Nonconstant Function Research Articles

Hardy's Reduction for a Class of Liouville Integrals of Elementary Functions

This paper is concerned with a class of integrals whose integrands are the product of a rational function times the exponential of a nonconstant rational function. We call these Liouville integrals. For these integrals, we provide a student-friendly algorithm producing a two-term decomposition with minimum transcendental and maximum elementary components. This decomposition fulfills the conditions of Hardy's reduction theory, determines whether these integrals are elementary functions, and when in the affirmative, finds them. To achieve our goal, we use partial fraction decomposition, simple notions of linear algebra, and a special case of an 1835 theorem of Liouville that we refer to as Liouville's criterion on integration. There is in the literature a complete algorithm to decide if the integral of an elementary function is also elementary. Ours is a gentle alternative for the class of Liouville integrals.

Enhancing the accuracy of the Fowler method for monitoring non-constant work functions.

The Fowler method is a prominent non-invasive technique to determine the absolute work function of a surface based on the photoelectric effect. The evaluation procedure relies on the correlation of the photocurrent with the incident photon energy hν which is mainly dependent on the surface work function χ. Applying Fowler's theory of the photocurrent, the measurements can be fitted by the theoretical curve near the threshold hν⪆χ yielding the work function χ and a parameter A. The straightforward experimental implementation of the Fowler method is to use several particular photon energies, e.g. via interference filters. However, with a realization like that the restriction hν ≈ χ can easily be violated, especially when the work function of the material is decreasing during the measurements as, for instance, with coating or adsorption processes. This can lead to an overestimation of the evaluated work function value of typically some 0.1 eV, reaching up to more than 0.5 eV in an unfavorable case. A detailed analysis of the Fowler theory now reveals the background of that effect and shows that the fit-parameter A can be used to assess the accuracy of the determined value of χ conveniently during the measurements. Moreover, a scheme is introduced to quantify a potential overestimation and to perform a correction to χ to a certain extent. The issues are demonstrated exemplarily at the monitoring of the work function reduction of a stainless steel sample surface due to caesiation.

On rational functions of first-class complexity

It is proved that, for every rational function of two variables P(x, y) of analytic complexity one, there is either a representation of the form f(a(x) + b(y)) or a representation of the form f(a(x)b(y)), where f(x), a(x), b(x) are nonconstant rational functions of a single variable. Here, if P(x, y) is a polynomial, then f(x), a(x), and b(x) are nonconstant polynomials of a single variable.

Omega diffusion risk model with surplus-dependent tax and capital injections

In this paper, we propose and study an Omega risk model with a constant bankruptcy function, surplus-dependent tax payments and capital injections in a time-homogeneous diffusion setting. The surplus value process is both refracted (paying tax) at its running maximum and reflected (injecting capital) at a lower constant boundary. The new model incorporates practical features from the Omega risk model (Albrecher et al., 2011), the risk model with tax (Albrecher and Hipp, 2007), and the risk model with capital injections (Albrecher and Ivanovs, 2014). The study of this new risk model is closely related to the Azéma–Yor process, which is a process refracted by its running maximum. We explicitly characterize the Laplace transform of the occupation time of an Azéma–Yor process below a constant level until the first passage time of another Azéma–Yor process or until an independent exponential time. We also consider the case when the process has a lower reflecting boundary. This result unifies and extends recent results of Li and Zhou (2013) and Zhang (2015). We explicitly characterize the Laplace transform of the time of bankruptcy in the Omega risk model with tax and capital injections up to eigen-functions, and determine the expected present value of tax payments until default. We also discuss a further extension to occupation functionals through stochastic time-change, which handles the case of a non-constant bankruptcy function. Finally we present examples using a Brownian motion with drift, and discuss the pricing of quantile options written on the Azéma–Yor process.

ON A DIVIDEND-PAYING STOCK OPTIONS PRICING MODEL (SOPM) USING CONSTANT ELASTICITY OF VARIANCE STOCHASTIC DYNAMICS

In this paper, we propose a pricing model for stock option valuation. The model is derived from the classical Black-Scholes option pricing equation via the application of the constant elasticity of variance (CEV) model with dividend yield. This modifies the Black- Scholes equation by incorporating a non-constant volatility power function of the underlying stock price, and a dividend yield parameter.

On the volume growth of Kähler manifolds with nonnegative bisectional curvature

Let $M$ be a complete K\"ahler manifold with nonnegative bisectional curvature. Suppose the universal cover does not split and $M$ admits a nonconstant holomorphic function with polynomial growth, we prove $M$ must be of maximal volume growth. This confirms a conjecture of Ni. There are two essential ingredients in the proof: The Cheeger-Colding theory on Gromov-Hausdorff convergence of manifolds; the three circle theorem for holomorphic functions.

Cosmological blueshifting may explain the gamma ray bursts

It is shown that the basic observed properties of the gamma-ray bursts (GRBs) are accounted for if one assumes that the GRBs arise by blueshifting the emission radiation of hydrogen and helium generated during the last scattering epoch. The blueshift generator for a single GRB is a region with a nonconstant bang-time function $t_B(r)$ (described by a Lema\^{\i}tre -- Tolman (L--T) exact solution of Einstein's equations) matched into a homogeneous and isotropic (Friedmann) background. Blueshift visible to the present observer arises \textit{only on those rays that are emitted radially in an L--T region}. The paper presents three L--T models with different Big Bang profiles, adapted for the highest and the lowest end of the GRB frequency range. The models account for: (1) The observed frequency range of the GRBs; (2) Their limited duration; (3) The afterglows; (4) Their hypothetical collimation into narrow jets; (5) The large distances to their sources; (6) The multitude of the observed GRBs. Properties (2), (3) and (6) are accounted for only qualitatively. With a small correction of the parameters of the model, the implied perturbations of the CMB radiation will be consistent with those actually caused by the GRBs. A complete model of the Universe would consist of many L--T regions with different $t_B(r)$ profiles, matched into the same Friedmann background. This paper is meant to be an initial exploration of the possibilities offered by models of this kind; the actual fitting of all parameters to observational results requires fine-tuning of several interconnected variables and is left for a separate study.

Absence of Non-Constant Harmonic Functions with ℓ p -gradient in some Semi-Direct Products

To obtain groups with bounded harmonic functions (which are amenable), one of the most frequent way is to look at some semi-direct products (e.g. lamplighter groups). The aim here is to show that many of these semi-direct products do not admit harmonic functions with gradient in l p , for \(p\in [1,\infty [\).

Three circles theorems for harmonic functions

We proved two Three Circles Theorems for harmonic functions on manifolds in integral sense. As one application, on manifold with nonnegative Ricci curvature, whose tangent cone at infinity is the unique metric cone with unique conic measure, we showed the existence of nonconstant harmonic functions with polynomial growth. This existence result recovered and generalized the former result of Y. Ding, and led to a complete answer of L. Ni's conjecture. Furthermore in similar context, combining the techniques of estimating the frequency of harmonic functions with polynomial growth, which were developed by Colding and Minicozzi, we confirmed their conjecture about the uniform bound of frequency.

Adjoining a Constant Function ton-Dimensional Chebyshev Space

This paper is concerned with extending a Chebyshev system ofncontinuous nonconstant functions into a set ofn+1functions including a constant function. Necessary and sufficient conditions for the new set to be a Chebyshev system are discussed and some results are obtained.

Hausdorff Dimension of Sets of Escaping Points and Escaping Parameters for Elliptic Functions

AbstractLetbe a non-constant elliptic function. We prove that the Hausdorff dimension of the escaping set offequals 2q/(q+1), whereqis the maximal multiplicity of poles off. We also consider theescaping parametersin the familyfβ=βf, i.e. the parametersβfor which the orbit of one critical value offβescapes to infinity. Under additional assumptions onfwe prove that the Hausdorff dimension of the set of escaping parametersεin the familyfβis greater than or equal to the Hausdorff dimension of the escaping set in the dynamical space. This demonstrates an analogy between the dynamical plane and the parameter plane in the class of transcendental meromorphic functions.

Boundary values, random walks, and $\ell^p$-cohomology in degree one

The vanishing of reduced \ell^2 -cohomology for amenable groups can be traced to the work of Cheeger and Gromov in [10]. The subject matter here is reduced \ell^p -cohomology for p \in ]1,\infty[ , particularly its vanishing. Results for the triviality of \underline {\ell^pH}^1(G) are obtained, for example: when p \in ]1,2] and G is amenable; when p \in ]1,\infty[ and G is Liouville (e.g. of intermediate growth). This is done by answering a question of Pansu in [34, §1.9] for graphs satisfying certain isoperimetric profile. Namely, the triviality of the reduced \ell^p -cohomology is equivalent to the absence of non-constant harmonic functions with gradient in \ell^q ( q depends on the profile). In particular, one reduces questions of non-linear analysis ( p -harmonic functions) to linear ones (harmonic functions with a very restrictive growth condition).

Modeling and simulation of the spread of H1N1 flu with periodic vaccination

Influenza H1N1 has been found to exhibit oscillatory levels of incidence in large populations. Clear peaks for influenza H1N1 are observed in several countries including Vietnam each year [M. F. Boni, B. H. Manh, P. Q. Thai, J. Farrar, T. Hien, N. T. Hien, N. Van Kinh and P. Horby, Modelling the progression of pandemic influenza A (H1N1) in Vietnam and the opportunities for reassortment with other influenza viruses, BMC Med. 7 (2009) 43, Doi: 10.1186/1741-7015-7-43]. So it is important to study seasonal forces and factors which can affect the transmission of this disease. This paper studies an SIRS epidemic model with seasonal vaccination rate. This SIRS model has a unique disease-free solution (DFS). The value R0, the basic reproduction number is obtained when the vaccination is a periodic function. Stability results for the DFS are obtained when R0 < 1. The disease persists in the population and remains endemic if R0 > 1. Also when R0 > 1 existence of a nonzero periodic solution is proved. These results obtained for our model when the vaccination strategy is a non-constant time-dependent function.

Quantitative expressions for the connectedness of sets in ℝ n

We prove that, for two arbitrary points a and b of a connected set E ⊂ Rn (n ≥ 2) and for any e > 0, there exist points x0 = a, x2,..., xp = b in E such that $$\parallel {x_1} - {x_0}{\parallel ^n} + \cdots + \parallel {x_p} - {x_{p - 1}}{\parallel ^n} < \varepsilon$$ . We prove that the exponent n in this assertion is sharp. The nonexistence of a chain of points in E with $$\parallel {x_1} - {x_0}{\parallel ^\alpha } + \cdots + \parallel {x_p} - {x_{p - 1}}{\parallel ^\alpha } < \varepsilon$$ for some α ∈ (1, n) proves to be equivalent to the existence of a nonconstant function f: E → R in the class Lipα(E). For each such α, we construct a curve E(α) of Hausdorff dimension α in Rn and a nonconstant function f: E(α) → R such that f ∈ Lipα(E(α)).

SHAPE OPERATOR AND GAUSS MAP OF POINTWISE 1-TYPE

Abstract. We examine the relationship of the shape operator of a sur-face of Euclidean 3-space with its Gauss map of pointwise 1-type. Sur-faces with constant mean curvature and right circular cones with respectto some properties of the shape operator are characterized when theirGauss map is of pointwise 1-type. 1. IntroductionFor last thirty years or so, the notion of finite type immersions has beenwidely applied in classifying and characterizing the submanifolds in Euclideanspace since it was introduced in the late 1970’s: Let x : M → E m be anisometric immersion of a Riemannian manifold M into the Euclidean space ofdimension m. If we identify x with the position vector at each point p of M, itis a vector valued function. Let ∆ be the Laplacian operator defined on M. Ifx is decomposed as x = x 0 +x 1 +x 2 +··· + x k , it is said to be of finite type,where x 0 is a constant vector and x i non-constant maps satisfying ∆x i = λ i x i for some constant λ i . In particular, if λ i ’s are different, we say that it is ofk-type ([3, 4, 15]). It is well known that a submanifold of Euclidean space of1-type is either a minimal submanifold of E

ON THE BRÜCK CONJECTURE

The Brück conjecture states that if a nonconstant entire function $f$ with hyper-order ${\it\sigma}_{2}(f)\in [0,+\infty )\setminus \mathbb{N}$ shares one finite value $a$ (counting multiplicities) with its derivative $f^{\prime }$, then $f^{\prime }-a=c(f-a)$, where $c$ is a nonzero constant. The conjecture has been established for entire functions with order ${\it\sigma}(f)&lt;+\infty$ and hyper-order ${\it\sigma}_{2}(f)&lt;{\textstyle \frac{1}{2}}$. The purpose of this paper is to prove the Brück conjecture for the case ${\it\sigma}_{2}(f)=\frac{1}{2}$ by studying the infinite hyper-order solutions of the linear differential equations $f^{(k)}+A(z)f=Q(z)$. The shared value $a$ is extended to be a ‘small’ function with respect to the entire function $f$.

On the design of switching circuits admitting small detection test sets

It is established that, for an arbitrary nonconstant Boolean function f(x1,..., xn), there exist the following testable switching circuits that admit a single fault detection test set of cardinality O(n): (a) a three-pole switching circuit (with one input and two output poles) that realizes a system of functions (f, \(\bar f\)) and (b) a two-pole switching circuit that realizes a function f(x1,..., xn) ⊕ xn+1. It is also proved that almost all Boolean functions f(x1,..., xn) can be realized by two-pole switching circuits that admit small detection test sets (test sets of cardinality O(n)) under homogeneous faults of switches (closures or breakings).

Robust virtual implementation under common strong belief in rationality

Bergemann and Morris (2009b) show that static mechanisms cannot robustly virtually implement non-constant social choice functions if preferences are sufficiently interdependent. Without any knowledge of how agents revise their beliefs this impossibility result extends to dynamic mechanisms. In contrast, we show that if the agents revise their beliefs according to the forward-induction logic embedded in strong rationalizability, admitting dynamic mechanisms leads to considerable gains. We show that all ex-post incentive compatible social choice functions are robustly virtually implementable in private consumption environments satisfying a weak sufficient condition, regardless of the level of preference interdependence. This result derives from the key insight that in such environments, in every belief-complete type space under common strong belief in rationality (Battigalli and Siniscalchi, 2002), dynamic mechanisms can distinguish all payoff type profiles by their strategic choices. Notably, dynamic mechanisms can robustly virtually implement the efficient allocation of an object in cases that static mechanisms cannot.

Disorder, entropy and harmonic functions

We study harmonic functions on random environments with particular emphasis on the case of the infinite cluster of supercritical percolation on $\mathbb{Z}^d$. We prove that the vector space of harmonic functions growing at most linearly is $(d+1)$-dimensional almost surely. Further, there are no nonconstant sublinear harmonic functions (thus implying the uniqueness of the corrector). A main ingredient of the proof is a quantitative, annealed version of the Avez entropy argument. This also provides bounds on the derivative of the heat kernel, simplifying and generalizing existing results. The argument applies to many different environments; even reversibility is not necessary.

A high-order numerical manifold method with nine-node triangular meshes

The numerical manifold method (NMM) is a unified framework that is used to describe continuous and discontinuous problems. The NMM is derived based on the finite cover approximation theory and gains its name after the mathematical notion of manifold. It is also a method based on the partition of unity (PU) and it introduces two cover systems: the mathematical cover (MC) and the physical cover (PC). There are two approaches for constructing high-order approximations. The first approach involves a non-constant PU function and non-constant local approximations. This results in the linear dependence (LD) problem and leads to the singularity in a global matrix. The second approach involves a higher PU function and constant local approximations. The increase in the order of approximations should go along with the increase in star but the LD problem can be avoided completely in theory. In this paper, a new high-order NMM with nine-node triangular meshes is proposed. The upgrade from first-order NMM to high-order NMM is illustrated in detail. Moreover, the initial stress matrix is analyzed in detail. The effectiveness and accuracy of the proposed high-order NMM are validated using several typical examples. The proposed high-order NMM supplements the existing family of non-LD high- and low-order NMM under MC with triangular meshes.