scholarly journals On analogue of one problem of Erdös for non-periodic splines on the real domain

2013 ◽  
Vol 21 ◽  
pp. 125
Author(s):  
V.A. Kofanov
Keyword(s):  
Uniform Norm ◽  
Fixed Interval ◽  
The Real ◽  
Minimal Defect ◽  
Periodic Spline ◽  
Maximal Arc ◽  
Real Domain

We solve the analog of some problem of Erdös about the characterization of the non-periodic spline of order r and of minimal defect, with knots at the points $kh$, $k\in \mathbb{Z}$ and fixed uniform norm that has maximal arc lens over any fixed interval.

scholarly journals Extremal problems for non-periodic splines on real domain and their derivatives

2019 ◽  
Vol 27 (1) ◽  
pp. 28
Author(s):  
K.A. Danchenko ◽  
V.A. Kofanov
Keyword(s):  
Extremal Problem ◽  
Extremal Problems ◽  
Fixed Interval ◽  
Arc Length ◽  
Minimal Defect ◽  
Periodic Spline ◽  
Maximal Arc ◽  
Real Domain ◽  
Erdős Problem

We consider the Bojanov-Naidenov problem over the set $\sigma_{h,r}$ of all non-periodic splines $s$ of order $r$ and minimal defect with knots at the points $kh$, $k \in \mathbb{Z}$. More exactly, for given $n, r \in \mathbb{N}$; $p, A > 0$ and any fixed interval $[a, b] \subset \mathbb{R}$ we solve the following extremal problem $\int\limits_a^b |x(t)|^q dt \rightarrow \sup$, $q \geqslant p$, over the classes $\sigma_{h,r}^p(A) := \bigl\{ s(\cdot + \tau) \colon s \in \sigma_{h,r}, \| s \|_{p, \delta} \leqslant A \| \varphi_{\lambda, r} \|_{p, \delta}, \delta \in (0, h], \tau \in \mathbb{R} \bigr\}$, where $\| x \|_{p, \delta} := \sup \bigl\{ \| x \|_{L_p[a,b]} \colon a, b \in \mathbb{R}, 0 < b - a \leqslant \delta \bigr\}$, and $\varphi_{\lambda, r}$ is $(2\pi / \lambda)$-periodic spline of Euler of order $r$. In particularly, for $k = 1, ..., r - 1$ we solve the extremal problem $\int\limits_a^b |x^{(k)}(t)|^q dt \rightarrow \sup$, $q \geqslant 1$, over the classes $\sigma_{h,r}^p (A)$. Note that the problems (1) and (2) were solved earlier on the classes $\sigma_{h,r}(A, p) := \bigl\{ s(\cdot + \tau) \colon s \in \sigma_{h,r}, L(s)_p \leqslant AL(\varphi_{n,r})_p, \tau \in \mathbb{R} \bigr\}$, where $L(x)_p := \sup \bigl\{ \| x \|_{L_p[a, b]} \colon a, b \in \mathbb{R}, |x(t)| > 0, t \in (a, b) \bigr\}$. We prove that the classes $\sigma_{h,r}^p (A)$ are wider than the classes $\sigma_{h,r}(A,p)$. Similarly we solve the analog of Erdös problem about the characterisation of the spline $s \in \sigma_{h,r}^p(A)$ that has maximal arc length over fixed interval $[a, b] \subset \mathbb{R}$.

scholarly journals The Bojanov-Naidenov problem for trigonometric polynomials and periodic splines

2019 ◽  
Vol 27 (1) ◽  
pp. 3
Author(s):  
E.V. Asadova ◽  
V.A. Kofanov
Keyword(s):  
Extremal Problem ◽  
Fixed Interval ◽  
Trigonometric Polynomials ◽  
Minimal Defect ◽  
Periodic Spline

For given $n, r \in \mathbb{N}$; $p, A > 0$ and any fixed interval $[a,b] \subset \mathbb{R}$ we solve the extremal problem $\int\limits_a^b |x(t)|^q dt \rightarrow \sup$, $q \geqslant p$, over sets of trigonometric polynomials $T$ of order $\leqslant n$ and $2\pi$-periodic splines $s$ of order $r$ and minimal defect with knots at the points $k\pi / n$, $k \in \mathbb{Z}$, such that $\| T \| _{p, \delta} \leqslant A \| \sin n (\cdot) \|_{p, \delta} \leqslant A \| \varphi_{n,r} \|_{p, \delta}$, $\delta \in (0, \pi / n]$, where $\| x \|_{p, \delta} := \sup \{ \| x \|_{L_p[a,b]} \colon a, b \in \mathbb{R}, 0 < b - a < \delta\}$ and $\varphi_{n, r}$ is the $(2\pi / n)$-periodic spline of Euler of order $r$. In particular, we solve the same problem for the intermediate derivatives $x^{(k)}$, $k = 1, ..., r-1$, with $q \geqslant 1$.

On the quartic surface

1944 ◽  
Vol 40 (2) ◽  
pp. 121-145 ◽  
Author(s):  
B. Segre
Keyword(s):  
Geometric Characterization ◽  
Discontinuous Group ◽  
Quartic Surface ◽  
The Real ◽  
Birational Transformations ◽  
Projective Transformations ◽  
The Third ◽  
Real Domain ◽  
Different Types

Summary1. The projective transformations of F into itself … 1212. The flecnodal curve, and the lines of F … 1223. A geometric characterization of F, and the six different types of F in the real domain … 1234. The τ-points and τ-planes, and a notation for the lines of F … 1245. The incidence conditions for the lines of F … 1256. The tetrads of the first kind … 1267. The tetrads of the second kind … 1268. The pairs of lines of F … 1279. The tetrads of the third kind … 12910. The 16-tangent quadrics of F … 13011. The conics of the first kind … 13112. The conics of the second kind … 13213. No other irreducible conics lie on F … 13314. The tangent planes of F of multiplicity greater than 3 … 13615. On twisted curves, especially cubics and quartics, lying on F … 13816. The T-transformations … 13917. Construction of an infinite discontinuous group of birational transformations of F into itself … 14218. Deduction of an infinity of unicursal curves lying on F … 143

scholarly journals Bernstein type inequalities of different metrics for splines defined on the real domain

2013 ◽  
Vol 21 ◽  
pp. 26
Author(s):  
V.F. Babenko ◽  
V.A. Zontov
Keyword(s):  
Bernstein Type ◽  
The Real ◽  
Minimal Defect ◽  
Integrable Functions ◽  
Equidistant Nodes ◽  
Real Domain ◽  
Bernstein Type Inequalities

New sharp Bernstein type inequalities of different metrics in spaces of integrable functions for non-periodic splines of order m and minimal defect, having equidistant nodes, are obtained.

scholarly journals A Tool for the Analysis and Characterization of School Mathematical Models

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1569
Author(s):  
Jesús Montejo-Gámez ◽  
Elvira Fernández-Ahumada ◽  
Natividad Adamuz-Povedano
Keyword(s):  
Mathematical Modeling ◽  
Mathematical Models ◽  
Educational Research ◽  
Analysis Procedure ◽  
Real System ◽  
The Real ◽  
School Model ◽  
Corresponding Analysis ◽  
Three Components

This paper shows a tool for the analysis of written productions that allows for the characterization of the mathematical models that students develop when solving modeling tasks. For this purpose, different conceptualizations of mathematical models in education are discussed, paying special attention to the evidence that characterizes a school model. The discussion leads to the consideration of three components, which constitute the main categories of the proposed tool: the real system to be modeled, its mathematization and the representations used to express both. These categories and the corresponding analysis procedure are explained and illustrated through two working examples, which expose the value of the tool in establishing the foci of analysis when investigating school models, and thus, suggest modeling skills. The connection of this tool with other approaches to educational research on mathematical modeling is also discussed.

Rim—A System that Maps Human Intelligence in the Real Domain

1995 ◽  
Vol 1 (3) ◽  
pp. 279-290
Author(s):  
C. P. Kwong
Keyword(s):  
Human Intelligence ◽  
The Real ◽  
Real Domain

Linear Differential Equations in the Real Domain

1967 ◽  
Vol 51 (378) ◽  
pp. 364
Author(s):  
R. P. Gillespie ◽  
Kenneth S. Miller
Keyword(s):  
Differential Equations ◽  
Linear Differential Equations ◽  
The Real ◽  
Real Domain ◽  
Linear Differential

Kleinian characterization of asymptotic symmetries of real general relativity

1993 ◽  
Vol 441 (1913) ◽  
pp. 465-471 ◽  
Keyword(s):  
General Relativity ◽  
Automorphism Group ◽  
Group Action ◽  
Radon Transform ◽  
Symmetry Groups ◽  
Asymptotic Symmetry ◽  
The Real ◽  
The Given ◽  
Asymptotic Symmetries

According to Klein’s Erlanger programme, one may (indirectly) specify a geometry by giving a group action. Conversely, given a group action, one may ask for the corresponding geometry. Recently, I showed that the real asymptotic symmetry groups of general relativity (in any signature) have natural ‘projective’ classical actions on suitable ‘Radon transform’ spaces of affine 3-planes in flat 4-space. In this paper, I give concrete models for these groups and actions. Also, for the ‘atomic’ cases, I give geometric structures for the spaces of affine 3-planes for which the given actions are the automorphism group.

Practical Steps and Reasons for Action

1997 ◽  
Vol 27 (1) ◽  
pp. 17-45 ◽  
Author(s):  
Philip Clark
Keyword(s):  
Reasons For Action ◽  
Reasons For Belief ◽  
The Real ◽  
General Characterization ◽  
Practical Inference

There is an idea, going back to Aristotle, that reasons for action can be understood on a parallel with reasons for belief. Not surprisingly, the idea has almost always led to some form of inferentialism about reasons for action. In this paper I argue that reasons for action can be understood on a parallel with reasons for belief, but that this requires abandoning inferentialism about reasons for action. This result will be thought paradoxical. It is generally assumed that if there is to be a useful parallel, there must be some such thing as a practical inference. As we shall see, that assumption tends to block the fruitful exploration of the real parallel. On the view I shall defend, the practical analogue of an ordinary inference is not an inference, but something I shall call a practical step. Nevertheless, the practical step will do, for a theory of reasons for action, what ordinary inference does for an inferentialist theory of reasons for belief. The result is a general characterization of reasons, practical and theoretical, in terms of the correctness conditions of the relevant sorts of step.

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