On condition numbers for the canonical generalized polar
... an orthosymmetric pair. Then A ∈ Rm×n has a unique canonical generalized polar decomposition A = W S with S being nonsingular if and only if A⋆M,N A has no nonpositive eigenvalues. In the rest of this paper, we always assume that K = R and A⋆M,N A has no nonpositive eigenvalues when we refer to the ...
... an orthosymmetric pair. Then A ∈ Rm×n has a unique canonical generalized polar decomposition A = W S with S being nonsingular if and only if A⋆M,N A has no nonpositive eigenvalues. In the rest of this paper, we always assume that K = R and A⋆M,N A has no nonpositive eigenvalues when we refer to the ...
Some Generalizations of Mulit-Valued Version of
... Q : X → P cl,cv ( X ) be a closed and k-set-contraction. Then Q has a fixed point. Before stating the generalization of Theorem 2.1of Sadovskii [14] type, we give a useful definition. Definition 2.6. A multi-valued mapping Q : E → P (E ) is called µ-condensing if for any bounded subset A of E , Q ( ...
... Q : X → P cl,cv ( X ) be a closed and k-set-contraction. Then Q has a fixed point. Before stating the generalization of Theorem 2.1of Sadovskii [14] type, we give a useful definition. Definition 2.6. A multi-valued mapping Q : E → P (E ) is called µ-condensing if for any bounded subset A of E , Q ( ...
Lesson 7.6 Properties of Systems of Linear Equations Exercises
... The coefficient of x is 1, so the slope is 1. ii) –x – y = 10 –y = x + 10 y = –x – 10 The coefficient of x is –1, so the slope is –1. iii) –2x + 2y = 10 2y = 2x + 10 y=x+5 The coefficient of x is 1, so the slope is 1. iv) x + y = 5 y = –x + 5 The coefficient of x is –1, so the slope is –1. b) Parall ...
... The coefficient of x is 1, so the slope is 1. ii) –x – y = 10 –y = x + 10 y = –x – 10 The coefficient of x is –1, so the slope is –1. iii) –2x + 2y = 10 2y = 2x + 10 y=x+5 The coefficient of x is 1, so the slope is 1. iv) x + y = 5 y = –x + 5 The coefficient of x is –1, so the slope is –1. b) Parall ...
Frölicher versus differential spaces: A Prelude to Cosmology
... Proof. We use the notation in the definition of differential space. Let c : IR → X be a contour. As hi ◦ c is smooth, so is g|Ui ◦ c|c−1 (Ui ) . Since the sets c−1 (Ui ) for i ∈ K cover IR, g ◦ c must be smooth. But, since (X, C, F) is a Frölicher space, g ∈ F . Suppose that c is again a contour. S ...
... Proof. We use the notation in the definition of differential space. Let c : IR → X be a contour. As hi ◦ c is smooth, so is g|Ui ◦ c|c−1 (Ui ) . Since the sets c−1 (Ui ) for i ∈ K cover IR, g ◦ c must be smooth. But, since (X, C, F) is a Frölicher space, g ∈ F . Suppose that c is again a contour. S ...
Complex Numbers
... It can be shown with algebraic tools that the only n dimensional vector space which is a field is C, i.e., it can be shown that for n ≥ 3 no multiplication exists to obtain a field. Euler introduced the imaginary unit as i := (0, 1)T . This leads to i2 = (0, 1)T · (0, 1)T = (−1, 0)T Furthermore e = ...
... It can be shown with algebraic tools that the only n dimensional vector space which is a field is C, i.e., it can be shown that for n ≥ 3 no multiplication exists to obtain a field. Euler introduced the imaginary unit as i := (0, 1)T . This leads to i2 = (0, 1)T · (0, 1)T = (−1, 0)T Furthermore e = ...
1 Weakly Perfect Generalized Ordered Spaces by Harold R Bennett
... subset X × {0} contains a dense subset S that is a Gδ -set in X ∗ , say S = {G(n) : n ≥ 1}. Then S would necessarily be a subset of the set of end points of the convex components of the sets G(n), so that S would be countable. But that is impossible because X × {0} is a topological copy of the Sorge ...
... subset X × {0} contains a dense subset S that is a Gδ -set in X ∗ , say S = {G(n) : n ≥ 1}. Then S would necessarily be a subset of the set of end points of the convex components of the sets G(n), so that S would be countable. But that is impossible because X × {0} is a topological copy of the Sorge ...
Stable isomorphism and strong Morita equivalence of C*
... obtained by completing X with respect to the norm \\(x9 %}B\\U2> as discussed in Proposition 2.10 of [8] and Proposition 3.1 of [11]. If X and Y are E — J5-imprimitivity bimodules, we say that X and Y are equivalent if their Hausdorff completions are isomorphic as E — JS-imprimitivity bimodules. In ...
... obtained by completing X with respect to the norm \\(x9 %}B\\U2> as discussed in Proposition 2.10 of [8] and Proposition 3.1 of [11]. If X and Y are E — J5-imprimitivity bimodules, we say that X and Y are equivalent if their Hausdorff completions are isomorphic as E — JS-imprimitivity bimodules. In ...
NOTES FOR MATH 4510, FALL 2010 1. Metric Spaces The
... The function d is called the metric, it is also called the distance function. 1.1. Examples of metric spaces. We now give examples of metric spaces. In most of the examples the conditions (1) and (2) of Definition 1.1 are easy to verify, so we mention these conditions only if there is some difficult ...
... The function d is called the metric, it is also called the distance function. 1.1. Examples of metric spaces. We now give examples of metric spaces. In most of the examples the conditions (1) and (2) of Definition 1.1 are easy to verify, so we mention these conditions only if there is some difficult ...
DEHN FUNCTION AND ASYMPTOTIC CONES
... 3.B. Subgroups of G with contracting elements. An easy way to prove quadratic filling is the use of elements whose action by conjugation on the unipotent part is contracting. Although G itself does not contain such elements, we will show that it contains large enough such subgroups. More precisely, ...
... 3.B. Subgroups of G with contracting elements. An easy way to prove quadratic filling is the use of elements whose action by conjugation on the unipotent part is contracting. Although G itself does not contain such elements, we will show that it contains large enough such subgroups. More precisely, ...
lecture notes as PDF
... • the rationals (Q, +, ∗), reals (R, +, ∗) and complex numbers (C, +, ∗) form fields; • the set of 2 × 2 matrices with real entries forms a non-commutative ring with identity w.r.t. matrix addition and multiplication. • the group Zn with addition as before and multiplication defined by [a][b] := [ab ...
... • the rationals (Q, +, ∗), reals (R, +, ∗) and complex numbers (C, +, ∗) form fields; • the set of 2 × 2 matrices with real entries forms a non-commutative ring with identity w.r.t. matrix addition and multiplication. • the group Zn with addition as before and multiplication defined by [a][b] := [ab ...
Finite Fields
... • the rationals (Q, +, ∗), reals (R, +, ∗) and complex numbers (C, +, ∗) form fields; • the set of 2 × 2 matrices with real entries forms a non-commutative ring with identity w.r.t. matrix addition and multiplication. • the group Zn with addition as before and multiplication defined by [a][b] := [ab ...
... • the rationals (Q, +, ∗), reals (R, +, ∗) and complex numbers (C, +, ∗) form fields; • the set of 2 × 2 matrices with real entries forms a non-commutative ring with identity w.r.t. matrix addition and multiplication. • the group Zn with addition as before and multiplication defined by [a][b] := [ab ...
The space of sections of a sphere-bundle I
... For example if 0 is the zero vector bundle over X then [0 + ;C + ] x is n0T(,+, since a fibrewise pointed map 0 + ->{ + is determined by its restriction to 0 and this is simply a section of (+. Similarly nor(X, y;C+) = [0 + ;C] ( A : . y ) where r(X, Y;(+) is as in (3.2). Now we stabilize [Qo; Q{]B. ...
... For example if 0 is the zero vector bundle over X then [0 + ;C + ] x is n0T(,+, since a fibrewise pointed map 0 + ->{ + is determined by its restriction to 0 and this is simply a section of (+. Similarly nor(X, y;C+) = [0 + ;C] ( A : . y ) where r(X, Y;(+) is as in (3.2). Now we stabilize [Qo; Q{]B. ...
Basis (linear algebra)
Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.A set of vectors in a vector space V is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set. In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space V, every element of V can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.