Arbitrary Moments Research Articles

ДЕЯКІ ЗАДАЧІ ПРО РУХ ТВЕРДОГО ТІЛА У СЕРЕДОВИЩІ З ОПОРОМ

The dynamics of rotating rigid bodies is a classical topic of study in mechanics. In the eighteenth and nineteenth centuries, several aspects of a rotating rigid body motion were studied by famous mathematicians as Euler, Jacobi, Poinsot, Lagrange, and Kovalevskya. However, the study of the dynamics of rotating bodies of still important for aplications such as the dynamics of satellite-gyrostat, spacecraft, re-entry vehicles, theory of gyroscopes, modern technology, navigation, space engineering and many other areas. A number of studies are devoted to the dynamics of a rigid body in a resistive medium. The presence of the velocity of proper rotation of the rigid body leads to the apearance of dissipative torques causing the braking of the body rotation. These torques depend on the properties of resistant medium in which the rigid body motions occur, on the body shape, on the properties of the surface of the rigid body and the distribution of mass in the body and on the characters of the rigid body motion. Therefore, the dependence of the resistant torque on the orientation of the rigid body and its angular velocity can de quite complicated and requires consideration of the motion of the medium around the body in the general case. We confine ourselves in this paper to some simple relations that can qualitative describe the resistance to rigid body rotation at small angular velocities and are used in the literature. In setting up the equations of motion of a rigid body moving in viscous medium, we need to consider the nature of the resisting force generated by the motion of the rigid body. The evolution of rotations of a rigid body influenced by dissipative disturbing torques were studied in many papers and books. The problems of motion of a rigid body about fixed point in a resistive medium described by nonlinear dynamic Euler equations. An analytical solution of the problem when the torques of external resistance forces are proportional to the corresponding projections of the angular velocity of the rigid body is obtain in several works. The dependence of the dissipative torque of the resistant forces on the angular velocity vector of rotation of the rigid body is assumed to be linear. We consider dynamics of a rigid body with arbitrary moments of inertia subjected to external torques include small dissipative torques.

Bootstrapping Bloch bands

Bootstrap methods, initially developed for solving statistical and quantum field theories, have recently been shown to capture the discrete spectrum of quantum mechanical problems, such as the single particle Schrödinger equation with an anharmonic potential. The core of bootstrap methods builds on exact recursion relations of arbitrary moments of some quantum operator and the use of an adequate set of positivity criteria. We extend this methodology to models with continuous Bloch band spectra, by considering a single quantum particle in a periodic cosine potential. We find that the band structure can be obtained accurately provided the bootstrap uses moments involving both position and momentum variables. We also introduce several new techniques that can apply generally to other bootstrap studies. First, we devise a trick to reduce by one unit the dimensionality of the search space for the variables parametrizing the bootstrap. Second, we employ statistical techniques to reconstruct the distribution probability allowing to compute observables that are analytic functions of the canonical variables. This method is used to extract the Bloch momentum, a quantity that is not readily available from the bootstrap recursion itself.

Figure-of-Merit for a Long-Term Survivorship of a Species Determined from the Short-Term Mortality Rate of Its Individual Organisms

The two analytical (“mathematical”) probabilistic predictive models considered in this analysis suggest that (1) the nonrandom time-derivative of the long-term mortality rate at a rather arbitrary initial moment of time for a particular type of species of interest can be viewed as a suitable physical or biological criterion, a sort of a figure of merit (FoM), of its long-term viability/survivorship and that (2) this derivative can be determined as the variance of the random mortality rate for the significantly shorter, of course, lifespan of the individual organisms that the type of species as a whole, addressed by the first model, is comprised of. This suggestion is obtained as a modification and extension of and as an “analogy” to a concept that the author developed earlier in application to microelectronics products. So, it is assumed in our approach that the long-term survivorship of a species comprised of numerous individual organisms is analogous to the long-term performance of an electronic product comprised of numerous mass-produced components. In the original research, it was shown that the time-derivative at the initial moment of time of the nonrandom infant mortality portion (IMP) of the bathtub curve (BTC) for an electronic product is, in effect, the variance of the random failure rate (RFR) of the mass-produced components that this product is comprised of, and it is assumed that such an analogy is applicable also to the long-term survivorship of a species comprised of numerous individual organisms. The larger this variance, the shorter is the expected long-term lifetime (survivorship) of the species as a whole. Future work should be focused, first of all, on the verification of the trustworthiness of our basic assumption for different species, including humans, and on the accumulation of statistical data for long-term survivorship of various species and their existing or future habitats, with consideration of the roles of gravity, temperature, level of radiation, attributes of the atmosphere, if any, etc., as well as on calculating lifespan variances for the organisms that the species of interest are comprised of.

From Ball's cube slicing inequality to Khinchin-type inequalities for negative moments

We establish a sharp moment comparison inequality between an arbitrary negative moment and the second moment for sums of independent uniform random variables, which extends Ball's cube slicing inequality.

Response Time Distribution in a Tandem Pair of Queues with Batch Processing

Response time density is obtained in a tandem pair of Markovian queues with both batch arrivals and batch departures. The method uses conditional forward and reversed node sojourn times and derives the Laplace transform of the response time probability density function in the case that batch sizes are finite. The result is derived by a generating function method that takes into account that the path is not overtake-free in the sense that the tagged task being tracked is affected by later arrivals at the second queue. A novel aspect of the method is that a vector of generating functions is solved for, rather than a single scalar-valued function, which requires investigation of the singularities of a certain matrix. A recurrence formula is derived to obtain arbitrary moments of response time by differentiation of the Laplace transform at the origin, and these can be computed rapidly by iteration. Numerical results for the first four moments of response time are displayed for some sample networks that have product-form solutions for their equilibrium queue length probabilities, along with the densities themselves by numerical inversion of the Laplace transform. Corresponding approximations are also obtained for (non-product-form) pairs of “raw” batch-queues—with no special arrivals—and validated against regenerative simulation, which indicates good accuracy. The methods are appropriate for modeling bursty internet and cloud traffic and a possible role in energy-saving is considered.

Schur\u2013Weyl Duality for the Clifford Group with Applications: Property Testing, a Robust Hudson Theorem, and de Finetti Representations

Schur–Weyl duality is a ubiquitous tool in quantum information. At its heart is the statement that the space of operators that commute with the t-fold tensor powers U^{otimes t} of all unitaries Uin U(d) is spanned by the permutations of the t tensor factors. In this work, we describe a similar duality theory for tensor powers of Clifford unitaries. The Clifford group is a central object in many subfields of quantum information, most prominently in the theory of fault-tolerance. The duality theory has a simple and clean description in terms of finite geometries. We demonstrate its effectiveness in several applications:We resolve an open problem in quantum property testing by showing that “stabilizerness” is efficiently testable: There is a protocol that, given access to six copies of an unknown state, can determine whether it is a stabilizer state, or whether it is far away from the set of stabilizer states. We give a related membership test for the Clifford group.We find that tensor powers of stabilizer states have an increased symmetry group. Conversely, we provide corresponding de Finetti theorems, showing that the reductions of arbitrary states with this symmetry are well-approximated by mixtures of stabilizer tensor powers (in some cases, exponentially well).We show that the distance of a pure state to the set of stabilizers can be lower-bounded in terms of the sum-negativity of its Wigner function. This gives a new quantitative meaning to the sum-negativity (and the related mana) – a measure relevant to fault-tolerant quantum computation. The result constitutes a robust generalization of the discrete Hudson theorem.We show that complex projective designs of arbitrary order can be obtained from a finite number (independent of the number of qudits) of Clifford orbits. To prove this result, we give explicit formulas for arbitrary moments of random stabilizer states.

Minimum‐series twist distributions for yawing‐moment control during pure roll*

AbstractThe main wings of most aircraft produce adverse yaw during roll. In order to control the lateral direction of the aircraft during the roll, the rudder is often mixed with the aileron. Lifting‐line theory is used here to develop spanwise lift distributions that require the minimum number of terms in the Fourier‐series solution for controlling the yawing moment during pure rolling motion using only the main wing. It is shown that the yawing moment can be controlled for arbitrary rolling moments and/or rolling rates by adding symmetric twist in the main wing. The induced‐drag penalty for using this method to control the yawing moment is significant and discussed in detail. For example, it is shown that if zero yawing moment is prescribed, the induced drag can increase by 108% for a prescribed rolling moment or by 300% during a steady rolling rate relative to the induced drag in steady level flight. Because this is the minimum‐series solution, it does not represent the solution for yaw control with minimum induced drag, since more terms could be used in the Fourier series describing the lift distribution to control yaw with less induced drag. However, the solutions presented here can be useful for aircraft with continuous trailing‐edge technologies that are limited in spanwise deflection gradients, and provide a theoretical upper bound on the minimum induced‐drag that can be expected if the main wing is used for yawing‐moment control during pure roll.

Dynamic Voronoi Diagram for Moving Disks.

Voronoi diagrams are powerful for understanding spatial properties. However, few reports have been made for moving generators despite their important applications. We present a topology-oriented event-increment (TOI-E) algorithm for constructing a Voronoi diagram of moving circular disks in the plane over the time horizon [0, t∞). The proposed TOI-E algorithm computes the event history of the Voronoi diagram over the entire time horizon in O(kF logn + kC n logn) time with O(n logn) preprocessing time and O(n + kF + kC) memory for n disk generators, kF edge flips, and kC disk collisions during the time horizon. Given an event history, the Voronoi diagram of an arbitrary moment can be constructed in O(k∗ + n) time where k∗ represents the number of events in [0, t∗). An example of the collision avoidance problem among moving disks is given by predicting future conjunctions among the disks using the proposed algorithm. Dynamic Voronoi diagrams will be very useful as a platform for the planning and management of the traffics of unmanned vehicles such as cars on street, vessels on surface, drones and airplanes in air, and satellites in geospace.

КАЛИБРОВОЧНЫЕ АЛГОРИТМЫ ОПРЕДЕЛЕНИЯ ЧИСЛА РАДИОГЕННЫХ СМЕРТЕЙ В КОГОРТЕ

Background: Development of algorithms for estimating the number of radiogenic deaths in a cohort that happened or may happen before an arbitrary moment in time. Algorithms should be based on continuous lifelong risk models and minimize the errors of estimates based on deformable non-radiation models ("seed" models) reflecting the general patterns of Russian mortality, but with the possibility of calibrating according to selected data.
 Materials and methods: For the study, the databases of the register of liquidators of the consequences of the Chernobyl accident – employees of the enterprises of the Ministry of Atomic Energy of Russia were used. Persons with a non-zero dose of ionizing radiation received as a result of the accident and who were registered in the register until 2018 were selected from the register. There are 7324 men and 655 women in this sample. The number of deaths from the beginning of the observation period until 1996, 2006, 2016 from all causes, as well as from solid cancers and leukemias, was used as a reference data for the calibration. Radiation risk was calculated using UNSCEAR 2006 models. Calibration was performed using the Levenberg – Marquardt method.
 Results: A general formula was obtained for assessing the excess lifetime risk of death, which is the basis of the developed algorithms. The "seed" models of the risk of death from all causes and from solid cancers and leukemia’s were obtained. In accordance with the purpose of the work, two algorithms have been developed. The first algorithm is intended for direct (without using radiation risk models) estimation of the number of radiation-induced deaths in a cohort. The second uses radiation risk models. The performance of the algorithms was tested using the industry register of liquidators of the consequences of the Chernobyl accident. In particular, a 30 % calculated increase in the estimated number of deaths from radiation-induced solid cancers was recorded using calibrated models.
 Conclusions: The proposed calculation algorithms for calibrated models provide reliable (if the radiation risk models are considered sufficiently adequate) estimates of the number of Chernobyl deaths for points in time preceding the current one and, presumably, more accurate predictive estimates compared to the calculation algorithm for uncalibrated models. The low sensitivity of the lifelong risk model to variations in the parameters of the background models and the total mortality model makes it possible to apply models calibrated according to the liquidator register to other closely related populations (NPP workers, etc.).

Analytical study on the evolutionary behavior of transfer matrix element moments in strongly coupled multimode systems.

In this paper, the evolutionary behavior of the transfer matrix element moments in strongly coupled multimode systems is studied analytically. The randomly coupled multimode system is modeled by a set of coupled stochastic differential equations (SDEs), which can be used to find the coupled ordinary differential equations (ODEs) for the arbitrary order moments of the matrix elements. Since the ODEs are with the constant coefficients, it is possible to obtain the analytical solutions. The asymptotic behaviors of the solutions are investigated by comparing with the existing results derived from the property of the Haar matrix, and a perfect agreement is observed. The evolutionary behaviors of the transfer matrix element moments computed by the analytical formulas have excellent match with the Monte Carlo simulation results. The analytical method can be highly beneficiary for the multimode system design and analysis.

Moment method for the Boltzmann equation of reactive quaternary gaseous mixture

We are interested in solving the Boltzmann equation of chemically reacting rarefied gas flows using the Grad’s-14 moment method. We first propose a novel mathematical model that describes the collision dynamics of chemically reacting hard spheres. Using the collision model, we present an algorithm to compute the moments of the Boltzmann collision operator. Our algorithm is general in the sense that it can be used to compute arbitrary order moments of the collision operator and not just the moments included in the Grad’s-14 moment system. For a first-order chemical kinetics, we derive reaction rates for a chemical reaction outside of equilibrium thereby, extending the Arrhenius law that is valid only in equilibrium. We show that the derived reaction rates (i) are consistent in the sense that at equilibrium, we recover the Arrhenius law and (ii) have an explicit dependence on the scalar fourteenth moment, highlighting the importance of considering a fourteen moment system rather than a thirteen one. Through numerical experiments we study the relaxation of the Grad’s-14 moment system to the equilibrium state.

Dynamics of a droplet driven by an internal active device

A liquid droplet, immersed into a Newtonian fluid, can be propelled solely by internal flow. In a simple model, this flow is generated by a collection of point forces, which represent externally actuated devices or model autonomous swimmers. We work out the general framework to compute the self-propulsion of the droplet as a function of the actuating forces and their positions within the droplet. A single point force, F with general orientation and position, r_0, gives rise to both, translational and rotational motion of the droplet. We show that the translational mobility is anisotropic and the rotational mobility can be nonmonotonic as a function of | r_0|, depending on the viscosity contrast. Due to the linearity of the Stokes equation, superposition can be used to discuss more complex arrays of point forces. We analyse force dipoles, such as a stresslet, a simple model of a biflagellate swimmer and a rotlet, representing a helical swimmer, driven by an external magnetic field. For a general force distribution with arbitrary high multipole moments the propulsion properties of the droplet depend only on a few low order multipoles: up to the quadrupole for translational and up to a special octopole for rotational motion. The coupled motion of droplet and device is discussed for a few exemplary cases. We show in particular that a biflagellate swimmer, modeled as a stresslet, achieves a steady comoving state, where the position of the device relative to the droplet remains fixed. In fact there are two fixpoints, symmetric with respect to the center of the droplet. A tiny external force selects one of them and allows to switch between forward and backward motion.

The cutoff phenomenon in total variation for nonlinear Langevin systems with small layered stable noise

This paper provides an extended case study of the cutoff phenomenon for a prototypical class of nonlinear Langevin systems with a single stable state perturbed by an additive pure jump Lévy noise of small amplitude ε>0, where the driving noise process is of layered stable type. Under a drift coercivity condition the associated family of processes Xε turns out to be exponentially ergodic with equilibrium distribution με in total variation distance which extends a result from [60] to arbitrary polynomial moments. The main results establish the cutoff phenomenon with respect to the total variation, under a sufficient smoothing condition of Blumenthal-Getoor index α>3 2. That is to say, in this setting we identify a deterministic time scale tεcut satisfying tεcut→∞, as ε→0, and a respective time window, tεcut±o(tεcut), during which the total variation distance between the current state and its equilibrium με essentially collapses as ε tends to zero. In addition, we extend the dynamical characterization under which the latter phenomenon can be described by the convergence of such distance to a unique profile function first established in [9] to the Lévy case for nonlinear drift. This leads to sufficient conditions, which can be verified in examples, such as gradient systems subject to small symmetric α-stable noise for α>3 2. The proof techniques differ completely from the Gaussian case due to the absence of a respective Girsanov transform which couples the nonlinear equation and the linear approximation asymptotically even for short times.

Arrow-Debreu Meets Kyle

We consider an extension of the Kyle (1985) setting where Arrow-Debreu securities are traded and the informed trader has private information regarding arbitrary higher moments of the asset payoff distribution.In this setting we analyze price discovery and informed demand of Arrow-Debreu securities---equivalently, options. Our model admits a simple equilibrium solution. The informed trader strategy in our model is consistent with options trading strategies used to trade on higher moments in practice. We find that information efficiency of Arrow-Debreu prices decreases with respect to the dispersion of the informed trader's private signals.

A ReMeDI for Microstructure Noise

We introduce the Realized moMents of Disjoint Increments (ReMeDI) paradigm to measure microstructure noise (the deviation of the observed asset prices from the fundamental values caused by market imperfections). We propose consistent estimators of arbitrary moments of the microstructure noise process based on highfrequency data, where the noise process could be serially dependent, endogenous, and nonstationary. We characterize the limit distributions of the proposed estimators and construct confidence intervals under infill asymptotics. Our simulation and empirical studies show that the ReMeDI approach is very effective to measure the scale and the serial dependence of microstructure noise. Moreover, the estimators are quite robust to model specifications, sample sizes and data frequencies.

Probabilistic memristive networks: Application of a master equation to networks of binary ReRAM cells

The possibility of using non-deterministic circuit components has been gaining significant attention in recent years. The modeling and simulation of their circuits require novel approaches, as now the state of a circuit at an arbitrary moment in time cannot be precisely predicted. Generally, these circuits should be described in terms of probabilities, the circuit variables should be calculated on average, and correlation functions should be used to explore interrelations among the variables. In this paper, we use, for the first time, a master equation to analyze the networks composed of probabilistic binary memristors. Analytical solutions of the master equation for the case of identical memristors connected in-series and in-parallel are found. Our analytical results are supplemented by results of numerical simulations that extend our findings beyond the case of identical memristors. The approach proposed in this paper facilitates the development of probabilistic/stochastic electronic circuits and advance their real-world applications.

Information Entropy-Based Leakage Profiling

An accurate leakage model is critical to side-channel attacks and evaluations. Leakage certification plays an important role to address the following question: “how good is my leakage model?” Moreover, most of the current leakage model profiling only exploit the information from lower orders of moments. They still need to tolerate assumption error and estimation error from unknown leakage models. There are many probability density functions (PDFs) satisfying given moment constraints. As such, finding an unbiased, objective, and reasonable model still remains an unresolved problem. In this article, we address a more fundamental question: “which model can approach the leakage infinitely and is the optimal in theory?” In particular, we extract information from higher order moments and propose maximum entropy distribution (MED) to estimate the leakage model as MED is an unbiased, objective, and theoretically the most reasonable PDF conditioned upon the available information. MED is a moment-based statistical PDF model in side-channel attacks. It can theoretically use information on arbitrary higher order moments to infinitely approximate the leakage distribution, and well compensates the theory vacancy of model profiling and evaluation. Experimental results demonstrate the superiority of our proposed method for approximating the leakage model using MED estimation.

A fast computational method for the landing footprints of space-to-ground vehicles

Fast computation of the landing footprint of a space-to- ground vehicle is a basic requirement for the deployment of parking orbits, as well as for enabling decision makers to develop real-time programs of transfer trajectories. In order to address the usually slow computational time for the determination of the landing footprint of a space-to-ground vehicle under finite thrust, this work proposes a method that uses polynomial equations to describe the boundaries of the landing footprint and uses back propagation (BP) neural networks to quickly determine the landing footprint of the space-to-ground vehicle. First, given orbital parameters and a manoeuvre moment, the solution model of the landing footprint of a space-to-ground vehicle under finite thrust is established. Second, given arbitrary orbital parameters and an arbitrary manoeuvre moment, a fast computational model for the landing footprint of a space-to-ground vehicle based on BP neural networks is provided. Finally, the simulation results demonstrate that under the premise of ensuring accuracy, the proposed method can quickly determine the landing footprint of a space-to-ground vehicle with arbitrary orbital parameters and arbitrary manoeuvre moments. The proposed fast computational method for determining a landing footprint lays a foundation for the parking-orbit configuration and supports the design of real-time transfer trajectories.

Expected Logarithm and Negative Integer Moments of a Noncentral χ2-Distributed Random Variable.

Closed-form expressions for the expected logarithm and for arbitrary negative integer moments of a noncentral -distributed random variable are presented in the cases of both even and odd degrees of freedom. Moreover, some basic properties of these expectations are derived and tight upper and lower bounds on them are proposed.

Extended η-μ Fading Models

This article proposes an extension to the η-μ fading model by including a parameter that quantifies the clustering imbalance between in-phase and quadrature components. The inclusion of such a parameter renders the model enormously flexible, extending its reach of application in a wider range of scenarios. Despite the inclusion of an additional parameter, the equations obtained remain as mathematically tractable as those for the original η-μ model. Several new closed-form expressions for important statistics are then obtained, including: (i) probability density functions for both envelope and phase; (ii) cumulative distribution functions for both envelope and phase; (iii) arbitrary moments for the envelope; (iv) moment generating function for the envelope; (v) average bit error rate for coherent binary systems; (v) level crossing rate, average fading duration, and phase crossing rate; and (vi) several other joint distributions encompassing envelope, phase, and their respective time derivatives. It is noteworthy that many of the closed-form formulations presented here are not available even for the simple η-μ model.