On the Modulii of Analytic Functions

1970 ◽  
Vol 13 (3) ◽  
pp. 325-327 ◽  
Author(s):  
Malcolm J. Sherman
Keyword(s):  
Analytic Functions ◽  
Point Of View ◽  
Sufficient Condition ◽  
Two Dimensional ◽  
Necessary And Sufficient Condition ◽  
Concrete Form ◽  
Norm Equality ◽  
Image Position ◽  
Necessary And Sufficient

The problem to be considered in this note, in its most concrete form, is the determination of all quartets f1, f2, g1, g2 of functions analytic on some domain and satisfying*where p > 0. When p = 2 the question can be reformulated in terms of finding a necessary and sufficient condition for (two-dimensional) Hilbert space valued analytic functions to have equal pointwise norms, and the answer (Theorem 1) justifies this point of view. If p ≠ 2, the problem is solved by reducing to the case p = 2, and the reformulation in terms of the norm equality of lp valued analytic functions gives no clue to the answer.

scholarly journals Lorentz Boosts and Wigner Rotations: Self-Adjoint Complexified Quaternions

Physics ◽  
2021 ◽  
Vol 3 (2) ◽  
pp. 352-366
Author(s):  
Thomas Berry ◽  
Matt Visser
Keyword(s):  
Point Of View ◽  
Sufficient Condition ◽  
Use Of Self ◽  
Necessary And Sufficient Condition ◽  
Wigner Rotation ◽  
Inertial Frames ◽  
Explicit Formulae ◽  
Specific Implementation ◽  
Necessary And Sufficient ◽  
Lorentz Boosts

In this paper, Lorentz boosts and Wigner rotations are considered from a (complexified) quaternionic point of view. It is demonstrated that, for a suitably defined self-adjoint complex quaternionic 4-velocity, pure Lorentz boosts can be phrased in terms of the quaternion square root of the relative 4-velocity connecting the two inertial frames. Straightforward computations then lead to quite explicit and relatively simple algebraic formulae for the composition of 4-velocities and the Wigner angle. The Wigner rotation is subsequently related to the generic non-associativity of the composition of three 4-velocities, and a necessary and sufficient condition is developed for the associativity to hold. Finally, the authors relate the composition of 4-velocities to a specific implementation of the Baker–Campbell–Hausdorff theorem. As compared to ordinary 4×4 Lorentz transformations, the use of self-adjoint complexified quaternions leads, from a computational view, to storage savings and more rapid computations, and from a pedagogical view to to relatively simple and explicit formulae.

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).

scholarly journals Certain subclasses of analytic functions involving complex order

Filomat ◽  
2008 ◽  
Vol 22 (2) ◽  
pp. 115-122 ◽  
Author(s):  
T.N. Shanmugam ◽  
S. Sivasubramanian ◽  
B.A. Frasin
Keyword(s):  
Analytic Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Complex Order ◽  
Necessary And Sufficient ◽  
Class Of Functions

In the present investigation, we consider an unified class of functions of complex order. Necessary and sufficient condition for functions to be in this class is obtained. The results obtained in this paper generalizes the results obtained by Srivastava and Lashin [10], and Ravichandran et al. [4]. .

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]).

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.

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