MathematicsMathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them.
Mathematical proofA mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation".
Mathematical analysisAnalysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space).
Philosophy of mathematicsThe philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people's lives. The logical and structural nature of mathematics makes this branch of philosophy broad and unique. The philosophy of mathematics has two major themes: mathematical realism and mathematical anti-realism. The origin of mathematics is of arguments and disagreements.
Mathematical modelA mathematical model is an abstract description of a concrete system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in applied mathematics and in the natural sciences (such as physics, biology, earth science, chemistry) and engineering disciplines (such as computer science, electrical engineering), as well as in non-physical systems such as the social sciences (such as economics, psychology, sociology, political science).
Mathematical physicsMathematical physics refers to the development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". An alternative definition would also include those mathematics that are inspired by physics (also known as physical mathematics).
Mathematics educationIn contemporary education, mathematics education—known in Europe as the didactics or pedagogy of mathematics—is the practice of teaching, learning, and carrying out scholarly research into the transfer of mathematical knowledge. Although research into mathematics education is primarily concerned with the tools, methods, and approaches that facilitate practice or the study of practice, it also covers an extensive field of study encompassing a variety of different concepts, theories and methods.
United StatesThe United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country primarily located in North America and consisting of 50 states, a federal district, five major unincorporated territories, nine Minor Outlying Islands, and 326 Indian reservations. It is the world's third-largest country by both land and total area. It shares land borders with Canada to its north and with Mexico to its south and has maritime borders with the Bahamas, Cuba, Russia, and other nations.
Los AngelesLos Angeles (USlɔːs_ˈændʒələs ; Los Ángeles los ˈaŋxeles, The Angels), often referred to by its initials L.A., and officially the City of Los Angeles, is the most populous city in the state of California. With roughly 3.9 million residents within the city limits , Los Angeles is the second-most populous city in the United States, behind only New York City, and is the commercial, financial, and cultural center of the Southern California region. Los Angeles has a Mediterranean climate, an ethnically and culturally diverse population, and a sprawling metropolitan area.
Cosmic neutrino backgroundThe cosmic neutrino background (CNB or CνB) is the universe's background particle radiation composed of neutrinos. They are sometimes known as relic neutrinos. The CνB is a relic of the Big Bang; while the cosmic microwave background radiation (CMB) dates from when the universe was 379,000 years old, the CνB decoupled (separated) from matter when the universe was just one second old. It is estimated that today, the CνB has a temperature of roughly 1.95K. As neutrinos rarely interact with matter, these neutrinos still exist today.
Goal orientationGoal orientation, or achievement orientation, is an "individual disposition towards developing or validating one's ability in achievement settings". In general, an individual can be said to be mastery or performance oriented, based on whether one's goal is to develop one's ability or to demonstrate one's ability, respectively. A mastery orientation is also sometimes referred to as a learning orientation. Goal orientation refers to how an individual interprets and reacts to tasks, resulting in different patterns of cognition, affect and behavior.
GoalA goal or objective is an idea of the future or desired result that a person or a group of people envision, plan and commit to achieve. People endeavour to reach goals within a finite time by setting deadlines. A goal is roughly similar to a purpose or aim, the anticipated result which guides reaction, or an end, which is an object, either a physical object or an abstract object, that has intrinsic value. Goal setting Goal-setting theory was formulated based on empirical research and has been called one of the most important theories in organizational psychology.
GoogleGoogle LLC (ˈɡuːɡəl) is an American multinational technology company focusing on artificial intelligence, online advertising, search engine technology, cloud computing, computer software, quantum computing, e-commerce, and consumer electronics. It has been referred to as "the most powerful company in the world" and as one of the world's most valuable brands due to its market dominance, data collection, and technological advantages in the field of artificial intelligence. Alongside Amazon, Apple Inc.
Gravitational wave backgroundThe gravitational wave background (also GWB and stochastic background) is a random background of gravitational waves permeating the Universe, which is detectable by gravitational-wave experiments, like pulsar timing arrays. The signal may be intrinsically random, like from stochastic processes in the early Universe, or may be produced by an incoherent superposition of a large number of weak independent unresolved gravitational-wave sources, like supermassive black-hole binaries.
Cosmic microwave backgroundThe cosmic microwave background (CMB, CMBR) is microwave radiation that fills all space in the observable universe. It is a remnant that provides an important source of data on the primordial universe. With a standard optical telescope, the background space between stars and galaxies is almost completely dark. However, a sufficiently sensitive radio telescope detects a faint background glow that is almost uniform and is not associated with any star, galaxy, or other object.
SingaporeSingapore (ˈsɪŋ(g)əpɔːr), officially the Republic of Singapore, is an island country and city-state in maritime Southeast Asia. It is located about one degree of latitude () north of the equator, off the southern tip of the Malay Peninsula, bordering the Strait of Malacca to the west, the Singapore Strait to the south, the South China Sea to the east, and the Straits of Johor to the north.
Goal settingGoal setting involves the development of an action plan designed in order to motivate and guide a person or group toward a goal. Goals are more deliberate than desires and momentary intentions. Therefore, setting goals means that a person has committed thought, emotion, and behavior towards attaining the goal. In doing so, the goal setter has established a desired future state which differs from their current state thus creating a mismatch which in turn spurs future actions.
Outer measureIn the mathematical field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. The theory of outer measures was first introduced by Constantin Carathéodory to provide an abstract basis for the theory of measurable sets and countably additive measures.
Σ-finite measureIn mathematics, a positive (or signed) measure μ defined on a σ-algebra Σ of subsets of a set X is called a finite measure if μ(X) is a finite real number (rather than ∞), and a set A in Σ is of finite measure if μ(A) < ∞. The measure μ is called σ-finite if X is a countable union of measurable sets each with finite measure. A set in a measure space is said to have σ-finite measure if it is a countable union of measurable sets with finite measure. A measure being σ-finite is a weaker condition than being finite, i.
Complete measureIn mathematics, a complete measure (or, more precisely, a complete measure space) is a measure space in which every subset of every null set is measurable (having measure zero). More formally, a measure space (X, Σ, μ) is complete if and only if The need to consider questions of completeness can be illustrated by considering the problem of product spaces. Suppose that we have already constructed Lebesgue measure on the real line: denote this measure space by We now wish to construct some two-dimensional Lebesgue measure on the plane as a product measure.
