we can we can find anything interesting. In particular, we investigate how macroeconomic data aff... more we can we can find anything interesting. In particular, we investigate how macroeconomic data affects equity prices. Our source of macroeconomic data is the St Louis Fed, via their very nice API. Our source of end of day S&P index data is Yahoo! Finance. Our source of older (lower frequency) S&P price data, as well as dividends data, corporate earnings, and the CPI is Robert Shillers wonderful resource 1.1 Introduction This paper is completely self-contained, First, we will set up our tools, then we will download and comment on macroeconoic data. Then, we will use some simple (but surprisingly effective) methods to develop a macro-based strategy, and then we will take a look at the longer-term data and make a couple of random comments.
We describe extensive computational experiments on spectral properties of random objects - random... more We describe extensive computational experiments on spectral properties of random objects - random cubic graphs, random planar triangulations, and Voronoi and Delaunay diagrams of random (uniformly distributed) point sets on the sphere). We look at bulk eigenvalue distribution, eigenvalue spacings, and locality properties of eigenvectors. We also look at the statistics of nodal domains of eigenvectors on these graphs. In all cases we discover completely new (at least to this author) phenomena. The author has tried to refrain from making specific conjectures, inviting the reader, instead, to meditate on the data.
We start by studying the distribution of (cyclically reduced) elements of the free groups F n wit... more We start by studying the distribution of (cyclically reduced) elements of the free groups F n with respect to their abelianization (or equivalently, their class in H 1 (F n , Z)). We derive an explicit generating function, and a limiting distribution, by means of certain results (of independent interest) on Chebyshev polynomials; we also prove that the reductions mod p (p-an arbitrary prime) of these classes are asymptotically equidistributed, and we study the deviation from equidistribution. We extend our techniques to a more general setting and use them to study the statistical properties of long cycles (and paths) on regular (directed and undirected) graphs. We return to the free group to study some growth functions of the number of conjugacy classes as a function of their cyclically reduced length.
doi:10.1093/imrn/rnq043 Zariski Density and Genericity
In this paper, we combine a number of results (some due to the author, some not) to show that Zar... more In this paper, we combine a number of results (some due to the author, some not) to show that Zariski density is, in a strong sense, a generic property of subgroups of SL(n,Z) and Sp(2n,Z). We also extend previous results of the author on generic properties of elements of the special linear and symplectic groups to generic elements in arbitrary Zariski-dense (as subgroups of the ambient complex algebraic group) subgroups of these groups. Jointly with Ilya Kapovich, we also show that a generic free group automor-phism is hyperbolic. 1
Abstract. We analyze completely the convergence speed of the batch learning algorithm, and compar... more Abstract. We analyze completely the convergence speed of the batch learning algorithm, and compare its speed to that of the memoryless learning algorithm and of learning with memory (as analyzed in [KR2001b]). We show that the batch learning algorithm is never worse than the memoryless learning algorithm (at least asymptotically). Its performance vis-a-vis learning with full memory is less clearcut, and depends on certain probabilistic assumptions.
Abstract. We carry out a numerical study of fluctuations in the spectrum of regular graphs. Our e... more Abstract. We carry out a numerical study of fluctuations in the spectrum of regular graphs. Our experiments indicate that the level spacing distribution of a generic k-regular graph approaches that of the Gaussian Orthogonal Ensemble of random matrix theory as we increase the number of vertices. A review of the basic facts on graphs and their spectra is included. 1.
Abstract. We study convex sets C of finite (but non-zero) volume inH n andE n. We show that the i... more Abstract. We study convex sets C of finite (but non-zero) volume inH n andE n. We show that the intersection C ∞ of any such set with the ideal boundary ofH n has Minkowski (and thus Hausdorff) dimension of at most (n−1)/2. and this bound is sharp, at least in some dimensions n. We also show a sharp bound when C ∞ is a smooth submanifold of∂∞H n. In the hyperbolic case, we show that for any k≤(n−1)/2 there is a bounded section S of C through any prescribed p, and we show an upper bound on the radius of the ball centered at p containing such a section. We show similar bounds for sections through the origin of a convex body inE n, and give asymptotic estimates as 1≪k≪n..
In this article we relate two different densities. Let Fk be the free group of finite rank k ≥ 2 ... more In this article we relate two different densities. Let Fk be the free group of finite rank k ≥ 2 and let α be the abelianization map from Fk onto Zk. We prove that if S ⊆ Zk is invariant under the natural action of SL(k, Z) then the asymptotic density of S in Zk and the annular density of its full preimage α−1(S) in Fk are equal. This implies, in particular, that for every integer t ≥ 1, the annular density of the set of elements in Fk that map to t-th powers of primitive elements in Zk is equal to 1 tkζ(k) , where ζ is the Riemann zeta-function. An element g of a group G is called a test element if every endomorphism of G which fixes g is an automorphism of G. As an application of the result above we prove that the annular density of the set of all test elements in the free group F (a, b) of rank two is 1 − 6 π2 . Equivalently, this shows that the union of all proper retracts in F (a, b) has annular density 6 π2 . Thus being a test element in F (a, b) is an “intermediate property” ...
We show that the Galois group of a random monic polynomial %of degree $d>12$ with integer coef... more We show that the Galois group of a random monic polynomial %of degree $d>12$ with integer coefficients between $-N$ and $N$ is NOT $S_d$ with probability $\ll \frac{\log^{\Omega(d)}N}{N}.$ Conditionally on NOTbeing the full symmetric group, we have a hierarchy of possibilities each of which has polylog probability of occurring. These results also apply to random polynomials with only a subset of the coefficients allowed to vary. This settles a question going back to 1936.
We take a look the changes of different asset prices over variable periods, using both traditiona... more We take a look the changes of different asset prices over variable periods, using both traditional and spectral methods, and discover universality phenomena which hold (in some cases) across asset classes.
In this note we show that the n-2-dimensional volumes of codimension 2 faces of an n-dimensional ... more In this note we show that the n-2-dimensional volumes of codimension 2 faces of an n-dimensional simplex are algebraically independent quantities of the volumes of its edge-lengths. The proof involves computation of the eigenvalues of Kneser graphs. We also construct families of non-congurent simplices not determined by their codimension-2 areas.
The Sharpe ratio is the most widely used risk metric in the quantitative finance community - amaz... more The Sharpe ratio is the most widely used risk metric in the quantitative finance community - amazingly, essentially everyone gets it wrong. In this note, we will make a quixotic effort to rectify the situation.
Probabilistic Methods in Geometry, Topology and Spectral Theory, 2019
We discuss some (numerical and theoretical) results about the coefficients and zeros of Tutte (di... more We discuss some (numerical and theoretical) results about the coefficients and zeros of Tutte (dichromatic) polynomial of graphs of bounded degree whose size increases. We also discuss related results for Bollobás-Riordan polynomials.
We obtain sharp bounds for the number of n-cycles in a finite graph as a function of the number o... more We obtain sharp bounds for the number of n-cycles in a finite graph as a function of the number of edges, and prove that the complete graph is optimal in more ways than could be imagined. En route, we prove some sharp estimates on power sums.
Content-Length : 21789 X-Lines : 501 Status : RO Simple curves on hyperbolic tori
Let T be a once punctured torus, equipped with a complete hyperbolic metric. Herein, we describe ... more Let T be a once punctured torus, equipped with a complete hyperbolic metric. Herein, we describe a new approach to the study of the set S of all simple geodesics on T. We introduce a valuation on the homology H1(T,ZZ), which associates to each homology class h the length l(h) of the unique simple geodesic homologous to h, and show that l extends to a norm on H1(T,R). We analyze the boundary of the unit ball B(l) and the variation of the area of B(l) over the moduli space of T . These results are applied to obtain sharp asymptotic estimates on the number of simple geodesics of length less than L. Courbes simples dans les tores Résumé. Soit T un tore troué, muni d’une métrique hyperbolique complète, d’aire finie. Nous présentons une nouvelle approche de l’étude de l’ensemble S de toutes les géodésiques fermées simples (sans points doubles) de T . Nous introduisons une application sur l’homologie H1(T,ZZ), qui associe à chaque classe h ∈ H1(T,ZZ) indivisible la longueur l(h) de l’uniqu...
A. We study the surface area of an ellipsoid in E as the function of the lengths of their ... more A. We study the surface area of an ellipsoid in E as the function of the lengths of their major semi-axes. We write down an explicit formula as an integral over Sn−1, use the formula to derive convexity properties of the surface area, to sharpen the estimates given [6], to produce asymptotic formulas in large dimensions, and to give an expression for the surface in terms of the Lauricella hypergeometric function.
We use the order complex corresponding to a symmetric matrix (defined by Giusti et al in 2015). I... more We use the order complex corresponding to a symmetric matrix (defined by Giusti et al in 2015). In this note, we use it to define a class of models of random graphs, and show some surprising experimental results, showing sharp phase transitions.
We start by studying the distribution of (cyclically reduced) elements of the free groups Fn with... more We start by studying the distribution of (cyclically reduced) elements of the free groups Fn with respect to their Abelianization (or equivalently, their class in H1(Fn, Z)). We derive an explicit generating function, and a limiting distribution, by means of certain results (of independent interest) on Chebyshev polynomials; we also prove that the reductions mod p (pan arbitrary prime) of these classes are asymptotically equidistributed, and we study the deviation from equidistribution. We extend our techniques to a more general setting and use them to study the statistical properties of long cycles (and paths) on regular (directed and undirected) graphs. We return to the free group to study some growth functions of the number of conjugacy classes as a function of their cyclically reduced length.
We describe extensive computational experiments on spectral properties of random objects-random c... more We describe extensive computational experiments on spectral properties of random objects-random cubic graphs, random planar triangulations, and Voronoi and Delaunay diagrams of random (uniformly distributed) point sets on the sphere). We look at bulk eigenvalue distribution, eigenvalue spacings, and locality properties of eigenvectors. We also look at the statistics of nodal domains of eigenvectors on these graphs. In all cases we discover completely new (at least to this author) phenomena. The author has tried to refrain from making specific conjectures, inviting the reader, instead, to meditate on the data.
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Papers by Igor Rivin