On two inequalities similar to Opial's inequality in two independent variables

In the present paper we establish two new integral inequalities similar to Opial's inequality in two independent variables. The inequalities established in this paper are similar to the analogues of Calvert's generalizations of Opial's inequality, in two independent variables and contains in the special case the analogue of Opial's inequality given by G. S. Yang in two independent variables.     

Rounding errors in numerical solutions of two linear equations in two unknowns

New condition numbers and stability constants for the numerical behaviour of Cramer's rule and Gaussian elimination for solving two linear equations in two unknowns under data perturbations and rounding errors of floating-point arithmetic are established. By these means fundamental error estimates and stability theorems are proved. The error estimates are illustrated by a series of numerical examples.     

Semiclassical orthogonal polynomials in two variables

In this work, semiclassical orthogonal polynomials in two variables are defined as the orthogonal polynomials associated with a quasi definite linear functional satisfying a matrix Pearson-type differential equation. Semiclassical functionals are characterized by means of the analogue of the structure relation in one variable. Moreover, non trivial examples of semiclassical orthogonal polynomials in two variables are given.     

About two characteristic points concerning two circles and their use in study of bicentric polygons

The article primarily deals with the connection between Fuss’ relation for bicentric n-gons where conics are two nested circles and Fuss’ relation for bicentric n-gons where conics are two separated circles. It is proved that one of them is quite determined by other. Also, the article deals with solutions of some old and difficult problems where characteristic points of two corresponding circles are defined and often used. Several interesting properties are established and some difficult problems are solved.     

Seshadri constants in two half-periods

We calculate the exact value of the multiple point Seshadri constant in two halfperiods of an abelian surface with Picard number one. Received: 3 December 2003; revised: 9 July 2005     

Discriminants in characteristic two

The Kronecker-Weierstrass theory of pencils is extended to give a necessary and sufficient condition that two 2×m×n tensors are equivalent. The connection between equivalence class representatives and the triple transitivity of PGL(2,F) is discussed. One consequence of the discussion is that the number of inequivalent 2×3×n tensors is finite. An efficient algorithm is given for testing the condition which ultimately depends on a fast pattern matching algorithm.     

The geometry of a quasilinear system of two partial differential equations containing the first and second partial derivatives of two functions in two independent variables

The geometry of a system of two partial differential equations containing the first and second partial derivatives of two functions in two independent variables is studied by using the Cartan method of invariant forms and the group-theoretic method of extensions and enclosings due to G. F. Laptev (for finite groups) and A. M. Vasil’ev (for infinite groups). Systems of quasilinear equations with the first and second partial derivatives of two functions u and v in two independent variables x and y are classified.     

On Koornwinder classical orthogonal polynomials in two variables

In 1975, Tom Koornwinder studied examples of two variable analogues of the Jacobi polynomials in two variables. Those orthogonal polynomials are eigenfunctions of two commuting and algebraically independent partial differential operators. Some of these examples are well known classical orthogonal polynomials in two variables, such as orthogonal polynomials on the unit ball, on the simplex or the tensor product of Jacobi polynomials in one variable, but the remaining cases are not considered classical by other authors. The definition of classical orthogonal polynomials considered in this work provides a different perspective on the subject. We analyze in detail Koornwinder polynomials and using the Koornwinder tools, new examples of orthogonal polynomials in two variables are given.     

The comparing of two methods in Microbiology

In many microbiological assays the presence or absence of microbes by two methods, sayA andB, is examined. In this paper the statistical method is developed giving the estimate of probability of detecting the microbes by the two methods used.     

The geometry of a quasilinear system of two partial differential equations containing the first and the second partial derivatives of two functions in two independent variables

The geometry of the system of two partial differential equations containing the first and second partial derivatives of two functions in two independent variables is studied by using Élie Cartan’s method of invariant forms and the group-theoretic method of extensions and enclosings due to G. F. Laptev (for finite groups) and A. M. Vasil’ev (for infinite groups). Systems of quasilinear equations with the first and second partial derivatives of two functions u and v in two independent variables x and y are classified.     

On the product of two skew-Hamiltonian or two skew-symmetric matrices

We show that the product C of two skew-Hamiltonian matrices obeys the Stenzel conditions. If at least one of the factors is nonsingular, then the Stenzel conditions amount to the requirement that every elementary divisor corresponding to a nonzero eigenvalue of C occurs an even number of times. The same properties are valid for the product of two skew-pseudosymmetric matrices. We observe that the method proposed by Van Loan for computing the eigenvalues of real Hamiltonian and skew-Hamiltonian matrices can be extended to complex skew-Hamiltonian matrices. Finally, we show that the computation of the eigenvalues of a product of two skew-symmetric matrices reduces to the computation of the eigenvalues of a similar skew-Hamiltonian matrix. Bibliography: 8 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 45–51.     

Sums of two squares in analytic rings

We study analytic singularities for which every positive semidefinite analytic function is a sum of two squares of analytic functions. This is a basic useful property of the plane, but difficult to check in other cases; in particular, what about , , or ? In fact, the unique positive examples we can find are the Brieskorn singularity, the union of two planes in 3-space and the Whitney umbrella. Conversely, we prove that a complete intersection with that property (other than the seven embedded surfaces already mentioned) must be a very simple deformation of the two latter, namely, In particular, except for the stems and , all singularities are real rational double points. Received April 4, 1997; in final form September 25, 1997     

Associative formal power series in two indeterminates

Investigating the associativity equation for formal power series in two variables we show that the transcendental associative formal power series are of order one or two and that they can be represented by an invertible formal power series in one variable. We also discuss the convergence of associative formal power series.     

The join of two minimal clones and the meet of two maximal clones

We construct two minimal clones on any finite set such that the join of the two clones contains all operations. Dually, we exhibit two maximal clones on any finite set with at least three elements such that the intersection of the two clones is the trivial clone containing projections only. Received October 20, 1998; accepted in final form December 16, 1998.     

On Opial-type inequalities in two variables

Summary In this paper we establish some new Opial-type inequalities in two variables which have a wide range of applications in the study of differential and integral equations.     

Total domination in planar graphs of diameter two

MacGillivary and Seyffarth [G. MacGillivray, K. Seyffarth, Domination numbers of planar graphs, J. Graph Theory 22 (1996) 213–229] proved that planar graphs of diameter two have domination number at most three. Goddard and Henning [W. Goddard, M.A. Henning, Domination in planar graphs with small diameter, J. Graph Theory 40 (2002) 1–25] showed that there is a unique planar graph of diameter two with domination number three. It follows that the total domination number of a planar graph of diameter two is at most three. In this paper, we consider the problem of characterizing planar graphs with diameter two and total domination number three. We say that a graph satisfies the domination-cycle property if there is some minimum dominating set of the graph not contained in any induced 5-cycle. We characterize the planar graphs with diameter two and total domination number three that satisfy the domination-cycle property and show that there are exactly thirty-four such planar graphs.     

A competition model for two resources in un-stirred chemostat

This paper studies a un-stirred chemostat with two species competing for two growth-limiting, non-reproducing resources. We determine the conditions for positive steady states of the two species, and then consider the global attractors of the model. In addition, we obtain the conditions under which the two populations uniformly strongly persist or go to extinction. Since the diffusion mechanism with homogeneous boundary conditions inhibits the growth of the organism species, it can be understood that the coexistence will be ensured by proportionally smaller diffusions for the two species. In particular, it is found that both instability and bi-stability subcases of the two semitrivial steady states are included in the coexistence region. The two populations will go to extinction when both possess large diffusion rates. If just one of them spreads faster with the other one diffusing slower, then the related semitrivial steady state will be globally attracting. The techniques used for the above results consist of the degree theory, the semigroup theory, and the maximum principle.     

Point capture of two evaders in succession

A constant-speed coplanar model with unlimited turn-rates leads to a rather simple geometrical solution of the problem of point capture of two successive evaders in minimum total time: the pursuer concentrates first on the nearer evader who runs in an appropriate direction; the second evader runs directly away from the predictable point of capture of the first evader. This simple solution is valid only if the second evader remains thereby the further of the two evaders. Otherwise, the solution must be modified to include a phase involving curved motion by all three players, during which the pursuer remains equidistant from both evaders.     

Orthogonal polynomials in two variables

Summary Some properties of orthogonal (and generalized orthogonal) polynomial sets in two variables are obtained, in particular a characterization of such sets based on generating functions. Then those linear homogeneous partial differential eqnations of the form L[w]+λw=0, having a set of polynomials as solution, are characterized; and a detailed study is made of all such equations of second order whose polynomial solutions form an orthogonal (or generalized orthogonal) set. Supported byN.S.F. Grant GP-5311.