Sharif Digital Repository

Rapid growth in science and technology has created vast diversity in products, services and needs of people and groups. As the Internet and web technologies have progressed, a web-based solution to explore products of either a specific domain or multiple domains is mandatory. One of the main challenges in these systems is learning users’ preferences to recommend items possibly interesting to the user. Social network of the users is one of the sources that can inject additional information about them which can be exploited to improve accuracy of the system. Depending on the method, the performance, accuracy and personalization may differ .One of the most popular methods to extract this... 

Knot theory is the study of ambient isotopy classes of compact 1–manifolds in a 3-manifold. In classical knot theory this 3-manifold is R3 or S3. This field has experienced a great transformative advances in recent years because of its strong connections with and a number of other mathematical disciplines including topology of 3-manifolds and 4-manifolds, gauge theory, representation theory, categorification, morse theory, symplectic geometry and the theory of pseudo-holomorphic curves. In this thesis we start with classical knot theory, introducing some of its (classical) invariants like unknotting number, Seifert genus and slice genus of a knot, knot group and finally Alexander Polynomial... 

Experiments conducted in recent two decades indicated critical behavior in neural activity at different scales. Theoretically occurrences of these critical and power-law behavior can significantly facilitate brain activities correspondent to computation and memory tasks, but attaining the critical point essentially demands externally fine-tuning which has not been established yet. This fine-tuning often lies with placing system at transition point. Recent studies of group showed that a transition from synchronous to asynchronous phase could be achievable by a change in external parameters. At the very transition point, neuronal avalanches statistically demonstrate a power-law behavior which... 

We study the Abelian Sandpile Model and its relation with surface growth. ese two models are related through their field theories and equations of motion. It has been shown that the different features of different sandpile models can be expressed in terms of the noise term in the surface growth equation. A mapping between the simplest sandpile model, the BTW model, and a surface growth has already been introduced. is surface growth has not been studied in details so far. In this thesis we study different features of this surface growth corresponding to the BTW model, continuous sandpile model and also massive abelian sandpile model. We also consider different boundary conditions  

The four-page article by Bak, Tang and Wiesenfeld in 1987 was a beginning to a new wave of physicists’ efforts to explain and describe the concept of complexity; a not-so-well-defined concept that resists against the reductionist tools and methods of physics. The Self-organized Criticality theory presented in that article via a simple model, known as sandpile model, was first of all an effort to explain the numerous occurrence of power law distribution in nature. SOC was introduced to tell us why so many natural phenomena like Earthquakes, landslides, forest fires, extinction and other seemingly non-related catastrophic events, more or less obey the scale-less power law distribution; A... 

In this research, we want to address the question of universality classes in BTW and Manna sandpile models. So far, number of works has been devoted to this issue but the the answer remained unsolved. We will try another approach to study this question by perturbing the original models. To this end, we introduce three models that have evolution rules between BTW model and Manna model. By simulating this models, we observe that in the presence of perturbation, the probability dis- tribution has two regimes of behaviour which are separated by a new characteristic scale. The regime of small avalanches is described by the exponent of BTW model and the regime of large avalanches by the exponent... 

Edwards-Wilkinson’s equation can be achieved from a Hamiltonian. When we have the Hamiltonian for the system, there are common approaches that makes it out of critical. In other words,the ”mass” should be added to the system. In this study we have tried to simulate and solve analytically these models that are involved mass term. We try to onstruct these mass terms in a way that have a minimum impact on the system and we study the quantities that characterize the out of critical behaviors  

Percolation is a phenomenon that can be found in many physical problems. Additionally, as a statistical model, it has a very rich physics, since many fundamental concepts in the context of critical phenomena and complex systems-such as phase transition, scaling laws etc can be found in the model. Percolation phenomenon can be defined on different lattices. In this thesis we study percolation on small-world networks. In small-world networks, in addition to local bonds that connects the neighbouring sites, there exist some long-ranged bonds that connect cites far from each other. Social networks, some networks of internet or the gene networks are examples of such networks. Therefore, to study... 

A complte set of characteristic parameters of the sandpile models is still unknown. We have studied the existence of ”weak chaos” critical exponent in different sandpile models and we have shown that it is a characteristic exponent of deterministic models. We have shown that BTW and Zhang models do not belong to the same universality class (contrary to Zhang’s previous conjecture and contrary to Ben-Hur & Biham’s results.) Also we have shown that directed models, specificly Ramaswamy-Dhar’s directed model form a different universality class. ”Weak chaos” exponent in also studied in massive models and we have shown that by increase of dissipation, the exponent decreases rapidly to an... 

Self-organized critical phenomena are interesting phenomena which are ubiquitous in nature. Examples include mountain ranges , coastlines and also activities in the hu-man's brain. In these processes, without fine-tuning of any external parameter such as the temperature, the system exhibits critical behavior. In other words, the dynamics of the system, drives it towards an state in which long range correlations in space and scaling behaviors can be seen.The first successful model which could characterize such systems was BTW model, introduced by Bak , Tang and Wiesenfeld in 1987. This model, later named Abelian sandpile model, was very simple and because of this simplicity, a large amount of... 

Since the concept of Self-Organized Criticality was introduced in terms of BTW Sandpiles model, its major features have been known as broad power law distributions without any tuning parameters. In some selforganized critical systems like brain and neural networks, some evidences and experiments show a periodic or non-power law distribution of avalanches in addition to the power-law distributions of avalanches. In this thesis we try to observe the same phenomenon in the well-known SOC models, namely the BTW and Manna sandpile models. We have considered small lattice sizes with periodic boundary conditions and a small amount of dissipation. Within such conditions we observe a periodic-like... 

Power law behavior of earthquakes has been a matter of interest for many scientists. One on these power laws known as Gutenberg-Richter law describes the magnitude distribution of earthquakes. The Burridge-Knopoff model of faults, produces the same power law distribution of events as the Gutenberg-Richter law for earthquakes. Olami, Feder and Christensen in 1992, introduced a 2-D, continues sand pile model Known as OFC that displays self-organized-criticality. They claimed that this model is equivalent to Burridge-Knopoff model. It means that criticality is the origin of power law behavior of the Burridge-Knopoff model. Nevertheless, there are some evidence against criticality in the... 

Surface growth have been one of the most interesting topics of research in non-equilibrium Statistical physics, due to their relevance in studying industrial growth processes. Many models such as Edwards-Wilkinson and KPZ have been proposed to study these systems where by incorporating renormalization group, numerical integration and computer simulations we can derive their critical exponents. In general, a thermal noise is implemented in these models, however, other types can be used as well. In particular for the case of Edwards-Wilkinson, it has been shown that a multiplicative noise changes the universality class of the model. In this thesis we want to investigate the effects of... 

The brain, as a complex system with various components working in concert, plays a fundamental role in many cognitive processes and human perception of the surrounding environment. Perception, in many cases, can differ from reality due to evolutionary processes, natural selection, or even flaws in any of the parameters within this complex intelligent system. One of the observed phenomena in perception is the heightened visibility of edges, often depicted with Mach bands. To describe this phenomenon, primarily, rate models are used, especially in regions where a linear approximation is suitable for neuronal responses. However, the response of neuronal populations in a range of external... 

The Manna sandpile model is a significant and widely-used model in the study of self-organized criticality. Various avalanche-related quantities, such as area, size, duration, and others, exhibit power-law distributions with finite-size effects. It has been demonstrated that this model exhibits simple monofractal behavior on both regular lattices and random networks, and the finite-size scaling (FSS) hypothesis holds for the avalanche distribution functions. On the other hand, it has been observed that a wide range of natural and human-made networks are small-world networks. Consequently, studying the Manna sandpile model on such networks and understanding its features and behavior can... 

Percolation is a fundamental topic in statistical physics with applications in modeling natural phenomena such as the spread of diseases and forest fires. It describes a geometric phase transition in which a system evolves from small-scale connectivity to extensive connectedness. At the percolation threshold, the system undergoes a continuous phase transition, exhibiting critical behavior. A recent study introduced a novel method for determining the percolation threshold by measuring the maximum change in the size of the largest cluster during the gradual increase of a control parameter. This maximum change, known as the ”gap,” signals the occurrence of the phase transition. The study also...