American Mathematical Society Research Articles

ON FREE ULTRAFILTERS ON $\omega $ WITH WELL-ORDERABLE BASES IN $\mathsf {ZF}$

Abstract In $\mathsf {ZF}$ (i.e., Zermelo–Fraenkel set theory minus the axiom of choice ( $\mathsf {AC}$ )), we investigate the open problem of the deductive strength of the principle UFwob(ω): “There exists a free ultrafilter on ω with a well-orderable base”, which was introduced by Herzberg, Kanovei, Katz, and Lyubetsky [(2018), Journal of Symbolic Logic, 83(1), 385–391]. Typical results are: (1) “ $\aleph _{1}\leq 2^{\aleph _{0}}$ ” is strictly weaker than $\mathsf {UF_{wob}}(\omega )$ in $\mathsf {ZF}$ . (2) “There exists a free ultrafilter on $\omega $ ” does not imply “ $\aleph _{1}\leq 2^{\aleph _{0}}$ ” in $\mathsf {ZF}$ , and thus (by (1)) neither does it imply $\mathsf {UF_{wob}}(\omega )$ in $\mathsf {ZF}$ . This fills the gap in information in Howard and Rubin [Mathematical Surveys and Monographs, American Mathematical Society, 1998], as well as in Herzberg et al. (2018). (3) Martin’s Axiom ( $\mathsf {MA}$ ) implies “no free ultrafilter on $\omega $ has a well-orderable base of cardinality $<2^{\aleph _{0}}$ ”, and the latter principle is not implied by $\aleph _{0}$ -Martin’s Axiom ( $\mathsf {MA}(\aleph _{0})$ ) in $\mathsf {ZF}$ . (4) $\mathsf {MA} + \mathsf {UF_{wob}}(\omega )$ implies $\mathsf {AC}(\mathbb {R})$ (the axiom of choice for non-empty sets of reals), which in turn implies $\mathsf {UF_{wob}}(\omega )$ . Furthermore, $\mathsf {MA}$ and $\mathsf {UF_{wob}}(\omega )$ are mutually independent in $\mathsf {ZF}$ . (5) For any infinite linearly orderable set X, each of “every filter base on X can be well ordered” and “every filter on X has a well-orderable base” is equivalent to “ $\wp (X)$ can be well ordered”. This yields novel characterizations of the principle “every linearly ordered set can be well ordered” in $\mathsf {ZFA}$ (i.e., Zermelo–Fraenkel set theory with atoms), and of $\mathsf {AC}$ in $\mathsf {ZF}$ . (6) “Every filter on $\mathbb {R}$ has a well-orderable base” implies “every filter on $\omega $ has a well-orderable base”, which in turn implies $\mathsf {UF_{wob}}(\omega )$ , and none of these implications are reversible in $\mathsf {ZF}$ . (7) “Every filter on $\omega $ can be extended to an ultrafilter with a well-orderable base” is equivalent to $\mathsf {AC}(\mathbb {R}),$ and thus is strictly stronger than $\mathsf {UF_{wob}}(\omega )$ in $\mathsf {ZF}$ . (8) “Every filter on $\omega $ can be extended to an ultrafilter” implies “there exists a free ultrafilter on $\omega $ which has no well-orderable base of cardinality ${<2^{\aleph _{0}}}$ ”. The former principle does not imply “there exists a free ultrafilter on $\omega $ which has no well-orderable base” in $\mathsf {ZF}$ , and the latter principle is true in the Basic Cohen Model.

Time almost periodicity for solutions of Toda lattice equation with almost periodic initial datum

Abstract This paper analyzes the initial value problem for the Toda lattice with almost periodic initial data: let $J(t; J_{0})$ denote the family of Jacobi matrices which are solutions of the Toda flow equation with initial condition $J(0; J_{0})=J_{0},$ then, the given almost periodic datum $J_{0}$ is a discrete linear Schrödinger operator with almost periodic potential, which plays a fundamental role in our considerations. We show that, under some given hypotheses, the spectrum of the Schrödinger operator is pure absolute continuous and homogeneous (measure-theoretically) by establishing exponential asymptotics on the size of spectral gaps. These two conclusions enable us to show the boundedness and almost periodicity in the time of solutions for Toda lattice equation with almost periodic initial data. As a consequence, our result presents a positive answer to the discrete Deift’s conjecture [Some open problems in random matrix theory and the theory of integrable systems. Integrable Systems and Random Matrices (Contemporary Mathematics, 458). American Mathematical Society, Providence, RI, 2008, pp. 419–430; Some open problems in random matrix theory and the theory of integrable systems. II. SIGMA Symmetry Integrability Geom. Methods Appl.13 (2017), Paper no. 016].

Tau Functions from Joyce Structures

We argued in [Proc. Sympos. Pure Math., Vol. 103, American Mathematical Society, Providence, RI, 2021, 1-66, arXiv:1912.06504] that, when a certain sub-exponential growth property holds, the Donaldson-Thomas invariants of a 3-Calabi-Yau triangulated category should give rise to a geometric structure on its space of stability conditions called a Joyce structure. In this paper, we show how to use a Joyce structure to define a generating function which we call the $\tau$-function. When applied to the derived category of the resolved conifold, this reproduces the non-perturbative topological string partition function of [J. Differential Geom. 115 (2020), 395-435, arXiv:1703.02776]. In the case of the derived category of the Ginzburg algebra of the A$_2$ quiver, we obtain the Painlevé I $\tau$-function.

On a fibrational construction for optics, lenses, and Dialectica categories

Categories of lenses/optics and Dialectica categories are both comprised of bidirectional morphisms of basically the same form. In this work we show how they can be considered a special case of an overarching fibrational construction, generalizing Hofstra's construction of Dialectica fibrations and Spivak's construction of generalized lenses. This construction turns a tower of Grothendieck fibrations into another tower of fibrations by iteratively twisting each of the components, using the opposite fibration construction.Comment: v3: 18 pp. Project results from the American Mathematical Society's Math Research Community on Applied Category Theory 2022. Final version for proceedings of MFPS 2024. Updated author affiliation

Heavenly metrics, hyper‐Lagrangians and Joyce structures

AbstractIn [Proc. Sympos. Pure Math., American Mathematical Society, Providence, RI, 2021, pp. 1–66], Bridgeland defined a geometric structure, named a Joyce structure, conjectured to exist on the space of stability conditions of a triangulated category. Given a non‐degeneracy assumption, a feature of this structure is a complex hyper‐Kähler metric with homothetic symmetry on the total space of the holomorphic tangent bundle. Generalising the isomonodromy calculation which leads to the Joyce structure in [Math. Ann. 385 (2023), 193–258], we obtain an explicit expression for a hyper‐Kähler metric with homothetic symmetry via construction of the isomonodromic flows of a Schrödinger equation with deformed polynomial oscillator potential of odd‐degree . The metric is defined on a total space of complex dimension and fibres over a ‐dimensional manifold which can be identified with the unfolding of the ‐singularity. The hyper‐Kähler structure is shown to be compatible with the natural symplectic structure on in the sense of admitting an affine symplectic fibration as defined in [Lett. Math. Phys. 111 (2021), 54]. Separately, using the additional conditions imposed by a Joyce structure, we consider reductions of Plebański's heavenly equations that govern the hyper‐Kähler condition. We introduce the notion of a projectable hyper‐Lagrangian foliation and show that in dimension four such a foliation of leads to a linearisation of the heavenly equation. The hyper‐Kähler metrics constructed here are shown to admit such a foliation.

Double-Anonymous Peer Review in Mathematics: Implementation for American Mathematical Society Journals

Double-Anonymous Peer Review in Mathematics: Implementation for American Mathematical Society Journals

How to Measure the Researcher Impact with the Aid of its Impactable Area: A Concrete Approach Using Distance Geometry

AbstractAssuming that the subject of each scientific publication can be identified by one or more classification entities, we address the problem of determining a similarity function (distance) between classification entities based on how often two classification entities are used in the same publication. This similarity function is then used to obtain a representation of the classification entities as points of an Euclidean space of a suitable dimension by means of optimization and dimensionality reduction algorithms. This procedure allows us also to represent the researchers as points in the same Euclidean space and to determine the distance between researchers according to their scientific production. As a case study, we consider as classification entities the codes of the American Mathematical Society Classification System.

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Answer to a 1971 Question of Grätzer and Lakser on Pseudocomplemented Lattices

Grätzer and Lakser asked in the 1971 Transactions of the American Mathematical Society if the pseudocomplemented distributive lattices in the amalgamation class of the subvariety generated by 𝟐𝑛 ⊕ 𝟏 can be characterized by the property of not having a *-homomorphism onto 𝟐𝑖 ⊕ 𝟏 for 1 < 𝑖 < 𝑛.In this article, their question from 1971 is answered.

Notice of Self-Retraction and Replacement: A 1971 Question of Grätzer and Lakser on Pseudocomplemented Lattices

Grätzer and Lakser asked in the 1971 Transactions of the American Mathematical Society if the pseudocomplemented distributive lattices in the amalgamation class of the subvariety generated by 2n ⊕ 1 can be characterized by the property of not having a * homomorphism onto 2i ⊕ 1 for 1 < i < n.In this article, this question is answered.

Using Data to Sustain and Measure Student Success Initiatives

This is a follow-up to an earlier case study [Faudree, J. 2021. Courage by experiment, rescue by data. PRIMUS. 31(3–5): 483–491.] describing one Math Department's attempt to improve pass rates in Calculus I by implementing the recommendations from national studies of successful calculus programs [Bressoud, D. and C. Rasmussen. 2015. Seven characteristics of successful calculus programs. Notices of the American Mathematical Society. 62(2): 144–146; Rasmussen, C., J. Ellis, and D. Zazkis. 2014. Lessons learned from case studies of successful calculus programs at five doctoral degree granting institutions. In Proceedings of the 17th Annual Conference on Research in Undergraduate Mathematics Education, pp. 999–1005. Denver, CO.]. This article, along with the earlier one, provide concrete, detailed strategies for promoting departmental change. We describe how the collection and analysis of baseline data has formed the backbone of a sustained, department-driven effort to improve outcomes for undergraduate students and one that weathered the pandemic. For context, our public university is the major research institution of the state, but it is also open-admissions, small (7500 undergraduates) and physically isolated. The effort described here started in 2016 with few resources: no faculty with expertise in mathematics education at the post-secondary level, no support from administration, and no external funding.

Reading the log canonical threshold of a plane curve singularity from its Newton polyhedron

There is a proposition due to Kollár as reported by Kollár (Proceedings of the summer research institute, Santa Cruz, CA, USA, July 9–29, 1995, American Mathematical Society, Providence, 1997) on computing log canonical thresholds of certain hypersurface germs using weighted blowups, which we extend to weighted blowups with non-negative weights. Using this, we show that the log canonical threshold of a convergent complex power series is at most 1/c, where (c,…,c)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(c, \\ldots , c)$$\\end{document} is a point on a facet of its Newton polyhedron. Moreover, in the case n=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n = 2$$\\end{document}, if the power series is weakly normalised with respect to this facet or the point (c, c) belongs to two facets, then we have equality. This generalises a theorem of Varchenko 1982 to non-isolated singularities.

Defining “Ethical Mathematical Practice” Through Engagement with Discipline-Adjacent Practice Standards and the Mathematical Community

This project explored what constitutes “ethical practice of mathematics”. Thematic analysis of ethical practice standards from mathematics-adjacent disciplines (statistics and computing), were combined with two organizational codes of conduct and community input resulting in over 100 items. These analyses identified 29 of the 52 items in the 2018 American Statistical Association Ethical Guidelines for Statistical Practice, and 15 of the 24 additional (unique) items from the 2018 Association of Computing Machinery Code of Ethics for inclusion. Three of the 29 items synthesized from the 2019 American Mathematical Society Code of Ethics, and zero of the Mathematical Association of America Code of Ethics, were identified as reflective of “ethical mathematical practice” beyond items already identified from the other two codes. The community contributed six unique items. Item stems were standardized to, “The ethical mathematics practitioner…”. Invitations to complete the 30-min online survey were shared nationally (US) via Mathematics organization listservs and other widespread emails and announcements. We received 142 individual responses to the national survey, 75% of whom endorsed 41/52 items, with 90–100% endorsing 20/52 items on the survey. Items from different sources were endorsed at both high and low rates. A final thematic analysis yielded 44 items, grouped into “General” (12 items), “Profession” (10 items) and “Scholarship” (11 items). Moreover, for the practitioner in a leader/mentor/supervisor/instructor role, there are an additional 11 items (4 General/7 Professional). These results suggest that the community perceives a much wider range of behaviors by mathematicians to be subject to ethical practice standards than had been previously included in professional organization codes. The results provide evidence against the argument that mathematics practitioners engaged in “pure” or “theoretical” work have minimal, small, or no ethical obligations.

Mathematicians Confront Political Tests: The American Mathematical Society and the Red Scare in 1954

Mathematicians Confront Political Tests: The American Mathematical Society and the Red Scare in 1954

Learning more about derivative: leveraging online resources for varied realizations

Recent literature underlines the increasing use of online platforms in learning undergraduate mathematics, where students refer to these as supplementary resources to develop their mathematical understanding. Through an intrinsic case study, we focus on a highly viewed YouTube learning resource for learning derivative. The selected case is from 3Blue1Brown, a YouTube channel whose founder has received an award from the American Mathematical Society. The video has garnered more than 3.3 million views in the past couple of years. Reflecting on the relevant literature, a realization tree for derivative is developed and then used as an analytical tool to analyze this resource to explore what realizations have been used in it to facilitate students’ understanding of derivative. The findings indicate that the analyzed YouTube resource discusses various realizations of derivative, including all its five main realizations, and effectively utilizes new digital technology for discussing these realizations. Such an exceptional resource for learning mathematics leads us to suggest that mathematics lecturers raise their awareness about such online free resources and incorporate them into their teaching packages when appropriate to facilitate student learning.

Trigonometric approximation of signals (functions) belonging to certain Lipschitz spaces using C δ.C operator

Abstract In 2015, Srivastava and Singh [S. K. Srivastava and U. Singh, Trigonometric approximation of periodic functions belonging to weighted Lipschitz class W ⁢ ( L p , Ψ ⁢ ( t ) , β ) W(L^{p},\Psi(t),\beta) , Function Spaces in Analysis, Contemp. Math. 645, American Mathematical Society, Providence 2015, 283–291] determined the order of approximation of periodic functions belonging to W ⁢ ( L p , Ψ ⁢ ( u ) , β ) {W(L^{p},\Psi(u),\beta)} -class, which is a weighted version of Lip ⁢ ( ω ⁢ ( u ) , p ) {\mathrm{Lip}(\omega(u),p)} -class with weight function sin β ⁢ p ⁡ ( y 2 ) {\sin^{\beta p}(\frac{y}{2})} through matrix means of their trigonometric Fourier series. It is a well-known fact that the product summability methods are stronger than the single summability methods, and they can approximate a wider class of functions. Therefore, in this article, an effort is made to determine the degree of approximation for periodic functions from the same weighted Lipschitz class W ⁢ ( L p , Ψ ⁢ ( u ) , β ) {W(L^{p},\Psi(u),\beta)} , p ≥ 1 {p\geq 1} , using C δ . T {C^{\delta}.T} -means of their trigonometric Fourier series.

A look at FPF rings

An error in Corollary 9.32 of [C. Faith, Rings and Things and a Fine Array of Twentieth Century Associative Algebra, Mathematical Surveys and Monographs, Vol. 65 (American Mathematical Society, Providence, RI, 2004)], motivated us to consider again FPF rings which were initiated by Faith in the 1970s. In this paper, it is shown that a commutative ring [Formula: see text] is reduced FPF if and only if it is [Formula: see text]-semihereditary. We show that when a semiperfect ring with a strongly right bounded basic ring with right and left Ore conditions, is an FPF ring. After some general results, the article focuses on rings of continuous functions. We give some algebraic characterizations for a [Formula: see text] to be FPF and retrieve a result of Jorge Martinez. Also, we show that a space [Formula: see text] is fraction-dense if and only if [Formula: see text] is a continuous ring.

ON EXTERIOR POWERS OF REFLECTION REPRESENTATIONS

AbstractIn 1968, Steinberg [Endomorphisms of Linear Algebraic Groups, Memoirs of the American Mathematical Society, 80 (American Mathematical Society, Providence, RI, 1968)] proved a theorem stating that the exterior powers of an irreducible reflection representation of a Euclidean reflection group are again irreducible and pairwise nonisomorphic. We extend this result to a more general context where the inner product invariant under the group action may not necessarily exist.

American Mathematical Society grants for young scholars

American Mathematical Society grants for young scholars

In the Shadow of the Palms: The Selected Works of David Eugene Smith

In the Shadow of the Palms: The Selected Works of David Eugene Smith