Prime Producing Quadratic Polynomials and Class-Numbers of Real Quadratic Fields

1990 ◽  
Vol 42 (2) ◽  
pp. 315-341 ◽  
Author(s):  
Stéphane Louboutin
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Quadratic Polynomials ◽  
Real Quadratic Fields ◽  
Consecutive Integers ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Real Quadratic

Frobenius-Rabinowitsch's theorem provides us with a necessary and sufficient condition for the class-number of a complex quadratic field with negative discriminant D to be one in terms of the primality of the values taken by the quadratic polynomial with discriminant Don consecutive integers (See [1], [7]). M. D. Hendy extended Frobenius-Rabinowitsch's result to a necessary and sufficient condition for the class-number of a complex quadratic field with discriminant D to be two in terms of the primality of the values taken by the quadratic polynomials and with discriminant D (see [2], [7]).

scholarly journals Use of the rabinowitsch polynomial to determine the class groups of a real quadratic field

1994 ◽  
Vol 50 (3) ◽  
pp. 435-443
Author(s):  
R.A. Mollin
Keyword(s):  
Class Group ◽  
Quadratic Field ◽  
Sufficient Condition ◽  
Real Quadratic Field ◽  
Necessary And Sufficient Condition ◽  
Class Groups ◽  
Necessary And Sufficient ◽  
Real Quadratic

The main result is a necessary and sufficient condition for the class group of a real quadratic field to be determined by primality properties of the well-known Rabinowitsch polynomial.

scholarly journals Existence of nonoscillatory solutions of first order nonlinear neutral equations

1990 ◽  
Vol 32 (2) ◽  
pp. 180-192 ◽  
Author(s):  
Lu Wudu
Keyword(s):  
Nonoscillatory Solution ◽  
Sufficient Condition ◽  
Neutral Equation ◽  
Necessary And Sufficient Condition ◽  
Nonoscillatory Solutions ◽  
Neutral Equations ◽  
First Order ◽  
Image Position ◽  
Necessary And Sufficient

AbstractConsider the nonlinear neutral equationwhere pi(t), hi(t), gj(t), Q(t) Є C[t0, ∞), limt→∞hi(t) = ∞, limt→∞gj(t) = ∞ i Є Im = {1, 2, …, m}, j Є In = {1, 2, …, n}. We obtain a necessary and sufficient condition (2) for this equation to have a nonoscillatory solution x(t) with limt→∞ inf|x(t)| > 0 (Theorems 5 and 6) or to have a bounded nonoscillatory solution x(t) with limt→∞ inf|x(t)| > 0 (Theorem 7).

On Global Inverse Theorems of Szász and Baskakov Operators

1979 ◽  
Vol 31 (2) ◽  
pp. 255-263 ◽  
Author(s):  
Z. Ditzian
Keyword(s):  
Rate Of Convergence ◽  
Exponential Growth ◽  
Polynomial Growth ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Baskakov Operators ◽  
Inverse Theorems ◽  
Image Position ◽  
Necessary And Sufficient

The Szász and Baskakov approximation operators are given by1.11.2respectively. For continuous functions on [0, ∞) with exponential growth (i.e. ‖ƒ‖A ≡ supx\ƒ(x)e–Ax\ < M) the modulus of continuity is defined by1.3where ƒ ∈ Lip* (∝, A) for some 0 < ∝ ≦ 2 if w2(ƒ, δ, A) ≦ Mδ∝ for all δ < 1. We shall find a necessary and sufficient condition on the rate of convergence of An(ƒ, x) (representing Sn(ƒ, x) or Vn(ƒ, x)) to ƒ(x) for ƒ(x) ∈ Lip* (∝, A). In a recent paper of M. Becker [1] such conditions were found for functions of polynomial growth (where (1 + \x\N)−1 replaced e–Ax in the above). M. Becker explained the difficulties in treating functions of exponential growth.

scholarly journals Algebraic independence properties of the Fredholm series

1978 ◽  
Vol 26 (1) ◽  
pp. 31-45 ◽  
Author(s):  
J. H. Loxton ◽  
A. J. van der Poorten
Keyword(s):  
Algebraic Independence ◽  
Rational Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Algebraic Numbers ◽  
Independence Properties ◽  
Image Position ◽  
Necessary And Sufficient

AbstractWe consider algebraic independence properties of series such as We show that the functions fr(z) are algebraically independent over the rational functions Further, if αrs (r = 2, 3, 4, hellip; s = 1, 2, 3, hellip) are algebraic numbers with 0 < |αrs|, we obtain an explicit necessary and sufficient condition for the algebraic independence of the numbers fr(αrs) over the rationals.

scholarly journals Sufficient conditions for matchings

1972 ◽  
Vol 18 (2) ◽  
pp. 129-136 ◽  
Author(s):  
Ian Anderson
Keyword(s):  
Simple Proof ◽  
Perfect Matching ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Number Of Components ◽  
Image Position ◽  
Necessary And Sufficient

A graph G is said to possess a perfect matching if there is a subgraph of G consisting of disjoint edges which together cover all the vertices of G. Clearly G must then have an even number of vertices. A necessary and sufficient condition for G to possess a perfect matching was obtained by Tutte (3). If S is any set of vertices of G, let p(S) denote the number of components of the graph G – S with an odd number of vertices. Then the conditionis both necessary and sufficient for the existence of a perfect matching. A simple proof of this result is given in (1).

On inequalities for powers of linear operators and for quadratic forms

1981 ◽  
Vol 89 (1-2) ◽  
pp. 25-50 ◽  
Author(s):  
Vũ Qúôc Phóng
Keyword(s):  
Quadratic Forms ◽  
Linear Operators ◽  
Dissipative Operator ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Dense Domain ◽  
The Best Possible Constant ◽  
Bilinear Functional ◽  
Image Position ◽  
Necessary And Sufficient

SynopsisLetHbe a Hilbert space in which a symmetric operatorSwith a dense domainDsis given and letShave a finite deficiency index (r, s). This paper contains a necessary and sufficient condition for validity of the following inequalities of Kolmogorov typeand a method for calculating the best possible constantsCn,m(S).Moreover, let φ be a symmetric bilinear functional with a dense domainDφsuch thatDs⊂Dφand φ(f, g) = (Sf, g) for allf∈Ds,g∈Dφ. A necessary and sufficient condition for validity of the inequalityas well as a method for calculating the best possible constantKare obtained. Then an analogous approach is worked out in order to obtain the best possible additive inequalities of the formThe paper is concluded by establishing the best possible constants in the inequalitieswhereTis an arbitrary dissipative operator. The theorems are extensions of the results of Ju. I. Ljubič, W. N. Everitt, and T. Kato.

Oscillation and Global Attractivity in a Periodic Delay Equation

1996 ◽  
Vol 39 (3) ◽  
pp. 275-283 ◽  
Author(s):  
J. R. Graef ◽  
C. Qian ◽  
P. W. Spikes
Keyword(s):  
Global Attractivity ◽  
Positive Periodic Solution ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
All Solutions ◽  
Periodic Delay ◽  
Image Position ◽  
Delay Differential ◽  
Necessary And Sufficient

AbstractConsider the delay differential equationwhere α(t) and β(t) are positive, periodic, and continuous functions with period w > 0, and m is a nonnegative integer. We show that this equation has a positive periodic solution x*(t) with period w. We also establish a necessary and sufficient condition for every solution of the equation to oscillate about x*(t) and a sufficient condition for x*(t) to be a global attractor of all solutions of the equation.

scholarly journals Remarks on a Problem of Obreanu

1963 ◽  
Vol 6 (2) ◽  
pp. 267-273 ◽  
Author(s):  
P. Erdös ◽  
A. Rényi
Keyword(s):  
Infinite Sequence ◽  
Greatest Common Divisor ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
De Bruijn ◽  
Image Position ◽  
Necessary And Sufficient

Let a1 < a2 < … be any sequence of integers. Assume that the infinite sequence of numbers un satisfies the following condition: To every ɛ > 0 there is an no = no (ɛ) such that for all n > no and all k1Obreanu asked (Problem P. 35 Can. Math. Bull.) under what conditions on the sequence a1 < a2 < … does (1) imply that the sequence u is convergent. N. G. de Bruijn and P. Erdos proved that a necessary and sufficient condition for (1) to imply the convergence of un is that the sequence {an} be infinite and that the greatest common divisor of the a1 should be 1.

The Associated Ultraspherical Polynomials and their q-Analogues

1982 ◽  
Vol 34 (3) ◽  
pp. 718-736 ◽  
Author(s):  
Joaquin Bustoz ◽  
Mourad E. H. Ismail
Keyword(s):  
Orthogonal Polynomials ◽  
Positive Measure ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Ultraspherical Polynomials ◽  
Associated Polynomials ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Precise Degree

A sequence of polynomials {Pn(x)} is orthogonal if Pn(x) is of precise degree n and there is a finite positive measure dμ such that1.1A necessary and sufficient condition for orthogonality [9] is that {Pn(x)} satisfies a three term recurrence1.2with1.3Given a sequence of orthogonal polynomials {Pn(x)} satisfying (1.2), the associated polynomials {Pn(γ)(x)}, γ > 0, are defined by1.4with P(γ)-1(x) = 0, P0(γ)(x) = 1, when An+γ and Bn+γ are well-defined.

Sign in / Sign up

Export Citation Format

Share Document