Sharif Digital Repository

The Onsager's conjecture is concerned with the dichotomy between rigidity and flexibility of weak solutions of incompressible Euler equations. Lars Onsager conjectured that weak solutions of Euler equations that are not smooth enough could be dissipative, even without the help of viscosity. On the other hand, it is well known that $C^1$ solutions conserve energy. Onsager conjectured that C^(1/3) regularity marks the threshold for this dichotomy. In other words, Hölder continuous solutions with Hölder exponent greater than 1/3 conserve the energy, while for every Hölder exponent less than 1/3, there are dissipative Hölder continuous solutions. The threshold 1/3 is intimately tied with... 

This thesis is devoted to the study of suitable functional frameworks for proving propagation of chaos and mean-field limit of nonlinear Vlasov-type equations for indistinguishable particles. The Kac model is a classic example and the motivation of this functional structure. On the one hand, Kac considered a spatially homogeneous gas, and on the other hand, he reduced the problem to one-dimensional collisions and dropped the conservation of momentum assumption. Therefore, he obtained a simplified version of the Boltzmann equation, which is named after him, the Kac equation. The main results which we study are quantitative estimates
on the decay of fluctuations around the deterministic... 

In 1952, Choquet-Bruhat proved the well-posedness of the Cauchy problem for Einstein’s equation, and demonstrated that a given initial data on a Cauchy hypersurface in spacetime propagates forward in time as a solution to the wave equation. It took nearly half a century for mathematicians to build the necessary tools and techniques for proving the stability of the most basic solution of Einstein’s vacuum equations (EVE), namely Minkowski spacetime. In 1993, Christodoulou and Klainerman showed the global nonlinear stability of this solution using the concept of a double null gauge. The Schwarzschild solution, introduced by Karl Schwarzschild in 1916 for spherically symmetric spacetimes as an... 

This research introduces temperature measurement by a non-intrusive pyrometric method of the ratio of two colors of soot. For pyrometry, the spectral sensitivity of the Nikon D7100 camera was obtained by developing a neural network based on the results of photographing colored papers and their spectrum, and this camera was used for temperature measurement. Based on the dependence of soot radiation on temperature and the spectral sensitivity of the camera, a calibration chart has been extracted on the ratio of the green channel to the red channel of the camera with the soot temperature, which is used to produce the temperature contour using the flame image. The temperature contour of... 

This research focuses on physics-informed neural networks, which are trained to solve supervised machine learning tasks while adhering to physical laws described by general nonlinear partial differential equations. Previous studies utilized Gaussian process regression to develop functional representations designed for a given linear operator. However, despite the flexibility of Gaussian processes, solving nonlinear problems presents two major limitations: first, authors had to linearize each nonlinear term over time, and second, the Bayesian nature of Gaussian process regression requires specific assumptions that may limit the model’s representational capacity. For these reasons, data-driven... 

Many statistical systems such as earthquakes, road trafcs, forest fres, neurocortical avalanches etc. exhibit self-organized criticality (SOC). In such systems without tuning extrenal parameters, the system arrives at criticality. During recent decades, a number of models are introduced which show the same charactristics. These models have made a platform to investigate the physics of self-organized criticality. Among them, sandpile models are the best known models. They exhibit critical behaviour such as scaling laws. Also in some of them conformal invariance is checked nummerically.Most of sandpile models deal with slope parameters, that is, the main dynamical parameters are the local... 

The aim of these notes is to describe an exciting chapter in the recent development of quantum cohomology. Guided by ideas from physics, a remarkable structure on the solutions of certain rational enumerative geometry problems has been found: the solutions are coefficients in the multiplication table of a quantum cohomology ring. Associativity of the ring yields non-trivial relations among the enumerative solutions. In many cases, these relations suffice to solve the enumerative problem. For example, let $N_d$ be the number of degree $d$, rational plane curves passing through $3d$ − $1$ general points in $\mathbb{P}^2$. Since there is a unique line passing through 2 points, $N_1 = 1$. The... 

Centrifuge is a method in which great gravitational-like forces are used. Gravitational forces which cause molecules to move and are relevant to molecular weight; can cause isotope separation. The separation of light isotope from heavy one is done with a centrifugal field which produces a pressure gradient for the gas mixture. Pressure gradient is different for each of the isotopes due to the dependence of pressure gradient to the mass of the materials. But the separation in this way is very limited. The capacity of separation can be brought up by producing an additional axial flow in the rotor. This rotational flow can be produced with internal or external incitement. Rotational flow with... 

Petroleum based products are the major sources of energy for industry and daily life, With the development of social economy, the demands of the oil have increased dramatically. Petroleum hydrocarbons can cause environmental pollution during various stages of production, transportation, refining and use. Releases of petroleum products(e.g., gasoline, diesel, fuel oil) from aboveground and underground storage tanks or transport pipelines are the major causes of the environment pollution.The major constituents of petroleum hydrocarbons are biodegradable. Many soil microorganisms transform oil hydrocarbons into nontoxic compounds or mineralize them to inorganic compounds. In situ... 

The main goal of this thesis is to study the Teichmüller space and its geometric properties. The Teichmüller space of a surface can be studied from complex geometric or hyperbolic geometric points of view, and in this thesis we refer to both of these viewpoints where needed. Another goal of this thesis is to review a method of compactification of the Teichmüller space. Therefore, we introduce the compactification method proposed by Thurston. To this end, we first briefly review hyperbolic geometry. Then we extend some tools, and use some techniques in hyperbolic geometry, to describe Thurston’s construction and pave the way for it. In the second phase of this study, we introduce... 

In this dissertation, we will study partially hyperbolic diffeomorphisms. First, we are going to introduce partially hyperbolic diffeomorphisms and construct some examples. One of the important questions in the study of partially hyperbolic diffeomorphisms is their classification problem, which provides a deeper understanding of manifolds and also of partially hyperbolic diffeomorphisms themselves. We will go through this problem by examining Hammerlindl, Potrie’s [1]’s work. They have proved that a partially hyperbolic diffeomorphism on a 3-manifold with a virtually solvable fundamental group, that has no periodic torus tangent to contraction-center or expansion-center, is dynamically... 

In many uranium mines such as gachin-bandarabbas, use sulfuric acid for leaching operation on ore. This leach liquor, containing uranium, is extracted by organic solvent, such as D2EHPA (dapex process ).althogh this process have being used for many years, there is a vacancy for detailed thermodynamical studying on uranium extraction of uranium from leach liquor. To have a thermodynamical model, we need information on equilibrium concentration of all component that may have effect on uranium extraction in addition we need concentration of solvent , pH, phase ratio and activity coefficient.such a model must have the ability to predict the equilibrium concentration in alloperation condition.... 

In many uranium mines such as gachin-bandarabbas, sulfuric acid is used for leaching operation on ore after crushing operation. This leach liquor, containing uranium, is extracted by specific solvent, including extractant like TOA. However, this operation, named Amex, have been used for many years, there is a vacancy for a detailed thermodynamical studying on uranium extraction from leach liquor.To have a thermodynamical model, we need information about the equilibrium concentration of all the components that may have effect on uranium extraction and also concentration of solvent, pH, phase ratio and activity coefficient alteration. This model must have the ability to predict the equilibrium... 

Nowadays environmental issues are of great importance and one of the most paramount issues is the quality of water, because clean water is vital to our existence. Besides different methods for water purification, water quality testing is a useful and needful tool. Traditional methods for determination of water quality parameters are mainly expensive, time consuming and require highly trained staff. In this study, we have described a centrifugal microfluidic platform (Lab- on – a- CD) for simultaneous determination of different physical and chemical parameters including Nitrite, Free Chlorine and pH. All of the processes from sample injection to detection are integrated on this low-cost and... 

Nowadays, cancer, which has been mentioned as the disease of the century, is the second leading cause of death throughout the world, and its incidence is constantly increasing. Isolation of circulating tumor cells is one of the most critical steps in diagnosing and controlling cancer progression. Due to the rarity of cancer cells compared to other cells in the blood sample, the isolation process requires optimal and high-precision devices. With the advent of inertial microfluidics, the ability to control the particles movement, the processing of blood samples as quickly and accurately as possible, and the viability of cells with a high percentage, introduced microfluidic systems as a...