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Can uncorrelated surrounding sound sources be used to generate extended diffuse sound fields? By definition, targets are a constant sound pressure level, a vanishing active sound intensity, and uncorrelated sound waves arriving isotropically from all directions. Are there ideal source layouts to synthesize a maximum diffuse sound field within? As methods, we employ numeric simulations and undertake a series of considerations based on uncorrelated source layouts at a finite radius. Statistically expected active sound intensity and sound energy density are insightful and highlight the relation of active sound intensity to potential theory. Correspondingly, both Gauß’ divergence and Newton’s spherical shell theorem apply, and they provide valuable insights. In a circular layout, uncorrelated elementary point-source fields decaying by 1/√r ideally compose an extended sound field of vanishing active sound intensity; in spherical layouts this is the case with a 1/r decay. None of the layouts synthesizes a perfectly constant sound energy density inside. Theory and simulation offer a broad basis for understanding the synthesis of diffuse sound fields with uncorrelated sources in the free sound field.

We prove that a parametric Lipschitz surface of codimension 1 in a smooth manifold induces a boundary in the sense of currents (roughly speaking, surrounds a “domain” with an eventual multiplicity and together with it forms a pair for the Stokes theorem) if and only if it passes a test in terms of crossing the surface by “almost all” curves. We use the AM-modulus recently introduced in [22] to measure the exceptional family of curves.

Energy is an important material that promotes social development and maintains human life. Clean energy has become a scientific research theme of open world which appeal to many scientists to research it. This paper inspired by the experience of sunflower growth. So we proposed a mathematical model for automatic adjustment of the maximum luminous flux for a limited solar panel, which can help to adjust the best light position. The physical model added to the light detector in the practical application, real-time comparison of the difference in illumination at different positions to adjust the direction of solar panel, thereby enhancing the utilization of light energy. In addition, in order to save space and prevent severe weather attacks, we designed a autofolding circle disc and retractable rectangular solar panels. The intelligent system judgement and decision making based on the electrical signals fed back by the wind sensor or the vibration sensor. When the wind speed reaches the threshold, it can be folded or contracted automatically, reducing manual maintenance. The fully automatic solar panels are designed in this paper, the research results show that theis model has a high utilization, strong sensitivity, safety and durability, and easy control. This is also a process of theoretical exploration of interdisciplinary services for the energy system.

In the current article, we present an efficient and accurate numerical method for the integration of the system matrices in fictitious domain approaches such as the finite cell method (FCM). In the framework of the FCM, the physical domain is embedded in a geometrically larger domain of simple shape which is discretized using a regular Cartesian grid of cells. Therefore, a spacetree-based adaptive quadrature technique is normally deployed to resolve the geometry of the structure. Depending on the complexity of the structure under investigation this method accounts for most of the computational effort. To reduce the computational costs for computing the system matrices an efficient quadrature scheme based on the divergence theorem (Gauß–Ostrogradsky theorem) is proposed. Using this theorem the dimension of the integral is reduced by one, i.e. instead of solving the integral for the whole domain only its contour needs to be considered. In the current paper, we present the general principles of the integration method and its implementation. The results to several two-dimensional benchmark problems highlight its properties. The efficiency of the proposed method is compared to conventional spacetree-based integration techniques.

In this paper, by the use of the divergence theorem, we establish some integral inequalities of Hermite–Hadamard type for convex functions of several variables defined on closed and bounded convex bodies in the Euclidean space for any

We study complete manifolds satisfying a weighted Poincaré type property. We establish a splitting and vanishing theorem for L2L^2 harmonic forms provided that the weight function ρ\\rho is of exponential growth of the distance function. Our theory generalizes the results of Li-Wang, Lam and Chen-Sung.

We give an elementary proof of the Jordan Brouwer separation theorem for smooth hypersurfaces using the divergence theorem and the inverse function theorem.

The objective of this work is to study the electrostatic response of materials accounting for boundary surfaces with their own (electrostatic) constitutive behaviour. The electric response of materials with (electrostatic) energetic boundary surfaces (surfaces that possess material properties and constitutive structures different from those of the bulk) is formulated in a consistent manner using a variational framework. The forces and moments that appear due to bulk and surface electric fields are also expressed in a consistent manner. The theory is accompanied by numerical examples on porous materials using the finite-element method, where the influence of the surface electric permittivity on the electric displacement, the polarization stress and the Maxwell stress is examined.

In this paper, new forms of Maxwell’s equations in vector and scalar variants are presented. The new forms are based on the use of Gauss’s theorem for magnetic induction and electrical induction. The equations are formulated in both differential and integral forms. In particular, the new forms of the equations relate to the non-stationary expressions and their integral identities. The indicated methodology enables a thorough analysis of non-stationary boundary conditions on the behavior of electromagnetic fields in multiple continuous regions. It can be used both for qualitative analysis and in numerical methods (control volume method) and optimization. The last Section introduces an application to equations of magnetic fluid in both differential and integral forms.

In this paper, a sharp form of the Moser-Trudinger inequality is established on a compact Riemannian surface via the method of blow-up analysis, and the existence of an extremal function for such an inequality is proved.