2015
http://publications.mfo.de/handle/mfo/1347
Fri, 12 Apr 2024 11:54:37 GMT2024-04-12T11:54:37ZEuler Reflexion Formulas for Motivic Multiple Zeta Functions
http://publications.mfo.de/handle/mfo/193
Euler Reflexion Formulas for Motivic Multiple Zeta Functions
Thuong, Lê Quy; Nguyen, Hong Duc
We introduce a new notion of *-product of two integrable series with coeficients in distinct Grothendieck rings of algebraic varieties, preserving the integrability and commuting with the limit of rational series. In the same context, we define a motivic multiple zeta function with respect to an ordered family of regular functions, which is integrable and connects closely to Denef-Loeser's motivic zeta functions. We also show that the *-product is associative in the class of motivic multiple zeta functions. Furthermore, a version of the Euler reflexion formula for motivic zeta functions is nicely formulated to deal with the *-product and motivic multiple zeta functions, and it is proved for both univariate and multivariate cases by using the theory of arc spaces. As an application, taking the limit for the motivic Euler reflexion formula we recover the well known motivic Thom-Sebastiani theorem.
OWLF 2015
Wed, 18 Nov 2015 00:00:00 GMThttp://publications.mfo.de/handle/mfo/1932015-11-18T00:00:00ZThuong, Lê QuyNguyen, Hong DucWe introduce a new notion of *-product of two integrable series with coeficients in distinct Grothendieck rings of algebraic varieties, preserving the integrability and commuting with the limit of rational series. In the same context, we define a motivic multiple zeta function with respect to an ordered family of regular functions, which is integrable and connects closely to Denef-Loeser's motivic zeta functions. We also show that the *-product is associative in the class of motivic multiple zeta functions. Furthermore, a version of the Euler reflexion formula for motivic zeta functions is nicely formulated to deal with the *-product and motivic multiple zeta functions, and it is proved for both univariate and multivariate cases by using the theory of arc spaces. As an application, taking the limit for the motivic Euler reflexion formula we recover the well known motivic Thom-Sebastiani theorem.The algebra of differential operators for a Gegenbauer weight matrix
http://publications.mfo.de/handle/mfo/189
The algebra of differential operators for a Gegenbauer weight matrix
Ignacio Nahuel Zurrián
In this work we study in detail the algebra of differential operators $\mathcal{D}(W)$ associated with a Gegenbauer matrix weight. We prove that two second order operators generate the algebra, indeed $\mathcal{D}(W)$ is isomorphic to the free algebra generated by two elements subject to certain relations. Also, the center is isomorphic to the affine algebra of a singular rational curve. The algebra $\mathcal{\mathcal{D}}(W)$ is a finitely-generated torsion-free module over its center, but it is not at and therefore neither projective. After [Tir11], this is the second detailed study of an algebra $\mathcal{D}(W)$ and the first one coming from spherical functions and group representation theory.
OWLF 2015
Wed, 29 Jul 2015 00:00:00 GMThttp://publications.mfo.de/handle/mfo/1892015-07-29T00:00:00ZIgnacio Nahuel ZurriánIn this work we study in detail the algebra of differential operators $\mathcal{D}(W)$ associated with a Gegenbauer matrix weight. We prove that two second order operators generate the algebra, indeed $\mathcal{D}(W)$ is isomorphic to the free algebra generated by two elements subject to certain relations. Also, the center is isomorphic to the affine algebra of a singular rational curve. The algebra $\mathcal{\mathcal{D}}(W)$ is a finitely-generated torsion-free module over its center, but it is not at and therefore neither projective. After [Tir11], this is the second detailed study of an algebra $\mathcal{D}(W)$ and the first one coming from spherical functions and group representation theory.Time and band limiting for matrix valued functions, an example
http://publications.mfo.de/handle/mfo/190
Time and band limiting for matrix valued functions, an example
Grünbaum, F. A.; Pacharoni, I.; Zurrián, Ignacio Nahuel
The main purpose of this paper is to extend to a situation involving matrix valued orthogonal polynomials and spherical functions, a result that traces its origin and its importance to work of Claude Shannon in laying the mathematical foundations of information theory and to a remarkable series of papers by D. Slepian, H. Landau and H. Pollak. To our knowledge, this is the first example showing in a non-commutative setup that a bispectral property implies that the corresponding global operator of "time and band limiting" admits a commuting local operator.This is a noncommutative analog of the famous prolate spheroidal wave operator.
OWLF 2015
Wed, 29 Jul 2015 00:00:00 GMThttp://publications.mfo.de/handle/mfo/1902015-07-29T00:00:00ZGrünbaum, F. A.Pacharoni, I.Zurrián, Ignacio NahuelThe main purpose of this paper is to extend to a situation involving matrix valued orthogonal polynomials and spherical functions, a result that traces its origin and its importance to work of Claude Shannon in laying the mathematical foundations of information theory and to a remarkable series of papers by D. Slepian, H. Landau and H. Pollak. To our knowledge, this is the first example showing in a non-commutative setup that a bispectral property implies that the corresponding global operator of "time and band limiting" admits a commuting local operator.This is a noncommutative analog of the famous prolate spheroidal wave operator.Right Unimodal and Bimodal Singularities in Positive Characteristic
http://publications.mfo.de/handle/mfo/192
Right Unimodal and Bimodal Singularities in Positive Characteristic
Nguyen, Hong Duc
The problem of classification of real and complex singularities was initiated by Arnol'd in the sixties who classified simple, unimodal and bimodal w.r.t. right equivalence. The classification of right simple singularities in positive characteristic was achieved by Greuel and the author in 2014. In the present paper we classify right unimodal and bimodal singularities in positive characteristic by giving explicit normal forms. Moreover we completely determine all possible adjacency diagrams of simple,unimodal and bimodal singularities. As an application we prove that, for singularities of right modality at most 2, the $\mu$-constant stratum is smooth and its dimension is equal to the right modality. In contrast to the complex analytic case, there are, for any positive characteristic, only finitely many 1-dimensional (resp. 2-dimensional) families of right class of unimodal (resp. bimodal) singularities. We show that for fixed characteristic $p > 0 $ of the ground field, the Milnor number of f satisfies $\mu(f)<4p$, if the right modality of $f$ is at most 2.
OWLF 2015
Wed, 18 Nov 2015 00:00:00 GMThttp://publications.mfo.de/handle/mfo/1922015-11-18T00:00:00ZNguyen, Hong DucThe problem of classification of real and complex singularities was initiated by Arnol'd in the sixties who classified simple, unimodal and bimodal w.r.t. right equivalence. The classification of right simple singularities in positive characteristic was achieved by Greuel and the author in 2014. In the present paper we classify right unimodal and bimodal singularities in positive characteristic by giving explicit normal forms. Moreover we completely determine all possible adjacency diagrams of simple,unimodal and bimodal singularities. As an application we prove that, for singularities of right modality at most 2, the $\mu$-constant stratum is smooth and its dimension is equal to the right modality. In contrast to the complex analytic case, there are, for any positive characteristic, only finitely many 1-dimensional (resp. 2-dimensional) families of right class of unimodal (resp. bimodal) singularities. We show that for fixed characteristic $p > 0 $ of the ground field, the Milnor number of f satisfies $\mu(f)<4p$, if the right modality of $f$ is at most 2.Prediction and Quantification of Individual Athletic Performance
http://publications.mfo.de/handle/mfo/191
Prediction and Quantification of Individual Athletic Performance
Blythe, Duncan A. J.; Király, Franz J.
We present a novel, quantitative view on the human athletic performance of individuals. We obtain a predictor for athletic running performances, a parsimonious model, and a training state summary consisting of three numbers, by application of modern validation techniques and recent advances in machine learning to the thepowerof10 database of British athletes’ performances (164,746 individuals, 1,417,432 performances). Our predictor achieves a low average prediction error (out-of-sample), e.g., 3.6 min on elite Marathon performances, and a lower error than the state-of-the-art in performance prediction (30% improvement, RMSE). We are also the first to report on a systematic comparison of predictors for athletic running performance. Our model has three parameters per athlete, and three components which are the same for all athletes. The first component of the model corresponds to a power law with exponent dependent on the athlete which achieves a better goodness-of-fit than known power laws in athletics. Many documented phenomena in quantitative sports science, such as the form of scoring tables, the success of existing prediction methods including Riegel’s formula, the Purdy points scheme, the power law for world records performances and the broken power law for world record speeds may be explained on the basis of our findings in a unified way. We provide strong evidence that the three parameters per athlete are related to physiological and/or behavioural parameters, such as training state, event specialization and age, which allows us to derive novel physiological hypotheses relating to athletic performance. We conjecture on this basis that our findings will be vital in exercise physiology, race planning, the study of aging and training regime design.
OWLF 2014
Thu, 27 Aug 2015 00:00:00 GMThttp://publications.mfo.de/handle/mfo/1912015-08-27T00:00:00ZBlythe, Duncan A. J.Király, Franz J.We present a novel, quantitative view on the human athletic performance of individuals. We obtain a predictor for athletic running performances, a parsimonious model, and a training state summary consisting of three numbers, by application of modern validation techniques and recent advances in machine learning to the thepowerof10 database of British athletes’ performances (164,746 individuals, 1,417,432 performances). Our predictor achieves a low average prediction error (out-of-sample), e.g., 3.6 min on elite Marathon performances, and a lower error than the state-of-the-art in performance prediction (30% improvement, RMSE). We are also the first to report on a systematic comparison of predictors for athletic running performance. Our model has three parameters per athlete, and three components which are the same for all athletes. The first component of the model corresponds to a power law with exponent dependent on the athlete which achieves a better goodness-of-fit than known power laws in athletics. Many documented phenomena in quantitative sports science, such as the form of scoring tables, the success of existing prediction methods including Riegel’s formula, the Purdy points scheme, the power law for world records performances and the broken power law for world record speeds may be explained on the basis of our findings in a unified way. We provide strong evidence that the three parameters per athlete are related to physiological and/or behavioural parameters, such as training state, event specialization and age, which allows us to derive novel physiological hypotheses relating to athletic performance. We conjecture on this basis that our findings will be vital in exercise physiology, race planning, the study of aging and training regime design.Milnor fibre homology via deformation
http://publications.mfo.de/handle/mfo/1109
Milnor fibre homology via deformation
Siersma, Dirk; Tibăr, Mihai-Marius
In case of one-dimensional singular locus, we use deformations in order toget refined information about the Betti numbers of the Milnor fibre.
Research in Pairs 2015
Thu, 01 Jan 2015 00:00:00 GMThttp://publications.mfo.de/handle/mfo/11092015-01-01T00:00:00ZSiersma, DirkTibăr, Mihai-MariusIn case of one-dimensional singular locus, we use deformations in order toget refined information about the Betti numbers of the Milnor fibre.A nested family of k-total effective rewards for positional games
http://publications.mfo.de/handle/mfo/1108
A nested family of k-total effective rewards for positional games
Boros, Endre; Elbassioni, Khaled; Gurvich, Vladimir; Makino, Kazuhisa
We consider Gillette's two-person zero-sum stochastic games with perfect information. For each $k \in \mathbb{Z}_+$ we introduce an effective reward function, called $k$-total. For $k = 0$ and $1$ this function is known as mean payoff and total reward, respectively. We restrict our attention to the deterministic case. For all $k$, we prove the existence of a saddle point which can be realized by uniformly optimal pure stationary strategies. We also demonstrate that $k$-total reward games can be embedded into $(k+1)$-total reward games.
Research in Pairs 2015
Thu, 01 Jan 2015 00:00:00 GMThttp://publications.mfo.de/handle/mfo/11082015-01-01T00:00:00ZBoros, EndreElbassioni, KhaledGurvich, VladimirMakino, KazuhisaWe consider Gillette's two-person zero-sum stochastic games with perfect information. For each $k \in \mathbb{Z}_+$ we introduce an effective reward function, called $k$-total. For $k = 0$ and $1$ this function is known as mean payoff and total reward, respectively. We restrict our attention to the deterministic case. For all $k$, we prove the existence of a saddle point which can be realized by uniformly optimal pure stationary strategies. We also demonstrate that $k$-total reward games can be embedded into $(k+1)$-total reward games.A Pseudo-Polynomial Algorithm for Mean Payoff Stochastic Games with Perfect Information and Few Random Positions
http://publications.mfo.de/handle/mfo/1107
A Pseudo-Polynomial Algorithm for Mean Payoff Stochastic Games with Perfect Information and Few Random Positions
Boros, Endre; Elbassioni, Khaled; Gurvich, Vladimir; Makino, Kazuhisa
We consider two-person zero-sum stochastic mean payoff games with perfect information, or BWR-games, given by a digraph $G = (V,E)$, with local rewards $r : E \to \mathbb{Z}$, and three types of positions: black $V_B$, white $V_W$, and random $V_R$ forming a partition of $V$ . It is a longstanding open question whether a polynomial time algorithm for BWR-games exists, or not, even when $|V_R| = 0$. In fact, a pseudo-polynomial algorithm for BWR-games would already imply their polynomial solvability. In this paper, we show that BWR-games with a constant number of random positions can be solved in pseudo-polynomial time. More precisely, in any BWR-game with $|V_R| = O(1)$, a saddle point in uniformly optimal pure stationary strategies can be found in time polynomial in $|V_W| + |V_B|$, the maximum absolute local reward, and the common denominator of the transition probabilities.
Research in Pairs 2015
Thu, 01 Jan 2015 00:00:00 GMThttp://publications.mfo.de/handle/mfo/11072015-01-01T00:00:00ZBoros, EndreElbassioni, KhaledGurvich, VladimirMakino, KazuhisaWe consider two-person zero-sum stochastic mean payoff games with perfect information, or BWR-games, given by a digraph $G = (V,E)$, with local rewards $r : E \to \mathbb{Z}$, and three types of positions: black $V_B$, white $V_W$, and random $V_R$ forming a partition of $V$ . It is a longstanding open question whether a polynomial time algorithm for BWR-games exists, or not, even when $|V_R| = 0$. In fact, a pseudo-polynomial algorithm for BWR-games would already imply their polynomial solvability. In this paper, we show that BWR-games with a constant number of random positions can be solved in pseudo-polynomial time. More precisely, in any BWR-game with $|V_R| = O(1)$, a saddle point in uniformly optimal pure stationary strategies can be found in time polynomial in $|V_W| + |V_B|$, the maximum absolute local reward, and the common denominator of the transition probabilities.A potential reduction algorithm for two-person zero-sum mean payoff stochastic games
http://publications.mfo.de/handle/mfo/1106
A potential reduction algorithm for two-person zero-sum mean payoff stochastic games
Boros, Endre; Elbassioni, Khaled; Gurvich, Vladimir; Makino, Kazuhisa
We suggest a new algorithm for two-person zero-sum undiscounted stochastic games focusing on stationary strategies. Given a positive real $\epsilon$, let us call a stochastic game $\epsilon$-ergodic, if its values from any two initial positions differ by at most $\epsilon$. The proposed new algorithm outputs for every $\epsilon > 0$ in finite time either a pair of stationary strategies for the two players guaranteeing that the values from any initial positions are within an $\epsilon$-range, or identifies two initial positions $u$ and $v$ and corresponding stationary strategies for the players proving that the game values starting from $u$ and $v$ are at least $\epsilon/24$ apart.In particular, the above result shows that if a stochastic game is $0$-ergodic, then there are stationary strategies for the players proving $24\epsilon$-ergodicity. This result strengthens and provides a constructive version of an existential result by Vrieze (1980)claiming that if a stochastic game is $0$-ergodic, then there are $\epsilon$-optimal stationary strategies for every $\epsilon>0$. The suggested algorithm is based on a potential transformation technique that changes the range of local values at all positions without changing the normal form of the game.
Research in Pairs 2015
Thu, 01 Jan 2015 00:00:00 GMThttp://publications.mfo.de/handle/mfo/11062015-01-01T00:00:00ZBoros, EndreElbassioni, KhaledGurvich, VladimirMakino, KazuhisaWe suggest a new algorithm for two-person zero-sum undiscounted stochastic games focusing on stationary strategies. Given a positive real $\epsilon$, let us call a stochastic game $\epsilon$-ergodic, if its values from any two initial positions differ by at most $\epsilon$. The proposed new algorithm outputs for every $\epsilon > 0$ in finite time either a pair of stationary strategies for the two players guaranteeing that the values from any initial positions are within an $\epsilon$-range, or identifies two initial positions $u$ and $v$ and corresponding stationary strategies for the players proving that the game values starting from $u$ and $v$ are at least $\epsilon/24$ apart.In particular, the above result shows that if a stochastic game is $0$-ergodic, then there are stationary strategies for the players proving $24\epsilon$-ergodicity. This result strengthens and provides a constructive version of an existential result by Vrieze (1980)claiming that if a stochastic game is $0$-ergodic, then there are $\epsilon$-optimal stationary strategies for every $\epsilon>0$. The suggested algorithm is based on a potential transformation technique that changes the range of local values at all positions without changing the normal form of the game.Simulation of Multibody Systems with Servo Constraints through Optimal Control
http://publications.mfo.de/handle/mfo/1105
Simulation of Multibody Systems with Servo Constraints through Optimal Control
Altmann, Robert; Heiland, Jan
We consider mechanical systems where the dynamics are partially constrained to prescribed trajectories. An example for such a system is a building crane with a load and the requirement that the load moves on a certain path. Modelling the system using Newton's second law- "The force acting on an object is equal to the mass of that object times its acceleration."- and enforcing the servo constraints directly leads to differential-algebraic equations (DAEs) of arbitrarily high index. Typically,the model equations are of index 5 which already poses high regularity conditions. Also, common approaches for the numerical time-integration will likely fail. If one relaxes the servo constraints and considers the system from an optimal control point of view, the strong regularity conditions vanish and the solution can be obtained by standard techniques. By means of a spring-mass system, we illustrate the theoretical and expected numerical difficulties. We show how the formulation of the problem in an optimal control context works and address the solvability of the optimal control system. We discuss that the problematic DAE behavior is still inherent in the optimal control system and show how its evidences depend on the regularization parameters of the optimization.
Research in Pairs 2014
Thu, 01 Jan 2015 00:00:00 GMThttp://publications.mfo.de/handle/mfo/11052015-01-01T00:00:00ZAltmann, RobertHeiland, JanWe consider mechanical systems where the dynamics are partially constrained to prescribed trajectories. An example for such a system is a building crane with a load and the requirement that the load moves on a certain path. Modelling the system using Newton's second law- "The force acting on an object is equal to the mass of that object times its acceleration."- and enforcing the servo constraints directly leads to differential-algebraic equations (DAEs) of arbitrarily high index. Typically,the model equations are of index 5 which already poses high regularity conditions. Also, common approaches for the numerical time-integration will likely fail. If one relaxes the servo constraints and considers the system from an optimal control point of view, the strong regularity conditions vanish and the solution can be obtained by standard techniques. By means of a spring-mass system, we illustrate the theoretical and expected numerical difficulties. We show how the formulation of the problem in an optimal control context works and address the solvability of the optimal control system. We discuss that the problematic DAE behavior is still inherent in the optimal control system and show how its evidences depend on the regularization parameters of the optimization.