arXiv research

New conjectures link SU(r) Vafa-Witten invariants to Ramanujan's continued fractions.

In this paper we give describe a new connection between the dilogarithm function and solutions to Pell's equation $x^2-ny^2 = \pm 1$. For each solution $x,y$ to Pell's equation we obtain a dilogarithm identity whose terms are given by the continued fraction expansion of the associated unit $x+y\sqrt{n} \in \Z[\sqrt{n}]…

New bounds on diameters and generators for specific lattices and graphs.

The objectives of this article are three-fold. Firstly, we present for the first time explicit constructions of an infinite family of \textit{unbalanced} Ramanujan bigraphs. Secondly, we revisit some of the known methods for constructing Ramanujan graphs and discuss the computational work required in actually implement…

In this paper we introduce a link between geometry of ordinary continued fractions and trajectories of points that moves according to the second Kepler law. We expand geometric interpretation of ordinary continued fractions to the case of continued fractions with arbitrary elements.

Develops mixed quantization for graph vector bundles.

Quantum invariants of three-manifolds linked to mock theta functions.

The study finds new infinite dilogarithm identities related to number sequences and continued fractions.

We present a new and very concrete connection between cluster algebras and knot theory. This connection is being made via continued fractions and snake graphs. It is known that the class of 2-bridge knots and links is parametrized by continued fractions, and it has recently been shown that one can associate to each con…

Proposes RBGP framework for efficient block sparse neural networks.

Continuous time random walks impose a random waiting time before each particle jump. Scaling limits of heavy tailed continuous time random walks are governed by fractional evolution equations. Space-fractional derivatives describe heavy tailed jumps, and the time-fractional version codes heavy tailed waiting times. Thi…

We show that a Born-Infeld soliton can be realised either as a spacelike minimal graph or timelike minimal graph over a timelike plane or a combination of both away from singular points. We also obtain some exact solutions of the Born-Infeld equation from already known solutions to the maximal surface equation. Further…

This paper investigates integer multiplication of continued fractions using geometric structures. In particular, this paper shows that integer multiplication of a continued fraction can be represented by replacing one triangulation of an orbifold with another triangulation. This method is used to show that eventually p…

We present a construction of complete self-dual Einstein metrics of negative scalar curvature on an uncountable family of manifolds of infinite topological type, which are enumerated by continued fraction expansions of irrational numbers. These manifolds may be regarded as limits of the resolutions of cyclic quotient s…

The article recovers tensor fields from partial data using weighted divergent ray transforms.

Derive new Euler-Ramanujan-type identities and infinite decompositions for zero mean curvature graphs in various spaces.

New volume functions for random hyperbolic surfaces link to spectral gaps.

Research on unique continuation principles in medical and seismic imaging.

The height function of various surfaces decomposes into finite sums of scaled and translated versions of itself.

Survey of continuous volatility models, focusing on fractional and rough methods.

We survey the use of continued fraction expansions in the algebraical and topological study of complex analytic singularities. We also prove new results, firstly concerning a geometric duality with respect to a lattice between plane supplementary cones and secondly concerning the existence of a canonical plumbing struc…

In this paper we study the matrix completion problem: Suppose $X \in {\mathbb R}^{n_r \times n_c}$ is unknown except for a known upper bound $r$ on its rank. By measuring a small number $m \ll n_r n_c$ of elements of $X$, is it possible to recover $X$ exactly with noise-free measurements, or to construct a good approxi…

The paper shows how Scherk-type surfaces can be decomposed into helicoids.

The tail of the colored Jones polynomial of an alternating link is a $q$-series invariant whose first $n$ terms coincide with the first $n$ terms of the $n$-th colored Jones polynomial. Recently, it has been shown that the tail of the colored Jones polynomial of torus knots give rise to Ramanujan type identities. In th…

Researchers develop neural networks for approximating functions in Banach spaces.

This paper explores how rational numbers on the Stern-Brocot diagram map to lines when terms are extended.

New algorithms solve word and conjugacy problems in braid group B3.

We conjecture that the stable Khovanov homology of torus knots can be described as the Koszul homology of an explicit non-regular sequence of quadratic polynomials. The corresponding Poincare series turns out to be related to the Rogers-Ramanujan identity.

A rational link may be represented by any of the (infinitely) many link diagrams corresponding to various continued fraction expansions of the same rational number. The continued fraction expansion of the rational number in which all signs are the same is called a {\em nonalternating form} and the diagram corresponding…

There exists and is unique up to multiplication by a constant function a form of the highest dimension on the manifold of n-dimensional continued fractions in the sense of Klein, such that the form is invariant under the natural action of the group of projective transformations PGL(n+1). A measure corresponding to the …

Study evaluates discretized arbitrage strategies in fractional financial markets.

CFTM uses fractional Brownian motion for dynamic topic modeling.

Expander graphs have been a focus of attention in computer science in the last four decades. In recent years a high dimensional theory of expanders is emerging. There are several possible generalizations of the theory of expansion to simplicial complexes, among them stand out coboundary expansion and topological expand…

Answering a question asked by Agol and Wise, we show that a desired stronger form of Wise's malnormal special quotient theorem does not hold. The counterexamples are generalizations of triangle groups, built using the Ramanujan graphs constructed by Lubotzky--Phillips--Sarnak.

Jones polynomial coincidences explored for rational knots.

Defines a new process for financial modeling.

Study large deviations in fractional volatility models with non-Gaussian volatility.

Long and short memory in economic processes is usually described by the so-called discrete fractional differencing and fractional integration. We prove that the discrete fractional differencing and integration are the Grunwald-Letnikov fractional differences of non-integer order d. Equations of ARIMA(p,d,q) and ARFIMA(…

We study fractional stochastic volatility models in which the volatility process is a positive continuous function $σ$ of a continuous Gaussian process $\widehat{B}$. Forde and Zhang established a large deviation principle for the log-price process in such a model under the assumptions that the function $σ$ is globally…

According to a formula by Gordon and Litherland, the signature of a knot K can be computed as the signature of a Goeritz matrix of K minus a suitable correction term, read off from the diagram of K. In this article, we consider the family of two bridge knots K(p/q) and compute the signature of their Goeritz matrices in…

New q-deformed integers help compute Jones polynomials efficiently.

Study shows volumes of knot complements are bounded by linear functions of geodesic periods.

In this work we present a new local to global criterion for proving a form of high dimensional expansion, which we term cosystolic expansion. Applying this criterion on Ramanujan complexes, yields for every dimension, an infinite family of bounded degree complexes with the topological overlapping property. This answer …

Expander graphs have been intensively studied in the last four decades. In recent years a high dimensional theory of expanders has emerged, and several variants have been studied. Among them stand out coboundary expansion and topological expansion. It is known that for every $d$ there are unbounded degree simplicial co…

The study tackles rough noise in high-frequency financial data using fractional Brownian motion.

The paper computes lens spaces resulting from rational surgeries on Hopf links.

The paper evaluates integrals for fBm with various Hurst indices.

The continuous observation of the financial markets has identified some stylized facts which challenge the conventional assumptions, promoting the born of new approaches. On the one hand, the long-range dependence has been faced replacing the traditional Gauss-Wiener process (Brownian motion), characterized by stationa…