Characterizations of power indices based on null player free winning coalitions

Abstract

In this paper, we characterize two power indices introduced in [1 Alonso-Meijide JM, Ferreira F, Álvarez-Mozos M, Pinto AA. Two new power indices based on winning coalitions. J. Differ. Equ. Appl. 2011;17:1095–1100.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]] using two different modifications of the monotonicity property first stated by [2 Young HP. Monotonic solutions of cooperative games. Internat. J. Game Theory. 1985;14:62–72.[Crossref] , [Google Scholar]]. The sets of properties are easily comparable among them and with previous characterizations of other power indices.     

Local and Global Existence for an Aggregation Equation

The purpose of this work is to develop a satisfactory existence theory for a general class of aggregation equations. An aggregation equation is a non-linear, non-local partial differential equation that is a regularization of a backward diffusion process. The non-locality arises via convolution with a potential. Depending on how regular the potential is, we prove either local or global existence for the solutions. Aggregation equations have been used recently to model the dynamics of populations in which the individuals attract each other (Bodnar and Velazquez, 2005 Bodnar , M. , Velazquez , J. J. L. ( 2005 ). Derivation of macroscopic equations for individual cell-based models: a formal approach . Math. Methods Appl. Sci. 28 ( 15 ): 1757 – 1779 .[Crossref], [Web of Science ®] , [Google Scholar]; Holm and Putkaradze, 2005 Holm , D. D. , Putkaradze , V. ( 2005 ). Aggregation of finite size particles with variable mobility . Phys. Rev. Lett. 95 : 226106 . [Google Scholar]; Mogilner and Edelstein-Keshet, 1999 Mogilner , A. , Edelstein-Keshet , L. ( 1999 ). A non-local model for a swarm . J. Math. Biol. 38 ( 6 ): 534 – 570 .[Crossref], [Web of Science ®] , [Google Scholar]; Morale et al., 2005 Morale , D. , Capasso , V. , Oelschläger , K. ( 2005 ). An interacting particle system modelling aggregation behavior: from individuals to populations . J. Math. Biol. 50 ( 1 ): 49 – 66 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]; Topaz and Bertozzi, 2004 Topaz , C. M. , Bertozzi , A. L. ( 2004 ). Swarming patterns in a two-dimensional kinematic model for biological groups . SIAM J. Appl. Math. 65 ( 1 ): 152 – 174 (electronic) .[Crossref], [Web of Science ®] , [Google Scholar]; Topaz et al., 2006 Topaz , C. M. , Bertozzi , A. L. , Lewis , M. A. ( 2006 ). A nonlocal continuum model for biological aggregation . Bull. Math. Biol. 68 ( 7 ): 1601 – 1623 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]).     

Combined Optimal Stopping and Singular Stochastic Control

In this article, a simple of combined singular stochastic control and optimal stopping in the jump-diffusion model is formulated and solved. We give sufficient conditions for the existence of an optimal strategy which has the same form as in continuous case given by Davis and Zervos [3 Davis , M.H.A. , and Zervos , M. 1994. A problem of singular stochastic control with discretionary stopping. Annals of Applied Probility 4(1):226–240. [Google Scholar]] and also Karatzas et al. [5 Karatzas , I. , Ocone , D. , Wang , H. , and Zervos , M. 2000 . Finite-fuel singular control with discretionary stopping . Stochastics and Stochastics Reports 71 : 1 – 50 .[Taylor & Francis Online] , [Google Scholar]]. This result is applied to solve explicitly an example of such problem.     

Discrete-Time BSDEs with Random Terminal Horizon

This article studies the theory of discrete-time backward stochastic differential equations (also called BSDEs) with a random terminal time, which is not a stopping time. We follow Cohen and Elliott [2 Cohen , S.N. , and Elliott , R.J. 2010 . A general theory of finite state backward stochastic difference equations . Stochastic Processes and Their Applications 120 ( 4 ): 442 – 466 .[Crossref], [Web of Science ®] , [Google Scholar]] and consider a reference filtration generated by a general discrete-time finite-state process. The martingale representation theorem for essentially bounded martingales under progressively enlarged filtration is established. Then we prove the existence and uniqueness theorem of BSDEs under enlarged filtration using some weak assumptions of the driver. We also present conditions for a comparison theorem. Applications to nonlinear expectations and optimal design of dynamic default risk are explored.     

Splitting of operations for alternative and Malcev structures

In this paper, we define pre-Malcev algebras and alternative quadri-algebras and prove that they generalize pre-Lie algebras and quadri-algebras, respectively, to the alternative setting. We use the results and techniques from [4 Bai, C., Bellier, O., Guo, L., Ni, X. (2013). Splitting of operations, Manin products, and Rota-Baxter operators. Int. Math. Res. Not. 2013(3):485–524. [Google Scholar], 14 Gubarev, V. Y., Kolesnikov, P. S. (2013). Embedding of dendriform algebras into Rota-Baxter algebras. Cent. Eur. J. Math. 11(2):226–245.[Crossref], [Web of Science ®] , [Google Scholar]] to discuss and give explicit computations of different constructions in terms of bimodules, splitting of operations, and Rota–Baxter operators.     

Calderón problem for connections

In this paper, we consider the problem of identifying a connection ? on a vector bundle up to gauge equivalence from the Dirichlet-to-Neumann map of the connection Laplacian ?*? over conformally transversally anisotropic (CTA) manifolds. This was proved in [9 Dos Santos Ferreira, D., Kenig, C., Salo, M., Uhlmann, G. (2009). Limiting Carleman weights and anisotropic inverse problems. Invent. Math. 178:119–171.[Crossref], [Web of Science ®] , [Google Scholar]] for line bundles in the case of the transversal manifold being simple—we generalize this result to the case where the transversal manifold only has an injective ray transform. Moreover, the construction of suitable Gaussian beam solutions on vector bundles is given for the case of the connection Laplacian and a potential, following the works of [11 Dos Santos Ferreira, D., Kurylev, Y., Lassas, M., Salo, M. (2016). The Calderón problem in transversally anisotropic geometries. J. Eur. Math. Soc., 18:2579–2626.[Crossref], [Web of Science ®] , [Google Scholar]]. This in turn enables us to construct the Complex Geometrical Optics (CGO) solutions and prove our main uniqueness result. We also reduce the problem to a new non-abelian X-ray transform for the case of simple transversal manifolds and higher rank vector bundles. Finally, we prove the recovery of a flat connection in general from the DN map, up to gauge equivalence, using an argument relating the Cauchy data of the connection Laplacian and the holonomy.     

A ring with FT-rank two

We construct a ring which admits a 2-generated, faithful torsion module but lacks a cyclic faithful torsion module. This answers a question by Oman and Schwiebert [1 Oman, G., Schwiebert, R. (2012). Rings which admit faithful torsion modules. Commun. Algebra 40(6):2184–2198.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar], 2 Oman, G., Schwiebert, R. (2012). Rings which admit faithful torsion modules II. J. Algebra Appl. 11(3):1250054 (12 p.).[Crossref], [Web of Science ®] , [Google Scholar]].     

Extension of Goulden–Jackson cluster method on pattern occurrences in random sequences and comparison with Régnier–Szpankowski method

The Goulden–Jackson cluster method is a powerful method to find generating functions of pattern occurrences in random sequences [1 Goulden, I.P. and Jackson, D.M. 1979. An inversion theorem for cluster decompositions of sequences with distinguished subsequences. Journal of London Mathematical Society, Second Series, 20: 567–576. [Crossref], [Web of Science ®] , [Google Scholar]]. The method is clearly explained, extended and implemented by Noonan and Zeilberger [2 Noonan, J. and Zeilberger, D. 1999. The Goulden-Jackson cluster method: extensions, applications, and implementations. Journal of Difference Equations and Applications, 5: 355–377. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]]. In this paper, we elaborate on one of the several extensions in [2 Noonan, J. and Zeilberger, D. 1999. The Goulden-Jackson cluster method: extensions, applications, and implementations. Journal of Difference Equations and Applications, 5: 355–377. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]], namely the extension from symmetrical Bernoulli sequences where the occurrences of each symbol have equal probability, to asymmetrical Bernoulli sequences with different probabilities of symbol generations. An explicit formula is derived for the extension, which is implicitly embedded in the treatment of [2 Noonan, J. and Zeilberger, D. 1999. The Goulden-Jackson cluster method: extensions, applications, and implementations. Journal of Difference Equations and Applications, 5: 355–377. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]]. The extended result is then compared with the method of Régnier–Szpankowski [3 Régnier, M. and Szpankowski, W. 1997. On the approximate pattern occurrences in a text. Proceedings of the compression and complexity of sequences 1997, : 253–264.  [Google Scholar]], a method which was developed independently to tackle the same problem. By manipulating some matrix inversions, we show that the Régnier–Szpankowski method can be simplified to the extended Goulden–Jackson method.     

Stochastic motion under G-framework: II. Kinematics

Continuing the study of stochastic motion that we started [11 Zhang, H., and Kannan, D. 2014. Stochastic motion under G-framework: I. Nelson stochastic derivatives. Stoch Anal Appl 32:1067–1096.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]], we present in this article the kinematics of such a motion. We begin by defining the quadratic derivative of an S2-process, and show that this derivative of the Brownian motion captures the variance uncertainty. We show, under certain vanishing derivatives and independence conditions, the martingale properties of an S1-process. Starting with an S1-process, we derive the equation of motion, an Itô equation corresponding to a G-diffusion process.     

Determining an Unbounded Potential from Cauchy Data in Admissible Geometries

In [4 Dos Santos Ferreira , D. , Kenig , C.E. , Salo , M. , Uhlmann , G. ( 2009 ). Limiting Carleman weights and anisotropic inverse problems . Invent. Math. 178 : 119 – 171 .[Crossref], [Web of Science ®] , [Google Scholar]] anisotropic inverse problems were considered in certain admissible geometries, that is, on compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. In particular, it was proved that a bounded smooth potential in a Schrödinger equation was uniquely determined by the Dirichlet-to-Neumann map in dimensions n ≥ 3. In this article we extend this result to the case of unbounded potentials, namely those in L ^n/2. In the process, we derive L ^p Carleman estimates with limiting Carleman weights similar to the Euclidean estimates of Jerison and Kenig [8 Jerison , D. , Kenig , C.E. ( 1985 ). Unique continuation and absence of positive eigenvalues for Schrödinger operators . Ann. Math. 121 : 463 – 494 .[Crossref], [Web of Science ®] , [Google Scholar]] and Kenig et al. [9 Kenig , C.E. , Ruiz , A. , Sogge , C.D. ( 1987 ). Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators . Duke Math. J. 55 : 329 – 347 .[Crossref], [Web of Science ®] , [Google Scholar]].     

The Stochastic Reflection Problem in Hilbert Spaces

We construct a Markov process X associated with the stochastic reflection problem on a closed convex subset with non empty interior and smooth boundary in a Hilbert space, as a solution to a random convex control problem. The transition semigroup corresponding to X is exactly that defined by the Kolmogorov equation with Neumann homogeneous boundary conditions (see [3 Barbu , V. , Da Prato , G. , Tubaro , L. ( 2011 ). Kolmogorov equation associated to the stochastic reflection problem on a smooth convex set of a Hilbert space II . Ann. Inst. H. Poincaré 4 : 699 – 724 .[Crossref] , [Google Scholar]]).     

Self-Interacting Markov Chains: Some Asymptotics

Abstract

Guided by the self-interaction mechanisms introduced in Benaim et al. [2 Benaim , M. , Ledoux , M. , and Raimond , O. 2002 . Self-interacting diffusions . Probab. Theory Related Fields 122 : 1 – 41 . [Google Scholar]] and in [5 Del Moral , P. , and Miclo , L. 2006 . Self-interacting Markov chains . Stochastic Anal. Appl. 24 : 615 – 660 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]], we present a more general definition of self-interacting Markov chains (SIMCs) (than in Del Moral and Miclo [5 Del Moral , P. , and Miclo , L. 2006 . Self-interacting Markov chains . Stochastic Anal. Appl. 24 : 615 – 660 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]] and Benaim et al. [2 Benaim , M. , Ledoux , M. , and Raimond , O. 2002 . Self-interacting diffusions . Probab. Theory Related Fields 122 : 1 – 41 . [Google Scholar]]). We then establish, for particular self-interaction mechanisms, a stability theorem with error estimation, two central limit theorems, two functional central limit theorems, and the large deviation principle.     

A note on semicentral idempotents

In this note we answer the question raised by Han et al. in [3 Han, J., Lee, Y., Park, S. (2014). Semicentral idempotents in a ring. J. Korean Math. Soc. 51(3):463–472, MR3206399.[Crossref], [Web of Science ®] , [Google Scholar]] whether an idempotent isomorphic to a semicentral idempotent is itself semicentral. We show that rings with this property are precisely the Dedekind-finite rings. An application to module theory is given.     

Almost Exponential Decay Near Maxwellian

ABSTRACT

By direct interpolation of a family of smooth energy estimates for solutions near Maxwellian equilibrium and in a periodic box to several Boltzmann type equations in Guo (2002 Guo , Y. ( 2002 ). The Landau equation in a periodic box . Comm. Math. Phys. 231 ( 3 ): 391 – 434 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar] 2003a Guo , Y. ( 2003a ). Classical solutions to the Boltzmann equation for molecules with an angular cutoff . Arch. Ration. Mech. Anal. 169 ( 4 ): 305 – 353 . [CSA] [CROSSREF]  [Google Scholar] b Guo , Y. ( 2003b ). The Vlasov-Maxwell-Boltzmann system near maxwellians . Invent. Math. 153 ( 3 ): 593 – 630 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) and Strain and Guo (2004 Strain , R. M. , Guo , Y. ( 2004 ). Stability of the relativistic Maxwellian in a collisional plasma . Comm. Math. Phys. 251 ( 2 ): 263 – 320 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]), we show convergence to Maxwellian with any polynomial rate in time. Our results not only resolve the important open problem for both the Vlasov-Maxwell-Boltzmann system and the relativistic Landau-Maxwell system for charged particles, but also lead to a simpler alternative proof of recent decay results (Desvillettes and Villani, 2005 Desvillettes , L. , Villani , C. ( 2005 ). On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation . Invent. Math. 159 ( 2 ): 245 – 316 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) for soft potentials as well as the Coulombic interaction, with precise decay rate depending on the initial conditions.     

Asymptotic results for finite superpositions of Ornstein–Uhlenbeck processes

A model of intermittency based on superposition of Lévy driven Ornstein–Uhlenbeck processes is studied in [6 Grahovac, D., Leonenko, N., Sikorskii, A., and Te?niak, I. 2016. Intermittency of superpositions of Ornstein–Uhlenbeck type processes. J. Stat. Phys. 165:390–408.[Crossref], [Web of Science ®] , [Google Scholar]]. In particular, as shown in Theorem 5.1 in that paper, finite superpositions obey a (sample path) central limit theorem under suitable hypotheses. In this paper we prove large (and moderate) deviation results associated with this central limit theorem.     

Poisson deleting derivations algorithm and Poisson spectrum

Cauchon [5 Cauchon, G. (2003). Effacement des dérivations et spectres premiers des algèbres quantiques. J. Algebra 260(2):476–518.[Crossref], [Web of Science ®] , [Google Scholar]] introduced the so-called deleting derivations algorithm. This algorithm was first used in noncommutative algebra to prove catenarity in generic quantum matrices, and then to show that torus-invariant primes in these algebras are generated by quantum minors. Since then this algorithm has been used in various contexts. In particular, the matrix version makes a bridge between torus-invariant primes in generic quantum matrices, torus orbits of symplectic leaves in matrix Poisson varieties and totally non-negative cells in totally non-negative matrix varieties [12 Goodearl, K. R., Launois, S., Lenagan, T. (2011). Torus invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves. Math. Z. 269(1):29–45.[Crossref], [Web of Science ®] , [Google Scholar]]. This led to recent progress in the study of totally non-negative matrices such as new recognition tests [18 Launois, S., Lenagan, T. (2014). E?cient recognition of totally non-negative matrix cells. Found. Comput. Math. 14:371–387.[Crossref], [Web of Science ®] , [Google Scholar]]. The aim of this article is to develop a Poisson version of the deleting derivations algorithm to study the Poisson spectra of the members of a class 𝒫 of polynomial Poisson algebras. It has recently been shown that the Poisson Dixmier–Moeglin equivalence does not hold for all polynomial Poisson algebras [2 Bell, J., Launois, S., Sanchez, O. L., Moosa, R. Poisson algebras via model theory and differential-algebraic geometry. J. Eur. Math. Soc. (to appear). [Google Scholar]]. Our algorithm allows us to prove this equivalence for a significant class of Poisson algebras, when the base field is of characteristic zero. Finally, using our deleting derivations algorithm, we compare topologically spectra of quantum matrices with Poisson spectra of matrix Poisson varieties.     

Breadth and characteristic sequence of nilpotent Lie algebras

The notion of breadth of a nilpotent Lie algebra was introduced and used to approach problems of classification up to isomorphism in [5 Khuhirun, B., Misra, K. C., Stitzinger, E. (2015). On nilpotent Lie algebras of small breadth. J. Algebra 444:328–338.[Crossref], [Web of Science ®] , [Google Scholar]]. In the present paper, we study this invariant in terms of characteristic sequence, another invariant, introduced by Goze and Ancochea in [1 Ancochea-Bermúdez, J. M., Goze, M. (1986). Sur la classification des algèbres de Lie nilpotentes de dimension 7. C. R. Acad. Sci. Paris 302:611–613. [Google Scholar]]. This permits to complete the determination of Lie algebras of breadth 2 studied in [5 Khuhirun, B., Misra, K. C., Stitzinger, E. (2015). On nilpotent Lie algebras of small breadth. J. Algebra 444:328–338.[Crossref], [Web of Science ®] , [Google Scholar]] and to begin the work for Lie algebras with breadth greater than 2.     

Polynomial extensions of SCN rings need not be SCN

In this article, we show that there exists an SCN ring R such that the polynomial ring R[x] is not SCN. This answers a question posed by T. K. Kwak et al. in [2 Kwak, T. K., Lee, M. J., Lee, Y. (2014). On sums of coe?cients of products of polynomials. Comm. Algebra 42(9):4033–4046.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]].     

Approximations for Solutions of Lévy-Type Stochastic Differential Equations

Abstract

The problem of the construction of strong approximations with a given order of convergence for jump-diffusion equations is studied. General approximation schemes are constructed for Lévy-type stochastic differential equation. In particular, the article generalizes the results from [2 Gardoń , A. 2004 . The order of approximations for solutions of Ito-type stochastic differential equations with jumps . Stoch. Anal. Appl. 22 ( 3 ): 679 – 699 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar], 5 Kloeden , P.E. , and Platen , E. 1995 . Numerical Solutions of Stochastic Differential Equations . Springer-Verlag , Berlin . [Google Scholar]]. The Euler and the Milstein schemes are shown for finite and infinite Lévy measure.