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Hunter–Saxton Equation
In mathematical physics, the Hunter–Saxton equation : (u_t + u u_x)_x = \frac \, u_x^2 is an integrable PDE that arises in the theoretical study of nematic liquid crystals. If the molecules in the liquid crystal are initially all aligned, and some of them are then wiggled slightly, this disturbance in orientation will propagate through the crystal, and the Hunter–Saxton equation describes certain aspects of such orientation waves. Physical background In the models for liquid crystals considered here, it is assumed that there is no fluid flow, so that only the ''orientation'' of the molecules is of interest. Within the elastic continuum theory, the orientation is described by a field of unit vectors n(''x'',''y'',''z'',''t''). For nematic liquid crystals, there is no difference between orienting a molecule in the n direction or in the −n direction, and the vector field n is then called a ''director field''. The potential energy density of a director field is usually ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Physics
Mathematical physics is the development of mathematics, 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, known as physical mathematics. Scope There are several distinct branches of mathematical physics, and these roughly correspond to particular historical parts of our world. Classical mechanics Applying the techniques of mathematical physics to classical mechanics typically involves the rigorous, abstract, and advanced reformulation of Newtonian mechanics in terms of Lagrangian mechanics and Hamiltonian mechanics (including both approaches in the presence of constraints). Both formulations are embodied in analytical mechanics and lead ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. In topology, it is often denoted as because it is a one-dimensional unit -sphere. If is a point on the unit circle's circumference, then and are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, and satisfy the equation x^2 + y^2 = 1. Since for all , and since the reflection of any point on the unit circle about the - or -axis is also on the unit circle, the above equation holds for all points on the unit circle, not only those in the first quadrant. The interior of the unit circle is called the open unit disk, while the interior of the unit circle combined with the unit circle itself is called the closed unit disk. One may also use other notions of "dis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Of Computation
''Mathematics of Computation'' is a bimonthly mathematics journal focused on computational mathematics. It was established in 1943 as ''Mathematical Tables and Other Aids to Computation'', obtaining its current name in 1960. Articles older than five years are available electronically free of charge. Abstracting and indexing The journal is abstracted and indexed in Mathematical Reviews, Zentralblatt MATH, Science Citation Index, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. According to the '' Journal Citation Reports'', the journal has a 2020 impact factor of 2.417. References External links * Delayed open access journals English-language journals Mathematics journals Academic journals established in 1943 American Mathematical Society academic journals Bimonthly journals {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SIAM Journal On Mathematical Analysis
Thailand, officially the Kingdom of Thailand and historically known as Siam (the official name until 1939), is a country in Southeast Asia on the Indochinese Peninsula. With a population of almost 66 million, it spans . Thailand is bordered to the northwest by Myanmar, to the northeast and east by Laos, to the southeast by Cambodia, to the south by the Gulf of Thailand and Malaysia, and to the southwest by the Andaman Sea; it also shares maritime borders with Vietnam to the southeast and Indonesia and India to the southwest. Bangkok is the state capital and largest city. Thai peoples migrated from southwestern China to mainland Southeast Asia from the 6th to 11th centuries. Indianised kingdoms such as the Mon, Khmer Empire, and Malay states ruled the region, competing with Thai states such as the Kingdoms of Ngoenyang, Sukhothai, Lan Na, and Ayutthaya, which also rivalled each other. European contact began in 1511 with a Portuguese diplomatic mission to Ayutthaya, whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Geometry And Physics
The ''Journal of Geometry and Physics'' is a scientific journal in mathematical physics. Its scope is to stimulate the interaction between geometry and physics by publishing primary research and review articles which are of common interest to practitioners in both fields. The journal is published by Elsevier since 1984. The Journal covers the following areas of research: ''Methods of:'' * Algebraic and Differential Topology * Algebraic Geometry * Real and Complex Differential Geometry * Riemannian and Finsler Manifolds * Symplectic Geometry * Global Analysis, Analysis on Manifolds * Geometric Theory of Differential Equations * Geometric Control Theory * Lie Groups and Lie Algebras * Supermanifolds and Supergroups * Discrete Geometry * Spinors and Twistors ''Applications to:'' * Strings and Superstrings * Noncommutative Topology and Geometry * Quantum Groups * Geometric Methods in Statistics and Probability * Geometry Approaches to Thermodynamics * Classical and Quantum Dynamical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advances In Mathematics
''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed at publishing articles addressed to a broader "mathematical community", and not only to mathematicians in the author's field. Herbert Busemann writes, in the preface of the first issue, "The need for expository articles addressing either all mathematicians or only those in somewhat related fields has long been felt, but little has been done outside of the USSR. The serial publication ''Advances in Mathematics'' was created in response to this demand." Abstracting and indexing The journal is abstracted and indexed in: * [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physica D '', an international scientific journal for experimental and theoretical physics
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Physica may refer to: * Physics (Aristotle) * ''Physica'', a twelfth-century medical text by Hildegard of Bingen * ''Physica'' (journal), a Dutch scientific journal :* ''Physica A'' :* ''Physica B'' ;* ''Physica C'' :* ''Physica D'' :* ''Physica E'' * ''Physica Scripta ''Physica Scripta'' is an international scientific journal for experimental and theoretical physics. It was established in 1970 as the successor of ''Arkiv för Fysik'' and published by the Royal Swedish Academy of Sciences (KVA). Since 2006, it ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SIAM Journal On Applied Mathematics
The ''SIAM Journal on Applied Mathematics'' is a peer-reviewed academic journal in applied mathematics published by the Society for Industrial and Applied Mathematics (SIAM), with Paul A. Martin (Colorado School of Mines) as its editor-in-chief. It was founded in 1953 as SIAM's first journal, the ''Journal of the Society for Industrial and Applied Mathematics'', and was given its current name in 1966. In most years since 1999, it has been ranked by SCImago Journal Rank as a second-quartile journal in applied mathematics. Together with ''Communications on Pure and Applied Mathematics ''Communications on Pure and Applied Mathematics'' is a monthly peer-reviewed scientific journal which is published by John Wiley & Sons on behalf of the Courant Institute of Mathematical Sciences. It covers research originating from or solicited ...'' it has been called "one of the two greatest American entries in applied math".. References {{Society for Industrial and Applied Mathematics Applied ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flow (mathematics)
In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including engineering and physics. The notion of flow is basic to the study of ordinary differential equations. Informally, a flow may be viewed as a continuous motion of points over time. More formally, a flow is a group action of the real numbers on a set. The idea of a vector flow, that is, the flow determined by a vector field, occurs in the areas of differential topology, Riemannian geometry and Lie groups. Specific examples of vector flows include the geodesic flow, the Hamiltonian flow, the Ricci flow, the mean curvature flow, and Anosov flows. Flows may also be defined for systems of random variables and stochastic processes, and occur in the study of ergodic dynamical systems. The most celebrated of these is perhaps the Bernoulli flow. Formal definition A flow on a set is a group action of the additive group of real numbers on . More explicit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semi-metric
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are a general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance and therefor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inner Product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in \langle a, b \rangle. Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or ''scalar product'' of Cartesian coordinates. Inner product spaces of infinite dimension are widely used in functional analysis. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898. An inner product naturally induces an associated norm, (denoted , x, and , y, in the pictu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
