A Third-Order Irregular Boundary Value Problem and the Associated Series

Reduction of everywhere convergent power series with respect to Gröbner bases

Normal matrices over hermitian discrete valuation rings

We solve the Laplace equation in an exterior infinite spherical domain with nonlinear (quadratic) boundary conditions on the spherical boundary. We linearize the problem and, under the additional assumption that the distinguishing function is spherically symmetric, write the solution by using the formal power series method with recursion of the series coefficients. Applying the Poincare--Perron theorem, we describe the space of convergent formal power series and calculate its dimension. Estimating the roots of the fourth-degree characteristic polynomial corresponding to the given problem, we also calculate the dimension of the space of functions whose gradient at each point of the sphere is orthogonal to the linear combination of an axially symmetric dipole and a quadrupole. In conclusion, we state several unsolved problems arising in geophysical applications.

In this chapter we prepare the ground for the proof of the Jung-Abhyankar theorem in section 2 and the study of quasiordinary power series in section 4 of chapter V. We assume that the reader is acquainted with the notion of power series over a field; in section 1, for the convenience of the reader, we give some background, introduce convergent power series over the real and complex numbers in section 2, and prove Weierstraß division and preparation theorem in section 3. The category of formal (resp. analytic) algebras—these are homomorphic images of formal (resp. convergent) rings of power series—is treated in section 4. The last section considers, in particular, the completion of an analytic algebra; these results are needed to prove the convergent case of the Jung-Abhyankar theorem.

Joseph Becker and Leonard Lipshitz. Remarks on the elementary theories of formal and convergent power series. Fundament a mathematicae, vol. 105 (1980), pp. 229–239. - Françoise Delon. Indécidabilité de la théorie des anneaux de séries formelles à plusiers indéterminées. Fundament a mathematicae, vol. 112 (1981), pp. 215–229. - J. Becker, J. Denef, and L. Lipshitz. Further remarks on the elementary theory of formal power series rings. Model theory of algebra and arithmetic, Proceedings of the Conference on Applications of Logic to Algebra and Arithmetic held at Karpacz, Poland, September 1–7, 1979, edited by L. Pacholski, J. Wierzejewski, and A. J. Wilkie, Lecture notes in mathematics, vol. 834, Springer-Verlag, Berlin, Heidelberg, and New York, 1980, pp. 1–9. - Françoise Delon. Hensel fields in equal characteristic p > 0. Model theory of algebra and arithmetic, Proceedings of the Conference on Applications of Logic to Algebra and Arithmetic held at Karpacz, Poland, September 1–7, 1979, edited by L. Pacholski, J. Wierzejewski, and A. J. Wilkie, Lecture notes in mathematics, vol. 834, Springer-Verlag, Berlin, Heidelberg, and New York, 1980, pp. 108–116. - Volume 50 Issue 3

This article is devoted to study the boundary value problems of the first and second kind with respect to the spatial variable for a mixed inhomogeneous differential equation of parabolic-hyperbolic type with a fractional Caputo operator in a rectangular domain. In the study of such boundary value problems, we abandoned the boundary value condition with respect to the first argument and instead it is used additional gluing condition. In this case, in the justification of the unique solvability of the problems, the conditions on the boundary domain are removed. This allowed us to weaken the criterion for the unique solvability of boundary value problems under consideration. The solution is constructed in the form of Fourier series with eigenfunctions corresponding to homogeneous spectral problems. Estimates for the convergence of Fourier series are obtained as a regular solution of this mixed equation.

Remarks on the elementary theories of formal and convergent power series

This chapter treats the theory of ordinary differential equations, both linear and nonlinear.Sections 1–4 establish existence and uniqueness theorems for ordinary differential equations. The first section gives some examples of first-order equations, mostly nonlinear, to illustrate certain kinds of behavior of solutions. The second section shows, in the presence of continuity for a vectorvalued F satisfying a “Lipschitz condition,” that the first-order system y′ = F(t, y) has a unique local solution satisfying an initial condition y (t0) = y0. Since higher-order equations can always be reduced to first-order systems, these results address existence and uniqueness for nth-order equations as a special case. Section 3 shows that the solutions to a system depend well on the initial condition and on any parameters that are present in F. Section 4 applies these results to existence of integral curves for a vector field and to construction of coordinate systems from families of integral curves.Sections 5–8 concern linear systems. Section 5 shows that local solutions of linear systems may be extended to global solutions and that in the homogeneous case the vector space of global solutions has dimension equal to the size of the system. The method of variation of parameters reduces the solution of any linear system to the solution of a homogeneous linear system. Sections 6–7 identify explicit solutions to nth-order linear equations and first-order linear systems. The “Jordan canonical form” of a square matrix plays a role in the case of a system. Section 8 discusses power-series solutions to second-order homogeneous linear equations whose coefficients are given by convergent power series, as well as solutions that arise in the case of regular singular points. Two kinds of special functions are mentioned that result from this study-Legendre polynomials and Bessel functions.KeywordsOrdinary Differential EquationUniqueness TheoremDiagonal EntryLipschitz ConditionIntegral CurfThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In the paper, a nonlocal initial-boundary value problem for a non-homogeneous one-dimensional heat equation is considered. The domain under consideration is a rectangle. The classical initial condition with respect to t is put. A nonlocal periodic boundary condition by a spatial variable x is put. It is well-known that a solution of problem can be constructed in the form of convergent orthonormal series according to eigenfunctions of a spectral problem for an operator of multiple differentiation with periodic boundary conditions. Therefore Green's function can be also written in the form of an infinite series with respect to trigonometric functions (Fourier series). For classical first and second initial-boundary value problems there also exists a second representation of the Green's function by Jacobi function. In this paper we find the representation of the Green's function of the nonlocal initial-boundary value problem with periodic boundary conditions in the form of series according to exponents.

The formal Laplace‐Borel transform of an analytic integral operator, known as a Fliess operator, is defined and developed. Then, in conjunction with the composition product over formal power series, the formal Laplace‐Borel transform is shown to provide an isomorphism between the semigroup of all Fliess operators under operator composition and the semigroup of all locally convergent formal power series under the composition product. Finally, the formal Laplace‐Borel transform is applied in a systems theory setting to explicitly derive the relationship between the formal Laplace transform of the input and output functions of a Fliess operator. This gives a compact interpretation of the operational calculus of Fliess for computing the output response of an analytic nonlinear system.

The Green’s function is widely used in solving boundary value problems for differential equations, to which many mathematical and physical problems are reduced. In particular, solutions of partial differential equations by the Fourier method are reduced to boundary value problems for ordinary differential equations. Using the Green's function for a homogeneous problem, one can calculate the solution of an inhomogeneous differential equation. Knowing the Green's function makes it possible to solve a whole class of problems of finding eigenvalues in quantum field theory. The developed method for constructing the Green’s function of boundary value problems for ordinary linear differential equations is described. An algorithm and program for calculating the Green's function of boundary value problems for differential equations of the second and third orders in an explicit analytical form are presented. Examples of computing the Green's function for specific boundary value problems are given. The fundamental system of solutions of ordinary differential equations with singular points needed to construct the Green's function is calculated in the form of generalized power series with the help of the developed programs in the Maple environment. An algorithm is developed for constructing the Green's function in the form of power series for second-order and third-order differential equations with given boundary conditions. Compiled work programs in the Maple environment for calculating the Green functions of arbitrary boundary value problems for differential equations of the second and third orders. Calculations of the Green's function for specific third-order boundary value problems using the developed program are presented. The obtained approximate Green’s function is compared with the known expressions of the exact Green’s function and very good agreement is found

Expansions in series of functions are one of the most important tools of the applied mathematician, particularly expansions in series of the classical orthogonal polynomials, e.g. Laguerre, Jacobi and Hermite polynomials. In applied problems, the uniqueness of the particular expansion is usually intrinsic to the analysis, and often implicitly assumed. Indeed, in those cases where the functions in the series are orthogonal, uniqueness can often be proved by an argument that runs as follows. Let {φn(x)} (n = 0, 1, 2, …) be a sequence of functions orthogonal with respect to the weight function ρ(x) over the interval [0, 1], and suppose thatthe series being boundedly convergent for 0 ≤ x ≤ 1.

Transactions of the American Mathematical Society, vol. i. No. 3.—Wave propagation over non-uniform electrical conductors, by M. I. Pupin, is a paper read before the society in December last. The main object of it is the solution of a problem which, looked at from a purely mathematical point of view, can be stated as follows:—Find the integral of the partial differential equation Ld2y/dt2+Rdy/dt = 1δ2y/Cδs2 and determine it so as to satisfy k + 2 boundary conditions, where k + 1 is the number of coils. The principal difficulty is to determine the proper mathematical formulation of these sundry conditions so as to obtain a system of equations which can be readily solved. The paper is illustrated by diagrams which put the problems discussed in a clear light.—“Ueber systeme von differentialgleichungen dessen vierfach periodische functionen genüge leisten,” by M. Krause, was presented at the Chicago (April) meeting of the present year. References are given to Hermite (“Sur quelques applications de la théorie des fonctions elliptiques,” 1885), and to a paper by Picard (Comptes rendus, Band 89), and to previous work by the author.—E. B. van Vleck follows with a paper on linear criteria for the determination of the radius of convergence of a power series. Its object is to establish criteria for the convergence of a power series when the (n+1)th coefficient An is connected with the preceding coefficients by a linear relation which tends to take, a limiting form as n increases indefinitely. The criteria include Cauchy's ratio-test as a special case, and may be looked upon as an extension of the test, and are applicable in cases in which the simple ratio-test fails. The paper closes with two theorems which are an extension for the case of two variables, criteria for the convergence of power series in such a case are stated to be very rare.—On the existence of the Green's function for the most general simply connected plane region, by W. F. Osgood.—A short but suggestive note “D” lines on quadrics, by A. Pell. These lines, so named by Cosserat, were originally considered by Darboux. They are the lines drawn upon a surface in such a way that the osculating sphere at every point is tangent to the surface at that point. In addition to the above, the lines have been studied by Enneper and Ribaucour (for surfaces in general). In the present paper the, author applies, the theory of elliptic functions to the integration of Darboux's differential equation, and obtains an idea of the appearance of the lines and also some of their properties.—Starting from an article, by Prof. F. Morley, in the previous number of the Transactions, F. H. Loud gives sundry metric theorems concerning n lines in a plane. By giving a different interpretation to formulæ got by Prof. Morley, Mr. Loud obtains a new series of theorems and other results of some interest.—An application of group theory to hydrodynamics, by E. J. Wilczynski. It was observed by Sophus Lie that the stationary motion of a fluid can serve as a perfect picture of a one-parameter group in three variables. Apparently this fact has not been utilised for the purposes of hydrodynamics. This paper does this. Amongst other advantages, the treatment, from the new standpoint, leads to special cases of exceptional interest and importance, which otherwise appear to be difficult and unpromising.—Dr. L E. Dickson, following up work recently published in the Proceedings of the London Mathematical Society (vol. xxxi. pp. 30, 351), contributes an article on the determination of an abstract simple group of order 27.36.5.7, holohedrically isomorphic with a certain orthogonal group and with a certain hyperabelian group (contributed to the Chicago [April] meeting of the society).

General expressions are derived for the characteristic function and probability density function of the output of a cross crrelator utilizing a hard clipper in one input channel. The inputs are assumed to be <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A_{1} \cos(\omega_{o}t)+n_{l}(t)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A_{2} \cos(\omega_{o}t+\phi)+n_{2}(t)</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n_{1}</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n_{2}</tex> are independent narrowband Gaussian noises. From the characteristic function, which is expressed in the form of a Fourier series in the relative phase angle <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\phi</tex> , a double series involving cosines of multiples of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\phi</tex> and Hermite polynomials of all orders is obtained as the probability density function of the output. Both the characteristic function and the density function pertain to the output at only a single instant of time. The first and second moments of the output are also calculated from the characteristic function, and lead to a closed-form expression for the output signal-to-noise ratio as a function of the input signal-to-noise ratios. Limiting cases of the output signal-to-noise ratio are computed and compared with similar cases involving the more conventional analog correlator without clippers and with the polarity-coincidence correlator.

We introduce a new local sine transform that can completely localize image information in both the space and spatial frequency domains. Instead of constructing a basis, we first segment an image into local pieces using the characteristic functions, then decompose each piece into two component: the <i>polyharmonic component </i>and the <i>residual</i>. The polyharmonic component is obtained by solving the elliptic boundary value problem associated with the so-called polyharmonic equation (e.g., Laplace equation, biharmonic equation, etc.) given the boundary values (the pixel values along the borders created by the characteristic functions) possibly with the estimates of normal derivatives at the boundaries. Once this component is obtained this is subtracted from the original local piece to obtain the residual, whose Fourier sine series expansion has quickly decaying coefficients since the boundary values of the residual (possibly with their normal derivatives) vanish. Using this transform, we can distinguish intrinsic singularities in the data from the artificial discontinuities created by the local windowing. We will demonstrate the superior performance of this new transform in terms of image compression to some of the conventional methods such as JPEG/DCT and the lapped orthogonal transform using actual examples.