9,126 research outputs found
Asymptotics of Harish-Chandra expansions, bounded hypergeometric functions associated with root systems, and applications
A series expansion for Heckman-Opdam hypergeometric functions
is obtained for all
As a consequence, estimates for away from the walls of a Weyl
chamber are established. We also characterize the bounded hypergeometric
functions and thus prove an analogue of the celebrated theorem of Helgason and
Johnson on the bounded spherical functions on a Riemannian symmetric space of
the noncompact type. The -theory for the hypergeometric Fourier transform
is developed for . In particular, an inversion formula is proved when
A continuum treatment of growth in biological tissue: The coupling of mass transport and mechanics
Growth (and resorption) of biological tissue is formulated in the continuum
setting. The treatment is macroscopic, rather than cellular or sub-cellular.
Certain assumptions that are central to classical continuum mechanics are
revisited, the theory is reformulated, and consequences for balance laws and
constitutive relations are deduced. The treatment incorporates multiple
species. Sources and fluxes of mass, and terms for momentum and energy transfer
between species are introduced to enhance the classical balance laws. The
transported species include: (\romannumeral 1) a fluid phase, and
(\romannumeral 2) the precursors and byproducts of the reactions that create
and break down tissue. A notable feature is that the full extent of coupling
between mass transport and mechanics emerges from the thermodynamics.
Contributions to fluxes from the concentration gradient, chemical potential
gradient, stress gradient, body force and inertia have not emerged in a unified
fashion from previous formulations of the problem. The present work
demonstrates these effects via a physically-consistent treatment. The presence
of multiple, interacting species requires that the formulation be consistent
with mixture theory. This requirement has far-reaching consequences. A
preliminary numerical example is included to demonstrate some aspects of the
coupled formulation.Comment: 29 pages, 11 figures, accepted for publication in Journal of the
Mechanics and Physics of Solids. See journal for final versio
In silico estimates of the free energy rates in growing tumor spheroids
The physics of solid tumor growth can be considered at three distinct size
scales: the tumor scale, the cell-extracellular matrix (ECM) scale and the
sub-cellular scale. In this paper we consider the tumor scale in the interest
of eventually developing a system-level understanding of the progression of
cancer. At this scale, cell populations and chemical species are best treated
as concentration fields that vary with time and space. The cells have
chemo-mechanical interactions with each other and with the ECM, consume glucose
and oxygen that are transported through the tumor, and create chemical
byproducts. We present a continuum mathematical model for the biochemical
dynamics and mechanics that govern tumor growth. The biochemical dynamics and
mechanics also engender free energy changes that serve as universal measures
for comparison of these processes. Within our mathematical framework we
therefore consider the free energy inequality, which arises from the first and
second laws of thermodynamics. With the model we compute preliminary estimates
of the free energy rates of a growing tumor in its pre-vascular stage by using
currently available data from single cells and multicellular tumor spheroids.Comment: 27 pages with 5 figures and 2 tables. Figures and tables appear at
the end of the pape
Biological remodelling: Stationary energy, configurational change, internal variables and dissipation
Remodelling is defined as an evolution of microstructure or variations in the
configuration of the underlying manifold. The manner in which a biological
tissue and its subsystems remodel their structure is treated in a continuum
mechanical setting. While some examples of remodelling are conveniently
modelled as evolution of the reference configuration (Case I), others are more
suited to an internal variable description (Case II). In this paper we explore
the applicability of stationary energy states to remodelled systems. A
variational treatment is introduced by assuming that stationary energy states
are attained by changes in microstructure via one of the two mechanisms--Cases
I and II. An example is presented to illustrate each case. The example
illustrating Case II is further studied in the context of the thermodynamic
dissipation inequality.Comment: 24 pages, 4 figures. Replaced version has corrections to typos in
equations, and the corresponding correct plot of the solution--all in Section
Domain-wall fermions with dynamical gauge fields
We have carried out a numerical simulation of a domain-wall model in
-dimensions, in the presence of a dynamical gauge field only in an extra
dimension, corresponding to the weak coupling limit of a ( 2-dimensional )
physical gauge coupling. Using a quenched approximation we have investigated
this model at 0.5 ( ``symmetric'' phase),
1.0, and 5.0 (``broken'' phase), where is the gauge coupling constant of
the extra dimension. We have found that there exists a critical value of a
domain-wall mass which separates a region with a fermionic zero
mode on the domain-wall from the one without it, in both symmetric and broken
phases. This result suggests that the domain-wall method may work for the
construction of lattice chiral gauge theories.Comment: 27 pages (11 figures), latex (epsf style-file needed
On the equisummability of Hermite and Fourier expansions
We prove an equisummability result for the Fourier expansions and Hermite expansions as well as special Hermite expansions. We also prove the uniform boundedness of the Bochner-Riesz means associated to the Hermite expansions for polyradial functions
Spontaneous Flavor and Parity Breaking with Wilson Fermions
We discuss the phase diagram of Wilson fermions in the -- plane for
two-flavor QCD. We argue that, as originally suggested by Aoki, there is a
phase in which flavor and parity are spontaneously broken. Recent numerical
results on the spectrum of the overlap Hamiltonian have been interpreted as
evidence against Aoki's conjecture. We show that they are in fact consistent
with the presence of a flavor-parity broken ``Aoki phase''. We also show how,
as the continuum limit is approached, one can study the lattice theory using
the continuum chiral Lagrangian supplemented by additional terms proportional
to powers of the lattice spacing. We find that there are two possible phase
structures at non-zero lattice spacing: (1) there is an Aoki phase of width
with two massless Goldstone pions; (2) there is no
symmetry breaking, and all three pions have an equal non-vanishing mass of
order . Present numerical evidence suggests that the former option is
realized for Wilson fermions. Our analysis then predicts the form of the pion
masses and the flavor-parity breaking condensate within the Aoki phase. Our
analysis also applies for non-perturbatively improved Wilson fermions.Comment: 22 pages, LaTeX, 5 figures (added several references and a comment
- …