Dimensional Real Vector Space Research Articles

A DIFFERENTIABLE STRUCTURE ON A FINITE DIMENSIONAL REAL VECTOR SPACE AS A MANIFOLD

There are three conditions for a topological space to be said a topological manifold of dimension : Hausdorff space, second-countable, and the existence of homeomorphism of a neighborhood of each point to an open subset of or -dimensional locally Euclidean. The differentiable structure is given if the intersection of two charts is an empty chart or its transition map is differentiable. In this article, we study a differentiable manifold on finite dimensional real vector spaces. The aim is to prove that any finite-dimensional vector space is a differentiable manifold. First of all, it is proved that a finite dimensional vector space is a topological manifold by constructing a norm as its topology. Given a metric which is induced by a norm. Two norms on a finite dimensional vector space are always equivalent and they are determine the same topology. Secondly, it is proved that the transition map in the finite dimensional vector space is differentiable. As conclusion, we have that any finite dimensional vector space with independent norm topology choice is a differentiable manifold. As a matter of discussion, it can be studied that the vector space of all linear operators of a finite dimensional vector space has a differentiable manifold structure as well.

Persistence and the Sheaf-Function Correspondence

Abstract The sheaf-function correspondence identifies the group of constructible functions on a real analytic manifold M with the Grothendieck group of constructible sheaves on M. When M is a finite dimensional real vector space, Kashiwara-Schapira have recently introduced the convolution distance between sheaves of $\mathbf {k}$ -vector spaces on M. In this paper, we characterize distances on the group of constructible functions on a real finite dimensional vector space that can be controlled by the convolution distance through the sheaf-function correspondence. Our main result asserts that such distances are almost trivial: they vanish as soon as two constructible functions have the same Euler integral. We formulate consequences of our result for Topological Data Analysis: there cannot exist nontrivial additive invariants of persistence modules that are continuous for the interleaving distance.

Ruled surfaces corresponding to hyper-dual curves

In this paper, we give the definition of the concept of unit hyper-dual sphere. We take a subset of this sphere and show that each curve on this subset represents two ruled surfaces in three dimensional real vector space such that these ruled surfaces have a common base curve and their rulings are perpendicular. Finally, we give some examples to illustrate the applications of our main results.

Bollobás-type theorems for hemi-bundled two families

Let {(Ai,Bi)}i=1m be a collection of pairs of sets with |Ai|=a and |Bi|=b for 1≤i≤m. Suppose that Ai∩Bj=0̸ if and only if i=j, then by the famous Two Families Theorem of Bollobás, we have the size of this collection m≤a+ba. In this paper, we consider a variant of this problem by requiring {Ai}i=1m to be intersecting additionally. Using exterior algebra method, we prove a weighted Bollobás-type theorem for finite dimensional real vector spaces under these constraints. As a consequence, we obtain a similar theorem for finite sets, which settles a recent conjecture of Gerbner et al. (2020). Moreover, we also determine the unique extremal structure of {(Ai,Bi)}i=1m for the primary case of the theorem for finite sets.

Exact dimensionality and Ledrappier-Young formula for the Furstenberg measure

Assuming strong irreducibility and proximality, we prove that the Furstenberg measure, corresponding to a finitely supported measure on the general linear group of a finite dimensional real vector space, is exact dimensional. We also establish a Ledrappier-Young type formula for its dimension. The general strategy of the proof is based on the argument given by Feng for the exact dimensionality of self-affine measures.

Ruled surfaces constructed by quaternions

In this paper, we define a quaternionic operator whose scalar part is a real parameter and vector part is a curve in three dimensional real vector space R3. We prove that quaternion product of this operator and a spherical curve represents a ruled surface in R3 if the vector part of the quaternionic operator is perpendicular to the position vector of the spherical curve. We express this surface as 2-parameter homothetic motion using the matrix representation of the operator. Furthermore, we define another quaternionic operator and show that each ruled surface in R3 can be obtained by this operator. Finally, we give the geometric interpretations of these operators with some examples.

Another Proof for the Continuity of the Lipsman Mapping

UDC 515.1 We consider the semidirect product $G = K \ltimes V$ where $K$ is a connected compact Lie group acting by automorphisms on a finite dimensional real vector space $V$ equipped with an inner product $\langle , \rangle$. By $\hat G $ we denote the unitary dual of $G$ and by ${\mathfrak{g}^{ \ddagger} /} G$ the space of admissible coadjoint orbits, where $\mathfrak{g}$ is the Lie algebra of $G$. It was pointed out by Lipsman that the correspondence between $\hat{G} $ and ${\mathfrak{g}^{ \ddagger} /} G$ is bijective. Under some assumption on $G$, we give another proof for the continuity of the orbit mapping (Lipsman mapping)$$\Theta : {\mathfrak{g}^{ \ddagger} /} G - \rightarrow \hat{G} .$$

Criteria for periodicity and an application to elliptic functions

AbstractLet P and Q be relatively prime integers greater than 1, and let f be a real valued discretely supported function on a finite dimensional real vector space V. We prove that if $f_{P}(x)=f(Px)-f(x)$ and $f_{Q}(x)=f(Qx)-f(x)$ are both $\Lambda $ -periodic for some lattice $\Lambda \subset V$ , then so is f (up to a modification at $0$ ). This result is used to prove a theorem on the arithmetic of elliptic function fields. In the last section, we discuss the higher rank analogue of this theorem and explain why it fails in rank 2. A full discussion of the higher rank case will appear in a forthcoming work.

Sharp sectional curvature bounds and a new proof of the spectral theorem

We algebraically compute all possible sectional curvature values for canonical algebraic curvature tensors, and use this result to give a method for constructing general sectional curvature bounds. We use a well-known method to geometrically realize these results to produce a hypersurface with prescribed sectional curvatures at a point. By extending our methods, we give a relatively short proof of the Spectral Theorem for self-adjoint operators on a finite dimensional real vector space.

ON UNIFORM CONTRACTIONS OF BALLS IN MINKOWSKI SPACES

Let $N$ balls of the same radius be given in a $d$-dimensional real normed vector space, i.e., in a Minkowski $d$-space. Then apply a uniform contraction to the centers of the $N$ balls without changing the common radius. Here a uniform contraction is a contraction where all the pairwise distances in the first set of centers are larger than all the pairwise distances in the second set of centers. The main results of this paper state that a uniform contraction of the centers does not increase (resp., decrease) the volume of the union (resp., intersection) of $N$ balls in Minkowski $d$-space, provided that $N\geq 2^d$ (resp., $N\geq 3^d$ and the unit ball of the Minkowski $d$-space is a generating set). Some improvements are presented in Euclidean spaces.

Weighted Lattice Point Sums in Lattice Polytopes, Unifying Dehn–Sommerville and Ehrhart–Macdonald

Let V be a real vector space of dimension n and let $$M\subset V$$ be a lattice. Let $$P\subset V$$ be an n-dimensional polytope with vertices in M, and let $$\varphi :V\rightarrow {\mathbb {C}}$$ be a homogeneous polynomial function of degree d. For $$q\in {\mathbb {Z}}_{>0}$$ and any face F of P, let $$D_{\varphi ,F} (q)$$ be the sum of $$\varphi $$ over the lattice points in the dilate qF. We define a generating function $$G_{\varphi }(q,y) \in {\mathbb {Q}}[q] [y]$$ packaging together the various $$D_{\varphi ,F} (q)$$ , and show that it satisfies a functional equation that simultaneously generalizes Ehrhart–Macdonald reciprocity and the Dehn–Sommerville relations. When P is a simple lattice polytope (i.e., each vertex meets n edges), we show how $$G_{\varphi }$$ can be computed using an analogue of Brion–Vergne’s Euler–Maclaurin summation formula.

Re-parameterizing and reducing families of normal operators

We present a new proof of results of Kurdyka & Paunescu, and of Rainer, about real-analytic multi-parameters generalizations of classical results by Rellich and Kato about the reduction in families of univariate deformations of normal operators over real or complex vector spaces of finite dimensions. Given a real analytic family of normal operators over a finite dimensional real or complex vector space, there exists a locally finite composition of blowings-up with smooth centers re-parameterizing the given family such that at each point of the source space of the re-parameterizing mapping, there exists a neighbourhood of any given point over which exists a real analytic orthonormal frame in which the pull back of the operator is in reduced form at every point of the neighbourhood. A free by-product of our proof is the local real analyticity of the eigen-values, which in all prior works was a prerequisite step to get local regular reducing bases.

Lipsman mapping and dual topology of semidirect products

We consider the semidirect product $G = K \ltimes V$ where $K$ is a connected compact Lie group acting by automorphisms on a finite dimensional real vector space $V$ equipped with an inner product $\langle,\rangle$. We denote by $\widehat{G}$ the unitary dual of $G$ (note that we identify each representation $\pi\in\widehat{G}$ to its classes $[\pi]$) and by $\mathfrak{g}^\ddag/G$ the space of admissible coadjoint orbits, where $\mathfrak{g}$ is the Lie algebra of $G.$ It was pointed out by Lipsman that the correspondence between $\mathfrak{g}^\ddag/G$ and $\widehat{G}$ is bijective. Under some assumption on $G,$ we prove that the Lipsman mapping \begin{eqnarray*} \Theta:\mathfrak{g}^\ddag/G &\longrightarrow&\widehat{G}\\ \mathcal{O}&\longmapsto&\pi_\mathcal{O} \end{eqnarray*} is a homeomorphism.

Invariant Whitney functions

Theorem 1 of [G. W. Schwarz, Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975) 63–68.] says that for a linear action of a compact Lie group [Formula: see text] on a finite dimensional real vector space [Formula: see text], any smooth [Formula: see text]-invariant function on [Formula: see text] can be written as a composite with the Hilbert map. We prove a similar statement for the case of Whitney functions along a subanalytic set [Formula: see text] fulfilling some regularity assumptions. In order to deal with the case when [Formula: see text] is not [Formula: see text]-stable, we use the language of groupoids.

On the essential spectrum of elliptic differential operators

Let A be a C⁎-algebra of bounded uniformly continuous functions on a finite dimensional real vector space X such that A is stable under translations and contains the continuous functions that have a limit at infinity. Denote A† the boundary of X in the character space of A. Then to each operator A in the crossed product A⋊X one may naturally associate a family of bounded operators Aϰ on L2(X) indexed by the characters ϰ∈A†. We show that the essential spectrum of A is the union of the spectra of the operators Aϰ. The applications cover very general classes of singular elliptic operators.

Inner structure in real vector spaces

Abstract Internal points were introduced in the literature of topological vector spaces to characterize the finest locally convex vector topology. In this manuscript we generalize the concept of internal point in real vector spaces by introducing a type of points, called inner points, that allows us to provide an intrinsic characterization of linear manifolds, which was not possible by using internal points. We also characterize infinite dimensional real vector spaces by means of the inner points of convex sets. Finally, we prove that in convex sets containing internal points, the set of inner points coincides with the one of internal points.

On almost polynomial structures from classical linear connections

Let \(\mathcal{M}f_m\) be the category of \(m\)-dimensional manifolds and local diffeomorphisms and let \(T\) be the tangent functor on \(\mathcal{M}f_m\). Let \(\mathcal{V}\) be the category of real vector spaces and linear maps and let \(\mathcal{V}_m\) be the category of \(m\)-dimensional real vector spaces and linear isomorphisms. Let \(w\) be a polynomial in one variable with real coefficients. We describe all regular covariant functors \(F\colon \mathcal{V}_m\to\mathcal{V}\) admitting \(\mathcal{M}f_m\)-natural operators \(\tilde{P}\) transforming classical linear connections \(\nabla\) on \(m\)-dimensional manifolds \(M\) into almost polynomial \(w\)-structures \(\tilde{P}(\nabla)\) on \(F(T)M=\bigcup_{x\in M}F(T_xM)\).<br /><br />

On almost complex structures from classical linear connections

Let \(\mathcal{M} f_m\) be the category of \(m\)-dimensional manifolds and local diffeomorphisms and let \(T\) be the tangent functor on \(\mathcal{M} f_m\). Let \(\mathcal{V}\) be the category of real vector spaces and linear maps and let \(\mathcal{V}_m\) be the category of \(m\)-dimensional real vector spaces and linear isomorphisms. We characterize all regular covariant functors \(F:\mathcal{V}_m\to\mathcal{V}\) admitting \(\mathcal{M} f_m\)-natural operators \(\tilde J\) transforming classical linear connections \(\nabla\) on \(m\)-dimensional manifolds \(M\) into almost complex structures \(\tilde J(\nabla)\) on \(F(T)M=\bigcup_{x\in M}F(T_xM)\).

On the essential spectrum of $N$-body Hamiltonians with asymptotically homogeneous interactions

We determine the essential spectrum of N-body Hamiltonians with 2-body (or, more generally, k-body) potentials that have radial limits at infinity. The classical N-body Hamiltonians appearing in the well known HVZ-theorem are a particular case of this type of potentials corresponding to zero limits at infinity. Our result thus extends the HVZ-theorem that describes the essential spectrum of the usual N-body Hamiltonians. More precisely, if the configuration space of the system is a finite dimensional real vector space X, then let e(X) be the C *-algebra of functions on X generated by the algebras C(X/Y), where Y runs over the set of all linear subspaces of X and C(X/Y) is the space of continuous functions on X/Y that have radial limits at infinity. This is the algebra used to define the potentials in our case, while in the classical case the C(X/Y) are replaced by C0 (X/Y). The proof of our main results is based on the study of the structure of the algebra e(X), in particular, we determine its character space and the structure of its cross-product e(X) := e(X) ⋊ X by the natural action τ of X on e(X). Our techniques apply also to more general classes of Hamiltonians that have a many-body type structure. We allow, in particular, potentials with local singularities and more general behaviours at infinity. We also develop general techniques that may be useful for other operators and other types of questions, such as the approximation of eigenvalues.

Largest Minimal Inversion-Complete and Pair-Complete Sets of Permutations

We solve two related extremal problems in the theory of permutations. A set 𝑄 of permutations of the integers 1 to 𝑛 is inversion-complete (resp., pair-complete) if for every inversion (𝑗, 𝑖), where 1 ≤ 𝑖 < 𝑗 ≤ 𝑛, (resp., for every pair (𝑖, 𝑗), where 𝑖 ≠ 𝑗) there exists a permutation in 𝑄 where 𝑗 is before 𝑖. It is minimally inversion-complete if in addition no proper subset of 𝑄 is inversion-complete; and similarly for pair completeness. The problems we consider are to determine the maximum cardinality of a minimal inversion-complete set of permutations, and that of a minimal pair-complete set of permutations. The latter problem arises in the determination of the Caratheodory numbers for certain abstract convexity structures on the (𝑛 − 1)-dimensional real and integer vector spaces. Using Mantel's Theorem on the maximum number of edges in a triangle-free graph, we determine these two maximum cardinalities and we present a complete description of the optimal sets of permutations for each problem. Perhaps surprisingly (since there are twice as many pairs to cover as inversions), these two maximum cardinalities coincide whenever 𝑛 ≥ 4.