arXiv subjects

Farzad Fathizadeh

Publications and source records attributed to Farzad Fathizadeh.

At least 19 recordsLinked to original sources

arXiv2021

Fuzzy Gaussian mixture optimization of the newsvendor problem: mixing online reviews and judgemental demand data

Motivated by the increasing exposition of decision makers to both statistical and judgemental based sources of demand information, we develop in this paper a fuzzy Gaussian Mixture Model (GMM) for the newsvendor permitting to mix probabilistic inputs with a subjective weight modelled as a fuzzy number. The developed framework can model for instance situations where sales are impacted by customers sensitive to online review feedbacks or expert opinions. It can also model situations where a marketing campaign leads to different stochastic alternatives for the demand with a fuzzy weight. Thanks to a tractable mathematical application of the fuzzy machinery on the newsvendor problem, we derived the optimal ordering strategy taking into account both probabilistic and fuzzy components of the demand. We show that the fuzzy GMM can be rewritten as a classical newsvendor problem with an associated density function involving these stochastic and fuzzy components of the demand. The developed model enables to relax the single modality of the demand distribution usually used in the newsvendor literature and to encode the risk attitude of the decision maker.

math.OC↗

arXiv2020

Curvature in Noncommutative Geometry

Our understanding of the notion of curvature in a noncommutative setting has progressed substantially in the past ten years. This new episode in noncommutative geometry started when a Gauss-Bonnet theorem was proved by Connes and Tretkoff for a curved noncommutative two torus. Ideas from spectral geometry and heat kernel asymptotic expansions suggest a general way of defining local curvature invariants for noncommutative Riemannian type spaces where the metric structure is encoded by a Dirac type operator. To carry explicit computations however one needs quite intriguing new ideas. We give an account of the most recent developments on the notion of curvature in noncommutative geometry in this paper.

math.QA↗

arXiv2019

Signal detection in extracellular neural ensemble recordings using higher criticism

Information processing in the brain is conducted by a concerted action of multiple neural populations. Gaining insights in the organization and dynamics of such populations can best be studied with broadband intracranial recordings of so-called extracellular field potential, reflecting neuronal spiking as well as mesoscopic activities, such as waves, oscillations, intrinsic large deflections, and multiunit spiking activity. Such signals are critical for our understanding of how neuronal ensembles encode sensory information and how such information is integrated in the large networks underlying cognition. The aforementioned principles are now well accepted, yet the efficacy of extracting information out of the complex neural data, and their employment for improving our understanding of neural networks, critically depends on the mathematical processing steps ranging from simple detection of action potentials in noisy traces - to fitting advanced mathematical models to distinct patterns of the neural signal potentially underlying intra-processing of information, e.g. interneuronal interactions. Here, we present a robust strategy for detecting signals in broadband and noisy time series such as spikes, sharp waves and multi-unit activity data that is solely based on the intrinsic statistical distribution of the recorded data. By using so-called higher criticism - a second-level significance testing procedure comparing the fraction of observed significances to an expected fraction under the global null - we are able to detect small signals in correlated noisy time-series without prior filtering, denoising or data regression. Results demonstrate the efficiency and reliability of the method and versatility over a wide range of experimental conditions and suggest the appropriateness of higher criticism to characterize neuronal dynamics without prior manipulation of the data.

stat.AP↗

arXiv2019

A twisted local index formula for curved noncommutative two tori

We consider the Dirac operator of a general metric in the canonical conformal class on the noncommutative two torus, twisted by an idempotent (representing the $K$-theory class of a general noncommutative vector bundle), and derive a local formula for the Fredholm index of the twisted Dirac operator. Our approach is based on the McKean-Singer index formula, and explicit heat expansion calculations by making use of Connes' pseudodifferential calculus. As a technical tool, a new rearrangement lemma is proved to handle challenges posed by the noncommutativity of the algebra and the presence of an idempotent in the calculations in addition to a conformal factor.

math.QA↗

arXiv2018

Bell polynomials and Brownian bridge in Spectral Gravity models on multifractal Robertson-Walker cosmologies

We obtain an explicit formula for the full expansion of the spectral action on Robertson-Walker spacetimes, expressed in terms of Bell polynomials, using Brownian bridge integrals and the Feynman-Kac formula. We then apply this result to the case of multifractal Packed Swiss Cheese Cosmology models obtained from an arrangement of Robertson-Walker spacetimes along an Apollonian sphere packing. Using Mellin transforms, we show that the asymptotic expansion of the spectral action contains the same terms as in the case of a single Robertson-Walker spacetime, but with zeta-regularized coefficients, given by values at integers of the zeta function of the fractal string of the radii of the sphere packing, as well as additional log-periodic correction terms arising from the poles (off the real line) of this zeta function.

math-ph↗

arXiv2017

Motives and periods in Bianchi IX gravity models

We show that, when considering the anisotropic scaling factors and their derivatives as affine variables, the coefficients of the heat kernel expansion of the Dirac-Laplacian on $SU(2)$ Bianchi IX metrics are algebro-geometric periods of motives of complements in affine spaces of unions of quadrics and hyperplanes. We show that the motives are mixed Tate and we provide an explicit computation of their Grothendieck classes.

math-ph↗

arXiv2016

The term a_4 in the heat kernel expansion of noncommutative tori

We consider the Laplacian associated with a general metric in the canonical conformal structure of the noncommutative two torus, and calculate a local expression for the term a_4 that appears in its corresponding small-time heat kernel expansion. The final formula involves one variable functions and lengthy two, three and four variable functions of the modular automorphism of the state that encodes the conformal perturbation of the flat metric. We confirm the validity of the calculated expressions by showing that they satisfy a family of conceptually predicted functional relations. By studying these functional relations abstractly, we derive a partial differential system which involves a natural action of cyclic groups of order two, three and four and a flow in parameter space. We discover symmetries of the calculated expressions with respect to the action of the cyclic groups. In passing, we show that the main ingredients of our calculations, which come from a rearrangement lemma and relations between the derivatives up to order four of the conformal factor and those of its logarithm, can be derived by finite differences from the generating function of the Bernoulli numbers and its multiplicative inverse. We then shed light on the significance of exponential polynomials and their smooth fractions in understanding the general structure of the noncommutative geometric invariants appearing in the heat kernel expansion. As an application of our results we obtain the a_4 term for noncommutative four tori which split as products of two tori. These four tori are not conformally flat and the a_4 term gives a first hint of the Riemann curvature and the higher dimensional modular structure.

math.QA↗

arXiv2016

Periods and motives in the spectral action of Robertson-Walker spacetimes

We show that, when considering the scaling factor as an affine variable, the coefficients of the asymptotic expansion of the spectral action on a (Euclidean) Robertson-Walker spacetime are periods of mixed Tate motives, involving relative motives of complements of unions of hyperplanes and quadric hypersurfaces and divisors given by unions of coordinate hyperplanes.

math-ph↗

arXiv2016

On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for $C^*$-Dynamical Systems

The analog of the Chern-Gauss-Bonnet theorem is studied for a $C^*$-dynamical system consisting of a $C^*$-algebra $A$ equipped with an ergodic action of a compact Lie group $G$. The structure of the Lie algebra $\mathfrak{g}$ of $G$ is used to interpret the Chevalley-Eilenberg complex with coefficients in the smooth subalgebra $\mathcal{A} \subset A$ as noncommutative differential forms on the dynamical system. We conformally perturb the standard metric, which is associated with the unique $G$-invariant state on $A$, by means of a Weyl conformal factor given by a positive invertible element of the algebra, and consider the Hermitian structure that it induces on the complex. A Hodge decomposition theorem is proved, which allows us to relate the Euler characteristic of the complex to the index properties of a Hodge-de Rham operator for the perturbed metric. This operator, which is shown to be selfadjoint, is a key ingredient in our construction of a spectral triple on $\mathcal{A}$ and a twisted spectral triple on its opposite algebra. The conformal invariance of the Euler characteristic is interpreted as an indication of the Chern-Gauss-Bonnet theorem in this setting. The spectral triples encoding the conformally perturbed metrics are shown to enjoy the same spectral summability properties as the unperturbed case.

math.OA↗

arXiv2015

Modular forms in the spectral action of Bianchi IX gravitational instantons

Following our result on rationality of the spectral action for Bianchi type-IX cosmological models, which suggests the existence of a rich arithmetic structure in the action, we study the arithmetic and modular properties of the action for an especially interesting family of metrics, namely $SU(2)$-invariant Bianchi IX gravitational instantons. A variant of the previous rationality result is first presented for a time-dependent conformal perturbation of the triaxial Bianchi IX metric which serves as a general form of Bianchi IX gravitational instantons. By exploiting a parametrization of the gravitational instantons in terms of theta functions with characteristics, we show that each Seeley-de Witt coefficient $\tilde a_{2n}$ appearing in the expansion of their spectral action gives rise to vector-valued and ordinary modular functions of weight 2. We investigate the modular properties of the first three terms $\tilde a_0$, $\tilde a_2$ and $\tilde a_4$ by preforming direct calculations. This guides us to provide spectral arguments for general terms $\tilde a_{2n}$, which are based on isospectrality of the involved Dirac operators. Special attention is paid to the relation of the new modular functions with well-known modular forms by studying explicitly two different cases of metrics with pairs of rational parameters that belong to two different general families. In the first case, it is shown that by running a summation over a finite orbit of the rational parameters, up to multiplication by the cusp form $Δ$ of weight 12, each term $\tilde a_{2n}$ lands in the 1-dimensional linear space of modular forms of weight 14. In the second case, it is proved that, after a summation over a finite orbit of the parameters and multiplication by $G_4^4$, where $G_4$ is the Eisenstein series of weight 4, each $\tilde a_{2n}$ lands in the 1-dimensional linear space of cusp forms of weight 18.

math.DG↗

arXiv2015

Spectral Action for Bianchi Type-IX Cosmological Models

A rationality result previously proved for Robertson-Walker metrics is extended to a homogeneous anisotropic cosmological model, namely the Bianchi type-IX minisuperspace. It is shown that the Seeley-de Witt coefficients appearing in the expansion of the spectral action for the Bianchi type-IX geometry are expressed in terms of polynomials with rational coefficients in the cosmic evolution factors $w_1(t), w_2(t), w_3(t),$ and their higher derivates with respect to time. We begin with the computation of the Dirac operator of this geometry and calculate the coefficients $a_0, a_2, a_4$ of the spectral action by using heat kernel methods and parametric pseudodifferential calculus. An efficient method is devised for computing the Seeley-de Witt coefficients of a geometry by making use of Wodzicki's noncommutative residue, and it is confirmed that the method checks out for the cosmological model studied in this article. The advantages of the new method are discussed, which combined with symmetries of the Bianchi type-IX metric, yield an elegant proof of the rationality result.

hep-th↗

arXiv2014

Rationality of Spectral Action for Robertson-Walker Metrics

We use pseudodifferential calculus and heat kernel techniques to prove a conjecture by Chamseddine and Connes on rationality of the coefficients of the polynomials in the cosmic scale factor $a(t)$ and its higher derivatives, which describe the general terms $a_{2n}$ in the expansion of the spectral action for general Robertson-Walker metrics. We also compute the terms up to $a_{12}$ in the expansion of the spectral action by our method. As a byproduct, we verify that our computations agree with the terms up to $a_{10}$ that were previously computed by Chamseddine and Connes by a different method.

math-ph↗

arXiv2014

On the Scalar Curvature for the Noncommutative Four Torus

The scalar curvature for the noncommutative four torus $\mathbb{T}_Θ^4$, where its flat geometry is conformally perturbed by a Weyl factor, is computed by making the use of a noncommutative residue that involves integration over the 3-sphere. This method is more convenient since it does not require the rearrangement lemma and it is advantageous as it explains the simplicity of the final functions of one and two variables, which describe the curvature with the help of a modular automorphism. In particular, it readily allows to write the function of two variables as the sum of a finite difference and a finite product of the one variable function. The curvature formula is simplified for dilatons of the form $sp$, where $s $ is a real parameter and $p \in C^\infty(\mathbb{T}_Θ^4)$ is an arbitrary projection, and it is observed that, in contrast to the two dimensional case studied by A. Connes and H. Moscovici, unbounded functions of the parameter $s$ appear in the final formula. An explicit formula for the gradient of the analog of the Einstein-Hilbert action is also calculated.

math.QA↗

arXiv2013

A Two Dimensional Adler-Manin Trace and Bi-singular Operators

Motivated by the theory of bi-singular pseudodifferential operators, we introduce a two dimensional version of the Adler-Manin trace. Our construction is rather general in the sense that it involves a twist afforded by an algebra automorphism. That is, starting from an algebra equipped with an automorphism, two twisted derivations, and a twisted invariant trace, we construct an algebra of formal twisted pseudodifferential symbols and define a noncommutative residue. Also, we provide related examples.

math.AP↗

arXiv2013

Scalar Curvature for Noncommutative Four-Tori

In this paper we study the curved geometry of noncommutative 4-tori $\mathbb{T}_θ^4$. We use a Weyl conformal factor to perturb the standard volume form and obtain the Laplacian that encodes the local geometric information. We use Connes' pseudodifferential calculus to explicitly compute the terms in the small time heat kernel expansion of the perturbed Laplacian which correspond to the volume and scalar curvature of $\mathbb{T}_θ^4$. We establish the analogue of Weyl's law, define a noncommutative residue, prove the analogue of Connes' trace theorem, and find explicit formulas for the local functions that describe the scalar curvature of $\mathbb{T}_θ^4$. We also study the analogue of the Einstein-Hilbert action for these spaces and show that metrics with constant scalar curvature are critical for this action.

math.QA↗

arXiv2012

Weyl's Law and Connes' Trace Theorem for Noncommutative Two Tori

We prove the analogue of Weyl's law for a noncommutative Riemannian manifold, namely the noncommutative two torus $\mathbb{T}_θ^2$ equipped with a general translation invariant conformal structure and a Weyl conformal factor. This is achieved by studying the asymptotic distribution of the eigenvalues of the perturbed Laplacian on $\mathbb{T}_θ^2$. We also prove the analogue of Connes' trace theorem by showing that the Dixmier trace and a noncommutative residue coincide on pseudodifferential operators of order -2 on $\mathbb{T}_θ^2$.

math.QA↗

arXiv2011

A Noncommutative Residue on Tori and a Semiclassical Limit

We define a noncommutative residue for classical Euclidean pseudodifferential operators on a torus of arbitrary dimension. We prove that, up to multiplication by a constant, it is the unique trace on the algebra of classical pseudodifferential operators modulo infinitely smoothing operators. In the case of the two torus, we show that the noncommutative residue is the semiclassical limit of a noncommutative residue defined on classical pseudodifferential operators on noncommutative two tori.

math.AP↗

arXiv2011

Scalar Curvature for the Noncommutative Two Torus

We give a local expression for the {\it scalar curvature} of the noncommutative two torus $ A_θ = C(\mathbb{T}_θ^2)$ equipped with an arbitrary translation invariant complex structure and Weyl factor. This is achieved by evaluating the value of the (analytic continuation of the) {\it spectral zeta functional} $ζ_a(s): = \text{Trace}(a \triangle^{-s})$ at $s=0$ as a linear functional in $a \in C^{\infty}(\mathbb{T}_θ^2)$. A new, purely noncommutative, feature here is the appearance of the {\it modular automorphism group} from the theory of type III factors and quantum statistical mechanics in the final formula for the curvature. This formula coincides with the formula that was recently obtained independently by Connes and Moscovici in their recent paper.

math.QA↗