Field Theory on q=−1q=-1 Quantum Plane

We build the $q=-1$ defomation of plane on a product of two copies of algebras of functions on the plane. This algebra constains a subalgebra of functions on the plane. We present general scheme (which could be used as well to construct quaternion from pairs of complex numbers) and we use it to derive differential structures, metric and discuss sample field theoretical models.Comment: LaTeX, 10 page

Noncommutative Riemannian Geometry of the Alternating Group A_4

We study the noncommutative Riemannian geometry of the alternating group A_4=(Z_2 \times Z_2)\cross Z_3 using a recent formulation for finite groups. We find a unique `Levi-Civita' connection for the invariant metric, and find that it has Ricci-flat but nonzero Riemann curvature. We show that it is the unique Ricci-flat connection on $A_4$ with the standard framing (we solve the vacuum Einstein's equation). We also propose a natural Dirac operator for the associated spin connection and solve the Dirac equation. Some of our results hold for any finite group equipped with a cyclic conjugacy class of 4 elements. In this case the exterior algebra $\Omega(A_4)$ has dimensions $1:4:8:11:12:12:11:8:4:1$ with top-form 9-dimensional. We also find the noncommutative cohomology $H^1(A_4)=C$.Comment: 28 pages Latex no figure

Metric On Quantum Spaes

We introduce the analogue of the metric tensor in case of $q$-deformed differential calculus. We analyse the consequences of the existence of such metric, showing that this enforces severe restrictions on the parameters of the theory. We discuss in detail the examples of the Manin plane and the $q$-deformation of $SU(2)$. Finally we touch the topic of relations with the Connes' approach.Comment: 7 pages (LaTeX), preprint TPJU 14/9

Multiple noncommutative tori and Hopf algebras

We derive the Kac-Paljutkin finite-dimensional Hopf algebras as finite fibrations of the quantum double torus and generalize the construction for quantum multiple tori.Comment: 18 pages; AMSLaTeX (major revision, the construction of dual rewritten using approach of multiplier Hopf algebras, references added

Almost commutative Riemannian geometry: wave operators

Associated to any (pseudo)-Riemannian manifold $M$ of dimension $n$ is an $n+1$-dimensional noncommutative differential structure (\Omega^1,\extd) on the manifold, with the extra dimension encoding the classical Laplacian as a noncommutative `vector field'. We use the classical connection, Ricci tensor and Hodge Laplacian to construct (\Omega^2,\extd) and a natural noncommutative torsion free connection $(\nabla,\sigma)$ on $\Omega^1$. We show that its generalised braiding \sigma:\Omega^1\tens\Omega^1\to \Omega^1\tens\Omega^1 obeys the quantum Yang-Baxter or braid relations only when the original $M$ is flat, i.e their failure is governed by the Riemann curvature, and that \sigma^2=\id only when $M$ is Einstein. We show that if $M$ has a conformal Killing vector field $\tau$ then the cross product algebra $C(M)\rtimes_\tau\R$ viewed as a noncommutative analogue of $M\times\R$ has a natural $n+2$-dimensional calculus extending $\Omega^1$ and a natural spacetime Laplacian now directly defined by the extra dimension. The case $M=\R^3$ recovers the Majid-Ruegg bicrossproduct flat spacetime model and the wave-operator used in its variable speed of light preduction, but now as an example of a general construction. As an application we construct the wave operator on a noncommutative Schwarzschild black hole and take a first look at its features. It appears that the infinite classical redshift/time dilation factor at the event horizon is made finite.Comment: 39 pages, 4 pdf images. Removed previous Sections 5.1-5.2 to a separate paper (now ArXived) to meet referee length requirements. Corresponding slight restructure but no change to remaining conten

Disorders of the Optic Nerve in Mitochondrial Cytopathies: New Ideas on Pathogenesis and Therapeutic Targets

Mitochondrial cytopathies are a heterogeneous group of human disorders triggered by disturbed mitochondrial function. This can be due to primary mitochondrial DNA mutations or nuclear defects affecting key components of the mitochondrial machinery. Optic neuropathy is a frequent disease manifestation and the degree of visual failure can be profound, with a severe impact on the patient’s quality of life. This review focuses on the major mitochondrial disorders exhibiting optic nerve involvement, either as the defining clinical feature or as an additional component of a more extensive phenotype. Over the past decade, significant progress has been achieved in our basic understanding of Leber hereditary optic neuropathy and autosomal-dominant optic atrophy—the two classical paradigms for these mitochondrial optic neuropathies. There are currently limited treatments for these blinding ocular disorders and, ultimately, the aim is to translate these major advances into tangible benefits for patients and their families

Thermal resistance of PCD materials with borides bonding phase

In these studies, one group of PCD materials was prepared using diamond powder and 10 wt % of TiB₂ and the second batch of the PCD material was prepared using a mixture of diamond powder with 5 wt % of TiB₂ and 2 wt % of Co. The materials have been sintered using a Bridgman-type high-pressure apparatus at 8.0±0.2 GPa, at a temperature of 2000±50 °C. Thermogravimetric (TG) measurements and Differential Thermal Analysis (DTA) have been carried out for diamond micropowders, TiB₂ bonding phase, and sintered composites. The coefficients of friction for diamond composites in a sliding contact with an Al₂O₃ ceramic ball have been determined from the room temperature up to 800 °C. Material phase compositions were analyzed for initial samples and after wear tests, at the temperature of 800 °C. Raman spectra of diamond composites with borides bonding phases, observed for the first-order zone centre modes of diamond and graphite during the heating up to 800 °C in air have been presented. Thermal properties have been compared with the commercial diamond-cobalt PCD. It has been found that diamond with TiB₂ and Co is the most resistant to the hardness changes at elevated temperatures and this material maintains the high hardness value up to 800 °C but it has a high coefficient of friction.Досліджено полікристалічні алмазні композити – одну групу матеріалів було приготовано з використанням алмазного порошку і 10 % (за масою) TiB₂, а другу – з алмазного порошку, 5 % (за масою) TiB₂ і 2 % (за масою) Co. Матеріали було спечено в апараті високого тиску типу Бріджмена при тиску 8,0±0,2 ГПа і температурі 2000±50 °С. Термогравіметричні вимірювання та диференційний термічний аналіз було проведено для алмазних мікропорошків, зв’язуючої фази TiB₂ і спечених композітов. Визначено коефіцієнти тертя для алмазних композитів при ковзному контакті з кулькою з кераміки Al₂O₃ при температурі від кімнатної до 800 °С. Фазові склади матеріалів проаналізовано для вихідних зразків і після їх випробування на знос при температурі 800 °С. Представлено спектри комбінаційного розсіювання алмазних композитів зі зв’язуючими фазами боридів, що спостерігаються в центрі зони першого порядку алмазу і графіту в процесі нагрівання до 800 °С на повітрі. Порівнювали термічні властивості отриманих полікристалічних алмазних композитів і промислового композита алмаз–кобальт. Було виявлено, що алмаз з TiB₂ і Co є найбільш стійким до змін твердості при підвищених температурах і зберігає високу твердість до 800 °С, але має високий коефіцієнт тертя.Исследованы поликристаллические алмазных композиты – одна группа материалов была приготовлена с использованием алмазного порошка и 10 % (по массе) TiB₂, а вторая – из алмазного порошка, 5 % (по массе) TiB₂ и 2 % (по массе) Co. Материалы были спечены в аппарате высокого давления типа Бриджмена при давлении 8,0±0,2 ГПа и температуре 2000±50 °С. Термогравиметрические измерения и дифференциальный термический анализ были проведены для алмазных микропорошков, связующей фазы TiB₂ и спеченных композитов. Определены коэффициенты трения для алмазных композитов при скользящем контакте с шариком из керамики Al₂O₃ при температуре от комнатной до 800 °С. Фазовые составы материалов проанализированы для исходных образцов и после их испытания на износ при температуре 800 °С. Представлены спектры комбинационного рассеяния алмазных композитов со связующими фазами боридов, наблюдаемые в центре зоны первого порядка алмаза и графита в процессе нагрева до 800 °С на воздухе. Сравнивали термические свойства полученных поликристаллических алмазных композитов и промышленного поликристаллического композита алмаз–кобальт. Было обнаружено, что алмаз с TiB₂ и Co является наиболее устойчивым к изменениям твердости при повышенных температурах и сохраняет высокую твердость до 800 °С, но имеет высокий коэффициент трения

Reconstruction of the spontaneously broken gauge theory in non-commutative geometry

The scheme previously proposed by the present authors is modified to incorporate the strong interaction by affording the direct product internal symmetry. We do not need to prepare the extra discrete space for the color gauge group responsible for the strong interaction to reconstruct the standard model and the left-right symmetric gauge model(LRSM). The approach based on non-commutative geometry leads us to presents many attractive points such as the unified picture of the gauge and Higgs field as the generalized connection on the discrete space; Minkowski space multipied by N-points discrete space. This approach leads us to unified picture of gauge and Higgs fields as the generalized connection. The standard model needs N=2 discrete space for reconstruction in this formalism. \lr is still alive as a model with the intermediate symmetry of the spontaneously broken SO(10) grand unified theory(GUT). N=3 discrete space is needed for the reconstruction of LRSM to include two Higgs bosons $\phi$ and $\xi$ which are as usual transformed as (2,2*,0)$ and (1,3,-2) under left-handed SU(2)x right-handed SU(2)x U(1), respectively. xi is responsible to make the right handed-neutrino Majorana fermion and so well explains the seesaw mechanism. Up and down quarks have the different masses through the vacuum expectation value of phi.Comment: 21 page

κ\kappa-Deformation of Poincar\'e Superalgebra with Classical Lorentz Subalgebra and its Graded Bicrossproduct Structure

The $\kappa$-deformed $D=4$ Poincar{\'e} superalgebra written in Hopf superalgebra form is transformed to the basis with classical Lorentz subalgebra generators. We show that in such a basis the $\kappa$-deformed $D=4$ Poincare superalgebra can be written as graded bicrossproduct. We show that the $\kappa$-deformed $D=4$ superalgebra acts covariantly on $\kappa$-deformed chiral superspace.Comment: 13 pages, late