Dimension Dim Research Articles

On the Riemannian metric on the space of density matrices

The concept of purification of mixed states of quantum systems leads naturally to an interesting Riemannian metric on the space of normalized density matrices D n . More precisely, this Riemannian metric introduced by Uhlmann is defined on submanifolds (of all hermitean matrices) D nk of density matrices of fixed rank k, but not for vectors transversal to this submanifolds (at least using definitions known in the literature). A natural question is, whether there exists a manifold M of dimension dim D nn = n 2 − 1, which contains ∪ k = 1 n D nk as a topological subspace such that D nk , k = 1,…, n, is isometrically embedded. For n = 2 this is true, as Uhlmann observed. We show that for higher n such a manifold does not exist, because the sectional curvature at ρ ϵ D nn diverges if ρ tends to a state of rank less than n − 1. Roughly speaking, the metrics on the manifolds D nk cannot be glued together, because D nn contains geodesically complete submanifolds which look like conical singularities in the neighbourhood of ∂D nn .

Complexes that Arise in Cohomological Dimension Theory: A Unified Approach

Let π: P ˜ → P be a combinatorial map (that is, π−1(L) is a subcomplex of P ˜ whenever L is a subcomplex of P) between CW-complexes. A map f: X → P is said to approximately lift with respect to π provided that there is a map f ˜ : X → P ˜ such that, for each x€ X, there is a cell in P containing both πo f ˜ (x) andf(x). A characteristic property of a compact metric space X having covering dimension dim X ⩽ n is that each map from X to a CW-complex P has an approximate lift with respect to the inclusion P(n) ↪ P. An analogous characterization of compacta X having integral cohomological dimension dimz X ⩽ n emerged from work of R. D. Edwards [12] and was introduced in [19]. Complexes and maps π: EWz(P,n) → P are associated to each simplicial complex P so that a compactum X has dimz X ⩾ n if and only if every map f : X → P to a simplicial complex P can be approximately lifted to EWz(P,n). These complexes provide a l‘combinatorial’ approach to cohomological dimension theory that has supported many of the recent developments in the area. Historically, cohomological dimension theory with respect to groups other than Z has provided computational machinery for determining covering dimension. Hence, it is not surprising that it has been useful to consider comparable complexes EWG(L, n) for other groups G. The goal of this paper is to present a unified exposition of these complexes. As an application, they are used to provide an alternative construction to that of Dranishnikov [4, 6] of compact metric spaces realizing the Bockstein functions.

Computing View Graphs of Algebraic Surfaces

The "bad" directions or centres of projection, which yield degenerate projections of a smooth surface S embedded in 3-space, lie on a bifurcation set B of positive codimension in view space V (where V = P2 or R3 \\ S). The connected components of V \\ B are the nodes in the view graph of S, and two nodes are connected by an edge if the corresponding components are separated by a branch of B of dimension dim V - 1. The view graph of an algebraic surface of degree d has at most O(d10dim V) nodes. We describe an algorithm for computing the view graphs of surfaces defined as zero sets of polynomials with rational coefficients and present some examples.

Corrections to "On Changing Fixed Points and Coincidences to Roots"

We are grateful to Professor Boju Jiang who pointed out to us after we sent the proofs to the editor that Theorem 2.3 of [1] is false in general. The main results, however, remain valid provided we give an alternate argument to replace the role of 2.3 played in Lemma 3.1 of [1]. Let f, g: M --+ M be two maps of a compact connected nilmanifold. Recall from the proof of [1, 3.2] that we may assume without loss of generality that f and g are fiber preserving maps of the principal T-bundle T -+ M P N where T is a torus and N is a compact connected nilmanifold of dimension dim N < dim M. In order to complete the proof of [1, 3.1], it suffices to use induction on dim M together with the following

Compactifications and cohomological dimension

The following theorems follow from results proved in the paper: Theorem 1. For each Abelian group G≠0 there is a separable metric space X such that dim GX≤3 and all Hausdorff compactifications X′ of S n× X, n≥0, have cohomological dimension dim GX′ strictly greater than dim G(S n×X). Theorem 2. If G≠0 is either a torsion group or G is torsion free and l={p|p· G=G, p prime} is infinite, then there is a separable metric space X such that dim GX=2 and all Hausdorff compactifications X′ of S n×X, n≥0, have cohomological dimension dim GX′; strictly greater than dim G(S n×X).

Comparison of algorithms calculating optimal embedding parameters for delay time coordinates

The success of noise reduction processes and forecasting methods as well as accurate calculations of fractal dimensions and Lyapunov exponents from scalar time series depends strongly on the quality of the reconstructed phase space. We present a comparison of four methods, which simultaneously estimate the two embedding parameters (delay time τ and sufficiently large embedding dimension dim E) for Takens' delay time coordinates. Recently we introduced a global geometrical measure, the fill-factor, and a local dynamical measure, the averaged integral local deformation. We will discuss these methods briefly and compare them to the topological wavering-product. In addition a simple algorithm is presented, which observes the spreading of trajectories at the transition from dim E to dim E + 1 and gives the opportunity to estimate the correlation entropy K 2. We applied these algorithms to experimental data series measured from a rotational Taylor-Couette experiment and verified the quality of the obtained embedding parameters by the estimation of correlation dimensions for different reconstructed attractors.

On weakly infinite-dimensional subspaces

We will construct weakly infinite-dimensional (in the sense of Y. Smirnov) spaces X and Y such that Y contains X topologically and $dim Y = ω_0$ and $dim X = ω_0 + 1$. Consequently, the subspace theorem does not hold for the transfinite dimension dim for

Approximate resolutions and covering dimension

It is proven that a topological space X has covering dimension dim X⩽ n if and only if X admits an approximate resolution of polyhedra X a with dim X a ⩽ n.

Imbeddings into topological groups preserving dimensions

We give a negative answer to the following question of Bel'nov: Can every Tychonoff space X be imbedded as a subspace of a topological group G so that dim G ⩽ dim X? We show that if n ≠ 0, 1, 3, 7, then the n-dimensional sphere S n cannot be imbedded into an n-dimensional topological group G (no matter which dimension function, ind, Ind or dim, is considered). However, in case dim X = 0 the answer to Bel'nov's question is “yes”. We prove that, for every Tychonoff space X, dim X =0 implies (in fact, equivalent to) dim F ∗(X) = 0 and dim A ∗(X) = 0 , where F ∗(X) (A ∗(X)) is the free precompact (Abelian) group of X. As a corollary we obtain that every precompact group G is a quotient group of a precompact group H such that dim H = 0 and w( H) = w( G). A complete metric space X 1 and a pseudocompact Tychonoff space X 2 are constructed such that ind X i = 0, while ind F ∗(X i) ≠ 0 and ind A ∗(X i) ≠ 0 (i = 1, 2) . The equivalence of ind G = 0 and dim G = 0 for a precompact group G is established. We prove that dim H ⩽ dim G whenever H is a precompact subgroup of a topological group G. We also show that for every Tychonoff topology T on a set X with ind(X, T) = 0 one can find a precompact Hausdorff group topology T ̃ on the free (Abelian) group G( X) of X such that w(G(X), T ̃ ) = w(X, T), T ̃ | x = T and dim(G(X), T ̃ ) = 0 .

On the Order Dimension of Convex Polytopes

Given a convex polytope P , how does the usual affine dimension dim p ( P ) correlate with dim O ( L ( P )), the order dimension of the face lattice of P ? It will be shown that dim P ( P ) + 1 ⩽ dim O ( P )), that equality holds for some of the standard examples, but that the order dimension of cyclic polytopes C ( n , d ) can be arbitrarily large for fixed d ⩾ 4. Using the Four Colour Theorem it is easy to show that a simple 3-dimensional convex polytope has order dimension at most 8.

Towards a classification of transitive group actions on finite metric spaces

Based on J. R. Isbell's “completion” of metric spaces (cf. Comment. Math. Helv. 39 (1964–1965), 65–74) the combinatorial dimension dim comb X of a metric space X has been defined in [A. W. M. Dress, Adv. in Math. 53(3)(1984), 321–402] and shown to satisfy dim comb X ⩽ ⌊ #X 2 ⌋ . In this paper finite metric spaces X with a transitive group G of isometries which satisfy in addition dim comb X = ⌊ #X 2 ⌋ are investigated. It is shown that—for almost trivial reasons—in case # X ≡ 0(2) there exists a fixed point free involution on X which commutes with the action of G and that vice versa any finite transitive G-space admitting a fixed point free G-automorphism of order 2 carries a G-invariant metric satisfying the above equality, while for not so trivial reasons in case # X ≡ 1(2) the group G must be either cyclic of order # X or dihedral of order 2 · # X. As a corollary the Thompson-Feit Theorem turns out to be actually equivalent to its simple consequence that any finite simple group acts transitively as a group of isometries on some finite metric space X satisfying dim comb X = ⌊ #X 2 ⌋ .

Smooth social choice

This paper reviews recent results on the analysis of voting rules in the discrete, continuous and smooth categories. A voting rule D is identified with a class of decisive coalitions. When the set of alternatives W has finite cardinality | W|, then there exists an integer [the Nakamura number, υ( D )] such that D has an equilibrium (or core) iff |W | ⩽ υ( D) − 1 . In the continuous or smooth categories, when W is a space of dimension dim( W), then D has a core iff dim(W) ⩽ υ( D) − 2 . For the smooth category, there is an integer [the instability dimension w( D )] which classifies D in the following way: for almost all smooth profiles it is the case that (i) when dim(W) ⩾ w( D) , the core is empty, (ii) when dim(W) ⩾ w( D) + 1 , then cycles fill almost all of the space W. In the latter case, voting outcomes can be expected to depend on accidental details of the political process. The theory is used to examine the possibility of stability in multiparty coalition systems.

Factorization theorems for cohomological dimension

The well-known factorization theorems for covering dimension dim and compact Hausdorff spaces are here established for the cohomological dimension dim Z using a new characterization of dim Z In particular, it is proved that every mapping f: X → Y from a compact Hausdorff space X with dim Z X ⩽ n to a compact metric space Y admits a factorization f = hg, where g: X → Z, h: Z → Y and Z is a metric compactum with dim Z Z ⩽ n . These results are applied to the well-known open problem whether dim X = dim Z X . It is shown that the problem has a positive answer for compact Hausdorff spaces X if and only if it has a positive answer for metric compacta X.

Inzidenzr�ume mit S�ss'scher Kollineationsgruppe

The following theorem will be proved: THEOREM. Let (P, ) be a linear space with dimension dim(P, ) ≥ 2, such that each line contains at least three points. Let further Γ ≤ Aut(P, ) be a two — transitive group of collineations on the point set P, such that every automorphism γ e Γ fixing one flag (p,L), fixes all lines through p. Then (P, ) is a desarguesian affine space and Γ contains the whole group of dilatations of (P, ). Moreover one can define an addition and a multiplication on the point set, such that (P,+,-) is a nearfield, the line\(K: = \overline {0,1}\) joining the neutral elements is a normal subfield and Γ consists of the affine transformations x → a + mx, m≠ 0 and = Γ(K).

The Hausdorff dimension of the limit set of a geometrically finite Kleinian group

We consider in this paper groups of MObius transformations of l~=RtJ {zc}. Such a group is called Kleinian if it acts discontinuously somewhere in 1~. The action of G extends to the (n+ D-dimensional hyperbolic space H-I=R• zc) and G is geometrically finite if there is a hyperbolic fundamental polyhedron with a finite number of faces for the action of G in H ' ' t (for a more precise definition see [15. 1 B]). We prove in this paper that the Hausdor[f dimension dim~L(G) of the limit set L(G) of a geometrically finite Kleinian group G of 1~ is less than n (Theorem D). Our proof of this theorem is based on the following obser~'ation, Assume that G is of compact t3'pe, i.e. if IzI-I=H-IuI~ ''. then ((-I-~\L(G))/G is compact. Then there is an integer such that if we divide any n-cube Q of R into q equal subcubes, then at least one of these subcubes does not touch L(G). Let L4~Q) be the family of these subcubes which touch L(G). Then the n-measures of Q' E,Le(Q) do not add up to the n-measure of Q and we get

Some consequences of 𝑑𝑖𝑚𝑝𝑟𝑜𝑗Ω(𝐴)&lt;∞

Let X X be an affine variety over a field k k and x x a point on X X . We are interested in relating the properties of Ω ( X ) x \Omega {(X)_x} , the Kähler module of differentials of x x , with geometric properties of X X at x x . Lipman has given necessary and sufficient conditions for Ω ( X ) x \Omega {(X)_x} to be respectively torsion free and reflexive in the case where X X is locally a complete intersection at x x . We give a generalization of these results for the case where the projective dimension (dim proj) of Ω ( X ) x \Omega {(X)_x} is finite.