Search results for: random walk approximation

Authors: Guillaume Leduc

Abstract:

In this paper, we describe a broad setting under which the error of the approximation can be quantified, controlled, and for which convergence occurs at a speed of n⁻¹ for European and American options. We describe how knowledge of the error allows for arbitrarily fast acceleration of the convergence.

Keywords: random walk approximation, European and American options, rate of convergence, option pricing

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Authors: Simeon E. H. Davies

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This study investigated whether selected metabolic responses and the perception of effort varied during four different walk protocols where speed increased progressively 3, 4, 5, 6, and 7 km/hr (progressive treadmill walk (PTW); and progressive land walk (PLW); or where the participant adjusted to random changes of speed e.g. 6, 3, 7, 4, and 5 km/hr during a randomized treadmill walk (RTW); and a randomized land walk (RLW). Mean stature and mass of the seven participants was 1.75m and 70kg respectively, with a mean body fat of 15%. Metabolic measures including heart rate, relative oxygen uptake, ventilation, increased in a linear fashion up to 6 km/hr, however at 7 km/hr there was a significant increase in metabolic response notably during the PLW, and to a similar, although lesser extent in RLW, probably as a consequence of the loss of kinetic energy when turning at each cone in order to maintain the speed during each shuttle. Respiration frequency appeared to be a more sensitive indicator of physical exertion, exhibiting a rapid elevation at 5 km/hr. The perception of effort during each mode and at each speed was largely congruent during each walk protocol.

Keywords: exertion, metabolic, progressive, random, walking

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Authors: Chaghoub Soraya, Zhang Xiaoyan

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This research presents the first constant approximation algorithm to the p-median network design problem with multiple cable types. This problem was addressed with a single cable type and there is a bifactor approximation algorithm for the problem. To the best of our knowledge, the algorithm proposed in this paper is the first constant approximation algorithm for the p-median network design with multiple cable types. The addressed problem is a combination of two well studied problems which are p-median problem and network design problem. The introduced algorithm is a random sampling approximation algorithm of constant factor which is conceived by using some random sampling techniques form the literature. It is based on a redistribution Lemma from the literature and a steiner tree problem as a subproblem. This algorithm is simple, and it relies on the notions of random sampling and probability. The proposed approach gives an approximation solution with one constant ratio without violating any of the constraints, in contrast to the one proposed in the literature. This paper provides a (21 + 2)-approximation algorithm for the p-median network design problem with multiple cable types using random sampling techniques.

Keywords: approximation algorithms, buy-at-bulk, combinatorial optimization, network design, p-median

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Authors: Razieh Khalafi

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Myocardial infarction (MI) or acute myocardial infarction (AMI), commonly known as a heart attack, occurs when blood flow stops to part of the heart causing damage to the heart muscle. An ECG can often show evidence of a previous heart attack or one that's in progress. The patterns on the ECG may indicate which part of your heart has been damaged, as well as the extent of the damage. In chaos theory, the correlation dimension is a measure of the dimensionality of the space occupied by a set of random points, often referred to as a type of fractal dimension. In this research by considering ECG signal as a random walk we work on forecasting the oncoming heart attack by analyzing the ECG signals using the correlation dimension. In order to test the model a set of ECG signals for patients before and after heart attack was used and the strength of model for forecasting the behavior of these signals were checked. Results shows this methodology can forecast the ECG and accordingly heart attack with high accuracy.

Keywords: heart attack, ECG, random walk, correlation dimension, forecasting

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Authors: Razi Khalafi

Abstract:

Myocardial Infarction (MI) or acute Myocardial Infarction (AMI), commonly known as a heart attack, occurs when blood flow stops to part of the heart causing damage to the heart muscle. An ECG can often show evidence of a previous heart attack or one that's in progress. The patterns on the ECG may indicate which part of your heart has been damaged, as well as the extent of the damage. In chaos theory, the correlation dimension is a measure of the dimensionality of the space occupied by a set of random points, often referred to as a type of fractal dimension. In this research by considering ECG signal as a random walk we work on forecasting the oncoming heart attack by analysing the ECG signals using the correlation dimension. In order to test the model a set of ECG signals for patients before and after heart attack was used and the strength of model for forecasting the behaviour of these signals were checked. Results show this methodology can forecast the ECG and accordingly heart attack with high accuracy.

Keywords: heart attack, ECG, random walk, correlation dimension, forecasting

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Authors: Valeria Bondarenko

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We proposed algorithms for: estimation of parameters fBm (volatility and Hurst exponent) and for the approximation of random time series by functional of fBm. We proved the consistency of the estimators, which constitute the above algorithms, and proved the optimal forecast of approximated time series. The adequacy of estimation algorithms, approximation, and forecasting is proved by numerical experiment. During the process of creating software, the system has been created, which is displayed by the hierarchical structure. The comparative analysis of proposed algorithms with the other methods gives evidence of the advantage of approximation method. The results can be used to develop methods for the analysis and modeling of time series describing the economic, physical, biological and other processes.

Keywords: mathematical model, random process, Wiener process, fractional Brownian motion

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Authors: Valeria Bondarenko

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In this paper, we propose two problems related to fractal Brownian motion. First problem is simultaneous estimation of two parameters, Hurst exponent and the volatility, that describe this random process. Numerical tests for the simulated fBm provided an efficient method. Second problem is approximation of the increments of the observed time series by a power function by increments from the fractional Brownian motion. Approximation and estimation are shown on the example of real data, daily deposit interest rates.

Keywords: fractional Brownian motion, Gausssian processes, approximation, time series, estimation of properties of the model

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Authors: Kankeyanathan Kannan

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On approximation properties of group C* algebras is everywhere; it is powerful, important, backbone of countless breakthroughs. For a discrete group G, let A(G) denote its Fourier algebra, and let M₀A(G) denote the space of completely bounded Fourier multipliers on G. An approximate identity on G is a sequence (Φn) of finitely supported functions such that (Φn) uniformly converge to constant function 1 In this paper we prove that approximation property pass to free product.

Keywords: approximation property, weakly amenable, strong invariant approximation property, invariant approximation property

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Authors: Po-Wen Chen, Jin-Yu Wu, Md. Manirul Ali, Yang Peng, Chen-Te Chang, Der-Jun Jan

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Dynamics of cathode spot has become a major part of vacuum arc discharge with its high academic interest and wide application potential. In this article, using a three-dimensional statistical model, we simulate the distribution of the ignition probability of a new cathode spot occurring in different magnetic pressure on old cathode spot surface and at different arcing time. This model for the ignition probability of a new cathode spot was proposed in two typical situations, one by the pure isotropic random walk in the absence of an external magnetic field, other by the retrograde motion in external magnetic field, in parallel with the cathode surface. We mainly focus on developed relationship between the ignition probability density distribution of a new cathode spot and the external magnetic field.

Keywords: cathode spot, vacuum arc discharge, transverse magnetic field, random walk

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Authors: Nikolaos Simantiris, Markos Avlonitis

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This work builds a simulation platform, modeling the spatial diffusion of the invasive species Callinectes sapidus (blue crab) as a random walk, incorporating also generation, fatality, and fishing rates modeling the time evolution of its population. Antinioti lagoon in West Greece was used as a testbed for applying the simulation model. Field measurements from June 2020 to June 2021 on the lagoon’s setting, bathymetry, and blue crab juveniles provided the initial population simulation of blue crabs, as well as biological parameters from the current literature were used to calibrate simulation parameters. The scope of this study is to render the authors able to predict the evolution of the blue crab population in confined environments of the Ionian Islands region in West Greece. The first result of the simulation experiments shows the possibility for a robust prediction for blue crab population evolution in the Antinioti lagoon.

Keywords: antinioti lagoon, blue crab, stochastic simulation, random walk

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Authors: Behnaz Moradijamei, Michael Higgins

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Identifying community structure is a critical task in analyzing social media data sets often modeled by networks. Statistical models such as the stochastic block model have proven to explain the structure of communities in real-world network data. In this work, we develop a goodness-of-fit test to examine community structure's existence by using a distinguishing property in networks: cyclic structures are more prevalent within communities than across them. To better understand how communities are shaped by the cyclic structure of the network rather than just the number of edges, we introduce a novel method for deciding on the existence of communities. We utilize these structures by using renewal non-backtracking random walk (RNBRW) to the existing goodness-of-fit test. RNBRW is an important variant of random walk in which the walk is prohibited from returning back to a node in exactly two steps and terminates and restarts once it completes a cycle. We investigate the use of RNBRW to improve the performance of existing goodness-of-fit tests for community detection algorithms based on the spectral properties of the adjacency matrix. Our proposed test on community structure is based on the probability distribution of eigenvalues of the normalized retracing probability matrix derived by RNBRW. We attempt to make the best use of asymptotic results on such a distribution when there is no community structure, i.e., asymptotic distribution under the null hypothesis. Moreover, we provide a theoretical foundation for our statistic by obtaining the true mean and a tight lower bound for RNBRW edge weights variance.

Keywords: hypothesis testing, RNBRW, network inference, community structure

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Authors: Herjuno Bagus Wicaksono, Emma Almira Fauni, Salma Amelia Dina

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The Efficient Market Hypothesis (EMH) states that, in an efficient capital market, price fully reflects the information available in the market. This theory has influenced many investors behavior in trading in the stock market. Advanced researches have been conducted to test the efficiency of the stock market in particular countries. Indonesia, as one of the emerging countries, has performed substantial growth in the past years. Hence, this paper aims to examine the efficiency of Islamic stock market in Indonesia in its weak form. The daily stock price data from Indonesia Sharia Stock Index (ISSI) for the period October 2015 to October 2016 were used to do the statistical tests: Run Test and Serial Correlation Test. The results show that there is no serial correlation between the current price with the past prices and the market follows the random walk. This research concludes that Indonesia Islamic stock market is weak form efficient.

Keywords: efficient market hypothesis, Indonesia sharia stock index, random walk, weak form efficiency

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Authors: N. Guruprasad

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This paper presents a modification of approximation method for transportation problems. The initial basic feasible solution can be computed using either Russel's or Vogel's approximation methods. Russell’s approximation method provides another excellent criterion that is still quick to implement on a computer (not manually) In most cases Russel's method yields a better initial solution, though it takes longer than Vogel's method (finding the next entering variable in Russel's method is in O(n1*n2), and in O(n1+n2) for Vogel's method). However, Russel's method normally has a lesser total running time because less pivots are required to reach the optimum for all but small problem sizes (n1+n2=~20). With this motivation behind we have incorporated a variation of the same – what we have proposed it has TMC (Total Modified Cost) to obtain fast and efficient solutions.

Keywords: computation, efficiency, modified cost, Russell’s approximation method, transportation, Vogel’s approximation method

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Authors: Jeremie Coullon, Robert J. Webber

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We introduce a Markov chain Monte Carlo (MCMC) sam-pler for infinite-dimensional inverse problems. Our sam-pler is based on the affine invariant ensemble sampler, which uses interacting walkers to adapt to the covariance structure of the target distribution. We extend this ensem-ble sampler for the first time to infinite-dimensional func-tion spaces, yielding a highly efficient gradient-free MCMC algorithm. Because our ensemble sampler does not require gradients or posterior covariance estimates, it is simple to implement and broadly applicable. In many Bayes-ian inverse problems, Markov chain Monte Carlo (MCMC) meth-ods are needed to approximate distributions on infinite-dimensional function spaces, for example, in groundwater flow, medical imaging, and traffic flow. Yet designing efficient MCMC methods for function spaces has proved challenging. Recent gradi-ent-based MCMC methods preconditioned MCMC methods, and SMC methods have improved the computational efficiency of functional random walk. However, these samplers require gradi-ents or posterior covariance estimates that may be challenging to obtain. Calculating gradients is difficult or impossible in many high-dimensional inverse problems involving a numerical integra-tor with a black-box code base. Additionally, accurately estimating posterior covariances can require a lengthy pilot run or adaptation period. These concerns raise the question: is there a functional sampler that outperforms functional random walk without requir-ing gradients or posterior covariance estimates? To address this question, we consider a gradient-free sampler that avoids explicit covariance estimation yet adapts naturally to the covariance struc-ture of the sampled distribution. This sampler works by consider-ing an ensemble of walkers and interpolating and extrapolating between walkers to make a proposal. This is called the affine in-variant ensemble sampler (AIES), which is easy to tune, easy to parallelize, and efficient at sampling spaces of moderate dimen-sionality (less than 20). The main contribution of this work is to propose a functional ensemble sampler (FES) that combines func-tional random walk and AIES. To apply this sampler, we first cal-culate the Karhunen–Loeve (KL) expansion for the Bayesian prior distribution, assumed to be Gaussian and trace-class. Then, we use AIES to sample the posterior distribution on the low-wavenumber KL components and use the functional random walk to sample the posterior distribution on the high-wavenumber KL components. Alternating between AIES and functional random walk updates, we obtain our functional ensemble sampler that is efficient and easy to use without requiring detailed knowledge of the target dis-tribution. In past work, several authors have proposed splitting the Bayesian posterior into low-wavenumber and high-wavenumber components and then applying enhanced sampling to the low-wavenumber components. Yet compared to these other samplers, FES is unique in its simplicity and broad applicability. FES does not require any derivatives, and the need for derivative-free sam-plers has previously been emphasized. FES also eliminates the requirement for posterior covariance estimates. Lastly, FES is more efficient than other gradient-free samplers in our tests. In two nu-merical examples, we apply FES to challenging inverse problems that involve estimating a functional parameter and one or more scalar parameters. We compare the performance of functional random walk, FES, and an alternative derivative-free sampler that explicitly estimates the posterior covariance matrix. We conclude that FES is the fastest available gradient-free sampler for these challenging and multimodal test problems.

Keywords: Bayesian inverse problems, Markov chain Monte Carlo, infinite-dimensional inverse problems, dimensionality reduction

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Authors: Phyo Kyaw Kyaw

Abstract:

In the Yangon urban landscape, one could not help, but notice old buildings from the colonial period along with condominium developments recently, and many walk-up apartment buildings to accommodate the urbanization, growing population and social-economic status of Myanmar people. Walk-up apartments were built and popular after the British colonial period (around 1950s) and are still built up to today due to its cost-effectiveness and to accommodate low to mid-income residents in the metropolitan Yangon. Approximately 90% of apartment buildings are walk-up apartments. The common impression of walk-up apartments in Yangon appears to be old rectangular box shape, homogenous envelope and limited square feet dull interior small space. In other words, the buildings are full of constraints, lack of good user experiences, and they are not well-fitted in the modern days. Therefore, the resident suffers consequently many years, some may live in the apartment their entire lives. Thousands of people living in the walk-up apartment on a daily basis are being shaped by the space and its inadequate quality of living. Can it be called “Home” by the dwellers or is the place a temporary shelter?. Online semi-structured interviews of 15 apartments’ residents and online questionnaire surveys of 70 apartment residents are conducted. This research aims to explore what makes “Home” “A sense of Home” for walk-up apartment users in Yangon, Myanmar by studying subjective responses shaped by the interior and experience of the spaces in apartment to understand the perception of the residents and improve the quality of living. The result reflects the priority level of important factors in relation to the sense of home framework.

Keywords: home, living quality, space, perception, residents, walk-up apartment, Yangon

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Authors: Serge Provost, Yishan Zang

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Copulas are currently utilized in finance, reliability theory, machine learning, signal processing, geodesy, hydrology and biostatistics, among several other fields of scientific investigation. It follows from Sklar's theorem that the joint distribution function of a multidimensional random vector can be expressed in terms of its associated copula and marginals. Since marginal distributions can easily be determined by making use of a variety of techniques, we address the problem of securing the distribution of the copula. This will be done by using several approaches. For example, we will obtain bivariate least-squares approximations of the empirical copulas, modify the kernel density estimation technique and propose a criterion for selecting appropriate bandwidths, differentiate linearized empirical copulas, secure Bernstein polynomial approximations of suitable degrees, and apply a corollary to Sklar's result. Illustrative examples involving actual observations will be presented. The proposed methodologies will as well be applied to a sample generated from a known copula distribution in order to validate their effectiveness.

Keywords: copulas, Bernstein polynomial approximation, least-squares polynomial approximation, kernel density estimation, density approximation

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Authors: Smita Sonker, Uaday Singh

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Various investigators have determined the degree of approximation of functions belonging to the classes W(L r , ξ(t)), Lip(ξ(t), r), Lip(α, r), and Lipα using different summability methods with monotonocity conditions. Recently, Lal has determined the degree of approximation of the functions belonging to Lipα and W(L r , ξ(t)) classes by using Ces`aro-N¨orlund (C 1 .Np)- summability with non-increasing weights {pn}. In this paper, we shall determine the degree of approximation of 2π - periodic functions f belonging to the function classes Lipα and W(L r , ξ(t)) by C 1 .T - means of Fourier series of f. Our theorems generalize the results of Lal and we also improve these results in the light off. From our results, we also derive some corollaries.

Keywords: Lipschitz classes, product matrix operator, signals, trigonometric Fourier approximation

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Authors: N. Lebga, Kh. Bouamama, K. Kassali

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We report first-principles calculation results on the structural and elastic properties of ZnS x Se1−x alloy for which we employed the virtual crystal approximation provided with the ABINIT program. The calculations done using density functional theory within the local density approximation and employing the virtual-crystal approximation, we made a comparative study between the numerical results obtained from ab-initio calculation using ABINIT or Wien2k within the Density Functional Theory framework with either Local Density Approximation or Generalized Gradient approximation and the pseudo-potential plane-wave method with the Hartwigzen Goedecker Hutter scheme potentials. It is found that the lattice parameter, the phase transition pressure, and the elastic constants (and their derivative with respect to the pressure) follow a quadratic law in x. The variation of the elastic constants is also numerically studied and the phase transformations are discussed in relation to the mechanical stability criteria.

Keywords: density functional theory, elastic properties, ZnS, ZnSe,

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Authors: Ana M. Korol, Bibiana Riquelme, Osvaldo A. Rosso

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Since erythrocyte deformability analysis is mostly qualitative, the development of quantitative nonlinear methods is crucial for restricting subjectivity in the study of cell behaviour. An electro-optic mechanic system called erythrodeformeter has been developed and constructed in our laboratory in order to evaluate the erythrocytes' viscoelasticity. A numerical method formulated on the basis of fractal approximation for ordinary (OBM) and fractionary Brownian motion (FBM), as well as wavelet transform analysis, are proposed to distinguish chaos from noise based on the assumption that diffractometric data involves both deterministic and stochastic components, so it could be modelled as a system of bounded correlated random walk. Here we report studies on 25 donors: 4 alpha thalassaemic patients, 11 beta thalassaemic patients, and 10 healthy controls non-alcoholic and non-smoker individuals. The Correlation Coefficient, a nonlinear parameter, showed evidence of the changes in the erythrocyte deformability; the Wavelet Entropy could quantify those differences which are detected by the light diffraction patterns. Such quantifiers allow a good deal of promise and the possibility of a better understanding of the rheological erythrocytes aspects and also could help in clinical diagnosis.

Keywords: red blood cells, deformability, nonlinear dynamics, chaos theory, wavelet trannsform

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Authors: Melvin Ballera, Aldrich Olivar, Mary Soriano

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Digital computers nowadays must be able to have a utility that is capable of generating random numbers. Usually, computer-generated random numbers are not random given predefined values such as starting point and end points, making the sequence almost predictable. There are many applications of random numbers such business simulation, manufacturing, services domain, entertainment sector and other equally areas making worthwhile to design a unique method and to allow unpredictable random numbers. Applying stochastic simulation using linear congruential algorithm, it shows that as it increases the numbers of the seed and range the number randomly produced or selected by the computer becomes unique. If this implemented in an environment where random numbers are very much needed, the reliability of the random number is guaranteed.

Keywords: stochastic simulation, random numbers, linear congruential algorithm, pseudorandomness

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Authors: Oluwayemisi O. Alaba, John O. Olaomi

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The 2013 Nigerian Demographic Health Survey (NDHS) data was used to investigate the determinants of family size in Nigeria using the geo-additive model. The fixed effect of categorical covariates were modelled using the diffuse prior, P-spline with second-order random walk for the nonlinear effect of continuous variable, spatial effects followed Markov random field priors while the exchangeable normal priors were used for the random effects of the community and household. The Negative Binomial distribution was used to handle overdispersion of the dependent variable. Inference was fully Bayesian approach. Results showed a declining effect of secondary and higher education of mother, Yoruba tribe, Christianity, family planning, mother giving birth by caesarean section and having a partner who has secondary education on family size. Big family size is positively associated with age at first birth, number of daughters in a household, being gainfully employed, married and living with partner, community and household effects.

Keywords: Bayesian analysis, family size, geo-additive model, negative binomial

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Authors: Anusuya Ghosh, Vishnu Narayanan

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The approximation of convex set by semidefinite representable set plays an important role in semidefinite programming, especially in modern convex optimization. To optimize a linear function over a convex set is a hard problem. But optimizing the linear function over the semidefinite representable set which approximates the convex set is easy to solve as there exists numerous efficient algorithms to solve semidefinite programming problems. So, our approximation technique is significant in optimization. We develop a technique to approximate any closed convex set, say K by compactly semidefinite representable set. Further we prove that there exists a sequence of compactly semidefinite representable sets which give tighter approximation of the closed convex set, K gradually. We discuss about the convergence of the sequence of compactly semidefinite representable sets to closed convex set K. The recession cone of K and the recession cone of the compactly semidefinite representable set are equal. So, we say that the sequence of compactly semidefinite representable sets converge strongly to the closed convex set. Thus, this approximation technique is very useful development in semidefinite programming.

Keywords: semidefinite programming, semidefinite representable set, compactly semidefinite representable set, approximation

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Authors: Augustine Okolie, Johannes Müller, Mirjam Kretzchmar

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We adopt a maximum-likelihood framework to estimate parameters of a stochastic susceptible-infected-recovered (SIR) model with contact tracing on a rooted random tree. Given the number of detectees per index case, our estimator allows to determine the degree distribution of the random tree as well as the tracing probability. Since we do not discover all infectees via contact tracing, this estimation is non-trivial. To keep things simple and stable, we develop an approximation suited for realistic situations (contract tracing probability small, or the probability for the detection of index cases small). In this approximation, the only epidemiological parameter entering the estimator is the basic reproduction number R0. The estimator is tested in a simulation study and applied to covid-19 contact tracing data from India. The simulation study underlines the efficiency of the method. For the empirical covid-19 data, we are able to compare different degree distributions and perform a sensitivity analysis. We find that particularly a power-law and a negative binomial degree distribution meet the data well and that the tracing probability is rather large. The sensitivity analysis shows no strong dependency on the reproduction number.

Keywords: stochastic SIR model on graph, contact tracing, branching process, parameter inference

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Authors: Xin Chen

Abstract:

It has been widely observed that marine food webs have significantly larger predator–prey body-size ratios compared with their terrestrial counterparts. A number of hypotheses have been proposed to account for such difference on the basis of primary productivity, trophic structure, biophysics, bioenergetics, habitat features, energy efficiency, etc. In this study, an alternative explanation is suggested based on the difference in the spatial dimensions of foraging arenas: terrestrial animals primarily forage in two dimensional arenas, while marine animals mostly forage in three dimensional arenas. Using 2-dimensional and 3-dimensional random walk simulations, it is shown that marine predators with 3-dimensional foraging would normally have a greater foraging efficiency than terrestrial predators with 2-dimensional foraging. Marine prey with 3-dimensional dispersion usually has greater swarms or aggregations than terrestrial prey with 2-dimensional dispersion, which again favours a greater predator foraging efficiency in marine animals. As an analytical tool, a Lotka-Volterra based adaptive dynamical model is developed with the predator-prey ratio embedded as an adaptive variable. The model predicts that high predator foraging efficiency and high prey conversion rate will dynamically lead to the evolution of a greater predator-prey ratio. Therefore, marine food webs with 3-dimensional foraging space, which generally have higher predator foraging efficiency, will evolve a greater predator-prey ratio than terrestrial food webs.

Keywords: predator-prey, body size, lotka-volterra, random walk, foraging efficiency

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Authors: D. S. Palimkar

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The FRIGDI (functional random integrodifferential inclusion) seems to be new and includes several known random differential inclusions already studied in the literature as special cases have been discussed in the literature for various aspects of the solutions. In this paper, we prove the existence result for FIGDI under the non-convex case of multi-valued function involved in it.Using random fixed point theorem of B. C. Dhage and caratheodory condition. This result is new to the theory of differential inclusion.

Keywords: caratheodory condition, random differential inclusion, random solution, integro-differential inclusion

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Authors: Tulin Coskun

Abstract:

We investigate the approximation of analytic functions of several variables in polydisc by the sequences of linear k-positive operators in Gadjiev sence. The approximation of analytic functions of complex variable by linear k-positive operators was tackled, and k-positive operators and formulated theorems of Korovkin's type for these operators in the space of analytic functions on the unit disc were introduced in the past. Recently, very general results on convergence of the sequences of linear k-positive operators on a simply connected bounded domain within the space of analytic functions were proved. In this presentation, we extend some of these results to the approximation of analytic functions of several complex variables by sequences of linear k-positive operators.

Keywords: analytic functions, approximation of analytic functions, Linear k-positive operators, Korovkin type theorems

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Authors: Anand K. Patel

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Background: Patients with COPD have decreased pulmonary functions, which in turn reflect on their day-to-day activities. Objectives: To assess the correlation between functional vital capacity (FVC) and forced expiratory volume in one second (FEV1) with 6 minutes walk test (6MWT). To correlate the Borg rating for perceived exertion scale (Borg scale) and Modified medical research council (MMRC) dyspnea scale with the 6MWT, FVC and FEV1. Method: In this prospective study total 72 patients with COPD diagnosed by the GOLD guidelines were enrolled after taking written consent. They were first asked to rate physical exertion on the Borg scale as well as the modified medical research council dyspnea scale and then were subjected to perform pre and post bronchodilator spirometry followed by 6 minute walk test. The findings were correlated by calculating the Pearson coefficient for each set and obtaining the p-values, with a p < 0.05 being clinically significant. Result: There was a significant correlation between spirometry and 6MWT suggesting that patients with lower measurements were unable to walk for longer distances. However, FVC had the stronger correlation than FEV1. MMRC scale had a stronger correlation with 6MWT as compared to the Borg scale. Conclusion: The study suggests that 6MWT is a better test for monitoring the patients of COPD. In spirometry, FVC should be used in monitoring patients with COPD, instead of FEV1. MMRC scale shows a stronger correlation than the Borg scale, and we should use it more often.

Keywords: spirometry, 6 minute walk test, MMRC, Borg scale

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Authors: Hare Krishna Nigam

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In this paper, for the first time, (E,q)(C,2) product summability method is introduced and two quite new results on degree of approximation of the function f belonging to Lip (alpha,r)class and W(L(r), xi(t)) class by (E,q)(C,2) product means of Fourier series, has been obtained.

Keywords: Degree of approximation, (E, q)(C, 2) means, Fourier series, Lebesgue integral, Lip (alpha, r)class, W(L(r), xi(t))class of functions

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Authors: Kairat T. Mynbaev, Elena N. Lomakina

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We consider a Hardy-type operator generated by a family of open subsets of a Hausdorff topological space. The family is indexed with non-negative real numbers and is totally ordered. For this operator, we obtain two-sided bounds of its norm, a compactness criterion, and bounds for its approximation numbers. Previously, bounds for its approximation numbers have been established only in the one-dimensional case, while we do not impose any restrictions on the dimension of the Hausdorff space. The bounds for the norm and conditions for compactness earlier have been found using different methods by G. Sinnamon and K. Mynbaev. Our approach is different in that we use domain partitions for all problems under consideration.

Keywords: approximation numbers, boundedness and compactness, multidimensional Hardy operator, Hausdorff topological space

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Authors: Rajkumar N. Ingle

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In this paper, the existence of a solution of nonlinear functional random differential equations of the first order is proved under caratheodory condition. The study of the functional random differential equation has got importance in the random analysis of the dynamical systems of universal phenomena. Objectives: Nonlinear functional random differential equation is useful to the scientists, engineers, and mathematicians, who are engaged in N.F.R.D.E. analyzing a universal random phenomenon, govern by nonlinear random initial value problems of D.E. Applications of this in the theory of diffusion or heat conduction. Methodology: Using the concepts of probability theory, functional analysis, generally the existence theorems for the nonlinear F.R.D.E. are prove by using some tools such as fixed point theorem. The significance of the study: Our contribution will be the generalization of some well-known results in the theory of Nonlinear F.R.D.E.s. Further, it seems that our study will be useful to scientist, engineers, economists and mathematicians in their endeavors to analyses the nonlinear random problems of the universe in a better way.

Keywords: Random Fixed Point Theorem, functional random differential equation, N.F.R.D.E., universal random phenomenon

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