Physical Sciences and Mathematics Commons^™

Why Probability Appears In Quantum Mechaincs, Jerome Blackman, Wu Teh Hsiang

Mathematics - All Scholarship

Early in the development of quantum theory Bohr introduced what came to be called the Copenhagen interpretation. Specifically, the square of the absolute value of the wave function was to be used as a probability density. There followed lengthy arguments about this ranging from alternative universes to Schrodinger's cat. Einstein famously remarked "I am convinced that He (God) does not play dice." The purpose of this paper is to present a mathematical model of the measuring process that shows that the Copenhagen interpretation can actually follow from the fact that the time development of quantum systems is governed by the …

Small And Large Time Stability Of The Time Taken For A Lévy Process To Cross Curved Boundaries, Philip S. Griffin, Ross A. Maller

Mathematics - All Scholarship

This paper is concerned with the small time behaviour of a Levy process X. In particular, we investigate the stabilities of the times, Tb(r) and Tb*(r), at which X, started with X₀ = 0, first leaves the space-time regions {(t, y) ∈ R² : y ≤ rt^b, t ≥ 0} (one-sided exit), or {(t, y) in R² :|y| ≤ rt^b, t ≥ 0} (two-sided exit), 0 ≤ b < 1, as r -> 0. Thus essentially we determine whether or not these passage times behave like deterministic functions in the sense of different modes of convergence; specifically convergence …

Small And Large Time Stability Of The Time Taken For A Lévy Process To Cross Curved Boundaries, Philip S. Griffin, Ross A. Maller

Mathematics - All Scholarship

This paper is concerned with the small time behaviour of a Levy process X. In particular, we investigate the stabilities of the times, T_b(r) and T^*_b (r), at which X, started with X₀ = 0, first leaves the space-time regions {(t, y) in R² : y ≤ rt^b, t ≥ 0} (one-sided exit), or {(t, y) in R² :|y| ≤ rt^b, t ≥ 0} (two-sided exit), 0 ≤ b < 1, as r ↓ 0. Thus essentially we determine whether or not these passage times behave like deterministic functions in the sense of different modes of convergence; specifically convergence in probability, almost surely and in L^p. In many instances these are seen to be equivalent to relative stability of the process X itself. The …

Warped Product Rigidity, Chenxu He, Peter Petersen, William Wylie

Mathematics - All Scholarship

In this paper we study the space of solutions to an overdetermined linear system involving the Hessian of functions. We show that if the solution space has dimension greater than one, then the underlying manifold has a very rigid warped product structure. This warped product structure will be used to study warped product Einstein structures in our paper "The space of virtual solutions to the warped product Einstein equation".

The Space Of Virtual Solutions To The Warped Product Einstein Equation, Chenxu He, Peter Petersen, William Wylie

Mathematics - All Scholarship

In this paper we introduce a vector space of virtual warping functions that yield Einstein metrics over a fixed base. There is a natural quadratic form on this space and we study how this form interacts with the geometry. We use this structure along with the results in our earlier paper "Warped product rigidity" to show that essentially every warped product Einstein manifold admits a particularly nice warped product structure that we call basic. As applications we give a sharp characterization of when a homogeneous Einstein metric can be a warped product and also generalize a construction of Lauret showing …

A Branching Process For Virus Survival, J. Theodore Cox, Rinaldo B. Schinazi

Mathematics - All Scholarship

Quasispecies theory predicts that there is a critical mutation probability above which a viral population will go extinct. Above this threshold the virus loses the ability to replicate the best adapted genotype, leading to a population composed of low replicating mutants that is eventually doomed. We propose a new branching model that shows that this is not necessarily so. That is, a population composed of ever changing mutants may survive.

Adjoint Functors, Projectivization, And Differentiation Algorithms For Representations Of Partially Ordered Sets, Mark Kleiner, Markus Reitenbach

Mathematics - All Scholarship

Adjoint functors and projectivization in representation theory of partially ordered sets are used to generalize the algorithms of differentiation by a maximal and by a minimal point. Conceptual explanations are given for the combinatorial construction of the derived set and for the differentiation functor.

Lipschitz Regularity For Inner-Variational Equations, Tadeusz Iwaniec, Leonid V. Kovalev, Jani Onninen

Mathematics - All Scholarship

We obtain Lipschitz regularity results for a fairly general class of nonlinear first-order PDEs. These equations arise from the inner variation of certain energy integrals. Even in the simplest model case of the Dirichlet energy the inner-stationary solutions need not be differentiable everywhere; the Lipschitz continuity is the best possible. But the proofs, even in the Dirichlet case, turn out to relay on topological arguments. The appeal to the inner-stationary solutions in this context is motivated by the classical problems of existence and regularity of the energy-minimal deformations in the theory of harmonic mappings and certain mathematical models of nonlinear …

Equidistribution Results For Singular Metrics On Line Bundles, Dan Coman, George Marinescu

Mathematics - All Scholarship

Let L be a holomorphic line bundle with a positively curved singular Hermitian metric over a complex manifold X. One can define naturally the sequence of Fubini-Study currents associated to the space of square integrable holomorphic sections of the p-th tensor powers of L. Assuming that the singular set of the metric is contained in a compact analytic subset of X and that the logarithm of the Bergman kernel function associated to the p-th tensor power of L (defined outside the singular set) grows like o(p) as p tends to infinity, we prove the following:
1) the k-th power of …

Non-Commutative Crepant Resolutions: Scenes From Categorical Geometry, Graham J. Leuschke

Mathematics - All Scholarship

Non-commutative crepant resolutions are algebraic objects defined by Van den Bergh to realize an equivalence of derived categories in birational geometry. They are motivated by tilting theory, the McKay correspondence, and the minimal model program, and have applications to string theory and representation theory. In this expository article I situate Van den Bergh's definition within these contexts and describe some of the current research in the area.

Strong Approximation Of Homeomorphisms Of Finite Dirichlet Energy, Tadeusz Iwaniec, Leonid V. Kovalev, Jani Onninen

Mathematics - All Scholarship

Let X and Y be planar Jordan domains of the same finite connectivity, Y being inner chordarc regular (such are Lipschitz domains). Every homeomorphism h:X->Y in the Sobolev space W^1,2 extends to a continuous map between closed domains. We prove that there exist homeomorphisms between closed domains which converge to h uniformly and in W^1,2. The problem of approximation of Sobolev homeomorphisms, raised by J. M. Ball and L. C. Evans, is deeply rooted in a study of energy-minimal deformations in nonlinear elasticity. The new feature of our main result is that approximation takes place also …

On Khovanov-Seidel Quiver Algebras And Bordered Floer Homology, Denis Auroux, J. Elisenda Grigsby, Stephan M. Wehrli

Mathematics - All Scholarship

We discuss a relationship between Khovanov- and Heegaard Floer-type homology theories for braids. Explicitly, we define a filtration on the bordered Heegaard-Floer homology bimodule associated to the double-branched cover of a braid and show that its associated graded bimodule is equivalent to a similar bimodule defined by Khovanov and Seidel.

Quasisymmetric Graphs And Zygmund Functions, Leonid V. Kovalev, Jani Onninen

Mathematics - All Scholarship

A quasisymmetric graph is a curve whose projection onto a line is a quasisymmetric map. We show that this class of curves is related to solutions of the reduced Beltrami equation and to a generalization of the Zygmund class lambda. This relation makes it possible to use the tools of harmonic analysis to construct nontrivial examples of quasisymmetric graphs and of quasiconformal maps.

Path Decomposition Of Ruinous Behaviour For A General Lévy Insurance Risk Process, Philip S. Griffin, Ross A. Maller

Mathematics - All Scholarship

We analyse the general Levy insurance risk process for Levy measures in the convolution equivalence class

S^(alpha), alpha> 0, via a new kind of path decomposition. This yields a very general functional limit theorem as the initial reserve level u → ∞, and a host of new results for functionals of interest in insurance risk. Particular emphasis is placed on the time to ruin, which is shown to have a proper limiting distribution, as u → ∞, conditional on ruin occurring, under our assumptions. Existing asymptotic results under the S^(alpha)

assumption are synthesised and extended, and …

The Time At Which A Lévy Process Creeps, Philip S. Griffin, Ross A. Maller

Mathematics - All Scholarship

We show that if a Levy process creeps then, as a function of u, the renewal function V (t, u) of the bivariate ascending ladder process (L⁻¹,H) is absolutely continuous on [0,∞) and left differentiable on (0,∞), and the left derivative at u is proportional to the (improper) distribution function of the time at which the process creeps over level u, where the constant of proportionality is d⁻¹_H, the reciprocal of the (positive) drift of H. This yields the (missing) term due to creeping in the recent quintuple law of Doney and Kyprianou (2006). As …

Path Decomposition Of Ruinous Behaviour For A General Lévy Insurance Risk Process, Philip S. Griffin, Ross A. Maller

Mathematics - All Scholarship

We analyse the general Levy insurance risk process for Levy measures in the convolution equivalence class S^(alpha), alpha > 0, via a new kind of path decomposition. This yields a very general functional limit theorem as the initial reserve level u → ∞, and a host of new results for functionals of interest in insurance risk. Particular emphasis is placed on the time to ruin, which is shown to have a proper limiting distribution, as u -> ∞, conditional on ruin occurring, under our assumptions. Existing asymptotic results under the S^(alpha) assumption are synthesised and extended, and proofs …

The Time At Which A Lévy Process Creeps, Philip S. Griffin, Ross A. Maller

Mathematics - All Scholarship

We show that if a Levy process creeps then, as a function of u, the renewal function V (t, u) of the bivariate ascending ladder process (L⁻¹,H) is absolutely continuous on [0,∞) and left differentiable on (0,∞), and the left derivative at u is proportional to the (improper) distribution function of the time at which the process creeps over level u, where the constant of proportionality is d⁻¹_H , the reciprocal of the (positive) drift of H. This yields the (missing) term due to creeping in the recent quintuple law of Doney and Kyprianou (2006). As …

Distinguishability Of Infinite Groups And Graphs, Simon M. Smith, Thomas W. Tucker, Mark E. Watkins

Mathematics - All Scholarship

The distinguishing number of a group G acting faithfully on a set V is the least number of colors needed to color the elements of V so that no non-identity element of the group preserves the coloring. The distinguishing number of a graph is the distinguishing number of its full automorphism group acting on its vertex set. A connected graph T is said to have connectivity 1 if there exists a vertex alpha in VT such that T \ {alpha} is not connected. For alpha in V , an orbit of the point stabilizer G_alpha is called a suborbit …

Asymptotic Distributions Of The Overshoot And Undershoots For The Lévy Insurance Risk Process In The Cramér And Convolution Equivalent Cases, Philip S. Griffin, Ross A. Maller, Kees Van Schaik

Mathematics - All Scholarship

Recent models of the insurance risk process use a Levy process to generalise the traditional Cramer-Lundberg compound Poisson model. This paper is concerned with the behaviour of the distributions of the overshoot and undershoots of a high level, for a Levy process which drifts to -infinity and satis es a Cramer or a convolution equivalent condition. We derive these asymptotics under minimal conditions in the Cramer case, and compare them with known results for the convolution equivalent case, drawing attention to the striking and unexpected fact that they become identical when certain parameters tend to equality. Thus, at least regarding …

Non-Commutative Desingularization Of Determinantal Varieties, Ii, Ragar-Olaf Buchweitz, Graham J. Leuschke, Michel Van Den Bergh

Mathematics - All Scholarship

In our paper "Non-commutative desingularization of determinantal varieties, I" we constructed and studied non-commutative resolutions of determinantal varieties defined by maximal minors. At the end of the introduction we asserted that the results could be generalized to determinantal varieties defined by non-maximal minors, at least in characteristic zero. In this paper we prove the existence of non-commutative resolutions in the general case in a manner which is still characteristic free. The explicit description of the resolution by generators and relations is deferred to a later paper. As an application of our results we prove that there is a fully faithful …

Pruitt's Estimates In Banach Space, Philip S. Griffin

Mathematics - All Scholarship

Pruitt's estimates on the expectation and the distribution of the time taken by a random walk to exit a ball of radius r are extended to the infinite dimensional setting. It is shown that they separate into two pairs of estimates depending on whether the space is type 2 or cotype 2. It is further shown that these estimates characterize type 2 and cotype 2 spaces.

A Direct Limit For Limit Hilbert-Kunz Multiplicity For Smooth Projective Curves, Holger Brenner, Jinjia Li, Claudia Miller

Mathematics - All Scholarship

This paper concerns the question of whether a more direct limit can be used to obtain the limit Hilbert Kunz multiplicity, a possible candidate for a characteristic zero Hilbert-Kunz multiplicity. The main goal is to establish an affirmative answer for one of the main cases for which the limit Hilbert Kunz multiplicity is even known to exist, namely that of graded ideals in the homogeneous coordinate ring of smooth projective curves. The proof involves more careful estimates of bounds found independently by Brenner and Trivedi on the dimensions of the cohomologies of twists of the syzygy bundle as the characteristic …

Efficient First Order Methods For Linear Composite Regularizers, Andreas Argyriou, Charles A. Micchelli, Massimiliano Pontil, Lixin Shen, Yuesheng Xu

Mathematics - All Scholarship

A wide class of regularization problems in machine learning and statistics employ a regularization term which is obtained by composing a simple convex function omega with a linear transformation. This setting includes Group Lasso methods, the Fused Lasso and other total variation methods, multi-task learning methods and many more. In this paper, we present a general approach for computing the proximity operator of this class of regularizers, under the assumption that the proximity operator of the function \omega is known in advance. Our approach builds on a recent line of research on optimal first order optimization methods and uses fixed …

Stable Generalized Finite Element Method (Sgfem), I. Babuska, U. Banerjee

Mathematics - All Scholarship

The Generalized Finite Element Method (GFEM) is a Partition of Unity Method (PUM), where the trial space of standard Finite Element Method (FEM) is augmented with non-polynomial shape functions with compact support. These shape functions, which are also known as the enrichments, mimic the local behavior of the unknown solution of the underlying variational problem. GFEM has been successfully used to solve a variety of problems with complicated features and microstructure. However, the stiffness matrix of GFEM is badly conditioned (much worse compared to the standard FEM) and there could be a severe loss of accuracy in the computed solution …

Voter Model Perturbations And Reaction Diffusion Equations, J. Theodore Cox, Richard Durrett, Edwin A. Perkins

Mathematics - All Scholarship

We consider particle systems that are perturbations of the voter model and show that when space and time are rescaled the system converges to a solution of a reaction diffusion equation in dimensions d > 3. Combining this result with properties of the PDE, some methods arising from a low density super-Brownian limit theorem, and a block construction, we give general, and often asymptotically sharp, conditions for the existence of non-trivial stationary distributions, and for extinction of one type. As applications, we describe the phase diagrams of three systems when the parameters are close to the voter model: (i) a stochastic …

Wild Hypersurfaces, Andrew Crabbe, Graham J. Leuschke

Mathematics - All Scholarship

Complete hypersurfaces of dimension at least 2 and multiplicity at least 4 have wild Cohen-Macaulay type.

Refinement Of Operator-Valued Reproducing Kernels, Yuesheng Xu, Haizhang Zhang, Qinghui Zhang

Mathematics - All Scholarship

This paper studies the construction of a refinement kernel for a given operator-valued reproducing kernel such that the vector-valued reproducing kernel Hilbert space of the refinement kernel contains that of the given one as a subspace. The study is motivated from the need of updating the current operator-valued reproducing kernel in multi-task learning when underfitting or overfitting occurs. Numerical simulations confirm that the established refinement kernel method is able to meet this need. Various characterizations are provided based on feature maps and vector-valued integral representations of operator-valued reproducing kernels. Concrete examples of refining translation invariant and finite Hilbert-Schmidt operator-valued reproducing …

N-Harmonic Mappings Between Annuli, Tadeusz Iwaniec, Jani Onninen

Mathematics - All Scholarship

The central theme of this paper is the variational analysis of homeomorphisms h: X onto −→ Y between two given domains X,Y ⊂ Rⁿ. We look for the extremal mappings in the Sobolev space W^1,n(X,Y) which minimize the energy integral Eh =ZX ||Dh(x) ||ⁿ dx Because of the natural connections with quasiconformal mappings this n harmonic alternative to the classical Dirichlet integral (for planar domains) has drawn the attention of researchers in Geometric Function Theory. Explicit analysis is made here for a pair of concentric spherical annuli where many unexpected phenomena about minimal n -harmonic …

An Essay On The Interpolation Theorem Of Józef Marcinkiewicz - Polish Patriot, Tadeusz Iwaniec

Mathematics - All Scholarship

In memory of Polish mathematicians murdured by the Soviets and the Nazis. The total record of accomplishments of Marcinkiewicz in his short life, his talent, perceptions rich in concepts, and technical novelties, go far beyond my ability to give full play within the confines of one article. The importance of Marcinkiewicz's short paper is reflected in the myriad applications and generalizations which earns the right to be called Marcinkiewicz Interpolation Theory Marcinkiewicz interpolation theorem came after the celebrated convexity theorem of M. Riesz and his student G.O. Thorin. These fundamental works by M. Riesz, G.O. Thorin and J. Marcinkiewicz deal …

On The Classification Of Warped Product Einstein Metrics, Chenxu He, Peter Petersen, William Wylie

Mathematics - All Scholarship

In this paper we take the perspective introduced by Case-Shu-Wei of studying warped product Einstein metrics through the equation for the Ricci curvature of the base space. They call this equation on the base the m-Quasi Einstein equation, but we will also call it the (lambda,n+m)-Einstein equation. In this paper we extend the work of Case-Shu-Wei and some earlier work of Kim-Kim to allow the base to have non-empty boundary. This is a natural case to consider since a manifold without boundary often occurs as a warped product over a manifold with boundary, and in this case we get some …