We consider Kolmorogov operator $-\Delta +b \cdot \nabla $ with drift b in the class of form-bounded vector fields (containing vector fields having critical-order singularities). We characterize quantitative dependence of the Sobolev and Hölder regularity of solutions to the corresponding elliptic equation on the value of the form-bound of b.

We consider the problem of finding two free export/import sets $E^+$ and $E^-$ that minimize the total cost of some export/import transportation problem (with export/import taxes $g^\pm $ ), between two densities $f^+$ and $f^-$ , plus penalization terms on $E^+$ and $E^-$ . First, we prove the existence of such optimal sets under some assumptions on $f^\pm $ and $g^\pm $ . Then we study some properties of these sets such as convexity and regularity. In particular, we show that the optimal free export (resp. import) region $E^+$ (resp. $E^-$ ) has a boundary of class $C^2$ as soon as $f^+$ (resp. $f^-$ ) is continuous and $\partial E^+$ (resp. $\partial E^-$ ) is $C^{2,1}$ provided that $f^+$ (resp. $f^-$ ) is Lipschitz.

We show that the only way of changing the framing of a link by ambient isotopy in an oriented $3$ -manifold is when the manifold has a properly embedded non-separating $S^{2}$ . This change of framing is given by the Dirac trick, also known as the light bulb trick. The main tool we use is based on McCullough’s work on the mapping class groups of $3$ -manifolds. We also relate our results to the theory of skein modules.

We set up the sharp Trudinger–Moser inequality under arbitrary norms. Using this result and the $L_{p}$ Busemann-Petty centroid inequality, we will provide a new proof to the sharp affine Trudinger–Moser inequalities without using the well-known affine Pólya–Szegö inequality.

We prove that if f and g are holomorphic functions on an open connected domain, with the same moduli on two intersecting segments, then $f=g$ up to the multiplication of a unimodular constant, provided the segments make an angle that is an irrational multiple of $\pi $ . We also prove that if f and g are functions in the Nevanlinna class, and if $|f|=|g|$ on the unit circle and on a circle inside the unit disc, then $f=g$ up to the multiplication of a unimodular constant.

We show that the space $G^r_{\underline d}(X)$ of linear series of certain multi-degree $\underline d=(d_1,d_2)$ (including the balanced ones) and rank r on a general genus-g binary curve X has dimension $\rho _{g,r,d}=g-(r+1)(g-d+r)$ if nonempty, where $d=d_1+d_2$ . This generalizes Caporaso’s result from the case $r\leq 2$ to arbitrary rank, and shows that the space of Osserman-limit linear series on a general binary curve has the expected dimension, which was known for $r\leq 2$ . In addition, we show that the space $G^r_{\underline d}(X)$ is still of expected dimension after imposing certain ramification conditions with respect to a sequence of increasing effective divisors supported on two general points $P_i\in Z_i$ , where $i=1,2$ and $Z_1,Z_2$ are the two components of X. Our result also has potential application to the lifting problem of divisors on graphs to divisors on algebraic curves.

We consider the graph $\Gamma _{\text {virt}}(G)$ whose vertices are the elements of a finitely generated profinite group G and where two vertices x and y are adjacent if and only if they topologically generate an open subgroup of G. We investigate the connectivity of the graph $\Delta _{\text {virt}}(G)$ obtained from $\Gamma _{\text {virt}}(G)$ by removing its isolated vertices. In particular, we prove that for every positive integer t, there exists a finitely generated prosoluble group G with the property that $\Delta _{\operatorname {\mathrm {virt}}}(G)$ has precisely t connected components. Moreover, we study the graph $\widetilde \Gamma _{\operatorname {\mathrm {virt}}}(G)$ , whose vertices are again the elements of G and where two vertices are adjacent if and only if there exists a minimal generating set of G containing them. In this case, we prove that the subgraph $\widetilde \Delta _{\operatorname {\mathrm {virt}}}(G)$ obtained removing the isolated vertices is connected and has diameter at most 3.

We exhibit a set of generating relations for the modular invariant ring of a vector and a covector for the two-dimensional general linear group over a finite field.

Many authors define an isometry of a metric space to be a distance-preserving map of the space onto itself. In this note, we discuss spaces for which surjectivity is a consequence of the distance-preserving property rather than an initial assumption. These spaces include, for example, the three classical (Euclidean, spherical, and hyperbolic) geometries of constant curvature that are usually discussed independently of each other. In this partly expository paper, we explore basic ideas about the isometries of a metric space, and apply these to various familiar metric geometries.

We consider autocorrelation functions for supersymmetric quantum mechanical systems (consisting of a fermion and a boson) confined in trigonometric Pöschl–Teller partner potentials. We study the limit of rescaled autocorrelation functions (at random time) as the localization of the initial state goes to infinity. The limiting distribution can be described using pairs of Jacobi theta functions on a suitably defined homogeneous space, as a corollary of the work of Cellarosi and Marklof. A construction by Contreras-Astorga and Fernández provides large classes of Pöschl-Teller partner potentials to which our analysis applies.

Let G be a locally compact group and let $\omega $ be a continuous weight on G. In this paper, we first characterize bicontinuous biseparating algebra isomorphisms between weighted $L^p$ -algebras. As a result, we extend previous results of Edwards, Parrott, and Strichartz on algebra isomorphisms between $L^p$ -algebras to the setting of weighted $L^p$ -algebras. We then study the automorphisms of certain weighted $L^p$ -algebras on integers, applying known results on composition operators to classical function spaces.

We demonstrate how recent work of Favre and Gauthier, together with a modification of a result of the author, shows that a family of polynomials with infinitely many post-critically finite specializations cannot have any periodic cycles with multiplier of very low degree, except those that vanish, generalizing results of Baker and DeMarco, and Favre and Gauthier.

We answer a question posed by Mordell in 1953, in the case of repeated radical extensions, and find necessary and sufficient conditions for $[F[\sqrt [m_1]{N_1},\dots ,\sqrt [m_\ell ]{N_\ell }]:F]=m_1\cdots m_\ell $ , where F is an arbitrary field of characteristic not dividing any $m_i$ .

We prove the existence of compact spacelike hypersurfaces with prescribed k-curvature in de Sitter space, where the prescription function depends on both space and the tilt function.

It is known that if $S(z)$ is a non-constant singular inner function defined on the unit disk, then $\min _{|z|\le r}|S(z)|\to 0$ as $r\to 1^-$ . We show that the convergence can be arbitrarily slow.

In this paper, we obtain a variation of the Pólya–Vinogradov inequality with the sum restricted to a certain height. Assume $\chi $ to be a primitive character modulo q, $ \epsilon>0$ and $N\le q^{1-\gamma }$ , with $0\le \gamma \le 1/3$ . We prove that

$$ \begin{align*} |\sum_{n=1}^N \chi(n) |\le c (\tfrac{1}{3} -\gamma+\epsilon )\sqrt{q}\log q \end{align*} $$

$c=2/\pi ^2$

$\chi $

$c=1/\pi $

$\chi $

Let $\Omega $ be homogeneous of degree zero and have mean value zero on the unit sphere ${S}^{d-1}$ , $T_{\Omega }$ be the convolution singular integral operator with kernel $\frac {\Omega (x)}{|x|^d}$ . In this paper, we prove that if $\Omega \in L\log L(S^{d-1})$ , and U is an operator which is bounded on $L^2(\mathbb {R}^d)$ and satisfies the weak type endpoint estimate of $L(\log L)^{\beta }$ type, then the composition operator $UT_{\Omega }$ satisfies a weak type endpoint estimate of $L(\log L)^{\beta +1}$ type.

We study a class of delta-like perturbations of the Laplacian on the half-line, characterized by Robin boundary conditions at the origin. Using the formalism of nonstandard analysis, we derive a simple connection with a suitable family of Schrödinger operators with potentials of very large (infinite) magnitude and very short (infinitesimal) range. As a consequence, we also derive a similar result for point interactions in the Euclidean space $\mathbb {R}^3$ , in the case of radial potentials. Moreover, we discuss explicitly our results in the case of potentials that are linear in a neighborhood of the origin.

A tight frame is the orthogonal projection of some orthonormal basis of $\mathbb {R}^n$ onto $\mathbb {R}^k.$ We show that a set of vectors is a tight frame if and only if the set of all cross products of these vectors is a tight frame. We reformulate a range of problems on the volume of projections (or sections) of regular polytopes in terms of tight frames and write a first-order necessary condition for local extrema of these problems. As applications, we prove new results for the problem of maximization of the volume of zonotopes.

Inequalities on partial traces of positive semidefinite matrices are studied. Extensions of several existing inequalities on the determinant of partial traces are then obtained. Particularly, we improve a determinantal inequality given by Lin [Canad. Math. Bull. 59(2016)].