Novel insights into the multiplier rule

We present the Lagrange multiplier rule, one of the basic optimization methods, in a new way. Novel features include:• Explanation of the true source of the power of the rule: reversal of tasks, but not the use of multipliers.• A natural proof based on a simple picture, but not the usual technical derivation from the implicit function theorem.• A practical method to avoid the cumbersome second order conditions.• Applications from various areas of mathematics, physics, economics.• Some hnts on the use of the rule.bargaining;dynamical systems;economics;finance;multiplier rule;second order condition

The Lagrange multiplier rule revisited

In this paper we give a short novel proof of the well-known Lagrange multiplier rule, discuss the sources of the power of this rule and consider several applications of this rule. The new proof does not use the implicit function theorem and combines the advantages of two of the most well-known proofs: it provides the useful geometric insight of the elimination approach based on differentiable curves and technically it is not more complicated than the simple penalty approach. Then we emphasize that the power of the rule is the reversal of order of the natural tasks, elimination and differentiation. This turns the hardest task,elimination, from a nonlinear problem into a linear one. This phenomenon is illustrated by several convincing examples of applications of the rule to various areas. Finally we give three hints on the use of the rule.Lagrange multiplier rule;compactness;optimization

Tail behaviour of Gaussian processes with applications to the Brownian pillow

In this paper we investigate the tail behaviour of a random variable S which maybe viewed as a functional T of a zero mean Gaussian process X, taking specialinterest in the situation where X obeys the structure whichis typical for limiting processes ocurring in nonparametric testing of[multivariate] indepencency and [multivariate] constancy over time.The tail behaviour of S is described by means of a constant aand a random variable R which is defined on the same probability spaceas S.The constant a acts as an upper bound, and is relevant for the computation ofthe efficiency of test statistics converging in distribution to S. The random variable R acts as a lower bound, andis instrumental in deriving approximations for the upper percentage points of S by simulation.Anderson-Darling type tests;Asymptotic distribution theory;Brownian pillow;Cramer-von Mises type tests;Gaussian processes;Kolmogorov type tests;Multivariate constancy;Tail behaviour;Multivariate independence

The correlation between the convergence of subdivision processes and solvability of refinement equations.

We consider a univariate two-scale difference equation,which is studied in approximation theory, curve design andwavelets theory. This paper analysis the correlation between the existence ofsmooth compactly supported solutions of this equation and the convergence of the corresponding cascade algorithm/subdivision scheme. We introduce a criterion thatexpresses this correlation in terms of mask of the equation. It was shown that the convergence of subdivision schemedepends on values that the mask takes at the points of its generalized cycles. In this paper we show that the criterion is sharp in thesense that an arbitrary generalized cycle causes the divergenceof a suitable subdivision scheme. To do this we constructa general method to produce divergent subdivision schemeshaving smooth refinable functions. The criterion thereforeestablishes a complete classification of divergent subdivision schemes.Cycles;Cascade algorithm;Refinement equations;Subdivision process;Rate of convergence

On Borel Probability Measures and Noncooperative Game Theory

In this paper the well-known minimax theorems of Wald, Ville and VonNeumann are generalized under weaker topological conditions onthepayoff function Æ’ and/or extended to the larger set of the Borelprobabilitymeasures instead of the set of mixed strategies.minimax theory;infinite dimensional separation;Borel measures;noncooperative game theory;weak compactness

On the Decay of Infinite Products of Trigonometric Polynomials

We consider infinite products of the form , where {mk} is an arbitrary sequence of trigonometric polynomials of degree at most n with uniformly bounded norms such that mk(0)=1 for all k. We show that can decrease at infinity not faster than and present conditions under which this maximal decay attains. This result proves the impossibility of the construction of infinitely differentiable nonstationary wavelets with compact support and restricts the smoothness of nonstationary wavelets by the length of their support. Also this generalizes well-known similar results obtained for stable sequences of polynomials (when all mk coincide). In several examples we show that by weakening the boundedness conditions one can achieve an exponential decay

The stability of subdivision operator at its fixed point

The paper analysis the correlation between the existence of smooth compactly supported solutions of the univariate two-scale refinement equation and the convergence of the corresponding cascade lgorithm/subdivision scheme. We introduce a criterion that expresses this correlation in terms of mask of the equation. We show that the convergence of subdivision scheme depends on values that the mask takes at the points of its generalized cycles. This means in particular that the stability of shifts of refinable function is not necessary for the convergence of the subdivision process. This also leads to some results on the degree of convergence of subdivision processes and on factorizations of refinable functions

The Stability of Subdivision Operator

We consider the univariate two-scale refinement equation. The paper analyzes the correlation between the existence of smooth compactly supported solutions of this equation and the convergence of the corresponding cascade algorithm/subdivision scheme. We introduce a criterion that expresses this correlation in terms of mask of the equation. We show that the convergence of subdivision scheme depends on values that the mask takes at the points of its generalized cycles. This means in particular that the stability of shifts of refinable function is not necessary for the convergence of the subdivision process. This also leads to some results on the degree of convergence of subdivision processes and on factorizations of refinable functions

The correlation between the convergence of subdivision processes and solvability of refinement equations.

We consider a univariate two-scale difference equation, which is studied in approximation theory, curve design and wavelets theory. This paper analysis the correlation between the existence of smooth compactly supported solutions of this equation and the convergence of the corresponding cascade algorithm/subdivision scheme. We introduce a criterion that expresses this correlation in terms of mask of the equation. It was shown that the convergence of subdivision scheme depends on values that the mask takes at the points of its generalized cycles. In this paper we show that the criterion is sharp in the sense that an arbitrary generalized cycle causes the divergence of a suitable subdivision scheme. To do this we construct a general method to produce divergent subdivision schemes having smooth refinable functions. The criterion therefore establishes a complete classification of divergent subdivision schemes

Novel insights into the multiplier rule

We present the Lagrange multiplier rule, one of the basic optimization methods, in a new way. Novel features include: • Explanation of the true source of the power of the rule: reversal of tasks, but not the use of multipliers. • A natural proof based on a simple picture, but not the usual technical derivation from the implicit function theorem. • A practical method to avoid the cumbersome second order conditions. • Applications from various areas of mathematics, physics, economics. • Some hnts on the use of the rule