On sums of seven cubes

We show that every integer between 1290741 and is a sum of 5 nonnegative cubes, from which we deduce that every integer which is a cubic residue modulo 9 and an invertible cubic residue modulo 37 is a sum of 7 nonnegative cubes.

     

On sums and products of combinatorial cubes

We obtain several lower bounds on the product set of combinatorial cubes, as well as some non–trivial upper estimates for the multiplicative energy of such sets.     

Slim exceptional set for sums of two squares,two cubes,and two biquadrates of primes

We prove that, with at most O\left(N^{\frac{\mathrm{17}}{\mathrm{192}}+\epsilon }\right) exceptions, all even positive integers up to Nare expressible in the form p_{\mathrm{1}}^{\mathrm{2}}+p_{\mathrm{2}}^{\mathrm{2}}+p_{\mathrm{3}}^{\mathrm{3}}+p_{\mathrm{4}}^{\mathrm{3}}+p_{\mathrm{5}}^{\mathrm{4}}+p_{\mathrm{6}}^{\mathrm{4}},where p_{\mathrm{1}},\mathrm{}{p}_{\mathrm{2}},.\mathrm{}.\mathrm{}.\mathrm{},\mathrm{}{p}_{\mathrm{6}} are prime numbers. This gives large improvement of a recent result O\left(N^{\frac{\mathrm{13}}{\mathrm{16}}+\epsilon }\right) due to M. Zhang and J. J. Li.     

On sums of two coprimek-th powers

Under the assumption of the Riemann Hypothesis, an asymptotic formula with a sharp error term is established for _{nx k} (n), where _k (n) denotes the number of ways to writen as a sum of twok-th powers of coprime positive integers (k3).     

On sums of two counter-monotonic risks

In risk management, capital requirements are most often based on risk measurements of the aggregation of individual risks treated as random variables. The dependence structure between such random variables has a strong impact on the behavior of the aggregate loss. One finds an extensive literature on the study of the sum of comonotonic risks but less, in comparison, has been done regarding the sum of counter-monotonic risks. A crucial result for comonotonic risks is that the Value-at-risk and the Tail Value-at-risk of their sum correspond respectively to the sum of the Value-at-risk and Tail Value-at-risk of the individual risks. In this paper, our main objective is to derive such simple results for the sum of counter-monotonic risks. To do so, we examine separately different contexts in the class of bivariate strictly continuous distributions for which we obtain closed-form expressions for the Value-at-risk and Tail Value-at-risk of the sum of two counter-monotonic risks. The expressions for the subadditive Tail Value-at risk allow us to quantify the maximal diversification benefit. Also, our findings allow us to analyze the tail of the distribution of the sum of two identically subexponentially distributed counter-monotonic random variables.     

Density of the sums of four cubes of primes

It is conjectured that all sufficiently large integers satisfying some necessary congruence conditions are the sum of four cubes of primes. In this paper, it is proved that the conjecture is true for at least 2.911% of the positive integers satisfying the necessary conditions. This improves the result 1.5% due to Ren (2003) [8].     

几乎相等的三次华林-哥德巴赫问题的例外集

本文证明了对5≤s≤8,几乎所有的满足某些同余条件的正整数N都可以表示为N=p31+···+p3s,|pi-(N/s)1/3|≤N1/3-θs,其中θ5=7261-2ε,θ6=5159-ε,θ7=11333-ε,θ8=19561-ε.     

On two supercongruences of double binomial sums

Periodica Mathematica Hungarica - In this note we confirm two conjectural supercongruences of double sums of binomial coefficients due to El Bachraoui.     

On Cantor cubes

Some decision making models are discussed from the point of view of neurophysiology and quantum mechanics. The main feature of these models is that a straight line segment is replaced by the Cantor set. In this direction, many interesting results have been obtained by methods of number theory, p-adic analysis, and the theory of dynamical systems. Some generalizations of existing models are also discussed, which are formulated in terms of the so-called Cantor cubes, that is, Cartesian products of infinitely many standard two-point spaces D (as is known, the Cantor cube $D^{\aleph _0 }$ is homeomorphic to the Cantor set). This approach involves difficulties caused by the nonmetrizability and nonseparability of the Cantor cubes D ^m for m > ?₀ and nonseparable for m > c, respectively.     

The probability that two random integers are coprime

An equivalence is proven between the Riemann hypothesis and the speed of convergence to of the probability that two independent random variables following the same geometric distribution are coprime integers, when the parameter of the distribution goes to 0.     

On the structure of rings which are sums of two subrings

We study the structure of rings that are sums of two subrings one of which is nil and the other reduced. Our main results concern the problem whether in this situation the nil subring must be an ideal.Received: 13 July 2001     

On sums of sums involving cube-full numbers

The Ramanujan Journal - Let $$f_{3}$$ denote the characteristic function of cube-full numbers, and let (n, q) be the greatest common divisor of positive integers n and q. For any positive...     

On the wedge product and coprime coalgebras

The wedge product of subcoalgebras of a coalgebra can be used to define coprime coalgebras. On the other hand, coprime elements in the big lattice of preradicals in module categories also lead to the definition of coprime modules. Considering a coalgebra C as a module over its dual algebra C*, this yields another notion of coprimeness for coalgebras. Under special conditions, the two definitions coincide. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 2, pp. 45–49, 2005.