Group Hn Research Articles

Universal inequalities for eigenvalues of quadratic polynomial operator of the Kohn Laplacian

In this paper, we investigate the Dirichlet weighted eigenvalue problem of quadratic polynomial operator of the Kohn Laplacian on a bounded domain in the Heisenberg group Hn . We establish two inequalities for eigenvalues of this problem. One of them implies an explicit estimate for the upper bound of the (k + 1) -th eigenvalue in terms of the first k eigenvalues. Moreover, as a special case, we give some universal inequalities and estimates for eigenvalues of the bi-Kohn Laplacian. Mathematics subject classification (2010): Primary 35P15, 53C17; Secondary 58C40.

The Brunn–Minkowski inequality in spaces with bitriangular laws of composition

The Brunn–Minkowski inequality and Prekopa–Leindler’s theorem are considered with respect to bitriangular laws of composition on Euclidean spaces R n . The result is illustrated by an example of the Heisenberg group H n . Bibliography: 11 titles.

The Hodge structure of the coloring complex of a hypergraph

Let G be a simple graph with n vertices. The coloring complex Δ(G) was defined by Steingrímsson, and the homology of Δ(G) was shown to be nonzero only in dimension n−3 by Jonsson. Hanlon recently showed that the Eulerian idempotents provide a decomposition of the homology group Hn−3(Δ(G)) where the dimension of the jth component in the decomposition, Hn−3(j)(Δ(G)), equals the absolute value of the coefficient of λj in the chromatic polynomial of G, χG(λ).Let H be a hypergraph with n vertices. In this paper, we define the coloring complex of a hypergraph, Δ(H), and show that the coefficient of λj in χH(λ) gives the Euler Characteristic of the jth Hodge subcomplex of the Hodge decomposition of Δ(H). We also examine conditions on a hypergraph, H, for which its Hodge subcomplexes are Cohen–Macaulay, and thus where the absolute value of the coefficient of λj in χH(λ) equals the dimension of the jth Hodge piece of the Hodge decomposition of Δ(H). We also note that the Euler Characteristic of the jth Hodge subcomplex of the Hodge decomposition of the intersection of coloring complexes is given by the coefficient of jth term in the associated chromatic polynomial.

The influence of benzamil hydrochloride on the evolution of hyponatremic brain edema as assessed by in vivo MRI study in rats

The present study was undertaken to reveal the influence of intracerebroventricular (ICV) benzamil on the dynamics of brain water accumulation in hyponatremic rats. Parameters of brain water homeostasis were continuously monitored, using in vivo magnetic resonance imaging (MRI) methods. The results were compared with those obtained in a previous study by tissue desiccation. A 3-T MRI instrument was applied to perform serial diffusion-weighted imaging to measure the apparent diffusion coefficient (ADC) and MR spectroscopy to determine water signal. A decrease of ADC is thought to represent an increase of intracellular water, whereas water signal is used to quantify brain water content. Five groups of male Wistar rats were studied as follows: normonatremic, native animals (group NN, n = 7), hyponatremic animals (group HN, n = 8), hyponatremic animals treated with ICV benzamil (group HNB, n = 8), hyponatremic animals treated with ICV saline (group HNS, n = 5) and normonatremic animals treated with ICV benzamil (group NNB, n = 5). Hyponatremia was induced by intraperitoneal administration of 140mmol/l dextrose solution in a dose of 20% of body weight. Benzamil hydrochloride (4μg) was injected ICV to the treated animals. During the course of hyponatemia, ADC declined steadily from the baseline (100%) to reach a minimum of 92.32 ± 3.20% at 90 min (p < 0.0005). This process was associated with an increase in water signal to a maximum of 5.95 ± 2.62% at 100 min (p < 0.0005). After pretreatment with benzamil, no consistent changes occurred either in ADC or in water signal. These findings suggest that sodium channel blockade with ICV benzamil has an immediate protective effect against the development of hyponatremic brain edema. Sodium channels, therefore, appear to be intimately involved in the initiation and progression of brain water accumulation in severe hyponatremia.

On Riesz Mean Inequalities for Subelliptic Laplacian

In this paper, we mainly focus on the Riesz means of eigenvalues of the subelliptic Laplacian on the Heisenberg group Hn. We establish a trace formula of associated eigenvalues, then we prove differential inequalities, difference inequalities and monotonicity formulas for the Riesz means of eigenvalues of the subelliptic Laplacian.

Broué's abelian defect group conjecture holds for the Harada–Norton sporadic simple group HN

In representation theory of finite groups, there is a well-known and important conjecture due to M. Broué. He conjectures that, for any prime p, if a p-block A of a finite group G has an abelian defect group P, then A and its Brauer corresponding block B of the normaliser NG(P) of P in G are derived equivalent (Rickard equivalent). This conjecture is called Broué's abelian defect group conjecture. We prove in this paper that Broué's abelian defect group conjecture is true for a non-principal 3-block A with an elementary abelian defect group P of order 9 of the Harada–Norton simple group HN. It then turns out that Broué's abelian defect group conjecture holds for all primes p and for all p-blocks of the Harada–Norton simple group HN.

The C*-algebras of the Heisenberg group and of thread-like Lie groups

We describe the C*-algebras of the Heisenberg group Hn, n ≥ 1, and the thread-like Lie groups GN, N ≥ 3, in terms of C*-algebras of operator fields.

Effects of different concentrations of nitroglycerin on ischemia-reperfusion injury in isolated rat hearts

Objective To investigate the effects of different concentrations of nitroglycerin on ischemia-reperfusion (IR) injury in the isolated rat hearts. Methods Healthy adult male SD rats 7 weeks old weighing 220-330 g were anesthetized with intraperitoneal pentobarbital 50 mg/kg. Their hearts were excised and perfused in a Langendorff apparatus. Thirty isolated hearts were randomly divided into 3 groups (n = 10 each) : group 1R, low-concentration nitroglycerin group (group LN) and high-concentration nitroglycerin group (group HN). The hearts in group IR were perfnsed with plain modified K-H solution. The hearts in group LN were perfused with K-H solution containing nitroglycerin 2.5 μg/L. The hearts in group HN were perfnsed with K-H solution containing nitroglycerin 25 μg/L. After 15 min of post-preparation equilibration, IR was induced by 20 min suspension of perfusion followed by 60 min reperfusion. HR, left ventricular end-diastolic pressure (LVEDP), coronary flow (CF), left ventricular developed pressure (LVDP), + dp/dtmax, and - dp/dtmax were measured at 10, 20, 30, 40, 50 and 60 min of reperfusion. To determine the activities of myocardial lactate dehydrogenase (LDH) and creatine kinase (CK), 1.5 ml of coronary effluent was collected at the end of equilibration, and at 5, 30 and 60 min of reperfusion. At 60 min of reperfusion, myocardial tissues were obtained to determine myocardial infarct size (IS) (by Trc staining) and assess the apoptosis rate (by TUNAL technique) .Results Compared with group IR, HR, CF, LVDP, + dp/dtmax and - dp/dtmax were significantly increased in group LN, while significantly decreased in group HN, and LVEDP, activities of CK and LDH in the coronary effluent, myocardial IS and apoptosis rate were significantly decreased in group LN, while significantly increased in group HN (P < 0.05). Compared with group LN, HR, CF, LVDP, + dp/dtmax and - dp/dtmax were significantly decreased, and LVEDP, activities of CK and LDH in the coronary effluent, myocardial IS and apoptosis rate significantly increased in group HN (P < 0. 05). Conclusion Low-concentration nitroglycerin can attenuate myocardial IR injury in isolated rat hearts, but high-concentration nitroglycerin can exacerbate the injury. Key words: Nitroglycerin; Myocardial reperfusion injury

A local central limit theorem on the Laguerre hypergroup

We consider here the Laguerre hypergroup (K,*α), where K=[0,+∞[×R and *α a convolution product on K coming from the product formula satisfied by the Laguerre functions Lm(α) (m∈N, α⩾0). We set on this hypergroup a local central limit theorem which consists to give a weakly estimate of the asymptotic behavior of the convolution powers μ*αk=μ*α⋯*αμ (k times), μ being a given probability measure satisfying some regularity conditions on this hypergroup. It is also given a central local limit theorem for some particular radial probability measures on the (2n+1)-dimensional Heisenberg group Hn.

Fischer-Clifford Matrices of the Non-Split Group Extension 26˙U 4(2)

The Harada-Norton group HN is a sporadic simple group of order 273030912000000 equal to 214 × 36 × 56 × 7 × 11 × 19. It has 14 conjugacy classes of maximal subgroups and in the present paper we construct the Fischer-Clifford matrices and hence character table for one of these maximal subgroups, that is the non-split extension 26˙U 4(2), of index 164587500. There is not that many known examples of the application of Fischer-Clifford theory to non-split extensions, and hence this maximal subgroup of Hard-Norton's group serves as a good candidate. Most of the computations were carried out using the computer algebra systems GAP and MAGMA.

The 5-modular characters of the sporadic simple Harada–Norton group [formula omitted] and its automorphism group [formula omitted

We determine the 5-modular character table of the sporadic simple Harada–Norton group HN and its automorphism group HN . 2 using The Meat-Axe and condensation.

Strongly singular convolution operators on the Heisenberg group

We consider the L2 mapping properties of a model class of strongly singular integral operators on the Heisenberg group Hn; these are convolution operators on Hn whose kernels are too singular at the origin to be of Calderon-Zygmund type. This strong singularity is compensated for by introducing a suitably large oscillation. Our results are obtained by utilizing the group Fourier transform and uniform asymptotic forms for Laguerre functions due to Erdelyi.

A convolution type characterization for Lp ‐ multipliers for the Heisenberg group

It is well known that if m is an Lp ‐ multiplier for the Fourier transform on ℝn(1 &lt; p &lt; ∞) , then there exists a pseudomeasure σ such that Tmf = σ*f. A similar result is proved for the group Fourier transform on the Heisenberg group Hn. Though this result is already known in generality for amenable groups, a simple proof is provided in this paper.

Global Poincaré Inequalities on the Heisenberg Group and Applications

Let f be in the localized nonisotropic Sobolev space \( W^{{1,p}}_{{{\text{loc}}}} {\left( {\mathbb{H}^{n} } \right)} \) on the n-dimensional Heisenberg group ℍn = ℂn × ℝ, where 1 = p < Q and Q = 2n + 2 is the homogeneous dimension of ℍn. Suppose that the subelliptic gradient is gloablly Lp integrable, i.e., \( {\int_{\mathbb{H}^{n} } {{\left| {\nabla _{{\mathbb{H}^{n} }} f} \right|}} }^{p} du \) is finite. We prove a Poincare inequality for f on the entire space ℍn. Using this inequality we prove that the function f subtracting a certain constant is in the nonisotropic Sobolev space formed by the completion of \( C^{\infty }_{0} {\left( {\mathbb{H}^{n} } \right)} \) under the norm of $$ {\left( {{\int_{\mathbb{H}^{n} } {{\left| f \right|}} }^{{\frac{{Qp}} {{Q - p}}}} } \right)}^{{\frac{{Q - p}} {{Qp}}}} + {\left( {{\int_{\mathbb{H}^{n} } {{\left| {\nabla _{{\mathbb{H}^{n} }} f} \right|}} }^{p} } \right)}^{{\frac{1} {p}}} . $$ We will also prove that the best constants and extremals for such Poincare inequalities on ℍn are the same as those for Sobolev inequalities on ℍn. Using the results of Jerison and Lee on the sharp constant and extremals for L2 to \( L\frac{{2Q}} {{Q - 2}} \) Sobolev inequality on the Heisenberg group, we thus arrive at the explicit best constant for the aforementioned Poincare inequality on ℍn when p = 2. We also derive the lower bound of the best constants for local Poincare inequalities over metric balls on the Heisenberg group ℍn.

Laguerre expansion on the Heisenberg group and Fourier-Bessel transform on ℂ n

Given a principal value convolution on the Heisenberg group H n = ℂ n × ℝ, we study the relation between its Laguerre expansion and the Fourier-Bessel expansion of its limit on ℂ n . We also calculate the Dirichlet kernel for the Laguerre expansion on the group H n .

On the ranks of the Harada-Norton sporadic simple group HN

Let G be a finite group and X a conjugacy class of G. We denote rank(G:X) to be the minimum number of elements of X generating G. In the present article, we determine the ranks for the Harada-Norton sporadic group HN. Computations were carried with the aid of computer algebra system GAP [21].

Odd radical subgroups of some sporadic simple groups

The radical p-subgroups of the Monster, the Baby Monster and the Harada–Norton group HN for p = 3 , 5 , 7 are determined up to conjugacy, which completes the classification problem of odd radical subgroups for all sporadic simple groups together with [M. Kitazume, S. Yoshiara, The radical subgroups of the Fischer simple groups, J. Algebra 255 (2002) 22–58] and the other works.

EXISTENCE OF MINIMISERS FOR A CLASS OF FREE DISCONTINUITY PROBLEMS IN THE HEISENBERG GROUP Hn

The purpose of this paper is to prove existence of minimisers of the functional J(K,u):=∫f(Lu)dx+α∫‖u−g‖qdx+βDdQ−1(K∩Ω),where Ω is an open set of the Heisenberg group Hn, K runs over all closed sets of Hn, u varies in CH1(Ω/K),α,β>0,q≥1,g∈Lq(Ω)∩L∞(Ω) and f : R2n → R is a convex function satisfying some structure conditions (H1)(H2)(H3) (see below).

Nil-Manifolds Cannot be Immersed as Hypersurfaces in Euclidean Spaces

We prove that the \(2n + 1\)-dimensional Heisenberg group Hn and the 4-manifolds \(Nil^4 \) and \(Nil^3 \times \mathbb{R}\) endowed with an arbitrary left-invariant metric admit no C3-regular immersions into Euclidean spaces \(\mathbb{R}^{2n + 2} \) and \(\mathbb{R}^5 \), respectively.

A Note on Hermite and Subelliptic Operators

In this note, we compute the fundamental solution for the Hermite operator with singularity at an arbitrary point y∈ℝ n . We also apply this result to obtain the fundamental solutions for the Grushin operator in ℝ2 and the sub-Laplacian in the Heisenberg group H n .