Spectral convergence of graph Laplacian and Heat kernel reconstruction in L∞ from random samples

2021 ◽  
Author(s):  
David B. Dunson ◽  
Hau-Tieng Wu ◽  
Nan Wu
Keyword(s):  
Heat Kernel ◽  
Graph Laplacian ◽  
Random Samples ◽  
Spectral Convergence

The Recovery of Factor VIII from Fresh-Frozen, Indated and Outdated Human Plasma

1976 ◽  
Vol 36 (01) ◽  
pp. 071-077 ◽  
Author(s):  
Daniel E. Whitman ◽  
Mary Ellen Switzer ◽  
Patrick A. McKee
Keyword(s):  
Factor Viii ◽  
Polyacrylamide Gel Electrophoresis ◽  
Sodium Dodecyl ◽  
The United States ◽  
Fresh Frozen Plasma ◽  
Red Cross ◽  
Random Samples ◽  
Fresh Frozen ◽  
Factor Viii Concentrates ◽  
Frozen Plasma

SummaryThe availability of factor VIII concentrates is frequently a limitation in the management of classical hemophilia. Such concentrates are prepared from fresh or fresh-frozen plasma. A significant volume of plasma in the United States becomes “indated”, i. e., in contact with red blood cells for 24 hours at 4°, and is therefore not used to prepare factor VIII concentrates. To evaluate this possible resource, partially purified factor VIII was prepared from random samples of fresh-frozen, indated and outdated plasma. The yield of factor VIII protein and procoagulant activity from indated plasma was about the same as that from fresh-frozen plasma. The yield from outdated plasma was substantially less. After further purification, factor VIII from the three sources gave a single subunit band when reduced and analyzed by sodium dodecyl sulfate polyacrylamide gel electrophoresis. These results indicate that the approximately 287,000 liters of indated plasma processed annually by the American National Red Cross (ANRC) could be used to prepare factor VIII concentrates of good quality. This resource alone could quadruple the supply of factor VIII available for therapy.

The Resolvent Operator

2017 ◽  
Author(s):  
Charles L. Epstein ◽  
Rafe Mazzeo
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Laplace Transform ◽  
Heat Kernel ◽  
Resolvent Operator ◽  
Contour Integration ◽  
Resolvent Kernel ◽  
The Laplace Transform ◽  
Elliptic Estimates ◽  
The Resolvent Operator

This chapter describes the construction of a resolvent operator using the Laplace transform of a parametrix for the heat kernel and a perturbative argument. In the equation (μ‎-L) R(μ‎) f = f, R(μ‎) is a right inverse for (μ‎-L). In Hölder spaces, these are the natural elliptic estimates for generalized Kimura diffusions. The chapter first constructs the resolvent kernel using an induction over the maximal codimension of bP, and proves various estimates on it, along with corresponding estimates for the solution operator for the homogeneous Cauchy problem. It then considers holomorphic semi-groups and uses contour integration to construct the solution to the heat equation, concluding with a discussion of Kimura diffusions where all coefficients have the same leading homogeneity.

Hypoelliptic Laplacian and Orbital Integrals (AM-177)

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Fixed Point ◽  
Symmetric Space ◽  
Heat Kernel ◽  
Geodesic Flow ◽  
Casimir Operator ◽  
Index Theory ◽  
Weighted Sum ◽  
Orbital Integrals ◽  
Hypoelliptic Heat Kernel ◽  
Wave Kernel

This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula. The hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. It is essentially the weighted sum of a harmonic oscillator along the fiber of the tangent bundle, and of the generator of the geodesic flow. In this book, semisimple orbital integrals associated with the heat kernel of the Casimir operator are shown to be invariant under a suitable hypoelliptic deformation, which is constructed using the Dirac operator of Kostant. Their explicit evaluation is obtained by localization on geodesics in the symmetric space, in a formula closely related to the Atiyah-Bott fixed point formulas. Orbital integrals associated with the wave kernel are also computed. Estimates on the hypoelliptic heat kernel play a key role in the proofs, and are obtained by combining analytic, geometric, and probabilistic techniques. Analytic techniques emphasize the wavelike aspects of the hypoelliptic heat kernel, while geometrical considerations are needed to obtain proper control of the hypoelliptic heat kernel, especially in the localization process near the geodesics. Probabilistic techniques are especially relevant, because underlying the hypoelliptic deformation is a deformation of dynamical systems on the symmetric space, which interpolates between Brownian motion and the geodesic flow. The Malliavin calculus is used at critical stages of the proof.

Mathematics of networks

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Random Walks ◽  
Adjacency Matrix ◽  
Graph Partitioning ◽  
Dynamic Networks ◽  
Graph Laplacian ◽  
Network Visualization ◽  
Final Part ◽  
Bipartite Networks ◽  
Component Structure ◽  
Basic Properties

An introduction to the mathematical tools used in the study of networks. Topics discussed include: the adjacency matrix; weighted, directed, acyclic, and bipartite networks; multilayer and dynamic networks; trees; planar networks. Some basic properties of networks are then discussed, including degrees, density and sparsity, paths on networks, component structure, and connectivity and cut sets. The final part of the chapter focuses on the graph Laplacian and its applications to network visualization, graph partitioning, the theory of random walks, and other problems.

scholarly journals Spectral Gap of the Largest Eigenvalue of the Normalized Graph Laplacian

2021 ◽  
Author(s):  
Jürgen Jost ◽  
Raffaella Mulas ◽  
Florentin Münch
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Spectral Gap ◽  
Complete Bipartite Graph ◽  
Graph Laplacian ◽  
New Method ◽  
Single Edge ◽  
Largest Eigenvalue ◽  
Complement Graph ◽  
The Largest Eigenvalue

AbstractWe offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least $$\frac{n+1}{n-1}$$ n + 1 n - 1 provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size $$\frac{n-1}{2}$$ n - 1 2 . With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most $$\frac{n-1}{2}$$ n - 1 2 .

scholarly journals Estimating epidemiologic dynamics from cross-sectional viral load distributions

Science ◽  
2021 ◽  
pp. eabh0635
Author(s):  
James A. Hay ◽  
Lee Kennedy-Shaffer ◽  
Sanjat Kanjilal ◽  
Niall J. Lennon ◽  
Stacey B. Gabriel ◽  
...  
Keyword(s):  
Population Distribution ◽  
Quantitative Polymerase Chain Reaction ◽  
Cross Sectional ◽  
Outbreak Management ◽  
Viral Loads ◽  
Time Estimates ◽  
Load Distributions ◽  
Random Samples ◽  
Combining Data ◽  
Polymerase Chain

Estimating an epidemic’s trajectory is crucial for developing public health responses to infectious diseases, but case data used for such estimation are confounded by variable testing practices. We show that the population distribution of viral loads observed under random or symptom-based surveillance, in the form of cycle threshold (Ct) values obtained from reverse-transcription quantitative polymerase chain reaction testing, changes during an epidemic. Thus, Ct values from even limited numbers of random samples can provide improved estimates of an epidemic’s trajectory. Combining data from multiple such samples improves the precision and robustness of such estimation. We apply our methods to Ct values from surveillance conducted during the SARS-CoV-2 pandemic in a variety of settings and offer alternative approaches for real-time estimates of epidemic trajectories for outbreak management and response.

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