Recursive Set Research Articles

Online Estimation and Optimization of Utility-Based Shortfall Risk

Utility-based shortfall risk (UBSR) is a risk metric that is increasingly popular in financial applications, owing to certain desirable properties that it enjoys. We consider the problem of estimating UBSR in a recursive setting, in which samples from the underlying loss distribution are available one at a time. We cast the UBSR estimation problem as a root-finding problem and propose stochastic approximation-based estimation schemes. We derive nonasymptotic bounds on the estimation error in the number of samples. We also consider the problem of UBSR optimization within a parameterized class of random variables. We propose a stochastic gradient descent–based algorithm for UBSR optimization and derive nonasymptotic bounds on its convergence.

Аксиоматизируемость и разрешимость универсальных теорий наследственных классов моделей конечных и бесконечных языков

In the paper, hereditary classes of L-structures are studied with language of the form L = Lfin ∪ L∞, where Lfin = ⟨R1, R2, . . . , Rm, =⟩ and L∞ = ⟨Rm+1, Rm+2, . . .⟩, and also in L∞ the number of predicates of each arity is finite, all predicates are ordered in ascending of their arities and satisfy the non-element repetition property. A class of L-structures is called hereditary if it is closed under substructures. It is proved that the class of L-structures is hereditary if and only if it can be defined in terms of forbidden substructures. A class of L-structures is called universally axiomatizable if there is a set Z of universal L-sentences such that the class consists of all structures satisfying Z. The problems of the universal axiomatizability of hereditary classes of L-structures are considered in the paper. It is shown that hereditary class of L-structures is universally axiomatizable if and only if it can be defined in terms of finite forbidden substructures. It is proved that the universal theory of any axiomatizable hereditary class of L-structures with a recursive set of minimal forbidden substructures is decidable.

On the relationships between some meta-mathematical properties of arithmetical theories

Abstract In this work, we aim at understanding incompleteness in an abstract way via metamathematical properties of formal theories. We systematically examine the relationships between the following twelve important metamathematical properties of arithmetical theories: Rosser, EI (effectively inseparable), RI (recursively inseparable), TP (Turing persistent), EHU (essentially hereditarily undecidable), EU (essentially undecidable), Creative, $\textbf{0}^{\prime }$ (theories with Turing degree $\textbf{0}^{\prime }$), REW (all RE sets are weakly representable), RFD (all recursive functions are definable), RSS (all recursive sets are strongly representable), RSW (all recursive sets are weakly representable). Given any two properties $P$ and $Q$ in the above list, we examine whether $P$ implies $Q$.

Modern perspectives in Proof Theory.

Modern perspectives in Proof Theory.

ON THE EXISTENCE OF UNIVERSAL NUMBERINGS

The paper is devoted to research existence property of universal numberings for different computable families. A numbering α is reducible to a numbering β if there is computable function ƒ such that α = β ◦ ƒ. A computable numbering α for some family S is universal if any computable numbering β for the family S is reducible to α. It is well known that the family of all computably enumerable (c.e.) sets has a computable universal numbering. In this paper, we study families of almost all c.e. sets, recursive sets, and almost all differences of c.e. sets, namely questions about the existence of universal numberings for given families. We proved that there is no universal numbering for the family of all recursive sets. For families of c.e. sets without an empty set or a finite number of finite sets, there still exists a universal numbering. However, for families of all c.e. sets without an infinite set, there is no universal numbering. Also, we proved that family ∑2-1 \ Β and the family ∑1-1 has no universal ∑2-1-computable numbering for any Β ∈ ∑2-1.

An asymptotic consistent couple stress theory for the three‐dimensional free vibration analysis of functionally graded microplates resting on an elastic medium

Based on the three‐dimensional (3D) consistent couple stress theory (CCST), we develop an asymptotic theory for the free vibration analysis of functionally graded (FG) microplates resting on an elastic medium. The material properties of the FG microplate are assumed to obey a power‐law distribution of the volume fractions of the constituents in the thickness direction, for which the effective material properties are estimated using the rule of mixtures. The interactions between the FG microplate and its foundation medium are simulated using a Winkler–Pasternak foundation. Carrying out the asymptotic expansion method, we obtain recursive sets of motion equations for various order problems. The CCST‐based classical plate theory is derived as a first‐order approximation of the 3D CCST. The asymptotic theory is shown to converge rapidly and to be in excellent agreement with the exact solutions for FG macroplates reported in the literature when the value of the material length scale parameter is assigned to be zero.

Computability of sets in Euclidean space

Abstract We consider several concepts of computability (recursiveness) for sets in Euclidean space. A list of four ideal properties for such sets is proposed and it is shown in a very elementary way that no notion can satisfy all four desiderata. Most notions introduced here are essentially based on separability of ℝ n and this is natural when thinking about operations on an actual digital computer where, in fact, rational numbers are the basis of everything. We enumerate some properties of some naïve but practical notions of recursive sets and contrast these with others, including the widely used and accepted notion of computable set developed by Weihrauch, Brattka and others which is based on the “Polish school” notion of a computable real function. We also offer a conjecture about the Mandelbrot set.

Comparing machine learning techniques for predicting glassy dynamics.

In the quest to understand how structure and dynamics are connected in glasses, a number of machine learning based methods have been developed that predict dynamics in supercooled liquids. These methods include both increasingly complex machine learning techniques and increasingly sophisticated descriptors used to describe the environment around particles. In many cases, both the chosen machine learning technique and choice of structural descriptors are varied simultaneously, making it hard to quantitatively compare the performance of different machine learning approaches. Here, we use three different machine learning algorithms-linear regression, neural networks, and graph neural networks-to predict the dynamic propensity of a glassy binary hard-sphere mixture using as structural input a recursive set of order parameters recently introduced by Boattini et al. [Phys. Rev. Lett. 127, 088007 (2021)]. As we show, when these advanced descriptors are used, all three methods predict the dynamics with nearly equal accuracy. However, the linear regression is orders of magnitude faster to train, making it by far the method of choice.

Computability of finite quotients of finitely generated groups

Abstract We systematically study groups whose marked finite quotients form a recursive set. We give several definitions, and prove basic properties of this class of groups, and in particular emphasize the link between the growth of the depth function and solvability of the word problem. We give examples of infinitely presented groups whose finite quotients can be effectively enumerated. Finally, our main result is that a residually finite group can fail to be recursively presented and still have computable finite quotients, and that, on the other hand, it can have solvable word problem but not have computable finite quotients.

A Prevailing-Decree Verifier intended for Cryptographic Etiquettes

In recent years, a number of cryptographic etiquettes have been mechanically verified using a selection of inductive methods. These attestations typically want central a figure of recursive sets of messages, and need deep intuition into why the etiquette is correct. As a result, these proofs frequently require days to weeks of expert effort. We ensure advanced an involuntary verifier, which seems to overawe these glitches for many cryptographic protocols. The code of behavior text to concept a number of first-order invariant the proof commitments mitigating these invariants, along with any user-specified protocol properties are showed from the invariants with a tenacity theorem proved. The individual litheness in construction these invariants is to guesstimate, for each type of nonce and encryption engendered by the protocol, a formulary arresting conditions compulsory for that nonce encryption to be published.

Event-Triggered Forecasting-Aided State Estimation for Active Distribution System With Distributed Generations

In this study, the forecasting-aided state estimation (FASE) problem for the active distribution system (ADS) with distributed generations (DGs) is investigated, considering the constraint of data transmission. First of all, the system model of the ADS with DGs is established, which expands the scope of the ADS state estimation from the power network to the DGs. Moreover, in order to improve the efficiency of data transmission under the limited communication bandwidth, a component-based event-triggered mechanism is employed to schedule the data transmission from the measurement terminals to the estimator. It can efficiently reduce the amount of data transmission while guaranteeing the performance of system state estimation. Second, an event-triggered unscented Kalman filter (ET-UKF) algorithm is proposed to conduct the state estimation of the ADS with mixed measurements. To this end, the unscented transform (UT) technique is employed to approximate the probability distribution of the state variable after nonlinear transformation, which can reach more than second order, and then, an upper bound of the filtering error covariance is derived and subsequently minimized at each iteration. The gain of the desired filter is obtained recursively by following a certain set of recursions. Finally, the effectiveness of the proposed method is demonstrated by using the IEEE-34 distribution test system.

COMPOSITION SERIES OF THE SOLVABLE ABELIAN FACTOR GROUP SOURCE OF EQUATION OF ALL POLYNOMIAL EQUATIONS

This work penciled down the Composition Series of Factor Abelian Group over the source of all polynomial equations gleaned through the nth roots of unity regular gons on a unit circle, a circle of radius one and centered at zero. To get the composition series, the third isomorphism theorem has to be passed through. But, the third isomorphism theorem itself gleaned via the first which is a deduction of the naturally existing canonical map. The solution of the source atom of the equation of all equation of polynomials are solvable by the intertwine of the Euler’s Formula and the De Moivre’s Theorem which after the inter-math, they become within the domain of complex analysis. For the source root of the equations, there is a recursive set of homomorphisms and ontoness of the mappings geneting the sequential terms in the composition series.
 

The Equation of the Set of Natural Numbers Just to Sum

Systems of equations of the form X = Y + Z and X = C, in which the unknowns are sets of integers,”+” denotes pairwise sum of sets S + T = m + n m S, n T , and C is an ultimately periodic constant. When restricted to sets of natural numbers, such equations can be equally seen as language equations over a one-letter alphabet with concatenation and regular constants, and it is shown that such systems are computationally universal, in the sense that for every recursive set S N there exists a system with a unique solution containing T with S = n 16n + 13 T. For systems over sets of all integers, both positive and negative, there is a similar construction of a system with a unique solution S = {n|16n ∈ T} representing any hyper-arithmetical set S ⊆ N. Keywords: Language equations, Natural numbers, Equations of natural number.

Exact Hausdorff and packing measures for random self-similar code-trees with necks

Random code-trees with necks were introduced recently to generalise the notion of $V$-variable and random homogeneous sets. While it is known that the Hausdorff and packing dimensions coincide irrespective of overlaps, their exact Hausdorff and packing measure has so far been largely ignored. In this article we consider the general question of an appropriate gauge function for positive and finite Hausdorff and packing measure. We first survey the current state of knowledge and establish some bounds on these gauge functions. We then show that self-similar code-trees do not admit a gauge functions that simultaneously give positive and finite Hausdorff measure almost surely. This surprising result is in stark contrast to the random recursive model and sheds some light on the question of whether $V$-variable sets interpolate between random homogeneous and random recursive sets. We conclude by discussing implications of our results.

Bayes-Optimal Set-Valued Tracking of Single Point Targets

Conventional single-point-target tracking algorithms are recursive point estimators with point-measurement input data. In less well-known approaches, the tracking algorithm is a recursive set estimator with point-measurement or set-measurement input data. This paper provides a very general, systematic, and theoretically rigorous foundation for Bayes-optimal single-target trackers of the latter kind; as well as specific trackers arising from that foundation. This foundation is intuitive and conceptually simple and, in particular, requires no measure-theoretic complexities.

Set-theoretic reflection is equivalent to induction over well-founded classes

We show that induction over $\Delta(\mathbb R)$-definable well-founded classes is equivalent to the reflection principle which asserts that any true formula of first order set theory with real parameters holds in some transitive set. The equivalence is proved in primitive recursive set theory (which is weaker than Kripke-Platek set theory) extended by the axiom of dependent choice.

A nonlocal continuum mechanics-based asymptotic theory for the buckling analysis of SWCNTs embedded in an elastic medium subjected to combined hydrostatic pressure and axial compression

Based on nonlocal continuum mechanics, the authors develop an asymptotic theory by which to examine the buckling behavior of simply supported, single-walled carbon nanotubes (SWCNTs) embedded in an elastic medium under combined hydrostatic pressure and axial compression using the perturbation method. In the formulation, the Eringen nonlocal constitutive relations are used to account for the small length scale effect; the interactions between the SWCNT and its surrounding medium are modelled as a Winkler foundation model, and the half-thickness- to- the mid-surface radius ratio is selected as the perturbation parameter. After performing some mathematical processes, including non-dimensionalization, asymptotic expansion, and successive integration, among others, the authors separate the three-dimensional (3D) nonlocal elasticity equations into recursive sets of governing equations (GEs) based on the Donnell-type nonlocal classical shell theory (CST) for various order problems. The Donnell-type nonlocal CST is derived as a first-order approximation of the 3D nonlocal elasticity problem, where the GEs for higher-order problems retain the same differential operators as those of the Donnell-type nonlocal CST although with different nonhomogeneous terms. The current asymptotic solutions for the critical load parameters of the embedded SWCNT are obtained in order to assess the accuracy of various nonlocal CSTs and nonlocal beam theories available in the literature.

Recursive distributed filtering over sensor networks on Gilbert–Elliott channels: A dynamic event-triggered approach

In this paper, the recursive distributed filtering problem is investigated for a class of discrete nonlinear time-varying systems under a dynamic event-triggered mechanism. The system outputs are collected through a sensor network subject to a time-varying topology that is connected via Gilbert–Elliott channels and governed by a set of Markov chains. In order to save communication cost, a dynamic event-triggered strategy is utilized to decide when a particular sensor node should broadcast the corresponding measurement output to its neighbors. For the addressed time-varying systems, our aim is to design a distributed filter for each sensor node such that an upper bound on the filtering error variance is guaranteed and subsequently minimized at each iteration under the dynamic event-triggered transmission protocol. The parameters of the desired filter are obtained recursively by following a certain set of recursions. Finally, an illustrative example is employed to verify the effectiveness of the proposed dynamic event-triggered distributed filtering algorithm.

Principal subspaces of twisted modules for certain lattice vertex operator algebras

This is the third in a series of papers studying the vertex-algebraic structure of principal subspaces of twisted modules for lattice vertex operator algebras. We focus primarily on lattices [Formula: see text] whose Gram matrix contains only non-negative entries. We develop further ideas originally presented by Calinescu, Lepowsky, and Milas to find presentations (generators and relations) of the principal subspace of a certain natural twisted module for the vertex operator algebra [Formula: see text]. We then use these presentations to construct exact sequences involving this principal subspace, which give a set of recursions satisfied by the multigraded dimension of the principal subspace and allow us to find the multigraded dimension of the principal subspace.

An optimization model of retiree decisions under recursive utility with housing

We investigate both of analytical and numerical solutions of retirees’ spending and investment decisions. We use a dynamic and realistic recursive utility setting which includes the standard expected utility setting as a special case. We find that recursive utility is superior to expected utility in terms of predicting retirees’ consumption data. In addition to stock and bond investment decisions, we explicitly include housing decision. Our setting includes the setting without housing as a special case. We estimate retiree decisions numerically through simulations. We provide both of analytical and numerical comparative analysis which shows that some of the analytical dependencies are found to be weak numerically. For example, although marginal propensity to consume depends on the parameter of intertemporal substitution analytically, this dependence is found to be weak numerically. These differences show the importance of providing both of analytical and the numerical solutions. Our analytical solution could be useful for future studies to estimate some model parameters, to evaluate different elderly related policies, to quantify the welfare effects of different decisions and to analyze the parameter related issues such as the interchangeability of some parameters.