Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gauß count the remaining ones, approximately and exactly. For polynomials in two or more variables, the situation changes dramatically. Most multivariate polynomials are irreducible. This survey presents counting results for some special classes of mu...

The reconstruction of discrete sets from four projections is in general an NP-hard problem. In this paper we study the class of decomposable discrete sets and give an efficient reconstruction algorithm for this class using four projections. It is also shown that an arbitrary discrete set which is Q-convex along the horizontal and vertical directions and consists of several components is decompo...

4 We prove that the weak k-linkage problem is polynomial for every fixed k for totally Φ5 decomposable digraphs, under appropriate hypothesis on Φ. We then apply this and recent results 6 by Fradkin and Seymour (on the weak k-linkage problem for digraphs of bounded independence 7 number or bounded cut-width) to get polynomial algorithms for some class of digraphs like quasi8 transitive digraphs...

In this paper, we introduce the 1− K robotic-cell scheduling problem, whose solution can be reduced to solving a TSP on specially structured permuted Monge matrices, we call b-decomposable matrices. We also review a number of other scheduling problems which all reduce to solving TSP-s on permuted Monge matrices. We present the important insight that the TSP on b-decomposable matrices can be sol...

Many combinatorial problems arising in machine learning can be reduced to the problem of minimizing a submodular function. Submodular functions are a natural discrete analog of convex functions, and can be minimized in strongly polynomial time. Unfortunately, state-of-the-art algorithms for general submodular minimization are intractable for larger problems. In this paper, we introduce a novel ...

D−dimensional central and complex potentials of a Coulomb plus quartic-polynomial form are considered in a PT −symmetrized radial Schrödinger equation. Arbitrarily large finite multiplets of bound states are shown obtainable in an elementary form. Relations between their energies and couplings are determined by a finite-dimensional secular equation. The Bender’s and Boettcher’s one-dimensional ...

We consider digraphs – called extended locally semicomplete digraphs, or extended LSD’s, for short – that can be obtained from locally semicomplete digraphs by substituting independent sets for vertices. We characterize Hamiltonian extended LSD’s as well as extended LSD’s containing Hamiltonian paths. These results as well as some additional ones imply polynomial algorithms for finding a longes...

Decomposition-based index calculus methods are currently efficient only for elliptic curves E defined over non-prime finite fields of very small extension degree n. This corresponds to the fact that the Semaev summation polynomials, which encode the relation search (or “sieving”), grows over-exponentially with n. Actually, even their computation is a first stumbling block and the largest Semaev...

ABSTRACT. The notion of symmetrization, also known as Davenport’s reflection principle, is well known in the area of the discrepancy theory and quasiMonte Carlo (QMC) integration. In this paper we consider applying a symmetrization technique to a certain class of QMC point sets called digital nets over Zb. Although symmetrization has been recognized as a geometric technique in the multi-dimensi...