On the Exceptional Set for Goldbach’s Problemin Short Intervals

 Let θ be a constant satisfying . We prove that there exists , such that the number of even integers in the interval which cannot be written as a sum of two primes is . References     

On the Exceptional Set of Goldbach’s Problem in Short Intervals

Denote by E[X,X+H] the set of even integers in [X,X+H] that are not a sum of two primes (i.e. that are not Goldbach numbers). Here we prove that there exists a (small) positive constant such that for we have .     

Corrigendum: “On the Exceptional Set for Goldbach’s Problem in Short Intervals”

We correct the proof of the log-free hybrid zero-density estimate given in On the Exceptional Set for Goldbachs Problem in Short Intervals, Monatsh Math 132: 49–65.     

Zero-free Regions for Dirichlet L-functions and the Exceptional Set of Goldbach Numbers

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Primes and Goldbach Numbers in Short Intervals

ＰｒｉｍｅｓａｎｄＧｏｌｄｂａｃｈＮｕｍｂｅｒｓｉｎＳｈｏｒｔＩｎｔｅｒｖａｌｓＬｉＨｏｎｇｚｅ（李红泽）（Ｄｅｐｔ．ｏｆＭａｔｈ．ＳｈａｎｄｏｘｇＵｎｉｖｅｒｓｉｔｙ，Ｊｉｎａｎ，Ｓｈａｎｄｏｎｇ，２５０１００）ＣｏｍｍｕｎｉｃａｔｅｄｂｙＰａｎＣ...     

On Linnik’s almost Goldbach theorem

It is proved that every sufficiently large even integer can be represented as a sum of two primes and at most 2250 powers of 2. Project supported partially by the National Natural Science Foundation of China (Grant No. 19771029).     

On the Equation n =p k~2 in Short Intervals

ＯｎｔｈｅＥｑｕａｔｉｏｎｎ＝ｐ＋ｋ~２ｉｎＳｈｏｒｔＩｎｔｅｒｖａｌｓ￥ＷａｎｇＴｉａｎｚｅ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＨｅｎａｎＵｎｉｖｅｒｓｉｔｙ，Ｋａｉｆｅｎｇ４７５００１）Ａｂｓｔｒａｃｔ：ＤｅｎｏｔｅｂｙＥ（Ｘ，Ｙ）?..     

On Wilking’s criterion for the Ricci flow

Wilking has recently shown that one can associate a Ricci flow invariant cone of curvature operators $C(S)$ , which are nonnegative in a suitable sense, to every $Ad_{SO(n,\mathbb{C })}$ invariant subset $S \subset \mathbf{so}(n,\mathbb{C })$ . In this article we show that if $S$ is an $Ad_{SO(n,\mathbb{C })}$ invariant subset of $\mathbf{so}(n,\mathbb{C })$ such that $S\cup \{0\}$ is closed and $C_+(S)\subset C(S)$ denotes the cone of curvature operators which are positive in the appropriate sense then one of the two possibilities holds: (a) The connected sum of any two Riemannian manifolds with curvature operators in $C_+(S)$ also admits a metric with curvature operator in $C_+(S)$ (b) The normalized Ricci flow on any compact Riemannian manifold $M$ with curvature operator in $C_+(S)$ converges to a metric of constant positive sectional curvature. We also point out that if $S$ is an arbitrary $Ad_{SO(n,\mathbb{C })}$ subset, then $C(S)$ is contained in the cone of curvature operators with nonnegative isotropic curvature.     

On Goulden-Jackson’s Determinantal Formula for the Immanant

In 1992, Goulden and Jackson found a beautiful determinantal expression for the immanant of a matrix. This paper proves the same result combinatorially. We also present a β-extension of the theorem and a simple determinantal expression for the irreducible characters of the symmetric group.     

The Exceptional Set in Hua＇s Theorem for Three Squares of Primes

It is proved that with at most O(N^11/12 c) exceptions, all positive integers n ≤ N satisfying some necessary congruence conditions are the sum of three squares of primes. This improves substantially the previous results in this direction.     

On the Scope of Averaging for Frankl’s Conjecture

Let be a union-closed family of subsets of an m-element set A. Let . For b ∈ A let w(b) denote the number of sets in containing b minus the number of sets in not containing b. Frankl’s conjecture from 1979, also known as the union-closed sets conjecture, states that there exists an element b ∈ A with w(b) ≥ 0. The present paper deals with the average of the w(b), computed over all b ∈ A. is said to satisfy the averaged Frankl’s property if this average is non-negative. Although this much stronger property does not hold for all union-closed families, the first author (Czédli, J Comb Theory, Ser A, 2008) verified the averaged Frankl’s property whenever n ≥ 2 ^m  − 2 ^m/2 and m ≥ 3. The main result of this paper shows that (1) we cannot replace 2 ^m/2 with the upper integer part of 2 ^m /3, and (2) if Frankl’s conjecture is true (at least for m-element base sets) and then the averaged Frankl’s property holds (i.e., 2 ^m/2 can be replaced with the lower integer part of 2 ^m /3). The proof combines elementary facts from combinatorics and lattice theory. The paper is self-contained, and the reader is assumed to be familiar neither with lattices nor with combinatorics. This research was partially supported by the NFSR of Hungary (OTKA), grant no. T 049433, T 48809 and K 60148.     

On the semilocal convergence behavior for Halley’s method

The present paper is concerned with the semilocal convergence problems of Halley’s method for solving nonlinear operator equation in Banach space. Under some so-called majorant conditions, a new semilocal convergence analysis for Halley’s method is presented. This analysis enables us to drop out the assumption of existence of a second root for the majorizing function, but still guarantee Q-cubic convergence rate. Moreover, a new error estimate based on a directional derivative of the twice derivative of the majorizing function is also obtained. This analysis also allows us to obtain two important special cases about the convergence results based on the premises of Kantorovich and Smale types.     

Goldbach’s conjecture in max-algebra

The Goldbach conjecture is one of the best known open problems in number theory. It claims that every even integer greater than 2 can be written as the sum of two primes. The present paper formulates a max-algebraic claim that is equivalent to Goldbach’s conjecture. The max-algebraic analogue allows examination of the conjecture by the methods of max-algebra. A max-algebra is an algebraic structure in which classical addition \(+\) and multiplication \(\times \) are replaced by the operations maximum \(\oplus \) and addition \(\otimes \), in other words \(a\oplus b=\max \{a,b\}\) and \(a\otimes b=a+b\).     

Rényi＇s Problem in Short Intervals

In this paper, we study Rényi＇s problem and other related problems about the additive functions in short intervals. As corollaries, we improve Ivic＇s results.     

On the Waring–Goldbach problem for seventh and higher powers

We apply recent progress on Vinogradov’s mean value theorem to improve bounds for the function H(k) in the Waring–Goldbach problem. We obtain new results for all exponents \(k\ge 7\), and in particular establish that for large k one has

$$\begin{aligned} H(k)\le (4k-2)\log k-(2\log 2-1)k-3. \end{aligned}$$

     

On waring’s problem for cubes

Proceedings - Mathematical Sciences -     

Instability Intervals for Hill’s Equation with Symmetric Single-Well Potential

Ukrainian Mathematical Journal - With the help of an auxiliary eigenvalue problem, we deduce some explicit estimates for the periodic and semiperiodic eigenvalues and the lengths of instability...     

On the Ostrowski’s inequality for Riemann-Stieltjes integral and applications

An Ostrowski type integral inequality for the Riemann-Stieltjes integral ∫_a^b ƒ (t) du (t), where ƒ is assumed to be of bounded variation on [a,b] andu is ofr-H- H?lder type on the same interval, is given. Applications to the approximation problem of the Riemann-Stieltjes integral in terms of Riemann-Stieltjes sums are also pointed out.     

On Kolmogorov’s ε-Entropy for a Compact Set of Infinitely Differentiable Aperiodic Functions (Babenko’s Problem)

The asymptotics of Kolmogorov’s ε-entropy for a compact set of infinitely differentiable aperiodic functions that are boundedly embedded in the space of continuous functions on a finite interval is calculated.