Bounded Functions Research Articles

Growth of analytic functions in an ultrametric open disk and branched values

Let $D$ be the open disk $|x|< R$ of a complete ultrametric algebraically closed field $\mathbb K$. We define the growth order $\rho(f)$, the growth type $\sigma(f)$ the cotype $\psi(f)$ and another expression $\theta(f)$ of an analytic function in $D$ and we show relations between the number of zeros of $f$ and the ``maximum modulus'' of $f$ involving $\rho,\sigma,\psi$. Then $\rho(f)$ lies in $[\theta(f)-1, \theta(f)]$. Moreover, if $0< \rho(f)< +\infty$ and $0< \psi(f)< +\infty$, then $\theta(f)=\rho(f)$ and $\sigma(f)=0$. Suppose $\mathbb K$ has characteristic zero and consider two unbounded analytic functions $f,g$ in $D$. The number of perfectly branched values admits overbounds linked to $\rho(f), \rho(g), \sigma(f), \sigma(g)$. In residue characteristic zero, then $\rho(f')=\rho(f),\sigma(f')=\sigma(f),$ $ \psi(f')=\psi(f)$, which can be applied to Levi-Civta fields.

Retraction Note: Ambient air quality prediction and unbounded variational continuous function based on big data

Retraction Note: Ambient air quality prediction and unbounded variational continuous function based on big data

Best One-Sided Multiplier Approximation of Unbounded Functions by Trigonometric Polynomials

The purpose of this paper is to find the best one –sided multiplier approximation of unbounded functions in 퐿푝,휓푛–space by trigonometric polynomials. Also, we will estimate the degree of the best one –sided multiplier approximation in term of averaged modulus.

On the limits of real-valued functions in sets involving ψ-density, and applications

New results on upper and lower limits of real-valued functions are proved by means of ψ-densities introduced by P. D. Barry in 1962. This leads to improvements of several existing results on the growth of non-decreasing and unbounded real-valued functions in sets of positive density. The ψ-densities allow us to introduce a new concept of a limit for real-valued functions, which is used to reveal further properties on the behavior of integrable functions at infinity.

PAC-Bayes Unleashed: Generalisation Bounds with Unbounded Losses.

We present new PAC-Bayesian generalisation bounds for learning problems with unbounded loss functions. This extends the relevance and applicability of the PAC-Bayes learning framework, where most of the existing literature focuses on supervised learning problems with a bounded loss function (typically assumed to take values in the interval [0;1]). In order to relax this classical assumption, we propose to allow the range of the loss to depend on each predictor. This relaxation is captured by our new notion of HYPothesis-dependent rangE (HYPE). Based on this, we derive a novel PAC-Bayesian generalisation bound for unbounded loss functions, and we instantiate it on a linear regression problem. To make our theory usable by the largest audience possible, we include discussions on actual computation, practicality and limitations of our assumptions.

Integral representation of unbounded variational functionals on Sobolev spaces

In this paper we establish an unbounded version of the integral representation theorem by Buttazzo and Dal Maso [see Buttazzo Dal Maso (Nonlinear Anal 9(6):515–532, 1985) and also Bouchitté et al. (Arch Ration Mech Anal 165(3), 187–242, 2002)]. More precisely, we prove an integral representation theorem (with a formula for the integrand) for functionals defined on $$W^{1,p}$$ with $$p>N$$ (N being the dimension) that do not satisfy a standard p-growth condition from above and can take infinite values. Applications to $$\Gamma $$ -convergence, relaxation and homogenization are also developed.

RETRACTED ARTICLE: Ambient air quality prediction and unbounded variational continuous function based on big data

RETRACTED ARTICLE: Ambient air quality prediction and unbounded variational continuous function based on big data

Relative (p, q)-φ order and relative (p, q)-φ type oriented some growth investigations of composite p-adic entire functions

Let is the -algebra of entire functions defined on a complete ultrametric algebraically closed field . For a p-adic entire function and for r > 0, sup{| f (x)|:|x|= r} is denoted by | f |(r), where |⋅|(r) is a multiplicative norm on . Taking φ(r): [0, +∞) → (0, +∞) as a non-decreasing unbounded function of r, in this paper we develop some results of composite p-adic entire functions in terms of their relative (p, q) – φ order and relative (p, q) - φ lower order along with relative (p, q) - φ type and relative (p, q) - φ weak type, where p, q are two positive integers.

Robust cost-sensitive kernel method with Blinex loss and its applications in credit risk evaluation

Credit risk evaluation is a crucial yet challenging problem in financial analysis. It can not only help institutions reduce risk and ensure profitability, but also improve consumers’ fair practices. The data-driven algorithms such as artificial intelligence techniques regard the evaluation as a classification problem and aim to classify transactions as default or non-default. Since non-default samples greatly outnumber default samples, it is a typical imbalanced learning problem and each class or each sample needs special treatment. Numerous data-level, algorithm-level and hybrid methods are presented, and cost-sensitive support vector machines (CSSVMs) are representative algorithm-level methods. Based on the minimization of symmetric and unbounded loss functions, CSSVMs impose higher penalties on the misclassification costs of minority instances using domain specific parameters. However, such loss functions as error measurement cannot have an obvious cost-sensitive generalization. In this paper, we propose a robust cost-sensitive kernel method with Blinex loss (CSKB), which can be applied in credit risk evaluation. By inheriting the elegant merits of Blinex loss function, i.e., asymmetry and boundedness, CSKB not only flexibly controls distinct costs for both classes, but also enjoys noise robustness. As a data-driven decision-making paradigm of credit risk evaluation, CSKB can achieve the “win-win” situation for both the financial institutions and consumers. We solve linear and nonlinear CSKB by Nesterov accelerated gradient algorithm and Pegasos algorithm respectively. Moreover, the generalization capability of CSKB is theoretically analyzed. Comprehensive experiments on synthetic, UCI and credit risk evaluation datasets demonstrate that CSKB compares more favorably than other benchmark methods in terms of various measures.

High energy bounds on wave operators

The wave operators W±(H1,H0) of two selfadjoint operators H0 and H1 are analyzed at asymptotic spectral values. Sufficient conditions for ∥(W±(H1,H0)−Pac1Pac0)f(H0)∥&lt;∞ are given, where Pacj projects onto the subspace of absolutely continuous spectrum of Hj and f is an unbounded function (f-boundedness), both in the case of trace-class perturbations and in terms of the high-energy behaviour of the boundary values of the resolvent of H0 (smooth method). Examples include f-boundedness for the perturbed polyharmonic operator and for Schr\"odinger operators with matrix-valued potentials. We discuss an application to the problem of quantum backflow.

THE TWO-SIDED ESTIMATES OF THE FREDHOLM RADIUS AND COMPACTNESS CONDITIONS FOR THE OPERATOR ASSOCIATED WITH A SECOND-ORDER DIFFERENTIAL EQUATION

In this paper we consider the properties of the resolvent of a linear operator corresponding to a degenerate singular second-order differential equation with variable coefficients, considered in the Lebesgue space. The singularity of the specified differential equation means that it is defined in a noncompact domain - on the whole set of real numbers, and its coefficients are unbounded functions. The conditions for the compactness of the resolvent were obtained, as well as a double-sided estimate of its fredgolm radius. The previously known compactness conditions of the resolvent were obtained under the assumption that the intermediate-term of the differential operator either is missing or, in the operator sense, is subordinate to the sum of the extreme terms. In the current paper these conditions are not met due to the rapid growth at infinity of the intermediate coefficient of the differential equation, and the minor coefficient can change sign. The property of compactness of the resolvent allows, in particular, to justify the process of finding an approximate solution of the associated equation. The Fredholm radius of a bounded operator characterizes its closeness to the Fredholm operator. The operator coefficients are assumed to be smooth functions, but we do not impose any constraints on their derivatives. The result on the invertibility of the operator and the estimation of its maximum regularity obtained by the authors earlier is essentially used in this paper.

Perturbative diagonalization for Maryland-type quasiperiodic operators with flat pieces

We consider quasiperiodic operators on Zd with unbounded monotone sampling functions (“Maryland-type”), which are not required to be strictly monotone and are allowed to have flat segments. Under several geometric conditions on the frequencies, lengths of the segments, and their positions, we show that these operators enjoy Anderson localization at large disorder.

Averaging Principles for Markovian Models of Plasticity

In this paper we consider a stochastic system with two connected nodes, whose unidirectional connection is variable and depends on point processes associated to each node. The input node is represented by an homogeneous Poisson process, whereas the output node jumps with an intensity that depends on the jumps of the input nodes and the connection intensity. We study a scaling regime when the rate of both point processes is large compared to the dynamics of the connection. In neuroscience, this system corresponds to a neural network composed by two neurons, connected by a single synapse. The strength of this synapse depends on the past activity of both neurons, the notion of synaptic plasticity refers to the associated mechanism. A general class of such stochastic models has been introduced in Robert and Vignoud (Stochastic models of synaptic plasticity in neural networks, 2020, arxiv: 2010.08195) to describe most of the models of long-term synaptic plasticity investigated in the literature. The scaling regime corresponds to a classical assumption in computational neuroscience that cellular processes evolve much more rapidly than the synaptic strength. The central result of the paper is an averaging principle for the time evolution of the connection intensity. Mathematically, the key variable is the point process, associated to the output node, whose intensity depends on the past activity of the system. The proof of the result involves a detailed analysis of several of its unbounded additive functionals in the slow-fast limit, and technical results on interacting shot-noise processes.

Analysis of VIX-linked fee incentives in variable annuities via continuous-time Markov chain approximation

We consider the pricing of variable annuities (VAs) with general fee structures under a class of stochastic volatility models which includes the Heston, Hull-White, Scott, α-Hypergeometric, 3/2, and 4/2 models. In particular, we analyze the impact of different VIX-linked fee structures on the optimal surrender strategy of a VA contract with guaranteed minimum maturity benefit (GMMB). Under the assumption that the VA contract can be surrendered before maturity, the pricing of a VA contract corresponds to an optimal stopping problem with an unbounded, time-dependent, and discontinuous payoff function. We develop efficient algorithms for the pricing of VA contracts using a two-layer continuous-time Markov chain approximation for the fund value process. When the contract is kept until maturity and under a general fee structure, we show that the value of the contract can be approximated by a closed-form matrix expression. We also provide a quick and simple way to determine the value of early surrenders via a recursive algorithm and give an easy procedure to approximate the optimal surrender surface. We show numerically that the optimal surrender strategy is more robust to changes in the volatility of the account value when the fee is linked to the VIX index.

Minimally deformed wormholes

This work is devoted to the study of wormhole solutions in the framework of gravitational decoupling by means of the minimal geometric deformation scheme. As an example, to analyze how this methodology works in this scenario, we have minimally deformed the well-known Morris–Thorne model. The decoupler function f(r) and the theta -sector are determined considering the following approaches: (i) the most general linear equation of state relating the theta _{mu nu } components is imposed and (ii) the generalized pseudo-isothermal dark matter density profile is mimicked by the temporal component of the theta -sector. It is found that the first approach leads to a non-asymptotically flat space-time with an unbounded mass function. To address this issue we have matched both the wormhole and the Schwarzschild vacuum solutions, via a thin-shell at the junction surface. Using the second approach, it can be seen that, on one hand, the solution for gamma =1 does not give place to a bounded mass and it presents a topological defect at large distances; on the other hand, the wormhole manifold is asymptotically flat in the gamma =2 case. In order to satisfy the flare-out condition, we have found restrictions on the value of the alpha parameter, which is related with the amount of exotic matter distribution. Finally, the averaged weak energy condition has been analyzed by using the volume integral quantifier.

Multiplier Approximation of Functions by q-Bernstein-Kantorovich Operator

The purpose of this paper is to find best multiplier approximation of unbounded functions in L p,ϕm –space by q-Bernstein-Kantorovich Aperator, in terms of the modulus of smoothness of one order using Korevkin Theorem.

The Direct and Converse Theorems for Best Approximation of Algebraic Polynomial in Lp, α (X)

The direct and inverse algebraic polynomials approximation theorems in weighted spaces of unbounded functions are proved by using modulus of smoothness. Also, we obtain sharp Jackson ( direct ) inequality of algebraic approximation of unbounded functions in terms modulus of smoothness. In addition, constructive characterization of modulus of smoothness are considered.

Supercaloric functions for the parabolic p-Laplace equation in the fast diffusion case

We study a generalized class of supersolutions, so-called p-supercaloric functions, to the parabolic p-Laplace equation. This class of functions is defined as lower semicontinuous functions that are finite in a dense set and satisfy the parabolic comparison principle. Their properties are relatively well understood for pge 2, but little is known in the fast diffusion case 1<p<2. Every bounded p-supercaloric function belongs to the natural Sobolev space and is a weak supersolution to the parabolic p-Laplace equation for the entire range 1<p<infty . Our main result shows that unbounded p-supercaloric functions are divided into two mutually exclusive classes with sharp local integrability estimates for the function and its weak gradient in the supercritical case frac{2n}{n+1}<p<2. The Barenblatt solution and the infinite point source solution show that both alternatives occur. Barenblatt solutions do not exist in the subcritical case 1<ple frac{2n}{n+1} and the theory is not yet well understood.

One-sided extendability of bounded functions

One-sided extendability of bounded functions

Multivariate multifractal formalism for simultaneous pointwise regularities

Recently, a multivariate multifractal analysis for pointwise regularities based on hierarchical multiresolution quantities was developed. General bounds between the Hausdorff dimension of the intersection of single fractal sets and that of the original sets were derived. Equalities were checked for some synthetic signals that include multiplicative cascades. In this paper, we focus on the setting supplied by simultaneous pointwise regularities. The regularity for was first introduced in order to better study elliptic partial differential equations where the natural function space setting is or a Sobolev space which includes unbounded functions. We will prove that both corresponding multivariate multifractal formalism and equalities above hold Baire generically in a given product of Besov spaces , such that . We therefore extend previous results where only cases ( and for all i) and (, and for all i) for simultaneous pointwise Hölder regularities have been proved.