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1
Content available remote

Analytic and $C^k$ approximations of norms in separable Banach spaces

100%
Deville R., Fonf V., Hájek P.
Studia Mathematica
|
1996
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tom 120
|
nr 1
61-74
EN
We prove that in separable Hilbert spaces, in $ℓ_{p}(ℕ)$ for p an even integer, and in $L_{p}[0,1]$ for p an even integer, every equivalent norm can be approximated uniformly on bounded sets by analytic norms. In $ℓ_{p}(ℕ)$ and in $L_{p}[0,1]$ for p ∉ ℕ (resp. for p an odd integer), every equivalent norm can be approximated uniformly on bounded sets by $C^[p]}$-smooth norms (resp. by $C^{p-1}$-smooth norms).
2
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Approximation of Jacobian inverse kinematics algorithms

100%
Tchoń K., Karpińska J., Janiak M.
International Journal of Applied Mathematics and Computer Science
|
2009
|
tom 19
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nr 4
519-531
EN
This paper addresses the synthesis problem of Jacobian inverse kinematics algorithms for stationary manipulators and mobile robots. Special attention is paid to the design of extended Jacobian algorithms that approximate the Jacobian pseudoinverse algorithm. Two approaches to the approximation problem are developed: one relies on variational calculus, the other is differential geometric. Example designs of the extended Jacobian inverse kinematics algorithm for 3DOF manipulators as well as for the unicycle mobile robot illustrate the theoretical concepts.
3
Content available remote

Volume approximation of convex bodies by polytopes - a constructive method

100%
Gordon Y., Meyer M., Reisner S.
Studia Mathematica
|
1994
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tom 111
|
nr 1
81-95
EN
Algorithms are given for constructing a polytope P with n vertices (facets), contained in (or containing) a given convex body K in $ℝ^d$, so that the ratio of the volumes |K∖P|/|K| (or |P∖K|/|K|) is smaller than $f(d)/n^{2/(d-1)}$.
4
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Optimal approximation simulation and analog realization of the fundamental fractional order transfer function

100%
Djouambi A., Charef A., Besancon A. V.
International Journal of Applied Mathematics and Computer Science
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2007
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tom 17
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nr 4
455-462
EN
This paper provides an optimal approximation of the fundamental linear fractional order transfer function using a distribution of the relaxation time function. Simple methods, useful in systems and control theories, which can be used to approximate the irrational transfer function of a class of fractional systems fora given frequency band by a rational function are presented. The optimal parameters of the approximated model are obtained by minimizing simultaneously the gain and the phase error between the irrational transfer function and its rational approximation. A simple analog circuit, which can serve as a fundamental analog fractional system is obtained. Illustrative examples are presented to show the quality and usefulness of the approximation method.
5
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Recognition of atherosclerotic plaques and their extended dimensioning with computerized tomography angiography imaging

100%
Markiewicz T., Dziekiewicz M., Maruszyński M., Bogusławska-Walecka R., Kozłowski W.
International Journal of Applied Mathematics and Computer Science
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2014
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tom 24
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nr 1
33-47
EN
In this paper the authors raise the issue of automatic discrimination of atherosclerotic plaques within an artery lumen based on numerical and statistical thresholding of Computerized Tomography Angiographic (CTA) images and their advanced dimensioning as a support for preoperative vessel assessment. For the study, a set of tomograms of the aorta, as well as the ilio-femoral and femoral arteries were examined. In each case a sequence of about 130-480 images of the artery cutoff planes were analyzed prior to their segmentation based on morphological image transformation. A crucial step in the staging of atherosclerotic alteration is recognition of the plaque in the CTA image. To solve this problem, statistical and linear fitting methods, including the least-squares approximation by polynomial and spline polynomial functions, as well as the error fitting function were used. Also, new descriptors of atherosclerotic changes, such as the lumen decrease factor, the circumference occupancy factor, and the convex plaque area factor, are proposed as a means of facilitating preoperative vessel examination. Finally, ways to reduce the computational time are discussed. The proposed methods can be very useful for automatic quantification of atherosclerotic changes visualized by CTA imaging.
6
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On some generalization of box splines

100%
Wronicz Z.
Annales Polonici Mathematici
|
1999
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tom 72
|
nr 3
261-271
EN
We give a generalization of box splines. We prove some of their properties and we give applications to interpolation and approximation of functions.
7
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The operation of infimal convolution

90%
Strömberg T.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS 1. Introduction and preliminaries....................................................5  1.1. Introduction............................................................................5  1.2. Organization..........................................................................6  1.3. Prerequisites.........................................................................6  1.4. Introductory examples...........................................................9 2. Elementary properties.............................................................14  2.1. Basic facts...........................................................................14  2.2. Infimal convolution of subadditive functions.........................17  2.3. Semicontinuity, continuity, and exactness............................19  2.4. Two examples......................................................................22 3. The convex case.....................................................................23  3.1. Basic results........................................................................23  3.2. Differential calculus, and first order differentiability..............28  3.3. Formulas on f ▫ g.................................................................31  3.4. Loss of differentiability.........................................................32 4. Continuity of the operation of infimal convolution....................33  4.1. Introduction.........................................................................34  4.2. Epi-convergence.................................................................35  4.3. The Mosco topology and the slice topology.........................36  4.4. The affine topology..............................................................38  4.5. The Attouch-Wets topology.................................................39 5. Regularization.........................................................................41  5.1. Introduction and first results................................................41  5.2. Approximation in Hilbert spaces...........................................47  5.3. Generalized Moreau-Yosida approximation.........................52 References..................................................................................55
8
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An application of the Fourier transform to optimization of continuous 2-D systems

88%
Dymkou V., Dymkov M.
International Journal of Applied Mathematics and Computer Science
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2003
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tom 13
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nr 1
45-54
EN
This paper uses the theory of entire functions to study the linear quadratic optimization problem for a class of continuous 2D systems. We show that in some cases optimal control can be given by an analytical formula. A simple method is also proposed to find an approximate solution with preassigned accuracy. Some application to the 1D optimization problem is presented, too. The obtained results form a theoretical background for the design problem of optimal controllers for relevant processes.
9
Content available remote

Approximation polynomiale et extension holomorphe avec croissance sur une variété algébrique

88%
Zeriahi A.
Annales Polonici Mathematici
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1996
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tom 63
|
nr 1
35-50
EN
We first give a general growth version of the theorem of Bernstein-Walsh-Siciak concerning the rate of convergence of the best polynomial approximation of holomorphic functions on a polynomially convex compact subset of an affine algebraic manifold. This can be considered as a quantitative version of the well known approximation theorem of Oka-Weil. Then we give two applications of this theorem. The first one is a generalization to several variables of Winiarski's theorem relating the growth of an entire function to the rate of convergence of its best polynomial approximation; the second application concerns the extension with growth of an entire function from an algebraic submanifold to the whole space.
10
Content available remote

Pointwise inequalities for Sobolev functions and some applications

88%
Bojarski B., Hajłasz P.
Studia Mathematica
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1993
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tom 106
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nr 1
77-92
EN
We get a class of pointwise inequalities for Sobolev functions. As a corollary we obtain a short proof of Michael-Ziemer's theorem which states that Sobolev functions can be approximated by $C^m$ functions both in norm and capacity.
11
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Determining the weights of a Fourier series neural network on the basis of the multidimensional discrete Fourier transform

88%
Halawa K.
International Journal of Applied Mathematics and Computer Science
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2008
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tom 18
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nr 3
369-375
EN
This paper presents a method for training a Fourier series neural network on the basis of the multidimensional discrete Fourier transform. The proposed method is characterized by low computational complexity. The article shows how the method can be used for modelling dynamic systems.
12
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The divergence phenomena of interpolation type operators in $L^p$ space

88%
Xie T. F., Zhou S. P.
Colloquium Mathematicum
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1992
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tom 63
|
nr 2
323-328
13
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A survey of subpixel edge detection methods for images of heat-emitting metal specimens

88%
Fabijańska A.
International Journal of Applied Mathematics and Computer Science
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2012
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tom 22
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nr 3
695-710
EN
In this paper the problem of accurate edge detection in images of heat-emitting specimens of metals is discussed. The images are provided by the computerized system for high temperature measurements of surface properties of metals and alloys. Subpixel edge detection is applied in the system considered in order to improve the accuracy of surface tension determination. A reconstructive method for subpixel edge detection is introduced. The method uses a Gaussian function in order to reconstruct the gradient function in the neighborhood of a coarse edge and to determine its subpixel position. Results of applying the proposed method in the measurement system considered are presented and compared with those obtained using different methods for subpixel edge detection.
14
Content available remote

Smooth approximations without critical points

88%
Hájek P., Johanis M.
Open Mathematics
|
2003
|
tom 1
|
nr 3
284-291
EN
In any separable Banach space containing c 0 which admits a C k-smooth bump, every continuous function can be approximated by a C k-smooth function whose range of derivative is of the first category. Moreover, the approximation can be constructed in such a way that its derivative avoids a prescribed countable set (in particular the approximation can have no critical points). On the other hand, in a Banach space with the RNP, the range of the derivative of every smooth bounded bump contains a set residual in some neighbourhood of zero.
15
Content available remote

Introduction to Diophantine Approximation

88%
Watase Y.
Formalized Mathematics
|
2015
|
tom 23
|
nr 2
101-106
EN
In this article we formalize some results of Diophantine approximation, i.e. the approximation of an irrational number by rationals. A typical example is finding an integer solution (x, y) of the inequality |xθ − y| ≤ 1/x, where 0 is a real number. First, we formalize some lemmas about continued fractions. Then we prove that the inequality has infinitely many solutions by continued fractions. Finally, we formalize Dirichlet’s proof (1842) of existence of the solution [12], [1].
16
Content available remote

Polynomial asymptotics and approximation of Sobolev functions

88%
Hajłasz P., Kałamajska A.
Studia Mathematica
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1995
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tom 113
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nr 1
55-64
EN
We prove several results concerning density of $C_{0}^{∞}$, behaviour at infinity and integral representations for elements of the space $L^{m,p} = {⨍ | ∇^{m}⨍ ∈ L^p}$.
17
Content available remote

"Counterexamples" to the harmonic Liouville theorem and harmonic functions with zero nontangential limits

75%
Bonilla A.
Colloquium Mathematicum
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2000
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tom 83
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nr 2
155-160
EN
We prove that, if μ>0, then there exists a linear manifold M of harmonic functions in $ℝ^N$ which is dense in the space of all harmonic functions in $ℝ^N$ and lim_{{‖x‖→∞} {x ∈ S}} ‖x‖^{μ}D^{α}v(x) = 0 for every v ∈ M and multi-index α, where S denotes any hyperplane strip. Moreover, every nonnull function in M is universal. In particular, if μ ≥ N+1, then every function v ∈ M satisfies ∫_H vdλ =0 for every (N-1)-dimensional hyperplane H, where λ denotes the (N-1)-dimensional Lebesgue measure. On the other hand, we prove that there exists a linear manifold M of harmonic functions in the unit ball 𝔹 of $ℝ^N$, which is dense in the space of all harmonic functions and each function in M has zero nontangential limit at every point of the boundary of 𝔹.
18
Content available remote

A note on convergence of semigroups

75%
Bobrowski A.
Annales Polonici Mathematici
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1998
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tom 69
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nr 2
107-127
EN
Convergence of semigroups which do not converge in the Trotter-Kato-Neveu sense is considered.
19
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Almost sure approximation of unbounded operators in $L_2 (X,A,μ)$

75%
Jajte R., Paszkiewicz A.
Studia Mathematica
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1998
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tom 128
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nr 2
103-120
EN
The possibilities of almost sure approximation of unbounded operators in $L_2(X,A,μ)$ by multiples of projections or unitary operators are examined.
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