Behaviour of solutions to the 1D focusing stochastic L2-critical and supercritical nonlinear Schrödinger equation with space-time white noise

We study the focusing stochastic nonlinear Schrödinger equation in 1D in the $L^2$-critical and supercritical cases with an additive or multiplicative perturbation driven by space-time white noise. Unlike the deterministic case, the Hamiltonian (or energy) is not conserved in the stochastic setting nor is the mass (or the $L^2$-norm) conserved in the additive case. Therefore, we investigate the time evolution of these quantities. After that, we study the influence of noise on the global behaviour of solutions. In particular, we show that the noise may induce blow up, thus ceasing the global existence of the solution, which otherwise would be global in the deterministic setting. Furthermore, we study the effect of the noise on the blow-up dynamics in both multiplicative and additive noise settings and obtain profiles and rates of the blow-up solutions. Our findings conclude that the blow-up parameters (rate and profile) are insensitive to the type or strength of the noise: if blow up happens, it has the same dynamics as in the deterministic setting; however, there is a (random) shift of the blow-up centre, which can be described as a random variable normally distributed.

Robust Group Synchronization via Cycle-Edge Message Passing

We propose a general framework for solving the group synchronization problem, where we focus on the setting of adversarial or uniform corruption and sufficiently small noise. Specifically, we apply a novel message passing procedure that uses cycle consistency information in order to estimate the corruption levels of group ratios and consequently solve the synchronization problem in our setting. We first explain why the group cycle consistency information is essential for effectively solving group synchronization problems. We then establish exact recovery and linear convergence guarantees for the proposed message passing procedure under a deterministic setting with adversarial corruption. These guarantees hold as long as the ratio of corrupted cycles per edge is bounded by a reasonable constant. We also establish the stability of the proposed procedure to sub-Gaussian noise. We further establish exact recovery with high probability under a common uniform corruption model.

Tasks of inter-project unification of spacecraft on-board systems for remote sensing of the Earth

The article discusses and presents the formulation of problems of inter-project unification of on-board systems in the development of modifications of spacecraft that are part of space systems for remote sensing of the Earth. It is shown that when developing a complex of advanced space systems, it is possible to partially combine unified on-board systems and finished products, which, under given constraints, provides a minimum of total costs. The formulation of the main task of inter-project unification of spacecraft for remote sensing of the Earth using finished products and partially unified on-board systems and a special case of the problem — conducting an economically justified inter-project unification from completely unified on-board systems (aggregates) of promising modifications of spacecraft is given. The initial data and limitations for solving the main and particular problems are determined. The tasks are presented in a deterministic setting. The concept of optimality of the choice of areas of unification of each on-board system is formulated, which is characterized by the minimum of a criterion having an additive structure, this is the total economic effect for areas of unification. It is believed that the analysis of the results of solving the problems of inter-project unification in the development of promising modifications of spacecraft will reveal the directions of inter-project unification of those on-board systems for which it is most appropriate; to formulate the fundamental principles of modernization of space systems and creation of modifications of spacecraft for remote sensing of the Earth in the planned period.

Gene duplication and subsequent diversification strongly affect phenotypic evolvability and robustness

We study the effects of non-determinism and gene duplication on the structure of genotype–phenotype (GP) maps by introducing a non-deterministic version of the Polyomino self-assembly model. This model has previously been used in a variety of contexts to model the assembly and evolution of protein quaternary structure. Firstly, we show the limit of the current deterministic paradigm which leads to built-in anti-correlation between evolvability and robustness at the genotypic level. We develop a set of metrics to measure structural properties of GP maps in a non-deterministic setting and use them to evaluate the effects of gene duplication and subsequent diversification. Our generalized versions of evolvability and robustness exhibit positive correlation for a subset of genotypes. This positive correlation is only possible because non-deterministic phenotypes can contribute to both robustness and evolvability. Secondly, we show that duplication increases robustness and reduces evolvability initially, but that the subsequent diversification that duplication enables has a stronger, inverse effect, greatly increasing evolvability and reducing robustness relative to their original values.

On Some Important Ordinary Differential Equations of Dynamic Economics

Mathematical modeling in economics became central to economic theory during the decade of the Second World War. The leading figure in that period was Paul Anthony Samuelson whose 1947 book, Foundations of Economic Analysis, formalized the problem of dynamic analysis in economics. In this brief chapter some seminal applications of differential equations in economic growth, capital and business trade cycles are outlined in deterministic setting. Chaos and bifurcations in economic dynamics are not considered. Explicit analytical solutions are presented only in relatively straightforward cases and in more complicated cases a path to the solution is outlined. Differential equations in modern dynamic economic modeling are extensions and modifications of these classical works. Finally we would like to stress that the differential equations presented in this chapter are of the “stand-alone” type in that they were solely introduced to model economic growth and trade cycles. Partial differential equations such as those which arise in related fields, like Bioeconomics and Differential Games, from optimizing the Hamiltonian of the problem, and stochastic differential equations of Finance and Macroeconomics are not considered here.

Control of connectivity and rigidity in prismatic assemblies

How can we manipulate the topological connectivity of a three-dimensional prismatic assembly to control the number of internal degrees of freedom and the number of connected components in it? To answer this question in a deterministic setting, we use ideas from elementary number theory to provide a hierarchical deterministic protocol for the control of rigidity and connectivity. We then show that it is possible to also use a stochastic protocol to achieve the same results via a percolation transition. Together, these approaches provide scale-independent algorithms for the cutting or gluing of three-dimensional prismatic assemblies to control their overall connectivity and rigidity.

A stochastic design optimization methodology to reduce emission spread in combustion engines

This paper introduces a method for efficiently solving stochastic optimization problems in the field of engine calibration. The main objective is to make more conscious decisions during the base engine calibration process by considering the system uncertainty due to component tolerances and thus enabling more robust design, low emissions, and avoiding expensive recalibration steps that generate costs and possibly postpone the start of production. The main idea behind the approach is to optimize the design parameters of the engine control unit (ECU) that are subject to uncertainty by considering the resulting output uncertainty. The premise is that a model of the system under study exists, which can be evaluated cheaply, and the system tolerance is known. Furthermore, it is essential that the stochastic optimization problem can be formulated such that the objective function and the constraint functions can be expressed using proper metrics such as the value at risk (VaR). The main idea is to derive analytically closed formulations for the VaR, which are cheap to evaluate and thus reduce the computational effort of evaluating the objective and constraints. The VaR is therefore learned as a function of the input parameters of the initial model using a supervised learning algorithm. For this work, we employ Gaussian process regression models. To illustrate the benefits of the approach, it is applied to a representative engine calibration problem. The results show a significant improvement in emissions compared to the deterministic setting, where the optimization problem is constructed using safety coefficients. We also show that the computation time is comparable to the deterministic setting and is orders of magnitude less than solving the problem using the Monte-Carlo or quasi-Monte-Carlo method.

Mice adaptively generate choice variability in a deterministic task

Can decisions be made solely by chance? Can variability be intrinsic to the decision-maker or is it inherited from environmental conditions? To investigate these questions, we designed a deterministic setting in which mice are rewarded for non-repetitive choice sequences, and modeled the experiment using reinforcement learning. We found that mice progressively increased their choice variability. Although an optimal strategy based on sequences learning was theoretically possible and would be more rewarding, animals used a pseudo-random selection which ensures high success rate. This was not the case if the animal is exposed to a uniform probabilistic reward delivery. We also show that mice were blind to changes in the temporal structure of reward delivery once they learned to choose at random. Overall, our results demonstrate that a decision-making process can self-generate variability and randomness, even when the rules governing reward delivery are neither stochastic nor volatile.

On Robust Fractional 0-1 Programming

We study single- and multiple-ratio robust fractional 0-1 programming problems (RFPs). In particular, this work considers RFPs under a wide range of disjoint and joint uncertainty sets, where the former implies separate uncertainty sets for each numerator and denominator and the latter accounts for different forms of interrelatedness between them. We first demonstrate that unlike the deterministic case, a single-ratio RFP is nondeterministic polynomial-time hard under general polyhedral uncertainty sets. However, if the uncertainty sets are imbued with a certain structure, variants of the well-known budgeted uncertainty, the disjoint and joint single-ratio RFPs are polynomially solvable when the deterministic counterpart is. We also propose mixed-integer linear programming (MILP) formulations for multiple-ratio RFPs. We conduct extensive computational experiments using test instances based on real and synthetic data sets to evaluate the performance of our MILP reformulations as well as to compare the disjoint and joint uncertainty sets. Finally, we demonstrate the value of the robust approach by examining the robust solution in a deterministic setting and vice versa.

Mice adaptively generate choice variability in a deterministic task

Can our choices just be driven by chance? To investigate this question, we designed a deterministic setting in which mice reinforce non-repetitive choice sequences, and modeled it using reinforcement learning. Mice progressively increased their choice variability using a memory-free, pseudo-random selection, rather than by learning complex sequences. Our results demonstrate that a decision-making process can self-generate variability and randomness even when the rules governing reward delivery are not stochastic.