A deformation lemma on aC 1 manifold

The classical construction of deformations by mean of pseudo-gradient vector fields requires theC ^1,1 regularity. Here, we are concerned with a deformation lemma for aC ¹ function on a manifold defined by aC ¹ functional. We will assume some coupled Palais-Smale conditions between the two functions. The deformation is constructed with the help of integral lines of pseudo-gradient vector fields on a foliation of the manifold. Three different constructions are used for a sub-manifold of codimension 1 in finite dimension, then in infinite dimension and lastly a sub-manifold of any finite codimension in an infinite dimensional Banach space.     

Spectral properties of metastable Markov semigroups

It is shown that if a symmetric Markov semigroup e^?Ht on the Hilbert space L²(X) is hypercontractive, then the approximate degeneracy of the ground state has several consequences concerning other parts of the spectrum of H and concerning the unitary group e^?iHt. In particular, in the presence of a space inversion symmetry, all the eigenvalues occur in pairs with gaps comparable to the gap between the bottom two eigenvalues.     

The eigenvalues of Kac's master equation

In this paper we derive formulae for the eigenvalues and spectral gap of the master equation for general collision kernels. We prove a conjecture of Mark Kac's on the existence of a spectral gap independent of the number of particles. We relate the eigenvalues to the “nonlinear” eigenvalues that occur in the exact solutions of model Boltzmann equations due to M. Ernst. Received: 30 November 2001; in final form: 26 March 2002/Published online: 2 December 2002     

Eigenvalues of kinematical conservation laws (KCL) based 3-D weakly nonlinear ray theory (WNLRT)

System of kinematical conservation laws (KCL) govern evolution of a curve in a plane or a surface in space, even if the curve or the surface has singularities on it. In our recent publication [K.R. Arun, P. Prasad, 3-D kinematical conservation laws (KCL): evolution of a surface in R³-in particular propagation of a nonlinear wavefront, Wave Motion 46 (2009) 293-311] we have developed a mathematical theory to study the successive positions and geometry of a 3-D weakly nonlinear wavefront by adding an energy transport equation to KCL. The 7 × 7 system of equations of this KCL based 3-D weakly nonlinear ray theory (WNLRT) is quite complex and explicit expressions for its two nonzero eigenvalues could not be obtained before. In this short note, we use two different methods: (i) the equivalence of KCL and ray equations and (ii) the transformation of surface coordinates, to derive the same exact expressions for these eigenvalues. The explicit expressions for nonzero eigenvalues are important also for checking stability of any numerical scheme to solve 3-D WNLRT.     

Spectral Gap for Stable Process on Convex Planar Double Symmetric Domains

We study the semigroup of the symmetric α-stable process in bounded domains in R ². We obtain a variational formula for the spectral gap, i.e. the difference between two first eigenvalues of the generator of this semigroup. This variational formula allows us to obtain lower bound estimates of the spectral gap for convex planar domains which are symmetric with respect to both coordinate axes. For rectangles, using “midconcavity” of the first eigenfunction (Bañuelos et al., Potential Anal. 24(3): 205–221, 2006), we obtain sharp upper and lower bound estimates of the spectral gap.     

On the conformal, concircular, and spin mappings of gravitational fields

A mapping ρ: of two Riemannian or pseudo-Riemannian spaces is called aspin mapping if for each geodesic curve γ in M_n its image ρ^oγ is a spin-curve in the space . In gravitational fields spin-curves describe the trajectories of uniformly accelerated particles of constant mass with simultaneous self-rotation. We prove: 1) a conformal mapping is a spin mapping only when it is concircular; 2) every conformal mapping of Einstein space is a spin mapping. The latter makes it possible to give a local representation of the metrics of all gravitational fields that admit spin mappings. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 44–47. Original article submitted March 17, 1993.     

Monotonicity inequalities for ther-area and a degeneracy theorem forr-minimal graphs

We establish monotonicity inequalities for the r-area of a complete oriented properly immersed r-minimal hypersurface in Euclidean space under appropriate quasi-positivity assumptions on certain invariants of the immersion. The proofs are based on the corresponding first variational formula. As an application, we derive a degeneracy theorem for an entire r-minimal graph whose defining function ƒ has first and second derivatives decaying fast enough at infinity: Its Hessian operator D² ƒ has at least n − r null eigenvalues everywhere.     

Lifts, Discrepancy and Nearly Optimal Spectral Gap*

We present a new explicit construction for expander graphs with nearly optimal spectral gap. The construction is based on a series of 2-lift operations. Let G be a graph on n vertices. A 2-lift of G is a graph H on 2n vertices, with a covering map π :H →G. It is not hard to see that all eigenvalues of G are also eigenvalues of H. In addition, H has n “new” eigenvalues. We conjecture that every d-regular graph has a 2-lift such that all new eigenvalues are in the range (if true, this is tight, e.g. by the Alon–Boppana bound). Here we show that every graph of maximal degree d has a 2-lift such that all “new” eigenvalues are in the range for some constant c. This leads to a deterministic polynomial time algorithm for constructing arbitrarily large d-regular graphs, with second eigenvalue . The proof uses the following lemma (Lemma 3.3): Let A be a real symmetric matrix with zeros on the diagonal. Let d be such that the l₁ norm of each row in A is at most d. Suppose that for every x,y ∈{0,1}ⁿ with ‹x,y›=0. Then the spectral radius of A is O(α(log(d/α)+1)). An interesting consequence of this lemma is a converse to the Expander Mixing Lemma. * This research is supported by the Israeli Ministry of Science and the Israel Science Foundation.     

Asymptotic behavior of eigenvalues and eigenfunctions of the Fourier problem in a thick multilevel junction

A spectral boundary-value problem is considered in a plane thick two-level junction Ω_ε formed as the union of a domain Ω₀ and a large number 2N of thin rods with thickness of order ε = O(N ⁻¹). The thin rods are split into two levels depending on their length. In addition, the thin rods from the indicated levels are ε-periodically alternating. The Fourier conditions are given on the lateral boundaries of the thin rods. The asymptotic behavior of the eigenvalues and eigenfunctions is investigated as ε → 0, i.e., when the number of thin rods infinitely increases and their thickness approaches zero. The Hausdorff convergence of the spectrum is proved as ε → 0, the leading terms of asymptotics are constructed, and the corresponding asymptotic estimates are justified for the eigenvalues and eigenfunctions. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 195–216, February, 2006.     

Approximation and asymptotics of eigenvalues of unbounded self-adjoint Jacobi matrices acting in <Emphasis Type="Italic">l</Emphasis><Superscript>2</Superscript> by the use of finite submatrices

We consider the problem of approximation of eigenvalues of a self-adjoint operator J defined by a Jacobi matrix in the Hilbert space l ²(ℕ) by eigenvalues of principal finite submatrices of an infinite Jacobi matrix that defines this operator. We assume the operator J is bounded from below with compact resolvent. In our research we estimate the asymptotics (with n → ∞) of the joint error of approximation for the eigenvalues, numbered from 1 to N; of J by the eigenvalues of the finite submatrix J _n of order n × n; where N = max{k ∈ ℕ: k ≤ rn} and r ∈ (0; 1) is arbitrary chosen. We apply this result to obtain an asymptotics for the eigenvalues of J. The method applied in this research is based on Volkmer’s results included in [23].     

On systems of two singularly perturbed quasilinear second-order equations

A system of two quasilinear second-order equations with a small parameter next to the second derivatives is studied. The cases where the matrix of coefficients next to the first derivatives has the following eigenvalues are considered: (a) both of them have negative real parts; (b) they are of opposite sign; (c) one of them is equal to zero. To find the solution and its asymptotics, the initial-value or boundary-value problems are posed depending on the form of these eigenvalues. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 5, pp. 21–28, 2006.     

Schrödinger Operators on Zigzag Nanotubes

We consider the Schr?dinger operator with a periodic potential on quasi-1D models of zigzag single-wall carbon nanotubes. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite number of eigenvalues with infinite multiplicity. We describe all compactly supported eigenfunctions with the same eigenvalue. We define a Lyapunov function, which is analytic on some Riemann surface. On each sheet, the Lyapunov function has the same properties as in the scalar case, but it has branch points, which we call resonances. We prove that all resonances are real. We determine the asymptotics of the periodic and antiperiodic spectrum and of the resonances at high energy. We show that there exist two types of gaps: i) stable gaps, where the endpoints are periodic and anti-periodic eigenvalues, ii) unstable (resonance) gaps, where the endpoints are resonances (i.e., real branch points of the Lyapunov function). We describe all finite gap potentials. We show that the mapping: potential all eigenvalues is a real analytic isomorphism for some class of potentials. Submitted: October 5, 2006. Accepted: December 15, 2006.     

Estimates of operator polynomials in the space Lp with respect to the multiplier norm

The paper is devoted to the determination of an analog of J, von Neumann's inequality for the space L^p.The fundamental result of the paper is: If T is an absolute contraction in the space L^p(X,F,μ) (i.e., |T|_L1≤1) and |T|_L∞≤1) then for every polynomial ϕ one has where S is the shift operator in the space. On the basis of this theorem, one finds a theorem on substitutions in the space of multipliers. One gives applications of the inequality (1) to the weighted shift operators in the space It turns out that under some natural restrictions on the weight, inequality (1) becomes an equality for such operators. One also presents a proof of J. von Neumann's inequality based on the approximation of a contraction in a Hilbert space by unitary operators. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 65, pp. 133–148, 1976. The author wishes to express his thanks to N. K. Nikol'skii for suggesting the problem and for his interest in the paper.     

Non-special divisors on real algebraic curves and embeddings into real projective spaces

We show that there is a large class of non-special divisors of relatively small degree on a given real algebraic curve. If the real algebraic curve has many real components, such a divisor gives rise to an embedding (birational embedding, resp.) of the real algebraic curve into the real projective space ℙ ^r for r≥3 (r=2, resp.). We study these embeddings in quite some detail. Received: October 17, 2001?Published online: February 20, 2003     

Perturbation Theory for the Two-Particle Schrodinger Operator on a One-Dimensional Lattice

We consider the two-particle Schrodinger operator H(k) on the one-dimensional lattice ℤ. The operator H(π) has infinitely many eigenvalues z_m(π) = v(m), m ∈ ℤ₊. If the potential v increases on ℤ₊, then only the eigenvalue z₀(π) is simple, and all the other eigenvalues are of multiplicity two. We prove that for each of the doubly degenerate eigenvalues z_m(π), m ∈ ℕ, the operator H(π) splits into two nondegenerate eigenvalues z _m ⁻ (k) and z _m ⁺ (k) under small variations of k ∈ (π − δ, π). We show that z _m ⁻ (k) < z _m ⁺ (k) and obtain an estimate for z _m ⁺ (k) − z _m ⁻ (k) for k ∈ (π − δ, π). The eigenvalues z₀(k) and z ₁ ⁻ (k) increase on [π − δ, π]. If (Δv)(m) > 0, then z _m ^± (k) for m ≥ 2 also has this property. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 2, pp. 212–220, November, 2005.     

Isoperimetric inequalities and the gap between the first and second eigenvalues of an Euclidean domain

In this paper we introduce a weighted Cheeger constant and show that the gap between the first two eigenvalues of a Riemannian manifold given Dirichlet conditions can be bounded from below in terms of this constant. When the Riemannian manifold is a bounded Euclidean domain satisfying an interior rolling sphere condition we give an estimate on the weighted Cheeger constant in terms of the rolling sphere radius, volume, a bound on the principal curvatures of the boundary and the dimension. This yields a lower bound on the nontrivial gap for Euclidean domains. S-Y. Cheng’s research partially supported by the CUHK direct grant A/C # 220600260. K. Oden’s research partially supported by the Department of Education Graduate Fellowship     

Some topological properties of vector measures with bounded variation and its applications

Summary In this paper, we study the duality between ℜ(T, E′) and , where ℜ(T, E′) is the space of vector measures with bounded variation, defined on a completely regular Suslin space T, with values in the weak dual space E′ of a separable Fréchet space E; is the space of continuous bounded mappings defined on T, with values in a finite dimensional subspace of E. We show that ℜ(T, E′) is a Suslin space when equipped with the weak topology . Some applications are given: a weak closure's theorem for integrable vector mappings (a generalization of a recent result of Olech), a theorem of integral representation, a theorem of existence of conditional expectation, and finally a theorem of density related to the optimal control theory. A new application of these results is the study of comparison of measures on the unit sphere Sⁿ ofR ⁿ, which will be published elsewhere. Entrata in Redazione il 20 ottobre 1976.     

The inverse eigenvalue problem for a weighted helmholtz equation

Given the m lowest eigenvalues, we seek to recover an approximation to the density function ρ in the weighted Helmholtz equation -Δ=λρu on a rectangle with Dirchlet boundary conditions. The density ρ is assumed to be symmetric with respect to the midlines of the rectangle. Projection of the boundary value problem and the unknown density function onto appropriate vector spaces leads to a matrix inverse problem. Solutions of the matrix inverse problem exist provided that the reciprocals of the prescribed eigenvalues are close to the reciprocals of the simple eigenvalues of the base problem with ρ = 1. The matrix inverse problem is solved by a fixed—point iterative method and a density function ρ^* is constructed which has the same m lowest eigenvalues as the unknown ρ. The algorithm can be modified when multiple base eigenvalues arise, although the success of the modification depends on the symmetry properties of the base eigenfunctions.     

The Spectrum of Annihilation Plus Orthogonal Creation

Let ℌ be a Hilbert space and F(ℌ) be the full Fock space generated by ℌ. For v ∈ ℌ, the (left) creation operator l(v) : F(ℌ) → F(ℌ),f↦ v ⊗ f, and its adjoint, the (left) annihilation operator l(v)^*, are defined. If v, w ∈ℌ are orthonormal, it is proved that the spectrum of the operator l(v) + l(w)^* is purely continuous and conincides with the closed unit disk. This revised version was published online in July 2006 with corrections to the Cover Date.     

Geometry of Local Adaptive Galerkin Bases

The local adaptive Galerkin bases for large-dimensional dynamical systems, whose long-time behavior is confined to a finite-dimensional manifold, are optimal bases chosen by a local version of a singular decomposition analysis. These bases are picked out by choosing directions of maximum bending of the manifold restricted to a ball of radius ɛ . We show their geometrical meaning by analyzing the eigenvalues of a certain self-adjoint operator. The eigenvalues scale according to the information they carry, the ones that scale as ɛ ² have a common factor that depends only on the dimension of the manifold, the ones that scale as ɛ ⁴ give the different curvatures of the manifold, the ones that scale as ɛ ⁶ give the third invariants, as the torsion for curves, and so on. In this way we obtain a decomposition of phase space into orthogonal spaces E _m , where E _m is spanned by the eigenvectors whose corresponding eigenvalues scale as ɛ ^m . This decomposition is analogous to the Frenet frames for curves. We also discover a practical way to compute the dimension and local structure of the invariant manifold. Accepted 14 October 1998