Betti and Tachibana numbers

The Tachibana numbers t _r (M), the Killing numbers k _r (M), and the planarity numbers p _r (M) are considered as the dimensions of the vector spaces of, respectively, all, coclosed, and closed conformal Killing r-forms with 1 ≤ r ≤ n ? 1 “globally” defined on a compact Riemannian n-manifold (M,g), n >- 2. Their relationship with the Betti numbers b _r (M) is investigated. In particular, it is proved that if b _r (M) = 0, then the corresponding Tachibana number has the form t _r (M) = k _r (M) + p _r (M) for t _r (M) > k _r (M) > 0. In the special case where b ₁(M) = 0 and t ₁(M) > k ₁(M) > 0, the manifold (M,g) is conformally diffeomorphic to the Euclidean sphere.     

More coverings by rook domains

The set V_kⁿ of all n-tuples (x₁, x₂,…, x_n) with x_i?, $\text{Z}$_k is considered. The problem treated in this paper is determining σ(n, k), the minimum size of a set W ? V_kⁿ such that for each x in V_kⁿ, there is an element in W that differs from x in at most one coordinate. By using a new constructive method, it is shown that σ(n, p) ? (p ? t + 1)p^n?r, where p is a prime and $\text{n = 1 +}\text{t(p}{}^{\mathrm{r?1}}\text{? 1)}\text{(p ? 1)}$ for some integers t and r. The same method also gives σ(7, 3) ? 216. Another construction gives the inequality σ(n, kt) ? σ(n, k)t^n?1 which implies that σ(q + 1, qt) = q^q?1t^q when q is a prime power. By proving another inequality σ(np + 1, p) ? σ(n, p)p^n(p?1), σ(10, 3) ? 5 · 3⁶ and σ(16, 5) ? 13 · 5¹² are obtained.     

THE EXTREME POINTS OF SEVERAL CLASSES OF ANALYTIC FUNCTIONS

1Intr0ducti0nLetAden0tethesetofallfunctionsanalyticinA={z:Izl<1}.LetB={W:WEAandIW(z)l51}.Aisalocallyconvexlineaztop0l0gicalspacewithrespecttothetopologyofuniformconvergenceon`c0mpact8ubsetsofA-LetTh(c1,'tc.-1)={p(z):p(z)EA,Rop(z)>0,p(z)=1 clz czzz ' c.-lz"-l 4z" ',wherecl,',cn-1areforedcomplexconstants}.LetTh,.(b,,-..,b,-,)={p(z):P(z)'EAwithReP(z)>Oandp(z)=1 blz ' b.-lz"-l 4z" '-,wherebl,-'-jbu-1areffeedrealconstantsanddkarerealnumbersf0rk=n,n 1,'--}-LetTu(l1,'i'tI.-1)={…     

Heilbronn's conjecture on Waring's number (mod p)

Let p be a prime k|p−1, t=(p−1)/k and γ(k,p) be the minimal value of s such that every number is a sum of s kth powers . We prove Heilbronn's conjecture that γ(k,p)?k1/2 for t>2. More generally we show that for any positive integer q, γ(k,p)?C(q)k1/q for ?(t)?q. A comparable lower bound is also given. We also establish exact values for γ(k,p) when ?(t)=2. For instance, when t=3, γ(k,p)=a+b−1 where a>b>0 are the unique integers with a²+b²+ab=p, and when t=4, γ(k,p)=a−1 where a>b>0 are the unique integers with a²+b²=p.     

Nontrivial lower bounds for the least common multiple of some finite sequences of integers

We present here a method which allows to derive a nontrivial lower bounds for the least common multiple of some finite sequences of integers. We obtain efficient lower bounds (which in a way are optimal) for the arithmetic progressions and lower bounds less efficient (but nontrivial) for quadratic sequences whose general term has the form un=an(n+t)+b with (a,t,b)∈Z³, a?5, t?0, gcd(a,b)=1. From this, we deduce for instance the lower bound: lcm{¹²+1,²²+1,…,n²+1}?0,32n(1,442) (for all n?1). In the last part of this article, we study the integer lcm(n,n+1,…,n+k) (k∈N, n∈N^∗). We show that it has a divisor dn,k simple in its dependence on n and k, and a multiple mn,k also simple in its dependence on n. In addition, we prove that both equalities: lcm(n,n+1,…,n+k)=dn,k and lcm(n,n+1,…,n+k)=mn,k hold for an infinitely many pairs (n,k).     

The diagnonal multivariate natural exponential families and their classification

A natural exponential family (NEF)F in ? ⁿ ,n>1, is said to be diagonal if there existn functions,a ₁,...,a _n , on some intervals of ?, such that the covariance matrixV _F (m) ofF has diagonal (a ₁(m ₁),...,a _n (m _n )), for allm=(m ₁,...,m _n ) in the mean domain ofF. The familyF is also said to be irreducible if it is not the product of two independent NEFs in ? ^k and ? ^n-k , for somek=1,...,n?1. This paper shows that there are only six types of irreducible diagonal NEFs in ? ⁿ , that we call normal, Poisson, multinomial, negative multinomial, gamma, and hybrid. These types, with the exception of the latter two, correspond to distributions well established in the literature. This study is motivated by the following question: IfF is an NEF in ? ⁿ , under what conditions is its projectionp(F) in ? ^k , underp(x ₁,...,x _n )∶=(x ₁,...,x _k ),k=1,...,n?1, still an NEF in ? ^k ? The answer turns out to be rather predictable. It is the case if, and only if, the principalk×k submatrix ofV _F (m ₁,...,m _n ) does not depend on (m _k+1,...,m _n ).     

Strong Sperner property of the subgroup lattice of an Abelianp-group

Letn andk be arbitrary positive integers,p a prime number and L_(k ⁿ)(p) the subgroup lattice of the Abelianp-group (Z/p ^k ) ⁿ . Then there is a positive integerN(n,k) such that whenp N(n,k),L _(k ^N )(p) has the strong Sperner property.     

Asymptotic behavior of solutions of the functional differential equation x′(t) = Ax(λt) + Bx(t), λ>0

The author discusses the asymptotic behavior of the solutions of the functional differential equation x′(t) = Ax(λt) + Bx(t), λ>0 (1) where x(t) is an n-dimensional column vector and A, B are n × n matrices with complex constant entries. He obtains the following results for the case 0 < λ < 1: (i) If B is diagonalizable with eigenvalues b_i such that Re b_i < 0 for all i, then there is a constant α such that every solution of (1) is O(t^α) as t → ∞. (ii) If B is diagonalizable with eigenvalues b_i such that 0 < Re b₁ ? Re b₂ ? ··· ? Re b_n and λ times Re b_n < Re b₁, then every solution of (1) is O(e^bn^t) as t → ∞. For the case λ>1, he has the following results: (i) If B is diagonalizable with eigenvalues b_i such that Re b_i>0 for all i, then there is a constant α such that no solution x(t) of (1), except the identically zero solution, is 0(t^α) as t → ∞. (ii) If B is diagonalizable with eigenvalues b_i such that Re b₁ ? Re b₂ ? ··· ? Re b_n < 0 and λ Re b_n < Re b₁, then no solution x(t) of (1), except the identically zero solution, is 0(e^b1^t) as t → ∞.     

Some reproducing identities for families of mean values

Gini, Lehmer, Beckenbach, and others studied the meanG _s (a, b) = (a ^s +b ^s )/(a ^{s 1} +b ^s-1 ) We proveTheorem 1 The identity (ina, b)G _s (G _t ,G _u ) =G _v holds if and only if (s, t, u, v) is (s, t, t, t) (the trivial solution) or one of (1, 1 –k, 1 +k, 1), (1/2, 1 –k, k, 1/2), or (0,–k, k, 0) (the exotic solutions,k is any real number)Theorem 2 IfP _s (a, b) is the power mean [(a ^s +b ^s )/2]^1/s , thenP _s (P _t ,P _u ) =P _v has only the trivial solution (s, t, u, v) = (s, t, t, t) and the exotic solution (0,t, –t, 0) The family of meansG _s (respP _s ) includes the classical arithmetic, geometric, and harmonic means     

Accurate approximation of time-varying matrices of linear differential systems

We consider the problem of the identification of the time-varying matrix A(t) of a linear m-dimensional differential system y′ = A(t)y. We develop an approximation A_n,k = ∑ⁿ_{j ? 1}c_j{Y(t_k + τ_j) Y^?1(t_k) ? I} to A(t_k) for grid points t_k = a + kh, k = 0,…, N using specified τ_j = θ_jh, 0 < θ_j < 1, j = 1, …, n, and show that for each t_k, the L₁ norm of the error matrix is $\text{O}$(hⁿ). We demonstrate an efficient scheme for the evaluation of A_n,k and treat sample problems.     

Polynomial expansions and generating functions

We study expansions in polynomials {P_n(x)}^∞_o generated by ∑^∞_{n = o}P_n(x)tⁿ = A(t) φ(xt^kθ(t)), θ(0) ≠ 0, and ∑^∞_{n = 0}P_n(x)tⁿ = ∑^k_{j = 1}A_j(t) φ(xt?_j), ?₁,…,?_k being the k roots of unity. The case k = 1 is contained in a recent work by Fields and Ismail. We also prove a new generalization of Vandermond's inverse relations.     

Non-uniform Turán-type problems

Given positive integers n,k,t, with 2?k?n, and t<2^k, let m(n,k,t) be the minimum size of a family F of (nonempty distinct) subsets of [n] such that every k-subset of [n] contains at least t members of F, and every (k-1)-subset of [n] contains at most t-1 members of F. For fixed k and t, we determine the order of magnitude of m(n,k,t). We also consider related Turán numbers T_?r(n,k,t) and T_r(n,k,t), where T_?r(n,k,t) (T_r(n,k,t)) denotes the minimum size of a family such that every k-subset of [n] contains at least t members of F. We prove that T_?r(n,k,t)=(1+o(1))T_r(n,k,t) for fixed r,k,t with and n→∞.     

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 distribution of reducible polynomials

Let ℛ _n (t) denote the set of all reducible polynomials p(X) over ℤ with degree n ≥ 2 and height ≤ t. We determine the true order of magnitude of the cardinality |ℛ _n (t)| of the set ℛ _n (t) by showing that, as t → ∞, t ² log t ≪ |ℛ₂(t)| ≪ t ² log t and t ⁿ ≪ |ℛ _n (t)| ≪ t _n for every fixed n ≥ 3. Further, for 1 < n/2 < k < n fixed let ℛ _k,n (t) ⊂ ℛ _n (t) such that p(X) ∈ ℛ _k,n (t) if and only if p(X) has an irreducible factor in ℤ[X] of degree k. Then, as t → ∞, we always have t ^k+1 ≪ |ℛ _k,n (t)| ≪ t ^k+1 and hence |ℛ _n−1,n (t)| ≫ |ℛ _n (t)| so that ℛ _n−1,n (t) is the dominating subclass of ℛ _n (t) since we can show that |ℛ _n (t)∖ℛ _n−1,n (t)| ≪ t ⁿ⁻¹(log t)².On the contrary, if R _n ^s (t) is the total number of all polynomials in ℛ _n (t) which split completely into linear factors over ℤ, then t ²(log t) ⁿ⁻¹ ≪ R _n ^s (t) ≪ t ² (log t) ⁿ⁻¹ (t → ∞) for every fixed n ≥ 2.      

Domino tilings of rectangles with fixed width

Let t(k,n) denote the number of ways to tile a 1 × n rectangle with 1 × 2 rectangles (called dominoes). We show that for each fixed k the sequence t_k=(t(k,0), t(k,1),…) satisfies a difference equation (linear, homogeneous, and with constant coefficients). Furthermore, a computational method is given for finding this difference equation together with the initial terms of the sequence. This gives rise to a new way to compute t(k,n) which differs completely with the known Pfaffian method. The generating function of t_k is a rational function F_k, and F_k is given explicitly for k=1,…,8. We end with some conjectures concerning the form of F_k based on our computations.     

A combinatorial problem involving graphs and matrices

In this paper we discuss a combinatorial problem involving graphs and matrices. Our problem is a matrix analogue of the classical problem of finding a system of distinct representatives (transversal) of a family of sets and relates closely to an extremal problem involving 1-factors and a long standing conjecture in the dimension theory of partially ordered sets. For an integer n ?1, let n denote the n element set {1,2,3,…, n}. Then let A be a k×t matrix. We say that A satisfies property P(n, k) when the following condition is satisfied: For every k-taple (x₁,x₂,…,x_k?n^k there exist k distinct integers j₁,j₂,…,j_k so that x_i= a_ii for i= 1,2,…,k. The minimum value of t for which there exists a k × t matrix A satisfying property P(n,k) is denoted by f(n,k). For each k?1 and n sufficiently large, we give an explicit formula for f(n, k): for each n?1 and k sufficiently large, we use probabilistic methods to provide inequalities for f(n,k).     

Universally bad integers and the 2-adics

In his 1964 paper, de Bruijn (Math. Comp. 18 (1964) 537) called a pair (a,b) of positive odd integers good, if , where is the set of nonnegative integers whose 4-adic expansion has only 0's and 1's, otherwise he called the pair (a,b) bad. Using the 2-adic integers we obtain a characterization of all bad pairs. A positive odd integer u is universally bad if (ua,b) is bad for all pairs of positive odd integers a and b. De Bruijn showed that all positive integers of the form u=2^k+1 are universally bad. We apply our characterization of bad pairs to give another proof of this result of de Bruijn, and to show that all integers of the form u=φ_p^k(4) are universally bad, where p is prime and φ_n(x) is the nth cyclotomic polynomial. We consider a new class of integers we call de Bruijn universally bad integers and obtain a characterization of such positive integers. We apply this characterization to show that the universally bad integers u=φ_p^k(4) are in fact de Bruijn universally bad for all primes p>2. Furthermore, we show that the universally bad integers φ_2^k(4), and more generally, those of the form 4^k+1, are not de Bruijn universally bad.     

On the fourier-stieltjes coefficients of cantor-type distributions

For a givenn-tuple of non-negative numbers (p ₀,p ₁,...,p _n?1) whose sum is equal to unity let μ(t) denote the probability that Σ _j ^{= 1/∞} X _j /n ^j ≦t, where the independent random variablesX _j assume the values 0,1,...,n?1 with probabilitiesp ₀,p ₁,...,p _n?1 respectively. For mostn-tuples we obtain upper and lower bounds on |û(m)|; these estimates involve then-ary representation ofm, or in some cases of 2m, so that a very simple and explicit characterization of the sequences on whichû(m) approaches zero can be given. In particular, for the Cantor middle-third measure, corresponding to the triple (1/2, 0, 1/2), the following criterion is obtained.û(m) approaches zero on a sequenceT of integers if and only if Ω(2m) approaches infinity onT, where Ω(k) is the sum of the following three quantities associated with the ternary representation ofk: the number of runs of zeros, the number of runs of twos and the number of ones. The results obtained are easily extended to the case when then-tuple varies withj (subject to certain mild restrictions).