Singular limits for Liouville-type equations on the flat two-torus

We consider the problem $$\begin{aligned} -\Delta u=\varepsilon ^{2}e^{u}- \frac{1}{|\Omega |}\int _\Omega \varepsilon ^{2} e^{u}+ {4\pi N\over |\Omega |} - 4 \pi N\delta _p, \quad \text{ in} {\Omega }, \quad \int _\Omega u=0 \end{aligned}$$ in a flat two-torus $\Omega $ with periodic boundary conditions, where $\varepsilon >0,\,|\Omega |$ is the area of the $\Omega $ , $N>0$ and $\delta _p$ is a Dirac mass at $p\in \Omega $ . We prove that if $1\le m<N+1$ then there exists a family of solutions $\{u_\varepsilon \}_{\varepsilon }$ such that $\varepsilon ^{2}e^{u_\varepsilon }\rightharpoonup 8\pi \sum _{i=1}^m\delta _{q_i}$ as $\varepsilon \rightarrow 0$ in measure sense for some different points $q_{1}, \ldots , q_{m}$ . Furthermore, points $q_i$ , $i=1,\dots ,m$ are different from $p$ .     

Dimension Reduction for Finite Trees in \varvec{\ell _1}

We show that every $n$ -point tree metric admits a $(1+\varepsilon )$ -embedding into $\ell _1^{C(\varepsilon ) \log n}$ , for every $\varepsilon > 0$ , where $C(\varepsilon ) \le O\big ((\frac{1}{\varepsilon })^4 \log \frac{1}{\varepsilon })\big )$ . This matches the natural volume lower bound up to a factor depending only on $\varepsilon $ . Previously, it was unknown whether even complete binary trees on $n$ nodes could be embedded in $\ell _1^{O(\log n)}$ with $O(1)$ distortion. For complete $d$ -ary trees, our construction achieves $C(\varepsilon ) \le O\big (\frac{1}{\varepsilon ^2}\big )$ .     

Singular perturbation method for inhomogeneous nonlinear free boundary problems

In this paper, we study solutions of one phase inhomogeneous singular perturbation problems of the type: $ F(D^2u,x)=\beta _{\varepsilon }(u) + f_{\varepsilon }(x) $ and $ \Delta _{p}u=\beta _{\varepsilon }(u) + f_{\varepsilon }(x)$ , where $\beta _{\varepsilon }$ approaches Dirac $\delta _{0}$ as $\varepsilon \rightarrow 0$ and $f_{\varepsilon }$ has a uniform control in $L^{q}, q>N.$ Uniform local Lipschitz regularity is obtained for these solutions. The existence theory for variational (minimizers) and non variational (least supersolutions) solutions for these problems is developed. Uniform linear growth rate with respect to the distance from the $\varepsilon -$ level surfaces are established for these variational and nonvaritional solutions. Finally, letting $\varepsilon \rightarrow 0$ basic properties such as local Lipschitz regularity and non-degeneracy property are proven for the limit and a Hausdorff measure estimate for its free boundary is obtained.     

1D Schrödinger Operators with Short Range Interactions: Two-Scale Regularization of Distributional Potentials

For real ${L_\infty(\mathbb{R})}$ -functions ${\Phi}$ and ${\Psi}$ of compact support, we prove the norm resolvent convergence, as ${\varepsilon}$ and ${\nu}$ tend to 0, of a family ${S_{\varepsilon \nu}}$ of one-dimensional Schrödinger operators on the line of the form $$S_{\varepsilon \nu} = -\frac{d^2}{dx^2} + \frac{\alpha}{\varepsilon^2} \Phi \left( \frac{x}{\varepsilon} \right) + \frac{\beta}{\nu} \Psi \left(\frac{x}{\nu} \right),$$ provided the ratio ${\nu/\varepsilon}$ has a finite or infinite limit. The limit operator S ₀ depends on the shape of ${\Phi}$ and ${\Psi}$ as well as on the limit of ratio ${\nu/\varepsilon}$ . If the potential ${\alpha\Phi}$ possesses a zero-energy resonance, then S ₀ describes a non trivial point interaction at the origin. Otherwise S ₀ is the direct sum of the Dirichlet half-line Schrödinger operators.     

Tunneling for a Class of Difference Operators

We analyze a general class of difference operators ${H_\varepsilon = T_\varepsilon + V_\varepsilon}$ on ${\ell^2((\varepsilon \mathbb {Z})^d)}$ where ${V_\varepsilon}$ is a multi-well potential and ${\varepsilon}$ is a small parameter. We decouple the wells by introducing certain Dirichlet operators on regions containing only one potential well, and we shall treat the eigenvalue problem for ${H_\varepsilon}$ as a small perturbation of these comparison problems. We describe tunneling by a certain interaction matrix, similar to the analysis for the Schr?dinger operator [see Helffer and Sj?strand in Commun Partial Differ Equ 9:337–408, 1984], and estimate the remainder, which is exponentially small and roughly quadratic compared with the interaction matrix.     

Rational matrix pseudodifferential operators

The skewfield $\mathcal{K }(\partial )$ of rational pseudodifferential operators over a differential field $\mathcal{K }$ is the skewfield of fractions of the algebra of differential operators $\mathcal{K }[\partial ]$ . In our previous paper, we showed that any $H\in \mathcal{K }(\partial )$ has a minimal fractional decomposition $H=AB^{-1}$ , where $A,B\in \mathcal{K }[\partial ],\,B\ne 0$ , and any common right divisor of $A$ and $B$ is a non-zero element of $\mathcal{K }$ . Moreover, any right fractional decomposition of $H$ is obtained by multiplying $A$ and $B$ on the right by the same non-zero element of $\mathcal{K }[\partial ]$ . In the present paper, we study the ring $M_n(\mathcal{K }(\partial ))$ of $n\times n$ matrices over the skewfield $\mathcal{K }(\partial )$ . We show that similarly, any $H\in M_n(\mathcal{K }(\partial ))$ has a minimal fractional decomposition $H=AB^{-1}$ , where $A,B\in M_n(\mathcal{K }[\partial ]),\,B$ is non-degenerate, and any common right divisor of $A$ and $B$ is an invertible element of the ring $M_n(\mathcal{K }[\partial ])$ . Moreover, any right fractional decomposition of $H$ is obtained by multiplying $A$ and $B$ on the right by the same non-degenerate element of $M_n(\mathcal{K } [\partial ])$ . We give several equivalent definitions of the minimal fractional decomposition. These results are applied to the study of maximal isotropicity property, used in the theory of Dirac structures.     

The Roles Played by Order of Convexity or Starlikeness and the Bloch Condition in the Extension of Mappings from the Disk to the Ball

Let ${G: \mathbb {C}^{n-1} \rightarrow \mathbb {C}}$ be holomorphic such that G(0)?=?0 and DG(0)?=?0. When f is a convex (resp. starlike) normalized (f(0)?=?0, f??(0)?=?1) univalent mapping of the unit disk ${\mathbb {D}}$ in ${\mathbb {C}}$ , then the extension of f to the Euclidean unit ball ${\mathbb {B}}$ in ${\mathbb {C}^n}$ given by ${\Phi_G(f)(z)=(f(z_1)+G(\sqrt{f^{\prime}(z_1)} \, \hat{z}),\sqrt{f^{\prime}(z_1)}\, \hat{z})}$ , ${\hat{z}=(z_2,\dots,z_n) \in \mathbb {C}^{n-1}}$ , is known to be convex (resp. starlike) if G is a homogeneous polynomial of degree 2 with sufficiently small norm. Conversely, it is known that G cannot have terms of degree greater than 2 in its expansion about 0 in order for ${\Phi_G(f)}$ to be convex (resp. starlike), in general. We examine whether the restriction that f be either convex or starlike of a certain order ${\alpha \in (0,1]}$ allows, in general, for G to contain terms of degree greater than 2 and still have ${\Phi_G(f)}$ maintain the respective geometric property. Related extension results for convex and starlike Bloch mappings are also given.     

On the boundary of the attainable set of the dirichlet spectrum

Denoting by ${\varepsilon\subseteq\mathbb{R}^2}$ the set of the pairs ${(\lambda_1(\Omega),\,\lambda_2(\Omega))}$ for all the open sets ${\Omega\subseteq\mathbb{R}^N}$ with unit measure, and by ${\Theta\subseteq\mathbb{R}^N}$ the union of two disjoint balls of half measure, we give an elementary proof of the fact that ${\partial\varepsilon}$ has horizontal tangent at its lowest point ${(\lambda_1(\Theta),\,\lambda_2(\Theta))}$ .     

Sharp geometric rigidity of isometries on Heisenberg groups

We prove sharp geometric rigidity estimates for isometries on Heisenberg groups. Our main result asserts that every $(1+\varepsilon )$ -quasi-isometry on a John domain of the Heisenberg group $\mathbb H ^n, n>1,$ is close to some isometry up to proximity order $\sqrt{\varepsilon }+\varepsilon $ in the uniform norm, and up to proximity order $\varepsilon $ in the $L_p^1$ -norm. We give examples showing the asymptotic sharpness of our results.     

Universal geometric cluster algebras

We consider, for each exchange matrix $B$ , a category of geometric cluster algebras over $B$ and coefficient specializations between the cluster algebras. The category also depends on an underlying ring $R$ , usually $\mathbb {Z},\,\mathbb {Q}$ , or $\mathbb {R}$ . We broaden the definition of geometric cluster algebras slightly over the usual definition and adjust the definition of coefficient specializations accordingly. If the broader category admits a universal object, the universal object is called the cluster algebra over $B$ with universal geometric coefficients, or the universal geometric cluster algebra over $B$ . Constructing universal geometric coefficients is equivalent to finding an $R$ -basis for $B$ (a “mutation-linear” analog of the usual linear-algebraic notion of a basis). Polyhedral geometry plays a key role, through the mutation fan ${\mathcal {F}}_B$ , which we suspect to be an important object beyond its role in constructing universal geometric coefficients. We make the connection between ${\mathcal {F}}_B$ and $\mathbf{g}$ -vectors. We construct universal geometric coefficients in rank $2$ and in finite type and discuss the construction in affine type.     

Real Clifford Algebras as Tensor Products over Centers

Denote by ${\mathcal{C}\ell_{p,q}}$ the Clifford algebra on the real vector space ${\mathbb{R}^{p,q}}$ . This paper gives a unified tensor product expression of ${\mathcal{C}\ell_{p,q}}$ by using the center of ${\mathcal{C}\ell_{p,q}}$ . The main result states that for nonnegative integers p, q, ${\mathcal{C}\ell_{p,q} \simeq \otimes^{\kappa-\delta}\mathcal{C}_{1,1} \otimes Cen(\mathcal{C}\ell_{p,q}) \otimes^{\delta} \mathcal{C}\ell_{0,2},}$ where ${p + q \equiv \varepsilon}$ mod 2, ${\kappa = ((p + q) - \varepsilon)/2, p - |q - \varepsilon| \equiv i}$ mod 8 and ${\delta = \lfloor i / 4 \rfloor}$ .     

Decomposable and indecomposable algebras of degree 8 and exponent 2

We study the decomposition of central simple algebras of exponent 2 into tensor products of quaternion algebras. We consider in particular decompositions in which one of the quaternion algebras contains a given quadratic extension. Let $B$ be a biquaternion algebra over $F(\sqrt{a})$ with trivial corestriction. A degree 3 cohomological invariant is defined and we show that it determines whether $B$ has a descent to $F$ . This invariant is used to give examples of indecomposable algebras of degree $8$ and exponent 2 over a field of 2-cohomological dimension 3 and over a field $\mathbb M(t)$ where the $u$ -invariant of $\mathbb M$ is $8$ and $t$ is an indeterminate. The construction of these indecomposable algebras uses Chow group computations provided by Merkurjev in “Appendix”.     

Existence of semiclassical states for a quasilinear Schrödinger equation with critical exponent in {\mathbb{R}^N}

We consider the following perturbed version of quasilinear Schrödinger equation $$\begin{array}{lll}-\varepsilon^2\Delta u +V(x)u-\varepsilon^2\Delta (u^2)u=h(x,u)u+K(x)|u|^{22^*-2}u\end{array}$$ in ${\mathbb{R}^N}$ , where N ≥ 3, 22* = 4N/(N ? 2), V(x) is a nonnegative potential, and K(x) is a bounded positive function. Using minimax methods, we show that this equation has at least one positive solution provided that ${\varepsilon \leq \mathcal{E}}$ ; for any ${m\in\mathbb{N}}$ , it has m pairs of solutions if ${\varepsilon \leq \mathcal{E}_m}$ , where ${\mathcal{E}}$ and ${\mathcal{E}_m}$ are sufficiently small positive numbers. Moreover, these solutions ${u_\varepsilon \to 0}$ in ${H^1(\mathbb{R}^N)}$ as ${\varepsilon \to 0}$ .     

On the Distribution of Atkin and Elkies Primes

Given an elliptic curve $E$ over a finite field $\mathbb {F}_q$ of $q$ elements, we say that an odd prime $\ell \not \mid q$ is an Elkies prime for $E$ if $t_E^2 - 4q$ is a square modulo  $\ell $ , where $t_E = q+1 - \#E(\mathbb {F}_q)$ and $\#E(\mathbb {F}_q)$ is the number of $\mathbb {F}_q$ -rational points on $E$ ; otherwise, $\ell $ is called an Atkin prime. We show that there are asymptotically the same number of Atkin and Elkies primes $\ell < L$ on average over all curves $E$ over $\mathbb {F}_q$ , provided that $L \ge (\log q)^\varepsilon $ for any fixed $\varepsilon >0$ and a sufficiently large $q$ . We use this result to design and analyze a fast algorithm to generate random elliptic curves with $\#E(\mathbb {F}_p)$ prime, where $p$ varies uniformly over primes in a given interval $[x,2x]$ .     

Stability results for the N-dimensional Schiffer conjecture via a perturbation method

Given a eigenvalue $\mu _{0m}^2$ of $-\Delta $ in the unit ball $B_1$ , with Neumann boundary conditions, we prove that there exists a class $\mathcal{D}$ of $C^{0,1}$ -domains, depending on $\mu _{0m} $ , such that if $u$ is a no trivial solution to the following problem $ \Delta u+\mu u=0$ in $\Omega , u=0$ on $\partial \Omega $ , and $ \int \nolimits _{\partial \Omega }\partial _{\mathbf{n}}u=0$ , with $\Omega \in \mathcal{D}$ , and $\mu =\mu _{0m}^2+o(1)$ , then $\Omega $ is a ball. Here $\mu $ is a eigenvalue of $-\Delta $ in $\Omega $ , with Neumann boundary conditions.     

Weakly B-orthodox semigroups

We introduce the notions of a band category and of a weakly orthodox category over a band. Our focus is to describe a class of weakly $B$ -orthodox semigroups, where $B$ denotes a band of idempotents. In particular, we investigate orthodox semigroups, by using orthodox groupoids. Weakly $B$ -orthodox semigroups are analogues of orthodox semigroups, where the relations $\widetilde{\mathcal {R}}_B$ and $\widetilde{\mathcal {L}}_B$ play the role that ${\mathcal {R}}$ and $\mathcal {L}$ take in the regular case. We show that the category of weakly $B$ -orthodox semigroups and admissible morphisms is equivalent to the category of weakly orthodox categories over bands and orthodox functors. The same class of weakly $B$ -orthodox semigroups was studied in an earlier article by Gould and the author using generalised categories. Our approach here is more akin to that of Nambooripad. The significant difference in strategy is that it is more convenient to consider categories equipped with pre-orders, rather than with partial orders.     

Convergence rate of the Allen-Cahn equation to generalized motion by mean curvature

We investigate the singular limit, as ${\varepsilon \to 0}$ , of the Allen-Cahn equation ${u^\varepsilon_t=\Delta u^\varepsilon+\varepsilon^{-2}f(u^\varepsilon)}$ , with f a balanced bistable nonlinearity. We consider rather general initial data u ₀ that is independent of ${{\varepsilon}}$ . It is known that this equation converges to the generalized motion by mean curvature ?? in the sense of viscosity solutions??defined by Evans, Spruck and Chen, Giga, Goto. However, the convergence rate has not been known. We prove that the transition layers of the solutions ${u^{\varepsilon}}$ are sandwiched between two sharp ??interfaces?? moving by mean curvature, provided that these ??interfaces?? sandwich at t?=?0 an ${\mathcal O({\varepsilon}|\,{\rm ln}\,{\varepsilon}|)}$ neighborhood of the initial layer. In some special cases, which allow both extinction and pinches off phenomenon, this enables to obtain an ${\mathcal O({\varepsilon}|\,{\rm ln}\,{\varepsilon}|)}$ estimate of the location and the thickness measured in space-time of the transition layers. A result on the regularity of the generalized motion by mean curvature is also provided in the Appendix.     

Approximations of periodic functions to \mathbb R ^n by curvatures of closed curves

We show that for any $n$ real periodic functions $f_1,\ldots , f_n$ with the same period, such that $f_i>0$ for $i<n$ , and a real number $\varepsilon >0$ , there is a closed curve in $\mathbb R ^{n+1}$ with curvatures $\kappa _1, \ldots , \kappa _n$ such that $\left| \kappa _{i(t)}-f_{i(t)}\right|<\varepsilon $ for all $i$ and $t$ . This does not hold for parametric families of closed curves in $\mathbb R ^{n+1}$ .