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Michael Aizenman
Michael Aizenman (; born 28 August 1945) is an American-Israeli mathematician and a physicist at Princeton University, working in the fields of mathematical physics, statistical mechanics, functional analysis and probability theory. The highlights of his work include: the quantum triviality, triviality of a class of scalar (physics), scalar quantum field theory, quantum field theories in extra dimensions, more than three dimensions; a description of the phase transition in the Ising model in three and more dimensions; the sharpness of the phase transition in percolation theory; a method for the study of spectral and dynamical localization for random Schrödinger operators; and insights concerning Riemann surface, conformal invariance in two-dimensional percolation. Biography Aizenman is a Jewish American - Israeli who was born in Russia. He was an undergraduate at the Hebrew University of Jerusalem. He was awarded his PhD in 1975 at Yeshiva University (Belfer Graduate School of S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nizhny Tagil
Nizhny Tagil ( rus, Нижний Тагил, p=ˈnʲiʐnʲɪj tɐˈgʲil) is a classification of inhabited localities in Russia, city in Sverdlovsk Oblast, Russia, located east of the Boundaries between the continents#Asia and Europe, boundary between Asia and Europe. Demographics History The history of Nizhny Tagil dates back to the mid-16th century, when the Stroganovs received the right to possess land by the Kama (river), Kama and Chusovaya basins. In 1579 they founded the first settlement, the Utkin sloboda (settlement), sloboda, by the river Utka, the mouth of Chusoya. Fateyevo, the first Russian village in the Tagil region, was founded in 1665. In 1696, by the order of Tsar Peter the Great, the Vysokogorsky iron ore quarry was opened. Voevode Dmitry Protasyev was elected to search for iron and magnetic ores. The deposits were particularly rich, and included lodes of pure magnetic iron. The surrounding landscape provided everything needed for a successful and product ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, or Topological space#Definitions, topology) and the linear transformation, linear functions defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining, for example, continuous function, continuous or unitary operator, unitary operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
New York City
New York, often called New York City (NYC), is the most populous city in the United States, located at the southern tip of New York State on one of the world's largest natural harbors. The city comprises five boroughs, each coextensive with a respective county. The city is the geographical and demographic center of both the Northeast megalopolis and the New York metropolitan area, the largest metropolitan area in the United States by both population and urban area. New York is a global center of finance and commerce, culture, technology, entertainment and media, academics, and scientific output, the arts and fashion, and, as home to the headquarters of the United Nations, international diplomacy. With an estimated population in 2024 of 8,478,072 distributed over , the city is the most densely populated major city in the United States. New York City has more than double the population of Los Angeles, the nation's second-most populous city. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hebrew University Of Jerusalem
The Hebrew University of Jerusalem (HUJI; ) is an Israeli public university, public research university based in Jerusalem. Co-founded by Albert Einstein and Chaim Weizmann in July 1918, the public university officially opened on 1 April 1925. It is the second-oldest Israeli university, having been founded 30 years before the Israeli Declaration of Independence, establishment of the State of Israel but six years after the older Technion university. The HUJI has three campuses in Jerusalem: one in Rehovot, one in Rishon LeZion and one in Eilat. Until 2023, the world's largest library for Jewish studies—the National Library of Israel—was located on its Edmond Safra, Edmond J. Safra campus in the Givat Ram neighbourhood of Jerusalem. The university has five affiliated teaching hospitals (including the Hadassah Medical Center), seven faculties, more than 100 research centers, and 315 academic departments. , one-third of all the doctoral candidates in Israel were studying at the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jewish
Jews (, , ), or the Jewish people, are an ethnoreligious group and nation, originating from the Israelites of History of ancient Israel and Judah, ancient Israel and Judah. They also traditionally adhere to Judaism. Jewish ethnicity, religion, and community are highly interrelated, as Judaism is their ethnic religion, though it is not practiced by all ethnic Jews. Despite this, religious Jews regard Gerim, converts to Judaism as members of the Jewish nation, pursuant to the Conversion to Judaism, long-standing conversion process. The Israelites emerged from the pre-existing Canaanite peoples to establish Kingdom of Israel (Samaria), Israel and Kingdom of Judah, Judah in the Southern Levant during the Iron Age.John Day (Old Testament scholar), John Day (2005), ''In Search of Pre-Exilic Israel'', Bloomsbury Publishing, pp. 47.5 [48] 'In this sense, the emergence of ancient Israel is viewed not as the cause of the demise of Canaanite culture but as its upshot'. Originally, J ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together. Examples of Riemann surfaces include Graph of a function, graphs of Multivalued function, multivalued functions such as √''z'' or log(''z''), e.g. the subset of pairs with . Every Riemann surface is a Surface (topology), surface: a two-dimensional real manifold, but it contains more structure (specifically a Complex Manifold, complex structure). Conversely, a two-dimensional real manifold can be turned into a Riemann surface (usually in several inequivalent ways) if and only if it is orientable and Metrizabl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schrödinger Operator
In quantum mechanics, the Hamiltonian of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. Its spectrum, the system's ''energy spectrum'' or its set of ''energy eigenvalues'', is the set of possible outcomes obtainable from a measurement of the system's total energy. Due to its close relation to the energy spectrum and time-evolution of a system, it is of fundamental importance in most formulations of quantum theory. The Hamiltonian is named after William Rowan Hamilton, who developed a revolutionary reformulation of Newtonian mechanics, known as Hamiltonian mechanics, which was historically important to the development of quantum physics. Similar to vector notation, it is typically denoted by \hat, where the hat indicates that it is an operator. It can also be written as H or \check. Introduction The Hamiltonian of a system represents the total energy of the system; that is, the sum of the kinetic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Percolation Theory
In statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are added. This is a geometric type of phase transition, since at a critical fraction of addition the network of small, disconnected clusters merge into significantly larger Glossary of graph theory, connected, so-called spanning clusters. The applications of percolation theory to materials science and in many other disciplines are discussed here and in the articles Network theory and Percolation (cognitive psychology). Introduction A representative question (and the etymology, source of the name) is as follows. Assume that some liquid is poured on top of some porosity, porous material. Will the liquid be able to make its way from hole to hole and reach the bottom? This physical question is mathematical model, modelled mathematically as a Grid graph, three-dimensional network of graph (discrete mathematics), vertices, usually called "sites", in which the graph (dis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ising Model
The Ising model (or Lenz–Ising model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical models in physics, mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent Nuclear magnetic moment, magnetic dipole moments of atomic "spins" that can be in one of two states (+1 or −1). The spins are arranged in a Graph (abstract data type), graph, usually a lattice (group), lattice (where the local structure repeats periodically in all directions), allowing each spin to interact with its neighbors. Neighboring spins that agree have a lower energy than those that disagree; the system tends to the lowest energy but heat disturbs this tendency, thus creating the possibility of different structural phases.The two-dimensional square-lattice Ising model is one of the simplest statistical models to show a phase transition. Though it is a highly simplified model of a magnetic material, the Ising model can sti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phase Transition
In physics, chemistry, and other related fields like biology, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic State of matter, states of matter: solid, liquid, and gas, and in rare cases, plasma (physics), plasma. A phase of a thermodynamic system and the states of matter have uniform physical property, physical properties. During a phase transition of a given medium, certain properties of the medium change as a result of the change of external conditions, such as temperature or pressure. This can be a discontinuous change; for example, a liquid may become gas upon heating to its boiling point, resulting in an abrupt change in volume. The identification of the external conditions at which a transformation occurs defines the phase transition point. Types of phase transition States of matter Phase transitions commonly refer to when a substance tran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extra Dimensions
In physics, extra dimensions or extra-dimensional spaces are proposed as additional space or time dimensions beyond the (3 + 1) typical of observed spacetime — meaning 5-dimensional or higher. such as the first attempts based on the Kaluza–Klein theory. Among theories proposing extra dimensions are: * Large extra dimension, mostly motivated by the ADD model, by Nima Arkani-Hamed, Savas Dimopoulos, and Gia Dvali in 1998, in an attempt to solve the hierarchy problem. This theory requires that the fields of the Standard Model are confined to a four-dimensional membrane, while gravity propagates in several additional spatial dimensions that are large compared to the Planck scale.For a pedagogical introduction, see * Warped extra dimensions, such as those proposed by the Randall–Sundrum model (RS), based on warped geometry where the universe is a five-dimensional anti-de Sitter space and the elementary particles except for the graviton are localized on a (3&n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Field Theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles. The current standard model of particle physics is based on QFT. History Quantum field theory emerged from the work of generations of theoretical physicists spanning much of the 20th century. Its development began in the 1920s with the description of interactions between light and electrons, culminating in the first quantum field theory—quantum electrodynamics. A major theoretical obstacle soon followed with the appearance and persistence of various infinities in perturbative calculations, a problem only resolved in the 1950s with the invention of the renormalization procedure. A second major barrier came with QFT's apparent inabili ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
