Global Poincaré Inequalities on the Heisenberg Group and Applications

Let f be in the localized nonisotropic Sobolev space on the n-dimensional Heisenberg group ℍ ⁿ = ℂ ⁿ × ℝ, where 1 = p < Q and Q = 2n + 2 is the homogeneous dimension of ℍn. Suppose that the subelliptic gradient is gloablly L ^p integrable, i.e., is finite. We prove a Poincaré inequality for f on the entire space ℍ ⁿ . Using this inequality we prove that the function f subtracting a certain constant is in the nonisotropic Sobolev space formed by the completion of under the norm of

A Remark on the Steklov–Poincaré Inequality

Mathematical Notes - In an $$n$$ -dimensional bounded domain $$\Omega_n$$ , $$n\ge 2$$ , we prove the Steklov–Poincaré inequality with the best constant in the case where $$\Omega_n$$ is...     

A Sawyer-type sufficient condition for the weighted Poincaré inequality

In this paper we prove the Poincaré-type weighted inequality

$$\begin{aligned} \Vert v^{1/q} f \Vert _{L^q(\Omega )} \le C \Vert \omega ^{1/p} \nabla f \Vert _{L^p(\Omega )}, \quad q\ge p>1, \end{aligned}$$

for a locally Lipschitz function f with a weighted mean equal to zero over a convex bounded domain \(\Omega \); here the weights v, \(\omega \) are positive measurable functions which satisfy a certain compatibility condition. This result is a generalization of the well-known weighted Poincaré inequality to the case of more general weights in the sense that we do not use the traditional conditions of high summability \(v,\, \omega ^{-\frac{1}{p-1}}\in L^{r,loc}\) with \(r>1\) for \(q=p\) or the reverse doubling condition on the function v for \(q>p\) . In other words, a Sawyer type sufficient condition on weight functions is established.

     

On the Uniform Poincaré Inequality

We give a proof of the Poincaré inequality in W ^{1, p} (Ω) with a constant that is independent of Ω ? , where  is a set of uniformly bounded and uniformly Lipschitz domains in ? ⁿ . As a byproduct, we obtain the following: The first non vanishing eigenvalues λ₂(Ω) of the standard Neumann (variational) boundary value problem on Ω for the Laplace operator are bounded below by a positive constant if the domains Ω vary and remain uniformly bounded and uniformly Lipschitz regular.     

Poincaré transformations in nonholonomic mechanics

We report on the application of the Poincaré transformation (from the theory of adaptive geometric integrators) to nonholonomic systems—mechanical systems with non-integrable velocity constraints. We prove that this transformation can be used to express the dynamics of certain nonholonomic systems at a fixed energy value in Hamiltonian form; examples and potential applications are also discussed.     

Maxwell Spaces with Zero Current that Admit Three-Dimensional Subgroups of the Poincaré Group

We describe classes of Maxwell spaces with zero current (in particular, electromagnetic waves) that admit three-dimensional subgroups of the Poincar´e group and find representatives of these classes.     

A Fractional Orlicz–Sobolev Poincaré Inequality in John Domains

Let n ≥ 2, β∈(0, n) and ■ Rnbe a bounded domain. Support that φ : [0, ∞) → [0, ∞)is a Young function which is doubling and satisfies ■If Ω is a John domain, then we show that it supports a(φ~(n/(n-β)), φ)β-Poincaré inequality. Conversely,assume that Ω is simply connected domain when n = 2 or a bounded domain which is quasiconformally equivalent to some uniform domain when n ≥ 3. If Ω supports a((φ~(n/(n-β)), φ)β-Poincaré inequality,then we show that it is a John domain.     

A Comparison of Preconditioners for the Steklov–Poincaré Formulation of the Fluid-Structure Coupling in Hemodynamics

A Fluid–Structure Interaction (FSI) problem can be reinterpreted as a heterogeneous problem with two subdomains. It is possible to describe the coupled problem at the interface between the fluid and the structure, yielding a nonlinear Steklov–Poincaré problem. The linear system can be linearized by Newton iterations on the interface and the resulting linear problem can be solved by the preconditioned GMRES method. In this work we investigate the behavior of preconditioners of Neumann–Neumann and Dirichlet–Neumann type. We find that, in the context of hemodynamics, the Dirichlet– Neumann, i.e., using Dirichlet boundary conditions on the fluid side and Neumann on the structure side, outperforms the Neumann–Neumann method, except when a weighting is used such that it basically reduces to the Dirichlet–Neumann method. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Poincaré series

Let N_α denote the number of solutions to the congruence F(x_i,..., x_m) ≡ 0 (mod p^α) for a polynomial F(x_i,..., x_m) with integral p-adic coefficients. We examine the series \(\varphi (t) = \sum\nolimits_{\alpha = 0}^\infty {N_{\alpha ^{t^\alpha } } } \) . called the Poincaré series for the polynomial F. In this work we prove the rationality of the series ?(t) for a class of isometrically equivalent polynomials of m variables, m ≥ 2, containing the sum of two forms ?_n(x, y) + ?_n+1(x, y) respectively of degrees n and n+1, n ≥ 2. In particular the Poincaré series for any third degree polynomial F₃(x, y) (over the set of unknowns) with integral p-adic coefficients is a rational function of t.     

On Approximate Summation of Poincaré Series in the Schottky Model

Doklady Mathematics - For approximate summation of Poincaré theta series in the Schottky model of real hyperelliptic curves, modifications of the Bogatyrev and Schmies algorithms are proposed...     

Strong Poincaré recurrence theorem in MV-algebras

The classical Poincaré strong recurrence theorem states that for any probability space (Ω, ℒ, P), any P-measure preserving transformation T, and any A ∈ ℒ, almost all points of A return to A infinitely many times. In the present paper the Poincaré theorem is proved when the σ-algebra ℒ is substituted by an MV-algebra of a special type. Another approach is used in [RIEČAN, B.: Poincaré recurrence theorem in MV-algebras. In: Proc. IFSA-EUSFLAT 2009 (To appear)], where the weak variant of the theorem is proved, of course, for arbitrary MV-algebras. Such generalizations were already done in the literature, e.g. for quantum logic, see [DVUREČENSKIJ, A.: On some properties of transformations of a logic, Math. Slovaca 26 (1976), 131–137.     

Poincaré duality isomorphisms in tensor categories

If for a vector space V of dimension g over a characteristic zero field we denote by ${\wedge }^{i}V$ its alternating powers, and by $V^{\vee }$ its linear dual, then there are natural Poincaré isomorphisms:

${\wedge }^i{V}^{\vee }?{\wedge }^{g?i}V.$

We describe an analogous result for objects in rigid pseudo-abelian $\mathbb{Q}$-linear ACU tensor categories.     

Poincaré type inequalities on the discrete cube and in the CAR algebra

We prove L ^p Poincaré inequalities with suitable dimension free constants for functions on the discrete cube {?1, 1} ⁿ . As well known, such inequalities for p an even integer allow to recover an exponential inequality hence the concentration phenomenon first obtained by Bobkov and Götze. We also get inequalities between the L ^p norms of $ \left\vert \nabla f\right\vert We prove L ^p Poincaré inequalities with suitable dimension free constants for functions on the discrete cube {−1, 1} ⁿ . As well known, such inequalities for p an even integer allow to recover an exponential inequality hence the concentration phenomenon first obtained by Bobkov and G?tze. We also get inequalities between the L ^p norms of and moreover L ^p spaces may be replaced by more general ones. Similar results hold true, replacing functions on the cube by matrices in the *-algebra spanned by n fermions and the L ^p norm by the Schatten norm C _p .     

A Generalization of Poincaré and Log-Sobolev Inequalities

A generalized Beckner-type inequality interpolating the Poincaré and the log-Sobolev inequalities is studied. This inequality possesses the additivity property and characterizes certain exponential convergence of the corresponding Markov semi-group. A correspondence between this inequality and the so-called F-Sobolev inequality is presented, with the known criteria of the latter applying also to the former. In particular, an important result of Lataa and Oleszkiewicz is generalized.     

A problem of Kreck on Poincaré manifolds

An 1-connected closed manifold M is called a Poincaré manifold if any other 1-connected closed manifold with the same homology as that of M is homeomorphic to M. In the metastable range 3pq<2p–3 we answer the question raised by Kreck : for which p and q is the product of two spheres S ^p ×S ^q a Poincaré manifold ?. The second named author is partially supported by the 973 program of China.     

On the homotopy type of Poincaré spaces

We study the homotopy type of finite-oriented Poincaré spaces (and, in particular, of closed topological manifolds) in even dimension. Our results relate polarized homotopy types over a stage of the Postnikov tower with the concept of CW-tower of categories due to Baues. This fact allows us to obtain a new formula for the top-dimensional obstruction for extending maps to homotopy equivalences. Then we complete the paper with an algebraic characterization of high-dimensional handlebodies. Received: April 14, 1999?Published online: October 2, 2001