Cyclical functions and permutation matrices

In studying the population living in some patches, a single patch model usually generalizes into a meta-population model. Roughly speaking a meta-population is population consisting of some sub-populations, such as sub-population reflecting different life stages of the population or different habitat of patches. In a meta-population framework the dynamics of population within and between patches can be easily studied. This paper will discuss the matrix meta-population model that describes the growth and reproduction within the patches denoted by F. The matrix F is obtained by multiplying three matrices; the transpose of a permutation matrix, a block diagonal matrix in which its diagonal blocks is a matrix Aj, and the permutation matrix. The matrix Aj is the real non-negative matrix that represents the dynamics of population growth in the patch j. The permutation matrix has a special form that has properties that are more specific than the general permutation matrix. This paper will also gives a brief application for the matrix F. This will provide a space for exploring the characterization of the matrix F from algebraic point of view.

We compute the mean two-point spectral form factor and the spectral number variance for permutation matrices of large order. The two-point correlation function is expressed in terms of generalized divisor functions, which are frequently discussed in number theory. Using classical results from number theory and casting them in a convenient form, we derive expressions which include the leading and next to leading terms in the asymptotic expansion, thus providing a new point of view on the subject, and improving some known results.

We define some important properties of the permutation matrices of t-shuffle type which can be used to synthesize omega and inverse omega multistage interconnection networks. We also address some features of these networks compared to other networks of delta type. Finally, we propose one modification of omega network which decreases the number of stages in the network.

Escherichia coli hns, encoding the abundant nucleoid protein H-NS, was subjected to site-directed mutagenesis either to delete Pro115 or to replace it with alanine. Unlike the wild-type protein, hyperproduction of the mutant proteins did not inhibit macromolecular syntheses, was not toxic to cells and caused a less drastic compaction of the nucleoid. Gel shift and ligase-mediated circularization tests demonstrated that the mutant proteins retained almost normal affinity for non-curved DNA, but lost the wild-type capacity to recognize preferentially curved DNA and to actively bend non-curved DNA, a property of wild-type H-NS demonstrated here for the first time. DNase I foot-printing and in vitro transcription experiments showed that the mutant proteins also failed to recognize the intrinsically bent site of the hns promoter required for H-NS transcription autorepression and to inhibit transcription from the same promoter. The failure of the Pro115 mutant proteins to recognize curved DNA and to bend DNA despite their near normal affinity for non-curved DNA can be attributed to a defect in protein-protein interaction resulting in a reduced capacity to form oligomers observed in vitro and by a new in vivo test based on functional replacement by H-NS of the oligomerization domain (C-domain) of bacteriophage lambda cI repressor.

A two-stage fourth order time-accurate discretization for Lax–Wendroff type flow solvers II. High order numerical boundary conditions

On the matrix of rotation symmetric Boolean functions

We develop an algorithm for computing the solution of a large system of linear ordinary differential equations (ODEs) with polynomial inhomogeneity. This is equivalent to computing the action of a certain matrix function on the vector representing the initial condition. The matrix function is a linear combination of the matrix exponential and other functions related to the exponential (the so-called ϕ -functions). Such computations are the major computational burden in the implementation of exponential integrators, which can solve general ODEs. Our approach is to compute the action of the matrix function by constructing a Krylov subspace using Arnoldi or Lanczos iteration and projecting the function on this subspace. This is combined with time-stepping to prevent the Krylov subspace from growing too large. The algorithm is fully adaptive: it varies both the size of the time steps and the dimension of the Krylov subspace to reach the required accuracy. We implement this algorithm in the matlab function phipm and we give instructions on how to obtain and use this function. Various numerical experiments show that the phipm function is often significantly more efficient than the state-of-the-art.

In the memoir En metod att komma i analytisk besittning af de elliptiska funktionernat we have shown that the method given by Abel in his Recherches sur les fonctions elliptiques,+ which is historically the earliest method of deriving the elliptic functions, is also mathematically sound and leads to the desired result with perfect rigor. The method of Abel has the advantage over all other methods that it does not presuppose any theorems from the general theory of functions. It is consequently the most elementary of all mathematical processes which lead to a real command of the analytical properties of the elliptic functions. A few serious objections can be raised against Abel's own presentation, however. The most important of these is that the passage from the multiplication theorems to the infinite double series and double products is not sufficiently justified; in fact, it is based upon an argument which it is scarcely possible to make mathematically rigorous. This has been pointed out by Broch in his paper Om de elliptiske Funktioners Roekkeudvikling.? In this paper, as well as in his learned and suggestive book Traite Elementaire des Fonctions Elliptiques, ff there is developed a new derivation of this limiting passage which is perfectly distinct from that of Abel.

A list of a number of natural developments for the field of algebraic manipulation is given. Then the prospects for a general theory of functions defined by ordinary differential equations are discussed. The claim is made that recent developments in mathematics indicate that it should be possible to algorithmically generate many properties of solutions to differential equations. Such a theory is preferable to a less general effort to make algebraic manipulation systems knowledgeable about the usual special functions (e.g. exponential, hypergeometric).

NUMBERS OF experiments have indicated learning of nonsense material to be less efficient than learning of meaningful material. Marks and 01 lie (7) used second, third, fourth, and fifth order approximations of sentences as compared with reg ular textual material. They found that as material became more meaningful, the ability of college stu dents to learn the material was markedly increased. Meaning, or at least degree of familiarity, seems then to be a variable in learning to repeat written material. Meaning as considered in terms of com munication is not a simple variable. There are basically three points of view on the relation between the referent or signif?cate of a symbol, and difficulty in the communication proc ess. The first of these positions, multi-ordinality, is concerned with the accuracy of meaning. In es sence it indicates that a word does not communicate well if the symbol does not have a specific signifi cate. Thus in so far as the symbol will represent a greater number of significates the probability of confusion, and hence error, in communication is in creased. In essence this view espoused by Korzybski (4) and Johnson (3) says that each word is a homonym of itself and the greater the number of homonyms that any given sign represents, the more likely there is to be confusion concerning its meaning. Zipf's argument (14) that the number of meanings of a word increase as a function of the frequency of use of the word mitigates against a similar conclusion for learning to read or spell words. These functions are based upon repetitions, and as the number of indi vidual repetitions approximates the number of rep etitions in the population of words, the greater the number of meanings the easier the word should be to learn. The second position, concreteness or its inverse abstraction, indicates that the difficulty in commu nication lies in the tangibility of the signif?cate. Thus, the lowest level of difficulty or abstraction, is a concrete signif?cate which can be contacted by the senses. Words which are contacted by a smaller number of senses, or are defined only in terms of other words, are less concrete, and hence repre sent greater communication difficulty. Lorge (5) used a five point scale measuring the child's familiarity with the signif?cate and found that this predicted the readability of a passage. This conclusion, drawn from the area of communication, might be applied to learning language materials where specific meanings are required for compre hension of the language. The third position is that familiarity with the symbol implies familiarity with the signif?cate. Thus, those words which occur most frequently in the language will tend, all things being equal, to communicate more specifically than those words which do not occur in the language. There has been considerable work done with frequency of occurrence, particularly in relation to the organization of children's material. The most significant work was perhaps done by Thorndike (10) (11) and Lorge and Thorndike (6) in the Teachers Word Books. Thorndike is basically responsible for the notion that generalized familiarity makes for ease in learning. Whether this notion was originally true or not cannot be tested within public school pop ulations, for these children have been exposed to words in relation to their frequency of occurrence, and hence, because they have had greater learning exposure, to more frequent words. The work of Noble (8) seems to answer this question for adults. His sample tended to know the meanings of words which were familiar to them when meaning was measured in word associations.

The bending-torsional flutter characteristics of a cantilever composite wing subjected to engine thrust are presented. The engine thrust modeled as a follower force and the wing modeled as a two degree of freedom beam. In order to consider the spanwise and chordwise location and the properties of the engine location, the generalized function theory is used. Unsteady Theodorsen aerodynamic theory in time domain, is used. The general laminate composite theory is used for modeling the effects of the ply angles on the flutter boundary. The Ritz method is subsequently applied to convert the partial differential equations into a set of ordinary differential equations. In order to precisely consider the location of the engine Dirac delta function is used. Moreover, the numerical results are compared with the published results and excellent agreement is observed. Numerical results highlighting the effect of engine thrust on the wing instability, is presented.

Dynamic elastography holds great promise for biological tissues characterization. Resulting from the radiation force induced by focused ultrasound beam, plane shear waves are generated within the medium and imaged with an ultrafast ultrasound scanner. Known as Supersonic Shear Imaging (SSI) technique, this method allows, from the measurements of shear wave velocities, to compute shear modulus (mu) maps. Beside, in order to improve tissue diagnostic, the evaluation of the nonlinear elastic moduli could be of some interest. Recently a new formulation of the nonlinear equation describing the propagation of plane shear waves in isotropic soft incompressible solids have been developed using a new expression, up to the fourth order, of the strain energy density (e): e = muI2 + A/3I3, + DI2 2. Where I2, I3 are invariants defined by Landau of the strain tensor and A, D the third and fourth order shear elastic constants. It has been shown that the nonlinearity parameter depends only on three coefficients betas = betas (mu, A, D). To date, no measurement of the parameter D have been carried out in incompressible media. In order to estimate the nonlinear parameter A, this theoretical background on soft incompressible solids is applied to the acoustoelasticity theory. Such analysis gives the variations of shear wave speed as a function of the applied stress and leads to measure both the linear shear modulus (mu) and the third order shear modulus (A). Taking advantages of the SSI technique, an acoustoelasticity experiment is performed in different incompressible soft media (agar-gelatin based phantoms). In addition, to create finite amplitude plane shear waves, the SSI technique is replaced by a vibrator applied at the surface of the phantoms. Thanks to the ultrasound ultrafast imaging system, the third harmonic component is generated by nonlinearity is measured as a function of the propagation distance. Then by comparing experiments and analytical expression of the third harmonic component given by a perturbation method, the nonlinear parameter betas is deduced. Finally, the combination of these experiments with results obtained in acoustoelasticity leads to the determination of the fourth order elastic modulus (D). First, measurements of the A modulus reveal that while the behavior of phantoms is quite close from a linear point of view, their nonlinear modulus A are quite different. Applied to acoustoelasticity, the SSI technique provides potential medical applications in in vivo conditions for nonlinear characterization of biological tissues. Second, results from the complete procedure reveal a variation of the nonlinear behavior as a function of the gelatin concentration increasing. This set of experiments provides the characterization, up to the fourth order, of the nonlinear shear elastic moduli in incompressible soft media.

The volume of the parallelepiped spanned by functions of a square complex matrix defined on its spectrum is considered. The sharp lower bound of such volumes is found.

Network is considered naturally as a wide range of different contexts, such as biological systems, social relationships as well as various technological scenarios. Investigation of the dynamic phenomena taking place in the network, determination of the structure of the network and community and description of the interactions between various elements of the network are the key issues in network analysis. One of the huge network structure challenges is the identification of the node(s) with an outstanding structural position within the network. The popular method for doing this is to calculate a measure of centrality. We examine node centrality measures such as degree, closeness, eigenvector, Katz and subgraph centrality for undirected networks. We show how the Katz centrality can be turned into degree and eigenvector centrality by considering limiting cases. Some existing centrality measures are linked to matrix functions. We extend this idea and examine the centrality measures based on general matrix functions and in particular, the logarithmic, cosine, sine, and hyperbolic functions. We also explore the concept of generalised Katz centrality. Various experiments are conducted for different networks generated by using random graph models. The results show that the logarithmic function in particular has potential as a centrality measure. Similar results were obtained for real-world networks.

To investigate the errors associated with extrapolating the topography of the central cornea into the peripheral cornea. Corneal topography data previously acquired from 92 young adult subjects was used. These data were collected and analyzed by using a method that allows central and peripheral maps to be combined to produce an extended corneal topography map that reaches close to the limbus. By using the central 5-, 6-, and 7-mm corneal topography that would normally be acquired in a single measurement, various mathematical techniques were applied to extrapolate the central topography into the periphery. These mathematical techniques included the use of simple conics, generalized conics, cosine hyperbolic functions, and a set of fourth, sixth, and eighth radial order polynomials. The root-mean-square error between the extrapolated topography and true extended topography at various diameters was then compared. The root-mean-square errors in the outer extrapolated region ranged from 30 to 220 microm, depending on the fit function used. The lowest errors were achieved with the conic fits, cosine hyperbolic functions, and fourth and sixth radial order polynomials, with root-mean-square error levels of approximately 30 to 40 microm. The variables of global versus meridional fitting, centering the data on the geometric center versus vertex normal, the regional quadrant, and the diameter of the central topography used had relatively little influence on the fit errors. Extrapolation from central corneal topography into the periphery by using radial polynomials of the fourth order provides the lowest levels of fit errors for normal corneas.