in this paper, we study the convexity of the integral operator

where q~ is a c losed, convex func t ion def ined on R n, having values in ~ and S is a c losed, convex set in ~n. We write S for the op t ima l so lu t ion set o f (1), S : = arg minxes cb(x) and assume this set to be non-empty , in o rde r tha t a p ro jec t ion o p e r a t i o n on to this set is well def ined. In o rde r to s impl i fy our analysis , let us define , b ~ ( x ) := ¢~(x )+~, (...

We obtain residue formulae for certain functions of several vari-ables. As an application, we obtain closed formulae for vector partition func-tions and for their continuous analogs. They imply an Euler-MacLaurin sum-mation formula for vector partition functions, and for rational convex poly-topes as well: we express the sum of values of a polynomial function at alllattice p...

We discuss whether on not it is possible to have interpolatory estimates in the approximation of a function Sobolev`s space by polynomials. The prob-lem positive estimate pointwise degree approxi-mation r times continuously differentiable and func-tions [0, 1]. Estimates form (1) for are known ([1, 2]). problem monotone that estimating de-gree nondecreasing bymonotone were proved [3, 4, 7]. In ...

Recently, research on accelerated stochastic gradient descentmethods (e.g., SVRG) has made exciting progress (e.g., lin-ear convergence for strongly convex problems). However,the best-known methods (e.g., Katyusha) requires at leasttwo auxiliary variables and two momentum parameters. Inthis paper, we propose a fast stochastic variance reductiongradient (FSVRG) method...

In a previous paper, the authors defined a new class ofmultiple-valued logic functions, called multiple-valued majority func-tions. This correspondence clarifies the distinction of multiple-valuedmajority functions from multiple-valued threshold functions through thedifference between a number function and an inner product of an inputvector and a weight vector. Zndcx Terns-I...

We d e r i v e t h e L i p s c h i t z dependence of t h e s e t of s o l u t i o n s of a convex minimizat ion problem and i t s Lagrange m u l t i p l i e r s upon t h e n a t u r a l parameters from an Inve r se Funct ion Theorem f o r se t -valued maps. Th i s r e q u i r e s t h e use of con t ingen t and Clarke d e r i v a t i v e s of se t -valued maps, a s w e l l a s gene ra l i zed se...

Over the next few years, LEDs are likely to be responsible for all of vehicle's exterior lighting func-tions. Everything is focused on maximum security. For this reason, there more and automatic color detection systems in car. Proper separation key optimal operation proper evaluation these systems. An example correct traffic light colors. The automotive industry dependent consistency predictabi...

In this paper, we study the convexity of the integral operator