On the spectrum of the Page and the Chen–LeBrun–Weber metrics

We give bounds on the first non-zero eigenvalue of the scalar Laplacian for both the Page and the Chen–LeBrun–Weber Einstein metrics. One notable feature is that these bounds are obtained without explicit knowledge of the metrics or numerical approximation to them. Our method also allows the estimation of the invariant part of the spectrum for both metrics. We go on to discuss an application of these bounds to the linear stability of the metrics. We also give numerical evidence to suggest that the bounds for both metrics are extremely close to the actual eigenvalue.     

On the Lazarev–Lieb extension of the Hobby–Rice theorem

O. Lazarev and E.H. Lieb proved that, given _f1,…,_fn∈L¹([0,1];C)$f_1,\dots ,f_n\in L^1([0,1];\mathbb{C})$, there exists a smooth function Φ$\Phi$ that takes values on the unit circle and annihilates span{_f1,…,_fn}$\text{span}\{f_1,\dots ,f_n\}$. We give an alternative proof of that fact that also shows the W^1,1$W^{1,1}$ norm of Φ$\Phi$ can be bounded by 5πn+1$5\pi n+1$. Answering a question raised by Lazarev and Lieb, we show that if p>1$p>1$ then there is no bound for the W^1,p$W^{1,p}$ norm of any such multiplier in terms of the norms of _f1,…,_fn$f_1,\dots ,f_n$.     

Proof of the Andrews–Dyson–Rhoades conjecture on the spt-crank

The spt-crank of a vector partition, or an S  -partition, was introduced by Andrews, Garvan and Liang. Let _NS(m,n)$N_S(m,n)$ denote the net number of S-partitions of n with spt-crank m, that is, the number of S  -partitions (_π1,_π2,_π3)$({\pi }_1,{\pi }_2,{\pi }_3)$ of n with spt-crank m   such that the length of _π1${\pi }_1$ is odd minus the number of S  -partitions (_π1,_π2,_π3)$({\pi }_1,{\pi }_2,{\pi }_3)$ of n with spt-crank m   such that the length of _π1${\pi }_1$ is even. Andrews, Dyson and Rhoades conjectured that _{NS(m,n)}m${\{N_S(m,n)\}}_m$ is unimodal for any n  , and they showed that this conjecture is equivalent to an inequality between the rank and crank of ordinary partitions. They obtained an asymptotic formula for the difference between the rank and crank of ordinary partitions, which implies _NS(m,n)≥_NS(m+1,n)$N_S(m,n)\ge N_S(m+1,n)$ for sufficiently large n and fixed m. In this paper, we introduce a representation of an ordinary partition, called the m-Durfee rectangle symbol, which is a rectangular generalization of the Durfee symbol introduced by Andrews. We give a proof of the conjecture of Andrews, Dyson and Rhoades. We also show that this conjecture implies an inequality between the positive rank and crank moments obtained by Andrews, Chan and Kim.     

An explicit form of the solution of the Cauchy problem for the Barenblatt–Zheltov–Kochina equation

A solution of the Cauchy problem in an isotropic medium and in an anisotropic medium with clearly expressed vertical or horizontal permeability is constructed for the Barenblatt–Zheltov–Kochina model representation of the seepage of a fluid in fissured porous rock by reducing the seepage problems under consideration to solving an abstract Cauchy problem in a Banach space.     

Inversion of the Cauchy–Bunyakovskii–Schwarz inequality

In the present paper, the inequality inverse to the Cauchy–Bunyakovskii–Schwarz inequality and generalizing other well-known inversions of this inequality is proved.     

Arithmetics of the numbers of orbits of the Fermat–Euler dynamical systems

Given positive integers a and n with (a,n)=1, we consider the Fermat–Euler dynamical system defined by the multiplication by a acting on the set of residues modulo n relatively prime to n. Given an integer M>1, the integers n for which the number of orbits of this dynamical system is a multiple of M form an ideal in the multiplicative semigroup of odd integers. We provide new results on the arithmetical properties of these ideals by using the topological properties of some directed graphs (the monads).      

Analyticity of the Cox–Ingersoll–Ross semigroup

Positivity - We study the analyticity of the Cox–Ingersoll–Ross semigroup generated by $$\begin{aligned} A_r u=\nu ^2 xu''+\gamma u'+\beta x u'-rx u, \end{aligned}$$ in...     

On the vanishing of the Lannes–Zarati homomorphism

The conjecture on spherical classes states that the Hopf invariant one and the Kervaire invariant one classes are the only elements in H_?(Q₀S⁰)$H_{?}(Q_0{S}^0)$ belonging to the image of the Hurewicz homomorphism. The Lannes–Zarati homomorphism is a map that corresponds to an associated graded (with a certain filtration) of the Hurewicz map. The algebraic version of the conjecture predicts that the s  -th Lannes–Zarati homomorphism vanishes in any positive stems for s>2$s>2$. In the article, we prove the conjecture for the fifth Lannes–Zarati homomorphism.     

On the zeros of the Davenport–Heilbronn function

Let N ₀(T) be the number of zeros of the Davenport–Heilbronn function in the interval [1/2, 1/2+ i T]. It is proved that N ₀(T) ? T (ln T)^1/2+1/16?ε, where ε is an arbitrarily small positive number.     

On the inconsistency of the Malmquist–Luenberger index

Apart from the well-known weaknesses of the standard Malmquist productivity index related to infeasibility and not accounting for slacks, already addressed in the literature, we identify a new and significant drawback of the Malmquist–Luenberger index decomposition that questions its validity as an empirical tool for environmental productivity measurement associated with the production of bad outputs. In particular, we show that the usual interpretation of the technical change component in terms of production frontier shifts can be inconsistent with its numerical value, thereby resulting in an erroneous interpretation of this component that passes on to the index itself. We illustrate this issue with a simple numerical example. Finally, we propose a solution for this inconsistency issue based on incorporating a new postulate for the technology related to the production of bad outputs.     

On the Dual of the Procesi–Formanek Lattice

Let G be the symmetric group on n letters. Procesi and Formanek have shown that C _n , the center of the generic division algebra of degree n defined over a field F, is stably isomorphic to F(B_n)^GF(B_{n})^{G} where B _n is a specific ZG-lattice. We refer to B _n as the Procesi–Formanek lattice. The question of the stable rationality of C _n is a long standing problem for which few results are known. Let F be an algebraically closed field of characteristic zero, let p be an odd prime, and let B_p^*=Hom_Z(B_p,Z)B_{p}^{*}=Hom_{Z}(B_{p},Z) be the dual of the Procesi–Formanek lattice. We show that F(B_p^*)^GF(B_{p}^{*})^{G} is stably rational over F. An interesting question is whether there exists a connection between C _p and F(B_p^*)^GF(B_{p}^{*})^{G}.     

Global well-posedness of the velocity–vorticity-Voigt model of the 3D Navier–Stokes equations

The velocity–vorticity formulation of the 3D Navier–Stokes equations was recently found to give excellent numerical results for flows with strong rotation. In this work, we propose a new regularization of the 3D Navier–Stokes equations, which we call the 3D velocity–vorticity-Voigt (VVV) model, with a Voigt regularization term added to momentum equation in velocity–vorticity form, but with no regularizing term in the vorticity equation. We prove global well-posedness and regularity of this model under periodic boundary conditions. We prove convergence of the model's velocity and vorticity to their counterparts in the 3D Navier–Stokes equations as the Voigt modeling parameter tends to zero. We prove that the curl of the model's velocity converges to the model vorticity (which is solved for directly), as the Voigt modeling parameter tends to zero. Finally, we provide a criterion for finite-time blow-up of the 3D Navier–Stokes equations based on this inviscid regularization.     

Orbital stability of the sum of N peakons for the Dullin–Gottwald–Holm equation

Analogous to the Camassa–Holm equation, the Dullin–Gottwald–Holm equation also possesses peaked solitary waves, which are called peakons. We prove in this paper the stability of ordered trains of peakons for the Dullin–Gottwald–Holm equation.     

On the Uniformity of Distribution of the Naor–Reingold Pseudo-Random Function

We show that the new pseudo-random number function, introduced recently by M. Naor and O. Reingold, possesses one more attractive and useful property. Namely, it is proved that for almost all values of parameters it produces a uniformly distributed sequence. The proof is based on some recent bounds of character sums with exponential functions.     

A particle approximation of the solution of the Kushner–Stratonovitch equation

We construct a sequence of branching particle systems α _n convergent in measure to the solution of the Kushner–Stratonovitch equation. The algorithm based on this result can be used to solve numerically the filtering problem. We prove that the rate of convergence of the algorithm is of order n ^?. This paper is the third in a sequence, and represents the most efficient algorithm we have identified so far. Received: 4 February 1997 / Revised version: 26 October 1998