The twisted Connes-Moscovici higher index theorem is generalized to the case of good orbifolds. The higher index is shown to be a rational number, and in fact non-integer in specific examples of 2-orbifolds. This results in a non-commutative geometry model that predicts the occurrence of fractional quantum numbers in t…
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A geometric model for twisted -homology is introduced. It is modeled after the Mathai-Melrose-Singer fractional analytic index theorem in the same way as the Baum-Douglas model of -homology was modeled after the Atiyah-Singer index theorem. A natural transformation from twisted geometric -homology to the new g…
First time projective elliptic genera constructed for oriented manifolds.
problem Defining and proving properties of projective elliptic genera.
method Topological construction and analytic interpretation via fractional index theorem.
result Modularity properties of projective elliptic genera proven.
G-framework is presented by Peng [41] for measure risk under uncertainty. In this paper, we define fractional G-Brownian motion (fGBm). Fractional G-Brownian motion is a centered G-Gaussian process with zero mean and stationary increments in the sense of sub-linearity with Hurst index . This process has sta…
Paper defines multi-dimensional fractional Brownian motion under volatility uncertainty.
problem Volatility uncertainty in fractional Brownian motion.
method Definition and study of multi-dimensional fractional Brownian motion (G-fBm) with Hurst index.
result First results on stochastic calculus for G-fBm with Hurst index > 0.5.
In this note the fractional analytic index, for a projective elliptic operator associated to an Azumaya bundle, of DG/0402329 is related to the equivariant index of Atiyah and Singer for an associated transversally elliptic operator.
The paper converts nonalternating forms of rational links into all-even forms and derives formulas for their braid index and HOMFLY polynomial.
problem Finding formulas for the braid index and HOMFLY polynomial of rational links.
method Algorithm to transform nonalternating forms into all-even forms and derivation of formulas.
result Formulas for the braid index and HOMFLY polynomial of rational links in terms of their reduced alternating form.
We calculate the equivariant index formula for an infinite dimensional Clifford module canonically associated to any Riemannian manifold. It encompasses the fractional index formula of the projective Dirac operator by Mathai--Melrose--Singer.
Scaling in the Bombay Stock Exchange Indexcond-mat.stat-mech
In this paper we study BSE Index financial time series for fractal and multifractal behaviour. We show that Bombay stock Exchange (BSE)Index time series is mono-fractal and can be represented by a fractional Brownian motion.
The paper studies inequalities for fractional GJMS operators on conformal infinity.
problem Deriving comparison inequalities for fractional Yamabe constants.
method Using Poincaré-Einstein manifolds and fractional GJMS operators.
result Two comparison inequalities for fractional Yamabe constants are derived.
Socio-economic inequality is characterized from data using various indices. The Gini () index, giving the overall inequality is the most common one, while the recently introduced Kolkata () index gives a measure of fraction of population who possess top fraction of wealth in the society. Here, we show t…
We characterize the fractional Dehn twist coefficient of a braid in terms of a slope of the homogenization of the Upsilon function, where Upsilon is the function-valued concordance homomorphism defined by Ozsváth, Stipsicz, and Szabó. We use this characterization to prove that -braids with fractional Dehn twist coef…
The paper extends Heintze-Karcher inequalities to fractional Q-curvature.
problem Extending Heintze-Karcher inequalities to fractional Q-curvature.
method Generalization of Heintze-Karcher inequalities to fractional Q-curvature on conformally compact Einstein manifolds.
result Rigidity theorems for specific values of γ.
New method uses fractional posteriors for semiparametric inference with improved uncertainty quantification.
problem Semiparametric inference with nonparametric priors and fractional posteriors.
method Established a general Bernstein--von Mises theorem for fractional posterior distributions, proposed shifted-and-rescaled credible sets.
result Fractional posterior credible sets provide reliable uncertainty quantification but have inflated size; shifted-and-rescaled set is an efficient confidence set.
Mathai, Melrose, and Singer introduced the notion of projective elliptic operators on manifolds equipped with an Azumaya bundle. In this note we compute the equivariant index of transversally elliptic operators that are the pullback of projective elliptic operators on the trivialization of the Azumaya bundle. It encomp…
The spectral eta-invariant of a self-adjoint elliptic differential operator on a closed manifold is rigid, provided that the parity of the order is opposite to the parity of dimension of the manifold. The paper deals with the calculation of the fractional part of the eta-invariant in this case. The method used to obtai…
The paper proposes methods to directly optimize complex classification metrics.
problem Handling class-imbalanced cases with non-decomposable metrics.
method Calibrated surrogate maximization of linear-fractional utility.
result Calibrated surrogate maximization can coincide with true utility maximization under certain conditions.
Develops fractional de Rham theory for Maxwell equations.
problem Formulating fractional calculus for Maxwell equations.
method Fractional tangent functionals, Riemann-Liouville integral, polynomial algebra, exterior algebra.
result Fractional de Rham complex for Maxwell equations.
This paper is the continuation of Part I, expanding previous results of math.DG/9803051. This paper uses techniques in noncommutative geometry as developed by Alain Connes in order to study the twisted higher index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant u…
Study on symmetric operators on non-compact manifolds, focusing on their index modulo 2.
problem Investigating elliptic operators with a specific symmetry and their index modulo 2.
method Analysis of Callias-type operators on non-compact manifolds, establishing mod 2 versions of index theorems.
result Established mod 2 versions of the Gromov-Lawson relative index theorem, Callias index theorem, and Boutet de Monvel's index theorem for Toeplitz operators.
Proves a lattice version of the Atiyah-Singer index theorem.
problem Index problems of Wilson-Dirac operators on lattice approximations of manifolds.
method Formulates and proves a -theoretic formula for an index-type invariant.
result Main theorem gives a formula for an index-type invariant of operators on lattice approximations of closed integral affine manifolds.
Formulates Index III lemma and Rauch III theorem with applications.
problem Develops new mathematical theorems based on existing ones.
method Formulation of Index III lemma and Rauch III theorem based on Index I, II lemmas and Rauch I, II theorems.
result Presented Rauch's type theorem and volume comparison result as applications.
Explain Arnold's proof of the Morse index theorem using Maslov index.
problem Proving the Morse index theorem in Riemannian geometry.
method Using symplectic arguments and the Maslov index.
result Self-contained exposition of Arnold's proof.
Deep learning predicts path-dependent processes from historical data.
problem Predicting path-dependent processes using historical data.
method Nonparametric regression with deep neural networks.
result Deep learning method converges to theoretical predictions as observation frequency increases.
An extremal fractional Gaussian with a possible application to option-pricing with skew and smileq-fin.PR
We derive an extremal fractional Gaussian by employing the Lévy-Khintchine theorem and Lévian noise. With the fractional Gaussian we then generalize the Black-Scholes-Merton option-pricing formula. We obtain an easily applicable and exponentially convergent option-pricing formula for fractional markets. We also carry o…
This study compares Bitcoin and S&P 500 returns using a new GTS distribution method.
problem Analyzing the daily return distributions and tail probabilities of Bitcoin and S&P 500.
method Used advanced Fast Fractional Fourier transform (FRFT) to fit the seven-parameter General Tempered Stable (GTS) distribution.
result Bitcoin has heavier tails and higher prevalence of high returns compared to S&P 500.
We give a short proof of the Morse index theorem for geodesics in semi-Riemannian manifolds by using K-theory. This makes the Morse index theorem reminiscent of the Atiyah-Singer index theorem for families of selfadjoint elliptic operators.
Paper develops Euler scheme for fractional delay diff. eqs with additive noise.
problem Developing a consistent Euler-Maruyama scheme for fractional stochastic delay diff. eqs.
method Euler-Maruyama scheme for fractional Brownian motion with additive noise.
result Achieved convergence rate of H+1/2 for smooth delays when H>1/2.
Researchers prove an equivariant index theorem on Euclidean space.
problem Calculating the equivariant index of the Bott-Dirac operator on .
method Continuous field of -algebras and equivariant index theorem.
result Explicit calculation of the equivariant index of the Bott-Dirac operator on .
Dynamics of the Warsaw Stock Exchange index as analysed by the nonhomogeneous fractional relaxation equationphysics.soc-ph
We analyse the dynamics of the Warsaw Stock Exchange index WIG at a daily time horizon before and after its well defined local maxima of the cusp-like shape decorated with oscillations. The rising and falling paths of the index peaks can be described by the Mittag-Leffler function superposed with various types of oscil…
On the Fractional Mean Valuemath.DG
This work, dealt with the classical mean value theorem and took advantage of it in the fractional calculus. The concept of a fractional critical point is introduced. Some sufficient conditions for the existence of a critical point is studied and an illustrative example rele- vant to the concept of the time dilation eff…
Extends a theorem for first-order elliptic operators on manifolds.
problem Proving the relative index theorem for general first-order elliptic operators.
method Using boundary value problems and graphical decomposition of elliptically regular boundary conditions.
result Proves the relative index theorem for general first-order elliptic operators.
The paper proves an index theorem for loop spaces of compact manifolds.
problem Defining an index theorem for loop spaces of compact manifolds.
method Formulated and proved an equivariant index theorem for non-compact manifolds with -actions, using a ring of formal power series.
result Found an appropriate form of the index theorem for loop spaces.
In this paper, we prove a local equivariant index theorem for sub-signature operators which generalizes the Zhang's index theorem for sub-signature operators.
Proves an equivariant version of index theorem for geometric families.
problem Index theorem for geometric families with group action.
method Apply equivariance --> families principle to Clifford module bundles.
result Equivariant version of Bismut's families index theorem.
We prove an index theorem for families of linear periodic Hamiltonian systems, which is reminiscent of the Atiyah-Singer index theorem for selfadjoint elliptic operators. For the special case of one-parameter families, we compare our theorem with a classical result of Salamon and Zehnder. Finally, we use the index theo…
Socio-economic inequality is measured using various indices. The Gini () index, giving the overall inequality is the most commonly used, while the recently introduced Kolkata () index gives a measure of fraction of population who possess top fraction of wealth in the society. This article reviews the ch…
We establish a mod 2 index theorem for real vector bundles over 8k+2 dimensional compact pin manifolds. The analytic index is the reduced invariant of (twisted) Dirac operators and the topological index is defined through -theory. Our main result extends the mod 2 index theorem of Atiyan and Singer to non-o…
Extends index theorem to domain walls with discontinuous Riemannian connections.
problem Index theorem for domain walls with discontinuous Yang-Mills and Riemannian connections.
method Extension of index theorem to new conditions.
result Validates index theorem for more complex discontinuities.
In this note, we prove an index theorem on Galois covering for Heisenberg elliptic differential operators, which is not elliptic, analogous to Atiyah's -index theorem. This note also contains an example of Heisenberg differential operators with non-trivial -index.
Atiyah-Singer theorem links math fields, predicts topological insights.
problem Understanding the interplay between analysis, geometry, and topology.
method Analyzes and generalizes topological invariants in differential geometry.
result Predicts the index of elliptic operators based on topology.
Massive fermions help understand index theorems without chiral symmetry.
problem Understanding index theorems in massive fermion systems.
method Reformulate chiral anomaly and index theorems with massive Dirac operators.
result Nontrivial mathematical relations between massless and massive fermions.
This paper investigates analytic properties of American option prices under the finite moment log-stable (FMLS) model. Under this model the price of American options is characterised by the free boundary problem of a fractional partial differential equation (FPDE) system. Using the technique of approximation we prove t…
Abstract: Review of Index theorem and its applications.
problem Index theorem applications in differential geometry.
method Review of Atiyah-Singer Index theorem.
result Basic knowledge required for understanding.
A recent anomaly computation of Horava and Witten is proved and generalized in the form of two index theorems in odd dimensions. Theorem A is a fixed point formula for orientation-reversing involutions. Theorem B is an index theorem for manifolds with boundary using local boundary conditions. Both hold for families of …
New rough stochastic volatility models using log-modulated fractional Brownian motion.
problem Analyzing rough stochastic volatility models over the range .
method Introducing log-modulated fractional Brownian motion (log-fBm) to handle and analyze over the full range.
result Obtained skew asymptotics of as for , no flattening of skew as .
Proves Morse index theorem for geodesics in conic Finsler manifolds.
problem Geodesic index theorem in conic Finsler manifolds with variable endpoints.
method Proves Morse index theorem for geodesics connecting submanifolds in a manifold with a conic pseudo-Finsler metric.
result Establishes the Morse index theorem for geodesics in conic Finsler manifolds.
Paper introduces Lie algebroid index theory and a generalized Riemann-Roch theorem.
problem Generalizing Riemann-Roch theorem for manifolds with regular foliations.
method Developed Lie algebroid index theory and applied it to obtain a generalized Riemann-Roch theorem.
result Obtained a generalized Riemann-Roch theorem for manifolds with regular foliations.