### Article

## On integer programming with bounded determinants

Let *A* be an \((m \times n)\) integral matrix, and let \(P=\{ x :A x \le b\}\) be an *n*-dimensional polytope. The width of *P* is defined as \( w(P)=min\{ x\in \mathbb {Z}^n{\setminus }\{0\} :max_{x \in P} x^\top u - min_{x \in P} x^\top v \}\). Let \(\varDelta (A)\) and \(\delta (A)\) denote the greatest and the smallest absolute values of a determinant among all \(r(A) \times r(A)\) sub-matrices of *A*, where *r*(*A*) is the rank of the matrix *A*. We prove that if every \(r(A) \times r(A)\) sub-matrix of *A* has a determinant equal to \(\pm \varDelta (A)\) or 0 and \(w(P)\ge (\varDelta (A)-1)(n+1)\), then *P* contains *n* affine independent integer points. Additionally, we present similar results for the case of *k-modular* matrices. The matrix *A* is called *totally* *k*-modular if every square sub-matrix of *A* has a determinant in the set \(\{0,\, \pm k^r :r \in \mathbb {N} \}\). When *P* is a simplex and \(w(P)\ge \delta (A)-1\), we describe a polynomial time algorithm for finding an integer point in *P*.(k−1)(n+1).

In this paper, we will show that the width of simplices defined by systems of linear inequalities can be computed in polynomial time if some minors of their constraint matrices are bounded. Additionally, we present some quasi-polynomial-time and polynomial-time algorithms to solve the integer linear optimization problem defined on simplices minus all their integer vertices assuming that some minors of the constraint matrices of the simplices are bounded.

The central question that motives this paper is the problem of making up a freight train and the routes on the railway. It is necessary from the set of orders available at the stations to determine time-scheduling and destination routing by railways in order to minimize the total completion time. In this paper it was suggested formulation of this problem by applying integer programming.

We consider the problem of trainings planning on ISS. Shown that the problem is a combination of a k Partition Problem and an Assignment Problem. NP-compleeteness is proofed. A heuristic and an exact algorithms are proposed.

The material of the present paper is grounded on the holist algebraic method (Q-analysis) proposed by English mathematician and physicist R.H.Atkin. At its core, the approach is aimed at both analysis of systems structures (in the form of simplicial complexes K, which is formed by a set of properly adjoined objects called simplexes) and calculation of numeric estimates of structural complexity of systems based on the results of such analysis.

Turning complexity estimate of system’s structure into a real number creates additional difficulties in the comparison of two different complexes because there is no real verbal scale, which would have been accustomed to human beings and would allow a group of experts to express opinions and draw easily conclusions about degree of complexity of K at each particular dimensional level of its analysis. Therefore, the present paper deals with consideration of the approach that is more focused on human perception of characteristics obtained, mental comprehension and formation (comparison) of personal constructs in psychological space (or, P-space) – modified structural complexity estimate is based right on notions of distance and similarity within psychological space.

We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.

We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.

We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.