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Doubling Measures, Monotonicity, And Quasiconformality, Leonid V. Kovalev, Diego Maldonado, Jang-Mei Wu
Doubling Measures, Monotonicity, And Quasiconformality, Leonid V. Kovalev, Diego Maldonado, Jang-Mei Wu
Mathematics - All Scholarship
We construct quasiconformal mappings in Euclidean spaces by integration of a discontinuous kernel against doubling measures with suitable decay. The differentials of mappings that arise in this way satisfy an isotropic form of the doubling condition. We prove that this isotropic doubling condition is satisfied by the distance functions of certain fractal sets. Finally, we construct an isotropic doubling measure that is not absolutely continuous with respect to the Lebesgue measure.
Overinterpolation, Dan Coman, Evgeny A. Poletsky
Overinterpolation, Dan Coman, Evgeny A. Poletsky
Mathematics - All Scholarship
In this paper we study the consequences of overinterpolation, i.e., the situation when a function can be interpolated by polynomial, or rational, or algebraic functions in more points that normally expected. We show that in many cases such a function has specific forms.
Some Heuristics About Elliptic Curves, Mark Watkins
Some Heuristics About Elliptic Curves, Mark Watkins
Mathematics - All Scholarship
We give some heuristics for counting elliptic curves with certain properties. In particular, we re-derive the Brumer-McGuinness heuristic for the number of curves with positive/negative discriminant up to X, which is an application of lattice-point counting. We then introduce heuristics (with refinements from random matrix theory) that allow us to predict how often we expect an elliptic curve E with even parity to have L(E,1)=0. We find that we expect there to be about c1X19/24(log X)3/8 curves with |Delta|< X with even parity and positive (analytic) rank; since Brumer and McGuinness predict cX5/6 total curves, this implies that asymptotically almost all even parity curves have rank 0. We …
Preprojective Representations Of Valued Quivers And Reduced Words In The Weyl Group Of A Kac-Moody Algebra, Mark Kleiner, Allen Pelley
Preprojective Representations Of Valued Quivers And Reduced Words In The Weyl Group Of A Kac-Moody Algebra, Mark Kleiner, Allen Pelley
Mathematics - All Scholarship
This paper studies connections between the preprojective representations of a valued quiver, the (+)-admissible sequences of vertices, and the Weyl group by associating to each preprojective representation a canonical (+)-admissible sequence. A (+)-admissible sequence is the canonical sequence of some preprojective representation if and only if the product of simple reflections associated to the vertices of the sequence is a reduced word in the Weyl group. As a consequence, for any Coxeter element of the Weyl group associated to an indecomposable symmetrizable generalized Cartan matrix, the group is infinite if and only if the powers of the element are reduced …
Categorification Of The Colored Jones Polynomial And Rasmussen Invariant Of Links, Anna Beliakova, Stephan Wehrli
Categorification Of The Colored Jones Polynomial And Rasmussen Invariant Of Links, Anna Beliakova, Stephan Wehrli
Mathematics - All Scholarship
We define a family of formal Khovanov brackets of a colored link depending on two parameters. The isomorphism classes of these brackets are invariants of framed colored links. The Bar-Natan functors applied to these brackets produce Khovanov and Lee homology theories categorifying the colored Jones polynomial. Further, we study conditions under which framed colored link cobordisms induce chain transformations between our formal brackets. We conjecture that, for special choice of parameters, Khovanov and Lee homology theories of colored links are functorial (up to sign). Finally, we extend the Rasmussen invariant to links and give examples, where this invariant is a …
Sequences Of Reflection Functors And The Preprojective Component Of A Valued Quiver, Mark Kleiner, Helene R. Tyler
Sequences Of Reflection Functors And The Preprojective Component Of A Valued Quiver, Mark Kleiner, Helene R. Tyler
Mathematics - All Scholarship
This paper concerns preprojective representations of a finite connected valued quiver without oriented cycles. For each such representation, an explicit formula in terms of the geometry of the quiver gives a unique, up to a certain equivalence, shortest (+)-admissible sequence such that the corresponding composition of reflection functors annihilates the representation. The set of equivalence classes of the above sequences is a partially ordered set that contains a great deal of information about the preprojective component of the Auslander-Reiten quiver. The results apply to the study of reduced words in the Weyl group associated to an indecomposable symmetrizable generalized Cartan …
Admissible Sequences, Preprojective Modules, And Reduced Words In The Weyl Group Of A Quiver, Mark Kleiner, Allen Pelley
Admissible Sequences, Preprojective Modules, And Reduced Words In The Weyl Group Of A Quiver, Mark Kleiner, Allen Pelley
Mathematics - All Scholarship
This paper studies connections between the preprojective modules over the path algebra of a finite connected quiver without oriented cycles, the (+)-admissible sequences of vertices, and the Weyl group. For each preprojective module, there exists a unique up to a certain equivalence shortest (+)-admissible sequence annihilating the module. A (+)-admissible sequence is the shortest sequence annihilating some preprojective module if and only if the product of simple reflections associated to the vertices of the sequence is a reduced word in the Weyl group. These statements have the following application that strengthens known results of Howlett and Fomin-Zelevinsky. For any fixed …
The Elementary Theory Of Normed Linear Spaces And Linear Functionals, Suresh Eswarthasan
The Elementary Theory Of Normed Linear Spaces And Linear Functionals, Suresh Eswarthasan
Honors Capstone Projects - All
Abstract not Included
Factoring The Adjoint And Maximal Cohen-Macaulay Modules Over The Generic Determinant, Ragnar-Olaf Buchweitz, Graham J. Leuschke
Factoring The Adjoint And Maximal Cohen-Macaulay Modules Over The Generic Determinant, Ragnar-Olaf Buchweitz, Graham J. Leuschke
Mathematics - All Scholarship
A question of Bergman asks whether the adjoint of the generic square matrix over a field can be factored nontrivially as a product of square matrices. We show that such factorizations indeed exist over any coefficient ring when the matrix has even size. Establishing a correspondence between such factorizations and extensions of maximal Cohen-Macaulay modules over the generic determinant, we exhibit all factorizations where one of the factors has determinant equal to the generic determinant. The classification shows not only that the Cohen-Macaulay representation theory of the generic determinant is wild in the tame-wild dichotomy, but that it is quite …
Symmetric Powers Of Elliptic Curve L-Functions, Phil Martin, Mark Watkins
Symmetric Powers Of Elliptic Curve L-Functions, Phil Martin, Mark Watkins
Mathematics - All Scholarship
The conjectures of Deligne, Beuilinson, and Bloch-Kato assert that there should be relations between the arithmetic of algebro-geometric objects and the special values of their L-functions. We make a numerical study for symmetric power L-functions of elliptic curves, obtaining data about the validity of their functional equations, frequency of vanishing of central values, and divisibility of Bloch-Kato quotients.
Some Remarks On Heegner Point Computations, Mark Watkins
Some Remarks On Heegner Point Computations, Mark Watkins
Mathematics - All Scholarship
We explain how to find a rational point on a rational elliptic curve of rank 1 using Heegner points. We give some examples, and list new algorithms that are due to Cremona and Delaunay. These are notes from a short course given at the Institut Henri Poincare in December 2004.
On The Growth Of The Betti Sequence Of The Canonical Module, David A. Jorgensen, Graham J. Leuschke
On The Growth Of The Betti Sequence Of The Canonical Module, David A. Jorgensen, Graham J. Leuschke
Mathematics - All Scholarship
We study the growth of the Betti sequence of the canonical module of a Cohen-Macaulay local ring. It is an open question whether this sequence grows exponentially whenever the ring is not Gorenstein. We answer the question of exponential growth affirmatively for a large class of rings, and prove that the growth is in general not extremal. As an application of growth, we give criteria for a Cohen-Macaulay ring possessing a canonical module to be Gorenstein.
Secondary Terms In The Number Of Vanishings Of Quadratic Twists Of Elliptic Curve L-Functions, J. Brian Conrey, Atul Pocharel, Michael O. Rubinstein, Mark Watkins
Secondary Terms In The Number Of Vanishings Of Quadratic Twists Of Elliptic Curve L-Functions, J. Brian Conrey, Atul Pocharel, Michael O. Rubinstein, Mark Watkins
Mathematics - All Scholarship
We examine the number of vanishings of quadratic twists of the L-function associated to an elliptic curve. Applying a conjecture for the full asymptotics of the moments of critical L-values we obtain a conjecture for the first two terms in the ratio of the number of vanishings of twists sorted according to arithmetic progressions.
Influence Of Surface Tension On The Conical Miniscus Of A Magnetic Fluid In The Field Of A Current-Carrying Wire, Thomas John, Dirk Rannacher, Adreas Engel
Influence Of Surface Tension On The Conical Miniscus Of A Magnetic Fluid In The Field Of A Current-Carrying Wire, Thomas John, Dirk Rannacher, Adreas Engel
Mathematics - All Scholarship
We study the influence of surface tension on the shape of the conical miniscus built up by a magnetic fluid surrounding a current-carrying wire. Minimization of the total energy of the system leads to a singular second order boundary value problem for the function zeta(r) describing the axially symmetric shape of the free surface. An appropriate transformation regularizes the problem and allows a straightforward numerical solution. We also study the effects a superimposed second liquid, a nonlinear magnetization law of the magnetic fluid, and the influence of the diameter of the wire on the free surface profile.
Universal Kernels, Charles A. Micchelli, Yuesheng Xu, Haizhang Zhang
Universal Kernels, Charles A. Micchelli, Yuesheng Xu, Haizhang Zhang
Mathematics - All Scholarship
In this paper we investigate conditions on the features of a continuous kernel so that it may approximate an arbitrary continuous target function uniformly on any compact subset of the input space. A number of concrete examples are given of kernels with this universal approximating property.
Testing For Cointegrating Rank Via Model Selection: Evidence From 165 Data Sets, Badi H. Baltagi, Zijun Wang
Testing For Cointegrating Rank Via Model Selection: Evidence From 165 Data Sets, Badi H. Baltagi, Zijun Wang
Center for Policy Research
The model selection approach has been proposed as an alternative to the popular tests for cointegration such as the residual-based ADF test and the system-based trace test. Using information criteria, we conduct cointegration tests on 165 data sets used in published studies. The empirical results demonstrate the usefulness of the model selection approach for applied researchers.
Random Effects And Spatial Autocorrelations With Equal Weights, Badi H. Baltagi
Random Effects And Spatial Autocorrelations With Equal Weights, Badi H. Baltagi
Center for Policy Research
This note considers a panel data regression model with spatial autoregressive disturbances and random effects where the weight matrix is normalized and has equal elements. This is motivated by Kelejian et al. (2005), who argue that such a weighting matrix, having blocks of equal elements, might be considered when units are equally distant within certain neighborhoods but unrelated between neighborhoods. We derive a simple weighted least squares transformation that obtains GLS on this model as a simple OLS. For the special case of a spatial panel model with no random effects, we obtain two sufficient conditions where GLS on this …
Prediction In The Panel Data Model With Spatial Correlation: The Case Of Liquor, Badi H. Baltagi, Dong Li
Prediction In The Panel Data Model With Spatial Correlation: The Case Of Liquor, Badi H. Baltagi, Dong Li
Center for Policy Research
This paper considers the problem of prediction in a panel data regression model with spatial autocorrelation in the context of a simple demand equation for liquor. This is based on a panel of 43 states over the period 1965-1994. The spatial autocorrelation due to neighboring states and the individual heterogeneity across states is taken explicitly into account. We compare the performance of several predictors of the states demand for liquor for one year and five years ahead. The estimators whose predictions are compared include OLS, fixed effects ignoring spatial correlation, fixed effects with spatial correlation, random effects GLS estimator ignoring …
Estimating Heterogeneous Capacity And Capacity Utilization In A Multi-Species Fishery, Ronald G. Felthoven, William C. Horrace, Kurt E. Schnier
Estimating Heterogeneous Capacity And Capacity Utilization In A Multi-Species Fishery, Ronald G. Felthoven, William C. Horrace, Kurt E. Schnier
Center for Policy Research
We use a stochastic production frontier model to investigate the presence of heterogeneous production and its impact on fleet capacity and capacity utilization in a multi-species fishery. Furthermore, we propose a new fleet capacity estimate that incorporates complete information on the stochastic differences between each vessel-specific technical efficiency distribution. Results indicate that ignoring heterogeneity in production technologies within a multi-species fishery, as well as the complete distribution of a vessel's technical efficiency score, may yield erroneous fleet-wide production profiles and estimates of capacity. Furthermore, our new estimate of capacity enables out-of-sample production predictions predicated on either homogeneity or heterogeneity modeling …