scholarly journals A new proof of the Hardy–Rellich inequality in any dimension

2019 ◽  
Vol 150 (6) ◽  
pp. 2894-2904 ◽  
Author(s):  
Cristian Cazacu
Keyword(s):  
Symmetry Breaking ◽  
Spherical Harmonics ◽  
Functional Space ◽  
Higher Dimensions ◽  
Minimizing Sequences ◽  
Best Constant ◽  
Rellich Inequality ◽  
Whole Space

The Hardy-Rellich inequality in the whole space with the best constant was firstly proved by Tertikas and Zographopoulos in Adv. Math. (2007) in higher dimensions N ⩾ 5. Then it was extended to lower dimensions N ∈ {3, 4} by Beckner in Forum Math. (2008) and Ghoussoub-Moradifam in Math. Ann. (2011) by applying totally different techniques.In this note, we refine the method implemented by Tertikas and Zographopoulos, based on spherical harmonics decomposition, to give an easy and compact proof of the optimal Hardy–Rellich inequality in any dimension N ⩾ 3. In addition, we provide minimizing sequences which were not explicitly mentioned in the quoted papers in lower dimensions N ∈ {3, 4}, emphasizing their symmetry breaking. We also show that the best constant is not attained in the proper functional space.

scholarly journals Fermion fields in the (non)symmetric Kaluza–Klein theory

2018 ◽  
Vol 96 (5) ◽  
pp. 529-554 ◽  
Author(s):  
M.W. Kalinowski
Keyword(s):  
Symmetry Breaking ◽  
Spontaneous Symmetry Breaking ◽  
Higher Dimensions ◽  
Higgs Mechanism ◽  
Fermion Fields ◽  
Klein Theory ◽  
Spin Fields ◽  
Mass Terms ◽  
Bosonic Part ◽  
Kaluza Klein

The paper is devoted to the unification of fermions within nonsymmetric Kaluza–Klein theories. We obtain a Lagrangian for fermions in non-Abelian Kaluza–Klein theory and non-Abelian Kaluza–Klein theory with spontaneous symmetry breaking and Higgs’ mechanism. A Lagrangian for fermions for geometrized bosonic part of GSW (Glashow–Salam–Weinberg) model in our approach has been derived. Yukawa-type terms and mass terms coming from higher dimensions have been obtained. In the paper, 1/2-spin fields and 3/2-spin fields are considered.

Spin nematics revisited

2002 ◽  
Vol 12 (9) ◽  
pp. 265-265
Author(s):  
V. Cvetkovic ◽  
J. Zaanen ◽  
Z. Nussinov
Keyword(s):  
Form Factor ◽  
Symmetry Breaking ◽  
Gauge Invariance ◽  
Degrees Of Freedom ◽  
Higher Dimensions ◽  
Energy Scale ◽  
Spin Systems ◽  
Heisenberg Spin ◽  
Nematic Order ◽  
Dynamical Algebra

Biquadratic spin 1 Heisenberg spin systems can be constructed exhibiting spin nematic order in higher dimensions. In terms of the original spin degrees of freedom, the spontaneous nematic symmetry breaking generates a Z2 gauge invariance. Using a generalized Holstein-Primakoff transformation for the underlying U(3) dynamical algebra, we calculate the spin dynamical form factor. Although spin is not a gauge singlet, we find form factor to be finite at finite q and $$\backslash$omega$\backslash$$, contrary to our expectations regarding the presence of a energy scale protecting the gauge invariance. This result appears to be perturbatively stable.

SYMMETRY AND SYMMETRY BREAKING FOR MINIMIZERS IN THE TRACE INEQUALITY

2005 ◽  
Vol 07 (06) ◽  
pp. 727-746 ◽  
Author(s):  
ENRIQUE J. LAMI DOZO ◽  
OLAF TORNÉ
Keyword(s):  
Symmetry Breaking ◽  
Minimization Problem ◽  
Radial Function ◽  
Radial Symmetry ◽  
Numerical Results ◽  
Trace Inequality ◽  
Variational Characterization ◽  
Best Constant ◽  
Symmetry Properties

We consider symmetry properties of minimizers in the variational characterization of the best constant in the trace inequality [Formula: see text] in the ball Bρ of radius ρ. When p is fixed, minimizers in this problem can be radial or non-radial depending on the parameters q and ρ. We prove that there is a global radial function u0 > 0, with u0 independent of q, such that any radial minimizer is a multiple of the restriction of u0 to Bρ. Next, we prove that if either q or ρ is sufficiently large, then the minimizers are non-radial. In the case when p = 2, we consider a generalization of the minimization problem and improve some of the above symmetry results. We also present some numerical results describing the exact values of q and ρ for which radial symmetry breaking occurs.

scholarly journals Random Harmonic Functions in Growth Spaces and Bloch-type Spaces

2014 ◽  
Vol 66 (2) ◽  
pp. 284-302
Author(s):  
Kjersti Solberg Eikrem
Keyword(s):  
Unit Ball ◽  
Unit Disk ◽  
Spherical Harmonics ◽  
Harmonic Functions ◽  
Higher Dimensions ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Doubling Condition ◽  
Similar Characterization ◽  
Type Spaces

Abstract. Let h∞v (D) and h∞v (B) be the spaces of harmonic functions in the unit disk and multidimensional unit ball admitting a two-sided radial majorant v(r). We consider functions v that fulfill a doubling condition. In the two-dimensional case letwhere ξ ={ξji} is a sequence of random subnormal variables and aji are real. In higher dimensions we consider series of spherical harmonics. We will obtain conditions on the coefficients aji that imply that u is in h∞v (B) almost surely. Our estimate improves previous results by Bennett, Stegenga, and Timoney, and we prove that the estimate is sharp. The results for growth spaces can easily be applied to Bloch-type spaces, and we obtain a similar characterization for these spaces that generalizes results by Anderson, Clunie, and Pommerenke and by Guo and Liu.

scholarly journals Improved Hardy-Rellich inequalities

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Biagio Cassano ◽  
Lucrezia Cossetti ◽  
Luca Fanelli
Keyword(s):  
Magnetic Field ◽  
Hardy Inequality ◽  
Best Constant ◽  
Rellich Inequality ◽  
Free Case ◽  
Optimal Inequality ◽  
Free Operator ◽  
The Hardy Inequality

<p style='text-indent:20px;'>We investigate Hardy-Rellich inequalities for perturbed Laplacians. In particular, we show that a non-trivial angular perturbation of the free operator typically improves the inequality, and may also provide an estimate which does not hold in the free case. The main examples are related to the introduction of a magnetic field: this is a manifestation of the diamagnetic phenomenon, which has been observed by Laptev and Weidl in [<xref ref-type="bibr" rid="b21">21</xref>] for the Hardy inequality, later by Evans and Lewis in [<xref ref-type="bibr" rid="b9">9</xref>] for the Rellich inequality; however, to the best of our knowledge, the so called Hardy-Rellich inequality has not yet been investigated in this regards. After showing the optimal inequality, we prove that the best constant is not attained by any function in the domain of the estimate.</p>

scholarly journals A Sharp Rellich Inequality on the Sphere

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 288
Author(s):  
Songting Yin
Keyword(s):  
Type Inequality ◽  
Best Constant ◽  
Rellich Inequality

We obtain a Rellich type inequality on the sphere and give the corresponding best constant. The result complements some related inequalities in recent literatures.

scholarly journals On the loss of compactness in the vectorial heteroclinic connection problem

2016 ◽  
Vol 146 (3) ◽  
pp. 595-608 ◽  
Author(s):  
Nikos Katzourakis
Keyword(s):  
A Priori ◽  
Functional Space ◽  
Alternative Proof ◽  
Local Minima ◽  
Minimizing Sequences ◽  
Connection Problem ◽  
Heteroclinic Connection ◽  
Image Position ◽  
Constraint Method ◽  
Loss Of Compactness

We give an alternative proof of the theorem of Alikakos and Fusco concerning existence of heteroclinic solutions U : ℝ → ℝN to the systemHere a± are local minima of a potential W ∈ C2(ℝN) with W(a±) = 0. This system arises in the theory of phase transitions. Our method is variational but differs from the original artificial constraint method of Alikakos and Fusco and establishes existence by analysing the loss of compactness in minimizing sequences of the action in the appropriate functional space. Our assumptions are slightly different from those considered previously and also imply a priori estimates for the solution.

Zonal and tesseral harmonic perturbations of an artificial satellite

1966 ◽  
Vol 25 ◽  
pp. 323-325 ◽  
Author(s):  
B. Garfinkel
Keyword(s):  
Angular Velocity ◽  
Spherical Harmonics ◽  
Artificial Satellite ◽  
Compact Form ◽  
Earth’S Rotation ◽  
Earth's Rotation ◽  
Secular Variations ◽  
Tesseral Harmonics

The paper extends the known solution of the Main Problem to include the effects of the higher spherical harmonics of the geopotential. The von Zeipel method is used to calculate the secular variations of orderJmand the long-periodic variations of ordersJm/J2andnJm,λ/ω. HereJmandJm,λare the coefficients of the zonal and the tesseral harmonics respectively, withJm,0=Jm, andωis the angular velocity of the Earth's rotation. With the aid of the theory of spherical harmonics the results are expressed in a most compact form.

Symmetry breaking in convergent-beam diffraction

1989 ◽  
Vol 47 ◽  
pp. 480-481
Author(s):  
D.J. Eaglesham
Keyword(s):  
Symmetry Breaking ◽  
Point Group ◽  
X Ray Diffraction ◽  
Multiple Diffraction ◽  
Space Groups ◽  
Atomic Displacements ◽  
Small Probe ◽  
Phase Sensitive ◽  
Convergent Beam

Convergent Beam Electron Diffraction is now almost routinely used in the determination of the point- and space-groups of crystalline samples. In addition to its small-probe capability, CBED is also postulated to be more sensitive than X-ray diffraction in determining crystal symmetries. Multiple diffraction is phase-sensitive, so that the distinction between centro- and non-centro-symmetric space groups should be trivial in CBED: in addition, the stronger scattering of electrons may give a general increase in sensitivity to small atomic displacements. However, the sensitivity of CBED symmetry to the crystal point group has rarely been quantified, and CBED is also subject to symmetry-breaking due to local strains and inhomogeneities. The purpose of this paper is to classify the various types of symmetry-breaking, present calculations of the sensitivity, and illustrate symmetry-breaking by surface strains.CBED symmetry determinations usually proceed by determining the diffraction group along various zone axes, and hence finding the point group. The diffraction group can be found using either the intensity distribution in the discs

Centrosome Aurora A gradient ensures single polarity axis in C. elegans embryos

2020 ◽  
Vol 48 (3) ◽  
pp. 1243-1253 ◽  
Author(s):  
Sukriti Kapoor ◽  
Sachin Kotak
Keyword(s):  
Symmetry Breaking ◽  
Cell Fate ◽  
Cell Cortex ◽  
Axis Formation ◽  
Aurora A Kinase ◽  
Aurora A ◽  
C Elegans ◽  
Cell Embryo ◽  
Polarity Establishment

Cellular asymmetries are vital for generating cell fate diversity during development and in stem cells. In the newly fertilized Caenorhabditis elegans embryo, centrosomes are responsible for polarity establishment, i.e. anterior–posterior body axis formation. The signal for polarity originates from the centrosomes and is transmitted to the cell cortex, where it disassembles the actomyosin network. This event leads to symmetry breaking and the establishment of distinct domains of evolutionarily conserved PAR proteins. However, the identity of an essential component that localizes to the centrosomes and promotes symmetry breaking was unknown. Recent work has uncovered that the loss of Aurora A kinase (AIR-1 in C. elegans and hereafter referred to as Aurora A) in the one-cell embryo disrupts stereotypical actomyosin-based cortical flows that occur at the time of polarity establishment. This misregulation of actomyosin flow dynamics results in the occurrence of two polarity axes. Notably, the role of Aurora A in ensuring a single polarity axis is independent of its well-established function in centrosome maturation. The mechanism by which Aurora A directs symmetry breaking is likely through direct regulation of Rho-dependent contractility. In this mini-review, we will discuss the unconventional role of Aurora A kinase in polarity establishment in C. elegans embryos and propose a refined model of centrosome-dependent symmetry breaking.

Sign in / Sign up

Export Citation Format

Share Document