340 research outputs found
Weights and L log L
Let ω(x) be positive locally integrable weight on [0,1]. Discussed are conditions on ω necessary and sufficient for the (dyadic) Hardy-Littlewood maximal function to map L log L(w dx) into L1(ω dx) or into weak L1
A negative mass theorem for surfaces of positive genus
We define the "sum of squares of the wavelengths" of a Riemannian surface
(M,g) to be the regularized trace of the inverse of the Laplacian. We normalize
by scaling and adding a constant, to obtain a "mass", which is scale invariant
and vanishes at the round sphere. This is an anlaog for closed surfaces of the
ADM mass from general relativity. We show that if M has positive genus then on
each conformal class, the mass attains a negative minimum. For the minimizing
metric, there is a sharp logarithmic Hardy-Littlewood-Sobolev inequality and a
Moser-Trudinger-Onofri type inequality.Comment: 8 page
Uniqueness and Nondegeneracy of Ground States for in
We prove uniqueness of ground state solutions for the
nonlinear equation in , where
and for and for . Here denotes the fractional Laplacian
in one dimension. In particular, we generalize (by completely different
techniques) the specific uniqueness result obtained by Amick and Toland for
and in [Acta Math., \textbf{167} (1991), 107--126]. As a
technical key result in this paper, we show that the associated linearized
operator is nondegenerate;
i.\,e., its kernel satisfies .
This result about proves a spectral assumption, which plays a central
role for the stability of solitary waves and blowup analysis for nonlinear
dispersive PDEs with fractional Laplacians, such as the generalized
Benjamin-Ono (BO) and Benjamin-Bona-Mahony (BBM) water wave equations.Comment: 45 page
Asymptotic behavior of solutions to the -Yamabe equation near isolated singularities
-Yamabe equations are conformally invariant equations generalizing
the classical Yamabe equation. In an earlier work YanYan Li proved that an
admissible solution with an isolated singularity at to the
-Yamabe equation is asymptotically radially symmetric. In this work
we prove that an admissible solution with an isolated singularity at to the -Yamabe equation is asymptotic to a radial
solution to the same equation on . These results
generalize earlier pioneering work in this direction on the classical Yamabe
equation by Caffarelli, Gidas, and Spruck. In extending the work of Caffarelli
et al, we formulate and prove a general asymptotic approximation result for
solutions to certain ODEs which include the case for scalar curvature and
curvature cases. An alternative proof is also provided using
analysis of the linearized operators at the radial solutions, along the lines
of approach in a work by Korevaar, Mazzeo, Pacard, and Schoen.Comment: 55 page
The Supremum Norm of the Discrepancy Function: Recent Results and Connections
A great challenge in the analysis of the discrepancy function D_N is to
obtain universal lower bounds on the L-infty norm of D_N in dimensions d \geq
3. It follows from the average case bound of Klaus Roth that the L-infty norm
of D_N is at least (log N) ^{(d-1)/2}. It is conjectured that the L-infty bound
is significantly larger, but the only definitive result is that of Wolfgang
Schmidt in dimension d=2. Partial improvements of the Roth exponent (d-1)/2 in
higher dimensions have been established by the authors and Armen Vagharshakyan.
We survey these results, the underlying methods, and some of their connections
to other subjects in probability, approximation theory, and analysis.Comment: 15 pages, 3 Figures. Reports on talks presented by the authors at the
10th international conference on Monte Carlo and Quasi-Monte Carlo Methods in
Scientific Computing, Sydney Australia, February 2011. v2: Comments of the
referee are incorporate
Spin-current modulation and square-wave transmission through periodically stubbed electron waveguides
Ballistic spin transport through waveguides, with symmetric or asymmetric
double stubs attached to them periodically, is studied systematically in the
presence of a weak spin-orbit coupling that makes the electrons precess. By an
appropriate choice of the waveguide length and of the stub parameters injected
spin-polarized electrons can be blocked completely and the transmission shows a
periodic and nearly square-type behavior, with values 1 and 0, with wide gaps
when only one mode is allowed to propagate in the waveguide. A similar behavior
is possible for a certain range of the stub parameters even when two-modes can
propagate in the waveguide and the conductance is doubled. Such a structure is
a good candidate for establishing a realistic spin transistor. A further
modulation of the spin current can be achieved by inserting defects in a
finite-number stub superlattice. Finite-temperature effects on the spin
conductance are also considered.Comment: 19 pages, 8 figure
Sharp Trace Hardy-Sobolev-Maz'ya Inequalities and the Fractional Laplacian
In this work we establish trace Hardy and trace Hardy-Sobolev-Maz'ya
inequalities with best Hardy constants, for domains satisfying suitable
geometric assumptions such as mean convexity or convexity. We then use them to
produce fractional Hardy-Sobolev-Maz'ya inequalities with best Hardy constants
for various fractional Laplacians. In the case where the domain is the half
space our results cover the full range of the exponent of the
fractional Laplacians. We answer in particular an open problem raised by Frank
and Seiringer \cite{FS}.Comment: 42 page
Finite SU(N)^k Unification
We consider N=1 supersymmetric gauge theories based on the group SU(N)_1 x
SU(N)_2 x ... x SU(N)_k with matter content (N,N*,1,...,1) + (1,N,N*,...,1) +
>... + (N*,1,1,...,N) as candidates for the unification symmetry of all
particles. In particular we examine to which extent such theories can become
finite and we find that a necessary condition is that there should be exactly
three families. We discuss further some phenomenological issues related to the
cases (N,k) = (3,3), (3,4), and (4,3), in an attempt to choose those theories
that can become also realistic. Thus we are naturally led to consider the
SU(3)^3 model which we first promote to an all-loop finite theory and then we
study its additional predictions concerning the top quark mass, Higgs mass and
supersymmetric spectrum.Comment: 15 page
Smooth metric measure spaces, quasi-Einstein metrics, and tractors
We introduce the tractor formalism from conformal geometry to the study of
smooth metric measure spaces. In particular, this gives rise to a
correspondence between quasi-Einstein metrics and parallel sections of certain
tractor bundles. We use this formulation to give a sharp upper bound on the
dimension of the vector space of quasi-Einstein metrics, providing a different
perspective on some recent results of He, Petersen and Wylie.Comment: 33 pages; final versio
Mass spectra of doubly heavy Omega_QQ' baryons
We evaluate the masses of baryons composed of two heavy quarks and a strange
quark with account for spin-dependent splittings in the framework of potential
model with the KKO potential motivated by QCD with a three-loop beta-function
for the effective charge consistent with both the perturbative limit at short
distances and linear confinement term at long distances between the quarks. The
factorization of dynamics is supposed and explored in the nonrelativistic
Schroedinger equation for the motion in the system of two heavy quarks
constituting the doubly heavy diquark and the strange quark interaction with
the diquark. The limits of approach, its justification and uncertainties are
discussed. Excited quasistable states are classified by the quantum numbers of
heavy diquark composed by the heavy quarks of the same flavor.Comment: 14 pages, revtex4-file, 3 eps-figures, 5 tables, typos correcte
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