p . It follows that inequality (1.2) holds for any a ∈ lp when U1/p ≥ ||C||p,p and fails to hold for some a ∈ lp when U1/p < ||C||p,p. Hardy’s inequality thus asserts that the Cesáro matrix operator C, given by cn,k = 1/n, k ≤ n and 0 otherwise, is bounded on l p and has norm ≤ p/(p−1). (The norm is in fact p/(p− 1).) We say a matrix A = (an,k) is a lower triangular matrix if an,k = 0 for n < k...

Let Ω ⊂ R 2 be a smooth bounded domain, and H 1 0 (Ω) be the standard Sobolev space. Define for any p > 1, λp(Ω) = inf u∈H 1 0 (Ω),u ≡0 ∇u 2 2 /u 2 p , where · p denotes L p norm. We derive in this paper a sharp form of the following improved Moser-Trudinger inequality involving the L p-norm using the method of blow-up analysis: sup u∈H 1 0 (Ω),∇u 2 =1 Ω e 4π(1+αu 2 p)u 2 dx < +∞ for 0 ≤ α < λp...

For iterative sequences that converge to the solution set of a linear matrix inequality, we show that the distance of the iterates to the solution set is at most O(2 ?d). The nonnegative integer d is the so{called degree of singularity of the linear matrix inequality, and denotes the amount of constraint violation in the iterate. For infeasible linear matrix inequalities, we show that the minim...

The main result of this paper is the inequality d(u,N(A))/31 ≤ ‖EA − EuAu∗ ‖∞,2 ≤ 4d(u,N(A)), where A is a masa in a separably acting type II1 factor N , u ∈ N is a unitary, N(A) is the group of normalizing unitaries, d is the distance measured in the ‖ ·‖2-norm, and ‖ · ‖∞,2 is a norm defined on the space of bounded maps on N by ‖φ‖∞,2 = sup{‖φ(x)‖2 : ‖x‖ ≤ 1}. This result implies that a unita...

p . It follows that inequality (1.2) holds for any a ∈ lp when U1/p ≥ ||C||p,p and fails to hold for some a ∈ lp when U1/p < ||C||p,p. Hardy’s inequality thus asserts that the Cesáro matrix operator C, given by cn,k = 1/n, k ≤ n and 0 otherwise, is bounded on l p and has norm ≤ p/(p−1). (The norm is in fact p/(p− 1).) We say a matrix A = (an,k) is a lower triangular matrix if an,k = 0 for n < k...

In order to avoid overfitting, it is common practice to regularize linear prediction models using squared or absolute-value norms of the model parameters. In our article we consider a new method of regularization: Huber-norm regularization imposes a combination of `1 and `2-norm regularization on the model parameters. We derive the dual optimization problem, prove an upper bound on the statisti...

Considerable interest exists in understanding the framework of weighted inequalities for differential operators and the Fourier transform, and the application of quantitative information drawn from these inequalities to varied problems in analysis and mathematical physics, including nonlinear partial differential equations, spectral theory, fluid mechanics, stability of matter, stellar dynamics...

In this paper we consider the problem of approximating a class of quadratic optimization problems that contain orthogonality constraints, i.e. constraints of the form X X = I, where X ∈ Rm×n is the optimization variable. This class of problems, which we denote by (Qp–Oc), is quite general and captures several well–studied problems in the literature as special cases. In a recent work, Nemirovski...

Tensor completion is a technique of filling missing elements of the incomplete data tensors. It being actively studied based on the convex optimization scheme such as nuclear-norm minimization. When given data tensors include some noises, the nuclear-norm minimization problem is usually converted to the nuclear-norm ‘regularization’ problem which simultaneously minimize penalty and error terms ...

model order reduction is known as the problem of minimizing the -norm of the difference between the transfer function of the original system and the reduced one. in many papers, linear matrix inequality (lmi) approach is utilized to address the minimization problem. this paper deals with defining an extra matrix inequality constraint to guarantee that the minimum phase characteristic of the sys...