Abstract:
Hopf algebra structure on the differential algebra of the extended $q$-plane is defined. An algebra of forms which is obtained from the generators of the extended $q$-plane is introduced and its Hopf algebra structure is given.

Abstract:
We introduce a noncommutative differential calculus on the two-parameter $h$-superplane via a contraction of the (p,q)-superplane. We manifestly show that the differential calculus is covariant under $GL_{h_1,h_2}(1| 1)$ transformations. We also give a two-parameter deformation of the (1+1)-dimensional phase space algebra.

Abstract:
In this paper properties of the quantum supermatrices in the quantum supergroup $GL_{p,q}(1|1)$ are discussed. It is shown that any element of $GL_{p,q}(1|1)$ can be expressed as the exponential of a matrix of non-commuting elements, like the group $GL_q(1|1)$. An explicit construction of this exponential representation is presented.

Abstract:
We introduce a two-parameter deformation of 2x2 matrices without imposing any condition on the matrices and give the universal R-matrix of the nonstandard quantum group which satisfies the quantum Yang-Baxter relation. Although in the standard two-parameter deformation the quantum determinant is not central, in the nonstandard case it is central. We note that the quantum group thus obtained is related to the quantum supergroup $GL_{p,q}(1|1)$ by a transformation.

Abstract:
In this paper, we give the quantum analogue of the dual matrices for the quantum supergroup $GL_q(1|1)$ and discuss these properties of the quantum dual supermatrices.

Abstract:
A non-commutative differential calculus on the $h$-superplane is presented via a contraction of the $q$-superplane. An R-matrix which satisfies both ungraded and graded Yang-Baxter equations is obtained and a new deformation of the $(1+1)$ dimensional classical phase space (the super-Heisenberg algebra) is introduced.

Abstract:
We give a two-parameter quantum deformation of the exterior plane and its differential calculus without the use of any R-matrix and relate it to the differential calculus with the R-matrix. We prove that there are two types of solutions of the Yang-Baxter equation whose symmetry group is $GL_{p,q}(2)$. We also give a two-parameter deformation of the fermionic oscillator algebra.

Abstract:
In this work, Z$_3$-graded quantum $(h,j)$-superplane is introduced with a help of proper singular $g$ matrix and a Z$_3$-graded calculus is constructed over this new $h$-superplane. A new Z$_3$-graded $(h,j)$-deformed quantum (super)group is constructed via the obtained calculus.

Abstract:
In this study, whether lycopene has preventive effect on chromosome aberrations or not was investigated. For this purpose, 2 different concentrations (0.02 and 0.03 M) of Ethyl Methane Sulfonate (EMS) which is known to induce chromosomal damage were treated on Allium cepa for 2 h. Afterwards, plant roots were treated with lycopene extracts in 4 different concentrations (1, 3, 5 and 10 μM) for 24 h. 0.02 and 0.03 M EMS were used as positive control and tap water was used as negative control. Root tip meristematic cells were observed under the microscope. Mitotic index (MI) and chromosomal aberrations (fragments, bridges, stickness, polar deviation) and micronucleus formation were evaluated and statistically analyzed. As a result; it was determined that depending on applied lycopene concentrations both in MI and chromosomal aberrations lycopene-caused some changes. It was seen that lycopene had preventive effect on chromosome aberrations particularly at 1 and 3 μM concentrations, but this effect decreased at 5 and 10 μM concentrations.

Abstract:
Using the vortex filament model with the full Biot-Savart law, we show that non-straight bundles of quantized vortex lines in HeII are structurally robust and can reconnect with each other maintaining their identity. We discuss vortex stretching in superfluid turbulence in many cases. We show that, during the bundle reconnection process, Kelvin waves of large amplitude are generated, in agreement with previous work and with the finding that helicity is produced by nearly singular vortex interactions in classical Euler flows. The reconnection events lead to changes in velocities, radius, number of points and total length. The existence of reconnections was confirmed by other authors using the model of nonlinear Schr?dinger equation (NLSE). Our results are agreed with the finding of other authors and extension to our numerical experiments.