Geometric Sequence Research Articles

Some Fundamental Results on Base Station Movement Problem for Wireless Sensor Networks

The benefits of using a mobile base station to prolong sensor network lifetime have been well recognized. However, due to the complexity of the problem (time-dependent network topology and traffic routing), theoretical performance limits and provably optimal algorithms remain difficult to develop. This paper fills this important gap by contributing some theoretical results regarding the optimal movement of a mobile base station. Our main result hinges upon two key intermediate results. In the first result, we show that a time-dependent joint base station movement and flow routing problem can be transformed into a location-dependent problem. In the second result, we show that, for $(1- \varepsilon)$ optimality, the infinite possible locations for base station movement can be reduced to a finite set of locations via several constructive steps [i.e., discretization of energy cost through a geometric sequence, division of a disk into a finite number of subareas, and representation of each subarea with a fictitious cost point (FCP)]. Subsequently, for each FCP, we can obtain the optimal sojourn time for the base station (as well as the corresponding location-dependent flow routing) via a simple linear program. We prove that the proposed solution can guarantee the achieved network lifetime is at least $(1- \varepsilon)$ of the maximum (unknown) network lifetime, where $\varepsilon$ can be made arbitrarily small depending on the required precision.

Simultaneous estimation of T1 and the flip angle in hyperpolarized NMR experiments using acquisition at non-regular time intervals

In NMR spectroscopy of the liquid state T1 is typically measured using an inversion recovery pulse sequence; but with hyperpolarized spins use is made of a sequence of multiple small radiofrequency (RF) induced nutations, α. Depending on the values of α and τ, the time interval between the pulses, the estimate of T1 can be artifactually smaller than the real value; so without knowing the value of α the estimate of T1 can be incorrect. Thus, we propose a method that involves a series of pulses with timing governed by a geometric sequence (or in general, any mathematically specified non-uniformly spaced sequence). This approach enables the simultaneous estimation of both the intrinsic T1 value and α. The method was successfully applied to obtain T1=(44.9±0.3)s and α=(4.0±0.2)° (n=3) for a sample of hyperpolarized 13C-urea in solution, matching with the inversion recovery pulse sequence estimate of T1=44±2s using non-hyperpolarized 13C-urea in solution.

Packing Coxeter honeycombs with sequences of spheres

In Odehnal (2011) packings with geometric sequences of spheres to regular polyhedra in the Euclidean d-space with d ≥ 2 have been introduced. The packing densities depending on the dimension and the type of polyhedron have been determined. Some of these packings can also be performed in d-dimensional Euclidean space. In this paper we generalize the above notion of sphere sequences packed into hyperbolic Coxeter honeycombs. We describe a method that determines the data and the density of each considered non-congruent sphere packing to every Coxeter tiling. Moreover, we apply our method to some 2- and 3-dimensional totally asymptotic honeycombs.

Sequence of Numbers with Three Alternate Common Differences and Common Ratios

This paper talks about two types of special sequences. The firstis the arithmetic sequence of numbers with three alternate common differences;and the other, is the geometric sequence of numbers with three alternate commondifferences. The formulas for the general term an and the sum of the firstn terms, denoted by Sn, are given respectively.

On nondecreasing sequences of regularization parameters for nonstationary iterated Tikhonov

Nonstationary iterated Tikhonov is an iterative regularization method that requires a strategy for defining the Tikhonov regularization parameter at each iteration and an early termination of the iterative process. A classical choice for the regularization parameters is a decreasing geometric sequence which leads to a linear convergence rate. The early iterations compute quickly a good approximation of the true solution, but the main drawback of this choice is a rapid growth of the error for later iterations. This implies that a stopping criteria, e.g. the discrepancy principle, could fail in computing a good approximation. In this paper we show by a filter factor analysis that a nondecreasing sequence of regularization parameters can provide a rapid and stable convergence. Hence, a reliable stopping criteria is no longer necessary. A geometric nondecreasing sequence of the Tikhonov regularization parameters into a fixed interval is proposed and numerically validated for deblurring problems.

On Legendre Multiplier Sequences

In this paper we give a complete characterization of linear, quadratic, and geometric Legendre multiplier sequences. We also prove that all Legendre multiplier sequences must be Hermite multiplier sequences, and describe the relationship between the Legendre and generalized Laguerre multiplier sequences. We conclude with a list of open questions for further research.

Applications of extension grey prediction model for power system forecasting

Although the grey forecasting model has been successfully adopted in various fields and demonstrated promising results, the literatures show its performance could be further improved. For this purpose, this paper proves that the growth rate of the simulated value of the grey model GM(1,1) is a fixed value. If the growth rates of the primary sequence are equate, the fitted value deriving from GM(1,1) is the same as the primary sequence, otherwise greater error would occur. In order to overcome shortcoming of the fixed growth rates, extend the traditional GM(1,1) model by introducing linear time-varying terms, which can predict more accurately on non geometric sequences, termed EGM(1,1). Finally, compared with the other improved grey model and ARIMA model, experimental results indicate that the proposed model obviously can improve the prediction accuracy.

First Digits of Squares and Cubes

SummaryBenford's Law is the observation that in many lists of numbers that arise in the real world, for d = 1, 2, …, 9, the likelihood that the first digit is d is log10 ((d + 1)/d). A similar phenomenon is known for essentially all geometric sequences ark, namely the relative frequencies RN(d) of the first digit d of the first N terms in such a sequence tend to log10((d + 1)/d). In this note, it is shown that the analogous statements for the sequence of squares and the sequence of cubes do not hold. However, in these cases, interesting subsequences of RN(d) do converge, but not to log10((d + 1)/d).

Computational Analysis of the Spatial Distribution of Pre-Christian Slavic Sacred Sites

This paper proposes a method to find out the units of length used by the Slavs prior to their Christianization. The method relies on a previous research of the tripartite structure of pre-Christian Slavic sacred sites discovered by Andrej Pleterski. Such spatial structures represent myth in space and are in correlation with the central Slavic myth of the divine battle between Perun and Veles as presented by Radoslav Katieiae. By measuring the angles which Sun takes through the year cycle and comparing them with the angles between the sacred sites, the pagan priests had means to accurately determine the days of religious festivals and the calendar in general. The angles likely had an important role, but this article tries to examine weather the absolute values of the distance between the sacred sites were also important. The method relies on the mathematical properties of arithmetic and geometric sequences. If for some of the inital parameters of the sequences the probability function for a given distribution of sacred sites in some area shows to be significantly smaller than the average, then it is an indication that the respective distribution is not random. The parameters in such case may point to the system of measures used during the creation of the structure. The proposed method really detected some common modules, but in this phase it is only experimental and still can not be used as a proof of common Pan-Slavic system of units.

Assembly Model Based on Dynamic Transition Graph for Satellite General Assembly Process Planning

Beginning with the disassembly of the semi-finished modules, the general assembly process of satellites is quite different from that of other mechanical products. This kind of disassembly-reassembly process cannot be solved by the widely used cut-set theory that only deals with linear problems, thus an approach called Dynamic Transition Graph (DTG) is proposed to solve this nonlinear assembly sequence planning problem. By analyzing and verifying possible assembly statuses hierarchically through matrix operations, a DTG is produced on the basis of Liaison Graph, where the DTG includes all the geometric feasible assembly sequences for the general assembly of satellite. Finally, the effectiveness of this method is verified by an example.

The mathematical relationship between Zipf’s law and the hierarchical scaling law

The empirical studies of city-size distribution show that Zipf’s law and the hierarchical scaling law are linked in many ways. The rank-size scaling and hierarchical scaling seem to be two different sides of the same coin, but their relationship has never been revealed by strict mathematical proof. In this paper, the Zipf’s distribution of cities is abstracted as a q-sequence. Based on this sequence, a self-similar hierarchy consisting of many levels is defined and the numbers of cities in different levels form a geometric sequence. An exponential distribution of the average size of cities is derived from the hierarchy. Thus we have two exponential functions, from which follows a hierarchical scaling equation. The results can be statistically verified by simple mathematical experiments and observational data of cities. A theoretical foundation is then laid for the conversion from Zipf’s law to the hierarchical scaling law, and the latter can show more information about city development than the former. Moreover, the self-similar hierarchy provides a new perspective for studying networks of cities as complex systems. A series of mathematical rules applied to cities such as the allometric growth law, the 2n principle and Pareto’s law can be associated with one another by the hierarchical organization.

Using Restricted Natural Language for Geometric Construction

Human-Computer Interaction is crucial for Instructional Software. It is significant to make Geometry Instructional Software understand geometric proposition described by natural language and construct geometric figure automatically. However, it is difficult for computer to accurately understand the flexible and variable geometric proposition because of the immaturity of Natural Language Processing Technology. This paper defines a restricted geometric proposition, presents a corresponding comprehension model based on ontology, and designs a matching algorithm for geometric relation pattern, finally transforms the restricted geometric proposition to geometric construction command sequence. The experiment results suggest that the average accuracy of understanding clauses of geometric proposition is almost to 88.9%, and the percentage of constructing correct geometric figure is to 82.5%.

The query complexity of estimating weighted averages

The query complexity of estimating the mean of some [0, 1] variables is understood. Inspired by some work by Carterette et al. on evaluating retrieval systems, and by Moffat and Zobel’s new proposal for such evaluation, we examine the query complexity of weighted average calculation. In general, determining an answer within accuracy $${\varepsilon}$$, with high probability, requires $${\Omega(\varepsilon^{-2})}$$ queries, as the mean is a special case. There is a matching upper bound for the weighted mean. If the weights are a normalized prefix of a divergent series, the same result holds. However, if the weights follow a geometric sequence, a sample of size $${\Omega(\log (1/\varepsilon))}$$ suffices. Our principal contribution is the investigation of power-law sequences of weights. We show that if the ith largest weight is proportional to i −p , for p > 1, then the query complexity is in $${\Omega(\varepsilon^{2/(1-2p)})}$$.

Basin-Wide Distribution of Land Use and Human Population: Stream Order Modeling and River Basin Classification in Japan

This paper presents a mathematical model developed using Horton-Strahler's stream order to describe basin-wide distributions of human activities, i.e., land use and human population, across several river basins with different geomorphologic features. We assume that for successive stream orders, the mean area of each land use type-paddy field, forest, city, village, etc.-and the human population form a geometric sequence, which is the same mathematical relationship as stated in Horton's laws of river geomorphology. This geometric sequence modeling implies fractal nature of human activity distributions within a river basin. GIS datasets for the land use and human population in 109 large river basins in Japan were used to verify the model. Herein, we examine the relationships between the Horton ratios and the common ratios obtained from the model to explore links between basin geomorphology and human activities. Furthermore, we quantitatively compare the human activity distributions across the 109 river basins on the basis of results obtained from the model with descriptive statistics. Further, we attempt to classify the river basins into several categories through multivariate statistical analysis.

Model of discrete dynamics of asset price relations based on the minimal arbitrage principle

Abstract In this paper we present a deterministic and a probabilistic model of the dynamics of the price relations for a number of assets on the market. The formalism is based on the asset space introduced in a theory by Illinski. We derive, from an action functional for the system of price relations in that space, the corresponding difference equations, which constitute the deterministic description. Furthermore, we obtain the probability density function of the probabilistic model of market dynamics from the same action functional. The deterministic solution corresponds to a geometric sequence for the interest, whereas the derived probability density describes the probability of the next value of the price relations in dependence on their prior value. The formalism is completely developed for systems (markets) with two and three assets, but exactly the same approach is applicable to the systems consisting of an arbitrary number of assets.

THE INFRARED PROPERTIES OF EMBEDDED SUPER STAR CLUSTERS: PREDICTIONS FROM THREE-DIMENSIONAL RADIATIVE TRANSFER MODELS

With high-resolution infrared data becoming available that can probe the formation of high-mass stellar clusters for the first time, appropriate models that make testable predictions of these objects are necessary. We utilize a three-dimensional radiative transfer code, including a hierarchically clumped dusty envelope, to study the earliest stages of super star cluster (SSC) evolution. We explore a range of parameter space in geometric sequences that mimic the hypothesized evolution of an embedded SSC. The inclusion of a hierarchically clumped medium can make the envelope porous, in accordance with previous models and supporting observational evidence. The infrared luminosity inferred from observations can differ by a factor of two from the true value in the clumpiest envelopes depending on the viewing angle. The infrared spectral energy distribution also varies with viewing angle for clumpy envelopes, creating a range in possible observable infrared colors and magnitudes, silicate feature depths, and dust continua. General observable features of cluster evolution differ between envelopes that are relatively opaque or transparent to mid-infrared photons. For optically thick envelopes, evolution is marked by a gradual decline of the 9.8 μm silicate absorption feature depth and a corresponding increase in the visual/ultraviolet flux. For the optically thin envelopes, clusters typically begin with a strong hot dust component and silicates in emission, and these features gradually fade until the mid-infrared polycyclic aromatic hydrocarbon features are predominant. For the models with a smooth dust distribution, the Spitzer MIPS or Herschel PACS [70]−[160] color is a good probe of the stellar mass relative to the total mass or star formation efficiency (SFE). Likewise, the IRAC/MIPS [3.6]−[24] color can be used to constrain the Rin and Rout values of the envelope. However, clumpiness confuses the general trends seen in the smooth dust distribution models, making it harder to determine a unique set of envelope properties. Nevertheless, good diagnostic colors were found for each of the input parameters: again, the [70]−[160] color can be used to separate models with different SFEs; the Spitzer IRAC/MIPS [8.0]−[24] color is able to constrain Rin and Rout values; and the IRAC [3.6]−[5.8] color is sensitive to the fraction of the dust distributed in clumps. Finally, in a comparison of this model set to IRAS data of ultracompact H ii regions, we find good agreement, suggesting that these models are physically relevant, and will provide useful diagnostic ability for data sets of resolved, embedded SSCs with the advent of high-resolution infrared telescopes like James Webb Space Telescope.

GALOIS THEORY OF THE CANONICAL THETA STRUCTURE

In this article, we give a Galois-theoretic characterization of the canonical theta structure. The Galois property of the canonical theta structure translates into certain p-adic theta relations which are satisfied by the canonical theta null point of the canonical lift. As an application, we prove some 2-adic theta identities which describe the set of canonical theta null points of the canonical lifts of ordinary abelian varieties in characteristic 2. The latter theta relations are suitable for explicit canonical lifting. Using the theory of canonical theta null points, we are able to give a theoretical foundation to Mestre's point counting algorithm which is based on the computation of the generalized arithmetic geometric mean sequence.

Fibonacci Primes and Topological Biomolecular Mechanics

Leonardo Fibonacci (c.1170-c.1250) in his book Liber Abaci (1202) presented two sequences: Fibonacci (Arithmetic) Sequence, Fa, ([1], p.260), and Fibonacci (Geometric) Sequence, Fg, ([1], p.404). We show that when Fg {Congruence (mod Fg6)} prime factorized, reveals 11 primes, which we named Fibonacci Primes identifying Fibonacci Primes Hexagon and Fibonacci Primes Spider. The center of the spider consists of Fibonacci primes 17 and 19 which move in the direction of increasing Fg numbers as congruence (mod 3), and the spider enlarges in a systematic pattern. Fa and Fg sequences determine specific positions in applications of Fibonacci primes to the two asymmetric halves of tRNA, mRNA, as well as protein biomolecules. [1] L.E. Sigler, Fibonacci's Liber Abaci: A Translation in Modern English of Leonardo Pisano's Book of Calculations, New York, Springer-Verlag, 2002.

Towards a new potential field theory of fractal objects

The Potential Field Anomaly (PFA) data of the self similar Fractal Objects (FOs) include gravity and magnetic fields and potentials along with the related derivatives. These elements are calculated on grids due to buried FOs at different fractal orders. The objects have variable physical property distributions; while in magnetic, the orientation and magnitude of polarization or earth magnetic field is arbitrary. Using the structural index as Universal Fractal Order Invariant Measure, one of the contributions of this work is expressing the elements of the PFA data at any measuring point on a grid as geometric sequences in terms of the fractal order. We found that the common ratio of the sequences is equal to the Fractal Mass Ratio (FMR), a physical quantity characterizing the object. Therefore, we can interpolate the PFA data backward or forward from one fractal order to the other. This in turn allows us to directly calculate PFA data of FOs from the zero order objects equivalent to the solid sources or initiators. We conclude that the patterns of PFA data due to a self similar FO are scale-invariant and reflect the nature of this object. We express the FMR of a FO in a new equation describing the difference between the topological and fractal dimensions in terms of a linear scale.

A study of the population structure of the Pacific sardine Sardinops sagax (Jenyns, 1842) in Mexico based on morphometric and genetic analyses

Several studies on the Pacific sardine Sardinops sagax have focused on the identification of stock composition and boundaries, using morphometric and genetic analysis. In this study, geometric morphometric body landmarks and control region mtDNA sequences were used to examine the population structure of sardines along the Pacific coast of the Baja California Peninsula. Samples from commercial landings in Ensenada (ENS), Baja California, and Bahia Magdalena (BM), Baja California Sur, were obtained during 2006–2007. The population hypotheses tested were based on the distribution of sea surface temperature (SST) along the coast, which was previously used to define stocks. A total of 275 sardines from ENS and 119 from BM were used in morphometric analysis. Fifty-three sequences from ENS and 106 from BM were used for genetic comparisons. Morphometric results showed differences among the three groups based on SST, suggesting the existence of different morphotypes. Percentage of molecular variance explained by the differences among three groups was significantly different from zero. However, the distribution of haplotypes in the groups did not show a clear phylogeographic pattern. Additionally, mismatch distributions supported relatively similar historical demographic events in the three groups. Although evidence of phenotypic groups along the Pacific coast of the Peninsula was found, current molecular data did not clearly support the existence of a phylogeographically structured population.