Real Entries Research Articles

Harnack inequality and maximum principle for degenerate Kolmogorov operators in divergence form

We prove the parabolic strong maximum principle and the Harnack inequality for classical solutions to degenerate second order partial differential equations of the formLu:=∑i,j=1m0∂xi(aij∂xju)+∑j=1m0bj∂xju+cu+∑i,j=1Nbijxj∂xiu−∂tu=f, where 1≤m0≤N, A0=(aij)i,j=1,…,m0 is a bounded, symmetric and uniformly positive matrix and the matrix B:=(bij)i,j=1,…,N has real constant entries. Moreover, we assume low regularity (i.e. Hölder continuity) on the coefficients aij, bj and c, for i,j=1,…,m0. We point out the proofs of our main results only rely on the classical theory developed for harmonic functions.

Matrix Transforms Into the Set of a-absolutely Convergent Sequences with Speed and the Regularity of Matrices on the Sub-spaces of C

Let α > 1. The α-absolute convergence with speed, where the speed is defined by a monotonically increasing positive sequence µ, has been studied in the present paper. Let l µ α be the set of all α-absolutely µconvergent sequences and X a sequence space defined by another speed λ. Necessary and sufficient conditions for a matrix A (with real or complex entries) to map X into l µ α have been presented. It is proved as an example that the Zweier matrix Z1/2 satisfies these necessary and sufficient conditions for certain speeds λ and µ. The notion of regularity on the subspace X of the set c of converging sequences is defined, and also, necessary and sufficient conditions for a matrix A to be regular on certain X ⊂ c are presented. It has also been shown that there exists an irregular matrix, which is regular on the subspace X of c.

On The Algebraic Properties of 3-Cyclic Refined Neutrosophic Matrices with Real Entries

On The Algebraic Properties of 3-Cyclic Refined Neutrosophic Matrices with Real Entries

The Mathematical Formulas for Inverting Plithogenic Matrices of Special Orders Between 20 and 24

This paper is dedicated to study the Invertibility properties of all plithogenic square matrices of high orders between 20 and 24, where we present the mathematical conditions and formulas for computing the inverses of all plithogenic square matrices with real plithogenic entries. This goal will be completed by proving many theorems that describe the Invertibility of all classic matrix parts of the corresponding symbolic m-plithogenic matrix.

On The Computational Properties of 3-Cyclic and 4-Cyclic Refined Matrices and the Diagonalization Algorithm

This paper is concerned with studying the matrix computations of 3-cyclic refined neutrosophic matrices and 4-cyclic refined neutrosophic matrices with 3cyclic4-cyclic real entries, where we introduce a novel method to compute eigenvalues and vectors of these matrix classes. Also, we provide a novel algorithm for diagonalization these matrices and to determine whether an n-cyclic refined matrix is diagonalizable or not for n=3, 4.

Algorithms for Computing M-Plithogenic Eigen Values and Vectors for Some Special Values of M

Plithogenic matrices are considered as advanced mathematical generalizations of classical square matrices. One of the most research problems that is related to them is finding the eigen-values and vectors. This paper aims to present an easy algorithm to find all eigen-values and eigen-vectors for 10 different symbolic square n-plithogenic matrices with plithogenic real entries, where n is between 20 and 29. All algorithms are explained through clear theorems and proofs.

MATRIX TRANSFORMS OF SPEED-SUMMABLE AND SPEED-BOUNDED SEQUENCES

First we recall the notions of $A^{\lambda}$-boundedness, $A^{\lambda}$-summability and the absolute $A^{\lambda}$-summability of sequences, and the notion of $\lambda$-reversibility of $A$, where $A$ is a matrix with real or complex entries and $\lambda$ is the speed of convergence, \ie; a monotonically increasing positive sequence. Let $B$ be a lower triangular matrix with real or complex entries, and $\mu=(\mu_{n})$ be another speed of convergence. We find necessary and sufficient conditions for a matrix $M$ (with real or complex entries) to map the set of all $A^{\lambda}$-bounded sequences (for a normal matrix $A$) into the set of all absolutely $B^{\mu}$-summable sequences, and the set of all $A^{\lambda}$-summable sequences (for a $\lambda$-reversible matrix $A$) into the set of all absolutely $B^{\mu}$-summable sequences.

On Novel Results about the Algebraic Properties of Symbolic 3-Plithogenic and 4-Plithogenic Real Square Matrices

Symbolic n-plithogenic sets are considered to be modern concepts that carry within their framework both an algebraic and logical structure. The concept of symbolic n-plithogenic algebraic rings is considered to be a novel generalization of classical algebraic rings with many symmetric properties. These structures can be written as linear combinations of many symmetric elements taken from other classical algebraic structures, where the square symbolic k-plithogenic real matrices are square matrices with real symbolic k-plithogenic entries. In this research, we will find easy-to-use algorithms for calculating the determinant of a symbolic 3-plithogenic/4-plithogenic matrix, and for finding its inverse based on its classical components, and even for diagonalizing matrices of these types. On the other hand, we will present a new algorithm for calculating the eigenvalues and eigenvectors associated with matrices of these types. Also, the exponent of symbolic 3-plithogenic and 4-plithogenic real matrices will be presented, with many examples to clarify the novelty of this work.

Replica approach to the generalized Rosenzweig-Porter model

The generalized Rosenzweig-Porter model with real (GOE) off-diagonal entries arguably constitutes the simplest random matrix ensemble displaying a phase with fractal eigenstates, which we characterize here by using replica methods. We first derive analytical expressions for the average spectral density in the limit in which the size NN of the matrix is large but finite. We then focus on the number of eigenvalues in a finite interval and compute its cumulant generating function as well as the level compressibility, i.e., the ratio of the first two cumulants: these are useful tools to describe the local level statistics. In particular, the level compressibility is shown to be described by a universal scaling function, which we compute explicitly, when the system is probed over scales of the order of the Thouless energy. Interestingly, the same scaling function is found to describe the level compressibility of the complex (GUE) Rosenzweig-Porter model in this regime. We confirm our results with numerical tests.

Exchange graphs for mutation-finite non-integer quivers

Skew-symmetric non-integer matrices with real entries can be viewed as quivers with non-integer arrow weights. Such quivers can be mutated following the usual rules of quiver mutation. Felikson and Tumarkin show that mutation-finite non-integer quivers admit geometric realisations by partial reflections. This allows us to define a geometric notion of seeds and thus to define the exchange graphs for mutation classes. In this paper we study exchange graphs of mutation-finite quivers. The concept of finite type generalises naturally to mutation-finite non-integer quivers. We show that for all non-integer quivers of finite type there is a well-defined notion of an exchange graph, and this notion is consistent with the classical notion of exchange graph of integer mutation types coming from cluster algebras. In particular, exchange graphs of finite type quivers are finite. We also show that exchange graphs of rank 3 affine quivers are finite modulo the action of a finite-dimensional lattice (but unlike the integer case, the rank of the lattice is higher than 1 for non-integer quivers).

On polynomials satisfying power inequality for numerical radius

Let A be a unital C⁎ algebra and for every a∈A, r(a) denote the numerical radius of a∈A. The power inequality for numerical radius states that for every polynomial P(z)=zn and a∈A the inequality P(r(a))≥r(P(a)) holds. In this paper, we get a characterization of polynomials with real coefficients that satisfy the power inequality on all 2×2 matrices with real entries. We also characterize all polynomials that satisfy the power inequality on every commutative unital C⁎ algebra.

On The Algebraic Properties of Symbolic n-Plithogenic Matrices For n=5, n=6

The main goal of this paper is to study the properties of symbolic 5-plithogenic matrices and symbolic 6-plithogenic matrices with real entries, where an algebraic view of their properties and relations will be presented and discussed. Also, we present many theorems that concern the computing of their eigenvalues and eigenvectors and their connection with classical ordinary matrices. Many related examples will be provided to clarify the validity of our work.

On The 15-Plithogenic Square Real Matrices

This paper is dedicated to study the algebraic properties and meta-structures that are related to symbolic 15-plithogenic with symbolic plithogenic real entries, where symbolic 15-plithogenic eigenvectors and values will be discussed and presented in terms of theorems. As well as, the computation of determinants, inverses, and scalar values.

Asymptotics of the inertia moments and the variance conjecture in Schatten balls

We study the first and second orders of the asymptotic expansion, as the dimension goes to infinity, of the moments of the Hilbert-Schmidt norm of a uniformly distributed matrix in the p-Schatten unit ball. We consider the case of matrices with real, complex or quaternionic entries, self-adjoint or not. When p>3, this asymptotic expansion allows us to establish a generalized version of the variance conjecture for the family of p-Schatten unit balls of self-adjoint matrices.

Alliance chain-based simulation on a new clinical research data pricing model.

BackgroundMulticenter clinical research faces many challenges, including how to quantitatively evaluate the data contribution of each research center. However, few data pricing model meets the requirements to the scenario. Thus, a suitable mechanism to measure the data value for clinical research is required.MethodsExtensive documents were acquired and analyzed, including a rare disease list from the National Health Commission, data structures of the electronic medical records (EMR) system, diagnosis-related groups (DRGs) regulations from the Health Commission of Zhejiang Province, and the Clinical Service Price List of Zhejiang Province. Nine senior experts were invited as consultants from hospital and enterprises with professional field of clinical research, data governance, and health economics. After brainstorming and expert evaluation, seven data attributes were identified as the main factors affecting the value of medical data. Different weights were assigned for each attribute based on its influence on data value. Each attribute was quantized to an index based on proposed algorithms. The data value models for chronic diseases and other diseases were distinguished given the different sensitivity of data timeliness. A simulation system using blockchain and federated learning techniques was constructed to verify the data pricing model in the scenario of clinical research.ResultsA comprehensive clinical data pricing model is proposed and the simulation of three research centers with 50 million real clinical data entries was conducted to verify its effectiveness. It demonstrates that the proposed model can compute medical data value quantitatively.ConclusionsQuantitative evaluation of the value of medical data for multicenter clinical research based on the proposed data pricing model works well in simulation. This model will be improved by real-world applications in the near future.

A new perspective on cooperative control of multi-agent systems through different types of graph Laplacians

In the field of systems and control, many cooperative control problems of multi-agent systems have been actively studied in the past two decades. This article aims to organize extensive existing work on different cooperative control problems into three categories, based on three different types of graph Laplacian matrices involved. A Laplacian matrix is an important representation of graph topology, and depending on the field of its entries, there are three types: ordinary Laplacian (non-negative diagonal entries and non-positive off-diagonal entries), signed Laplacian (arbitrary real entries), and complex Laplacian (arbitrary complex entries). Each type of graph Laplacian is useful in modeling and solving a different set of cooperative control problems. In particular, their algebraic properties are fundamental in characterizing stability and performance of the respective solution algorithms. To our best knowledge, organizing the literature on multi-agent cooperative control through the lens of different graph Laplacians is new.

Fluctuations in the spectrum of non-Hermitian i.i.d. matrices

We consider large non-Hermitian random matrices X with independent identically distributed real or complex entries. In this paper, we review recent results about the eigenvalues of X: (i) universality of local eigenvalue statistics close to the edge of the spectrum of X [Cipolloni et al., “Edge universality for non-Hermitian random matrices,” Probab. Theory Relat. Fields 179, 1–28 (2021)], which is the non-Hermitian analog of the celebrated Tracy–Widom universality; (ii) central limit theorem for linear eigenvalue statistics of macroscopic test functions having 2 + ϵ derivatives [Cipolloni et al., “Central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices,” Commun. Pure Appl. Math. (published online) (2021) and Cipolloni et al., “Fluctuation around the circular law for random matrices with real entries,” Electron. J. Probab. 26, 1–61 (2021)]. The main novel ingredients in the proof of these results are local laws for products of two resolvents of the Hermitization of X at two different spectral parameters, coupling of weakly dependent Dyson Brownian motions, and the lower tail estimate for the smallest singular value of X − z in the transitional regime |z| ≈ 1 [Cipolloni et al., “Optimal lower bound on the least singular value of the shifted Ginibre ensemble,” Probab. Math. Phys. 1, 101–146 (2020)].

Four-qubit states generated by Clifford gates

The Clifford group is the set of gates generated by controlled not (CNOT) gates and the two local gates [Formula: see text] and [Formula: see text]. We will say that an [Formula: see text]-qubit state is a Clifford state if it can be prepared using Clifford gates, this is, [Formula: see text] is Clifford if [Formula: see text] where [Formula: see text] is a Clifford gate. In this paper, we study the set of all 4-qubit Clifford states. We prove that there are 293760 states, each of which has entanglement entropy equal to 0, [Formula: see text], 1, [Formula: see text], or [Formula: see text]. We also show that any pair of these states can be connected using local gates and at most 3 CNOT gates. We also study the Clifford states with real entries under the action of the subgroup [Formula: see text] of Clifford gates with real entries. This time we show that every pair of Clifford states with real entries can be connected with at most 5 CNOT gates and local gates in [Formula: see text]. To understand the action of the 12 different CNOT gates, we partition the Clifford states into orbits using the equivalence relation: two states are equivalent if they differ by a local Clifford gate. We label each orbit in such a way that it is easy to see the effect of the CNOT gates. Diagrams and tables explaining the action of the CNOT gates on all the orbits are presented in the paper. The link https://youtu.be/42MI6ks2_eU leads to a YouTube video that explains the most important results in this paper.

Extremal eigenvalues of sample covariance matrices with general population

We consider the eigenvalues of sample covariance matrices of the form Q=(Σ1/2X)(Σ1/2X)∗. The sample X is an M×N rectangular random matrix with real independent entries and the population covariance matrix Σ is a positive definite diagonal matrix independent of X. Assuming that the limiting spectral density of Σ exhibits convex decay at the right edge of the spectrum, in the limit M,N→∞ with N/M→d∈(0,∞), we find a certain threshold d+ such that for d>d+ the limiting spectral distribution of Q also exhibits convex decay at the right edge of the spectrum. In this case, the largest eigenvalues of Q are determined by the order statistics of the eigenvalues of Σ, and in particular, the limiting distribution of the largest eigenvalue of Q is given by a Weibull distribution. In case d<d+, we also prove that the limiting distribution of the largest eigenvalue of Q is Gaussian if the entries of Σ are i.i.d. random variables. While Σ is considered to be random mostly, the results also hold for deterministic Σ with some additional assumptions.

Inverse spectral problems for arrowhead matrices

The problem of constructing a matrix by its spectral information is called inverse eigenvalue problem (IEP) which arises in a variety of applications. In this paper, we study an IEP for arrowhead matrices in different cases. The problem involves constructing of the matrix by some eigenvalues of each of the leading principal submatrices and one eigenpair. We will also investigate this problem and its variants in the cases of matrix entries being real, nonnegative, positive definite, complex and equal diagonal entries. To solve the problems, a new method to establish a relationship between the IEP and properties of symmetric and general form of matrices is developed. The necessary and sufficient conditions of the solvability of the problems are obtained. Finally, some numerical examples are presented.