2,596 research outputs found
Universality of slow decorrelation in KPZ growth
There has been much success in describing the limiting spatial fluctuations
of growth models in the Kardar-Parisi-Zhang (KPZ) universality class. A proper
rescaling of time should introduce a non-trivial temporal dimension to these
limiting fluctuations. In one-dimension, the KPZ class has the dynamical
scaling exponent , that means one should find a universal space-time
limiting process under the scaling of time as , space like
and fluctuations like as .
In this paper we provide evidence for this belief. We prove that under
certain hypotheses, growth models display temporal slow decorrelation. That is
to say that in the scalings above, the limiting spatial process for times and are identical, for any . The hypotheses are known
to be satisfied for certain last passage percolation models, the polynuclear
growth model, and the totally / partially asymmetric simple exclusion process.
Using slow decorrelation we may extend known fluctuation limit results to
space-time regions where correlation functions are unknown.
The approach we develop requires the minimal expected hypotheses for slow
decorrelation to hold and provides a simple and intuitive proof which applied
to a wide variety of models.Comment: Exposition improved, typos correcte
Limit processes for TASEP with shocks and rarefaction fans
We consider the totally asymmetric simple exclusion process (TASEP) with
two-sided Bernoulli initial condition, i.e., with left density rho_- and right
density rho_+. We consider the associated height function, whose discrete
gradient is given by the particle occurrences. Macroscopically one has a
deterministic limit shape with a shock or a rarefaction fan depending on the
values of rho_{+/-}. We characterize the large time scaling limit of the
fluctuations as a function of the densities rho_{+/-} and of the different
macroscopic regions. Moreover, using a slow decorrelation phenomena, the
results are extended from fixed time to the whole space-time, except along the
some directions (the characteristic solutions of the related Burgers equation)
where the problem is still open.
On the way to proving the results for TASEP, we obtain the limit processes
for the fluctuations in a class of corner growth processes with external
sources, of equivalently for the last passage time in a directed percolation
model with two-sided boundary conditions. Additionally, we provide analogous
results for eigenvalues of perturbed complex Wishart (sample covariance)
matrices.Comment: 46 pages, 3 figures, LaTeX; Extended explanations in the first two
section
Modeling Course-Based Undergraduate Research Experiences: An Agenda for Future Research and Evaluation
Course-based undergraduate research experiences (CUREs) are being championed as scalable ways of involving undergraduates in science research. Studies of CUREs have shown that participating students achieve many of the same outcomes as students who complete research internships. However, CUREs vary widely in their design and implementation, and aspects of CUREs that are necessary and sufficient to achieve desired student outcomes have not been elucidated. To guide future research aimed at understanding the causal mechanisms underlying CURE efficacy, we used a systems approach to generate pathway models representing hypotheses of how CURE outcomes are achieved. We started by reviewing studies of CUREs and research internships to generate a comprehensive set of outcomes of research experiences, determining the level of evidence supporting each outcome. We then used this body of research and drew from learning theory to hypothesize connections between what students do during CUREs and the outcomes that have the best empirical support. We offer these models as hypotheses for the CURE community to test, revise, elaborate, or refute. We also cite instruments that are ready to use in CURE assessment and note gaps for which instruments need to be developed.Howard Hughes Medical InstituteScience and Mathematics Educatio
Height fluctuations for the stationary KPZ equation
We compute the one-point probability distribution for the stationary KPZ
equation (i.e. initial data H(0,X)=B(X), for B(X) a two-sided standard Brownian
motion) and show that as time T goes to infinity, the fluctuations of the
height function H(T,X) grow like T^{1/3} and converge to those previously
encountered in the study of the stationary totally asymmetric simple exclusion
process, polynuclear growth model and last passage percolation.
The starting point for this work is our derivation of a Fredholm determinant
formula for Macdonald processes which degenerates to a corresponding formula
for Whittaker processes. We relate this to a polymer model which mixes the
semi-discrete and log-gamma random polymers. A special case of this model has a
limit to the KPZ equation with initial data given by a two-sided Brownian
motion with drift beta to the left of the origin and b to the right of the
origin. The Fredholm determinant has a limit for beta>b, and the case where
beta=b (corresponding to the stationary initial data) follows from an analytic
continuation argument.Comment: 91 pages, 8 figure
Extensions of Johnson's and Morita's homomorphisms that map to finitely generated abelian groups
We extend each higher Johnson homomorphism to a crossed homomorphism from the
automorphism group of a finite-rank free group to a finite-rank abelian group.
We also extend each Morita homomorphism to a crossed homomorphism from the
mapping class group of once-bounded surface to a finite-rank abelian group.
This improves on the author's previous results [Algebr. Geom. Topol. 7
(2007):1297-1326]. To prove the first result, we express the higher Johnson
homomorphisms as coboundary maps in group cohomology. Our methods involve the
topology of nilpotent homogeneous spaces and lattices in nilpotent Lie
algebras. In particular, we develop a notion of the "polynomial straightening"
of a singular homology chain in a nilpotent homogeneous space.Comment: 34 page
Anemia and blood transfusion in the critically ill patient: role of erythropoietin
Critically ill patients receive an extraordinarily large number of blood transfusions. Between 40% and 50% of all patients admitted to intensive care units receive at least 1 red blood cell (RBC) unit during their stay, and the average is close to 5 RBC units. RBC transfusion is not risk free. There is little evidence that 'routine' transfusion of stored allogeneic RBCs is beneficial to critically ill patients. The efficacy of perioperative recombinant human erythropoietin (rHuEPO) has been demonstrated in a variety of elective surgical settings. Similarly, in critically ill patients with multiple organ failure, rHuEPO therapy will also stimulate erythropoiesis. In a randomized, placebo-controlled trial, therapy with rHuEPO resulted in a significant reduction in RBC transfusions. Despite receiving fewer RBC transfusions, patients in the rHuEPO group had a significantly greater increase in hematocrit. Strategies to increase the production of RBCs are complementary to other approaches to reduce blood loss in the intensive care unit, and they decrease the transfusion threshold in the management of all critically ill patients
Integral formulas with distribution kernels for irreducible projections in L2 of a nilmanifold
AbstractLet N be a simply connected nilpotent Lie group and Γ a discrete uniform subgroup. The authors consider irreducible representations σ in the spectrum of the quasi-regular representation N × L2(Γ/N) → L2(Γ→) which are induced from normal maximal subordinate subgroups M ⊆ N. The primary projection Pσ and all irreducible projections P ⩽ Pσ are given by convolutions involving right Γ-invariant distributions D on Γ→, Pf(Γn) = D ∗ f(Γn) = <D, n · f>all f ϵ C∞(Γ/N), where n · f(ζ) = f(ζ · n). Extending earlier work of Auslander and Brezin, and L. Richardson, the authors give explicit character formulas for the distributions, interpreting them as sums of characters on the torus Tκ = (Γ ∩ M) · [M, M]⧹M. By examining these structural formulas, they obtain fairly sharp estimates on the order of the distributions: if σ is associated with an orbit O ⊆ n∗ and if V ⊆ n∗ is the largest subspace which saturates θ in the sense that f ϵ O ⇒ f + V ⊆ O. As a corollary they obtain Richardson's criterion for a projection to map C0(Γ→) into itself. The authors also resolve a conjecture of Brezin, proving a Zero-One law which says, among other things, that if the primary projection Pσ maps Cr(Γ→) into C0(Γ→), so do all irreducible projections P ⩽ Pσ. This proof is based on a classical lemma on the extent to which integral points on a polynomial graph in Rn lie in the coset ring of Zn (the finitely additive Boolean algebra generated by cosets of subgroups in Zn). This lemma may be useful in other investigations of nilmanifolds
Thermodynamic phase-field model for microstructure with multiple components and phases: the possibility of metastable phases
A diffuse-interface model for microstructure with an arbitrary number of
components and phases was developed from basic thermodynamic and kinetic
principles and formalized within a variational framework. The model includes a
composition gradient energy to capture solute trapping, and is therefore suited
for studying phenomena where the width of the interface plays an important
role. Derivation of the inhomogeneous free energy functional from a Taylor
expansion of homogeneous free energy reveals how the interfacial properties of
each component and phase may be specified under a mass constraint. A diffusion
potential for components was defined away from the dilute solution limit, and a
multi-obstacle barrier function was used to constrain phase fractions. The
model was used to simulate solidification via nucleation, premelting at phase
boundaries and triple junctions, the intrinsic instability of small particles,
and solutal melting resulting from differing diffusivities in solid and liquid.
The shape of metastable free energy surfaces is found to play an important role
in microstructure evolution and may explain why some systems premelt at phase
boundaries and phase triple junctions while others do not.Comment: 14 pages, 8 figure
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