Mathematical Conditions Research Articles

A modified version of diagonal systematic sampling in the presence of linear trend

Systematic sampling is one of the simplest and popular methods for selecting a random sample from a finite population. The diagonal systematic sampling scheme is a type of systematic sampling design which has gained the attention of researchers during the last two decades. In this paper, a modification to the conventional diagonal systematic sampling design is proposed for use in situations where population units follow a linear trend. It is found that the proposed strategy reduces the variance of the diagonal systematic sampling thus resulting in an efficient sampling design. The mathematical conditions under which the suggested modified diagonal systematic sampling design is more precise than some of the available sampling designs are derived. With the help of a numerical illustration using milk yield data, it is shown that the proposed sampling scheme is more efficient than some of the available sampling schemes.

On the role of eviction in group living sex changers

In most sex-changing fishes in coral reefs, a dominant male and multiple females form a mating group (harem). In a few species, the subordinates are simultaneous hermaphrodites that may act as sneakers. In this paper, we ask whether the subordinates in most sex changers choose to be female or whether they are forced to give up their male function to avoid eviction by the harem holder. We consider a game model in which (1) the dominant male evicts some hermaphroditic subordinates if the risk of sperm competition in regard to fertilizing eggs is high, and (2) each subordinate individual chooses its own sex allocation considering the risk of being evicted. In the evolutionarily stable state, the dominant male evicts subordinates only when the subordinates vary greatly in their reproductive resources. All the subordinate individuals are female if the summed male function of the subordinates is smaller than that of the dominant male. Otherwise, all the subordinates are hermaphrodites, and the large individuals have the same male investment but a greatly different female investment, while small individuals have a reduced male investment to avoid eviction risk. We conclude that situations in which the sex allocation of subordinates is affected by the possibility of eviction by the harem holder are rather limitedSignificance statementWe studied the role of eviction in social evolution. In most sex-changing fishes in coral reefs, a dominant male and multiple females form a mating group. In a few species, subordinates are simultaneous hermaphrodites. We asked whether the subordinates are forced to give up their male function to avoid eviction by the harem holder. We examined a game model in which the dominant male evicts hermaphroditic subordinates with a high risk of sperm competition, and each subordinate chooses its own sex allocation considering the eviction risk. We derived mathematical conditions for when subordinates are females or hermaphrodites in the ESS. The model demonstrated that the control by the dominant over subordinate reproductive decisions is rather limited.

Dynamics of meandering rivers in finite-length channels: linear theory

Meandering channels are dynamic landforms that arise as a result of fluid mechanic and sedimentary processes. Their evolution has been described by the meander-morphodynamic equations, which dictate that channel curvature and bed topography give rise to local perturbations in streamwise fluid velocity, prompting the preferential erosion and sediment deposition that constitute meander behaviour. Novel mathematical conditions are presented to guarantee unique solutions for the linearized equations in non-periodic domains with finite boundaries. With the boundary condition specification sufficient for the uniqueness proof one finds a well-posed initial-boundary-value problem amenable to standard numerical techniques for partial differential equations. This provides a pathway for improved numerical algorithms for simulations of meandering river dynamics. Previous theoretical analysis for linear stability theory in meandering dynamics has been restricted to spatially periodic systems. The present effort develops new results for linear stability theory in non-periodic systems with temporal driving at system boundaries as well as non-homogeneous initial conditions. Predictions for temporal driving at the inlets for non-periodic finite domains provide clarification for observed behaviour in laboratory flumes where driven conditions at the inlet avoids the long-term decay of all meanders observed in flumes with fixed entry conditions. Linear stability theory for finite domains confirms that a continuous perturbation is required for sustained meandering. Original scaling arguments are presented for the dependence of the meander migration rate on geological parameters, showing that the rate of channel migration increases with increased width, downreach slope and bank erodibility, and decreases with increased volumetric flow rate.

Monotonicity violations under plurality with a runoff: the case of French presidential elections

A voting rule is monotonic if a winning candidate never becomes a loser by being raised in voters’ rankings of candidates, ceteris paribus. Plurality with a runoff is known to fail monotonicity. To see how widespread this failure is, we focus on French presidential elections since 1965. We identify mathematical conditions that allow a logically conceivable scenario of vote shifts between candidates that may lead to a monotonicity violation. We show that eight among the ten elections held since 1965 (those in 1965 and 1974 being the exceptions) exhibit this theoretical vulnerability. To be sure, the conceived scenario of vote shifts that enables a monotonicity violation may not be plausible under the political context of the considered election. Thus, we analyze the political landscape of these eight elections and argue that for two of them (2002 and 2007 elections), the monotonicity violation scenario was plausible within the conjuncture of the time.

On Quadratic Interpolation of Image Cross-Correlation for Subpixel Motion Extraction.

Digital image correlation techniques are well known for motion extraction from video images. Following a two-stage approach, the pixel-level displacement is first estimated by maximizing the cross-correlation between two images, then the estimation is refined in the vicinity of the cross-correlation peak. Among existing subpixel refinement methods, quadratic surface fitting (QSF) provides good performances in terms of accuracy and computational burden. It estimates subpixel displacement by interpolating cross-correlation values with a quadratic surface. The purpose of this paper is to analytically investigate the QSF method. By means of counterexamples, it is first shown in this paper that, contrary to a widespread intuition, the quadratic surface fitted to the pixel-level cross-correlation values in the neighborhood of the cross-correlation peak does not always have a maximum. The main contribution of this paper then consists in establishing the mathematical conditions ensuring the existence of a maximum of this fitted quadratic surface, based on a rigorous analysis. Algorithm modifications for handling the failure cases of the QSF method are also proposed in this paper, in order to consolidate it for subpixel motion extraction. Experimental results based on two typical types of images are also reported.

Storage Minimization of Marine Energy Grids Using Polyphase Power

Multiple wave energy converter (WEC) buoys can be used to establish a WEC array-powered microgrid collectively forming a Marine Energy Grid (MEG). An oceanic domain with gravity waves will have significant spatial variability in phase, causing the power produced by a WEC array to have high peak-to-average ratios. Minimizing these power fluctuations reduces the demand for large energy storage by WEC array-powered DC microgrids while also reducing losses in the undersea cable to the shore. Designs that reduce energy storage requirements are desirable to reduce deployment and maintenance costs. This work demonstrates that polyphase power in conjunction with an energy storage system can be used to maintain constant power. This work shows that an N WEC array geometry can be designed to reduce the energy storage requirements needed to mitigate the power fluctuations if the WEC array produces constant, polyphase power. Additionally, the conditions that identify the wave frequencies and control the effort needed to produce polyphase power are developed. This paper also shows that increasing the number of WECs in an array reduces aggregate power fluctuations. Finally, WEC array power profiles are investigated using simulation results to verify the mathematical conditions developed for the three and six WEC cases.

Spin-orbit interactions may relax the rigid conditions leading to flat bands

Flat bands are of extreme interest in a broad spectrum of fields since, given their high degeneracy, a small perturbation introduced in the system is able to push the ground state in the direction of an ordered phase of interest. Hence, the flat-band engineering in real materials attracts huge attention. However, manufacturing a flat band represents a difficult task because its appearance in a real system is connected to rigid mathematical conditions relating a part of Hamiltonian parameters. Consequently, whenever a flat band is to be manufactured, these Hamiltonian parameters must be tuned exactly to the values fixed by these rigid mathematical conditions. Here we demonstrate that taking the many-body spin-orbit interaction into account, which can be continuously tuned, e.g., by external electric fields, these rigid mathematical conditions can be substantially relaxed. Consequently, we show that a $\ensuremath{\sim}20%--30%$ variation in the Hamiltonian parameters rigidly fixed by the flat-band conditions can also lead to flat bands in the same, or in a bit displaced, position on the energy axis. This percentage can even increase to $\ensuremath{\sim}80%$ in the presence of an external magnetic field. This study is made for the case of conducting polymers. These systems are relevant not only because they have broad application possibilities, but also because they can be used to present the mathematical background of the flat-band conditions in full generality, in a concise, clear, and understandable manner applicable everywhere in itinerant systems.

Biological emergent properties in non-spiking neural networks

<abstract><p>A central goal of neuroscience is to understand the way nervous systems work to produce behavior. Experimental measurements in freely moving animals (<italic>e.g.</italic> in the <italic>C. elegans</italic> worm) suggest that ON- and OFF-states in non-spiking nervous tissues underlie many physiological behaviors. Such states are defined by the collective activity of non-spiking neurons with correlated up- and down-states of their membrane potentials. How these network states emerge from the intrinsic neuron dynamics and their couplings remains unclear. In this paper, we develop a rigorous mathematical framework for better understanding their emergence. To that end, we use a recent simple phenomenological model capable of reproducing the experimental behavior of non-spiking neurons. The analysis of the stationary points and the bifurcation dynamics of this model are performed. Then, we give mathematical conditions to monitor the impact of network activity on intrinsic neuron properties. From then on, we highlight that ON- and OFF-states in non-spiking coupled neurons could be a consequence of bistable synaptic inputs, and not of intrinsic neuron dynamics. In other words, the apparent up- and down-states in the neuron's bimodal voltage distribution do not necessarily result from an intrinsic bistability of the cell. Rather, these states could be driven by bistable presynaptic neurons, ubiquitous in non-spiking nervous tissues, which dictate their behaviors to their postsynaptic ones.</p></abstract>

Statistical Hypothesis Testing Based on Machine Learning: Large Deviations Analysis

We study the performance -- and specifically the rate at which the error probability converges to zero -- of Machine Learning (ML) classification techniques. Leveraging the theory of large deviations, we provide the mathematical conditions for a ML classifier to exhibit error probabilities that vanish exponentially, say $\sim \exp\left(-n\,I + o(n) \right)$, where $n$ is the number of informative observations available for testing (or another relevant parameter, such as the size of the target in an image) and $I$ is the error rate. Such conditions depend on the Fenchel-Legendre transform of the cumulant-generating function of the Data-Driven Decision Function (D3F, i.e., what is thresholded before the final binary decision is made) learned in the training phase. As such, the D3F and, consequently, the related error rate $I$, depend on the given training set, which is assumed of finite size. Interestingly, these conditions can be verified and tested numerically exploiting the available dataset, or a synthetic dataset, generated according to the available information on the underlying statistical model. In other words, the classification error probability convergence to zero and its rate can be computed on a portion of the dataset available for training. Coherently with the large deviations theory, we can also establish the convergence, for $n$ large enough, of the normalized D3F statistic to a Gaussian distribution. This property is exploited to set a desired asymptotic false alarm probability, which empirically turns out to be accurate even for quite realistic values of $n$. Furthermore, approximate error probability curves $\sim \zeta_n \exp\left(-n\,I \right)$ are provided, thanks to the refined asymptotic derivation (often referred to as exact asymptotics), where $\zeta_n$ represents the most representative sub-exponential terms of the error probabilities.

Iterative Control Framework with Application to Guidance Control of Rockets with Impulsive Thrusters

In contrast with traditional control techniques that require a priori verification of complex mathematical conditions, in this paper, we develop a fundamentally new framework of control logic that can be readily applied to a general class of nonlinear systems. Our approach, called the Iterative Control Framework (ICF), uses a novel control logic to guarantee the stability of a closed-loop system without a priori verification of Lyapunov-like conditions. The underlying idea is in repetitive reconfiguration of the control vector provided by a computational routine running in the background in real time during each iteration. The control approach is applicable to a broad class of complex nonlinear systems but is particularly suitable for systems inherently admitting only control action of finite-time duration such as short- and long-range missiles, satellites, and spacecrafts. In this work, we focus on the application of ICF to guidance control of rockets and missiles where actuation is provided via single-use thrusters.

Extremum-Seeking Adaptive-Droop for Model-Free and Localized Volt-VAR Optimization

In an active power distribution system, Volt-VAR optimization (VVO) methods are employed to achieve network-level objectives such as minimization of network power losses. The commonly used model-based centralized and distributed VVO algorithms perform poorly in the absence of a communication system and with model and measurement uncertainties. In this paper, we proposed a model-free local Volt-VAR control approach for network-level optimization that does not require communication with other decision-making agents. The proposed algorithm is based on extremum-seeking approach that uses only local measurements to minimize the network power losses. To prove that the proposed extremum-seeking controller converges to the optimum solution, we also derive mathematical conditions for which the loss minimization problem is convex with respect to the control variables. Local controllers pose stability concerns during highly variable scenarios. Thus, the proposed extremum-seeking controller is integrated with an adaptive-droop control module to provide a stable local control response. The proposed approach is validated using IEEE 4-bus and IEEE 123-bus systems and achieves the loss minimization objective while maintaining the voltage within the pre-specific limits even during highly variable DER generation scenarios.

Transition Therapy: Tackling the Ecology of Tumor Phenotypic Plasticity

Phenotypic switching in cancer cells has been found to be present across tumor types. Recent studies on Glioblastoma report a remarkably common architecture of four well-defined phenotypes coexisting within high levels of intra-tumor genetic heterogeneity. Similar dynamics have been shown to occur in breast cancer and melanoma and are likely to be found across cancer types. Given the adaptive potential of phenotypic switching (PHS) strategies, understanding how it drives tumor evolution and therapy resistance is a major priority. Here we present a mathematical framework uncovering the ecological dynamics behind PHS. The model is able to reproduce experimental results, and mathematical conditions for cancer progression reveal PHS-specific features of tumors with direct consequences on therapy resistance. In particular, our model reveals a threshold for the resistant-to-sensitive phenotype transition rate, below which any cytotoxic or switch-inhibition therapy is likely to fail. The model is able to capture therapeutic success thresholds for cancers where nonlinear growth dynamics or larger PHS architectures are in place, such as glioblastoma or melanoma. By doing so, the model presents a novel set of conditions for the success of combination therapies able to target replication and phenotypic transitions at once. Following our results, we discuss transition therapy as a novel scheme to target not only combined cytotoxicity but also the rates of phenotypic switching.

Statistical modeling and inference in the era of Data Science and Graphical Causal modeling

AbstractThe paper discusses four paradigm shifts in statistics since the 1920s with a view to compare their similarities and differences, and evaluate their effectiveness in giving rise to ‘learning from data' about phenomena of interest. The first is Fisher's 1922 recasting of Karl Pearson's descriptive statistics into a model‐based [] statistical induction that dominates current statistics (frequentist and Bayesian). A crucial departure was Fisher's replacing the curve‐fitting perspective guided by goodness‐of‐fit measures with a model‐based perspective guided by the statistical adequacy: the validity of the probabilistic assumptions comprising . Statistical adequacy is pivotal in securing trustworthy evidence since it underwrites the reliability of inference. The second is the nonparametric turn in the 1970s aiming to broaden by replacing its distribution assumption with weaker mathematical conditions relating to the unknown density function underlying . The third is a two‐pronged development initiated in Artificial Intelligence (AI) in the 1990s that gave rise to Data Science (DS) and Graphical Causal (GC) modeling. The primary objective of the paper is to compare and evaluate the other competing approaches with a refined/enhanced version of Fisher's model‐based approach in terms of their effectiveness in giving rise to genuine “learning from data;” excellent goodness‐of‐fit/prediction is neither necessary nor sufficient for statistical adequacy, or so it is argued.

Lubrication\u2013Contact Interface Conditions and Novel Mixed/Boundary Lubrication Modeling Methodology

Under severe conditions, solid contacts take place even when parts are lubricated. Precise mathematical conditions are needed to describe the interior interface between fluid lubrication and solid-contact zones. In order to distinguish the conditions for this interface from conventional lubrication boundary conditions, they are named lubrication–contact interface conditions (LCICs). In this work, mathematical LCICs are derived with local flow continuity from the continuum mechanics point of view and pressure inequality across the interface. Numerical implementations are developed and tested with problems having simple geometries and configurations, and they are integrated into a new mixed/boundary elastohydrodynamic lubrication solver that uses a new method to determine solid-contact pressures. This solver is capable of capturing film thickness and pressure behaviors involving solid contacts.

Qualitative analysis of a discrete-time phytoplankton\u2013zooplankton model with Holling type-II response and toxicity

The interaction among phytoplankton and zooplankton is one of the most important processes in ecology. Discrete-time mathematical models are commonly used for describing the dynamical properties of phytoplankton and zooplankton interaction with nonoverlapping generations. In such type of generations a new age group swaps the older group after regular intervals of time. Keeping in observation the dynamical reliability for continuous-time mathematical models, we convert a continuous-time phytoplankton–zooplankton model into its discrete-time counterpart by applying a dynamically consistent nonstandard difference scheme. Moreover, we discuss boundedness conditions for every solution and prove the existence of a unique positive equilibrium point. We discuss the local stability of obtained system about all its equilibrium points and show the existence of Neimark–Sacker bifurcation about unique positive equilibrium under some mathematical conditions. To control the Neimark–Sacker bifurcation, we apply a generalized hybrid control technique. For explanation of our theoretical results and to compare the dynamics of obtained discrete-time model with its continuous counterpart, we provide some motivating numerical examples. Moreover, from numerical study we can see that the obtained system and its continuous-time counterpart are stable for the same values of parameters, and they are unstable for the same parametric values. Hence the dynamical consistency of our obtained system can be seen from numerical study. Finally, we compare the modified hybrid method with old hybrid method at the end of the paper.

Analytical and Numerical Bifurcation Analysis of Circuits Based on Nonlinear Resonators

This work presents new frequency-domain methods for the analysis and simulation of circuits based on a nonlinear resonator, with operation ranges delimited by turning points. Insightful analytical conditions fulfilled at these turning points are derived, which will enable an identification of the effect of each circuit element on their location in the solution curve. The cusp points, or co-dimension two bifurcations at which two turning points merge into one, thus delimiting the multivalued intervals, are directly calculated for the first time to our knowledge. In addition, a new numerical method, compatible with the use of commercial harmonic balance, is presented for the straightforward tracing of the multivalued solution curves, together with a new procedure to determine the locus of turning points in terms of any two analysis parameters. This relies on the use of a new mathematical condition to obtain a surface of turning points in the space defined by the two parameters and the input power. The methods have been applied to a wireless power-transfer system based on a recently proposed configuration, obtaining very good agreement with the experimental results.

Empirical Tail Risk Management with Model-Based Annealing Random Search

Tail risk measures such as Value at Risk (VaR) and Conditional Value at Risk (CVaR) are popularly accepted criteria for financial risk management, but are usually difficult to optimize. Especially for VaR, it generally leads to a non-convex NP-hard problem which is computationally challenging. In this paper we propose the use of model-based annealing random search (MARS) method in tail risk optimization problems. The MARS, which is a gradient-free and flexible method, can widely be applied to solving many financial and insurance problems under mild mathematical conditions. We use a weather index insurance design problem with tail risk measures including VaR, CVaR and Entropic Value at Risk (EVaR) as the objective function to demonstrate the viability and effectiveness of MARS. We conduct an empirical analysis in which we use a set of weather variables to hedge against corn production losses in Illinois. Numerical results show that the proposed optimization scheme effectively helps corn producers to manage their tail risk.

Conditionally Exact Closed-Form Solution for Moving Boundary Problems in Heat and Mass Transfer in the Presence of Advection

Moving boundary problems occur in a variety of heat and mass transfer processes. While significant literature already exists on the mathematical analysis of such problems in the presence of diffusion, there is a lack of general solutions for problems in which advective transport of heat/mass due to fluid flow also occurs. This paper presents an error function based analytical solution for a one-dimensional phase change problem in the presence of advection with constant velocity. While this solution is not universally exact, however, a mathematical condition to ensure exactness of the solution is derived. Good agreement with numerical simulations, as well as with past work for special cases is shown. Even outside the condition to ensure exactness, the present method is shown to offer improved accuracy compared to other approximate analytical methods. In particular, the method offers greater accuracy at large value of the Stefan number, where other approximate analytical methods usually perform poorly. The impact of Peclet number that represents advection on the accuracy of the method is investigated. It is shown, as expected, that the t dependence of phase change front propagation is not valid in the presence of advection in general. This work improves the theoretical understanding of an important phase change problem, and may find applications in the design and optimization of engineering processes and systems involving phase change.

Three-Stage Numerical Solution for Optimal Control of COVID-19

In this paper, we present a three-stage algorithm for finding numerical solutions for optimal control problems. The algorithm first performs an exhaustive search through a discrete set of widely dispersed solutions which are representative of large subregions of the search space; then, it uses the search results to initialize a Monte Carlo process that searches quasi-randomly for a best solution; then, it finally uses a Newton-type iteration to converge to a solution that satisfies mathematical conditions of local optimality. We demonstrate our methodology on an epidemiological model of the coronavirus disease with testing and distancing controls applied over a period of 180 days to two different subpopulations (low-risk and high-risk), where model parameters are chosen to fit the city of Houston, Texas, USA. In order to enable the user to select his/her preferred trade-off between (number of deaths) and (herd immunity) outcomes, the objective function includes costs for deaths and non-immunity. Optimal strategies are estimated for a grid of (death cost) × (non-immunity cost) combinations, in order to obtain a Pareto curve that represents optimum trade-offs. The levels of the four controls for the different Pareto-optimal solutions over the 180-day period are visually represented and their characteristics discussed. Three different variants of the algorithm are run in order to determine the relative importance of the three stages in the optimization. Results from the three algorithm variants are fairly consistent, indicating that solutions are robust. Results also show that the Monte Carlo stage plays an especially prominent role in the optimization, but that all three stages of the process make significant contributions towards finding lower-cost, more effective control strategies.

Stability aware spatial cut of metapopulations ecological networks

Ecological complex networks are common in the study of patched ecological systems where evolving populations interact within and among the patches. The loss of the dispersal connections between patches due to reasons such as erosion of migration corridors and road construction can cause an undesirable partitioning of such networks resulting in instability or negative impact on the metapopulations. A partitioning or spatial cut that is aware of the stability of the dynamics in the resulting daughter sub-networks can be an effective tool in dealing with the situation like proposing road alignment through a metapopulations network. This paper provides some mathematical conditions along with an heuristic graph partitioning algorithm that can help in finding ecologically suitable partitions of the metapopulations networks. Our study noted the crucial role of network connectivity (measured by Fiedler value) in stabilizing the metapopulations. That is, a sufficiently connected metapopulations network along with constrained internal patch dynamics has stable dynamics around its homogeneous co-existential equilibrium solution. With the considered mathematical model in this paper, network partitioning does not alter the internal patch dynamics around its homogeneous equilibrium point, but it can change the connectivity levels in the partitioned subnetworks. Thus, the proposed partitioning problem for an already stable metapopulations network is reduced to finding its subnetworks with desirable connectivity levels.