Nonzero Constant Research Articles

Global solutions to stochastic reaction–diffusion equations with super-linear drift and multiplicative noise

Let $\xi (t,x)$ denote space–time white noise and consider a reaction–diffusion equation of the form \[\dot{u}(t,x)=\frac{1}{2}u"(t,x)+b\big(u(t,x)\big)+\sigma \big(u(t,x)\big)\xi (t,x),\] on $\mathbb{R}_{+}\times [0,1]$, with homogeneous Dirichlet boundary conditions and suitable initial data, in the case that there exists $\varepsilon >0$ such that $\vert b(z)\vert \ge \vert z\vert (\log \vert z\vert )^{1+\varepsilon }$ for all sufficiently-large values of $\vert z\vert $. When $\sigma \equiv 0$, it is well known that such PDEs frequently have nontrivial stationary solutions. By contrast, Bonder and Groisman [Phys. D 238 (2009) 209–215] have recently shown that there is finite-time blowup when $\sigma $ is a nonzero constant. In this paper, we prove that the Bonder–Groisman condition is unimprovable by showing that the reaction–diffusion equation with noise is “typically” well posed when $\vert b(z)\vert =O(\vert z\vert \log_{+}\vert z\vert )$ as $\vert z\vert \to \infty $. We interpret the word “typically” in two essentially-different ways without altering the conclusions of our assertions.

A result on a question of Lü, Li and Yang

Let f be a transcendental meromorphic function of finite order with finitely many poles, c ? C\{0} and n,k ? N. Suppose fn(z)-Q1(z) and (fn(z+c))(k)- Q2(z) share (0,1) and f(z), f(z+c) share 0 CM. If n ? k + 1, then (fn(z+c))(k) ? Q2(z)/Q1(z)fn(z), where Q1, Q2 are polynomials with Q1Q2 ?/ 0. Furthermore, if Q1 = Q2, then f(z)=c1e?/n z, where c1 and ? are non-zero constants such that e?c = 1 and ?k = 1. Also we exhibit some examples to show that the conditions of our result are the best possible.

Band structure and spatial electromagnetic field distribution in the 2D photonic crystals with nontrivial elementary cell

Periodic systems with the silver-based waveguides are discussed. The elementary cell in such systems consists of several waveguides. We employ the multiple scattering formalism based on solving the Maxwell equations solution for a single cylinder. The method involving the lattice-sum transformation into a fast convergent series is implemented. This enables us not only to increase the computational accuracy but also to reduce the necessary time for performing the numerical simulation. It is shown that under condition of nonzero propagation constant, the guided modes located closely in the frequency region appear. The number of these modes equals the number of waveguides in the elementary cell of the system. As a quasi-wave vector approaches the edge of the Brillouin zone, the frequency of these modes decreases. Increasing the propagation constant affects the modes shifting them towards the high frequency region. It is also shown that it is possible to match simultaneously wave vector and frequency of guided modes in the zigzag shaped systems.

Contact Resistance of a Square Contact

We consider a conductive body in the form of a parallelepiped with small square contacts attached to its ends. The potential of the electric current is modeled by a boundary value problem for the Laplace equation in a parallelepiped. The zero normal derivative is assigned on the boundary except for the areas under the contacts, where the derivative is a nonzero constant. Physically, this condition corresponds to the presence of a low-conductivity film on the surface of the contacts. The problem is solved by separation of variables, and then the electrical resistance is found as a functional of the solution in the form of the sum of a double series. Our main aim is to study the dependence of the resistance on a small parameter characterizing the size of the contacts. The leading term of the asymptotics that expresses this dependence is the contact resistance. The mathematical problem is to treat the singular dependence of the sum of the series corresponding to the resistance on the small parameter: the series diverges as the small parameter vanishes. We solve this problem by replacing the series with a two-dimensional integral. We find the leading term of the asymptotics and estimate the remainder. It turns out that the main contribution to the remainder is made by the difference between the two-dimensional integral and the double sum.

Inhomogeneous boundary value problems for steady compressible magnetohydrodynamic equations

This paper will be concerned with the compressible perturbation to steady magnetohydrodynamic equations near a uniform flow. In particular, the velocity of the basic flow under consideration can be any non-zero constant. We prove that there exists a stationary strong solution around a given basic flow with inhomogeneous boundary condition.

Students’ ability to correctly apply differentiation rules to structurally different functions

The derivative concept is studied in first-year university mathematics. In this study, we focused on students’ ability to correctly apply the rules for derivatives of functions with the different structures that they encounter in their university studies. This was done by investigating the online responses of first-year students at the University of KwaZulu-Natal to online quizzes that contributed to their assessment. Based on this investigation, we then interviewed eight students to gain an insight into the thinking behind their responses. We report on the analysis of students’ responses to five items on the online quizzes based on the derivative concept. The categories in which those items were based are: condition for existence of derivative at a point; rules for derivatives of standard functions; application of chain rule to different function structures; the application of multiple rules; and application of derivatives to optimise a function. Our findings indicate that students had difficulty in detecting that multiple rules for derivatives were required to differentiate certain types of functions represented in symbolic form. Furthermore, students had difficulty in finding the derivative of a function when more than one application of the chain rule was required. However, there were students who had the ability to apply the rules for derivatives of functions without difficulty. In particular, most of the students were able to correctly recall the differentiation rules for functions with standard structures f(x)=xn, h(x)=ekx and y=[g(x)]n, n ≠ 0 and k is a non-zero constant. Students were also able to correctly apply the chain rule to an exponential function with base e, raised to 4x. The majority of students were able to correctly apply the chain rule together with differentiation rules for logarithmic and exponential (with bases a >1) function structures, and function structures that required the application of the product rule together with the chain rule. Most of the students were able to apply derivatives to optimise a function.Significance:
 
 A significant percentage of students who took online quizzes experienced difficulties with applying multiple differentiation rules in the context of a single function. The difficulties stemmed from their inability to detect from the structure of the function which rules should be applied and also the order in which those relevant rules should be applied.

Asymptotics for Optimal Design Problems for the Schrödinger Equation with a Potential

We study the problem of optimal observability and prove time asymptotic observability estimates for the Schrödinger equation with a potential in L∞Ω, with Ω⊂Rd, using spectral theory. An elegant way to model the problem using a time asymptotic observability constant is presented. For certain small potentials, we demonstrate the existence of a nonzero asymptotic observability constant under given conditions and describe its explicit properties and optimal values. Moreover, we give a precise description of numerical models to analyze the properties of important examples of potentials wells, including that of the modified harmonic oscillator.

A mathematical analysis of a circular pipe in rate type fluid via Hankel transform

In this paper, helices of a generalized Oldroyd-B fluid have been analyzed through a horizontal circular pipe. The circular pipe is taken in the form of a circular cylinder. The analytical solutions are determined for velocities and shear stresses due to the unsteady helical flow of a generalized Oldroyd-B fluid. The general solutions are derived by using finite Hankel and discrete Laplace transforms to satisfy the imposed conditions and the governing equations. The special cases of our general solutions are also perused performing the same motion for fractional and ordinary Maxwell fluid, fractional and ordinary second-grade fluid and fractional and ordinary viscous fluid as well. The graphical illustration is depicted in order to explore how the two velocities and shear stresses profiles are impacted by different rheological parameters, for instance, fractional parameter, relaxation time, retardation time, material non-zero constant, dynamic viscosity and few others. Finally, ordinary and fractional operators have various similarities and differences on a circular pipe for helicoidal behavior of fluid flow.

Abundant exact solutions to the discrete complex mKdV equation by Darboux transformation

In this paper, an N-fold Darboux transformation is constructed for the discrete complex modified Korteweg-de Vries equation of focusing type, in terms of determinants. Through the obtained one-fold and two-fold Darboux transformations, a variety of new exact solutions, including an anti-dark soliton solution, a breather solution, a periodic solution, and a two-soliton solution, are derived from a nonzero constant and plane-wave seed solution. Via numerical simulation, a new kind of dynamical behavior of the two-soliton solution is exhibited, which tells that the two-soliton solution includes an anti-dark solitary wave and a w-shaped solitary wave.

Dark energy with w→−1 : Asymptotic Λ versus pseudo- Λ

If the dark energy density asymptotically approaches a nonzero constant, $\rho_{DE} \rightarrow \rho_0$, then its equation of state parameter $w$ necessarily approaches $-1$. The converse is not true; dark energy with $w \rightarrow -1$ can correspond to either $\rho_{DE} \rightarrow \rho_0$ or $\rho_{DE} \rightarrow 0$. This provides a natural division of models with $w \rightarrow -1$ into two distinct classes: asymptotic $\Lambda$ ($\rho_{DE} \rightarrow \rho_0$) and pseudo-$\Lambda$ ($\rho_{DE} \rightarrow 0$). We delineate the boundary between these two classes of models in terms of the behavior of $w(a)$, $\rho_{DE}(a)$, and $a(t)$. We examine barotropic and quintessence realizations of both types of models. Barotropic models with positive squared sound speed and $w \rightarrow -1$ are always asymptotically $\Lambda$; they can never produce pseudo-$\Lambda$ behavior. Quintessence models can correspond to either asymptotic $\Lambda$ or pseudo-$\Lambda$ evolution, but the latter is impossible when the expansion is dominated by a background barotropic fluid. We show that the distinction between asymptotic $\Lambda$ and pseudo-$\Lambda$ models for $w> -1$ is mathematically dual to the distinction between pseudo-rip and big/little rip models when $w < -1$.

Outlier detection and variable selection via difference based regression model and penalized regression

This paper studies an efficient procedure for the outlier detection and variable selection problem in linear regression. The effect of outliers is added in linear regression as a mean shift parameter, nonzero or zero constant. To t this mean shift model, most penalized regressions have used some adaptive penalties on the parameters to shrink most of the parameters to zero. Such penalized models do select the true variables well, but do not detect the outliers correctly. To overcome this problem, we first determine a group of possibly suspected outliers using difference-based regression model (DBRM) and add the group to the linear model as the parameters of the effect of each suspected outlier. Then, we perform outlier detection and variable selection simultaneously using Lasso regression or Elastic net regression for the linear regression with the effect term of each suspected outlier added. The proposed method is more efficient than the previous penalized regression. We compare the proposed procedure with other methods using a simulation study and apply this procedure to the real data.

Meromorphic functions that share a polynomial with their difference operators

In this paper, we prove the following result: Let f be a nonconstant meromorphic function of finite order, p be a nonconstant polynomial, andc be a nonzero constant. If f, Delta _{c}f, and Delta_{c}^{n}f (nge 2) share ∞ and p CM, then fequiv Delta_{c}f. Our result provides a difference analogue of the result of Chang and Fang in 2004 (Complex Var. Theory Appl. 49(12):871–895, 2004).

Current interactions from the one-form sector of nonlinear higher-spin equations

The form of higher-spin current interactions in the sector of one-forms is derived from the nonlinear higher-spin equations in AdS4. Quadratic corrections to higher-spin equations are shown to be independent of the phase of the parameter η=exp⁡iφ in the full nonlinear higher-spin equations. The current deformation resulting from the nonlinear higher-spin equations is represented in the canonical form with the minimal number of space-time derivatives. The non-zero spin-dependent coupling constants of the resulting currents are determined in terms of the higher-spin coupling constant ηη¯. Our results confirm the conjecture that (anti-)self-dual nonlinear higher-spin equations result from the full system at (η=0) η¯=0.

On entire solutions of some type of nonlinear difference equations

In this article, the existence of finite order entire solutions of nonlinear difference equations fn+Pd(z,f)=p1eα1z+p2eα2z are studied, where n ≥ 2 is an integer, Pd(z, f) is a difference polynomial in f of degree d(≤ n − 2), p1, p2 are small meromorphic functions of ez, and α1, α2 are nonzero constants. Some necessary conditions are given to guarantee that the above equation has an entire solution of finite order. As its applications, we also find some type of nonlinear difference equations having no finite order entire solutions.

A class of new exact solutions of the system of PDE for the plane motion of viscous incompressible fluids in the presence of body force

The purpose of this paper is to indicate a class of exact solutions of the system of partial differential equations governing the steady, plane motion of incompressible fluid of variable viscosity with body force term to the right-hand side of Navier-Stokes equations. The class consists of the stream function characterized by the equation  in polar coordinates  and  where  and  are continuously differentiable functions and the function  is such that  where a non-zero constant is  and overhead prime represents derivative with respect to . When  or  we show exact solutions for given one component of the body force for both the cases when the function  is arbitrary and when it is not. For the arbitrary function case,  appears in the coefficient of a linear second order ordinary differential equation showing a large numbers of solutions of this equation. This in turn establishes an infinite set of exact solutions to the problem concerned however; we show three examples of such exact solutions. The alternate case fixes  and provides viscosity as derivative of temperature function for  and . Anyhow, we find an infinite set of streamlines, the velocity components, viscosity function, generalized energy function and temperature distribution.

On the Formation of Shock for Quasilinear Wave Equations with Weak Intensity Pulse

In this paper we continue to study the shock formation for the 3-dimensional quasilinear wave equation with $$G''(0)$$ being a non-zero constant. Since ( $$\star $$ ) admits global-in-time solution with small initial data, to present shock formation, we consider a class of large data. Moreover, no symmetric assumption is imposed on the data. Compared to our previous work (Miao and Yu in Invent Math 207(2):697–831, 2017), here we pose data on the hypersurface $$\{(t,x)|t=-\,r_{0}\}$$ instead of $$\{(t,x)|t=-\,2\}$$ , with $$r_{0}$$ being arbitrarily large. We prove an a priori energy estimate independent of $$r_{0}$$ . Therefore a complete description of the solution behavior as $$r_{0}\rightarrow \infty $$ is obtained. This allows us to relax the restriction on the profile of initial data which still guarantees shock formation. Since ( $$\star $$ ) can be viewed as a model equation for describing the propagation of electromagnetic waves in nonlinear dielectric, the result in this paper reveals the possibility to use wave pulse with weak intensity to form electromagnetic shocks in laboratory. A main new feature in the proof is that all estimates in the present paper do not depend on the parameter $$r_{0}$$ , which requires different methods to obtain energy estimates. As a byproduct, we prove the existence of semi-global-in-time solutions which lead to shock formation by showing that the limits of the initial energies exist as $$r_{0}\rightarrow \infty $$ . The proof combines the ideas in Christodoulou (in: EMS monographs in mathematics, European Mathematical Society (EMS), Zurich, 2007) where the formation of shocks for 3-dimensional relativistic compressible Euler equations with small initial data is established, and the short pulse method introduced in Christodoulou (in: EMS monographs in mathematics, European Mathematical Society (EMS), Zurich, 2009) and generalized in Klainerman and Rodnianski (Acta Math 208(2):211–333, 2012), where the formation of black holes in general relativity is proved.

Impedance-based temperature measurement method for organic light-emitting diodes (OLEDs)

This short communication presents a method to measure the integral temperature of organic light-emitting diodes (OLEDs). Based on electrochemical impedance measurements at OLEDs, a non-zero intercept frequency (NZIF) can be determined which is related to the OLED temperature. The NZIF is defined as the frequency at which the imaginary part of the impedance is equal to a predefined (non-zero) constant. The advantage of using an impedance-based temperature indication method through an NZIF is that no hardware temperature sensors are required and that temperature measurements can be performed relatively fast. An experimental analysis reveals that the NZIF is clearly temperature dependent and, moreover, also DC current dependent. Since the NZIF can readily be measured this impedance-based temperature indication method is therefore simple and convenient for many applications using OLEDs and offers an alternative for traditional temperature sensing.

A Novel Self-Interference Cancellation Scheme for Channel-Unaware Differential Space-Time Two-Way Relay Networks

This paper considers channel-unaware two-way relay networks in which two single-antenna nodes exchange information via multiple non-regenerative relays, each with multiple antennas. A novel self-interference cancellation scheme for distributed differential space-time signalling is developed. Despite the absence of channel-state information, this scheme enables self-interference to be completely eliminated, thereby maximizing the signal-to-interference-plus-noise-ratio of the nodes. First, we obtain a lower bound on the pairwise error probability (PEP) under residual self-interference and we show that this bound approaches a non-zero constant at high signal-to-noise ratios (SNRs), indicating a zero diversity order and an asymptotic error floor. Second, we derive a necessary and sufficient condition for the proposed scheme to eliminate self-interference perfectly. Proper operation of this scheme requires the relays to have an even number of active antennas and for relays with odd number of active antennas, such a scheme does not exist. Third, we show that, when self-interference is cancelled perfectly, the error floor vanishes and an upper bound on the PEP approaches zero at high SNRs. In this case, it is shown that the diversity gain is equal to the number of relays and is independent of the number of antennas per relay. Finally, it is shown that the coding gain increases with increasing the number of antennas per relay and converges to a constant as the number of relay antennas becomes large.

Global Well-Posedness of the Incompressible Magnetohydrodynamics

This paper studies the Cauchy problem of the incompressible magnetohydro dynamic systems with or without viscosity ν. Under the assumption that the initial velocity field and the displacement of the initialmagnetic field froma non-zero constant are sufficiently small in certain weighted Sobolev spaces, the Cauchy problem is shown to be globally well-posed for all ν ≧ 0 and all spaces with dimension n ≧ 2. Such a result holds true uniformly in nonnegative viscosity parameters. The proof is based on the inherent strong null structure of the systems introduced by Lei (Commun Pure Appl Math 69(11):2072–2106, 2016) and the ghost weight technique introduced by Alinhac (Invent Math 145(3):597–618, 2001).

A generalized Cox-Ingersoll-Ross equation with growing initial conditions

In this paper we solve the problem of the existence and strong continuity of the semigroup associated with the initial value problem for a generalized Cox-Ingersoll-Ross equation for the price of a zero-coupon bond (see [ 8 ]), on spaces of continuous functions on \begin{document}$ [0, \infty) $\end{document} which can grow at infinity. We focus on the Banach spaces \begin{document}$ Y_{s} = \left\{f\in C[0,\infty): \dfrac{f(x)}{1+x^{s}}\in C_0[0,\infty)\right\},\qquad s\ge 1, $\end{document} which contain the nonzero constants very common as initial data in the Cauchy problems coming from financial models. In addition, a Feynman-Kac type formula is given.