scholarly journals Linear Maps that Preserve Any Two Term Ranks on Matrix Spaces over Anti-Negative Semirings

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 41
Author(s):  
Kyung Tae Kang ◽  
Seok-Zun Song ◽  
Young Bae Jun
Keyword(s):  
Linear Operators ◽  
Linear Map ◽  
Linear Maps ◽  
Term Rank ◽  
Q Matrix ◽  
Matrix Spaces

There are many characterizations of linear operators from various matrix spaces into themselves which preserve term rank. In this research, we characterize the linear maps which preserve any two term ranks between different matrix spaces over anti-negative semirings, which extends the previous results on characterizations of linear operators from some matrix spaces into themselves. That is, a linear map T from p × q matrix spaces into m × n matrix spaces preserves any two term ranks if and only if T preserves all term ranks if and only if T is a ( P , Q , B )-block map.

scholarly journals Linear Maps Preserving Rank-additivity and Rank-sum-minimal on Tensor Products of Matrix Spaces

2019 ◽  
pp. 1-10
Author(s):  
Lele Gao ◽  
Yang Zhang ◽  
Jinli Xu
Keyword(s):  
Matrix Theory ◽  
Tensor Products ◽  
Linear Mapping ◽  
Linear Map ◽  
The Core ◽  
Research Areas ◽  
Linear Maps ◽  
Matrix Spaces

The problems of characterizing maps that preserve certain invariant on given sets are called the preserving problems, which have become one of the core research areas in matrix theory. If for any a linear map, , as established, there is we say that  preserves the rank-additivity. If for any , and a linear map,  established, there is  we say that rank-sum-miminal. In this paper, we characterize the form of linear mapping .

scholarly journals Linear k-power Preservers on Tensor Products of Matrices

2020 ◽  
Vol 12 (6) ◽  
pp. 110
Author(s):  
Le Yan ◽  
Yang Zhang
Keyword(s):  
Tensor Products ◽  
Linear Operators ◽  
General Matrix ◽  
Linear Map ◽  
Preserver Problems ◽  
Matrix Spaces ◽  
Products Of Matrices

Invariants and the study of the map preserving a certain invariant play vital roles in the study of the theoretical mathematics. The preserver problems are the researches on linear operators that preserve certain invariants between matrix sets. Based on the result of linear $k$-power preservers on general matrix spaces, in terms of the advantages of matrix tensor products which is not limited by the size of matrices as well as the immense actual background, the study of the structure of the linear $k$-power preservers on tensor products of matrices is essential, which is coped with in this paper. That is to characterize a linear map $f:M_{m_{1}\cdots m_{l}}\rightarrow M_{m_{1}\cdots m_{l}}$ satisfying $f(X_{1}\otimes \cdots \otimes X_{l})^{k}=f\left( (X_{1}\otimes \cdots \otimes X_{l})^{k}\right) $ for all $X_{1}\otimes \cdots \otimes X_{l}\in M_{m_{1}\cdots m_{l}}$.

scholarly journals How Many Inflections are There in the Lyapunov Spectrum?

2021 ◽  
Author(s):  
O. Jenkinson ◽  
M. Pollicott ◽  
P. Vytnova
Keyword(s):  
Piecewise Linear ◽  
Lyapunov Spectrum ◽  
Stat Phys ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
Expanding Map ◽  
Lyapunov Spectra ◽  
Linear Maps ◽  
Inflection Points ◽  
Piecewise Linear Maps

AbstractIommi and Kiwi (J Stat Phys 135:535–546, 2009) showed that the Lyapunov spectrum of an expanding map need not be concave, and posed various problems concerning the possible number of inflection points. In this paper we answer a conjecture in Iommi and Kiwi (2009) by proving that the Lyapunov spectrum of a two branch piecewise linear map has at most two points of inflection. We then answer a question in Iommi and Kiwi (2009) by proving that there exist finite branch piecewise linear maps whose Lyapunov spectra have arbitrarily many points of inflection. This approach is used to exhibit a countable branch piecewise linear map whose Lyapunov spectrum has infinitely many points of inflection.

scholarly journals Linear maps on *-algebras acting on orthogonal elements like derivations or anti-derivations

Filomat ◽  
2018 ◽  
Vol 32 (13) ◽  
pp. 4543-4554 ◽  
Author(s):  
H. Ghahramani ◽  
Z. Pan
Keyword(s):  
Linear Map ◽  
Unital Algebra ◽  
C Algebras ◽  
Orthogonality Conditions ◽  
Linear Maps ◽  
Unital Simple

Let U be a unital *-algebra and ? : U ? U be a linear map behaving like a derivation or an anti-derivation at the following orthogonality conditions on elements of U: xy = 0, xy* = 0, xy = yx = 0 and xy* = y*x = 0. We characterize the map ? when U is a zero product determined algebra. Special characterizations are obtained when our results are applied to properly infinite W*-algebras and unital simple C*-algebras with a non-trivial idempotent.

scholarly journals Physical Implementability of Linear Maps and Its Application in Error Mitigation

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 600
Author(s):  
Jiaqing Jiang ◽  
Kun Wang ◽  
Xin Wang
Keyword(s):  
General Linear ◽  
Local Error ◽  
Decomposition Technique ◽  
Completely Positive ◽  
Linear Map ◽  
Quantum Operations ◽  
Error Mitigation ◽  
Linear Maps ◽  
Analytic Expressions ◽  
Sampling Cost

Completely positive and trace-preserving maps characterize physically implementable quantum operations. On the other hand, general linear maps, such as positive but not completely positive maps, which can not be physically implemented, are fundamental ingredients in quantum information, both in theoretical and practical perspectives. This raises the question of how well one can simulate or approximate the action of a general linear map by physically implementable operations. In this work, we introduce a systematic framework to resolve this task using the quasiprobability decomposition technique. We decompose a target linear map into a linear combination of physically implementable operations and introduce the physical implementability measure as the least amount of negative portion that the quasiprobability must pertain, which directly quantifies the cost of simulating a given map using physically implementable quantum operations. We show this measure is efficiently computable by semidefinite programs and prove several properties of this measure, such as faithfulness, additivity, and unitary invariance. We derive lower and upper bounds in terms of the Choi operator's trace norm and obtain analytic expressions for several linear maps of practical interests. Furthermore, we endow this measure with an operational meaning within the quantum error mitigation scenario: it establishes the lower bound of the sampling cost achievable via the quasiprobability decomposition technique. In particular, for parallel quantum noises, we show that global error mitigation has no advantage over local error mitigation.

scholarly journals On the surjectivity of linear maps on locally convex spaces

1982 ◽  
Vol 32 (2) ◽  
pp. 187-211 ◽  
Author(s):  
Sadayuki Yamamuro
Keyword(s):  
Locally Convex Spaces ◽  
Continuous Linear ◽  
Basic Notion ◽  
Linear Map ◽  
Linear Maps ◽  
Locally Convex ◽  
Do So ◽  
Convex Spaces

AbstractThe aim of this note is to investigate the structure of general surjectivity problem for a continuous linear map between locally convex spaces. We shall do so by using the method introduced in Yamamuro (1980). Its basic notion is that of calibrations which has been introduced in Yamamuro (1975), studied in detail in Yamamuro (1979) and appliced to several problems in Yamamuro (1978) and Yamamuro (1979a).

scholarly journals About Zero Torsion Linear Maps on Lie Algebras

ISRN Algebra ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-16
Author(s):  
Louis Magnin
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Equivalence Classes ◽  
Explicit Description ◽  
Complex Structures ◽  
Linear Map ◽  
3 Dimensional ◽  
Linear Maps

We prove that any zero torsion linear map on a nonsolvable real Lie algebra is an extension of some CR-structure. We then study the cases of (2, ) and the 3-dimensional Heisenberg Lie algebra . In both cases, we compute up to equivalence all zero torsion linear maps on , and deduce an explicit description of the equivalence classes of integrable complex structures on .

Linear Maps Transforming the Unitary Group

2003 ◽  
Vol 46 (1) ◽  
pp. 54-58 ◽  
Author(s):  
Wai-Shun Cheung ◽  
Chi-Kwong Li
Keyword(s):  
Linear Transformation ◽  
Unitary Group ◽  
Linear Operators ◽  
Unitary Matrices ◽  
Linear Maps

AbstractLet U(n) be the group of n × n unitary matrices. We show that if ϕ is a linear transformation sending U(n) into U(m), then m is a multiple of n, and ϕ has the formfor some V, W ∈ U(m). From this result, one easily deduces the characterization of linear operators that map U(n) into itself obtained by Marcus. Further generalization of the main theorem is also discussed.

Kendall’s Tau Based Correlation Analysis of Chaotic Sequences Generated by Piecewise Linear Maps

2015 ◽  
Vol 25 (13) ◽  
pp. 1550177
Author(s):  
Zouhair Ben Jemaa ◽  
Daniele Fournier-Prunaret ◽  
Safya Belghith
Keyword(s):  
Correlation Analysis ◽  
Chaotic Maps ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Chaotic Map ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
Chaotic Sequences ◽  
Linear Maps ◽  
Piecewise Linear Maps

In many applications, sequences generated by chaotic maps have been considered as pseudo-random sequences. This paper deals with the correlation between chaotic sequences generated by a given piecewise linear map; we have based the measure of the correlation on the statistics of the Kendall tau, which is usually used in the field of statistics. We considered three piecewise linear maps to generate chaotic sequences and computed the statistics of the Kendall tau of couples of sequences obtained from randomly chosen couples of initial conditions. We essentially found that the results depend on the considered chaotic map and that it is possible to approach the uncorrelated case.

Linear preservers of Hadamard majorization

2016 ◽  
Vol 31 ◽  
pp. 593-609 ◽  
Author(s):  
Sara Motlaghian ◽  
Ali Armandnejad ◽  
Frank Hall
Keyword(s):  
Stochastic Matrix ◽  
Doubly Stochastic Matrix ◽  
Graph Theoretic ◽  
Doubly Stochastic ◽  
Linear Preservers ◽  
Linear Maps ◽  
Term Rank

Let $\textbf{M}_{n }$ be the set of all $n \times n $ realmatrices. A matrix $D=[d_{ij}]\in\textbf{M}_{n } $ with nonnegative entries is called doubly stochastic if $\sum_{k=1}^{n} d_{ik}=\sum_{k=1}^{n} d_{kj}=1$ for all $1\leq i,j\leq n$. For $ X,Y \in \textbf{M}_{n}$ we say that $X$ is Hadamard-majorized by $Y$, denoted by $ X\prec_{H} Y$, if there exists an $n \times n$ doubly stochastic matrix $D$ such that $X=D\circ Y$.In this paper, some properties of$\prec_{H}$ on $\textbf{M}_{n}$ are first obtained, and then the (strong) linear preservers of$\prec_{H}$ on $\textbf{M}_{n }$ are characterized. For $n\geq3$, it is shown that the strong linear preservers of Hadamard majorization on $\textbf{M}_{n}$ are precisely the invertible linear maps on $\textbf{M}_{n}$ which preserve the set of matrices of term rank 1.An interesting graph theoretic connection to the linear preservers of Hadamard majorization is exhibited. A number of examples are also provided in the paper.

Sign in / Sign up

Export Citation Format

Share Document