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34th Brazilian Conference on Intelligent Systems

作者： Viana, Joaquim Matos, Helder Mota, Marcelle Santos, Reginaldo Fed Univ Para Inst Ciencias Exatas Naturais Belem Para Brazil

ISBN: (纸本)9783031790287;9783031790294

The classification of images of elementary mathematical function graphs presents a significant challenge in computer vision;this is due to the varied shapes and formats of each functions curves. This classification is crucial for identifying function graphs, which have important applications in text and mathematical symbol recognition technologies, aiding visually impaired individuals by providing access to printed content. In educational environments, this identification helps obtain the analytical expression of drawn graphs, facilitating the extraction of information from educational materials. This article investigates various convolutional neural network (CNN) architectures to identify the most suitable model for classifying images of elementary mathematical function graphs. We compare our model with other renowned architectures, such as ResNet, MobileNet, and EfficientNet, using a custom dataset of function graphs. Our experiments show that the proposed architecture significantly outperforms networks of general purpose, achieving an accuracy of 98.51% in classifying elementary mathematical function graphs.

关键词： graph of a function Curve recognition Image classification Convolutional Neural Networks Deep Learning

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A Lyapunov-type approach to convergence of the Douglas-Rachford algorithm for a nonconvex setting

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JOURNAL OF GLOBAL OPTIMIZATION 2019年第1期73卷 83-112页

作者： Dao, Minh N. Tam, Matthew K. Univ Newcastle CARMA Callaghan NSW 2308 Australia Univ Gottingen Inst Numer & Angew Math D-37083 Gottingen Germany

The Douglas-Rachford projection algorithm is an iterative method used to find a point in the intersection of closed constraint sets. The algorithm has been experimentally observed to solve various nonconvex feasibility problems;an observation which current theory cannot sufficiently explain. In this paper, we prove convergence of the Douglas-Rachford algorithm in a potentially nonconvex setting. Our analysis relies on the existence of a Lyapunov-type functional whose convexity properties are not tantamount to convexity of the original constraint sets. Moreover, we provide various nonconvex examples in which our framework proves global convergence of the algorithm.

关键词： Douglas-Rachford algorithm Feasibility problem Global convergence graph of a function Linear convergence Lyapunov function Method of alternating projections Newton's method Nonconvex set Projection Stability Zero of a function

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Sum of graphs of continuous functions and boundedness of additive operators

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2005年第2期312卷 527-534页

作者： Jablonski, W Univ Rzeszow Dept Math PL-35310 Rzeszow Poland

Assume that f : D-1 -> R and g : D-2 -> R are uniformly continuous functions, where D-1, D-2 C X are nonempty open and arc-connected subsets of a real normed space X. We prove that then either f and g are affine functions, that is f(x) = x*(x) + a and g (x) = x*(x) + b with some x* is an element of X* and a, b is an element of R or the algebraic sum of graphs of functions f and g has a nonempty interior in a product space X x R treated as a normed space with a norm vertical bar vertical bar(x, a)vertical bar vertical bar = root vertical bar vertical bar x vertical bar vertical bar(2) + vertical bar a vertical bar(2). (c) 2005 Elsevier Inc. All rights reserved.

关键词： Jensen function additive function graph of a function

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On two conjectures about the intersection distribution

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JOURNAL OF ALGEBRAIC COMBINATORICS 2022年第4期55卷 1289-1298页

作者： Li, Yubo Li, Kangquan Qu, Longjiang Natl Univ Def Technol Coll Liberal Arts & Sci Changsha 410073 Peoples R China Hunan Engn Res Ctr Commercial Cryptog Theory & Te Changsha 410073 Peoples R China

Recently, Li and Pott proposed a new concept of intersection distribution concerning the interaction between the graph {(x, f (x)) vertical bar x is an element of F-q } of f and the lines in the classical affine plane AG(2, q). Later, Kyureghyan et al. proceeded to consider the next simplest case, and derived the intersection distribution for all degree three polynomials over F-q with q both odd and even. They also proposed several conjectures therein. In this paper, we completely solve two conjectures of Kyureghyan et al. Namely, we prove two classes of power functions having intersection distribution: v(0)(f) = q(q-1)/3, v(1)(f) = q(q+1)/2, v(2)(f) = 0, v(3)(f) = q(q-1)/6. We mainly make use of the multivariate method and a certain type of equivalence on 2-to-1 mappings. The key point of our proof is to consider the number of the solutions of some low-degree equations.

关键词： graph of a function Intersection distribution Polynomial 2-to-1 mapping

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Almost-Linear Segments of graphs of functions

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MATHEMATICAL NOTES 2019年第5-6期106卷 720-726页

作者： Zubkov, A. M. Orlov, O. P. Russian Acad Sci Steklov Math Inst Moscow 119991 Russia Lomonosov Moscow State Univ Moscow 119991 Russia

Let f: Double-struck capital R -> Double-struck capital R be a function whose graph {(x, f(x))}(x is an element of Double-struck capital R) in Double-struck capital R-2 is a rectifiable curve. It is proved that, for all L < infinity and e > 0, there exist points A = (a, f(a)) and B = (b, f(b)) such that the distance between A and B is greater than L and the distances from all points (x, f(x)), a <= x <= b, to the segment AB do not exceed epsilon|AB|. An example of a plane rectifiable curve for which this statement is false is given. It is shown that, given a coordinate-wise nondecreasing sequence of integer points of the plane with bounded distances between adjacent points, for any r < infinity, there exists a straight line containing at least r points of this sequence.

关键词： rectifiable curve graph of a function discrete geometry

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A ONE-ONE SELECTION THEOREM

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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 1986年第2期97卷 320-322页

作者： SARBADHIKARI, H Indian Statistical Institute Division of Theoretical Statistics and Mathematics Calcutta 700 035 203 Barrackpore Trunk Road India

Abstract: Let $X$, $Y$ be Polish spaces without isolated points and $B \subseteq X \times Y$ a Borel set such that $x:{B_x}$ is nonmeager is comeager in $X$ and $y:{B^y}$ is nonmeager is comeager in $Y$. There is a comeager Borel $E \subseteq X$, a comeager Borel $F \subseteq Y$ and a Borel isomorphism $f$ from $E$ onto $F$ such that graph of $f \subseteq B$.

关键词： Comeager set Borel isomorphism graph of a function

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n-point sets and graphs of functions

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TOPOLOGY AND ITS APPLICATIONS 2008年第16期155卷 1804-1807页

作者： Cobb, John Univ Idaho Dept Math Moscow ID 83844 USA

We prove that, for every n >= 2, there exists an n-point set (a plane set which hits every line in exactly n points) that is homeomorphic to the graph of a function from R to R: for n >= 4, there exist both 0-dimensional and 1-dimensional examples. This raises the question (which we do not answer) of whether n-point sets for different n's could be homeomorphic. (C) 2008 Elsevier B.V. All rights reserved.

关键词： n-point set planar set graph of a function

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CONDITION FOR A function SPACE TO BE LOCALLY COMPACT

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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 1972年第2期36卷 615-617页

作者： FULLER, RV UNIV S CAROLINA DEPT MATHCOLUMBIASC 29208

Let F be an equicontinuous family of functions from a compact Hausdorff space to a locally compact Hausdorff uniform space. In this paper we prove that the pointwise closure of F is locally compact relative to the top... 详细信息

关键词： function space hyperspace finite topology upper semifinite topology lower semifinite topology topology of pointwise convergence topology of uniform convergence equicontinuous family of functions graph of a function graph topology

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Visualization of functions of two variables using Mathematica

Visualization of functions of two variables using Mathematic...

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9th Iberian Conference on Information Systems and Technologies (CISTI)

作者： Ramos, Higinio Nieto, Susana Univ Salamanca EPS Dept Matemat Aplicada Zamora Spain

ISBN: (纸本)9781479961115

functions of two variables appear frequently in engineering studies. Students have particular difficulties in understanding how the graphs of these functions are. This paper presents a sequential mode to analyze and reconstruct the graphical information in a function of two variables, up to display its three-dimensional image. This is done by representing the cuts of the graph of the function with a family of horizontal planes, but placing each contour line in the corresponding sectional plane. Cuts of the graph of the function with different vertical planes are also shown: those with parallel planes to the axial ones, OYZ and OXZ, and those with vertical planes through the axis OZ. The combined display of all these curves produces a much better idea of how the graph of the function under study is.

关键词： functions of two variables graph of a function Level sets Mathematica system

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On almost continuous functions and peculiar points

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EUROPEAN JOURNAL OF MATHEMATICS 2019年第1期5卷 106-115页

作者： Loranty, Anna Pawlak, Ryszard J. Terepeta, Malgorzata Univ Lodz Fac Math & Comp Sci Banacha 22 PL-90238 Lodz Poland Ctr Math & Phys Al Politech 11 PL-90924 Lodz Poland Lodz Univ Technol Inst Math Ul Wolczanska 215 PL-90924 Lodz Poland

We introduce the concept of a peculiar point (of the first and second kind), which combines stability of functions around a given point on a large set in the sense of Lebesgue measure with strong chaos of a function (in the sense of its entropy value) around this point. We prove that almost continuity of a function is equivalent to the fact that in every Gamma-neighbourhood of this function one can find a continuous function having a peculiar point either of the first or second kind.

关键词： Almost continuity graph of a function Dynamical system Entropy point (Strongly) O-approximate continuity (Almost) stable point Odd point Peculiar point

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