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**Authors: **Neer Bhardwaj and Lou van den Dries

**On the Pila-Wilkie theorem**

https://arxiv.org/abs/2010.14046

**Abstract:** In this expository paper we give an account of the Pila-Wilkie counting theorem and some of its extensions and generalizations. We use semialgebraic cell decomposition to simplify part of the original proof. We include a full treatment of a result due to Pila and Bombieri, and of a variant of the Yomdin-Gromov theorem that are used in this proof.

**Authors: **Neer Bhardwaj and Minh Chieu Tran

**The additive groups of ℤ and ℚ with predicates for being square-free
Journal: **

*The Journal of Symbolic Logic*

https://doi.org/10.1017/jsl.2020.30

**Abstract:**We consider the four structures (

**ℤ**;SF

^{ℤ}), (

**ℤ**; <,SF

^{ℤ}), (

**ℚ**;SF

^{ℚ}), and (

**ℚ**; <,SF

^{ℚ}) where

**ℤ**is the additive group of integers, SF

^{ℤ}is the set of

*a*>

**ℤ**such that

*v*(

_{p}*a*) < 2 for every prime

*p*and corresponding

*p*-adic valuation

*v*,

_{p}**ℚ**and SF

^{ℚ}are defined likewise for rational numbers, and < denotes the natural ordering on each of these domains. We prove that the second structure is model-theoretically wild while the other three structures are model-theoretically tame. Moreover, all these results can be seen as examples where number-theoretic randomness yields model-theoretic consequences.

**Author:** Robert Dicks

**A higher weight analogue of Ogg's theorem on Weierstrass points**

https://arxiv.org/abs/2006.09520

**Abstract**: For a positive integer N, we say that ∞ is a Weierstrass point on the modular curve X0(N) if there is a non-zero cusp form of weight 2 on Γ0(N) which vanishes at ∞ to order greater than the genus of X0(N). If p is a prime with p∤N, Ogg proved that ∞ is not a Weierstrass point on X0(pN) if the genus of X0(N) is 0. We prove a similar result for even weights k≥4. We also study the space of weight k cusp forms on Γ0(N) vanishing to order greater than the dimension.

**Author:** Robert Dicks

**Congruences for Level 1 cusp forms of half-integral weight**

https://arxiv.org/abs/2012.10587

**Abstract:** Suppose that \ell ≥ 5 is prime. For a positive integer N with 4 | N, previous works studied properties of half-integral weight modular forms on Γ0(N) which are supported on ﬁnitely many square classes modulo \ell, in some cases proving that these forms are congruent to the image of a single variable theta series under some number of iterations of the Ramanujan Θ-operator. Here, we study the analogous problem for modular forms of half-integral weight on SL2(Z). Let η be the Dedekind eta function. For a wide range of weights, we prove that every half-integral weight modular form on SL2(Z) which is supported on ﬁnitely many square classes modulo \ell can be written modulo \ell in terms of η\ell and an iterated derivative of η.

**Author:** Oscar E. González

**Title:** An observation of Rankin on Hankel determinants

**Journal:** *Integers*

**Abstract:** While studying the location of the zeros of the Eisenstein series *E _{k}*(

*z*), Rankin considered the determinants n of an associated Hankel matrix. He observed that the first few possess remarkable factorizations, and expressed the hope that a general theorem explaining these factorizations could be found. In this note we provide such a theorem by giving an explicit formula for Delta

_{n}using work of Kaneko and Zagier on Atkin polynomials.

**Author:** Oscar E. González

**Title: **Effective estimates for the smallest parts function (https://arxiv.org/pdf/2006.15504.pdf)

**Abstract:** We give a substantial improvement for the error term in the asymptotic formula for the smallest parts function spt(n) of Andrews. Our methods depend on an explicit bound for sums of Kloosterman sums of half integral weight on the full modular group.

Authors: J. Connor Grady, Ching Hung Lam, James E. Tener, and Hiroshi Yamauchi

**Classification of extremal vertex operator algebras with two simple modules
Journal:**

*Journal of Mathematical Physics*

https://aip.scitation.org/doi/full/10.1063/1.5121446

**Abstract:**In recent work, Wang and Tener defined a class of “extremal” vertex operator algebras (VOAs), consisting of those with at least two simple modules and conformal dimensions as large as possible for the central charge. In this article, we show that there are exactly 15 character vectors of extremal VOAs with two simple modules. All but one of the 15 character vectors are realized by a previously known VOA. The last character vector is realized by a new VOA with central charge 33.

**Authors:** Chiun-Chuan Chen, Ting-Yang Hsiao, and Li-Chang Hung

**Discrete N-barrier maximum principle for a lattice dynamical system arising in competition models**

**Journal:** *Discrete and Continuous Dynamical Systems*

**Abstract:** We show that an analogous N-barrier maximum principle remains true for lattice systems. This extends the results from continuous equations to discrete equations. In order to overcome the difficulty induced by a discretized version of the classical diffusion in the lattice systems, we propose a more delicate construction of the N-barrier which is appropriate for the proof of the N-barrier maximum principle for lattice systems. As an application of the discrete N-barrier maximum principle, we study a coexistence problem of three species arising from biology and show that the three species cannot coexist under certain conditions.

**Authors: **József Balogh, Béla Csaba, Yifan Jing, and András Pluhár

**Title: On the discrepancies of graphs**

**Journal: ***Electronic Journal of Combinatorics*

https://www.combinatorics.org/ojs/index.php/eljc/article/view/v27i2p12

**Abstract:** In the literature, the notion of discrepancy is used in several contexts, even in the theory of graphs. Here, for a graph $G$, $\{−1, 1\}$ labels are assigned to the edges, and we consider a family $\mathcal{S}_G$ of (spanning) subgraphs of certain types, among others spanning trees, Hamiltonian cycles. As usual, we seek for bounds on the sum of the labels that hold for all elements of $\mathcal{S}_G$, for every labeling.

**Authors: **Yifan Jing and Bojan Mohar

**Title: The genus of complete 3-uniform hypergraphs**

**Journal: ***Journal of Combinatorial Theory Series B*

https://www.sciencedirect.com/science/article/pii/S0095895619300796

**Abstract:** In 1968, Ringel and Youngs confirmed the last open case of the Heawood Conjecture by determining the genus of every complete graph $K_n$. In this paper, we determine both the orientable and the non-orientable genus of $K_n^{(3)}$ when n is even, generalizing Ringel–Youngs Theorems to hypergraphs. Moreover, it is shown that the number of non-isomorphic minimum genus embeddings of $K_n^{(3)}$ is at least $2^{\frac{1}{4}n^2\log n(1-o(1))}$.

**Authors: **Gennian Ge,** **Yifan Jing, Zixiang Xu, and Tao Zhang

**Title: Color isomorphic even cycles and a related Ramsey problem**

**Journal: ***SIAM Journal on Discrete Mathematics*

https://epubs.siam.org/doi/abs/10.1137/20M1329652

**Abstract:** Given a graph $H$ and an integer $k\geq 2$, let $f_{k}(n,H)$ be the smallest number of colors $c$ such that there exists a proper edge coloring of $K_{n}$ with $c$ colors containing no $k$ vertex-disjoint color-isomorphic copies of $H$. Using algebraic properties of polynomials over finite fields, we construct an explicit proper edge coloring of $K_{n}$ and show that $f_{k}(n, C_{4})=\Theta(n)$ when $k\geq 3$. We also consider a related generalized Ramsey problem. For given graphs $G$ and $H,$ let $r(G,H,q)$ be the minimum number of edge colors (not necessarily proper) of $G$, such that the edges of every copy of $H\subseteq G$ together receive at least $q$ distinct colors. We obtain some general lower bounds for $r(K_{n,n},K_{s,t},q)$ with a broad range of $q$.

**Authors: **** **Yifan Jing and Bojan Mohar

**Title: The genus of a random bipartite graph**

**Journal: ***Canadian Journal of Mathematics*

https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/genus-of-a-random-bipartite-graph/AB463CE386957B88BE99FD1C9D2C42D5

**Abstract:** Archdeacon and Grable (1995) proved that the genus of the random graph $G(n,p)$ is almost surely close to $pn^2/2$ if $p\geq 3(\log n)^2 n^{-1/2}$. In this paper we prove an analogous result for random bipartite graphs in $G(n, p_1, p_2)$. If $n_1\geq n_2\gg 1$, phase transitions occur for every positive integer $i$ when $p = \Theta( (n_1n_2)^{-i/(2i+1)} )$. A different behaviour is exhibited when one of the bipartite parts has constant size, i.e., $n_1\gg 1$ and $n_2$ is a constant. In that case, phase transitions occur when $p= \Theta( n_1^{-1/2} )$ and when $p=\Theta( n_1^{-1/3} )$.

**Authors: **** **Yifan Jing, Alexandr Kostochka, Fuhong Ma, Pongpat Sittitrai, and Jingwei Xu

**Title: Defective DP-colorings of sparse multigraphs**

**Journal: ***European Journal of Combinatorics*

https://www.sciencedirect.com/science/article/abs/pii/S0195669820301876

**Abstract:** DP-coloring is a generalization of list coloring developed recently by Dvorak and Postle. We introduce and study (i, j)-defective DP-colorings of multigraphs. We concentrate on sparse multigraphs and consider f_DP (i, j, n) — the minimum number of edges that may have an n-vertex (i, j)-critical multigraph, that is, a multigraph G that has no (i, j)-defective DP-coloring but whose every proper subgraph has such a coloring. For every i and j, we find linear lower bounds on f_DP (i, j, n) that are exact for infinitely many n.

**Authors: **** **Yifan Jing and Shukun Wu

**Title: **The largest (k,l)-sum-free subsets

**Journal: ***Transactions of the American Mathematical Society (to appear)*

https://arxiv.org/abs/2001.05632

Abstract: An old conjecture in additive combinatorics asks: what is the largest sum-free subset of any set of N positive integers? Here the word "largest" should be understood in terms of cardinality. For example, the largest sum-free subset of the first N positive integers has cardinality [(N+1)/2], which is the number of odd integers smaller than N, as well as the number of integers lie in the interval [N/2,N]. In this paper, we study the analogous conjecture on (k,l)-sum-free sets, and confirm the conjecture for infinitely many pairs (k,l).

**Authors: **Vaibhav Karve and Anil N. Hirani

**The complete set of minimal simple graphs that support unsatisfiable 2-CNFs**

**Journal: ***Discrete Applied Mathematics*

https://doi.org/10.1016/j.dam.2019.12.017

**Abstract:** A propositional logic sentence in conjunctive normal form that has clauses of length at most two (a 2-CNF) can be associated with a multigraph in which the vertices correspond to the variables and edges to clauses. We show that every 2-CNF that has been reduced under the application of certain tautologies, is equisatisfiable to a 2-CNF whose associated multigraph is, in fact, a simple graph. Our main result is a complete characterization of graphs that can support unsatisfiable 2-CNF sentences. We show that a simple graph can support an unsatisfiable reduced 2-CNF sentence if and only if it contains any one of four specific small graphs as a topological minor. Equivalently, all reduced 2-CNF sentences supported on a given simple graph are satisfiable if and only if all subdivisions of those four graphs are forbidden as subgraphs of the original graph.

**Authors:** Yu Wu, Gabriel Shindes, Vaibhav Karve, Derrek Yager, Daniel B. Work, Arnab Khakraborty, and Richard B. Sowers

**Congestion barcodes: Exploring the topology of urban congestion using persistent homology**

https://arxiv.org/abs/1707.08557

**Abstract:** This work presents a new method to quantify connectivity in transportation networks. Inspired by the field of topological data analysis, we propose a novel approach to explore the robustness of road network connectivity in the presence of congestion on the roadway. The robustness of the pattern is summarized in a congestion barcode, which can be constructed directly from traffic datasets commonly used for navigation. As an initial demonstration, we illustrate the main technique on a publicly available traffic dataset in a neighborhood in New York City.

**Authors:** Vaibhav Karve, Derrek Yager, Marzieh Abolhelm, Daniel B. Work, and Richard B. Sowers

**Seasonal disorder in urban traffic patterns: A low rank analysis**

**Journal:** *Journal of Big Data Analytics in Transportation*

https://link.springer.com/article/10.1007/s42421-021-00033-4

**Abstract: **This article proposes several advances to sparse nonnegative matrix factorization (SNMF) as a way to identify large-scale patterns in urban traffic data. The input to our model is traffic counts organized by time and location. Nonnegative matrix factorization additively decomposes this information, organized as a matrix, into a linear sum of temporal signatures. Penalty terms encourage this factorization to concentrate on only a few temporal signatures, with weights which are not too large. Our interest here is to quantify and compare the regularity of traffic behavior, particularly across different broad temporal windows. In addition to the rank and error, we adapt a measure introduced by Hoyer to quantify sparsity in the representation. Combining these, we construct several curves which quantify error as a function of rank (the number of possible signatures) and sparsity; as rank goes up and sparsity goes down, the approximation can be better and the error should decreases. Plots of several such curves corresponding to different time windows leads to a way to compare disorder/order at different time scalewindows. In this paper, we apply our algorithms and procedures to study a taxi traffic dataset from New York City. In this dataset, we find weekly periodicity in the signatures, which allows us an extra framework for identifying outliers as significant deviations from weekly medians. We then apply our seasonal disorder analysis to the New York City traffic data and seasonal (spring, summer, winter, fall) time windows. We do find seasonal differences in traffic order.

**Authors:** Vivek Kaushik and Daniele Ritelli

**Evaluation of harmonic sums with integrals
Journal:**

*AMS Quarterly of Applied Mathematics*

**Abstract:**We consider the sums and with being a positive integer. We evaluate these sums with multiple integration, a modern technique. First, we start with three different double integrals that have been previously used in the literature to show which implies Euler's identity Then, we generalize each integral in order to find the considered sums. The dimensional analogue of the first integral is the density function of the quotient of independent, nonnegative Cauchy random variables. In seeking this function, we encounter a special logarithmic integral that we can directly relate to The dimensional analogue of the second integral, upon a change of variables, is the volume of a convex polytope, which can be expressed as a probability involving certain pairwise sums of independent uniform random variables. We use combinatorial arguments to find the volume, which in turn gives new closed formulas for and The dimensional analogue of the last integral, upon another change of variables, is an integral of the joint density function of Cauchy random variables over a hyperbolic polytope. This integral can be expressed as a probability involving certain pairwise products of these random variables, and it is equal to the probability from the second generalization. Thus, we specifically highlight the similarities in the combinatorial arguments between the second and third generalizations.

**Authors: **Marissa Miller

**Stable subgroups of the genus two handlebody group**

https://arxiv.org/abs/2009.05067

**Abstract:** We show that a nitely generated subgroup of the genus two handlebody group is stable if and only if the orbit map to the disk graph is a quasi-isometric embedding. To this end, we prove that the genus two handlebody group is a hierarchically hyperbolic group, and that the maximal hyperbolic space in the hierarchy is quasi-isometric to the disk graph of a genus two handlebody by appealing to a construction of Hamenstdt-Hensel. We then utilize the characterization of stable subgroups of hierarchically hyperbolic groups provided by Abbott-Behrstock-Durham. We also provide a counterexample for the higher genus analogue of the main theorem.

**Author:** M. A. Tursi

**A separable universal homogeneous Banach lattice**

https://arxiv.org/abs/2008.06658

**Abstract:** We prove the existence of a separable approximately ultra-homogeneous Banach lattice \mathfrak{BL} that is isometrically universal for separable Banach lattices. This is done by showing that the class of Banach lattices has the Amalgamation Property, and thus finitely generated Banach lattices form a metric Fraïssé class. Some additional results about the structural properties of \mathfrak{BL} are also proven.

**Authors: **** **Yifan Jing and Shukun Wu

**Title: **The largest (k,l)-sum-free subsets

**Journal: ***Transactions of the American Mathematical Society (to appear)*

https://arxiv.org/abs/2001.05632

**Abstract: **An old conjecture in additive combinatorics asks: what is the largest sum-free subset of any set of N positive integers? Here the word "largest" should be understood in terms of cardinality. For example, the largest sum-free subset of the first N positive integers has cardinality [(N+1)/2], which is the number of odd integers smaller than N, as well as the number of integers lie in the interval [N/2,N]. In this paper, we study the analogous conjecture on (k,l)-sum-free sets, and confirm the conjecture for infinitely many pairs (k,l).

**Authors: **Leonid V. Kovalev and Xuerui Yang

**Algebraic structure of the range of a trigonometric polynomial**

**Journal: ***Bulletin of the Australian Mathematical Society*

https://www.cambridge.org/core/journals/bulletin-of-the-australian-mathematical-society/article/algebraic-structure-of-the-range-of-a-trigonometric-polynomial/91E169A34C4CD273B266C023E2943A8C

**Abstract:** The range of a trigonometric polynomial with complex coefficients can be interpreted as the image of the unit circle under a Laurent polynomial. We show that this range is contained in a real algebraic subset of the complex plane. Although the containment may be proper, the difference between the two sets is finite, except for polynomials with certain symmetry.

**Authors: **Leonid V. Kovalev and Xuerui Yang

**Near-isometric duality of Hardy norms with applications to harmonic mappings**

**Journal: ***Journal of Mathematical Analysis and Applications*

https://www.sciencedirect.com/science/article/abs/pii/S0022247X2030202X?via%3Dihub

**Abstract: **Hardy spaces in the complex plane and in higher dimensions have natural finite-dimensional subspaces formed by polynomials or by linear maps. We use the restriction of Hardy norms to such subspaces to describe the set of possible derivatives of harmonic self-maps of a ball, providing a version of the Schwarz lemma for harmonic maps. These restricted Hardy norms display unexpected near-isometric duality between the exponents 1 and 4, which we use to give an explicit form of harmonic Schwarz lemma.

**Authors: **Leonid V. Kovalev and Xuerui Yang

**Extreme values of the derivative of Blaschke products and hypergeometric polynomials**

https://arxiv.org/abs/2007.09760

**Abstract:** A finite Blaschke product, restricted to the unit circle, is a smooth covering map. The maximum and minimum values of the derivative of this map reflect the geometry of the Blaschke product. We identify two classes of extremal Blaschke products: those that maximize the difference between the maximum and minimum of the derivative, and those that minimize it. Both classes turn out to have the same algebraic structure, being the quotient of two hypergeometric polynomials.

**Authors: **Leonid V. Kovalev and Xuerui Yang

**Fourier series of circle embeddings **

**Journal: ***Computational Methods and Function Theory*

https://link.springer.com/article/10.1007/s40315-019-00263-2

**Abstract: **We study the Fourier series of circle homeomorphisms and circle embeddings, with an emphasis on the Blaschke product approximation and the vanishing of Fourier coefficients. The analytic properties of the Fourier series are related to the geometry of the circle embeddings, and have implications for the curvature of minimal surfaces.

**Authors: **Sachin Shetty, Michael McShane, Linfeng Zhang, Jay P. Kesan, Charles A. Kamhoua, Kevin Kwiat, and Laurent L. Njilla

**Title: **Reducing informational disadvantages to improve cyber risk management

**Journal: ***The Geneva Papers on Risk and Insurance--Issues and Practice*

https://link.springer.com/article/10.1057/s41288-018-0078-3

**Abstract: **Effective cyber risk management should include the use of insurance not only to transfer cyber risk but also to provide incentives for insured enterprises to invest in cyber self-protection. Research indicates that asymmetric information, correlated loss, and interdependent security issues make this difficult if insurers cannot monitor the cybersecurity efforts of the insured enterprises. To address this problem, this paper proposes the Cyber Risk Scoring and Mitigation (CRISM) tool, which estimates cyberattack probabilities by directly monitoring and scoring cyber risk based on assets at risk and continuously updated software vulnerabilities. CRISM also produces risk scores that allow organisations to optimally choose mitigation policies that can potentially reduce insurance premiums.

**Authors:** J.P. Kesan and Linfeng Zhang

**T****itle: **An empirical investigation of the relationship between local government budgets, IT expenditures and cyber losses

**Journal: ***IEEE Transactions on Emerginy Topics in Computing*

https://ieeexplore.ieee.org/document/8708927

**Abstract: **Information technology (IT) is the key component of e-government infrastructures, but at the same time, it makes governments more exposed to the cyber risk. In this study, we take an empirical approach to investigate the cyber risk in the public sector. We describe the most common cyber threats facing local governments and build linear models to explain the relationships between cyber losses, local government budgets and IT expenditures. We find that local governments are affected by cyber incidents more frequently, and disruption incidents that lead to the malfunction of e-government services are on the rise. In addition, the magnitude of cyber losses used to have a strong positive relationship with the affected governments' budget size. However, in recent years, this relationship is weakening, and small local governments are more heavily impacted by cyber incidents than before. Our findings further suggest that investing in information technology is becoming more effective in terms of lowering the loss-to-budget ratio. However, this also means local governments with small budgets do not benefit from this change as much as the large ones do.

**Authors: **Gina Tonn, Jay P. Kesan, Linfeng Zhang, and Jeffrey Czajkowski

**Title: **Cyber risk and insurance for transportation infrastructure

**Journal: ***Transport Policy*

https://doi.org/10.1016/j.tranpol.2019.04.019

**Abstract: **While advances in information technology and interconnectivity have improved efficiency for transportation infrastructure, they have also created higher risk associated with cyber systems. The objective of this study is to inform transportation policy and management in the U.S. by identifying barriers to a robust cyber insurance market and improved cyber resilience for transportation infrastructure. This is accomplished through a mixed-methods approach involving analysis of U.S. cyber incident data for transportation systems and a series of interviews with transportation infrastructure managers and insurers. Contributions include new insights into the nature of cyber risk for transportation infrastructure and recommendations on research needs to improve cyber risk management and insurance. Results indicate that the annual number of transport-related companies affected by cyber incidents and the associated costs are on the rise. The most common incidents involve data breaches, while incidents involving privacy violation have the highest average loss per incident. Cyber risk assessment, mitigation and security measures, and insurance are being implemented to varying degrees in transportation infrastructure systems but are generally inadequate. Infrastructure managers do not currently have the tools to rigorously assess and manage cyber risk. Limited data and models also inhibit the accurate modeling of cyber risk for insurance purposes. Even after improved tools and modeling are developed, insurance purchase can be an important risk management strategy to allow transportation infrastructure systems to recover from cyber incidents.

**Authors: **Jay P. Kesan and Linfeng Zhang

**Title: **Analysis of cyber incident categories based on losses

**Journal: ***ACM Transactions on Management Information Systems*

https://dl.acm.org/doi/10.1145/3418288

**Abstract: **The fact that “cyber risk” is indeed a collective term for various distinct risks creates great difficulty in communications. For example, policyholders of “cyber insurance” contracts often have a limited or inaccurate understanding about the coverage that they have. To address this issue, we propose a cyber risk categorization method using clustering techniques. This method classifies cyber incidents based on their consequential losses for insurance and risk management purposes. As a result, it also reveals the relationship between the causes and the outcomes of incidents. Our results show that similar cyber incidents, which are often not properly distinguished, can lead to very different losses. We hope that our work can clarify the differences between cyber risks and provide a set of risk categories that is feasible in practice and for future studies.

**Authors: **Xiaowei Chen, Wing Fung Chong, Runhuan Feng, and Linfeng Zhang

**Title: **Pandemic risk management: resources contingency planning and allocation

**Journal: **arXiv e-print (https://arxiv.org/abs/2012.03200)

**Abstract: **Repeated history of pandemics, such as SARS, H1N1, Ebola, Zika, and COVID-19, has shown that pandemic risk is inevitable. Extraordinary shortages of medical resources have been observed in many parts of the world. Some attributing factors include the lack of sufficient stockpiles and the lack of coordinated efforts to deploy existing resources to the location of greatest needs. The paper investigates contingency planning and resources allocation from a risk management perspective, as opposed to the prevailing supply chain perspective. The key idea is that the competition of limited critical resources is not only present in different geographical locations but also at different stages of a pandemic. This paper draws on an analogy between risk aggregation and capital allocation in finance and pandemic resources planning and allocation for healthcare systems. The main contribution is to introduce new strategies for optimal stockpiling and allocation balancing spatio-temporal competitions of medical supply and demand.