Fast Algorithm of Square Rooting in Some Finite Fields of Odd Characteristic

It was proved that the complexity of square root computation in the Galois field GF(3^s), s = 2^kr, is equal to O(M(2^k)M(r)k + M(r) log₂r) + 2^kkr^1+o(1), where M (n) is the complexity of multiplication of polynomials of degree n over fields of characteristics 3. The complexity of multiplication and division in the field GF(3^s) is equal to O(M(2^k)M(r)) and O(M(2^k)M(r)) + r^1+o(1), respectively. If the basis in the field GF(3^r) is determined by an irreducible binomial over GF(3) or is an optimal normal basis, then the summands 2^kkr^1+o(1) and r^1+o(1) can be omitted. For M(n) one may take n log₂nψ(n) where ψ(n) grows slower than any iteration of the logarithm. If k grow and r is fixed, than all the estimates presented here have the form O_r (M (s) log ₂s) = s (log ₂s)²ψ(s).     

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.     

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.     

Disjoint cliques in claw-free graphs

A graph is said to be claw-free if it does not contain an induced subgraph isomorphic to K_1,3. Let s and k be two integers with 0 ≤ s ≤ k and let G be a claw-free graph of order n. In this paper, we investigate clique partition problems in claw-free graphs. It is proved that if n ≥ 3s+4(k?s) and d(x)+d(y) ≥ n?2s+2k+1 for any pair of non-adjacent vertices x, y of G, then G contains s disjoint K₃s and k ? s disjoint K₄s such that all of them are disjoint. Moreover, the degree condition is sharp in some cases.     

Properties of quasi-Boolean function on quasi-Boolean algebra

In this paper, we investigate the following problem: give a quasi-Boolean function Ψ(x ₁, …, x _n ) = (a ∧ C) ∨ (a ₁ ∧ C ₁) ∨ … ∨ (a _p ∧ C _p ), the term (a ∧ C) can be deleted from Ψ(x ₁, …, x _n )? i.e., (a ∧ C) ∨ (a ₁ ∧ C ₁) ∨ … ∨ (a _p ∧ C _p ) = (a ₁ ∧ C ₁) ∨ … ∨ (a _p ∧ C _p )? When a = 1: we divide our discussion into two cases. (1) ?₁(Ψ,C) = ø, C can not be deleted; ?₁(Ψ,C) ≠ ø, if S _i ⁰ ≠ ø (1 ≤ i ≤ q), then C can not be deleted, otherwise C can be deleted. When a = m: we prove the following results: (m∧C)∨(a ₁∧C ₁)∨…∨(a _p ∧C _p ) = (a ₁∧C ₁)∨…∨(a _p ∧C _p ) ? (m ∧ C) ∨ C ₁ ∨ … ∨C _p = C ₁ ∨ … ∨C _p . Two possible cases are listed as follows, (1) ?₂(Ψ,C) = ø, the term (m∧C) can not be deleted; (2) ?₂(Ψ,C) ≠ ø, if (?i ₀) such that \(S'_{i_0 } \) = ø, then (m∧C) can be deleted, otherwise ((m∧C)∨C ₁∨…∨C _q )(v ₁, …, v _n ) = (C ₁ ∨ … ∨ C _q )(v ₁, …, v _n )(?(v ₁, …, v _n ) ∈ L ₃ ⁿ ) ? (C ₁ ^′ ∨ … ∨ C _q ^′ )(u ₁, …, u _q ) = 1(?(u ₁, …, u _q ) ∈ B ₂ ⁿ ).     

An improvement of the five halves theorem of J. Boardman

Let (M ^m , T) be a smooth involution on a closed smooth m-dimensional manifold and F = ∪ _j=0 ⁿ F ^j (n ≤ m) its fixed point set, where F ^j denotes the union of those components of F having dimension j. The famous Five Halves Theorem of J. Boardman, announced in 1967, establishes that, if F is nonbounding, then m ≤ 5/2n. In this paper we obtain an improvement of the Five Halves Theorem when the top dimensional component of F, F ⁿ , is nonbounding. Specifically, let ω = (i ₁, i ₂, …, i _r ) be a non-dyadic partition of n and s _ω (x ₁, x ₂, …, x _n ) the smallest symmetric polynomial over Z ₂ on degree one variables x ₁, x ₂, …, x _n containing the monomial \(x_1^{i_1 } x_2^{i_2 } \cdots x_r^{i_r }\). Write s _ω (F ⁿ ) ∈ H ⁿ (F ⁿ , Z ₂) for the usual cohomology class corresponding to s _ω (x ₁, x ₂, …, x _n ), and denote by ?(F ⁿ ) the minimum length of a nondyadic partition ω with s _ω (F ⁿ ) ≠ 0 (here, the length of ω = (i ₁, i ₂, …, i _r ) is r). We will prove that, if (M ^m , T) is an involution for which the top dimensional component of the fixed point set, F ⁿ , is nonbounding, then m ≤ 2n + ?(F ⁿ ); roughly speaking, the bound for m depends on the degree of decomposability of the top dimensional component of the fixed point set. Further, we will give examples to show that this bound is best possible.     

An algorithm to compute <Emphasis Type="Italic">ω</Emphasis>-primality in a numerical monoid

Let M be a commutative, cancellative, atomic monoid and x a nonunit in M. We define ω(x)=n if n is the smallest positive integer with the property that whenever x∣a ₁???a _t , where each a _i is an atom, there is a T?{1,2,…,t} with |T|≤n such that x∣∏_k∈T a _k . The ω-function measures how far x is from being prime in M. In this paper, we give an algorithm for computing ω(x) in any numerical monoid. Simple formulas for ω(x) are given for numerical monoids of the form 〈n,n+1,…,2n?1〉, where n≥3, and 〈n,n+1,…,2n?2〉, where n≥4. The paper then focuses on the special case of 2-generator numerical monoids. We give a formula for computing ω(x) in this case and also necessary and sufficient conditions for determining when x is an atom. Finally, we analyze the asymptotic behavior of ω(x) by computing \(\lim_{x\rightarrow \infty}\frac{\omega(x)}{x}\).     

The communication complexity of addition

Suppose each of k≤n ^o(1) players holds an n-bit number x _i in its hand. The players wish to determine if ∑ _i≤k x _i =s. We give a public-coin protocol with error 1% and communication O(k logk). The communication bound is independent of n, and for k≥3 improves on the O(k logn) bound by Nisan (Bolyai Soc. Math. Studies; 1993).     

Complexities of normal bases constructed from Gauss periods

Let q be a power of a prime p, and let \(r=nk+1\) be a prime such that \(r\not \mid q\), where n and k are positive integers. Under a simple condition on q, r and k, a Gauss period of type (n, k) is a normal element of \({\mathbb {F}}_{q}^{n}\) over \({\mathbb {F}}_q\); the complexity of the resulting normal basis of \({\mathbb {F}}_{q}^{n}\) over \({\mathbb {F}}_q\) is denoted by C(n, k; p). Recent works determined C(n, k; p) for \(k\le 7\) and all qualified n and q. In this paper, we show that for any given \(k>0\), C(n, k; p) is given by an explicit formula except for finitely many primes \(r=nk+1\) and the exceptional primes are easily determined. Moreover, we describe an algorithm that allows one to compute C(n, k; p) for the exceptional primes \(r=nk+1\). Our numerical results cover C(n, k; p) for \(k\le 20\) and all qualified n and q.     

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 ?.     

On the k-polygonal numbers and the mean value of Dedekind sums

For any positive integer k ≥ 3, it is easy to prove that the k-polygonal numbers are a_n(k) = (2n+n(n?1)(k?2))/2. The main purpose of this paper is, using the properties of Gauss sums and Dedekind sums, the mean square value theorem of Dirichlet L-functions and the analytic methods, to study the computational problem of one kind mean value of Dedekind sums S(a_n(k)ā_m(k), p) for k-polygonal numbers with 1 ≤ m, n ≤ p ? 1, and give an interesting computational formula for it.     

On the additive complexity of GCD and LCM matrices

In the paper, the additive complexity of matrices formed by positive integer powers of greatest common divisors and least common multiples of the indices of the rows and columns is considered. It is proved that the complexity of the n × n matrix formed by the numbers GCD^r(i, k) over the basis {x + y} is asymptotically equal to rn log²n as n→∞, and the complexity of the n × n matrix formed by the numbers LCM^r(i, k) over the basis {x + y,?x} is asymptotically equal to 2rn log₂n as n→∞.     

Holomorphic factorization of polynomials

We give necessary and sufficient conditions for a holomorphic factorization of an irreducible polynomial P(s, λ), s ∈ Cⁿ, λ ∈ C, in a domain Ω ? Cⁿ which is connected with the ordering of the real part of the roots of the equation P(s, λ) = 0, s ∈ Ω.     

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.     

On the phase transitions of (<Emphasis Type="Italic">k</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-SAT

Call a sequence of k Boolean variables or their negations a k-tuple. For a set V of n Boolean variables, let T _k (V) denote the set of all 2 ^k n ^k possible k-tuples on V. Randomly generate a set C of k-tuples by including every k-tuple in T _k (V) independently with probability p, and let Q be a given set of q “bad” tuple assignments. An instance I = (C,Q) is called satisfiable if there exists an assignment that does not set any of the k-tuples in C to a bad tuple assignment in Q. Suppose that θ, q > 0 are fixed and ε = ε(n) > 0 be such that εlnn/lnlnn→∞. Let k ≥ (1 + θ) log₂ n and let \({p_0} = \frac{{\ln 2}}{{q{n^{k - 1}}}}\). We prove that

$$\mathop {\lim }\limits_{n \to \infty } P\left[ {I is satisfiable} \right] = \left\{ {\begin{array}{*{20}c} {1,} & {p \leqslant (1 - \varepsilon )p_0 ,} \\ {0,} & {p \geqslant (1 + \varepsilon )p_0 .} \\ \end{array} } \right.$$

     

On the Problem of the Optimal Choice of Record Values

Let the independent random variables X1, X2, … have the same continuous distribution function. The upper record values X(1) = X₁ < X(2) < … generated by this sequence of variables, as well as the lower record values x(1) = X₁ > x(2) > …, are considered. It is known that in this situation, the mean value c(n) of the total number of the both types of records among the first n variables X is given by the equality c(n)=2(1+1/2+…+1/n), n = 1, 2, …. The problem considered here is following: how, sequentially obtaining the observed values x₁, x₂, … of variables X and selecting one of them as the initial point, to obtain the maximal mean value e(n) of the considered numbers of records among the rest random variables. It is not possible to come back to rejected elements of the sequence. Some procedures of the optimal choice of the initial element X _r are discussed. The corresponding tables for the values e(n) and differences δ(n)= e(n)–c(n) are presented for different values of n. The value of δ= lim_n→∞δ(n)is also given. In some sense, the considered problem and optimization procedure presented in this paper are quite similar to the classical “secretary problem,” in which the probability of selecting the last record value in the set of independent identically distributed X is maximized.     

On the number of edges in a uniform hypergraph with a range of permitted intersections

The paper studies the quantity p(n, k, t ₁, t ₂) equal to the maximum number of edges in a k-uniform hypergraph with the property that the size of the intersection of any two edges lies in the interval [t ₁, t ₂]. Previously known upper and lower bounds are given. New bounds for p(n, k, t ₁, t ₂) are obtained, and the relationship between these bounds and known estimates is studied. For some parameter values, the exact values of p(n, k, t ₁, t ₂) are explicitly calculated.     

On the summability factors of infinite series

The author has established that if [λ_n] is a convex sequence such that the series Σn ^-1λ_n is convergent and the sequence {K _n} satisfies the condition |K _n|=O[log(n+1)]^k(C, 1),k?0, whereK _n denotes the (R, logn, 1) mean of the sequence {n log (n+1)a _n}, then the series Σlog(n+1)^1-kλ_n a _n is summable |R, logn, 1|. The result obtained for the particular casek=0 generalises a previous result of the author [1].     

Consecutive integers in high-multiplicity sumsets

Sharpening (a particular case of) a result of Szemerédi and Vu [4] and extending earlier results of Sárközy [3] and ourselves [2], we find, subject to some technical restrictions, a sharp threshold for the number of integer sets needed for their sumset to contain a block of consecutive integers, whose length is comparable with the lengths of the set summands.A corollary of our main result is as follows. Let k,l≥1 and n≥3 be integers, and suppose that A ₁,…,A _k ?[0,l] are integer sets of size at least n, none of which is contained in an arithmetic progression with difference greater than 1. If k≥2?(l?1)/(n?2)?, then the sumset A ₁+???+A _k contains a block of at least k(n?1)+1 consecutive integers.     

Ck-regularity for the $$\bar \partial $$ -equation with a support condition

Let D be a C ^d q-convex intersection, d ≥ 2, 0 ≤ q ≤ n ? 1, in a complex manifold X of complex dimension n, n ≥ 2, and let E be a holomorphic vector bundle of rank N over X. In this paper, C ^k -estimates, k = 2, 3,...,∞, for solutions to the \(\bar \partial \)-equation with small loss of smoothness are obtained for E-valued (0, s)-forms on D when n ? q ≤ s ≤ n. In addition, we solve the \(\bar \partial \)-equation with a support condition in C ^k -spaces. More precisely, we prove that for a \(\bar \partial \)-closed form f in C _0,q ^k (X D,E), 1 ≤ q ≤ n ? 2, n ≥ 3, with compact support and for ε with 0 < ε < 1 there exists a form u in C _0,q?1 ^k?ε (X D,E) with compact support such that \(\bar \partial u = f\) in \(X\backslash \bar D\). Applications are given for a separation theorem of Andreotti-Vesentini type in C ^k -setting and for the solvability of the \(\bar \partial \)-equation for currents.