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Ehrenpreis's Fundamental Principle
In mathematical analysis, Ehrenpreis's fundamental principle, introduced by Leon Ehrenpreis, states: :Every solution of a system (in general, overdetermined) of homogeneous partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to h ...s with constant coefficients can be represented as the integral with respect to an appropriate Radon measure over the complex “ characteristic variety” of the system. References Mathematical analysis {{analysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leon Ehrenpreis
Eliezer 'Leon' Ehrenpreis (May 22, 1930 – August 16, 2010, Brooklyn) was a mathematician at Temple University who proved the Malgrange–Ehrenpreis theorem, the fundamental theorem about differential operators with constant coefficients. He previously held tenured positions at Yeshiva University and at the Courant Institute at New York University. Early life and education Leon was born in New York City to a family of Jewish immigrants from Eastern Europe. He graduated from Stuyvesant High School and studied Mathematics as an undergraduate at City College of New York. Afterward, he enrolled as a doctoral student at Columbia University where he studied under mathematician Claude Chevalley, obtaining his PhD in 1953 at the age of 23. His doctoral thesis was entitled "Theory of Distributions in Locally Compact Spaces". Religion Ehrenpreis was also a Rabbi, having received his ordination from the renowned Rabbi Moshe Feinstein. He was the author of a work on the Chumash ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity, and stability. Among the many open questions are the ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves variables, they may also be called parameters. For example, the polynomial 2x^2-x+3 has coefficients 2, −1, and 3, and the powers of the variable x in the polynomial ax^2+bx+c have coefficient parameters a, b, and c. The constant coefficient is the coefficient not attached to variables in an expression. For example, the constant coefficients of the expressions above are the number 3 and the parameter ''c'', respectively. The coefficient attached to the highest degree of the variable in a polynomial is referred to as the leading coefficient. For example, in the expressions above, the leading coefficients are 2 and ''a'', respectively. Terminology and definition In mathematics, a coefficient is a multiplicative factor in some term ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic Variety
In mathematical analysis, the characteristic variety of a microdifferential operator ''P'' is an algebraic variety that is the zero set of the principal symbol of ''P'' in the cotangent bundle. It is invariant under a quantized contact transformation. The notion is also defined more generally in commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom .... A basic theorem says a characteristic variety is involutive. References * M. Sato, T. Kawai, and M. Kashiwara: Microfunctions and Pseudo-differential Equations. Lecture note in Math., No. 287, Springer, Berlin-Heidelberg-New York, pp. 265–529 (1973) {{analysis-stub Algebraic varieties Mathematical analysis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
