Diffraction in a cylindrically orthotropic elastic solid containing a stress free crack

A cylindrically orthotropic elastic solid is excited by a point impulsive body force. The solid contains a semi-infinite stress free crack. The resulting anti-plane wave motion problem has been solved in the form of a finite series representing the incident and reflected pulses plus an integral representing the diffraction pulse. The series part of the solution has been previously treated. In the present investigation the diffraction integral is integrated when λ (which measures the anisotropy of the solid) is an odd integer number. The diffraction integral is also integrated when λ is half an odd integer, for the special case in which the source lies in the plane of the crack and parallel to the crack edge. The displacement jump across the circular diffraction wave front is given for unrestricted (positive) values of λ.     

Fractional dimensions in semifields of odd order

A finite semifield D is considered a fractional dimensional semifield if it contains a subsemifield E such that λ := log_|E||D| is not an integer. We develop spread-theoretic tools to determine when finite planes admit coordinatization by fractional semifields, and to find such semifields when they exist. We use our results to show that such semifields exist for prime powers 3 ⁿ whenever n is an odd integer divisible by 5 or 7.     

A Shy Invariant of Graphs

 Moving from a well known result of Hammer, Hansen, and Simeone, we introduce a new graph invariant, say λ(G) referring to any graph G. It is a non-negative integer which is non-zero whenever G contains particular induced odd cycles or, equivalently, admits a particular minimum clique-partition. We show that λ(G) can be efficiently evaluated and that its determination allows one to reduce the hard problem of computing a minimum clique-cover of a graph to an identical problem of smaller size and special structure. Furthermore, one has α(G)≤θ(G)−λ(G), where α(G) and θ(G) respectively denote the cardinality of a maximum stable set of G and of a minimum clique-partition of G. Received: April 12, 1999 Final version received: September 15, 2000     

On ohtsuki’s invariants of integral homology 3-spheres

We provide some more explicit formulae to facilitate the computation of Ohtsuki’s rational invariants λ _n of integral homology 3-spheres extracted from Reshetikhin-TuraevSU(2) quantum invariants. Several interesting consequences will follow from our computation of λ₂. One of them says that λ₂ is always an integer divisible by 3. It seems interesting to compare this result with the fact shown by Murakami that λ₁ is 6 times the Casson invariant. Other consequences include some general criteria for distinguishing homology 3-spheres obtained from surgery on knots by using the Jones polynomial. The first author is supported in part by NSF and the second author is supported by an NSF Postdoctoral Fellowship.     

Estimating a density on the positive half line by the method of orthogonal series

The kernel method of density estimation is not so attractive when the density has its support confined to (0, ∞), particularly when the density is unsmooth at the origin. In this situation the method of orthogonal series is competitive. We consider three essentially different orthogonal series—those based on the even and odd Hermite functions, respectively, and that based on Laguerre functions—and compare them from the point of view of mean integrated square error.     

On diffraction in a piezoelectric medium by a half-plane: The Sommerfeld problem

This paper is concerned with the diffraction problem in a transversely isotropic piezoelectric medium by a half-plane. The half-plane obstacle considered here is a semi-infinite slit, or a crack; both its surfaces are traction free and electric absorbent screens. In a generalized sense, we are dealing with the Sommerfeld problem in a piezoelectric medium.¶The coupled diffraction fields between acoustic wave and electric wave are excited by both incident acoustic wave as well as incident electric wave; and the sound soft and electric "blackness" conditions on the screens are characterized by a system of simultaneous Wiener-Hopf equations. Closed form solutions are sought by employing special techniques. Some interesting results have been obtained, such as mode conversions between acoustic wave and electric wave, novel diffraction patterns in the scattering fields, and the effect of electroacoustic head wave, as well as of surface wave-Bleustein-Gulyaev wave.¶Unlike the classical Sommerfeld problem, in which the only concern is the scattering field of electric wave, the strength of material, e.g. material toughness, is another concern here. From this perspective, relevant dynamic field intensity factors at the crack tip are derived explicitly.     

Some remarks about the limit point and limit circle theory

LetL be a formally selfadjoint differential operator andp a real-valued function, both ona≤x<∞. The deficiency indices are the numbers of solutions ofLu=λpu for Im λ>0 and for Im λ<0 which have a certain regularity atx=∞. (A) Ifp(x)≥0 this regularity means that the integral ofp(x)│u│² converges at infinity. (B) Ifp changes its sign for arbitrarily large values ofx butL has a positive definite Dirichlet integral it is natural to relate the regularity to this integral. Weyl’s classical study of the deficiency indices is reviewed for (A) with the help of elementary theory of quadratic forms. Individual bounds are found for the deficiency indices also whenL is of odd order. It is then indicated how the method carriers over to (B).     

Nonlinear localized waves in a medium with nonlocal interaction

Soliton solutions are found for nonlinear integro-differential equations with a type λ/(τ-τ′) kernel used to describe particle tunneling and magnetic and superconducting vortices in a medium with nonlocal interaction. The Fourier transform method is applied to derive asymptotic formulas for even and odd localized solutions. Analytical solutions are found for particular parameter values. A complete pattern is constructed for the behavior of soliton solutions in an arbitrary range of the interaction parameter λ by means of numerical calculations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 114, No. 3, pp. 366–379, March, 1998.     

Optimization problems with algebraic solutions: Quadratic fractional programs and ratio games

A mathematical program with a rational objective function may have irrational algebraic solutions even when the data are integral. We suggest that for such problems the optimal solution will be represented as follows: If λ* denotes the optimal value there will be given an intervalI and a polynomialP(λ) such thatI contains λ* and λ* is the unique root ofP(λ) inI. It is shown that with this representation the solutions to convex quadratic fractional programs and ratio games can be obtained in polynomial time.     

On the Lüroth semigroup of curves lying on a smooth quadric

For a smooth irreducible complete algebraic curveC the “gaps” are the integersn such that every linear series of degreen has at least a base point. The Lüroth semigroup S_C of a curveC is the subsemigroup ofN whose elements are not gaps. In this paper we deal with irreducible smooth curves of type (a, b) on a smooth quadricQ. The main result is an algorithm by which we can say if some integer λ∈N is a gap or is in S_C. In the general case there are integers λ which are undecidable. For curves such as complete intersection, arithmetically Cohen-Macaulay or Buchsbaum, we are able to describe explicitly “intervals” of gaps and “intervals” of integers which belong to S_C. For particular cases we can completely determine S_C, by giving just the type of the curve (in particular the degree and the genus). Work done with financial support of M.U.R.S.T. while the authors were members of G.N.S.A.G.A. of C.N.R.     

A new explicit solution method for the diffraction through a slit – Part 2

In the paper [N. Gorenflo, A new explicit solution method for the diffraction through a slit, ZAMP 53 (2002), 877–886] the problem of diffraction through a slit in a screen has been considered for arbitrary Dirichlet data, prescribed in the slit, and under the assumption that the normal derivative of the diffracted wave vanishes on the screen itself. For this problem certain functions with the following properties have been constructed: Each function f is defined on the whole of R and on the screen the values f(x), |x|  ≥  1, are the Dirichlet data of the diffracted wave which takes on the Dirichlet data f(x), |x|  ≤  1, in the slit. The problem of expanding arbitrary Dirichlet data, prescribed in the slit, into a series of functions of the considered form has been addressed, but not solved in a satisfactory way (only the application of the Gram-Schmidt orthogonalization process to such functions has been proposed). In this continuation of the aforementioned paper we choose the remaining degrees of freedom in the earlier given representations of such functions in a certain way. The resulting concrete functions can be expressed by Hankel functions and explicitly given coefficients. We suggest the expansion of arbitrary Dirichlet data, prescribed in the slit, into a series of these functions, here the expansion coefficients can be expressed explicitly by certain moments of the expanded data. Using this expansion, the diffracted wave can be expressed in an explicit form. In the future it should be examined whether similar techniques as those which are presented in the present paper can be used to solve other canonical diffraction problems, inclusively vectorial diffraction problems.     

Sistemi ellittici di tipo A.D.N. a coefficienti variabili e condizioni di radiazione

Summary I study an elliptic system, in the sense of Agmon-Douglis-Nirenberg, of partial differential equations with variable coefficients. The matrix operator is of type P(D) + + λR(x, D) where λ εC, P(D) has constant coefficients, is elliptic, and his determinant admits a special elementary solution. On the coefficients in R(x, D), sufficiently smooth, a certain behaviour at the infinity is assumed. For suitable known vectors f, the problem P(D)u - λR(x, D)u=f is shown to be equivalent to a system of singular integral equations in special subspaces of [W^l,p]^N, if N is the rank of the system, as is studied in [5]. This is possible when the unknown vector u belongs to a class that, generally, is stricter than the one of existence and uniquencess for P(D) [4]. Then results on the solvability of the system follow when λ is such that P(D+λR(x, D) is elliptic.

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Entrata in Redazione l'8 giugno 1977.

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Lavoro svolto nell'ambito del ? Laboratorio per la matematica applicata ? del C.N.R. presso l'Università di Genova.     

Integral trees of odd diameters

A graph is called integral if all eigenvalues of its adjacency matrix consist entirely of integers. Recently, Csikvári proved the existence of integral trees of any even diameter. In the odd case, integral trees have been constructed with diameter at most 7. In this article, we show that for every odd integer n>1, there are infinitely many integral trees of diameter n. © 2011 Wiley Periodicals, Inc. J Graph Theory     

Predicting eddy detachment for an equivalent barotropic thin jet

Summary The time-dependent meandering of thin ocean jets in reduced gravity models has recently been shown to obey a natural coordinate version of the standard modified Korteweg-deVries (mKdV) equation. The detachment of eddies from such a jet begins when different segments of the jet path come into contact, causing the initially simply connected jet to “pinch” together. It is shown that this pinching process is effected primarily by breather solutions to the mKdV equation. For a given initial condition the solution will evolve into a dispersive wave train plus a finite number of breathers, the connectivity of which is determined by a steepness parameter λ. Using the scattering transform for the mKdV equation the value(s) of λ can be calculated in a straightforward manner, and the detachment (or lack thereof) of meanders can be forecast to a high degree of confidence by calculating λ. Examples with simple meander disturbances show a remarkable degree of stability and resistance to detachment.     

On acoustic and electric waves diffraction in piezoelectric medium by a permeable half-plane crack

The problem of electric and acoustic waves diffraction by a half-plane crack in a transversal isotropic piezoelectric medium is investigated. The crack is assumed to be electric permeable and free of tractions. The so-called “quasi-hyperbolic approximation” [15] is adopted. Applying Laplace transformations and Wiener–Hopf technique a closed form solution is obtained. By the means of Cagniard–de Hoop method a detailed dynamic full electroacoustic wavefield’s investigation is conducted. Mode conversion between electric and acoustic waves, effect of electroacoustic head wave, Bleustein–Gulyaev surface wave and the wavefield structure depending on the type of the incident wave (acoustic or electric) and its angle of incidence are analyzed in details. The dynamic field intensity factors at the crack tip depending on the angle of incidence and on time are derived explicitly. Numerical analysis is presented.     

On acoustic and electric waves diffraction in piezoelectric medium by a permeable half-plane crack

The problem of electric and acoustic waves diffraction by a half-plane crack in a transversal isotropic piezoelectric medium is investigated. The crack is assumed to be electric permeable and free of tractions. The so-called “quasi-hyperbolic approximation” [15] is adopted. Applying Laplace transformations and Wiener–Hopf technique a closed form solution is obtained. By the means of Cagniard–de Hoop method a detailed dynamic full electroacoustic wavefield’s investigation is conducted. Mode conversion between electric and acoustic waves, effect of electroacoustic head wave, Bleustein–Gulyaev surface wave and the wavefield structure depending on the type of the incident wave (acoustic or electric) and its angle of incidence are analyzed in details. The dynamic field intensity factors at the crack tip depending on the angle of incidence and on time are derived explicitly. Numerical analysis is presented.     

A lower bound for the r -order of a matrix modulo N

For a positive integer N, we define the N-rank of a non singular integer d × d matrix A to be the maximum integer r such that there exists a minor of order r whose determinant is not divisible by N. Given a positive integer r, we study the growth of the minimum integer k, such that A ^k − I has N-rank at most r, as a function of N. We show that this integer k goes to infinity faster than log N if and only if for every eigenvalue λ which is not a root of unity, the sum of the dimensions of the eigenspaces relative to eigenvalues which are multiplicatively dependent with λ and are not roots of unity, plus the dimensions of the eigenspaces relative to eigenvalues which are roots of unity, does not exceed d − r − 1. This result will be applied to recover a recent theorem of Luca and Shparlinski [6] which states that the group of rational points of an ordinary elliptic curve E over a finite field with q ⁿ elements is almost cyclic, in a sense to be defined, when n goes to infinity. We will also extend this result to the product of two elliptic curves over a finite field and show that the orders of the groups of \Bbb F_qⁿ-{\Bbb F}_{q^n}- rational points of two non isogenous elliptic curves are almost coprime when n approaches infinity.     

Matching-perfect and cover-perfect graphs

It is shown that a graphG has all matchings of equal size if and only if for every matching setλ inG, G\V(λ) does not contain a maximal open path of odd length greater than one, which is not contained in a cycle. (V(λ) denotes the set of vertices incident with some edge ofλ.) Subsequently edge-coverings of graphs are discussed. A characterization is supplied for graphs all whose minimal covers have equal size.     

Families of Spectral Sets for Bernoulli Convolutions

We study the harmonic analysis of Bernoulli measures μ _λ , a one-parameter family of compactly supported Borel probability measures on the real line. The parameter λ is a fixed number in the open interval (0,1). The measures μ _λ may be understood in any one of the following three equivalent ways: as infinite convolution measures of a two-point probability distribution; as the distribution of a random power series; or as an iterated function system (IFS) equilibrium measure determined by the two transformations λ(x±1). For a given λ, we consider the harmonic analysis in the sense of Fourier series in the Hilbert space L ²(μ _λ ). For L ²(μ _λ ) to have infinite families of orthogonal complex exponential functions e ^2πis(⋅), it is known that λ must be a rational number of the form \fracm2n\frac{m}{2n}, where m is odd. We show that L²(m_\frac12n)L^{2}(\mu_{\frac{1}{2n}}) has a variety of Fourier bases; i.e. orthonormal bases of exponential functions. For some other rational values of λ, we exhibit maximal Fourier families that are not orthonormal bases.     

A lower bound for the <Emphasis Type="Italic">r</Emphasis>-order of a matrix modulo <Emphasis Type="Italic">N</Emphasis>

For a positive integer N, we define the N-rank of a non singular integer d × d matrix A to be the maximum integer r such that there exists a minor of order r whose determinant is not divisible by N. Given a positive integer r, we study the growth of the minimum integer k, such that A ^k − I has N-rank at most r, as a function of N. We show that this integer k goes to infinity faster than log N if and only if for every eigenvalue λ which is not a root of unity, the sum of the dimensions of the eigenspaces relative to eigenvalues which are multiplicatively dependent with λ and are not roots of unity, plus the dimensions of the eigenspaces relative to eigenvalues which are roots of unity, does not exceed d − r − 1. This result will be applied to recover a recent theorem of Luca and Shparlinski [6] which states that the group of rational points of an ordinary elliptic curve E over a finite field with q ⁿ elements is almost cyclic, in a sense to be defined, when n goes to infinity. We will also extend this result to the product of two elliptic curves over a finite field and show that the orders of the groups of rational points of two non isogenous elliptic curves are almost coprime when n approaches infinity. Author’s address: Dipartimento di Matematica e Informatica, Via Delle Scienze 206, 33100 Udine, Italy