arXiv subjects

Juan B. Gil

Publications and source records attributed to Juan B. Gil.

46 records · Page 3Linked to original sources

arXiv2006

Complete Padovan sequences in finite fields

Given a prime $p\ge 5$, and given $1<κ<p-1$, we call a sequence $(a_n)_{n}$ in $\mathbb{F}_p$ a $Φ_κ$-sequence if it is periodic with period $p-1$, and if it satisfies the linear recurrence $a_n+a_{n+1}=a_{n+κ}$ with $a_0=1$. Such a sequence is said to be a complete $Φ_κ$-sequence if in addition $\{a_0,a_1,...,a_{p-2}\}=\{1,...,p-1\}$. For instance, every primitive root $b$ mod $p$ generates a complete $Φ_κ$-sequence $a_n=b^n$ for some (unique) $κ$. A natural question is whether every complete $Φ_κ$-sequence is necessarily defined by a primitive root. For $κ=2$ the answer is known to be positive. In this paper we reexamine that case and investigate the case $κ=3$ together with the associated cases $κ=p-2$ and $κ=p-3$.

math.NT↗

arXiv2006

Resolvents of cone pseudodifferential operators, asymptotic expansions and applications

We study the structure and asymptotic behavior of the resolvent of elliptic cone pseudodifferential operators acting on weighted Sobolev spaces over a compact manifold with boundary. We obtain an asymptotic expansion of the resolvent as the spectral parameter tends to infinity, and use it to derive corresponding heat trace and zeta function expansions as well as an analytic index formula.

math.SP↗

arXiv2005

On rays of minimal growth for elliptic cone operators

We present an overview of some of our recent results on the existence of rays of minimal growth for elliptic cone operators and two new results concerning the necessity of certain conditions for the existence of such rays.

math.SP↗

arXiv2005

Geometry and spectra of closed extensions of elliptic cone operators

We study the geometry of the set of closed extensions of index 0 of an elliptic cone operator and its model operator in connection with the spectra of the extensions, and give a necessary and sufficient condition for the existence of rays of minimal growth for such operators.

math.AP↗

arXiv2004

Resolvents of elliptic cone operators

We prove the existence of sectors of minimal growth for general closed extensions of elliptic cone operators under natural ellipticity conditions. This is achieved by the construction of a suitable parametrix and reduction to the boundary. Special attention is devoted to the clarification of the analytic structure of the resolvent.

math.AP↗

arXiv2003

Hecke operators on rational functions

We define Hecke operators U_m that sift out every m-th Taylor series coefficient of a rational function in one variable, defined over the reals. We prove several structure theorems concerning the eigenfunctions of these Hecke operators, including the pleasing fact that the point spectrum of the operator U_m is simply the set {+/- m^k, k in N} U {0}. It turns out that the simultaneous eigenfunctions of all of the Hecke operators involve Dirichlet characters mod L, giving rise to the result that any arithmetic function of m that is completely multiplicative and also satisfies a linear recurrence must be a Dirichlet character times a power of m. We also define the notions of level and weight for rational eigenfunctions, by analogy with modular forms, and we show the existence of some interesting finite-dimensional subspaces of rational eigenfunctions (of fixed weight and level), whose union gives all of the rational functions whose coefficients are quasi-polynomials.

math.NT↗

arXiv2002

Full asymptotic expansion of the heat trace for non-self-adjoint elliptic cone operators

The operator $e^{-tA}$ and its trace are investigated in the case when $A$ is a non-self-adjoint elliptic differential operator on a manifold with conical singularities. Under a certain spectral condition (parameter-ellipticity) we obtain a full asymptotic expansion in $t$ of the heat trace as $t\to 0^+$. As in the smooth compact case, the problem is reduced to the investigation of the resolvent $(A-λ)^{-1}$. The main step will consist in approximating this operator family by a parametrix to $A-λ$ using a suitable parameter-dependent calculus.

math.AP↗

arXiv2001

A note on the index of cone differential operators

We prove that the index formula for $b$-elliptic cone differential operators given by M. Lesch holds verbatim for operators whose coefficients are not necessarily independent of the normal variable near the boundary. We also show that, for index purposes, the operators can always be considered on weighted Sobolev spaces.

math.AP↗

arXiv2001

On the Noncommutative Residue and the Heat Trace Expansion on Conic Manifolds

Given a cone pseudodifferential operator $P$ we give a full asymptotic expansion as $t\to 0^+$ of the trace $\Tr Pe^{-tA}$, where $A$ is an elliptic cone differential operator for which the resolvent exists on a suitable region of the complex plane. Our expansion contains $\log t$ and new $(\log t)^2$ terms whose coefficients are given explicitly by means of residue traces. Cone operators are contained in some natural algebras of pseudodifferential operators on which unique trace functionals can be defined. As a consequence of our explicit heat trace expansion, we recover all these trace functionals.

math.SP↗

arXiv2001

Adjoints of elliptic cone operators

We study the adjointness problem for the closed extensions of a general b-elliptic operator A in x^{-ν}Diff^m_b(M;E), ν>0, initially defined as an unbounded operator A:C_c^\infty(M;E)\subset x^μL^2_b(M;E)\to x^μL^2_b(M;E), μ\in \R. The case where A is a symmetric semibounded operator is of particular interest, and we give a complete description of the domain of the Friedrichs extension of such an operator.

math.AP↗