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Consider an elliptic curve E without complex multiplication defined over the rationals.The absolute Galois group of Q acts on the group of torsion points of E, and this action can be expressed in term...
For a non-CM elliptic curve E/Q, Lang and Trotter made very deep conjectures concerning the number of primes p ≤ x for which ap(E) is a fixed integer (and for which the Frobeniusfield at p is a fixed ...
HYBRID SUP-NORM BOUNDS FOR HECKE-MAASS CUSP FORMS
Cutting-edge characteristic value hyperbolic measure
2015/8/25
Let f be a Hecke–Maass cusp form of eigenvalue λ and square-free level N. Normalize the hyperbolic measure such that vol(Y0(N)) = 1 and the form f such that kfk2 = 1. It is shown that kfk1 ≪1...
Nonlinear Estimators and Tail Bounds for Dimension Reduction in l1 Using Cauchy Random Projections
dimension reduction l1 norm Johnson-Lindenstrauss (JL) lemma Cauchy random projections
2015/8/21
For1 dimension reduction in the l1 norm, the method of Cauchy random projections multiplies the original data matrix A ∈ Rn×D with a random matrix R ∈ RD×k (k D) whose entries are i.i.d. samples of ...
Subharmonic functions and performance bounds on linear time-invariant feedback systems
Harmonic function linear the performance of the single output system a linear control system the harmonic function
2015/8/13
In this paper we study multi-input multi-output (MIMO) linear time-invariant (LTI) control systems. We show that some well known constraints on the performance of single-input single-output (SISO) lin...
Computing bounds for the structured singular value via an interior point algorithm
Calculation structure singular value the interior point algorithm
2015/8/12
We describe an interior point algorithm for computing the upper bound for the structured singular value described in the paper by Fan, Tits and Doyle, IEEE Trans AC, Jan. 1991. We demonstrate the perf...
Upper bounds on algebraic connectivity via convex optimization
Laplace operator matrix algebra connection Laplacian invariant subset matrix convex function
2015/8/10
The second smallest eigenvalue of the Laplacian matrix L of a graph is called its algebraic connectivity. We describe a method for obtaining an upper bound on the algebraic connectivity of a family of...
Generalized Chebyshev bounds via semidefinite programming
Quadratic inequality convex optimization computation single variable and random variable chebyshev inequality
2015/8/10
A sharp lower bound on the probability of a set defined by quadratic inequalities, given the first two moments of the distribution, can be efficiently computed using convex optimization. This result g...
Performance bounds for linear stochastic control
Stochastic control Model predictive control Linear matrix inequality Convex optimization
2015/8/10
We develop computational bounds on performance for causal state feedback stochastic control with linear dynamics, arbitrary noise distribution, and arbitrary input constraint set. This can be very use...
Performance bounds and suboptimal policies for linear stochastic control via LMIs
dynamic programming stochastic control convex optimization
2015/8/7
In a recent paper, the authors showed how to compute performance bounds for infinite horizon stochastic control problems with linear system dynamics and arbitrary constraints, objective, and noise dis...
Bounds on the speed of propagation of the KPP fronts in a cellular flow
propagation KPP fronts cellular flow
2015/7/14
We consider a reaction-diffusion-advection equation with a nonlinearity of the KPP type in a cellular flow. We show that the minimal pulsating traveling front speed c∗(A) in a flow ofamplitude A...
We develop bounds for the second largest eigenvalue and spectral gap
of a reversible Markov chain. The bounds depend on geometric quantities
such as the maximum degree, diameter and covering numbe...
On the nonlocal Fisher-KPP equation:steady states,spreading speed and global bounds
nonlocal Fisher-KPP equation steady states spreading speed global bounds
2015/7/14
We consider the Fisher-KPP equation with a non-local interaction term. We establish a condition on the interaction that allows for existence of non-constant periodic solutions,and prove uniform upper ...
BOUNDS FOR KAC’S MASTER EQUATION
Markov chain the n ball random rotation selected at random
2015/7/14
Mark Kac considered a Markov Chain on the n–sphere based on random rotations
in randomly chosen coordinate planes. This same walk was used by Hastings on
the orthogonal group. We show that the walk ...
Tight Oracle Bounds for Low-rank Matrix Recovery from a Minimal Number of Random Measurements
Matrix completion The Dantzig selector oracle inequalities norm of random matrices convex optimization and semidefinite programming
2015/6/17
This paper presents several novel theoretical results regarding the recovery of a low-rank matrix from just a few measurements consisting of linear combinations of the matrix entries. We show that pro...