Low Regularity Well-Posedness for the 3D Generalized Hall-MHD System

In this paper, we obtain the local well-posedness for the 3D incompressible Hall-magnetohydrodynamics (Hall-MHD) system with \(\varLambda^{2\alpha }u\) and \(\varLambda^{2\beta }B\), \(0<\alpha \le 1\), \(\frac{1}{2}<\beta \le 1\). Our results improve regularity conditions on the initial data of previous works.     

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.     

On quasi-inversions

Given a bounded domain \(D \subset {\mathbb R}^n\) strictly starlike with respect to \(0 \in D\,,\) we define a quasi-inversion w.r.t. the boundary \(\partial D \,.\) We show that the quasi-inversion is bi-Lipschitz w.r.t. the chordal metric if and only if every “tangent line” of \(\partial D\) is far away from the origin. Moreover, the bi-Lipschitz constant tends to 1,  when \(\partial D\) approaches the unit sphere in a suitable way. For the formulation of our results we use the concept of the \(\alpha \)-tangent condition due to Gehring and Väisälä (Acta Math 114:1–70,1965). This condition is shown to be equivalent to the bi-Lipschitz and quasiconformal extension property of what we call the polar parametrization of \(\partial D\). In addition, we show that the polar parametrization, which is a mapping of the unit sphere onto \(\partial D\), is bi-Lipschitz if and only if D satisfies the \(\alpha \)-tangent condition.     

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 \(\le 8\).     

Toroidal Geometry Stabilizing a Latitudinal Ring of Point Vortices on a Torus

We carry out the linear stability analysis of a polygonal ring configuration of N point vortices, called an N-ring, along the line of latitude \(\theta _0\) on a torus with the aspect ratio \(\alpha \). Deriving a criterion for the stability depending on the parameters N, \(\theta _0\) and \(\alpha \), we reveal how the aspect ratio \(\alpha \) contributes to the stability of the N-ring. While the N-ring necessarily becomes unstable when N is sufficiently large for fixed \(\alpha \), the stability is closely associated with the geometric property of the torus for variable \(\alpha \); for low aspect ratio \(\alpha \sim 1\), \(N=7\) is a critical number determining the stability of the N-ring when it is located along a certain range of latitudes, which is an analogous result to those in a plane and on a sphere. On the other hand, the stability is determined by the sign of curvature for high aspect ratio \(\alpha \gg 1\). That is to say, the N-ring is neutrally stable if it is located on the inner side of the toroidal surface with a negative curvature, while the N-ring on its outer side with a positive curvature is unstable. Furthermore, based on the linear stability analysis, we describe nonlinear evolution of the N-ring when it becomes unstable. It is difficult to deal with this problem, since the evolution equation of the N point vortices is formulated as a Hamiltonian system with N degrees of freedom, which is in general non-integrable. Thus, we reduce the Hamiltonian system to a simple integrable system by introducing a cyclic symmetry. Owing to this reduction, we successfully find some periodic orbits in the reduced system, whose local bifurcations and global transitions for variable \(\alpha \) are characterized in terms of the fundamental group of the torus.     

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.     

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 \(\mathbb {C}^n\).     

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 \(\alpha \).     

The Sobolev Stability Threshold for 2D Shear Flows Near Couette

We consider the 2D Navier–Stokes equation on \(\mathbb T \times \mathbb R\), with initial datum that is \(\varepsilon \)-close in \(H^N\) to a shear flow (U(y), 0), where \(\Vert U(y) - y\Vert _{H^{N+4}} \ll 1\) and \(N>1\). We prove that if \(\varepsilon \ll \nu ^{1/2}\), where \(\nu \) denotes the inverse Reynolds number, then the solution of the Navier–Stokes equation remains \(\varepsilon \)-close in \(H^1\) to \((e^{t \nu \partial _{yy}}U(y),0)\) for all \(t>0\). Moreover, the solution converges to a decaying shear flow for times \(t \gg \nu ^{-1/3}\) by a mixing-enhanced dissipation effect, and experiences a transient growth of gradients. In particular, this shows that the stability threshold in finite regularity scales no worse than \(\nu ^{1/2}\) for 2D shear flows close to the Couette flow.     

Lifting Maps from the Symmetrized Polydisc in Small Dimensions

The spectral unit ball \(\Omega _n\) is the set of all \(n\times n\) matrices M with spectral radius less than 1. Let \(\pi (M) \in \mathbb {C}^n\) stand for the coefficients of the characteristic polynomial of a matrix M (up to signs), i.e. the elementary symmetric functions of its eigenvalues. The symmetrized polydisc is \({{\mathbb {G}}}_n:=\pi (\Omega _n)\). When investigating Nevanlinna–Pick problems for maps from the disk to the spectral ball, it is often useful to project the map to the symmetrized polydisc (for instance to obtain continuity results for the Lempert function): if \(\Phi \in {\mathrm {Hol}}(\mathbb {D}, \Omega _n)\), then \(\pi \circ \Phi \in {\mathrm {Hol}}(\mathbb {D}, {{\mathbb {G}}}_n)\). Given a map \(\varphi \in {\mathrm {Hol}}(\mathbb {D}, {{\mathbb {G}}}_n)\), we are looking for necessary and sufficient conditions for this map to “lift through given matrices”, i.e. find \(\Phi \) as above so that \(\pi \circ \Phi = \varphi \) and \(\Phi (\alpha _j) = A_j\), \(1\le j \le N\). A natural necessary condition is \(\varphi (\alpha _j)=\pi (A_j)\), \(1\le j \le N\). When the matrices \(A_j\) are derogatory (i.e. do not admit a cyclic vector) new necessary conditions appear, involving derivatives of \(\varphi \) at the points \(\alpha _j\). We prove that those conditions are necessary and sufficient for a local lifting. We give a formula which performs the global lifting in small dimensions (\(n \le 5\)), and a counter-example to show that the formula fails in dimensions 6 and above.     

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.     

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 \(1<\alpha \le 2\).     

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.     

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.     

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.     

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.     

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 \(\phi (s,t)=\beta _{a,b}(s)+\beta _{a,b}(t)\).     

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 \(\Omega \).