Impedance profile inversion via the first transport equation

The subject of this paper is the inverse reflection problem for a stratified elastic half-space. That is, a linear elastic medium, whose elastic properties depend only on depth from a planar free surface, is stimulated at t = 0 by a plane wave impulsive source. The motion of a typical surface element is recorded for 0 ? t ? 2T. It is shown that this surface trace determines the acoustic impedance of the medium as a function of travel time, to (travel-time) depth T. Moreover, we give a precise characterization of those functions which may appear as surface traces, and show uniqueness, existence, and continuous dependence of the logarithm of the impedance as a function of the surface trace in the Sobolev H¹ topology.     

Harmonic analysis associated with a discrete Laplacian

It is well known that the fundamental solution of

$${u_t}\left( {n,t} \right) = u\left( {n + 1,t} \right) - 2u\left( {n,t} \right) + u\left( {n - 1,t} \right),n \in \mathbb{Z},$$

with u(n, 0) = δ _nm for every fixed m ∈ Z is given by u(n, t) = e ^?2t I _n?m (2t), where I _k (t) is the Bessel function of imaginary argument. In other words, the heat semigroup of the discrete Laplacian is described by the formal series W _t f(n) = Σ _m∈Z e ^?2t I _n?m (2t)f(m). This formula allows us to analyze some operators associated with the discrete Laplacian using semigroup theory. In particular, we obtain the maximum principle for the discrete fractional Laplacian, weighted ? ^p (Z)-boundedness of conjugate harmonic functions, Riesz transforms and square functions of Littlewood-Paley. We also show that the Riesz transforms essentially coincide with the so-called discrete Hilbert transform defined by D. Hilbert at the beginning of the twentieth century. We also see that these Riesz transforms are limits of the conjugate harmonic functions. The results rely on a careful use of several properties of Bessel functions.

     

On the trace of finite sets

For a family $\text{T}$ of subsets of an n-set X we define the trace of it on a subset Y of X by T_$\text{T}$(Y) = {F∩Y:F?$\text{T}$}. We say that (m,n) → (r,s) if for every $\text{T}$ with |$\text{T| ?m}$ we can find a Y?X|Y| = s such that |T_$\text{T}$(Y)| ? r. We give a unified proof for results of Bollobàs, Bondy, and Sauer concerning this arrow function, and we prove a conjecture of Bondy and Lovász saying $\text{(?}\text{n}{}^2\text{4}\text{? + n + 2,n)→ (3,7)}$, which generalizes Turán's theorem on the maximum number of edges in a graph not containing a triangle.     

On the weighted enumeration of alternating sign matrices and descending plane partitions

We prove a conjecture of Mills, Robbins and Rumsey [Alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A 34 (3) (1983) 340-359] that, for any n, k, m and p, the number of n×n alternating sign matrices (ASMs) for which the 1 of the first row is in column k+1 and there are exactly m −1?s and m+p inversions is equal to the number of descending plane partitions (DPPs) for which each part is at most n and there are exactly k parts equal to n, m special parts and p nonspecial parts. The proof involves expressing the associated generating functions for ASMs and DPPs with fixed n as determinants of n×n matrices, and using elementary transformations to show that these determinants are equal. The determinants themselves are obtained by standard methods: for ASMs this involves using the Izergin-Korepin formula for the partition function of the six-vertex model with domain-wall boundary conditions, together with a bijection between ASMs and configurations of this model, and for DPPs it involves using the Lindström-Gessel-Viennot theorem, together with a bijection between DPPs and certain sets of nonintersecting lattice paths.     

On the Fourier transforms of retarded Lorentz-invariant functions

In this article we evaluate the Fourier transforms of retarded Lorentz-invariant functions (and distributions) as limits of Laplace transforms. Our method works generally for any retarded Lorentz-invariant functions φ(t) (t?Rⁿ) which is, besides, a continuous function of slow growth. We give, among others, the Fourier transform of G_R(t, α, m², n) and G_A(t, α, m², n), which, in the particular case α = 1, are the characteristic functions of the volume bounded by the forward and the backward sheets of the hyperboloid u = m² and by putting α = ?k are the derivatives of k-order of the retarded and the advanced-delta on the hyperboloid u = m². We also obtain the Fourier transform of the function W(t, α, m², n) introduced by M. Riesz (Comm. Sem. Mat. Univ. Lund4 (1939)). We finish by evaluating the Fourier transforms of the distributional functions G_R(t, α, m², n), G_A(t, α, m², n) and W(t, α, m², n) in their singular points.     

On the Selberg trace formula for strictly hyperbolic groups

We show that, for the case of strictly hyperbolic groups, the right-hand side of the Selberg trace formula admits a representation in the form of a series in the eigenvalues of the Laplacian. The behavior of the Minakshisundaram function as t → 0 and t → ∞ is studied. Countably many conditions satisfied by the spectrum of the Laplacian are obtained in explicit form.     

On the existence of Hadamard matrices

Given any natural number q > 3 we show there exists an integer t ? [2log₂(q ? 3)] such that an Hadamard matrix exists for every order 2^sq where s > t. The Hadamard conjecture is that s = 2.This means that for each q there is a finite number of orders 2^υq for which an Hadamard matrix is not known. This is the first time such a statement could be made for arbitrary q.In particular it is already known that an Hadamard matrix exists for each 2^sq where if q = 2^m ? 1 then s ? m, if q = 2^m + 3 (a prime power) then s ? m, if q = 2^m + 1 (a prime power) then s ? m + 1.It is also shown that all orthogonal designs of types (a, b, m ? a ? b) and (a, b), 0 ? a + b ? m, exist in orders m = 2^t and 2^t+2 · 3, t ? 1 a positive integer.     

On the cycle structure of certain classes of nonlinear shift registers

When m = qt, g(x_t+1, x_2t+1,…, x_(q?1)t+1) is a linear combination of only odd (or only even) elementary symmetric functions, then every cycle of the nonlinear shift register with feedback function f(x₁, x₂,…, x_m) = x₁ + g(x_t+1, x_2t+1,…, x_(q?1)t+1) has a minimal period dividing m(q+1). It is also shown that when g is derived from a cyclic code with minimum distance ?3, every cycle of this shift register has a minimal period dividing m(q + 1).     

On some analogs of Sturm's and Kneser's theorems for nonlinear systems

Analogs of certain conjugate point properties in the calculus of variations are developed for optimal control problems. The main result in this direction is concerned with the characterization of a parameterized family of extremals going through the first backward conjugate point, t_c. A corollary of this result is that for the linear quadratic problem (LQP) there exists at least a one-parameter family of extremals going though the conjugate point which gives the same cost as the candidate extremal, i.e., the extremal control is optimal but nonunique on [t_c, t_f]. An analysis of the effect on the conjugate point of employing penalty functions for terminal equality constraints in the LQP is presented, also. It is shown that the sequence of approximate conjugate points is always conservative, and it converges to the conjugate point of the constrained problem. Furthermore, it is proved that the addition of terminal constraints has the effect of causing the conjugate point to move backward (or remain the same).     

On the noncommutative residue and the heat trace expansion on conic manifolds

 Given a cone pseudodifferential operator P we give a full asymptotic expansion as t → 0 ⁺ of the trace Tr Pe ^-tA , where A is an elliptic cone differential operator for which the resolvent exists on a suitable region of the complex plane. Our expansion contains log t and new (log t)² terms whose coefficients are given explicitly by means of residue traces. Cone operators are contained in some natural algebras of pseudodifferential operators on which unique trace functionals can be defined. As a consequence of our explicit heat trace expansion, we recover all these trace functionals. Received: 12 November 2001 / Revised version: 26 June 2002 Mathematics Subject Classification (2000): Primary 58J35; Secondary 35C20, 58J42     

A symmetrical plane partition correspondence

MacMahon conjectured the form of the generating function for symmetrical plane partitions, and as a special case deduced the following theorem. The set of partitions of a number n whose part magnitude and number of parts are both no greater than m is equinumerous with the set of symmetrical plane partitions of 2n whose part magnitude does not exceed 2 and whose largest axis does not exceed m. This theorem, together with a companion theorem for the symmetrical plane partitions of odd numbers, are proved by establishing 1-1 correspondences between the sets of partitions.     

Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations

We investigate qualitative properties of local solutions u(t,x)?0 to the fast diffusion equation, t_∂u=Δ(um)/m with m<1, corresponding to general nonnegative initial data. Our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of the form [0,T]×Ω, with Ω⊆Rd. They combine into forms of new Harnack inequalities that are typical of fast diffusion equations. Such results are new for low m in the so-called very fast diffusion range, precisely for all m?mc=(d−2)/d. The boundedness statements are true even for m?0, while the positivity ones cannot be true in that range.     

Lower bounds on mt(r, s)

Useful lower bounds are obtained for m_t(r, s), s a prime power, r ? t ? 3, by relating (t ? 1)-flats of PG(r ? 1, s) to (t ? 1) and lower dimensional flats of its subgeometries. Explicit expressions in general are obtained for these bounds when t = 3. A general method is developed for deriving these bounds when t > 3.     

Construction of universal one-way hash functions: Tree hashing revisited

We present a binary tree based parallel algorithm for extending the domain of a universal one-way hash function (UOWHF). For t?2, our algorithm extends the domain from the set of all n-bit strings to the set of all ((2^t-1)(n-m)+m)-bit strings, where m is the length of the message digest. The associated increase in key length is 2m bits for t=2; m(t+1) bits for 3?t?6 and m×(t+⌊log₂(t-1)⌋) bits for t?7.     

Some general existence conditions for balanced arrays of strength t and 2 symbols

This paper gives necessary and sufficient conditions for the existence of a balanced array (B-array) (“partially balanced” array, in the terminology of Chakravarti [7]) of 2 symbols, strength t, and m rows (with m ? t + 2). Necessary existence conditions, for all m, are also derived in the form of many classes of diophantine equations.     

On the trace of symmetric stable processes on Lipschitz domains

It is shown that the second term in the asymptotic expansion as t→0 of the trace of the semigroup of symmetric stable processes (fractional powers of the Laplacian) of order α, for any 0<α<2, in Lipschitz domains is given by the surface area of the boundary of the domain. This brings the asymptotics for the trace of stable processes in domains of Euclidean space on par with those of Brownian motion (the Laplacian), as far as boundary smoothness is concerned.     

Nonzero decomposable symmetrized tensors

Suppose k₁ ? ? ? k_t ? 1, m₁ ? ?? m_r ? 1, k₁+ ? +k_t = m₁+ ? +m_r = m. Let λ=(k₁,…,k_t) be a character of the symmetric group S_m. The restriction of λ to S_m1X…XS_mr contains the principal character as a component if and only if λ majorizes (m₁,…,m_r). This result is used to characterize the index set of the nonzero decomposable symmetrized tensors, corresponding to S_m and λ, which are induced from a basis of the underlying vector space.     

Two-sided estimates for the trace of the difference of two semigroups

This paper deals with the derivation of two-sided estimates for the trace of the difference of two semigroups generated by two Schrödinger operators in L ₂(?³) with trace class difference of the resolvents. Use is made of a purely operator-theoretic technique. The results are stated in a rather general abstract form. The sharpness of our estimates is confirmed by the fact that they imply the asymptotic behavior of the trace of the difference of the semigroups as t → +0. Our considerations are substantially based on the Krein-Lifshits formula and on the Birman-Solomyak representation for the spectral shift function.     

Perturbation analysis for the trace quotient problem

Given a symmetric matrix B?∈?? ^m×m and a symmetric and positive-definite matrix W?∈?? ^m×m , maximizing the ratio trace(V ^? BV)/trace(V ^? WV) with respect to V?∈?? ^m×? (??≤?m) subject to the orthogonal constraint V ^? V?=?I _? is called the trace quotient problem or the trace ratio problem (TRP). TRP arises originally from the linear discriminant analysis (LDA), which is a popular approach for feature extraction and dimension reduction. It has been known that TRP is equivalent to a nonlinear extreme eigenvalue problem and very efficient method has been proposed to find a global optimal solution successfully. The matrices B and W arising in LDA are constructed from samples, and thereby are contaminated by noises and errors. In this article, we perform a perturbation analysis for TRP assuming the original B and W are perturbed. The upper perturbation bounds of both the global optimal value and the set of global optimal solutions are derived, and numerical investigation is carried out to illustrate these perturbation estimates.     

On a nonlinear Volterra integral equation in a Banach space

The equation u(t) + ∝₀^tk(t ? s)g(s) ds?f(t), t ? 0, is studied in a real Banach space with uniformly convex dual. Conditions, sufficient for the existence of a unique solution, are given for the operatorvalued kernel k, the nonlinear m-accretive operators g(t) and the function f. The case when k is realvalued, g(t) ≡ g and X a reflexive Banach space is also considered. These results extend earlier results by Barbu, Londen and MacCamy.