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Articles 1 - 30 of 596
Full-Text Articles in Physical Sciences and Mathematics
Regular Solutions To Elliptic Equations, Alfonso Castro, Jon T. Jacobsen
Regular Solutions To Elliptic Equations, Alfonso Castro, Jon T. Jacobsen
All HMC Faculty Publications and Research
A review of results and techniques on the existence of regular radial solutions to second order elliptic boundary value problems driven by linear and quasilinear operators is presented. Of particular interest are results where the solvability of a given elliptic problem can be analyzed by the relationship between the spectrum of the operator and the behavior of the nonlinearity near infinity and at zero. Energy arguments and Pohozaev type identities are used extensively in that analysis. An appendix with a proof of the contraction mapping principle best suited for using continuous dependence to ordinary differential equations on initial conditions is …
Stable Trace Ideals And Applications, Haydee Lindo, Hai Long Dao
Stable Trace Ideals And Applications, Haydee Lindo, Hai Long Dao
All HMC Faculty Publications and Research
We study stable trace ideals in one dimensional local Cohen–Macaulay rings and give numerous applications.
Infinitely Many Radial Solutions For A P-Laplacian Problem With Indefinite Weight, Alfonso Castro, Jorge Cossio, Sigifredo Herrón, Carlos Vélez
Infinitely Many Radial Solutions For A P-Laplacian Problem With Indefinite Weight, Alfonso Castro, Jorge Cossio, Sigifredo Herrón, Carlos Vélez
All HMC Faculty Publications and Research
We prove the existence of infinitely many sign changing radial solutions for a p-Laplacian Dirichlet problem in a ball. Our problem involves a weight function that is positive at the center of the unit ball and negative in its boundary. Standard initial value problems-phase plane analysis arguments do not apply here because solutions to the corresponding initial value problem may blow up near the boundary due to the fact that our weight function is negative at the boundary. We overcome this difficulty by connecting the solutions to a singular initial value problem with those of a regular initial value problem …
Infinitely Many Stability Switches In A Problem With Sublinear Oscillatory Boundary Conditions, Alfonso Castro, Rosa Pardo
Infinitely Many Stability Switches In A Problem With Sublinear Oscillatory Boundary Conditions, Alfonso Castro, Rosa Pardo
All HMC Faculty Publications and Research
We consider the elliptic equation −u+u = 0 with nonlinear boundary condition ∂u ∂n = λu + g(λ, x, u), where g(λ,x,s) s → 0, as |s|→∞ and g is oscillatory. We provide sufficient conditions on g for the existence of unbounded sequences of stable solutions, unstable solutions, and turning points, even in the absence of resonant solutions.
Existence Of Solutions To A Semilinear Elliptic Boundary Value Problem With Augmented Morse Index Bigger Than Two, Alfonso Castro, Ivan Ventura
Existence Of Solutions To A Semilinear Elliptic Boundary Value Problem With Augmented Morse Index Bigger Than Two, Alfonso Castro, Ivan Ventura
All HMC Faculty Publications and Research
Building on the construction of least energy sign-changing solutions to variational semilinear elliptic boundary value problems introduced in [A. Castro, J. Cossio and J.M. Neuberger, Sign changing solutions for a superlinear Dirichlet problem, Rocky Mountain J. Math. 27 (1997), 1041--1053], we prove the existence of a solution with augmented Morse index at least three when a sublevel of the corresponding action functional has nontrivial topology. We provide examples where the set of least energy sign changing solutions is disconnected, hence has nontrivial topology.
Mathematics For Human Flourishing, Francis Su
Mathematics For Human Flourishing, Francis Su
All HMC Faculty Publications and Research
Why does the practice of mathematics often fall short of our ideals and hopes? How can the deeply human themes that drive us to do mathematics be channeled to build a more beautiful and just world in which all can truly flourish?
First Passage Statistics For The Capture Of A Brownian Particle By A Structured Spherical Target With Multiple Surface Traps, Alan E. Lindsay, Andrew Bernoff, Michael J. Ward
First Passage Statistics For The Capture Of A Brownian Particle By A Structured Spherical Target With Multiple Surface Traps, Alan E. Lindsay, Andrew Bernoff, Michael J. Ward
All HMC Faculty Publications and Research
We study the first passage time problem for a diffusing molecule in an enclosed region to hit a small spherical target whose surface contains many small absorbing traps. This study is motivated by two examples of cellular transport. The first is the intracellular process through which proteins transit from the cytosol to the interior of the nucleus through nuclear pore complexes that are distributed on the nuclear surface. The second is the problem of chemoreception, in which cells sense their surroundings through diffusive contact with receptors distributed on the cell exterior. Using a matched asymptotic analysis in terms of small …
Race, Space, And The Conflict Inside Us, Francis Su
Race, Space, And The Conflict Inside Us, Francis Su
All HMC Faculty Publications and Research
Talking about race is hard. Our nation is wrestling with some open wounds about race. These sores have been around a while, but they have been brought to light recently by technology, politics, and an increasingly diverse population. And regardless of the outcome of the U.S. presidential election, we will all need to work at healing these sores, not just in our personal lives, but in our classrooms and in our profession.
A Sampling Kaczmarz-Motzkin Algorithm For Linear Feasibility, Jesus A. De Loera, Jamie Haddock, Deanna Needell
A Sampling Kaczmarz-Motzkin Algorithm For Linear Feasibility, Jesus A. De Loera, Jamie Haddock, Deanna Needell
CMC Faculty Publications and Research
We combine two iterative algorithms for solving large-scale systems of linear inequalities, the relaxation method of Agmon, Motzkin et al. and the randomized Kaczmarz method. We obtain a family of algorithms that generalize and extend both projection-based techniques. We prove several convergence results, and our computational experiments show our algorithms often outperform the original methods.
Biquasiles And Dual Graph Diagrams, Deanna Needell, Sam Nelson
Biquasiles And Dual Graph Diagrams, Deanna Needell, Sam Nelson
CMC Faculty Publications and Research
We introduce dual graph diagrams representing oriented knots and links. We use these combinatorial structures to define corresponding algebraic structures we call biquasiles whose axioms are motivated by dual graph Reidemeister moves, generalizing the Dehn presentation of the knot group analogously to the way quandles and biquandles generalize the Wirtinger presentation. We use these structures to define invariants of oriented knots and links. In particular, we identify an example of a finite biquasile whose counting invariant distinguishes the chiral knot 9-32 from its mirror image, demonstrating that biquasile counting invariants are distinct from biquandle counting invariants.
Batched Stochastic Gradient Descent With Weighted Sampling, Deanna Needell, Rachel Ward
Batched Stochastic Gradient Descent With Weighted Sampling, Deanna Needell, Rachel Ward
CMC Faculty Publications and Research
We analyze a batched variant of Stochastic Gradient Descent (SGD) with weighted sampling distribution for smooth and non-smooth objective functions. We show that by distributing the batches computationally, a significant speedup in the convergence rate is provably possible compared to either batched sampling or weighted sampling alone. We propose several computationally efficient schemes to approximate the optimal weights, and compute proposed sampling distributions explicitly for the least squares and hinge loss problems. We show both analytically and experimentally that substantial gains can be obtained
Tolerant Compressed Sensing With Partially Coherent Sensing Matrices, Tobias Birnbaum, Yonina C. Eldar, Deanna Needell
Tolerant Compressed Sensing With Partially Coherent Sensing Matrices, Tobias Birnbaum, Yonina C. Eldar, Deanna Needell
CMC Faculty Publications and Research
We consider compressed sensing (CS) using partially coherent sensing matrices. Most of CS theory to date is focused on incoherent sensing, that is, columns from the sensing matrix are highly uncorrelated. However, sensing systems with naturally occurring correlations arise in many applications, such as signal detection, motion detection and radar. Moreover, in these applications it is often not necessary to know the support of the signal exactly, but instead small errors in the support and signal are tolerable. In this paper, we focus on d-tolerant recovery, in which support set reconstructions are considered accurate when their locations match the true …
A Practical Study Of Longitudinal Reference Based Compressed Sensing For Mri, Samuel Birns, Bohyun Kim, Stephanie Ku, Kevin Stangl, Deanna Needell
A Practical Study Of Longitudinal Reference Based Compressed Sensing For Mri, Samuel Birns, Bohyun Kim, Stephanie Ku, Kevin Stangl, Deanna Needell
CMC Faculty Publications and Research
Compressed sensing (CS) is a new signal acquisition paradigm that enables the reconstruction of signals and images from a low number of samples. A particularly exciting application of CS is Magnetic Resonance Imaging (MRI), where CS significantly speeds up scan time by requiring far fewer measurements than standard MRI techniques. Such a reduction in sampling time leads to less power consumption, less need for patient sedation, and more accurate images. This accuracy increase is especially pronounced in pediatric MRI where patients have trouble being still for long scan periods. Although such gains are already significant, even further improvements can be …
Freedom Through Inquiry, Francis Su
Freedom Through Inquiry, Francis Su
All HMC Faculty Publications and Research
I delivered this speech at the Inquiry‐Based Learning Forum & 19th Annual Legacy of R.L. Moore Conference on August 4, 2016. It is partly an homage to an influential teacher, partly an excuse to articulate what makes some styles of teaching so effective, and partly an excuse to talk about difficult issues facing our nation and our classrooms today.
The Problems Of Contemporariness And Voice: Review Of Literacy & Mathematics: A Contemporary Approach To Quantitative Literacy By Jay P. Abramson And Matthew A. Isom (2005), Gizem Karaali
Pomona Faculty Publications and Research
The book under review covers the traditional content of a typical mathematical literacy text. After a brief overview of the book contents, the review then focuses on two specific challenges that QL textbooks have to meet: the timeliness of the contexts used and the subjective author voice that inevitably colors any contextualized discussion. Both issues noticeably arise in the text reviewed. Nonetheless instructors may find it a helpful resource.
One-Bit Compressive Sensing Of Dictionary-Sparse Signals, Richard Baraniuk, Simon Foucart, Deanna Needell, Yaniv Plan, Mary Wootters
One-Bit Compressive Sensing Of Dictionary-Sparse Signals, Richard Baraniuk, Simon Foucart, Deanna Needell, Yaniv Plan, Mary Wootters
CMC Faculty Publications and Research
One-bit compressive sensing has extended the scope of sparse recovery by showing that sparse signals can be accurately reconstructed even when their linear measurements are subject to the extreme quantization scenario of binary samples—only the sign of each linear measurement is maintained. Existing results in one-bit compressive sensing rely on the assumption that the signals of interest are sparse in some fixed orthonormal basis. However, in most practical applications, signals are sparse with respect to an overcomplete dictionary, rather than a basis. There has already been a surge of activity to obtain recovery guarantees under such a generalized sparsity model …
Optimizing Quantization For Lasso Recovery, Xiaoyi Gu, Shenyinying Tu, Hao-Jun Michael Shi, Mindy Case, Deanna Needell, Yaniv Plan
Optimizing Quantization For Lasso Recovery, Xiaoyi Gu, Shenyinying Tu, Hao-Jun Michael Shi, Mindy Case, Deanna Needell, Yaniv Plan
CMC Faculty Publications and Research
This letter is focused on quantized Compressed Sensing, assuming that Lasso is used for signal estimation. Leveraging recent work, we provide a framework to optimize the quantization function and show that the recovered signal converges to the actual signal at a quadratic rate as a function of the quantization level. We show that when the number of observations is high, this method of quantization gives a significantly better recovery rate than standard Lloyd-Max quantization. We support our theoretical analysis with numerical simulations.
Math Education: A Messy Problem, Gizem Karaali
Math Education: A Messy Problem, Gizem Karaali
Pomona Faculty Publications and Research
The current state of math education in America is certainly not ideal, writes Gizem Karaali, but mathematicians, researchers, policy makers and others are working on it -- and it is definitely a problem worth working on.
Collaboration And Creativity In Southern Califonia: An Offering, Gizem Karaali, Ami Radunskaya
Collaboration And Creativity In Southern Califonia: An Offering, Gizem Karaali, Ami Radunskaya
Pomona Faculty Publications and Research
WiMSoCal (Women in Math in Southern California) is a regional conference in its ninth incarnation. The conference is the result of the efforts of Professor Cymra Haskell (USC) to create a supportive local community for women mathematicians. At our first meeting in 2007, a confluence of Ami’s EDGE regional cluster and Cymra’s WISE group at USC, we socialized, got to know each other and brainstormed about what we, as a group, would like to see happen. It was clear that our younger colleagues wanted to meet as mathematicians, sharing intellectual ideas as well as anecdotes from the trenches.
The Power Of Two: Two Tips For Mathematicians, Gizem Karaali
The Power Of Two: Two Tips For Mathematicians, Gizem Karaali
Pomona Faculty Publications and Research
This post is about two great tips involving the number 2 that I learned along the way. They will perhaps not double your happiness or fortune, but I promise you that you will not regret it if you do decide to take them along for the ride.
Review: On Complex Symmetric Toeplitz Operators, Stephan Ramon Garcia
Review: On Complex Symmetric Toeplitz Operators, Stephan Ramon Garcia
Pomona Faculty Publications and Research
No abstract provided.
Multi-Year Optimization Of Malaria Intervention: A Mathematical Model, Harry J. Dudley, Abhishek Goenka '15, Cesar J. Orellana '17, Susan E. Martonosi
Multi-Year Optimization Of Malaria Intervention: A Mathematical Model, Harry J. Dudley, Abhishek Goenka '15, Cesar J. Orellana '17, Susan E. Martonosi
All HMC Faculty Publications and Research
Malaria is a mosquito-borne, lethal disease that affects millions and kills hundreds of thousands of people each year, mostly children. There is an increasing need for models of malaria control. In this paper, a model is developed for allocating malaria interventions across geographic regions and time, subject to budget constraints, with the aim of minimizing the number of person-days of malaria infection.
Review: On Rank One Perturbations Of Complex Symmetric Operators, Stephan Ramon Garcia
Review: On Rank One Perturbations Of Complex Symmetric Operators, Stephan Ramon Garcia
Pomona Faculty Publications and Research
No abstract provided.
Review: A C*-Algebra Approach To Complex Symmetric Operators, Stephan Ramon Garcia
Review: A C*-Algebra Approach To Complex Symmetric Operators, Stephan Ramon Garcia
Pomona Faculty Publications and Research
No abstract provided.
Simulating Surfactant Spreading: Impact Of A Physically Motivated Equation Of State, Dina Sinclair '17, Rachel Levy, Karen E. Daniels
Simulating Surfactant Spreading: Impact Of A Physically Motivated Equation Of State, Dina Sinclair '17, Rachel Levy, Karen E. Daniels
All HMC Faculty Publications and Research
For more than two decades, a single model for the spreading of a surfactant-driven thin liquid film has dominated the applied mathematics literature on the subject. Recently, through the use of fluorescently-tagged lipids, it has become possible to make direct, quantitative comparisons between experiments and models. These comparisons have revealed two important discrepancies between simulations and experiments: the spatial distribution of the surfactant layer, and the timescale over which spreading occurs. In this paper, we present numerical simulations that demonstrate the impact of the particular choice of the equation of state (EoS) relating the surfactant concentration to the surface tension. …
Review: Transitivity And Bundle Shifts, Stephan Ramon Garcia
Review: Transitivity And Bundle Shifts, Stephan Ramon Garcia
Pomona Faculty Publications and Research
No abstract provided.
Existence Of Positive Solutions For A Semipositone P-Laplacian Problem, Alfonso Castro, Djairo G. De Figueredo, Emer Lopera
Existence Of Positive Solutions For A Semipositone P-Laplacian Problem, Alfonso Castro, Djairo G. De Figueredo, Emer Lopera
All HMC Faculty Publications and Research
We prove the existence of positive solutions to a semipositone p-Laplacian problem combining mountain pass arguments, comparison principles, regularity principles and a priori estimates.
Bernstein’S Lethargy Theorem In Fréchet Spaces, Asuman Güven Aksoy, Grzegorz Lewicki
Bernstein’S Lethargy Theorem In Fréchet Spaces, Asuman Güven Aksoy, Grzegorz Lewicki
CMC Faculty Publications and Research
In this paper we consider Bernstein’s Lethargy Theorem (BLT) in the context of Fréchet spaces. Let X be an infinite-dimensional Fréchet space and let V = {Vn} be a nested sequence of subspaces of X such that Vn ⊆ Vn+1 for any n ∈ N and X = S∞ n=1 Vn. Let en be a decreasing sequence of positive numbers tending to 0. Under an additional natural condition on sup{dist(x, Vn)}, we prove that there exists x ∈ X and no ∈ N such that
en/3 ≤ dist(x, V …
Constrained Adaptive Sensing, Mark A. Davenport, Andrew K. Massimino, Deanna Needell, Tina Woolf
Constrained Adaptive Sensing, Mark A. Davenport, Andrew K. Massimino, Deanna Needell, Tina Woolf
CMC Faculty Publications and Research
Suppose that we wish to estimate a vector x∈Cn from a small number of noisy linear measurements of the form y=Ax+z, where z represents measurement noise. When the vector x is sparse, meaning that it has only s nonzeros with s≪n, one can obtain a significantly more accurate estimate of x by adaptively selecting the rows of A based on the previous measurements provided that the signal-to-noise ratio (SNR) is sufficiently large. In this paper we consider the case where we wish to realize the potential of adaptivity but where the rows of A are subject to physical constraints. In …
On Arithmetic Lattices In The Plane, Lenny Fukshansky, Pavel Guerzhoy, Florian Luca
On Arithmetic Lattices In The Plane, Lenny Fukshansky, Pavel Guerzhoy, Florian Luca
CMC Faculty Publications and Research
We investigate similarity classes of arithmetic lattices in the plane. We introduce a natural height function on the set of such similarity classes, and give asymptotic estimates on the number of all arithmetic similarity classes, semi-stable arithmetic similarity classes, and well-rounded arithmetic similarity classes of bounded height as the bound tends to infinity. We also briefly discuss some properties of the j-invariant corresponding to similarity classes of planar lattices.