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Of all publications in the section: 482
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Working paper
Kolesnikov A., Roeckner M. arxiv.org. math. Cornell University, 2012
Let $\gamma$ be a Gaussian measure on a locally convex space and $H$ be the corresponding Cameron-Martin space. It has been recently shown by L. Ambrosio and A. Figalli that the linear first-order PDE $$\dot{\rho} + \mbox{div}_{\gamma} (\rho \cdot {b}) =0, \ \ \rho|_{t=0} = \rho_0,$$ where $\rho_0 \cdot \gamma$ is a probability measure, admits a weak solution, in particular, under the following assumptions: $$\|b\|_{H} \in L^p(\gamma), \ p>1, \ \ \ \exp\bigl(\varepsilon(\mbox{\rm div}_{\gamma} b)_{-} \bigr) \in L^1(\gamma).$$ Applying transportation of measures via triangular maps we prove a similar result for a large class of non-Gaussian probability measures $\nu$ on $\R^{\infty}$, under the main assumption that $\beta_i \in \cap_{n \in \Nat} L^{n}(\nu)$ for every $i \in \Nat$, where $\beta_i$ is the logarithmic derivative of $\nu$ along the coordinate $x_i$. We also show uniqueness of the solution for a wide class of measures. This class includes uniformly log-concave Gibbs measures and certain product measures.
Working paper
Khanin K., Sobolevski A. arxiv.org. math. Cornell University, 2012. No. arXiv:1211.7084.
Characteristic curves of a Hamilton–Jacobi equation can be seen as action minimizing trajectories of fluid particles. However this description is valid only for smooth solutions. For nonsmooth “viscosity” solutions, which give rise to discontinuous velocity fields, this picture holds only up to the moment when trajectories hit a shock and cease to minimize the Lagrangian action. In this paper we show that for any convex Hamiltonian, a viscous regularization allows to construct a nonsmooth flow that extends particle tra- jectories and determines dynamics inside the shock manifolds. This flow consists of integral curves of a particular velocity field, which is uniquely defined everywhere in the flow domain and is discontinuous on shock manifolds.
Working paper
Kochetkov Y. arxiv.org. math. Cornell University, 2016. No. 1608.02510.
We consider planar cubic maps, i.e. connected cubic graphs, embedded into plane, with marked spanning tree and marked directed edge (not in this tree). The number of such objects with 2n vertices is C2n∙ Cn+1, where Ck is Catalan number.
Working paper
Alexey Elagin. arxiv.org. math. Cornell University, 2014. No. 1403.7027.
Consider a finite group $G$ acting on a triangulated category $\TTT$. In this paper we try to understand when the category $\TTT^G$ of $G$-equivariant objects in $\TTT$ is triangulated. We prove that it is so in two cases: the action on the derived category $\D^b(\AA)$ induced by an action on an abelian category $\AA$ and the action on the homotopy category $H^0(\AA)$ of a pretriangulated DG-category $\AA$ induced by an action on $\AA$. Also, we show that the relation to be an equivariant category with respect to a finite abelian group action'' is symmetric on idempotent complete additive categories.
Working paper
Zlotnik Alexander. arxiv.org. math. Cornell University, 2015
We deal with an initial-boundary value problem for the generalized time-dependent Schr\"odinger equation with variable coefficients in an unbounded $n$--dimensional parallelepiped ($n\geq 1$). To solve it, the Crank-Nicolson in time and the polylinear finite element in space method with the discrete transpa\-rent boundary conditions is considered. We present its stability properties and derive new error estimates $O(\tau^2+|h|^2)$ uniformly in time in $L^2$ space norm, for $n\geq 1$, and mesh $H^1$ space norm, for $1\leq n\leq 3$ (a superconvergence result), under the Sobolev-type assumptions on the initial function. Such estimates are proved for methods with the discrete TBCs for the first time.
Working paper
Roman Avdeev. arxiv.org. math. Cornell University, 2020. No. 2005.05234.
Given a connected reductive complex algebraic group G and a spherical subgroup H⊂G, the extended weight monoid Λˆ+G(G/H) encodes the G-module structures on spaces of regular sections of all G-linearized line bundles on G/H. Assuming that G is semisimple and simply connected and H is specified by a regular embedding in a parabolic subgroup P⊂G, in this paper we obtain a description of Λˆ+G(G/H) via the set of simple spherical roots of~G/H together with certain combinatorial data explicitly computed from the pair (P,H). As an application, we deduce a new proof of a result of Avdeev and Gorfinkel describing Λˆ+G(G/H) in the case where H is strongly solvable.
Working paper
Alexey Elagin, Lunts V. arxiv.org. math. Cornell University, 2015. No. 1505.06477.
Working paper
Markaryan N. S. arxiv.org. math. Cornell University, 2020
We introduce a family of linear relations between cell-zeta values that have a form similar to product map relations and jointly with them imply stuffle relations between multiple zeta values.
Working paper
Khoroshkin A., Piontkovski D. arxiv.org. math. Cornell University, 2014. No. arXiv:1202.5170.
Given an operad P with a finite Grobner basis of relations, we study the generating functions for the dimensions of its graded components P(n). Under moderate assumptions on the relations we prove that the exponential generating function for the sequence {dim P(n)} is differential algebraic, and in fact algebraic for P is a symmetrization of a non-symmetric operad. If, in addition, the growth of the dimensions P(n) is bounded by an exponent of n (or a polynomial of n, in the non-symmetric case) then, Moreover, the ordinary generating function for the above sequence {dim P(n)} is rational. We give a number of examples of calculations and discuss conjectures about the above generating functions for more general classes of operads.
Working paper
Yurii Burman, Shapiro B. arxiv.org. math. Cornell University, 2016. No. 06935.
For a point p in a complex projective plane and a triple (g,d,l) of non-negative integers we define a plane Hurwitz number of the Severi variety W_{g,d,l} consisting of all reduced irreducible plane curves of genus g and degree d+l having an l-fold node at p and at most ordinary nodes as singularities at the other points. In the cases d+l >= g+2 and d+2l >= g+2 > d+l we express the plane Hurwitz numbers via appropriate ordinary Hurwitz numbers. The remaining case d+2l<g+2 is still widely open.
Working paper
Grines V., Kurenkov E. arxiv.org. math. Cornell University, 2017
Working paper
Vladimir L. Popov. arxiv.org. math. Cornell University, 2014. No. 1401.0278.
We explore orbits, rational invariant functions, and quotients of the natural actions of connected, not necessarily finite dimensional subgroups of the automorphism groups of irreducible algebraic varieties. The applications of the results obtained are given.
Working paper
Alexander Kuznetsov. arxiv.org. math. Cornell University, 2015
We describe the geometry of K\"uchle varieties (i.e. Fano 4-folds of index 1 contained in the Grassmannians as zero loci of equivariant vector bundles) with Picard number greater than 1 and the structure of their derived categories.
Working paper
Vladimir Lebedev. arxiv.org. math. Cornell University, 2013. No. 1303.5384v2.
We consider bounded analytic functions in domains generated by sets that have Littlewood--Paley property. We show that each such function is an lp -multiplier.
Working paper
Timorin V., Petrushchenko S. arxiv.org. math. Cornell University, 2014. No. 1409.3403.
We study cubic rational maps that take lines to plane curves. A complete description of such cubic rational maps concludes the classification of all planarizations, i.e., maps taking lines to plane curves.
Working paper
Anton Fonarev. arxiv.org. math. Cornell University, 2011
We construct two Lefschetz decompositions of the derived category of coherent sheaves on the Grassmannian of $k$-dimensional subspaces in a vector space of dimension $n$. Both of them admit a Lefschetz basis consisting of equivariant vector bundles. We prove fullness of the first decomposition and conjecture it for the second one. In the case when $n$ and $k$ are coprime these decompositions coincide and are minimal. In general, we conjecture minimality of the second decomposition.
Working paper
Kuznetsov A., Ingalls C. arxiv.org. math. Cornell University, 2010. No. 1012.3530.
We consider the class of singular double coverings $X \to \PP^3$ ramified in the degeneration locus $D$ of a family of 2-dimensional quadrics. These are precisely the quartic double solids constructed by Artin and Mumford as examples of unirational but nonrational conic bundles. With such quartic surface $D$ one can associate an Enriques surface $S$ which is the factor of the blowup of $D$ by a natural involution acting without fixed points (such Enriques surfaces are known as nodal Enriques surfaces or Reye congruences). We show that the nontrivial part of the derived category of coherent sheaves on this Enriques surface $S$ is equivalent to the nontrivial part of the derived category of a minimal resolution of singularities of $X$.
Working paper
Kalmynin A. B. arxiv.org. math. Cornell University, 2016. No. 1611.00417.
In this work, we obtain some new lower bounds for the number N_B(x) of Novak numbers less than or equal to x. We also prove, conditionally on Generalized Riemann Hypothesis, the upper estimates for the number of primes dividing at least one Nov\'ak number and give description for the prime factors of Nov\'ak numbers N such that 2N is a Novak-Carmichael number.