Mathematical Analysis of a Soccer Game. Part I: Individual and Collective Behaviors

Detailed time series analysis of a soccer match is given based on the detailed data of the 2D motions of all 22 players and of the ball for the match. The whole analysis includes two parts. In Part I, the individual and collective behaviors of the players of the two teams as well as the motion of the ball are presented as various time series. Geometrical centers, radii, expansion speeds, possession functions of the two teams are defined and calculated as functions of time. Major ranges of all players as well as of different groups of players (defenders, midfielders, forwards) of the two teams during the entire first half, the attacking phase of team A and the attacking phase of team B are calculated, respectively, showing the structures of the two teams during different phases. Distance coverage of each player and the mean distances covered by different groups of players (defenders, midfielders, forwards) during different phases are calculated. The time portions of possession of the ball by each team and the time portions of different phases are also calculated. In Part II, energy and spectral analysis and various correlations will be derived. The relation between various parameters and potential indicators will be discussed. The major purpose of the present study is to offer some general mathematical tools for the detailed analysis and to reveal some general features of soccer match when the detailed 2D data are available. The results would offer the raw materials for various potential indicators which may eventually be used in the coaching process to enhance the performance and in the prediction of the results of soccer matches.     

Rankings and values for team games

In this article we attack several problems that arise when a group of individuals is organized in several teams with equal number of players in each one (e.g., for company work, in sports leagues, etc). We define a team game as a cooperative game v that can have non-zero values only on coalitions of a given cardinality; it is further shown that, for such games, there is essentially a unique ranking among the players. We also study the way the ranking changes after one or more players retire. Also, we characterize axiomatically different ways of ranking the players that intervene in a cooperative game.     

Two-Person Second-Order Games,Part 1: Formulation and Transition Anatomy

It is well known that human psychology determines his/her action and behavior. This fact has not been fully incorporated in game theory. This paper intends to incorporate human psychology in formulating games as people play them. In Part 1 of the paper, we formulate a two-person game by the habitual domain theory and the Markov chain theory. Using the habitual domains theory, we present a new model describing the evolution of the states of mind of players over time, the two-person second-order game. We introduce the concept of the focal mind profile as well as the solution concept of the win-win mind profile. In addition, we provide also a method to predict the average number of steps needed for a game to reach a focal or win-win mind profile. Then, in Part 2 of the paper, under some reasonable assumptions, we derive the possibility theorem stating that it is always possible to reach a win-win mind profile when suitable conditions are satisfied. This research was partially supported by the National Science Council, Taiwan, NSC96-2416-H009-013.     

How to Allocate Network Centers

This paper deals with the issue of allocating and utilizing centers in a distributed network, in its various forms. The paper discusses the significant parameters of center allocation, defines the resulting optimization problems, and proposes several approximation algorithms for selecting centers and for distributing the users among them. We concentrate mainly on balanced versions of the problem, i.e., in which it is required that the assignment of clients to centers be as balanced as possible. The main results are constant ratio approximation algorithms for the balanced κ-centers and balanced κ-weighted centers problems, and logarithmic ratio approximation algorithms for the ρ-dominating set and the k-tolerant set problems.     

Two-Person Second-Order Games,Part 2: Restructuring Operations to Reach a Win-Win Profile

In Part 1 of the paper, using habitual domains theory and finite Markov chain theory, we have introduced a new model for describing the evolution of the states of mind of players over time, the two-person second-order game. The concepts of focal mind profile as well as the solution concept of win-win mind profile have been introduced as solution concepts for these games. In Part 2 of the paper, we address the problem of restructuring a game where the focal profile (1,1) is not reachable or is not a win-win profile into a game where the profile (1,1) is a reachable win-win profile. Precisely, under some reasonable assumptions, we derive the possibility theorem that it is always possible to reach a win-win mind profile in a two-person second-order game. Moreover, we provide practical operations for restructuring games for reaching a win-win profile. This research was partially supported by the National Science Council, Taiwan, NSC96-2416-H009-013.     

On the geometry of generalized Gaussian distributions

In this paper we consider the space of those probability distributions which maximize the q-Rényi entropy. These distributions have the same parameter space for every q, and in the q=1 case these are the normal distributions. Some methods to endow this parameter space with a Riemannian metric is presented: the second derivative of the q-Rényi entropy, the Tsallis entropy, and the relative entropy give rise to a Riemannian metric, the Fisher information matrix is a natural Riemannian metric, and there are some geometrically motivated metrics which were studied by Siegel, Calvo and Oller, Lovri?, Min-Oo and Ruh. These metrics are different; therefore, our differential geometrical calculations are based on a new metric with parameters, which covers all the above-mentioned metrics for special values of the parameters, among others. We also compute the geometrical properties of this metric, the equation of the geodesic line with some special solutions, the Riemann and Ricci curvature tensors, and the scalar curvature. Using the correspondence between the volume of the geodesic ball and the scalar curvature we show how the parameter q modulates the statistical distinguishability of close points. We show that some frequently used metrics in quantum information geometry can be easily recovered from classical metrics.     

Z-cyclic triplewhist tournaments—The noncompatible case,part I

Two odd primes odd, are said to be noncompatible if b₁ ≠ b₂. For all noncompatible (ordered) pairs of primes (p₁, p₂) such that p_i ≡ p_i < 200, i = 1,2 we establish the existence of Z-cyclic triplewhist tournaments on 3p₁ p₂ + 1 players. It is believed that these results are the first examples of such tournaments, indeed the first examples of Z-cyclic whist tournaments for such players. In Part 2 we extend the results of this study and establish the existence of Z-cyclic triplewhist tournaments on players for all α₁ ≥ 1, α₂ ≥ 1 and p₁, p₂ as described above. © 1997 John Wiley & Sons, Inc.     

A characterization of square banach spaces

Square Banach spaces are characterized among real Banach spaces in terms of the Alfsen-Effros structure topology on the extreme points of the dual ball. As a corollary, one has that the class of separable square spaces coincides with the class of separableG-spaces. It is also shown that for aG-space (hence for a square space) regularity of the quotient structure topology is equivalent to complete regularity, and that square spaces exist for which this topology is not regular. Part of this paper is from the author’s Ph.D. thesis prepared at Bryn Mawr College under the direction of Professor Frederic Cunningham, Jr., whose valuable guidance is greatly appreciated by the author.     

Higher Intersection Theory on Algebraic Stacks: I

In this paper and the sequel we establish a theory of Chow groups and higher Chow groups on algebraic stacks locally of finite type over a field and establish their basic properties. This includes algebraic stacks in the sense of Deligne–Mumford as well as Artin. An intrinsic difference between our approach and earlier approaches is that the higher Chow groups of Bloch enter into our theory early on and depends heavily on his fundamental work. Our theory may be more appropriately called the (Lichtenbaum) motivic homology and cohomology of algebraic stacks. One of the main themes of these papers is that such a motivic homology does provide a reasonable intersection theory for algebraic stacks (of finite type over a field), with several key properties holding integrally and extending to stacks locally of finite type. While several important properties of our higher Chow groups, like covariance for projective representable maps (that factor as the composition of a closed immersion into the projective space associated to a locally free coherent sheaf and the obvious projection), an intersection pairing and contravariant functoriality for all smooth algebraic stacks, are shown to hold integrally, our theory works best with rational coefficients.The main results of Part I are the following. The higher Chow groups are defined in general with respect to an atlas, but are shown to be independent of the choice of the atlas for smooth stacks if one uses finite coefficients with torsion prime to the characteristics or in general for Deligne–Mumford stacks. (Using some results on motivic cohomology, we extend this integrally to all smooth algebraic stacks in Part II.) Using cohomological descent, we extend Bloch's fundamental localization sequence for quasi-projective schemes to long exact localization sequences of the higher Chow groups modulo torsion for all Artin stacks: this is one of the main results of the paper. We show that these higher Chow groups modulo torsion are covariant for all proper representable maps between stacks of finite type while being contravariant for all representable flat maps and, in Part II, that they are independent of the choice of an atlas for all stacks of finite type over the given field k. The comparison with motivic cohomology, as is worked out in Part II, enables us to provide an explicit comparison of our theory for quotient stacks associated to actions of linear algebraic groups on quasi-projective schemes with the corresponding Totaro–Edidin–Graham equivariant intersection theory. As an application of our theory we compute the higher Chow groups of Deligne–Mumford stacks and show that they are isomorphic modulo torsion to the higher Chow groups of their coarse moduli spaces. As a by-product of our theory we also produce localization sequences in (integral) higher Chow groups for all schemes locally of finite type over a field: these higher Chow groups are defined as the Zariski hypercohomology with respect to the cycle complex.     

James' Theorem Fails for Starlike Bodies

Starlike bodies are interesting in nonlinear functional analysis because they are strongly related to bump functions and to n-homogeneous polynomials on Banach spaces, and their geometrical properties are thus worth studying. In this paper we deal with the question whether James' theorem on the characterization of reflexivity holds for (smooth) starlike bodies, and we establish that a feeble form of this result is trivially true for starlike bodies in nonreflexive Banach spaces, but a reasonable strong version of James' theorem for starlike bodies is never true, even in the smooth case. We also study the related question as to how large the set of gradients of a bump function can be, and among other results we obtain the following new characterization of smoothness in Banach spaces: a Banach space X has a C¹ Lipschitz bump function if and only if there exists another C¹ smooth Lipschitz bump function whose set of gradients contains the unit ball of the dual space X*. This result might also be relevant to the problem of finding an Asplund space with no smooth bump functions.     

On the interaction of geometry,analysis and stochastics on a body

It is reasonable to expect that the geometrical feature of a body has influence to spectral asymptotics of its “natural” Laplacian as well as to the behavior of its “natural” Brownian motion. In fact, such an interaction can be expressed by a so–called “Einstein relation” implicating Hausdorff, spectral and walk dimension. These quantities express geometric, analytic and stochastic aspects of a set. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Normalising geometrical figures: dynamic manipulation and construction of meanings for ratio and proportion

Enlarging-shrinking geometrical figures by 13 year-olds is studied during the implementation of proportional geometric tasks in the classroom. Students worked in groups of two using ‘Turtleworlds’, a piece of geometrical construction software which combines symbolic notation, through a programming language, with dynamic manipulation of geometrical objects by dragging on sliders representing variable values. In this paper we study the students’ normalising activity, as they use this kind of dynamic manipulation to modify ‘buggy’ geometrical figures while developing meanings for ratio and proportion. We describe students’ normative actions in terms of four distinct Dynamic Manipulation Schemes (Reconnaissance, Correlation, Testing, Verification). We discuss the potential of dragging for mathematical insight in this particular computational environment, as well as the purposeful nature of the task which sets up possibilities for students to appreciate the utility of proportional relationships.     

Functional properties of privalov spaces of holomorphic functions in several variables

We consider Privalov classes of degreeq>1 in the unit ball and the polydisk in ℂⁿ. They are defined, say, for the ball, as the sets of functionsf(z) such that the average of ln ₊ ^q |f(z)| over a sphere centered at the origin remains bounded as the radius increases to 1. These classes, which were introduced (in the one-dimensional case) by Privalov before 1941, were often used in the foreign literature in the last 10–20 years; typically, the notation varied and Privalov was not mentioned. We discuss various equivalent definitions of these classes as well as the most general properties, such as growth estimates, properties of the natural metric, and boundedness or total boundedness of subsets. Translated fromMatematicheskie Zametki Vol. 65, No. 2, pp. 280–288, February, 1999.     

Dynamical states of bubbling in vertically vibrated granular materials. Part I: Collective processes

Granular materials deform plastically like a solid under weak shear and they flow like a fluid under high shear. These materials exhibit other unusual kinds of behavior, including pattern formation in shaking of granular materials for which the onset characteristics of the various patterns are not well understood. Vertically shaken granular materials undergo a transition to a convective motion which can result in the formation of bubbles. In Part I, a detailed overview is presented of collective processes in gas-particle flows useful for developing a simplified model for molecular dynamic type simulations of dense gas-particle flows. The large eddy simulation method (LES) has been employed for simulating fluid flows through irregular array of particles. The results obtained may lead to scale-dependent closures for quantities such as the drag, stresses and effective dispersion. These are of use for developing a continuum approach for describing the deformation and flow of dense gas-particle mixtures described in Part II.     

Partially finite convex programming,Part II: Explicit lattice models

In Part I of this work we derived a duality theorem for partially finite convex programs, problems for which the standard Slater condition fails almost invariably. Our result depended on a constraint qualification involving the notion ofquasi relative interior. The derivation of the primal solution from a dual solution depended on the differentiability of the dual objective function: the differentiability of various convex functions in lattices was considered at the end of Part I. In Part II we shall apply our results to a number of more concrete problems, including variants of semi-infinite linear programming,L ¹ approximation, constrained approximation and interpolation, spectral estimation, semi-infinite transportation problems and the generalized market area problem of Lowe and Hurter (1976). As in Part I, we shall use lattice notation extensively, but, as we illustrated there, in concrete examples lattice-theoretic ideas can be avoided, if preferred, by direct calculation.     

Continuous Algorithms in n-Term Approximation and Non-Linear Widths

In the present paper we investigate optimal continuous algorithms in n-term approximation based on various non-linear n-widths, and n-term approximation by the dictionary V formed from the integer translates of the mixed dyadic scales of the tensor product multivariate de la Vallée Poussin kernel, for the unit ball of Sobolev and Besov spaces of functions with common mixed smoothness. The asymptotic orders of these quantities are given. For each space the asymptotic orders of non-linear n-widths and n-term approximation coincide. Moreover, these asymptotic orders are achieved by a continuous algorithm of n-term approximation by V, which is explicitly constructed.     

Information geometry of small diffusions

Information geometrical quantities such as metric tensors and connection coefficients for small diffusion models are obtained. Asymptotic properties of bias-corrected estimators for small diffusion models are investigated from the viewpoint of information geometry. Several results analogous to those for independent and identically distributed (i.i.d.) models are obtained by using the asymptotic normality of the statistics appearing in asymptotic expansions. In contrast to the asymptotic theory for i.i.d.models, the geometrical quantities depend on the magnitude of noise.

---

     

Congruence,similarity, and symmetries of geometric objects

We consider the problem of computing geometric transformations (rotation, translation, reflexion) that map a point setA exactly or approximately into a point setB. We derive efficient algorithms for various cases (Euclidean or maximum metric, translation or rotation, or general congruence).Part of this research was supported by the DFG under Grants Me 620/6-1 and Al 253/1-1.     

On the reissner-naghdi elasticity relationships : PMM vol. 38, n 6, 1974, pp. 1063–1071

Shell theory equations are constructed by the method in [1] to the accuracy of quantities of the order of h_*^2+k, where and k = 2−4t for (h_* is the relative semithickness of the shell and t is the index of the state of stress variation). Without being within the framework of the Lovetype theory, the equations obtained are compared with the Reissner-Naghdi equations. [2, 3] in which the transverse shear is taken into account, and it is shown that from the asymptotic viewpoint these latter are inconsistent. It is also shown that if the shell resists shear weakly, then from the asymptotic viewpoint the Reissner-Naghdi theory is completely well founded.The three-dimensional equations of elasticity theory are reduced to two-dimensional equations in [1] by using an asymptotic method, i.e. all members of the same order relative to the small parameter h_* are taken into account at each stage of the calculations. It has been shown that without going outside the framework of the ordinary concepts of the Love-type theory of shells (in particular, without taking account of transverse shear), the shell theory equations can be constructed to the accuracy of quantities of the order of h^2−2t_*, but it is impossible to exceed this limit without a qualitative complication in the theory.