CONTINUOUS ON RAYS SOLUTIONS OF A GOŁA̧B–SCHINZEL TYPE EQUATION

2014 ◽  
Vol 91 (2) ◽  
pp. 273-277 ◽  
Author(s):  
JACEK CHUDZIAK
Keyword(s):  
Linear Space ◽  
Type Equation ◽  
Composite Equation ◽  
Continuity On Rays ◽  
Image Position

AbstractWe show that if the pair $(f,g)$ of functions mapping a linear space $X$ over the field $\mathbb{K}=\mathbb{R}\text{ or }\mathbb{C}$ into $\mathbb{K}$ satisfies the composite equation $$\begin{eqnarray}f(x+g(x)y)=f(x)f(y)\quad \text{for }x,y\in X\end{eqnarray}$$ and $f$ is nonconstant, then the continuity on rays of $f$ implies the same property for $g$. Applying this result, we determine the solutions of the equation.

On a Normal Form of the Orthogonal Transformation III

1958 ◽  
Vol 1 (3) ◽  
pp. 183-191 ◽  
Author(s):  
Hans Zassenhaus
Keyword(s):  
Normal Form ◽  
Linear Space ◽  
Bilinear Form ◽  
Matrix Equation ◽  
Orthogonal Transformation ◽  
The Matrix ◽  
Image Position

Under the assumptions of case of theorem 1 we derive from (3.32) the matrix equationso that there corresponds the matrix B to the bilinear form4.1on the linear space4.2and fP,μ, is symmetric if ɛ = (-1)μ+1, anti-symmetric if ɛ = (-1)μ.The last statement remains true in the case a) if P is symmetric irreducible because in that case fP,μ is 0.

scholarly journals Nearly directed subspaces of partially ordered linear spaces

1968 ◽  
Vol 16 (2) ◽  
pp. 135-144
Author(s):  
G. J. O. Jameson
Keyword(s):  
Linear Space ◽  
Transitive Relation ◽  
Linear Spaces ◽  
Real Linear Space ◽  
Partially Ordered ◽  
Real Linear ◽  
Image Position ◽  
Ordered Linear Spaces

Let X be a partially ordered linear space, i.e. a real linear space with a reflexive, transitive relation ≦ such that

scholarly journals Operators with a Single Spectrum

1966 ◽  
Vol 15 (1) ◽  
pp. 11-18 ◽  
Author(s):  
T. T. West
Keyword(s):  
Linear Space ◽  
Normed Linear Space ◽  
Linear Operators ◽  
Complex Field ◽  
Unique Extension ◽  
Single Spectrum ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Resolvent Set ◽  
Image Position

Let X be an infinite dimensional normed linear space over the complex field Z. X will not be complete, in general, and its completion will be denoted by . If ℬ(X) is the algebra of all bounded linear operators in X then T ∈ ℬ(X) has a unique extension and . The resolvent set of T ∈ ℬ(X) is defined to beand the spectrum of T is the complement of ρ(T) in Z.

scholarly journals Pythagorean Orthogonality in a Normed Linear Space

1958 ◽  
Vol 9 (4) ◽  
pp. 168-169
Author(s):  
Hazel Perfect
Keyword(s):  
Euclidean Space ◽  
Linear Space ◽  
Normed Linear Space ◽  
Image Position ◽  
Pythagorean Orthogonality

This note presents a proof of the following proposition:Theorem. If Pythagorean orthogonality is homogeneous in a normed linear space T then T is an abstract Euclidean space.The theorem was originally stated and proved by R. C. James ([1], Theorem 5. 2) who systematically discusses various characterisations of a Euclidean space in terms of concepts of orthogonality. I came across the result independently and the proof which I constructed is a simplified version of that of James. The hypothesis of the theorem may be stated in the form:Since a normed linear space is known to be Euclidean if the parallelogram law:is valid throughout the space (see [2]), it is evidently sufficient to show that (l) implies (2).

A modified constitutive model based on Arrhenius-type equation to predict the flow behavior of Fe–36%Ni Invar alloy

2017 ◽  
Vol 32 (20) ◽  
pp. 3831-3841 ◽  
Author(s):  
Shuai He ◽  
Chang-sheng Li ◽  
Zhen-yi Huang ◽  
Jian-jun Zheng
Keyword(s):  
Constitutive Model ◽  
Flow Behavior ◽  
Type Equation ◽  
Invar Alloy ◽  
Model Based ◽  
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Abstract

scholarly journals Existence and Concentration of Positive Solutions for Nonlinear Kirchhoff-Type Problems with a General Critical Nonlinearity

2018 ◽  
Vol 61 (4) ◽  
pp. 1023-1040 ◽  
Author(s):  
Jianjun Zhang ◽  
David G. Costa ◽  
João Marcos do Ó
Keyword(s):  
Positive Solutions ◽  
Local Minimum ◽  
Type Equation ◽  
Bound State ◽  
State Solution ◽  
Critical Nonlinearity ◽  
Bound State Solution ◽  
Kirchhoff Type ◽  
Image Position ◽  
Kirchhoff Type Problems

AbstractWe are concerned with the following Kirchhoff-type equation$$ - \varepsilon ^2M\left( {\varepsilon ^{2 - N}\int_{{\open R}^N} {\vert \nabla u \vert^2{\rm d}x} } \right)\Delta u + V(x)u = f(u),\quad x \in {{\open R}^N},\quad N{\rm \ges }2,$$whereM ∈ C(ℝ+, ℝ+),V ∈ C(ℝN, ℝ+) andf(s) is of critical growth. In this paper, we construct a localized bound state solution concentrating at a local minimum ofVasε → 0 under certain conditions onf(s),MandV. In particular, the monotonicity off(s)/sand the Ambrosetti–Rabinowitz condition are not required.

scholarly journals On Exposed and Farthest Points in Normed Linear Spaces

1971 ◽  
Vol 12 (3) ◽  
pp. 301-308 ◽  
Author(s):  
M. Edelstein ◽  
J. E. Lewis
Keyword(s):  
Linear Space ◽  
Normed Linear Space ◽  
Nonempty Subset ◽  
Linear Spaces ◽  
Farthest Points ◽  
Normed Linear Spaces ◽  
Farthest Point ◽  
Image Position

Let S be a nonempty subset of a normed linear space E. A point s0 of S is called a farthest point if for some x ∈ E, . The set of all farthest points of S will be denoted far (S). If S is compact, the continuity of distance from a point x of E implies that far (S) is nonempty.

Open induction and the true theory of rationals

1987 ◽  
Vol 52 (3) ◽  
pp. 793-801
Author(s):  
Zofia Adamowicz
Keyword(s):  
Linear Space ◽  
Usual Sense ◽  
Saturated Model ◽  
The Real ◽  
Maximal Subset ◽  
True Theory ◽  
Real Closure ◽  
Open Induction ◽  
Infinite Sets ◽  
Image Position

In the paper we prove the following theorem:Theorem. There is a model N of open induction in which the set of primes is bounded and N is such that its field of fractions 〈N*, +, ·, <〉 is elementarily equivalent to 〈Q, +, ·, <〉 (the standard rationals).We fix an ω1-saturated model 〈M, +, ·, <〉 of PA. Let 〈M*, +, ·, <〉 denote the field of fractions of M. The model N that we are looking for will be a substructure of 〈M*, +, ·, <〉.If A ⊆ M* then let Ā denote the ring generated by A within M*, Ậ the real closure of A, and A* the field of fractions generated by A. We haveLet J ⊆ M. Then 〈M*, +, ·〉 is a linear space over J*. If x1,…,xk ∈ M*, we shall say that x1,…,xk are J-independent if 〈1, x1,…, xk〉 are J*-independent in the usual sense. As usual, we extend the notion of J-independence to the case of infinite sets.If A ⊆ M* and X ⊆ A, then we say that X is a J-basis of A if X is a maximal subset of A which is J-independent.Definition 1.1. By a J-form ρ we mean a function from (M*)k into M*, of the formwhere q0,…, qk ∈ J*If υ ∈ M, we say that ρ is a υ-form if the numerators and denominators of the qi's have absolute values ≤ υ.

On Matrix Commutators

1960 ◽  
Vol 12 ◽  
pp. 269-277 ◽  
Author(s):  
Marvin Marcus ◽  
Nisar A. Khan
Keyword(s):  
Linear Space ◽  
Elementary Divisor ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Elementary Divisors ◽  
Image Position

Let A, B, and X be n-square matrices over an algebraically closed field F of characteristic 0. Let [A, B] = AB — BA and set (A, B) = [A, [A, B]]. Recently several proofs (1; 3; 5) of the following result have appeared: if det (AB) ≠ 0 and (A,B) = 0 then A-1B-1AB - I is nilpotent. In (2) McCoy determined the general form of any X satisfying1.1in the case that A has a single elementary divisor corresponding to each eigenvalue, that is, A is non-derogatory. In Theorem 1 we determine the structure of any matrix X satisfying (1.1) and also give a formula for the dimension of the linear space of all such X in terms of the degrees of the elementary divisors of A.

The Strong ϕ Topology on Symmetric Sequence Spaces

1985 ◽  
Vol 37 (6) ◽  
pp. 1112-1133
Author(s):  
William H. Ruckle
Keyword(s):  
Finite Number ◽  
Linear Space ◽  
Natural Duality ◽  
Sequence Spaces ◽  
Image Position

The strong ϕ topology. Let S be a linear space of real sequences written in functional notationThere is a natural duality between S and the space ϕ of sequences which are eventually ϕ given by the equationThe series has only a finite number of nonzero terms since t is in ϕ.A subset B of ϕ is called S-bounded iffor each s in S.

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