* Indicates student co-author
Aksoy, Asuman. Review of “A Remark on Entropy Numbers,” by Vilademir Temlyakov. MathSciNet Mathematical Reviews, 2022, MR4373864.
Aksoy, Asuman Güven and Yunied Puig de Dios. “Ideals of Hypercyclic Operators that Factor Through ℓP.” Proceedings of the American Mathematical Society, vol. 150, no. 2, 2022, pp. 691-700.
Abstract: We study the injective and surjective hull of operator ideals generated by hypercyclic backward weighted shifts that factor through $\ell ^p$.
Blanca, Antonio, Sarah Cannon, and Will Perkins. "Fast and Perfect Sampling of Subgraphs and Polymer Systems." Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques (APPROX/RANDOM 2022), 2022, article no. 4, pp. 4:1-4:18.
Abstract: We give an efficient perfect sampling algorithm for weighted, connected induced subgraphs (or graphlets) of rooted, bounded degree graphs. Our algorithm utilizes a vertex-percolation process with a carefully chosen rejection filter and works under a percolation subcriticality condition. We show that this condition is optimal in the sense that the task of (approximately) sampling weighted rooted graphlets becomes impossible in finite expected time for infinite graphs and intractable for finite graphs when the condition does not hold. We apply our sampling algorithm as a subroutine to give near linear-time perfect sampling algorithms for polymer models and weighted non-rooted graphlets in finite graphs, two widely studied yet very different problems. This new perfect sampling algorithm for polymer models gives improved sampling algorithms for spin systems at low temperatures on expander graphs and unbalanced bipartite graphs, among other applications.
Forst, Maxwell* and Lenny Fukshansky. “Counting Basis Extensions in a Lattice.” Proceedings of the American Mathematical Society, vol. 150, no. 8, 2022, pp. 3199-3213.
Abstract: Given a primitive collection of vectors in the integer lattice, we count the number of ways it can be extended to a basis by vectors with sup-norm bounded by T, producing an asymptotic estimate as T → ∞. This problem can be interpreted in terms of unimodular matrices, as well as a representation problem for a class of multilinear forms. In the 2-dimensional case, this problem is also connected to the distribution of Farey fractions. As an auxiliary lemma we prove a counting estimate for the number of integer lattice points of bounded sup-norm in a hyperplane in R^n. Our main result on counting basis extensions also generalizes to arbitrary lattices in R^n. Finally, we establish some basic properties of sparse representations of integers by multilinear forms.
Fukshansky, Lenny and David Kogan. “Cyclic and Well-Rounded Lattices.” Moscow Journal of Combinatorics and Number Theory, vol. 11, no. 1, 2022, pp. 79-96.
Abstract: We focus on two important classes of lattices, the well-rounded and the cyclic. We show that every well-rounded lattice in the plane is similar to a cyclic lattice and use this cyclic parametrization to count planar well-rounded similarity classes defined over a fixed number field with respect to height. We then investigate cyclic properties of the irreducible root lattices in arbitrary dimensions, in particular classifying those that are simple cyclic, i.e., generated by rotation shifts of a single vector. Finally, we classify cyclic, simple cyclic and well-rounded cyclic lattices coming from rings of integers of Galois algebraic number fields.
Fukshansky, Lenny and Siki Wang*. “Positive Semigroups in Lattices and Totally Real Number Fields.” Advances in Geometry, vol. 22, no. 4, 2022, pp. 503-512.
Abstract: Let L be a full-rank lattice in ℝd and write L+ for the semigroup of all vectors with nonnegative coordinates in L. We call a basis X for L positive if it is contained in L+. There are infinitely many such bases, and each of them spans a conical semigroup S(X) consisting of all nonnegative integer linear combinations of the vectors of X. Such S(X) is a sub-semigroup of L+, and we investigate the distribution of the gaps of S(X) in L+, i.e. the points in L+ ∖ S(X). We describe some basic properties and counting estimates for these gaps. Our main focus is on the restrictive successive minima of L+ and of L+ ∖ S(X), for which we produce bounds in the spirit of Minkowski’s successive minima theorem and its recent generalizations. We apply these results to obtain analogous bounds for the successive minima with respect to Weil heights of totally positive sub-semigroups of ideals in totally real number fields.
External Grant: Fukshansky, Lenny. Avery Teaching Fellow, Claremont Graduate University, Fall 2022.
External Grant: Fukshansky, Lenny. Institute of Mathematical Sciences Award, Claremont Graduate University, Fall 2022.
Huber, Mark. “Generating from the Strauss Process Using Stitching.” International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, vol. 387, 2022, pp. 241-251.
Abstract: The Strauss process is a point process with unnormalized density with respect to a Poisson point process of rate 𝜆, where each pair of points within a specified distance r of each other contributes a factor γ∈[0,1] to the density. Basic acceptance-rejection works spectacularly poorly for this problem, which is why several other perfect simulation methods have been developed. These methods, however, also work poorly for reasonably large values of 𝜆. Recursive Acceptance Rejection Stitching is a new method that works much faster, allowing the simulation of point processes with values of 𝜆 much larger than ever before.
Huber, Mark and Gizem Karaali. “Doughnuts and Ice Cream Cones: Sweet Mathematics.” Journal of Humanistic Mathematics, vol. 12, no. 2, 2022, pp. 1-2.
Abstract: Our issue begins with but does not end with these Ethics in Mathematics articles! As you scroll down our main page or table of contents, you will also find a selection of eclectic essays, two neat short stories, some exquisite poetry, as well as a new Open Call for Poetry to round out the issue. COVID continues to be a presence in our lives, and this issue brings us three articles on this theme. Jane Friedman starts us off by showing how pandemic data can be used to help students understand both exponential growth and social justice concerns. Keith Gallagher and Whitney George then take a look at survey data from 43 students enrolled in an asynchronous online Precalculus course to identify strategies for moving students towards active learning. Finally, Zareen Rahman, Rani Satyam, and Younggon Bae reflect on ways mathematics educators can better prepare their courses during COVID-19.
Huber, Mark and Gizem Karaali. “Mathematical Constants Beyond the Half-Circle: An Open Call for Poetry.” Journal of Humanistic Mathematics, vol. 12, no. 2, 2022, pp. 579-580.
Huber, Mark and Gizem Karaali. “Seeing Mathematics and Seeing Mathematicians.” Journal of Humanistic Mathematics, vol. 12, no. 1, 2022, pp. 1-3.
Abstract: Is mathematics purely abstract thought? If so, there seem to be an awful lot of drawing and visualizations involved. Graphs and cographs, number theory proofs with pattern blocks, even simple proportions used to find heights using shadows. All these are ways of seeing mathematics in action, and all heighten our mathematical intuitions.
Izbicki, Mike. “Aligning Word Vectors on Low-Resource Languages with Wiktionary.” Proceedings of the Fifth Workshop on Technologies for Machine Translation of Low-Resource Languages (LoResMT 2022), 2022, pp. 107-117.
Abstract: Aligned word embeddings have become a popular technique for low-resource natural language processing. Most existing evaluation datasets are generated automatically from machine translations systems, so they have many errors and exist only for high-resource languages. We introduce the Wiktionary bilingual lexicon collection, which provides high-quality human annotated translations for words in 298 languages to English. We use these lexicons to train and evaluate the largest published collection of aligned word embeddings on 157 different languages. All of our code and data is publicly available at https://github.com/mikeizbicki/wiktionary_bli.
Anderson, Heather A., Melissa D. Bailey, Ruth E. Manny, and Chiu-Yen Kao. “Ciliary Muscle Thickness in Adults with Down Syndrome.” Opthalmic and Physiological Optics, vol. 42, issue 4, 2022, pp. 897-903.
Abstract: Purpose: The relationship between ciliary muscle thickness (CMT), age and refractive error was investigated to determine if CMT, like other anterior ocular anatomy, differs in adults with Down syndrome (DS). Methods: The CMT of 33 adults with DS was imaged using anterior segment optical coherence tomography. Images from the right eye obtained 45 minutes after cycloplegia (1% tropicamide, 2.5% phenylephrine) were analysed to calculate thickness at 1, 2 and 3 mm posterior to the scleral spur (CMT1, CMT2, CMT3), maximum thickness (CMTMAX) and apical thickness (AT = CMT1 – CMT2). Spherical equivalent refractive error was determined by clinical refraction using both non-dilated and dilated measures. Multivariate regression analysis evaluated the relationship between CMT and refractive error while controlling for subject age. Results: Images were analysed from 26 subjects (mean age (SD) 29 years; mean refractive error (SD): −0.90 (5.03) D, range: −15.75 to +5.13D). Mean (SD) CMT decreased with posterior position (CMT1: 804 (83) μm; CMT2: 543 (131) μm; CMT3: 312 (100) μm). Mean (SD) CMTMAX and AT was 869 (57) μm and 260 (84) μm, respectively. There was a significant linear correlation indicating thinning CMT with increasing age for CMT1 and CMT2 (p ≤0.05). CMT2 and CMT3 had a significant negative correlation (thicker muscle with increasing myopic refractive error) (p ≤0.01). AT had a significant positive correlation (thicker muscle with increasing hyperopic refractive error) (p <0.01). Conclusions: Ciliary muscle thickness in participants with DS was found to be in a similar range with similar refractive error trends to previous reports of individuals without DS. However, it is important to note that the refractive error trends were driven by individuals with moderate to high levels of myopia.
Ashby, Paul, Max Baroi, Ghanshyam Bhatt, Chiu-Yen Kao, Mikko Kivelä, Peter Kramer, Taras Lakoba, Sean Matz, and Hamza Rusayqat. “Development of Image Processing Algorithms for an AFM Scanner.” Cambridge University Press, 2022.
Abstract: The Lawrence Berkeley National Laboratory working group was presented with three problems involving the improvement of images from an AFM scanner. The first concerned the inpainting of an image for which data is collected via a spiral scan waveform. The second concerned the exploitation of data obtained at different times to obtain better spatial resolution. The third concerned the correction of horizontal positions reported by the sensor, particularly when the scanner is operating at high frequency. Herein is a brief report on the progress and ideas generated by the working group on each of these problems.
Kao, Chiu-Yen and Seyyed Abbas Mohammadi. “Maximal Total Population of Species in a Diffusive Logical Model.” Journal of Mathematical Biology, vol. 85, 2022, article 47.
Abstract: In this paper, we investigate the maximization of the total population of a single species which is governed by a stationary diffusive logistic equation with a fixed amount of resources. For large diffusivity, qualitative properties of the maximizers like symmetry will be addressed. Our results are in line with previous findings which assert that for large diffusion, concentrated resources are favorable for maximizing the total population. Then, an optimality condition for the maximizer is derived based upon rearrangement theory. We develop an efficient numerical algorithm applicable to domains with different geometries in order to compute the maximizer. It is established that the algorithm is convergent. Our numerical simulations give a real insight into the qualitative properties of the maximizer and also lead us to some conjectures about the maximizer.
Kao, Chiu-Yen and Seyyed Abbas Mohammadi. “A Rearrangement Minimization Problem Corresponding to p-Laplacian Equation.” ESAIM: Control, Optimisation and Calculus Variations, vol. 20, 2022, article 11.
Abstract: In this paper a rearrangement minimization problem corresponding to solutions of the p-Laplacian equation is considered. The solution of the minimization problem determines the optimal way of exerting external forces on a membrane in order to have a minimum displacement. Geometrical and topological properties of the optimizer is derived and the analytical solution of the problem is obtained for circular and annular membranes. Then, we find nearly optimal solutions which are shown to be good approximations to the minimizer for specific ranges of the parameter values in the optimization problem. A robust and efficient numerical algorithm is developed based upon rearrangement techniques to derive the solution of the minimization problem for domains with different geometries in ℝ2 and ℝ3.
External Grant: Kao, Chiu-Yen, “DMS 2208373 RUI: Geometric Optimization Involving Partial Differential Equations,” NSF Grant, 2022, $244,985.00.
Abstract: Optimal geometric design provides a vast number of interesting and challenging mathematical problems. One of the famous problems goes back to 18th Century. J.-L. Lagrange formulated the problem to maximize the critical load of a rod of variable cross-sectional area with given length and volume. Another famous classic example is that L. Rayleigh conjectured that the disk should minimize the fundamental frequency of a membrane among all shapes of equal area, more than a century ago. Other recent applications include mechanical vibration, design of optical resonator, photonic crystal waveguides, determination of favorable and unfavorable regions in population dynamics, soap films and minimal surfaces, drug design, and image segmentation. Numerical approaches for these kinds of problems require both forward solvers and optimization solvers. The forward solvers are numerical approaches to solve problems on a given setting of geometric parameters or domain. The optimization solvers aim to find the optimal geometric design, which maximizes the design objective.
In this proposal, the aim is to study geometric optimization of p-Laplacian Poisson’s equations, Laplace Beltrami operator, Steklov problems, and their applications in optimal radiotherapy design and free boundary minimal surfaces. The forward solvers are based on finite element methods and methods of particular solutions while the optimization solvers are based on rearrangement methods, shape derivatives, and sensitivity analysis of conformal factor, conformal classes, and metrics. The PI will study a wide class of problems arising from many applications including (1) optimization of total displacement, (2) convergence rate study of rearrangement methods for optimization problems, (3) optimal radiotherapy design, (4) maximizing conformal and topological Laplace-Beltrami eigenvalues on closed Manifolds, and (5) extremal Steklov eigenvalue problems and free boundary minimal surfaces. The project will advance the development of optimization solvers based on rearrangement methods, shape derivatives, and sensitivity analysis and provide tools to solve aforementioned applications. Also, the obtained results will be integrated to develop new curriculums on numerical analysis and partial differential equations at Claremont McKenna College. The PI will supervise both undergraduate and graduate students. In addition, the PI will organize applied math seminars, working group seminars, and a series of minisymposium at coming AIMS, ICIAM, SIAM, and other international conferences to engage interested scientists, including those from underrepresented groups. The PI and her students will also outreach to K-12 students via Gateway to Exploring Mathematical Sciences (GEMS) program at Claremont.
Wang, Yue, Bhaven A. Mistry, and Tom Chou. “Discrete Stochastic Models of SELEX: Aptamer Capture Probabilities and Protocol Optimization.” Journal of Chemical Physics, vol. 156, 2022, 244103.
Abstract: Antibodies are important biomolecules that are often designed to recognize target antigens. However, they are expensive to produce and their relatively large size prevents their transport across lipid membranes. An alternative to antibodies is aptamers, short (∼15−60 bp) oligonucleotides (and amino acid sequences) with specific secondary and tertiary structures that govern their affinity to specific target molecules. Aptamers are typically generated via solid phase oligonucleotide synthesis before selection and amplification through Systematic Evolution of Ligands by EXponential enrichment (SELEX), a process based on competitive binding that enriches the population of certain strands while removing unwanted sequences, yielding aptamers with high specificity and affinity to a target molecule. Mathematical analyses of SELEX have been formulated in the mass action limit, which assumes large system sizes and/or high aptamer and target molecule concentrations. In this paper, we develop a fully discrete stochastic model of SELEX. While converging to a mass-action model in the large system-size limit, our stochastic model allows us to study statistical quantities when the system size is small, such as the probability of losing the best-binding aptamer during each round of selection. Specifically, we find that optimal SELEX protocols in the stochastic model differ from those predicted by a deterministic model.
Nelson, Sam and Yuqi Zhao*. “Twisted Virtual Bikeigebras and Twisted Virtual Handlebody-Knots.” Journal of Knot Theory and Its Ramifications, vol. 31, no. 2, 2022, 2250010.
Abstract: In this paper, we generalize unoriented handlebody-links to the twisted virtual case, obtaining Reidemeister moves for handlebody-links in ambient spaces of the form Σ×[0,1] for Σ a compact closed 2-manifold up to stable equivalence. We introduce a related algebraic structure known as twisted virtual bikeigebras whose axioms are motivated by the twisted virtual handlebody-link Reidemeister moves. We use twisted virtual bikeigebras to define 𝑋-colorability for twisted virtual handlebody-links and define an integer-valued invariant Φℤ𝑋 of twisted virtual handlebody-links. We provide example computations of the new invariants and use them to distinguish some twisted virtual handlebody-links.
Aksoy, Asuman Güven and Yunied Puig de Dios. “Ideals of Hypercyclic Operators that Factor Through ℓP.” Proceedings of the American Mathematical Society, vol. 150, no. 2, 2022, pp. 691-700.
Abstract: We study the injective and surjective hull of operator ideals generated by hypercyclic backward weighted shifts that factor through ℓP.
Martin, Özgür, Quentin Menet, and Yunied Puig. “Disjoint Frequently Hypercyclic Pseudo-Shifts.” Journal of Functional Analysis, vol. 283, no. 1, 2022, 109474.
Abstract: We obtain a Disjoint Frequent Hypercyclicity Criterion and show that it characterizes disjoint frequent hypercyclicity for a family of unilateral pseudo-shifts on c0(N) and ℓp(N), 1≤p<∞. As an application, we characterize disjoint frequently hypercyclic weighted shifts. We give analogous results for the weaker notions of disjoint upper frequent and reiterative hypercyclicity. Finally, we provide counterexamples showing that, although the frequent hypercyclicity, upper frequent hypercyclicity, and reiterative hypercyclicity coincide for weighted shifts on ℓp(N), this equivalence fails for disjoint versions of these notions.