results for au:Jiang_D in:math

- To improve signal-to-interference ratio (SIR) and make better use of file diversity provided by random caching, we consider two types of linear receivers, i.e., maximal ratio combining (MRC) receiver and partial zero forcing (PZF) receiver, at users in a large-scale cache-enabled single-input multi-output (SIMO) network. First, for each receiver, by utilizing tools from stochastic geometry, we derive a tractable expression and a tight upper bound for the successful transmission probability (STP). In the case of the MRC receiver, we also derive a closed-form expression for the asymptotic outage probability in the low SIR threshold regime. Then, for each receiver, we maximize the STP. In the case of the MRC receiver, we consider the maximization of the tight upper bound on the STP by optimizing the caching distribution, which is a non-convex problem. We obtain a stationary point, by solving an equivalent difference of convex (DC) programming problem using concave-convex procedure (CCCP). We also obtain a closed-form asymptotically optimal solution in the low SIR threshold regime. In the case of the PZF receiver, we consider the maximization of the tight upper bound on the STP by optimizing the caching distribution and the degrees of freedom (DoF) allocation (for boosting the signal power), which is a mixed discrete-continuous problem. Based on structural properties, we obtain a low-complexity near optimal solution by using an alternating optimization approach. The analysis and optimization results reveal the impact of antenna resource at users on random caching. Finally, by numerical results, we show that the random caching design with the PZF receiver achieves significant performance gains over the random caching design with the MRC receiver and some baseline caching designs.
- Jan 03 2018 math.OC arXiv:1801.00561v1The subgradient extragradient method for solving the variational inequality (VI) problem, which is introduced by Censor et al. \citeCGR, replaces the second projection onto the feasible set of the VI, in the extragradient method, with a subgradient projection onto some constructible half-space. Since the method has been introduced, many authors proposed extensions and modifications with applications to various problems. In this paper, we introduce a modified subgradient extragradient method by improving the stepsize of its second step. Convergence of the proposed method is proved under standard and mild conditions and primary numerical experiments illustrate the performance and advantage of this new subgradient extragradient variant.
- Nov 07 2017 math.OC arXiv:1711.01936v3In this paper we study the bounded perturbation resilience of projection and contraction algorithms for solving variational inequality (VI) problems in real Hilbert spaces. Under typical and standard assumptions of monotonicity and Lipschitz continuity of the VI's associated mapping, convergence of the perturbed projection and contraction algorithms is proved. Based on the bounded perturbed resilience of projection and contraction algorithms, we present some inertial projection and contraction algorithms. In addition we show that the perturbed algorithms converges at the rate of $O(1/t)$.
- Nov 07 2017 math.OC arXiv:1711.01937v3In this paper we study the bounded perturbation resilience of the extragradient and the subgradient extragradient methods for solving variational inequality (VI) problem in real Hilbert spaces. This is an important property of algorithms which guarantees the convergence of the scheme under summable errors, meaning that an inexact version of the methods can also be considered. Moreover, once an algorithm is proved to be bounded perturbation resilience, superiorizion can be used, and this allows flexibility in choosing the bounded perturbations in order to obtain a superior solution, as well explained in the paper. We also discuss some inertial extragradient methods. Under mild and standard assumptions of monotonicity and Lipschitz continuity of the VI's associated mapping, convergence of the perturbed extragradient and subgradient extragradient methods is proved. In addition we show that the perturbed algorithms converges at the rate of $O(1/t)$. Numerical illustrations are given to demonstrate the performances of the algorithms.
- Monte Carlo Tree Search (MCTS), most famously used in game-play artificial intelligence (e.g., the game of Go), is a well-known strategy for constructing approximate solutions to sequential decision problems. Its primary innovation is the use of a heuristic, known as a default policy, to obtain Monte Carlo estimates of downstream values for states in a decision tree. This information is used to iteratively expand the tree towards regions of states and actions that an optimal policy might visit. However, to guarantee convergence to the optimal action, MCTS requires the entire tree to be expanded asymptotically. In this paper, we propose a new technique called Primal-Dual MCTS that utilizes sampled information relaxation upper bounds on potential actions, creating the possibility of "ignoring" parts of the tree that stem from highly suboptimal choices. This allows us to prove that despite converging to a partial decision tree in the limit, the recommended action from Primal-Dual MCTS is optimal. The new approach shows significant promise when used to optimize the behavior of a single driver navigating a graph while operating on a ride-sharing platform. Numerical experiments on a real dataset of 7,000 trips in New Jersey suggest that Primal-Dual MCTS improves upon standard MCTS by producing deeper decision trees and exhibits a reduced sensitivity to the size of the action space.
- We study exact solutions of the quasi-one-dimensional Gross-Pitaevskii (GP) equation with the (space, time)-modulated potential and nonlinearity and the time-dependent gain or loss term in Bose-Einstein condensates. In particular, based on the similarity transformation, we report several families of exact solutions of the GP equation in the combination of the harmonic and Gaussian potentials, in which some physically relevant solutions are described. The stability of the obtained matter-wave solutions is addressed numerically such that some stable solutions are found. Moreover, we also analyze the parameter regimes for the stable solutions. These results may raise the possibility of relative experiments and potential applications.
- We investigate the local descents for special orthogonal groups over p-adic local fields of characteristic zero, and obtain an explicit spectral decomposition of the local descents at the first occurrence index in terms of the local Langlands data via the explicit local Langlands correspondence and explicit calculations of relevant local root numbers. The main result can be regarded as a refinement of the local Gan-Gross-Prasad conjecture.
- Existing designs for content dissemination do not fully explore and exploit potential caching and computation capabilities in advanced wireless networks. In this paper, we propose two partition-based caching designs, i.e., a coded caching design based on Random Linear Network Coding and an uncoded caching design. We consider the analysis and optimization of the two caching designs in a large-scale successive interference cancelation (SIC)-enabled wireless network. First, under each caching design, by utilizing tools from stochastic geometry and adopting appropriate approximations, we derive a tractable expression for the successful transmission probability in the general file size regime. To further obtain design insights, we also derive closed-form expressions for the successful transmission probability in the small and large file size regimes, respectively. Then, under each caching design, we consider the successful transmission probability maximization in the general file size regime, which is an NP-hard problem. By exploring structural properties, we successfully transform the original optimization problem into a Multiple-Choice Knapsack Problem (MCKP), and obtain a near optimal solution with 1/2 approximation guarantee and polynomial complexity. We also obtain closed-form asymptotically optimal solutions. The analysis and optimization results show the advantage of the coded caching design over the uncoded caching design, and reveal the impact of caching and SIC capabilities. Finally, by numerical results, we show that the two proposed caching designs achieve significant performance gains over some baseline caching designs.
- Jul 21 2016 math.AP arXiv:1607.05917v1In this paper, we first establish a weak unique continuation property for time-fractional diffusion-advection equations. The proof is mainly based on the Laplace transform and the unique continuation properties for elliptic and parabolic equations. The result is weaker than its parabolic counterpart in the sense that we additionally impose the homogeneous boundary condition. As a direct application, we prove the uniqueness for an inverse problem on determining the spatial component in the source term in by interior measurements. Numerically, we reformulate our inverse source problem as an optimization problem, and propose an iteration thresholding algorithm. Finally, several numerical experiments are presented to show the accuracy and efficiency of the algorithm.
- May 11 2016 math.OC arXiv:1605.02848v2We consider the sequential decision problem faced by the manager of an electric vehicle (EV) charging station, who aims to satisfy the charging demand of the customer while minimizing cost. Since the total time needed to charge the EV up to capacity is often less than the amount of time that the customer is away, there are opportunities to exploit electricity spot price variations within some reservation window. We formulate the problem as a finite horizon Markov decision process (MDP) and consider a risk-averse objective function by optimizing under a dynamic risk measure constructed using a convex combination of expected value and conditional value at risk (CVaR). It has been recognized that the objective function of a risk-averse MDP lacks a practical interpretation. Therefore, in both academic and industry practice, the dynamic risk measure objective is often not of primary interest; instead, the risk-averse MDP is used as a computational tool for solving problems with predefined "practical" risk and reward objectives (termed the base model). In this paper, we study the extent to which the two sides of this framework are compatible with each other for the EV setting -- roughly speaking, does a "more risk-averse" MDP provide lower risk in the practical sense as well? In order to answer such a question, the effect of the degree of dynamic risk-aversion on the optimal MDP policy is analyzed. Based on these results, we also propose a principled approximation approach to finding an instance of the risk-averse MDP whose optimal policy behaves well under the practical objectives of the base model. Our numerical experiments suggest that EV charging stations can be operated at a significantly higher level of profitability if dynamic charging is adopted and a small amount of risk is tolerated.
- Heterogeneous wireless networks (HetNets) provide a powerful approach to meet the dramatic mobile traffic growth, but also impose a significant challenge on backhaul. Caching and multicasting at macro and pico base stations (BSs) are two promising methods to support massive content delivery and reduce backhaul load in HetNets. In this paper, we jointly consider caching and multicasting in a large-scale cache-enabled HetNet with backhaul constraints. We propose a hybrid caching design consisting of identical caching in the macro-tier and random caching in the pico-tier, and a corresponding multicasting design. By carefully handling different types of interferers and adopting appropriate approximations, we derive tractable expressions for the successful transmission probability in the general region as well as the high signal-to-noise ratio (SNR) and user density region, utilizing tools from stochastic geometry. Then, we consider the successful transmission probability maximization by optimizing the design parameters, which is a very challenging mixed discrete-continuous optimization problem due to the sophisticated structure of the successful transmission probability. By using optimization techniques and exploring the structural properties, we obtain a near optimal solution with superior performance and manageable complexity. This solution achieves better performance in the general region than any asymptotically optimal solution, under a mild condition. The analysis and optimization results provide valuable design insights for practical cache-enabled HetNets.
- In heterogeneous networks (HetNets), strong interference due to spectrum reuse affects each user's signal-to-interference ratio (SIR), and hence is one limiting factor of network performance. In this paper, we propose a user-centric interference nulling (IN) scheme in a downlink large-scale HetNet to improve coverage/outage probability by improving each user's SIR. This IN scheme utilizes at most maximum IN degree of freedom (DoF) at each macro-BS to avoid interference to uniformly selected macro (pico) users with signal-to-individual-interference ratio (SIIR) below a macro (pico) IN threshold, where the maximum IN DoF and the two IN thresholds are three design parameters. Using tools from stochastic geometry, we first obtain a tractable expression of the coverage (equivalently outage) probability. Then, we analyze the asymptotic coverage/outage probability in the low and high SIR threshold regimes. The analytical results indicate that the maximum IN DoF can affect the order gain of the outage probability in the low SIR threshold regime, but cannot affect the order gain of the coverage probability in the high SIR threshold regime. Moreover, we characterize the optimal maximum IN DoF which optimizes the asymptotic coverage/outage probability. The optimization results reveal that the IN scheme can linearly improve the outage probability in the low SIR threshold regime, but cannot improve the coverage probability in the high SIR threshold regime. Finally, numerical results show that the proposed scheme can achieve good gains in coverage/outage probability over a maximum ratio beamforming scheme and a user-centric almost blank subframes (ABS) scheme.
- In [Ar13], Arthur classifies the automorphic discrete spectrum of symplectic groups up to global Arthur packets, based on the theory of endoscopy. It is an interesting and basic question to ask: which global Arthur packets contain no cuspidal automorphic representations? The investigation on this question can be regarded as a further development of the topics originated from the classical theory of singular automorphic forms. The results obtained yield a better understanding of global Arthur packets and of the structure of local unramified components of the cuspidal spectrum, and hence are closely related to the generalized Ramanujan problem as posted by Sarnak in [Sar05].
- Caching and multicasting at base stations are two promising approaches to support massive content delivery over wireless networks. However, existing analysis and designs do not fully explore and exploit the potential advantages of the two approaches. In this paper, we consider the analysis and optimization of caching and multicasting in a large-scale cache-enabled wireless network. We propose a random caching and multicasting scheme with a design parameter. By carefully handling different types of interferers and adopting appropriate approximations, we derive a tractable expression for the successful transmission probability in the general region, utilizing tools from stochastic geometry. We also obtain a closed-form expression for the successful transmission probability in the high signal-to-noise ratio (SNR) and user density region. Then, we consider the successful transmission probability maximization, which is a very complex non-convex problem in general. Using optimization techniques, we develop an iterative numerical algorithm to obtain a local optimal caching and multicasting design in the general region. To reduce complexity and maintain superior performance, we also derive an asymptotically optimal caching and multicasting design in the asymptotic region, based on a two-step optimization framework. Finally, numerical simulations show that the asymptotically optimal design achieves a significant gain in successful transmission probability over some baseline schemes in the general region.
- In this paper, we investigate the inverse problem on determining the spatial component of the source term in a hyperbolic equation with time-dependent principal part. Based on a newly established Carleman estimate for general hyperbolic operators, we prove a local stability result of HÃ¶lder type in both cases of partial boundary and interior observation data. Numerically, we adopt the classical Tikhonov regularization to transform the inverse problem into an output least-squares minimization, which can be solved by the iterative thresholding algorithm. The proposed algorithm is computationally easy and efficient: the minimizer at each step has explicit solution. Abundant amounts of numerical experiments are presented to demonstrate the accuracy and efficiency of the algorithm.
- In this paper, we consider a finite-horizon Markov decision process (MDP) for which the objective at each stage is to minimize a quantile-based risk measure (QBRM) of the sequence of future costs; we call the overall objective a dynamic quantile-based risk measure (DQBRM). In particular, we consider optimizing dynamic risk measures where the one-step risk measures are QBRMs, a class of risk measures that includes the popular value at risk (VaR) and the conditional value at risk (CVaR). Although there is considerable theoretical development of risk-averse MDPs in the literature, the computational challenges have not been explored as thoroughly. We propose data-driven and simulation-based approximate dynamic programming (ADP) algorithms to solve the risk-averse sequential decision problem. We address the issue of inefficient sampling for risk applications in simulated settings and present a procedure, based on importance sampling, to direct samples toward the "risky region" as the ADP algorithm progresses. Finally, we show numerical results of our algorithms in the context of an application involving risk-averse bidding for energy storage.
- The endoscopic classification via the stable trace formula comparison provides certain character relations between irreducible cuspidal automorphic representations of classical groups and their global Arthur parameters, which are certain automorphic representations of general linear groups. It is a question of J. Arthur and W. Schmid that asks: How to construct concrete modules for irreducible cuspidal automorphic representations of classical groups in term of their global Arthur parameters? In this paper, we formulate a general construction of concrete modules, using Bessel periods, for cuspidal automorphic representations of classical groups with generic global Arthur parameters. Then we establish the theory for orthogonal and unitary groups, based on certain well-expected conjectures. Among the consequences of the theory in this paper is that the global Gan-Gross-Prasad conjecture for those classical groups is proved in full generality in one direction and with a global assumption in the other direction.
- Jul 14 2015 math.NT arXiv:1507.03297v2In this paper, we introduce a new family of period integrals attached to irreducible cuspidal automorphic representations $\sigma$ of symplectic groups $\mathrm{Sp}_{2n}(\mathbb{A})$, which detects the right-most pole of the $L$-function $L(s,\sigma\times\chi)$ for some character $\chi$ of $F^\times\backslash\mathbb{A}^\times$ of order at most $2$, and hence the occurrence of a simple global Arthur parameter $(\chi,b)$ in the global Arthur parameter $\psi$ attached to $\sigma$. We also give a characterisation of first occurrences of theta correspondence by (regularised) period integrals of residues of certain Eisenstein series.
- The Local Converse Problem is to determine how the family of the local gamma factors $\gamma(s,\pi\times\tau,\psi)$ characterizes the isomorphism class of an irreducible admissible generic representation $\pi$ of $\mathrm{GL}_n(F)$, with $F$ a non-archimedean local field, where $\tau$ runs through all irreducible supercuspidal representations of $\mathrm{GL}_r(F)$ and $r$ runs through positive integers. The Jacquet conjecture asserts that it is enough to take $r=1,2,\ldots,\left[\frac{n}{2}\right]$. Based on arguments in the work of Henniart and of Chen giving preliminary steps towards the Jacquet conjecture, we formulate a general approach to prove the Jacquet conjecture. With this approach, the Jacquet conjecture is proved under an assumption which is then verified in several cases, including the case of level zero representations.
- The existence of the well-known Jacquet-Langlands correspondence was established by Jacquet and Langlands via the trace formula method in 1970. An explicit construction of such a correspondence was obtained by Shimizu via theta series in 1972. In this paper, we extend the automorphic descent method of Ginzburg-Rallis-Soudry to a new setting. As a consequence, we recover the classical Jacquet-Langlands correspondence for PGL(2) via a new explicit construction.
- We study wave-front sets of representations of reductive groups over global or non-archimedean local fields.
- In the theory of automorphic descents developed by Ginzburg, Rallis and Soudry in [GRS11], the structure of Fourier coefficients of the residual representations of certain special Eisenstein series plays important roles. Started from [JLZ13], the authors are looking for more general residual representations, which may yield more general theory of automorphic descents. In this paper, we investigate the structure of Fourier coefficients of certain residual representations of symplectic groups, corresponding to certain interesting families of global Arthur parameters. On one hand, the results partially confirm a conjecture proposed by the first named author in [J14] on relations between the global Arthur parameters and the structure of Fourier coefficients of the automorphic representations in the corresponding global Arthur packets. On the other hand, the results of this paper can be regarded as a first step towards more general automorphic descents for symplectic groups, which will be considered in our future work.
- In [J14], a conjecture was proposed on a relation between the global Arthur parameters and the structure of Fourier coefficients of the automorphic representations in the corresponding global Arthur packets. In this paper, we discuss the recent progress on this conjecture and certain problems which lead to better understanding of Fourier coefficients of automorphic forms. At the end, we extend a useful technical lemma to a few versions, which are more convenient for future applications.
- Sep 03 2014 math.NT arXiv:1409.0767v3Following the idea of [GJS09] for orthogonal groups, we introduce a new family of period integrals for cuspidal automorphic representations $\sigma$ of unitary groups and investigate their relation with the occurrence of a simple global Arthur parameter $(\chi,b)$ in the global Arthur parameter $\psi_\sigma$ associated to $\sigma$, by the endoscopic classification of Arthur ([Art13], [Mok13], [KMSW14]). The argument uses the theory of theta correspondence. This can be viewed as a part of the $(\chi,b)$-theory outlined in [Jia14] and can be regarded as a refinement of the theory of theta correspondences and poles of certain $L$-functions, which was outlined in [Ral91].
- Jul 07 2014 math.PR arXiv:1407.1263v2In probability theory, equalities are much less than inequalities. In this paper, we find a series of equalities which characterize the symmetry of the forming times of a family of similar cycles for discrete-time and continuous-time Markov chains. Moreover, we use these cycle symmetries to study the circulation fluctuations for Markov chains. We prove that the empirical circulations of a family of cycles passing through a common state satisfy a large deviation principle with a rate function which has an highly non-obvious symmetry. Finally, we discuss the applications of our work in statistical physics and biochemistry.
- Feb 17 2014 math.OC arXiv:1402.3575v4There is growing interest in the use of grid-level storage to smooth variations in supply that are likely to arise with increased use of wind and solar energy. Energy arbitrage, the process of buying, storing, and selling electricity to exploit variations in electricity spot prices, is becoming an important way of paying for expensive investments into grid-level storage. Independent system operators such as the NYISO (New York Independent System Operator) require that battery storage operators place bids into an hour-ahead market (although settlements may occur in increments as small as 5 minutes, which is considered near "real-time"). The operator has to place these bids without knowing the energy level in the battery at the beginning of the hour, while simultaneously accounting for the value of leftover energy at the end of the hour. The problem is formulated as a dynamic program. We describe and employ a convergent approximate dynamic programming (ADP) algorithm that exploits monotonicity of the value function to find a revenue-generating bidding policy; using optimal benchmarks, we empirically show the computational benefits of the algorithm. Furthermore, we propose a distribution-free variant of the ADP algorithm that does not require any knowledge of the distribution of the price process (and makes no assumptions regarding a specific real-time price model). We demonstrate that a policy trained on historical real-time price data from the NYISO using this distribution-free approach is indeed effective.
- Jan 09 2014 math.OC arXiv:1401.1590v6Many sequential decision problems can be formulated as Markov Decision Processes (MDPs) where the optimal value function (or cost-to-go function) can be shown to satisfy a monotone structure in some or all of its dimensions. When the state space becomes large, traditional techniques, such as the backward dynamic programming algorithm (i.e., backward induction or value iteration), may no longer be effective in finding a solution within a reasonable time frame, and thus we are forced to consider other approaches, such as approximate dynamic programming (ADP). We propose a provably convergent ADP algorithm called Monotone-ADP that exploits the monotonicity of the value functions in order to increase the rate of convergence. In this paper, we describe a general finite-horizon problem setting where the optimal value function is monotone, present a convergence proof for Monotone-ADP under various technical assumptions, and show numerical results for three application domains: optimal stopping, energy storage/allocation, and glycemic control for diabetes patients. The empirical results indicate that by taking advantage of monotonicity, we can attain high quality solutions within a relatively small number of iterations, using up to two orders of magnitude less computation than is needed to compute the optimal solution exactly.
- Nov 12 2013 math.PR arXiv:1311.2203v4Cyclic structure and dynamics are of great interest in both the fields of stochastic processes and nonequilibrium statistical physics. In this paper, we find a new symmetry of the Brownian motion named as the quasi-time-reversal invariance. It turns out that such an invariance of the Brownian motion is the key to prove the cycle symmetry for diffusion processes on the circle, which says that the distributions of the forming times of the forward and backward cycles, given that the corresponding cycle is formed earlier than the other, are exactly the same. With the aid of the cycle symmetry, we prove the strong law of large numbers, functional central limit theorem, and large deviation principle for the sample circulations and net circulations of diffusion processes on the circle. The cycle symmetry is further applied to obtain various types of fluctuation theorems for the sample circulations, net circulation, and entropy production rate.
- Fourier coefficients of automorphic representations $\pi$ of $\Sp_{2n}(\BA)$ are attached to unipotent adjoint orbits in $\Sp_{2n}(F)$, where $F$ is a number field and $\BA$ is the ring of adeles of $F$. We prove that for a given $\pi$, all maximal unipotent orbits, which gives nonzero Fourier coefficients of $\pi$ are special, and prove, under a well acceptable assumption, that if $\pi$ is cuspidal, then the stabilizer attached to each of those maximal unipotent orbits is $F$-anisotropic as algebraic group over $F$. These results strengthen, refine and extend the earlier work of Ginzburg, Rallis and Soudry on the subject. As a consequence, we obtain constraints on those maximal unipotent orbits if $F$ is totally imaginary, further applications of which to the discrete spectrum with the Arthur classification will be considered in our future work.
- We study the structures of Fourier coefficients of automorphic forms on symplectic groups based on their local and global structures related to Arthur parameters. This is a first step towards the general conjecture on the relation between the structure of Fourier coefficients and Arthur parameters given by the first named author in [J14].
- A family of global integrals representing a product of tensor product (partial) $L$-functions: $ L^S(s,\pi\times\tau_1)L^S(s,\pi\times\tau_2)... L^S(s,\pi\times\tau_r) $ are established in this paper, where $\pi$ is an irreducible cuspidal automorphic representation of a quasi-split classical group of Hermitian type and $\tau_1,...,\tau_r$ are irreducible unitary cuspidal automorphic representations of $\GL_{a_1},...,\GL_{a_r}$, respectively. When $r=1$ and the classical group is an orthogonal group, this was studied by Ginzburg, Piatetski-Shapiro and Rallis in 1997 and when $\pi$ is generic and $\tau_1,...,\tau_r$ are not isomorphic to each other, this is considered by Ginzburg, Rallis and Soudry in 2011. In this paper, we prove that the global integrals are eulerian and finish the explicit calculation of unramified local $L$-factors in general. The remaining local and global theory for this family of global integrals will be considered in our future work.
- Dec 31 2012 math.RT arXiv:1212.6525v1A general framework of constructions of endoscopy correspondences via automorphic integral transforms for classical groups is formulated in terms of the Arthur classification of the discrete spectrum of square-integrable automorphic forms. This suggests a principle, which is called the $(\tau,b)$-theory of automorphic forms of classical groups, to reorganize and extend the series of work of Piatetski-Shapiro, Rallis, Kudla and others on standard $L$-functions of classical groups and theta correspondence.
- Dec 26 2012 math.RT arXiv:1212.6015v1Determination of quasi-invariant generalized functions is important for a variety of problems in representation theory, notably character theory and restriction problems. In this note, we review some new and easy-to-use techniques to show vanishing of quasi-invariant generalized functions, developed in the recent work of the authors (Uniqueness of Ginzburg-Rallis models: the Archimedean case, Trans. Amer. Math. Soc. 363, (2011), 2763-2802). The first two techniques involve geometric notions attached to submanifolds, which we call metrical properness and unipotent $\chi$-incompatibility. The third one is analytic in nature, and it arises from the first occurrence phenomenon in Howe correspondence. We also highlight how these techniques quickly lead to two well-known uniqueness results, on trilinear forms and Whittaker models.
- For a multivariate linear model, Wilk's likelihood ratio test (LRT) constitutes one of the cornerstone tools. However, the computation of its quantiles under the null or the alternative requires complex analytic approximations and more importantly, these distributional approximations are feasible only for moderate dimension of the dependent variable, say $p\le 20$. On the other hand, assuming that the data dimension $p$ as well as the number $q$ of regression variables are fixed while the sample size $n$ grows, several asymptotic approximations are proposed in the literature for Wilk's $\bLa$ including the widely used chi-square approximation. In this paper, we consider necessary modifications to Wilk's test in a high-dimensional context, specifically assuming a high data dimension $p$ and a large sample size $n$. Based on recent random matrix theory, the correction we propose to Wilk's test is asymptotically Gaussian under the null and simulations demonstrate that the corrected LRT has very satisfactory size and power, surely in the large $p$ and large $n$ context, but also for moderately large data dimensions like $p=30$ or $p=50$. As a byproduct, we give a reason explaining why the standard chi-square approximation fails for high-dimensional data. We also introduce a new procedure for the classical multiple sample significance test in MANOVA which is valid for high-dimensional data.
- We analytically investigate the nonautonomous discrete rogue wave solutions and their interaction in the generalized Ablowitz-Ladik-Hirota lattice with variable coefficients, which possess complicated wave propagations in time and are beyond the usual discrete rogue waves. When the amplitude of the tunnel coupling coefficient between sites decreases, these nonautonomous discrete rogue wave solutions become localized in time after they propagate over some certain large critical values. Moreover, we find that the interaction between nonautonomous discrete rogue waves is elastic. In particular, these results can reduce to the usual discrete rogue wave solutions when the gain or loss term is ignored.
- Aug 13 2009 math.RT arXiv:0908.1728v2In the archimedean case, we prove uniqueness of Bessel models for general linear groups, unitary groups and orthogonal groups.
- Mar 10 2009 math.RT arXiv:0903.1411v2In this paper, we prove the uniqueness of Ginzburg-Rallis models in the archimedean case. As a key ingredient, we introduce a new descent argument based on two geometric notions attached to submanifolds, which we call metrical properness and unipotent $\chi$-incompatibility.
- In this paper, we give an explanation to the failure of two likelihood ratio procedures for testing about covariance matrices from Gaussian populations when the dimension is large compared to the sample size. Next, using recent central limit theorems for linear spectral statistics of sample covariance matrices and of random F-matrices, we propose necessary corrections for these LR tests to cope with high-dimensional effects. The asymptotic distributions of these corrected tests under the null are given. Simulations demonstrate that the corrected LR tests yield a realized size close to nominal level for both moderate p (around 20) and high dimension, while the traditional LR tests with chi-square approximation fails. Another contribution from the paper is that for testing the equality between two covariance matrices, the proposed correction applies equally for non-Gaussian populations yielding a valid pseudo-likelihood ratio test.
- In this paper we characterize irreducible generic representations of $\SO_{2n+1}(k)$ where $k$ is a $p$-adic field) by means of twisted local gamma factors (the Local Converse Theorem). As applications, we prove that two irreducible generic cuspidal automorphic representations of $\SO_{2n+1}({\Bbb A})$ (where ${\Bbb A}$ is the ring of adeles of a number field) are equivalent if their local components are equivalent at almost all local places (the Rigidity Theorem);and prove the Local Langlands Reciprocity Conjecture for generic supercuspidal representations of $\SO_{2n+1}(k)$.
- Dec 05 2002 math.AT arXiv:math/0212054v4In this article, we give some conditions on the structure of an unstable module, which are satisfied whenever this module is the reduced cohomology of a space or a spectrum. First, we study the structure of the sub-modules of Sigma^sH^*(B(Z/2)^oplus d;Z/2), i.e., the unstable modules whose nilpotent filtration has length 1. Next, we generalise this result to unstable modules whose nilpotent filtration has a finite length, and which verify an additional condition. The result says that under certain hypotheses, the reduced cohomology of a space or a spectrum does not have arbitrary large gaps in its structure. This result is obtained by applying Adams' theorem on the Hopf invariant and the classification of the injective unstable modules. This work was carried out under the direction of L. Schwartz. Resume Dans cet article, on donne des restrictions sur la structure d'un module instable, qui doivent etre verifiees pour que celui-ci soit la cohomologie reduite d'un espace ou d'un spectre. On commence par une etude sur la structure des sous-modules de Sigma^sH^*(B(Z/2)^oplus d;Z/2), i.e., les modules instables dont la filtration nilpotente est de longueur 1. Ensuite, on generalise le resultat aux modules instables dont la filtration nilpotente est de longueur finie, et qui verifient une condition supplementaire. Le resultat dit que sous certaines hypotheses, la cohomologie reduite d'un espace ou d'un spectre ne contient pas de lacunes de longueur arbitrairement grande. Ce resultat est obtenu par application du celebre theoreme d'Adams sur l'invariant de Hopf et de la classification des modules instables injectifs. Ce travail est effectue sous la direction de L. Schwartz.