Combinatorics & Statistics

En este capítulo estudiamos diversas técnicas para contar conjuntos finitos de objetos matemáticos, tales como tuplas, funciones, funciones inyectivas y subconjuntos. Finalizamos el capítulo con una breve introducción a la probabilidad finita, la cual involucra muchas de las técnicas presentadas previamente.

For a long time, one of my dreams was to describe the nature of uncertainty axiomatically, and it looks like I've finally done it in my co∼eventum mechanics! Now it remains for me to explain to everyone the co∼eventum mechanics in the most approachable way. This is what I'm trying to do in this work. The co∼eventum mechanics is another name for the co∼event theory, i.e., for the theory of experience and chance which I axiomatized in 2016 [1, 2]. In my opinion, this name best reflects the co∼event-based idea of the new dual theory of uncertainty, which combines the probability theory as a theory of chance, with its dual half, the believability theory as a theory of experience. In addition, I like this new name indicates a direct connection between the co∼event theory and quantum mechanics, which is intended for the physical explanation and description of the conict between quantum observers and quantum observations [4]. Since my theory of uncertainty satises the Kolmogorov axioms of probability theory, to explain this co∼eventum mechanics I will use a way analogous to the already tested one, which explains the theory of probability as a theory of chance describing the results of a random experiment. The simplest example of a random experiment in probability theory is the " tossing a coin ". Therefore, I decided to use this the simplest random experiment itself, as well as the two its analogies: the " "flipping a coin " and the " spinning a coin " to explain the co∼eventum mechanics, which describes the results of a combined experienced random experiment. I would like to resort to the usual for the probability theory " coin-based " analogy to explain (and first of all for myself) the logic of the co∼eventum mechanics as a logic of experience and chance. Of course, this analogy one may seem strange if not crazy. But I did not come up with a better way of tying the explanations of the logic of the co∼eventum mechanics to the coin-based explanations that are commonly used in probability theory to explain at least for myself the logic of the chance through a simple visual " coin-based " model that clarifies what occurs as a result of a combined experienced random experiment in which the experience of observer faces the chance of observation. I hope this analogy can be useful not only for me in understanding the co∼eventum mechanics.

You yourself, or what is the same, your experience is such ``coin'' that, while you aren't questioned, it rotates all the time in ``free flight''. And only when you answer the question the ``coin'' falls on one of the sides: ``Yes'' or ``No'' with the believability that your experience tells you.

Data yang telah didapat pada penelitian atau studi kasus sebelumnya ,meliputi penyebaran data, pola data pemusatandata dan sebagainya, dapat digunakan untuk memprediksi kejadian yang akan datang, seberapa besar kemungkinannya, dan sebagainya. Hal tersebut termasuk kedalam distribusi peluang. Distribusi peluang merupakan pemetaan kemungkinan atau variabel yang bersifat acak kedalam sebuah fungsi. Distribusi peluang dibagi menjadi 2 yaitu distribusi diskrit dan distribusi kontinu. Fungsi distribusi diskrit adalah distribusi peluang acak yang dapat menjalankan bilangan bulat, sedangkan fungsi disribusi kontinue adalah distribusi peluang acak yang mampu menjalankan bilangan real. Contoh fungsi distribusi diskrit adalah distribusi binomial dan untuk ditibusi kontinu contohnya adalah distribusi normal. Fungsi distribusi diskret menggunakan perhitungan yang komplit meliputi konsep faktorial, exponen , kombinasi, permutasi, dan juga desimal. Opersi-operasi perhitungan tersebut membutuhkan waktu yang banyak dan tingkat ketelitian tinggi. Selain dilakukan secara manual, perhitungan Fungsi distribusi diskrit dapat di lakukan menggunakan software MS. Excel. Menggunakan MS.Excel tentu lebih praktis dibandingkan jika menghitung secara manual. Namun, ada beberapa hal yang harus diperhatikan dalam mengoperasikan MS Excel diantaranya pengguanaan tanda baca, urutan elemen dan sebagainya. Pada laporan praktikum ke 4 ini di tuliskan bagaimana hubungan antara distribusi diskrit dengan pengoperasiannya menggunakan MS Excel. Dalam mencari peluang berdistribusi diskrit, banyak formula yang dapat digunakan. Penggunaan Formula-nya akan dijelaskan dalam laporan ini.

Eventology of multivariate statistics
Eventology and mathematical eventology
Philosophical eventology and philosophy of probability
Practical eventology
Eventology of safety
Eventological economics and psychology
Mathematics in the humanities, socio-economic and natural sciences
Financial and actuarial mathematics
Multivariate statistical analysis
Multivariate complex analysis
Decision-making under risk and uncertainty
Risk measurement and risk models
Theory of fuzzy events and generalized theory of uncertainty
System analysis and events management

EEC'2016 ~ workshop on axiomatizing experience and chance, and Hilbert's sixth problem

With topics from quantum physics, probability and believability to economics, sociology, and psychology, the workshop will be intended for an interdisciplinary discussion on mathematical theories of experience and chance. Topics of discussion include the results, thoughts, and ideas on the axiomatization of the eventological theory of experience and chance in the framework of the decision of Hilbert sixth problem.
Eventology of experience and chance
Believability theory and statistics of experience
Probability theory and statistics of chance
Axiomatizing experience and chance

The book about the eventology ~ a science about events and its applications to problems of management of sets of events ~ a new direction in philosophy, mathematics and event management (see English version of the book: https://www.academia.edu/38605343/).
This book outlines ideas that inspired me while I was trying to build a new event science, eventology. Not each of these ideas was developed, but all of them inspired me in the construction of eventology, and then in the discovery of a revolutionary theory of experience and chance, co~eventum mechanics. (see, for example, https://www.academia.edu/34357251,
https://www.academia.edu/34373279,
https://www.academia.edu/34390291,
https://www.academia.edu/38368776,
https://www.academia.edu/38154263 and so on).

The logic of uncertainty is not the logic of experience and as well as it is not the logic of chance. It is the logic of experience and chance. Experience and chance are two inseparable poles. These are two dual reflections of one essence, which is called co∼event. The theory of experience and chance is the theory of co∼events. To study the co∼events, it is not enough to study the experience and to study the chance. For this, it is necessary to study the experience and chance as a single entire, a co∼event. In other words, it is necessary to study their interaction within a co∼event. The new co∼event axiomatics and the theory of co∼events following from it were created precisely for these purposes. In this work, I am going to demonstrate the effectiveness of the new theory of co∼events in a studying the logic of uncertainty. I will do this by the example of a co∼event splitting of the logic of the Bayesian scheme, which has a long history of fierce debates between Bayesionists and frequentists. I hope the logic of the theory of experience and chance will make its modest contribution to the application of these old dual debaters.

Keywords: Eventology, event, probability, probability theory, Kolmogorov’s axiomatics, experience, chance, cause, consequence, co∼event, set of co∼events, bra-event, set of bra-events, ket-event, set of ket-events, believability, certainty, believability theory, certainty theory, theory of co∼events, theory of experience and chance, co∼event dualism, co∼event axiomatics, logic of uncertainty, logic of experience and chance, logic of cause and consequence, logic of the past and the future, Bayesian scheme.

For certain algorithms such as sorting and searching, the parameters of the input probability distribution, in addition to the size of the input, have been found to influence the complexity of the underlying algorithm. The present paper makes a statistical comparative study on parameterized complexity between linear and binary search algorithms for binomial inputs.

Eventology of multivariate statistics
Eventology and mathematical eventology
Philosophical eventology and philosophy of probability
Practical eventology
Eventology of safety
Eventological economics and psychology
Mathematics in the humanities, socio-economic and natural sciences
Financial and actuarial mathematics
Multivariate statistical analysis
Multivariate complex analysis
Decision-making under risk and uncertainty
Risk measurement and risk models
Theory of fuzzy events and generalized theory of uncertainty
System analysis and events management

EEC'2016 ~ workshop on axiomatizing experience and chance, and Hilbert's sixth problem

With topics from quantum physics, probability and believability to economics, sociology and psychology, the workshop will be intended for an interdisciplinary discussion on mathematical theories of experience and chance. Topics of discussion include the results, thoughts and ideas on axiomatization of the eventological theory of experience and chance in the framework of the decision of Hilbert sixth problem.
Eventology of experience and chance
Beleivability theory and statistics of experience
Probability theory and statistics of chance
Axiomatizing experience and chance

The logic of uncertainty is not the logic of experience and as well as it is not the logic of chance. It is the logic of experience and chance. Experience and chance are two inseparable poles. These are two dual reflections of one essence, which is called co∼event. The theory of experience and chance is the theory of co∼events. To study the co∼events, it is not enough to study the experience and to study the chance. For this, it is necessary to study the experience and chance as a single entire, a co∼event. In other words, it is necessary to study their interaction within a co∼event. The new co∼event axiomatics and the theory of co∼events following from it were created precisely for these purposes. In this work, I am going to demonstrate the effectiveness of the new theory of co∼events in a studying the logic of uncertainty. I will do this by the example of a co∼event splitting of the logic of the Bayesian scheme, which has a long history of fierce debates between Bayesionists and frequentists. I hope the logic of the theory of experience and chance will make its modest contribution to the application of these old dual debaters.

Keywords: Eventology, event, probability, probability theory, Kolmogorov’s axiomatics, experience, chance, cause, consequence, co∼event, set of co∼events, bra-event, set of bra-events, ket-event, set of ket-events, believability, certainty, believability theory, certainty theory, theory of co∼events, theory of experience and chance, co∼event dualism, co∼event axiomatics, logic of uncertainty, logic of experience and chance, logic of cause and consequence, logic of the past and the future, Bayesian scheme.

Este artículo propone un nuevo modelo aplicativo a la ciencia básica, principalmente a la teoría de conjuntos donde los constantes cambios que actualmente existe en nuestra sociedad, nos conlleva a realizar modificaciones en nuestra base académica buscando de esta manera proveer de mayores y mejores aplicaciones del conocimiento adquirido en las diferentes áreas del saber. En este caso buscamos incansablemente la mayor cantidad de posibles soluciones que se puedan dar en las diferentes áreas de la ciencia básica donde la matemática resulta ser la principal herramienta de trabajo para los diferentes profesionales que buscan resolver día a día la problemática que se les llega a presentar en el desenvolvimiento de sus funciones.

La teoría de conjuntos no solo fue en sí misma un acontecimiento muy destacado en el avance de la matemática, también estableció el método axiomático de la ciencia elemental siendo así la manera más clara y precisa en la que se pueda dar una representación del conocimiento teórico, desde un punto de vista particular sabemos que todo conocimiento impartido en forma teórica y aplicada debe estar al alcance de todos los niveles educativos para una mayor aceptabilidad de sus fundamentos, es por ello que presento una nueva propuesta denominada como la intersección de conjuntos al infinito mediante el acoplamiento de graficas de discos con el ánimo de proveer una posible solución al planteamiento de problemas concretos y ampliados.

Con la ayuda de los elementos clásicos de la teoría de conjuntos vamos a resolver intersecciones de conjuntos superiores a 3 para lo cual vamos a utilizar el cuadro de doble entrada, el análisis combinatorio y gráficas de disco, que nos permitirá demostrar en óptimas condiciones la posibilidad de realizar intersecciones de 2, 3, 4, 5,…, “n” conjuntos, más aun nos dará la posibilidad de realizar aplicaciones prácticas en áreas afines que la vayan a requerir.

Analíticamente se demuestra que los elementos utilizados en la investigación expresa claramente su utilidad en la intersección de conjuntos al infinito, donde el cuadro de doble entrada nos ayudará a comprobar todas las posibles intersecciones existentes, siendo consecutivo para obtener intersecciones superiores a 3 conjuntos siendo además elemento de trabajo para comprobar las diferentes intersecciones que debe existir en los discos acoplados, el análisis combinatorio nos ayudará a saber exactamente cuántas intersecciones debe existir según la cantidad de conjuntos que vamos a trabajar según el criterio a desarrollar y finalmente son los discos acoplados que dan el soporte didáctico de toda la información recopilada en la intersección de conjuntos al infinito, se puede decir que al realizar las diferentes intersecciones se va a obtener discos por niveles siendo esta a su vez un nuevo modelo aplicativo ajustándose a las necesidades del usuario.

Todo el análisis desarrollado ha requerido de una minuciosa observación en cuanto a su estructura, que busca ser lo más sencillo posible para una fácil accesibilidad del conocimiento que se esté adquiriendo, cuando estamos hablando de cuadros de doble entrada se podrá observar que es instrumental al igual que el análisis combinatorio, permitiéndonos obtener cada intersección que debe existir por nivel y la cantidad de conjuntos trabajados respectivamente, finalmente esta nueva propuesta busca ser un soporte más del conocimiento general de la matemática básica, cumpliendo con el objetivo de dar respuestas a problemas ampliados en la teoría de conjuntos.

The purpose of this paper is to introduce and investigate new unification of unified family of Apostol-type polynomials and numbers based on results given in [1] [2]. Also, we derive some properties for these polynomials and obtain some relationships between the Jacobi polynomials, Laguerre polynomials, Hermite polynomials, Stirling numbers and some other types of generalized polynomials.

The author has developed a general theory for measuring the statistical interconnections between events, called ``mathematical eventology'' \cite[2007]{Vorobyev2007}, \cite[2011]{Vorobyev2011em}. The paper represents fresh ``eventological'' ideas that radically rethinking the basis of previous approaches to measuring statistical interconnections of events beyond the traditional measurements in the probability theory and mathematical statistics, adding to the already well-known  methods of measuring association, or statistical dependency, another new group of methods of measuring agreement of a set of events.
It shows by useful ``total'' quantities (called agreement, association, covariance, set-distance, set-nearness, and set-closeness of a set of events) formulas in this context can be simplified and elegantly summarized for many applications, including for the correct generalization of the traditional indicators of statistical interconnections of events as Pearson correlation, Yule association, as well as a popular index of agreement --- Cohen kappa \cite[1960]{Cohen1960}.
Proposed new methods demonstrate the growing potential of ``eventological'' technologies in the study of structures of interconnections (agreement, association, or statistical dependency) of events, which applies to the measurement in the nominal (categorical), and the ordered and numeric (difference, ratio) scales.
The development and use in a variety of applications of the new concept of the set-distance for a set of events must have necessarily come into contact with the area of modern multivariate statistics that deals with measuring the commonality (agreement) of experts in estimates within categorical (nominal) scales. The results of this dramatic contact are set forth below. As the ``whipping boy'', the most popular group of such statistics is chosen: the kappa-indicators of the commonality (agreement) of experts in estimates within categorical scales.

A fixed company of players observes a person selected from a fixed queue. After each observation, players are asked to bet the dollar secret from others, either on the fact that person is bald, or on what is not. A denite formula of the gain is suggested, such that every time after bets the gain of each player from a given company are completely determined by this formula. However, before bets player's gain is an uncertain value. Is it possible for a given company of players and a given queue of people before bets to build a correct mathematical model of uncertaint gain of each player within the framework of Kolmogorov's probability theory? If not, what else do you need to add to the foundations of probability theory so that before bets to be able to use this model for decision making? The paper answers these questions within the framework of the new theory of experience and of chance (the certainty theory) [1] that consists of two dual halves: the believability theory and the probability theory, and that is intended for the mathematical description of experienced-random experiments, the uncertainty in outcomes of which is generated by the observer. A xed company of players observes a person selected from a xed queue. After each observation, players are asked to bet the dollar secret from others, either on the fact that the person is bald, or on what is not. If, as a result of the bets made, the player's choice is in the minority, the player loses the bet. If, on the other hand, in the majority, the player is returned his bet plus an equal share of the loser players' bets. With a draw, the equality of the number of betting on the bald and the number of betting on the not bald, all players remain at their own: they are returned their bets 1. Every time after the bets, the gain of each player from this company is fully certain. But before the bets, the gain is an uncertain value. Is it possible for a given company of players and a given queue of persons before bets to build a correct mathematical model of uncertain gain of each player within the framework of Kolmogorov's probability theory?
c ⃝ 2016 O.Yu.Vorobyev Oleg Vorobyev (ed.), Proc. XV FAMEMS'2016, Krasnoyarsk: SFU

The logic of uncertainty is not the logic of experience and as well as it is not the logic of chance. It is the logic of experience and chance. Experience and chance are two inseparable poles. These are two dual reections of one essence, which is called co∼event. The theory of experience and chance is the theory of co∼events. To study the co∼events, it is not enough to study the experience and to study the chance. For this, it is necessary to study the experience and chance as a single entire, a co∼event. In other words, it is necessary to study their interaction within a co∼event. The new co∼event axiomatics and the theory of co∼events following from it were created precisely for these purposes. In this work, I am going to demonstrate the effectiveness of the new theory of co∼events in a studying the logic of uncertainty. I will do this by the example of a co∼event splitting of the logic of the Bayesian scheme, which has a long history of erce debates between Bayesionists and frequentists. I hope the logic of the theory of experience and chance will make its modest contribution to the application of these old dual debaters., theory of experience and chance, co∼event dualism, co∼event axiomatics, logic of uncertainty, logic of experience and chance, logic of cause and consequence, logic of the past and the future, Bayesian scheme.

This is a further formation of the theory of experience and chance. My previous works are devoted to the elements of this new theory, which can also be called the theory of co∼events, or the certainty theory. The theory is a synthesis of two dual theories: Kolmogorov's theory of probability and the believability theory based on the axiomatics, which is a non-trivial extension of Kolmogorov's axiomatics. In my paper " Theory of dual co∼event-based means " (2016), the concepts of two dual co∼event-based means of one co∼event are introduced (the bra-mean and the ket-mean). Here I first suggest the definition of the unifying mean of one co∼event (the bra-ket-mean), and then I enter the general definition of the bra-ket-mean for a finite set of co∼events. The mean co∼event-based characteristics of co∼events play the same role in the theory of experience and chance as the mean numerical characteristics of random variables — in the probability theory, or the mean characteristics of events — in the eventological theory.

In my recent works, I propose a new theory of experience and chance, or the theory of certainties, which is a synthesis of two dual theories: Kolmogorov's theory of probabilities and a new theory of believabilities. The theory of experience and chance is based on axiomatics, adding to the Kolmogorov axiomatics of probability theory an axiom of co~event, a new concept defined as a pair of dual objects, one of which serves as a Kolmogorov event in probability space and the other - as an event in the dualistic believability space. One of my papers gives a theory of dual co~event-based means and examples of its use for processing
statistics of ``approving voting'' in elections and in forestry. In this paper, I will go into more detail in examples of processing the statistics
of ``approving voting'' in forestry, where a group of forestry experts is actually engaged in ``approving voting'' when they label the trees of the
forest, labelling those that according to their opinions must be felling. My goal is to show in these examples that the theory of experience and
chance is an effective tool for constructing co~evnts models of experts' behavior in different fields.

The purpose of this paper is to introduce and investigate new unification of unified family of Apostol-type polynomials and numbers based on results given in [1] [2]. Also, we derive some properties for these polynomials and obtain some relationships between the Jacobi polynomials, Laguerre polynomials, Hermite polynomials, Stirling numbers and some other types of genera-lized polynomials.

The main theorem of the games theory of random coalitions is reformulated in the random set language which generalizes the classical maximin theorem but unlike it defines a coalition imputation also.

The theorem about maximin random coalitions has been introduced as a random set form of classical maximin theorem. This interpretation of the maximin theorem indicate the characteristic function of the game and its close connection with optimal random coalitions. So we can write the apparent natural formula of coalition imputation generalizing the strained formulas of imputation have been in the game theory till now. Those formulas of imputation we call the strained formulas because it is unknown from where the characteristic function of the game appears and because it is necessary to make additional suppositions about a type of distributions of random coalitions. The reformulated maximin theorem has both as its corollaries. The main outputs are two results of the games theory were united and the type of characteristic function of game defined by the game matrix was discovered.

Imagine that you are conducting an experiment, as a result of which the event occurs or does not occur. You must determine whether this event occurred or not. Your experience tells you that many times in the past you, being exactly in the same circumstances, only in half of the cases came to the conclusion this event happened. In this situation I would say your believability of this event happened is equaled to the half. But you can't answer in this way. You are required to answer only either ``Yes'' or ``No''. What would you do in this situation? You must not think such situations are big rare. Refresh your memory what are you doing every day under uncertainty? Every day you are doing only it and no more. The question arises how you do this action?
If I say you ``toss a coin in your mind, or in your thoughts'' I'll do the mistake. The point is every one of us has an ability to make the believabilistic choice. You yourself, or what is the same, your experience is such ``coin'' that, while you aren't questioned, it rotates all the time in ``free flight''. And only when you answer the question the ``coin'' falls on one of the sides: ``Yes'' or ``No'' with the believability that your experience tells you.
Therefore, to denote this action, I will choose the new expression ``tossing an experienced coin'' in an experienced experiment that becomes dual to the usual expression ``tossing a coin'' in a random experiment.

Very briefly on the new axiomatics of co∼events, it is intended for the rigorous mathematical description of an entanglement between observer and observation. Here we try to consider the new axiomatics in the normal human non-mathematical language, although it uses a new but quite natural terminology: ``a co∼event occurs (as an entanglement of observer and observation), a ket-event happens (as chance of observation), a bra-event is experienced (as an experience of observer) ''. And finally, it seems to me that the new theory in a not very rigorous mathematical language is most conveniently set out in the form of questions and answers that my students, graduate students, and colleagues often asked me during numerous internal and external discussions.

Any set of events has the unique own ordered set of half-rare events (o.s.h-r.e.) that characterizes the set of events in the following sense: its event probability distribution (e.p.d.) is connected to the e.p.d. of the o.s.h-r.e. by mutually inverse renumbering and event permutations.

In this paper, we introduce a new multivariate discrete distribution which called multivariate unification of generalized Apostol type distribution (UGAT). Several prop- erties are studied as, moments, probability generating function and other properties. Also, reliability study of distribution are introduced. Maximum likelihood method is used to estimate parameters and numerical method is used to obtain (MLEs).

In this paper, we introduce a new multivariate discrete distribution which called multivariate unification of generalied Apostol type distribution (UGAT). Several properties are studied as, moments, probability generating function and other properties. Also, reliability study of distribution are introduced. Maximum likelihood method is used to estimate parameters and numerical method is used to obtain (MLEs).