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Of all publications in the section: 482
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Working paper
Pirkovskii A. Y. arxiv.org. math. Cornell University, 2012. No. 1204.4936.
We introduce and study holomorphically finitely generated (HFG) Fr\'echet algebras, which are analytic counterparts of affine (i.e., finitely generated) C-algebras. Using a theorem of O. Forster, we prove that the category of commutative HFG algebras is anti-equivalent to the category of Stein spaces of finite embedding dimension. We also show that the class of HFG algebras is stable under some standard constructions. This enables us to give a series of concrete examples of HFG algebras, including Arens-Michael envelopes of affine algebras (such as the algebras of holomorphic functions on the quantum affine space and on the quantum torus), the algebras of holomorphic functions on the free polydisk, on the quantum polydisk, and on the quantum ball. We further concentrate on the algebras of holomorphic functions on the quantum polydisk and on the quantum ball and show that they are isomorphic, in contrast to the classical case. Finally, we interpret our algebras as Fr\'echet algebra deformations of the classical algebras of holomorphic functions on the polydisk and on the ball in C^n.
Working paper
Sergey Slavnov. arxiv.org. math. Cornell University, 2018
We propose a model of full propositional linear logic based on topological vector spaces. The essential feature is that we consider partially ordered vector spaces, or, rather, positive cones in such spaces. Thus we introduce a category whose objects are dual pairs of normed cones satisfying certain specific completeness properties, such as existence of norm-bounded monotone suprema, and whose morphisms are bounded (adjointable) positive maps. Norms allow us distinct interpretation of dual additive connectives as product and coproduct; in this sense the model is nondegenerate. Also, unlike the familiar case of probabilistic coherence spaces, there is no reference or need for preferred basis; in this sense the model is invariant. Probabilistic coherence spaces form a full subcategory. whose objects, seen as posets, are lattices. Thus we get a model fitting in the tradition of interpreting linear logic in linear algebraic setting, which arguably is free from the drawbacks of its predecessors.  We choose the somewhat show-offy title "noncommutative coherence spaces", hinting of noncommutative geometry, because the passage from probabilistic coherence spaces with their preferred bases to general partially ordered cones seems analogous to the passage from commutative algebras of functions on topological spaces to general, not necessarily commutative algebras. Also, natural non-lattice examples of our spaces come from self-adjoint parts of noncommutative operator algebras. Probabilistic coherence spaces then appear as a "commutative" subcategory.
Working paper
Piontkovski D. arxiv.org. math. Cornell University, 2014. No.  arXiv:1401.6549.
A noncommutative Grassmanian NGr(m,n) is introduced by Efimov, Luntz, and Orlov in `Deformation theory of objects in homotopy and derived categories III: Abelian categories' as a noncommutative algebra associated to an exceptional collection of n-m+1 coherent sheaves on P^n. It is a graded Calabi--Yau Z-algebra of dimension n-m+1. We show that this algebra is coherent provided that the codimension d=n-m of the Grassmanian is two. According to op. cit., this gives a t-structure on the derived category of the coherent sheaves on the noncommutative Grassmanian. The proof is quite different from the recent proofs of the coherence of some graded 3-dimensional Calabi--Yau algebras and is based on properties of a PBW-basis of the algebra.
Working paper
Romanov I., Shamaev A. arxiv.org. math. Cornell University, 2016. No.  arXiv:1408.0382.
We will consider the exact controllability of the distributed system governed by two-dimensional Gurtin-Pipkin equation. It will be proved that this mechanical system can not be driven to an equilibrium point in a finite time if the distributed control has a compact support properly contained in the considered domain. In this case the memory kernel is a twice continuously differentiable function such that its Laplace transformation has at least one root.
Working paper
Feigin E., Makedonskyi I. arxiv.org. math. Cornell University, 2014. No. 1407.6316.
The Cherednik-Orr conjecture expresses the t\to\infty limit of the nonsymmetric Macdonald polynomials in terms of the PBW twisted characters of the affine level one Demazure modules. We prove this conjecture in several special cases.
Working paper
Kalmynin A. B. arxiv.org. math. Cornell University, 2017. No. 1706.07343.
We prove some new lower bounds for the counting function N_C(x) of the set of Novák-Carmichael numbers. Our estimates depend on the bounds for the number of shifted primes without large prime factors. In particular, we prove that N_C(x)>>x^0.7039-o(1) unconditionally and that N_C(x)>>x*exp(-(7+o(1))*log x logloglog x/loglog x), under some reasonable hypothesis.
Working paper
Vladimir L. Popov. arxiv.org. math. Cornell University, 2015. No. 1503.08303.
For every pair (G, V ) where G is a connected simple linear algebraic group and V is a simple algebraic G-module with a free algebra of invariants, the number of irreducible components of the nullcone of unstable vectors in V is found.
Working paper
Kalmynin A. B. arxiv.org. math. Cornell University, 2017. No. 1706.07364.
We introduce a generalization of the method of S. P. Zaitsev. This generalization allows us to prove omega-theorems for the Riemann zeta function and its derivatives in some regions near the line Re s=1.
Working paper
Glutsyuk A. arxiv.org. math. Cornell University, 2014. No. 1405.5990.
The famous conjecture of V.Ya.Ivrii  says that in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero. In the present paper we study its   complex analytic version for quadrilateral orbits in two dimensions, with reflections from  holomorphic curves.  We present the complete classification of 4-reflective analytic counterexamples: billiards formed by four holomorphic curves in the projective plane that have open set of quadrilateral orbits. This extends the   author's result  classifying 4-reflective planar algebraic counterexamples. We provide applications to real billiards: classification of 4-reflective real planar analytic pseudo-billiards; solution of the piecewise-analytic case of Tabachnikov's commuting planar billiard problem; solution of a particular case of Plakhov's Invisibility Conjecture. In particular, we retrieve the solution of Ivrii's Conjecture for quadrilateral orbits in planar billiards  in piecewise-analytic case previously obtained in a joint paper of the author with Yu.Kudryashov.
Working paper
Zlotnik A., Koltsova N. arxiv.org. math. Cornell University, 2012. No. arXiv:1211.3613 [math.NA].
An initial-boundary value problem for the 1D self-adjoint parabolic equation on the half-axis is solved. We study a broad family of two-level finite-difference schemes with two parameters related to averagings both in time and space. Stability in two norms is proved by the energy method. Also discrete transparent boundary conditions are rigorously derived for schemes by applying the method of reproducing functions. Results of numerical experiments are included as well.
Working paper
Vladimir L. Popov. arxiv.org. math. Cornell University, 2021. No. 2102.08032.
Several results on presenting an affine algebraic group variety as a product of algebraic varieties are obtained.
Working paper
Kazakov A. arxiv.org. math. Cornell University, 2017. No. 1801.00150.
n this paper we present the scenario of the occurrence of strongly dissipative mixed dynamics in two-dimensional reversible diffeomorphisms, using as an example the system describing a motion of two point vortices under the influence of wave perturbation and shear flow. For mixed dynamics of this type the chaotic attractor intersects with the chaotic repeller, but their intersection forms a "thin" set. The main stage of this scenario is the appearance of homoclinic structures for a symmetric saddle orbit which arise after crisis of a homoclinic attractor and repeller.
Working paper
Bitoun T. arxiv.org. math. Cornell University, 2014
We introduce a theory of the b-function (or Bernstein-Sato polynomial) in positive characteristic. Let  be a field of characteristic and let be a nonconstant polynomial. The b-function of is an ideal of a nonnoetherian commutative algebra of characteristic but it has "roots" in. We prove the existence of the b-function as well as the rationality of its roots. The framework of the theory is that of unit D-modules. There is a close connection with test ideals. In particular, we prove that the roots of the bD-function are the opposites of the Fjumping exponents of f which are in (0,1]∩Z(p).
Working paper
Galkin S., Karzhemanov I., Shinder E. arxiv.org. math. Cornell University, 2016. No. 1602.06107.
On the projective plane there is a unique cubic root of the canonical bundle and this root is acyclic. On fake projective planes such root exists and is unique if there are no 3-torsion divisors (and usually exists, but not unique, otherwise). Earlier we conjectured that any such cubic root must be acyclic. In the present note we give two short proofs of this statement and show acyclicity of some other line bundles on the fake projective planes with at least 9 automorphisms. Similarly to our earlier work we employ simple representation theory for non-abelian finite groups. The novelty stems from the idea that if some line bundle is non-linearizable with respect to a finite abelian group, then it should be linearized by a finite, \emph{non-abelian}, Heisenberg group. For the second proof, we also demonstrate vanishing of odd Betti numbers for a class of abelian covers, and use a linearization of an auxiliary line bundle as well.
Working paper
Kulakova E., Lando S., Mukhutdinova T. et al. arxiv.org. math. Cornell University, 2013. No. 1307.4933.
We introduce a new series Rk, k=2,3,4,…, of integer valued weight systems. The value of the weight system Rk on a chord diagram is a signed number of cycles of even length 2k in the intersection graph of the diagram. We show that this value depends on the intersection graph only. We check that for small orders of the diagrams, the value of the weight system Rk on a diagram of order exactly 2k coincides with the coefficient of ck in the value of the sl2-weight system on the projection of the diagram to primitive elements.
Working paper
Kulakova E., Lando S., Mukhutdinova T. et al. arxiv.org. math. Cornell University, 2013
We introduce a new series~$R_k$, $k=2,3,4,\dots$, of integer valued weight systems. The value of the weight system~$R_k$ on a chord diagram is a signed number of cycles of even length~$2k$ in the intersection graph of the diagram. We show that this value depends on the intersection graph only. We check that for small orders of the diagrams, the value of the weight system~$R_k$ on a diagram of order exactly~$2k$ coincides with the coefficient of~$c^k$ in the value of the $\sl_2$-weight system on the projection of the diagram to primitive elements.
Working paper
Galkin S., Shinder E. arxiv.org. math. Cornell University, 2015. No. 1506.05831.
We define a zeta-function of a pre-triangulated dg-category and investigate its relationship with the motivic zeta-function in the geometric case.
Working paper
Rybakov S. arxiv.org. math. Cornell University, 2014