Asymptotics of sloshing eigenvalues for a triangular prism

We consider the three-dimensional sloshing problem on a triangular prism whose angles with the sloshing surface are of the form ${\pi}/{2q}$ , where q is an integer. We are interested in finding a two-term asymptotic expansion of the eigenvalue counting function. When both angles are ${\pi}/{4}$ , we compute the exact value of the second term. As for the general case, we conjecture an asymptotic expansion by constructing quasimodes for the problem and computing the counting function of the related quasi-eigenvalues. These quasimodes come from solutions of the sloping beach problem and correspond to two kinds of waves, edge waves and surface waves. We show that the quasi-eigenvalues are exponentially close to real eigenvalues of the sloshing problem. The asymptotic expansion of their counting function is closely related to a lattice counting problem inside a perturbed ellipse where the perturbation is in a sense random. The contribution of the angles can then be detected through that perturbation.

An Intelligent Optimization Strategy Based on Deep Reinforcement Learning for Step Counting

With the popularity of Internet of things technology and intelligent devices, the application prospect of accurate step counting has gained more and more attention. To solve the problems that the existing algorithms use threshold to filter noise, and the parameters cannot be updated in time, an intelligent optimization strategy based on deep reinforcement learning is proposed. In this study, the counting problem is transformed into a serialization decision optimization. This study integrates the noise recognition and the user feedback to update parameters. The end-to-end processing is direct, which alleviates the inaccuracy of step counting in the follow-up step counting module caused by the inaccuracy of noise filtering in the two-stage processing and makes the model parameters continuously updated. Finally, the experimental results show that the proposed model achieves superior performance to existing approaches.

Combinatorial characterization of a certain class of words and a conjectured connection with general subclasses of phylogenetic tree-child networks

The combinatorial study of phylogenetic networks has attracted much attention in recent times. In particular, one class of them, the so-called tree-child networks, are becoming the most prominent ones. However, their combinatorial properties are largely unknown. In this paper we address the problem of exactly counting them. We conjecture a relationship with the cardinality of a certain class of words. By solving the counting problem for the words, and on the basis of the conjecture, several simple recurrence formulas for general cases arise. Moreover, a precise asymptotic analysis is provided. Our results coincide with all current formulas in the literature for particular subclasses of tree-child networks, as well as with numerical results obtained for small networks. We expect that the study of the relationship between the newly defined words and the networks will lead to further combinatoric characterizations of this class of phylogenetic networks.

The Model Counting Competition 2020

Many computational problems in modern society account to probabilistic reasoning, statistics, and combinatorics. A variety of these real-world questions can be solved by representing the question in (Boolean) formulas and associating the number of models of the formula directly with the answer to the question. Since there has been an increasing interest in practical problem solving for model counting over the past years, the Model Counting Competition was conceived in fall 2019. The competition aims to foster applications, identify new challenging benchmarks, and promote new solvers and improve established solvers for the model counting problem and versions thereof. We hope that the results can be a good indicator of the current feasibility of model counting and spark many new applications. In this article, we report on details of the Model Counting Competition 2020, about carrying out the competition, and the results. The competition encompassed three versions of the model counting problem, which we evaluated in separate tracks. The first track featured the model counting problem, which asks for the number of models of a given Boolean formula. On the second track, we challenged developers to submit programs that solve the weighted model counting problem. The last track was dedicated to projected model counting. In total, we received a surprising number of nine solvers in 34 versions from eight groups.

A Biophysical Counting Mechanism for Keeping Time

The ability to estimate and produce appropriately timed responses is central to many behaviors including speaking, dancing, and playing a musical instrument. A classical framework for estimating or producing a time interval is the pacemaker-accumulator model in which pulses of a pacemaker are counted and compared to a stored representation. However, the neural mechanisms for how these pulses are counted remains an open question. The presence of noise and stochasticity further complicate the picture. We present a biophysical model of how to keep count of a pacemaker in the presence of various forms of stochasticity using a system of bistable Wilson-Cowan units asymmetrically connected in a one-dimensional array; all units receive the same input pulses from a central clock but only one unit is active at any point in time. With each pulse from the clock, the position of the activated unit changes thereby encoding the total number of pulses emitted by the clock. This neural architecture maps the counting problem into the spatial domain, which in turn translates count to a time estimate. We further extend the model to a hierarchical structure to be able to robustly achieve higher counts.

Semi-Supervised Domain Adaptation for Holistic Counting under Label Gap

This paper proposes a novel approach for semi-supervised domain adaptation for holistic regression tasks, where a DNN predicts a continuous value y∈R given an input image x. The current literature generally lacks specific domain adaptation approaches for this task, as most of them mostly focus on classification. In the context of holistic regression, most of the real-world datasets not only exhibit a covariate (or domain) shift, but also a label gap—the target dataset may contain labels not included in the source dataset (and vice versa). We propose an approach tackling both covariate and label gap in a unified training framework. Specifically, a Generative Adversarial Network (GAN) is used to reduce covariate shift, and label gap is mitigated via label normalisation. To avoid overfitting, we propose a stopping criterion that simultaneously takes advantage of the Maximum Mean Discrepancy and the GAN Global Optimality condition. To restore the original label range—that was previously normalised—a handful of annotated images from the target domain are used. Our experimental results, run on 3 different datasets, demonstrate that our approach drastically outperforms the state-of-the-art across the board. Specifically, for the cell counting problem, the mean squared error (MSE) is reduced from 759 to 5.62; in the case of the pedestrian dataset, our approach lowered the MSE from 131 to 1.47. For the last experimental setup, we borrowed a task from plant biology, i.e., counting the number of leaves in a plant, and we ran two series of experiments, showing the MSE is reduced from 2.36 to 0.88 (intra-species), and from 1.48 to 0.6 (inter-species).

On the people counting problem in smart homes: undirected graphs and theoretical lower-bounds

Smart homes of the future will have to deal with multi-occupancy scenarios. Multi-occupancy systems entail a preliminary and critical feature: the capability of counting people. This can be fulfilled by means of simple binary sensors, cheaper and more privacy preserving than other sensors, such as cameras. However, it is currently unclear how many people can be counted in a smart home, given the set of available sensors. In this paper, we propose a graph-based technique that allows to map a smart home to an undirected graph G and discover the lower-bound of certainly countable people, also defined as certain count. We prove that every independent set of n vertices of an undirected graph G represents a minimum count of n people. We also prove that the maximum number of certainly countable people corresponds to the maximum independent sets of G, and that the maximal independent sets of G provide every combination of active sensors that ensure different minimum count. Last, we show how to use this technique to identify and optimise suboptimal deployment of sensors, so that the assumptions can be tightened and the theoretical lower-bound improved.

(Non)-escape of mass and equidistribution for horospherical actions on trees

Let G be a large group acting on a biregular tree T and $$\Gamma \le G$$ Γ ≤ G a geometrically finite lattice. In an earlier work, the authors classified orbit closures of the action of the horospherical subgroups on $$G/\Gamma $$ G / Γ . In this article we show that there is no escape of mass and use this to prove that, in fact, dense orbits equidistribute to the Haar measure on $$G/\Gamma $$ G / Γ . On the other hand, we show that new dynamical phenomena for horospherical actions appear on quotients by non-geometrically finite lattices: we give examples of non-geometrically finite lattices where an escape of mass phenomenon occurs and where the orbital averages along a Følner sequence do not converge. In the last part, as a by-product of our methods, we show that projections to $$\Gamma \backslash T$$ Γ \ T of the uniform distributions on large spheres in the tree T converge to a natural probability measure on $$\Gamma \backslash T$$ Γ \ T . Finally, we apply this equidistribution result to a lattice point counting problem to obtain counting asymptotics with exponential error term.

Harmonic analysis and statistics of the first Galois cohomology group

We utilize harmonic analytic tools to count the number of elements of the Galois cohomology group $$f\in H^1(K,T)$$ f ∈ H 1 ( K , T ) with discriminant-like invariant $$\text {inv}(f)\le X$$ inv ( f ) ≤ X as $$X\rightarrow \infty $$ X → ∞ . Specifically, Poisson summation produces a canonical decomposition for the corresponding generating series as a sum of Euler products for a very general counting problem. This type of decomposition is exactly what is needed to compute asymptotic growth rates using a Tauberian theorem. These new techniques allow for the removal of certain obstructions to known results and answer some outstanding questions on the generalized version of Malle’s conjecture for the first Galois cohomology group.