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UNITARY SPACE
Unitary space. A vector space over the field C of complex numbers, on which there is given an inner product of vectors (where the product ( a, b) of two vectors a and b is, in general, a complex number) that satisfies the following axioms: 4) if a ≠ 0, then ( a, a) > 0, i.e. the scalar square of a non-zero vector is a positive real number.GREEN FORMULAS
Green's formulas play an important role in analysis and, particularly, in the theory of boundary value problems for differential operators (both ordinary and partial differential operators) of the second or higher orders. For functions , which are sufficiently smooth in , Green's formulas (2) and (4) serve as the source of several relationsREGULAR FUNCTION
This article was adapted from an original article by Yu.D. Maksimov (originator), which appeared in Encyclopedia of Mathematics - ISBN1402006098.
HÖLDER SPACE
Hölder space. A Banach space of bounded continuous functions f ( x) = f ( x 1 x n) defined on a set E of an n - dimensional Euclidean space and satisfying a Hölder condition on E . The Hölder space C m ( E) , where m ≥ 0 is an integer, consists of the functions that are m times continuously differentiable on E ( continuous for m = 0 ).FUBINI-STUDY METRIC
The Fubini–Study metric is, up to proportionality, the unique Riemannian metric on C P n that is invariant under the unitary group U ( n + 1) , which preserves the scalar product. The space C P n , endowed with the Fubini–Study metric, is a compact Hermitian symmetric space of rank 1. It is also called an elliptic Hermitianspace.
DIAGONAL OPERATOR
Diagonal operator. From Encyclopedia of Mathematics. Jump to: navigation , search. An operator D defined on the (closed) linear span of a basis { e k } k ≥ 1 in a normed (or only locally convex) space X by the equations D e k = λ k e k , where k ≥ 1 and where λ k are complex numbers. If D is a continuous operator, one has. sup k ≥ 1GEVREY CLASS
Gevrey class. An intermediate space between the spaces of smooth (i.e. C∞ -) functions and real-analytic functions. In fact, the name is given in honour of M. Gevrey, who gave the first motivating example (see , in which regularity estimates of the heat kernel are deduced). Given Ω ⊂ Rn and s ≥ 1, the Gevrey class GS(Ω) (ofindex s
WREATH PRODUCT
The wreath product of semi-groups is a construction assigning to two semi-groups a third in the following way: The wreath product W of A and B has as underlying set F(B, A) × B , where F(B, A) is the semi-group of all mappings from B into A under pointwise multiplication, and the operation on W is given by the formula: (f, b)(g, c) = (f bg, bc FRITZ JOHN CONDITION The Fritz John condition describes local optimality of a feasible point x ∗ using the gradients of the objective function and the "active" constraints P ( x ∗) = { i ∈ P: g i ( x ∗) = 0 }, . The basic Fritz John condition is as follows. Consider the problem (1) where all STRATONOVICH INTEGRAL There is no universally agreed notation for the integral, but the above is the most common (see , , for example). It is also known as the Fisk, Fisk–Stratonovich or symmetrized stochastic integral, the latter in view of the property that. \tag {a2 } \int\limits _ { 0 } ^ { t } Y ( s) \circ dX ( s) =.UNITARY SPACE
Unitary space. A vector space over the field C of complex numbers, on which there is given an inner product of vectors (where the product ( a, b) of two vectors a and b is, in general, a complex number) that satisfies the following axioms: 4) if a ≠ 0, then ( a, a) > 0, i.e. the scalar square of a non-zero vector is a positive real number.GREEN FORMULAS
Green's formulas play an important role in analysis and, particularly, in the theory of boundary value problems for differential operators (both ordinary and partial differential operators) of the second or higher orders. For functions , which are sufficiently smooth in , Green's formulas (2) and (4) serve as the source of several relationsREGULAR FUNCTION
This article was adapted from an original article by Yu.D. Maksimov (originator), which appeared in Encyclopedia of Mathematics - ISBN1402006098.
HÖLDER SPACE
Hölder space. A Banach space of bounded continuous functions f ( x) = f ( x 1 x n) defined on a set E of an n - dimensional Euclidean space and satisfying a Hölder condition on E . The Hölder space C m ( E) , where m ≥ 0 is an integer, consists of the functions that are m times continuously differentiable on E ( continuous for m = 0 ).FUBINI-STUDY METRIC
The Fubini–Study metric is, up to proportionality, the unique Riemannian metric on C P n that is invariant under the unitary group U ( n + 1) , which preserves the scalar product. The space C P n , endowed with the Fubini–Study metric, is a compact Hermitian symmetric space of rank 1. It is also called an elliptic Hermitianspace.
DIAGONAL OPERATOR
Diagonal operator. From Encyclopedia of Mathematics. Jump to: navigation , search. An operator D defined on the (closed) linear span of a basis { e k } k ≥ 1 in a normed (or only locally convex) space X by the equations D e k = λ k e k , where k ≥ 1 and where λ k are complex numbers. If D is a continuous operator, one has. sup k ≥ 1GEVREY CLASS
Gevrey class. An intermediate space between the spaces of smooth (i.e. C∞ -) functions and real-analytic functions. In fact, the name is given in honour of M. Gevrey, who gave the first motivating example (see , in which regularity estimates of the heat kernel are deduced). Given Ω ⊂ Rn and s ≥ 1, the Gevrey class GS(Ω) (ofindex s
WREATH PRODUCT
The wreath product of semi-groups is a construction assigning to two semi-groups a third in the following way: The wreath product W of A and B has as underlying set F(B, A) × B , where F(B, A) is the semi-group of all mappings from B into A under pointwise multiplication, and the operation on W is given by the formula: (f, b)(g, c) = (f bg, bc FRITZ JOHN CONDITION The Fritz John condition describes local optimality of a feasible point x ∗ using the gradients of the objective function and the "active" constraints P ( x ∗) = { i ∈ P: g i ( x ∗) = 0 }, . The basic Fritz John condition is as follows. Consider the problem (1) where all STRATONOVICH INTEGRAL There is no universally agreed notation for the integral, but the above is the most common (see , , for example). It is also known as the Fisk, Fisk–Stratonovich or symmetrized stochastic integral, the latter in view of the property that. \tag {a2 } \int\limits _ { 0 } ^ { t } Y ( s) \circ dX ( s) =.GREEN FORMULAS
Green's formulas play an important role in analysis and, particularly, in the theory of boundary value problems for differential operators (both ordinary and partial differential operators) of the second or higher orders. For functions , which are sufficiently smooth in , Green's formulas (2) and (4) serve as the source of several relationsSEMI-DIRECT PRODUCT
Semi-direct product. A group G = A B which is the product of its subgroups A and B, where B is normal in G and A ∩ B = { 1 }. If A is also normal in G, then the semi-direct product becomes a direct product. The semi-direct product of two groups A and B is not uniquelydetermined.
TOTALLY-DISCONNECTED SPACE A topological space in which any subset containing more than one point is disconnected (cf. connected space).An equivalent condition is that the connected component of any point in the space is that point itself. The topological product and the topological sum of totally-disconnected spaces, as well as any subspace of a totally-disconnected space, are totally disconnected. STRATONOVICH INTEGRAL There is no universally agreed notation for the integral, but the above is the most common (see , , for example). It is also known as the Fisk, Fisk–Stratonovich or symmetrized stochastic integral, the latter in view of the property that. \tag {a2 } \int\limits _ { 0 } ^ { t } Y ( s) \circ dX ( s) =. CONVEX FUNCTION (OF A REAL VARIABLE) N. Bourbaki, "Elements of mathematics. Functions of a real variable" , Addison-Wesley (1976) pp. Chapt. 1 Sect. 4 (Translatedfrom French)
MACKEY INTERTWINING NUMBER THEOREM C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962) pp. §44 G. Warner, "Harmonic analysis on semi-simple Lie GEOMETRY OF IMMERSED MANIFOLDS A theory that deals with the extrinsic geometry and the relation between the extrinsic and intrinsic geometry (cf. also Interior geometry) of submanifolds in a Euclidean or Riemannian space.The geometry of immersed manifolds is a generalization of the classical differential geometry of surfaces in the Euclidean space $ \mathbf R ^{3} $.
OBERBECK-BOUSSINESQ EQUATIONS Equations giving an approximate description of the thermo-mechanical response of linearly viscous fluids (Navier–Stokes fluids or Newtonian fluids) that can only sustain volume-preserving motions (isochoric motions) in isothermal processes, but can undergo motions that are not volume-preserving during non-isothermal processes. GALOIS THEORY, INVERSE PROBLEM OF Comments. In recent years much progress has been made on the inverse problem of Galois theory over the base field $K=\mathbf Q$, see –.The technique is to realizeCATEGORY:TEX DONE
Pages in category "TeX done" The following 200 pages are in this category, out of 7,033 total. (previous page) ()UNITARY SPACE
Unitary space. A vector space over the field C of complex numbers, on which there is given an inner product of vectors (where the product ( a, b) of two vectors a and b is, in general, a complex number) that satisfies the following axioms: 4) if a ≠ 0, then ( a, a) > 0, i.e. the scalar square of a non-zero vector is a positive real number.LINEAR VARIETY
Linear variety. A subset M of a (linear) vector space E that is a translate of a linear subspace L of E, that is, a set M of the form x 0 + L for some x 0. The set M determines L uniquely, whereas x 0 is defined only modulo L : if and only if L = N and x 1 − x 0 ∈ L. The dimension of M is the dimension of L. A linear varietycorresponding
HÖLDER SPACE
Hölder space. A Banach space of bounded continuous functions f ( x) = f ( x 1 x n) defined on a set E of an n - dimensional Euclidean space and satisfying a Hölder condition on E . The Hölder space C m ( E) , where m ≥ 0 is an integer, consists of the functions that are m times continuously differentiable on E ( continuous for m = 0 ).REGULAR FUNCTION
This article was adapted from an original article by Yu.D. Maksimov (originator), which appeared in Encyclopedia of Mathematics - ISBN1402006098.
GEVREY CLASS
Gevrey class. An intermediate space between the spaces of smooth (i.e. C∞ -) functions and real-analytic functions. In fact, the name is given in honour of M. Gevrey, who gave the first motivating example (see , in which regularity estimates of the heat kernel are deduced). Given Ω ⊂ Rn and s ≥ 1, the Gevrey class GS(Ω) (ofindex s
DIAGONAL OPERATOR
Diagonal operator. From Encyclopedia of Mathematics. Jump to: navigation , search. An operator D defined on the (closed) linear span of a basis { e k } k ≥ 1 in a normed (or only locally convex) space X by the equations D e k = λ k e k , where k ≥ 1 and where λ k are complex numbers. If D is a continuous operator, one has. sup k ≥ 1ADJUGATE MATRIX
adjoint matrix. The signed transposed matrix of cofactors for a given square matrix A. The ( i, j) entry of a d j A is. a d j A i j = ( − 1) i + j det A ( j, i) where A ( j, i) is the minor formed by deleting the row and column through the matrix entry A j i . The expansion of the determinant in cofactors is expressed as.FUBINI-STUDY METRIC
The Fubini–Study metric is, up to proportionality, the unique Riemannian metric on C P n that is invariant under the unitary group U ( n + 1) , which preserves the scalar product. The space C P n , endowed with the Fubini–Study metric, is a compact Hermitian symmetric space of rank 1. It is also called an elliptic Hermitianspace.
NOWHERE-DENSE SET
A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) J.C. Oxtoby, "Measure and category FRITZ JOHN CONDITION The Fritz John condition describes local optimality of a feasible point x ∗ using the gradients of the objective function and the "active" constraints P ( x ∗) = { i ∈ P: g i ( x ∗) = 0 }, . The basic Fritz John condition is as follows. Consider the problem (1) where allUNITARY SPACE
Unitary space. A vector space over the field C of complex numbers, on which there is given an inner product of vectors (where the product ( a, b) of two vectors a and b is, in general, a complex number) that satisfies the following axioms: 4) if a ≠ 0, then ( a, a) > 0, i.e. the scalar square of a non-zero vector is a positive real number.LINEAR VARIETY
Linear variety. A subset M of a (linear) vector space E that is a translate of a linear subspace L of E, that is, a set M of the form x 0 + L for some x 0. The set M determines L uniquely, whereas x 0 is defined only modulo L : if and only if L = N and x 1 − x 0 ∈ L. The dimension of M is the dimension of L. A linear varietycorresponding
HÖLDER SPACE
Hölder space. A Banach space of bounded continuous functions f ( x) = f ( x 1 x n) defined on a set E of an n - dimensional Euclidean space and satisfying a Hölder condition on E . The Hölder space C m ( E) , where m ≥ 0 is an integer, consists of the functions that are m times continuously differentiable on E ( continuous for m = 0 ).REGULAR FUNCTION
This article was adapted from an original article by Yu.D. Maksimov (originator), which appeared in Encyclopedia of Mathematics - ISBN1402006098.
GEVREY CLASS
Gevrey class. An intermediate space between the spaces of smooth (i.e. C∞ -) functions and real-analytic functions. In fact, the name is given in honour of M. Gevrey, who gave the first motivating example (see , in which regularity estimates of the heat kernel are deduced). Given Ω ⊂ Rn and s ≥ 1, the Gevrey class GS(Ω) (ofindex s
DIAGONAL OPERATOR
Diagonal operator. From Encyclopedia of Mathematics. Jump to: navigation , search. An operator D defined on the (closed) linear span of a basis { e k } k ≥ 1 in a normed (or only locally convex) space X by the equations D e k = λ k e k , where k ≥ 1 and where λ k are complex numbers. If D is a continuous operator, one has. sup k ≥ 1ADJUGATE MATRIX
adjoint matrix. The signed transposed matrix of cofactors for a given square matrix A. The ( i, j) entry of a d j A is. a d j A i j = ( − 1) i + j det A ( j, i) where A ( j, i) is the minor formed by deleting the row and column through the matrix entry A j i . The expansion of the determinant in cofactors is expressed as.FUBINI-STUDY METRIC
The Fubini–Study metric is, up to proportionality, the unique Riemannian metric on C P n that is invariant under the unitary group U ( n + 1) , which preserves the scalar product. The space C P n , endowed with the Fubini–Study metric, is a compact Hermitian symmetric space of rank 1. It is also called an elliptic Hermitianspace.
NOWHERE-DENSE SET
A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) J.C. Oxtoby, "Measure and category FRITZ JOHN CONDITION The Fritz John condition describes local optimality of a feasible point x ∗ using the gradients of the objective function and the "active" constraints P ( x ∗) = { i ∈ P: g i ( x ∗) = 0 }, . The basic Fritz John condition is as follows. Consider the problem (1) where all CHARACTERISTIC OF A FIELD Characteristic of a field. An invariant of a field which is either a prime number or the number zero, uniquely determined for a given field in the following way. If for some n > 0 , where e is the unit element of the field F, then the smallest such n is a prime number; it is called the characteristic of F. If there are no such numbers n, thenRADON MEASURE
Radon space. A topological space X is called a Radon space if every finite measure defined on the σ -algebra B ( X) of Borel sets is a Radon measure. For instance the Euclidean space is a Radon space (cp. with Theorem 1.11 and Corollary 1.12 of ). If B ( X) is countably generated, X is a Radon space if and only if it is Borel isomorphic GEOMETRY OF IMMERSED MANIFOLDS A theory that deals with the extrinsic geometry and the relation between the extrinsic and intrinsic geometry (cf. also Interior geometry) of submanifolds in a Euclidean or Riemannian space.The geometry of immersed manifolds is a generalization of the classical differential geometry of surfaces in the Euclidean space $ \mathbf R ^{3} $.
BURKHOLDER-DAVIS-GUNDY INEQUALITY N.L. Bassily, "A new proof of the right hand side of the Burkholder–Davis–Gundy inequality" , Proc. 5th Pannonian Symp. Math. Statistics, Visegrad, Hungary (1985) pp. 7–21 D.L. Burkholder, B. Davis, R.F. Gundy, "Integral inequalities for convex functions of operators on martingales" , Proc. 6th Berkeley Symp. Math. Statistics and Probability, 2 (1972) pp. 223–240 COMPLETELY-CONTINUOUS OPERATOR Completely-Continuous Operator. A bounded linear operator f, acting from a Banach space X into another space Y, that transforms weakly-convergent sequences in X to norm-convergent sequences in Y. Equivalently, an operator f is completely-continuous if it maps every relatively weakly compact subset of X into a relatively compact subsetof Y.
CONVEX FUNCTION (OF A REAL VARIABLE) N. Bourbaki, "Elements of mathematics. Functions of a real variable" , Addison-Wesley (1976) pp. Chapt. 1 Sect. 4 (Translatedfrom French)
MACKEY INTERTWINING NUMBER THEOREM C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962) pp. §44 G. Warner, "Harmonic analysis on semi-simple Lie OBERBECK-BOUSSINESQ EQUATIONS Equations giving an approximate description of the thermo-mechanical response of linearly viscous fluids (Navier–Stokes fluids or Newtonian fluids) that can only sustain volume-preserving motions (isochoric motions) in isothermal processes, but can undergo motions that are not volume-preserving during non-isothermal processes. GALOIS THEORY, INVERSE PROBLEM OF Comments. In recent years much progress has been made on the inverse problem of Galois theory over the base field $K=\mathbf Q$, see –.The technique is to realizeCATEGORY:TEX DONE
Pages in category "TeX done" The following 200 pages are in this category, out of 7,033 total. (previous page) ()UNITARY SPACE
Unitary space. A vector space over the field C of complex numbers, on which there is given an inner product of vectors (where the product ( a, b) of two vectors a and b is, in general, a complex number) that satisfies the following axioms: 4) if a ≠ 0, then ( a, a) > 0, i.e. the scalar square of a non-zero vector is a positive real number.LINEAR VARIETY
Linear variety. A subset M of a (linear) vector space E that is a translate of a linear subspace L of E, that is, a set M of the form x 0 + L for some x 0. The set M determines L uniquely, whereas x 0 is defined only modulo L : if and only if L = N and x 1 − x 0 ∈ L. The dimension of M is the dimension of L. A linear varietycorresponding
HÖLDER SPACE
Hölder space. A Banach space of bounded continuous functions f ( x) = f ( x 1 x n) defined on a set E of an n - dimensional Euclidean space and satisfying a Hölder condition on E . The Hölder space C m ( E) , where m ≥ 0 is an integer, consists of the functions that are m times continuously differentiable on E ( continuous for m = 0 ).REGULAR FUNCTION
This article was adapted from an original article by Yu.D. Maksimov (originator), which appeared in Encyclopedia of Mathematics - ISBN1402006098.
GEVREY CLASS
Gevrey class. An intermediate space between the spaces of smooth (i.e. C∞ -) functions and real-analytic functions. In fact, the name is given in honour of M. Gevrey, who gave the first motivating example (see , in which regularity estimates of the heat kernel are deduced). Given Ω ⊂ Rn and s ≥ 1, the Gevrey class GS(Ω) (ofindex s
DIAGONAL OPERATOR
Diagonal operator. From Encyclopedia of Mathematics. Jump to: navigation , search. An operator D defined on the (closed) linear span of a basis { e k } k ≥ 1 in a normed (or only locally convex) space X by the equations D e k = λ k e k , where k ≥ 1 and where λ k are complex numbers. If D is a continuous operator, one has. sup k ≥ 1ADJUGATE MATRIX
adjoint matrix. The signed transposed matrix of cofactors for a given square matrix A. The ( i, j) entry of a d j A is. a d j A i j = ( − 1) i + j det A ( j, i) where A ( j, i) is the minor formed by deleting the row and column through the matrix entry A j i . The expansion of the determinant in cofactors is expressed as.FUBINI-STUDY METRIC
The Fubini–Study metric is, up to proportionality, the unique Riemannian metric on C P n that is invariant under the unitary group U ( n + 1) , which preserves the scalar product. The space C P n , endowed with the Fubini–Study metric, is a compact Hermitian symmetric space of rank 1. It is also called an elliptic Hermitianspace.
NOWHERE-DENSE SET
A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) J.C. Oxtoby, "Measure and category FRITZ JOHN CONDITION The Fritz John condition describes local optimality of a feasible point x ∗ using the gradients of the objective function and the "active" constraints P ( x ∗) = { i ∈ P: g i ( x ∗) = 0 }, . The basic Fritz John condition is as follows. Consider the problem (1) where allUNITARY SPACE
Unitary space. A vector space over the field C of complex numbers, on which there is given an inner product of vectors (where the product ( a, b) of two vectors a and b is, in general, a complex number) that satisfies the following axioms: 4) if a ≠ 0, then ( a, a) > 0, i.e. the scalar square of a non-zero vector is a positive real number.LINEAR VARIETY
Linear variety. A subset M of a (linear) vector space E that is a translate of a linear subspace L of E, that is, a set M of the form x 0 + L for some x 0. The set M determines L uniquely, whereas x 0 is defined only modulo L : if and only if L = N and x 1 − x 0 ∈ L. The dimension of M is the dimension of L. A linear varietycorresponding
HÖLDER SPACE
Hölder space. A Banach space of bounded continuous functions f ( x) = f ( x 1 x n) defined on a set E of an n - dimensional Euclidean space and satisfying a Hölder condition on E . The Hölder space C m ( E) , where m ≥ 0 is an integer, consists of the functions that are m times continuously differentiable on E ( continuous for m = 0 ).REGULAR FUNCTION
This article was adapted from an original article by Yu.D. Maksimov (originator), which appeared in Encyclopedia of Mathematics - ISBN1402006098.
GEVREY CLASS
Gevrey class. An intermediate space between the spaces of smooth (i.e. C∞ -) functions and real-analytic functions. In fact, the name is given in honour of M. Gevrey, who gave the first motivating example (see , in which regularity estimates of the heat kernel are deduced). Given Ω ⊂ Rn and s ≥ 1, the Gevrey class GS(Ω) (ofindex s
DIAGONAL OPERATOR
Diagonal operator. From Encyclopedia of Mathematics. Jump to: navigation , search. An operator D defined on the (closed) linear span of a basis { e k } k ≥ 1 in a normed (or only locally convex) space X by the equations D e k = λ k e k , where k ≥ 1 and where λ k are complex numbers. If D is a continuous operator, one has. sup k ≥ 1ADJUGATE MATRIX
adjoint matrix. The signed transposed matrix of cofactors for a given square matrix A. The ( i, j) entry of a d j A is. a d j A i j = ( − 1) i + j det A ( j, i) where A ( j, i) is the minor formed by deleting the row and column through the matrix entry A j i . The expansion of the determinant in cofactors is expressed as.FUBINI-STUDY METRIC
The Fubini–Study metric is, up to proportionality, the unique Riemannian metric on C P n that is invariant under the unitary group U ( n + 1) , which preserves the scalar product. The space C P n , endowed with the Fubini–Study metric, is a compact Hermitian symmetric space of rank 1. It is also called an elliptic Hermitianspace.
NOWHERE-DENSE SET
A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) J.C. Oxtoby, "Measure and category FRITZ JOHN CONDITION The Fritz John condition describes local optimality of a feasible point x ∗ using the gradients of the objective function and the "active" constraints P ( x ∗) = { i ∈ P: g i ( x ∗) = 0 }, . The basic Fritz John condition is as follows. Consider the problem (1) where all CHARACTERISTIC OF A FIELD Characteristic of a field. An invariant of a field which is either a prime number or the number zero, uniquely determined for a given field in the following way. If for some n > 0 , where e is the unit element of the field F, then the smallest such n is a prime number; it is called the characteristic of F. If there are no such numbers n, thenRADON MEASURE
Radon space. A topological space X is called a Radon space if every finite measure defined on the σ -algebra B ( X) of Borel sets is a Radon measure. For instance the Euclidean space is a Radon space (cp. with Theorem 1.11 and Corollary 1.12 of ). If B ( X) is countably generated, X is a Radon space if and only if it is Borel isomorphic GEOMETRY OF IMMERSED MANIFOLDS A theory that deals with the extrinsic geometry and the relation between the extrinsic and intrinsic geometry (cf. also Interior geometry) of submanifolds in a Euclidean or Riemannian space.The geometry of immersed manifolds is a generalization of the classical differential geometry of surfaces in the Euclidean space $ \mathbf R ^{3} $.
BURKHOLDER-DAVIS-GUNDY INEQUALITY N.L. Bassily, "A new proof of the right hand side of the Burkholder–Davis–Gundy inequality" , Proc. 5th Pannonian Symp. Math. Statistics, Visegrad, Hungary (1985) pp. 7–21 D.L. Burkholder, B. Davis, R.F. Gundy, "Integral inequalities for convex functions of operators on martingales" , Proc. 6th Berkeley Symp. Math. Statistics and Probability, 2 (1972) pp. 223–240 COMPLETELY-CONTINUOUS OPERATOR Completely-Continuous Operator. A bounded linear operator f, acting from a Banach space X into another space Y, that transforms weakly-convergent sequences in X to norm-convergent sequences in Y. Equivalently, an operator f is completely-continuous if it maps every relatively weakly compact subset of X into a relatively compact subsetof Y.
CONVEX FUNCTION (OF A REAL VARIABLE) N. Bourbaki, "Elements of mathematics. Functions of a real variable" , Addison-Wesley (1976) pp. Chapt. 1 Sect. 4 (Translatedfrom French)
MACKEY INTERTWINING NUMBER THEOREM C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962) pp. §44 G. Warner, "Harmonic analysis on semi-simple Lie OBERBECK-BOUSSINESQ EQUATIONS Equations giving an approximate description of the thermo-mechanical response of linearly viscous fluids (Navier–Stokes fluids or Newtonian fluids) that can only sustain volume-preserving motions (isochoric motions) in isothermal processes, but can undergo motions that are not volume-preserving during non-isothermal processes. GALOIS THEORY, INVERSE PROBLEM OF Comments. In recent years much progress has been made on the inverse problem of Galois theory over the base field $K=\mathbf Q$, see –.The technique is to realizeCATEGORY:TEX DONE
Pages in category "TeX done" The following 200 pages are in this category, out of 7,033 total. (previous page) ()* Log in
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From Encyclopedia of Mathematics Jump to: navigation, search IMPORTANT NOTIFICATION MOVE OF THE ENCYCLOPEDIA OF MATHEMATICS FROM SPRINGER VERLAG TO EMSPRESS
* _The Encyclopedia of Mathematics (EoM) has moved from Springer Verlag to EMS Press , the Berlin-based mathematics publisher, owned by the European Mathematical Society ._ * _Therefore, the software of this server was updated - see the Special:Version for details._ * _In case you encounter any problems with the new software just drop a note on the discussion page of this page._ * _Further Information will be posted here soon, in particular concerning the licensing agreement._ PLEASE STAY TUNED WITH EOM! The Encyclopedia of Mathematics wiki is an open access resource designed specifically for the mathematics community. The original articles are from the online Encyclopaedia of Mathematics, published by Kluwer Academic Publishers in 2002. With more than 8,000 entries, illuminating nearly 50,000 notions in mathematics, the Encyclopaedia of Mathematics was the most up-to-date graduate-level reference work in the field of mathematics. Springer, in cooperation with the European Mathematical Society, has made the content of this Encyclopedia freely open to the public. It is hoped that the mathematics community will find it useful and will be motivated to update those topics that fall within their own expertise or add new topics enabling the wiki to become yet again the most comprehensive and up-to-date online mathematics reference work. The original articles from the Encyclopaedia of Mathematics remain copyrighted to Springer but any new articles added and any changes made to existing articles within encyclopediaofmath.org will come under the Creative Commons Attribution Share-Alike License . An editorial board , under the management of the European Mathematical Society, monitors any changes to articles and has full scientific authority over alterations and deletions. This wiki is a MediaWiki that uses the MathJax extension, making it possible to insert mathematical equations in TEXTEXand LATEXLATEX
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