The generalized Laguerre inequalities and functions in the Laguerre-Pólya class

2013 ◽  
Vol 11 (9) ◽  
Author(s):  
George Csordas ◽  
Anna Vishnyakova
Keyword(s):  
Entire Function ◽  
Open Problem ◽  
A Priori ◽  
Sufficient Conditions ◽  
Class Definition ◽  
Lindelöf Theorem ◽  
Order And Type ◽  
Geometric Result ◽  
Real Entire Function ◽  
Inequality Theorem

AbstractThe principal goal of this paper is to show that the various sufficient conditions for a real entire function, φ(x), to belong to the Laguerre-Pólya class (Definition 1.1), expressed in terms of Laguerre-type inequalities, do not require the a priori assumptions about the order and type of φ(x). The proof of the main theorem (Theorem 2.3) involving the generalized real Laguerre inequalities, is based on a beautiful geometric result, the Borel-Carathédodory Inequality (Theorem 2.1), and on a deep theorem of Lindelöf (Theorem 2.2). In case of the complex Laguerre inequalities (Theorem 3.2), the proof is sketched for it requires a slightly more delicate analysis. Section 3 concludes with some other cognate results, an open problem and a conjecture which is based on Cardon’s recent, ingenious extension of the Laguerre-type inequalities.

scholarly journals Existence and global stability of periodic solution for delayed discrete high-order Hopfield-type neural networks

2005 ◽  
Vol 2005 (3) ◽  
pp. 281-297 ◽  
Author(s):  
Hong Xiang ◽  
Ke-Ming Yan ◽  
Bai-Yan Wang
Keyword(s):  
Neural Networks ◽  
Periodic Solution ◽  
Global Stability ◽  
A Priori Estimates ◽  
A Priori ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
High Order ◽  
Priori Estimates

By using coincidence degree theory as well as a priori estimates and Lyapunov functional, we study the existence and global stability of periodic solution for discrete delayed high-order Hopfield-type neural networks. We obtain some easily verifiable sufficient conditions to ensure that there exists a unique periodic solution, and all theirs solutions converge to such a periodic solution.

scholarly journals On dimensions supporting a rational projective plane

2019 ◽  
Vol 11 (03) ◽  
pp. 535-555 ◽  
Author(s):  
Lee Kennard ◽  
Zhixu Su
Keyword(s):  
Projective Plane ◽  
Open Problem ◽  
Sufficient Conditions ◽  
Quadratic Residue ◽  
Bernoulli Numbers ◽  
Existence Results ◽  
Necessary And Sufficient ◽  
Simply Connected ◽  
Existence Question ◽  
Smooth Closed Manifold

A rational projective plane ([Formula: see text]) is a simply connected, smooth, closed manifold [Formula: see text] such that [Formula: see text]. An open problem is to classify the dimensions at which such a manifold exists. The Barge–Sullivan rational surgery realization theorem provides necessary and sufficient conditions that include the Hattori–Stong integrality conditions on the Pontryagin numbers. In this paper, we simplify these conditions and combine them with the signature equation to give a single quadratic residue equation that determines whether a given dimension supports a [Formula: see text]. We then confirm the existence of a [Formula: see text] in two new dimensions and prove several non-existence results using factorization of the numerators of the divided Bernoulli numbers. We also resolve the existence question in the Spin case, and we discuss existence results for the more general class of rational projective spaces.

scholarly journals Bank-Laine Functions with Real Zeros

2020 ◽  
Vol 20 (3-4) ◽  
pp. 653-665
Author(s):  
J. K. Langley
Keyword(s):  
Entire Function ◽  
Finite Order ◽  
Explicit Representation ◽  
Trigonometric Functions ◽  
Exponent Of Convergence ◽  
Real Zeros ◽  
Real Entire Function ◽  
Bounded Below ◽  
Quasiconformal Surgery

AbstractSuppose that E is a real entire function of finite order with zeros which are all real but neither bounded above nor bounded below, such that $$E'(z) = \pm 1$$ E ′ ( z ) = ± 1 whenever $$E(z) = 0$$ E ( z ) = 0 . Then either E has an explicit representation in terms of trigonometric functions or the zeros of E have exponent of convergence at least 3. An example constructed via quasiconformal surgery demonstrates the sharpness of this result.

scholarly journals General Local Convergence Theorems about the Picard Iteration in Arbitrary Normed Fields with Applications to Super–Halley Method for Multiple Polynomial Zeros

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1599
Author(s):  
Stoil I. Ivanov
Keyword(s):  
Error Estimates ◽  
Local Convergence ◽  
A Priori ◽  
A Posteriori Error Estimates ◽  
Sufficient Conditions ◽  
Posteriori Error ◽  
Picard Iteration ◽  
Polynomial Zeros ◽  
Convergence Theorems ◽  
Simple And Multiple Roots

In this paper, we prove two general convergence theorems with error estimates that give sufficient conditions to guarantee the local convergence of the Picard iteration in arbitrary normed fields. Thus, we provide a unified approach for investigating the local convergence of Picard-type iterative methods for simple and multiple roots of nonlinear equations. As an application, we prove two new convergence theorems with a priori and a posteriori error estimates about the Super-Halley method for multiple polynomial zeros.

scholarly journals The pseudoidentity problem and reducibility for completely regular semigroups

2001 ◽  
Vol 63 (3) ◽  
pp. 407-433 ◽  
Author(s):  
Jorge Almedia ◽  
Peter G. Trotter
Keyword(s):  
Word Problem ◽  
Finite Groups ◽  
Open Problem ◽  
Sufficient Conditions ◽  
90Th Birthday ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Strengthened Version ◽  
Completely Regular Semigroups ◽  
Necessary And Sufficient

Dedicated to George Szekeres on the occasion of his 90th birthdayNecessary and sufficient conditions for equality over the pseudovariety CR of all finite completely regular semigroups are obtained. They are inspired by the solution of the word problem for free completely regular semigroups and clarify the role played by groups in the structure of such semigroups. A strengthened version of Ash's inevitability theorem (κ-reducibility of the pseudovariety G of all finite groups) is proposed as an open problem and it is shown that, if this stronger version holds, then CR is also κ-reducible and, therefore, hyperdecidable.

ON THE MATABILITY OF POLYGONS

2008 ◽  
Vol 18 (05) ◽  
pp. 469-506
Author(s):  
GILL BAREQUET ◽  
AYA STEINER
Keyword(s):  
Information Systems ◽  
Medical Imaging ◽  
Surface Reconstruction ◽  
Geographical Information Systems ◽  
Piecewise Linear ◽  
A Priori ◽  
Sufficient Conditions ◽  
Geographical Information ◽  
Triangulated Surface ◽  
The Past

Interpolating a piecewise-linear triangulated surface between two polygons lying in parallel planes has attracted a lot of attention in the literature over the past three decades. This problem is the simplest variant of interpolation between parallel slices, which may contain multiple polygons with unrestricted geometries and/or topologies. Its solution has important applications to medical imaging, digitization of objects, geographical information systems, and more. Practically all currently-known methods for surface reconstruction from parallel slices assume a priori the existence of a non-self-intersecting triangulated surface defined over the vertices of the two polygons, which connects them. Gitlin et al. were the first to specify a nonmatable pair of polygons. In this paper we provide proof of the nonmatability of a “simpler” pair of polygons, which is less complex than the example given by Gitlin et al. Furthermore, we provide a family of polygon pairs with unbounded complexity, which we believe to be nonmatable. We also give a few sufficient conditions for polygon matability.

scholarly journals An inverse spectral problem for Sturm-Liouville-type integro-differential operators with robin boundary conditions

2019 ◽  
Vol 50 (3) ◽  
pp. 207-221 ◽  
Author(s):  
Sergey Buterin
Keyword(s):  
Boundary Condition ◽  
Inverse Problem ◽  
Boundary Conditions ◽  
Differential Operators ◽  
A Priori ◽  
Sufficient Conditions ◽  
Robin Boundary Conditions ◽  
Sturm Liouville ◽  
Necessary And Sufficient ◽  
Robin Boundary

The perturbation of the Sturm--Liouville differential operator on a finite interval with Robin boundary conditions by a convolution operator is considered. The inverse problem of recovering the convolution term along with one boundary condition from the spectrum is studied, provided that the Sturm--Liouville potential as well as the other boundary condition are known a priori. The uniqueness of solution for this inverse problem is established along with necessary and sufficient conditions for its solvability. The proof is constructive and gives an algorithm for solving the inverse problem.

On the Growth of Entire Functions Bounded on Large Sets

1977 ◽  
Vol 29 (6) ◽  
pp. 1287-1291
Author(s):  
Lowell J. Hansen
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Maximum Modulus ◽  
Modulus Function ◽  
Classical Theorem ◽  
Minimum Modulus ◽  
Rate Of Growth ◽  
Large Sets ◽  
Growth Of Entire Functions ◽  
Lindelöf Theorem

There have been many indications of a relationship between the rate of growth of an entire function and the “size” of the set, E(c), where the modulus of the function is larger than the constant, c. Theorems of this type include the classical theorem of Wiman on functions of bounded minimum modulus, the Phragmén-Lindelöf Theorem, the Denjoy-Carleman-Ahlfors Theorem, and its many subsequent improvements. These theorems can all be understood as quantitative versions of the statement that if ƒ is an entire function such that, for some c > 0, the set E(c) is ‘'small”, then the maximum modulus function M(R, f) is forced to grow rapidly with R.

Growth Estimates of Entire Function Solutions of Generalized Bi-Axially Symmetric Helmholtz Equation

2017 ◽  
Vol 8 ◽  
pp. 12-26 ◽  
Author(s):  
Devendra Kumar
Keyword(s):  
Entire Function ◽  
Helmholtz Equation ◽  
Axially Symmetric ◽  
Order And Type

Growth estimates for entire function solutions of the generalized bi-axially symmetric Helmholtz equation ∂2u/∂x2 + ∂2u/∂y2 + (2µ/x)·(∂u/∂x) + (2ν/y)·(∂u/∂y) +k2u = 0, (µ,ν Є R+), in terms of their Jacobi Bessel coefficients and ratio of these coefficients have been studied. Some relations for order and type also have been obtained in terms of Taylor and Neumann coefficients. Our results generalize and extend some results of Gilbert and Howard, McCoy, Kumar and Singh.

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