M-dimensional Space Research Articles

BOOST INVARIANT SURFACES WITH POINTWISE 1-TYPE GAUSS MAP IN MINKOWSKI 4-SPACE E41

Abstract. In this paper, we study spacelike rotational surfaces whichare called boost invariant surfaces in Minkowski 4-space E 41 . We givenecessary and sufficient condition for flat spacelike rotational surface tohave pointwise 1-type Gauss map. Also, we obtain a characterization forboost invariant marginally trapped surface with pointwise 1-type Gaussmap. 1. IntroductionThe notion of finite type mapping was introduced by B. Y. Chen in late1970’s. A smooth map φ from a compact Riemannian manifold M into m-dimensional Euclidean space E m is said to be of finite type if it is a finitesum of eigenfunctions of Laplacian ∆ of M. More precisely, the smooth mapφ : M → E m is of finite type if it can be expressed as a finite sumφ = φ 0 +X ki=1 φ i ,where φ 0 is a constant map, φ 1 ,...,φ k are non-constant maps such that ∆φ i =λ i φ i , λ i ∈ R, i = 1,2,...,k. If λ 1 ,λ 2 ,...,λ k are all different, then the map φis said to be of k-type.Similarly, an isometric immersion x of a submanifold in the Euclidean spaceE

ANTHROPOMETRIC PARAMETERS OF STUDENTS IN RURAL AREAS OF KHANTY-MANSIYSK AUTONOMOUS OKRUG - YUGRA

There has been held an anthropometric study of 322 pupils, the Middle Ob natives, who permanently lived in the rural areas of the Surgut region of the Khanty-Mansiysk Autonomus Okrug - Ugra. The results were analyzed with use of the standard methods of mathematical statistics and the author’s program “The Identification of parameters of behavior attractors of the biosystems state vector in the m-dimensional phase space”, designed for study of systems with chaotic organization. The complex analysis of the key indices of physical development has revealed a number of differences reflecting the process of formation of opposite-sex somatic types in the initial stage of phylogenesis with formation of new adaptive signs in modern economic and environmental conditions. The age-related correspondence of the main anthropometric parameters with some enlargement of the dimensional characteristics, which was revealed in the examination of pupils, can be considered as a fortunate trend which reduces the risk of functional deviations in healthy children of Ugra.

Dimensionality of social networks using motifs and eigenvalues.

We consider the dimensionality of social networks, and develop experiments aimed at predicting that dimension. We find that a social network model with nodes and links sampled from an m-dimensional metric space with power-law distributed influence regions best fits samples from real-world networks when m scales logarithmically with the number of nodes of the network. This supports a logarithmic dimension hypothesis, and we provide evidence with two different social networks, Facebook and LinkedIn. Further, we employ two different methods for confirming the hypothesis: the first uses the distribution of motif counts, and the second exploits the eigenvalue distribution.

Assessing the convolutedness of multivariate physiological time series.

A feature of time-series variability that may reveal underlying complex dynamics is the degree of "convolutedness". For multivariate series of m components, convolutedness can be defined as the propensity of the trail of the time-series samples to fill the m-dimensional space. This work proposes different convolutedness indices and compare them on synthesized and real physiological signals. The indices are based on length L and planar extension d of the trail in m dimensions. The classical ones are: the L/d ratio, and the Mandelbrot's fractal dimension (FD) of a curve: FDM =log(L)/log(d). In this work we also consider a correction of the Katz's estimator of FDM, i.e., FDKC =log(N)/(log(N)+log(d/L)), with N the number of samples; and FDMC, an estimator of FDM based on FDKC calculated over a shorter running window Nw<;N appropriately selected to reduce estimation bias. Synthesized fractional Brownian motions indicated that all the indices increase with FD, but differ for other aspects, namely the dependence on N; the capacity to estimate FD; or to distinguish between true bivariate and degenerate bivariate time series. Application on real multivariate recordings of muscular activity before and after exercise-induced fatigue suggests that these indices can be profitably used to identify complex changes in the dynamics of physiological signals.

Nonlinear prediction of river flow in different watershed acreage

Analysis of river flow is important in water resources management. For this purpose, river flow data at three measurement stations, Lubok Paku at Pahang River, Ladang Victoria at Muda River and Tanjung Tualang at Kinta River in Peninsular Malaysia have been studied using chaos theory. The analysis involved rivers which are located at different watershed acreage. Pahang River watershed are categorized as large, Muda River as medium and small for Kinta River based on watershed area. The analysis shows that the size of a large watershed has high average annual river flow while a small acreage watershed has low average annual river flow. Referring to the matter, analysis and prediction of river flow was carried out using nonlinear prediction method based on chaos theory. This method involves the reconstruction of phase space and prediction. Reconstruction of phase space has been performed by reconstruction of a single variable (river flow data) in m-dimensional phase space. Then, the prediction is done by using local linear approximation method. Overall, the prediction result is closed to agreement for all stations with high correlation coefficient. These results proved that nonlinear prediction method is suitable for river flow prediction of rivers that located in the area of different watershed acreage which is significant with different average annual flow. The results also indicate that the presence of chaotic behavior of the system as well as more than six variables affected the dynamics of the Pahang River, Muda River and the Kinta River.

K-Capture in multiagent pursuit evasion, or the lion and the hyenas

We consider the following generalization of the classical pursuit–evasion problem, which we call k-capture. A group of n pursuers (hyenas) wish to capture an evader (lion) who is free to move in an m-dimensional Euclidean space, the pursuers and the evader can move with the same maximum speed, and at least k pursuers must simultaneously reach the evaderʼs location to capture it. If fewer than k pursuers reach the evader, then those pursuers get destroyed by the evader. Under what conditions can the evader be k-captured? We study this problem in the discrete time, continuous space model and prove that k-capture is possible if and only if there exists a time when the evader lies in the interior of the pursuersʼ k-Hull. When the pursuit occurs inside a compact, convex subset of the Euclidean space, we show through an easy constructive strategy that k-capture is always possible.

Uniformly invariant normed spaces

In this work, we introduce the concepts of compactly invariant and uniformly invariant. Also we define sometimes C-invariant closed subspaces and then prove every m-dimensional normed space with m &gt; 1 has a nontrivial sometimes C-invariant closed subspace. Sequentially C-invariant closed subspaces are also introduced. Next, An open problem on the connection between compactly invariant and uniformly invariant normed spaces has been posed. Finally, we prove a theorem on the existence of a positive operator on a strict uniformly invariant Hilbert space. DOI: http://dx.doi.org/10.3126/bibechana.v10i0.7555 BIBECHANA 10 (2014) 31-33

Heat Content and Inradius for Regions with a Brownian Boundary

In this paper we consider β[0, s], Brownian motion of time length s > 0, in m-dimensional Euclidean space ${\mathbb R}^m$ and on the m-dimensional torus ${\mathbb T}^m$ . We compute the expectation of (i) the heat content at time t of ${\mathbb R}^m \backslash \beta [0,s]$ for fixed s and m = 2, 3 in the limit t ↓ 0, when β[0, s] is kept at temperature 1 for all t > 0 and ${\mathbb R}^m \backslash \beta [0,s]$ has initial temperature 0, and (ii) the inradius of ${\mathbb T}^m \backslash \beta [0,s]$ for m = 2, 3, ⋯ in the limit s → ∞.

Correlation dimension estimates of human postural sway

Human postural sway during quiet standing demonstrates a complex structured dynamics, which has been studied by applying numerous methods, such as linear system identification methods, stochastic analysis, and nonlinear system dynamics tools. Although each of the methods applied revealed some particular features of the sway data none of them have succeeded to present a global picture of the quiet stance dynamics, which probably has both stochastic and deterministic properties. In this study we have started applying ergodic theory of dynamical systems to explore statistical characteristic of the sway dynamics observed in successive trials of a subject, different subjects in an age group, and finally different age groups constituted by children, adults, and elderly subjects. Five successive 180-s long trials were performed by each of 28 subjects in four age groups at quiet stance with eyes open. Stationary and ergodic signal characteristics of five successive center of pressure time series collected from a subject in antero-posterior direction (CoPx) were examined. 97% of the trials were found to be stationary by applying Run Test while children and elderly groups demonstrated significant nonstationary behavior. On the other hand 13 out of 24 subjects were found to be nonergodic. We expected to observe differences in complexity of CoPx dynamics due to aging (Farmer, Ott, & Yorke, 1983). However linear metrics such as standard deviation and Fourier spectra of CoPx signals did not show differences due to the age groups. Correlation dimension (Dk) estimates of stationary CoPx signals being an invariant measure of nonlinear system dynamics were computed by using the average displacement method (Eckmann & Ruelle, 1985). Postural dynamics was expanded in m-dimensional space through CoPx signal by introducing optimum time delays, τcritical. 112 out of 136 stationary CoPx signals for 24 stationary subjects converged to Dk estimates. Average of Dk estimates for children and adult groups was 3.67±0.28, whereas mean of Dk estimates for elderly subjects was 4.12±0.59. Nonlinear metrics of postural sway (τcritical, msaturated, and Dk estimates) showed significant differences with respect to the age groups. Dk estimates computed from ergodic subjects’ CoPx sway trajectories revealed that human quiet standing demonstrates multiple degree of freedom dynamics having a fractal structure with a considerable level of noise embedded in the signal whose characteristics is determined individually for each subject. Furthermore by using ergodic theory of complex systems, we have been able to show that the ability to independently control multiple degrees of freedom has been affected by aging.

Closed-form and iterative weighted least squares solutions of Helmert transformation parameters

The Helmert transformation is the most common transformation between different geodetic systems. In 2-D, in contrast to higher dimensions, it is a well-known procedure how to determine the 4 transformation parameters in a closed form. Here we derive the closed-form weighted least squares solution in m-dimensional space for an arbitrary number (≥ m) of coordinate set-ups in two related systems. The solution employs singular value decomposition (SVD) for the rotation matrix, while the translation vector and scale parameters are obtained in simpler ways. To avoid the SVD routine, we also present an iterative approach to solve for the rotation matrix. The paper is completed with a test procedure for detecting outlying coordinate pairs.

Quasi-chaotic behaviors of narrow-band response of a non-deterministic resonant system: application to analysis of ship motion in irregular seas

This paper describes similarities between the narrow-band response of a resonant system excited by random inputs and low-dimensional chaos, with particular emphasis on the geometric characteristics of trajectories reconstructed in m-dimensional phase space from measured scalar time series data with a time-delay coordinate system. In this study, the time series data of ship roll angle in irregular waves were analyzed as an example of the narrow-band response of a resonant system. These time series data were measured by one of the authors in Tokyo Bay. The similarities between the narrow-band response of a resonant system excited by random inputs and low-dimensional chaos are verified by numerical simulation data.

Simplicial resolutions and spaces of algebraic maps between real projective spaces

We show that the space A˜d(m,n) consisting of all real projective classes of (n+1)-tuples of real coefficients homogeneous polynomials of degree d in (m+1) variables, without common real roots except zero, has the same homology as the space Map(RPm,RPn) of continuous maps from the m-dimensional real projective space RPm into the n real dimensional projective space RPn up to dimension (n−m)(d+1)−1. This considerably improves the main result of Adamaszek et al. (2011) [1].

Arrangements of codimension-one submanifolds

We study the number f of connected components in the complement to a finite set (arrangement) of closed submanifolds of codimension 1 in a closed manifold M. In the case of arrangements of closed geodesics on an isohedral tetrahedron, we find all possible values for the number fof connected components. We prove that the set of numbers that cannot be realized by the number f of an arrangement of n ≥ 71 projective planes in the three-dimensional real projective space is contained in the similar known set of numbers that are not realizable by arrangements of n lines on the projective plane. For Riemannian surfaces M we express the number f via a regular neighbourhood of a union of immersed circles and the multiplicities of their intersection points. For m-dimensional Lobachevskii space we find the set of all possible numbers f for hyperplane arrangements.Bibliography: 18 titles.

Difference bodies in complex vector spaces

A complete classification is obtained of continuous, translation invariant, Minkowski valuations on an m-dimensional complex vector space which are covariant under the complex special linear group.

Generic planes conjecture

In some particular cases we prove the density of the set of mappings of an n-dimensional compactum into an m-dimensional Euclidean space such that the set of all d-dimensional planes with the preimage cardinality ≥ q has the dimension ≤ qn - (q − d − 1)(m − d).

SU-E-J-144: Recurrence Quantification Analysis of Lung Cancer Patients' Breathing Pattern.

To demonstrate that Recurrence quantification analysis (RQA) can be used as a quantitative decision making tool to classify patients breathing pattern and select treatment strategy for maneuvering the tumor motion : (a) MIP based treatment (b) 4D treatment using non-linear prediction only (c) 4D treatment non-linear control prediction and breathing control. In our previous work we established that breathing patterns can be described as a 5-6 dimensional nonlinear, stationary and deterministic system that exhibits sensitive dependence on initial conditions. Recurrence plots enable one to investigate an m-dimensional state space trajectory through a two-dimensional representation of its recurrences where the value of a specific pixel is 1 if the distance between the two corresponding trajectory points is less than a threshold value ε. Important measures calculated are: Recurrence Rate (RR), %Determinism, Divergence, Shannon Entropy, LMean, and Renyi entropy (K2). Time Resolved RQA: By implementing a sliding window design, each of the above calculated parameters is computed multiple times. Alignment of those parameters with the time series reveals details not obvious in the 1 -dimensional time series data. The breathing pattern for seven randomly chosen volunteers were recorded using the RPM system for 15 minutes. Non-linear prediction was performed and the normalized root mean square error (NRMSE) was calculated for each of the volunteer data. The threshold value ε was chosen such that the Recurrence Rate was equal to 1%. There is a strong correlation of NRMSE with Entropy, Determinism and LMean. Time resolved RR locates strong Unstable Periodic Orbits(UPOs), i.e. patterns of uninterrupted equally spaced diagonal lines. RQAs could prove to be a very powerful tool for design of personalized treatment regimen. Entropy, Determinism in conjunction with strong UPOs can be used to determine if patients are suitable candidates for prediction and chaos control.

Process modelling for contingency tables with ordered categories

We consider the setting of a multi-way contingency table with ordinal classifications. The contribution of this paper is to propose a joint probability model for the uncensored variables that is apart from the imposed categorization. Specifically, for an m-way table, we assume that the cell counts arise as a binned point pattern over a bounded set in m-dimensional Euclidean space where the point pattern is a realization of a non-homogeneous Poisson process. The intensity which drives the point pattern is itself viewed as a realization of a log Gaussian process over the set. With such an approach we achieve full inference regarding the underlying joint distribution, in particular, inference for familiar associations between the ordinal variables in the absence of interval censoring. Additionally, inference can be provided for any newly created cells where such creation is achieved through redefinition of the ordinal classifications. That is, rather than ad hoc reallocation, we achieve a fully model-based reallocation enabling quantification of uncertainty. For a contingency table with nominal classifications as well, our approach creates an intensity for the ordinal variables for each level of the nominal variables. The methodology is detailed within a hierarchical framework, showing associated computation and convenient dimension reduction techniques to facilitate model fitting. We illustrate with both simulated data and a real census dataset.

Convexity and multi-dimensional screening for spaces with different dimensions

Abstract We study the principal–agent problem. We show that b-convexity of the space of products, a condition which appears in a recent paper by Figalli, Kim and McCann (2011) [9] , is necessary to formulate the problem as a maximization over a convex set. We then show that when the dimension m of the space of types is larger than the dimension n of the space of products, this condition implies that the extra dimensions do not encode independent economic information. When m is smaller than n, we show that under b-convexity of the space of types, it is always optimal for the principal to offer goods only from a certain prescribed subset. We show that this is equivalent to offering an m-dimensional space of goods.

On Hawaiian groups of some topological spaces

The paper is devoted to study the structure of Hawaiian groups of some topological spaces. We present some behaviors of Hawaiian groups with respect to product spaces, weak join spaces, cone spaces, covering spaces and locally trivial bundles. In particular, we determine the structure of the n-dimensional Hawaiian group of the m-dimensional Hawaiian earring space, for all 1⩽m⩽n.

N-to-M JOINT REMOTE STATE PREPARATION OF 2-LEVEL STATES

In this paper, we investigate novel protocols for the joint remote state preparation involving several senders and receivers. The highlight of our paper lies in two aspects. First, we focus on the distribution of information among multiple senders and receivers. Second, each receiver can simultaneously reconstruct a qubit state containing the joint information from all senders. These properties imply that our protocols may have many extensive applications in burgeoning quantum network communication. Our protocols cover a variety of class, i.e. two-to-three, N-to-three, and N-to-M. Through introducing the quantum fourier transform, the set of orthogonal basis in the M-dimensional Hilbert space is ingeniously constructed. Moreover, we present all the recovery operations in details. The success probability and the classical communication costs are also given.