Harmonic morphisms from four-dimensional Lie groups

We consider 4-dimensional Lie groups with left-invariant Riemannian metrics. For such groups we classify left-invariant conformal foliations with minimal leaves of codimension 2. These foliations produce local complex-valued harmonic morphisms.     

Lie groups and gravity theories

Generalized theories of gravitation using the group manifold approach are outlined. It is suggested that free differential algebras should take the place of Lie algebras in current physical theory.     

Einstein-like Lorentzian Lie groups of dimension four

Einstein-like examples of four-dimensional Lorentzian Lie groups are listed and geometric properties of each class have been investigated.     

Hopf fibration and monopole connection over the contact quantum spheres

Non-commutative geometry of quantised contact spheres introduced by Omori et al. in [J. Math. Soc. Jpn. 50 (1998) 915; Noncommutative 3-sphere as an Example of Noncommutative Contact Algebras, Banach Center Publications, vol. 40, 1997, pp. 329–334] is studied. In particular it is proven that these spheres form a non-commutative Hopf fibration in the sense of Hopf–Galois extensions. The monopole (strong) connection is constructed, and projectors describing projective modules of all monopole charges are computed.     

Poisson structures on double Lie groups

Lie bialgebra structures are reviewed and investigated in terms of the double Lie algebra, of Manin- and Gauss-decompositions. The standard R-matrix in a Manin decomposition then gives rise to several Poisson structures on the correponding double group, which is investigated in great detail.     

A very strong difference property for semisimple compact connected lie groups

Let G be a topological group. For a function f: G → ℝ and h ∈ G, the difference function Δ _h f is defined by the rule Δ _h f(x) = f(xh) − f(x) (x ∈ G). A function H: G → ℝ is said to be additive if it satisfies the Cauchy functional equation H(x + y) = H(x) + H(y) for every x, y ∈ G. A class F of real-valued functions defined on G is said to have the difference property if, for every function f: G → ℝ satisfying Δ _h f ∈ F for each h ∈ G, there is an additive function H such that f − H ∈ F. Erdős’ conjecture claiming that the class of continuous functions on ℝ has the difference property was proved by N. G. de Bruijn; later on, F. W. Carroll and F. S. Koehl obtained a similar result for compact Abelian groups and, under the additional assumption that the other one-sided difference function ∇ _h f defined by ∇ _h f(x) = f(xh) − f(x) (x ∈ G, h ∈ G) is measurable for any h ∈ G, also for noncommutative compact metric groups. In the present paper, we consider a narrower class of groups, namely, the family of semisimple compact connected Lie groups. It turns out that these groups admit a significantly stronger difference property. Namely, if a function f: G → ℝ on a semisimple compact connected Lie group has continuous difference functions Δ _h f for any h ∈ G (without the additional assumption concerning the measurability of the functions of the form ∇ _h f), then f is automatically continuous, and no nontrivial additive function of the form H is needed. Some applications are indicated, including difference theorems for homogeneous spaces of compact connected Lie groups.     

The application of proton nuclear magnetic resonance imaging for the in vivo characterisation of chemically induced renal lesions in rats over a prolonged time study

Renal cortical and medullary spin-lattice (T₁) relaxation times were measured at various time points over a period of 56 days following the administration of a single i.p. injection of 100 mg/kg 2-bromoethanamine hydrobromide (BEA), 200 mg/kg hexachloro-1,3-butadiene (HCBD) or 100 mg/kg puromycin aminonucleoside (PAN) to male Wistar rats. Administration of a single injection of HCBD caused a dramatic, immediate rise in the cortical T₁ values above control values, and these levels remained elevated until, by Day 28 postinjection the levels were back to control values. Administration of BEA also caused an elevation in cortical T₁ values, but in this case these values remained above control values for the rest of the study. The administration of PAN did not produce any significant increases in cortical T₁ values until 14 days postinjection. The elevated T₁ values remained above control values for the rest of the study. These increases observed in cortical T₁ values appeared to be mirrored by decreases in medullary T₁ values. Increases in cortical T₁ values were accompanied by visual changes in the NMR images and enlargement of the kidneys. The histological findings were consistent with the NMR data, confirming that morphologically the tissues did show a full recovery by Day 28 in the HCBD-treated animals. This was not the case following injection of both BEA and PAN, where necrosis was not reversible and there was no recovery of the tissues.     

Comparison of direct and indirect calculations of laryngeal airway resistance in connected speech

Direct measures of subglottal pressure obtained through a tracheal puncture were used to calculate laryngeal airway resistance. Six subjects completed tasks including syllable trains and more natural speech samples produced at three loudness levels. Direct calculations of natural speech resistance values were compared with indirect estimates obtained during syllable train production. The degree of correspondence between direct and indirect calculations varied by subject. Overall, the smallest relative errors among calculations occurred for syllable trains, with higher relative errors for the monologue and sentence. For loudness conditions, the smallest and largest relative errors occurred for soft and loud productions, respectively. The clinical utility of indirect estimation is questioned and suggestions for improving its validity are provided.     

Frames built on fractal sets

Frames of finite dimensional Hilbert spaces have recently been of great interest in applications to modern communication networks transport packets. In this note, continuous and discrete frames, living on fractal sets, of both finite and infinite dimensional separable abstract Hilbert spaces are found. In particular, we find discrete frames, robust to erasures, of finite dimensional Hilbert spaces using iterated function systems.     

Waves propagating along a density gradient in a dusty plasma

The waves propagating in the direction of a density gradient in a dusty plasma are analyzed. The analysis includes the recently predicted electrostatic dust Rossby mode that can develop in a rotating dusty plasma with charged grains, as well as waves in magnetized plasmas, and dust acoustic waves. The spatial behavior of the mode amplitude is determined for a few realistic plasma density profiles.     

Filamentation of a rotating electron beam in a magnetized plasma

Filamentation of a nonrelativistic rotating electron beam in a magnetized plasma is investigated by solving the kinetic equation and finding its dielectric permitivity. The period and the establishment time of the filamentation structure and threshold for instability development are obtained. It will be shown that only when the external magnetic field strength becomes smaller than a characteristic value, filamentation appears.     

On the Wigner-Eckart theorem for tensor operators connected with compact matrix quantum groups

In this Letter, the notion of a tensor operator connected with a unitary, smooth, finite-dimensional representation of a compact, matrix quantum group is introduced and investigated. It is proved that, for compact matrix, simply reducible quantum groups, there exists a theorem analogous to the famous Wigner-Eckart theorem.     

Magnetic resonance imaging of soft-tissue tumors: Comparison with computed tomography

Twenty-seven patients with soft-tissue tumors were examined with a Picker 0.15-tesla resistive magnet and by computed tomography (CT). In all but one patient, MRI was better than or equal to CT in defining the anatomic extent of the tumor. We could determine whether major vascular structures were engulfed by the tumor in 80% of the MRI examinations but only in 62% of the CT scans. MRI and CT were equally effective in determining the presence or absence of bony invasion. The MRI images of all the tumors showed increased signal intensity relative to normal muscle when spin-echo (SE) sulse sequences with long repeat times were used (SE: echo time [TE], 60 ms; repetition time [TR], 2,000 ms). When T1 weighted pulse sequences were used (SE: TE, 30 ms; TR, 500 ms or inversion recovery: inversion time, 500 ms; TE, 40 ms; TR, 2,000 ms) the malignant tumors showed decreased signal intensity compared to normal muscle. Only lipomas showed high signal intensity on both T1 and T2 weighted pulse sequences.     

Pseudo-Kähler metrics on six-dimensional nilpotent Lie algebras

It is well known that any symplectic manifold (M,Ω) has an almost complex structure J which is compatible with Ω. In this paper, we deal with the existence of compatible pairs (J,Ω) on nilpotent Lie algebras of dimension ≤6, J being an integrable almost complex structure. We prove that if such a pair exists, J must satisfy some extra conditions, namely J must be nilpotent in the sense of [Trans. Am. Math. Soc. 352 (2000) 5405]. Associated to any such a compatible pair, there is a pseudo-Kähler metric g which cannot be positive definite unless be abelian. All these metrics are Ricci flat, although many of them are nonflat, and we study the behaviour of its curvature tensor under deformation.     

Varied MR appearance of autism: Fifty-three pediatric patients having the full autistic syndrome

Fifty-three autism patients ranging in age from 2 to 22 yr with a mean age of 9 yr were evaluated by MR imaging over a 3-yr study period. Sagittal, axial, and coronal spin-echo and short TI inversion recovery scans were performed on a 0.5 Tesla (Picker Inc., Cleveland, OH) system. Results were compared to 32 control patients age range 1 to 17 yr, mean 8.5 yr. MR scans were evaluated by three neuroradiologists. Measurements of midsagittal vermian height and AP diameter were performed. Subjective estimates were made of ventricular size, amygdala size, fourth ventricular size, and vermian shape. Results were correlated with clinical presentation, course, and lab analyses by a pediatric neurologist. MR findings did not present a single pattern capable of predicting the presence or severity of autism. The constellation of MR findings in this group of 53 patients was highly variable, thus we advise caution in the interpretation of MR images in autistic patients. Autism is a heterogeneous disease entity containing different clinical subgroups, which do not manifest similar radiologic pictures.     

Effects of radiation therapy on the human normal brain (white matter) visualized by MR imaging

Sequential MR imaging could provide information related to the pathological changes due to the application of external cytotoxic agents such as radiotherapy on the central nervous system. This paper describes the results of our attempt to demonstrate short-term changes associated with normal brain during and immediately following radiotherapy when the whole brain is irradiated for malignant conditions. No observable changes were found in the normal brain parenchyma in any of the patients (n = 8) in T₁-, T₂-, and proton-density-weighted images in the sequential scans in the first and second weeks and immediately following radiotherapy. Also, no changes were observed in the normal brain appearance at 2 mo (n = 6), up to 6 mo (n = 1), and up to 15 mo (n = 1) after completion of radiotherapy.     

On the security of a class of discrete-time chaotic cryptosystems

In this Letter we analyze the security of some recently proposed chaotic cryptosystems and give methods to break the cipher to reveal the encrypted information. The method exploits the dependencies between the parameters and the output sequence of a dynamical system to reveal the secretly shared system parameters. For each encryption scheme a known plaintext attack and a ciphertext only attack are given.