M-dimensional Space Research Articles

Kinematic formulas for mean curvature powers of hypersurfaces and Hadwiger’s theorem in 𝑅²ⁿ

We first discuss the theory of hypersurfaces and submanifolds in the m-dimensional Euclidean space leading up to high dimensional analogues of the classical Euler’s and Meusnier’s theorems. Then we deduce the kinematic formulas for powers of mean curvature of the ( m − 2 ) (m - 2) -dimensional intersection submanifold S 0 ∩ g S 1 {S_0} \cap g{S_1} of two C 2 {C^2} -smooth hypersurfaces S 0 {S_0} , S 1 {S_1} , i.e., ∫ G ( ∫ S 0 ∩ g S 1 H 2 k d σ ) d g {\smallint _G}({\smallint _{{S_0} \cap g{S_1}}}{H^{2k}}d\sigma )dg . Many well-known results, for example, the C-S. Chen kinematic formula and Crofton type formulas are easy consequences of our kinematic formulas. As direct applications of our formulas, we obtain analogues of Hadwiger’s theorem in R 2 n {\mathbb {R}^{2n}} , i.e., sufficient conditions for one domain K β {K_\beta } to contain, or to be contained in, another domain K α {K_\alpha } .

Estimating correlation dimension from a chaotic time series: when does plateau onset occur?

Suppose that a dynamical system has a chaotic attractor A with a correlation dimension D 2. A common technique to probe the system is by measuring a single scalar function of the system state and reconstructing the dynamics in an m-dimensional space using the delay-coordinate technique. The estimated correlation dimension of the reconstructed attractor typically increases with m and reaches a plateau (on which the dimension estimate is relatively constant) for a range of large enough m values. The plateaued dimension value is then assumed to be an estimate of D 2 for the attractor in the original full phase space. In this paper we first present rigorous results which state that, for a long enough data string with low enough noise, the plateau onset occurs at m = Ceil( D 2), where Ceil( D 2), standing for ceiling of D 2, is the smallest integer greater than or equal to D 2. We then numerical examples illustrating the theoretical prediction. In addition, we discuss new findings showing how practical factors such as a lack of data and observational noise can produce results that may seem to be inconsistent with the theoretically predicted plateau onset at m = Ceil( D 2).

Analysis of function duplicating capabilities of fuzzy controllers

Abstract Fuzzy controllers have been successfully applied to many cases to which conventional control algorithms are difficult to be applied. Even though the representations and the processings of data and information in the fuzzy controller are quite different from other control algorithms, the information processing operation that it carries out is basically a function f:A ⊂ Rn→Rm, from a bounded subset A of n-dimensional Euclidean space to a bounded subset f[A] of m-dimensional Euclidean space, where n and m are the number of measured states and the number of control inputs of controlled system, respectively. Under the assumptions of Mamdani's direct reasoning method, and C.O.G. (center of gravity) defuzzification method, the fuzzy controllers are proven to perform the mappings of any given functions with appropriately defined fuzzy sets. The mapping capabilities of the fuzzy controllers are analyzed in detail for two cases; f:R1 → R1 and g:R2 → R1. Also, it will be shown that the results can be extended to multiple dimensional cases.

Melhor Estimativa da Posição de um Ponto Emissor no Espaço M-Dimensional e Determinação dos Dominios Ótimos de Busca

The best estimate of the position of an emitting point located in the m-dimensional space and conditional to the observation of the emitting point's propagation trajectories - added to gaussian distortion, is obtained. The analytical and matrix form solution developed is based on the implementation of the Maximum Likelihood Hypotesis Testing as one considers a continous m-dimensional field of hipotesis. Refering to the position of the emitting point in the m-dimensional space, gaussian conditional probability densities are derived. The optimum detection/search domains are also implemented, as junctions of the required probabilities (P) of detection or success. The results directly apply to the radio direction finding problem, electronic support measures (ESM) problem and search for submersed emitting objects.

Buffons Nadelproblem und verwandte probleme behandelt mit der Theorie der Gleichverteilung: Teil II

We denote with PC m the m-dimensional complex projective space, with U the unitary group acting on it with z i(j=0, 1,..., m) the homogenous coordinates of a point [z] of PC m and assume that the z i are normalized such that z 0z0 +...+z mzm=1. Furthermore we denote the U-invariant metric on PC m with d. We consider now a uniformly distributed sequence ([z] k ; k=1,2,...) of points on PC m and study the sequence (d l([z] k , [z]0)), l≥0, [z]0 a fixed point. We prove with the help of the theory of uniform distribution properties of this sequence. We consider furthermore a dual sequence suggested by the theory of H. Weyl and L. V. Ahlfors on meromorphic curves.

A note on redundancy averaging

It is well known that the performance of direction-of-arrival estimation and beamforming algorithms degrades in the presence of correlated signals. Since techniques like spatial and weighted smoothing, which were developed for decorrelating the signals, suffer from reduced effective aperture, several authors have recently proposed redundancy averaging as an alternative spatial averaging method. Considering a two-source signal model and the asymptotic case, the authors show analytically that this technique results in an eigenstructure which is inconsistent with that of the underlying signal model. In particular, the resulting signal subspace fills up the whole M-dimensional space, where M is the array size, and the resulting covariance matrix is not guaranteed to be nonnegative definite. It is shown that complete decorrelation is not possible with this method even if the array size is infinitely large.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Enumeration of subspaces by dimension sequence

Let F be the finite field with q elements and let V be an m-dimensional vector space over F. Fix a linear endomorphism A of V. Suppose that X ⊆ V is a subspace. Let X (0) = 0, X (1) = X, and X ( k) = X ( k − 1) + A k − 1 ( X) for k > 1. For k ⩾ 1 let j k be the dimension of the quotient space X (k) X (k−1) . The m-tuple j = ( j 1, j 2, …, j m ) is called the dimension sequence of the subspace X. In this paper we consider the problem of determining the numbers C(j) of subspaces of V which have dimension sequence j. We derive two surprisingly simple formulas. First, if A is a shift operator (nilpotent with 1-dimensional null space) then C(j)= ∏ k=2 m q j 2 k j k−1 j k . Second, if A is simple (no non-trivial invariant F-subspaces) then C(j)= q m−1 q j1−1 ∏ k=2 m q jk(jk−1) j k−1 j k . We known of no simple counting proof of these formulas. Our derivation finds the C(j)'s as the solution to a set of linear equations obtained with Möbius inversion on the lattice of subspaces of V.

Multicomponent formalism in the mean-field approximation: a geometric interpretation of Chebychev polynomials

A simple geometrical interpretation is given for the Chebychev polynomials used in the definition of the correlation functions for multicomponent Ising models. This interpretation is based on the fact that each chemical species is associated with one basis unit vector of the canonical reference frame of a M-dimensional space where M is the number of chemical components of the system. Under this scheme the Flinn operator attached to each lattice site is written simply as a scalar product from which multisite Flinn operators are easily derived. The method is exemplified in the framework of the cluster variation method approximation for the practical case of ternary systems.

Properties of solutions of linear functional-differential equations depending on a parameter

On the segment 0 ⩽t⩽1 we study the equation A(d/dt, ρ)x(t) + [F(ρ)x](t)=f(t), whereA (d/dt, ρ) x=x( n )+ρA1x(n−1 +...+ρ n A n x, the matrices A1,...,An are of size m × m, x is an unknown and f a given function with values in the m-dimensional space ℂ m , F(ρ) is a linear operator acting from a Holder space to a Lebesgue space of vectorfunctions with values in ℂm and depending on a complex parameter ρ. We find the set of those ρ at which a one-to-one correspondence is established between the solutions of the given equation and the solutions of the equation A(d/dt, ρ)x(t)=0.

Least-squares estimation of transformation parameters between two point patterns

In many applications of computer vision, the following problem is encountered. Two point patterns (sets of points) (x/sub i/) and (x/sub i/); i=1, 2, . . ., n are given in m-dimensional space, and the similarity transformation parameters (rotation, translation, and scaling) that give the least mean squared error between these point patterns are needed. Recently, K.S. Arun et al. (1987) and B.K.P. Horn et al. (1987) presented a solution of this problem. Their solution, however, sometimes fails to give a correct rotation matrix and gives a reflection instead when the data is severely corrupted. The proposed theorem is a strict solution of the problem, and it always gives the correct transformation parameters even when the data is corrupted.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Complete space-like hypersurfaces of a de Sitter space with constant mean curvature

Let Mf(c) be an m-dimensional connected semi-Riemannian manifold ofindex sand of constant curvature c, which is calledan indefinitespace form ofindex s or simply a space form according as s>0 or s=0. An m-dimensional space form of constant curvature cis only denoted by Mm(c). The study of hypersurfaces with constant mean curvature of Mn+＼c) was initiatedby Nomizu and Smyth [13], who proved some excellent results. Itis seen that a complete space-like hypersurface of a Minkowski space

Comparisons of the nonlinear dynamics of electroencephalograms under various task loading conditions: A preliminary report

Comparison of the characteristics of electroencephalogram (EEG) records treated as realizations from a nonlinear process were compared under four different conditions: eyes shut resting, and three silent observation instructions to predict the patterns of randomly generated lights which illuminated every 10 seconds. The correlation dimension of the EEG was calculated by a method involving finding the correlation integral in m-dimensional space, and found to show some variations within time series. The degree and directions of changes in the dimensionality of the process varied between observers and did not clearly confirm some earlier reported findings, but it is demonstrable that the measures of nonlinear brain dynamics can be correlated with psychological variables. Reasons for this are discussed.

A divisive scheme for constructing minimal spanning trees in coordinate space

An algorithm to generate a minimal spanning tree is presented when the nodes with their coordinates in some m-dimensional Euclidean space and the corresponding metric are given. This algorithm is tested on manually generated data sets. The worst case time complexity of this algorithm is O( n log 2 n) for a collection of n data samples.

The computational measurement of apparent motion: A recurrent pattern recognition strategy as an approach to solve the correspondence problem

In short, the model consists of a two-dimensional set of edge detecting units, modelled according to the zero-crossing detectors introduced first by Marr and Ullman (1981). These detectors are located peripherally in our synthetic vision system and are the input elements for an intelligent recurrent network. The purpose of that network is to recognize and categorize the previously detected contrast changes in a multi-resolution representation of the original image in such a manner that the original information will be decomposed into a relatively small number N of well-defined edge primitives. The advantage of such a construction is that time-consuming pattern recognition has no longer to be done on the originally complex motion-blurred images of moving objects, but on a limited number of categorized forms. Based on a number M of elementary feature attributes for each individual edge primitive, the model is then able to decompose each edge pattern into certain features. In this way an M-dimensional vector can be constructed for each edge. For each sequence of two successive frames a tensor can be calculated containing the distances (measured in M-dimensional feature space) between all features in both images. This procedure yields a set of K-1 tensors for a sequence of K images. After cross-correlation of all N x M feature attributes from image (i) with those from image (i + 1), where i = 1,...,K-1, probability distributions can be computed. The final step is to search for maxima in these probability functions and then to construct from these extremes an optimal motion field. A number of simulation examples will be presented.

On the asymptotic behavior of subharmonic functions with a regular mass distribution

0. Let f(z) be an entire function of order p and of completely regular growth for the proximate order p(r) [i, p. 47 of the Russian edition]. Then there exists a C~ outside of which in If(z)[ has a definite asymptotics, expressed in terms of the growth indicator of the function f(z). In [2, p. 37], this exceptional set has been characterized in an exhaustive manner in terms of the distribution of the zeros of f(z). In this paper these results are generalized to subharmonic functions in the m-dimensional (m~2) space R m. We mention that the nontriviality of such a generalization is connected first of all with the fact that the indicator of a function, subharmonic in R m, for m~3, is neither bounded nor continuous anymore; this implies the modification of the basic formulations in comparison with [2] and leads to the necessity of the improvement of the applied method. We recall the fundamental definitions and results. We shall denote by p, q .... ,PQ vectors from R m. If O is the origin, then r 0 = Jl OQ H, rp o = ii PQ I!. If u:Rm-+R, then we shall write u(O),u~), u(Rx), where llxll = 1 and Let u(rx) be a subharmonic function in R m and let M(r, u) be its maximum on the sphere S, = {rx: II xll = I}. We recall that the function u(rx) has normal type for the proximate order p(r) u6SH(p(O] [p(r) ~ p for r + ~] if the quantity a, : = l imsupM (r, u)r, t" r a a

Invariant recovery of curves in m-dimensional space from sparse data

We address the problem of reconstructing a smooth curve from sparse and noisy information that is invariant to the choice of the coordinate system. Tikhonov regularization is used to form a well-posed mathematical problem statement, and conditions for an invariant reconstruction are given. The resulting functional minimization problem is shown to be nonconvex. Approximations to the invariant functional are often used to form a convex problem that can be solved efficiently. Two common approximations, those of cubic and weighted cubic splines, are detailed, and examples are given to show that the approximations are often invalid. To form a valid approximation to the invariant functional we propose a two-step algorithm. The first step forms a piecewise-linear curve, which is invariant to the coordinate system. This piecewise-linear curve is then used to construct a parameterization of the curve for which we can make a valid approximation to the invariant functional. Examples are given to demonstrate the effectiveness of the algorithm, and two example applications for which the invariant property is important are given.

On the quantum cosmology of the superstring theory including the effects of higher-derivative terms

In a previous paper the quantum cosmology of the D-dimensional superstring theory was examined, retaining only the lowest-order terms in the effective lagrangian, namely the Ricci scalar R and the dilaton field π. The (anisotropic) mini-superspace approximation was employed, the interval being ds 2 = dt 2 − a 2(t) d x 2 − b 2(t) d y 2 , where t is comoving time, a( t) ≡ e α( t) is the scale factor of the M-dimensional physical space d x 2 and b(t) ≡ e β(t) is the scale factor of the N-dimensional internal space d y 2 , where D = M + N + 1. The Wheeler-DeWitt equation for the wave function of the universe Ψ was derived, and it was found that Ψ was a constant in the “no-scale” metric for which α = − β, provided that ( M − N) 2 = M + N (eq. (1)). This condition requires that ( D − 1) must be a perfect square. It allows the cases both of the bosonic string ( D = 26, M = 10, N = 15) and the superstring ( D = 10, M = 3, N = 6), as well as the original Kaluza-Klein theory ( D = 5, M = 3, N = 1). The analysis is extended here to include the effects of higher-derivative terms α′ R 2, where α′ is the string slope parameter (the string tension being T = 1 2 πα′ ). The Wheeler-DeWitt equation for the bosonic string and for the superstring is derived as a Schrödinger equation, and it is explained why no corresponding equation can be derived for the heterotic string, at this level of approximation. The question is raised, why the expansion of the universe at the onset of the classical era is anisotropic rather than isotropic. The Schrödinger equation contains a “potential” V(α, β; α dot , β dot ) , which, to lowest order in α′, vanishes in the “no-scale” metric, provided that eq. (1) holds, but is positive definite in the isotropic space-time, for which α = β. The anisotropic solution is thus associated with a lower energy, and may be favoured for this reason.

Spherically symmetric solutions in dimensionally reduced spacetimes

The author studies static solutions of the vacuum Einstein equations in D=m+n+2 dimensions which have the following symmetries: the solutions are spherically symmetric in (m+2) dimensions, (or more generally the m-sphere is replace by an arbitrary m-dimensional Einstein space), while the internal space is any arbitrary n-dimensional Einstein space. The global properties of all such solutions are derived by considering the equivalent dimensionally reduced system in (m+2) dimensions, and by using techniques from the theory of dynamical systems after a judicious choice of variables. Apart from the trivial case of the Schwarzschild solution with constant scalar field, all solutions are found to contain naked singularities or else not to be asymptotically flat, as would be expected from the 'no hair' theorems.

On the partitioning of squared Euclidean distance and its applications in cluster analysis

The partitioning of squared Eucliean distance between two vectors in M-dimensional space into the sum of squared lengths of vectors in mutually orthogonal subspaces is discussed and applications given to specific cluster analysis problems. Examples of how the partitioning idea can be used to help describe and interpret derived clusters, derive similarity measures for use in cluster analysis, and to design Monte Carlo studies with carefully specified types and magnitudes of differences between the underlying population mean vectors are presented. Most of the example applications presented in this paper involve the clustering of longitudinal data, but their use in cluster analysis need not be limited to this arena.

An angular momentum analysis of symmetric products of paired nucleon states

Pair correlated angular momentum projected standard Weyl tableau states spanning an m-dimensional paired shell-model space have been obtained for a system of 2N identical nucleons. A simple procedure has been developed for carrying out the restriction of the unitary group U(m) to the rotation group O(3) for configurations of both single and multilevel distributions of the single particle states. The mapping of the correlated pair configuration space onto the antisymmetric states allowed by Pauli principle is achieved using the particle antisymmetrizer. The procedure for determining the shell-model Hamiltonian matrix using the above basis is outlined.