Showing papers in "Rendiconti Del Circolo Matematico Di Palermo in 1972"

Shock waves in a mixture of linear elastic materials

Abstract: This work concerns the behavior of shock waves propagating in a mixture of an isotropic elastic solid and an elastic fluid. The displacement of each constituent are presumed to be small and the resulting field equations are linear. Contrary to the usual behavior of shocks predicted by linear theories, we find that the shock amplitudes are influenced by strain gradients behind the waves. Also, we find that the difference of the amplitude of the shock associated with the solid and that associated with the fluid are damped due to the presence of diffusion, but a certain weighted sum of the amplitudes is unaffected.

On the growth and decay of wave fronts in a mixture of linear elastic materials

Abstract: This work concerns the propagation of wave fronts in a mixture of fluids and isotropic solids. Only mechanical effects are included. In the statement of the field equations the importance of buoyancy forces is stressed. These forces arise from a local interaction among the constituents of the mixture and are present even in the absence of diffusion. It is shown that there are as many longitudinal waves possible as there are constituents and that there are as many transverse waves possible as there are solid constitutents. Rules for the evolution of discontinuities on wave fronts are deduced, and it is shown that diffusion causes a decay in the wave strength in addition to the usual geometric decay familiar from classical elasticity.

Invariants associated with the epsilon algorithm and its first confluent form

Abstract: It is shown that if numbers e r (m) (r=0, 1, ..., 2n;m=0, 1, ...) can be constructed from the initial values e r (m) =0 (m=1, 2, ...), e r (m) =S m (m=0, 1, ...) by systematic use of the relationships e r+1 (m) =e r-1 (m+1) +(e r (m+D) -e r (m) )–1 (r=0, 1, ..., 2n-1;m=0, 1, …) and an irreducible relationship of the form $$\sum\limits_{v = o}^n {C_v S_{m + v} = G} $$ . (m=0, 1, ...) withG=0 holds among the members of the sequence {S m}, then $$\sum\limits_{r = o}^{2n - 2} {( - 1)^r \varepsilon _r^{(m)} \varepsilon _{r + 1}^{(m)} = - \frac{{\left\{ {\sum\limits_{v = 1}^n {vCv} } \right\}}}{{\left\{ {\sum\limits_{v = 0}^n {Cv} } \right\}}}} $$ (m=0, 1, ...) Furthermore, if functions er(μ) (r=0, 1, ..., 2n) are constructed from the initial values e–1(μ)=0, e0(μ)=S(μ) by systematic use of the relationships $$\varepsilon _{r + 1} (\mu ) = \varepsilon _{r - 1} (\mu ) + \left\{ {\frac{{d\varepsilon _r (\mu )}}{{d\mu }}} \right\}^{ - 1} $$ (r=0, 1, ..., 2n–1) and the functionsS (μ) satisfies an irreducible equation of the form $$\sum\limits_{v = 0}^n {c_v \frac{{d^v S(\mu )}}{{d\mu ^v }} = G} $$ withG=0, then $$\sum\limits_{r = 0}^{2n - 2} {( - 1)^r \varepsilon _r (\mu )\varepsilon _{r + 1} (\mu ) = - \frac{{c_1 }}{{c_0 }}} $$ identically.

An Introduction to Nachbin Spaces.

Abstract: This paper studies from the viewpoint of vector fibrations and vector spaces of cross-sections, endowed with appropriate topologies, the realm of ideas and results centered around the Weierstrass-Stone Theorem for algebras and modules of continuous functions in the real and self-adjoint complex cases. The proof of the equivalence between the separating and the general cases of such results by means of a quotient construction motivates the use of vector fibrations.

Sulla differenziabilità delle funzioni

Abstract: On caracterise les fonctions, de plusieurs variables partialement derivables p.p. par des conditions locales de type variationel.

Harmonic spaces with positive potentials and nonconstant harmonic functions

Abstract: On a Harmonic Space Satisfying the axioms 1, 2, 3 of M. Brelot and having positive potentials, the existence of nonconstant harmonic functions is studied, following the lines of the classification of a hyperbolic Riemann Surface.

Sulla sommabilità delle derivate di una funzione arbitraria d’insieme

Abstract: LetF be a real-valued function defined, on a familyF of measurable subsets ofRn. In an earlier paper [1] we have considered the question of almost everywhere existence of the derivativesFf′, For′, Fa′ Here we investigate the problem of summability ofFf′, For′ Fa′. Main results are theorems 5.3, 8.3, 9.3 and corollaries 6.3, 7.3, 4.4.

Sulla derivabilità delle funzioni arbitrarie d’insieme

Abstract: LetF be a real-valued function defined on a familyF of measurable subsets ofR n. The purpose of this paper is to give necessary and sufficient conditions for the existence almost everywhere of aF-strong, or aF-ordinary, or aF-regular derivative ofF. Main results are theorems 7.2, 7.3, 8.3 and corollaries 8.2, 9.3.

On the existence of ψ-integrals

Abstract: The approximating sums for the Riemann-Stieltjes integral are formed with the integrand function evaluated at an arbitrary point in each subdivision interval. The ψ-integral uses the idea of achoice function to select a particular point in each of these intervals.

On periodic entire functions

Abstract: In this paper criteria for deciding on the periodicity of entire functions which have certain forms or satisfy certain functional equations have been provided.

Stronger forms of connectivity

Abstract: This paper is a comparative study of stronger forms of connectivity introduced by N. Levine and Steen and Seebach. Many of the standard theorems on connectivity are also valid for these stronger forms of connectivity, with some notable exceptions which are pointed out.

Matrix representations of polynomial operators

Abstract: LetH be a separable Hilbert space. Every bounded,n-linear operatorL onH n toH(n=0,1,2,…) is shown to have a unique matrix representation with respect to each complete orthonormal sequence {ϕk} 1 ∞ . Conversely, every operator onH n toH possessing a matrix representation is proved to be a bounded,n-linear operator. The foregoing conclusions then apply to polynomial operatorsP onH toH wherePx=L 0+L1x+L2x2+…+Lnxn and eachL k is ak-linear operator.

On the classes of differential systems with the desired behavior

Abstract: A discussion of the affect of perturbations on systems of differential equations is presented. In particular, given a property of the solutions of a differential system, we are interested in finding a class of differential equations related to the original system that have a certain prescribed property. Lyapunov theory is the main technique employed.

Concerning and certain uniform structure

Abstract: A regular structure for a topological space is defined. Also, a relation between a regular structure and an Abian’s structure is investigated.

On certain expansions of generalized hypergeometric functions involving free parameters

Abstract: A large number of new expansions of hypergeometric functions in terms of hypergeometric functions have been established by numerous mathematicians from time to time, generalizing the previous ones. Here an attempt has been made to establish certain new expansions, with the help of Laplace transform and its inverse which extend almost all such expansions besides being the most general ones.

Strongly bounded differential systems

Abstract: SummarySee Introduction.

A singular boundary problem for -u"+qu=λu

Abstract: In this paper an expansion theorem is derived for a functionf(t) integrable on (−∞, ∞). It is shown thatf(t) is represented by an expansion of the form\(\sum\limits_{i = 1}^n {h(t,\lambda _i )\int {_{ - \infty }^\infty } h(\tau ,\lambda _i )f(\tau )d\tau + } \int {_0^\infty \left[ {\int {_{ - \infty }^\infty } F(t,\tau ,s^2 )f(\tau )d\tau } \right]} sds\) where the functionsh(t, λ1) are solutions of -x"(t)+q(tx(t=λx(t for a finite set of values {λi} of λ, andF (t, τ,s2) is a function of certain solutions of this equation. The representation converges to 1/2[f(t+0)+f(t-0)] is any interval wheref(t) is of bounded variation. The result may be regarded as a generalization of the Fourier integral theorem, to which it reduces in the special caseq(t)=0.

Alcune considerazioni sulle transformazioni lineari di una retta “ennonaria”

Abstract: In due precedenti lavori [5], [6] ai quali ci riferiremo per la terminologia e rinviamo per maggiori dettagli, s’e fatto vedere come e possibile, partendo da certi tipi di ennioni, definire una Geometria “ennonaria” (in un piano ennonario conm=n(n+1)/2) e studiare le trasformazioni lineari della retta “ennonaria”. Scopo del presente lavoro e quello di far vedere come le trasformazioni lineari della retta ennonaria possono determinarsi come prodotto di un numero finito di prospettivita.

Some generalizations of the contraction mapping theorem

Abstract: The Contraction Mapping Theorem is extended to the class of completea-metric spaces. Also, results of Edelstein on local contractions are generalized.

Hyperbolic potential theory in two dimensions

Abstract: In this paper we show that the classical approach through “single” and “double” layer potentials based on the fundamental solution as a kernel leads in a natural way to simple solutions of the boundary value problems for hyperbolic equations in two independent variables. The “hyperbolic potentials” defined here are closely analogous to the thermal potentials as well as to the potentials arising in the study of elliptic equations. This extension of the use of potentials for studying boundary value problems for hyperbolic equations thus establishes the idea of potential as one of the unifying concepts of partial differential equations: it is now applicable to all three types of second order equations.

Ein Beweis Des Satzes von Plancherel

Abstract: Man betrachtet fur eine Funktion ϕ(x)∈L2(−π, π) die vom Parameter α abhangige Entwicklung\(\varphi (x) = \frac{1}{{\sqrt {2\pi } }}\sum\limits_{k = - \infty }^\infty {(F_\varphi )(\alpha + k)e^{ - ix(\alpha + k)} } \). Durch Integration nach α wird es moglich, von den Fourierreihen zur Fouriertransformation zu gelangen und den Satz von Plancherel zu beweisen.

Absolute Cesàro summability factors of trigonometric series

Abstract: The author has extended the definition of |C, α|k (α > -1,k ≥1) and has obtained the summability |C, α|k (α > 1/2, 1≥k >0) of the factored trigonometric series which generalises a result of SinghN. [Proc. Edin. Math. Soc., 16 (1968), 71–76].

On symmetric flow of gas following a shock of variable strength

Abstract: This paper makes a comparative study of one dimensional, spherical and cylindrical flows following a normal shock of variable strength.