Activities

In this seminar, we first recap the α-energy functional, Euler-Lagrange operator and α stress-energy tensor. Second, it is shown that the critical points of the α-energy functional are explicitly related to harmonic maps through conformal deformation. In addition, an α-harmonic map is constructed from any smooth map between Riemannian manifolds under certain assumptions. Next, we determine the conditions under which the fibers of horizontally conformal α-harmonic maps are minimal submanifolds. Then, the stability of any α-harmonic map from a Riemannian manifold to a Riemannian manifold with non-positive Riemannian curvature is demonstrated. Finally, the instability of α-harmonic maps from a compact manifold to a standard unit sphere is investigated.

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We look at a class of nonassociative algebras that can be defined using skew polynomials, and discuss their structure and applications. These algebras can be seen as canonical generalizations of central simple algebras, and some behave surprisingly similar to their classical "cousins". They are employed to build codes used for wireless digital data transmission, e.g. in mobile phones, laptops or portable TVs. To make the talk accessible to a broad audience, we will explain the main ideas using the example of Hamilton's quaternion algebra. We will explain how to generalize the construction of Hamilton's quaternion algebra to nonassociative four-dimensional algebras, and how to use them in space-time block coding.

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Arithmetic Kleinian group is an important branch of low dimensional geometry and topology. One of the significant side of it, is two generating arithmetic groups. In our research, we will complete Maclaclan and Gaven martin work for all possible Arithmetic groups which generated by two elements $(f; g)$ of order $(p; q)$. They calculated $(p; q)$- cases for $p; q < 6$. But for $p; q < 6$, cases still have remainded. Here we try to introduce some Theorems and Methods to complete our work.

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For two Banach algebras A and B and a nonzero multiplicative linear functional ϴ on B, Monfared introduced the ϴ-Lau product B. In this paper, we investigate and study the notions of ϕ-biprojectivity, ϕ-biflatness and ϕ-Johnson amenability of B and their relations with A and B. As an application, we characterize ϕ-biflatness and ϕ-biprojectivity for ϴ-Lau product of Banach algebras related to locally compact groups and discrete semigroups. 

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Imagine a mature enough quantum computer is available in the near future. What can it do? There are several tasks for which a quantum computer will be useful. It can also be a threat to our privacy. The one that is mentioned most frequently is that quantum computers will be able to read secret messages communicated over the internet using the current cryptographic schemes. Which simply means that none of us would be able to securely and privately communicate via internet if we stay with our current online devices. In this talk we will address this issue. We will also describe the actions that we can do in advance in order to stay secure in the world with quantum computers.

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For a given finitely generated multiplicative subgroup of the rationals which possibly contain negative numbers, we will demonstrate formulas for the densities of primes subject to GRH for which the index of the reduction group has a given value. Moreover, we will completely classify the cases of rank one, torsion groups for which the density vanishes and the set of primes for which the index of the reduction group has a given value, is finite. For higher rank groups we will mention some partial results.

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Transferring geometric problems into the languages of permutations groups. Our goal is to describe the geometric problem about spaces and then transfer it into the theory of permutation groups by using the fundamental groups. The Riemann existence theorem provides a one to one correspondence between them. As a result, we can deal with finite primitive groups instead of indecomposable meromorphic functions.

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 This research is primarily concerned with the convexity properties of distortion functionals -particularly the linear distortion - defined on quasiconformal homeomorphisms of domains in Euclidean n-spaces, though we will mainly stick to three-dimensions. The principal application is in identifying the lower semi-continuity of distortion on uniformly convergent limits of sequence of quasiconformal mappings. For example, given the curve family or analytic definitions of quasiconformality – discussed in this research – it is known that if {𝒇𝑛} is a sequence of K-quasiconformal mappings (and here K depends on the particular distortion but is the same for every element of the sequence) which converges to a function 𝒇, then the limit function is also K-quasiconformal. 

Despite a widespread belief that this was also true the geometric definition of quasiconformal-ity (via the linear distortion 𝐻(𝒇) defined below) Tadeusz Iwaniec gave a specific surprising example to show that the linear distortion function is not lower semicontinuous. The main aim of this research is to show that this failure of lower semicontinuity is actually far more common. Perhaps even generic in the sense that it might be true under mild restrictions on a quasiconformal 𝒇, there may be a sequence if {𝒇𝑛} with 𝒇𝑛→𝒇 uniformly and with the below property 

lim𝑛→∞sup𝐻(𝒇𝑛)→𝐻(𝒇). 

The main result in my thesis is to show this is true for wide class of linear mapping. 

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This work presented some novel systems for chaotic systems‎, ‎the so-called parallel systems‎. ‌‎Indeed ‎we will define a parallel system for a chaotic system where the‎ ‎attractor obtained from the new system exhibits complex chaotic dynamics as well‎. The application of these systems is discussed.  For instance, It is shown with the help of these systems, the synchronisation problem between two systems can be transformed into an optimal control problem‎.

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