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Of all publications in the section: 484
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Working paper
Rybnikov G. arxiv.org. math. Cornell University, 2014
We study a special type of $E_\infty$-operads that govern strictly unital $E_\infty$-coalgebras (and algebras) over the ring of integers. Morphisms of coalgebras over such an operad are defined by using universal $E_\infty$-bimodules. Thus we obtain a category of $E_\infty$-coalgebras. It turns out that if the homology of an $E_\infty$-coalgebra have no torsion, then there is a natural way to define an $E_\infty$-coalgebra structure on the homology so that the resulting coalgebra be isomorphic to the initial $E_\infty$-coalgebra in our category. We also discuss some invariants of the $E_\infty$-coalgebra structure on homology and relate them to the invariant formerly used by the author to distinguish the fundamental groups of the complements of combinatorially equivalent complex hyperplane arrangements.
Working paper
Malyshev D. arxiv.org. math. Cornell University, 2013. No. 1307.0278v1.
The coloring problem is studied in the paper for graph classes deﬁned by two small forbidden induced subgraphs. We prove some suﬃcient conditions for eﬀective solvability of the problem in such classes. As their corollary we determine the computational complexity for all sets of two connected forbidden induced subgraphs with at most ﬁve vertices except 13 explicitly enumerated cases.
Working paper
Blokh A., Oversteegen L., Ptacek R. et al. arxiv.org. math. Cornell University, 2015
We interpret the combinatorial Mandelbrot set in terms of quadratic laminations (equivalence relations ∼ on the unit circle invariant under σ2). To each lamination we associate a particular geolamination (the collection L∼ of points of the circle and edges of convex hulls of ∼-equivalence classes) so that the closure of the set of all of them is a compact metric space with the Hausdorff metric. Two such geolaminations are said to be minor equivalent if their minors (images of their longest chords) intersect. We show that the corresponding quotient space of this topological space is homeomorphic to the boundary of the combinatorial Mandelbrot set. To each equivalence class of these geolaminations we associate a unique lamination and its topological polynomial so that this interpretation can be viewed as a way to endow the space of all quadratic topological polynomials with a suitable topology
Working paper
Shitov Y. arxiv.org. math. Cornell University, 2012
The tropical arithmetic operations are the minimum and the sum of two numbers. Let A be a tropical matrix and k a positive integer, the problem of Tropical Matrix Factorization (TMF) asks whether there exist an m-by-k tropical matrix B and a k-by-n tropical matrix C whose tropical product is A. We show that no algorithm for TMF is likely to work in polynomial time for every fixed k, thus resolving a problem proposed by Barvinok in 1993. TMF is also shown to be hard for matrices with bounded tropical rank. Proving that TMF can be solved by a polynomial-time algorithm if k is less than 4, we answer a question posed by Develin, Santos, and Sturmfels. Another question they have posed asks whether every tropical matrix of factor rank k has a rank-k submatrix of size at most N(k)-by-N(k); we answer this question in the negative for every k greater than 4.
Working paper
Vladimir L. Popov. arxiv.org. math. Cornell University, 2010. No. 1009.6107.
Working paper
Galkin S. arxiv.org. math. Cornell University, 2014. No. 1404.7388.
Consider a Laurent polynomial with real positive coefficients such that the origin is strictly inside its Newton polytope. Then it is strongly convex as a function of real positive argument. So it has a distinguished Morse critical point --- the unique critical point with real positive coordinates.  As a consequence we obtain a positive answer to a question of Ostrover and Tyomkin: the quantum cohomology algebra of a toric Fano manifold contains a field as a direct summand. Moreover, it gives a good evidence that the same statement holds for any Fano manifold.
Working paper
Adler D., Gritsenko V. arxiv.org. math. Cornell University, 2019
We construct a tower of arithmetic generators of the bigraded polynomial ring J_{*,*}^{w, O}(D_n) of weak Jacobi modular forms invariant with respect to the full orthogonal group O(D_n) of the root lattice D_n for 2\le n\le 8. This tower corresponds to the tower of strongly reflective modular forms on the orthogonal groups of signature (2,n) which determine the Lorentzian Kac-Moody algebras related to the BCOV (Bershadsky-Cecotti-Ooguri-Vafa)-analytic torsions. We prove that the main three generators of index one of the graded ring satisfy a special system of modular differential equations. We found also a general modular differential equation of the generator of weight 0 and index 1 which generates the automorphic discriminant of the moduli space of Enriques surfaces.
Working paper
Bogomolov F. A., Silberstein A. M. arxiv.org. math. Cornell University, 2015
Given an infinite collection of pairwise-disjoint, Zariski-closed, connected, codimension-1 subvarieties of a complete, normal, irreducible algebraic variety over an algebraically-closed field, we prove that there exists a regular, nonconstant morphism to a complete curve so that each of these divisors is contained in a fiber of this morphism.
Working paper
Galkin S., Shinder E. arxiv.org. math. Cornell University, 2014. No. 1405.5154.
We find a relation between a cubic hypersurface Y and its Fano variety of lines F(Y) in the Grothendieck ring of varieties. We prove that if the class of an affine line is not a zero-divisor in the Grothendieck ring of varieties, then Fano variety of lines on a smooth rational cubic fourfold is birational to a Hilbert scheme of two points on a K3 surface; in particular, general cubic fourfold is irrational.
Working paper
Nadezhda Kodaneva. arxiv.org. math. Cornell University, 2019. No. arXiv:2002.12440.
In this work, we study the interlace polynomial as a generalization of a graph invariant to delta-matroids. We prove that the interlace polynomial satisfies the four-term relation for delta-matroids and determines thus a finite type invariant of links in the 3-sphere.
Working paper
Vyacheslav V. Chistyakov, Svetlana A. Chistyakova. arxiv.org. math. Cornell University, 2016. No. 1601.07298.
Given a subset T of real numbers and a metric space M, we introduce a nondecreasing sequence {v_n} of pseudometrics on the set M^T of all functions from T into M, called the joint modulus of variation. We prove that if two sequences of functions {f_j} and {g_j} from M^T are such that {f_j} is pointwise precompact, {g_j} is pointwise convergent, and the limit superior of v_n(f_j,g_j) as j → ∞ is o(n) as n → ∞, then {f_j} admits a pointwise convergent subsequence whose limit is a conditionally regulated function. We illustrate the sharpness of this result by examples (in particular, the assumption on the limsup is necessary for uniformly convergent sequences {f_j} and {g_j}, and ‘almost necessary’ when they converge pointwise) and show that most of the known Helly-type pointwise selection theorems are its particular cases.
Working paper
Vladimir L. Popov. arxiv.org. math. Cornell University, 2018. No. 1804.00323v1.
We prove that the family of all connected n-dimensional real Lie groups is uniformly Jordan for every n. This  implies that all algebraic groups (not  necessarily affine) over fields of cha\-racte\-ristic zero and some transformation groups of complex spaces and Riemannian manifods are Jordan.
Working paper
Timorin V., Oversteegen L., Blokh A. et al. arxiv.org. math. Cornell University, 2013. No. 1305.5798.
We discuss different analogs of the main cardioid in the parameter space of cubic polynomials, and establish relationships between them.
Working paper
Blokh A., Oversteegen L., Ptacek R. et al. arxiv.org. math. Cornell University, 2015
Thurston parameterized quadratic invariant laminations with a non-invariant lamination, the quotient of which yields a combinatorial model for the Mandelbrot set. As a step toward generalizing this construction to cubic polynomials, we consider slices of the family of cubic invariant laminations defined by a fixed critical leaf with non-periodic endpoints. We parameterize each slice by a lamination just as in the quadratic case, relying on the techniques of smart criticality previously developed by the authors.
Working paper
Konakov V., Markova A. arxiv.org. math. Cornell University, 2016. No. 1610.08715.
We consider the diffusion process and its approximation by Markov chain with nonlinear increasing trends. The usual parametrix method is not appliable because these models have unbounded trends. We describe a procedure that allows to exclude nonlinear growing trend and move to stochastic differential equation with reduced drift and diffusion coefficients. A similar procedure is considered for a Markov chain
Working paper
V.L. Chernyshev, Tolchennikov A. arxiv.org. math. Cornell University, 2011. No. 1111.3945.
The article deals with the description of the statistical behavior of Gaussian packets on a metric graph. Semiclassical asymptotics of solutions of the Cauchy problem for the Schr\"{o}dinger equation with initial data concentrated in the neighborhood of one point on the edge, generates a classical dynamical system on a graph. In a situation where all times for packets to pass over edges ("edge travel times") are linearly independent over the rational numbers, a description of the behavior of such systems is related to the number-theoretic problem of counting the number of lattice points in an expanding polyhedron. In this paper we show that for a finite compact graph packets almost always are distributed evenly. A formula for the leading coefficient of the asymptotic behavior of the number of packets with an increasing time is obtained. The article also discusses a situation where the times of passage over the edges are not linearly independent over the rationals.
Working paper
Bitoun T. arxiv.org. math. Cornell University, 2010
We prove an analog of the involutivity of the characteristic variety theorem for the reduction to positive characteristic of holonomic D-modules, involving the p-curvature. The proof given is independent of O. Gabber's theorem.
Working paper
Kurnosov N. arxiv.org. math. Cornell University, 2014
Let M be a compact irreducible hyperkahler manifold, from Bogomolov inequality [V1] we obtain forbidden values of the second Betti number b_2 in arbitrary dimension.