A first course in Local arithmetic
Notes for a course at the H.-C. R. I., Allahabad, 15 August 2008 -- 26 January 2009
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Ribet's modular construction of unramified p-extensions of Q(μ_p)
An expository account of Ribet's modular construction
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Bidding chess
An expository introduction to bidding chess and other bidding games. To appear in Mathematical Intelligencer.
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Notes on solving and playing peg solitaire on a computer
We consider the one-person game of peg solitaire played on a computer. Two popular board shapes are the 33-hole cross-shaped board, and the 15-hole triangle board---we use them as examples throughout. The basic game begins from a full board with one peg missing and the goal is to finish at a board position with one peg. First, we discuss ways to solve the basic game on a computer. Then we consider the problem of quickly distinguishing board positions where the goal can still be reached ("winning" board positions) from those where it cannot. This enables a computer to alert the player if a jump under consideration leads to a dead end. On the 15-hole triangle board, it is possible to identify all winning board positions (from any single vacancy start) by storing a key set of 437 board positions. For the "central game" on the 33-hole cross-shaped board, we can identify all winning board positions by storing 839,536 board positions. By viewing a successful game as a traversal of a directed graph of winning board positions, we apply a simple algorithm to count the number of ways to traverse this graph, and calculate that the total number of solutions to the central game is 40,861,647,040,079,968. Our analysis can also determine how quickly we can reach a "dead board position", where a one peg finish is no longer possible.
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The optimal control and its multiple applications
In this work we refer to motivations, applications, and relations of control theory with other areas of mathematics. We present a brief historical review of optimal control theory, from its roots in the calculus of variations and the classical theory of control to the present time, giving particular emphasis to the Pontryagin maximum principle.
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The trace of First World War on mathematics in Brno
In the present paper, we study the evolution of the mathematical community in Brno (Brünn in German), the Moravian maintown, in the years between 1900 and 1930. In particular, we want to discuss how the Great War and its consequences (creation of Czechoslovakia and shift of power from the Germans to the Czechs) had an effect on this evolution.
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A note on growth of Fourier transforms and Moduli of continuity on Damek Ricci spaces
We obtain results related to boundedness of the growth of Fourier transform by the modulus of continuity on Damek-Ricci spaces. For noncompact riemannian symmetric spaces of rank one, analogues of all the results follow the same way.
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A Geometric Approach to Saddle Points of Surfaces
We outline an alternative approach to the geometric notion of a saddle point for real-valued functions of two variables. It is argued that this is more natural compared to the usual treatment of this topic in standard texts on Calculus.
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The area of a sector at centre of a planar quadric
Three linearly dependent and pairwise linearly independent vectors of an euclidian space uniquely determine a planar quadric with symmetry centre in the origin. A rather simple formula for the area of an arbitrary sector at centre of such a quadric will be shown by classical methods. The formula describes that area in dependence of 1. the lengths of the two straight lines that bound the sector at two sides, 2. the length of an arbitrary straight line from the centre to the quadric arc that bounds the sector at the third side, 3. the two angles in between these three straight lines.
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A New Approach to Rational Values of Trigonometric Functions
It is well known that sin(aπ/b), cos(aπ/b), etc., are only rational numbers for a few select integers a and b. We show that this is equivalent to the fact that only for d = 1,2,3,4, and 6 is the primitive dth root of unity of degree 2 over Q.
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Early Record of Divisibility and Primality
We provide textual evidence on divisibility and primality in the ancient Vedic texts of India. Concern with divisibility becomes clear from the listing of all the fifteen pairs of divisors of the number 720. The total number of pairs of divisors of 10,800 is also given. The motivation behind finding the divisors was the theory that the number of divisors of a certain periodic process is related to the count associated with some other periodic process. For example, 720 (days and nights of the year) has 15 pairs of divisors, and this was related to the 15 days of the waxing and waning of the moon. Numbers that have no divisors appeared to have been used to symbolize the "transcendent" that is beyond periodicity and change.
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Ruled quartic surfaces, models and classification
New historical aspects of the classification, by Cayley and Cremona, of ruled quartic surfaces and the relation to string models and plaster models are presented. In a `modern' treatment of the classification of ruled quartic surfaces the classical one is corrected and completed. A conceptual proof is presented of a result of Rohn concerning curves in $\mathbb{P}^1\times \mathbb{P}^1$ of bi-degree $(2,2)$. The string models of Series XIII (of some ruled quartic surfaces) are based on Rohn's result.
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The Riemann Conjecture and the advanced Calculus Methods for Physics
We present a set of lectures on topics of advanced calculus in one real and complex variable with several new results and proofs on the subject, specially with detailed proof-always missing in the literature - of the Cissoti explicitly integral formula conformally representing a polygon onto a disc.Besides we present-in the paper appendix-a new study embodied with a mathematical physicist perspective,on the famous Riemann conjecture on the zeros of the Zeta function, reducing its proof to a conjecture on the positivity of a numerical series.
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The ultimate question
We present a planar hypohamiltonian graph on 42 vertices and show some consequences.
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The Seven Messengers and the "Buzzati sequence"
A young Prince decides to explore the Father's Kingdom and aims to reach its furthermost boundaries. He starts from the City with a Caravan and seven fast and strong Messengers. They have the task to maintain the communications between the Caravan and the City during the exploration, going back and forth between the Caravan and the City while the Caravan inexorably goes away. Drawing inspiration from a fantasy tale by the Italian writer Dino Buzzati, I derive the geometric progression (named "Buzzati sequence" in his honor) which governs the duration of the Messenger's trips, renewing further the fascination of the tale. I also note with wonder how all this apparently hidden mathematical structure was already known to the author. An extension of the "Buzzati sequence" to relativistic velocities of the Caravan and the Messengers is finally presented as exercise.
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Axiomatizing mathematical conceptualism in third order arithmetic
We review the philosophical framework of mathematical conceptualism as an alternative to set-theoretic foundations and show how mainstream mathematics can be developed on this basis. The paper includes an explicit axiomatization of the basic principles of conceptualism in a formal system CM set in the language of third order arithmetic.
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The concept of a set
Metaphysical interpretations of set theory are either inconsistent or incoherent. The uses of sets in mathematics actually involve three distinct kinds of collections (surveyable, definite, and heuristic), which are governed by three different kinds of logic (classical, intuitionistic, and minimal). A foundational system incorporating this analysis and based on the principles of mathematical conceptualism accords better with actual mathematical practice than Zermelo-Fraenkel set theory does.
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Is set theory indispensable?
Although Zermelo-Fraenkel set theory (ZFC) is generally accepted as the appropriate foundation for modern mathematics, proof theorists have known for decades that virtually all mainstream mathematics can actually be formalized in much weaker systems which are essentially number-theoretic in nature. Feferman has observed that this severely undercuts a famous argument of Quine and Putnam according to which set theoretic platonism is validated by the fact that mathematics is "indispensable" for some successful scientific theories (since in fact ZFC is not needed for the mathematics that is currently used in science). I extend this critique in three ways: (1) not only is it possible to formalize core mathematics in these weaker systems, they are in important ways better suited to the task than ZFC; (2) an improved analysis of the proof-theoretic strength of predicative theories shows that most if not all of the already rare examples of mainstream theorems whose proofs are currently thought to require metaphysically substantial set-theoretic principles actually do not; and (3) set theory itself, as it is actually practiced, is best understood in formalist, not platonic, terms, so that in a real sense *set theory is not even indispensable for set theory*. I also make the point that even if ZFC is consistent, there are good reasons to suspect that some number-theoretic assertions provable in ZFC may be false. This suggests that set theory should not be considered central to mathematics.
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