Generalized separants of differential polynomials

Let f ∈ K{y} be an element of the ring of differential polynomials in one differential variable y with one differential operator δ. For any variable y _k , the polynomial g = δ ⁿ (f) can be represented in the form g = A _k y _k + go, where go does not depend on y _k . If y _k is the leader of g, then A _k is a separant of the polynomial f. A formula for A _k is obtained for sufficiently large numbers n and k and some applications of this formula are presented.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates for the Schrödinger type operators

Let L _k = (?Δ) ^k + V ^k be a Schrödinger type operator, where k ≥ 1 is a positive integer and V is a nonnegative polynomial. We obtain the L ^p estimates for the operators ?^2k L _k ^?1 and ? ^k L _k ^?1/2 .     

A probabilistic approach to polynomial inequalities

We define a probability measure on the space of polynomials over ? ⁿ in order to address questions regarding the attainment of the norm at given points and the validity of polynomial inequalities.Using this measure, we prove that for all degrees k ≥ 3, the probability that a k-homogeneous polynomial attains a local extremum at a vertex of the unit ball of ? ₁ ⁿ tends to one as the dimension n increases. We also give bounds for the probability of some general polynomial inequalities.     

Membership of distributions of polynomials in the Nikolskii–Besov class

The main result of this paper asserts that the distribution density of any non-constant polynomial f(ξ₁,ξ₂,...) of degree d in independent standard Gaussian random variables ξ₁ (possibly, in infinitely many variables) always belongs to the Nikol’skii–Besov space B^1/d (R¹) of fractional order 1/d (see the definition below) depending only on the degree of the polynomial. A natural analog of this assertion is obtained for the density of the joint distribution of k polynomials of degree d, also with a fractional order that is independent of the number of variables, but depends only on the degree d and the number of polynomials. We also give a new simple sufficient condition for a measure on R^k to possess a density in the Nikol’skii–Besov class B^α(R)^k. This result is applied for obtaining an upper bound on the total variation distance between two probability measures on R^k via the Kantorovich distance between them and a certain Nikol’skii–Besov norm of their difference. Applications are given to estimates of distributions of polynomials in Gaussian random variables.     

New Cases of the Polynomial Solvability of the Independent Set Problem for Graphs with Forbidden Paths

The independent set problem is solvable in polynomial time for the graphs not containing the path P _k for any fixed k. If the induced path P _k is forbidden then the complexity of this problem is unknown for k > 6. We consider the intermediate cases that the induced path P _k and some of its spanning supergraphs are forbidden. We prove the solvability of the independent set problem in polynomial time for the following cases: (1) the supergraphs whose minimal degree is less than k/2 are forbidden; (2) the supergraphs whose complementary graph has more than k/2 edges are forbidden; (3) the supergraphs from which we can obtain P _k by means of graph intersection are forbidden.     

Some irreducibility and indecomposability results for truncated binomial polynomials of small degree

In this paper, we show that the truncated binomial polynomials defined by \(P_{n,k}(x)={\sum }_{j=0}^{k} {n \choose j} x^{j}\) are irreducible for each k≤6 and every n≥k+2. Under the same assumption n≥k+2, we also show that the polynomial P _n,k cannot be expressed as a composition P _n,k (x) = g(h(x)) with \(g \in \mathbb {Q}[x]\) of degree at least 2 and a quadratic polynomial \(h \in \mathbb {Q}[x]\). Finally, we show that for k≥2 and m,n≥k+1 the roots of the polynomial P _m,k cannot be obtained from the roots of P _n,k , where m≠n, by a linear map.     

Polynomial dynamical systems and the Korteweg—de Vries equation

We explicitly construct polynomial vector fields L_k, k = 0, 1, 2, 3, 4, 6, on the complex linear space C⁶ with coordinates X = (x₂, x₃, x₄) and Z = (z₄, z₅, z₆). The fields L_k are linearly independent outside their discriminant variety Δ ? C⁶ and are tangent to this variety. We describe a polynomial Lie algebra of the fields L_k and the structure of the polynomial ring C[X,Z] as a graded module with two generators x₂ and z₄ over this algebra. The fields L₁ and L₃ commute. Any polynomial P(X,Z) ∈ C[X,Z] determines a hyperelliptic function P(X,Z)(u₁, u₃) of genus 2, where u₁ and u₃ are the coordinates of trajectories of the fields L₁ and L₃. The function 2x₂(u₁, u₃) is a two-zone solution of the Korteweg–de Vries hierarchy, and ?z₄(u₁, u₃)/?u₁ = ?x₂(u₁, u₃)/?u₃.     

Systems of dilated functions: Completeness,minimality, basisness

The completeness, minimality, and basis property in L ²[0, π] and L ^p[0, π], p ≠ 2, are considered for systems of dilated functions u _n (x) = S(nx), n ∈ N, where S is the trigonometric polynomial S(x) = Σ _k=0 ^m a _k sin(kx), a ₀ a _m ≠ 0. A series of results are presented and several unanswered questions are mentioned.     

The asymptotic formula in Waring’s problem for one square and seventeen fifth powers

Let R _k,s(n) denote the number of solutions of the equation \({n= x^2 + y_1^k + y_2^k + \cdots + y_s^k}\) in natural numbers x, y ₁, . . . , y _s . By a straightforward application of the circle method, an asymptotic formula for R _k,s(n) is obtained when k ≥ 3 and s ≥ 2^k–1 + 2. When k ≥ 6, work of Heath-Brown and Boklan is applied to establish the asymptotic formula under the milder constraint s ≥ 7 · 2^k–4 + 3. Although the principal conclusions provided by Heath-Brown and Boklan are not available for smaller values of k, some of the underlying ideas are still applicable for k = 5, and the main objective of this article is to establish an asymptotic formula for R _5,17(n) by this strategy.     

Distance domination of generalized de Bruijn and Kautz digraphs

Let G = (V,A) be a digraph and k ≥ 1 an integer. For u, v ∈ V, we say that the vertex u distance k-dominate v if the distance from u to v at most k. A set D of vertices in G is a distance k-dominating set if each vertex of V D is distance k-dominated by some vertex of D. The distance k-domination number of G, denoted by γ _k (G), is the minimum cardinality of a distance k-dominating set of G. Generalized de Bruijn digraphs G _B (n, d) and generalized Kautz digraphs G _K (n, d) are good candidates for interconnection networks. Denote Δ _k := (∑ _j=0 ^k d ^j )^?1. F. Tian and J. Xu showed that ?nΔ _k ? γ _k (G _B (n, d)) ≤?n/d ^k? and ?nΔ _k ? ≤ γ _k (G _K (n, d)) ≤ ?n/d ^k ?. In this paper, we prove that every generalized de Bruijn digraph G _B (n, d) has the distance k-domination number ?nΔ _k ? or ?nΔ _k ?+1, and the distance k-domination number of every generalized Kautz digraph G _K (n, d) bounded above by ?n/(d ^k?1+d ^k )?. Additionally, we present various sufficient conditions for γ _k (G _B (n, d)) = ?nΔ _k ? and γ _k (G _K (n, d)) = ?nΔ _k ?.     

Hirzebruch Functional Equation: Classification of Solutions

The Hirzebruch functional equation is \(\sum\nolimits_{i = 1}^n {\prod\nolimits_{j \ne i} {(1/f({z_j} - {z_i})) = c} } \) with constant c and initial conditions f(0) = 0 and f'(0) = 1. In this paper we find all solutions of the Hirzebruch functional equation for n ≤ 6 in the class of meromorphic functions and in the class of series. Previously, such results have been known only for n ≤ 4. The Todd function is the function determining the two-parameter Todd genus (i.e., the χ_a,b-genus). It gives a solution to the Hirzebruch functional equation for any n. The elliptic function of level N is the function determining the elliptic genus of level N. It gives a solution to the Hirzebruch functional equation for n divisible by N. A series corresponding to a meromorphic function f with parameters in U ? ?^k is a series with parameters in the Zariski closure of U in ?^k, such that for the parameters in U it coincides with the series expansion at zero of f. The main results are as follows: (1) Any series solution of the Hirzebruch functional equation for n = 5 corresponds either to the Todd function or to the elliptic function of level 5. (2) Any series solution of the Hirzebruch functional equation for n = 6 corresponds either to the Todd function or to the elliptic function of level 2, 3, or 6. This gives a complete classification of complex genera that are fiber multiplicative with respect to ?P^n?1 for n ≤ 6. A topological application of this study is an effective calculation of the coefficients of elliptic genera of level N for N = 2,..., 6 in terms of solutions of a differential equation with parameters in an irreducible algebraic variety in ?⁴.     

On Representation of a Solution to the Cauchy Problem by a Fourier Series in Sobolev-Orthogonal Polynomials Generated by Laguerre Polynomials

We consider the problem of representing a solution to the Cauchy problem for an ordinary differential equation as a Fourier series in polynomials l _r,k ^α (x) (k = 0, 1,...) that are Sobolev-orthonormal with respect to the inner product

$$\left\langle {f,g} \right\rangle = \sum\limits_{v = 0}^{r - 1} {{f^{(v)}}(0){g^{(v)}}} (0) + \int\limits_0^\infty {{f^{(r)}}(t)} {g^{(r)}}(t){t^\alpha }{e^{ - t}}dt$$

, and generated by the classical orthogonal Laguerre polynomials L _k ^α (x) (k = 0, 1,...). The polynomials l _r,k ^α (x) are represented as expressions containing the Laguerre polynomials L _n ^α?r (x). An explicit form of the polynomials l _r,k+r ^α (x) is established as an expansion in the powers x ^r+l , l = 0,..., k. These results can be used to study the asymptotic properties of the polynomials l _r,k ^α (x) as k→∞and the approximation properties of the partial sums of Fourier series in these polynomials.

     

Search-extension method for finding multiple solutions of nonlinear problem

Based on the eigensystem {λj,φj}of -Δ, the multiple solutions for nonlinear problem Δu f(u) = 0 in Ω, u = 0 on (?)Ωare approximated. A new search-extension method (SEM) is proposed, which consists of three algorithms in three level subspaces. Numerical experiments for f(u) = u3 in a square and L-shape domain are presented. The results show that there exist at least 3k - 1 distinct nonzero solutions corresponding to each κ-ple eigenvalue of -Δ(Conjecture 1).     

New algorithm for computing the Hermite interpolation polynomial

Let x ₀, x ₁,? , x _n , be a set of n + 1 distinct real numbers (i.e., x _i ≠ x _j , for i ≠ j) and y _{i, k} , for i = 0,1,? , n, and k = 0 ,1 ,? , n _i , with n _i ≥ 1, be given of real numbers, we know that there exists a unique polynomial p _{N ? 1}(x) of degree N ? 1 where \(N={\sum }_{i=0}^{n}(n_{i}+1)\), such that \(p_{N-1}^{(k)}(x_{i})=y_{i,k}\), for i = 0,1,? , n and k = 0,1,? , n _i . P _N?1(x) is the Hermite interpolation polynomial for the set {(x _i , y _{i, k} ), i = 0,1,? , n, k = 0,1,? , n _i }. The polynomial p _N?1(x) can be computed by using the Lagrange polynomials. This paper presents a new method for computing Hermite interpolation polynomials, for a particular case n _i = 1. We will reformulate the Hermite interpolation polynomial problem and give a new algorithm for giving the solution of this problem, the Matrix Recursive Polynomial Interpolation Algorithm (MRPIA). Some properties of this algorithm will be studied and some examples will also be given.     

On the Extremal Zagreb Indices of Graphs with Cut Edges

For a (molecular) graph, the first Zagreb index M ₁ is equal to the sum of squares of the vertex degrees, and the second Zagreb index M ₂ is equal to the sum of products of degrees of pairs of adjacent vertices. In this paper, we show that all connected graphs with n vertices and k cut edges, the maximum (resp. minimum) M ₁- and M ₂-value are obtained, respectively, and uniquely, at K _n ^k (resp. P _n ^k ), where K _n ^k is a graph obtained by joining k independent vertices to one vertex of K _n?k and P _n ^k is a graph obtained by connecting a pendent path P _k+1 to one vertex of C _n?k.     

Conditional connectivity of bubble sort graphs

A subset F ? V (G) is called an R ^k -vertex-cut of a graph G if G ? F is disconnected and each vertex of G ? F has at least k neighbors in G ? F. The R ^k -vertex-connectivity of G, denoted by κ ^k (G), is the cardinality of a minimum R ^k -vertex-cut of G. Let B _n be the bubble sort graph of dimension n. It is known that κ _k (B _n ) = 2 ^k (n ? k ? 1) for n ≥ 2k and k = 1, 2. In this paper, we prove it for k = 3 and conjecture that it is true for all k ∈ N. We also prove that the connectivity cannot be more than conjectured.     

Nonlinear trigonometric approximations of multivariate function classes

Order-sharp estimates are established for the best N-term approximations of functions from Nikol’skii–Besov type classes B_pq^sm(T^k) with respect to the multiple trigonometric system T^(k) in the metric of L_r(T^k) for a number of relations between the parameters s, p, q, r, and m (s = (s₁,..., s_n) ∈ R₊ⁿ, 1 ≤ p, q, r ≤ ∞, m = (m₁,..., m_n) ∈ Nⁿ, k = m₁ +... + m_n). Constructive methods of nonlinear trigonometric approximation—variants of the so-called greedy algorithms—are used in the proofs of upper estimates.     

On generalized Heawood inequalities for manifolds: a van Kampen–Flores-type nonembeddability result

The fact that the complete graph K₅ does not embed in the plane has been generalized in two independent directions. On the one hand, the solution of the classical Heawood problem for graphs on surfaces established that the complete graph K_n embeds in a closed surface M (other than the Klein bottle) if and only if (n?3)(n?4) ≤ 6b₁(M), where b₁(M) is the first Z₂-Betti number of M. On the other hand, van Kampen and Flores proved that the k-skeleton of the n-dimensional simplex (the higher-dimensional analogue of K_n+1) embeds in R^2k if and only if n ≤ 2k + 1.Two decades ago, Kühnel conjectured that the k-skeleton of the n-simplex embeds in a compact, (k ? 1)-connected 2k-manifold with kth Z₂-Betti number b_k only if the following generalized Heawood inequality holds: ( _k+1 ^n?k?1 ) ≤ ( _k+1 ^2k+1 )b_k. This is a common generalization of the case of graphs on surfaces as well as the van Kampen–Flores theorem.In the spirit of Kühnel’s conjecture, we prove that if the k-skeleton of the n-simplex embeds in a compact 2k-manifold with kth Z₂-Betti number bk, then n ≤ 2b_k( _k ^2k+2 )+2k+4. This bound is weaker than the generalized Heawood inequality, but does not require the assumption that M is (k?1)-connected. Our results generalize to maps without q-covered points, in the spirit of Tverberg’s theorem, for q a prime power. Our proof uses a result of Volovikov about maps that satisfy a certain homological triviality condition.     

Sum-free sets in abelian groups

An IP system is a functionn taking finite subsets ofN to a commutative, additive group Ω satisfyingn(α∪β)=n(α)+n(β) whenever α∩β=ø. In an extension of their Szemerédi theorem for finitely many commuting measure preserving transformations, Furstenberg and Katznelson showed that ifS _i ,1≤i≤k, are IP systems into a commutative (possibly infinitely generated) group Ω of measure preserving transformations of a probability space (X, B, μ, andA∈B with μ(A)>0, then for some ø≠α one has μ(? _i=1 ^k S _i({α})A>0). We extend this to so-called FVIP systems, which are polynomial analogs of IP systems, thereby generalizing as well joint work by the author and V. Bergelson concerning special FVIP systems of the formS(α)=T(p(n(α))), wherep:Z ^t →Z ^d is a polynomial vanishing at zero,T is a measure preservingZ ^d action andn is an IP system intoZ ^t . The primary novelty here is potential infinite generation of the underlying group action, however there are new applications inZ ^d as well, for example multiple recurrence along a wide class ofgeneralized polynomials (very roughly, functions built out of regular polynomials by iterated use of the greatest integer function).     

General compactly supported solution of an integral equation of the convolution type

We find the general form of solutions of the integral equation ∫k(t ? s)u₁(s) ds = u₂(t) of the convolution type for the pair of unknown functions u₁ and u₂ in the class of compactly supported continuously differentiable functions under the condition that the kernel k(t) has the Fourier transform \(\widetilde {{P_2}}\), where \(\widetilde {{P_1}}\) and \(\widetilde {{P_2}}\) are polynomials in the exponential e^iτx, τ > 0, with coefficients polynomial in x. If the functions \({P_l}\left( x \right) = \widetilde {{P_l}}\left( {{e^{i\tau x}}} \right)\), l = 1, 2, have no common zeros, then the general solution in Fourier transforms has the form U_l(x) = P_l(x)R(x), l = 1, 2, where R(x) is the Fourier transform of an arbitrary compactly supported continuously differentiable function r(t).