Mathematical Modeling in Social Network Analysis： Using TOPSIS to Find Node Influences in a Social Network

In a social network analysis the output provided includes many measures and metrics. For each of these measures and metric, the output provides the ability to obtain a rank ordering of the nodes in terms of these measures. We might use this information in decision making concerning disrupting or deceiving a given network. All is fine when all the measures indicate the same node as the key or influential node. What happens when the measures indicate different key nodes？ Our goal in this paper is to explore two methodologies to identify the key players or nodes in a given network. We apply TOPSIS to analyze these outputs to find the most influential nodes as a function of the decision makers＇ inputs as a process to consider both subjective and objectives inputs through pairwise comparison matrices. We illustrate our results using two common networks from the literature： the Kite network and the Information flow network from Knoke and Wood. We discuss some basic sensitivity analysis can may be applied to the methods. We find the use of TOPSIS as a flexible method to weight the criterion based upon the decision makers＇ inputs or the topology of the network.

Using Sobel Operator for Automatic Edge Detection in Medical Images

The problems of installation and integration of complex suite of software for processing medical images. Based analysis of the situation is realized in an easier integration of an automated system using the latest information technologies using the web - environment for analysis and segmentation of DICOM - images.

The Generalized Roper-Suffridge Extension Operator on the Reinhardt Domain

Let pj ∈ N and pj ≥-1, j = 2,...,n be a fixed positive integer. In this paper a generalized Roper-Suffridge extension operator F（z） ={f（Z1）＋f＇（z1）} on Reinhardt domain is defined. Some different conditions for Pj areestablished under which the operator preserves an almost spirallike mapping of type fl and order a and spirallike mapping of type β and order α, respectively. In particular, our results reduce to many well-known results.

Issues in the Influence of Ito-type Noise on the Oscillation of Solutions of Delay Differential Pantograph Equations

In this paper, a deterministic delay differential pantograph equation （DDPE） with an unbounded memory is stochastically perturbed by an Ito-type noise. The contribution of white noise to the oscillatory behaviour of the new stochastic delay differential pantograph equation （SDDPE） is investigated. It is established that under certain conditions and with a highly positive probability, the new stochastic delay differential pantograph equation has an oscillatory solution influenced by the presence of the noise. This is not possible with the original deterministic system which has a non-oscillatory solution due to the absence of noise.

Geometrical Langlands Ramifications and Differential Operators Classification by Coherent D-Modules in Field Theory

Spaces of equivalence modulo a relation of congruence are constructed on field solutions to establish a theory of the universe that includes the theory QFT （Quantum Field theory）, the SUSY （Super-symmetry theory） and HST （heterotic string theory） using the sheaves correspondence of differential operators of the field equations and sheaves of coherent D - Modules [1]. The above mentioned correspondence use a Zuckerman functor that is a factor of the universal functor of derived sheaves of Harish-Chandra to the Langlands geometrical program in mirror symmetry [2, 3]. The obtained development includes complexes of D - modules of infinite dimension, generalizing for this way, the BRST-cohomology in this context. With it, the class of the integrable systems is extended in mathematical physics and the possibility of obtaining a general theory of integral transforms for the space - time （integral operator cohomology [4]）, and with it the measurement of many of their observables [5]. Also tends a bridge to complete a classification of the differential operators for the different field equations using on the base of Verma modules that are classification spaces of SO（l, n ＋ 1）, where elements of the Lie algebra al（1, n ＋ 1）, are differential operators, of the equations in mathematical physics [1]. The cosmological problem that exists is to reduce the number of field equations that are resoluble under the same gauge field （Verma modules） and to extend the gauge solutions to other fields using the topological groups symmetries that define their interactions. This extension can be given by a global Langlands correspondence between the Hecke sheaves category on an adequate moduli stack and the holomorphic L G - bundles category with a special connection （Deligne connection）. The corresponding D - modules may be viewed as sheaves of conformal blocks （or co-invariants） （images under a version of the Penrose transform [1, 6]） naturally arising in the framework of conformal field theory.

A New Proposal to Adjust a Straight Line to a Normal Q-Q Plot

In this paper a new proposal of a straight line, the ＂modified Tukey＇s line＂, for fitting to a normal quantile-quantile Plot, or normal Q-Q plot, is presented. This probability plot allows us to determine whether a set of sample observations is distributed according to a normal distribution. For this, the sample quantiles are represented against the quantiles of a theoretical probability model, which in this case is the normal distribution. If the data set follows the above mentioned distribution, the plotted points will have a rectilinear configuration. To verify this, there are different alternatives for fitting a straight line to the plotted points. Among the straight lines which can be fitted to a Q-Q plot, in this paper, besides the proposed straight line, the following straight lines are considered： straight line that passes through the first and third quartiles, straight line that passes through the 10th and 90th percentiles, straight line fitted by the method of least squares, straight line with slope s and constant the average of the data set, Theil＇s line and Tukey＇s line. In addition, an example, in which there are represented the different straight lines considered and the proposed straight line on a normal Q-Q plot obtained for the same set of observations, is developed. In this example the existing differences among the straight lines are observed.

The Degenerate Form of the Adomian Polynomials in the Power Series Method for Nonlinear Ordinary Differential Equations

In this paper, we propose a new variation of the Adomian polynomials, which we call the degenerate Adomian polynomials, for the power series solutions of nonlinear ordinary differential equations with nonseparable nonlinearities. We establish efficient algorithms for the degenerate Adomian polynomials. Next we compare the results by the Adomian decomposition method using the classic Adomian polynomials with the results by the Rach-Adomian-Meyers modified decomposition method incorporating the degenerate Adomian polynomials, which itself has been shown to be a confluence of the Adomian decomposition method and the power series method. Convergence acceleration techniques including the diagonal Pade approximants are considered, and new numeric algorithms for the multistage decomposition are deduced using the degenerate Adomian polynomials. Our new technique provides a significant advantage for automated calculations when computing the power series form of the solution for nonlinear ordinary differential equations. Several expository examples are investigated to demonstrate its reliability and efficiency.

A Bayesian Theory of Games： An Analysis of Strategic Interactions with Statistical Decision Theoretic Foundation

This paper introduces a new game theoretic equilibrium which is based upon the Bayesian subjective view of probability, BEIC （Bayesian equilibrium iterative conjectures）. It requires players to make predictions, starting from first order uninformative predictive distribution functions （or conjectures） and keep updating with statistical decision theoretic and game theoretic reasoning until a convergence of conjectures is achieved. Information known by the players such as the reaction functions are thereby incorporated into their higher order conjectures and help to determine the convergent conjectures and the equilibrium. In a BEIC, conjectures are consistent with the equilibrium or equilibriums they supported and so rationality is achieved for actions, strategies and conjectures. The BEIC approach is capable of analyzing a larger set of games than current Nash Equilibrium based games theory, including games with inaccurate observations, games with unstable equilibrium and games with double or multiple sided incomplete information games. On the other hand, for the set of games analyzed by the current games theory, it generates far lesser equilibriums and normally generates only a unique equilibrium. It treats games with complete and perfect information as special cases of games with incomplete information and noisy observation whereby the variance of the prior distribution function on type and the variance of the observation noise term tend to zero. Consequently, there is the issue of indeterminacy in statistical inference and decision making in these games as the equilibrium solution depends on which variances tends to zero first. It therefore identifies equilibriums in these games that have so far eluded the classical theory of games. Finally, it also resolves inconsistencies in equilibrium results by different solution concepts in current games theory such as that between Nash Equilibrium and iterative elimination of dominated strategies and that between Perfect Bayesian Equilibrium and backward induction （Subgame Perfect Equilibrium）.

A Double Shot Noise Process and Its Application in Insurance

The authors consider a compound Cox model of insurance risk with the additional economic assumption of a positive interest rate. As the authors note a duality result relating a compound Cox model of insurance risk with a positive interest rate and a double shot noise process, the authors analyze a double shot noise process systematically for its theoretical distributional properties, based on the piecewise deterministic Markov process theory, and the martingale methodology. The authors also obtain the moments of aggregate accumulated/discounted claims where the claim arrival process follows a Cox process with shot noise intensity. Removing the parameters in a double shot noise process gradually, the authors show that it becomes a compound Cox process with shot noise intensity, a single shot noise process and a compound Poisson process. Numerical comparisons are shown between the moments （i.e. means and variances） of a compound Poisson model and their counterparts of a compound Cox model with/without considering a positive interest rate. For that purpose, the authors assume that claim sizes and primary event sizes follow an exponential distribution, respectively.

Reconstructed Elzaki Transform Method for Delay Differential Equations with Mamadu-Njoseh Polynomials

One of the solution techniques used for ordinary differential equations, partial and integral equations is the Elzaki Transform. This paper is an extension of Mamadu and Njoseh [1] numerical procedure (Elzaki transform method (ETM)) for computing delay differential equations (DDEs). Here, a reconstructed Elzaki transform method (RETM) is proposed for the solution of DDEs where Mamadu-Njoseh polynomials are applied as basis functions in the approximation of the analytic solution. Using this strategy, a numerical illustration as in Ref.[1] is provided to the RETM as a basis for comparison to guarantee accuracy and consistency of the method. All numerical computations were performed with MAPLE 18 software.

A Robust Method for Ordering Performances of Multi-assets, Based Purely on Their Return Series

This study propose a new robust method to rank the performances of multi-assets （portfolios）, based purely on their return time series. This method makes no assumption on the distributions. Topsoe distance is symmetrized Kullback-Leibler divergence by average of the probabilities. The square root of Topsoe distance is a metric. We extend this metric from probability density functions to real number series on （0, 1 ]. We call it ST-metric. We show the consistency of ST-metric with mean-variance theory and stochastic dominance method of order one and two. We demonstrate the advantages of ST-metric over mean-variance rule and stochastic dominance method of order one and two.

Availability Assessment of Electric Power based on Switch Reliability Modelling with Dynamic Bayesian Networks： Case Study of Electrical Distribution Networks

As a generalization of the successful hidden Markov models, Dynamic Bayesian Networks （DBNs） are a natural basis for the general temporal action interpretation task. This document provides a conditional probabilistic approach to analyze the energy availability in electrical distribution networks by using Bayesian networks （BN）. Firstly a static BN modelling is presented to show the influence of the switch behaviour on the energy availability. Then, the dynamic behaviour of the switch is cared by switch reliability modelling using DBN which permits to predict the energy availability. The prediction by DBNs discussed in the case study of this paper gives a strong contribution on electrical network supervisory control and it can also be applied to transportation networks.

On Error Analysis Comparison of Some Numerical Experimental Results

In numerical analysis,truncation error is the error made by truncating an infinite sum and approximating it by a finite sum which is present even with infinite-precision arithmetic often caused by truncation of the infinite Taylor seriesto form the algorithm [1].

Determination of Causal Effect in Observational Studies： Analysis of Correlated Data with Binary End-Points

Identifying the causal impact of＇ some intervention challenging when one is faced with correlated binary end-points in observational studies is a challenging task, and it is even more The statistical literature on analyzing such data is well documented. Dependence between observations from the same study subject in correlated data renders invalid the usual chi-square tests of independence and inflates the variance ofparameter estimates. Disaggregated approaches such as hierarchical linear models which are able to adjust for individual level covariate：s are favoured in the analysis of such data, thereby gaining power over aggregated and individual-level analyses. In this article the authors, therefore, address the issue of analyzing correlated data with dichotomous end-points by using hierarchical logistic regression, a generalization of the standard logistic regression model for independent outcomes.

Project-based Learning in Calculus on The Use of Maple Software Technology

In this article, authors describe how to use the project-based learning （PBL） pedagogy to enhance students＇ Calculus learning based on the first author＇s experimental teaching experience. The ＂2014 BMCC Polar Art Calendar＂ project was completed by Calculus students at Borough of Manhattan Community College （BMCC） during the fall 2013 semester. Students were requested to apply graphs of polar equations to create computer-generated images with a variety of flower patterns by using the Maple technology in a math lab. At the end of this project, students were requested to submit and present their written reports to express their mathematical thinking. Authors also explain in details how to create projects compatible with textbook knowledge learning objectives, how to prepare scaffolding materials for students to use, how to utilize a math lab and to work with lab technicians in Maple Software, and how to design a rubric for project evaluations. Students＇ artwork created in the Polar Art Calendar are presented. Students＇ positive outcomes have proven a success of this project design as well as its execution as an example of PBL. Benefits to students and challenges to teachers on the use of PBL approach have been discussed at the end of this article.

Bernoulli Numbers ＆ Vandermonde Matrices

Matrix structuring is a very beautiful way to place Bernoulli numbers, by which a new view to the numbers is opened. Natural Numbers are mathematics seeds and Natural Number System （NNS） breeds the whole world mathematically.

AVLINK： Robust Clustering Algorithm based on Average Link Applied to Protein Sequence Analysis

Robust Clustering methods are aimed at avoiding unsatisfactory results resulting from the presence of certain amount of outlying observations in the input data of many practical applications such as biological sequences analysis or gene expressions analysis. This paper presents a fuzzy clustering algorithm based on average link and possibilistic clustering paradigm termed as AVLINK. It minimizes the average dissimilarity between pairs of patterns within the same cluster and at the same time the size of a cluster is maximized by computing the zeros of the derivative of proposed objective function. AVLINK along with the proposed initialization procedure show a high outliers rejection capability as it makes their membership very low furthermore it does not requires the number of clusters to be known in advance and it can discover clusters of non convex shape. The effectiveness and robustness of the proposed algorithms have been demonstrated on different types of protein data sets.

Investigation of a Network Failure Problem with a Significant Path, from the Perspective of Crisis Management

The objective of this study is to investigate a network failure problem with a significant path, emerging from the context of crisis management, such as in the case of natural disasters. For a given tree with m failed edges, we assume that we have sufficient resources to recover k edges of the m edges. Each node has a positive weight. In this situation, we consider which k edges should be fixed in order to maximize the sum of the weights of the nodes reachable from the significant path. In this paper, we formulate such a problem as a combinatorial problem. Further, we show that a part of our problem may be solved by translating it into the terms of the so-called tree knapsack problem.

Rearrangement Inequality and Chebyshev＇s Sum Inequality on Positive Tensor Products of Orlicz Sequence Space with Banach Lattice

Let φ be an Orlicz function that has a complementary function φ＊ and let lφ be an Orlicz sequence space. We prove a similar version of Rearrangement Inequality and Chebyshev＇s Sum Inequality in lφ FX, the Fremlin projective tensor product of lφ with a Banach lattice X, and in lφ iX, the Wittstock injective tensor product of lφ with a Banach lattice X.

Determining Bookkeeping Cash Maximum of Serbian Army Units by Using Multicriteria Optimization

Normal practice of financial management in the defense system is crucial for the performance of assigned tasks. Payment transactions in cash, in addition to non-cash payment system are very important if we take into account the specificity of the defense system. With limited financial resources optimization level of bookkeeping cash limit should provide continuous funding of units and institutions of the defense system. The aim of this paper is to show that using the method of analytic hierarchy process （AHP） we can help optimize the allocation of cash financial funds within the defense system.