Cantor Function Research Articles

Lebesgue Measure and Integration on Subsets of R^d

Henri Lebesgue, a French mathematician, discovered centuries ago that the Riemann Integral does not work well on unbounded functions. It prompts him to consider another way to integration known as Lebesgue Integral. This paper discusses the Riemann integral's shortcomings and introduce a more thorough concept of integration, the Lebesgue integral, repeated integration. There is also some debate about the Lebesgue measure, which determines the Lebesgue integral. Some examples are given, such as F_σ -set, G_δ -set and Cantor function. In this article, we first look into a unified theory for d-dimensional volume based on the concept of a measure, and then we will use that theory to build a stronger and more flexible theory for integration.

A Devil’s Staircase to Urysohn’s Lemma

Summary We show that the same recursive construction that produces the Cantor function also proves Urysohn’s lemma.

A Classification of Elements of Function Space F(R,R)

In this paper, we present a constructive description of the function space of all real-valued functions on R(F(R,R)) by presenting a partition of it into 28 distinct blocks and a closed-form formula for the representative function of each of them. Each block contains elements that share common features in terms of the cardinality of their sets of continuity and differentiability. Alongside this classification, we introduce the concept of the Connection, which reveals a special relationship structure between the well-known representatives of four of the blocks: the Cantor function, the Dirichlet function, the Thomae function, and the Weierstrass function. Despite the significance of this field, several perspectives remain unexplored.

The Regularization Technique in Modeling of the Plane E-Polarized EM Wave Scattering by Coplanar System of Electrically Conducting Flat Strips

The integral equations technique has been successfully modified for studying the plane E-polarized electromagnetic wave scattering by multilevel coplanar systems of zero-thickness impedance strips. Reformulation of the scattering problem in the form of the second-kind regular integral equations has been realized on the base of the Carleman regularization technique. Two novel and original classes of specific Cantor functions have been presented and analyzed. Using new Cantor functions, it is easy to create a lot of non-classical orderings for specific multilevel coplanar strips systems. That sort of system can be useful for modeling irregular natural processes and zones in the future. Considerable attention was focused on the plane E-polarized electromagnetic wave scattering by sparsely filled coplanar systems of electrically narrow impedance strips. An explicit solution of the scattering problem has been obtained for such case of strips system.

Seshadri functions on abelian surfaces

So far, Seshadri constants on abelian surfaces are completely understood only in the cases of Picard number one and on principally polarized abelian surfaces with real multiplication. Beyond that, there are partial results for products of elliptic curves. In this paper, we show how to compute the Seshadri constant of any nef line bundle on any abelian surface over the complex numbers. We develop an effective algorithm depending only on the basis of the Néron-Severi group to compute not only the Seshadri constants but also the numerical data of their Seshadri curves. Access to the Seshadri curves allows us to plot Seshadri functions and better understand their structure. We show that already in the case of Picard number two the complexity of Seshadri functions can vary to a great degree. Our results indicate that aside from finitely many cases the complexity of the Seshadri function is at least as high as in the Cantor function.

Compact Tree Structures for Mining High Utility Itemsets

High Utility Item set Mining (HUIM) from large transaction databases has garnered significant attention as it accounts for the revenue of the items purchased in a transaction. Existing tree-based HUIM algorithms discard unpromising items and require at most two database scans for their construction. Hence, whenever utility threshold is changed, the trees have to be reconstructed from scratch. In this regard, the current study proposes to not only incorporate all the items in the tree structure but compactly represent transaction information. The proposed trees namely- Utility Prime Tree (UPT), Prime Cantor Function Tree (PCFT), and String based Utility Prime Tree (SUPT) store transaction-level information in a node unlike item-based prefix trees. Experiments conducted on both real and synthetic datasets compare the execution time and memory of these tree structures with a proposed Utility Count Tree (UCT) and existing IHUP, UP-Growth trees. Due to transaction-level encoding, these structures consume significantly less memory when compared to the tree structures in the literature.

Seshadri constants on principally polarized abelian surfaces with real multiplication

Seshadri constants on abelian surfaces are fully understood in the case of Picard number one. Little is known so far for simple abelian surfaces of higher Picard number. In this paper we investigate principally polarized abelian surfaces with real multiplication. They are of Picard number two and might be considered the next natural case to be studied. The challenge is to not only determine the Seshadri constants of individual line bundles, but to understand the whole Seshadri function on these surfaces. Our results show on the one hand that this function is surprisingly complex: on surfaces with real multiplication in mathbb {Z}[sqrt{e}] it consists of linear segments that are never adjacent to each other—it behaves like the Cantor function. On the other hand, we prove that the Seshadri function is invariant under an infinite group of automorphisms, which shows that it does have interesting regular behavior globally.

Cantor集与Cantor函数性质探究

Cantor集与Cantor函数性质探究

Асимптотична поведiнка модуля характеристичної функцiї розподiлу Кантора

The asymptotic behavior of the modulus of a characteristic function of a random variable, the distribution function of which is the classical singular Cantor function, is investigated. The emphasis is on calculating the upper bound of the modulus of the characteristic Cantor distribution function. The probabilistic measure corresponding to Cantor's distribution belongs to the class of Bernoulli's symmetric convolutions, the interest in which is considerable today. Bernoulli's symmetrical convolutions were actively studied by both domestic mathematicians: M. Pratsovyty, G. Turbin, G. Torbin, J. Honcharenko, O. Baranovsky and others, and foreign ones: Erdos P, Peres Y, Schlag W, Solomyak B, Albeverio, S and other. The value of the upper bound of the modulus of the characteristic function plays an important role in the problem of determining the Lebesgue structure of distributions of sums of probably convergent random series with independent discrete terms (random values of the Jessen-Winter type). The exact value of the upper bound of the module of the characteristic Cantor distribution function is found in the article.

PROOFS OF URYSOHN’S LEMMA AND THE TIETZE EXTENSION THEOREM VIA THE CANTOR FUNCTION

AbstractUrysohn’s lemma is a crucial property of normal spaces that deals with separation of closed sets by continuous functions. It is also a fundamental ingredient in proving the Tietze extension theorem, another property of normal spaces that deals with the existence of extensions of continuous functions. Using the Cantor function, we give alternative proofs for Urysohn’s lemma and the Tietze extension theorem.

Generalized Cantor functions: random function iteration

We provide a generalization of the classical Cantor function. One characterization of the Cantor function is generated by a sequence of real numbers that starts with a seed value and at each step randomly applies one of two different linear functions. The Cantor function is defined as the probability that this sequence approaches infinity. We generalize the Cantor function to instead use a set of any number of linear functions with integer coefficients. We completely describe the resulting probability function and give a full explanation of which intervals of seed values lead to a constant probability function value.

Enhancing the Security in ElGamal Cryptosystem using Paring Functions

The potential disadvantage of ElGamal cryptosystem is the ciphertext produced is always twice as long as the plaintext i.e., the message expansion by a factor of two takes place during encryption. When the message is too long the ciphertext produced by the ElGamal cryptosystem is also too long. i.e., when the ciphertexts are transmitted through the communication channel which lead to provide less security because if anyone of the ciphertext from two ciphertexts for each character of the plaintext is intercepted by the adversary, the other may be retrieved easily because there is a relationship between the two ciphertexts. If two ciphertexts are reduced to one, the adversary may not be able to predict the two ciphertexts from one. To enhance the security of ElGamal cryptosystem, the binary Cantor function , Rosenberg pairing function and Elegant pairing functions are used in this paper. When the said functions are used, the two ciphertexts produced by each plaintext character are reduced to one, so that the adversary cannot easily be recovered the plaintext. Experimental results clearly revealed enhancing the security of ElGamal cryptosystem after incorporating the pairing functions into it.

Dimension of the Non-Differentiability Subset of the Cantor Function

The main purpose of this note is to estimate the size of the set Tμλ of points, at which the Cantor function is not differentiable and we find that the Hausdorff dimension of Tμλ is [log2/log3]2. Moreover, the Packing dimension of Tμλ is log2/log3. The log2 = loge2 is that if ax = N (a >0, and a≠1), then the number x is called the logarithm of N with a base, recorded as x = logaN, read as the logarithm of N with a base, where a is called logarithm Base number, N is called true number.

Characterization of the Local Growth of Two Cantor-Type Functions

The Cantor set and its homonymous function have been frequently utilized as examples for various physical phenomena occurring on discontinuous sets. This article characterizes the local growth of the Cantor’s singular function by means of its fractional velocity. It is demonstrated that the Cantor function has finite one-sided velocities, which are non-zero of the set of change of the function. In addition, a related singular function based on the Smith–Volterra–Cantor set is constructed. Its growth is characterized by one-sided derivatives. It is demonstrated that the continuity set of its derivative has a positive Lebesgue measure of 1/2.

Five Practices for Supporting Inquiry in Analysis

This article describes an inquiry-oriented real analysis classroom in which students were guided to discover mathematics for themselves. To support student inquiry, the framework of “five practices” from K-12 education was used. To illustrate this framework, two case examples are given from actual discussions that took place in this classroom. These examples focus on student construction of sequences of functions, both the space-filling Z-order curve and the Cantor staircase function. The article closes with practical guidance for other instructors who would use the five practices in their higher-level mathematics courses.

Invariant Submanifolds of Genus 5 and a Cantor Staircase in the Nonholonomic Model of a Snakeboard

In this paper, we address the free (uncontrolled) dynamics of a snakeboard consisting of two wheel pairs fastened to a platform. The snakeboard is one of the well-known sports vehicles on which the sportsman executes necessary body movements. From the theoretical point of view, this system is a direct generalization of the classical nonholonomic system of the Chaplygin sleigh. We carry out a topological and qualitative analysis of trajectories of this dynamical system. An important feature of the problem is that the common level set of first integrals is a compact two-dimensional surface of genus 5. We specify conditions under which the reaction forces infinitely increase during motion and the so-called phenomenon of nonholonomic jamming is observed. In this case, the nonholonomic model ceases to work and it is necessary to use more complex mechanical models incorporating sliding, elasticity, etc.

Distribution of values of classic singular Cantor function of random argument

Abstract Let X be a random variable with independent ternary digits and let y = F ⁢ ( x ) {y=F(x)} be a classic singular Cantor function. For the distribution of the random variable Y = F ⁢ ( X ) {Y=F(X)} , the Lebesgue structure (i.e., the content of discrete, absolutely continuous and singular components), the structure of its point and the continuous spectra are exhaustively studied.

Generalized riemann integral on fractal sets

The theory of integration to mathematical analysis is so important that many mathematicians continue to develop new theory to enlarge the class of integrable functions and simplify the Lebesgue theory integration. In this paper, by slight modifying the definition of the Henstock integral which was introduced by Jaroslav Kurzweil and Ralph Henstock, we present a new definition of integral on fractal sets. Furthermore, its integrability has been discussed, and the relationship between differentiation and integral is also established. As an example, the integral of Cantor function on Cantor set is calculated.

The infinite derivatives of Okamoto's self-affine functions: an application of $\beta$-expansions

Okamoto's one-parameter family of self-affine functions F_a: [0,1] \to [0,1] , where 0 < a < 1 , includes the continuous nowhere differentiable functions of Perkins ( a=5/6 ) and Bourbaki/Katsuura ( a=2/3 ), as well as the Cantor function ( a=1/2 ). The main purpose of this article is to characterize the set of points at which F_a has an infinite derivative. We compute the Hausdorff dimension of this set for the case a \leq 1/2 , and estimate it for a > 1/2 . For all a , we determine the Hausdorff dimension of the sets of points where: (i) F_a'=0 ; and (ii) F_a has neither a finite nor an infinite derivative. The upper and lower densities of the digit 1 in the ternary expansion of x \in [0,1] play an important role in the analysis, as does the theory of \beta -expansions of real numbers.

SINGULARITY ORDER OF THE RIESZ-NÁGY-TAKÁCS FUNCTION

Abstract. We give the characterization of H¨older differentiability pointsand non-differentiability points of the Riesz-N´agy-Taka´cs (RNT) singularfunction Ψ a,p satisfying Ψ a,p (a) = p. It generalizes recent multifractaland metric number theoretical results associated with the RNT function.Besides, we classify the singular functions using the singularity order de-duced from the H¨older derivative giving the information that a strictlyincreasing smooth function having a positive derivative Lebesgue almosteverywhere has the singularity order 1 and the RNT function Ψ a,p hasthe singularity order g(a,p) = alogp+(1−a)log(1−p)aloga+(1−a)log(1−a) ≥1. 1. IntroductionRecently many ([8, 9, 10, 16]) studied the Cantor function, a singular func-tion which is not strictly increasing and the Minkowski’s Question Mark func-tion which is a strictly increasing singular function. Also J. Parad´isetal. ([17])studied some conditions of the null and infinite derivatives of the RNT strictlyincreasing singular function using metric number theory.Recently we ([4]) also studied multifractal characterization of the null andinfinite derivative sets and the non-differentiability set of the RNT singularfunction, which is the typical singular function related to mutifractal theory.In this paper, we employ the Ho¨lder derivative, which is a generalized form ofthe usual derivative, of the RNT function on the unit interval. This definitionextends the concept of the singularity to the general singularity for a strictlyincreasing continuous function. For every point in the unit interval, we ([4])can give some code or dyadic expansion using digit 0 and 1, generating thedistribution set determined by the frequency of the zero in its expansion. Thedistribution sets in the unit interval are the local dimension sets by a self-similar