10,361 research outputs found
The strength of the Weak Lefschetz Property
We study a number of conditions on the Hilbert function of a level artinian
algebra which imply the Weak Lefschetz Property (WLP). Possibly the most
important open case is whether a codimension 3 SI-sequence forces the WLP for
level algebras. In other words, does every codimension 3 Gorenstein algebra
have the WLP? We give some new partial answers to this old question: we prove
an affirmative answer when the initial degree is 2, or when the Hilbert
function is relatively small. Then we give a complete answer to the question of
what is the largest socle degree forcing the WLP.Comment: A few minor corrections; to appear in the Illinois J. Mat
Electron-phonon coupling close to a metal-insulator transition in one dimension
We consider a one-dimensional system of electrons interacting via a
short-range repulsion and coupled to phonons close to the metal-insulator
transition at half filling. We argue that the metal-insulator transition can be
described as a standard one dimensional incommensurate to commensurate
transition, even if the electronic system is coupled to the lattice distortion.
By making use of known results for this transition, we prove that low-momentum
phonons do not play any relevant role close to half-filling, unless their
coupling to the electrons is large in comparison with the other energy scales
present in the problem. In other words the effective strength of the
low-momentum transferred electron-phonon coupling does not increase close to
the metal-insulator transition, even though the effective velocity of the
mobile carriers is strongly diminished.Comment: 20 pages, REVTEX styl
A New Approach to Equations with Memory
In this work, we present a novel approach to the mathematical analysis of
equations with memory based on the notion of a state, namely, the initial
configuration of the system which can be unambiguously determined by the
knowledge of the future dynamics. As a model, we discuss the abstract version
of an equation arising from linear viscoelasticity. It is worth mentioning that
our approach goes back to the heuristic derivation of the state framework,
devised by L.Deseri, M.Fabrizio and M.J.Golden in "The concept of minimal state
in viscoelasticity: new free energies and applications to PDEs", Arch. Ration.
Mech. Anal., vol. 181 (2006) pp.43-96. Starting from their physical
motivations, we develop a suitable functional formulation which, as far as we
know, is completely new.Comment: 39 pages, no figur
On the degree two entry of a Gorenstein -vector and a conjecture of Stanley
In this note we establish a (non-trivial) lower bound on the degree two entry
of a Gorenstein -vector of any given socle degree and any
codimension .
In particular, when , that is for Gorenstein -vectors of the form
, our lower bound allows us to prove a conjecture of Stanley
on the order of magnitude of the minimum value, say , that may
assume. In fact, we show that
In general, we wonder whether our lower bound is sharp for all integers
and .Comment: A few minor changes. To appear in Proc. of the AM
Entire slice regular functions
Entire functions in one complex variable are extremely relevant in several
areas ranging from the study of convolution equations to special functions. An
analog of entire functions in the quaternionic setting can be defined in the
slice regular setting, a framework which includes polynomials and power series
of the quaternionic variable. In the first chapters of this work we introduce
and discuss the algebra and the analysis of slice regular functions. In
addition to offering a self-contained introduction to the theory of
slice-regular functions, these chapters also contain a few new results (for
example we complete the discussion on lower bounds for slice regular functions
initiated with the Ehrenpreis-Malgrange, by adding a brand new Cartan-type
theorem).
The core of the work is Chapter 5, where we study the growth of entire slice
regular functions, and we show how such growth is related to the coefficients
of the power series expansions that these functions have. It should be noted
that the proofs we offer are not simple reconstructions of the holomorphic
case. Indeed, the non-commutative setting creates a series of non-trivial
problems. Also the counting of the zeros is not trivial because of the presence
of spherical zeros which have infinite cardinality. We prove the analog of
Jensen and Carath\'eodory theorems in this setting
Bounds and asymptotic minimal growth for Gorenstein Hilbert functions
We determine new bounds on the entries of Gorenstein Hilbert functions, both
in any fixed codimension and asymptotically.
Our first main theorem is a lower bound for the degree entry of a
Gorenstein -vector, in terms of its entry in degree . This result carries
interesting applications concerning unimodality: indeed, an important
consequence is that, given and , all Gorenstein -vectors of
codimension and socle degree (this function being
explicitly computed) are unimodal up to degree . This immediately gives a
new proof of a theorem of Stanley that all Gorenstein -vectors in
codimension three are unimodal.
Our second main theorem is an asymptotic formula for the least value that the
-th entry of a Gorenstein -vector may assume, in terms of codimension,
, and socle degree, . This theorem broadly generalizes a recent result of
ours, where we proved a conjecture of Stanley predicting that asymptotic value
in the specific case and , as well as a result of Kleinschmidt which
concerned the logarithmic asymptotic behavior in degree .Comment: Several minor changes; to appear in J. Algebr
A characterization of Gorenstein Hilbert functions in codimension four with small initial degree
The main goal of this paper is to characterize the Hilbert functions of all
(artinian) codimension 4 Gorenstein algebras that have at least two independent
relations of degree four. This includes all codimension 4 Gorenstein algebras
whose initial relation is of degree at most 3. Our result shows that those
Hilbert functions are exactly the so-called {\em SI-sequences} starting with
(1,4,h_2,h_3,...), where . In particular, these Hilbert functions
are all unimodal.
We also establish a more general unimodality result, which relies on the
values of the Hilbert function not being too big, but is independent of the
initial degree.Comment: A few changes. Final version, to appear in Math. Res. Let
Sequential multi-photon strategy for semiconductor-based terahertz detectors
A semiconductor-based terahertz-detector strategy, exploiting a
bound-to-bound-to-continuum architecture, is presented and investigated. In
particular, a ladder of equidistant energy levels is employed, whose step is
tuned to the desired detection frequency and allows for sequential multi-photon
absorption. Our theoretical analysis demonstrates that the proposed
multi-subband scheme could represent a promising alternative to conventional
quantum-well infrared photodetectors in the terahertz spectral region.Comment: Submitted to Journal of Applied Physic
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