Translation Surfaces in Euclidean Space

A translation surface in Euclidean space is a surface that is the sum of two regular curves \(\alpha \) and \(\beta \). In this paper we characterize all minimal translation surfaces. In the case that \(\alpha \) and \(\beta \) are non-planar curves, we prove that the curvature \(\kappa \) and the torsion \(\tau \) of both curves must satisfy the equation \(\kappa ^2 \tau = C\) where C is constant. We show that, up to a rigid motion and a dilation in the Euclidean space and, up to reparametrizations of the curves generating the surfaces, all minimal translation surfaces are described by two real parameters \(a,b\in \mathbb {R}\) where the surface is of the form \(\phi (s,t)=\beta _{a,b}(s)+\beta _{a,b}(t)\).     

The First Passage Time Problem Over a Moving Boundary for Asymptotically Stable Lévy Processes

We study the asymptotic tail behaviour of the first passage time over a moving boundary for asymptotically \(\alpha \)-stable Lévy processes with \(\alpha <1\). Our main result states that if the left tail of the Lévy measure is regularly varying with index \(- \alpha \), and the moving boundary is equal to \(1 - t^{\gamma }\) for some \(\gamma <1/\alpha \), then the probability that the process stays below the moving boundary has the same asymptotic polynomial order as in the case of a constant boundary. The same is true for the increasing boundary \(1 + t^{\gamma }\) with \(\gamma <1/\alpha \) under the assumption of a regularly varying right tail with index \(-\alpha \).     

Typical Behavior of the Harmonic Measure in Critical Galton–Watson Trees with Infinite Variance Offspring Distribution

We study the typical behavior of the harmonic measure in large critical Galton–Watson trees whose offspring distribution is in the domain of attraction of a stable distribution with index \(\alpha \in (1,2]\). Let \(\mu _n\) denote the hitting distribution of height n by simple random walk on the critical Galton–Watson tree conditioned on non-extinction at generation n. We extend the results of Lin (Typical behavior of the harmonic measure in critical Galton–Watson trees, arXiv:1502.05584, 2015) to prove that, with high probability, the mass of the harmonic measure \(\mu _n\) carried by a random vertex uniformly chosen from height n is approximately equal to \(n^{-\lambda _\alpha }\), where the constant \(\lambda _\alpha >\frac{1}{\alpha -1}\) depends only on the index \(\alpha \). In the analogous continuous model, this constant \(\lambda _\alpha \) turns out to be the typical local dimension of the continuous harmonic measure. Using an explicit formula for \(\lambda _\alpha \), we are able to show that \(\lambda _\alpha \) decreases with respect to \(\alpha \in (1,2]\), and it goes to infinity at the same speed as \((\alpha -1)^{-2}\) when \(\alpha \) approaches 1.     

Compact $$\lambda $$-translating Solitons with Boundary

A \(\lambda \)-translating soliton with density vector \(\mathbf {v}\) is a surface \(\varSigma \) in Euclidean space \(\mathbb {R}^3\) whose mean curvature H satisfies \(2H=2\lambda +\langle N,\mathbf {v}\rangle \), where N is the Gauss map of \(\varSigma \). In this article, we study the shape of a compact \(\lambda \)-translating soliton in terms of its boundary. If \(\varGamma \) is a given closed curve, we deduce under what conditions on \(\lambda \) there exists a compact \(\lambda \)-translating soliton \(\varSigma \) with boundary \(\varGamma \) and we provide estimates of the surface area depending on the height of \(\varSigma \). Finally, we study the shape of \(\varSigma \) related with the geometry of \(\varGamma \), in particular, we give conditions that assert that \(\varSigma \) inherits the symmetries of its boundary \(\varGamma \).     

rthogonality Property $$\mathcal {}$$ and Compact Perturbations

A bounded linear operator T acting on a Hilbert space is said to have orthogonality property \(\mathcal {O}\) if the subspaces \(\ker (T-\alpha )\) and \(\ker (T-\beta )\) are orthogonal for all \(\alpha , \beta \in \sigma _p(T)\) with \(\alpha \ne \beta \). In this paper, the authors investigate the compact perturbations of operators with orthogonality property \(\mathcal {O}\). We give a sufficient and necessary condition to determine when an operator T has the following property: for each \(\varepsilon >0\), there exists \(K\in \mathcal {K(H)}\) with \(\Vert K\Vert <\varepsilon \) such that \(T+K\) has orthogonality property \(\mathcal {O}\). Also, we study the stability of orthogonality property \(\mathcal {O}\) under small compact perturbations and analytic functional calculus.     

Counting exceptional points for rational numbers associated to the Fibonacci sequence

If \(\alpha \) is a non-zero algebraic number, we let \(m(\alpha )\) denote the Mahler measure of the minimal polynomial of \(\alpha \) over \(\mathbb Z\). A series of articles by Dubickas and Smyth, and later by the author, develop a modified version of the Mahler measure called the t-metric Mahler measure, denoted \(m_t(\alpha )\). For fixed \(\alpha \in \overline{\mathbb Q}\), the map \(t\mapsto m_t(\alpha )\) is continuous, and moreover, is infinitely differentiable at all but finitely many points, called exceptional points for \(\alpha \). It remains open to determine whether there is a sequence of elements \(\alpha _n\in \overline{\mathbb Q}\) such that the number of exceptional points for \(\alpha _n\) tends to \(\infty \) as \(n\rightarrow \infty \). We utilize a connection with the Fibonacci sequence to formulate a conjecture on the t-metric Mahler measures. If the conjecture is true, we prove that it is best possible and that it implies the existence of rational numbers with as many exceptional points as we like. Finally, with some computational assistance, we resolve various special cases of the conjecture that constitute improvements to earlier results.     

New lower bounds for the Shannon capacity of odd cycles

The Shannon capacity of a graph G is defined as \(c(G)=\sup _{d\ge 1}(\alpha (G^d))^{\frac{1}{d}},\) where \(\alpha (G)\) is the independence number of G. The Shannon capacity of the cycle \(C_5\) on 5 vertices was determined by Lovász in 1979, but the Shannon capacity of a cycle \(C_p\) for general odd p remains one of the most notorious open problems in information theory. By prescribing stabilizers for the independent sets in \(C_p^d\) and using stochastic search methods, we show that \(\alpha (C_7^5)\ge 350\), \(\alpha (C_{11}^4)\ge 748\), \(\alpha (C_{13}^4)\ge 1534\), and \(\alpha (C_{15}^3)\ge 381\). This leads to improved lower bounds on the Shannon capacity of \(C_7\) and \(C_{15}\): \(c(C_7)\ge 350^{\frac{1}{5}}> 3.2271\) and \(c(C_{15})\ge 381^{\frac{1}{3}}> 7.2495\).