Some additive relations in the Pascal triangle
We derive some, seemingly new, curious additive relations in the Pascal triangle. They arise in summing up the numbers in the triangle along some vertical line up to some place.
math.HO↗
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Extended finite operator calculus as an example of algebraization of analysis
A calculus of sequences started by professor morgan ward constitutes the general scheme for extensions of classical operator calculus of the distinguished gian carlo rota considered by many afterwards and after ward morgan. Because of the historically now established notation we call the wardian calculus of sequences in its afterwards elaborated form a psi calculus. The psi calculus in parts appears to be almost automatic, natural extension of classical operator calculus or equivalently of umbral calculus . This is a review article based on the turn of the centuries author relevant contributions.
math.HO↗
An Unusual Proof that the Reals are Uncountable
This somewhat unusual proof for the fact that the reals are uncountable, which is adapted from one of Bourbaki's proofs in "Fonctions d'une variable reelle", may be of some interest.
math.HO↗
The mathematics of Donald Gordon Higman
This is about the mathematics and life of Donald Gordon Higman, 1928-2006. He did important work in representation theory of groups and algebras and in algebraic combinatorics. Charles C. Sims and Donald Higman discovered and constructed one of the sporadic simple groups.
math.HO↗
The beginnings of the theory of Hopf algebras
We consider issues related to the origins, sources and initial motivations of the theory of Hopf algebras. We consider the two main sources of primeval development: algebraic topology and algebraic group theory. Hopf algebras are named from the work of Heinz Hopf in the 1940's. In this note we trace the infancy of the subject back to papers from the 40's, 50's and 60's in the two areas mentioned above. Many times we just describe -- and/or transcribe parts of -- some of the relevant original papers on the subject.
math.HO↗
How the work of Gian Carlo Rota had influenced my group research and life
One outlines here in brief how the work of $Gian Carlo Rota$ had influenced my group research and life, starting from the end of the last century up to present time state of $The Internet Gian Carlo Rota Polish Seminar$. This note has been written for the $Rota Memorial Conference$ to be held on 16-18 Feb 2009, Milan, Italy.
math.HO↗
Poincaré et les quanta
We present some reflections concerning two papers by H. Poincaré concerning the theory of quanta.
math.HO↗
Book Review: "Geometry of Quantum States" by Ingemar Bengtsson and Karol Zyczkowski (Cambridge University Press, 2006)
Invited book review, as submitted to the electronic database MathSciNet, of the 2006 Cambridge University Press book, "Geometry of Quantum States," by Ingemar Bengtsson and Karol Zyczkowski
math.HO↗
Book Review: "A beautiful math. John Nash, game theory, and the modern quest for a code of nature" by Tom Siegfried (Joseph Henry Press, 2006)
Invited book review, as submitted to the electronic database MathSciNet, of the 2006 book by Tom Siegfried, "A beautiful math. John Nash, game theory, and the modern quest for a code of nature" (Joseph Henry Press, Washington, DC).
math.HO↗
Optimal profiles in variable speed flows
Where a 2D problem of optimal profile in variable speed flow is resolved in a class of convex Bezier curves, using symbolic and numerical computations.
math.HO↗
Minimal counting systems and commutative monoids
These notes present an approach to obtaining the basic operations of addition and multiplication on the natural numbers in terms of elementary results about commutative monoids.
math.HO↗
Representation of One as the Sum of Unit Fractions
One is expressed as the sum of the reciprocals of a certain set of integers. We give an elegant proof to the fact applying the polynomial theorem and basic calculus.
math.HO↗
Counting domino trains
In this paper we present a way to count the number of trains that we can construct with a given set of domino pieces. As an application we obtain a new method to compute the total number of eulerian paths in an undirected graph as well as their starting and ending vertices.
math.CO↗
Zeno at the Steering Wheel
This paper collects some reflections about an apparent incongruity between the usual (third-person) understanding of the probability of an event calculated for an extended period of time in the future (e.g., the expected probability of a driver to meet with a car accident in the next $M$ years) and the subjective perception of the same probability/risk that the person involved in that event has, instant by instant during that period of time. Similarities with the classical Zeno's paradoxes come to mind.
math.HO↗
The mathematization of the individual sciences - revisited
We recall major findings of a systematic investigation of the mathematization of the individual sciences, conducted by the author in Bielefeld some 35 years ago under the direction of Klaus Krickeberg, and confront them with recent developments in physics, medicine, economics, and spectral geometry.
math.HO↗
Existence of monoids compatible with a family of mappings
These notes present an approach to obtaining monoid operations which are compatible with a given family of mappings in the sense that the mappings become left translations in the monoid. This can be applied to various situations such as the addition on the natural numbers and the integers as well as the concatenation of lists.
math.HO↗
Young's Inequality and Fermat's Last Theorem
This paper has been withdrawn by the author for further investigation.
math.NT↗
A brief history of information-based complexity
This paper was presented on the occasion of an honorary doctoral degree for Henryk Wozniakowski at Friedrich Schiller University in Jena, Germany on June 6, 2008. Information-based complexity (IBC) is the study of algorithms and computational complexity of continuous problems. Examples of such problems include partial differential equations (in particular, the Schrodinger equation) very high dimensional integration, approximation, continuous optimization, and path integration. Because the computer has only partial information about the continuous mathematical problem adversary arguments at the information level often lead to tight complexity bounds. This may be contrasted with discrete problems where only conjectures that the complexity hierarchy does not collapse are available. This paper discusses precursors to IBC. It reports on the beginning of optimal iteration theory in the early 60s which was published in Traub's 1964 monograph. In the 70s Traub and Wozniakowski began to formulate the foundations of IBC leading to their 1980 monograph. The paper continues with the development of IBC to the present.
math.HO↗