De nition. The reason for the title MDS is because of the tie with maximum distance separable codes. Indeed, a [n; k; d] linear code C is said to be maximum distance separable if it meets the Singleton Bound k n d + 1 at equality. This bound is simply proved by noting that in a code of distance d, any choice of n d+ 1 entries can determine at most one codeword of C, so in the case of equality, ...

We consider a general class of problems of minimization of convex integral functionals subject to linear constraints. Using Fenchel duality, we prove the equality of the values of the minimization problem and its associated dual problem. This equality is a variational criterion for the existence of solution to a large class of inverse problems entering the class of generalized Fredholm integral...

Abstract: In the analysis of contingency tables, often one faces two difficult criteria: sampled and target populations are not identical and prior information translates to the presence of general linear inequality restrictions. Under these situations, we present new models of estimating cell probabilities related to four well-known methods of estimation. We prove that each model yields maximu...

In the second equality, we use (1). In the third equality, we use E‖xi−Exi‖ = E‖xi‖−‖Exi‖. In the second inequality, we use ‖a+b‖ ≤ 2‖a‖+2‖b‖. In the last inequality, we employ the following fact [Xiao and Zhang, 2014]: 1 n ∑n i=1 ‖∇fi(x)− ∇fi(x∗)‖2 ≤ 2Lmax ( f(x)− f(x∗)−∇f(x∗) (x− x∗) ) . .2 Proof of Theorem 1 First, we introduce the following Lemma. Lemma 1. u∗ = − 1 ρ (A T )∇f(x∗). Proof. Co...

Linear constraints occur naturally in many reasoning problems and the information that they represent is often uncertain. There is a difficulty in applying many AI uncertainty formalisms to this situation, as their representation of the underlying logic, either as a mutually exclusive and exhaustive set of possibilities, or with a propositional or a predicate logic, is inappropriate (or at leas...

Note: This paper was reconstructed into L A T E X. The formatting is different and there might be errors created during reconstruction. reduce the amount of redundancy generated at each step, Fourier's method becomes practical for a class of parametric linear programs. More recently, [12], [5] and [6] discuss further applications. Acknowledgement: We thank Tien Huynh for his comments. Sometimes...

In this series of lectures the basic ideas of the 1/mQ expansion in QCD (mQ is the mass of a heavy quark) are outlined. Applications to exclusive and inclusive decays are given.