Weighted arcs,the finite radon transform and a Ramsey problem

We establish a link between the theory of (k, v)-arcs in affine planes and a graph theoretic Ramsey problem: A (n, k)-coloring of the complete graphK _u is a coloring of the edges ofK _u withk colours such that monochromatic connected subgraphs have at mostn vertices. The Ramsey numberf(n, k) is the smallestu such thatK _u does not admit a (n, k)-coloring. Let be an affine plane of orderq. Aweighted v-arc is a function w: such that line integrals are at mostv. Themass ofw is the sum of all the weights.In certain cases the maximum massM _q (v) can be determined (maximum over all weightedv-arcs and all affine planes of orderq). This yields the exact value off(n, q + 1) for all large enoughn.     

Co-elementary equivalence, co-elementary maps, and generalized arcs

By a generalized arc we mean a continuum with exactly two non-separating points; an arc is a metrizable generalized arc. It is well known that any two arcs are homeomorphic (to the real closed unit interval); we show that any two generalized arcs are co-elementarily equivalent, and that co-elementary images of generalized arcs are generalized arcs. We also show that if is a function between compacta and if is an arc, then is a co-elementary map if and only if is an arc and is a monotone continuous surjection.

     

Regularization of the solution of the cauchy problem for the generalized Moisil-Theodoresco system

We consider the problem of analytic continuation of a solution of the generalized Moisil-Theodoresco system in a spatial domain on the basis of its values on part of the boundary of this domain, that is, the Cauchy problem. We construct an approximate solution of this problem on the basis of the Carleman matrix method.     

Polynomial modules and a generalized cauchy problem for systems of holomorphic partial differential operators

In this paper, we study holomorphic solutions of overdetermined systems of partial differential equatoins with constant coefficients. We parametrize the solution space, using an ideal of holomorphic functions generated by certain polynomials associated to the principal symbols of the operators, by proving a Cauchy-Kowalevsky type theorem for these systems. In this theorem, we introduce a non-characteristicity condition for systems which is a direct generalization of a condition for single equations introduced by the author and H.S. Shapiro in a previous paper.

As a tool for studying the systems mentioned above, certain estimates, which may be of independent interest, for elements of analytic modules generated by polynomials are obtained.     

The inversion problem and applications of the generalized radon transform

We prove that under certain conditions the inversion problem for the generalized Radon transform reduces to solving a Fredholm integral equation and we obtain the asymptotic expansion of the symbol of the integral operator in this equation. We consider applications of the generalized Radon transform to partial differential equations with variable coefficients and provide a solution to the inversion problem for the attenuated and exponential Radon transforms.     

n次积分C半群与抽象柯西问题的强解

给出了n次积分C半群的几个性质及其证明,讨论了它与一类抽象柯西问题存在强解的关系及强解的表示式.     

Approximated cauchy problem for linear,first order,overdetermined systems

Sunto Le soluzioni di certi sistemi differenziali sono dense tra le funzioni C^∞ di una superficie non caratteristica.

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Entrata in Redazione l'11 ottobre 1978.

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Supported by C.N.R. Research groups.     

Smoothness of generalized solutions of the first boundary-value problem for an elliptic difference-differential equation

Mathematical Notes -     

On the riemann zeta-function and the divisor problem II

Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of . If E ^*(t)=E(t)-2πΔ^*(t/2π) with , then we obtain

An underdetermined problem of integral geometry for the generalized radon transform

Under study is a new problem of integral geometry. All kinds of planes are considered in the three-dimensional Euclidean space. The available data are given by the integrals over all these planes of an unknown piecewise-smooth function depending both on the spatial variables and the variables characterizing the planes. The sought object is a first kind discontinuity surface of the integrand. The uniqueness theorem of a desired surface is proved. The above results are related to one of the aspects of the theory of testing unknown media by various physical signals.     

On regularity of generalized solutions of the third boundary-value problem for an elliptic difference-differential equation

Unlike the case of elliptic differential equations, generalized solutions of elliptic difference-differential equations may be not smooth in a domainQ but remain smooth only in certain subdomainsQ _r Q Conditions are considered which are necessary and sufficient for generalized solutions of the third boundary-value problem to preserve smoothness on the boundary of adjacent subdomainsQ _r .Translated from Ukrainskii Matematicheskii Zhurnal, Vol.47, No. 1, pp. 93–104, January, 1995.The author expresses his deep gratitude to A. D. Myshkis and A. L. Skubachevskii for helpful discussions.     

A problem on arcs without bypasses in tournaments

An arc $\text{xy}$ in a tournament is bad if there exists no path of length two from x to y. Formulas are found for the number of tournaments T_n whose bad arcs determine a spanning cycle or path.     

Metric regularity and systems of generalized equations

The paper is devoted to a revision of the metric regularity property for mappings between metric or Banach spaces. Some new concepts are introduced: uniform metric regularity and metric multi-regularity for mappings into product spaces, when each component is perturbed independently. Regularity criteria are established based on a nonlocal version of Lyusternik-Graves theorem due to Milyutin. The criteria are applied to systems of generalized equations producing some “error bound” type estimates.     

Global classical discontinuous solutions to a class of generalized riemann problem for general quasilinear hyperbolic systems of conservation laws

In this paper, the authors prove the global existence and uniqueness of piecewise C¹ solution u = u(t, x) containing only n contact discontinuities with small amplitude to the generalized Riemann problem for general linearly degenerate quasilinear hyperbolic systems of conservation laws with small decay initial data. This solution has a global structure similar to the similarity solution u=U(x/t) to the corresponding Riemann problem. The result shows that the similarity solution u=U(x/t) possesses a global nonlinear structural stability.     

Uniqueness of the solutions of the cauchy problem and of boundary-value problems in unbounded domains for certain classes of nonlinear degenerate parabolic equations

The Cauchy problem and boundary value problems in unbounded domains are considered for certain classes of nonlinear degenerate parabolic equations, which contain as a particular case the equations of nonstationary filtration.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 11, pp. 12–31, 1986.The author expresses her deep gratitude to Olga Arsenovna Oleinik for the formulation of the problem and for her constant interest in the paper.