Parametric Family Research Articles

A classification of nilpotent compatible Lie algebras

Working over an arbitrary field of characteristic different from 2, we extend the Skjelbred-Sund method to compatible Lie algebras and give a full classification of nilpotent compatible Lie algebras up to dimension 4. In case the base field is cubically closed, we find that there are three isomorphism classes and a one-parameter family in dimension 3, and 12 isomorphism classes, 6 one-parameter families and one 2-parameter family in dimension 4.

Sparse Wasserstein Barycenters and Application to Reduced Order Modeling

We develop a general theoretical and algorithmic framework for sparse approximation and structured prediction in P2(Ω) with Wasserstein barycenters. The barycenters are sparse in the sense that they are computed from an available dictionary of measures, but the approximations only involve a reduced number of atoms. We show that the best reconstruction from the class of sparse barycenters is characterized by a notion of best n-term barycenter which we introduce, and which can be understood as a natural extension of the classical concept of best n-term approximation in Banach spaces. We show that the best n-term barycenter is the minimizer of a highly non-convex, bi-level optimization problem, and we develop algorithmic strategies for practical numerical computation. We next leverage this approximation tool to build interpolation strategies that involve a reduced computational cost, and that can be used for structured prediction, and metamodeling of parametrized families of measures. We illustrate the potential of the method through the specific problem of Model Order Reduction (MOR) of parametrized PDEs. Since our approach is sparse, adaptive and preserves mass by construction, it has potential to overcome known bottlenecks of classical linear methods in hyperbolic conservation laws transporting discontinuities. It also paves the way towards MOR for measure-valued PDE problems such as gradient flows.

Diversification Quotients: Quantifying Diversification via Risk Measures

We establish the first axiomatic theory for diversification indices using six intuitive axioms: nonnegativity, location invariance, scale invariance, rationality, normalization, and continuity. The unique class of indices satisfying these axioms, called the diversification quotients (DQs), are defined based on a parametric family of risk measures. A further axiom of portfolio convexity pins down DQs based on coherent risk measures. The DQ has many attractive properties, and it can address several theoretical and practical limitations of existing indices. In particular, for the popular risk measures value at risk and expected shortfall, the corresponding DQ admits simple formulas, and it is efficient to optimize in portfolio selection. Moreover, it can properly capture tail heaviness and common shocks, which are neglected by traditional diversification indices. When illustrated with financial data, the DQ is intuitive to interpret, and its performance is competitive against other diversification indices. This paper was accepted by Manel Baucells, behavioral economics and decision analysis. Funding: X. Han is supported by the National Natural Science Foundation of China [Grants 12301604, 12371471, and 12471449). L. Lin is supported by the Hickman Scholarship from the Society of Actuaries. R. Wang is supported by the Natural Sciences and Engineering Research Council of Canada [Grants CRC-2022-00141 and RGPIN-2024-03728] and the Sun Life Research Fellowship. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.00513 .

Automated Credit Card Risk Assessment using Fuzzy Parameterized Neutrosophic Hypersoft Expert Set

In the financial industry, financial fraud is an ever-evolving risk with extreme consequences. Data mining has been instrumental in the recognition of credit card fraud (CCF) during online transactions. CCF recognition, which is a data mining problem, become a challenge owing to its two main reasons - firstly, the profiles of fraudulent and normal behaviors modify continually and then, CCF dataset is extremely lopsided. The implementation of fraud recognition in credit card transactions is tremendously influenced by the sampling methodology on data, detection approach and variable selection utilized. The conception of the neutrosophic hypersoft set (NHSS) is a parameterized family that handles the sub-attributes of the parameter and is an appropriate extension of the NHSS to correctly evaluate the uncertainty, deficiencies, and anxiety in decision-making. In comparison to previous research, NHSS can accommodate additional uncertainty, which is the crucial approach to describe fuzzy datasets in the decision-making algorithm. This study introduces an Automated Credit Card Risk Assessment using Fuzzy Parameterized Neutrosophic Hypersoft Expert Set (ACCRA-FPNHES) technique. In the ACCRA-FPNHES technique, a three-step process is involved. As a primary step, the ACCRA-FPNHES technique designs sparrow search algorithm (SSA) for choosing features. In the second step, the detection of CCF takes place using FPNHES technique. Finally, in the third step, the parameters related to the FPNHES technique can be adjusted by arithmetic optimization algorithm (AOA). The simulation validation of the ACCRA-FPNHES technique can be studied on credit card dataset. The obtained values indicate that the ACCRA-FPNHES technique showcases better performance

Orbits of families of discrete dynamical systems evolving in the natural numbers

In this paper, we give a class of one-dimensional discrete dynamical systems with state space N+. This class of systems is defined by two parameters: one of them sets the number of nearest neighbors that determine the rule of evolution, and the other parameter determines a segment of natural numbers Λ={1,2,…,b}. In particular, we investigate the behavior of a class of one-dimensional maps where an integer moves to an other integer given by the sum of the nearest neighbors minus a multiple of b∈N+. We find the coexistence of fixed points and periodic cycles. Two single parameter families of maps are introduced and their dynamics in the segment of natural sequence Λ. Furthermore, an order of the numbers of the set Λ−b is given by these families. Last, we present a connection of the N+ generated by the orbits of a particular case.

CALCULATING POWER INTEGRAL BASES IN SOME QUARTIC FIELDS CORRESPONDING TO MONOGENIC FAMILIES OF POLYNOMIALS

Harrington and Jones characterized monogenity of four new parametric families of quartic polynomials with various Galois groups. A short time later Voutier added a cyclic family. In this note, we intend to describe all generators of power integral bases in the number fields generated by a root of the monogenic polynomials.

New Andrews–Curtis trivializations for Miller–Schupp group presentations

We present recent developments in the applications of automated theorem proving in the investigation of the Andrews-Curtis conjecture. We demonstrate previously unknown trivializations of group presentations from a parametric family MSn(w∗) of trivial group presentations for n=3,4,5,6,7,8 (subset of well-known Miller–Schupp family). Based on the human analysis of these trivializations we formulate two conjectures on the structure of simplifications for the infinite family MSn(w∗), n≥3.

X-vine models for multivariate extremes

Abstract Regular vine sequences permit the organization of variables in a random vector along a sequence of trees. Vine-based dependence models have become greatly popular as a way to combine arbitrary bivariate copulas into higher-dimensional ones, offering flexibility, parsimony, and tractability. In this project, we use regular vine sequences to decompose and construct the exponent measure density of a multivariate extreme value distribution, or, equivalently, the tail copula density. Although these densities pose theoretical challenges due to their infinite mass, their homogeneity property offers simplifications. The theory sheds new light on existing parametric families and facilitates the construction of new ones, called X-vines. Computations proceed via recursive formulas in terms of bivariate model components. We develop simulation algorithms for X-vine multivariate Pareto distributions as well as methods for parameter estimation and model selection on the basis of threshold exceedances. The methods are illustrated by Monte Carlo experiments and a case study on US flight delay data.

Gill and Massar type bound for estimation of SU(2) channel

Abstract In the estimation for a parametric family of quantum state on a Hilbert space H, the Gill and Massar bound is known as a lower bound of weighted traces of covariances of unbiased estimators. The Gill and Massar bound is derived by considering the convexity of the set of classical Fisher information matrices, and the bound is locally achievable by using randomized strategies when H=C2. In this paper, we show that estimation for a parametric SU(2) unitary channel model has a similar convex structure as qubit state model, and a Gill and Massar type lower bound of weighted traces of covariances of unbiased estimators can be derived for any weight matrix. We show that the Gill and Massar type lower bound is achievable by using randomized strategies when certain conditions are satisfied. To derive a convex strucutre of the set of classical Fisher information matrices, we introduce a Fisher information matrix J(U) for a SU(2) unitary channel model, and we show a upper bound of inverse J(U) weighted trace of classical Fisher information matrix. The optimal randomized strategy we construct in this paper does not require ancilla systems in many cases.

Computational aspects of likelihood-based inference for the univariate generalized hyperbolic distribution

The generalized hyperbolic distribution is among the more often adopted parametric families in a wide range of application areas, thanks to its high flexibility as the parameters vary and also to a plausible stochastic mechanism for its genesis. This high flexibility comes at some cost, however, namely the frequent difficulty of estimating its parameters due to the presence of flat areas of the log-likelihood function, so that selected points of the parameter space, while very distant, can be essentially equivalent as for data fitting. This phenomenon affects not only maximum likelihood estimation, but Bayesian methods too, since the target function is little affected by the introduction of a prior distribution. Our interest focuses in fact on maximum likelihood estimation of the Generalized hyperbolic distribution, working in the univariate case. This paper improves upon currently employed computational techniques by presenting an alternative proposal that works effectively in reaching the global maximum of the likelihood function. The paper further illustrates the above mentioned problems in a number of cases, using both simulated and real data.

Two New Families of Local Asymptotically Minimax Lower Bounds in Parameter Estimation.

We propose two families of asymptotically local minimax lower bounds on parameter estimation performance. The first family of bounds applies to any convex, symmetric loss function that depends solely on the difference between the estimate and the true underlying parameter value (i.e., the estimation error), whereas the second is more specifically oriented to the moments of the estimation error. The proposed bounds are relatively easy to calculate numerically (in the sense that their optimization is over relatively few auxiliary parameters), yet they turn out to be tighter (sometimes significantly so) than previously reported bounds that are associated with similar calculation efforts, across many application examples. In addition to their relative simplicity, they also have the following advantages: (i) Essentially no regularity conditions are required regarding the parametric family of distributions. (ii) The bounds are local (in a sense to be specified). (iii) The bounds provide the correct order of decay as functions of the number of observations, at least in all the examples examined. (iv) At least the first family of bounds extends straightforwardly to vector parameters.

An Algorithm for Producing Fuzzy Implications via Conical Sections

In the present study, a novel parametric family of fuzzy implications is introduced and its properties are examined. The parametric family of implications is produced only via a fuzzy negation. This in turn enables the effortless production of a wide range of implications from which to select the one that best fits a given problem, for example in fuzzy inference systems or fuzzy neural networks. The fuzzy negations that have been selected as a basis for the proposed methodology are strong, i.e., involutions, thus leading, in general, to the generated fuzzy implications possessing many desirable additional properties. We have examined which of these properties hold for the implications produced by our algorithm and under which conditions. Finally, it is demonstrated that the family of implications generated via the proposed methodology generalizes other well-established implications, including the Łukasiewicz implication.

Scalable Empirical Bayes Inference and Bayesian Sensitivity Analysis.

Consider a Bayesian setup in which we observe , whose distribution depends on a parameter , that is, . The parameter is unknown and treated as random, and a prior distribution chosen from some parametric family , is to be placed on it. For the subjective Bayesian there is a single prior in the family which represents his or her beliefs about , but determination of this prior is very often extremely difficult. In the empirical Bayes approach, the latent distribution on is estimated from the data. This is usually done by choosing the value of the hyperparameter that maximizes some criterion. Arguably the most common way of doing this is to let be the marginal likelihood of , that is, , and choose the value of that maximizes . Unfortunately, except for a handful of textbook examples, analytic evaluation of is not feasible. The purpose of this paper is two-fold. First, we review the literature on estimating it and find that the most commonly used procedures are either potentially highly inaccurate or don't scale well with the dimension of , the dimension of , or both. Second, we present a method for estimating , based on Markov chain Monte Carlo, that applies very generally and scales well with dimension. Let be a real-valued function of , and let be the posterior expectation of when the prior is . As a byproduct of our approach, we show how to obtain point estimates and globally-valid confidence bands for the family , . To illustrate the scope of our methodology we provide three detailed examples, having different characters.

Dimension of Planar Non-conformal Attractors with Triangular Derivative Matrices

We study the dimension of the attractor and quasi-Bernoulli measures of parametrized families of iterated function systems of non-conformal and non-affine maps. We introduce a transversality condition under which, relying on a weak Ledrappier-Young formula, we show that the dimensions equal to the root of the subadditive pressure and the Lyapunov dimension, respectively, for almost every choice of parameters. We also exhibit concrete examples satisfying the transversality condition with respect to the translation parameters.

A generic classification of locally free representations of affine GLS algebras

Throughout, let K be an algebraically closed field of characteristic 0. We provide a generic classification of locally free representations of Geiß-Leclerc-Schröer's algebras HK(C,D,Ω) associated to affine Cartan matrices C with minimal symmetrizer D and acyclic orientation Ω. Affine GLS algebras are “smooth” degenerations of tame hereditary algebras and as such their representation theory is presumably still tractable. Indeed, we observe several “tame” phenomena of affine GLS algebras even though they are in general representation wild. For the GLS algebras of type BC˜1 we achieve a classification of all stable representations. For general GLS algebras of affine type, we construct a 1-parameter family of representations stable with respect to the defect. Our construction is based on a generalized one-point extension technique. This confirms in particular τ-tilted versions of the second Brauer-Thrall Conjecture recently raised by Mousavand and Schroll-Treffinger-Valdivieso for the class of GLS algebras. Finally, we show that generically every locally free H-module is isomorphic to a direct sum of τ-rigid modules and modules from our 1-parameter family. This generalizes Kac's canonical decomposition from the symmetric to the symmetrizable case in affine types and we obtain such a decomposition by folding the canonical decomposition of dimension vectors over path algebras. As a corollary we obtain that affine GLS algebras are E-tame in the sense of Derksen-Fei and Asai-Iyama.

Integrable flows on null curves in the Anti-de Sitter 3-space

Abstract We formulate integrable flows related to the Korteweg–De Vries (KdV) hierarchy on null curves in the anti-de Sitter 3-space ( AdS ). Exploiting the specific properties of the geometry of AdS , we analyze their interrelationships with Pinkall flows in centro-affine geometry. We show that closed stationary solutions of the lower order flow can be explicitly found in terms of periodic solutions of a Lamé equation. In addition, we study the evolution of non-stationary curves arising from a 3-parameter family of periodic solutions of the KdV equation.

Limiting Behavior of Mixed Coherent Systems With Lévy‐Frailty Marshall–Olkin Failure Times

ABSTRACTIn this article, we show a limit result for the reliability function of a system—that is, the probability that the whole system is still operational after a certain given time—when the number of components of the system grows to infinity. More specifically, we consider a sequence of mixed coherent systems whose components are homogeneous and non‐repairable, with failure‐times governed by a Lévy‐Frailty Marshall–Olkin (LFMO) distribution—a distribution that allows simultaneous component failures. We show that under integrability conditions the reliability function converges to the probability of a first‐passage time of a Lévy subordinator process. To the best of our knowledge, this is the first result to tackle the asymptotic behavior of the reliability function as the number of components of the system grows. To illustrate our approach, we give an example of a parametric family of reliability functions where the system failure time converges in distribution to an exponential random variable, and give computational experiments testing convergence.

Risk measures based on weak optimal transport

In this paper, we study convex risk measures with weak optimal transport penalties. In a first step, we show that these risk measures allow for an explicit representation via a nonlinear transform of the loss function. In a second step, we discuss computational aspects related to the nonlinear transform as well as approximations of the risk measures using, for example, neural networks. Our setup comprises a variety of examples, such as classical optimal transport penalties, parametric families of models, divergence risk measures, uncertainty on path spaces, moment constraints, and martingale constraints. In a last step, we show how to use the theoretical results for the numerical computation of worst-case losses in an insurance context and no-arbitrage prices of European contingent claims after quoted maturities in a model-free setting.

Semiclassical Equivalence of Two White Dwarf Models as Ground States of the Relativistic Hartree–Fock and Vlasov–Poisson energies

AbstractWe are concerned with the semi-classical limit for ground states of the relativistic Hartree–Fock energies (HF) under a mass constraint, which are considered as the quantum mean-field model of white dwarfs (Lenzmann and Lewin in Duke Math J 152:257–315, 2010). In Jang and Seok (Kinet Relat Models 15:605–620, 2022), fermionic ground states of the relativistic Vlasov–Poisson energy (VP) are constructed as a classical mean-field model of white dwarfs, and are shown to be equivalent to the classical Chandrasekhar model. In this paper, we prove that as the reduced Planck constant $$\hbar $$ ħ goes to the zero, the $$\hbar $$ ħ -parameter family of the ground energies and states of (HF) converges to the fermionic ground energy and state of (VP) with the same mass constraint.

Cosmological correlators for Bogoliubov initial states

We consider late-time correlators in de Sitter (dS) space for initial states related to the Bunch-Davies vacuum by a Bogoliubov transformation. We propose to study such late-time correlators by reformulating them in the familiar language of Witten diagrams in Euclidean anti-de Sitter space (EAdS), showing that they can be perturbatively re-cast in terms of corresponding dS boundary correlators in the Bunch-Davies vacuum and in turn, Witten diagrams in EAdS. Unlike the standard relationship between late-time correlators in the Bunch-Davies vacuum and EAdS Witten diagrams, this involves points on the upper and lower sheet of the EAdS hyperboloid which account for antipodal singularities of the two-point functions. Such Bogoliubov states include an infinite one parameter family of de Sitter invariant vacua as a special case, where the late-time correlators are constrained by conformal Ward identities. In momentum space, it is well known that their late-time correlators exhibit singularities in collinear (“folded”) momentum configurations. We give a position space interpretation of such solutions to the conformal Ward identities, where in embedding space they can be generated from the solution without collinear singularities by application of the antipodal map. We also discuss the operator product expansion (OPE) limit of late-time correlators in a generic dS invariant vacuum. Many results are derived using the Mellin space representation of late-time correlators, which in this work we extend to accommodate generic dS invariant vacua.