An equivalence of two families of quantum integrable systems

An equivalence of two quantum integrable systems with complete set of quadratic integrals of motion is established. Bibliography: 9 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 199, 1992, pp. 114–131. Translated by I. V. Komarov.     

Homotopy types of box complexes

In [14] Matoušek and Ziegler compared various topological lower bounds for the chromatic number. They proved that Lovász’s original bound [9] can be restated as X(G) ≥ ind(B(G)) + 2. Sarkaria’s bound [15] can be formulated as X(G) ≥ ind(B₀(G)) + 1. It is known that these lower bounds are close to each other, namely the difference between them is at most 1. In this paper we study these lower bounds, and the homotopy types of box complexes. The most interesting result is that up to ℤ₂-homotopy the box complex B(G) can be any ℤ₂-space. This together with topological constructions allows us to construct graphs showing that the mentioned two bounds are different. Some of the results were announced in [14]. Supported by the joint Berlin/Zürich graduate program Combinatorics, Geometry, and Computation, financed by ETH Zürich and the German Science Foundation (DFG).     

Homotopy equivalence of coverings and the spectral sequence of a fibration

Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 32, No. 1, pp. 141–147, January–February, 1991.     

Homotopy groups of Hom complexes of graphs

The notion of ×-homotopy from [Anton Dochtermann, Hom complexes and homotopy theory in the category of graphs, European J. Combin., in press] is investigated in the context of the category of pointed graphs. The main result is a long exact sequence that relates the higher homotopy groups of the space Hom_∗(G,H) with the homotopy groups of Hom_∗(G,HI). Here Hom_∗(G,H) is a space which parameterizes pointed graph maps from G to H (a pointed version of the usual Hom complex), and HI is the graph of based paths in H. As a corollary it is shown that πi(Hom_∗(G,H))≅_×[G,ΩiH], where ΩH is the graph of based closed paths in H and _×[G,K] is the set of ×-homotopy classes of pointed graph maps from G to K. This is similar in spirit to the results of [Eric Babson, Hélène Barcelo, Mark de Longueville, Reinhard Laubenbacher, Homotopy theory of graphs, J. Algebraic Combin. 24 (1) (2006) 31-44], where the authors seek a space whose homotopy groups encode a similarly defined homotopy theory for graphs. The categorical connections to those constructions are discussed.     

Homotopy properties of certain complexes associated to spherical buildings

We show that the complex Opp Δ of ordered pairs of opposite simplices in a spherical building Δ is spherical, i.e., (n-1)-connected, wheren is the common dimension of Δ and Opp Δ. As a corollary we obtain that Opp Δ is even homotopy Cohen-Macaulay.     

On equivalence between known families of quadratic APN functions

This paper is dedicated to a question whether the currently known families of quadratic APN polynomials are pairwise different up to CCZ-equivalence. We reduce the list of these families to those CCZ-inequivalent to each other. In particular, we prove that the families of APN trinomials (constructed by Budaghyan and Carlet in 2008) and multinomials (constructed by Bracken et al. 2008) are contained in the APN hexanomial family introduced by Budaghyan and Carlet in 2008. We also prove that a generalization of these trinomial and multinomial families given by Duan et al. (2014) is contained in the family of hexanomials as well.     

The equivalence of two tax processes

We introduce two models of taxation, the latent and natural tax processes, which have both been used to represent loss-carry-forward taxation on the capital of an insurance company. In the natural tax process, the tax rate is a function of the current level of capital, whereas in the latent tax process, the tax rate is a function of the capital that would have resulted if no tax had been paid. Whereas up to now these two types of tax processes have been treated separately, we show that, in fact, they are essentially equivalent. This allows a unified treatment, translating results from one model to the other. Significantly, we solve the question of existence and uniqueness for the natural tax process, which is defined via an integral equation. Our results clarify the existing literature on processes with tax.     

Homotopy type of circle graph complexes motivated by extreme Khovanov homology

It was proven by González-Meneses, Manchón and Silvero that the extreme Khovanov homology of a link diagram is isomorphic to the reduced (co)homology of the independence simplicial complex obtained from a bipartite circle graph constructed from the diagram. In this paper, we conjecture that this simplicial complex is always homotopy equivalent to a wedge of spheres. In particular, its homotopy type, if not contractible, would be a link invariant (up to suspension), and it would imply that the extreme Khovanov homology of any link diagram does not contain torsion. We prove the conjecture in many special cases and find it convincing to generalize it to every circle graph (intersection graph of chords in a circle). In particular, we prove it for the families of cactus, outerplanar, permutation and non-nested graphs. Conversely, we also give a method for constructing a permutation graph whose independence simplicial complex is homotopy equivalent to any given finite wedge of spheres. We also present some combinatorial results on the homotopy type of finite simplicial complexes and a theorem shedding light on previous results by Csorba, Nagel and Reiner, Jonsson and Barmak. We study the implications of our results to knot theory; more precisely, we compute the real-extreme Khovanov homology of torus links T(3, q) and obtain examples of H-thick knots whose extreme Khovanov homology groups are separated either by one or two gaps as long as desired.     

On the equivalence of two variational problems

We consider the following problems where is a convex function, is an open bounded subset of is a closed convex subset of such that and and are suitable obstacles. We give conditions on the function {\it g} under which the two problems are equivalent. Received March 24, 1999/ Accepted January 14, 2000 / Published online June 28, 2000     

Homotopy perturbation method for two dimensional time-fractional wave equation

The aim of this paper is to present an efficient and reliable treatment of the homotopy perturbation method (HPM) for two dimensional time-fractional wave equation (TFWE) with the boundary conditions. The fractional derivative is described in the Caputo sense. The initial approximation can be determined by imposing the boundary conditions. The method provides approximate solutions in the form of convergent series with easily computable components. The obtained results shown that the technique introduced here is efficient and easy to implement.     

Topological realizations of families of ergodic automorphisms, multitowers and orbit equivalence

We study minimal topological realizations of families of ergodic measure preserving automorphisms (e.m.p.a.'s). Our main result is the following theorem. Theorem: Let {T_p:p∈I} be an arbitrary finite or countable collection of e.m.p.a.'s on nonatomic Lebesgue probability spaces (Y _p v _p ). Let S be a Cantor minimal system such that the cardinality of the set ε _S of all ergodic S-invariant Borel probability measures is at least the cardinality of I. Then for any collection {μ _p :pεI} of distinct measures from ε _S there is a Cantor minimal system S′ in the topological orbit equivalence class of S such that, as a measure preserving system, (S ¹,μ _p ) is isomorphic to T_p for every p∈I. Moreover, S′ can be chosen strongly orbit equivalent to S if and only if all finite topological factors of S are measure-theoretic factors of T_p for all p∈I. This result shows, in particular, that there are no restrictions at all for the topological realizations of countable families of e.m.p.a.'s in Cantor minimal systems. Namely, for any finite or countable collection {T ₁,T₂,…} of e.m.p.a.'s of nonatomic Lebesgue probability spaces, there is a Cantor minimal systemS, whose collection {μ1,μ2…} of ergodic Borel probability measures is in one-to-one correspondence with {T ₁,T₂,…}, and such that (S,μ _i ) is isomorphic toT _i for alli. Furthermore, since realizations are taking place within orbit equivalence classes of a given Cantor minimal system, our results generalize the strong orbit realization theorem and the orbit realization theorem of [18]. Those theorems are now special cases of our result where the collections {T _p}, {T _p }{μ _p } consist of just one element each. Research of I.K. was supported by NSF grant DMS 0140068.     

Geometry of chain complexes and outer automorphisms under derived equivalence

The two main theorems proved here are as follows: If is a finite dimensional algebra over an algebraically closed field, the identity component of the algebraic group of outer automorphisms of is invariant under derived equivalence. This invariance is obtained as a consequence of the following generalization of a result of Voigt. Namely, given an appropriate geometrization of the family of finite -module complexes with fixed sequence of dimensions and an ``almost projective' complex , there exists a canonical vector space embedding

where is the pertinent product of general linear groups acting on , tangent spaces at are denoted by , and is identified with its image in the derived category .

     

Degenerating families of rank two bundles

We construct families of rank two bundles on , in characteristic two, where for , is a sum of line bundles, and is non-split. We construct families of rank two bundles on , in characteristic , where for , is a sum of line bundles, and is non-split.