Integral Theorem Research Articles

Time-varying IBLFs-based adaptive control of uncertain nonlinear systems with full state constraints

This paper presents an adaptive control design for nonlinear systems with time-varying full state constraints. It is the first time to introduce the novel time-varying Integral Barrier Lyapunov functions (TVIBLFs) into the adaptive control design, which not only overcomes the limitation of conservatism existing in the traditional BLFs, but also guarantees that the full state time-varying constraint bounds are not violated. The TVIBLFs are combined with the backstepping design procedure to construct the controllers and adaptation laws, and the integral mean value theorem is used to differentiate TVIBLFs. It can be proven that all the states are forced in the time-varying regions and the stability of the closed-loop system is achieved. The effectiveness of the proposed adaptive control strategy can be illustrated through a simulation example.

Daily leaf area index from photosynthetically active radiation for long term records of canopy structure and leaf phenology

Leaf area index (LAI) is a critical biophysical indicator that describes foliage abundance in ecosystems. An accurate and continuous estimation of LAI is therefore desirable to quantify ecosystem status and function (e.g. carbon and water exchange between the land surface and the atmosphere). However, deriving accurate LAI measurements at regular temporal intervals remains challenging, requiring either destructive sampling or manual collection of canopy gap fraction measurements at discrete time intervals. In this study, we present four methods to obtain continuous LAI data, simply derived from above and below canopy measurements of photosynthetically active radiation (PAR) at the Borden Forest Research Station from 1999 to 2018. We compared LAI derived using the four PAR-based methods to independent measurements of LAI from optical methods and the MODIS satellite LAI product. LAI derived from all four PAR-based methods captured the seasonal changes in observed and remotely sensed LAI and showed a close linear correspondence with one another (R2 of 0.55 to 0.76 compared to MODIS LAI, and R2 of 0.78 to 0.84 compared to LAI-2000 measurements). A PAR-based method using Miller's Integral theorem showed the strongest linear relationship with LAI-2000 measurements (R2=0.84, p<0.001, SE=0.40). In many years MODIS LAI indicated an earlier start of season and earlier end of season than the daily PAR-based LAI datasets showing systematic biases in the MODIS assessment of growing season.The four PAR-based LAI methods outlined in this study provide an LAI dataset of unprecedented temporal resolution. These methods will allow precise determination of phenological events, improve leaf to canopy scaling in process-based models, and provide valuable insight into dynamic vegetation responses to global climate change.

On the application of the beam model for linear dynamic analysis of pile and suction caisson foundations for offshore wind turbines

Piles and suction caissons are the most common foundation solutions for fixed Offshore Wind Turbines at intermediate water depths. They are generally used as a single element, presenting large diameters and short aspect ratios. These specific dimensions drastically differ from the ones of classical applications (offshore platforms, bridges, tall buildings etc.). Thus, in this paper the validity of their modelling as beam elements for the particular problem of OWT is revised. The results of a soil-beam model, based on the integral Reciprocity Theorem in Elastodynamics and specific Green’s functions for the layered half-space for the soil behaviour coupled with Timoshenko’s beam Finite Elements, are benchmarked against the ones of a soil-shell model, based on Boundary Elements for the soil coupled with shell Finite Elements. The comparative study is conducted in terms of foundation characterization variables (impedance functions and kinematic interaction factors). Their influence on the OWT seismic response is also studied through a substructuring procedure. From the results, some expressions for determining the applicability range of the beam simplification are proposed as functions of the relative foundation-soil stiffness ratio. It is observed that this applicability range goes beyond that the one commonly considered. .

Option pricing formulas based on uncertain fractional differential equation

Uncertain fractional differential equations have been playing an important role in modelling complex dynamic systems. Early researchers have presented the extreme value theorems and time integral theorem on uncertain fractional differential equation. As applications of these theorems, this paper investigates the pricing problems of American option and Asian option under uncertain financial markets based on uncertain fractional differential equations. Then the analytical solutions and numerical solutions of these option prices are derived, respectively. Finally, some numerical experiments are performed to verify the effectiveness of our results.

Integral Comparison Criteria for Half-Linear Differential Equations Seen as a Perturbation

In this paper, we present further developed results on Hille–Wintner-type integral comparison theorems for second-order half-linear differential equations. Compared equations are seen as perturbations of a given non-oscillatory equation, which allows studying the equations on the borderline of oscillation and non-oscillation. We bring a new comparison theorem and apply it to the so-called generalized Riemann–Weber equation (also referred to as a Euler-type equation).

APPLICATION OF OPTIMIZED FRACTIONAL GREY MODEL-BASED VARIABLE BACKGROUND VALUE TO PREDICT ELECTRICITY CONSUMPTION

In recent years, the grey-based models with fractional accumulation have received extensive attention by scholars and have been widely used in various fields. However, the existing rough construction of the background value in the fractional grey model impairs its predictive performance to some extent. To address this problem, this paper reconstructs a dynamic background value for the fractional grey model by the composite integral median theorem, as a result, a novel fractional grey model-based variable background value (denoted as OFAGM(1,1) for short) is proposed. In particular, the Particle Swarm Optimization algorithm (PSO) is then employed to determine the optimum for the fractional order and the background value coefficients. Based on electricity consumption of two regions of China (i.e. Beijing, Inner Mongolia), the superiority of the proposed model has been verified in comparison with other benchmark models, the electricity consumption of Beijing is predicted to reach [Formula: see text][Formula: see text]Kwh in 2022 and that of Inner Mongolia will reach [Formula: see text][Formula: see text]Kwh in 2023. This study also takes the relative growth rate and the doubling time into account for these two regions that facilities comparison of the regions’ performance.

A note on simultaneous approximation on Vitushkin sets

Given a planar Jordan domain $G$ with rectifiable boundary, it is well known that smooth functions on the closure of $G$ do not always admit smooth extensions to $\mathbb C$. Further conditions on the boundary are necessary to guarantee such extensions. On the other hand, Weierstrass’ approximation theorem yields polynomials converging uniformly to $f \in C(\overline{G}, \mathbb C)$. In this note we show that for Vitushkin sets $K$ with $K = \overline{K^\circ}$ it is always possible to uniformly approximate on $K$ the smooth function $f \in C^1(K, \mathbb C)$ by smooth functions $f_n$ in $\mathbb C$ so that also $\overline{\partial}f_n$ converges uniformly to $\overline{\partial}f$ on $K$. As a byproduct we deduce from its ‘‘smooth in a neighborhood version’’ the general Gauss integral theorem for functions whose partial derivatives in $G$ merely admit continuous extensions to its boundary.

On a class of analytic functions related to Robertson\u2019s formula and subordination

In this paper, we define and study a class of analytic functions in the unit disc by modification of the well-known Robertson’s analytic formula for starlike functions with respect to a boundary point combined with subordination. An integral representation and growth theorem are proved. Early coefficients and the Fekete–Szegö functional are also estimated.

On Hausdorff integrations of Lie algebroids

We present Hausdorff versions for Lie Integration Theorems 1 and 2 and apply them to study Hausdorff symplectic groupoids arising from Poisson manifolds. To prepare for these results we include a discussion on Lie equivalences and propose an algebraic approach to holonomy. We also include subsidiary results, such as a generalization of the integration of subalgebroids to the non-wide case, and explore in detail the case of foliation groupoids.

An accurate and novel numerical simulation with convergence analysis for nonlinear partial differential equations of Burgers–Fisher type arising in applied sciences

Abstract In this study, the second-order nonlinear partial differential equations of Burgers–Fisher type are considered under a unique formulation by introducing a novel highly accurate numerical method based on the Nörlund rational polynomial and matrix-collocation computational system. The method aims to obtain a sustainable approach since it contains the rational structure of the Nörlund polynomial. A unique computer program module, which involves very few routines, is constructed to discuss the precision and efficiency of the method and these few steps are described via an algorithm. A residual function is employed in both the error and convergence analyses with mean value theorem for double integrals. The considered equations in the numerical tests stand for model phenomena arising widely in applied sciences. Graphical and numerical comparisons provide a clear observation about the consistency of the method. All results prove that the method is highly accurate, eligible, and provides the ultimate operation for aforementioned problems.

Explicit fundamental solution for the operator $$L+\alpha |T|$$ on the Gelfand pair $$(\mathbb {H}_{n},U(n))$$.

By means of the spherical functions associated to the Gelfand pair \((\mathbb {H}_{n},U(n))\) we define the operator \(L+\alpha |T|\), where L denotes the Heisenberg sublaplacian and T denotes the central element of the Heisenberg Lie algebra, we establish a notion of fundamental solution and explicitly compute in terms of the Gauss hypergeometric function. For \(\alpha <n\) we use the Integral Representation Theorem to obtain a more detailed expression. Finally, we remark that when \(\alpha =0\) we recover the fundamental solution for the Heisenberg sublaplacian given by Folland.

Research on Transmission Capacity Characteristics of the Cluster Flight Spacecraft Network

In order to analyze the transmission capacity performance of the cluster flight spacecraft network, there are two different types of outage performance theory which are derived in this paper. First of all, by applying the mean value theorem of integrals, the expression of the outage probability of decode-and-forward relaying is derived. Subsequently, according to the Macdonald random variable form, the expression of the outage probability of amplify-and-forward is derived. By simulating the transmission capacity of decode-and-forward, the transmission capacity characteristics of a single hop and dual hops are analyzed. The simulation results showed that transmission capacity performance changes with the change of the time slot in the orbital hyperperiod, and the transmission capacity of a dual-hop relay has better performance than a single-hop transmission in the cluster flight spacecraft network.

On the stratification and induced flow in an emptying–filling box driven by a plane vertically distributed source of buoyancy

A theoretical model is presented for the steady multi-layered flow induced by a plane vertically distributed buoyancy source producing a turbulent wall plume in a ventilated box. While aspects of the stratification and rate of fluid exchange between box and exterior have been studied previously, the streamline pattern and velocity field have not been considered until now, despite having potentially important practical implications for achieving comfort in naturally ventilated buildings and for the indoor spread of airborne contagions. The boundary condition at the wall for each layer is established by deducing the turbulent entrainment rate. Using conformal mapping techniques and Poisson's integral theorem, closed-form solutions for the streamfunction of the induced flow in each layer are established. While the flow near the ceiling was overlooked in the classic model for the multi-layered stratification, after considering the possible flow scenarios, the stratification is re-evaluated herein by incorporating an entraining ceiling current. With a markedly thinner top layer, the refined stratification matches well with the available experimental observations, the restrictions we place on the applicability of the model overcoming the previous over-prediction in the number of interfaces. The magnitude of the dimensionless flow velocity, independent of the wall buoyancy flux and physical scale of the box, decreases significantly with the number of layers. Three types of layer, each with a distinct induced flow pattern, are distinguished and their implications for room airflow considered. Notably, the flow in the base layer represents a continual and smooth flushing of air between the inlet opening and the wall plume, whereas an intermediate layer is almost entirely comprised of near-stagnant air.

Propagation of surface waves past asymmetric elastic plates

Water wave scattering by two asymmetric thin elastic plates with arbitrary inclinations is investigated using integral equations. The plates are submerged in finite depth water. The assumption of Euler–Bernoulli beam model for the plates, the use of the appropriate Euclidean transformations to handle the fifth-order plate conditions and the application of Green’s function technique allow us to obtain the expressions of normal velocities at arbitrary points over the plates. On the other hand, an application of Green’s integral theorem on the scattered potential and the source potential functions gives us the alternative expressions of the above-mentioned normal velocities. The comparison of these alternative forms provides two coupled integral equations involving the unknown potential differences across the plates. Kernels of the integral equations have regular as well as hypersingular parts so that the resulting integral equations are hypersingular in nature. These are solved numerically and the solutions are utilized to compute the numerical estimates for different physical quantities. Published results are recovered for different arrangements of the plates and new results are presented graphically for various parametric values.

Analysis of the Propagation in High-Speed Interconnects for MIMICs by Means of the Method of Analytical Preconditioning: A New Highly Efficient Evaluation of the Coefficient Matrix

The method of analytical preconditioning combines the discretization and the analytical regularization of a singular integral equation in a single step. In a recent paper by the author, such a method has been applied to a spectral domain integral equation formulation devised to analyze the propagation in polygonal cross-section microstrip lines, which are widely used as high-speed interconnects in monolithic microwave and millimeter waves integrated circuits. By choosing analytically Fourier transformable expansion functions reconstructing the behavior of the fields on the wedges, fast convergence is achieved, and the convolution integrals are expressed in closed form. However, the coefficient matrix elements are one-dimensional improper integrals of oscillating and, in the worst cases, slowly decaying functions. In this paper, a novel technique for the efficient evaluation of such kind of integrals is proposed. By means of a procedure based on Cauchy integral theorem, the general coefficient matrix element is written as a linear combination of fast converging integrals. As shown in the numerical results section, the proposed technique always outperforms the analytical asymptotic acceleration technique, especially when highly accurate solutions are required.

An operator-valued T(1) theorem for symmetric singular integrals in UMD spaces

The natural BMO (bounded mean oscillation) conditions suggested by scalar-valued results are known to be insufficient for the boundedness of operator-valued paraproducts. Accordingly, the boundedness of operator-valued singular integrals has only been available under versions of the classical “T(1)∈BMO” assumptions that are not easily checkable. Recently, Hong, Liu and Mei (J. Funct. Anal. 2020) observed that the situation improves remarkably for singular integrals with a symmetry assumption, so that a classical T(1) criterion still guarantees their L2-boundedness on Hilbert space -valued functions. Here, these results are extended to general UMD (unconditional martingale differences) spaces with the same natural BMO condition for symmetrised paraproducts, and requiring in addition only the usual replacement of uniform bounds by R-bounds in the case of general singular integrals. In particular, under these assumptions, we obtain boundedness results on non-commutative Lp spaces for all 1<p<∞, without the need to replace the domain or the target by a related non-commutative Hardy space as in the results of Hong et al. for p≠2.

Approximation of CDF of Non-Central Chi-Square Distribution by Mean-Value Theorems for Integrals

The cumulative distribution function of the non-central chi-square distribution χn′2(λ) of n degrees of freedom possesses an integral representation. Here we rewrite this integral in terms of a lower incomplete gamma function applying two of the second mean-value theorems for definite integrals, which are of Bonnet type and Okamura’s variant of the du Bois–Reymond theorem. Related results are exposed concerning the small argument cases in cumulative distribution function (CDF) and their asymptotic behavior near the origin.

Модифицированный в рамках негауссовской каппа-статистики интегральный критерий устойчивости Чандрасекхара для равновесного сферического облака протозвезды

Within the framework of the non-extensive statistical mechanics of Kanyadakis, a generalization of the integral stability theorem of Chandrasekhar for the spherically symmetric distribution of matter and black radiation in an exoplanetary cloud in a state of gravitational equilibrium is obtained. For this purpose, the elements of deformed thermodynamics for an ideal gas, deformed canonical Gibbs distribution, as well as the effective gravitational constant, calculated in the formalisms of Kanyadakis and Verlinde, are used. In this, the deformation parameter κ (kappa) measures the so-called degree of nonextensiveness of the cloud system. In addition, the modified thermodynamic properties of blackbody radiation, in particular, the analogue of Stefan's law for radiation energy and generalized expressions for the entropy, heat capacity and radiation pressure, are discussed in the context of κ -statistics. The presented method of combining the indicated anomalous physical processes provides an alternative to the classical procedure of Chandrasekhar's derivation of the well-known integral theorems for gas configurations in gravitational equilibrium, and restores all standard expressions in the limit κ → 0. The results obtained will be able, according to the author, to explain some astrophysical problems of stellar-planetary cosmogony, associated, in particular, with modeling the processes of joint formation and evolution of a protosun and an exoplanetary cloud from a single nebula.

A series representation for Riemann's zeta function and some interesting identities that follow

Using Cauchy's Integral Theorem as a basis, what may be a new series representation for Dirichlet's function $\eta(s)$, and hence Riemann's function $\zeta(s)$, is obtained in terms of the Exponential Integral function $E_{s}(i\kappa)$ of complex argument. From this basis, infinite sums are evaluated, unusual integrals are reduced to known functions and interesting identities are unearthed. The incomplete functions $\zeta^{\pm}(s)$ and $\eta^{\pm}(s)$ are defined and shown to be intimately related to some of these interesting integrals. An identity relating Euler, Bernouli and Harmonic numbers is developed. It is demonstrated that a known simple integral with complex endpoints can be utilized to evaluate a large number of different integrals, by choosing varying paths between the endpoints.

A Note on the Mean Value Theorems

The mean value theorem for derivatives says that for a given function over a closed and bounded interval, there is a point P on the graph such that the tangent at P is parallel to the secant through the two endpoints. The mean value theorem for definite integrals says that the area under the function is equal to the area of a rectangle whose base is the length of the interval and height of some point Q on the graph. These two theorems have been studied and utilized extensively and they form the backbone of many important theorems in different branches of mathematics. In this note, we pose the question: for what functions do the two points P and Q always coincide? We find that the only analytic functions satisfying this condition are linear or exponential functions.