VECTOR-VALUED FUZZY MULTIFUNCTIONS

... further implies that the fuzzy multifunction f is lower semicontinuous. fuzzy multifunction f is continuous. Theorem 5.6: Let fi:XY (i 1,2,3,...,n) be (single-valued) fuzzy functions from a fuzzy topological space X into a locally convex fuzzy topological vector space Y, and I:X--Y be given by f(x)- ...

... further implies that the fuzzy multifunction f is lower semicontinuous. fuzzy multifunction f is continuous. Theorem 5.6: Let fi:XY (i 1,2,3,...,n) be (single-valued) fuzzy functions from a fuzzy topological space X into a locally convex fuzzy topological vector space Y, and I:X--Y be given by f(x)- ...

Embeddings of compact convex sets and locally compact cones

... The general problem for cones is whether the topology of every locally compact cone C[τ] in a real vector space V - C - C can be extended to a locally convex separated topological vector space topology T* on V such that r* | C = r. The finite-dimensional case was disposed of by D. R. Brown and M. Fr ...

... The general problem for cones is whether the topology of every locally compact cone C[τ] in a real vector space V - C - C can be extended to a locally convex separated topological vector space topology T* on V such that r* | C = r. The finite-dimensional case was disposed of by D. R. Brown and M. Fr ...

Differential geometry with SageMath

... In SageManifolds, an arbitrary number of charts can be introduced To fully specify the manifold, one shall also provide the transition maps on overlapping chart domains (SageManifolds class CoordChange) ...

... In SageManifolds, an arbitrary number of charts can be introduced To fully specify the manifold, one shall also provide the transition maps on overlapping chart domains (SageManifolds class CoordChange) ...

as a PDF - Universität Bonn

... There are two definitions, one using index theory and the other one using elementary differential topology and linear algebra over F2 . We give an easy new proof of Atiyah’s theorem that any diffeomorphism of a surface fixes a spin structure (Corollary 4.1.21). In the section 4.2, we apply the gener ...

... There are two definitions, one using index theory and the other one using elementary differential topology and linear algebra over F2 . We give an easy new proof of Atiyah’s theorem that any diffeomorphism of a surface fixes a spin structure (Corollary 4.1.21). In the section 4.2, we apply the gener ...

A geometric introduction to K-theory

... of the higher terms as “corrections” to this initial term; in a certain sense these corrections get smaller as j increases (this is not obvious). An algebraist who looks at (1.5) will immediately notice some possible generalizations. The R/(f ) and R/(g) terms can be replaced by any finitely-generat ...

... of the higher terms as “corrections” to this initial term; in a certain sense these corrections get smaller as j increases (this is not obvious). An algebraist who looks at (1.5) will immediately notice some possible generalizations. The R/(f ) and R/(g) terms can be replaced by any finitely-generat ...

Projective Geometry on Manifolds - UMD MATH

... Other “weaker” geometries are obtained by removing some of these concepts. Similarity geometry is the geometry of Euclidean space where the equivalence relation of congruence is replaced by the broader equivalence relation of similarity. It is the geometry invariant under similarity transformations. ...

... Other “weaker” geometries are obtained by removing some of these concepts. Similarity geometry is the geometry of Euclidean space where the equivalence relation of congruence is replaced by the broader equivalence relation of similarity. It is the geometry invariant under similarity transformations. ...

Smooth Manifolds

... However, many (perhaps most) important applications of manifolds involve calculus. For example, applications of manifold theory to geometry involve such properties as volume and curvature. Typically, volumes are computed by integration, and curvatures are computed by differentiation, so to extend th ...

... However, many (perhaps most) important applications of manifolds involve calculus. For example, applications of manifold theory to geometry involve such properties as volume and curvature. Typically, volumes are computed by integration, and curvatures are computed by differentiation, so to extend th ...

pdf lecture notes

... Proof. It is straightforward to see that Cb (X) is a normed linear space. If {fn } is a Cauchy sequence in Cb (X), then there exists B such that for all n, ||fn || ≤ B. Thus any pointwise limit of {fn } is also bounded. The rest of this proof is just like for C(X). Definition 3.6 Let X be a topologi ...

... Proof. It is straightforward to see that Cb (X) is a normed linear space. If {fn } is a Cauchy sequence in Cb (X), then there exists B such that for all n, ||fn || ≤ B. Thus any pointwise limit of {fn } is also bounded. The rest of this proof is just like for C(X). Definition 3.6 Let X be a topologi ...

Manifolds and Topology MAT3024 2011/2012 Prof. H. Bruin

... An equivalence relation ∼ on a space X is a relation on a set which is 1. Reflexive: x ∼ x. 2. Symmetric: x ∼ y if and only if y ∼ x. 3. Transitive: x ∼ y and y ∼ z imply x ∼ z. The set {y ∈ X : y ∼ x} is the equivalence class of x; it is denoted as [x]. Definition 20 Given an equivalence relation ∼ ...

... An equivalence relation ∼ on a space X is a relation on a set which is 1. Reflexive: x ∼ x. 2. Symmetric: x ∼ y if and only if y ∼ x. 3. Transitive: x ∼ y and y ∼ z imply x ∼ z. The set {y ∈ X : y ∼ x} is the equivalence class of x; it is denoted as [x]. Definition 20 Given an equivalence relation ∼ ...

Solutions to homework problems

... We recall from Calculus that t → sintπt is a continuous function on R \ {0} which extends to a continuous function on all of R (by defining its value at t = 0 to be π). It follows that f is a continuous function since all its components are continuous. It’s easy to check that f (v) belongs to S n ⊂ ...

... We recall from Calculus that t → sintπt is a continuous function on R \ {0} which extends to a continuous function on all of R (by defining its value at t = 0 to be π). It follows that f is a continuous function since all its components are continuous. It’s easy to check that f (v) belongs to S n ⊂ ...

Lecture Notes (unique pdf file)

... space X is the only dense set in itself. If X has the discrete topology then every subset is equal to its own closure (because every subset is closed), so the closure of a proper subset is always proper. Conversely, if X is the only dense subset of itself, then for every proper subset A its closure ...

... space X is the only dense set in itself. If X has the discrete topology then every subset is equal to its own closure (because every subset is closed), so the closure of a proper subset is always proper. Conversely, if X is the only dense subset of itself, then for every proper subset A its closure ...

HOMEWORK 1 – SOLUTION

... 4. Find a vector with the magnitude 4 in the same direction as the vector v = 2i − j . Answer. Let u be such a vector, i.e., the one with the magnitude 4 in the same direction as the vector v = 2i − j . Since u is parallel to v , there should be a positive scalar s (negative scalar gives the opposit ...

... 4. Find a vector with the magnitude 4 in the same direction as the vector v = 2i − j . Answer. Let u be such a vector, i.e., the one with the magnitude 4 in the same direction as the vector v = 2i − j . Since u is parallel to v , there should be a positive scalar s (negative scalar gives the opposit ...

Locally Convex Vector Spaces III: The Metric Point of View

... But (∗∗) is clearly true, since all p ∈ P are T0 -continuous (so the B(p)’s are in fact open neighborhoods of 0.) Remark 8. With note notations as above, if we consider, for every p ∈ P, ε > 0, the set Wpε = {x ∈ X : p(x) ≤ ε} = εB(p), then the collection W = {Wpε : p ∈ P, ε > 0} also constitutes a ...

... But (∗∗) is clearly true, since all p ∈ P are T0 -continuous (so the B(p)’s are in fact open neighborhoods of 0.) Remark 8. With note notations as above, if we consider, for every p ∈ P, ε > 0, the set Wpε = {x ∈ X : p(x) ≤ ε} = εB(p), then the collection W = {Wpε : p ∈ P, ε > 0} also constitutes a ...

# Vector field

In vector calculus, a vector field is an assignment of a vector to each point in a subset of space. A vector field in the plane, for instance, can be visualized as a collection of arrows with a given magnitude and direction each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from point to point.The elements of differential and integral calculus extend to vector fields in a natural way. When a vector field represents force, the line integral of a vector field represents the work done by a force moving along a path, and under this interpretation conservation of energy is exhibited as a special case of the fundamental theorem of calculus. Vector fields can usefully be thought of as representing the velocity of a moving flow in space, and this physical intuition leads to notions such as the divergence (which represents the rate of change of volume of a flow) and curl (which represents the rotation of a flow).In coordinates, a vector field on a domain in n-dimensional Euclidean space can be represented as a vector-valued function that associates an n-tuple of real numbers to each point of the domain. This representation of a vector field depends on the coordinate system, and there is a well-defined transformation law in passing from one coordinate system to the other. Vector fields are often discussed on open subsets of Euclidean space, but also make sense on other subsets such as surfaces, where they associate an arrow tangent to the surface at each point (a tangent vector). More generally, vector fields are defined on differentiable manifolds, which are spaces that look like Euclidean space on small scales, but may have more complicated structure on larger scales. In this setting, a vector field gives a tangent vector at each point of the manifold (that is, a section of the tangent bundle to the manifold). Vector fields are one kind of tensor field.