arXiv research

Delisle's projection explained by Euler in 18th century.

This paper explores historical and philosophical aspects of angles and solid angles, inspired by Euler's work.

Euler explored spherical geometry using trigonometric formulae and solid geometry methods.

We present the results of study of a possible relationship between the space weather and terrestrial markets of agricultural products. It is shown that to implement the possible effect of space weather on the terrestrial harvests and prices, a simultaneous fulfillment of three conditions is required: 1) sensitivity of …

This paper describes several key discoveries in the 19th century that led to the modern theory of manifolds in the twentieth century: intrinsic differential geometry, projective geometry and higher dimensional manifolds and Riemannian geometry.

This paper gives an overview of several key innovations in the 19th century which led to complex geometry in the 20th century. This includes the creation of the complex plane, the work of Abel on addition theorems for generalized elliptic integrals, the theory of elliptic functions, holomorphic functions, and the creat…

This paper describes the evolution of aspects of differential and algebraic gometry from the mid 17th century till the end of the 18th century.

Capital usually leads to income, and income is more accurately and easily measured. Thus we summarize income distributions in USA, Germany, etc.

The twentieth century was a period of outstanding economic growth together with an unequal income distribution. This paper analyses the international distribution of growth rates and its dynamics during the twentieth century. We show that the whole century is characterized by a high heterogeneity in the distribution of…

Golden age of mathematical finance in the late 20th century.

During this last decades, several attempts to construct slow invariant manifold of the Lorenz-Krishnamurthy five-mode model of slow-fast interactions in the atmosphere have been made by various authors. Unfortunately, as in the case of many two-time scales singularly perturbed dynamical systems the various asymptotic p…

We give a brief review of a research made in the field of differential geometry in Estonia in the period from the beginning of the 19th century to the present time. The biographic data of mathematicians who made a valuable contribution to the development of differential geometry in Estonia in mentioned period are prese…

We survey the 19th century development of the signature of a quadratic form, and the applications in the 20th and 21st century to the topology of manifolds and dynamical systems. Version 2 is an expanded and corrected version of Version 1, including an Appendix by the second named author "Algebraic L-theory of rings wi…

This overview paper is intended as a quick introduction to Lie algebras of vector fields. Originally introduced in the late 19th century by Sophus Lie to capture symmetries of ordinary differential equations, these algebras, or infinitesimal groups, are a recurring theme in 20th-century research on Lie algebras. I will…

We apply fixed-point techniques to compute the coefficient ring of semifree geometric circle-equivariant complex cobordism with isolated fixed points, recovering a 2004 result of Sinha through 19th-century methods.

H. Weyl's proposal of 1918 for generalizing Riemannian geometry by local scale gauge (later called {\em Weyl geometry}) was motivated by mathematical, philosophical and physical considerations. It was the starting point of his unified field theory of electromagnetism and gravity. After getting disillusioned with this r…

Explains Conway's tangle trick and its mathematical origins.

Study refines trend-following strategy to improve adaptability.

Revisits the Gauss-Bonnet formula using double forms.

Confocal quadrics capture (encode) and geometrize spectral properties of symmetric operators. Certain metric-projective properties of confocal quadrics (most of them established in the first half of the XIX$^{\mathrm{th}}$ century) {\it carry out} (stick and transfer) by rolling to and influence surfaces {\it applicabl…

This paper is a review of the book "Knots" by Alexei Sossinsky. The review includes a short personal history of knot theory at the end of the twentieth century.

We present a systematic algorithm testing for the existence of collective self-organization in the behavior of agents in social systems, with a concrete empirical implementation on the Dow Jones Industrial Average index (DJIA) over the 20th century and on Hong Kong Hang Seng composite index (HSI) since 1969. The algori…

We use equivariant surgery to classify all involutions on closed surfaces, up to isomorphism. Work on this problem is classical, dating back to the nineteenth century, but some questions seem to have been left unanswered. We give a modern treatment that leads to a complete classification.

First the title could be also understood as ``3-manifolds related by non-zero degree maps" or "Degrees of maps between 3-manifolds" for some aspects in this survey talk. The topology of surfaces was completely understood at the end of 19th century, but maps between surfaces kept to be an active topic in the 20th centur…

It has been shown that the long term evolution of the Gross Product of the World after World War II can be well portrayed by the exponential function with the crossover at the year 1973, cinsiding with the Oil Crisis onset. For the the Standard and Poor 500 index the single exponential behavior extends down to at least…

Mathematical analysis shows Delisle-Euler map methods are optimal.

Schwartz functions smoothly extend to real projective spaces.

Modern excavations yielded a distribution of the house areas in the ancient Egyptian city Akhetaten, which was populated for a short period during the 14th century BC. Assuming that the house area is a measure of the wealth of its inhabitants allows us to make a comparison of the wealth distributions in ancient and mod…

We study affine immersions as introduced by Nomizu and Pinkall. We classify those affine immersions of a surface in 4-space which are degenerate and have vanishing cubic form (i.e. parallel second fundamental form). This completes the classification of parallel surfaces of which the first results were obtained in the b…

We present in this chapter (Chapter II) the history of ideas which lead up to the development of modern knot theory. We are more detailed when pre-XX century history is reported. With more recent times we are more selective, stressing developments related to Jones type invariants of links. In the Appendix, A.Przybyszew…

Long-range climate forecasts use integrated assessment models to link the global economy to greenhouse gas emissions. This paper evaluates an alternative economic framework outlined in part 1 of this study (Garrett, 2014) that approaches the global economy using purely physical principles rather than explicitly resolve…

Historically, the banking multiplier has been in a range of 4 to 100, with 25% to 1% reserve ratios at most layers of the banking system encompassing the majority of its range in recent centuries. Here it is shown that multipliers over 1 000 can occur from a new mechanism in banking. This new multiplier uses a default …

Markov chain Monte Carlo (MCMC) is widely regarded as one of the most important algorithms of the 20th century. Its guarantees of asymptotic convergence, stability, and estimator-variance bounds using only unnormalized probability functions make it indispensable to probabilistic programming. In this paper, we introduce…

These lecture notes in the De Rham-Hodge theory are designed for a 1-semester undergraduate course (in mathematics, physics, engineering, chemistry or biology). This landmark theory of the 20th Century mathematics gives a rigorous foundation to modern field and gauge theories in physics, engineering and physiology. The…

We introduce a local algorithm for Khovanov Homology computations - that is, we explain how it is possible to "cancel" terms in the Khovanov complex associated with a ("local") tangle, hence canceling the many associated "global" terms in one swoosh early on. This leads to a dramatic improvement in computational effici…

Discretization of curves is an ancient topic. Even discretization of curves with an eye toward differential geometry is over a century old. However there is no general theory or methodology in the literature, despite the ubiquitous use of discrete curves in mathematics and science. There are conflicting definitions of …

We produce a new general family of flat tori in R^4, the first one since Bianchi's classical works in the 19th century. To construct these flat tori, obtained via small perturbation of certain Hopf tori in S^3, we first present a global description of all isometric immersions of R^2 into R^4 with flat normal bundle.

We call attention against what seems to a widely held misconception according to which large crashes are the largest events of distributions of price variations with fat tails. We demonstrate on the Dow Jones Industrial index that with high probability the three largest crashes in this century are outliers. This result…

Study financial crises using mathematical techniques to compare equity performance.

This article offers an introductory look at Khovanov homology and its historical context.

We solve a century-old conjecture about Alexander polynomials of special alternating links.

A new systolic inequality with a remainder for the real projective plane.

The set of osculating circles of a given curve in $\SS^3$ forms a curve in the set of oriented circles in $\SS^3$. We show that its "${\frac12}$-dimensional measure" with respect to the pseudo-Riemannian structure of the set of circles is proportional to the conformal arc-length of the original curve, which is a confor…

Econophysics embodies the recent upsurge of interest by physicists into financial economics, driven by the availability of large amount of data, job shortage in physics and the possibility of applying many-body techniques developed in statistical and theoretical physics to the understanding of the self-organizing econo…

These lecture notes in Lie Groups are designed for a 1--semester third year or graduate course in mathematics, physics, engineering, chemistry or biology. This landmark theory of the 20th Century mathematics and physics gives a rigorous foundation to modern dynamics, as well as field and gauge theories in physics, engi…

We give a survey of our joint ongoing work with Ali Chamseddine, Slava Mukhanov and Walter van Suijlekom. We show how a problem purely motivated by "how geometry emerges from the quantum formalism" gives rise to a slightly noncommutative structure and a spectral model of gravity coupled with matter which fits with expe…

Yoshikawa moves were introduced at least quarter-century ago and are still actively used by researchers. For any marked graph diagram we will define its twisted diagram and its mirror cut surface. By using a surface-link group of a mirror cut surface of a twisted diagram we will prove the independence of Yoshikawa eigh…

New proof shows how to fill a square with Fibonacci curve.