Largest eigenvalue of large random block matrices: A combinatorial approach

2017 ◽  
Vol 06 (02) ◽  
pp. 1750008
Author(s):  
Debapratim Banerjee ◽  
Arup Bose
Keyword(s):  
Block Structure ◽  
Extreme Points ◽  
Block Size ◽  
Relative Growth ◽  
Largest Eigenvalue ◽  
Block Matrices ◽  
Variance Formula ◽  
The Matrix ◽  
Random Block ◽  
Symmetric Block

We study the largest eigenvalue of certain block matrices where the number of blocks and the block size both increase with suitable conditions on their relative growth. In one of them, we employ a symmetric block structure with large independent Wigner blocks and in the other we have the Wigner block structure with large independent symmetric blocks. The entries are assumed to be independent and identically distributed with mean [Formula: see text] variance [Formula: see text] with an appropriate growth condition on the moments. Under our conditions the limit spectral distribution of these matrices is the standard semi-circle law. It is natural to ask if the extreme eigenvalues converge to the extreme points of its support, namely [Formula: see text]. We exhibit models where this indeed happens as well as models where the spectral norm converges to [Formula: see text]. Our proofs are based on combinatorial analysis of the behavior of the trace of large powers of the matrix.

scholarly journals Complex-analytic and matrix-analytic solutions for a queueing system with group service controlled by arrivals

2000 ◽  
Vol 13 (4) ◽  
pp. 415-427
Author(s):  
Lev Abolnikov ◽  
Alexander Dukhovny
Keyword(s):  
Block Size ◽  
Iteration Process ◽  
Analytic Solutions ◽  
Analytic Method ◽  
Service Time Distribution ◽  
Matrix Analytic Method ◽  
Queueing Process ◽  
The Matrix ◽  
Matrix Iteration ◽  
Necessary And Sufficient

A bulk M/G/1 system is considered that responds to large increases (decreases) of the queue during the service act by alternating between two service modes. The switching rule is based on two “up” and “down” thresholds for total arrivals over the service act. A necessary and sufficient condition for the ergodicity of a Markov chain embedded into the main queueing process is found. Both complex-analytic and matrix-analytic solutions are obtained for the steady-state distribution. Under the assumption of the same service time distribution in both modes, a combined complex-matrix-analytic method is introduced. The technique of “matrix unfolding” is used, which reduces the problem to a matrix iteration process with the block size much smaller than in the direct application of the matrix-analytic method.

Some exact results for dams with Markovian inputs

1976 ◽  
Vol 13 (02) ◽  
pp. 329-337
Author(s):  
Pyke Tin ◽  
R. M. Phatarfod
Keyword(s):  
Probability Distribution ◽  
Stationary Distribution ◽  
Special Form ◽  
Geometric Distribution ◽  
Transition Probabilities ◽  
Exact Results ◽  
Input Process ◽  
Largest Eigenvalue ◽  
The Matrix ◽  
Finite Dam

In the theory of dams with Markovian inputs explicit results are not usually obtained, as the theory depends very heavily on the largest eigenvalue of the matrix (pijzj ) where p ij are the transition probabilities of the input process. In this paper we show that explicit results can be obtained if one considers an input process of a special form. The probability distribution of the time to first emptiness is obtained for both the finite and the infinite dam, as well as the stationary distribution of the dam content for the finite dam. Explicit results are given for the case where the stationary distribution of the input process has a geometric distribution.

𝑉-law for random block matrices under the generalized Lindeberg condition

2019 ◽  
Vol 27 (3) ◽  
pp. 177-198
Author(s):  
Vyacheslav L. Girko
Keyword(s):  
Random Matrices ◽  
Stochastic Matrix ◽  
Block Matrices ◽  
Lindeberg Condition ◽  
Random Block ◽  
Random Block Matrices

Abstract The V-law under generalized Lindeberg condition for the independent blocks of random matrices having double stochastic matrix of covariances and different expectations of their array is proven.

V-density for eigenvalues of random block matrices with independent blocks whose entries have different variances and expectations

2019 ◽  
Vol 27 (3) ◽  
pp. 161-165
Author(s):  
Vyacheslav L. Girko ◽  
L. D. Shevchuk
Keyword(s):  
Random Matrices ◽  
Block Matrices ◽  
Lindeberg Condition ◽  
Random Block ◽  
Random Block Matrices

Abstract V-density under Lindeberg condition for the independent blocks of random matrices having different variances and expectations is found.

The Application of Matrix Compression Algorithm in Network Education Platform for the User Security Authentication

2014 ◽  
Vol 889-890 ◽  
pp. 1730-1736
Author(s):  
Nong Zheng
Keyword(s):  
Open Channel ◽  
Block Size ◽  
Matrix Decomposition ◽  
Compression Algorithm ◽  
Network Education ◽  
Cipher Text ◽  
Security Authentication ◽  
Matrix Compression ◽  
The Matrix ◽  
Education Platform

Using the matrix compression algorithm in the network education platform for the user information security certification is a good way. The sensitive user information is transfer in an open channel and it can be authentication for using the matrix compression/decompression, matrix decomposition/reduction algorithm. The client conduct random capture and compression by the user information been divided into a number of rectangular block size, corresponding to generate a key and cipher text. The server take out the corresponding data from the data queue of receiving and to extract In accordance with the key rules, then to reduce of information in corresponding positions. Thus the user identity information can be authenticity and integrity verification.

Ordered vacancy distribution in 2/1 mullite: a superspace model

2017 ◽  
Vol 73 (3) ◽  
pp. 377-388 ◽  
Author(s):  
Paul B. Klar ◽  
Noelia de la Pinta ◽  
Gabriel A. Lopez ◽  
Iñigo Etxebarria ◽  
Tomasz Breczewski ◽  
...  
Keyword(s):  
Single Crystal ◽  
Block Structure ◽  
Block Size ◽  
X Ray Diffraction ◽  
X Ray ◽  
Scale Factors ◽  
Vacancy Distribution

A mullite single crystal with composition Al4.84Si1.16O9.58 (2)exhibiting sharp satellite reflections was investigated by means of X-ray diffraction. For the refinement of a superspace model in the superspace groupPbam(α0½)0ssdifferent scale factors for main and satellite reflections were used in order to describe an ordered mullite structure embedded in a disordered polymorph. The ordered fraction of the mullite sample exhibits a completely ordered vacancy distribution and can be described as a block structure of vacancy blocks (VBs) that alternate with vacancy-free blocks (VFBs) alongaandc. The incommensurate nature of mullite originates from a modulation of the block size, which depends on the composition. The displacive modulation is analyzed with respect to the vacancy distribution and a possible Al/Si ordering scheme is derived, although the measurement itself is not sensitive to the Al/Si distribution. An idealized, commensurate approximation for 2/1 mullite is also presented. Comparison of the ordered superspace model with different preceding models reconciles many key investigations of the last decades with partly contradicting conclusions, where mullite was usually treated as either ordered or disordered instead of considering simultaneously different states of order.

Some exact results for dams with Markovian inputs

1976 ◽  
Vol 13 (2) ◽  
pp. 329-337 ◽  
Author(s):  
Pyke Tin ◽  
R. M. Phatarfod
Keyword(s):  
Probability Distribution ◽  
Stationary Distribution ◽  
Special Form ◽  
Geometric Distribution ◽  
Transition Probabilities ◽  
Exact Results ◽  
Input Process ◽  
Largest Eigenvalue ◽  
The Matrix ◽  
Finite Dam

In the theory of dams with Markovian inputs explicit results are not usually obtained, as the theory depends very heavily on the largest eigenvalue of the matrix (pijzj) where pij are the transition probabilities of the input process. In this paper we show that explicit results can be obtained if one considers an input process of a special form. The probability distribution of the time to first emptiness is obtained for both the finite and the infinite dam, as well as the stationary distribution of the dam content for the finite dam. Explicit results are given for the case where the stationary distribution of the input process has a geometric distribution.

An Extensive Simplex Method Mapping the Global Feasibility

2002 ◽  
Author(s):  
Liang Zhu ◽  
David Kazmer
Keyword(s):  
Performance Measures ◽  
Simplex Method ◽  
Extreme Points ◽  
Rational Engineering ◽  
Active Constraints ◽  
Decision Variables ◽  
The Matrix ◽  
Model Generator ◽  
And Performance ◽  
Feasible Space

Understanding the global feasibility of engineering decision-making problems is fundamental to the synthesis of rational engineering decisions. An Extensive Simplex Method is presented to solve the global feasibility for a linear decision model relating multiple decision variables to multiple performance measures, and constrained by corresponding limits. The developed algorithm effectively traverses all extreme points in the feasible space and establishes the graph structure reflecting the active constraints and their connectivity. The algorithm demarcates basic and nonbasic variables at each extreme point, which is exploited to traverse the active constraints and merge the degenerate extreme points. Finally, a random model generator is presented with the capability to control the matrix sparseness and the model degeneracy for an arbitrary number of decision variables and performance measures. The results indicate that all these model properties are significant factors affect the total number of extreme points, their connected graph, and the global feasibility.

Matrix Refinement in Mass Transport Across Fracture-Matrix Interface: Application to Improved Oil Recovery in Fractured Reservoirs

2021 ◽  
Author(s):  
Sarah Abdullatif Alruwayi ◽  
Ozan Uzun ◽  
Hossein Kazemi
Keyword(s):  
Fractured Reservoirs ◽  
Block Size ◽  
Grid Cell ◽  
Matrix Interface ◽  
Unsteady State ◽  
Matrix Block ◽  
Matrix Blocks ◽  
The Matrix ◽  
Saturation Distribution ◽  
State Pressure

Abstract In this paper, we will show that it is highly beneficial to model dual-porosity reservoirs using matrix refinement (similar to the multiple interacting continua, MINC, of Preuss, 1985) for water displacing oil. Two practical situations are considered. The first is the effect of matrix refinement on the unsteady-state pressure solution, and the second situation is modeling water-oil, Buckley-Leverett (BL) displacement in waterflooding a fracture-dominated flow domain. The usefulness of matrix refinement will be illustrated using a three-node refinement of individual matrix blocks. Furthermore, this model was modified to account for matrix block size variability within each grid cell (in other words, statistical distribution of matrix size within each grid cell) using a discrete matrix-block-size distribution function. The paper will include two mathematical models, one unsteady-state pressure solution of the pressure diffusivity equation for use in rate transient analysis, and a second model, the Buckley-Leverett model to track saturation changes both in the reservoir fractures and within individual matrix blocks. To illustrate the effect of matrix heterogeneity on modeling results, we used three matrix bock sizes within each computation grid and one level of grid refinement for the individual matrix blocks. A critical issue in dual-porosity modeling is that much of the fluid interactions occur at the fracture-matrix interface. Therefore, refining the matrix block helps capture a more accurate transport of the fluid in-and-out of the matrix blocks. Our numerical results indicate that the none-refined matrix models provide only a poor approximation to saturation distribution within individual matrices. In other words, the saturation distribution is numerically dispersed; that is, no matrix refinement causes unwarranted large numerical dispersion in saturation distribution. Furthermore, matrix block size-distribution is more representative of fractured reservoirs.

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