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Jacobi Bound Problem
The Jacobi bound problem concerns the veracity of Jacobi's inequality which is an inequality on the absolute dimension of a differential algebraic variety in terms of its defining equations. This is one of Kolchin's Problems. The inequality is the differential algebraic analog of Bézout's theorem in affine space. Although first formulated by Jacobi, In 1936 Joseph Ritt recognized the problem as non-rigorous in that Jacobi didn't even have a rigorous notion of absolute dimension (Jacobi and Ritt used the term "order" - which Ritt first gave a rigorous definition for using the notion of transcendence degree). Intuitively, the absolute dimension is the number of constants of integration required to specify a solution of a system of ordinary differential equations. A mathematical proof of the inequality has been open since 1936. Statement Let (K,\partial) be a differential field In mathematics, differential algebra is, broadly speaking, the area of mathematics consisting i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inequality (mathematics)
In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. The main types of inequality are less than and greater than (denoted by and , respectively the less-than sign, less-than and greater-than sign, greater-than signs). Notation There are several different notations used to represent different kinds of inequalities: * The notation ''a'' ''b'' means that ''a'' is greater than ''b''. In either case, ''a'' is not equal to ''b''. These relations are known as strict inequalities, meaning that ''a'' is strictly less than or strictly greater than ''b''. Equality is excluded. In contrast to strict inequalities, there are two types of inequality relations that are not strict: * The notation ''a'' ≤ ''b'' or ''a'' ⩽ ''b'' or ''a'' ≦ ''b'' means that ''a'' is less than or equal to ''b'' (or, equivalently, at most ''b'', or no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolute Dimension
Absolute may refer to: Companies * Absolute Entertainment, a video game publisher * Absolute Radio, (formerly Virgin Radio), independent national radio station in the UK * Absolute Software Corporation, specializes in security and data risk management * Absolut Vodka, a brand of Swedish vodka Mathematics and science * Absolute (geometry), the quadric at infinity * Absolute (perfumery), a fragrance substance produced by solvent extraction * Absolute infinite or Tav (number), a number that is bigger than any other conceivable or inconceivable quantity * Absolute magnitude, the brightness of a star * Absolute value, a notion in mathematics, commonly a number's numerical value without regard to its sign * Absolute pressure, the pressure in a fluid, measured relative to a vacuum *Absolute temperature, a temperature on the thermodynamic temperature scale * Absolute zero, the lower limit of the thermodynamic temperature scale, -273.15 °C * Absoluteness (logic), a concept in ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Algebraic Variety
In mathematics, differential algebra is, broadly speaking, the area of mathematics consisting in the study of differential equations and differential operators as algebraic objects in view of deriving properties of differential equations and operators without computing the solutions, similarly as polynomial algebras are used for the study of algebraic varieties, which are solution sets of systems of polynomial equations. Weyl algebras and Lie algebras may be considered as belonging to differential algebra. More specifically, ''differential algebra'' refers to the theory introduced by Joseph Ritt in 1950, in which differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations. A natural example of a differential field is the field of rational functions in one variable over the complex numbers, \mathbb(t), where the derivation is differentiation with respect to t. More generally, every differential eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kolchin's Problems
Kolchin's problems are a set of unsolved problems in differential algebra, outlined by Ellis Kolchin at the International Congress of Mathematicians in 1966 (Moscow) Kolchin Catenary Conjecture The Kolchin Catenary Conjecture is a fundamental open problem in differential algebra related to dimension theory. Statement "Let \Sigma be a differential algebraic variety of dimension d. By a ''long gap chain'' we mean a chain of irreducible differential subvarieties \Sigma_0 \subset \Sigma_1 \subset \Sigma_2 \subset \cdots of ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ... length \omega^m \cdot d." Given an irreducible differential variety \Sigma of dimension d > 0 and an arbitrary point p \in \Sigma , does there exist a long gap chain beginning at p and ending at \Sig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bézout's Theorem
In algebraic geometry, Bézout's theorem is a statement concerning the number of common zeros of polynomials in indeterminates. In its original form the theorem states that ''in general'' the number of common zeros equals the product of the degrees of the polynomials. It is named after Étienne Bézout. In some elementary texts, Bézout's theorem refers only to the case of two variables, and asserts that, if two plane algebraic curves of degrees d_1 and d_2 have no component in common, they have d_1d_2 intersection points, counted with their multiplicity, and including points at infinity and points with complex coordinates. In its modern formulation, the theorem states that, if is the number of common points over an algebraically closed field of projective hypersurfaces defined by homogeneous polynomials in indeterminates, then is either infinite, or equals the product of the degrees of the polynomials. Moreover, the finite case occurs almost always. In the case of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carl Gustav Jacob Jacobi
Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants and number theory. Biography Jacobi was born of Ashkenazi Jewish parentage in Potsdam on 10 December 1804. He was the second of four children of a banker, Simon Jacobi. His elder brother, Moritz, would also become known later as an engineer and physicist. He was initially home schooled by his uncle Lehman, who instructed him in the classical languages and elements of mathematics. In 1816, the twelve-year-old Jacobi went to the Potsdam Gymnasium, where students were taught all the standard subjects: classical languages, history, philology, mathematics, sciences, etc. As a result of the good education he had received from his uncle, as well as his own remarkable abilities, after less than half a year Jacobi was moved to the senior year despite his young age. However, as the Unive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph Ritt
Joseph Fels Ritt (August 23, 1893 – January 5, 1951) was an American mathematician at Columbia University in the early 20th century. He was born and died in New York. Biography After beginning his undergraduate studies at City College of New York, Ritt received his B.A. from George Washington University in 1913. He then earned a doctorate in mathematics from Columbia University in 1917 under the supervision of Edward Kasner. After doing calculations for the war effort in World War I, he joined the Columbia faculty in 1921. He served as department chair from 1942 to 1945, and in 1945 became the Davies Professor of Mathematics.. In 1932, George Washington University honored him with a Doctorate in Science,. and in 1933 he was elected to join the United States National Academy of Sciences. He has 905 academic descendants listed in the Mathematics Genealogy Project, mostly through his student Ellis Kolchin, as of May 2024. Ritt was an Invited Speaker with talk ''Elementary fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transcendence Degree
In mathematics, a transcendental extension L/K is a field extension such that there exists an element in the field L that is transcendental over the field K; that is, an element that is not a root of any univariate polynomial with coefficients in K. In other words, a transcendental extension is a field extension that is not algebraic. For example, \mathbb and \mathbb are both transcendental extensions of \mathbb. A transcendence basis of a field extension L/K (or a transcendence basis of L over K) is a maximal algebraically independent subset of L over K. Transcendence bases share many properties with bases of vector spaces. In particular, all transcendence bases of a field extension have the same cardinality, called the transcendence degree of the extension. Thus, a field extension is a transcendental extension if and only if its transcendence degree is nonzero. Transcendental extensions are widely used in algebraic geometry. For example, the dimension of an algebraic varie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Differential Equation
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematics), function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differential equation, ''partial'' differential equations (PDEs) which may be with respect to one independent variable, and, less commonly, in contrast with stochastic differential equations, ''stochastic'' differential equations (SDEs) where the progression is random. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where a_0(x),\ldots,a_n(x) and b(x) are arbitrary differentiable functions that do not need to be linea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Proof
A mathematical proof is a deductive reasoning, deductive Argument-deduction-proof distinctions, argument for a Proposition, 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 that establish logical certainty, to be distinguished from empirical evidence, empirical arguments or non-exhaustive inductive reasoning that establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in ''all'' possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Field
In mathematics, differential algebra is, broadly speaking, the area of mathematics consisting in the study of differential equations and differential operators as algebra, algebraic objects in view of deriving properties of differential equations and operators without computing the solutions, similarly as polynomial algebras are used for the study of algebraic varieties, which are solution sets of systems of polynomial equations. Weyl algebras and Lie algebras may be considered as belonging to differential algebra. More specifically, ''differential algebra'' refers to the theory introduced by Joseph Ritt in 1950, in which differential rings, differential fields, and differential algebras are ring (mathematics), rings, field (mathematics), fields, and algebra over a field, algebras equipped with finitely many derivation (differential algebra), derivations. A natural example of a differential field is the field of rational functions in one variable over the complex numbers, \math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unsolved Problems In Mathematics
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations. Some problems belong to more than one discipline and are studied using techniques from different areas. Prizes are often awarded for the solution to a long-standing problem, and some lists of unsolved problems, such as the Millennium Prize Problems, receive considerable attention. This list is a composite of notable unsolved problems mentioned in previously published lists, including but not limited to lists considered authoritative, and the problems listed here vary widely in both difficulty and importance. Lists of unsolved problems in mathematics Various mathematicians and organiz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
