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

This 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.

An implicit relation, relational theoretic approach under w-distance and application to nonlinear matrix equations

We propose a new class of implicit relations and an implicit type contractive condition based on it in the relational metric spaces under w-distance functional. Further we derive fixed points results based on them. Useful examples illustrate the applicability and effectiveness of the presented results. We apply these results to discuss sufficient conditions ensuring the existence of a unique positive definite solution of the nonlinear matrix equation (NME) of the form $\mathcal{U}=\mathcal{Q} + \sum_{i=1}^{k}\mathcal{A}_{i}^{*} \mathcal{G}\mathcal{(U)}\mathcal{A}_{i}$ U = Q + ∑ i = 1 k A i ∗ G ( U ) 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 which are continuous in the trace norm. In order to demonstrate the obtained conditions, we consider an example together with convergence and error analysis and visualisation of solutions in a surface plot.

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

We 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

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.

On (Λ,Υ,ℜ)-Contractions and Applications to Nonlinear Matrix Equations

In this paper, we study the behavior of Λ , Υ , ℜ -contraction mappings under the effect of comparison functions and an arbitrary binary relation. We establish related common fixed point theorems. We explain the significance of our main theorem through examples and an application to a solution for the following nonlinear matrix equations: X = D + ∑ i = 1 n A i ∗ X A i − ∑ i = 1 n B i ∗ X B i X = D + ∑ i = 1 n A i ∗ γ X A i , where D is an Hermitian positive definite matrix, A i , B i are arbitrary p × p matrices and γ : H ( p ) → P ( p ) is an order preserving continuous map such that γ ( 0 ) = 0 . A numerical example is also presented to illustrate the theoretical findings.

Inequalities for the eigenvalues of the positive definite solutions of the nonlinear matrix equation

In this paper, the Hermitian positive definite solutions of the matrix equation Xs + A*X-tA = Q are considered, where Q is a Hermitian positive definite matrix, s and t are positive integers. Bounds for the sum of eigenvalues of the solutions to the equation are given. The equivalent conditions for solutions of the equation are obtained. The eigenvalues of the solutions of the equation with the case AQ = QA are investigated.

The Hermitian Positive Definite Solution of the Nonlinear Matrix Equation

In this paper, we study the nonlinear matrix equation $X^{s}\pm\sum^{m}_{i=1}A^{T}_{i}X^{\delta_{i}}A_{i}=Q$, where $A_{i}\;(i=1,2,\ldots,m)$ is $n\times n$ nonsingular real matrix and $Q$ is $n\times n$ Hermitian positive definite matrix. It is shown that the equation has an unique Hermitian positive definite solution under some conditions. Iterative algorithms for obtaining the Hermitian positive definite solution of the equation are proposed. Finally, numerical examples are reported to illustrate the effectiveness of algorithms.