224 research outputs found
Geometric Phase Integrals and Irrationality Tests
Let be an analytical, real valued function defined on a compact domain
. We prove that the problem of establishing the
irrationality of evaluated at can be stated with
respect to the convergence of the phase of a suitable integral , defined
on an open, bounded domain, for that goes to infinity. This is derived as a
consequence of a similar equivalence, that establishes the existence of
isolated solutions of systems equations of analytical functions on compact real
domains in , if and only if the phase of a suitable ``geometric''
complex phase integral converges for . We finally
highlight how the method can be easily adapted to be relevant for the study of
the existence of rational or integer points on curves in bounded domains, and
we sketch some potential theoretical developments of the method
Stem-Like Adaptive Aneuploidy and Cancer Quasispecies
We analyze and reinterpret experimental evidence from the literature to argue
for an ability of tumor cells to self-regulate their aneuploidy rate. We
conjecture that this ability is mediated by a diversification factor that
exploits molecular mechanisms common to embryo stem cells and, to a lesser
extent, adult stem cells, that is eventually reactivated in tumor cells.
Moreover, we propose a direct use of the quasispecies model to cancer cells
based on their significant genomic instability (i.e. aneuploidy rate), by
defining master sequences lengths as the sum of all copy numbers of physically
distinct whole and fragmented chromosomes. We compute an approximate error
threshold such that any aneuploidy rate larger than the threshold would lead to
a loss of fitness of a tumor population, and we confirm that highly aneuploid
cancer populations already function with aneuploidy rates close to the
estimated threshold
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
Positivity, rational Schur functions, Blaschke factors, and other related results in the Grassmann algebra
We begin a study of Schur analysis in the setting of the Grassmann algebra,
when the latter is completed with respect to the -norm. We focus on the
rational case. We start with a theorem on invertibility in the completed
algebra, and define a notion of positivity in this setting. We present a series
of applications pertaining to Schur analysis, including a counterpart of the
Schur algorithm, extension of matrices and rational functions. Other topics
considered include Wiener algebra, reproducing kernels Banach modules, and
Blaschke factors.Comment: 35 page
Regular Functions on the Space of Cayley Numbers
In this paper we present a new definition of regularity on the space Ç of Cayley numbers (often referred to as octonions), based on a Gateaux-like notion of derivative. We study the main properties of regular functions, and we develop the basic elements of a function theory on Ç. Particular attention is given to the structure of the zero sets of such functions
Extension results for slice regular functions of a quaternionic variable
In this paper we prove a new representation formula for slice regular
functions, which shows that the value of a slice regular function at a
point can be recovered by the values of at the points and
for any choice of imaginary units This result allows us to
extend the known properties of slice regular functions defined on balls
centered on the real axis to a much larger class of domains, called axially
symmetric domains. We show, in particular, that axially symmetric domains play,
for slice regular functions, the role played by domains of holomorphy for
holomorphic functions
A Sheaf Theoretic Approach to Consciousness
A new fundamental mathematical model of consciousness based on category theory is presented. The model is based on two philosophical-theological assumptions: a) the universe is a sea of consciousness, and b) time is multi-dimensional and non-linear
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