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 $$\mathbb {Z}[\sqrt{e}]$$ Z [ 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.

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

Urysohn’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.

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.

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.

Distribution of values of classic singular Cantor function of random argument

Let X be a random variable with independent ternary digits and let {y=F(x)} be a classic singular Cantor function. For the distribution of the random variable {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.

NON-ARCHIMEDEAN SCALE INVARIANCE AND CANTOR SETS

The framework of a new scale invariant analysis on a Cantor set C ⊂ I = [0,1], presented recently1 is clarified and extended further. For an arbitrarily small ε > 0, elements [Formula: see text] in I\C satisfying [Formula: see text], x ∈ C together with an inversion rule are called relative infinitesimals relative to the scale ε. A non-archimedean absolute value [Formula: see text], ε → 0 is assigned to each such infinitesimal which is then shown to induce a non-archimedean structure in the full Cantor set C. A valued measure constructed using the new absolute value is shown to give rise to the finite Hausdorff measure of the set. The definition of differentiability on C in the non-archimedean sense is introduced. The associated Cantor function is shown to relate to the valuation on C which is then reinterpreated as a locally constant function in the extended non-archimedean space. The definitions and the constructions are verified explicitly on a Cantor set which is defined recursively from I deleting q number of open intervals each of length [Formula: see text] leaving out p numbers of closed intervals so that p + q = r.

ANALYSIS ON A FRACTAL SET

The formulation of a new analysis on a zero measure Cantor set C(⊂I = [0,1]) is presented. A non-Archimedean absolute value is introduced in C exploiting the concept of relative infinitesimals and a scale invariant ultrametric valuation of the form log ε-1 (ε/x) for a given scale ε > 0 and infinitesimals 0 < x < ε, x ∈ I\C. Using this new absolute value, a valued (metric) measure is defined on C and is shown to be equal to the finite Hausdorff measure of the set, if it exists. The formulation of a scale invariant real analysis is also outlined, when the singleton {0} of the real line R is replaced by a zero measure Cantor set. The Cantor function is realized as a locally constant function in this setting. The ordinary derivative dx/dt in R is replaced by the scale invariant logarithmic derivative d log x/d log t on the set of valued infinitesimals. As a result, the ordinary real valued functions are expected to enjoy some novel asymptotic properties, which might have important applications in number theory and in other areas of mathematics.