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Abstract

Consider the sequence s of the signs of the coefficients of a real univariate polynomial P of degree d. Descartes’ rule of signs gives compatibility conditions between s and the pair (r + ,r ), where r + is the number of positive roots and r the number of negative roots of P. It was recently asked if there are other compatibility conditions, and the answer was given in the form of a list of incompatible triples (s; r + ,r ) which begins at degree d = 4 and is known up to degree 8. In this paper we raise the question of the compatibility conditions for (s;r0+,r0;r1+,r1;;rd1+,rd1), whereri+ (resp.ri) is the number of positive (resp. negative) roots of the i-th derivative of P. We prove that up to degree 5, there are no other compatibility conditions than the Descartes conditions, the above recent incompatibilities for each i, and the trivial conditions given by Rolle’s theorem.

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Abstract

Let l,m,r be fixed positive integers such that 2| l, 3 lm, l > r and 3 | r. In this paper, using the BHV theorem on the existence of primitive divisors of Lehmer numbers, we prove that if min{rlm 2 − 1,(lr)lm 2 + 1} > 30, then the equation (rlm 2 − 1)x + ((lr)lm 2 + 1)y = (lm)z has only the positive integer solution (x,y,z) = (1,1,2).

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Abstract

In 1975 C. F. Chen and C. H. Hsiao established a new procedure to solve initial value problems of systems of linear differential equations with constant coefficients by Walsh polynomials approach. However, they did not deal with the analysis of the proposed numerical solution. In a previous article we study this procedure in case of one equation with the techniques that the theory of dyadic harmonic analysis provides us. In this paper we extend these results through the introduction of a new procedure to solve initial value problems of differential equations with not necessarily constant coefficients.

Open access

Abstract

Let p(z)=j=0najzj be a polynomial of degree n. Further, letM(p,R)=max|z|=R1|p(z)|, and p=M(p,1). Then according to the well-known Bernstein inequalities, we have pnp and M(p,R)Rnp. It is an open problem to obtain inequalities analogous to these inequalities for the class of polynomials satisfying p(z) ≡ z n p(1/z). In this paper we obtain some inequalites in this direction for polynomials that belong to this class and have all their coefficients in any sector of opening γ, where 0 _ γ < π. Our results generalize and sharpen several of the known results in this direction, including those of Govil and Vetterlein [3], and Rahman and Tariq [12]. We also present two examples to show that in some cases the bounds obtained by our results can be considerably sharper than the known bounds.

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Authors: Wei-Feng Xuan and Yan-Kui Song

Abstract

In this paper, we prove that if X is a space with a regular G δ-diagonal and X 2 is star Lindelöf then the cardinality of X is at most 2c. We also prove that if X is a star Lindelöf space with a symmetric g-function such that {g 2(n, x): nω} = {x} for each xX then the cardinality of X is at most 2c. Moreover, we prove that if X is a star Lindelöf Hausdorff space satisfying (X) = κ then e(X) 22κ; and if X is Hausdorff and we(X) = (X) = κsubset of a space then e(X) 2κ. Finally, we prove that under V = L if X is a first countable DCCC normal space then X has countable extent; and under MA+¬CH there is an example of a first countable, DCCC and normal space which is not star countable extent. This gives an answer to the Question 3.10 in Spaces with property (DC(ω 1)), Comment. Math. Univ. Carolin., 58(1) (2017), 131-135.

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Abstract

Fejes Tóth [] studied approximations of smooth surfaces in three-space by piecewise flat triangular meshes with a given number of vertices on the surface that are optimal with respect to Hausdorff distance. He proves that this Hausdorff distance decreases inversely proportional with the number of vertices of the approximating mesh if the surface is convex. He also claims that this Hausdorff distance is inversely proportional to the square of the number of vertices for a specific non-convex surface, namely a one-sheeted hyperboloid of revolution bounded by two congruent circles. We refute this claim, and show that the asymptotic behavior of the Hausdorff distance is linear, that is the same as for convex surfaces.

Open access

Abstract

Let H n be the n-th harmonic number and let v n be its denominator. It is known that v n is even for every integer n>=2. In this paper, we study the properties of H n and prove that for any integer n, v n = e n(1+o(1)). In addition, we obtain some results of the logarithmic density of harmonic numbers.

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Authors: Éva Czabarka, Inne Singgih, Laszlό Székely and Zhiyu Wang

Abstract

We verify an upper bound of Pach and Tóth from 1997 on the midrange crossing constant. Details of their89π2 upper bound have not been available. Our verification is different from their method and hinges on a result of Moon from 1965. As Moon’s result is optimal, we raise the question whether the midrange crossing constant is 89π2.

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Abstract

Let 0 < γ 1 < γ 2 < ··· ⩽ ··· be the imaginary parts of non-trivial zeros of the Riemann zeta-function. In the paper, we consider the approximation of analytic functions by shifts of the Hurwitz zeta-function ζ(s + k h, α), h > 0, with parameter α such that the set {log(m + α): m0} is linearly independent over the field of rational numbers. For this, a weak form of the Montgomery conjecture on the pair correlation of {γ k} is applied.

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Abstract

We study certain subgroups of the full group of Hopf algebra automorphisms of twisted tensor biproducts.

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