The conjugate trace and trace of a plane partition

The conjugate trace and trace of a plane partition are defined, and the generating function for the number of plane partitions π of n with ?r rows and largest part ?m, with conjugate trace t (or trace t, when m = ∞), is found. Various properties of this generating function are studied. One consequence of these properties is a formula which can be regarded as a q-analog of a well-known result arising in the representation theory of the symmetric group.     

Isoperimetric inequality in the Grushin plane

We prove a sharp isoperimetric inequality in the Grushin plane and compute the corresponding isoperimetric sets.     

The limiting distribution of the trace of a random plane partition

We study the asymptotic behaviour of the trace (the sum of the diagonal parts) τ _n = τ _n (ω) of a plane partition ω of the positive integer n, assuming that ω is chosen uniformly at random from the set of all such partitions. We prove that (τ _n − c ₀ n ^2/3)/c ₁ n ^1/3 log^1/2 n converges weakly, as n → ∞, to the standard normal distribution, where c ₀ = ζ(2)/ [2ζ(3)]^2/3, c ₁ = √(1/3/) [2ζ(3)]^1/3 and ζ(s) = Σ _j=1^∞ j ^−s . Partial support given by the National Science Fund of the Bulgarian Ministry of Education and Science, grant No. VU-MI-105/2005.     

A symmetrical plane partition correspondence

MacMahon conjectured the form of the generating function for symmetrical plane partitions, and as a special case deduced the following theorem. The set of partitions of a number n whose part magnitude and number of parts are both no greater than m is equinumerous with the set of symmetrical plane partitions of 2n whose part magnitude does not exceed 2 and whose largest axis does not exceed m. This theorem, together with a companion theorem for the symmetrical plane partitions of odd numbers, are proved by establishing 1-1 correspondences between the sets of partitions.     

A graph partition problem

ＡＧＲＡＰＨＰＡＲＴＩＴＩＯＮＰＲＯＢＬＥＭ￥ＬＩＵＴＡＮＰＥＩ（刘彦佩）（ＤｅｐａｒｆｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ,ＮｏｒｔｈｅｒｎＪｉａｏｔｏｎｙＵｎｉｖｅｒｓｉｔｙ,Ｂｅｉｊｉｎｇ１０００４４,Ｃｈｉｎａ）ＡＵＲＯＲＡＭＯＲＧＡＮＡ（Ｄｅ...     

A matrix partition problem

In the set of positive definite semi-integral symmetric matrices we propose a partition problem. Then by introducing the notion of “additively prime” we obtain the generating function for this problem. Finally we establish the analyticity of the generating function.     

A one-dimensional variational problem with gradient constraint

In this note we give an explicit characterization of the minimum value of one-dimensional integral variational problem with gradient constraint by a positive measurable function.     

A two-dimensional strip cutting problem with sequencing constraint

We study a strip cutting problem that arises in the production of corrugated cardboard. In this context, rectangular items of different sizes are obtained by machines, called corrugators, that cut strips of large dimensions according to particular schemes containing at most two types of items. Because of buffer restrictions, these schemes have to be sequenced in such a way that, at any moment, at most two types of items are in production and not completed yet (sequencing constraint). We show that the problem of finding a set of schemes of minimum trim loss that satisfies an assigned demand for each item size is strongly NP-hard, even if the sequencing constraint is relaxed. Then, we present two heuristics for the problem with the sequencing constraint, both based on a graph characterization of the feasible solutions. The first heuristic is a two-phase procedure based on a mixed integer linear programming model. The second heuristic follows a completely combinatorial approach and consists of solving a suitable sequence of minimum cost matching problems. For both procedures, an upper bound on the number of schemes (setups) is found. Finally, a computational study comparing the quality of the heuristic solutions with respect to an LP lower bound is reported.     

A Bernoulli problem with non-constant gradient boundary constraint

In this article, we present a result about the existence and convexity of solutions to a free boundary problem of Bernoulli type, with non-constant gradient boundary constraint depending on the outer unit normal. In particular, we prove that, in the convex case, the existence of a subsolution guarantees the existence of a classical solution, which is proved to be convex.     

A problem in plane topology

In a closed and bounded region of the plane there is given a field of curves satisfying natural continuity properties expressed in geometrical terms. We show that such a field is easy to construct: the ends of all curves which separate the region lie on its boundary.Translated from Matematicheskie Zametki, Vol. 5, No. 6, pp. 709–718, June, 1969.The author wishes to thank his scientific advisor S. G. Krein.     

A gallai-type transversal problem in the plane

LetK be a family of compact convex sets in the plane. We show that if every three members ofK admit a common line transversal, then there exist four lines which together meet all the members ofK.     

A solid transportation problem with mixed constraint in different environment

In this paper, we have introduced a Solid Transportation Problem where the constrains are mixed type. The model is developed under different environment like, crisp, fuzzy and intuitionistic fuzzy etc. Using the interval approximation method we defuzzify the fuzzy amount and for intuitionistic fuzzy set we use the ($\alpha,\beta$)-cut sets to get the corresponding crisp amount. To find the optimal transportation units a time and space based with order of convergence $O (MN^2)$ meta-heuristic Genetic Algorithm have been proposed. Also the equivalent crisp model so obtained are solved by using LINGO 13.0. The results obtained using GA treats as the best solution by comparing with LINGO results for this present study. The proposed models and techniques are finally illustrated by providing numerical examples. Degree of efficiency have been find out for both the algorithm.     

Approximation algorithms for k-partitioning problems with partition matroid constraint

The $k$ -partitioning problem with partition matroid constraint is to partition the union of $k$ given sets of size $m$ into $m$ sets such that each set contains exactly one element from each given set. With the objective of minimizing the maximum load, we present an efficient polynomial time approximation scheme (EPTAS) for the case where $k$ is a constant and a full polynomial time approximation scheme (FPTAS) for the case where $m$ is a constant; with the objective of maximizing the minimum load, we present a $\frac{1}{k-1}$ -approximation algorithm for the general case, an EPTAS for the case where $k$ is a constant; with the objective of minimizing the $l_p$ -norm of the load vector, we prove that the layered LPT algorithm (Wu and Yao in Theor Comput Sci 374:41–48, 2007) is an all-norm 2-approximation algorithm.     

A branch-and-price approach for the partition coloring problem

This work proposes a new integer programming model for the partition coloring problem and a branch-and-price algorithm to solve it. Experiments are reported for random graphs and instances originating from routing and wavelength assignment problems arising in telecommunication network design. We show that our method largely outperforms previously existing approaches.     

Universal number partition problem with divisibility

We examine a version of the Universal Number Partition Problem with a divisibility property referred to as the Universal Shelf Packing Problem (USPP). We show that if a shelf length is a product of powers of two primes the USPP is always partitionable. In the case where the shelf length is a product of three distinct primes we propose an efficient scheme to determine when such a case is not partitionable. We also show that a shelf length that is a product of powers of four or more primes always has at least one partition failure. Our analysis uses elementary number theory, known results related to the linear Diophantine Frobenius problem, and a new result on Frobenius gaps.     

A minimal area problem in conformal mapping

LetS denote the usual class of functionsf holomorphic and univalent in the unit diskU such thatf(0)=f′(0)−1=0. The main result of the paper is that area (f(U) ≥27π/7)(2-α)⁻² for allf∈S such that |f″(0)|=2α, 1/2<α<2. This solves a long-standing extremal problem for the class of functions considered.