# 2017 Mathematical Sciences Publications and Grants

*Indicates student co-author

Askoy, Asuman Güven. Review of “s-Numbers of Compact embeddings of Some Sequence and Function Spaces” by Alicja Dota. *MathSciNet Mathematical Reviews*, 2017, MR3682053.

**Abstract:** In this paper, asymptotic formulas for the behavior of s-numbers (approximation, Gelfand, Kolmogorov and Weyl numbers) of embeddings between weighted sequence spaces are obtained. The concept of uniformly E-porous domain was introduced by H. Triebel in [Function spaces and wavelets on domains, EMS Tracts Math., 7, Eur. Math. Soc., Zurich, 2008; MR2455724]; this concept allows one to use the wavelet characterization of function spaces. Using the estimates obtained in the first sections of the paper, the author derives estimates for the behavior of the s-numbers for Sobolev embeddings between Besov and Triebel-Lizorkin function spaces defined on quasibounded domains.

Askoy, Asuman Güven, Monairah al-Ansari*, Caleb Case*, and Qidi Peng. “Subspace Condition for Bernstein's Lethargy Theorem.”* Turkish Journal of Mathematics*, vol. 41, 2017, pp. 1101-1107.

**Abstract:** In this paper, we consider a condition on subspaces in order to improve bounds given in Bernstein’s lethargy theorem for Banach spaces. Let d1 ≥ d2 ≥ . . . dn ≥ · · · > 0 be an infinite sequence of numbers converging to 0, and let Y1 ⊂ Y2 ⊂ · · · ⊂ Yn ⊂ · · · ⊂ X be a sequence of closed nested subspaces in a Banach space X with the property that Y n ⊂ Yn+1 for all n ≥ 1. We prove that for any c ∈ (0, 1] there exists an element xc ∈ X such that cdn ≤ ρ(xc, Yn) ≤ min(4, a˜)c dn. Here, ρ(x, Yn) = inf{||x − y|| : y ∈ Yn}, a˜ = sup i≥1 sup {qi} { a −3 ni+1−1 } where the sequence {an} is defined as: for all n ≥ 1, an = inf l≥n inf q∈⟨ql ,ql+1,... ⟩ ρ(q, Yl) ||q|| in which each point qn is taken from Yn+1 \ Yn , and satisfies inf n≥1 an > 0. The sequence {ni}i≥1 is given by n1 = 1; ni+1 = min { n ≥ 1 : dn a 2 n ≤ dni } , i ≥ 1.

Aksoy, Asuman G., and Ellis Cumberbatch. “Uncovering GEMS of Mathematics.” *Journal of Humanistic Mathematics*, vol. 7, issue 2, 2017, pp. 384-393.

**Abstract:** Gateway to Exploring Mathematical Sciences (GEMS) is an outreach program offered by the six mathematics departments of the Claremont Colleges for eighth, ninth, and tenth graders. In this paper, we describe our program (in terms of format, participants, mathematical activities and topics involved) and share why we are so enthusiastic about it.

Boettcher, Albrecht, and Lenny Fukshansky. "Addendum to: Lattices from Tight Equiangular Frames." *Linear Algebra and its Applications*, vol. 531, 2017, pp. 592-601.

**Abstract:** This paper supplies additions to our paper in Linear Algebra Appl. 510 (2016) 395-420 on integral spans of tight frames in Euclidean spaces. In that previous paper, we considered the case of an equiangular tight frame (ETF), proving that if its integral span is a lattice then the frame must be rational, but overlooking a simple argument in the reverse direction. Thus our first result here is that the integral span of an ETF is a lattice if and only if the frame is rational. Further, we discuss conditions under which such lattices are eutactic and perfect and, consequently, are local maxima of the packing density function in the dimension of their span. In particular, the unit (276, 23) equiangular tight frame is shown to be eutactic and perfect. More general tight frames and their norm-forms are considered as well, and definitive results are obtained in dimensions two and three.

Fukshansky, Lenny, Oleg German and Nikolay Moshchevitin. "On an Effective Variation of Kronecker's Approximation Theorem Avoiding Algebraic Sets." *Oberwolfach Preprints*, ISSN 1864-7596, OWP 2017 - 28, 2017.

**Abstract:** Let Λ ⊂ ℝ^{n} be an algebraic lattice, coming from a projective module over the ring of integers of a number field Κ. Let Ζ ⊂ ℝ^{n} be the zero locus of a finite collection of polynomials such that Λ ⊈ Ζ or a finite union of proper full-rank sublattices of Λ. Let Κ_{1} be the number field generated over Κ by coordinates of vectors in Λ, and let L_{1},…, L_{t} be linear forms in n variables with algebraic coefficients satisfying an appropriate linear independence condition over Κ_{1}. For each ε > 0 and a ∈ ℝ^{n}, we prove the existence of a vector x ∈ Λ ∖ Ζ of explicitly bounded sup-norm such that

∥L_{i}(x)−a_{i} ∥ < ε

for each 1 ≤ *i* ≤ *t*, where ∥ ∥ stands for the distance to the nearest integer. The bound on sup-norm of *x* depends on *ε*, as well as on Λ, Κ, Z and heights of linear forms. This presents a generalization of Kronecker's approximation theorem, establishing an effective result on density of the image of Λ ∖ Ζ under the linear forms L_{1},…, L_{t} in the *t*-torus ~ ℝ^{t} / Ζ^{t}. In the appendix, we also discuss a construction of badly approximable matrices, a subject closely related to our proof of effective Kronecker's theorem, via Liouville-type inequalities and algebraic transference principles.

Fukshansky, Lenny, Pavel Guerzhoy, and Florian Luca. "On Arithmetic Lattices in the Plane." *Proceedings of the American Mathematical Society*, vol. 145, no. 4, 2017, pp. 1453-1465.

**Abstract:** 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.

**External Grant:** Fukshansky, Lenny. Simons Foundation. Mathematics and Physical Sciences-Collaboration Grants for Mathematicians. Project Title: *Lattices, Quadratic Forms, and Height Functions* Institution: Claremont McKenna College. Award Number: 519058. Period: 2017 - 2022. Amount: $42,000.

I work in Number Theory and Discrete Geometry with connections to Combinatorics and Arithmetic Geometry. Most of my work in the past five years focused on algebraic constructions and distribution of arithmetic lattices with good properties, as well as on distribution of zeros of algebraic quadratic forms with respect to height. These two seemingly different directions are connected by their common origins in the Minkowski's geometry of numbers, as well as the common nature of some algebraic, analytic and combinatorial techniques. The first direction has to do with algebraic constructions of lattices with good geometric properties, such as well-roundedness, eutaxy, perfection, extremality, and stability. I also investigate distribution properties and provide counting estimates on various classes of arithmetic and algebraic lattices. The second direction is the effective study of quadratic spaces with respect to a height function. This area grew out of a classical theorem of Cassels on small-height zeros of rational quadratic forms, and aims to establish general results on distribution of such zeros and effective structure of quadratic spaces over rather general fields of arithmetic interest.

Cloud, Kirkwood, and Mark Huber. "Fast perfect simulation of Vervaat perpetuities." *Journal of Complexity*, vol. 42, 2017, pp. 19-30.

**Abstract:** This work presents a new method of simulating exactly from a distribution known as a Vervaat perpetuity. This type of perpetuity is indexed by a parameter β. The new method has a bound on the expected run time which is polynomial in β (as β goes to infinity). This is much faster than the previously best known bound due to an earlier method of Fill and the second author, which ran in expected time exp(βln(β)+Θ(β)) as β→∞. The earlier method utilized dominated coupling from the past to place bounds on a stochastic process for perpetuities from above. By extending to an update function that changes based on the dominating process, it is possible to create a new method that bounds the perpetuities from both above and below. This new approach is shown to run in expected time O(βln(β)) as β→∞.

Huber, Mark. "A Bernoulli mean estimate with known relative error distribution." *Random Structures & Algorithms*, vol. 50, issue 2, 2017, pp. 173-182.

**Abstract:** Suppose that *X*_{1},*X*_{2} ... are independent identically distributed Bernoulli random variables with mean *p*, so ℙ(*X*_{i} = 1) = *p* and ℙ(*X*_{i} = 0) = 1 - *p*. Any estimate *p̂* of *p* has relative error *p̂*/*p* - 1. This paper builds a new estimate *p̂* of *p* with the remarkable property that the relative error of the estimate does not depend in any way on the value of *p*. This allows the easy construction of exact confidence intervals for *p* of any desired level without needing any sort of limit or approximation. In addition, *p̂* is unbiased. For ∊ and δ in (0, 1), to obtain an estimate where ℙ(|*p̂*/*p* - 1| > ∊) ≤ δ, the new algorithm takes on average at most 2∊^{-2}*p*^{-1}⌈n(2δ^{-1})(1 - (4/3)∊)^{-1} samples. It is also shown that any such algorithm that applies whenever *p* ≤ 1/2 requires at least (1/5)∊^{-2}(1 + 2∊)(1 - δ)⌈n((2 - δ)δ^{-1})*p*^{-1} samples on average. The same algorithm can also be applied to estimate the mean of any random variable that falls in [0,1]. The *p̂* used here employs randomness external to the sample, and has a small (but nonzero) chance of being above 1. It is shown that any nontrivial *p̂* where the relative error is independent of *p* must also have these properties. Applications of this methodology include finding exact *p*-values and randomized approximation algorithms for #*P* complete problems.

Huber, Mark. "Optimal linear Bernoulli factories for small mean problems." *Methodology and Computing in Applied Probability*, vol. 19, issue 2, 2017, pp. 631-645.

**Abstract:** Suppose a coin with unknown probability *p* of heads can be flipped as often as desired. A Bernoulli factory for a function *f* is an algorithm that uses flips of the coin together with auxiliary randomness to flip a single coin with probability *f(p)* of heads. Applications include perfect sampling from the stationary distribution of certain regenerative processes. When *f* is analytic, the problem can be reduced to a Bernoulli factory of the form *f(p)* = *Cp* for constant *C*. Presented here is a new algorithm that for small values of *Cp*, requires roughly only *C* coin flips. From information theoretic considerations, this is also conjectured to be (to first order) the minimum number of flips needed by any such algorithm. For large values of *Cp*, the new algorithm can also be used to build a new Bernoulli factory that uses only 80% of the expected coin flips of the older method. In addition, the new method also applies to the more general problem of a linear multivariate Bernoulli factory, where there are *k* coins, the *k*th coin has unknown probability *p*_{k} of heads, and the goal is to simulate a coin flip with probability *C*_{1}*p*_{1} + ⋯ + *C*_{k}*p*_{k} of heads.

Akhmetgaliyev, Eldar, Chiu-Yen Kao, and Braxton Osting. "Computational Methods for Extremal Steklov Problems." *SIAM Journal on Control and Optimization*, vol. 55, issue 2, 2017, pp. 1226-1240.

**Abstract:** We develop a computational method for extremal Steklov eigenvalue problems and apply it to study the problem of maximizing the *p*th Steklov eigenvalue as a function of the domain with a volume constraint. In contrast to the optimal domains for several other extremal Dirichlet- and Neumann-Laplacian eigenvalue problems, computational results suggest that the optimal domains for this problem are very structured. We reach the conjecture that the domain maximizing the *p*th Steklov eigenvalue is unique (up to dilations and rigid transformations), has *p*-fold symmetry, and has at least one axis of symmetry. The *p*th Steklov eigenvalue has multiplicity 2 if p is even and multiplicity 3 if *p* ≥ 3 is odd.

Kang, Di^{*}, and Chiu-Yen Kao. "Minimization of Inhomogeneous Biharmonic Eigenvalue Problems." *Applied Mathematical Modelling*, vol. 51, 2017, pp. 587-604.

**Abstract:** Biharmonic eigenvalue problems arise in the study of the mechanical vibration of plates. In this paper, we study the minimization of the first eigenvalue of a simplified model with clamped boundary conditions and Navier boundary conditions with respect to the coefficient functions which are of bang-bang type (the coefficient functions take only two different constant values). A rearrangement algorithm is proposed to find the optimal coefficient function based on the variational formula of the first eigenvalue. On various domains, such as square, circular and annular domains, the region where the optimal coefficient function takes the larger value may have different topologies. An asymptotic analysis is provided when two different constant values are close to each other. In addition, a symmetry breaking behavior is also observed numerically on annular domains.

Kao, Chiu-Yen, Rongjie Lai, and Braxton Osting. "Maximization of Laplaceâˆ' Beltrami Eigenvalues on Closed Riemannian Surfaces." *ESAIM: Control, Optimisation and Calculus of Variations*, vol. 23, no. 2, 2017, pp. 685-720.

**Abstract:** Let (*M,g*) be a connected, closed, orientable Riemannian surface and denote by λ_{k}(*M,g*) the *k*th eigenvalue of the Laplace−Beltrami operator on (*M,g*). In this paper, we consider the mapping (*M,g*) → λ_{k}(*M,g*). We propose a computational method for finding the conformal spectrum Λ^{c}_{k}(*M*,[g0]), which is defined by the eigenvalue optimization problem of maximizing λ_{k}(*M,g*) for *k* fixed as *g* varies within a conformal class [ g_{0} ] of fixed volume vol(*M,g*) = 1. We also propose a computational method for the problem where *M* is additionally allowed to vary over surfaces with fixed genus, γ. This is known as the topological spectrum for genus γ and denoted by Λ^{t}_{k}(γ). Our computations support a conjecture of [N. Nadirashvili, *J. Differ. Geom. 61* (2002) 335-340.] that Λ

^{t}

_{k}(0) = 8π

*k*, attained by a sequence of surfaces degenerating to a union of

*k*identical round spheres. Furthermore, based on our computations, we conjecture that Λ

^{t}

_{k}(1) = 8π2/√3 + 8π(

*k*− 1), attained by a sequence of surfaces degenerating into a union of an equilateral flat torus and

*k*− 1 identical round spheres. The values are compared to several surfaces where the Laplace−Beltrami eigenvalues are well-known, including spheres, flat tori, and embedded tori. In particular, we show that among flat tori of volume one, the

*k*th Laplace−Beltrami eigenvalue has a local maximum with value λ

_{k}= 4π

^{2}⌈k/2⌉

^{2}(⌈

*k*/2⌉

^{2}− 1/4)

^{-1/2}. Several properties are also studied computationally, including uniqueness, symmetry, and eigenvalue multiplicity.

**External Grant:** Kao, Chiu-Yen. Collaboration Grants for Mathematicians, Simons Foundation. Title: “Numerical Methods for PDEs and Shape Optimization,” September 1, 2017-August 30, 2022.

Needell, Deanna, and Sam Nelson. "Biquasiles and Dual Graph Diagrams." *Journal of Knot Theory and Its Ramifications*, vol. 26, no. 8, 2017, 1750048.

**Abstract:** 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 and provide examples.

Churchill, Indu R.U., Mohamed Elhamdadi, Mustafa Hajij, and Sam Nelson. "Singular knots and involutive quandles." *Journal of Knot Theory and Its Ramifications*, vol. 26, no. 14, 2017, 1750099.

**Abstract:** The aim of this paper is to define certain algebraic structures coming from generalized Reidemeister moves of singular knot theory. We give examples, show that the set of colorings by these algebraic structures is an invariant of singular links. As an application we distinguish several singular knots and links.

Ishii, Atsushi, and Sam Nelson. "Partially Multiplicative Biquandles and Handlebody-Knots." *Contemporary Mathematics*, vol. 689, 2017, pp. 159-176.

**Abstract:** We introduce several algebraic structures related to handlebody-knots, including G-families of biquandles, partially multiplicative biquandles and group decomposable biquandles. These structures can be used to color the semiarcs in Y-oriented spatial trivalent graph diagrams representing S1-oriented handlebody-knots to obtain computable invariants for handlebody-knots and handlebody-links. In the case of G-families of biquandles, we enhance the counting invariant using the group G to obtain a polynomial invariant of handlebody-knots.

Needell, Deanna, and Sam Nelson. "Biquasiles and Dual Graph Diagrams." *Journal of Knot Theory and Its Ramifications*, vol. 26, no. 8, 2017, 1750048.

**Abstract:** 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 and provide examples.

Nelson, Sam, Michael E. Orrison, and Veronica Rivera. "Quantum Enhancements and Biquandle Brackets." *Journal of Knot Theory and Its Ramifications*, vol. 26, no. 05, 2017, 1750034.

**Abstract:** We introduce a new class of quantum enhancements we call biquandle brackets, which are customized skein invariants for biquandle colored links.Quantum enhancements of biquandle counting invariants form a class of knot and link invariants that includes biquandle cocycle invariants and skein invariants such as the HOMFLY-PT polynomial as special cases, providing an explicit unification of these apparently unrelated types of invariants. We provide examples demonstrating that the new invariants are not determined by the biquandle counting invariant, the knot quandle, the knot group or the traditional skein invariants.

Nelson, Sam, and Jake Rosenfield. "Bikei Homology." *Homology, Homotopy, and Applications*, vol. 19, no. 1, 2017, pp. 23-35.

**Abstract:** We introduce a modified homology and cohomology theory for involutory biquandles (also known as bikei). We use bikei 2-cocycles to enhance the bikei counting invariant for unoriented knots and links as well as unoriented and non-orientable knotted surfaces in ℝ^{4}.

Nelson, Sam, and Natsumi Oyamaguchi. “Trace Diagrams and Biquandle Brackets.” *International Journal of Mathematics*, vol. 28, no. 14, 2017, 1750104.

**Abstract:** We introduce a method of computing biquandle brackets of oriented knots and links using a type of decorated trivalent spatial graphs we call trace diagrams. We identify algebraic conditions on the biquandle bracket coeﬃcients for moving strands over and under traces and identify a new stop condition for the recursive expansion. In the case of monochromatic crossings we show that biquandle brackets satisfy a Homﬂypt-style skein relation and we identify algebraic conditions on the biquandle bracket coeﬃcients to allow pass-through trace moves.