62 research outputs found
Geometric Path Integrals. A Language for Multiscale Biology and Systems Robustness
In this paper we suggest that, under suitable conditions, supervised learning
can provide the basis to formulate at the microscopic level quantitative
questions on the phenotype structure of multicellular organisms. The problem of
explaining the robustness of the phenotype structure is rephrased as a real
geometrical problem on a fixed domain. We further suggest a generalization of
path integrals that reduces the problem of deciding whether a given molecular
network can generate specific phenotypes to a numerical property of a
robustness function with complex output, for which we give heuristic
justification. Finally, we use our formalism to interpret a pointedly
quantitative developmental biology problem on the allowed number of pairs of
legs in centipedes
Pseudodifferential multi-product representation of the solution operator of a parabolic equation
By using a time slicing procedure, we represent the solution operator of a
second-order parabolic pseudodifferential equation on as an infinite
product of zero-order pseudodifferential operators. A similar representation
formula is proven for parabolic differential equations on a compact Riemannian
manifold. Each operator in the multi-product is given by a simple explicit
Ansatz. The proof is based on an effective use of the Weyl calculus and the
Fefferman-Phong inequality.Comment: Comm. Partial Differential Equations to appear (2009) 28 page
Boundedness of Pseudodifferential Operators on Banach Function Spaces
We show that if the Hardy-Littlewood maximal operator is bounded on a
separable Banach function space and on its associate space
, then a pseudodifferential operator
is bounded on whenever the symbol belongs to the
H\"ormander class with ,
or to the the Miyachi class
with ,
. This result is applied to the case of
variable Lebesgue spaces .Comment: To appear in a special volume of Operator Theory: Advances and
Applications dedicated to Ant\'onio Ferreira dos Santo
A time-frequency analysis perspective on Feynman path integrals
The purpose of this expository paper is to highlight the starring role of
time-frequency analysis techniques in some recent contributions concerning the
mathematical theory of Feynman path integrals. We hope to draw the interest of
mathematicians working in time-frequency analysis on this topic, as well as to
illustrate the benefits of this fruitful interplay for people working on path
integrals.Comment: 26 page
Finite Temperature Casimir Effect for a Massless Fractional Klein-Gordon field with Fractional Neumann Conditions
This paper studies the Casimir effect due to fractional massless Klein-Gordon
field confined to parallel plates. A new kind of boundary condition called
fractional Neumann condition which involves vanishing fractional derivatives of
the field is introduced. The fractional Neumann condition allows the
interpolation of Dirichlet and Neumann conditions imposed on the two plates.
There exists a transition value in the difference between the orders of the
fractional Neumann conditions for which the Casimir force changes from
attractive to repulsive. Low and high temperature limits of Casimir energy and
pressure are obtained. For sufficiently high temperature, these quantities are
dominated by terms independent of the boundary conditions. Finally, validity of
the temperature inversion symmetry for various boundary conditions is
discussed.Comment: 31 page
Regularizing effect and local existence for non-cutoff Boltzmann equation
The Boltzmann equation without Grad's angular cutoff assumption is believed
to have regularizing effect on the solution because of the non-integrable
angular singularity of the cross-section. However, even though so far this has
been justified satisfactorily for the spatially homogeneous Boltzmann equation,
it is still basically unsolved for the spatially inhomogeneous Boltzmann
equation. In this paper, by sharpening the coercivity and upper bound estimates
for the collision operator, establishing the hypo-ellipticity of the Boltzmann
operator based on a generalized version of the uncertainty principle, and
analyzing the commutators between the collision operator and some weighted
pseudo differential operators, we prove the regularizing effect in all (time,
space and velocity) variables on solutions when some mild regularity is imposed
on these solutions. For completeness, we also show that when the initial data
has this mild regularity and Maxwellian type decay in velocity variable, there
exists a unique local solution with the same regularity, so that this solution
enjoys the regularity for positive time
Structural resolvent estimates and derivative nonlinear Schrodinger equations
A refinement of uniform resolvent estimate is given and several smoothing
estimates for Schrodinger equations in the critical case are induced from it.
The relation between this resolvent estimate and radiation condition is
discussed. As an application of critical smoothing estimates, we show a global
existence results for derivative nonlinear Schrodinger equations.Comment: 21 page
A Conformally Invariant Holographic Two-Point Function on the Berger Sphere
We apply our previous work on Green's functions for the four-dimensional
quaternionic Taub-NUT manifold to obtain a scalar two-point function on the
homogeneously squashed three-sphere (otherwise known as the Berger sphere),
which lies at its conformal infinity. Using basic notions from conformal
geometry and the theory of boundary value problems, in particular the
Dirichlet-to-Robin operator, we establish that our two-point correlation
function is conformally invariant and corresponds to a boundary operator of
conformal dimension one. It is plausible that the methods we use could have
more general applications in an AdS/CFT context.Comment: 1+49 pages, no figures. v2: Several typos correcte
- …