Seeking a proof of Xie’s inequality: analogues for series and Fourier series

Seeking to free the existence and regularity theory for the Navier–Stokes equations from assumptions about the regularity of a fluid’s boundary, we continue efforts of Wenzheng Xie and myself to prove a certain domain independent inequality for solutions of the steady Stokes equations. For the Laplacian, Xie proved an analogue of the desired inequality by using the maximum principle in obtaining an intermediary result. His conjecture that an analogue of this intermediary result is also valid for the Stokes equations remains unproven. My efforts to circumvent the need for it have led, so far, only to further interesting conjectures. Here, we seek to better understand both Xie’s arguments and mine by applying them to simpler problems concerning series and Fourier series. First, a bound is proven for a series of real numbers that can be interpreted as a bound for the sup-norm of a Fourier cosine series, in terms of the \(L^{2}\) -norms of its fractional-order derivatives of orders 1/3 and 2/3. This is generalized to a bound for a weighted sum of a sequence of real numbers. We conjecture that the hypotheses concerning the weights are satisfied by the sequence of numbers \(\{ \sin ny\}\) , for any nonzero \(y\in (-\pi ,\pi )\) . If so, we obtain an inequality for the sup-norm of a Fourier sine series, similar to that for a cosine series. Remarkably, the hypotheses for the weights are analogous to those we have been seeking to verify in trying to prove the original inequality for the Stokes equations. We conclude with a remark showing that Xie’s central argument provides a possibly new, very straightforward, proof of Hölder’s inequality for series.     

Completing Book II of Archimedes’s On Floating Bodies

Conclusion  One need only glance at Archimedes’s Proposition 8 above to see thatOn Floating Bodies is several orders of magnitude more sophisticated than anything else found in ancient mathematics. It ranks with Newton’sPrincipia Mathematica as a work in which basic physical laws are both formulated and accompanied by superb applications.     

On sharp estimates of the convergence of double Fourier–Bessel series

The problem of approximation of a differentiable function of two variables by partial sums of a double Fourier–Bessel series is considered. Sharp estimates of the rate of convergence of the double Fourier–Bessel series on the class of differentiable functions of two variables characterized by a generalized modulus of continuity are obtained. The proofs of four theorems on this issue, which can be directly applied to solving particular problems of mathematical physics, approximation theory, etc., are presented.     

On Oliver’s p-group conjecture: II

Let p be an odd prime and S a finite p-group. B. Oliver’s conjecture arises from an open problem in the theory of p-local finite groups. It is the claim that a certain characteristic subgroup \mathfrakX(S){\mathfrak{X}(S)} of S always contains the Thompson subgroup. In previous work the first two authors and M. Lilienthal recast Oliver’s conjecture as a statement about the representation theory of the factor group S/\mathfrakX(S){S/\mathfrak{X}(S)}. We now verify the conjecture for a wide variety of groups S/\mathfrakX(S){S/\mathfrak{X}(S)}.     

On Thompson’s conjecture for alternating and symmetric groups of degree greater than 1361

Let G be a finite group G, and let N(G) be the set of sizes of its conjugacy classes. It is shown that, if N(G) equals N(Alt _n ) or N(Sym _n ), where n > 1361, then G has a composition factor isomorphic to an alternating group Altm with m ≤ n and the interval (m, n] contains no primes.     

Value distributions of analogues of Kloosterman’s sums

A theorem describing the value distribution of analogues of Kloosterman’s sums is proved. Asymptotic formulas for fractional moments are obtained.     

The evaluation of Tornheim double sums. Part 2

We provide an explicit formula for the Tornheim double series T(a,0,c) in terms of an integral involving the Hurwitz zeta function. For integer values of the parameters, a=m, c=n, we show that in the most interesting case of even weight N:=m+n the Tornheim sum T(m,0,n) can be expressed in terms of zeta values and the family of integrals

On Hilbert’s solution of Waring’s problem

In 1909, Hilbert proved that for each fixed k, there is a number g with the following property: Every integer N ≥ 0 has a representation in the form N = x ₁ ^k + x ₂ ^k + … + x _g ^k , where the x _i are nonnegative integers. This resolved a conjecture of Edward Waring from 1770. Hilbert’s proof is somewhat unsatisfying, in that no method is given for finding a value of g corresponding to a given k. In his doctoral thesis, Rieger showed that by a suitable modification of Hilbert’s proof, one can give explicit bounds on the least permissible value of g. We show how to modify Rieger’s argument, using ideas of F. Dress, to obtain a better explicit bound. While far stronger bounds are available from the powerful Hardy-Littlewood circle method, it seems of some methodological interest to examine how far elementary techniques of this nature can be pushed.     

Some extensions of Brioschi’s double alternant

We evaluate some determinants related to Brioschi’s extension of the classical double alternant of Cauchy, and Scott’s extension of Brioschi’s.     

Thompson’s conjecture for alternating group of degree 22

For a finite group G, it is denoted by N(G) the set of conjugacy class sizes of G. In 1980s, J. G. Thompson posed the following conjecture: if L is a finite nonabelian simple group, G is a finite group with trivial center, and N(G) = N(L), then L and G are isomorphic. In this paper, it is proved that Thompson’s conjecture is true for the alternating group A ₂₂ with connected prime graph.     

Composition analogues of Beck’s conjectures on partitions

In this work, we first present new bijections for the composition analogues of the partition theorems of Euler, Glaisher, and Franklin respectively. Then we generalize one composition analogue of Beck’s conjectures found by Huang recently, and give a purely combinatorial proof applying our bijection. Finally, a new and general composition analogue of Beck’s conjectures is established.     

On Analogues of Heilbronn’s Theorem

Mathematical Notes - Continued fractions with rational partial quotients arise in a natural way in the course of applying any $$k$$ -ary gcd algorithm to the ratio of natural numbers $$a$$ , $$b$$...     

On collapsibilities of Yule’s measure

Simpson’s paradox reminds people that the statistical inference in a low-dimensional space probably distorts the reality in a high one seriously.To study the paradox with respect to Yule’s measure, this paper discusses simple collapsibility, strong collapsibility and consecutive collapsibility, and presents necessary and sufficient conditions of them.In fact, these conditions are of great importance for observational and experimental designs, eliminating confounding bias, categorizing discrete variables and so on.     

On Sears’s general transformation formula for basic hypergeometric series

We present a new proof of a general transformation formula for basic hypergeometric series that was discovered by D.B. Sears (Proc. Lond. Math. Soc. 53(2): 181–191, 1951) and discuss some special cases. Next we apply Sears’s general transformation to give an extension of a result of Andrews et al. (Duke Math. J. 108: 395–419, 2001).     

On generalized Eisenstein series and Ramanujan’s formula for periodic zeta-functions

We construct the asymptotic formulas for solutions of a certain linear second-order delay differential equation as independent variable tends to infinity. When the delay equals zero this equation turns into the so-called one-dimensional Schrödinger equation at energy zero with Wigner–von Neumann type potential. The question of interest is how the behaviour of solutions changes qualitatively and quantitatively when the delay is introduced in this dynamical model. We apply the method of asymptotic integration that is based on the ideas of the centre manifold theory in its presentation with respect to the systems of functional differential equations with oscillatory decreasing coefficients.     

On Berndtsson’s generalization of Prékopa’s theorem

We present a new approach to Berndtssons complex extension of Prékopas theorem. This approach is inspired by the recent local proof of Prékopas theorem obtained by Ball, Barthe and Naor. In the complex setting, this approach leads to a formula expressing in terms of (derivatives of) where e^–(z)= As a consequence, we obtain new conditions ensuring that is subharmonic when is plurisubharmonic.