Representations of the necklace braid group $${{\mathcal {N}}{\mathcal {B}}}_n$$ of dimension 4 ($$n=2,3,4$$)

Hermitian Positive Definite Matrix ◽

Dimension 2 ◽

AbstractWe consider the irreducible representations each of dimension 2 of the necklace braid group $${\mathcal {N}}{\mathcal {B}}_n$$ N B n ($$n=2,3,4$$ n = 2 , 3 , 4 ). We then consider the tensor product of the representations of $${\mathcal {N}}{\mathcal {B}}_n$$ N B n ($$n=2,3,4$$ n = 2 , 3 , 4 ) and determine necessary and sufficient condition under which the constructed representations are irreducible. Finally, we determine conditions under which the irreducible representations of $${\mathcal {N}}{\mathcal {B}}_n$$ N B n ($$n=2,3,4$$ n = 2 , 3 , 4 ) of degree 2 are unitary relative to a hermitian positive definite matrix.

On the Unitary Representations of the Braid Group B6

Mathematics ◽

10.3390/math7111080 ◽

2019 ◽

Vol 7 (11) ◽

pp. 1080

Author(s):

Malak M. Dally ◽

Mohammad N. Abdulrahim

Keyword(s):

Braid Group ◽

Positive Definite Matrix ◽

Positive Definite ◽

Unitary Representations ◽

Irreducible Representations ◽

Dimensional Representation ◽

One Dimensional ◽

Low Degree ◽

Hermitian Positive Definite Matrix ◽

Group B

We consider a non-abelian leakage-free qudit system that consists of two qubits each composed of three anyons. For this system, we need to have a non-abelian four dimensional unitary representation of the braid group B 6 to obtain a totally leakage-free braiding. The obtained representation is denoted by ρ . We first prove that ρ is irreducible. Next, we find the points y ∈ C * at which the representation ρ is equivalent to the tensor product of a one dimensional representation χ ( y ) and μ ^ 6 ( ± i ) , an irreducible four dimensional representation of the braid group B 6 . The representation μ ^ 6 ( ± i ) was constructed by E. Formanek to classify the irreducible representations of the braid group B n of low degree. Finally, we prove that the representation χ ( y ) ⊗ μ ^ 6 ( ± i ) is a unitary relative to a hermitian positive definite matrix.

Finding Cut-Edges and the Minimum Spanning Tree via Semi-Tensor Product Approach

Journal of Systems Science and Information ◽

10.21078/jssi-2018-459-14 ◽

2018 ◽

Vol 6 (5) ◽

pp. 459-472

Author(s):

Xujiao Fan ◽

Yong Xu ◽

Xue Su ◽

Jinhuan Wang

Keyword(s):

Tensor Product ◽

Spanning Tree ◽

Minimum Spanning Tree ◽

Sufficient Condition ◽

The Matrix ◽

Product Approach ◽

Matrix Expression ◽

Necessary And Sufficient ◽

Logical Functions

Abstract Using the semi-tensor product of matrices, this paper investigates cycles of graphs with application to cut-edges and the minimum spanning tree, and presents a number of new results and algorithms. Firstly, by defining a characteristic logical vector and using the matrix expression of logical functions, an algebraic description is obtained for cycles of graph, based on which a new necessary and sufficient condition is established to find all cycles for any graph. Secondly, using the necessary and sufficient condition of cycles, two algorithms are established to find all cut-edges and the minimum spanning tree, respectively. Finally, the study of an illustrative example shows that the results/algorithms presented in this paper are effective.

GENERIC IRREDUCIBLE REPRESENTATIONS OF FINITE-DIMENSIONAL LIE SUPERALGEBRAS

International Journal of Mathematics ◽

10.1142/s0129167x9400022x ◽

1994 ◽

Vol 05 (03) ◽

pp. 389-419 ◽

Cited By ~ 33

Author(s):

IVAN PENKOV ◽

VERA SERGANOVA

Keyword(s):

Irreducible Module ◽

Lie Superalgebra ◽

Lie Superalgebras ◽

Irreducible Representations ◽

Sufficient Condition ◽

Highest Weight ◽

Finite Dimensional ◽

Finite Dimensionality ◽

A theory of highest weight modules over an arbitrary finite-dimensional Lie superalgebra is constructed. A necessary and sufficient condition for the finite-dimensionality of such modules is proved. Generic finite-dimensional irreducible representations are defined and an explicit character formula for such representations is written down. It is conjectured that this formula applies to any generic finite-dimensional irreducible module over any finite-dimensional Lie superalgebra. The conjecture is proved for several classes of Lie superalgebras, in particular for all solvable ones, for all simple ones, and for certain semi-simple ones.

Sur Le Produit Tensoriel D'algèbres

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-24181 ◽

2016 ◽

Vol 119 (1) ◽

pp. 5 ◽

Cited By ~ 1

Author(s):

Mohamed Tabaâ

Keyword(s):

Tensor Product ◽

Complete Intersection ◽

Noetherian Ring ◽

Regular Ring ◽

Noetherian Rings ◽

Sufficient Condition ◽

Produit Tensoriel ◽

Let $\sigma \colon A\rightarrow B$ and $\rho \colon A\rightarrow C$ be two homomorphisms of noetherian rings such that $B\otimes_{A}C$ is a noetherian ring. We show that if $\sigma$ is a regular (resp. complete intersection, resp. Gorenstein, resp. Cohen-Macaulay, resp. ($S_{n}$), resp. almost Cohen-Macaulay) homomorphism, so is $\sigma\otimes I_{C}$ and the converse is true if $\rho$ is faithfully flat. We deduce the transfer of the previous properties of $B$ and $C$ to $B\otimes_{A}C$, and then to the completed tensor product $B\mathbin{\hat\otimes}_{A}C$. If $B\otimes_{A}B$ is noetherian and $\sigma$ is flat, we give a necessary and sufficient condition for $B\otimes_{A}B$ to be a regular ring.

Extended Branciari quasi-b-distance spaces, implicit relations and application to nonlinear matrix equations

Journal of Inequalities and Applications ◽

10.1186/s13660-021-02736-2 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Reena Jain ◽

Hemant Kumar Nashine ◽

Vahid Parvaneh

Keyword(s):

Positive Definite Matrix ◽

Weak Limit ◽

Positive Definite ◽

Contractive Condition ◽

Sufficient Condition ◽

Nonlinear Matrix Equation ◽

Property A ◽

Underlying Space ◽

Hermitian Positive Definite Matrix ◽

Nonlinear Matrix

AbstractThis study introduces extended Branciari quasi-b-distance spaces, a novel implicit contractive condition in the underlying space, and basic fixed-point results, a weak well-posed property, a weak limit shadowing property and generalized Ulam–Hyers stability. The given notions and results are exemplified by suitable models. We apply these results to obtain a sufficient condition ensuring the existence of a unique positive-definite solution of a nonlinear matrix equation (NME) $\mathcal{X}=\mathcal{Q} + \sum_{i=1}^{k}\mathcal{A}_{i}^{*} \mathcal{G(X)}\mathcal{A}_{i}$ X = Q + ∑ i = 1 k A i ∗ G ( X ) A i , where $\mathcal{Q}$ Q is an $n\times n$ n × n Hermitian positive-definite matrix, $\mathcal{A}_{1}$ A 1 , $\mathcal{A}_{2}$ A 2 , …, $\mathcal{A}_{m}$ A m are $n \times n$ n × n matrices, and $\mathcal{G}$ G is a nonlinear self-mapping of the set of all Hermitian matrices that are continuous in the trace norm. We demonstrate this sufficient condition for the NME $\mathcal{X}= \mathcal{Q} +\mathcal{A}_{1}^{*}\mathcal{X}^{1/3} \mathcal{A}_{1}+\mathcal{A}_{2}^{*}\mathcal{X}^{1/3} \mathcal{A}_{2}+ \mathcal{A}_{3}^{*}\mathcal{X}^{1/3}\mathcal{A}_{3}$ X = Q + A 1 ∗ X 1 / 3 A 1 + A 2 ∗ X 1 / 3 A 2 + A 3 ∗ X 1 / 3 A 3 , and visualize this through convergence analysis and a solution graph.

WADA'S REPRESENTATIONS OF THE PURE BRAID GROUP

International Journal of Mathematics ◽

10.1142/s0129167x05003053 ◽

2005 ◽

Vol 16 (07) ◽

pp. 757-763

Author(s):

MOHAMMAD N. ABDULRAHIM

Keyword(s):

Braid Group ◽

Sufficient Condition ◽

Pure Braid Group ◽

Magnus Representation ◽

We consider the Magnus representation of the image of the pure braid group under the generalizations of the standard Artin representation, discovered by M. Wada. We will give a necessary and sufficient condition for the specialization of the reduced Wada's representation Gn(z) : Pn → GLn-1(ℂ) to be irreducible. It will be shown that for z = (z1,…,zn) ∈ (ℂ*)n, Gn(z) is irreducible if and only if z1k⋯znk ≠ 1. This is a generalization of our previous result concerning the irreducibility of the complex specialization of the reduced Gassner representation of Pn.

On the Irreducibility of Fourth Dimensional Tuba's Representation of the Pure Braid Group on Three Strands

European Journal of Pure and Applied Mathematics ◽

10.29020/nybg.ejpam.v11i3.3273 ◽

2018 ◽

Vol 11 (3) ◽

pp. 682-701

Author(s):

Hasan A. Haidar ◽

Mohammad N. Abdulrahim

Keyword(s):

Braid Group ◽

Sufficient Conditions ◽

Complex Numbers ◽

Closed Field ◽

Sufficient Condition ◽

Algebraically Closed Field ◽

Pure Braid Group ◽

We consider Tuba's representation of the pure braid group, $%P_{3} $, given by the map $\phi :P_{3}\longrightarrow GL(4,F)$, where $F$ is an algebraically closed field. After, specializing the indeterminates used in defining the representation to non- zero complex numbers, we find sufficient conditions that guarantee the irreducibility of Tuba's representation of the pure braid group $P_{3}$ with dimension $d=4$. Under further restriction for the complex specialization of the indeterminates, we get a necessary and sufficient condition for the irreducibility of $\phi

DOMAIN CONTINUITY FOR AN ELLIPTIC OPERATOR OF FOURTH ORDER

Communications in Contemporary Mathematics ◽

10.1142/s0219199702000543 ◽

2002 ◽

Vol 04 (01) ◽

pp. 1-14 ◽

Cited By ~ 2

Author(s):

MOHAMMED HAYOUNI ◽

MICHEL PIERRE

Keyword(s):

Boundary Value Problem ◽

Elliptic Operator ◽

Fourth Order ◽

Hausdorff Metric ◽

Sufficient Conditions ◽

Sufficient Condition ◽

Sobolev Embeddings ◽

Dimension 2 ◽

In this paper, we deal with the continuity with respect to the domain of the solutions of a first boundary value problem of fourth order in dimension 2 and 3. These dimensions are those involved in applications and are critical for this question of continuity. Indeed, continuity holds in dimension 1 thanks to Sobolev embeddings while homogenization may occur in dimensions higher than or equal to 4. Specific phenomena appear in dimensions 2 and 3. Here, we provide a necessary and sufficient condition for continuity with respect to the complementary Hausdorff metric. A main point is that the condition involves only the regularity of the limit domain and not the sequence of approximating domains. We then study various sufficient conditions for continuity in terms of the H2-capacity and we analyze a discontinuous case on an explicit example.

EMBEDDINGS AND -ENVELOPES OF EXACT OPERATOR SYSTEMS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972717000284 ◽

2017 ◽

Vol 96 (2) ◽

pp. 274-285

Author(s):

PREETI LUTHRA ◽

AJAY KUMAR

Keyword(s):

Tensor Product ◽

Sufficient Condition ◽

Operator System ◽

Operator Systems ◽

We prove a necessary and sufficient condition for embeddability of an operator system into ${\mathcal{O}}_{2}$. Using Kirchberg’s theorems on a tensor product of ${\mathcal{O}}_{2}$ and ${\mathcal{O}}_{\infty }$, we establish results on their operator system counterparts ${\mathcal{S}}_{2}$ and ${\mathcal{S}}_{\infty }$. Applications of the results, including some examples describing $C^{\ast }$-envelopes of operator systems, are also discussed.

On Weakly Positive Matrices

Canadian Journal of Mathematics ◽

10.4153/cjm-1963-036-4 ◽

1963 ◽

Vol 15 ◽

pp. 313-317 ◽

Cited By ~ 19

Author(s):

Eugene P. Wigner

Keyword(s):

Quadratic Form ◽

Present Article ◽

Positive Definite ◽

Sufficient Condition ◽

Hermitian Quadratic Form ◽

Positive Matrices ◽

Characteristic Values ◽

Image Position ◽

A matrix is said to be positive definite if it is hermitian and if all of its characteristic values are positive. It is well known, and easy to prove, that the necessary and sufficient condition for a matrix P to be positive definite is that its hermitian quadratic formwith any vector v ≠ 0 be positive. (This will imply, in the present article, that it is real.) It is easy to see from (1) that if P1 and P2 are positive definite, the same holds of a1P1 + a2P2 if a1 and a2 are positive numbers.