Square Root Method Research Articles

Quantum photonic communications with four-dimensional constellations of coherent states undergoing phase noise

We present the analysis of the quantum detection of modulated coherent states used in quantum communications systems that employ the degrees of freedom of both the complex amplitude and the state of polarization of the optical carrier, in the presence of optical phase fluctuations, unavoidable in realistic laser fields as optical sources in photonic communications.We analyze several photonic M-ary constellations with different four-dimensional (4D) formats with joint modulation of the field quadratures and the states of polarization of quantum coherent states affected by optical phase noise, of interest in quantum photonic communications, cryptography, and other applications.For the quantum state detection/discrimination at the receiver, we employ the square root method (SRM) for the received phase-diffused 4D constellations, constituted by a set of mixed quantum coherent states, and arrive at results on their performance in error probability, as well as their mutual information, spectral and photon number efficiencies.We find that the overall performances of our 4D Polarization-Shifted and Polarization-Multiplexed constellations are better than those of their single polarization counterparts; also, our binary and quaternary Phase-Shift-Keying formats perform better than the On-Off-Keying and Pulse-Position-Modulation schemes, with a trade-off with their sensitivity to phase noise, which is quantified in our analysis.

Penrose method for Kuramoto model with inertia and noise

Using the Penrose method of instability analysis, we consider the synchronization transition in the Kuramoto model with inertia and noise with all-to-all couplings. Analyzing the Penrose curves, we identify the appearance of cluster and chimera states in the presence of noise. We observe that noise can destroy chimera and biclusters states. The critical coupling describing bifurcation from incoherent to coherent state is found analytically. To confirm our propositions based on the Penrose method, we perform numerical simulations.

“Weighed, not counted”: territorial (mis)representation in Italian metropolitan council elections

Metropolitan cities were established by Italian Law no. 56/2014 (commonly known as the “Delrio Law”) as a new level of government, replacing and redefining functions previously performed by provinces in selected major urban areas. One of their three key governing bodies is the metropolitan council, a representative assembly the members of which are elected via an indirect, second-level, proportional, list-based system in which the electorate comprises all sitting mayors and councillors from metropolitan city municipalities. The election mechanism applies a differential weighting scheme that reflects the population size of the municipalities in which voters serve as mayors or councillors. Using the outcomes of the metropolitan council elections held in the years 2021 and 2022 in eight metropolitan cities, this study highlights the variety of ways (many of which appear to be largely unintended) in which demographic weighting bestows significantly greater (and, arguably, unwarranted) power to larger cities’ representatives, essentially disrupting the principles of territorial representation that the weighting scheme intended to embody. The study also focuses on how the legal framework for metropolitan council elections generates institutional instability via mandatory forfeitures and ensuing substitutions of seat vacancies. Finally, the author identifies potential adjustments to the electoral system – especially a proposal for the attenuation of disproportionate territorial representation via demographic weighting based on the so-called “square-root method”.

Optimal Routing to Parallel Servers in Heavy Traffic

Routing control is an important component in many engineering and management systems consisting of multiple and possibly heterogeneous servers. Imagine that upon the arrival of each job (or customer), a controller will evaluate the available (dynamic) state information and make a decision to dispatch the job to one of the servers. The state information can be queue length, arrival history, service history, and so on, depending on the nature of the application. How will the controller use the available state information to minimize the average waiting time an arriving job may experiences? In the paper, “Optimal Routing to Parallel Servers in Heavy Traffic,” Ye carries out the heavy traffic analysis to identify the routing policies that best use the available state information. For example, when there is no state information available for routing control, the best “blind” strategy is to dispatch the incoming jobs in a weighted round-robin fashion that exhibits certain form of the square-root rule. Although in the case that the job arrival history is available, the controller should use the information by closely chasing a kind of “arrival deviation,” which can reduce up to 50% of the waiting time compared with the best blind strategy. This study sheds new insights into the value of state information for routing control and provides new tools for engineering and service system design.

Optimal dividends under a drawdown constraint and a curious square-root rule

In this paper, we address the problem of optimal dividend payout strategies from a surplus process governed by Brownian motion with drift under a drawdown constraint, i.e., the dividend rate can never decrease below a given fraction a of its historical maximum. We solve the resulting two-dimensional optimal control problem and identify the value function as the unique viscosity solution of the corresponding Hamilton–Jacobi–Bellman equation. We then derive sufficient conditions under which a two-curve strategy is optimal, and we show how to determine its concrete form using calculus of variations. We establish a smooth-pasting principle and show how it can be used to prove the optimality of two-curve strategies for sufficiently large initial and maximum dividend rates. We also give a number of numerical illustrations in which the optimality of the two-curve strategy can be established for instances with smaller values of the maximum dividend rate and the concrete form of the curves can be determined. One observes that the resulting drawdown strategies nicely interpolate between the solution for the classical unconstrained dividend problem and that for a ratcheting constraint as recently studied in Albrecher et al. (SIAM J. Financial Math. 13:657–701, 2022). When the maximum allowed dividend rate tends to infinity, we show a surprisingly simple and somewhat intriguing limit result in terms of the parameter a for the surplus level above which, for a sufficiently large current dividend rate, a take-the-money-and-run strategy is optimal in the presence of the drawdown constraint.

Cross-Gramian-Based Model Reduction for Descriptor Systems

In this paper, we explore model order reduction for large-scale square descriptor systems. A balancing-free square-root method is proposed. The balancing-free square-root method is based on two cross Gramians, one of which is known as the proper cross Gramian and the other as the improper cross Gramian. The proper cross Gramian is the unique solution of a projected generalized continuous-time Sylvester equation, and the improper cross Gramian solves a projected generalized discrete-time Sylvester equation. In order to compute the low-rank factors of these two cross Gramians, we extend the low-rank iteration of the alternating direction implicit method and the Smith method to the projected generalized Sylvester equations. We illustrate the effectiveness of the balance truncation method with one numerical example.

Continuous–discrete unscented Kalman filtering framework by MATLAB ODE solvers and square-root methods

This paper addresses the problem of designing the continuous–discrete unscented Kalman filter (UKF) implementation methods. More precisely, the aim is to propose the MATLAB-based UKF algorithms for accurate and robust state estimation of stochastic dynamic systems. The accuracy of the continuous–discrete nonlinear filters heavily depends on how the implementation method manages the discretization error arisen at the filter prediction step. We suggest the elegant and accurate implementation framework for tracking the hidden states by utilizing the MATLAB built-in numerical integration schemes developed for solving ordinary differential equations (ODEs). The accuracy is boosted by the discretization error control involved in all MATLAB ODE solvers. This keeps the discretization error below the tolerance value provided by users, automatically. Meanwhile, the robustness of the UKF filtering methods is examined in terms of the stability to roundoff. In contrast to the pseudo-square-root UKF implementations established in engineering literature, which are based on the one-rank Cholesky updates, we derive the stable square-root methods by utilizing the J-orthogonal transformations for calculating the Cholesky square-root factors.

Gravitational waves in the extended theory of gravity

Extended theories of gravity are considered as a new approach for solving the infrared and ultraviolet scale problems; the standard theory of gravity (general relativity) and observational evidence of gravitational waves and subsequent identification of the number of existing polarizations are an effective tool for testing general relativity and extended theories of gravity. The Newman–Penrose method is used to characterize the polarization modes for specific forms of [Formula: see text] in the present study. Both the forms of the [Formula: see text] theory belong to far more general variational class of gravitational waves, capable of presenting up to six separate polarizations states. We have introduced a new [Formula: see text] gravity model as an attempt to have a theory with more parametric regulations so that the model can be used to describe existing issues and discover different directions in gravity physics. The primary goal of this research is to look into the properties of gravitational waves with new cases. The model shows the existence of scalar degrees of freedom in [Formula: see text] gravity metric notation.

NIRK-based Cholesky-factorized square-root accurate continuous-discrete unscented Kalman filters for state estimation in nonlinear continuous-time stochastic models with discrete measurements

This paper further advances the idea of accurate Gaussian filtering towards efficient unscented-type Kalman methods for estimating continuous-time nonlinear stochastic systems with discrete measurements. It implies that the differential equations evolving sigma points utilized in computations of the predicted mean and covariance in time-propagations of the Gaussian distribution are solved accurately, i.e. with negligible error. The latter allows the total error of the unscented Kalman filtering technique to be reduced significantly and gives rise to the novel accurate continuous-discrete unscented Kalman filtering algorithm. At the same time, this algorithm is rather vulnerable to round-off and numerical integration errors committed in each state estimation run because of the need for the Cholesky decomposition of covariance matrices involved. Such a factorization will always fail when the covariance's positivity is lost. This positivity lost issue is commonly resolved with square-root filtering implementations, which propagate not the full covariance matrix but its square root (Cholesky factor) instead. Unfortunately, negative weights encountered in applications of the accurate continuous-discrete unscented Kalman filter to high-dimensional stochastic systems preclude from designing conventional square-root methods. In this paper, we address this problem with low-rank Cholesky factor update procedures or with hyperbolic QR transforms used for yielding J-orthogonal square roots. Our novel square-root algorithms are justified theoretically and examined numerically in an air traffic control scenario.

Prediction of Availability of Packed Red Cells (PRC) at PMI Surabaya City Using Ensemble Kalman Filter as Management of Blood Transfusion Management

Blood transfusion in terms of both quality and quantity is needed by patients with various health problems they experience. Along with the increased need for blood transfusion, the management of blood transfusion arrangements (blood collection from donors, selection, distribution, storage), and the ability of nurses to provide transfusion to patients is needed. If there are a lot of mistakes in both the ability of management and the ability of nurses, it can not only lead to a fatal impact on a patientbut also increase amount of need for blood transfusion. Due to the importance of blood transfusion, maintaining the stability of blood stock must be done so as not to cause blood loss due to exces of the blood stock. To minimize such loss, blood prediction is needed. The objective of this study is to predict blood demand for blood type of Packed Red Cells (PRC) or concentrated red blood cells at PMI Surabaya by using the method of Ensemble Kalman Filter (EnKF) and Ensemble Kalman Filter Square Root (EnKF-SR). The simulation results show that both methods have high accuracy with an error of less than 1% and RMSE of EnKF-SR. The best simulation exhibited the error between the real data and the simulation with EnKF-SR was in the order of 0.0023574, whereas with EnKF was some 0.025566 with generated 400 ensembles.

Utility of novel serum biomarkers to predict subclinical atherosclerosis: A sub-analysis of the EISNER study

Background and aimsCertain novel biomarkers may predict atherosclerotic cardiovascular disease (ASCVD) events; however, data on their relationship to coronary atherosclerosis and its progression as measured by coronary artery calcium (CAC) scanning is lacking. We evaluated the association between novel biomarkers and presence or progression of CAC. MethodsThe EISNER study was a prospective trial of patients without known ASCVD. Data on CAC and several biomarkers (hs-CRP, LTβR, osteopontin [OPN], RAGE, TNFR1α and TROY) were available at baseline and 4-year follow-up. Biomarkers were standardized and summed for a composite score. CAC progression was defined by the square-root (CACSQRT) method and rapid (top decile) progression. Adjusted regression models created a final prediction model for baseline CAC and CAC progression. Results1207 subjects (mean age 58.4 ± 8 years, 53% male) were evaluated; 621 had a baseline CAC >0, in whom 323 progressed by CACSQRT, and 121 rapidly progressed. Baseline CAC was associated only with OPN (p = 0.03), TROY (p = 0.0058) and TNFR1α (p = 0.0039) in unadjusted analyses. In adjusted analyses, only OPN was independently related to CAC progression using CACSQRT (p = 0.04). ConclusionsOPN identifies progression of atherosclerosis in persons free of ASCVD at baseline and may be a useful predictive tool to guide ASCVD prevention management.

Model reduction of constrained mechanical systems in M-M.E.S.S.

We discuss balanced truncation model reduction of constrained mechanical systems. The incorporation of the algebraic constraints leads to linear differential-algebraic control systems of index 2 or 3. Then for the square-root method in balanced truncation, the solution of projected Lyapunov equations is required. In this paper we discuss how to avoid the explicit construction of the projectors and the formulation of a projection-avoiding balanced truncation method which is suitable for efficient numerical computations. We have implemented this method in the Matlab package M-M.E.S.S. and present some numerical results for illustration.

Assessment of Two Soil Fertility Indexes to Evaluate Paddy Fields for Rice Cultivation

Assessing soil fertility is essential to help identify strategies with less environmental impact in order to achieve more sustainable agricultural systems. The main objective of this research was to assess two soil fertility evaluation approaches in paddy fields for rice cultivation, in order to develop a user-friendly and credible soil fertility index (SFI). The Square-Root method was used as a parametric approach, while the Joint Fuzzy Membership functions as a fuzzy method with adapted criteria definition tables, were used to compute SFI. Results indicated that both of the methods determined the major soil limiting factors for rice cultivation clearly, and soil fertility maps established using GIS (Geographic Information System) could be helpful for decision makers. The coefficients of determination (R2) for the linear regression between the two SFI values and rice yields were relatively high (0.63 and 0.61, respectively). Additionally, the two SFI were significantly correlated to each other (r = 0.68, p < 0.05). The study results demonstrated that both of the methods provide reliable and valuable information. Compared to the fuzzy method, the procedure of the parametric method is easier but may be expensive and time-consuming. However, the fuzzy method, with carefully chosen indicators, can adequately evaluate soil fertility and provide useful information for decision making.

The UD-based Approach for Designing Pairwise Kalman Filtering Algorithms

Over the past few years we have observed a growing interest in the class of Pairwise Markov Models (PMMs). In this paper, we explore the Pairwise Kalman Filter (PKF) intended for the hidden state estimation in linear PMMs in the presence of Gaussian noises. Previous works produced the robust square-root PKF algorithm for improving a numerical stability of the estimator with respect to roundoff errors. The square-root approach is, definitely, the most popular technique for designing numerically stable filter implementations. However, the use of the modified Cholesky decomposition of corresponding error covariance matrix together with the modified weighted Gram-Schmidt orthogonalization at each iteration step of the Kalman filter was shown to improve the estimation accuracy with the reduced computational cost compared to the square-root methods. Here, we propose the UD-based approach for designing the PKF implementations. As for the classical Kalman filter, we may anticipate a high accuracy of the new UD-based PKF implementation with the reduced CPU cost. The methodology is derived in terms of covariance quantities.

Square‐root Kalman‐like filters for estimation of stiff continuous‐time stochastic systems with ill‐conditioned measurements

This study addresses the performance of the extended and extended-cubature Kalman filters for state estimation of ill-conditioned stiff continuous-discrete stochastic systems. Stiff models arise naturally in practise as, for instance, the Van der Pol oscillator in electrical engineering and the Oregonator reaction in chemistry. Their accurate state estimation is a challenging task, which is made even more difficult by including ill-conditioned measurements, here. The authors' goal is to study accuracy of these methods when the Van der Pol oscillator and the Oregonator reaction become increasingly ill-conditioned. Thus, the authors determine efficient tools for estimation of stiff stochastic systems in the presence of round-off and other disturbances. They examine both conventional filters and their square-root forms. The square-root methods are obtained in two ways, namely, by solving the square-root moment differential equations and by square-rooting the filter itself. They prove that only the square-root filters grounded in the second approach (with use of stable orthogonal decompositions) are numerically stable and provide the highest state estimation accuracy in the ill-conditioned stiff stochastic examples under consideration.

Managing capacity and inventory jointly for multi-server make-to-stock queues

We consider joint capacity---inventory management for multi-server make-to-stock queues operating under a base stock policy. The number of servers corresponds to the capacity decision, and the base stock level is the inventory decision. Our goal is to minimize a combination of capacity, inventory, and backordering costs. We develop a square-root rule for the joint decision and justify the rule analytically in a many-server queue asymptotic framework. We demonstrate the accuracy of the square-root rule, first via our derivation and numerical assessment of a novel corrected diffusion approximation and then more directly by conducting extensive numerical experiments. Finally, we provide operational insights into the trade-offs involved in such joint management problems, through various analysis based on the square-root rule as well as a comparison with analogous results for single-server make-to-stock queues.

An initial guess of Newton's method for the matrix square root based on a sphere constrained optimization problem

An initial guess of Newton's method for the matrix square root based on a sphere constrained optimization problem

Location Selection and Size Estimation of Resource Buffers in the Critical Chain Project Management Method

Critical Chain Project Management (CCPM) is based on a network planning and the Theory of Constraints. The importance of management of uncertainty and deviations that occur during a project is emphasized. The probability of the project completion in a predefined time limit, or even before the time limit is increased. Literature analysis indicates that other types of buffers must be applied (buffer supporting a project, buffer reacting on a critical path, buffer reacting on a non-critical path) apart of feeding, resource and project buffers. Applying the additional buffers keeps the critical path in the same position and maintains the unchanging due date of the project. Methods estimating size of feeding and project buffers are identified: half of the chain method, square root of fault method, modified method of square root of fault, method taking into account due date of the project. In this paper, a new method estimating a size of resource buffer and selecting a location of resource buffer based on the theory of probability is presented. In this paper, the CCPM method is applied to estimate due date of the mining machine production project.

Aggregation and Amplification of Wind-Turbine Harmonic Emission in a Wind Park

This paper presents measurements of harmonic and interharmonic current (with 5-Hz frequency resolution) in the complex plane. The complex currents are spread around a center point. For interharmonics, this center corresponds to the origin of the complex plane. For integer harmonics, this center is offset from the origin. A goodness-of-fit test reveals that phase angles of most interharmonics are uniformly distributed. A Monte Carlo simulation based on the measurements has been performed to study the aggregation of the emission from individual turbines to the public grid. Low-order integer harmonics show less cancellation compared to high-order harmonics. Interharmonics aggregate close to the square-root rule for uniform phase angles.

The Square-Root Rule for Reinsurance: Further Analysis and Consideration of China’s Market

Abstract China has been experiencing rapid development in its insurance market in recent years, with its reinsurance market also in a developing stage. Previous theoretical work using a game theory approach suggests that the appropriate number of companies in the insurance market should be the square root of the number of insurers (index = 0.5). To verify the validity of this square-root rule, we conduct an analysis using data over a period. To examine this relationship between the number of insurers and the number of reinsurers, we perform regression analyses using data from the Organization for Economic Cooperation and Development (OECD). We find that the level of the index fluctuates around an index level of 0.4, instead of 0.5, the optimal value by theory. We also observe that this index level fluctuation behaves similar to the price of risk and the property–liability combined ratio. In our analysis, we then replace the number of reinsurers and number of insurers by ceded premium and net premium, respectively, and find similar results. By comparing and calculating these theoretical relationships, we present two formulas of the average size of the reinsurance market with our interpretations on the gap between the theory and the data. In addition, we analyze the Chinese reinsurance market to offer more evidence of our interpretation.