A theorem of Roe and Strichartz on homogeneous trees

Let 𝔛 {\mathfrak{X}} be a homogeneous tree and let ℒ {\mathcal{L}} be the Laplace operator on 𝔛 {\mathfrak{X}} . In this paper, we address problems of the following form: Suppose that { f k } k ∈ ℤ {\{f_{k}\}_{k\in\mathbb{Z}}} is a doubly infinite sequence of functions in 𝔛 {\mathfrak{X}} such that for all k ∈ ℤ {k\in\mathbb{Z}} one has ℒ ⁢ f k = A ⁢ f k + 1 {\mathcal{L}f_{k}=Af_{k+1}} and ∥ f k ∥ ≤ M {\lVert f_{k}\rVert\leq M} for some constants A ∈ ℂ {A\in\mathbb{C}} , M > 0 {M>0} and a suitable norm ∥ ⋅ ∥ {\lVert\,\cdot\,\rVert} . From this hypothesis, we try to infer that f 0 {f_{0}} , and hence every f k {f_{k}} , is an eigenfunction of ℒ {\mathcal{L}} . Moreover, we express f 0 {f_{0}} as the Poisson transform of functions defined on the boundary of 𝔛 {\mathfrak{X}} .

Parameter estimation and signal reconstruction

In frequency analysis an often appearing problem is the reconstruction of a signal from given samples. Since the samples are usually noised, pure interpolating approaches are not recommended and appropriate approximation methods are more suitable as they can be interpreted as a kind of denoising. Two approaches are widely used. One uses the reflection coefficients of a finite sequence of Szegő polynomials and the other one the zeros of the so called Prony polynomial. We show that both approaches are closely related. As a kind of inverse problem, it’s not surprising that they have in common that both methods depend very sensitive on sampling errors. We use known properties of the signal to estimate the positions of the zeros of the corresponding Szegő- or Prony-like polynomials and construct adaptive algorithms to calculate these ones. Hereby, we get the corresponding parameters in the exponential parts of the signal, too. Then, the coefficients of the signal (as a linear combination of such exponential functions) can be obtained from a system of linear equations by minimizing the residuals with respect to a suitable norm as a kind of denoising.

On the Boussinesq Equations with Non-monotone Temperature Profiles

In this article, we consider the asymptotic stability of the two-dimensional Boussinesq equations with partial dissipation near a combination of Couette flow and temperature profiles T(y). As a first main result, we show that if $$T'$$ T ′ is of size at most $$\nu ^{1/3}$$ ν 1 / 3 in a suitable norm, then the linearized Boussinesq equations with only vertical dissipation of the velocity but not of the temperature are stable. Thus, mixing enhanced dissipation can suppress Rayleigh–Bénard instability in this linearized case. We further show that these results extend to the (forced) nonlinear equations with vertical dissipation in both temperature and velocity.

Scale Invariant Effective Hamiltonians for a Graph with a Small Compact Core

We consider a compact metric graph of size ε and attach to it several edges (leads) of length of order one (or of infinite length). As ε goes to zero, the graph G ε obtained in this way looks like the star-graph formed by the leads joined in a central vertex. On G ε we define an Hamiltonian H ε , properly scaled with the parameter ε . We prove that there exists a scale invariant effective Hamiltonian on the star-graph that approximates H ε (in a suitable norm resolvent sense) as ε → 0 . The effective Hamiltonian depends on the spectral properties of an auxiliary ε -independent Hamiltonian defined on the compact graph obtained by setting ε = 1 . If zero is not an eigenvalue of the auxiliary Hamiltonian, in the limit ε → 0 , the leads are decoupled.

Explicit exponential stabilization of nonautonomous linear parabolic-like systems by a finite number of internal actuators

An explicit feedback controller is proposed for stabilization of linear parabolic equations, with a time-dependent reaction–convection operator. The range of the feedback controller is finite-dimensional, and is typically modeled by indicator functions of small subdomains. Its dimension depends polynomially on a suitable norm of the reaction–convection operator. A sufficient condition for stabilizability is given, which involves the asymptotic behavior of the eigenvalues of the (time-independent) diffusion operator, the norm of the reaction–convection operator, and the norm of the nonorthogonal projection onto the controller’s range along a suitable infinite-dimensional (higher-modes) eigenspace. To construct the explicit feedback, the essential step consists in computing the nonorthogonal projection. Numerical simulations are presented, in 1D and 2D, showing the practicability of the controller and its response to measurement errors, where the actuators are indicator functions of suitable small subsets.

The Chicken Fallacy and the Ethics of Cruelty to Non-Human Animals

The ideological underpinning that guides our interaction with non-human animals needs revision. The traditional outlook, according to which humans have a higher moral status vis-à-vis non-human animals, is now otiose. If these claims are to be justified, what ideological framework would serve this end? What are the moral implications of endorsing the view that humans possess no higher moral status than non-human animals? This work takes as foundation Charles Darwin’s theory of evolution, which affirms that humans emerged from the long chain of evolutionary history, where non-human animals have been the carriers of the genes that shaped humans. A revisit to the discourse on the moral implication of humans’ cruelty to their ancestors and neighbours becomes pertinent. This essay goes against the mainstream and dominant perspective that non-human animals exist to serve human ends and as such can be treated with disdain. The thesis of this paper goes beyond Peter Singer’s submission that sentience is the basis for conferring moral worth on non-human animals. It affirms that in addition to sentience, good neighbourliness is a factor in determining the moral worth of non-human animals. It submits that cruelty to reared and domesticated animals may produce violent and wild species of these animals’ kind in a future evolution, thereby endangering the lives of future human generation, through negative alteration of genes. In the end, this paper proposes the principle of biological altruism as a suitable norm for determining the moral worth of non-human animals.

Analyticity for a class of non-linear evolutionary pseudo-differential equations

We study the analyticity properties of solutions for a class of non-linear evolutionary pseudo-differential equations possessing global attractors. In order to do this we utilise an analyticity criterion for spatially periodic functions, which involves the rate of growth of a suitable norm of the nth derivative of the solution, with respect to the spatial variable, as n tends to infinity. This criterion can be used to a wide class of dissipative-dispersive partial differential equations, provided they possess global attractors. Using this criterion and the spectral method developed in Akrivis et al. [1] we have improved previous results.

Penyusunan Kriteria Domestikasi Dan Evaluasi Praktek Pengasuhan Gajah: Studi Di Taman Nasional Way Kambas Kabupaten Lampung Timur

Sumatran elephants (Elephas maximus sumatranus) is one of the many wild animals been domesticated, but so far there is no criteria for that propose. This research is a prelimenary effort in providing those criteria. This research began in April to May 2013. The first step was to establish the norm of Suitable range of as many as 17 variables required for upholding the elephant life domestication area. The Suitable norm range we determined through an exhausted study of previous researchs or literature available. The next step was to determined the Less Suitable and the Not Suitable norms range for those 17 variables employed, e.i. by deviating by 25% and by 50% from the Suitable norm range respectively. The second step, we employed a Delphi Method in order to validate and refinary the three catagorical norms range of the 17 varibles concerned by mean of asking to some elaphant experts in our country. We, then applied the deemed criteria to evaluate a practice of ephants management at The Sumatra Elephant Conservation Center (PKG) and the Elephant Response Unit (ERU), Way Kambas National Park. The research results: (1) We obtained the prelimenary criteria as a rubric for Sumatran elephant domestication, and (2) The elephant nurturing practices at PKG and ERU were Suitable, except for the variables of population density, home range and elephant shading.

Symbolic and Numeric Computation of Optimal Initial Velocity in a Wave Equation

We get symbolic and numeric solutions developing a MAPLE® program which uses the initial velocity on the state variable of a wave equation as control function. Solution of this problem implies the minimization at the final time of the distance measured in a suitable norm between the solution of the problem and a given target. An iterative algorithm is constructed to compute the required optimal control as the limit of a suitable subsequence of controls. Results are tested with some numerical examples.