Fractional Brownian fields, duality, and martingales

In this paper the whole family of fractional Brownian motions is constructed as a single Gaussian field indexed by time and the Hurst index simultaneously. The field has a simple covariance structure and it is related to two generalizations of fractional Brownian motion known as multifractional Brownian motions. A mistake common to the existing literature regarding multifractional Brownian motions is pointed out and corrected. The Gaussian field, due to inherited ``duality'', reveals a new way of constructing martingales associated with the odd and even part of a fractional Brownian motion and therefore of the fractional Brownian motion. The existence of those martingales and their stochastic representations is the first step to the study of natural wavelet expansions associated to those processes in the spirit of our earlier work on a construction of natural wavelets associated to Gaussian-Markov processes.Comment: Published at http://dx.doi.org/10.1214/074921706000000770 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Concepts modelling using consistent fuzzy logic

Концепти су менталне репрезентације категорија. Теорија концепата треба да објасни како концепти функционишу као категорије и како се компонују у сложене концепте и мисли. Композиционалност објашњава креативност, како коначни ум може имати бесконачан когнитивни капацитет. Кључно питање у теорији је да ли је степен примене концепта бинаран или градиран. Теорија градације додатно објашњава репрезентативност инстанци категорија, али је теоретичари класичног погледа критикују да не може објаснити композиционалност те закључују да је погрешна. У дисертацији су помоћу конзистентне фази логике формализовани и објашњени концепти у случају градације. Прототип теорија концепата формализована је у стандардном и случају логичких интеракција између атрибута. Теорија егземплара формализована је на два начина. Теорија граница категорија формализована је у биполарном оквиру. Каузална теорија концепата формализована је у стандардном, случају каузалних повратних спрега између атрибута и/ или логичких интеракција између узрока. Математички је објашњена композиционалност Булових концепата у складу са прототип, егземплар и теоријом граница категорија, под претпоставком да концепти имају векторску природу. Одговорено је на критику прототип теорије и оповргнута су два аргумента којима се она поткрепљује, да закони искључења трећег и неконтрадикције нису загарантовани у прототип оквиру и да неодређено много Булових концепата нема прототип иако га све компоненте имају. Моделовани су конкретни Булови концепти за које теоретичари класичног погледа сматрају да не могу бити објашњени у прототип оквиру. У наставку дисертације генерализована су два експертска формализма за сложенe системе, Булове мреже и фази когнитивне мапе. Дескриптивна снага формализама драстично је увећана чиме они постају употребљиви у знатно ширем спектру ситуација...Concepts are mental representations of categories. Theory of concepts has to explain how concepts function as categories and how they compose into complex concepts and thoughts. The compositionality explains creativity – how a finite mind can have an infinite cognitive capacity. Fundamental question in theory of concepts is whether a degree of conceptual application is binary or graded. Theory of gradation additionally explains representativeness of category members, but it is criticized by classical theoreticians that it cannot explain compositionality, and thus, it is considered wrong. In this doctoral dissertation, concepts in the case of gradation are formalized and explained using consistent fuzzy logic. Prototype theory of concepts is formalized in standard case and in case of logical interactions between attributes. Exemplar theory of concepts is formalized in two ways. Boundary theory of concepts is formalized in bipolar framework. Causal theory of concepts is formalized in standard case, in case of causal feedback between attributes and/ or logical interactions between causes. The compositionality of Boolean concepts is mathematically explained according to the prototype, exemplar and boundary theory, under the assumption that concepts have vector nature. The critique of the prototype theory has been addressed, and two arguments that support the critique are refuted - that laws of excluded middle and noncontradiction are not secured in the prototype framework, and that there isn’t any prototype for indefinitely many Boolean concepts even though all components have one. Specific Boolean concepts, which are considered unexplainable in the prototype framework by classical theoreticians, are modelled. In the rest of the dissertation, two expert-based formalisms for complex systems, Boolean networks and fuzzy cognitive maps, are generalized. The descriptive power of two formalisms is drastically increased, which allows their usage in much broader spectrum of cases..

Pravna (ne)sigurnost kod ugovora o lizingu nepokretnosti u savremenom srpskom pravu

Predmet rada je istraživanje pravne nesigurnosti različitih oblika ugovora o lizingu nepokretnosti, koji se javljaju u savremenom srpskom pravu. Primenom pravnodogmatskog i uporednopravnog metoda, autori istražuju karakteristike domaćeg pravnog režima ugovora o lizingu nepokretnosti, sa posebnim naglaskom na pitanje mogućnosti primene pravila drugih, imenovanih ugovora na ugovore o lizingu nepokretnosti. U savremenom srpskom pravu uređen je samo oblik finansijskog lizinga, ali ne i drugi oblici lizinga nepokretnosti. Mogući problemi koji se javljaju zbog nedovoljne pravne regulative očigledni su i u odlukama sudova u sporu koji je proistekao iz jednog od neimenovanih oblika lizinga nepokretnosti u Srbiji. Analiza tih odluka ukazuje na pitanja koja su izvor pravne nesigurnosti ugovora o lizingu nepokretnosti, a koja se mogu prevazići odgovarajućim izmenama i dopunama zakonske regulative. Autori u radu daju predloge za regulisanje ugovora o lizingu nepokretnosti što je, kako zaključuju, najbolji način da se uočeni nedostaci prevaziđu

Formalization of Human Categorization Process Using Interpolative Boolean Algebra

Since the ancient times, it has been assumed that categorization has the basic form of classical sets. This implies that the categorization process rests on the Boolean laws. In the second half of the twentieth century, the classical theory has been challenged in cognitive science. According to the prototype theory, objects belong to categories with intensities, while humans categorize objects by comparing them to prototypes of relevant categories. Such categorization process is governed by the principles of perceived world structure and cognitive economy. Approaching the prototype theory by using truth-functional fuzzy logic has been harshly criticized due to not satisfying the complementation laws. In this paper, the prototype theory is approached by using structure-functional fuzzy logic, the interpolative Boolean algebra. The proposed formalism is within the Boolean frame. Categories are represented as fuzzy sets of objects, while comparisons between objects and prototypes are formalized by using Boolean consistent fuzzy relations. Such relations are directly constructed from a Boolean consistent fuzzy partial order relation, which is treated by Boolean implication. The introduced formalism secures the principles of categorization showing that Boolean laws are fundamental in the categorization process. For illustration purposes, the artificial cognitive system which mimics human categorization activity is proposed

Trend analysis of extreme temperature climatic indices in Belgrade (1961-2020)

Climate change are one of the most significant global challenges in the 21st century. Air temperature and precipitation as the most important climatic elements, are the most frequently analyzed parameters. Climate change is not only the cause of global warming, but also brings extreme low and high temperatures, heavy rainfall and other meteorological anomalies. One of the main signals indicating climate change is the change of extreme temperatures. In order to gain insight into the changes in the temperature regime in Belgrade, the number of characteristic days - frost days (FD), ice days (ID), summer days (SU), and the number of tropical nights (TR) over a period of 60 years - was analyzed. The extremes are examined, i.e. the change in the number of days when the daily minimum or maximum temperature is below zero (frost and ice days) and the change in the number of days when the maximum temperature exceeds 25℃ (summer days). Two non–parametric statistical tests were used to determine the trend and the trend magnitude of the selected indices. The Mann-Kendall test was used for trend analysis and the Pettit’s test was used to detect significant changes in the time series of climatological data. The study area is Belgrade, the capital of the Republic of Serbia. The data of the Hydrometeorological Institute of the Republic of Serbia (RHMZ) measured at the weather station Belgrade (Serbia) during two climatological periods 1961-1990 and 1991-2020 were used. The obtained results indicate that the number of frost and ice days decreases, while the number of summer and especially tropical nights increases. According to the applied Mann-Kendell test, statistically significant changes were found in all data series, while the Petitt's test indicated that sudden changes in parameter values began in the late 1980s

Trend analysis of extreme temperature climatic indices in Belgrade (1961-2020)

Climate change are one of the most significant global challenges in the 21st century. Air temperature and precipitation as the most important climatic elements, are the most frequently analyzed parameters. Climate change is not only the cause of global warming, but also brings extreme low and high temperatures, heavy rainfall and other meteorological anomalies. One of the main signals indicating climate change is the change of extreme temperatures. In order to gain insight into the changes in the temperature regime in Belgrade, the number of characteristic days - frost days (FD), ice days (ID), summer days (SU), and the number of tropical nights (TR) over a period of 60 years - was analyzed. The extremes are examined, i.e. the change in the number of days when the daily minimum or maximum temperature is below zero (frost and ice days) and the change in the number of days when the maximum temperature exceeds 25℃ (summer days). Two non–parametric statistical tests were used to determine the trend and the trend magnitude of the selected indices. The Mann-Kendall test was used for trend analysis and the Pettit’s test was used to detect significant changes in the time series of climatological data. The study area is Belgrade, the capital of the Republic of Serbia. The data of the Hydrometeorological Institute of the Republic of Serbia (RHMZ) measured at the weather station Belgrade (Serbia) during two climatological periods 1961-1990 and 1991-2020 were used. The obtained results indicate that the number of frost and ice days decreases, while the number of summer and especially tropical nights increases. According to the applied Mann-Kendell test, statistically significant changes were found in all data series, while the Petitt's test indicated that sudden changes in parameter values began in the late 1980s

Interpolative Boolean Networks

Boolean networks are used for modeling and analysis of complex systems of interacting entities. Classical Boolean networks are binary and they are relevant for modeling systems with complex switch-like causal interactions. More descriptive power can be provided by the introduction of gradation in this model. If this is accomplished by using conventional fuzzy logics, the generalized model cannot secure the Boolean frame. Consequently, the validity of the model’s dynamics is not secured. The aim of this paper is to present the Boolean consistent generalization of Boolean networks, interpolative Boolean networks. The generalization is based on interpolative Boolean algebra, the [0,1]-valued realization of Boolean algebra. The proposed model is adaptive with respect to the nature of input variables and it offers greater descriptive power as compared with traditional models. For illustrative purposes, IBN is compared to the models based on existing real-valued approaches. Due to the complexity of the most systems to be analyzed and the characteristics of interpolative Boolean algebra, the software support is developed to provide graphical and numerical tools for complex system modeling and analysis