Abstract:
We consider the complex specializations of Krammer’s representation of the pure braid group on three strings, namely K(q,t), where q and t are non-zero complex numbers. We then specialize the indeterminate t by one and replace by for simplicity. Then we present our main theorem that gives us sufficient conditions that guarantee the irreducibility of the tensor product of two irreducible complex specializations of Krammer’s repre- sentations .

Abstract:
We consider the Magnus representation of the image of the braid group under the generalizations of the standard Artin representation discovered by M. Wada. We show that the images of the generators of the braid group under the Magnus representation are unitary relative to a Hermitian matrix. As a special case, we get that the Burau representation is unitary, which was known and proved by C. C. Squier.

Abstract:
Following up on our result in [1], we find a milder sufficient condition for the tensor product of specializations of the reduced Gassner representation of the pure braid group to be irreducible. We prove that $G_n(x_1, ldots, x_n) otimes G_n(y_1, ldots, y_n) : P_n o GL(mathbb{C}^{n-1} otimes mathbb{C}^{n-1})$ is irreducible if $x_i eq pm y_i $ and $x_j eq pm {{y_j}^{-1}} $ for some $i$ and $j$.

Abstract:
We consider Wada's representation as a twisted version of the standard action of the braid group, Bn, on the free group with n generators. Constructing a free group, Gnm, of rank nm, we compose Cohen's map Bn→Bnm and the embedding Bnm→Aut(Gnm) via Wada's map. We prove that the composition factors of the obtained representation are one copy of Burau representation and m−1 copies of the standard representation after changing the parameter t to tk in the definitions of the Burau and standard representations. This is a generalization of our previous result concerning the standard Artin representation of the braid group.

Abstract:
We consider Krammer's representation of the pure braid group on three strings: , where and are indeterminates. As it was done in the case of the braid group, , we specialize the indeterminates and to nonzero complex numbers. Then we present our main theorem that gives us a necessary and sufficient condition that guarantees the irreducibility of the complex specialization of Krammer's representation of the pure braid group, . 1. Introduction Let be the braid group on strings. There are a lot of linear representations of . The earliest was the Artin representation, which is an embedding , the automorphism group of a free group on generators. Applying the free differential calculus to elements of sometimes gives rise to linear representations of and its normal subgroup, the pure braid group denoted by [1]. The Burau, Gassner, and Krammer's representations arise this way. In a previous paper, we considered Krammer's representation of the braid group on three strings and we specialized the indeterminates to nonzero complex numbers. We then found a necessary and sufficient condition that guarantees the irreducibility of such a representation. For more details, see [2]. In Section 2, we introduce some definitions of the pure braid group and Krammer's representation. In Sections 3 and 4, we present our work that leads to our main theorem, Theorem 4.2, which gives a necessary and sufficient condition for the specialization of Krammer's representation of to be irreducible. 2. Definitions Definition 2.1 (see [1]). The braid group on strings, , is the abstract group with presentation for if . The generators are called the standard generators of . Definition 2.2. The kernel of the group homomorphism is called the pure braid group on strands and is denoted by . It consists of those braids which connect the th item of the left set to the th item of the right set, for all . The generators of are , where . Let us recall the Lawrence-Krammer representation of braid groups. This is a representation of in , where and is the free module of rank over . The representation is denoted by . For simplicity we write instead of . What distinguishes this representation from others is that Krammer's representation defined on the braid group, , is a faithful representation for all [3]. The question of whether or not a specific linear representation of an abstract group is irreducible has always been a significant question to answer, especially those representations of the braid group and its normal subgroups. In a previous result, we determined a necessary and sufficient condition for

Abstract:
The reduced Gassner representation is a multi-parameter representation of $% P_{n},$ the pure braid group on n strings. Specializing the parameters $% t_{1},t_{2},...,t_{n}$ to nonzero complex numbers $x_{1},x_{2},...,x_{n}$ gives a representation $G_{n}(x_{1},ldots ,x_{n}):P_{n} ightarrow GL(mathbb{C}^{n-1})$ which is irreducible if and only if $x_{1}ldots x_{n} eq 1$. In a previous work, we found a sufficient condition for the irreducibility of the tensor product of two irreducible Gassner representations. In our current work, we find a sufficient condition that guarantees the irreducibility of the tensor product of three Gassner representations. Next, a generalization of our result is given by considering the irreducibility of the tensor product of $k$ representations (;$k geq 3;$).

Abstract:
This research aims at estimating the temperature of the aquifer that supplies Assammaqieh well at the depth of 550 m, on the basis of chemical analyses and geothermometric techniques which are one of the methods used for searching for the renewable geothermal energy and conserving the environment. In this study, about twenty-two geothermometric indicators have been used. For verifying the results, these results have been compared with data and estimates of temperature of fluids of deep typical wells in New Zealand, and it has been noticed that the theoretical and actual results approach the limits of 95% in many indicators. The study has been restricted to the relations of Cations because they are the most reliable, and the least affected by dissolution and evaporation. Most of the indicators that are based on the four chemical elements: Calcium (Ca), Potassium (K), Sodium (Na), Magnesium (Mg), have been adopted. The laboratory analysis data of Assammaqieh well confirmed that it was hot sulphurous water that acquired its chemical properties from complicated geochemical conditions, underground thermal conditions and volcanic rock nature. It also turned out that the underground heating process was basically due to thermal conductivity and rock adjacency, and that Assammaqieh well was supplied with water from adjacent groundwater tables whose source was the penetration of surface water. It also appeared that most of the equations used in the search for geothermal energy revealed the presence of an aquifer of hot and very hot water, and they were compatible with the high thermal gradient in volcanic rocks. It also tuned out that 86% of the used geothermometric equations estimated the aquifer temperature of Assammaqieh well as being hot and very hot with around 135.5 Celsius (±20). The study concluded with the hypothesis that Akkar possessed a huge geothermal energy, and benefiting from this energy might put an end to the chronic problem of electricity in Lebanon, and opened up many prospects and uses that could participate in a sustainable and comprehensive development of Akkar and Lebanon as a whole.

The WHO project for conforming PHC to requirements of all age groups has
resulted in publishing a toolkit for age-friendly PHC in 2008. The toolkit included
checklists for physical environment and signage properties. This study matched
the current physical environment properties of DHA’s PHC Health Centers
against WHO’s recommendations. This is a cross sectional descriptive study
that included visits to all 12 Primary Health Care Centers in Dubai city during
August-September 2016 with the objective to assess the degree of fulfillment
of current properties of Health Centers building to the recommendations of
WHO as listed in “Age-friendly Primary Health Care Centres Toolkit” [1]. The
study found that 81.86% of physical environment properties are matching the
recommendation of WHO, while signage matching was 44.6%. The study concluded
that most PHC properties have a physical design that met WHO’s
recommendations. The two major deviations were accessibility by public transportations
and presence of grab bars. Factors that had a significant impact on
design were compliance with multiple international and local standards, the
availability of private cars, and the availability of wheel chairs. Signage in DHA’s
health centers followed a central plan that differed from WHO’s recommendations.