No abstract

In the present paper, we study some properties of the Heisenberg Lie algebra of dimension . The main purpose of this research is to construct a real Frobenius Lie algebra from the Heisenberg Lie algebra of dimension . To achieve this, we exhibit  how to compute the derivation of the Heisenberg Lie algebra by following Oom’s result. In this research, we use a literature review method to some related papers corresponding to a derivation of a Lie algebra, Frobenius Lie algebras, and Plancherel measure. Determining a conjecture of a real Frobenius Lie algebra is obtained. As the main result, we prove that conjecture. Namely, for the given the Heisenberg Lie algebra, there exists a commutative subalgebra of dimension one such that its semi direct sum is a real Frobenius Lie algebra of dimension . Futhermore, in the notion of the Lie group of the Heisenberg Lie algebra which is called the Heisenberg Lie group, we compute the generalized character of its group  and we determine the Plancherel measure of the unitary dual of the Heisenberg Lie group. As our contributions, we complete some examples of Frobenius Lie algebras obtained from a nilpotent Lie algebra and we also give alternative computations to find the Plancherel measure of the Heisenberg Lie group.

Dalam artikel ini, didiskusikan bagaimana memotivasi siswa-siswa sekolah dasar khususnya para siswa di SDN Cikuda Jatinangor dalam memahami konsep matematika melalui alat-alat pembelajaran matematika. Tujuan utamanya adalah untuk menarik minat siswa dalam memahami konsep dasar matematika dengan lebih mudah. Metode yang digunakan untuk mencapai tujuan ini adalah student learning center. Selain itu, diberikan juga penjelasan kepada salah seorang perwakilan guru dan siswa melalui praktik penggunaan alat-alat pembelajaran matematika tersebut. Lebih jauh, karena kondisi COVID-19, kegiatan ini juga direalisasikan melalui pembuatan video pembelajaran yang dapat diakses di YouTube. Fokus utama dalam kegiatan ini adalah penekanan pada penguatan konsep aritmetika. Di sisi lain, penguatan konsep siswa dilakukan melalui problem solving di aplikasi zenius.

Functional materials are becoming an increasingly important part of our daily life, e.g. they used for sensing, actuation, computing, energy conversion. These materials often have  unique physical, chemical, and structural characteristic involving very complex phase.  Many mathematical model have been devised to study the complex behavior of functional materials. Some of the models have been proven powerful in predicting the behavior of new materials built upon the composites of existing materials. One of  mathematical methods used to model the behavior of the materials is the differential equation. Very often the resulting differential equations are very complicated so that most methods failed in obtaining the exact solutions of the problems. Fortunately, a relatively new approach via Lie symmetry gives a new hope in obtaining or at least understanding the behavior of the solutions, which is needed to understand the behavior of the materials being modeled. In this paper we present a survey on the use of Lie symmetry and related concepts (such as  Lie algebra, Lie group, etc) in modeling the behavior of functional materials and discuss some fundamental results of the Lie symmetry theory which often used in solving differential equations. The survey shows that the use of Lie symmetry and alike have been accepted in many field and gives an alternative approach in studying the complex behavior of functional materials.

<p><em>No fines concrete can be known such as porous concrete, no-fines concrete and pervious concrete, because not use of sand in the mixture causing the cavities between coarse aggregates. When the rainy season, especially in urban areas there are many flood because the water is difficult to infiltration into the ground. Because permeable nature of non-fines concrete which can accelerate the absorption of water to the soil and to the water channel, reduce run-off and increase groundwater reserves. In this research will study for compressive strength and infiltration in non-finnes concrete. This research uses coarse aggregates from the results of stone crushing machines with coarse aggregate sizes (5-10) mm. cement: aggregate ratio used 1: 2; 1: 3; 1: 4; 1: 5; 1: 6; 1: 7; 1: 8. The study began with material checking, planning of material requirements, making non-sand concrete, then testing compressive strength, and infiltration testing at the age of 28 days. The results showed that the compressive strength of non-finnes concrete with a variation of the ratio of cement : gravel 1: 2  is  33.19 MPa while for a mixture of 1: 8 it is 5.23 MPa. The Infiltration rate has increased along with the greater variation in the ratio of the mixture. The maximum infiltration rate for a mixture of 1: 8 is 9.44 mm/sec. The infiltration rates in no-finnes concrete can be used to accelerate the absorption of water into the soil and can function to reduce water on the surface of the yard.</em></p>