Matrix Ball Research Articles

Research Progress on Microstructure Design of CNTs Reinforced Al Matrix Composites

Carbon nanotubes (CNTs) have unique characteristics which make them a suitable reinforcing agent in Aluminum Metal Matrix composites. This review summarizes the work carried out in the field of the microstructure design of CNTs/Al composites. By designing the microstructure of CNTs, Al matrix and their composite interface, the problems of agglomeration, entanglement and dispersion of CNTs, wetting and bonding with Al matrix can be effectively solved, and mechanical properties such as Young’s modulus and yield strength can be improved. Such microstructural design includes metal plating on the surface of CNTs, deposition of nano-ceramic particles, Al matrix ball milling and flakes, grain refinement combined with bimodal-grain strengthening of coarse grains.

Orthonormal System for a Matrix Ball of the Second Type $\mathbb B_{m,n}^{(2)}$ and Its Skeleton (Shilov's Boundary) $\mathbb X_{m,n}^{(2)}$

Orthonormal System for a Matrix Ball of the Second Type $\mathbb B_{m,n}^{(2)}$ and Its Skeleton (Shilov's Boundary) $\mathbb X_{m,n}^{(2)}$

Laurent-Hua Luogeng Series with Respect to the Matrix Ball from Space Cn [m × m

The aim of this work is to obtain multidimensional analogs of the Laurent series with respect to the matrix ball from space Cn [m × m]. To do this, we first introduce the concept of a "layer of the matrix ball" from Cn [m × m], then in this layer of the matrix ball, we use the properties of integrals of the Bochner-Hua Luogeng type to obtain analogs of the Laurent ser

Holomorphic continuation into a matrix ball of functions defined on a piece of its skeleton

The question of the possibility of holomorphic continuation into some domain of functions defined on the entire boundary of this domain has been well studied. The problem of describing functions defined on a part of the boundary that can be extended holomorphically into a fixed domain is attracting more interest. In this article, we reformulate the problem under consideration: Under what conditions can we extend holomorphically to a matrix ball the functions given on a part of its skeleton? We describe the domains into which the integral of the Bochner—Hua Luogeng type for a matrix ball can be extended holomorphically. As the main result, we present the criterion of holomorphic continuation into a matrix ball of functions defined on a part of the skeleton of this matrix ball. The proofs of several results are briefly presented. Some recent advances are highlighted. The results obtained in this article generalize the results of L.A. Aizenberg, A.M. Kytmanov and G. Khudayberganov.

On volumes of matrix ball of third type and generalized Lie balls

On volumes of matrix ball of third type and generalized Lie balls

Maximum modulus principle for “holomorphic functions” on the quantum matrix ball

We describe the Shilov boundary ideal for a q-analog of the algebra of holomorphic functions on the unit ball in the space of n × n matrices and show that its C ⁎ -envelope is isomorphic to the C ⁎ -algebra of continuous functions on the quantum unitary group U q ( n ) .

Boundary Morera Theorem for the Matrix Ball of the Third Type

Boundary Morera Theorem for the Matrix Ball of the Third Type

Prediction of effect of reinforcement content, flake size and flake time on the density and hardness of flake AA2024-SiC nanocomposites using neural networks

Abstract It is known that the size and morphology of matrix powders and reinforcement content affects the physical and mechanical properties of ceramic nanoparticle reinforced metal matrix composites. Therefore, an artificial neural network model has been developed for the prediction of density and hardness of AA2024-SiC nanocomposites fabricated by flake powder metallurgy. The weight percentage of SiC nanoparticles, AA2024 matrix size and ball milling time were selected as the inputs and the sintered density and hardness of the flake AA2024-SiC nanocomposites as the output of the model. Prediction model ofAA2024-SiC nanocompositeswas able to predict the density and hardness with a mean absolute percentage error (MAPE) of 0.18% and 0.99%, respectively. The root mean score error (RMSE) of ANN model developed for AA2024-SiC nanocomposites were 0.06% and 0.835 for the density and hardness, respectively. The results of present study shows that the density and hardness of SiC nanoparticle reinforced AA2024 matrix flake nanocomposites can be predicted with high accuracy using neural network model.

Проектирование многозондовых устройств для тестирования электронных компонентов с шариковыми выводами

In the article design and technological features of multiprobe connecting device for testing the electronic components with matrix ball leads are described and substantiated. Such test fixture has probes made as two separated flatcontact lands that can be used for testing BGA/CSP components or microelectromechanical devices. Only in case, when two parts of probe contact lands are pressed to according lead of electronic component, electrical circuit between them closes. This fact confirms presence of contact between testing fixture probe and tested lead of BGA device and can be considered as way of testing reliability increasing. Due to the proposed new form of contact probe for electronic component testing it became possible to simplify the topology of connecting circuit board. Developed commutative board with ZIF connectors allows realizing multiprobe device connection to automated measuring systems, providing also the possibility of its future application to test other electronic components with more leads. Also the results of experimental and modeling research of developed device prototype are presented and explained. Obtained results substantiate the basic requirements for the multiprobe connecting device that should be observed during its contacting to the unit under test. Designed test fixture is more simple and cheap in comparison with its analogues. Also developed method of testing effectively provides the necessary contact pressure between test fixture and unit under test without hazard of its deformation which can appear in similar devices.

Boundary Version of the Morera Theorem for a Matrix Ball of the Second Type

Boundary Version of the Morera Theorem for a Matrix Ball of the Second Type

Representations of [formula omitted]-central [formula omitted]-Potapov functions in both nondegenerate and degenerate cases

Let J be an m × m signature matrix (i.e. J ∗ = J and J 2 = I m ) and let D : = { z ∈ C : | z | < 1 } . Denote P J ( D ) the class of all J -Potapov functions in D , i.e. the set of all meromorphic m × m matrix-valued functions f in D with J -contractive values at all points of D at which f is holomorphic. Further, denote P J , 0 ( D ) the subclass of all f ∈ P J ( D ) which are holomorphic at the origin. Let f ∈ P J , 0 ( D ) , and let f ( w ) = ∑ j = 0 ∞ A j w j be the Taylor series representation of f in some neighborhood of 0 . Then it was proved in [B. Fritzsche, B. Kirstein, U. Raabe, On the structure of J -Potapov sequences, Linear Algebra Appl., in press] that for each n ∈ N the matrix A n can be described by its position in a matrix ball depending on the sequence ( A j ) j = 0 n - 1 . The J -Potapov function f is called J -central if there exists some k ∈ N such that for each integer j ⩾ k the matrix A j coincides with the center of the corresponding matrix ball. In this paper, we derive left and right quotient representations of matrix polynomials for J -central J -Potapov functions in D . Moreover, we obtain recurrent formulas for the matrix polynomials involved in these quotient representations.

On the structure of J-Potapov sequences

Let J be an m × m signature matrix (i.e., J ∗ = J and J 2 = I m ) and let D ≔ { z ∈ C : | z | < 1 } . Denote P J , 0 ( D ) the set of all meromorphic m × m matrix-valued functions f in D which are holomorphic at 0 and take J -contractive values at all points of D at which f is holomorphic. Then it was proved in [B. Fritzsche, B. Kirstein, U. Raabe, On some interrelations between J -Potapov functions and J -Potapov sequences, OT Series, in press] that the Taylor coefficient sequences of the functions belonging to P J , 0 ( D ) are exactly the infinite J -Potapov sequences of complex m × m matrices. The main goal of this paper is to investigate the inner structure of infinite J -Potapov sequences. It turns out that such sequences have a clear geometric structure. Let ( A j ) j = 0 ∞ be a J -Potapov sequence. Then we will show that for each n ∈ N the matrix A n belongs to a certain matrix ball depending on the sequence ( A j ) j = 0 n - 1 . This observation leads us to the consideration of those J -Potapov sequences ( A j ) j = 0 ∞ for which there exists some k ∈ N such that for j ∈ { k , k + 1 , … } the matrix A j coincides with the center of the corresponding matrix ball. We will call these J -Potapov sequences J -central of order k . It turns out that J -central J -Potapov sequences have a recursive structure. We investigate connections between J -Potapov sequences and their J -Potapov–Ginzburg transforms, which are m × m Schur sequences. In particular, we derive formulas containing explicit interrelations between the parameters of the corresponding matrix balls. An essential consequence of these formulas is the observation that the concept of centrality is invariant with respect to J -Potapov–Ginzburg transform.

Toeplitz C*-Algebras on Super-Cartan Domains

We study Hilbert spaces of super-holomorphic functions (including anti-commuting Grassmann variables) in the setting of bounded symmetric domains, more precisely for the matrix ball of arbitrary size. Our main results concern the classication of irreducible representations of the associated Toeplitz C*- algebra and an explicit decomposition of the super-Bergman space as a direct sum of vector-valued (ordinary) Bergman spaces.

On a q-analog of the Wallach–Okounkov Formula

We obtain a q-analog of the well-known result on a joint spectrum of invariant differential operators with polynomial coefficients on a prehomogeneous vector space of complex n × n-matrices. We are motivated by applications to the problems of harmonic analysis in the quantum matrix ball: our main theorem can be used to prove the Plancherel formula (to be published).

Interrelations Between the Taylor Coefficients of a Matricial Carathéodory Function and Its Cayley Transform

Abstract The main goal of this paper is to discuss several interrelations between the Taylor coefficients of a q x q matrix-valued Carathéodory function and its Cayley transform which is a q x q matrix Schur function. Both Taylor coefficient sequences are described in terms of corresponding matrix balls. Hereby, we will obtain explicit formulas for the parameters of one matrix ball in terms of the other one. These expressions imply a one-to-one correspondence between central matricial Carathéodory functions and central matricial Schur functions which is established via Cayley transform.

L2-Poisson integral representations of solutions of the Hua system on the bounded symmetric domain SU( n, n)/ S( U( n)× U( n))

Let D={Z∈M(n; C );I n −ZZ ∗ positive definite} be the matrix ball of rank n and let H D be the associated Hua operator. For a complex number λ , such that Reiλ > n −1 we give a necessary and sufficient condition on solutions F of the following Hua system of differential equations on D : H D F(Z)=−(λ 2 +n 2 )F(Z).I n , to have an L 2 -Poisson integral representations over the Shilov boundary S of D . Namely, F is the Poisson integral of an L 2 -function on the Shilov boundary S if and only if its satisfies the following growth condition of Hardy type: ||F|| ∗,λ 2 = sup 0⩽r<1 [(1−r 2 ) −n(n−Reiλ) ∫ S |F(rU)| 2 dU]<+∞. In particular for λ =− in , this shows that every harmonic function F with respect to the Hua operator H D has an L 2 -Poisson representation over S if and only if its Hardy norm is finite.

Матричные аналоги $\text {\textrm {B}}$-функции и формула Планшереля для керн-представлений Березина

We obtain a family of matrix integrals which decompose to a product of Gamma-functions (they have some relations with S.G.Gindikin 'Beta', but generally speaking essentially differ from it). We obtain Plancherel formula for Berezin representations for all series of classical groups (for large values of parameters of representations). The Berezin representations are deformations of L^2 on Riemann noncompact symmetric spaces G/K defined by Berezin kernels, i.e powers of |det(1-zu^*)| (in matrix ball models). These representations also can be obtained by restrictions of holomorphic representations of some group Q containing G as symmetric subgroup. We also discuss models of noncompact Riemann symmetric spaces: matrix balls, matrix cones, matrix wedges, and sections of wedges.

Solutions du système d'équations différentielles de Hua et intégrales de Poisson-Shilov des fonctions de carré intégrable sur la frontière de Shilov de la boule matricielle de rang deux

Soit D la boule matricielle de rang deux et HD le système d'équations différentielles de Hua associé. Nous donnons une caractérisation des solutions F du système de Hua HDF(Z) = −(λ2 + 4)F(Z) · I, qui sont transformées de Poisson-Shilov de fonctions qui sont de carré intégrable sur le bord de Shilov du domaine symétrique borné D de type tube et de rang deux. We give a characterization of those solutions F of the Hua system of differential equations HDF(Z) = − (λ2 + 4)F(Z) · I, that are Poisson-Shilov integrals of L2-functions on the Shilov boundary of the rank two matrix ball.

Homogeneous Tuples of Multiplication Operators on Twisted Bergman Spaces

LetBthe Bergman kernel on the domainΩn, mofn×mcontractive complex matrices (m⩾n⩾1). Let W=Wn, mbe the associated Wallach set consisting of theλ⩾0 for whichBλ/(m+n)is (non-negative definite and hence) the reproducing kernel of a functional Hilbert space H(λ). Forλ∈W, we examine themn-tupleM(λ)of operators on H(λ)whose components are multiplications by themnco-ordinate functions. This tuple is homogeneous with respect to the group action ofPSU(n, m) on the matrix ball. Utilising this group action we are able to determine the set of allλ∈W for which (i) M(λ)is bounded, and for which (ii) M(λ)is (bounded and) jointly subnormal. Further, the joint Taylor spectrum ofM(λ)is determined for allλas in (i). The subnormality ofM(λ)turns out to be closely tied with the representation theory ofPSU(n, m). Namely,M(λ)is subnormal precisely when the natural (projective) representation ofPSU(n, m) on the twisted Bergman space H(λ)is a subrepresentation of an induced representation of multiplicity 1. Finally, we examine the values ofλfor whichM(λ)admits its Taylor spectrum as ak-spectral set, and obtain incomplete results on this question. This question remains open and interesting onn−1 gaps, that is, forλbelonging to the union ofn−1 pairwise disjoint open intervals. Most of the techniques developed in this paper are applicable to all bounded Cartan domains, though we stick to the matrix domainsΩn, mfor concreteness.

A new model for analysing thermal stress in granular composite

A double embedding model of inletting reinforcement grain and hollow matrix ball into the effective media of the particulate-reinforced composite is advanced. And with this model the distributions of thermal stress in different phases of the composite during cooling are studied. Various expressions for predicting elastic and elastoplastic thermal stresses are derived. It is found that the reinforcement suffers compressive hydrostatic stress and the hydrostatic stress in matrix zone is a tensile one when temperature decreases; when temperature further decreases, yield area in matrix forms; when the volume fraction of reinforcement is enlarged, compressive stress on grain and tensile hydrostatic stress in matrix zone decrease; the initial temperature difference of the interface of reinforcement and matrix yielding rises, while that for the matrix yielding overall decreases.