Markov-modulated Affine Processes

We study Markov-modulated affine processes (abbreviated MMAPs), a class of Markov processes that are created from affine processes by allowing some of their coefficients to be a function of an exogenous Markov process. MMAPs allow for richer models in various applications. At the same time MMAPs largely preserve the tractability of standard affine processes, as their characteristic function has a computationally convenient functional form. Our setup is a substantial generalization of earlier work, since we consider the case where the generator of the exogenous process $X$ is an unbounded operator (as is the case for diffusions or jump processes with infinite activity). We prove existence of MMAPs via a martingale problem approach, we derive the formula for their characteristic function and we study various mathematical properties of MMAPs. The paper closes with a discussion of several applications of MMAPs in finance.Comment: 30 page

Microbial carbon use efficiency: accounting for population, community, and ecosystem-scale controls over the fate of metabolized organic matter

Microbial carbon use efficiency (CUE) is a critical regulator of soil organic matter dynamics and terrestrial carbon fluxes, with strong implications for soil biogeochemistry models. While ecologists increasingly appreciate the importance of CUE, its core concepts remain ambiguous: terminology is inconsistent and confusing, methods capture variable temporal and spatial scales, and the significance of many fundamental drivers remains inconclusive. Here we outline the processes underlying microbial efficiency and propose a conceptual framework that structures the definition of CUE according to increasingly broad temporal and spatial drivers where (1) CUEP reflects population-scale carbon use efficiency of microbes governed by species-specific metabolic and thermodynamic constraints, (2) CUEC defines community-scale microbial efficiency as gross biomass production per unit substrate taken up over short time scales, largely excluding recycling of microbial necromass and exudates, and (3) CUEE reflects the ecosystem-scale efficiency of net microbial biomass production (growth) per unit substrate taken up as iterative breakdown and recycling of microbial products occurs. CUEE integrates all internal and extracellular constraints on CUE and hence embodies an ecosystem perspective that fully captures all drivers of microbial biomass synthesis and decay. These three definitions are distinct yet complementary, capturing the capacity for carbon storage in microbial biomass across different ecological scales. By unifying the existing concepts and terminology underlying microbial efficiency, our framework enhances data interpretation and theoretical advances

The arithmetic of genus two curves with (4,4)-split Jacobians

In this paper we study genus 2 curves whose Jacobians admit a polarized (4,4)-isogeny to a product of elliptic curves. We consider base fields of characteristic different from 2 and 3, which we do not assume to be algebraically closed. We obtain a full classification of all principally polarized abelian surfaces that can arise from gluing two elliptic curves along their 4-torsion and we derive the relation their absolute invariants satisfy. As an intermediate step, we give a general description of Richelot isogenies between Jacobians of genus 2 curves, where previously only Richelot isogenies with kernels that are pointwise defined over the base field were considered. Our main tool is a Galois theoretic characterization of genus 2 curves admitting multiple Richelot isogenies.Comment: 30 page

TikTok Tourette\u27s: Are we witnessing a rise in functional tic-like behavior driven by adolescent social media use?

Perceptions of Tourette syndrome (TS) and tic disorders are often driven by social media. During the COVID-19 pandemic, social media consumption greatly increased, particularly in the adolescent population. In parallel with increased social media consumption, there has also been an increase in tic severity and functional tic-like behavior (FTLB). Given that many of the tic videos posted on social media are misleading, perpetuate false beliefs about TS, or reinforce tic-like behaviors, there is increasing concern that these videos are driving the rapid increase in FTLBs. Several studies have reviewed newly presenting cases of FTLB and have found shared characteristics, including that a higher proportion of affected individuals are female, there is a low proportion with a history of childhood or family tics, and symptom onset is typically acute and develops in the teenage years. In addition, the quality of the tics seen in association with FTLB mirrors many of the tics seen on popular social media channels, with higher rates of coprophenomena, tic attacks, and involvement of the trunk and extremities than is seen with typical tics. FTLBs are likely a specific subgroup of functional tics largely influenced by the portrayal of and growing popularity of functional tics posted on social media during the COVID-19 pandemic. However, several factors, including increased anxiety, social isolation, and social media use in general during the pandemic are likely also contributing factors to the surge of FTLBs seen recently. In this era of increased social media consumption, it will become increasingly important for clinicians to educate patients about where and how medical information is spread, to ensure the best possible diagnosis, treatment, and outcomes for patients

Probabilistic n-Choose-k Models for Classification and Ranking

In categorical data there is often structure in the number of variables that take on each label. For example, the total number of objects in an image and the number of highly relevant documents per query in web search both tend to follow a structured distribution. In this paper, we study a probabilistic model that explicitly includes a prior distribution over such counts, along with a count-conditional likelihood that defines probabilities over all subsets of a given size. When labels are binary and the prior over counts is a Poisson-Binomial distribution, a standard logistic regression model is recovered, but for other count distributions, such priors induce global dependencies and combinatorics that appear to complicate learning and inference. However, we demonstrate that simple, efficient learning procedures can be derived for more general forms of this model. We illustrate the utility of the formulation by exploring applications to multi-object classification, learning to rank, and top-K classification.Engineering and Applied Science

Elliptic curves, modular forms, and sums of Hurwitz class numbers

Let H(N) denote the Hurwitz class number. It is known that if $p$ is a prime, then {equation*} \sum_{|r|<2\sqrt p}H(4p-r^2) = 2p. {equation*} In this paper, we investigate the behavior of this sum with the additional condition $r\equiv c\pmod m$. Three different methods will be explored for determining the values of such sums. First, we will count isomorphism classes of elliptic curves over finite fields. Second, we will express the sums as coefficients of modular forms. Third, we will manipulate the Eichler-Selberg trace for ula for Hecke operators to obtain Hurwitz class number relations. The cases $m=2,3$ and 4 are treated in full. Partial results, as well as several conjectures, are given for $m=5$ and 7.Comment: Preprint of an old pape

Combining System Introspection with User-Provided Description to Support Configuration and Understanding of Pervasive systems

Pervasive computing systems such as smart spaces typically combine multiple embedded and/or mobile sensing, computing and interaction devices. A variety of distributed computing approaches are used to integrate these devices to support coordinated applications. This paper describes how simple user descriptions of (primarily) physical aspects of such a system can be combined with information from system introspection to make the system and its log recordings more understandable to potential users, as well as supporting easier configuration and monitoring, and allowing the expression of certain kinds of system behaviour that are otherwise hard to achieve