Nonsingularity of matrices associated with classes of arithmetical functions on lcm-closed sets

Let S = {x₁, … , x_n} be a set of n distinct positive integers and f be an arithmetical function. Let [f(x_i, x_j)] denote the n × n matrix having f evaluated at the greatest common divisor (x_i, x_j) of x_i and x_j as its i, j-entry and (f[x_i, x_j]) denote the n × n matrix having f evaluated at the least common multiple [x_i, x_j] of x_i and x_j as its i, j-entry. The set S is said to be lcm-closed if [x_i, x_j] ∈ S for all 1 ? i, j ? n. For an integer x > 1, let ω(x) denote the number of distinct prime factors of x. Define ω(1) = 0. In this paper, we show that if S = {x₁, … , x_n} is an lcm-closed set satisfying , and if f is a strictly increasing (resp. decreasing) completely multiplicative function, or if f is a strictly decreasing (resp. increasing) completely multiplicative function satisfying (resp. f(p) ? p) for any prime p, then the matrix [f(x_i, x_j)] (resp. (f[x_i, x_j])) defined on S is nonsingular. By using the concept of least-type multiple introduced in [S. Hong, J. Algebra 281 (2004) 1-14], we also obtain reduced formulas for det(f(x_i, x_j)) and det(f[x_i, x_j]) when f is completely multiplicative and S is lcm-closed. We also establish several results about the nonsingularity of LCM matrices and reciprocal GCD matrices.     

A note on uniqueness problem for meromorphic mappings with 2N + 3 hyperplanes

In this paper,a uniqueness theorem for meromorphic mappings partially sharing 2N+3 hyperplanes is proved.For a meromorphic mapping f and a hyperplane H,set E(H,f) = {z|ν(f,H)(z) 0}.Let f and g be two linearly non-degenerate meromorphic mappings and {Hj}j2=N1+ 3be 2N + 3 hyperplanes in general position such that dim f-1(Hi) ∩ f-1(Hj) n-2 for i = j.Assume that E(Hj,f) E(Hj,g) for each j with 1 j 2N +3 and f = g on j2=N1+ 3f-1(Hj).If liminfr→+∞ 2j=N1+ 3N(1f,Hj)(r) j2=N1+ 3N(1g,Hj)(r) NN+1,then f ≡ g.     

Construction of a Jacobi matrix from spectral data

A proof is given for the existence and uniqueness of a correspondence between two pairs of sequences {a},{b} and {ω},{μ}, satisfying b_i>0 for i=1,…,n?1 and ω₁<μ₁<?<μ_n?1<ω_n, under which the symmetric Jacobi matrices J(n,a,b) and J(n?1,a,b) have eigenvalues {ω} and {μ} respectively. Here J(m,a,b) is symmetric and tridiagonal with diagonal elements a_i (i=1,…,m) and off diagonal elements b_i (i=1,…,m?1). A new concise proof is given for the known uniqueness result. The existence result is new.     

Factorization of entire solutions of some algebraic differential equations

We investigate the factorization of entire solutions of the following algebraic differential equations:

bn(z)finjn(f^′)+bn−1(z)fin−1jn−1(f^′)+?+b₀(z)fi₀j₀(f^′)=b(z),     

Relative probabilities of majority winners under partial information

Suppose the only observable characteristic of each of n random variables that is uniformly distributed on the six rankings of objects in a three-element set is its first-ranked object. Let ?(n₁,n₂,n₃) be the probability that one of the three objects has majorities over the other two within the rankings when n_j of the n rankings have the jth object in first place. It is assumed that n is odd, so that ?(n₁,n₂,n₃)=1 only if n_j≥(n+1)/2 for some j.It is shown that $\text{?(a+1,b,c)<?(a,b+1,c) if a <b,a ≤ c ≤ b+1}$ and max {b,c}≤(n?1)/2. It follows from this that ? is minimized for fixed n if and only if n_j?n_k≤1 for all j,k? {1,2,3}. However, ? does not necessarily increase when two of its arguments get farther apart. For example, ?(b,b,3)>?(b?1,b+1,3) for b≥28, and ?(b,b,2b?1)>?(b?1,b+1,2b?1) for b≥12.     

Divisibility of matrices associated with multiplicative functions

Let S={x₁,…,x_n} be a set of n distinct positive integers. For x,y∈S and y<x, we say the y is a greatest-type divisor of x in S if y∣x and it can be deduced that z=y from y∣z,z∣x,z<x and z∈S. For x∈S, let G_S(x) denote the set of all greatest-type divisors of x in S. For any arithmetic function f, let (f(x_i,x_j)) denote the n×n matrix having f evaluated at the greatest common divisor (x_i,x_j) of x_i and x_j as its i,j-entry and let (f[x_i,x_j]) denote the n×n matrix having f evaluated at the least common multiple [x_i,x_j] of x_i and x_j as its i,j-entry. In this paper, we assume that S is a gcd-closed set and . We show that if f is a multiplicative function such that (f∗μ)(d)∈Z whenever and f(a)|f(b) whenever a|b and a,b∈S and (f(x_i,x_j)) is nonsingular, then the matrix (f(x_i,x_j)) divides the matrix (f[x_i,x_j]) in the ring M_n(Z) of n×n matrices over the integers. As a consequence, we show that (f(x_i,x_j)) divides (f[x_i,x_j]) in the ring M_n(Z) if (f∗μ)(d)∈Z whenever and f is a completely multiplicative function such that (f(x_i,x_j)) is nonsingular. This confirms a conjecture of Hong raised in 2004.     

On monotone functions of tree structures

Let T be a rooted tree structure with n nodes a₁,…,a_n. A function f: {a₁,…,a_n} into {1 < ? < k} is called monotone if whenever a_i is a son of a_j, then f(a_i) ≥ f(a_j). The average number of monotone bijections is determined for several classes of tree structures. If k is fixed, for the average number of monotone functions asymptotic equivalents of the form c · ?^?nn^{?$\text{3}\text{2}$} (n → ∞) are obtained for several classes of tree structures.     

Probabilities of dominant candidates based on first-place votes

Suppose each of an odd number n of voters has a strict preference order on the three ‘candidates’ in {1,2,3} and votes for his most preferred candidate on a plurality ballot. Assume that a voter who votes for i is equally likely to have ijk and ikj as his preference order when {i,j,k} = {1,2,3}.Fix an integer m between $\text{1}\text{2}$(n + 1) and n inclusive. Then, given that n_i of the n voters vote for i, let f_m(n₁,n₂,n₃) be the probability that one of the three candidates is preferred by m or more voters to each of the other two.This paper examines the behavior of f_m over the lattice points in L_n, the set of triples of non-negative integers that sum to n. It identifies the regions in L_n where f_m is 1 and where f_m is 0, then shows that f_m(a,b + 1, c)>f_m(a + 1,b,c) whenever a + b + c + 1 = n, a≤c≤b, a<c<m and c≤n ? m. These results are used to partially identify the points in L_n where f_m is minimized subject to f_m>0. It is shown that at least two of the n_i are equal at minimizing points.     

A constructive method for uniform approximation by means of integrated Lagrange interpolation polynomials in the spaceC 1[a,b]

Пусть интерполяционные уз лыa <x ₁ <x ₂ < ... <х _n <b являются корнямиn-го полинома системы полиномов {w _n(x)} _n ^∞ =1, ортогонально й с весомρ(х), 0 < ρ(х) ∈L[a, b], причем 1/ρ(x) ∈L[a, b]. Доказано, что еслиf(х) ∈С ¹ [a, b], то проинтегрированн ые полиномы $$Q_n (x) = \smallint _a^x L_{n - 1} (f',t)dt + f(a)$$ интерполяционных по линомов Лагранжа $$L_{n - 1} (f,x) = \sum\limits_{k = 1}^n {f'(x_k )\frac{{\omega _n (x)}}{{\omega '_n (x_k )(x - x_k )}}}$$ порядкаn?1 функцииf′(x) сходятся равномерно к функцииf(х) в замкнуто м интервале [а,b]. Во второй части стать и, где вместо условия 1/ρ(х) ∈L[a, b] предполагается, чтоρ(х) ∈С ²[а, b],ρ(х) > 0,ρ′ (х) > 0, доказываются подобные утверждени я для проинтегрированных интерполяционных по линомов Лагранжа и для прибли жений рациональными дробя ми. В работе даются неулу чшаемые порядковые о ценки и описываются примене ния полученных результа тов в теории дифферен циальных уравнений.     

Bounds on absorption times of directionally biased random sequences

A sequence of random variables X₀,X₁, … with values in {0, 1, …, n} representing a general finite-state stochastic process with absorbing state 0 is said to be directionally biased towards 0, if, for all j > 0, ϵ_j: = inf_k>0 {j − E[X_k | X_k−1 = j]} > 0. For such sequences, let t be the expected value of the time to absorption at 0. For a fixed set of biases, the least upper bound for this time can be computed with an algorithm requiring O(n²) steps. Simple upper bounds are described. In particular, t ≤ E[b_x0], where b_i = Σ_j≤i 1/¯ϵ_j and ¯ϵ_j = min_l≥j {ϵ_l}. If all ϵ_j ≤ ϵ_{j + 1} (so ¯ϵ_j = ϵ_j) and ϵ_n < 1, this bound for t is the best possible. For certain finite stochastic processes which we term conditionally independent of X₀ = i, b(i) bounds the expected time given X₀ = i. Similar results are given for lower bounds. The results of this paper were designed to be a useful tool for determining rates of convergence of stochastic optimization algorithms. © 1996 John Wiley & Sons, Inc.     

Process convergence of self-normalized sums of i.i.d. random variables coming from domain of attraction of stable distributions

In this paper we show that the continuous version of the self-normalized process Y _n,p (t)?=?S _n (t)/V _n,p ?+?(nt???[nt])X _[nt]?+?1/V _n,p ,0?<?t?≤?1; p?>?0 where $S_n(t)=\sum_{i=1}^{[nt]} X_i$ and $V_{(n,p)}=(\sum_{i=1}^{n}|X_i|^p)^{1/p}$ and X _i i.i.d. random variables belong to DA(α), has a non-trivial distribution iff p?=?α?=?2. The case for 2?>?p?>?α and p?≤?α?<?2 is systematically eliminated by showing that either of tightness or finite dimensional convergence to a non-degenerate limiting distribution does not hold. This work is an extension of the work by Csörg? et al. who showed Donsker’s theorem for Y _n,2(·), i.e., for p?=?2, holds iff α?=?2 and identified the limiting process as a standard Brownian motion in sup norm.     

Multiplicative isometries on the Smirnov class

We show that T is a surjective multiplicative (but not necessarily linear) isometry from the Smirnov class on the open unit disk, the ball, or the polydisk onto itself, if and only if there exists a holomorphic automorphism Φ such that T(f)=f ○ Φ for every class element f or T(f) = [`(f<sup>°</sup> [`(j)] )]\overline {f^\circ \bar \varphi } for every class element f, where the automorphism Φ is a unitary transformation in the case of the ball and Φ(z ₁, ..., z _n ) = (l₁ z_i1 ,...,l_n z_in )(\lambda _1 z_{i_1 } ,...,\lambda _n z_{i_n } ) for |λ _j | = 1, 1 ≤ j ≤ n, and (i ₁; ..., i _n )is some permutation of the integers from 1through n in the case of the n-dimensional polydisk.     

Nonsingularity of least common multiple matrices on gcd-closed sets

Let n be a positive integer. Let S={x₁,…,x_n} be a set of n distinct positive integers. The least common multiple (LCM) matrix on S, denoted by [S], is defined to be the n×n matrix whose (i,j)-entry is the least common multiple [x_i,x_j] of x_i and x_j. The set S is said to be gcd-closed if for any x_i,x_j∈S,(x_i,x_j)∈S. For an integer m>1, let ω(m) denote the number of distinct prime factors of m. Define ω(1)=0. In 1997, Qi Sun conjectured that if S is a gcd-closed set satisfying max_x∈S{ω(x)}?2, then the LCM matrix [S] is nonsingular. In this paper, we settle completely Sun's conjecture. We show the following result: (i). If S is a gcd-closed set satisfying max_x∈S{ω(x)}?2, then the LCM matrix [S] is nonsingular. Namely, Sun's conjecture is true; (ii). For each integer r?3, there exists a gcd-closed set S satisfying max_x∈S{ω(x)}=r, such that the LCM matrix [S] is singular.     

A limit theorem for expectations conditional on a sum

Let ξ₁, ξ₂, ξ₃,... be a sequence of independent random variables, such that μ _j ?E[ξ _j ], 0<α?Var[ξ _j ] andE[|ξ _j ?μ _j |^2+δ] for some δ, 0<δ?1, and everyj?1. IfU and ξ₀ are two random variables such thatE[ξ ₀ ² ]<∞ andE[|U|ξ ₀ ² ]<∞, and the vector 〈U,ξ〉 is independent of the sequence {ξ _j :j?1}, then under appropriate regularity conditions $$E\left[ {U\left| {\xi _0 + S_n } \right. = \sum\limits_{j = 1}^n {\mu _j + c_n } } \right] = E[U] + O\left( {\frac{1}{{s_n^{1 + \delta } }}} \right) + O\left( {\frac{{|c_n |}}{{s_n^2 }}} \right)$$ whereS _n ?ξ₁+ξ₂+?+ξ _n ,μ _j ?E[ξ _j ],s _n ² ?Var[S _n ], andc _n =O(s _n ).     

Indecomposable modules of the intermediate series over W(a, b)

For any complex parameters a and b,W(a,b)is the Lie algebra with basis{Li,Wi|i∈Z}and relations[Li,Lj]=(j i)Li+j,[Li,Wj]=(a+j+bi)Wi+j,[Wi,Wj]=0.In this paper,indecomposable modules of the intermediate series over W(a,b)are classified.It is also proved that an irreducible Harish-Chandra W(a,b)-module is either a highest/lowest weight module or a uniformly bounded module.Furthermore,if a∈/Q,an irreducible weight W(a,b)-module is simply a Vir-module with trivial actions of Wk’s.     

Optimal labelling of a product of two paths

If G is a graph with p vertices and at least one edge, we set φ (G) = m n max |f(u) ? f(v)|, where the maximum is taken over all edges uv and the minimum over all one-to-one mappings f : V(G) → {1, 2, …, p}: V(G) denotes the set of vertices of G.P_n will denote a path of length n whose vertices are integers 1, 2, …, n with i adjacent to j if and only if |i ? j| = 1. P_m × P_n will denote a graph whose vertices are elements of {1, 2, …, m} × {1, 2, …, n} and in which (i, j), (r, s) are adjacent whenever either i = r and |j ? s| = 1 or j = s and |i ? r| = 1.Theorem.If max(m, n) ? 2, thenφ(P_m × P_n) = min(m, n).     

On friendly index sets of 2-regular graphs

Let G be a graph with vertex set V and edge set E, and let A be an abelian group. A labeling f:V→A induces an edge labeling f^∗:E→A defined by f^∗(xy)=f(x)+f(y). For i∈A, let v_f(i)=card{v∈V:f(v)=i} and e_f(i)=card{e∈E:f^∗(e)=i}. A labeling f is said to be A-friendly if |v_f(i)−v_f(j)|≤1 for all (i,j)∈A×A, and A-cordial if we also have |e_f(i)−e_f(j)|≤1 for all (i,j)∈A×A. When A=Z₂, the friendly index set of the graph G is defined as {|e_f(1)−e_f(0)|:the vertex labelingf is Z₂-friendly}. In this paper we completely determine the friendly index sets of 2-regular graphs. In particular, we show that a 2-regular graph of order n is cordial if and only if n?2 (mod 4).     

On the Bateman-Horn Conjecture

Let r be a positive integer and f₁,…,f_r be distinct polynomials in Z[X]. If f₁(n),…,f_r(n) are all prime for infinitely many n, then it is necessary that the polynomials f_i are irreducible in Z[X], have positive leading coefficients, and no prime p divides all values of the product f₁(n)···f_r(n), as n runs over Z. Assuming these necessary conditions, Bateman and Horn (Math. Comput.16 (1962), 363-367) proposed a conjectural asymptotic estimate on the number of positive integers n?x such that f₁(n),…,f_r(n) are all primes. In the present paper, we apply the Hardy-Littlewood circle method to study the Bateman-Horn conjecture when r?2. We consider the Bateman-Horn conjecture for the polynomials in any partition {f₁,…,f_s}, {f_s+1,…,f_r} with a linear change of variables. Our main result is as follows: If the Bateman-Horn conjecture on such a partition and change of variables holds true with some conjectural error terms, then the Bateman-Horn conjecture for f₁,…,f_r is equivalent to a plausible error term conjecture for the minor arcs in the circle method.     

A new inequality for symmetric functions

Let x_i ≥ 0, y_i ≥ 0 for i = 1,…, n; and let a_j(x) be the elementary symmetric function of n variables given by a_j(x) = ∑_{1 ≤ ii < … <ij ≤ n}x_ii … x_ij. Define the partical ordering x <y if a_j(x) ≤ a_j(y), j = 1,… n. We show that $\text{x}\text{\$}\text{?}\text{y ? x}{}^{\mathrm{\alpha }}\text{\$}\text{?}\text{y}{}^{\mathrm{\alpha }}\text{, 0}\text{\$}\text{?}\text{α ≤ 1}$, where {x^α}_i = x^α_i. We also give a necessary and sufficient condition on a function f(t) such that x <y ? f(x) <f(y). Both results depend crucially on the following: If x <y there exists a piecewise differentiable path z(t), with z_i(t) ≥ 0, such that z(0) = x, z(1) = y, and z(s) <z(t) if 0 ≤ s ≤ t ≤ 1.     

Note on the local growth of iterated polynomials

The local behavior of the iterates of a real polynomial is investigated. The fundamental result may be stated as follows: THEOREM. Let x_i, for i=1, 2, ..., n+2, be defined recursively by x_i+1=f(x_i), where x₁ is an arbitrary real number and f is a polynomial of degree n. Let x_i+1?x_i≧1 for i=1, ..., n + 1. Then for all i, 1 ≦i≦n, and all k, 1≦k≦n+1?i, $$ - \frac{{2^{k - 1} }}{{k!}}< f\left[ {x_1 ,... + x_{i + k} } \right]< \frac{{x_{i + k + 1} - x_{i + k} + 2^{k - 1} }}{{k!}},$$ where f[x_i, ..., x_i+k] denotes the Newton difference quotient. As a consequence of this theorem, the authors obtain information on the local behavior of the solutions of certain nonlinear difference equations. There are several cases, of which the following is typical: THEOREM. Let {x_i}, i = 1, 2, 3, ..., be the solution of the nonlinear first order difference equation x_i+1=f(x_i) where x₁ is an arbitrarily assigned real number and f is the polynomial \(f(x) = \sum\limits_{j = 0}^n {a_j x^j } ,n \geqq 2\) . Let δ be positive with δ^n?1=|2^n?1/n!a_n|. Then, if n is even and a_n<0, there do not exist n + 1 consecutive increments Δx_i=x_i+1?x_i in the solution {x_i} with Δx_i≧δ. The special case in which the iterated polynomial has integer coefficients leads to a “nice” upper bound on a generalization of the van der Waerden numbers. Ap _k -sequence of length n is defined to be a strictly increasing sequence of positive integers {x ₁, ...,x _n } for which there exists a polynomial of degree at mostk with integer coefficients and satisfyingf(x _j )=x _j+1 forj=1, 2, ...,n?1. Definep _k (n) to be the least positive integer such that if {1, 2, ...,p _k (n)} is partitioned into two sets, then one of the two sets must contain ap _k -sequence of lengthn. THEOREM. p_n?2(n)≦(n!)^(n?2)!/2.