scholarly journals Characteristic Points of Recursive Systems

10.37236/393 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Jason P. Bell ◽  
Stanley N. Burris ◽  
Karen A. Yeats
Keyword(s):  
Power Series ◽  
Extreme Point ◽  
Spectral Radius ◽  
Generating Function ◽  
Characteristic Point ◽  
Recursive Equation ◽  
Series System ◽  
Negative Power ◽  
Characteristic Points ◽  
Recursive Systems

Characteristic points have been a primary tool in the study of a generating function defined by a single recursive equation. We investigate the proper way to adapt this tool when working with multi-equation recursive systems. Given an irreducible non-negative power series system with $m$ equations, let $\rho$ be the radius of convergence of the solution power series and let $\pmb{\tau}$ be the values of the solution series evaluated at $\rho$. The main results of the paper include: (a) the set of characteristic points form an antichain in ${\mathbb R}^{m+1}$, (b) given a characteristic point $(a,\mathbf{b})$, (i) the spectral radius of the Jacobian of $\pmb \gamma$ at $(a, \mathbf{b})$ is $\ge 1$, and (ii) it is $=1$ iff $(a,\mathbf{b}) = (\rho,\pmb{\tau})$, (c) if $(\rho,\pmb{\tau})$ is a characteristic point, then (i) $\rho$ is the largest $a$ for $(a,\mathbf{b})$ a characteristic point, and (ii) a characteristic point $(a,\mathbf{b})$ with $a=\rho$ is the extreme point $(\rho,\pmb{\tau})$.

scholarly journals Blue as an Underrated Alternative to Green: Photoplethysmographic Heartbeat Intervals Estimation under Two Temperature Conditions

Sensors ◽  
2021 ◽  
Vol 21 (12) ◽  
pp. 4241
Author(s):  
Evgeniia Shchelkanova ◽  
Liia Shchapova ◽  
Alexander Shchelkanov ◽  
Tomohiro Shibata
Keyword(s):  
Blue Light ◽  
Characteristic Point ◽  
Biological Effects ◽  
Green Light ◽  
Estimation Accuracy ◽  
Lower Sensitivity ◽  
Light Spectrum ◽  
Cardiovascular Parameter ◽  
Temperature Conditions ◽  
Characteristic Points

Since photoplethysmography (PPG) sensors are usually placed on open skin areas, temperature interference can be an issue. Currently, green light is the most widely used in the reflectance PPG for its relatively low artifact susceptibility. However, it has been known that hemoglobin absorption peaks at the blue part of the spectrum. Despite this fact, blue light has received little attention in the PPG field. Blue wavelengths are commonly used in phototherapy. Combining blue light-based treatments with simultaneous blue PPG acquisition could be potentially used in patients monitoring and studying the biological effects of light. Previous studies examining the PPG in blue light compared to other wavelengths employed photodetectors with inherently lower sensitivity to blue, thereby biasing the results. The present study assessed the accuracy of heartbeat intervals (HBIs) estimation from blue and green PPG signals, acquired under baseline and cold temperature conditions. Our PPG system is based on TCS3472 Color Sensor with equal sensitivity to both parts of the light spectrum to ensure unbiased comparison. The accuracy of the HBIs estimates, calculated with five characteristic points (PPG systolic peak, maximum of the first PPG derivative, maximum of the second PPG derivative, minimum of the second PPG derivative, and intersecting tangents) on both PPG signal types, was evaluated based on the electrocardiographic values. The statistical analyses demonstrated that in all cases, the HBIs estimation accuracy of blue PPG was nearly equivalent to the G PPG irrespective of the characteristic point and measurement condition. Therefore, blue PPG can be used for cardiovascular parameter acquisition. This paper is an extension of work originally presented at the 42nd Annual International Conference of the IEEE Engineering in Medicine and Biology Society.

scholarly journals Identities Involving the Fourth-Order Linear Recurrence Sequence

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1476 ◽  
Author(s):  
Lan Qi ◽  
Zhuoyu Chen
Keyword(s):  
Power Series ◽  
Series Expansion ◽  
Generating Function ◽  
Fourth Order ◽  
Power Series Expansion ◽  
Linear Recurrence ◽  
Recurrence Sequence ◽  
Order Linear ◽  
Elementary Methods ◽  
Linear Recurrence Sequence

In this paper, we introduce the fourth-order linear recurrence sequence and its generating function and obtain the exact coefficient expression of the power series expansion using elementary methods and symmetric properties of the summation processes. At the same time, we establish some relations involving Tetranacci numbers and give some interesting identities.

scholarly journals Affine screening operators, affine Laumon spaces and conjectures concerning non-stationary Ruijsenaars functions

2019 ◽  
Vol 4 (1) ◽  
Author(s):  
Jun’ichi Shiraishi
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Generating Function ◽  
Vertex Operators ◽  
Poincaré Duality ◽  
Euler Characteristics ◽  
Poincare Duality ◽  
Main Conjecture ◽  
Formal Power

Abstract Based on the screened vertex operators associated with the affine screening operators, we introduce the formal power series $f^{\widehat{\mathfrak gl}_N}(x,p|s,\kappa|q,t)$ which we call the non-stationary Ruijsenaars function. We identify it with the generating function for the Euler characteristics of the affine Laumon spaces. When the parameters $s$ and $\kappa$ are suitably chosen, the limit $t\rightarrow q$ of $f^{\widehat{\mathfrak gl}_N}(x,p|s,\kappa|q,q/t)$ gives us the dominant integrable characters of $\widehat{\mathfrak sl}_N$ multiplied by $1/(p^N;p^N)_\infty$ (i.e. the $\widehat{\mathfrak gl}_1$ character). Several conjectures are presented for $f^{\widehat{\mathfrak gl}_N}(x,p|s,\kappa|q,t)$, including the bispectral and the Poincaré dualities, and the evaluation formula. The main conjecture asserts that (i) one can normalize $f^{\widehat{\mathfrak gl}_N}(x,p|s,\kappa|q,t)$ in such a way that the limit $\kappa\rightarrow 1$ exists, and (ii) the limit $f^{{\rm st.}\,\widehat{\mathfrak gl}_N}(x,p|s|q,t)$ gives us the eigenfunction of the elliptic Ruijsenaars operator. The non-stationary affine $q$-difference Toda operator ${\mathcal T}^{\widehat{\mathfrak gl}_N}(\kappa)$ is introduced, which comes as an outcome of the study of the Poincaré duality conjecture in the affine Toda limit $t\rightarrow 0$. The main conjecture is examined also in the limiting cases of the affine $q$-difference Toda ($t\rightarrow 0$), and the elliptic Calogero–Sutherland ($q,t\rightarrow 1$) equations.

scholarly journals Is the Sibuya distribution a progeny?

2019 ◽  
Vol 56 (01) ◽  
pp. 52-56
Author(s):  
Gérard Letac
Keyword(s):  
Power Series ◽  
Series Expansion ◽  
Generating Function ◽  
Branching Process ◽  
Power Series Expansion ◽  
Nonnegative Coefficients ◽  
Positive Integers

AbstractFor 0 < a < 1, the Sibuya distribution sa is concentrated on the set ℕ+ of positive integers and is defined by the generating function $$\sum\nolimits_{n = 1}^\infty s_a (n)z^{{\kern 1pt} n} = 1 - (1 - z)^a$$. A distribution q on ℕ+ is called a progeny if there exists a branching process (Zn)n≥0 such that Z0 = 1, such that $$(Z_1 ) \le 1$$, and such that q is the distribution of $$\sum\nolimits_{n = 0}^\infty Z_n$$. this paper we prove that sa is a progeny if and only if $${\textstyle{1 \over 2}} \le a < 1$$. The main point is to find the values of b = 1/a such that the power series expansion of u(1 − (1 − u)b)−1 has nonnegative coefficients.

scholarly journals Some problems of non-associative combinations (2)

1940 ◽  
Vol 32 ◽  
pp. vii-xii ◽  
Author(s):  
A. Erdelyi ◽  
I. M. H. Etherington
Keyword(s):  
Power Series ◽  
Algebraic Equation ◽  
Generating Function ◽  
Convex Polygon ◽  
Preceding Note ◽  
Image Position

§ 1. The preceding Note has shown the connection between the partition of a convex polygon by non-crossing diagonals and the insertion of brackets in a product, the latter being more commonly represented by the construction of a tree. It was shown that the enumeration of these entities leads to a generating function y = f(x) which satisfies an algebraic equation of the typeIn simple cases, the solution of the equation was found as a power series in x, the coefficient An of xn giving the required number of partitions of an (n + 1)-gon.

Subharmonic Coordination in Networks of Neurons with Slow Conductances

1994 ◽  
Vol 6 (1) ◽  
pp. 69-84 ◽  
Author(s):  
Thomas LoFaro ◽  
Nancy Kopell ◽  
Eve Marder ◽  
Scott L. Hooper
Keyword(s):  
Coupled Oscillators ◽  
Characteristic Point ◽  
Reciprocal Inhibition ◽  
Excitable Cell ◽  
Characteristic Points ◽  
Cell Firing ◽  
Voltage Activation ◽  
Phase Relationships ◽  
Excitable Neuron ◽  
Phase Differences

We study the properties of a network consisting of two model neurons that are coupled by reciprocal inhibition. The study was motivated by data from a pair of cells in the crustacean stomatogastric ganglion. One of the model neurons is an endogenous burster; the other is excitable but not bursting in the absence of phasic input. We show that the presence of a hyperpolarization activated inward current (ih) in the excitable neuron allows these neurons to fire in integer subharmonics, with the excitable cell firing once for every N ≥ 1 bursts of the oscillator. The value of N depends on the amount of hyperpolarizing current injected into the excitable cell as well as the voltage activation curve of ih. For a fast synapse, these parameter changes do not affect the characteristic point in the oscillator cycle at which the excitable cell bursts; for slower synapses, such a relationship is maintained within small windows for each N. The network behavior in the current work contrasts with the activity of a pair of coupled oscillators for which the interaction is through phase differences; in the latter case, subharmonics exist if the uncoupled oscillators have near integral frequency relationships, but the phase relationships of the oscillators in general change significantly with parameters. The mechanism of this paper provides a potential means of coordinating subnetworks acting on different time scales but maintaining fixed relationships between characteristic points of the cycles.

Stereo Matching Algorithm Based on Detecting Feature Points

2012 ◽  
Vol 433-440 ◽  
pp. 6190-6194
Author(s):  
Shuo Bo Xu ◽  
Di Shi Xu ◽  
Hua Fang
Keyword(s):  
Stereo Matching ◽  
Characteristic Point ◽  
Feature Points ◽  
Epipolar Line ◽  
Matching Problem ◽  
Matching Algorithm ◽  
Characteristic Points ◽  
Left And Right ◽  
Straight Lines ◽  
Verification Strategy

A new method for solving the stereo matching problem in the presence of large occlusion is presented. This method for stereo matching and occlusion detection is based on searching disparity point. In this paper, we suppose that a pair of epipolar-line images is a projection of a group of piece-wise straight lines on the left and right images respective. Therefore the disparity curve corresponding to a pair of epipolar-line images may be approximated by a group of piece-wise straight lines. Then the key of solving disparity curve is how to get the “characteristic points” on the group of piece-wise straight lines. Based on this view, we fetched out the conception “disparity point”, and three kinds of special disparity points are correctly corresponding to the “characteristic point”. By analyzing intensity property of a disparity point and its neighbor points, an approach which combines stepwise hypothesis-verification strategy with three constraint conditions is devised to extract the candidate disparity points from the epipolar images.

2-Adic valuations of coefficients of certain integer powers of formal power series

2019 ◽  
Vol 15 (04) ◽  
pp. 723-762
Author(s):  
Maciej Ulas
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Generating Function ◽  
Integer Sequence ◽  
Formal Power

Let [Formula: see text] be an integer sequence and [Formula: see text] be its ordinary generating function. In this paper, we study the behavior of 2-adic valuations of the sequence [Formula: see text], where [Formula: see text] is fixed and [Formula: see text] More precisely, we propose a method, which under suitable assumptions on the sequence [Formula: see text] allows us to prove boundedness of the sequence [Formula: see text] for certain values of [Formula: see text]. In particular, if [Formula: see text] is the classical Rudin–Shapiro sequence, then we prove that [Formula: see text] for given [Formula: see text] and all [Formula: see text]. A similar result is proved for a relative of the Rudin–Shapiro sequence recently introduced by Lafrance, Rampersad and Yee.

Detecting and Determining the Characteristic Points on the Parametric Surface/Surface Intersection

1995 ◽  
Author(s):  
Xianglin Zeng ◽  
Qifu Wang ◽  
Ji Zhou ◽  
Jun Yu
Keyword(s):  
Characteristic Point ◽  
New Method ◽  
Parametric Surface ◽  
Initial Interval ◽  
Surface Modelling ◽  
Surface Intersection ◽  
Characteristic Points ◽  
Numeric Solution ◽  
Modelling System ◽  
Significant Point

Abstract In this paper we present a new method for detecting and determining characteristic points on the surface/surface intersection while marching along the intersections. The initial interval which contains a potential characteristic point is first determined by certain criteria, then a numeric solution of the significant point is obtained by the binary subdivision method. Based on these ideas, a new marching algorithm is constructed, and it has been implemented in a surface modelling system (SurfCADM V1.0). Examples are also presented for illustrating the capability of our algorithm.

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