Convergence properties of sequences of functions with application to restricted derivative approximation

Convergence properties of sequences of continuous functions, with kth order divided differences bounded from above or below, are studied. It is found that for such sequences, convergence in a “monotone norm” (e.g., L_p) on [a, b] to a continuous function implies uniform convergence of the sequence and its derivatives up to order k − 1 (whenever they exist), in any closed subinterval of [a, b]. Uniform convergence in the closed interval [a, b] follows from the boundedness from below and above of the kth order divided differences. These results are applied to the estimation of the degree of approximation in Monotone and Restricted Derivative approximation, via bounds for the same problems with only one restricted derivative.     

Simultaneous best approximation by three polynomials

Let [a, b] be any interval and let p₀, p₁, p_k be any three polynomials of degrees 0, 1, k, respectively, where k 2. A set of necessary and sufficient conditions for the existence of an f in C[a, b] such that p_i is the best approximation to f from the space of all polynomials of degree less than or equal to i, for all i = 0, 1, k, is given.     

The best quadrature based on given Hermite information for the Sobolev class KW r[a, b]

As usual, denote by KW ^r[a, b] the Sobolev class consisting of every function whose (r ? 1)th derivative is absolutely continuous on the interval [a, b] and rth derivative is bounded by K a.e. in [a, b]. For a function f ∈ KW ^r [a, b], its values and derivatives up to r ? 1 order at a set of nodes x are known. These values are said to be the given Hermite information. This work reports the results on the best quadrature based on the given Hermite information for the class KW ^r [a, b]. Existence and concrete construction issue of the best quadrature are settled down by a perfect spline interpolation. It turns out that the best quadrature depends on a system of algebraic equations satisfied by a set of free nodes of the interpolation perfect spline. From our another new result, it is shown that the system can be converted in a closed form to two single-variable polynomial equations, each being of degree approximately r/2. As a by-product, the best interpolation formula for the class KW ^r [a, b] is also obtained.     

On the difference between an integer and its m-th power mod n

For any real constants λ ₁, λ ₂ ∈ (0, 1], let $n \geqslant \max \{ [\tfrac{1} {{\lambda _1 }}],[\tfrac{1} {{\lambda _2 }}]\} $ , m ? 2 be integers. Suppose integers a ∈ [1, λ ₁ n] and b ∈ [1, λ ₂ n] satisfy the congruence b ≡ a ^m (mod n). The main purpose of this paper is to study the mean value of (a ? b)^2k for any fixed positive integer k and obtain some sharp asymptotic formulae.     

Inverse indefinite Sturm-Liouville problems with three spectra

We prove that the potential q(x) of an indefinite Sturm-Liouville problem on the closed interval [a,b] with the indefinite weight function w(x) can be determined uniquely by three spectra, which are generated by the indefinite problem defined on [a,b] and two right-definite problems defined on [a,0] and [0,b], where point 0 lies in (a,b) and is the turning point of the weight function w(x).     

On Extending Peano Derivatives

Let E be a closed set with inf E = a and sup E = b, and k be a positive integer. Let f : E Rbe such that the k-th Peano derivative of f relative to E, f _(k) (x, E), exists. It is proved under certain condition on the function f, that an extension F : [a, b] Rof f exists such that the ordinary derivative of F of order k, F ^<k> (x) exists on [a, b] and is continuous on [a, b], and f _<> (x, E) = F ^<i> (x) on E, for i = 1, 2, &, k.     

Three-monotone spline approximation

For r≥3, n∈N and each 3-monotone continuous function f on [a,b] (i.e., f is such that its third divided differences [x₀,x₁,x₂,x₃]f are nonnegative for all choices of distinct points x₀,…,x₃ in [a,b]), we construct a spline s of degree r and of minimal defect (i.e., s∈C^r−1[a,b]) with n−1 equidistant knots in (a,b), which is also 3-monotone and satisfies ‖f−s‖_L∞[a,b]≤cω₄(f,n⁻¹,[a,b])_∞, where ω₄(f,t,[a,b])_∞ is the (usual) fourth modulus of smoothness of f in the uniform norm. This answers in the affirmative the question raised in [8, Remark 3], which was the only remaining unproved Jackson-type estimate for uniform 3-monotone approximation by piecewise polynomial functions (ppfs) with uniformly spaced fixed knots.Moreover, we also prove a similar estimate in terms of the Ditzian–Totik fourth modulus of smoothness for splines with Chebyshev knots, and show that these estimates are no longer valid in the case of 3-monotone spline approximation in the L_p norm with p<∞. At the same time, positive results in the L_p case with p<∞ are still valid if one allows the knots of the approximating ppf to depend on f while still being controlled.These results confirm that 3-monotone approximation is the transition case between monotone and convex approximation (where most of the results are “positive”) and k-monotone approximation with k≥4 (where just about everything is “negative”).     

Divisibility of determinants of power GCD matrices and power LCM matrices on finitely many quasi-coprime divisor chains

Let a, n ? 1 be integers and S = {x₁, … , x_n} be a set of n distinct positive integers. The matrix having the ath power (x_i, x_j)^a of the greatest common divisor of x_i and x_j as its i, j-entry is called ath power greatest common divisor (GCD) matrix defined on S, denoted by (S^a). Similarly we can define the ath power LCM matrix [S^a]. We say that the set S consists of finitely many quasi-coprime divisor chains if we can partition S as S = S₁ ∪ ? ∪ S_k, where k ? 1 is an integer and all S_i (1 ? i ? k) are divisor chains such that (max(S_i), max(S_j)) = gcd(S) for 1 ? i ≠ j ? k. In this paper, we first obtain formulae of determinants of power GCD matrices (S^a) and power LCM matrices [S^a] on the set S consisting of finitely many quasi-coprime divisor chains with gcd(S) ∈ S. Using these results, we then show that det(S^a)∣det(S^b), det[S^a]∣det[S^b] and det(S^a)∣det[S^b] if a∣b and S consists of finitely many quasi-coprime divisor chains with gcd(S) ∈ S. But such factorizations fail to be true if such divisor chains are not quasi-coprime.     

On the rational approximation ofe −x on [0, ∞)

In recent years the problem of uniform approximation ofe ^?x on [0, ∞) by rational functions has found wide interest. In this paper we present a method for the construction of rational approximations toe ^?x that seem to come arbitrarily near to the asymptotically best one. We start with a generalization of the integral form of the Padé approximant by introducing certain real parametersa _j ,b _i ,k and?. The corresponding error function has again an integral representation and is estimated for everyx∈[0,∞) by the Laplace method. This leads to the consideration of a finite number of new error functionsφ _i·j whose maxima are determined by a set of nonlinear equations and, under some restrictions on thea _j ,b _i ,k, and?, are also unique. Variation of these parameters according to a special algorithm and computation of the corresponding maxima of the functionsφ _i·j shows that forn→∞ the order of rational approximation ofe ^?x on [0, ∞) is at least 9.03^?n .     

[a,b]-factorization of a graph

Let a and b be integers such that 0 ? a ? b. Then a graph G is called an [a, b]-graph if a ? d_G(x) ? b for every x ? V(G), and an [a, b]-factor of a graph is defined to be its spanning subgraph F such that a ? d_F(x) ? b for every vertex x, where d_G(x) and d_F(x) denote the degrees of x in G and F, respectively. If the edges of a graph can be decomposed into [a.b]-factors then we say that the graph is [2a, 2a]-factorable. We prove the following two theorems: (i) a graph G is [2a, 2b)-factorable if and only if G is a [2am,2bm]-graph for some integer m, and (ii) every [8m + 2k, 10m + 2k]-graph is [1,2]-factorable.     

Completely strong path-connected tournaments

Let T = (V, A) be a tournament with p vertices. T is called completely strong path-connected if for each arc (a, b) ∈ A and k (k = 2, 3,…, p), there is a path from b to a of length k (denoted by P_k(a, b)) and a path from a to b of length k (denoted by P′_k(a, b)). In this paper, we prove that T is completely strong path-connected if and only if for each arc (a, b) ∈ A, there exist P₂(a, b), P′₂(a, b) in T, and T satisfies one of the following conditions: (a) T/T₀-type graph, (b) T is 2-connected, (c) for each arc (a, b) ∈ A, there exists a P′_p?1(a, b) in T.     

On a conjecture of Butler and Graham

Motivated by a hat guessing problem proposed by Iwasawa, Butler and Graham made the following conjecture on the existence of a certain way of marking the coordinate lines in [k] ⁿ : there exists a way to mark one point on each coordinate line in [k] ⁿ , so that every point in [k] ⁿ is marked exactly a or b times as long as the parameters (a, b, n, k) satisfies that there are nonnegative integers s and t such that s + t = k ⁿ and as + bt = nk ^n?1. In this paper we prove this conjecture for any prime number k. Moreover, we prove the conjecture for the case when a = 0 for general k.     

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.     

Sums and k-sums in abelian groups of order k

Let G be an abelian group of order k. How is the problem of minimizing the number of sums from a sequence of given length in G related to the problem of minimizing the number of k-sums? In this paper we show that the minimum number of k-sums for a sequence a₁,…,ar that does not have 0 as a k-sum is attained at the sequence b₁,…,br−k+1,0,…,0, where b₁,…,br−k+1 is chosen to minimise the number of sums without 0 being a sum. Equivalently, to minimise the number of k-sums one should repeat some value k−1 times. This proves a conjecture of Bollobás and Leader, and extends results of Gao and of Bollobás and Leader.     

A uniqueness theorem for indefinite Sturm-Liouville operators

In this paper, we discuss the inverse problem for indefinite Sturm-Liouville operators on the finite interval [a, b]. For a fixed index n(n = 0, 1, 2, ··· ), given the weight function ω(x), we will show that the spectral sets {λ n (q, h a , h k )} +∞ k=1 and {λ-n (q, h b , h k )} +∞ k=1 for distinct h k are sufficient to determine the potential q(x) on the finite interval [a, b] and coefficients h a and h b of the boundary conditions.     

Extended directed triple systems

Let {n;b₂,b₁} denote the class of extended directed triple systems of the order n in which the number of blocks of the form [a,b,a] is b₂ and the number of blocks of the form [b,a,a] or [a,a,b] is b₁. In this paper, we have shown that the necessary and sufficient condition for the existence of the class {n;b₂,b₁} is b₁≠1, 0?b₂+b₁?n and

(1)

for ;

(2)

for .

     

Maximum vertex and face degree of oblique graphs

Let G=(V,E,F) be a 3-connected simple graph imbedded into a surface S with vertex set V, edge set E and face set F. A face α is an 〈a₁,a₂,…,a_k〉-face if α is a k-gon and the degrees of the vertices incident with α in the cyclic order are a₁,a₂,…,a_k. The lexicographic minimum 〈b₁,b₂,…,b_k〉 such that α is a 〈b₁,b₂,…,b_k〉-face is called the type of α.Let z be an integer. We consider z-oblique graphs, i.e. such graphs that the number of faces of each type is at most z. We show an upper bound for the maximum vertex degree of any z-oblique graph imbedded into a given surface. Moreover, an upper bound for the maximum face degree is presented. We also show that there are only finitely many oblique graphs imbedded into non-orientable surfaces.     

On the asymptotic existence of Hadamard matrices

It is conjectured that Hadamard matrices exist for all orders 4t (t>0). However, despite a sustained effort over more than five decades, the strongest overall existence results are asymptotic results of the form: for all odd natural numbers k, there is a Hadamard matrix of order k²[a+blog₂k], where a and b are fixed non-negative constants. To prove the Hadamard Conjecture, it is sufficient to show that we may take a=2 and b=0. Since Seberry's ground-breaking result, which showed that we may take a=0 and b=2, there have been several improvements where b has been by stages reduced to 3/8. In this paper, we show that for all ?>0, the set of odd numbers k for which there is a Hadamard matrix of order k²2+[?log₂k] has positive density in the set of natural numbers. The proof adapts a number-theoretic argument of Erdos and Odlyzko to show that there are enough Paley Hadamard matrices to give the result.     

Brownian Gibbs property for Airy line ensembles

Consider a collection of N Brownian bridges $B_{i}:[-N,N] \to \mathbb{R} $ , B _i (?N)=B _i (N)=0, 1≤i≤N, conditioned not to intersect. The edge-scaling limit of this system is obtained by taking a weak limit as N→∞ of the collection of curves scaled so that the point (0,2^1/2 N) is fixed and space is squeezed, horizontally by a factor of N ^2/3 and vertically by N ^1/3. If a parabola is added to each of the curves of this scaling limit, an x-translation invariant process sometimes called the multi-line Airy process is obtained. We prove the existence of a version of this process (which we call the Airy line ensemble) in which the curves are almost surely everywhere continuous and non-intersecting. This process naturally arises in the study of growth processes and random matrix ensembles, as do related processes with “wanderers” and “outliers”. We formulate our results to treat these relatives as well. Note that the law of the finite collection of Brownian bridges above has the property—called the Brownian Gibbs property—of being invariant under the following action. Select an index 1≤k≤N and erase B _k on a fixed time interval (a,b)?(?N,N); then replace this erased curve with a new curve on (a,b) according to the law of a Brownian bridge between the two existing endpoints (a,B _k (a)) and (b,B _k (b)), conditioned to intersect neither the curve above nor the one below. We show that this property is preserved under the edge-scaling limit and thus establish that the Airy line ensemble has the Brownian Gibbs property. An immediate consequence of the Brownian Gibbs property is a confirmation of the prediction of M. Prähofer and H. Spohn that each line of the Airy line ensemble is locally absolutely continuous with respect to Brownian motion. We also obtain a proof of the long-standing conjecture of K. Johansson that the top line of the Airy line ensemble minus a parabola attains its maximum at a unique point. This establishes the asymptotic law of the transversal fluctuation of last passage percolation with geometric weights. Our probabilistic approach complements the perspective of exactly solvable systems which is often taken in studying the multi-line Airy process, and readily yields several other interesting properties of this process.     

The Product of Like-Indexed Terms in Binary Recurrences

In recent work by Hajdu and Szalay, Diophantine equations of the form (a^k−1)(b^k−1)=x² were completely solved for a few pairs (a, b). In this paper, a general finiteness theorem for equations of the form u_kv_k=xⁿ is proved, where u_k and v_k are terms in certain types of binary recurrence sequences. Also, a unified computational approach for solving equations of the type (a^k−1)(b^k−1)=x² is described, and this approach is used to completely solve such equations for almost all (a,b) in the range 1<a<b?100. In the final section of this paper, it is shown that the abc conjecture implies much stronger results on these types of Diophantine problems.