Steady flows of Jeffrey–Hamel type from the half-plane into an infinite channel. 2. Linearization on a symmetric solution

The steady flow of a Navier–Stokes fluid is analysed in a two-dimensional asymmetric unbounded domain $\text{Ω}$, having two outlets to infinity, namely a half-plane K and a semi-infinite channel Π₋. Assuming that $\text{Ω}$ differs from a symmetric domain $\text{Ω}$ only by a small perturbation, we show the existence of a unique solution to the Navier–Stokes system in $\text{Ω}$. The solution is obtained as a perturbation of the symmetric solution and, at large distances in K, it takes the Jeffrey–Hamel form. Curiously, our results are valid only if the flux Φ, besides being small, is directed from the half-plane towards the semi-infinite channel, i.e. Φ is negative.The main ingredients in our proofs are estimates in weighted spaces with detached asymptotics and the study of a model problem resulting from the linearization around the symmetric solution which, for non-zero flux, leads, in contrast to the linearization around the zero solution, to the absence of compatibility conditions for the convective term and, for Φ<0, to the domination of nonlinear terms by the linear ones. We also provide some explicit examples of the domain perturbation.     

Noncommutative geometry and conformal geometry. III. Vafa–Witten inequality and Poincaré duality

This paper is the third part of a series of papers whose aim is to use the framework of twisted spectral triples to study conformal geometry from a noncommutative geometric viewpoint. In this paper we reformulate the inequality of Vafa–Witten [42] in the setting of twisted spectral triples. This involves a notion of Poincaré duality for twisted spectral triples. Our main results have various consequences. In particular, we obtain a version in conformal geometry of the original inequality of Vafa–Witten, in the sense of an explicit control of the Vafa–Witten bound under conformal changes of metrics. This result has several noncommutative manifestations for conformal deformations of ordinary spectral triples, spectral triples associated with conformal weights on noncommutative tori, and spectral triples associated with duals of torsion-free discrete cocompact subgroups satisfying the Baum–Connes conjecture.     

Quantum Fractals on <Emphasis Type="Italic">n</Emphasis>–Spheres. Clifford Algebra Approach

Using the Clifford algebra formalism we extend the quantum jumps algorithm of the Event Enhanced Quantum Theory (EEQT) to convex state figures other than those stemming from convex hulls of complex projective spaces that form the basis for the standard quantum theory. We study quantum jumps on n-dimensional spheres, jumps that are induced by symmetric configurations of non-commuting state monitoring detectors. The detectors cause quantum jumps via geometrically induced conformal maps (M?bius transformations) and realize iterated function systems (IFS) with fractal attractors located on n-dimensional spheres. We also extend the formalism to mixed states, represented by “density matrices” in the standard formalism, (the n-balls), but such an extension does not lead to new results, as there is a natural mechanism of purification of states. As a numerical illustration we study quantum fractals on the circle (one-dimensional sphere and pentagon), two–sphere (octahedron), and on three-dimensional sphere (hypercubetesseract, 24 cell, 600 cell, and 120 cell). The attractor, and the invariant measure on the attractor, are approximated by the powers of the Markov operator. In the appendices we calculate the Radon-Nikodym derivative of the SO(n + 1) invariant measure on Sⁿ under SO(1, n + 1) transformations and discuss the Hamilton’s “icossian calculus” as well as its application to quaternionic realization of the binary icosahedral group that is at the basis of the 600 cell and its dual, the 120 cell. As a by-product of this work we obtain several Clifford algebraic results, such as a characterization of positive elements in a Clifford algebra as generalized Lorentz “spin–boosts”, and their action as M?bius transformation on n-sphere, and a decomposition of any element of Spin⁺(1, n + 1) into a spin–boost and a spin–rotation, including the explicit formula for the pullback of the SO(n + 1) invariant Riemannian metric with respect to the associated M?bius transformation.     

Corrigendum to “Gaussian bounds for degenerate parabolic equations” [J. Funct. Anal. 255 (2) (2008) 283–312]

We rectify an error in the proof of the Gaussian estimates for the heat kernel associated to certain weighted elliptic equations.     

Minimal submanifolds in ℝ<Superscript>4</Superscript> with a G.C.K. structure

In this paper we obtain all invariant, anti-invariant and CR submanifolds in (?⁴, g, J) endowed with a globally conformal Kähler structure which are minimal and tangent or normal to the Lee vector field of the g.c.K. structure.     

A translation of the T. Levi-Civita paper “Interpretazione Gruppale degli Integrali di un Sistema Canonico” (Rend. Acc. Lincei, 1899, s. 3<Superscript>a</Superscript>, vol. VII,pp. 235–238)

In this paper we provide a translation of a paper by T. Levi-Civita, published in 1899, about the correspondence between symmetries and conservation laws for Hamilton’s equations. We discuss the results of this paper and their relationship with the more general classical results by E. Noether.     

A new symmetric interior penalty discontinuous Galerkin formulation for the Serre–Green–Naghdi equations

In this work, we investigate the construction of a new discontinuous Galerkin discrete formulation to approximate the solution of Serre–Green–Naghdi (SGN) equations in the one-dimensional horizontal framework. Such equations describe the time evolution of shallow water free surface flows in the fully nonlinear and weakly dispersive asymptotic approximation regime. A new non-conforming discrete formulation belonging to the family of symmetric interior penalty discontinuous Galerkin methods is introduced to accurately approximate the solutions of the second order elliptic operator occurring in the SGN equations. We show that the corresponding discrete bilinear form enjoys some consistency and coercivity properties, thus ensuring that the corresponding discrete problem is well-posed. The resulting global discrete formulation is then validated through an extended set of benchmarks, including convergence studies and comparisons with data taken from experiments.     

Corrigendum to “The envelope theorem for multiobjective convex programming via contingent derivatives” [J. Math. Anal. Appl. 372 (1) (2010) 197–207]

The aim of this note is to formulate an envelope theorem for vector convex programs. This version corrects an earlier work, “The envelope theorem for multiobjective convex programming via contingent derivatives” by Jiménez Guerra et al. (2010) [3]. We first propose a necessary and sufficient condition allowing to restate the main result proved in the alluded paper. Second, we introduce a new Lagrange multiplier in order to obtain an envelope theorem avoiding the aforementioned error.