Classical limits of quantum toroidal and affine Yangian algebras

In this short article, we compute the classical limits of the quantum toroidal and affine Yangian algebras of ${\mathfrak{sl}}_n$ by generalizing our arguments for ${\mathfrak{gl}}_1$ from [7] (an alternative proof for $n>2$ is given in [10]). We also discuss some consequences of these results.     

Cardinal characteristics at κ in a small u(κ) model

We provide a model where $\mathfrak{u}(\kappa )<2^{\kappa }$ for a supercompact cardinal κ. [10] provides a sketch of how to obtain such a model by modifying the construction in [6]. We provide here a complete proof using a different modification of [6] and further study the values of other natural generalizations of classical cardinal characteristics in our model. For this purpose we generalize some standard facts that hold in the countable case as well as some classical forcing notions and their properties.     

Dispersive estimates for the wave equation inside cylindrical convex domains: A model case

In this work, we will establish local in time dispersive estimates for solutions to the model-case Dirichlet wave equation inside a cylindrical convex domain $\mathrm{\Omega }?{\mathbb{R}}^3$ with a smooth boundary $?\mathrm{\Omega }\ne ?$. Let us recall that dispersive estimates are key ingredients to prove Strichartz estimates. Nonoptimal Strichartz estimates for waves inside an arbitrary domain Ω have been proved by Blair–Smith–Sogge [1], [2]. Better estimates in strictly convex domains have been obtained in [4]. Our case of cylindrical domains is an extension of the result of [4] in the case where the curvature radius ≥0 depends on the incident angle and vanishes in some directions.     

Global,decaying solutions of a focusing energy-critical heat equation in R4

We study solutions of the focusing energy-critical nonlinear heat equation $u_t=\mathrm{\Delta }u?|u{|}^{2}u$ in ${\mathbb{R}}^4$. We show that solutions emanating from initial data with energy and ${\dot{H}}^1$-norm below those of the stationary solution W are global and decay to zero, via the “concentration-compactness plus rigidity” strategy of Kenig–Merle [33], [34]. First, global such solutions are shown to dissipate to zero, using a refinement of the small data theory and the $L^2$-dissipation relation. Finite-time blow-up is then ruled out using the backwards-uniqueness of Escauriaza–Seregin–Sverak [17], [18] in an argument similar to that of Kenig–Koch [32] for the Navier–Stokes equations.     

A presentation of the double Pieri algebra

The double Pieri algebra, constructed by Howe, Lee, and the author in [10], [12], encodes information on the decomposition of Pieri type tensor products of irreducible representations for complex classical groups. In this paper we study its finite presentation in terms of generators and relations, and then prove that it admits the structure of a cluster algebra. We also study its ${\mathfrak{sl}}_2$-module structure.     

Goldman–Turaev formality from the Knizhnik–Zamolodchikov connection

For an oriented 2-dimensional manifold Σ of genus g with n boundary components, the space $\mathbb{C}{\pi }_1(\mathrm{\Sigma })/[\mathbb{C}{\pi }_1(\mathrm{\Sigma }),\mathbb{C}{\pi }_1(\mathrm{\Sigma })]$ carries the Goldman–Turaev Lie bialgebra structure defined in terms of intersections and self-intersections of curves. Its associated graded Lie bialgebra (under the natural filtration) is described by cyclic words in $H_1(\mathrm{\Sigma })$ and carries the structure of a necklace Schedler Lie bialgebra. The isomorphism between these two structures in genus zero has been established in [13] using Kontsevich integrals and in [2] using solutions of the Kashiwara–Vergne problem.In this note, we give an elementary proof of this isomorphism over $\mathbb{C}$. It uses the Knizhnik–Zamolodchikov connection on $\mathbb{C}\backslash \{z_1,\dots z_n\}$. We show that the isomorphism naturally depends on the complex structure on the surface. The proof of the isomorphism for Lie brackets is a version of the classical result by Hitchin [9]. Surprisingly, it turns out that a similar proof applies to cobrackets.Furthermore, we show that the formality isomorphism constructed in this note coincides with the one defined in [2] if one uses the solution of the Kashiwara–Vergne problem corresponding to the Knizhnik–Zamolodchikov associator.     

On the kernel of the regulator map

By using the infinitesimal methods due to Bloch, Green, and Griffiths in [1], [4], we construct an infinitesimal form of the regulator map and verify that its kernel is ${\mathrm{\Omega }}_{\mathbb{C}/\mathbb{Q}}^1$, which suggests that Question 1.1 seems reasonable at the infinitesimal level.     

Stability of the sum of two solitary waves for (gDNLS) in the energy space

In this paper, we continue the study in [18]. We use the perturbation argument, modulational analysis and the energy argument in [15], [16] to show the stability of the sum of two solitary waves with weak interactions for the generalized derivative Schrödinger equation (gDNLS) in the energy space. Here (gDNLS) hasn't the Galilean transformation invariance, the pseudo-conformal invariance and the gauge transformation invariance, and the case $\sigma >1$ we considered corresponds to the $L^2$-supercritical case.     

Large subgroups in finite groups

Following Isaacs (see [6, p. 94]), we call a normal subgroup N of a finite group Glarge, if ${\mathbf{C}}_G(N)\le N$, so that N has bounded index in G. Our principal aim here is to establish some general results for systematically producing large subgroups in finite groups (see Theorem A, Theorem C). We also consider the more specialised problems of finding large (non-abelian) nilpotent as well as abelian subgroups in soluble groups.     

Finite-time blow-up for quasilinear degenerate Keller–Segel systems of parabolic–parabolic type

This paper deals with the quasilinear degenerate Keller–Segel systems of parabolic–parabolic type in a ball of ${\mathbb{R}}^N$ ($N\ge 2$). In the case of non-degenerate diffusion, Cie?lak–Stinner [3], [4] proved that if $q>m+\frac{2}{N}$, where m denotes the intensity of diffusion and q denotes the nonlinearity, then there exist initial data such that the corresponding solution blows up in finite time. As to the case of degenerate diffusion, it is known that a solution blows up if $q>m+\frac{2}{N}$ (see Ishida–Yokota [13]); however, whether the blow-up time is finite or infinite has been unknown. This paper gives an answer to the unsolved problem. Indeed, the finite-time blow-up of energy solutions is established when $q>m+\frac{2}{N}$.     

Trace formulas for relative Schatten class perturbations

We derive trace formulas for a pair of self-adjoint operators $H+V$ and H under the assumption that ${(H?\mathrm{i})}^{?1}V$ is in a Schatten class. This extends the trace formulas of [8], where V alone is assumed to be in a Schatten class. Our trace formulas apply, in particular, in the setting of differential operators and are based on Taylor-like approximations of operator functions. This significantly improves non-Taylor based trace formulas of [10].     

Regularity of the Eikonal equation with two vanishing entropies

Let $\mathrm{\Omega }?{\mathbb{R}}^2$ be a bounded simply-connected domain. The Eikonal equation $\left|\mathrm{?}u\right|=1$ for a function $u:\mathrm{\Omega }?{\mathbb{R}}^2\to \mathbb{R}$ has very little regularity, examples with singularities of the gradient existing on a set of positive $H^1$ measure are trivial to construct. With the mild additional condition of two vanishing entropies we show ?u is locally Lipschitz outside a locally finite set. Our condition is motivated by a well known problem in Calculus of Variations known as the Aviles–Giga problem. The two entropies we consider were introduced by Jin, Kohn [26], Ambrosio, DeLellis, Mantegazza [2] to study the Γ-limit of the Aviles–Giga functional. Formally if u satisfies the Eikonal equation and if

(1)$\mathrm{?}?({\stackrel{?}{\mathrm{\Sigma }}}_{e_1{e}_2}(\mathrm{?}{u}^{\perp }))=0\text{ and }\mathrm{?}?({\stackrel{?}{\mathrm{\Sigma }}}_{{?}_1{?}_2}(\mathrm{?}{u}^{\perp }))=0\text{ distributionally in }\mathrm{\Omega },$

where ${\stackrel{?}{\mathrm{\Sigma }}}_{e_1{e}_2}$ and ${\stackrel{?}{\mathrm{\Sigma }}}_{{?}_1{?}_2}$ are the entropies introduced by Jin, Kohn [26], and Ambrosio, DeLellis, Mantegazza [2], then ?u is locally Lipschitz continuous outside a locally finite set.Condition (1) is motivated by the zero energy states of the Aviles–Giga functional. The zero energy states of the Aviles–Giga functional have been characterized by Jabin, Otto, Perthame [25]. Among other results they showed that if ${\lim }_{n\to \infty }?I_{{?}_n}(u_n)=0$ for some sequence $u_n\in W_0^{2,2}(\mathrm{\Omega })$ and $u={\lim }_{n\to \infty }?u_n$ then ?u is Lipschitz continuous outside a finite set. This is essentially a corollary to their theorem that if u is a solution to the Eikonal equation $\left|\mathrm{?}u\right|=1$ a.e. and if for every “entropy” Φ (in the sense of [18], Definition 1) function u satisfies $\mathrm{?}?[\mathrm{\Phi }(\mathrm{?}{u}^{\perp })]=0$ distributionally in Ω then ?u is locally Lipschitz continuous outside a locally finite set. In this paper we generalize this result in that we require only two entropies to vanish.The method of proof is to transform any solution of the Eikonal equation satisfying (1) into a differential inclusion $DF\in K$ where $K?M^{2\times 2}$ is a connected compact set of matrices without Rank-1 connections. Equivalently this differential inclusion can be written as a constrained non-linear Beltrami equation. The set K is also non-elliptic in the sense of Sverak [32]. By use of this transformation and by utilizing ideas from the work on regularity of solutions of the Eikonal equation in fractional Sobolev space by Ignat [23], DeLellis, Ignat [15] as well as methods of Sverak [32], regularity is established.     

Curves containing all points of a finite projective Galois plane

In the projective plane $PG(2,q)$ over a finite field of order q, a Tallini curve is a plane irreducible (algebraic) curve of (minimum) degree $q+2$ containing all points of $PG(2,q)$. Such curves were investigated by G. Tallini [8], [9] in 1961, and by Homma and Kim [5] in 2013. Our results concern the automorphism groups, the Weierstrass semigroups, the Hasse–Witt invariants, and quotient curves of the Tallini curves.     

Convergence to nonlinear diffusion waves for solutions of Euler equations with time-depending damping

In this paper, we are concerned with the asymptotic behavior of solutions to the system of Euler equations with time-depending damping, in particular, include the constant coefficient damping. We rigorously prove that the solutions time-asymptotically converge to the diffusion wave whose profile is self-similar solution to the corresponding parabolic equation, which justifies Darcy's law. Compared with previous results about Euler equations with constant coefficient damping obtained by Hsiao and Liu (1992) [2], and Nishihara (1996) [9], we obtain a general result when the initial perturbation belongs to the same space, i.e. $H^3(\mathbb{R})\times H^2(\mathbb{R})$. Our proof is based on the classical energy method.     

Higher-order elliptic and parabolic equations with VMO assumptions and general boundary conditions

We prove mixed $L_p(L_q)$-estimates, with $p,q\in (1,\infty )$, for higher-order elliptic and parabolic equations on the half space ${\mathbb{R}}_{+}^{d+1}$ with general boundary conditions which satisfy the Lopatinskii–Shapiro condition. We assume that the elliptic operators A have leading coefficients which are in the class of vanishing mean oscillations both in the time variable and the space variable. In the proof, we apply and extend the techniques developed by Krylov [24] as well as Dong and Kim in [13] to produce mean oscillation estimates for equations on the half space with general boundary conditions.     

Topology and measure in logics for region-based theories of space

Space, as we typically represent it in mathematics and physics, is composed of dimensionless, indivisible points. On an alternative, region-based approach to space, extended regions together with the relations of ‘parthood’ and ‘contact’ are taken as primitive; points are represented as mathematical abstractions from regions.Region-based theories of space have been traditionally modeled in regular closed (or regular open) algebras, in work that goes back to [5] and [21]. Recently, logics for region-based theories of space were developed in [3] and [19]. It was shown that these logics have both a nice topological and relational semantics, and that the minimal logic for contact algebras, ${\mathbb{L}}_{min}^{cont}$ (defined below), is complete for both.The present paper explores the question of completeness of ${\mathbb{L}}_{min}^{cont}$ and its extensions for individual topological spaces of interest: the real line, Cantor space, the rationals, and the infinite binary tree. A second aim is to study a different, algebraic model of logics for region-based theories of space, based on the Lebesgue measure algebra (or algebra of Borel subsets of the real line modulo sets of Lebesgue measure zero). As a model for point-free space, the algebra was first discussed in [2]. The main results of the paper are that ${\mathbb{L}}_{min}^{cont}$ is weakly complete for any zero-dimensional, dense-in-itself metric space (including, e.g., Cantor space and the rationals); the extension ${\mathbb{L}}_{min}^{cont}+(Con)$ is weakly complete for the real line and the Lebesgue measure contact algebra. We also prove that the logic ${\mathbb{L}}_{min}^{cont}+(Univ)$ is weakly complete for the infinite binary tree.     

An LP empirical quadrature procedure for parametrized functions

We extend the linear program empirical quadrature procedure proposed in [9] and subsequently [3] to the case in which the functions to be integrated are associated with a parametric manifold. We pose a discretized linear semi-infinite program: we minimize as objective the sum of the (positive) quadrature weights, an ${?}_1$ norm that yields sparse solutions and furthermore ensures stability; we require as inequality constraints that the integrals of J functions sampled from the parametric manifold are evaluated to accuracy $\overline{\delta }$. We provide an a priori error estimate and numerical results that demonstrate that under suitable regularity conditions, the integral of any function from the parametric manifold is evaluated by the empirical quadrature rule to accuracy $\overline{\delta }$ as $J\to \infty$. We present two numerical examples: an inverse Laplace transform; reduced-basis treatment of a nonlinear partial differential equation.