Asymptotic Properties of One Mathematical Model in Food Engineering under Stochastic Perturbations

For the example of one nonlinear mathematical model in food engineering with several equilibria and stochastic perturbations, a simple criterion for determining a stable or unstable equilibrium is reported. The obtained analytical results are illustrated by detailed numerical simulations of solutions of the considered Ito stochastic differential equations. The proposed criterion can be used for a wide class of nonlinear mathematical models in different applications.

Learning to Diversify for Ensembling of An Amount of Classifiers

Not only common man but also intelligent machine always merge all available decisions to solve problems. However, given an amount of learned classifiers, how to select and combine diverse classifiers for machine is still a grand challenge in the literature for decades of history. In this paper, we introduce a novel approach for classifier ensemble, learning to diversify, which learns to adaptively combine classifiers by considering both accuracy and diversity. Specifically, our approach, learning to diversify via weighted kernels, performs classifier combination by optimizing a direct but simple criterion: maximizing ensemble accuracy and adaptive diversity simultaneously by minimizing a loss function. Given a measure formulation, the diversity is calculated with weighted kernels, i.e., the diversity is measured on the component classifiers’ outputs which are kernelized and weighted. Moreover, we propose an iterative training algorithm for weights optimization, where this loss function is iteratively minimized by estimating the kernel weights in conjunction with the classifier weights. Extensive experiments on a variety of classification benchmark datasets show that the proposed approach consistently outperforms state-of-the-art ensembles.

Decomposable-Net: Scalable Low-Rank Compression for Neural Networks

Compressing DNNs is important for the real-world applications operating on resource-constrained devices. However, we typically observe drastic performance deterioration when changing model size after training is completed. Therefore, retraining is required to resume the performance of the compressed models suitable for different devices. In this paper, we propose Decomposable-Net (the network decomposable in any size), which allows flexible changes to model size without retraining. We decompose weight matrices in the DNNs via singular value decomposition and adjust ranks according to the target model size. Unlike the existing low-rank compression methods that specialize the model to a fixed size, we propose a novel backpropagation scheme that jointly minimizes losses for both of full- and low-rank networks. This enables not only to maintain the performance of a full-rank network {\it without retraining} but also to improve low-rank networks in multiple sizes. Additionally, we introduce a simple criterion for rank selection that effectively suppresses approximation error. In experiments on the ImageNet classification task, Decomposable-Net yields superior accuracy in a wide range of model sizes. In particular, Decomposable-Net achieves the top-1 accuracy of 73.2% with 0.27xMACs with ResNet-50, compared to Tucker decomposition (67.4% / 0.30x), Trained Rank Pruning (70.6% / 0.28x), and universally slimmable networks (71.4% / 0.26x).

The Hilbert series of the irreducible quotient of the polynomial representation of the rational Cherednik algebra of type An−1 in characteristic p for p|n − 1

In this paper, we study the irreducible quotient [Formula: see text] of the polynomial representation of the rational Cherednik algebra [Formula: see text] of type [Formula: see text] over an algebraically closed field of positive characteristic [Formula: see text] where [Formula: see text]. In the [Formula: see text] case, for all [Formula: see text] we give a complete description of the polynomials in the maximal proper graded submodule [Formula: see text], the kernel of the contravariant form [Formula: see text], and subsequently find the Hilbert series of the irreducible quotient [Formula: see text]. In the [Formula: see text] case, we give a complete description of the polynomials in [Formula: see text] when the characteristic [Formula: see text] and [Formula: see text] is transcendental over [Formula: see text], and compute the Hilbert series of the irreducible quotient [Formula: see text]. In doing so, we prove a conjecture due to Etingof and Rains completely for [Formula: see text], and also for any [Formula: see text] and [Formula: see text]. Furthermore, for [Formula: see text], we prove a simple criterion to determine whether a given polynomial [Formula: see text] lies in [Formula: see text] for all [Formula: see text] with [Formula: see text] and [Formula: see text] fixed.

On the Interplay of Direct Topological Factorizations and Cellular Automata Dynamics on Beta-Shifts

We consider the range of possible dynamics of cellular automata (CA) on two-sided beta-shifts [Formula: see text] and its relation to direct topological factorizations. We show that any reversible CA [Formula: see text] has an almost equicontinuous direction whenever [Formula: see text] is not sofic. This has the corollary that non-sofic beta-shifts are topologically direct prime, i.e. they are not conjugate to direct topological factorizations [Formula: see text] of two nontrivial subshifts [Formula: see text] and [Formula: see text]. We also give a simple criterion to determine whether [Formula: see text] is conjugate to [Formula: see text] for a given integer [Formula: see text] and a given real [Formula: see text] when [Formula: see text] is a subshift of finite type. When [Formula: see text] is strictly sofic, we show that such a conjugacy is not possible at least when [Formula: see text] is a quadratic Pisot number of degree [Formula: see text]. We conclude by using direct factorizations to give a new proof for the classification of reversible multiplication automata on beta-shifts with integral base and ask whether nontrivial multiplication automata exist when the base is not an integer.

A simple criterion to design optimal non-pharmaceutical interventions for mitigating epidemic outbreaks

For mitigating the COVID-19 pandemic, much emphasis is made on implementing non-pharmaceutical interventions to keep the reproduction number below one. However, using that objective ignores that some of these interventions, like bans of public events or lockdowns, must be transitory and as short as possible because of their significant economic and societal costs. Here, we derive a simple and mathematically rigorous criterion for designing optimal transitory non-pharmaceutical interventions for mitigating epidemic outbreaks. We find that reducing the reproduction number below one is sufficient but not necessary. Instead, our criterion prescribes the required reduction in the reproduction number according to the desired maximum of disease prevalence and the maximum decrease of disease transmission that the interventions can achieve. We study the implications of our theoretical results for designing non-pharmaceutical interventions in 16 cities and regions during the COVID-19 pandemic. In particular, we estimate the minimal reduction of each region’s contact rate necessary to control the epidemic optimally. Our results contribute to establishing a rigorous methodology to design optimal non-pharmaceutical intervention policies for mitigating epidemic outbreaks.

Hypercalcemia as a marker of decreased bone mineral density in patients with peptic ulcer disease

Purpose of the study. To study calcium-phosphorus metabolism in patients with peptic ulcer (PU) in comparison with bone mineral density (BMD), its age and gender aspects. Materials and methods. 216 people with ulcer were in remission and 154 people without ulcer history and erosive-ulcerative changes during fibrogastroduodenoscopy (FGDS) were examined. All patients underwent clinical and endoscopic studies, dichromatic X-ray densitometry with the calculation of the T-test of the femoral neck (FN) and the lumbar spine (LS). Calcium and blood phosphorus have been studied. Results. In patients with PU compared with patients without this pathology there was a significant decrease of BMD. In 32% of patients with PU osteoporosis (OP) was diagnosed and in 34% of patients - osteopenia. In the group of persons who had no PU osteoporosis was found in 18%, osteopenia in 26% of cases. In both groups women prevailed. Blood calcium was significantly higher in patients with PU as compared to those examined without an ulcer history. A significant increase of calcium was observed in individuals with a comorbid course of PU and OP. Conclusion. The data obtained show that the study of blood calcium can serve as a simple criterion for diagnosing disorders of BMD and, if its values are high, conduct a more detailed examination of patients with PU for early detection of OP and the possibility of its treatment.

A Criterion for the Effective Work of Adhesion in Loading and Unloading of Adhesive Soft Solids from Rough Surfaces

Recently, Dalvi and co-authors have shown detailed experimental data of adhesion of soft spheres with rough substrates with roughness measured down to almost the atomic scale, finding that the Persson and Tosatti theory gave satisfactory predictions of the apparent work of adhesion during loading, once the increase of the surface area due to roughness is correctly computed at extremely small scales. We show that unloading data would show similar correlation with the Persson–Tosatti’s simple criterion, but for a much larger effective work of adhesion, which therefore becomes not an “intrinsic” property. This suggests either strong hysteresis even at apparently very low peeling velocities or the need to use a criterion that has different behavior during loading and unloading. We attempt this inspired by the results of Guduru for a simple case of axisymmetric waviness, and a much better fit of the experimental data by Dalvi and co-authors is obtained using the entire set of data at loading and unloading, even assuming a single work of adhesion value. However, we cannot rule out that both (viscoelastic) and (roughness-induced) enhancement effects coexist in these data.