One-variable Functions Research Articles

Submodules of the Hardy module over the polydisc

We say that a submodule S of \({H^2}({D^n})\) (n > 1) is co-doubly commuting if the quotient module \({H^2}({D^n})/S\) is doubly commuting. We show that a co-doubly commuting submodule of \({H^2}({D^n})\) is essentially doubly commuting if and only if the corresponding one-variable inner functions are finite Blaschke products or n = 2. In particular, a co-doubly commuting submodule S of \({H^2}({D^n})\) is essentially doubly commuting if and only if n = 2 or that S is of finite co-dimension. We obtain an explicit representation of the Beurling-Lax-Halmos inner functions for those submodules of \(H_{H^2 (\mathbb{D}^{n - 1} )}^2 (\mathbb{D})\) which are co-doubly commuting submodules of \({H^2}({D^n})\). Finally, we prove that a pair of co-doubly commuting submodules of \({H^2}({D^n})\) are unitarily equivalent if and only if they are equal.

Two remarks on polynomially bounded reducts of the restricted analytic field with exponentiation

AbstractThis article presents two constructions motivated by a conjecture of van den Dries and Miller concerning the restricted analytic field with exponentiation. The first construction provides an example of two o-minimal expansions of a real closed field that possess the same field of germs at infinity of one-variable functions and yet define different global one-variable functions. The second construction gives an example of a family of infinitely many distinct maximal polynomially bounded reducts (all this in the sense of definability) of the restricted analytic field with exponentiation.

Two remarks on polynomially bounded reducts of the restricted analytic field with exponentiation

AbstractThis article presents two constructions motivated by a conjecture of van den Dries and Miller concerning the restricted analytic field with exponentiation. The first construction provides an example of two o-minimal expansions of a real closed field that possess the same field of germs at infinity of one-variable functions and yet define different global one-variable functions. The second construction gives an example of a family of infinitely many distinct maximal polynomially bounded reducts (all this in the sense of definability) of the restricted analytic field with exponentiation.

Spanning with indexes

This paper presents several approximation theorems of a general contingent claim in terms of index options. We demonstrate that any contingent claim on the primitive securities in an infinite state economy can be approximated arbitrarily close by a portfolio of index options. In addition, these index options are associated with the same payout function, which belongs to a large and explicit class of one-variable measurable functions. I also characterize the layer structure of a general contingent claim.

Students’ images of two-variable functions and their graphs

This paper presents a conceptual analysis for students’ images of graphs and their extension to graphs of two-variable functions. We use the conceptual analysis, based on quantitative and covariational reasoning, to construct a hypothetical learning trajectory (HLT) for how students might generalize their understanding of graphs of one-variable functions to graphs of two-variable functions. To evaluate the viability of this learning trajectory, we use data from two teaching experiments based on tasks intended to support development of the schemes in the HLT. We focus on the schemes that two students developed in these teaching experiments and discuss their relationship to the original HLT. We close by considering the role of covariational reasoning in generalization, consider other ways in which students might come to conceptualize graphs of two-variable functions, and discuss implications for instruction.

Gradient-Orthonormal Bases of Momentum-Entire Wavelets in Odd Dimension

We construct gradient-orthonormal bases of momentum-entire wavelets in dimension n in the case where n is odd. The scale factor is 3 √ 2 and the coherence on a given scale is based on the propagator exp(~s · ~ 54). This propagator can be described in terms of two one-variable functions. One of them is the Airy function, while the other satisfies an ordinary differential equation less familiar than the Airy equation. This construction is part of our on-going quest for compactly supported, gradient-orthonormal wavelets that have some kind of coherence on a given scale.

On The Residual Effective Potential within Global One-Dimensional Quantum Gravity

The global one-dimensional quantum gravity is the model of quantum gravity which arises from the global one-dimensionality conjecture within quantum general relativity, first considered by the author in 2010 and then in 2012. In this model the global dimension is a determinant of a metric of three-dimensional space embedded into an enveloping Lorentizan four-dimensional spacetime. In 2012, it has already been presented by the author that this model can be extended to any Lorentzian D + 1-dimensional spacetime, where D is a dimension of space, and resulting in the global one-dimensional model of a higher dimensional quantum gravity. The purely quantum-mechanical part of this model is a minimal effective model within the quantum geometrodynamics, introduced by J.A. Wheeler and B.S. DeWitt in the 1960s, but the effective potential is manifestly different from the one considered by Wheeler & DeWitt. Moreover, in our model the wave functionals solving the quantum gravity are one-variable smooth functions and, therefore, the troublesome mathematical technique of the Feynman functional integration present in the Hawking formulation of quantum gravity, is absent is this model, what makes it a mathematically consistent theory of quantum gravitation. In this paper, we discuss in some detail a certain part of the global one-dimensional model already proposed in 2010, and then developed in 2012. The generalized functional expansion of the effective potential and the residual approximation, which describe the embedded spaces which are maximally symmetric three-dimensional Einstein's manifolds, whose lead to the Newton-Coulomb type potential in the quantum gravity model, are considered. Furthermore, scenarios related to few selected specific forms of the effective potential are suggested as physically interesting and discussed.

Geometric zeta functions for higher rank $p$-adic groups

The higher rank Lefschetz formula for p-adic groups is used to prove rationality of a several-variable zeta function attached to the action of a p-adic group on its Bruhat-Tits building. By specializing to certain lines one gets one-variable zeta functions, which then can be related to geometrically defined zeta functions

Some issues concerning approximations of functions by Fourier-Bessel sums

Some issues concerning the approximation of one-variable functions from the class $\mathbb{L}_2 $ by nth-order partial sums of Fourier-Bessel series are studied. Several theorems are proved that estimate the best approximation of a function characterized by the generalized modulus of continuity.

Absolute convergence of double series of Fourier-Haar coefficients for functions of bounded p-variation

We consider functions of two variables of bounded p-variation of the Hardy type on the unit square. For these functions we obtain a sufficient condition for the absolute convergence of series of positive powers of Fourier coefficients with power-type weights with respect to the double Haar system. This condition implies those for the absolute convergence of series of Fourier-Haar coefficients of one-variable functions which have a bounded Wiener p-variation or belong to the class Lip α. We show that the obtained results are unimprovable. We also formulate N-dimensional analogs of the main result and its corollaries.

A method of multivariable Hermite basis function approximation

A method of multivariable (multivariate) Hermite function based approximation is presented and discussed. The multivariable basis is constructed as a product of one-variable Hermite functions with adjustable scaling parameters. Thanks basis orthonormality, the approximated function expansion coefficients are calculated by using explicit, non-search formulae. The scaling parameters are determined via a search algorithm. Initially, an excessive number of functions in the basis is calculated, then a simple pruning method is applied. Only those are taken which contribute the most to error decrease, down to a desired level. The method ensures a very good generalization property. This claim is supported by both theoretical considerations and working examples.

Free Vibration Analysis for Step Tubular Structures of Tall Buildings Supported on Elastic Foundation Soil

A method based on Ordinary Differential Equations (ODE) solver for free vibration analysis of tubular structures of tall buildings is developed, considering the deformation of the foundation soil as well as the interactions between the foundation and soil, by means of a three dimensional model with continuously distributed mass and stiffness. The nodal lines employed to discretize the computational model of the structures are one-variable functions defined on the nodal lines selected by the analyst to describe the dynamic behavior of the model. The unknown functions determined numerically herein are actual vibration modes that can be also recognized as the deformation functions of a set of conceptual structural components. By a Hamiltonian principle, the governing equations of the free vibration analysis can be obtained, which are a set of ordinary differential equations (ODE) of the vibration modes with their corresponding boundary conditions. The desired frequencies and corresponding vibration modes can be obtained by numerically solving the ODEs with boundary conditions. The method is applied to the tubular structures of tall buildings. The results from the illustration example show that the method is rational and powerful for the free vibration analysis of tall buildings.

Control optimization in a hyperbolic linear system

An optimal control problem is considered for a distributed-parameter Goursat-Darboux system with controlled boundary conditions. For the numerical solution of the problem, an algorithm based on separability and minimax theorems is constructed, which reduces the problem to finding the maximum of a concave functional defined in the class of one-variable functions.

Quadrature formulas for functions with a boundary-layer component

Quadrature formulas for one-variable functions with a boundary-layer component are constructed and studied. It is assumed that the integrand can be represented as the sum of a regular and a boundary-layer component, the latter having high gradients that reduce the accuracy of classical quadrature formulas, such as the trapezoidal and Simpson rules. The formulas are modified so that their error is independent of the gradients of the boundary-layer component. Results of numerical experiments are presented.

Phase portraits of separable Hamiltonian systems

We study a generalization of potential Hamiltonian systems ( H ( x , y ) = y 2 + F ( x ) ) with one degree of freedom; namely, those with Hamiltonian functions of type H ( x , y ) = F ( x ) + G ( y ) , which will be denoted by X H . We present an algorithm to obtain the phase portrait (including the behaviour at infinity) of X H when F and G are arbitrary polynomials. Indeed, from the graphs of the one-variable functions F and G , we are able to give the full description on the Poincaré disk, therefore extending the well-known method to obtain the phase portrait of potential systems in the finite plane. The fact that the phase portraits can be fully described in terms of the two one-variable real functions F and G allows, as well, a complete study of the bifurcation diagrams in complete families of vector fields. The algorithm can be applied to study separable Hamiltonian systems with one degree of freedom, which include a vast amount of examples in physical applications.

Computationally Efficient Kalman Filtering for a Class of Nonlinear Systems

This paper deals with recursive state estimation for the class of discrete time nonlinear systems whose nonlinearity consists of one or more static nonlinear one-variable functions. This class contains several important subclasses. The special structure is exploited to permit accurate computations without an increase in computational cost. The proposed method is compared with standard Extended Kalman Filter, Unscented Kalman Filter and Gauss-Hermite Kalman Filter in three illustrative examples. The results show that it yields good results with small computational cost.

A new free vibration analysis method for space mega frames of super tall buildings

A free vibration analysis method for space mega frames of super tall buildings is developed. The physical model of a mega frame is idealized as a three-dimensional assemblage of stiffened close-thin-walled tubes with continuously distributed mass and stiffness. One-variable functions defined on the nodal lines selected by the analyst are unknown functions (eigenfunctions) employed to describe the dynamic behavior of the model. By a Hamiltonian principle, the governing equations of the free vibration analysis can be obtained, which are a set of Ordinary Differential Equations (ODE) of the eigenfunctions with their corresponding boundary conditions. The desired vibration frequencies and corresponding vibration modes (eigenfunctions) can be obtained by numerically solving the system of ODEs with boundary conditions. The free vibration analysis also includes the coupling vibration modes of the collection composed by several nodal lines. The results from the illustration example show that the method is rational and powerful for the free vibration analysis of space-mega-frame systems.

Adaptive detection of a high-variable function

A major difficulty arising in statistics of multi-variable functions is “the curse of dimensionality:” Rates of accuracy in estimation and separation rates in detection problems behave poorly as the number of variables increases. This difficulty arises for most popular functional classes such as Sobolev or Holder balls. In 2007, the authors considered functional classes of a new type, which were first introduced by Sloan and Woźniakowski. These classes are balls SWσ,s in weighed tensor product spaces, which are characterized by two parameters: σ> 0 is a “smoothness” parameter, and s> 0 determines the weight sequence that characterizes the “importance” of variables. In particular, it was shown that under the white Gaussian noise model, the log-asymptotics of separation rates in detection are similar to those for one-variable functions of smoothness σ* = min (s, σ) independently of the original problem dimensions; thus, the curse of dimensionality is “lifted.” However, the test procedure depends on parameters (σ, s), which are typically unknown. In the present paper, we propose a test procedure which does not depend on parameters (σ, s) and provides the same log-asymptotics of separation rates uniformly over any compact set of parameters (σ, s). In addition, we give an independent simple proof of log-asymptotics of separation rates in the problem. Bibliography: 16 titles.

Plurisubharmonic functions characterized by one-variable extremal functions

Plurisubharmonic functions characterized by one-variable extremal functions

Domain statistics in a finite Ising chain

We present a comprehensive study for the statistical properties of random variables that describe the domain structure of a finite Ising chain with nearest-neighbor exchange interactions and free boundary conditions. By use of extensive combinatorics we succeed in obtaining the one-variable probability functions for (i) the number of domain walls, (ii) the number of up domains, and (iii) the number of spins in an up domain. The corresponding averages and variances of these probability distributions are calculated and the limiting case of an infinite chain is considered. Analyzing the averages and the transition time between differing chain states at low temperatures, we also introduce a criterion of the ferromagnetic-like behavior of a finite Ising chain. The results can be used to characterize magnetism in monatomic metal wires and atomic-scale memory devices.