Elements Aij Research Articles

ALGORITHMS FOR SOLVING MATRIX GAMES IN THE SPACE OF MIXED STRATEGIES

A number of game issues solutions is performed based on the payment matrix. In the respective model, conditionally, the first player attempts to win, while the second player strives not to lose. It is known that the analysis of this game is carried out based on the saddle element in the payment matrix. The element aij is called the saddle element, if it is the minimum of row number i and the maximum of column number j. In this case, the pure strategy of the first player corresponds to the i-th row of the matrix, and the pure strategy of the second player corresponds to the j-th column of the matrix. When there is no saddle element in the matrix, the analysis of the game is carried out in the space of mixed strategy. Experience shows that it is more efficient to perform this work by removing surplus rows and columns from the matrix. Keywords: matrix games, saddle element, pure strategies, mixed strategies, Braun-Robinson method.

Explicit Runge–Kutta–Nyström methods for the numerical solution of second order linear inhomogeneous IVPs

Runge–Kutta–Nyström (RKN) methods for the numerical solution of inhomogeneous linear initial value problems with constant coefficients are considered.A general procedure to construct explicit s-stage RKN methods with maximal order p=s+1, similar to the developed by the authors (Montijano et al., 2023) for the class of second order IVP under consideration, depending on the nodes ci,i=1,…,s is presented. This procedure requires only the solution of successive linear equations in the elements aij, 1≤j<i≤s of the matrix of coefficients A of the RKN method and avoids the solution of non linear equations.The remarkable fact is that using as free parameters the nodes ci,i=1,…,s with a quadrature relation, the s(s−1)/2 elements of matrix A can be computed by solving successively linear systems with coefficients depending on the nodes, so that if they are non-singular we get a unique s-stage method with maximal order s+1.We obtain an optimized six-stage seventh-order RKN method in the sense that the nodes are chosen so that minimize the leading term of the local error. Finally, some numerical experiments are presented to test the behaviour of the optimized RKN method with others with Radau and Lobatto nodes.

Branch and Bound method for binary problems with the procedure that reduces dimension of problems

The relationships between elements aij of coefficient matrix, elements di of vector d and elements cj of vector c in general binary problem are considered. Some of them allow us to establish the values of selected elements of feasible or optimal vector x. This procedure reduces the dimension of basic problem and can be install in branch and bound method. It gives positive effects.

Parameterized solution of linear interval parametric systems

Let A(p)x=b(p),p∈p denote a linear interval parametric system whose elements aij(p) and bi(p) are given functions of the entries of the parameter vector p. In this paper, a new type of solution x(p) of the parametric system considered is suggested. The new solution x(p) is of the form x(p)=Lp+a,p∈p where L is a real n×m matrix (n and m are the sizes of the square matrix A and vector p, respectively) whereas a is an interval vector. It is shown that the parameterized solution can be employed for solving various constrained optimization problems related to the linear interval parametric system given. Thus, an iterative method for determining the interval hull solution of A(p)x=b(p),p∈p for the case of linear parametric dependencies has been proposed which is based on the use of x(p) and a simple constraint satisfaction technique.

A strange property of the determinant of minors

If Mi j are minors of an n × n determinant D with elements aij (1 ≤ i, j ≤ n), then we prove the following relationship where Mk is any square sub matrix of order k of the matrix of minors M, δk is the determinant of a submatrix of D of order (n − k) taking the complements of the row/column positions that was used in Mk , and 1 ≤ k ≤ n. The above relation generalizes the trivial case for k = 1 and is consistent with the proven relationship for k = n when δn is taken as 1.

Modified iterative aggregation algorithms

The iterative aggregation methods were developed in 1960s due to the necessity of practical solution of problems in mathematical economy; so they admit an economical interpretation. These methods are still insufficiently investigated from the mathematical point of view and not well-known. The development of the iterative aggregation methods, as well as the term, is connected with the works of L. M. Dudkin and E. B. Ershov (see [1], pp. 155–158). The following view on these methods is very significant ([1], p. 158): “Since the theory of the method is developed insufficiently and no convergence conditions are obtained for it, many numerical tests were performed; in many cases the method proved to be efficient”. An important result for an one-parameter case is established in [1]. As applied to a system of linear algebraic equations of the form x = Ax+ b, the convergence conditions for this method adduced in [1] (p. 156) require, in particular, that the elements aij (i = 1, . . . , n; j = 1, . . . ,m) of the matrix A be positive, the components bi of the vector of free terms be nonnegative, and the spectral radius of the matrix Ameets the inequality ρ(A) < 1. In this paper we study several modifications of the iterative aggregation methods for the linear equation

Outer Interval Solution of the Eigenvalue Problem under General Form Parametric Dependencies

The paper addresses the problem of determining an outer interval solution of the parametric eigenvalue problem A(p)x = λx, A(p) ∈ ℝn×n for the general case where the matrix elements aij(p) are continuous nonlinear functions of the parameter vector p, p belonging to the interval vector p. A method for computing an interval enclosure of each eigenpair (λμ, x(μ)), μ = 1, ..., n, is suggested for the case where λμ is a simple eigenvalue. It is based on the use of an affine interval approximation of a ij (p) in p and reduces, essentially, to setting up and solving a real system of n or 2n incomplete quadratic equations for each real or complex eigenvalue, respectively.

Universality of the Edge Distribution of Eigenvalues of Wigner Random Matrices with Polynomially Decaying Distributions of Entries

We consider an ensemble of Wigner symmetric random matrices An={aij}, i,j=1, . . . ,n with matrix elements aij, being i.i.d. symmetrically distributed random variables Open image in new window We assume that Open image in new window and that

Cancellation of MEG and EEG signals with distribute source activation on realistic cortical surface

INTRODUCTION The spatial patterns of magnetoencephalographic (MEG) and electroencephalographic (EEG) signals generated by source currents in different parts of the brain typically overlap extensively. For simultaneous sources, overlapping patterns result in reduced resolution in inverse estimates of the underlying source distributions. In particular, cancellation due to superposition of fields with opposite signs reduces the measurable signal magnitude, thus reducing the effective signal-to-noise ratio per source. To quantify this cancellation, we computed a cancellation index as a function of the number of simultaneous, randomly selected sources on a realistic reconstruction of the human cortical surface. METHODS The cortical surface was represented by a mesh of ~330000 vertices, constructed from the white-matter segmentation of anatomical MRI [1]. A subset (~7000) of these vertices were chosen for the allowed locations of MEG/EEG sources [2], with approximately even 10-mm distance between neighboring points. The source orientations were assumed normal to the surface. The elements aij of a forward matrix describe the signal strength in sensor i generated by unit activity in source dipole j. The forward matrix was constructed using a boundary element model of the head [3]. A configuration of 306 MEG sensors was assumed (VectorView, Neuromag Ltd.: 102 magnetometers and 204 1st-order planar gradiometers). For EEG, a montage of 64 electrodes was used. The magnitude of signal cancellation due to overlapping field patterns generated by simultaneously active sources was described with the cancellation index C = i | j aij | / j i | aij |, where j runs over a subset of n source locations. For independent signals, C=1; for complete cancellation C=0. We calculated C for subsets of n sources, selected randomly from the 7000. This random selection was repeated 1000 times for each n, and the values for C were averaged. RESULTS Fig. 1 shows the cancellation index as a function of the number of randomly selected point sources on the cortical surface. The index decreased monotonically for both MEG and EEG. Cancellation effects were prominent even for a small number of sources: with 5 sources the index was ~50%. When all sources were active simultaneously, the cancellation index was 1% for MEG and 4% for EEG. The index was smaller (i.e., cancellation was larger) for EEG than for MEG when the number of sources was less than 200. For 1000 dipoles (corresponding to 14% of the total cortical surface) or more, the index saturated to 4-5% for EEG, but for MEG it decreased until the total number of dipoles was reached. DISCUSSION The results suggested that cancellation of MEG and EEG signals can be substantial for even a small number of simultaneously activated sources. This implies that the effective signal-to-noise ratio per source is likely to be reduced for simultaneously active sources; this has to be taken into account when evaluating the resolving capabilities of MEG/EEG source analysis techniques. REFERENCES 1. Dale AM, Fischl B, and Sereno MI. Neuroimage 1999; 9:179-194. 2. Liu AK, Belliveau JW, and Dale AM. PNAS 1998; 95:8945-8950. 3. Oostendorp TF and van Oosterom A. IEEE Trans Biomed Eng 1989; 36:382-391.

Direct Perturbation Analysis for Better Conservation

Only demographic studies can link in a quantitative way observed trends in population growth rate to parts of the life cycle. Population growth rate has been used for some time to assess the overall status of populations in conservation management. More recently there has been an increasing interest in elaborating methods that can assess the relative importance of different life-cycle phases for observed population growth trends (Silvertown et al. 1993, 1996; Caswell 1996a; Royama 1996). This kind of information is essential to identifying critical life-cycle stages and concentrating management efforts on them. One tool that has become widely used to extract information from demographic data is population projection matrices (Caswell 1978, 1989a; van Groenendael et al. 1988; Silvertown et al. 1993, 1996; Schemske et al. 1994). Silvertown et al. (1996) discuss elasticity matrices as a way to identify such critical life-cycle stages and as an aid to constructing rules of thumb useful in plant conservation. They warn against a naive interpretation of elasticities and argue that the state of the populationits rate of increase-must be taken into account. We want to focus attention on some additional problems with the use of elasticity values for identifying key phases of the life cycle and to propose an alternative method that reduces these problems. The rate of increase of a population (X) depends on ageor stage-specific vital rates such as survival, growth, and reproduction. Sensitivity analysis explores this dependence by calculating the effects on X of infinitely small absolute changes to matrix element aij, which represents particular transitions in the life cycle (Caswell 1978, 1989a). Hence, the sensitivity sii of a matrix element aii is

An O(n2) algorithm for the maximum cycle mean of an n×n bivalent matrix

For a general n×n real matrix (aij), standard O(n3) algorithms exist to find its maximum cycle mean, i.e., to find λ(A)=max(ai1i2+ai2i3+aikil)\k over all cyclic permutations (il,…,ik) of subsets of the set {1,2,…,n}. We consider the case when all the elements aij take values in a binary set {a,b} and construct an algorithm of complexity O(n2) which then suffices to find λ(A).

Arithmetic-geometric means of positive matrices

AbstractWe prove the existence of unique limits and establish inequalities for matrix generalizations of the arithmetic–geometric mean of Lagrange and Gauss. For example, for a matrix A = (aij) with positive elements aij, define (contrary to custom) A½ elementwise by [A½]ij = (aij)½. Let A(0) and B(0) be d × d matrices (1 &lt; d &lt; ∞) with all elements positive real numbers. Let A(n + 1) = (A(n) + B(n))/2 and B(n + 1 ) = (d−1A(n)B(n))½. Then all elements of A(n) and B(n) approach a common positive limit L. When A(0) and B(0) are both row-stochastic or both column-stochastic, dL is less than or equal to the arithmetic average of the spectral radii of A(0) and B(0).

Differentiability of Inverse Matrices

Consider a m x m matrix A consisting of elements aij as independent variables. Let f(A) be a I x I non-singular matrix with its elements as functions of aij. Okamoto (l), Memon and Okamoto (2), and Siotani (3) evaluate the effects of differential operator ê/êaij on some matrices f-I (A) at A = I(m), m x m identity matrix, for purpose of obtaining asymptotic expansions of certain probability distributions. Normally one inverts a matrix before applying differential operator. This paper proposes a method to deal with problems of this nature without inverting matrices. Some illustrations for use in multivariate statistical analysis are given

Derivatives of the spectral radius as a function of non-negative matrix elements

Let A = (aij) be a non-negative n × n matrix, that is, aij ≥ 0, i, j = 1, …, n; n &gt; 1. We write A ≥ 0. Let r = r(A) be the spectral radius of A; assume r &gt; 0 throughout to avoid trivial cases. Let be the mth derivative of r with respect to the element aij, all other elements of A being held constant.

Stability Matrices and the Solvability of Certain Systems of Linear Inequalities†

Let A be a real n × nmatrix with non-negative non-diagonal elements aij(I ≠ j). Ais called a stability matrix if and only if all of its eigenvalues have strictly negative real parts. It is proved that A is a stability matrix if and only if the system Ax > 0, x > 0has no non-trivial solution. Further, one and only one of the systems Ax 0and Ax > 0, x > 0has a non-trivial solution

A theorem on infinite positive matrices

1. Let A = (aij) be an infinite matrix with positive elements aij > 0, i, j = 0, 1, *, (matrices (aij) with aij > 0 will be called in the sequel positive matrices). It was proved in [3 ], that (1) if A is a finite positive matrix, a unique doubly stochastic matrix T exists such that T=D1AD2 where D1 and D2 are diagonal matrices with all elements on the diagonal positive and are unique up to a scalar factor. The method used in [3], and introduced first in [4], is a constructive one and consists in alternate normalizing rows and columns of A and proving the convergence of this procedure. Another proof of (1) was given in [1]. This second proof uses besides Brouwer's fixed point theorem the fact, that (2) the set { x = (x0, x1, , xn); xi real numbers, E = 1 and x?>01 is homeomorphic to an n-dimensional ball. Although a purely existential one, this second proof contains a statement about the existence of directions of fixed points for some mapping defined by help of a finite matrix A. In this paper we note that statement (1) does not hold for infinite matrices and prove a theorem generalizing properly (1) to the case of infinite matrices. Essentially, both proofs in [1] and in [3] could be, with some nontrivial changes, applied to give the desired generalization. The difficulty in generalizing the proof given in [3 ] consists i.a. in the fact that for an infinite matrix Ej aij (or Es aij) is not always finite. The idea of our proof is similar to that of [1 ] except that (2) is not used and that Brouwer's theorem is replaced by the theorem of Schauder (see [2 ]). In the sequel a matrix A = (aij) with aij > O will be called a positive matrix and a diagonal matrix with positive diagonal elements will be called a positive diagonal matrix. Finally bij = 0, i 5j; 1, i =j, will denote the delta of Kronecker.

Bounds for the characteristic exponents of linear systems

For an n-vector x = (xi) and n × n matrix A = (aij) with complex elements, let |x|2 = Σi|xi|2,|A|2 = ΣiΣj|aij|2. Also, M(A), m(A) denoteℜλ1,ℜλn, respectively, where λ1,…,λA are the eigenvalues of A arranged so that ℜλ1 ≥ … ≥ ℜλn. Throughout this paper A(t) denotes a matrix whose elements aij(t) are complex valued Lebesgue integrable functions of t in (0, T) for all T &gt; 0. Then M(A(t)), m(A(t)) are also Lebesgue integrable in (0, T) for all T &gt; 0. The characteristic exponent μ of a non-zero solution x(t) of the n × n system of differential equationscan be defined, following Perron ((12)), aswhere ℒ denotes lim sup as t → + ∞. When |A(t)| is bounded in (0,∞), μ is finite; in other cases it could be ± ∞.

Labor Skills and International Trade: Evaluating Many Trade Flows with a Single Measuring Device

PURSUING the methods by which Leontief discovered his renowned paradox, previous empirical studies of the relationship among factors, production, and trade have computed the content of a single country's trade from the factor requirements of its production processes.' The present study introduces a somewhat different method. The assumption is made that all manufactured goods traded, not only by the United States but by other countries as well, are produced with a single set of technical coefficients, namely the combinations of labor skills observed in each industry in the United States. It is postulated that the availability of labor skills determines patterns of international location and trade for a broad group of manufactured products, those not closely tied to natural resources. It is further supposed that the relationship between trade and skills for these goods will reveal itself in American skill requirements for reproducing trade flows. The first expectation seems plausible. Direct capital requirements are not as high in manufacturing as in most other activities.2 Labor appears to be less mobile internationally than liquid capital or capital goods. Generations of industrial experience and education may be required to build a skilled labor force. Let me spell out the basic method before examining its rationale. As in Leontief's computations, production functions are assumed to involve simple linear combinations of factors. There are no scale effects, and factors are perfectly divisible. We define Si as a quantity of the ith factor, such as labor of a specified skill class; Xj is the quantity of the jth product traded, so that XI, X2, X3, . . . , Xm describes the composition of a trade flow such as Japanese exports.3 To determine the skills required to produce this trade flow with American coefficients, we multiply our m-item trade vector X by an m X n matrix A, in which the elements aij (i= 1, 2, . . . n; j=1, 2,... m) represent average United States direct requirements for labor of n skill classes to produce a unit of output of each product. The result, the vector SI, S2, S3, . . . , SX, shows United States skill requirements for producing the goods in the trade flow.

The Stability of Linear Time-dependent Control Systems†

ABSTRACT It is shown that the trivial solution of the equation x'' = A(t)x, where the dash denotes the time derivative, is asymptotically stable if the rate of change of the elements aij(t) of A(t) is sufficiently slow. An explicit bound for the aij(t) is obtained when the matrix A has a special form.

Stability of the Solution to a Linear Programming Problem

Much effort has been devoted to finding the solution to the linear programming problem, although it is often the case in practice that some of the parameters are not known exactly. In this paper it is shown how, starting from the Simplex tableau, quantitative descriptions can be obtained without much further calculation of the way in which the elements aij (i = 1, 2,..., m; j = s) may vary without affecting the choice of optimal basis.