Integral representation of local functionals depending on vector fields

Given an open and bounded set Ω ⊆ ℝ n {\Omega\subseteq\mathbb{R}^{n}} and a family 𝐗 = ( X 1 , … , X m ) {\mathbf{X}=(X_{1},\ldots,X_{m})} of Lipschitz vector fields on Ω, with m ≤ n {m\leq n} , we characterize three classes of local functionals defined on first-order X-Sobolev spaces, which admit an integral representation in terms of X, i.e. F ⁢ ( u , A ) = ∫ A f ⁢ ( x , u ⁢ ( x ) , X ⁢ u ⁢ ( x ) ) ⁢ 𝑑 x , F(u,A)=\int_{A}f(x,u(x),Xu(x))\,dx, with f being a Carathéodory integrand.

On density of compactly supported smooth functions in fractional Sobolev spaces

We describe some sufficient conditions, under which smooth and compactly supported functions are or are not dense in the fractional Sobolev space $$W^{s,p}(\Omega )$$ W s , p ( Ω ) for an open, bounded set $$\Omega \subset \mathbb {R}^{d}$$ Ω ⊂ R d . The density property is closely related to the lower and upper Assouad codimension of the boundary of $$\Omega$$ Ω . We also describe explicitly the closure of $$C_{c}^{\infty }(\Omega )$$ C c ∞ ( Ω ) in $$W^{s,p}(\Omega )$$ W s , p ( Ω ) under some mild assumptions about the geometry of $$\Omega$$ Ω . Finally, we prove a variant of a fractional order Hardy inequality.

On a Parallelised Diffusion Induced Stochastic Algorithm with Pure Random Search Steps for Global Optimisation

We propose a stochastic algorithm for global optimisation of a regular function, possibly unbounded, defined on a bounded set with regular boundary; a function that attains its extremum in the boundary of its domain of definition. The algorithm is determined by a diffusion process that is associated with the function by means of a strictly elliptic operator that ensures an adequate maximum principle. In order to preclude the algorithm to be trapped in a local extremum, we add a pure random search step to the algorithm. We show that an adequate procedure of parallelisation of the algorithm can increase the rate of convergence, thus superseding the main drawback of the addition of the pure random search step.

MENUJU SATU TUBUH DENGAN BERBAGAI ANGGOTA

In this article, I propose an gesture for Indonesian Evangelicals to mediate conflicts that have the potential to arise due to the diversity of the church denominations in Indonesia by using a centered-set approach, not bounded-set approach, by placing five Evangelicals commitments at the center: (1) conversionism, (2) activism, (3) biblicism, (4) cruci-centrism, and (5) transdenominationalism. I argue that these five commitments should become the identity of Indonesian Evangelicals. By confessing these five commitments with a centered-set approach, Indonesian Evangelicals can set aside other and less basic commitments and differences from the various church denominations in Indonesia in order to live and work together. The main purpose of mediating the conflict that could potentially arise due to the diversity of church denominations here is to achieve a vision of one body with multiple members as the church described in the Bible (1 Cor. 12:12-27; Eph. 4:4, 16). To achieve the objectives of this article, I will first describe the context of the diversity of church denominations in Indonesia. Further, I will analyze the identity of Evangelicals in Indonesia and describe a centered-set approach for the attitudes of Indonesian Evangelicals in responding to the diversity of church denominations. Keywords: evangelicals, church denominations, christianity in Indonesia, church unity. Abstrak: Dalam artikel ini, penulis bertujuan untuk mengusulkan suatu sikap bagi kaum Injili Indonesia untuk memediasi konflik yang berpotensi muncul akibat keragaman denominasi gereja di Indonesia dengan menggunakan suatu pendekatan yang terpusat, bukan dengan pendekatan yang membatasi, yaitu dengan menaruh lima komitmen kaum Injili sebagai pusatnya: (1) conversionism, (2) activism, (3) biblicism, (4) cruci-centrism, dan (5) transdenominationalism. Penulis berargumen bahwa lima komitmen inilah yang semestinya menjadi identitas kaum Injili Indonesia. Dengan mengakui lima komitmen tersebut dengan pendekatan yang terpusat, kaum Injili Indonesia dapat menyampingkan komitmen-komitmen dan perbedaan-perbedaan yang lain dan kurang mendasar dari beragam denominasi gereja di Indonesia agar dapat hidup bersama dan bekerja sama. Tujuan utama dari memediasi konflik yang berpotensi muncul akibat keragaman denominasi gereja di sini adalah untuk menuju visi satu tubuh dengan berbagai anggota sebagaimana gereja yang digambarkan di dalam Alkitab (1Kor. 12:12-27; Ef. 4:4, 16). Untuk mencapai tujuan dalam tulisan ini, pertama penulis akan mendeskripsikan konteks keragaman denominasi gereja di Indonesia. Selanjutnya, penulis menganalisis identitas kaum Injili di Indonesia dan menjelaskan pendekatan yang terpusat bagi sikap kaum Injili Indonesia di dalam merespons keragaman denominasi gereja. Kata-kata kunci: kaum Injili, denominasi gereja, kekristenan di Indonesia, kesatuan gereja.

Distinguished Property in Tensor Products and Weak* Dual Spaces

A local convex space E is said to be distinguished if its strong dual Eβ′ has the topology β(E′,(Eβ′)′), i.e., if Eβ′ is barrelled. The distinguished property of the local convex space CpX of real-valued functions on a Tychonoff space X, equipped with the pointwise topology on X, has recently aroused great interest among analysts and Cp-theorists, obtaining very interesting properties and nice characterizations. For instance, it has recently been obtained that a space CpX is distinguished if and only if any function f∈RX belongs to the pointwise closure of a pointwise bounded set in CX. The extensively studied distinguished properties in the injective tensor products CpX⊗εE and in Cp(X,E) contrasts with the few distinguished properties of injective tensor products related to the dual space LpX of CpX endowed with the weak* topology, as well as to the weak* dual of Cp(X,E). To partially fill this gap, some distinguished properties in the injective tensor product space LpX⊗εE are presented and a characterization of the distinguished property of the weak* dual of Cp(X,E) for wide classes of spaces X and E is provided.

Strong 1-boundedness of unimodular orthogonal free quantum groups

Recently, Brannan and Vergnioux showed that the orthogonal free quantum group factors [Formula: see text] have Jung’s strong [Formula: see text]-boundedness property, and hence are not isomorphic to free group factors. We prove an analogous result for the other unimodular case, where the parameter matrix is the standard symplectic matrix in [Formula: see text] dimensions [Formula: see text]. We compute free derivatives of the defining relations by introducing self-adjoint generators through a decomposition of the fundamental representation in terms of Pauli matrices, resulting in [Formula: see text]-boundedness of these generators. Moreover, we prove that under certain conditions, one can add elements to a [Formula: see text]-bounded set without losing [Formula: see text]-boundedness. In particular, this allows us to include the character of the fundamental representation, proving strong [Formula: see text]-boundedness.

Optimization of a k-covering of a bounded set with circles of two given radii

A numerical method for investigating k-coverings of a convex bounded set with circles of two given radii is proposed. Cases with constraints on the distances between the covering circle centers are considered. An algorithm for finding an approximate number of such circles and the arrangement of their centers is described. For certain specific cases, approximate lower bounds of the density of the k-covering of the given domain are found. We use either 0–1 linear programming or general integer linear programming models. Numerical results demonstrating the effectiveness of the proposed methods are presented.

Convexification-based globally convergent numerical method for a 1D coefficient inverse problem with experimental data

<p style='text-indent:20px;'>To compute the spatially distributed dielectric constant from the backscattering computationally simulated ane experimentally collected data, we study a coefficient inverse problem for a 1D hyperbolic equation. To solve this inverse problem, we establish a new version of the Carleman estimate and then employ this estimate to construct a cost functional, which is strictly convex on a convex bounded set of an arbitrary diameter in a Hilbert space. The strict convexity property is rigorously proved. This result is called the convexification theorem and it is the central analytical result of this paper. Minimizing this cost functional by the gradient descent method, we obtain the desired numerical solution to the coefficient inverse problems. We prove that the gradient descent method generates a sequence converging to the minimizer starting from an arbitrary point of that bounded set. We also establish a theorem confirming that the minimizer converges to the true solution as the noise in the measured data and the regularization parameter tend to zero. Unlike the methods, which are based on the optimization, our convexification method converges globally in the sense that it delivers a good approximation of the exact solution without requiring a good initial guess. Results of numerical studies of both computationally simulated and experimentally collected data are presented.</p>

Regularity results for a class of obstacle problems with p, q−growth conditions

In this paper we prove the the local Lipschitz continuity for solutions to a class of obstacle problems of the type min{ ∫ΩF(x, Dz) : z ∈ 𝛫ψ(Ω)}. Here 𝛫ψ(Ω) is the set of admissible functions z ∈ u0 + W1,p(Ω) for a given u0 ∈ W1,p(Ω) such that z ≥ ψ a.e. in Ω, ψ being the obstacle and Ω being an open bounded set of ℝn, n ≥ 2. The main novelty here is that we are assuming that the integrand F(x, Dz) satisfies (p, q)-growth conditions and as a function of the x-variable belongs to a suitable Sobolev class. We remark that the Lipschitz continuity result is obtained under a sharp closeness condition between the growth and the ellipticity exponents. Moreover, we impose less restrictive assumptions on the obstacle with respect to the previous regularity results. Furthermore, assuming the obstacle ψ is locally bounded, we prove the local boundedness of the solutions to a quite large class of variational inequalities whose principal part satisfies non standard growth conditions.

Approximate guaranteed estimates for matrices in linear regression problems with a small parameter

The problem of finding linear unbiased estimates of the linear operator of unknown matrices — components of the observations vector, is investigated. It is assumed that the observation vector additively depends on a random vector with zero expected value, and the unknown correlation matrix belongs to a known bounded set. For the introduced class of linear estimates, necessary and sufficient conditions for the existence of solutions of operator equations that determine the unknown parameters of the vector estimate, are proved. The form of the guaranteed mean square error of the estimate is introduced on the sets of constraints of the problem parameters. The influence on the linear unbiased estimate of small perturbations of known rectangular matrices, which are the composites of the observations vector components, is also investigated. The analytical form is given through the parameters of the perturbed set of singularities for the introduced special operators that depend on a small parameter, which determine the corresponding operator equations, as well as their approximate solutions, in the first approximation of the small parameter method. A test example of solving the problem of finding a linear unbiased estimate under the condition of perturbation of both linearly independent and linearly dependent known observation matrices is presented.