On the Alexander-Hirschowitz theorem

The Alexander-Hirschowitz theorem says that a general collection of k double points in imposes independent conditions on homogeneous polynomials of degree d with a well-known list of exceptions. Alexander and Hirschowitz completed its proof in 1995, solving a long standing classical problem, connected with the Waring problem for polynomials. We expose a self-contained proof based mainly on the previous works by Terracini, Hirschowitz, Alexander and Chandler, with a few simplifications. We claim originality only in the case d=3, where our proof is shorter. We end with an account of the history of the work on this problem.     

A New Proof for the Equivalence of Weak and Viscosity Solutions for the p-Laplace Equation

In this paper, we give a new proof for the fact that the distributional weak solutions and the viscosity solutions of the p-Laplace equation ?div(|Du| ^p?2 Du) = 0 coincide. Our proof is more direct and transparent than the original proof of Juutinen et al. [8 Juutinen , P. , Lindqvist , P. , Manfredi , J.J. ( 2001 ). On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation . SIAM J. Math. Anal. 33 : 699 – 717 .[Crossref], [Web of Science ®] , [Google Scholar]], which relied on the full uniqueness machinery of the theory of viscosity solutions. We establish a similar result also for the solutions of the non-homogeneous version of the p-Laplace equation.     

A simple and direct derivation for the number of noncrossing partitions

Kreweras considered the problem of counting noncrossing partitions of the set , whose elements are arranged into a cycle in its natural order, into parts of given sizes (but not specifying which part gets which size). He gave a beautiful and surprising result whose proof resorts to a recurrence relation. In this paper we give a direct, entirely bijective, proof starting from the same initial idea as Kreweras' proof.

     

Ramanujan-Fourier series and the conjecture D of Hardy and Littlewood

We give a heuristic proof of a conjecture of Hardy and Littlewood concerning the density of prime pairs to which twin primes and Sophie Germain primes are special cases. The method uses the Ramanujan-Fourier series for a modified von Mangoldt function and the Wiener-Khintchine theorem for arithmetical functions. The failing of the heuristic proof is due to the lack of justification of interchange of certain limits. Experimental evidence using computer calculations is provided for the plausibility of the result. We have also shown that our argument can be extended to the m-tuple conjecture of Hardy and Littlewood.     

On a Special Case of the Herbert Stahl Theorem

The BMV conjecture states that for n ×  n Hermitian matrices A and B the function f_A,B(t) = trace e^tA+B is exponentially convex. Recently the BMV conjecture was proved by Herbert Stahl. The proof of Herbert Stahl is based on ingenious considerations related to Riemann surfaces of algebraic functions. In the present paper we give a purely “matrix” proof of the BMV conjecture for the special case rank A = 1. This proof is based on the Lie product formula for the exponential of the sum of two matrices and does not require complex analysis.     

An elementary proof of the prime number theorem with a remainder term

An elementary proof of the prime number theorem in the form $$\psi (x) - x = O(x exp\{ - (\log x)^{\tfrac{1}{7}} (log log x)^{ - 2} \} )$$ is given. The proof uses a generalization of Selberg's formula and a tauberian argument.     

On the distribution of the length of the longest increasing subsequence of random permutations

The authors consider the length, , of the longest increasing subsequence of a random permutation of numbers. The main result in this paper is a proof that the distribution function for , suitably centered and scaled, converges to the Tracy-Widom distribution of the largest eigenvalue of a random GUE matrix. The authors also prove convergence of moments. The proof is based on the steepest descent method for Riemann-Hilbert problems, introduced by Deift and Zhou in 1993 in the context of integrable systems. The applicability of the Riemann-Hilbert technique depends, in turn, on the determinantal formula of Gessel for the Poissonization of the distribution function of .

     

A note on the Kobayashi-Royden metric for real ellipsoids

We give a proof of the fact proven by L.D. Kay that the Kobayashi-Royden metric of a real ellipsoid (of dimension at least ) at is hermitian exactly when the ellipsoid is the ball. The proof given by us is much simpler and shorter than that of Kay although it is based on the same results.

     

An elementary proof of the irrationality of Tschakaloff series

We present a new proof of the irrationality of values of the series in both qualitative and quantitative forms. The proof is based on a hypergeometric construction of rational approximations to T _q (z). __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 6, pp. 59–64, 2005.     

On the Baxter functional equation

Summary A new shorter proof is given for the Theorem of P. Volkmann and H. Weigel determining the continuous solutionsf:R R of the Baxter functional equationf(f(x)y + f(y)x – xy) = f(x)f(y). The proof is based on the well known theorem of J. Aczél describing the continuous, associative, and cancellative binary operations on a real interval.     

On the rigidity theorem for elliptic genera

We give a detailed proof of the rigidity theorem for elliptic genera. Using the Lefschetz fixed point formula we carefully analyze the relation between the characteristic power series defining the elliptic genera and the equivariant elliptic genera. We show that equivariant elliptic genera converge to Jacobi functions which are holomorphic. This implies the rigidity of elliptic genera. Our approach can be easily modified to give a proof of the rigidity theorem for the elliptic genera of level .

     

Dehn surgery on the figure 8 knot: Immersed surfaces

It is known that about 70% of surgeries on the figure 8 knot give manifolds which contain immersed incompressible surfaces. We improve this to about 80% by giving a very simple proof that all even surgeries give manifolds containing such a surface. Moreover, we give a quick proof that every surgery is virtually Haken, thereby partially dealing with some exceptional cases in Baker's results.

     

The Cremmer-Gervais solution of the Yang-Baxter equation

A direct proof is given of the fact that the Cremmer-Gervais -matrix satisfies the (Quantum) Yang-Baxter equation

     

Fixed points and powers of self-maps of -spaces

We give a new proof of the following result due to Duan: Let be any self-map of a finite connected -space. Then has a fixed point if . Our proof is based on an approach due to Steve Halperin, whereby the Lefschetz number of a self-map is expressed in terms of the eigenvalues of the induced homomorphism of rational homotopy groups. This allows us to give a considerably shorter proof which avoids most of the technicalities of the original proof.

     

Zeros of the Riemann zeta-function

A proof that the Riemann zeta-function (+ it) has no zeros in the region where R=9.65 and T=12.Translated from Matematicheskie Zametki, Vol. 8, No. 4, pp. 419–429, October, 1970.     

On analyticity rate estimates of the solutions to the Navier-Stokes equations in Bessel-potential spaces

On this paper spatial analyticity of solutions to the nonstationary incompressive Navier-Stokes flow in is established. The proof is based on the estimates for the higher order derivatives of solutions. These estimates imply not only the regularizing rates near t=0 but also decaying rates at t→∞, as long as the solution exists. Although basic strategy is similar to our previous work with Giga for Ln space, one can make the proof short using several tools from harmonic analysis.     

On the classifying space of Artin monoids

A theorem proved by Dobrinskaya [9 Dobrinskaya, N. E. (2006). Configuration spaces of labeled particles and finite Eilenberg-MacLane complexes. Proc. Steklov Inst. Math. 252(1):30–46.[Crossref] , [Google Scholar]] shows that there is a strong connection between the K(π,1) conjecture for Artin groups and the classifying spaces of Artin monoids. More recently Ozornova obtained a different proof of Dobrinskaya’s theorem based on the application of discrete Morse theory to the standard CW model of the classifying space of an Artin monoid. In Ozornova’s work, there are hints at some deeper connections between the above-mentioned CW model and the Salvetti complex, a CW complex which arises in the combinatorial study of Artin groups. In this work we show that such connections actually exist, and as a consequence, we derive yet another proof of Dobrinskaya’s theorem.     

On 4-transitivity in the Moufang plane

The purpose of this paper is to give a short proof of 4-transitivity in Moufang planes. This proof originated in the observation of the two first name authors that the standard Moufang identities, together with the identity (1) x^–1(y(xz)) = (x^–1(yx)z, which is asserted in [2, p. 103] to hold in Cayley-Dickson division algebras, can be applied to give a particularly simple algebraic proof of the fact that the collineation group of a Moufang plane is transitive on four-points. Unfortunately, as pointed out by H. Karzel and demonstrated here in Proposition 1, (1) does not hold in Cayley-Dickson algebras. Nevertheless, the algebraic proof of transitivity remains valid after slight modifications and is given here as Theorem 1.The authors wish to thank Professor Karzel for pointing out the error in [2] and for his suggestions in preparing the final version of this paper.Dedicated to Professor H. Karzel on the occasion of his 60th birthday.     

Orthogonal Invariance of the Dirac Operator and the Critical Index of Subharmonicity for Octonionic Analytic Functions

We show that the Dirac operator ${D = \sum_{0}^{7} e_k \frac{\partial}{\partial _{{x}_k}}}$ is orthogonal invariant. As an application, we give a new proof of the theorem in [7].     

One more simple proof of the Craig–Sakamoto theorem

We give one more elementary proof of the Craig-Sakamoto’s theorem: given such that ; then AB=0.