Extremal routing problem with internal losses

An extremal routing problem with constraints in the form of precedence conditions and with additional (internal) losses related to the trajectory staying within the goal sets is considered.     

Extremal routing problem with internal losses

We discuss an approach to studying Fano manifolds based on Homological Mirror Symmetry. We consider some classical examples from a new point of view. To the cherished memory of our unforgettable teacher V.A. Iskovskikh     

Iteration method in a routing problem with internal losses

The problem of a sequential traversal of sets is considered, which is complicated by the necessity of fulfilling (internal) tasks on the sets as well as by constraints in the form of precedence conditions. It is assumed that the method of aggregating the expenses is additive. For the appearing extremal problem with dependent variables, an equivalent transformation to an optimization problem on a Cartesian product is proposed. Based on this, an iteration method is constructed that uses a reconstructible model of the courier problem (a traveling salesman problem complicated by precedence conditions).     

An extremal problem for operators

Let A be a Banach algebra, F a compact set in the complex plane, and h a function holomorphic in some neighborhood of the set F. Thus h(a) is meaningful for each element a ε A whose spectrum σ(a) is contained in F, and it is possible to evaluate the norm |h(a)|. Problem: Compute the supremum of the norms |h(a) as a ranges over all elements of A with spectrum contained in F and whose norm does not exceed one; that is, compute sup{|h(a)|; a ε A, σ(a) ⊂ F, |a| ⩽ 1}. This problem was first formulated and treated by the author in the particular case where A is the algebra of all linear operators on a finite-dimensional Hilbert space and F is the disc {z; |z| ⩽ r} for a given positive number r<1. The paper discusses motivation, connections with complex function theory, convergence of iterative processes, critical exponents, and the infinite companion matrix.     

An extremal problem in convolutions

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 3, pp. 412–414, March, 1989.     

An extremal problem for polynomials

We give a solution to an extremal problem for polynomials, which asks for complex numbers α₀,…,α_n${\alpha }_0,\dots ,{\alpha }_n$ of unit magnitude that minimise the largest supremum norm on the unit circle for all polynomials of degree n whose k  -th coefficient is either α_k${\alpha }_k$ or −α_k$-{\alpha }_k$.     

An extremal problem in geodetic graphs

An upper bound for the number of lines in a geodetic block of diameter d on p points is obtained, using some new general properties of geodetic blocks which are also of independent interest.     

An extremal problem in graph theory

It is proved that the maximum number of cut-vertices in a connected graph withn vertices andm edges is $$max\left\{ {q:m \leqq (_2^{n - q} ) + q} \right\}$$ All the extremal graphs are determined and the corresponding problem for cut-edges is also solved.     

An extremal problem for trigonometric polynomials

Let be a trigonometric polynomial of degree The problem of finding the largest value for in the inequality is studied. We find exactly provided is the conjugate of an even integer and For general we get an interval estimate for where the interval length tends to as tends to

     

An extremal problem related to probability

Summary We consider a walk from a stateA ₁ to a stateA _n+1 in which the probability of remaining atA _i isp _i , and the probability of progressing fromA _i toA _i+1 is 1 –p _i . The probabilityW _nk of reachingA _n+1 fromA ₁ in exactlyn + k steps can then be expressed as a polynomial of degreen + k in then variablesp ₁,,p _n . We determine the maximum value ofW _nk and the (unique) choice (p ₁,,p _n ) for which this extremum occurs.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday     

An extremal problem of quasiconformal mappings

In this paper, the following problem is studied. Let and be two domains in the complex plane with . Suppose that are two quasiconformal mappings satisfying . Let be the mapping in defined by (). If both and are uniquely extremal, is always uniquely extremal? It is shown in this paper that the answer to this problem is no.