جستجوی مقالات مرتبط با کلیدواژه "نظریه کی" در نشریات گروه "ریاضی"

Functional analysis was an  important development in the viewpoint of mathematicians in examining the behavior of functions as a set (space of functions). The important concept of single space and the distance of its points, invented by ‎Fréchet‎ in the early 20th century, and the fusion of algebraic and topological concepts eventually led to the establishment of this branch of mathematics around 1933. The first Iranian mathematician to make a mark in this field was Heydar Radjavi, who contributed to the field in Iran from 1963 to 1968. Another branch of this field is called nonlinear functional analysis, and Behzad Djafari-Rouhani was the first specialist in this branch who worked in the country from 19‎87‎ onwards. Another key figure in the development and expansion of this branch in the country is Jafar Zafarani. This article, after outlining the history of functional analysis, will provide a brief overview of his contributions to this field. Through a conversation with him, we will delve into his experiences and insights from various aspects of his professional life.

The above abstract has been extracted by the translator from the original article (J. Klaška, Real-world applications of number theory, South Bohemia Mathematical Letters, 25 no. 1 (2017) 39–47.) The present paper is concerned with practical applications of the number theory and is intended for all readers interested in applied mathematics. Using examples we show how human creativity can change the results of the pure mathematics into a practical usable form. Some historical notes are also included.

German mathematician Johann Carl Friedrich Gauss (30 April 1777-23 February 1855), regarded as one of the greatest mathematicians of all time, claimed: "Mathematics is the queen of the sciences and number theory is the queen of mathematics." However, for many years number theory had only few practical applications. It is well known that the great English number theorist Godfrey Harold Hardy (7 February 1877-1 December 1947) believed that number theory had no practical applications. See his essay "A Mathematician's Apology" [16]. Over the 20th and 21st centuries, this situation has changed significantly. Contrary to Hardy's opinion, many practical and interesting applications of number theory have been discovered. The present paper brings some remarkable examples of number theory applications in the real world. The paper can be regarded as a loose continuation of the author's preceding work [19] and [20]. Let $n$ be a positive integer, $\geq 2$. Then, the equation (1.1)$$a_1x_1+\ldots+a_nx_n=m$$is said to be a linear Diophantine equation if all unknowns $x_1,\ldots,x_n$ and all coefficients $a_1,\ldots,a_n,m$ are integers. For general methods for solving (1.1), see for example [5], [25], and [24, pp. 27-31]. In the following we give some interesting examples of using Diophantine equations in the natural sciences.

As the first example we show some applications of a linear Diophantine equation to problems in chemistry. In particular, we will deal with the balancing of chemical equations. See [6]. Consider a chemical equation written in the form (2.1)$$x_1A_{a_1}B_{b_1}C_{c_1}\ldots +x_2A_{a_2}B_{b_2}C_{c_3}\cdots +\cdots\rightarrow x'_1A'_{a'_1}B'_{b'_1}C'_{c'_1}\cdots +x'_2A'_{a'_2}B'_{b'_2}C'_{c'_3}\cdots +\cdots$$where $A, B, C,\ldots$ are the elements occurring in the reaction, $a_1, b_1, c_1,\ldots,a'_1, b'_1, c'_1,\ldots$ are positive integers or $0$, and $x_1, x_2,\ldots,x'_1, x'_2,\ldots$ are the unknown coeffcients of the reactants and products. Then, we hav (2.2)\begin{array}{rcl}x_1a_1+x_2a_2+\cdots&=&x'_1a'_1+x'_2a'_2+\cdots\\x_1b_1+x_2b_2+\cdots&=&x'_1b'_1+x'_2b'_2+\cdots\\x_1c_1+x_2c_2+\cdots&=&x'_1c'_1+x'_2c'_2+\cdots\\&\cdots&\end{array} Clearly, each equation of (2.2) expresses the law of conservation of the number of atoms for any particular element $A,B,C,\ldots$. Finding all integer solutions $[x_1,x_2,\ldots,x'_1,x'_2,\ldots]$ of (2.2) is a nice elementary problem of Diophantine analysis. In the second example we show how linear Diophantine equations can be used to determine the molecular formula [6]. Assume that a substance with a molecular weight of $m$ contains elements $A, B, C,\ldots$ with atomic weights $a, b, c,\ldots$ and that $x,y,z,\ldots$ represent the numbers of atoms of $A, B, C,\ldots$ in a molecule. Then, we have (2.3)$$ax+by+cz+\ldots=m.$$Let $\alpha,\beta,\gamma,\ldots$ denote the integers nearest the values $a, b, c,\ldots$ and $\mu$ denote theinteger nearest m. Then, (2.3) can be replaced by the linear Diophantine equation (2.4)$$\alpha x+\beta y+\gamma z+\cdots=\mu.$$ If we require that the values $x, y, z,\ldots$ in (2.4) should be reasonably small, we can solve (2.4) under a condition (2.5)$$-\frac{1}{2}<(a-\alpha)x+(b-\beta)y+(c-\gamma)z+\cdots<\frac{1}{2}.$$If more solutions of (2.4) are obtained, the true values may be found by substituting into (2.3) and finding which of them satisfies (2.3) with minimum deviation from $m$. In the third example we focus on an interesting problem in virology. Recall, that virus particles consist of protein subunits ordered geometrically according to strict symmetry rules. These rules highly depend on the chemical properties of the protein. As the last example we establish the number $p(n)$ which is the number of partitions of $n$ and it is known under the name of \emph{partitio numerorum}. Two interesting connections between the problem \emph{partitio numerorum} and physics will now be mentioned. First recall that the Hardy-Ramanujan formula (2.6)$$p(n)\sim \frac{1}{4n\sqrt{3}}\cdot\exp(\pi\sqrt{\frac{2n}{3}})\ \text{for}\ n\rightarrow\infty$$has been used, with great success, in quantum physics. The formula (2.6) was discovered in 1917 by G. H. Hardy and the brilliant Indian mathematician Srinivasa Ramanujan (22 December 1887-26 April 1920). The second very important application of Hardy-Ramanujan formula can be found in the problems of statistical mechanics.

Finally, some further significant applications of the number theory will be shortly mentioned. Above all, it is well known that the theory of Fibonacci numbers has many applications in physics, chemistry, biology, economy, and architecture. Listing 163 chronological references to papers published from 1611 to 2011, paper [19] can serve as an introduction to this field. Further fields of number theory with important applications include the theory of sequences over finite fields [20]. This theory found an application in the testing of Einstein's general relativity or in testing the global warming of oceans. Furthermore, using methods of elementary number theory, practical problems have been solved concerning to the splicing of telephone cables [21]. Many further interesting applications can be found in the book Number Theory and the Periodicity of Matter [3]. Lastly, new attractive applications of the number theory include cryptography, coding theory, and random number generation. With the rise of computers, these fields develop very rapidly with their importance continuously increasing.

Data envelopment analysis technique is used to evaluate the relative efficiency of a set of decision-making units that have been studied in different fields. One of the important issues in data envelopment analysis is sensitivity analysis. Many articles have been presented in this field by researchers, sometimes managers are concentrating on issues that would be critical to allocate a fixed cost to decision-making units. Since in real problems the primary data are not precise but interval, ordinal, and qualitative therefore this study have been discussed this issue and present a model for assigning a fuzzy fixed cost to decision-making units. Moreover, the inputs and outputs of all units are assumed to be fuzzy and the allocation of new costs should be such that the highest number of inefficient units become efficient. At the end, this model has been utilized in two numerical examples and the results have been presented.

How important is the theory of categories? From the early days of its creation until today, this question has occupied the minds of mathematicians and different answers have been provided: For some categories are merely a tool. For others, they are a fundamental part of today’s mathematics. Generally, questions like these do not have definite and absolute answers. At least for this particular question, the mathematics community has not reached a common answer, and probably will never reach a resolution. Our goal in writing this article is to discuss the origins and developments of the theory of categories and also discuss its role in mathematics.

A new proof identifies precisely how large a mathematical graph must be before it contains a regular substructure.

Groupoid is a structure similar to a group, with more applications in differential geometry, differential equations, alge- braic geometry, mathematical analysis, and physics. In this paper, we introduce groupoids from two points of view: algebraic and the category theory. We review the definitions of groupoid C∗− algebra, Fourier algebra and Fourier-Stieltjes algebra. Introducing the theory of unitary representation of groupoids, we extend a clasical result in the representation theory of locally compact groups to groupoids.

Survival analysis, and in particular survival distribution estimation, are important issues in the statistical sciences. Various parametric and nonparametric methods have been proposed to estimate the survival distribution. In this respect, the theoretical survival distributions are specified and their parameters are obtained by methods such as the maximum likelihood estimator and the Bayesian estimator and we can mention to nonparametric methods such as the Kaplan-Meier method, Cox regression and the life table. In addition, another important issue in survival analysis, especially in actuarial and biostatistics, is graduation of data for which smoothness and goodness of fit are two fundamental requirements.On the other hand, in the probability theory, there are two basic approaches to estimate probability distributions by using the concept of entropy: Maximum Entropy Principle (ME) and Minimum Kullback-Leibler Principle (MKL) or Minimum Cross Entropy Principle.In this paper, we examine the approach of the above two optimization models to estimate survival and probability distributions, especially for the classification of the data. In these studies, in addition to investigating parametric models, in order to achieve a compromise between the conditions of smoothness and goodness of fit, we apply a new entropy optimization model by defining an objective function combined from both of the two above principles and adjusting a coefficient that is used to ensure the degree of goodness of fitting and smoothing the estimates, as well as to show their priority in the classification of the data. We use this model to estimate the mortality probability distribution, particularly the column related to the mortality probability of a certain age ( qx) in life table. Finally, with the help of this method, we set the life table for Iranian women and men in 2011.

In many applications, there are some fixed costs for constructing the common platform of an organization which must be shared by all decision making units (DMUs). It is important how one should allocate such costs among all competing DMUs. Data envelopment analysis (DEA), which is a useful tool to evaluate the relative efficiency of DMUs, has been successfully used in allocating fixed costs among DMUs. Two main approaches have been proposed to allocate fixed costs that are based on maintaining or improving the relative efficiency of DMUs. In fixed costs allocation, one however needs to take into account both competitive and cooperative aspects among all DMUs. Therefore, it seems more reasonable to apply a peer-evaluation method in evaluating the efficiency of DMUs. To this end, we use cross-efficiency evaluation in DEA to allocate fixed costs between DMUs. Using cross-efficiency method and some concepts of game theory, we propose a new fixed costs allocation approach to share costs between DMUs such that the vector of cross-efficiency scores of DMUs after allocation is Pareto. We use a real application to more illustrate the proposed method and compare it with some of the existing methods.

In recent years, social network analysis gains great deal of attention. Social networks have various applications in different areas, namely predicting disease epidemic, search engines and viral advertisements. A key property of social networks is that interpersonal relationships can influence the decisions that they make. Finding the most influential nodes is important in social networks because such nodes can greatly affect many other people. In this paper, diffusion among nodes is investigated using the centrality of Shapely value and by dividing a network into communities in linear threshold and by the independent cascade model. Furthermore, this algorithm is evaluated by different data sets and compared with benchmarks.

In the proposed algorithm, according to the cooperative game, the value of each coalition is equal to the nodes inside it or at least adjacent to n nodes inside the coalition. After calculating  the Shapley value for each node in the social network, the Shapley value of the community is allocated to each community by summing the Shapley values of nodes inside the community which is used as a criteria for community evaluation.It is worth mentioning that a social network is divided into non-overlapping communities. In other words, each node should belong to only one community. As each community has a unique Shapley value, the fair selection algorithm is utilized to select nodes from communities. Subsequently, the fair selection is applied to the network. Finally, the selected nodes are evaluated to find k influential nodes.

In this research, we propose a heuristic method called CSCS to tackle the influential maximization problem. This open problem focuses on influencing a larger number of individuals within a social network. The proposed algorithm is based on communities and centrality of the Shapley value.In order to evaluate the performance of our algorithm,  the CSCS algorithm has been applied to six different social networks and then results are compared with four benchmarks. Results of experiments demonstrate that in the linear threshold model, the CSCS algorithm has an acceptable performance by increasing the number of initial nodes. Furthermore, in the independent cascade model, the CSCS algorithm performance is often similar to the Community Based Degree algorithm and in some cases its even better.

The execution time of the algorithm is improved efficiently, and consequently, it can easily be applied on large social networks. In the linear threshold model, the CSCS algorithm has an acceptable performance by increasing the number of initial nodes. Furthermore, in the independent cascade model, the CSCS algorithm performance is often similar to the Community Based Degree algorithm and in some cases its even better. The CSCS algorithm also works well with disconnected social networks because it divides the network into communities.

Although TOPSIS is one of the widely used models in analyzing MCDM/MADM problems, there exists a necessary condition for its application that is the increasing or decreasing utility of the criteria. In the real world, in many cases, some of the criteria of decision making lack this property. In these cases an unrealistic assumption of the monotonic utility of the criteria is imposed to the model. However such an assumption may affect the accuracy of the results. This paper provides an extension of TOPSIS model which overcomes this limitation.