Sign patterns that require or allow power-positivity

Identifying and classifying the potentially eventually positive sign patterns and the potentially eventually exponentially positive sign patterns of orders greater than 3 have been raised as two open problems since 2010. In this article, we investigate the potential eventual positivity of the class of double star-like sign patterns S(n,m,1) whose underlying graph G(S(n,m,1)) is obtained from the underlying graph G(S(n,m)) of the (n+m)-by-(n+m) double star sign patterns S(n,m) by adding an additional vertex adjacent to the two center vertices and removing the edge between the center vertices. We firstly establish some necessary conditions for a double star-like sign pattern to be potentially eventually positive, and then identify all the minimal potentially eventually positive double star-like sign patterns. Secondly, we classify all the potentially eventually positive sign patterns in the class of double star-like sign patterns S(n,m,1). Finally, as an application of our results about the potentially eventually positive double star-like sign patterns, we identify all the minimal potentially eventually exponentially positive sign patterns and characterize all the potentially eventually exponentially positive sign patterns in the class of double star-like sign patterns S(n,m,1).

An n -by-n sign pattern A is said to be potentially power-positive if there exists some A ∈ Q(A ) such that A is power-positive, i.e., Ak > 0 for some positive integer k . Catral, Hogben, Olesky and van den Driessche [Sign patterns that require or allow power-positivity, Electron. J. Linear Algebra, 19 (2010), 121-128] investigated the sign patterns that require or allow power-positivity. It has been shown that an n -by-n sign pattern A is potentially power-positive if and only if either A or −A is potentially eventually positive. But as the identification of sufficient and necessary conditions for potentially eventually positive sign patterns remains open, the characterization of potentially power-positive sign patterns is still open. In this paper, we introduce the minimal potentially power-positive sign patterns to classify the potentially powerpositive sign patterns. Some properties of minimal potentially power-positive sign patterns are presented. It is shown that for an n -by-n sign pattern A with at most n+ 1 negative entries, A is minimal potentially power-positive if and only if either A or −A is minimal potentially eventually positive. Finally, we classify the minimal potentially power-positive sign patterns of order n 3 . Mathematics subject classification (2010): 15B35, 05C50.

Coefficients of ergodicity and the scrambling index

Primitive digraphs with the largest scrambling index

In this paper, we consider solving the integer linear systems, i.e., given a matrix A in R^{m*n}, a vector b in R^m, and a positive integer d, to compute an integer vector x in D^n such that Ax <= b, where m and n denote positive integers, R denotes the set of reals, and D={0,1,..., d-1}. The problem is one of the most fundamental NP-hard problems in computer science. For the problem, we propose a complexity index h which is based only on the sign pattern of A. For a real r, let ILS_=(r) denote the family of the problem instances I with h(I)=r. We then show the following trichotomy: - ILS_=(r) is linearly solvable, if r < 1, - ILS_=(r) is weakly NP-hard and pseudo-polynomially solvable, if r = 1, and - ILS_=(r) is strongly NP-hard, if r > 1. This, for example, includes the existing results that quadratic systems and Horn systems can be solved in pseudo-polynomial time.

A digraph is primitive if some positive degree of it is a complete digraph, i.e. has all possible arcs. The smallest degree of this kind is called the exponent of the digraph. Given a primitive digraph, the elementary local exponent for some vertices $$u$$ and $$v $$ is the least positive integer $$\gamma $$ such that there exists a path from $$u $$ to $$v$$ of any length not less than $$\gamma $$ . For the transformation on the binary $$n $$ -dimensional vector space which is given by a set of $$n $$ coordinate functions, there corresponds some $$n $$ vertex digraph such that a pair $$(u,v) $$ is an arc if the $$v $$ th coordinate component of the transformation depends essentially on the $$u$$ th variable. Such a digraph we call a mixing digraph of the transformation. Under study are the mixing digraphs of $$n$$ -bit shift registers with a nonlinear Boolean feedback function (NFSR), $$n>1 $$ , which are widely used in cryptography. We find the exact formulas for the exponent and elementary local exponents of the $$n $$ -vertex primitive mixing digraph associated to an NFSR. The above results can be applied to evaluating the length of the blank run of the pseudo-random sequences generators based on NFSRs.

A bound on the generalized competition index of a primitive matrix using Boolean rank

For positive integers $n, L$ and $s$, consider the following two sets that both contain partitions of $n$ with the difference between the largest and smallest parts bounded by $L$: the first set contains partitions with smallest part $s$, while the second set contains partitions with smallest part at least $s+1$. Let $G_{L,s}(q)$ be the generating series whose coefficient of $q^n$ is difference between the sizes of the above two sets of partitions. This generating series was introduced by Berkovich and Uncu (2019). Previous results concentrated on the nonnegativity of $G_{L,s}(q)$ in the cases $s=1$ and $s=2$. In the present paper, we show the eventual positivity of $G_{L,s}(q)$ for general $s$ and also find a precise nonnegativity result for the case $s=3$.

Matrices with totally signed powers

Reducible sign k-potent sign pattern matrices

Irreducible sign k-potent sign pattern matrices

Let n,k be the positive integers, and let S_{k}(n) be the sums of the k-th power of positive integers up to n. By means of that we consider the evaluation of the sum of more general series by Bernstein polynomials. Additionally we show the reality of our idea with some examples.

The parameter l(G) for a primitive digraph G introduced by Lewin is the minimum positive integer k for which there are walks of both lengths k and k + 1 from some vertex u to some vertex v. We obtain upper bounds on l(G) if G is primitive ministrong, or G is just primitive and not necessarily ministrong, or G is primitive symmetric. We also discuss the numbers attainable as l(G).

Scrambling index set of primitive digraphs

CHAPTER 1. AN INTRODUCTION TO DATA AND FUNCTIONS. 1.1 Describing Single-Variable Data. 1.2 Describing Relationships between Two Variables. 1.3 An Introduction to Functions. 1.4 The Language of Functions. 1.5 Visualizing Functions. CHAPTER 2. RATES OF CHANGE AND LINEAR FUNCTIONS. 2.1 Average Rates of Change. 2.2 Change in the Average Rate of Change. 2.3 The Average Rate of Change Is a Slope. 2.4 Putting a Slant on Data. 2.5 Linear Functions: When Rates of Change Are Constant. 2.6 Visualizing Linear Functions. 2.7 Constructing Graphs and Equations of Linear Functions. 2.8 Special Cases. 2.9 Breaking the Line: Piecewise Linear Functions. 2.10 Constructing Linear Models of Data. 2.11 Looking for Links between Education and Earnings: A Case Study on Using Regression Lines. CHAPTER 3. WHEN LINES MEET: LINEAR SYSTEMS. 3.1 Interpreting Intersection Points: Linear and Nonlinear Systems. 3.2 Visualizing and Solving Linear Systems. 3.3 Reading between the Lines: Linear Inequalities. 3.4 Systems with Piecewise Linear Functions: Tax Plans. CHAPTER 4. THE LAWS OF EXPONENTS AND LOGARITHMS: MEASURING THE UNIVERSE. 4.1 The Numbers of Science: Measuring Time and Space. 4.2 Positive Integer Exponents. 4.3 Zero, Negative, and Fractional Exponents. 4.4 Converting Units. 4.5 Orders of Magnitude. 4.6 Logarithms as Numbers. CHAPTER 5. GROWTH AND DECAY: AN INTRODUCTION TO EXPONENTIAL FUNCTIONS. 5.1 Exponential Growth. 5.2 Exponential Decay. 5.3 Comparing Linear and Exponential Functions. 5.4 Visualizing Exponential Functions. 5.5 Exponential Functions: A Constant Percent Change. 5.6 More Examples of Exponential Growth and Decay. 5.7 Compound Interest and the Number e. 5.8 Semi-Log Plots of Exponential Functions. CHAPTER 6. LOGARITHMIC LINKS: LOGARITHMIC AND EXPONENTIAL FUNCTIONS. 6.1 Using Logarithms to Solve Exponential Equations. 6.2 Using Natural Logarithms to Solve Exponential Equations Base e. 6.3 Visualizing and Applying Logarithmic Functions. 6.4 Using Semi-Log Plots to Construct Exponential Models for Data. C H A P T E R 7. POWER FUNCTIONS. 7.1 The Tension between Surface Area and Volume. 7.2 Direct Proportionality: Power Functions with Positive Powers. 7.3 Visualizing Positive Integer Power Functions. 7.4 Comparing Power and Exponential Functions. 7.5 Inverse Proportionality: Power Functions with Negative Powers. 7.6 Visualizing Negative Integer Power Functions. 7.7 Using Logarithmic Scales to Find the Best Functional Model. CHAPTER 8. QUADRATICS AND THE MATHEMATICS OF MOTION. 8.1 An Introduction to Quadratic Functions: The Standard Form. 8.2 Visualizing Quadratics: The Vertex Form. 8.3 The Standard Form vs. the Vertex Form. 8.4 Finding the Horizontal Intercepts: The Factored Form. 8.5 The Average Rate of Change of a Quadratic Function. 8.6 The Mathematics of Motion. CHAPTER 9. NEW FUNCTIONS FROM OLD. 9.1 Transformations. 9.2 The Algebra of Functions. 9.3 Polynomials: The Sum of Power Functions. 9.4 Rational Functions: The Quotient of Polynomials. 9.5 Composition and Inverse Functions. 9.6 Exploring, Extending & Expanding. APPENDIX Student Data Tables for Exploration 2.1. Data Dictionary for FAM1000 Data. SOLUTIONS For all Algebra Aerobics and Check Your Understanding problems for odd-numbered problems in the Exercises and Chapter Reviews. All solutions are grouped by chapter ANS-1. INDEX.