Piecewise linear definition of transformation functions for speaker de-identification

2016 ◽  
Author(s):  
Carmen Magarinos ◽  
Paula Lopez-Otero ◽  
Laura Docio-Fernandez ◽  
Eduardo R. Banga ◽  
Carmen Garcia-Mateo ◽  
...  
Keyword(s):  
Piecewise Linear ◽  
Transformation Functions ◽  
Definition Of

Mesh duality and Legendre duality

1990 ◽  
Vol 428 (1875) ◽  
pp. 351-377 ◽  
Keyword(s):  
Piecewise Linear ◽  
Voronoi Tessellation ◽  
Tangent Plane ◽  
Linear Functions ◽  
Piecewise Linear Functions ◽  
Dual Transformation ◽  
Delaunay Mesh ◽  
Definition Of

We describe a sense in which mesh duality is equivalent to Legendre duality. That is, a general pair of meshes, which satisfy a definition of duality for meshes, are shown to be the projection of a pair of piecewise linear functions that are dual to each other in the sense of a Legendre dual transformation. In applications the latter functions can be a tangent plane approximation to a smoother function, and a chordal plane approximation to its Legendre dual. Convex examples include one from meteorology, and also the relation between the Delaunay mesh and the Voronoi tessellation. The latter are shown to be the projections of tangent plane and chordal approximations to the same paraboloid.

scholarly journals Determination of the Starting Point in Time Series for Trend Detection Based on Overlapping Trend

2016 ◽  
Vol 16 (6) ◽  
pp. 98-110
Author(s):  
Gao Xuedong ◽  
Gu Kan
Keyword(s):  
Time Series ◽  
Piecewise Linear ◽  
Negative Influence ◽  
Trend Detection ◽  
Computational Accuracy ◽  
Starting Point ◽  
Time Series Studies ◽  
Definition Of ◽  
Stage Characteristic

Abstract The traditional time series studies consider the time series as a whole while carrying on the trend detection; therefore not enough attention is paid to the stage characteristic. On the other hand, the piecewise linear fitting type methods for trend detection are lacking consideration of the possibility that the same node belongs to multiple trends. The above two methods are affected by the start position of the sequence. In this paper, the concept of overlapping trend is proposed, and the definition of milestone nodes is given on its base; these way not only the recognition of overlapping trend is realized, but also the negative influence of the starting point of sequence is effectively reduced. The experimental results show that the computational accuracy is not affected by the improved algorithm and the time cost is greatly reduced when dealing with the processing tasks on dynamic growing data sequence.

Invariant densities for piecewise linear maps of the unit interval

2009 ◽  
Vol 29 (5) ◽  
pp. 1549-1583 ◽  
Author(s):  
PAWEŁ GÓRA
Keyword(s):  
Piecewise Linear ◽  
Unit Interval ◽  
Invariant Density ◽  
Piecewise Linear Map ◽  
Full Support ◽  
Linear Maps ◽  
Piecewise Linear Maps ◽  
The Matrix ◽  
Definition Of ◽  
Invariant Densities

AbstractWe find an explicit formula for the invariant densityhof an arbitrary eventually expanding piecewise linear mapτof an interval [0,1]. We do not assume that the slopes of the branches are the same and we allow arbitrary number of shorter branches touching zero or touching one or hanging in between. The construction involves the matrixSwhich is defined in a way somewhat similar to the definition of the kneading matrix of a continuous piecewise monotonic map. Under some additional assumptions, we prove that if 1 is not an eigenvalue ofS, then the dynamical system (τ,h⋅m) is ergodic with full support.

A Normalized Numerical Scaling Method for the Unbalanced Multi-Granular Linguistic Sets

2015 ◽  
Vol 23 (02) ◽  
pp. 221-243 ◽  
Author(s):  
Baoli Wang ◽  
Jiye Liang ◽  
Yuhua Qian ◽  
Chuangyin Dang
Keyword(s):  
Piecewise Linear ◽  
Linear Interpolation ◽  
Decision Makers ◽  
Decision Problems ◽  
Scaling Functions ◽  
Linguistic Terms ◽  
Scaling Method ◽  
Piecewise Linear Interpolation ◽  
Transformation Functions

Decision makers often express their evaluations on decision problems with multi-granular linguistic terms. This fact leads to the unification of the multi-granular linguistic terms into a single linguistic set in the literature. However, this unification process increases the complexity of computation and the subjectivity in the determination of transformation functions. To overcome this deficiency, this paper aims to develop a normalized numerical scaling method for determining the semantics of multi-granular linguistic terms in the same domain. We first introduce a class of numerical scaling functions to generate several balanced or unbalanced linguistic sets. Since these scaled linguistic sets have different domains, we then develop a normalized numerical scaling method to form them into the unique interval [0,1]. As a result of this development, two classes of normalized scaling functions are derived from the priori scale information and applications of piecewise linear interpolation and piecewise arc interpolation. Finally, an example is given to illustrate how the method works.

scholarly journals MEDIAL AXIS APPROXIMATION AND UNSTABLE FLOW COMPLEX

2008 ◽  
Vol 18 (06) ◽  
pp. 533-565
Author(s):  
JOACHIM GIESEN ◽  
EDGAR A. RAMOS ◽  
BARDIA SADRI
Keyword(s):  
Medial Axis ◽  
Piecewise Linear ◽  
Unstable Flow ◽  
Steepest Ascent ◽  
The Core ◽  
Discrete Sample ◽  
Bounded Open Subset ◽  
Point Set ◽  
Definition Of ◽  
Flow Map

The medial axis of a shape is known to carry a lot of information about the shape. In particular, a recent result of Lieutier establishes that every bounded open subset of ℝn has the same homotopy type as its medial axis. In this paper we provide an algorithm that computes a structure we call the core for the approximation of the medial axis of a shape with smooth boundary from a discrete sample of its boundary. The core is a piecewise linear cell complex that is guaranteed to capture the topology of the medial axis of the shape provided the sample of its boundary is sufficiently dense but not necessarily uniform. We also present a natural method for augmenting the core in order to extend it geometrically while maintaining the topological guarantees. The definition of the core and its extension are based on the steepest ascent flow map that results from the distance function induced by the sample point set. We also provide a geometric guarantee on the closeness of the core and the actual medial axis.

scholarly journals Formal Chaos Existing Conditions on a Transmission Line Circuit with a Piecewise Linear Resistor

2021 ◽  
Vol 11 (20) ◽  
pp. 9672
Author(s):  
Kazuya Ozawa ◽  
Kaito Isogai ◽  
Hideo Nakano ◽  
Hideaki Okazaki
Keyword(s):  
Transmission Line ◽  
Piecewise Linear ◽  
Sufficient Conditions ◽  
Voltage Source ◽  
One Dimensional ◽  
In Series ◽  
Load Resistor ◽  
Dc Bias ◽  
Definition Of ◽  
Bifurcation Behavior

By using one-dimensional (1-D) map methods, some lossless transmission line circuits with a short at one side terminal have been actively studied. Bifurcation results or chaotic states in the circuits have been reported. On the other hand, many weak or strong definitions such that a 1-D map is mathematically chaotic are still being studied. In such definitions, the definition of formal chaos is well known as being the most traditional and most definite. However, formal chaos existences have not been rigorously proven in such circuits. In this paper, a general lossless transmission circuit is considered first with a dc bias voltage source in series with a load resistor at one side terminal and with a three-segment piecewise linear resistor at another side terminal. Secondly, the method for deriving a 1-D map describing the behavior of the circuit is summarized. Thirdly, to provide a basis of chaotic application for the 1-D map, the mathematical definition of formal chaos and the sufficient conditions of the existence of formal chaos are discussed. Furthermore, by using Maple, formal chaos existences and bifurcation behavior of 1-D maps are presented. By using the Lyapunov exponent, the observability of formal chaos in such bifurcation processes is outlined. Finally, the principal results and the future works are summarized.

Continuous-time threshold AR(1) processes

1996 ◽  
Vol 28 (03) ◽  
pp. 728-746 ◽  
Author(s):  
O. Stramer ◽  
P. J. Brockwell ◽  
R. L. Tweedie
Keyword(s):  
Differential Equation ◽  
Weak Solution ◽  
Stochastic Differential Equation ◽  
Strong Solution ◽  
Piecewise Linear ◽  
Transition Probability ◽  
Unique Strong Solution ◽  
Direct Definition ◽  
Definition Of ◽  
Boundary Width

A threshold AR(1) process with boundary width 2δ > 0 was defined by Brockwell and Hyndman [5] in terms of the unique strong solution of a stochastic differential equation whose coefficients are piecewise linear and Lipschitz. The positive boundary-width is a convenient mathematical device to smooth out the coefficient changes at the boundary and hence to ensure the existence and uniqueness of the strong solution of the stochastic differential equation from which the process is derived. In this paper we give a direct definition of a threshold AR(1) process with δ = 0 in terms of the weak solution of a certain stochastic differential equation. Two characterizations of the distributions of the process are investigated. Both express the characteristic function of the transition probability distribution as an explicit functional of standard Brownian motion. It is shown that the joint distributions of this solution with δ = 0 are the weak limits as δ ↓ 0 of the distributions of the solution with δ > 0. The sense in which an approximating sequence of processes used by Brockwell and Hyndman [5] converges to this weak solution is also investigated. Some numerical examples illustrate the value of the latter approximation in comparison with the more direct representation of the process obtained from the Cameron–Martin–Girsanov formula and results of Engelbert and Schmidt [9]. We also derive the stationary distribution (under appropriate assumptions) and investigate stability of these processes.

A finite set of generators for the homeotopy group of a 2-manifold

1964 ◽  
Vol 60 (4) ◽  
pp. 769-778 ◽  
Author(s):  
W. B. R. Lickorish
Keyword(s):  
Real Number ◽  
Euclidean Plane ◽  
Piecewise Linear ◽  
Linear Homeomorphism ◽  
Finite Set ◽  
Definition Of ◽  
Image Position

The homeotopy group Λx of a space X is the group of all homeomorphisms of X to itself, modulo the subgroup of those homeomorphisms that are isotopic to the identity. In this paper X will be taken to be a closed oriented 2-manifold, together with a polyhedral structure, and the definition of Λx is then restricted to the consideration of piecewise-linear homeomorphisms and isotopies. Although this restriction to the polyhedral category is not really essential to what follows, it does tend to simplify some of the arguments. In (2) a homeomorphism of X was associated with every simple closed (polyhedral) curve c in X in the following way. First, let A be an annulus in the Euclidean plane parametrized by (r, θ) where 1 ≤ r ≤ 2 and θ is a real number mod 2 π. We define a homeomorphism H: A → A byH is then fixed on the boundary of A. If now e: A → X is an orientation-preserving embedding, and eA is a neighbourhood of c in X, then eHe−1|eA can be extended by the identity on X − eA to a homeomorphism h:X → X. Any piecewise linear homeomorphism hc which is isotopic to h will be called a twist about c or, if c is not specified, just a twist.

scholarly journals Fractional Line Integral

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1150
Author(s):  
Gabriel Bengochea ◽  
Manuel Ortigueira
Keyword(s):  
Fundamental Theorem ◽  
Piecewise Linear ◽  
Directional Derivatives ◽  
Definite Integral ◽  
Integral Calculus ◽  
Regular Curve ◽  
Line Integral ◽  
Linear Path ◽  
Definition Of

This paper proposed a definition of the fractional line integral, generalising the concept of the fractional definite integral. The proposal replicated the properties of the classic definite integral, namely the fundamental theorem of integral calculus. It was based on the concept of the fractional anti-derivative used to generalise the Barrow formula. To define the fractional line integral, the Grünwald–Letnikov and Liouville directional derivatives were introduced and their properties described. The integral was defined for a piecewise linear path first and, from it, for any regular curve.

scholarly journals Conservative discontinuous Galerkin schemes for nonlinear Dougherty–Fokker–Planck collision operators

2020 ◽  
Vol 86 (4) ◽  
Author(s):  
Ammar Hakim ◽  
Manaure Francisquez ◽  
James Juno ◽  
Gregory W. Hammett
Keyword(s):  
Discontinuous Galerkin ◽  
Piecewise Linear ◽  
Plasma Heating ◽  
Weak Form ◽  
Basis Set ◽  
Benchmark Problems ◽  
Reconstruction Procedure ◽  
Diffusion Term ◽  
Definition Of ◽  
Fokker Planck

We present a novel discontinuous Galerkin algorithm for the solution of a class of Fokker–Planck collision operators. These operators arise in many fields of physics, and our particular application is for kinetic plasma simulations. In particular, we focus on an operator often known as the ‘Lenard–Bernstein’ or ‘Dougherty’ operator. Several novel algorithmic innovations, based on the concept of weak equality, are reported. These weak equalities are used to define weak operators that compute primitive moments, and are also used to determine a reconstruction procedure that allows an efficient and accurate discretization of the diffusion term. We show that when two integrations by parts are used to construct the discrete weak form, and finite velocity-space extents are accounted for, a scheme that conserves density, momentum and energy exactly is obtained. One novel feature is that the requirements of momentum and energy conservation lead to unique formulas to compute primitive moments. Careful definition of discretized moments also ensure that energy is conserved in the piecewise linear case, even though the kinetic-energy term, $v^{2}$ is not included in the basis set used in the discretization. A series of benchmark problems is presented and shows that the scheme conserves momentum and energy to machine precision. Empirical evidence also indicates that entropy is a non-decreasing function. The collision terms are combined with the Vlasov equation to study collisional Landau damping and plasma heating via magnetic pumping.

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