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Working paper

Let $(M,I,J,K)$ be a hyperkahler manifold, and $Z\subset (M,I)$ a complex subvariety in $(M,I)$. We say that $Z$ is trianalytic if it is complex analytic with respect to $J$ and $K$, and absolutely trianalytic if it is trianalytic with respect to any hyperk\"ahler triple of complex structures $(M,I,J',K')$ containing $I$. For a generic complex structure $I$ on $M$, all complex subvarieties of $(M,I)$ are absolutely trianalytic. It is known that a normalization $Z'$ of a trianalytic subvariety is smooth; we prove that $b_2(Z')$ is no smaller than $b_2(M)$ when $M$ has maximal holonomy (that is, $M$ is IHS). To study absolutely trianalytic subvarieties further, we define a new geometric structure, called k-symplectic structure; this structure is a generalization of the hypersymplectic structure. A k-symplectic structure on a 2d-dimensional manifold $X$ is a k-dimensional space $R$ of closed 2-forms on $X$ which all have rank 2d or d. It is called non-degenerate if the set of all degenerate forms in $R$ is a smooth, non-degenerate quadric hypersurface in $R$. We consider absolutely trianalytic tori in a hyperkahler manifold $M$ of maximal holonomy. We prove that any such torus is equipped with a non-degenerate k-symplectic structure, where $k=b_2(M)$. We show that the tangent bundle $TX$ of a k-symplectic manifold is a Clifford module over a Clifford algebra $Cl(k-1)$. Then an absolutely trianalytic torus in a hyperkahler manifold $M$ with $b_2(M)\geq 2r+1$ is at least $2^{r-1}$-dimensional.

Added: Sep 5, 2014

Working paper

According to a recent paper \cite{bopt13}, polynomials from the closure $\ol{\phd}_3$ of the {\em Principal Hyperbolic Domain} ${\rm PHD}_3$ of the cubic connectedness locus have a few specific properties. The family $\cu$ of all polynomials with these properties is called the \emph{Main Cubioid}. In this paper we describe the set $\cu^c$ of laminations which can be associated to polynomials from $\cu$.

Added: Oct 6, 2013

Working paper

Let S={s1<s2<s3<…} be the sequence of all natural numbers which can be represented as a sum of two squares of integers. For X≥2 we denote by g(X) the largest gap between consecutive elements of S that do not exceed X. We prove that for X→+∞ the lower bound
g(X)≥(390/449−o(1))lnX
holds.
This estimate is twice the recent estimate by R. Dietmann and C. Elsholtz.

Added: Jul 17, 2019

Working paper

We establish a natural translation from word rewriting systems to strictly positive polymodal logics. Thereby, the latter can be considered as a generalization of the former. As a corollary we obtain examples of undecidable strictly positive normal modal logics. The translation has its counterpart on the level of proofs: we formulate a natural deep inference proof system for strictly positive logics generalizing derivations in word rewriting systems. We also formulate some open questions related to the theory of modal companions of superintuitionistic logics that was initiated by L.L. Maximova and V.V. Rybakov.

Added: Mar 13, 2016

Working paper

We study abstract versions of G\"odel's second incompleteness theorem and formulate generalizations of L\"ob's derivability conditions that work for logics weaker than the classical one. We isolate the role of contraction rule in G\"odel's theorem and give a (toy) example of a system based on modal logic without contraction invalidating G\"odel's argument.

Added: Mar 13, 2016

Working paper

Logachev D.,

. arxiv.org. math. Cornell University, 2016
We continue study of varieties started in a paper of A. Grishkov, D. Logachev "Resultantal varieties related to zeroes of L-functions of Carlitz modules". We give a (conjecturally) complete description of these varieties in terms of rooted binary trees, we find parametrizations of their irreducible components and their invariants: degrees, multiplicities, Jordan forms, Galois actions. Maybe a generalization of this research will give us a solution of the problem of boundedness of the analytic rank of twists of Carlitz modules.

Added: Nov 26, 2017

Working paper

Given a representation V of a group G, there are two natural ways of defining a representation of the group algebra k[G] in the external power V^{\wedge m}. The set L(V) of elements of k[G] for which these two ways give the same result is a Lie algebra and a representation of G. For the case when G is a symmetric group and V = C^n, a permutation representation, these spaces L(C^n) are naturally embedded into one another. We describe L(C^n) for small n and formulate some questions and conjectures. This is a note on research in progress.

Added: Nov 19, 2013

Working paper

Bienvenu L., Muchnik A. A., Shen A. et al. arxiv.org. math. Cornell University, 2012. No. 1204.0201.

The main goal of this article is to put some known results in a common perspective and to simplify their proofs. We start with a simple proof of a result of Vereshchagin saying that lim supnC(x|n) equals C0′(x). Then we use the same argument to prove similar results for prefix complexity, a priori probability on binary tree, to prove Conidis' theorem about limits of effectively open sets, and also to improve the results of Muchnik about limit frequencies. As a by-product, we get a criterion of 2-randomness proved by Miller: a sequence X is 2-random if and only if there exists c such that any prefix x of X is a prefix of some string y such that C(y)≥|y|−c. (In the 1960ies this property was suggested in Kolmogorov as one of possible randomness definitions.) We also get another 2-randomness criterion by Miller and Nies: X is 2-random if and only if C(x)≥|x|−c for some c and infinitely many prefixes x of X. This is a modified version of our old paper that contained a weaker (and cumbersome) version of Conidis' result, and the proof used low basis theorem (in quite a strange way). The full version was formulated there as a conjecture. This conjecture was later proved by Conidis. Bruno Bauwens (personal communication) noted that the proof can be obtained also by a simple modification of our original argument, and we reproduce Bauwens' argument with his permission.

Added: Dec 14, 2013

Working paper

In this paper we consider the behavior of Kalman Filter state estimates in the case of distribution with heavy tails .The simulated linear state space models with Gaussian measurement noises were used. Gaussian noises in state equation are replaced by components with alpha-stable distribution with di erent parameters alpha and beta. We consider the case when "all parameters are known"and two methods of parameters estimation are compared: the maximum likelihood estimator (MLE) and the expectation- maximization algorithm (EM). It was shown that in cases of large deviation from Gaussian distribution the total error of states estimation rises dramatically.We conjecture that it can be explained by underestimation of the state equation noises covariance matrix that can be taken into account through the EM parameters estimation and ignored in the case of ML estimation

Added: Jun 1, 2015

Working paper

Cerulli Irelli G., Fang X.,

et al. arxiv.org. math. Cornell University, 2019. No. 1901.11020.
We continue, generalize and expand our study of linear degenerations of flag varieties from [G. Cerulli Irelli, X. Fang, E. Feigin, G. Fourier, M. Reineke, Math. Z. 287 (2017), no. 1-2, 615-654]. We realize partial flag varieties as quiver Grassmannians for equi-oriented type A quivers and construct linear degenerations by varying the corresponding quiver representation. We prove that there exists the deepest flat degeneration and the deepest flat irreducible degeneration: the former is the partial analogue of the mf-degenerate flag variety and the latter coincides with the partial PBW-degenerate flag variety. We compute the generating function of the number of orbits in the flat irreducible locus and study the natural family of line bundles on the degenerations from the flat irreducible locus. We also describe explicitly the reduced scheme structure on these degenerations and conjecture that similar results hold for the whole flat locus. Finally, we prove an analogue of the Borel-Weil theorem for the flat irreducible locus.

Added: Feb 5, 2019

Working paper

In this note we discuss a variant of linear logic with idempotent exponential modalities. We propose a sequent calculus system and discuss its semantics. We also give a concrete relational model for this calculus.

Added: Dec 23, 2015

Working paper

It is shown that a generic bihamiltonian structure on an odd-dimensional manifold is flat if and only if it is locally unimodular.

Added: Nov 19, 2013

Working paper

We consider a sequence of Markov chains weakly convergent to a diffusion. We suppose that a drift term contains a linearly increasing component. The usual parametrix method fails because of this unbounded drift term. We show how to modify the parametrix method to obtain local theorems for this case.

Added: Jan 21, 2015

Working paper

Ornea L.,

arxiv.org. math. Cornell University, 2012
Added: Feb 6, 2013

Working paper

In the authors's previous work on symmetric differentials and their
connection to the topological properties of the ambient manifold, a class of
symmetric differentials was introduced: closed symmetric differentials
([BoDeO11] and [BoDeO13]). In this article we give a description of the local
structure of closed symmetric 2-differentials on complex surfaces, with an
emphasis towards the local decompositions as products of 1-differentials. We
show that a closed symmetric 2-differential $w$ of rank 2 (i.e. defines two
distinct foliations at the general point) has a subvariety $B_w\subset X$
outside of which $w$ is locally the product of closed holomorphic
1-differentials. The main result, theorem 2.6, gives a complete description of
a (locally split) closed symmetric 2-differential in a neighborhood of a
general point of $B_w$. A key feature of theorem 2.6 is that closed symmetric
2-differentials still have a decomposition as a product of 2 closed
1-differentials (in a generalized sense) even at the points of $B_w$. The
(possibly multi-valued) closed 1-differentials can have essential singularities
along $B_w$, but one still has a control on these essential singularities. The
essential singularities come from exponentials of meromorphic functions
acquiring poles along the irreducible components of $B_w$ of order bounded by
the order of contact of the 2 foliations defined by the symmetric
2-differential along that irreducible component.

Added: Nov 21, 2014

Working paper

We describe a class of integrable systems on Poisson submanifolds of the affine Poisson-Lie groups PGLˆ(N), which can be enumerated by cyclically irreducible elements the co-extended affine Weyl groups (Wˆ×Wˆ)♯. Their phase spaces admit cluster coordinates, whereas the integrals of motion are cluster functions. We show, that this class of integrable systems coincides with the constructed by Goncharov and Kenyon out of dimer models on a two-dimensional torus and classified by the Newton polygons. We construct the correspondence between the Weyl group elements and polygons, demonstrating that each particular integrable model admits infinitely many realisations on the Poisson-Lie groups. We also discuss the particular examples, including the relativistic Toda chains and the Schwartz-Ovsienko-Tabachnikov pentagram map.

Added: Oct 29, 2014

Working paper

Marcati C.,

, Ulander J. arxiv.org. math. Cornell University, 2020. No. 2010.06919.
We derive rank bounds on the quantized tensor train (QTT) compressed approximation of singularly perturbed reaction diffusion partial differential equations (PDEs) in one dimension. Specifically, we show that, independently of the scale of the singular perturbation parameter, a numerical solution with accuracy 0<ϵ<1 can be represented in QTT format with a number of parameters that depends only polylogarithmically on ϵ. In other words, QTT compressed solutions converge exponentially to the exact solution, with respect to a root of the number of parameters. We also verify the rank bound estimates numerically, and overcome known stability issues of the QTT based solution of PDEs by adapting a preconditioning strategy to obtain stable schemes at all scales. We find, therefore, that the QTT based strategy is a rapidly converging algorithm for the solution of singularly perturbed PDEs, which does not require prior knowledge on the scale of the singular perturbation and on the shape of the boundary layers.

Added: Oct 20, 2020

Working paper

We construct an analogue of the classical p-Wasserstein distance for the state space of a C*-algebra. Given an abstract Lipschitz gauge on a C*-algebra A in the sense of Rieffel, one can define the classical p-Wasserstein distance on the state space of each commutative C*-subalgebra of A. We consider a projective limit of these metric spaces, which appears to be the space of all quasi-linear states, equipped with a distance function. We call this distance the projective p-Wasserstein distance. It is easy to show, that the state space of a C*-algebra is naturally embedded in the space of its quasi-linear states, hence, the introduced distance is defined on the state space as well. We show that this distance is reasonable and well-behaved. We also formulate a sufficient condition for a Lipschitz gauge, such that the corresponding projective p-Wasserstein distance metricizes the weak*-topology on the state space.

Added: May 25, 2015

Working paper

Bennett M., Berenstein A., Chari V. et al. arxiv.org. math. Cornell University, 2012. No. 2446.

We study the category of graded representations with finite--dimensional graded pieces for the current algebra associated to a simple Lie algebra. This category has many similarities with the category $\cal O$ of modules for $\lie g$ and in this paper, we use the combinatorics of Macdonald polynomials to prove an analogue of the famous BGG duality in the case of $\lie{sl}_{n+1}$.

Added: Feb 20, 2013

Working paper

Braverman A.,

, Shiraishi J. arxiv.org. math. Cornell University, 2012
Added: Feb 6, 2013

Working paper

In many practical tasks it is needed to estimates effect of a treatemnt on individual level. For example in medicine it is essential to determine the patients that would benifit from a certain medicament. In marketing knowning the persons that are likely to buy a new product would reduce the amount of spam. In this chapter we review the methods to estimate individulize treatment effect from a randomized trial, i.e., an experiment when a part of individuals recieves a new treatment, while the others does not. Finally, it is shown that new efficient methods are needed in this domain.

Added: Oct 2, 2018