Miura opers and critical points of master functions

Critical points of a master function associated to a simple Lie algebra come in families called the populations [11]. We prove that a population is isomorphic to the flag variety of the Langlands dual Lie algebra . The proof is based on the correspondence between critical points and differential operators called the Miura opers. For a Miura oper D, associated with a critical point of a population, we show that all solutions of the differential equation DY=0 can be written explicitly in terms of critical points composing the population. Supported in part by NSF grant DMS-0140460 Supported in part by NSF grant DMS-0244579     

Detectability of critical points of smooth functionals from their finite-dimensional approximations

Given a critical point of a C²-functional on a separable Hilbert space, we obtain sufficient conditions for it to be detectable (i.e. ‘visible’) from finite-dimensional Rayleigh-Ritz-Galerkin (RRG) approximations. While examples show that even nondegenerate critical points are, without any further restriction, not visible, we single out relevant classes of smooth functionals, e.g. the Hamiltonian action on the loop space or the functionals associated with boundary value problems for some semilinear elliptic equations, such that their nondegenerate critical points are visible from their RRG approximations.     

Multiple critical points theorems without the Palais-Smale condition

In this paper multiple critical points theorems, where the Palais-Smale condition on the functional is not requested, are presented. As an application, multiple solutions for a quasilinear two point boundary value problem involving the one-dimensional p-Laplacian are obtained.     

The infinite server queue and heuristic approximations to the multi-server queue with and without retrials

In this paper, properties of the time-dependent state probabilities of theM _t /G/∞ queue, when the queue is assumed to start empty are studied. Those results are compared with corresponding time-dependent results for theM/M/1 queue. Approximation to the time-dependent state probabilities of theM/G/m/m queue by means of the corresponding time-dependent state probabilities of theM/G/∞ queue are discussed. Through a decomposition formula it is shown that the main performance characteristics of the ergodicM/M/m/m+d queue are sums of the corresponding random variables for the ergodicM/M/m/m andM/M/1/1+(d−1) queues, respectively, weighted by the 3-rd Erlang formula (stationary probability of waiting or being lost for theM/M/m/m+d queue). Successful exact and approximation extensions of this kind of decomposition formula to theM/M/m/m+d queue with retrials are presented.     

On regular polynomial endomorphisms of ℂ<Superscript>2</Superscript> without bounded critical orbitswithout bounded critical orbits

We study conditions involving the critical set of a regular polynomial endomorphism f∶ℂ²↦ℂ² under which all complete external rays from infinity for f have well defined endpoints.     

Smooth approximations

We prove, among other things, that a Lipschitz (or uniformly continuous) mapping f:X→Y can be approximated (even in a fine topology) by smooth Lipschitz (resp. uniformly continuous) mapping, if X is a separable Banach space admitting a smooth Lipschitz bump and either X or Y is a separable C(K) space (resp. super-reflexive space). Further, we show how smooth approximation of Lipschitz mappings is closely related to a smooth approximation of C¹-smooth mappings together with their first derivatives. As a corollary we obtain new results on smooth approximation of C¹-smooth mappings together with their first derivatives.     

Relative homology and the structure of relative approximations

The structure of rightF-^⊥ T-approximations of any finitely generated module over an artin algebra Λ is given, relative to an additive subbifunctorF of Ext _Λ ¹ (-, -) and anF-cotilting moduleT.     

Connected domination critical graphs with respect to relative complements

A dominating set in a graph G is a connected dominating set of G if it induces a connected subgraph of G. The minimum number of vertices in a connected dominating set of G is called the connected domination number of G, and is denoted by γ _c (G). Let G be a spanning subgraph of K _s,s and let H be the complement of G relative to K _s,s ; that is, K _s,s = G ⊕ H is a factorization of K _s,s . The graph G is k-γ _c -critical relative to K _s,s if γ _c (G) = k and γ _c (G + e) < k for each edge e ∈ E(H). First, we discuss some classes of graphs whether they are γ _c -critical relative to K _s,s . Then we study k-γ _c -critical graphs relative to K _s,s for small values of k. In particular, we characterize the 3-γ _c -critical and 4-γ _c -critical graphs.     

Cozero-accessible points

We define a point x to be cozero-accessible if for each dense open set U, there is a cozero-set CU such that . It is shown to be independent of Martin's Axiom that there are cozero-accessible points in .     

Differentiable mappings with an infinite number of critical points

In this paper we shall give some sufficient conditions in order that the so-called -category of a pair of differentiable manifolds be infinite.

     

On some properties of the functions from Sobolev-Morrey type spaces

In this paper the spaces of type Sobolev-Morrey-W _p,a,г,τ ^l (Q,G)-are constructed, the differential properties are studied and it is proved that the functions from these spaces satisfy Holder's condition, in the case, if the domain G∋R ⁿ satisfies the flexible λ-horn condition.     

Poisson approximations for 2-dimensional patterns

LetX=(X _ij) _n×n be a random matrix whose elements are independent Bernoulli random variables, taking the values 0 and 1 with probabilityq _ij andp _ij (p _ij+q _ij=1) respectively. Upper and lower bounds for the probabilities ofm non-overlapping occurrences of a square submatrix with all its elements being equal to 1, are obtained. Some Poisson convergence theorems are established forn . Numerical results indicate that the proposed bounds perform very well, even for moderate and small values ofn.This work is supported in part by the Natural Science and Engineering Research Council of Canada under Grant NSERC A-9216.     

Marino—Prodi perturbation type results and Morse indices of minimax critical points for a class of functionals in Banach spaces

In this work, we study a class of Euler functionals defined in Banach spaces, associated with quasilinear elliptic problems involving p-Laplace operator (p > 2). First we obtain perturbation results in the spirit of the remarkable paper by Marino and Prodi (Boll. U.M.I. (4) 11(Suppl. fasc. 3): 1–32, 1975), using the new definition of nondegeneracy given in (Ann. Inst. H. Poincaré: Analyse Non Linéaire. 2:271–292, 2003). We also extend Morse index estimates for minimax critical points, introduced by Lazer and Solimini (Nonlinear Anal. T.M.A. 12:761–775, 1988) in the Hilbert case, to our Banach setting. Mathematics Subject Classification (1991) 58E05, 35B20, 35J60, 35J70     

On quasistatic inelastic models of gradient type with convex composite constitutive equations

This article defines and presents the mathematical analysis of a new class of models from the theory of inelastic deformations of metals. This new class, containing so called convex composite models, enlarges the class containing monotone models of gradient type defined in [1]. This paper starts to establish the existence theory for models from this new class; we restrict our investigations to the coercive and linear self-controlling case.     

Invariance of the cone algebra without asymptotics

Let B be a manifold with conical singularities, and denote by the smooth bounded manifold with cylindrical ends obtained by blowing up near the singularities.B.-W. Schulze has developed a framework for a pseudodifferential calculus on B by defining various classes of distribution spaces and operator algebras, working in fixed coordinates on the manifold . I am showing here that the Mellin Sobolev spaces without asymptotics, the cone algebra without asymptotics, and its ideal of smoothing operators are independent of the choice of coordinates and therefore may be considered intrinsic objects for manifolds with conical singularities.     

Theory of center-focus for a class of higher-degree critical points and infinite points

For the real planar autonomous differential system, the questions of detection between center and focus, successor function, formal series, central integration, integration factor, focal values, values of singular point and bifurcation of limit cycles for a class of higher-degree critical points and infinite points are expounded.     

Zero-dimensional subschemes of ruled varieties

Here we study zero-dimensional subschemes of ruled varieties, mainly Hirzebruch surfaces and rational normal scrolls, by applying the Horace method and the Terracini method This research is part of the T.A.S.C.A. project of I.N.d.A.M., supported by P.A.T. (Trento) and M.I.U.R. (Italy)     

On quasi-solution to infeasible linear complementarity problem obtained by Lemke’s method

For a linear complementarity problem with inconsistent system of constraints a notion of quasi-solution of Tschebyshev type is introduced. It’s shown that this solution can be obtained automatically by Lemke’s method if the constraint matrix of the original problem is copositive plus or belongs to the intersection of matrix classes P ₀ and Q ₀.     

On the location of critical points of polynomials

Given a polynomial of degree and with at least two distinct roots let . For a fixed root we define the quantities and . We also define and to be the corresponding minima of and as runs over . Our main results show that the ratios and are bounded above and below by constants that only depend on the degree of . In particular, we prove that , for any polynomial of degree .