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arXivIntegration of the EulerPoinsot Problem in New Variables Martin Lara, Sebastián Ferrer 20101231 The essentially unique reduction of the EulerPoinsot problem may be performed in different sets of variables. Actionangle variables are usually preferred because of their suitability for approaching perturbed rigidbody motion. But they are just one among the variety of sets of canonical coordinates that integrate the problem. We present an alternate set of variables that, while allowing for similar performances than actionangles in the study of perturbed problems, show an important advantage over them: Their transformation from and to Andoyer variables is given in explicit form.
Subject 수학 Source arXiv URL https://arxiv.org/abs/1101.0229view Article Title Integration of the EulerPoinsot Problem in New VariablesAuthors Martin Lara; Sebastián FerrerAbstract The essentially unique reduction of the EulerPoinsot problem may be performed in different sets of variables. Actionangle variables are usually preferred because of their suitability for approaching perturbed rigidbody motion. But they are just one among the variety of sets of canonical coordinates that integrate the problem. We present an alternate set of variables that, while allowing for similar performances than actionangles in the study of perturbed problems, show an important advantage over them: Their transformation from and to Andoyer variables is given in explicit form.Is Part Of 20101231 Identifier ISSN: Category nlin.SI mathph math.DS math.MPLicense 
arXivYet Another Riemann Hypothesis Linas Vepstas 20101231 This short note presents a peculiar generalization of the Riemann hypothesis, as the action of the permutation group on the elements of continued fractions. The problem is difficult to attack through traditional analytic techniques, and thus this note focuses on providing a numerical survey. These results indicate a broad class of previously unexamined functions may obey the Riemann hypothesis in general, and even share the nontrivial zeros in particular.
Subject 수학 Source arXiv URL https://arxiv.org/abs/1101.0311view Article Title Yet Another Riemann HypothesisAuthors Linas VepstasAbstract This short note presents a peculiar generalization of the Riemann hypothesis, as the action of the permutation group on the elements of continued fractions. The problem is difficult to attack through traditional analytic techniques, and thus this note focuses on providing a numerical survey. These results indicate a broad class of previously unexamined functions may obey the Riemann hypothesis in general, and even share the nontrivial zeros in particular.Is Part Of 20101231 Identifier ISSN: Category math.NTLicense 
arXivOn the Sum of Reciprocals of Amicable Numbers Jonathan Bayless, Dominic Klyve 20101231 Two numbers $m$ and $n$ are considered amicable if the sum of their proper divisors, $s(n)$ and $s(m)$, satisfy $s(n) = m$ and $s(m) = n$. In 1981, Pomerance showed that the sum of the reciprocals of all such numbers, $P$, is a constant. We obtain both a lower and an upper bound on the value of $P$.
Subject 수학 Source arXiv URL https://arxiv.org/abs/1101.0259view Article Title On the Sum of Reciprocals of Amicable NumbersAuthors Jonathan Bayless; Dominic KlyveAbstract Two numbers $m$ and $n$ are considered amicable if the sum of their proper divisors, $s(n)$ and $s(m)$, satisfy $s(n) = m$ and $s(m) = n$. In 1981, Pomerance showed that the sum of the reciprocals of all such numbers, $P$, is a constant. We obtain both a lower and an upper bound on the value of $P$.Is Part Of 20101231 Identifier ISSN: Category math.NTLicense 
arXivExotic PDE's Agostino Prástaro 20101231 In the framework of the PDE's algebraic topology, previously introduced by A. Pr\'astaro, are considered {\em exotic differential equations}, i.e., differential equations admitting Cauchy manifolds $N$ identifiable with exotic spheres, or such that their boundaries $\partial N$ are exotic spheres. For such equations are obtained local and global existence theorems and stability theorems. In particular the smooth ($4$dimensional) Poincar\'e conjecture is proved. This allows to complete the previous Theorem 4.59 in \cite{PRA17} also for the case $n=4$.
Subject 수학 Source arXiv URL https://arxiv.org/abs/1101.0283view Article Title Exotic PDE'sAuthors Agostino PrástaroAbstract In the framework of the PDE's algebraic topology, previously introduced by A. Pr\'astaro, are considered {\em exotic differential equations}, i.e., differential equations admitting Cauchy manifolds $N$ identifiable with exotic spheres, or such that their boundaries $\partial N$ are exotic spheres. For such equations are obtained local and global existence theorems and stability theorems. In particular the smooth ($4$dimensional) Poincar\'e conjecture is proved. This allows to complete the previous Theorem 4.59 in \cite{PRA17} also for the case $n=4$.Is Part Of 20101231 Identifier ISSN: Category math.GMLicense 
arXivThe Degrees of Freedom Regions of TwoUser and Certain ThreeUser MIMO Broadcast Channels with Delayed CSIT Chinmay S. Vaze, Mahesh K. Varanasi 20101231 The degrees of freedom (DoF) region of the fastfading MIMO (multipleinput multipleoutput) Gaussian broadcast channel (BC) is studied when there is delayed channel state information at the transmitter (CSIT). In this setting, the channel matrices are assumed to vary independently across time and the transmitter is assumed to know the channel matrices with some arbitrary finite delay. An outerbound to the DoF region of the general $K$user MIMO BC (with an arbitrary number of antennas at each terminal) is derived. This outerbound is then shown to be tight for two classes of MIMO BCs, namely, (a) the twouser MIMO BC with arbitrary number of antennas at all terminals, and (b) for certain threeuser MIMO BCs where all three receivers have an equal number of antennas and the transmitter has no more than twice the number of antennas present at each receivers. The achievability results are obtained by developing an interference alignment scheme that optimally accounts for multiple, and possibly distinct, number of antennas at the receivers.
Subject 수학 Source arXiv URL https://arxiv.org/abs/1101.0306view Article Title The Degrees of Freedom Regions of TwoUser and Certain ThreeUser MIMO Broadcast Channels with Delayed CSITAuthors Chinmay S. Vaze; Mahesh K. VaranasiAbstract The degrees of freedom (DoF) region of the fastfading MIMO (multipleinput multipleoutput) Gaussian broadcast channel (BC) is studied when there is delayed channel state information at the transmitter (CSIT). In this setting, the channel matrices are assumed to vary independently across time and the transmitter is assumed to know the channel matrices with some arbitrary finite delay. An outerbound to the DoF region of the general $K$user MIMO BC (with an arbitrary number of antennas at each terminal) is derived. This outerbound is then shown to be tight for two classes of MIMO BCs, namely, (a) the twouser MIMO BC with arbitrary number of antennas at all terminals, and (b) for certain threeuser MIMO BCs where all three receivers have an equal number of antennas and the transmitter has no more than twice the number of antennas present at each receivers. The achievability results are obtained by developing an interference alignment scheme that optimally accounts for multiple, and possibly distinct, number of antennas at the receivers.Is Part Of 20101231 Identifier ISSN: Category cs.IT math.ITLicense 
arXivAleksandrovBakelmanPucci Type Estimates For IntegroDifferential Equations Nestor Guillen, Russell Schwab 20101231 In this work we provide an AleksandrovBakelmanPucci type estimate for a certain class of fully nonlinear elliptic integrodifferential equations, the proof of which relies on an appropriate generalization of the convex envelope to a nonlocal, fractionalorder setting and on the use of Riesz potentials to interpret second derivatives as fractional order operators. This result applies to a family of equations involving some nondegenerate kernels and as a consequence provides some new regularity results for previously untreated equations. Furthermore, this result also gives a new comparison theorem for viscosity solutions of such equations which only depends on the $L^\infty$ and $L^n$ norms of the right hand side, in contrast to previous comparison results which utilize the continuity of the right hand side for their conclusions. These results appear to be new even for the linear case of the relevant equations.
Subject 수학 Source arXiv URL https://arxiv.org/abs/1101.0279view Article Title AleksandrovBakelmanPucci Type Estimates For IntegroDifferential EquationsAuthors Nestor Guillen; Russell SchwabAbstract In this work we provide an AleksandrovBakelmanPucci type estimate for a certain class of fully nonlinear elliptic integrodifferential equations, the proof of which relies on an appropriate generalization of the convex envelope to a nonlocal, fractionalorder setting and on the use of Riesz potentials to interpret second derivatives as fractional order operators. This result applies to a family of equations involving some nondegenerate kernels and as a consequence provides some new regularity results for previously untreated equations. Furthermore, this result also gives a new comparison theorem for viscosity solutions of such equations which only depends on the $L^\infty$ and $L^n$ norms of the right hand side, in contrast to previous comparison results which utilize the continuity of the right hand side for their conclusions. These results appear to be new even for the linear case of the relevant equations.Is Part Of 20101231 Identifier ISSN: Category math.AP math.OC math.PRLicense 
arXivAn Asymptotic Formula For Counting Subset Sums Over Subgroups Of Finite Fields Guizhen Zhu, Daqing Wan 20101231 Let F_q be the finite field of q elements. Let H be a multiplicative subgroup of F_q^*. For a positive integer k and element b\in F_q, we give a sharp estimate for the number of kelement subsets of H which sum to b.
Subject 수학 Source arXiv URL https://arxiv.org/abs/1101.0289view Article Title An Asymptotic Formula For Counting Subset Sums Over Subgroups Of Finite FieldsAuthors Guizhen Zhu; Daqing WanAbstract Let F_q be the finite field of q elements. Let H be a multiplicative subgroup of F_q^*. For a positive integer k and element b\in F_q, we give a sharp estimate for the number of kelement subsets of H which sum to b.Is Part Of 20101231 Identifier ISSN: Category math.NTLicense 
arXivThe nonprojective part of the Lie module for the symmetric group Karin Erdmann, Kai Meng Tan 20101231 The Lie module of the group algebra $FS_n$ of the symmetric group is known to be not projective if and only if the characteristic $p$ of $F$ divides $n$. We show that in this case its nonprojective summands belong to the principal block of $FS_n$. Let $V$ be a vector space of dimension $m$ over $F$, and let $L^n(V)$ be the $n$th homogeneous part of the free Lie algebra on $V$; this is a polynomial representation of $GL_m(F)$ of degree $n$, or equivalently, a module of the Schur algebra $S(m,n)$. Our result implies that, when $m \geq n$, every summand of $L^n(V)$ which is not a tilting module belongs to the principal block of $S(m,n)$, by which we mean the block containing the $n$th symmetric power of $V$.
Subject 수학 Source arXiv URL https://arxiv.org/abs/1101.0254view Article Title The nonprojective part of the Lie module for the symmetric groupAuthors Karin Erdmann; Kai Meng TanAbstract The Lie module of the group algebra $FS_n$ of the symmetric group is known to be not projective if and only if the characteristic $p$ of $F$ divides $n$. We show that in this case its nonprojective summands belong to the principal block of $FS_n$. Let $V$ be a vector space of dimension $m$ over $F$, and let $L^n(V)$ be the $n$th homogeneous part of the free Lie algebra on $V$; this is a polynomial representation of $GL_m(F)$ of degree $n$, or equivalently, a module of the Schur algebra $S(m,n)$. Our result implies that, when $m \geq n$, every summand of $L^n(V)$ which is not a tilting module belongs to the principal block of $S(m,n)$, by which we mean the block containing the $n$th symmetric power of $V$.Is Part Of 20101231 Identifier ISSN: Category math.RTLicense 
arXivGeneralised Wishart Processes Andrew Gordon Wilson, Zoubin Ghahramani 20101231 We introduce a stochastic process with Wishart marginals: the generalised Wishart process (GWP). It is a collection of positive semidefinite random matrices indexed by any arbitrary dependent variable. We use it to model dynamic (e.g. time varying) covariance matrices. Unlike existing models, it can capture a diverse class of covariance structures, it can easily handle missing data, the dependent variable can readily include covariates other than time, and it scales well with dimension; there is no need for free parameters, and optional parameters are easy to interpret. We describe how to construct the GWP, introduce general procedures for inference and predictions, and show that it outperforms its main competitor, multivariate GARCH, even on financial data that especially suits GARCH. We also show how to predict the mean of a multivariate process while accounting for dynamic correlations.
Subject 수학 Source arXiv URL https://arxiv.org/abs/1101.0240view Article Title Generalised Wishart ProcessesAuthors Andrew Gordon Wilson; Zoubin GhahramaniAbstract We introduce a stochastic process with Wishart marginals: the generalised Wishart process (GWP). It is a collection of positive semidefinite random matrices indexed by any arbitrary dependent variable. We use it to model dynamic (e.g. time varying) covariance matrices. Unlike existing models, it can capture a diverse class of covariance structures, it can easily handle missing data, the dependent variable can readily include covariates other than time, and it scales well with dimension; there is no need for free parameters, and optional parameters are easy to interpret. We describe how to construct the GWP, introduce general procedures for inference and predictions, and show that it outperforms its main competitor, multivariate GARCH, even on financial data that especially suits GARCH. We also show how to predict the mean of a multivariate process while accounting for dynamic correlations.Is Part Of 20101231 Identifier ISSN: Category stat.ME math.PR qfin.CP qfin.ST stat.MLLicense 
arXivAn inverse scattering problem for the KleinGordon equation with a classical source in quantum field theory Hironobu Sasaki, Akito Suzuki 20101231 An inverse scattering problem for a quantized scalar field ${\bm \phi}$ obeying a linear KleinGordon equation $(\square + m^2 + V) {\bm \phi} = J \mbox{in $\mathbb{R} \times \mathbb{R}^3$}$ is considered, where $V$ is a repulsive external potential and $J$ an external source $J$. We prove that the scattering operator $\mathscr{S}= \mathscr{S}(V,J)$ associated with ${\bm \phi}$ uniquely determines $V$. Assuming that $J$ is of the form $J(t,x)=j(t)\rho(x)$, $(t,x) \in \mathbb{R} \times \mathbb{R}^3$, we represent $\rho$ (resp. $j$) in terms of $j$ (resp. $\rho$) and $\mathscr{S}$.
Subject 수학 Source arXiv URL https://arxiv.org/abs/1101.0310view Article Title An inverse scattering problem for the KleinGordon equation with a classical source in quantum field theoryAuthors Hironobu Sasaki; Akito SuzukiAbstract An inverse scattering problem for a quantized scalar field ${\bm \phi}$ obeying a linear KleinGordon equation $(\square + m^2 + V) {\bm \phi} = J \mbox{in $\mathbb{R} \times \mathbb{R}^3$}$ is considered, where $V$ is a repulsive external potential and $J$ an external source $J$. We prove that the scattering operator $\mathscr{S}= \mathscr{S}(V,J)$ associated with ${\bm \phi}$ uniquely determines $V$. Assuming that $J$ is of the form $J(t,x)=j(t)\rho(x)$, $(t,x) \in \mathbb{R} \times \mathbb{R}^3$, we represent $\rho$ (resp. $j$) in terms of $j$ (resp. $\rho$) and $\mathscr{S}$.Is Part Of 20101231 Identifier ISSN: Category mathph math.MPLicense