We enumerate smooth rational curves on very general Weierstrass fibrations over hypersurfaces in projective space. The generating functions for these numbers lie in the ring of classical modular forms. The method of proof uses topological intersection products on a period stack and the cohomological theta correspondence of Kudla and Millson for special cycles on a locally symmetric space of orthogonal type. The results here apply only in base degree 1, but heuristics for higher base degree match predictions from the topological string partition function.

This paper is concerned with a leader-follower problem for a multi-agent system with a switching interconnection topology. Distributed observers are designed for the second-order follower-agents, under the common assumption that the velocity of the active leader cannot be measured in real time. Some dynamic neighbor-based rules, consisting of distributed controllers and observers for the autonomous agents, are developed to keep updating the information of the leader. With the help of an explicitly constructed common Lyapunov function (CLF), it is proved that each agent can follow the active leader. Moreover, the tracking error is estimated even in a noisy environment. Finally, a numerical example is given for illustration.

AbstractRussia’s border regions have had moments as open ‘gateways’ to cooperation. More often, however, the border has been viewed as a ‘garrison’: an outpost of state power. This chapter places the Russian Far East in the broader context of Russia’s pursuit of economic development and security concerns, noting that Russian foreign policy is not necessarily uniform: there are elements of compartmentalization/disaggregation along geographical vectors. The chapter broaches the question that informs all case studies in this volume: has Russia intensified its diplomatic and economic outreach to its eastern border areas and beyond because of the recent breakdown in relations with the West—or would such a shift have taken place anyway, given the economic pull of the Asia-Pacific region?

We consider the problem of counting motifs in bipartite affiliation networks, such as author-paper, user-product, and actor-movie relations. We focus on counting the number of occurrences of a "butterfly", a complete $2 \times 2$ biclique, the simplest cohesive higher-order structure in a bipartite graph. Our main contribution is a suite of randomized algorithms that can quickly approximate the number of butterflies in a graph with a provable guarantee on accuracy. An experimental evaluation on large real-world networks shows that our algorithms return accurate estimates within a few seconds, even for networks with trillions of butterflies and hundreds of millions of edges.

We prove combinatorial theorems concerning the stick principle and cardinal characteristics.

In recent years, investigations of gravitational interactions has led us to discover new facets of the fundamental force. With these discoveries the general theory of relativity is under greater scrutiny now than it was 100 years ago. Development of a more advanced theory is needed for experimental tests and extensive predictions of gravitational interactions in local and cosmological settings. In this work, we present axioms of a novel theory that describes spacetime as a relativistic viscoelastic continuum; retaining the properties of the natural ${}^{+}\Lambda$-CDM vacuum. With these axioms, we provide a foundation for a tensor field theory of Gravitational Strain; introduced as an advanced formulation of an elastic interpretation of gravitational interactions. This viscoelastic definition of gravity is a natural advancement of general relativity to an observable measure of a proper relativistic gravitational field.

We solve explicitly the Goldstone modes in spontaneously symmetry breaking models with supersymmetry. We find that, when more than one fields or representations contribute to the symmetry breaking, there exist identities among the Clebsch-Gordan coefficients which can be used as consistent checks on the calculations.

Telephone call centers offer a convenient communication channel between businesses and their customers. Efficient management of call centers needs accurate modeling of customer waiting behavior, which contains important information about customer patience (how long a customer is willing to wait) and service quality (how long a customer needs to wait to get served). Hazard functions offer dynamic characterization of customer waiting behavior, and provide critical inputs for agent scheduling. Motivated by this application, we develop a two-way functional hazards (tF-Hazards) model to study customer waiting behavior as a function of two timescales, waiting duration and the time of day that a customer calls in. The model stems from a two-way piecewise constant hazard function, and imposes low-rank structure and smoothness on the hazard rates to enhance interpretability. We exploit an alternating direction method of multipliers (ADMM) algorithm to optimize a penalized likelihood function of the model. We carefully analyze the data from a US bank call center, and provide informative insights about customer patience and service quality patterns along waiting time and across different times of a day. The findings provide primitive inputs for call center agent staffing and scheduling, as well as for call center practitioners to understand the effect of system protocols on customer waiting behavior.

We improve the estimates for the Ekeland--Hofer--Zehnder capacity of convex bodies by Gluskin and Ostrover. In the course of our argument we show that a closed characteristic of minimal action on the boundary of a centrally symmetric convex body in $\mathbb R^{2n}$ must itself be centrally symmetric, with generalizations to some other types of symmetry.

Our confidence set quantifies the statistical uncertainty from data-driven cluster assignment in clustered panel models. It covers the true cluster memberships jointly for all units with pre-specified probability and is constructed by inverting many simultaneous unit-specific one-sided tests for group membership. We justify our approach under $N, T \to \infty$ asymptotics using tools from high-dimensional statistics, some of which we extend or develop in this paper. We provide an empirical application as well as Monte Carlo evidence that the confidence set has adequate coverage in finite samples.