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 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 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 preserving r-potents of tensor products of matrices

2017 ◽  
Vol 520 ◽  
pp. 67-76
Author(s):  
Jinli Xu ◽  
Ajda Fošner ◽  
Baodong Zheng ◽  
Yuting Ding
Keyword(s):  
Tensor Products ◽  
Linear Maps ◽  
Products Of Matrices

scholarly journals ASSESSMENT FORM - NEW IMPROVEMENT OF ACTIONS: CONCENTRATION AND RESEARCH AREAS / CURRICULUM STRUCTURE / FUNDRAISING

2015 ◽  
Vol 42 (suppl 1) ◽  
pp. 14-16
Author(s):  
Iracema MP Calderon
Keyword(s):  
Evaluation Criteria ◽  
Basic Structure ◽  
Research Projects ◽  
Current Analysis ◽  
Evaluation Report ◽  
The Core ◽  
Research Areas ◽  
Curriculum Structure ◽  
Core Subjects ◽  
Curricular Structure

Objective: This review aims to develop a critical and current analysis of the basic structure of a Postgraduate program for proposing improvement actions and new evaluation criteria. Method: To examine the items that are areas of concentration (AC), research lines (LP), research projects (PP), curricular structure and fundraising were consulted the Area Document, the 2013 Evaluation Report and the Assessment Sheets of Medicine III programs, evaluated in the 2010-2012 period. Results: Consistency is recommended especially among AC, LP and PP, with genuine link between activities and permanent teachers skills and based on structured curriculum in the education of the student. The Program Proposal interfere, and much, in qualifying a program. The curriculum should provide subsidy to the formation of the researcher, through the core subjects, and development of PP, being the concept of disciplines to support lines and research projects. Fundraise should be set out in research projects and in the CV-Lattes. The area recommended that at least 40-50% of permanent teachers present fundraising and the minimum 20-25% of these teachers to have productivity scholarship PQ / CNPq during the triennium. Conclusion: It is necessary to promote wide discussion and find a consensus denominator for these issues. The actions should contribute to the improvement of evaluation forms and certainly for the qualification of the programs but graduate.

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 .

scholarly journals First Steps in the Analysis of Prokaryotic Pan-Genomes

2020 ◽  
Vol 14 ◽  
pp. 117793222093806
Author(s):  
Sávio Souza Costa ◽  
Luís Carlos Guimarães ◽  
Artur Silva ◽  
Siomar Castro Soares ◽  
Rafael Azevedo Baraúna
Keyword(s):  
Genome Analysis ◽  
Core Genome ◽  
Bacterial Species ◽  
Genomic Analysis ◽  
Gene Families ◽  
Specific Group ◽  
The Core ◽  
Pan Genome ◽  
Research Areas ◽  
Key Concepts

Pan-genome is defined as the set of orthologous and unique genes of a specific group of organisms. The pan-genome is composed by the core genome, accessory genome, and species- or strain-specific genes. The pan-genome is considered open or closed based on the alpha value of the Heap law. In an open pan-genome, the number of gene families will continuously increase with the addition of new genomes to the analysis, while in a closed pan-genome, the number of gene families will not increase considerably. The first step of a pan-genome analysis is the homogenization of genome annotation. The same software should be used to annotate genomes, such as GeneMark or RAST. Subsequently, several software are used to calculate the pan-genome such as BPGA, GET_HOMOLOGUES, PGAP, among others. This review presents all these initial steps for those who want to perform a pan-genome analysis, explaining key concepts of the area. Furthermore, we present the pan-genomic analysis of 9 bacterial species. These are the species with the highest number of genomes deposited in GenBank. We also show the influence of the identity and coverage parameters on the prediction of orthologous and paralogous genes. Finally, we cite the perspectives of several research areas where pan-genome analysis can be used to answer important issues.

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