Continuous Functions Research Articles

Group invariant variational principles

In this paper we introduce a group invariant version of the well-known Ekeland variational principle. To achieve this, we define the concept of convexity with respect to a group and establish a version of the theorem within this framework. Additionally, we present several consequences of the group invariant Ekeland variational principle, including Palais-Smale minimizing sequences, the Brønsted-Rockafellar theorem, and a characterization of the linear and continuous group invariant functionals space. Moreover, we provide an alternative proof of the Bishop-Phelps theorem and proofs for the group-invariant Hahn-Banach separating theorems. Finally, we discuss some implications and applications of these results.

Hybrid impulses for almost sure quasi-synchronization of stochastic complex networks: an indefinite Lyapunov function method

. This article studies quasi-synchronization of stochastic complex networks under hybrid impulses. With parameter mismatches, different from the previous quasi-synchronization results in the sense of mean square on stochastic complex networks, almost sure quasi-synchronization is investigated, in which noises have a positive effect on the synchronization. By utilizing the Lyapunov method, stochastic analysis theory, average impulsive interval, and average impulsive gain, we formulate the almost sure quasi-synchronization criteria when the average impulsive interval z 0 satisfies z 0 < + ∞ and z 0 = + ∞ , respectively, where the synchronizing impulses and the desynchronizing impulses are simultaneously considered. It is worth noticing that in our criteria of quasi-synchronization, the mutual restraints among the average impulsive interval, average impulsive gain, and noise intensity are given. Moreover, for the criteria in this article, the piecewise continuous scalar functions cannot only replace the constant coefficient of the upper bound estimation for the diffusion operator of a Lyapunov function in available literature but also even be unbounded, which has wider applications than some existing works. Our corollaries regarding complete synchronization and analyses of the synchronization for stochastic complex networks with aperiodic intermittent noises provide sufficient conditions in closed form. Finally, the theoretical results are applied to a coupled chaotic Lorentz system, and several numerical examples are presented to demonstrate the advantages of the theoretical results.

Hierarchical Piezoelectric Composites for Noninvasive Continuous Cardiovascular Monitoring.

Continuous monitoring of blood pressure (BP) and multiparametric analysis of cardiac functions are crucial for the early diagnosis and therapy of cardiovascular diseases. However, existing monitoring approaches often suffer from bulky and intrusive apparatus, cumbersome testing procedures, and challenging data processing, hampering their applications in continuous monitoring. Here, a heterogeneously hierarchical piezoelectric composite is introduced for wearable continuous BP and cardiac function monitoring, overcoming the rigidity of ceramic and the insensitivity of polymer. By optimizing the hierarchical structure and components of the composite, the developed piezoelectric sensor delivers impressive performances, ensuring continuous and accurate monitoring of BP at Grade A level. Furthermore, the hemodynamic parameters are extracted from the detected signals, such as local pulse wave velocity, cardiac output, and stroke volume, all of which are in alignment with clinical results. Finally, the all-day tracking of cardiac function parameters validates the reliability and stability of the developed sensor, highlighting its potential for personalized healthcare systems, particularly in early diagnosis and timely intervention of cardiovascular disease.

An optimization method for ITER radial X-ray camera line-of-sight design basing on Bayesian uncertainty analysis

Abstract This paper presents a novel uncertainty optimization algorithm for the design of line-of-sight systems used in tomographic inversion. By extending Gaussian process tomography from discrete pixel space to continuous function space through Bayesian inference, we introduce an uncertainty function and analyze its typical distributions. We develop an algorithm to minimize the uncertainty, which is then applied to optimize the line-of-sight configuration of the internal camera in the ITER project. Uncertainty analysis and phantom testing are conducted to validate the effectiveness of the proposed algorithm. The results demonstrate improved accuracy and stability in tomographic reconstructions. This study contributes to the advancement of line-of-sight design for tomographic inversion, offering a practical solution for optimizing diagnostic systems in complex experimental settings.

Representations of groups on Banach spaces

We establish a general framework for representability of a metric group on a (well-behaved) class of Banach spaces. More precisely, let G \mathcal {G} be a topological group, and A \mathcal {A} a unital symmetric C ∗ C^* -subalgebra of U C ( G ) \mathrm {UC}(\mathcal {G}) , the algebra of bounded uniformly continuous functions on G \mathcal {G} . Generalizing the notion of a stable metric, we study A \mathcal {A} -metrics δ \delta , i.e., the function δ ( e , ⋅ ) \delta (e, \cdot ) belongs to A \mathcal {A} ; the case A = W A P ( G ) \mathcal {A}=W\hskip -0.7mm A\hskip -0.2mm P(\mathcal {G}) , the algebra of weakly almost periodic functions on G \mathcal {G} , recovers stability. If the topology of G G is induced by a left invariant metric d d , we prove that A \mathcal {A} determines the topology of G \mathcal {G} if and only if d d is uniformly equivalent to a left invariant A \mathcal {A} -metric. As an application, we show that the additive group of C [ 0 , 1 ] C[0,1] is not reflexively representable; this is a new proof of Megrelishvili [Topological transformation groups: selected topics, Elsevier, 2007, Question 6.7] (the problem was already solved by Ferri and Galindo [Studia Math. 193 (2009), pp. 99–108] with different methods and later the results were generalized by Yaacov, Berenstein, and Ferri [Math. Z. 267 (2011), pp.129–138]). Let now G \mathcal {G} be a metric group, and assume A ⊆ L U C ( G ) \mathcal {A}\subseteq \mathrm {LUC}(\mathcal {G}) , the algebra of bounded left uniformly continuous functions on G \mathcal {G} , is a unital C ∗ C^* -algebra which is the uniform closure of coefficients of representations of G \mathcal {G} on members of F \mathscr {F} , where F \mathscr {F} is a class of Banach spaces closed under ℓ 2 \ell _2 -direct sums. We prove that A \mathcal {A} determines the topology of G \mathcal {G} if and only if G \mathcal {G} embeds into the isometry group of a member of F \mathscr {F} , equipped with the weak operator topology. As applications, we obtain characterizations of unitary and reflexive representability.

Digital noise spectroscopy with a quantum sensor

We introduce and experimentally demonstrate a quantum sensing protocol to sample and reconstruct the autocorrelation of a noise process using a single-qubit sensor under digital control modulation. This Walsh noise spectroscopy method exploits simple sequences of spin-flip pulses to generate a complete basis of digital filters that directly sample the power spectrum of the target noise in the sequency domain, from which the autocorrelation function in the time domain, as well as the power spectrum in the frequency domain, can be reconstructed using linear transformations. Our method, which can also be seen as an implementation of frame-based noise spectroscopy, solves the fundamental difficulty in sampling continuous functions with digital filters by introducing a transformation that relates the arithmetic and logical time domains. In comparison to standard, frequency-based dynamical-decoupling noise spectroscopy protocols, the accuracy of our method is only limited by sampling and discretization in the time domain and can be easily improved, even under limited evolution time due to decoherence and hardware limitations. Finally, we experimentally reconstruct the autocorrelation function of the effective magnetic field produced by the nuclear-spin bath on the electronic spin of a single nitrogen-vacancy center in diamond, discuss practical limitations of the method, and avenues to further improve the reconstruction accuracy.

SOME RESULTS ON BOX DIMENSION ESTIMATION OF FRACTAL CONTINUOUS FUNCTIONS

In this paper, we explore upper box dimension of continuous functions on [Formula: see text] and their Riemann–Liouville fractional integral. Firstly, by comparing function limits, we prove that the upper box dimension of the Riemann–Liouville fractional order integral image of a continuous function will not exceed [Formula: see text], the result similar to [Y. S. Liang and W. Y. Su, Fractal dimensions of fractional integral of continuous functions, Acta Math. Appl. Sin. E 32 (2016) 1494–1508]. Secondly, we prove that upper box dimension of multiple algebraic sums of continuous functions does not exceed the largest box dimension among them, backing up our conclusion with an appropriate example. Finally, we draw the same conclusions for the product of multiple continuous functions.

FRACTAL DIMENSIONS FOR THE MIXED (κ,s)-RIEMANN–LIOUVILLE FRACTIONAL INTEGRAL OF BIVARIATE FUNCTIONS

The research object of this paper is the mixed [Formula: see text]-Riemann–Liouville fractional integral of bivariate functions on rectangular regions, which is a natural generalization of the fractional integral of univariate functions. This paper first indicates that the mixed integral still maintains the validity of the classical properties, such as boundedness, continuity and bounded variation. Furthermore, we investigate fractal dimensions of bivariate functions under the mixed integral, including the Hausdorff dimension and the Box dimension. The main results indicate that fractal dimensions of the graph of the mixed [Formula: see text]-Riemann–Liouville integral of continuous functions with bounded variation are still two. The Box dimension of the mixed integral of two-dimensional continuous functions has also been calculated. Besides, we prove that the upper bound of the Box dimension of bivariate continuous functions under [Formula: see text] order of the mixed integral is [Formula: see text] where [Formula: see text].

The Impact of the Limit q-Durrmeyer Operator on Continuous Functions

AbstractThe limit q-Durrmeyer operator, $$D_{\infty ,q}$$ D ∞ , q , was introduced and its approximation properties were investigated by Gupta (Appl. Math. Comput. 197(1):172–178, 2008) during a study of q-analogues for the Bernstein–Durrmeyer operator. In the present work, this operator is investigated from a different perspective. More precisely, the growth estimates are derived for the entire functions comprising the range of $$D_{\infty ,q}$$ D ∞ , q . The interrelation between the analytic properties of a function f and the rate of growth for $$D_{\infty ,q}f$$ D ∞ , q f are established, and the sharpness of the obtained results are demonstrated.

Exploring Iterated Implicit Function Systems: Existence and Properties of Attractors

This paper investigates a type of iterated implicit function systems composed of equations [Formula: see text], where [Formula: see text] is a continuous function, and [Formula: see text] is a constant. The existence of attractors of iterated implicit function systems is proved based on different equation conditions, including the equation [Formula: see text] containing the implicit function or being [Formula: see text]-contractive about [Formula: see text]. Meanwhile, we give definitions of implicit convergence of functions and monotone sequence of iterated implicit function systems. Finally, some properties of attractors of iterated implicit function systems are elucidated.

Dynamic event-triggered control of large-scale nonlinear systems with sensor uncertainty

In this article, the dynamic event-triggered control problem is investigated for a class of large-scale nonlinear systems subject to sensor uncertainty. The unknown nonlinearities involved in each subsystem are assumed to be bounded by time-varying continuous functions multiplied by unknown constants and unmeasured states. To estimate the unmeasured states of each subsystem, a time-varying high-gain observer is constructed. Then, a dynamic event-triggered mechanism is proposed by introducing an internal dynamic variable in triggering function. Combined with the dual-domination approach, a dynamic event-triggered output feedback controller is developed for each subsystem. Subsequently, it is proved the convergence of all states is ensured based on the Lyapunov theory and the Zeno behavior can be avoided. Eventually, an example is presented to demonstrate the effectiveness of the proposed dynamic event-triggered control scheme.

High frame rate acoustic navigation of AUV based on range ambiguity suppression

Abstract Underwater acoustic navigation is extensively utilized across many industries due to its exceptional dependability and continuous real-time functionality. Due to the swift advancement of technology in recent years, the velocity of autonomous underwater vehicles (AUVs) has progressively increased. A high frame rate is required for the underwater acoustic navigation system in order to precisely obtain the trajectory of the AUV. Nevertheless, the discrepancy lies in the fact that the high frame rate and the unambiguous measurement could be seen as contradictory. Higher frame rates greatly raise the probability of range ambiguity, leading to various potential pseudo solutions that exacerbate the difficulty of accurate positioning and noticeably impact navigation performance. To address the aforementioned issue, a high frame rate acoustic navigation technique that relies on the suppression of range ambiguity has been studied in this paper. Initially, the relationship between measurements were investigated, specifically the time delay and Doppler shift, and the confusing periodic leap. Subsequently, the coupling model was employed to develop a high-precision acoustic navigation model, taking into account the statistical characteristics of measurements. We also examined a high-precision solving approach. Ultimately, the proposed method was validated through both simulation analysis and field testing to confirm its practicality and efficiency. The results demonstrate that it can attain a high level of accuracy in resolving the ambiguous problem and achieve high-precise acoustic navigation.

New Auxiliary Principle Technique for General Harmonic Directional Variational Inequalities

This paper explores the utilisation of harmonic variational inequalities to establish the minimum value among two locally Lipschitz continuous harmonic convex functions. This investigation introduces novel classes of harmonic directed variational inequalities, particularly focusing on scenarios like harmonic complementarity and related optimization challenges. The study proposes and analyses various inertial iterative strategies for addressing harmonic directed variational inequalities through the auxiliary principle technique. It examines convergence criteria under specific weak conditions, emphasising the simplicity of the approach compared to other methodologies. The findings presented herein have broad applicability in the context of harmonic variational inequalities and optimization problems, though they are limited to theoretical exploration. Further research is required to implement these strategies numerically.

Sliding mode controller with neural network compensation for tank

The control effectiveness of the all-electric tank stabilizers directly affects the firing accuracy of the tank gun. In order to improve the firing accuracy of the moving tank, this paper proposes a sliding mode control (SMC) strategy using neural network feed-forward compensation for the tank bidirectional stabilizers. SMC technology can quickly and effectively deal with uncertainties, unmodeled terms, and external disturbances in complex tank systems. Neural networks possess the advantage of approximating arbitrary continuous functions in finite time, realizing the estimation of SMC control errors with feed-forward compensation. The Lyapunov stability analysis proves that the designed controller can achieve asymptotic stabilization in finite time. Finally, both co-simulation and physical experiments are conducted. The results show that the proposed control strategy possesses better tracking speed and tracking accuracy compared to the conventional controller and exhibits strong robustness.

An Explicit Form of Ramp Function

In this paper, an analytical exact form of the ramp function is presented. This seminal function constitutes a fundamental concept of the digital signal processing theory and is also involved in many other areas of applied sciences and engineering. In particular, the ramp function is performed in a simple manner as the pointwise limit of a sequence of real and continuous functions with pointwise convergence. This limit is zero for strictly negative values of the real variable x, whereas it coincides with the independent variable x for strictly positive values of the variable x. Here, one may elucidate beforehand that the pointwise limit of a sequence of continuous functions can constitute a discontinuous function, on the condition that the convergence is not uniform. The novelty of this work, when compared to other research studies concerning analytical expressions of the ramp function, is that the proposed formula is not exhibited in terms of miscellaneous special functions, e.g., gamma function, biexponential function, or any other special functions, such as error function, hyperbolic function, orthogonal polynomials, etc. Hence, this formula may be much more practical, flexible, and useful in the computational procedures, which are inserted into digital signal processing techniques and other engineering practices.

Age-varying associations between attempts to change weight and suicidal ideation among adolescents in the United States

PurposeThe goal of this study was to inform theory and suicide prevention efforts by examining how the link between attempts to change weight and suicidal ideation varies across adolescence, when it is strongest, and whether there are sex differences. MethodsData were from 13,518 youth ages 14–18 years who participated in the 2019 Youth Risk Behavior Survey, a nationally representative study of U.S. high school students. Time-varying effect modeling was used to estimate associations between attempts to change weight and suicidal ideation as continuous functions of age and to test sex differences in these links. ResultsThe link between attempts to lose weight and suicidal ideation was positive and significant between ages 14.0–17.3 years, with the strongest association at age 15.0 (OR = 2.07). There were no sex differences in this age-varying association. The association between attempts to gain weight and suicidal ideation was positive and age-varying for girls, with the strongest association at age 17.2 years (OR = 2.79). This association was not significant for boys at any age. ConclusionsGirls and boys who are trying to lose weight and girls who are trying to gain weight may be at elevated risk for suicidal ideation between ages 14 and 18 years. Findings illuminate patterns of association during adolescence and have implications for suicide prevention.

Small dataset augmentation with radial basis function approximation for causal discovery using constraint‐based method

AbstractCausal analysis involves analysis and discovery. We consider causal discovery, which implies learning and discovering causal structures from available data, owing to the significance of interpreting causal relationships in various fields. Research on causal discovery has been primarily focused on constraint‐ and score‐based interpretable methods rather than on methods based on complex deep learning models. However, identifying causal relationships in real‐world datasets remains challenging. Numerous studies have been conducted using small datasets with established ground truths. Moreover, constraint‐based methods are based on conditional independence tests. However, such tests have a lower statistical power when applied to small datasets. To solve the small sample size problem, we propose a model that generates a continuous function from available samples using radial basis function approximation. We address the problem by extracting data from the generated continuous function and evaluate the proposed method on both real and synthetic datasets generated by structural equation modeling. The proposed method outperforms constraint‐based methods using only small datasets.

Homomorphisms of the lattice of slowly oscillating functions on the half-line

We study the space H(SO) of all homomorphisms of the vector lattice of all slowly oscillating functions on the half-line ℍ = [ 0 , ∞ ) . In contrast to the case of homomorphisms of uniformly continuous functions, it is shown that a homomorphism in H(SO) which maps the unit to zero must be zero-homomorphism. Consequently, we show that the space H(SO) without zero-homomorphism is homeomorphic to ℍ x (0, ∞). By describing a neighborhood base of zero-homomorphism, we show that H(SO) is homeomorphic to the space ℍ x (0, ∞) with one point added.

General fractal dimensions of graphs of products and sums of continuous functions and their decompositions

This study takes a broad approach to the fractal geometry problem and proposes an intrinsic definition of the general box dimensions and the general Hausdorff and packing dimensions by taking into account sums of the type ∑ih−1(sg(ri)) for some prescribed functions h, g and for all positive real s. Our primary aim is to conduct a more comprehensive exploration of the fractal dimensions exhibited by graphs resulting from the products and sums of two continuous functions. This article is entirely devoted to the investigation of this specific problem domain. In the course of our research, we furnish explicit formulas for computing both the general upper box dimension and the general lower box dimension associated with the graphs formed by the products and sums of two continuous functions. Additionally, we shed light on the decomposition of continuous functions into the sum of two continuous functions, employing the framework of the general lower and upper box dimensions. Furthermore, we establish an upper bound for the general upper box dimension applicable to each element within a finite ring of polynomials derived from continuous functions defined over the real number field R. Our inquiry also extends to the examination of general Hausdorff and packing dimensions for the graphs generated through both the summation and multiplication of two continuous functions.

RESEARCH ON FRACTAL DIMENSIONS AND THE HÖLDER CONTINUITY OF FRACTAL FUNCTIONS UNDER OPERATIONS

Based on the previous studies, we make further research on how fractal dimensions of graphs of fractal continuous functions under operations change and obtain a series of new results in this paper. Initially, it has been proven that a positive continuous function under unary operations of any nonzero real power and the logarithm taking any positive real number that is not equal to one as the base number can keep the fractal dimension invariable. Then, a general method to calculate the Box dimension of two continuous functions under binary operations has been proposed. Using this method, the lower and upper Box dimensions of the product and the quotient of continuous functions without zero points have been investigated. On this basis, these conclusions will be generalized to the ring of rational functions. Furthermore, we discuss the Hölder continuity of continuous functions under operations and then prove that a Lipschitz function can be absorbed by any other continuous functions under certain binary operations in the sense of fractal dimensions. Some elementary results for vector-valued continuous functions have also been given.