Permutation endomorphisms and refinement of a theorem of Birkhoff

1960 ◽  
Vol 56 (4) ◽  
pp. 322-328 ◽  
Author(s):  
H. K. Farahat ◽  
L. Mirsky
Keyword(s):  
Abelian Group ◽  
Finite Number ◽  
Linear Combination ◽  
Finite Linear Combination ◽  
Usual Manner ◽  
Restricted Permutation ◽  
Image Position ◽  
Ring Of Endomorphisms ◽  
Finite Linear

Let be a free additive abelian group, and let be a basis of , so that every element of can be expressed in a unique way as a (finite) linear combination with integral coefficients of elements of . We shall be concerned with the ring of endomorphisms of , the sum and product of the endomorphisms φ, χ being defined, in the usual manner, by the equationsA permutation of a set will be called restricted if it moves only a finite number of elements. We call an endomorphism of a permutation endomorphism if it induces a restricted permutation of the basis .

scholarly journals On the p-norm of the truncated n-dimensional Hilbert transform

1991 ◽  
Vol 43 (2) ◽  
pp. 241-250 ◽  
Author(s):  
J.N. Pandey ◽  
O.P. Singh
Keyword(s):  
Linear Operator ◽  
Linear Combination ◽  
Hilbert Transform ◽  
Bounded Linear Operator ◽  
Measurable Subset ◽  
Bounded Linear ◽  
Finite Linear Combination ◽  
The Hilbert Transform ◽  
Finite Linear ◽  
Hilbert Type

It is shown that a bounded linear operator T from Lρ(Rn) to itself which commutes both with translations and dilatations is a finite linear combination of Hilbert-type transforms. Using this we show that the ρ-norm of the Hilbert transform is the same as the ρ-norm of its truncation to any Lebesgue measurable subset of Rn with non-zero measure.

On the number of representations of a positive integer as a sum of two binary quadratic forms

2014 ◽  
Vol 10 (06) ◽  
pp. 1395-1420 ◽  
Author(s):  
Şaban Alaca ◽  
Lerna Pehlivan ◽  
Kenneth S. Williams
Keyword(s):  
Positive Integer ◽  
Linear Combination ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Definite Integral ◽  
Binary Quadratic Forms ◽  
Finite Linear Combination ◽  
Positive Integers ◽  
Finite Linear

Let ℕ denote the set of positive integers and ℤ the set of all integers. Let ℕ0 = ℕ ∪{0}. Let a1x2 + b1xy + c1y2 and a2z2 + b2zt + c2t2 be two positive-definite, integral, binary quadratic forms. The number of representations of n ∈ ℕ0 as a sum of these two binary quadratic forms is [Formula: see text] When (b1, b2) ≠ (0, 0) we prove under certain conditions on a1, b1, c1, a2, b2 and c2 that N(a1, b1, c1, a2, b2, c2; n) can be expressed as a finite linear combination of quantities of the type N(a, 0, b, c, 0, d; n) with a, b, c and d positive integers. Thus, when the quantities N(a, 0, b, c, 0, d; n) are known, we can determine N(a1, b1, c1, a2, b2, c2; n). This determination is carried out explicitly for a number of quaternary quadratic forms a1x2 + b1xy + c1y2 + a2z2 + b2zt + c2t2. For example, in Theorem 1.2 we show for n ∈ ℕ that [Formula: see text] where N is the largest odd integer dividing n and [Formula: see text]

scholarly journals Examples of linear multi-box splines

2012 ◽  
Vol 15 ◽  
pp. 444-462
Author(s):  
Abdellatif Bettayeb
Keyword(s):  
Linear Combination ◽  
Piecewise Linear ◽  
Linear Functions ◽  
Order Derivative ◽  
Box Splines ◽  
Piecewise Linear Functions ◽  
Finite Linear Combination ◽  
First Order ◽  
Box Spline ◽  
Finite Linear

AbstractLet S1=S1(v0,…,vr+1) be the space of compactly supported C0 piecewise linear functions on a mesh M of lines through ℤ2 in directions v0,…,vr+1, possibly satisfying some restrictions on the jumps of the first order derivative. A sequence ϕ=(ϕ1,…,ϕr) of elements of S1 is called a multi-box spline if every element of S1 is a finite linear combination of shifts of (the components of) ϕ. We give some examples for multi-box splines and show that they are stable. It is further shown that any multi-box spline is not always symmetric

scholarly journals Weak and Strong Constraints Variational Data Assimilation with the NCOM-4DVAR in the Agulhas Region Using the Representer Method

2017 ◽  
Vol 145 (5) ◽  
pp. 1755-1764
Author(s):  
Hans Ngodock ◽  
Matthew Carrier ◽  
Scott Smith ◽  
Innocent Souopgui
Keyword(s):  
Data Assimilation ◽  
Linear Combination ◽  
Numerical Experiments ◽  
Single Observation ◽  
Finite Linear Combination ◽  
Method Formulation ◽  
The Difference ◽  
Representer Method ◽  
Weak Constraints ◽  
Finite Linear

Abstract The difference between the strong and weak constraints four-dimensional variational (4DVAR) analyses is examined using the representer method formulation, which expresses the analysis as the sum of a first guess and a finite linear combination of representer functions. The latter are computed analytically for a single observation under both strong and weak constraints assumptions. Even though the strong constraints representer coefficients are different from their weak constraints counterparts, that difference is unable to help the strong constraints compensate for the loss of information that the weak constraints includes. Numerical experiments carried out in the Agulhas retroflection for single and multiobservation assimilations clearly show that the weak constraint 4DVAR produces analyses that fit the observations with significantly higher accuracy than the strong constraints.

scholarly journals Some remarks on reconstruction from local weighted averages

2012 ◽  
Vol 44 (3) ◽  
pp. 217-226
Author(s):  
Devaraj Ponnaian
Keyword(s):  
Linear Combination ◽  
Convolution Equation ◽  
Indicator Function ◽  
Finite Linear Combination ◽  
Weighted Averages ◽  
Finite Linear

We solve the convolution equation of the type $f\star\mu=g,$ where $f\star \mu$ is the convolution of $f$ and $\mu$ defined by $(f\star \mu)(x)=\int_{{\mathbb{R}}}f(x-y)d\mu(y),$ $g$ is a given function and $\mu$ is a finite linear combination of translates of an indicator function on an interval.

Dimension of Null Spaces with Applications to Group Rings

1978 ◽  
Vol 30 (02) ◽  
pp. 289-300 ◽  
Author(s):  
David Promislow
Keyword(s):  
Linear Combination ◽  
Upper Bound ◽  
Null Space ◽  
Group Rings ◽  
Finite Linear Combination ◽  
Finite Linear

In this paper we investigate methods for estimating the dimension of the null space of operators in a finite W* algebra, the dimension being measured by the trace τ. For the most part we are concerned with operators A which are a finite linear combination of orthogonal unitaries. We give various results which show how certain information about the unitaries and the coefficients can be utilized to derive an upper bound for τ(N A ) where NA is the null space of A.

On polynomial expansions of analytic functions

1969 ◽  
Vol 10 (3-4) ◽  
pp. 335-340
Author(s):  
Fred Gross
Keyword(s):  
Linear Combination ◽  
Analytic Functions ◽  
Finite Linear Combination ◽  
Basic Set ◽  
Polynomial Expansions ◽  
Finite Linear

A set of polynomials p0(z), p1(z), … is said to form a basic set if every polynomial can be expressed in one and only one way as a finite linear combination of them.

ON PERTURBATION OF BANACH FRAMES

2006 ◽  
Vol 04 (03) ◽  
pp. 559-565 ◽  
Author(s):  
P. K. JAIN ◽  
S. K. KAUSHIK ◽  
L. K. VASHISHT
Keyword(s):  
Linear Combination ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Banach Frames ◽  
Finite Linear Combination ◽  
Linearly Independent ◽  
Banach Frame ◽  
Necessary And Sufficient ◽  
Finite Linear

A necessary and sufficient condition for the perturbation of a Banach frame by a non-zero functional to be a Banach frame has been obtained. Also a sufficient condition for the perturbation of a Banach frame by a sequence in E* to be a Banach frame has been given. Finally, a necessary condition for the perturbation of a Banach frame by a finite linear combination of linearly independent functionals in E* to be a Banach frame has been given.

scholarly journals THE INVARIANT MEASURE OF RANDOM WALKS IN THE QUARTER-PLANE: REPRESENTATION IN GEOMETRIC TERMS

2015 ◽  
Vol 29 (2) ◽  
pp. 233-251 ◽  
Author(s):  
Yanting Chen ◽  
Richard J. Boucherie ◽  
Jasper Goseling
Keyword(s):  
Random Walk ◽  
Invariant Measure ◽  
Random Walks ◽  
Linear Combination ◽  
Balance Equations ◽  
Linear Combinations ◽  
Finite Linear Combination ◽  
Quarter Plane ◽  
Geometric Term ◽  
Finite Linear

We consider the invariant measure of homogeneous random walks in the quarter-plane. In particular, we consider measures that can be expressed as a finite linear combination of geometric terms and present conditions on the structure of these linear combinations such that the resulting measure may yield an invariant measure of a random walk. We demonstrate that each geometric term must individually satisfy the balance equations in the interior of the state space and further show that the geometric terms in an invariant measure must have a pairwise-coupled structure. Finally, we show that at least one of the coefficients in the linear combination must be negative.

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