Moment vanishing problem and positivity: Some examples

We consider the problem of vanishing of the momentswith Ω a compact domain in Rn and P(x), q(x) complex polynomials in x∈Ω (MVP). The main stress is on relations of this general vanishing problem to the following conjecture which has been studied recently in Mathieu (1997) [22], Duistermaat and van der Kallen (1998) [17], Zhao (2010) [34] and [35] and in other publications in connection with the vanishing problem for differential operators and with the Jacobian conjecture:

Conjecture A. 

For positive μ ifmk(P,1)=0fork=1,2,… , thenmk(P,q)=0fork?1for any q.     

Some examples of clans

It is shown that every space obtained by identifying exactly two points of a two-cell supports a clan, including the so-called “French Army Hat”. Examples are given of clans on spaces obtained by sewing together ice-cream cones.     

Some examples of historical cryptanalysis

The term “historical cryptanalysis” is introduced and defined as the solving of cryptograms whose keys have been lost or misplaced. Numerous examples of such cryptograms, many undeciphered since their deposit in archives hundreds of years ago, still exist and may contain important historical and literary information. Credit is given to previous investigators in this field, who often performed individual cryptanalytic feats of great brilliance. A new approach of international coordinated research in historical cryptanalysis--bringing together the scholars who discover cryptographic materials and those who can help with their cryptanalysis--is discussed.A few examples of solved cryptograms are presented to demonstrate some of the methods used to encipher and cryptanalyze information. Included are a Papal cipher containing information about the election of the king of Poland in 1573, a German cryptogram written by a participant in a key event of the Counter-Reformation--the choice of the archbishop of Cologne in 1583, a Venetian cipher from 1630 whose key may have been eaten to preserve the cipher's security, a Russian cryptanalyst's worksheet from the time of Catherine the Great, and a military cipher from the American Civil War.     

Some examples of Hermitian surfaces

The Riemannian version of the Goldberg-Sachs theorem says that a compact Einstein Hermitian surface is locally conformal Kähler. In contrast to the compact case, we show that there exists an Einstein Hermitian surface which is not locally conformal Kähler. On the other hand, it is known that on a compact Hermitian surface M ⁴, the zero scalar curvature defect implies that M ⁴ is Kähler. Contrary to the compact case, we show that there exists a non-Kähler Hermitian surface with zero scalar curvature defect.     

Some examples of Tychonoff groups

In this paper we survey the Tychono. property for discrete groups. General results and relations with amenability, with discrete potential theory (random walks), and with the theory of weights and of characters are discussed. We then present a characterization of Tychonoff property for virtually abelian groups, for polycyclic groups, and for linear groups. We also consider the case of metabelian groups, and several examples are given. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 50, Functional Analysis, 2007.     

Some new examples of Gibbs measures

We investigate Gibbs measures on general subshifts. In particular we show the uniqueness of Gibbs measures as equilibrium states and we construct such measures on other spaces than mixing subshifts of finite type or sofic systems.     

Some examples inL p spectral geometry

We construct surfaces \(T_{x_0 }^p \) with volume, first eigenvalue, andL ^p norm of the curvature bounded independent ofx ₀, but whose Cheeger constant tends to 0 asx ₀ tends to infinity.     

Some algebraic examples of Frobenius manifolds

Using analytic methods of finite-gap integration, we construct quasihomogeneous algebraic solutions of the WDVV associativity equations and the nonsemisimple Frobenius manifolds associated with them. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 2, pp. 195–206, May, 2007.     

Some examples of singular fluid flows

We show, by an explicit construction, the existence of solutions to the Stokes system in dimension three that are singular on a fractal set. The singular points are understood in the sense given by Caffarelli, Kohn and Nirenberg [2]. As an application, we show the existence of suitable weak solutions to the Navier-Stokes equations, driven by rough forces, that are singular on a fractal set.     

Some examples related to knot sliceness

It is known that the linking form on the 2-cover of slice knots has a metabolizer. We show that several weaker conditions, or some other conditions related to sliceness, do not imply the existence of a metabolizer. We then show how the Rudolph-Bennequin inequality can be used indirectly to prove that some knots are not slice.