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Working paper

Chaudru de Raynal P.,

math. arxive. Cornell University, 2019
We establish well-posedness results for multidimensional non degenerate α-stable driven SDEs with time inhomogeneous singular drifts in Lr−B−1+γp,q with γ<1 and α in (1,2], where Lr and B−1+γp,q stand for Lebesgue and Besov spaces respectively. Precisely, we first prove the well-posedness of the corresponding martingale problem and then give a precise meaning to the dynamics of the SDE. Our results rely on the smoothing properties of the underlying PDE, which is investigated by combining a perturbative approach with duality results between Besov spaces.

Added: Oct 31, 2020

Working paper

The multistochastic (n,k)-Monge--Kantorovich problem on a product space ∏ni=1Xi is an extension of the classical Monge--Kantorovich problem. This problem is considered on the space of measures with fixed projections onto Xi1×…×Xik for all k-tuples {i1,…,ik}⊂{1,…,n} for a given 1≤k<n. In our paper we study well-posedness of the primal and the corresponding dual problem. Our central result describes a solution π to the following important model case: n=3,k=2,Xi=[0,1], the cost function c(x,y,z)=xyz, and the corresponding two--dimensional projections are Lebesgue measures on [0,1]2. We prove, in particular, that the mapping (x,y)→x⊕y, where ⊕ is the bitwise addition (xor- or Nim-addition) on [0,1]≅Z∞2, is the corresponding optimal transportation. In particular, the support of π is the Sierpiński tetrahedron. In addition, we describe a solution to the corresponding dual problem.

Added: Jul 31, 2018

Working paper

A celebrated conjecture of Kobayashi and Lang says that the canonical line bundle K_X of a Kobayashi hyperbolic compact complex manifold X is ample. In this note we prove that K_X is ample if X is projective and satisfies a stronger condition of nondegenerate negative total k-jet curvature. We use positivity of direct image sheaves and decomposition of jets in order to produce pluridifferentials on X.

Added: Sep 30, 2019

Working paper

Angella D., Tomassini A.,

math. arxive. Cornell University, 2016
We study cohomological properties of complex manifolds. In particular, we provide an upper bound for the Bott-Chern cohomology in terms of Betti numbers for compact complex surfaces, according to the dichotomy b1 even or odd. In higher dimension, a similar result is obtained at degree 1 under additional metric conditions (see Theorem 2.4).

Added: May 14, 2016

Working paper

We consider threefold del Pezzo fibrations over a curve germ whose central fiber is non-rational. Under the additional assumption that the singularities of the total space are at worst ordinary double points, we apply a suitable base change and show that there is a 1-to-1 correpspondence between such fibrations and certain non-singular del Pezzo fibrations equipped with a cyclic group action.

Added: Dec 1, 2018

Working paper

We study the question when a $*$-autonomous Mix-category has a representation as a $*$-autonomous Mix-subcategory of a compact one. We define certain partial trace-like operation on morphisms of a Mix-category, which we call a mixed trace, and show that any structure preserving embedding of a Mix-category into a compact one induces a mixed trace on the former. We also show that, conversely, if a Mix-category ${\bf K}$ has a mixed trace, then we can construct a compact category and structure preserving embedding of ${\bf K}$ into it, which induces the same mixed trace. Finally, we find a specific condition expressed in terms of interaction of Mix- and coevaluation maps on a Mix-category ${\bf K}$, which is necessary and sufficient for a structure preserving embedding of ${\bf K}$ into a compact one to exist. When this condition is satisfied, we construct a ``free'' or ``minimal'' mixed trace on ${\bf K}$ directly from the Mix-category structure, which gives us also a ``free'' compactification of ${\bf K}$. K.

Added: Nov 23, 2016

Working paper

We consider an initial-boundary value problem for the $n$-dimensional wave equation with the variable sound speed, $n\geq 1$. We construct three-level implicit in time compact in space (three-point in each space direction) 4th order finite-difference schemes on the uniform rectangular meshes including their one-parameter (for $n=2$) and three-parameter (for $n=3$) families. They are closely connected to some methods and schemes constructed recently by several authors. In a unified manner, we prove the conditional stability of schemes in the strong and weak energy norms together with the 4th order error estimate under natural conditions on the time step. We also give an example of extending a compact scheme for non-uniform in space and time rectangular meshes. We suggest simple effective iterative methods based on FFT to implement the schemes whose convergence rate, under the stability condition, is fast and independent on both the meshes and variable sound speed. A new effective initial guess to start iterations is given too. We also present promising results of numerical experiments.

Added: Feb 2, 2021

Working paper

In this paper we study an optimal control problem that is affine in two-dimensional bounded control. The problem is related to the stabilization of an inverted spherical pendulum in the vicinity of the upper unstable equilibrium. We find solutions stabilizing the pendulum in a finite time, wherein the corresponding optimal controls perform an infinite number of rotations along the circle S1.

Added: Oct 16, 2019

Working paper

We review recent advances in the analysis of the Wang--Landau algorithm, which is designed for the direct Monte Carlo estimation of the density of states (DOS). In the case of a discrete energy spectrum, we present an approach based on introducing the transition matrix in the energy space (TMES). The TMES fully describes a random walk in the energy space biased with the Wang-Landau probability. Properties of the TMES can explain some features of the Wang-Landau algorithm, for example, the flatness of the histogram. We show that the Wang--Landau probability with the true DOS generates a Markov process in the energy space and the inverse spectral gap of the TMES can estimate the mixing time of this Markov process. We argue that an efficient implementation of the Wang-Landau algorithm consists of two simulation stages: the original Wang-Landau procedure for the first stage and a 1/t modification for the second stage. The mixing time determines the characteristic time for convergence to the true DOS in the second simulation stage. The parameter of the convergence of the estimated DOS to the true DOS is the difference of the largest TMES eigenvalue from unity. The characteristic time of the first stage is the tunneling time, i.e., the time needed for the system to visit all energy levels.

Added: Aug 29, 2018

Working paper

We study the properties of sets Σ which are the solutions of the maximal distance minimizer problem, id est of sets having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets Σ ⊂ R 2 satisfying the inequality maxy∈M dist (y, Σ) ≤ r for a given compact set M ⊂ R 2 and some given r > 0. Such sets play the role of the shortest possible pipelines arriving at a distance at most r to every point of M, where M is the set of customers of the pipeline. In this work it is proved that each maximal distance minimizer is a union of finite number of curves, having one-sided tangent lines at each point. Moreover the angle between these lines at each point of a maximal distance minimizer is greater or equal to 2π/3. It shows that a maximal distance minimizer is isotopic to a finite Steiner tree even for a “bad” compact M, which differs it from a solution of the Steiner problem (an example of a Steiner tree with an infinite number of branching points can be find in [10]). Also we classify the behavior of a minimizer in a neighbourhood of any point of Σ. In fact, all the results are proved for more general class of local minimizer, i. e. sets which are optimal in any neighbourhood of its arbitrary point.

Added: Oct 21, 2019

Working paper

We define and discuss some general properties of residual categories of Lefschetz decompositions in triangulated categories. In the case of the derived category of coherent sheaves on the Grassmannian G(k,n) we conjecture that the residual category associated with Fonarev's Lefschetz exceptional collection is generated by a completely orthogonal exceptional collection. We prove this conjecture for k=p, a prime number, modulo completeness of Fonarev's collection (and for p=3 we check this completeness).

Added: Dec 3, 2018

Working paper

We develop the theory of semisimplifications of tensor categories defined by Barrett and Westbury. In particular, we compute the semisimplification of the category of representations of a finite group in characteristic p in terms of representations of the normnalizer of its Sylow p-subgroup. This allows us to compute the semisimplification of the representation category of the symmetric group S_n+p in characteristic p, where 0 ≤ n ≤ p−1, and of the Deligne category Rep^ab S_t, where t∈ℕ. We also compute the semisimplification of the category of representations of the Kac-De Concini quantum group of the Borel subalgebra of 𝔰𝔩_2. We also study tensor functors between Verlinde categories of semisimple algebraic groups arising from the semisimplification construction, and objects of finite type in categories of modular representations of finite groups (i.e., objects generating a fusion category in the semisimplification). Finally, we determine the semisimplifications of the tilting categories of GL(n), SL(n) and PGL(n) in characteristic 2. In the appendix, we classify categorifications of the Grothendieck ring of representations of SO(3) and its truncations.

Added: May 30, 2019

Working paper

We develop the theory of semisimplifications of tensor categories defined by Barrett and Westbury. In particular, we compute the semisimplification of the category of representations of a finite group in characteristic p in terms of representations of the normnalizer of its Sylow p-subgroup. This allows us to compute the semisimplification of the representation category of the symmetric group Sn+p in characteristic p, where 0≤n≤p−1, and of the Deligne category Rep^ab S_t, where t∈ℕ. We also compute the semisimplification of the category of representations of the Kac-De Concini quantum group of the Borel subalgebra of 𝔰𝔩2. We also study tensor functors between Verlinde categories of semisimple algebraic groups arising from the semisimplification construction, and objects of finite type in categories of modular representations of finite groups (i.e., objects generating a fusion category in the semisimplification). Finally, we determine the semisimplifications of the tilting categories of GL(n), SL(n) and PGL(n) in characteristic 2. In the appendix, we classify categorifications of the Grothendieck ring of representations of SO(3) and its truncations.

Added: Dec 6, 2018

Working paper

Consider a family of Fano varieties π:X⟶B∋o over a curve germ with a smooth total space X. Assume that the generic fiber is smooth and the special fiber F=π^{−1}(o) has simple normal crossings. Then F is called a semistable degeneration of Fano varieties. We show that the dual complex of F is a simplex of dimension ≤dim F. Simplices of any admissible dimension can be realized for any dimension of the fiber. Using this result and the Minimal Model Program in dimension 3 we reproduce the classification of the semistable degenerations of del Pezzo surfaces obtained by Fujita. We also show that the maximal degeneration is unique and has trivial monodromy in dimension ≤3.

Added: Oct 11, 2019

Working paper

The paper deals with a three-parameter family of special dou- ble confluent Heun equations that was introduced and studied by V. M. Buchstaber and S. I. Tertychnyi as an equivalent presentation of a model of overdamped Josephson junction in superconductivity. The parameters are l, λ, μ ∈ R. Buchstaber and Tertychnyi described those parameter values, for which the corresponding equation has a poly- nomial solution. They have shown that for μ ≠ 0 this happens ex- actly when l ∈ N and the parameters (λ, μ) lie on an algebraic curve Γl ⊂ C2(λ,μ) called the l-spectral curve and defined as zero locus of de- terminant of a remarkable three-diagonal l × l-matrix. They studied the real part of the spectral curve and obtained important results with applications to model of Josephson junction, which is a family of dy- namical systems on 2-torus depending on real parameters (B,A;ω); the parameter ω, called the frequency, is fixed. One of main problems on the above-mentioned model is to study the geometry of boundaries of its phase-lock areas in R2(B,A) and their evolution, as ω decreases to 0. An approach to this problem suggested in the present paper is to study the complexified boundaries. We prove irreducibility of the complex spectral curve Γl for every l ∈ N. We also calculate its genus for l ⩽ 20 and present a conjecture on general genus formula. We apply the irreducibility result to the complexified boundaries of the phase-lock areas of model of Josephson junction. The family of real boundaries taken for all ω > 0 yields a countable union of two-dimensional analytic surfaces in R3 . We show that, unexpectedly, its complexification is(B,A,ω−1) a complex analytic subset consisting of just four two-dimensional irreducible components, and we describe them. This is done by using the representation of some special points of the boundaries (the so-called generalized simple intersections) as points of the real spectral curves and the above irreducibility result. We also prove that the spectral curve has no real ovals. We present a Monotonicity Conjecture on the evolution of the phase-lock area portraits, as ω decreases, and a partial positive result towards its confirmation.

Added: Nov 12, 2019

Working paper

We study when kernels of inflation maps associated to extraspecial p-groups in stable group cohomology are generated by their degree two components. This turns out to be true if the prime is large enough compared to the rank of the elementary abelian quotient, but false in general.

Added: Dec 5, 2018

Working paper

We construct a mininal Lefschetz decomposition of the bounded derived category of coherent sheaves on the isotropic Grassmannian $\IGr(3,7)$. Moreover, we show that $\IGr(3, 7)$ admits a full exceptional collection consisting of equivariant vector bundles.

Added: Apr 20, 2018

Working paper

Let X be an irreducible holomorphic symplectic fourfold and D a smooth hypersurface in X. It follows from a result by Amerik and Campana that the characteristic foliation (that is the foliation given by the kernel of the restriction of the symplectic form to D) is not algebraic unless D is uniruled. Suppose now that the Zariski closure of its general leaf is a surface. We prove that X has a lagrangian fibration and D is the inverse image of a curve on its base.

Added: Nov 3, 2016

Working paper

We provide a stable rationality construction for some smooth complex Gushel-Mukai varieties of dimension 6. As a consequence, we compute the integral singular cohomology of any smooth Gushel-Mukai sixfold and in particular, show that it is torsion-free.

Added: Jul 14, 2016

Working paper

A description of rational Newton maps in terms of the partial fraction decomposition of rational functions is obtained. Dynamics on parabolic immediate basins for rational Newton maps of entire functions have been studied. It is proved that every parabolic immediate basin contains invariant accesses to the parabolic fixed point at infinity. Moreover, among these accesses there exists a unique dynamically defined access where dynamics are attracted towards the parabolic fixed point, whereas for other accesses, if there is any, the dynamics are repelled.

Added: Feb 4, 2019

Working paper

In this paper, we study the conjecture of Gardner and Zvavitch from \cite{GZ}, which suggests that the standard Gaussian measure γ enjoys 1n-concavity with respect to the Minkowski addition of \textbf{symmetric} convex sets. We prove this fact up to a factor of 2: that is, we show that for symmetric convex K and L,
γ(λK+(1−λ)L)12n≥λγ(K)12n+(1−λ)γ(L)12n.
Further, we show that under suitable dimension-free uniform bounds on the Hessian of the potential, the log-concavity of even measures can be strengthened to p-concavity, with p>0, with respect to the addition of symmetric convex se

Added: Jul 31, 2018