scholarly journals Connecting and Improving Direct Sum Masking and Inner Product Masking

2018 ◽  
pp. 123-141 ◽  
Author(s):  
Romain Poussier ◽  
Qian Guo ◽  
François-Xavier Standaert ◽  
Claude Carlet ◽  
Sylvain Guilley
Keyword(s):  
Inner Product ◽  
Direct Sum

scholarly journals TWO-INTERVAL EVEN-ORDER DIFFERENTIAL OPERATORS IN MODIFIED HILBERT SPACES

2011 ◽  
Vol 85 (2) ◽  
pp. 241-260
Author(s):  
JIANQING SUO ◽  
WANYI WANG
Keyword(s):  
Boundary Conditions ◽  
Hilbert Spaces ◽  
Differential Operators ◽  
Order Equation ◽  
Inner Product ◽  
Coupling Matrix ◽  
Direct Sum ◽  
Even Order

AbstractBy modifying the inner product in the direct sum of the Hilbert spaces associated with each of two underlying intervals on which an even-order equation is defined, we generate self-adjoint realisations for boundary conditions with any real coupling matrix which are much more general than the coupling matrices from the ‘unmodified’ theory.

scholarly journals Some Properties of Fuzzy Inner Product Space

2021 ◽  
pp. 2384-2392
Author(s):  
Jehad R. Kider
Keyword(s):  
Normed Space ◽  
Product Space ◽  
Closed Subspace ◽  
Cross Product ◽  
Inner Product Space ◽  
Inner Product ◽  
Orthogonal Complement ◽  
Fuzzy Normed Space ◽  
Direct Sum ◽  
New Type

     Our goal in the present paper is to introduce a new type of fuzzy inner product space. After that, to illustrate this notion, some examples are introduced. Then we prove that that every fuzzy inner product space is a fuzzy normed space. We also prove that the cross product of two fuzzy inner spaces is again a fuzzy inner product space. Next, we prove that the fuzzy inner product is a non decreasing function. Finally, if U is a fuzzy complete fuzzy inner product space and D is a fuzzy closed subspace of U, then we prove that U can be written as a direct sum of D and the fuzzy orthogonal complement    of D.

Two-Interval Sturm–Liouville Operators in Direct Sum Spaces with Inner Product Multiples

2007 ◽  
Vol 50 (1-2) ◽  
pp. 155-168 ◽  
Author(s):  
Jiong Sun ◽  
Aiping Wang ◽  
Anton Zettl
Keyword(s):  
Inner Product ◽  
Direct Sum ◽  
Sturm Liouville ◽  
Liouville Operators

scholarly journals THE STRUCTURE OF THE SPACE OF AFFINE KÄHLER CURVATURE TENSORS AS A COMPLEX MODULE

2011 ◽  
Vol 08 (08) ◽  
pp. 1849-1868 ◽  
Author(s):  
M. BROZOS-VÁZQUEZ ◽  
P. GILKEY ◽  
S. NIKČEVIĆ
Keyword(s):  
Complex Structure ◽  
Inner Product ◽  
Almost Complex Structure ◽  
The Real ◽  
Direct Sum ◽  
Irreducible Modules ◽  
Almost Complex ◽  
Affine Curvature ◽  
Curvature Tensors ◽  
The Given

In dimension m ≥ 4, results of Strichartz decompose the space 𝔄 of affine curvature tensors as a direct sum of 3 modules in the real setting and results of Bokan give a corresponding finer decomposition of 𝔄 in the Riemannian setting as the direct sum of 8 irreducible modules. In dimension m ≥ 8, results of Matzeu and Nikčević decompose the space 𝔎 of affine Kähler curvature tensors as the direct sum of 12 irreducible modules in the Hermitian setting (i.e. given an auxiliary inner product which is invariant under the given almost complex structure). In this paper, we decompose 𝔎 as a direct sum of six irreducible modules in the complex setting in dimension m ≥ 8. Corresponding decompositions into fewer modules are given in dimension m = 4 and m = 6.

scholarly journals Nonsingular bilinear forms on direct sums of ideals

2013 ◽  
Vol 63 (4) ◽  
Author(s):  
Beata Rothkegel
Keyword(s):  
Bilinear Form ◽  
Dedekind Domain ◽  
Bilinear Forms ◽  
Direct Sums ◽  
Direct Sum

AbstractIn the paper we formulate a criterion for the nonsingularity of a bilinear form on a direct sum of finitely many invertible ideals of a domain. We classify these forms up to isometry and, in the case of a Dedekind domain, up to similarity.

scholarly journals Dimension of the Exceptional Set in the Aronszajn–Donoghue Theorem for Finite Rank Perturbations

2020 ◽  
Author(s):  
Constanze Liaw ◽  
Sergei Treil ◽  
Alexander Volberg
Keyword(s):  
Finite Rank ◽  
Measure Zero ◽  
Adjoint Operator ◽  
Cyclic Vector ◽  
Spectral Measures ◽  
Hermitian Matrices ◽  
Exceptional Set ◽  
Direct Sum ◽  
Rank One Perturbation ◽  
Rank One

Abstract The classical Aronszajn–Donoghue theorem states that for a rank-one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank $d$ perturbations $A_{\boldsymbol{\alpha }}:= A + {\textbf{B}} {\boldsymbol{\alpha }} {\textbf{B}}^*$, ${\textbf{B}}:{\mathbb C}^d\to{{\mathcal{H}}}$, with ${\operatorname{Ran}}{\textbf{B}}$ being cyclic for $A$, parametrized by $d\times d$ Hermitian matrices ${\boldsymbol{\alpha }}$, the singular parts of the spectral measures of $A$ and $A_{\boldsymbol{\alpha }}$ are mutually singular for all ${\boldsymbol{\alpha }}$ except for a small exceptional set $E$. It was shown earlier by the 1st two authors, see [4], that $E$ is a subset of measure zero of the space $\textbf{H}(d)$ of $d\times d$ Hermitian matrices. In this paper, we show that the set $E$ has small Hausdorff dimension, $\dim E \le \dim \textbf{H}(d)-1 = d^2-1$.

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