Continuous Wavelets on Compact Manifolds

Let $\bf M$ be a smooth compact oriented Riemannian manifold, and let $\Delta_{\bf M}$ be the Laplace-Beltrami operator on ${\bf M}$. Say 0 \neq f \in \mathcal{S}(\RR^+), and that $f(0) = 0$. For $t > 0$, let $K_t(x,y)$ denote the kernel of $f(t^2 \Delta_{\bf M})$. We show that $K_t$ is well-localized near the diagonal, in the sense that it satisfies estimates akin to those satisfied by the kernel of the convolution operator $f(t^2\Delta)$ on \RR^n. We define continuous ${\cal S}$-wavelets on ${\bf M}$, in such a manner that $K_t(x,y)$ satisfies this definition, because of its localization near the diagonal. Continuous ${\cal S}$-wavelets on ${\bf M}$ are analogous to continuous wavelets on \RR^n in \mathcal{S}(\RR^n). In particular, we are able to characterize the H$\ddot{o}$lder continuous functions on ${\bf M}$ by the size of their continuous ${\mathcal{S}}-$wavelet transforms, for H$\ddot{o}$lder exponents strictly between 0 and 1. If $\bf M$ is the torus \TT^2 or the sphere $S^2$, and $f(s)=se^{-s}$ (the ``Mexican hat'' situation), we obtain two explicit approximate formulas for $K_t$, one to be used when $t$ is large, and one to be used when $t$ is small

Non-hermitian integrable quantum chains and their applications to equilibrium crystal shapes and reaction-diffusion problems

In the first part of this work the free energy of the asymmetric six-vertex model is computed analytically. Using the correspondence between this model and the surface shape of an fcc crystal, it is shown how the non-analytical behavior of the free energy leads to the appearance of singularities (edges) in the crystal facet. A new exponent describing such singularities is found, and its possible experimental measurement is discussed. In the second part, a list of reaction-diffusion processes for two species of molecules on a one-dimensional lattice is presented. The markovian evolution of such systems is shown to be governed by a Master Equation whose Hamiltonian is that of the U_qSU(P/M)-invariant Perk-Schultz chain. (orig.)Available from FIZ Karlsruhe / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

A refined modular approach to the Diophantine equation x2+y2n=z3x^2+y^{2n}=z^3

Let $n$ be a positive integer and consider the Diophantine equation of generalized Fermat type $x^2+y^{2n}=z^3$ in nonzero coprime integer unknowns $x,y,z$. Using methods of modular forms and Galois representations for approaching Diophantine equations, we show that for $n \in \{5, 31\}$ there are no solutions to this equation. Combining this with previously known results, this allows a complete description of all solutions to the Diophantine equation above for $n \leq 10^7$. Finally, we show that there are also no solutions for $n\equiv -1 \pmod{6}

Classical and modular methods applied to Diophantine equations

Deep methods from the theory of elliptic curves and modular forms have been used to prove Fermat's last theorem and solve other Diophantine equations. These so-called modular methods can often benefit from information obtained by other, classical, methods from number theory; and vice versa. In our work we are interested in explicitly solving Diophantine equations, especially generalized Fermat equations. We construct certain families of Frey curves and prove irreducibility results for the Galois representation associated to the p-torsion points of some of these curves for small primes p. This allows us to use modular methods to solve, amongst other equations, the generalized Fermat equations with coefficients 1 and signatures (3,3,5) and (2,10,3). By combining classical arguments from algebraic number theory with modular methods, we solve the generalized Fermat equation with coefficients 1 and signature (2,62,3). Using classical methods, we obtain an algorithm to solve the generalized Fermat equations with signature (2,3,5) and arbitrary nonzero integer coefficients. An algorithm for solving these equations was obtained earlier. However, using our algorithm we are able to prove that the local-to-global principle does not hold for signature (2,3,5). Furthermore, our algorithm allows input obtained by modular methods

On the residue class distribution of the number of prime divisors of an integer

Let Ω(n) denote the number of prime divisors of n counting multiplicity. One can show that for any positive integer m and all j =0, 1, . . . , m−1, we have #{n ≤ x : Ω(n) ≡ j(modm)} = x/m + o(xα), with α = 1. Building on work of Kubota and Yoshida, we show that for m>2 and any j =0, 1, . . . , m − 1, the error term is not o(xα) for any

Klein forms and the generalized superelliptic equation

If F(x; y) 2 Z[x; y] is an irreducible binary form of degree k 3, then a theorem of Darmon and Granville implies that the generalized superelliptic equation F(x; y) = zl has, given an integer l maxf2; 7 kg, at most nitely many solutions in coprime integers x; y and z. In this paper, for large classes of forms of degree k = 3; 4; 6 and 12 (including, heuristically, \most" cubic forms), we extend this to prove a like result, where the parameter l is now taken to be variable. In the case of irreducible cubic forms, this provides the rst examples where such a conclusion has been proven. The method of proof combines classical invariant theory, modular Galois representations, and properties of elliptic curves with isomorphic mod-n Galois representations

Andrew Wiles en de Abelprijs

Op 24 mei 2016 ontving Sir Andrew J. Wiles de Abelprijs ‘for his stunning proof of Fermat’s Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory’. In dit artikel schetsen Sander Dahmen en Arno Kret wat Wiles bewezen heeft en laten zien wat voor prachtig onderzoek er uit zijn werk voortgekomen is

Perfect powers expressible as sums of two fifth or seventh powers

We show that the generalized Fermat equations with signatures (5,5,7), (5,5,19), and (7,7,5) (and unit coefficients) have no non-trivial primitive integer solutions. Assuming GRH, we also prove the nonexistence of non-trivial primitive integer solutions for the signatures (5,5,11), (5,5,13), and (7,7,11). The main ingredients for obtaining our results are descent techniques, the method of Chabauty-Coleman, and the modular approach to Diophantine equations.Comment: The current version incorporates minor comments of the refere