On the asymptotics with respect to a parameter of solutions of differential systems with coefficients in the class <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>

We consider a system of first-order ordinary linear differential equations with coefficients depending on an arbitrary parameter λ. For large λ, if the coefficients are smooth with respect to x, then there are known classical exponentially asymptotic (with respect to λ) formulas for the solution of the system. We generalize such formulas to the case in which the coefficients belong to the class L _q , q > 1. We use a new method for the reduction of problems to an integral system of special form.     

Von Neumann algebras generated by translation-invariant Gibbs states of the Ising model on a Bethe lattice

The Ising model on a Bethe lattice of orderk≥2 is considered. For maximum or minimum translation-invariant Gibbs states of this model, the relations between the von Neumann algebras generated by these states for the Gelfand-Neimark-Segal representation are found. These algebras can be of types III_λ, λ∈(0, 1), and III₁. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 1, pp. 88–93, April, 2000.     

Eden-Staudacher and Beisert-Eden-Staudacher equations in the $$\mathcal{N} = 4$$ supersymmetric gauge theory

We investigate the Eden-Staudacher and Beisert-Eden-Staudacher equations for the anomalous dimension of twist-2 operators at a large spin s in the supersymmetric gauge theory. We reduce these equations to a set of linear algebraic equations and calculate their kernels analytically. We demonstrate that in the perturbation theory, the anomalous dimension is a sum of products of the Euler functions ζ(k) having the maximum transcendentality property. We also show that at a large coupling, the “singular” solution of the Beisert-Eden-Staudacher equation reproduces the anomalous dimension constants predicted from the string side of the AdS/CFT correspondence. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 1, pp. 117–129, April, 2008.     

Categoricity of an abstract elementary class in two successive cardinals

We investigate categoricity of abstract elementary classes without any remnants of compactness (like non-definability of well ordering, existence of E.M. models, or existence of large cardinals). We prove (assuming a weak version of GCH around λ) that if ℜ is categorical in λ, λ⁺,LS(ℜ) ≤ λ and has intermediate number of models in λ⁺⁺,then ℜ has a model in λ⁺⁺⁺. Partially supported by the United States-Israel Binational Science Foundation. I thank Alice Leonhardt for the excellent typing. Publication No. 576.     

Electron scattering at the domain wall

We study the transmission and reflection coefficients for the Hamiltonian of a spin-polarized electron passing through the domain wall in a ferromagnetic quantum wire. We prove that total reflection occurs for energies λ ∈ (−α, α) (−α is the boundary of the essential spectrum) for both sufficiently small and sufficiently large λ, which agrees with the ballistic magnetoresistance effect in ferromagnetic nanocontacts. For energies λ > α, almost total reflection becomes almost total transmission, and both effects occur without a spin flip.     

Topology of spaces of hyperbolic polynomials and combinatorics of resonances

In this paper we study the topology of the strata, indexed by number partitions λ, in the natural stratification of the space of monic hyperbolic polynomials of degreen. We prove stabilization theorems for removing an independent block or an independent relation in λ. We also prove contractibility of the ‘one-point compactifications of the strata indexed by a large class of number partitions, including λ=(k ^m , 1 ^r ), form≥2. Furthermore, we study the maps between the homology groups of the strata, induced by imposing additional relations (resonances) on the number partition λ, or by merging some of the blocks of λ.     

On the number of singularities for the obstacle problem in two dimensions

We study the obstacle problem in two dimensions. On the one hand we improve a result of L.A. Caffarelli and N.M. Rivière: we state that every connected component of the interior of the coincidence set has at most N ₀ singular points, where N ₀ is only dependent on some geometric constants. Moreover, if the component is small enough, then this component has at most two singular points. On the other hand, we prove in a simple case a conjecture of D.G. Schaeffer on the generic regularity of the free boundary: for a family of obstacle problems in two dimensions continuously indexed by a parameter λ, the free boundary of the solution u_λ is analytic for almost every λ. Finally we present a new monotonicity formula for singular points. Dedicated to Henri Berestycki and Alexis Bonnet.     

Ginzburg-Landau vortex analogues

We consider a static one-dimensional Ginzburg-Landau equation (on a line segment or a circle) involving a large parameter λ. We show that as λ→∞, there exist solutions whose asymptotic behavior resembles the behavior of the two-dimensional vortex solutions. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 1, pp. 18–35, July, 2000.     

On a class of semilinear weakly hyperbolic equations

Abstract  In this paper, we deal with some global existence results for the large data smooth solutions of the Cauchy Problem associated with the semilinear weakly hyperbolic equations

Absolute continuity for Banach space valued mappings

We consider the notion ofp, λ, δ-absolute continuity for Banach space valued mappings introduced in [2] for real valued functions and for λ=1. We investigate the validity of some basic properties that are shared byn, λ-absolutely continuous functions in the sense of Maly and hencl. We introduce the class δ-BV _λ,loc^p and we give a characterization of the functions belonging to this class.     

Existence and regularity properties of non-isotropic singular elliptic equations

We establish existence and sharp regularity results for solutions to singular elliptic equations of the order u ^−β , 0 < β < 1, with gradient dependence and involving a forcing term λ f(x, u). Our approach is based on a singularly perturbed technique. We show that if the forcing parameter λ > 0 is large enough, our solution is positive. For λ small solutions vanish on a nontrivial set and therefore they exhibit free boundaries. We also establish regularity results for the free boundary and study the asymptotic behavior of the problem as b\searrow 0{\beta\searrow 0} and b\nearrow 1{\beta\nearrow 1}. In the former, we show that our solutions u _β converge to a C ^1,1 function which is a solution to an obstacle type problem. When b\nearrow 1{\beta\nearrow 1} we recover the Alt-Caffarelli theory.     

On the eigenvalues of matrices

Summary Let A be an n×n matric with arbitrary complex elements and with eigen-values λ₁, λ₂, ..., λ_n. A method is described for the approximàte determination of max | λ_j | ; characteristical is that prescribing a percentual error the number of elementary operations of the process, necessary to reach such precision, depends only on n and not on the elements of A More general characteristical equations are also considered. To Enrico Bompiani on his scientific Jubilee     

On Relative Difference Sets in Dihedral Groups

In this paper, we study extensions of trivial difference sets in dihedral groups. Such relative difference sets have parameters of the form (uλ,u,uλ, λ) or (uλ+2,u, uλ+1, λ) and are called semiregular or affine type, respectively. We show that there exists no nontrivial relative difference set of affine type in any dihedral group. We also show a connection between semiregular relative difference sets in dihedral groups and Menon–Hadamard difference sets. In the last section of the paper, we consider (m, u, k, λ) difference sets of general type in a dihedral group relative to a non-normal subgroup. In particular, we show that if a dihedral group contains such a difference set, then m is neither a prime power nor product of two distinct primes.     

Differential-geometric structures on generalized Reidemeister and Bol three-webs

In this paper, we present the main results of the study of multidimensional three-websW(p, q, r) obtained by the method of external forms and moving Cartan frame. The method was developed by the Russian mathematicians S. P. Finikov, G. F. Laptev, and A. M. Vasiliev, while fundamentals of differential-geometric (p, q, r)-webs theory were described by M. A. Akivis and V. V. Goldberg. Investigation of (p, q, r)-webs, including algebraic and geometric theory aspects, has been continued in our papers, in particular, we found the structure equations of a three-web W(p, q, r), where p = λl, q = λm, and r = λ(l + m − 1). For such webs, we define the notion of a generalized Reidemeister configuration and proved that a three-web W(λl, λm, λ(l + m − 1)), on which all sufficiently small generalized Reidemeister configurations are closed, is generated by a λ-dimensional Lie group G. The structure equations of the web are connected with the Maurer–Cartan equations of the group G. We define generalized Reidemeister and Bol configurations for three-webs W(p, q, q). It is proved that a web W(p, q, q) on which generalized Reidemeister or Bol configurations are closed is generated, respectively, by the action of a local smooth q-parametric Lie group or a Bol quasigroup on a smooth p-dimensional manifold. For such webs, the structure equations are found and their differential-geometric properties are studied.     

Computing a closest bifurcation instability in multidimensional parameter space

Summary Engineering and physical systems are often modeled as nonlinear differential equations with a vector λ of parameters and operated at a stable equilibrium. However, as the parameters λ vary from some nominal value λ₀, the stability of the equilibrium can be lost in a saddle-node or Hopf bifurcation. The spatial relation in parameter space of λ₀ to the critical set of parameters at which the stable equilibrium bifurcates determines the robustness of the system stability to parameter variations and is important in applications. We propose computing a parameter vector λ_* at which the stable equilibrium bifurcates which is locally closest in parameter space to the nominal parameters λ₀. Iterative and direct methods for computing these locally closest bifurcations are described. The methods are extensions of standard, one-parameter methods of computing bifurcations and are based on formulas for the normal vector to hypersurfaces of the bifurcation set. Conditions on the hypersurface curvature are given to ensure the local convergence of the iterative method and the regularity of solutions of the direct method. Formulas are derived for the curvature of the saddle node bifurcation set. The methods are extended to transcritical and pitchfork bifurcations and parametrized maps, and the sensitivity to λ₀ of the distance to a closest bifurcation is derived. The application of the methods is illustrated by computing the proximity to the closest voltage collapse instability of a simple electric power system.     

First-Quantized Fermions in Compact Dimensions

We discuss a path integral representation for fermionic particles and strings in the spirit of V. Ya. Fainberg and the author (Nucl. Phys. B, 306, 659–676, (1998); Phys. Lett. B, 211, 81–85, (1988)). We concentrate on the problems arising when some target-space dimensions are compact. We consider the partition function for a fermionic particle at a finite temperature or in compact time in detail as an example. We demonstrate that a self-consistent definition of the path integral generally requires introducing nonvanishing background Wilson loops and that modulo some common problems for real fermions in the Grassmannian formulation, these loops can be interpreted as condensates of world-line fermions. Properties of the corresponding string-theory path integrals are also discussed.     

Iterated grafting and holonomy lifts of Teichmüller space

Let X be a closed hyperbolic surface and λ, η be weighted geodesic multicurves which are short on X. We show that the iterated grafting along λ and η is close in the Teichmüller metric to grafting along a single multicurve which can be given explicitly in terms of λ and η. Using this result, we study the holonomy lifts gr _λ ρ _X,λ of Teichmüller geodesics ρ _X,λ for integral laminations λ and show that all of them have bounded Teichmüller distance to the geodesic ρ _X,λ. We obtain analogous results for grafting rays. Finally we consider the asymptotic behaviour of iterated grafting sequences gr _nλ X and show that they converge geometrically to a punctured surface.     

Large sets of pure directed triple systems with index λ

A directed triple system of order v with index λ, briefly by DTS（v,λ）, is a pair （X, B） where X is a v-set and B is a collection of transitive triples （blocks） on X such that every ordered pair of X belongs to λ blocks of B. A simple DTS（v, λ） is a DTS（v, λ） without repeated blocks. A simple DTS（v, ）,） is called pure and denoted by PDTS（v, λ） if （x, y, z） ∈ B implies （z, y, x）, （z, x, y）, （y, x, z）, （y, z, x）, （x, z, y） B. A large set of disjoint PDTS（v, λ）, denoted by LPDTS（v, λ）, is a collection of 3（v - 2）/λ disjoint pure directed triple systems on X. In this paper, some results about the existence for LPDTS（v, λ） are presented. Especially, we determine the spectrum of LPDTS（v, 2）.     

A remark on absolutely continuous functions in ℝ n

We introduce the notion ofα, λ-absolute continuity for functions of several variables and we compare it with the Hencl’s definition. We obtain that eachα, λ-absolutely continuous function isn, λ-absolutely continuous in the sense of Hencl and hence is continuous, differentiable almost everywhere and satisfies change of variables results based on a coarea formula and an area formula.