Inhomogeneous Constraints Research Articles

On the maximizing problem associated with critical Sobolev inequality under inhomogeneous constraints

In this paper, we study a maximizing problem associated with the critical Sobolev inequality under inhomogeneous constraints. The problem of this type was previously studied by Ishiwata and Wadade [On the effect of equivalent constrain on a maximizing problem associated with the Sobolev type embedding in . Math Ann. 2016;364:1043–1068; On the maximizing problem associated with Sobolev type embeddings under inhomogeneous constraints. Appl Anal. 2019;98(10):1916–1934], Ishiwata [On variational problems associated with Sobolev type inequalities and related topics. Available from: http://www.rism.it/doc/Ishiwata.pdf] and Nguyen [Maximizers for the variational problems associated with Sobolev type inequalities under constraints. Math Ann. 2018;372(1–2):229–255]. Our results give a complete picture of the effect of the constraints on the attainability and non-attainability of the problem. The sharp Sobolev inequality plays a crucial role in our argument. Our method also provides a new and simple proof for the recent results of Ishiwata and Wadade [On the maximizing problem associated with Sobolev type embeddings under inhomogeneous constraints. Appl Anal. 2019;98(10):1916–1934] concerning the sub-critical Sobolev-type inequalities (or Gagliardo–Nirenberg inequalities).

Equivalence of Subcritical and Critical Adams Inequalities in $W^{m,2}(\mathbb {R}^{2m})$ and Existence and Non-existence of Extremals for Adams Inequalities under Inhomogeneous Constraints

Equivalence of Subcritical and Critical Adams Inequalities in $W^{m,2}(\mathbb {R}^{2m})$ and Existence and Non-existence of Extremals for Adams Inequalities under Inhomogeneous Constraints

On the effect of inhomogeneous constraints for a maximizing problem associated with the Sobolev embedding of the space of functions of bounded variation

We consider a maximizing problem associated with the Sobolev type embedding ${\rm BV} (\Bbb R^N)\hookrightarrow L^r(\Bbb R^N)$ for $1\leq r\leq 1^*:=\frac {N}{N-1}$ with $N\geq 2$. For given $\alpha \gt 0$, set $$ D_\alpha (a,b,q):=\sup _{\substack {u\in

New Designs on MVDR Robust Adaptive Beamforming Based on Optimal Steering Vector Estimation

The robust adaptive beamforming design problem based on estimation of the signal of interest steering vector is considered in the paper. In this case, the optimal beamformer is obtained by computing the sample matrix inverse and an optimal estimate of the signal of interest steering vector. The common criteria to find the best estimate of the steering vector are the beamformer output SINR and output power, while the constraints assume as little as possible prior inaccurate knowledge about the signal of interest, the propagation media, and the antenna array. Herein, a new beamformer output power maximization problem is formulated and solved subject to a double-sided norm perturbation constraint, a similarity constraint, and a quadratic constraint that guarantees that the direction-of-arrival (DOA) of the signal of interest is away from the DOA region of all linear combinations of the interference steering vectors. In the new robust design, the prior information required consists of some allowable error norm bounds, the approximate knowledge of the antenna array geometry, and the angular sector of the signal of interest. It turns out that the array output power maximization problem is a non-convex QCQP problem with inhomogeneous constraints. However, we show that the problem is still solvable, and develop efficient algorithms for finding globally optimal estimate of the signal of interest steering vector. The results are generalized to the case where an ellipsoidal constraint is considered, and sufficient conditions for the global optimality are derived. In addition, a new quadratic constraint on the actual signal steering vector is proposed in order to improve the array performance. To validate our results, simulation examples are presented, and they demonstrate the improved performance of the new robust beamformers in terms of the output SINR as well as the output power.

On the maximizing problem associated with Sobolev-type embeddings under inhomogeneous constraints

ABSTRACTWe consider the attainability of a maximizing problem associated with the Sobolev embedding of the form , where , , and . The existence and non-existence of a maximizer for D are related to the exponents and b. Indeed, for some range of α, we have a non-existence result in spite of the sub-critical setting and the radial nature of the problem. We show that the value is the threshold in terms of the attainability of D. We also consider the attainability of D with and compute the exact value for some range of the exponents.

Existence and non-existence of maximizers for the Moser–Trudinger type inequalities under inhomogeneous constraints

In this paper, we study the existence and non-existence of maximizers for the Moser-Trudinger type inequalities in $\Bbb R^N$ of the form \[ D_{N,\alpha}(a,b):= \sup_{u\in W^{1,N}(\Bbb R^N),\,\|\nabla u\|_{L^N(\Bbb R^N)}^a+\|u\|_{L^N(\Bbb R^N)}^b=1} \int_{\Bbb R^N}\Phi_N\left(\alpha|u|^{N'}\right)dx. \] Here $N\geq 2$, $N'=\frac{N}{N-1}$, $a,b>0$, $\alpha \in (0,\alpha_N]$ and $\Phi_N(t):=e^t-\sum_{j=0}^{N-2}\frac{t^j}{j!}$ where $\alpha_N:= N \omega_{N-1}^{1/(N-1)}$ and $\omega_{N-1}$ denotes the surface area of the unit ball in $\Bbb R^N$. We show the existence of the threshold $\alpha_\ast = \alpha_\ast(a,b,N) \in [0,\alpha_N]$ such that $D_{N,\alpha}(a,b)$ is not attained if $\alpha \in (0,\alpha_\ast)$ and is attained if $ \alpha \in (\alpha_\ast , \alpha_N)$. We also provide the conditions on $(a,b)$ in order that the inequality $\alpha_\ast < \alpha_N$ holds.

On the stabilization problems of non-holonomic systems with inhomogeneous constraints with incomplete information on the state

The systematic application of Voronetz vector-matrix equations in Routh variables is developed to describe the dynamics of non-holonomic systems with inhomogeneous constraints. The obtained form of the equations of disturbed motion in problems of stability and stabilization makes it possible to analyze the structure of the model for a reasonable choice of the application of control actions. The developed technique makes it possible to automate the solution of problems of stabilization of operating modes of complex technical devices.

The inhomogeneous Suslov problem

We consider the Suslov problem of nonholonomic rigid body motion with inhomogeneous constraints. We show that if the direction along which the Suslov constraint is enforced is perpendicular to a principal axis of inertia of the body, then the reduced equations are integrable and, in the generic case, possess a smooth invariant measure. Interestingly, in this generic case, the first integral that permits integration is transcendental and the density of the invariant measure depends on the angular velocities. We also study the Painlev\'e property of the solutions.

Minimal cut sets in a metabolic network are elementary modes in a dual network

Elementary modes (EMs) and minimal cut sets (MCSs) provide important techniques for metabolic network modeling. Whereas EMs describe minimal subnetworks that can function in steady state, MCSs are sets of reactions whose removal will disable certain network functions. Effective algorithms were developed for EM computation while calculation of MCSs is typically addressed by indirect methods requiring the computation of EMs as initial step. In this contribution, we provide a method that determines MCSs directly without calculating the EMs. We introduce a duality framework for metabolic networks where the enumeration of MCSs in the original network is reduced to identifying the EMs in a dual network. As a further extension, we propose a generalization of MCSs in metabolic networks by allowing the combination of inhomogeneous constraints on reaction rates. This framework provides a promising tool to open the concept of EMs and MCSs to a wider class of applications. utz-uwe.haus@math.ethz.ch; klamt@mpi-magdeburg.mpg.de Supplementary data are available at Bioinformatics online.

Enumerating constrained elementary flux vectors of metabolic networks

The concept of elementary vector is generalised to the case where the steady-state space of the metabolic network is not a flux cone but is a general polyhedron due to further inhomogeneous constraints on the flows through some of the reactions. On one hand, this allows to selectively enumerate elementary modes which satisfy certain optimality criteria and this can yield a large computational gain compared with full enumeration. On the other hand, in contrast to the single optimum found by executing a linear program, this enables a comprehensive description of the set of alternate optima often encountered in flux balance analysis. The concepts are illustrated on a metabolic network model of human cardiac mitochondria.

Mobile polymer connectors.

We study the junction between two solid objects, formed by polymer connectors which are supposed to freely explore the two grafting surfaces. We focus on the sphere-plane and sphere-sphere geometries. Because the chains are mobile, they are able to adapt both the local grafting density and their conformation to the position of the surfaces. Using a scaling approach, we show that, in chemical equilibrium, local grafting density is non-monotonic, and the junction can be divided into two different regions: the center, where connectors are compressed and exert a repulsive force between the objects (high excluded-volume interactions) and an external corona, where chains are stretched and attract the two objects. The relative importance of these regions varies for different separations and depends on surface geometries. When an external force is applied, this spatially inhomogeneous constraint exerted by the junction leads to specific force-distance profiles, which differ from profiles in the case of fixed connectors. An effective interaction potential and the adhesion energy are computed, showing a strong dependence on the geometry of surfaces.

A stable and optimal complexity solution method for mixed finite element discretizations

We outline a solution method for mixed finite element discretizations based on dissecting the problem into three separate steps. The first handles the inho- mogeneous constraint, the second solves the flux variable from the homogeneous problem, whereas the third step, adjoint to the first, finally gives the Lagrangian multiplier. We concentrate on aspects involved in the first and third step mainly, and advertise a multi-level method that allows for a stable computation of the intermediate and final quantities in optimal computational complexity.

Substructure analysis with linear constraints using the natural factor formulation

The purpose of this paper is to extend the natural factor substructuring method to include linear, inhomogeneous constraints on the nodal point freedoms. This algorithm, which retains the superior numerical accuracy of the natural factor method, is derived using Lagrange multipliers. A numerical example illustrates the improvement in accuracy over classical methods.