• Journal of the Chungcheong Mathematical Society
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• v.29 no.1
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• pp.73-82
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• 2016

The utility of exponential generating functions is that they are relevant for combinatorial problems involving sets and subsets. Sequences of polynomials play a fundamental role in applied mathematics, such sequences can be described using the exponential generating functions. The actuarial polynomials ${\alpha}^{({\beta})}_n(x)$, n = 0, 1, 2, ${\cdots}$, which was suggested by Toscano, have the following exponential generating function: $${\limits\sum^{\infty}_{n=0}}{\frac{{\alpha}^{({\beta})}_n(x)}{n!}}t^n={\exp}({\beta}t+x(1-e^t))$$. A linear functional on polynomial space can be identified with a formal power series. The set of formal power series is usually given the structure of an algebra under formal addition and multiplication. This algebra structure, the additive part of which agree with the vector space structure on the space of linear functionals, which is transferred from the space of the linear functionals. The algebra so obtained is called the umbral algebra, and the umbral calculus is the study of this algebra. In this paper, we investigate some umbral representations in the actuarial polynomials.

• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.535-539
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• 2007

Given operators X and Y acting on a separable complex Hilbert space $\mathcal{H}$, an interpolating operator is a bounded operator A such that AX=Y. In this article, we show the following: Let $Alg\mathcal{L}$ be a tridiagonal algebra on a separable complex Hilbert space $\mathcal{H}$ and let $X=(x_{ij})\;and\;Y=(y_{ij})$ be operators in $\mathcal{H}$. Then the following are equivalent: (1) There exists a normal operator $A=(a_{ij})\;in\;Alg\mathcal{L}$ such that AX=Y. (2) There is a bounded sequence $\{\alpha_n\}\;in\;\mathbb{C}$ such that $y_{ij}=\alpha_jx_{ij}\;for\;i,\;j\;{\in}\;\mathbb{N}$. Given vectors x and y in a separable complex Hilbert space $\mathcal{H}$, an interpolating operator is a bounded operator A such that Ax=y. We show the following: Let $Alg\mathcal{L}$ be a tridiagonal algebra on a separable complex Hilbert space $\mathcal{H}$ and let $x=(x_i)\;and\;y=(y_i)$ be vectors in $\mathcal{H}$. Then the following are equivalent: (1) There exists a normal operator $A=(a_{ij})\;in\;Alg\mathcal{L}$ such that Ax=y. (2) There is a bounded sequence $\{\alpha_n\}$ in $\mathbb{C}$ such that $y_i=\alpha_ix_i\;for\;i{\in}\mathbb{N}$.

• Journal for History of Mathematics
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• v.25 no.4
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• pp.69-82
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• 2012

The objective of this paper is to study De Morgan's perspective on teaching and learning complex numbers. De Morgan's didactical approaches reflect the process of development of his thoughts about algebra from universal arithmetic, symbolic algebra to meaning algebra. De Morgan develop his perspective on algebra by justifying and explaining complex numbers. This implies that teaching and learning complex numbers is a catalyst for mathematical development of De Morgan.

• Bulletin of the Korean Mathematical Society
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• v.59 no.1
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• pp.73-82
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• 2022

Let K be a field of characteristic zero. We first show that images of the linear derivations and the linear 𝓔-derivations of the polynomial algebra K[x] = K[x₁, x₂, …, x_n] are ideals if the products of any power of eigenvalues of the matrices according to the linear derivations and the linear 𝓔-derivations are not unity. In addition, we prove that the images of D and 𝛿 are Mathieu-Zhao spaces of the polynomial algebra K[x] if D = ∑ⁿ_i=1 (a_ix_i + b_i)∂_i and 𝛿 = I - 𝜙, 𝜙(x_i) = λ_ix_i + 𝜇_i for a_i, b_i, λ_i, 𝜇_i ∈ K for 1 ≤ i ≤ n. Finally, we prove that the image of an affine 𝓔-derivation of the polynomial algebra K[x₁, x₂] is a Mathieu-Zhao space of the polynomial algebra K[x₁, x₂]. Hence we give an affirmative answer to the LFED Conjecture for the affine 𝓔-derivations of the polynomial algebra K[x₁, x₂].

• Bulletin of the Korean Mathematical Society
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• v.60 no.6
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• pp.1687-1695
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• 2023

In this paper, we study the Green ring r(𝔴⁰_n) of the weak Hopf algebra 𝔴⁰_n based on Taft Hopf algebra H_n(q). Let R(𝔴⁰_n) := r(𝔴⁰_n) ⊗_ℤ ℂ be the Green algebra corresponding to the Green ring r(𝔴⁰_n). We first determine all finite dimensional simple modules of the Green algebra R(𝔴⁰_n), which is based on the observations of the roots of the generating relations associated with the Green ring r(𝔴⁰_n). Then we show that the nilpotent elements in r(𝔴⁰_n) can be written as a sum of finite dimensional indecomposable projective 𝔴⁰_n-modules. The Jacobson radical J(r(𝔴⁰_n)) of r(𝔴⁰_n) is a principal ideal, and its rank equals n - 1. Furthermore, we classify all finite dimensional non-simple indecomposable R(𝔴⁰_n)-modules. It turns out that R(𝔴⁰_n) has n² - n + 2 simple modules of dimension 1, and n non-simple indecomposable modules of dimension 2.

• Honam Mathematical Journal
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• v.27 no.4
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• pp.595-602
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• 2005

Comtrans algebras are modules equipped with two trilinear operations: a left alternative commutator and a translator satisfying the Jacobi identity, the commutator and translator being connected by the so-called comtrans identity. These identities have analogues for trilinear forms. On a given vector space, the set of all comtrans algebra structures itself forms a vector space. In this paper, the dimension of the space of comtrans algebra structures on a finite-dimensional vector space is determined.

• Journal of the Chungcheong Mathematical Society
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• v.26 no.2
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• pp.357-365
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• 2013

Fix $n-2$ elements $h_1,{\cdots},h_{n-2}$ of the quotient field B of the polynomial algebra $\mathbb{C}[x_1,x_2,{\cdots},x_n]$. It is proved that B is a Poisson algebra with Poisson bracket defined by $\{f,g\}=det(Jac(f,g,h_1,{\cdots},h_{n-2})$ for any $f,g{\in}B$, where det(Jac) is the determinant of a Jacobian matrix.

• Honam Mathematical Journal
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• v.4 no.1
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• pp.167-172
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• 1982

Let A be a Banach $^{\ast}$-algebra and C(t) be a closed $^{\ast}$-subalgebra of A gengerated by $t{\in}A$. A is locally $B^{\ast}$-equivalent [$B^{\ast}$-equivalent] if C(t) [A] for every hermitian element t is $^{\ast}$-isomorphic to some $B^{\ast}$-algebra. It was proved that the locally $B^{\ast}$-equivalent algebras with some conditions is $B^{\ast}$-equivalent by B. A. Barnes. In this paper, we obtain the some conditions for a locally $B^{\ast}$-equivalent algebra to be $B^{\ast}$-equivalent.

• Honam Mathematical Journal
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• v.28 no.1
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• pp.57-67
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• 2006

we apply the hyperstructures to BCC-algebras, and introduce the concept of a hyper BCC-algebra which is a generalization of a BCC-algebra, and investigates some related properties. We also introduce the notion of a hyper BCC-subalgebra, BCC-scalar element and a hyper BCC-ideal, and discuss related properties.

• Honam Mathematical Journal
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• v.37 no.2
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• pp.221-233
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• 2015

The notion of a mighty (vague) filter in a BE-algebra is introduced, and the relation between a (vague) filter and a mighty (vague) filter are given. We investigate an equivalent condition for a (vague) filter to be mighty, and state an extension property for mighty filter. Also we define the notion of an n-fold mighty filter which is an extended notion of a mighty filter in a BE-algebra. Characterizations of an n-fold mighty filter are given. Extension property for an n-fold mighty filter are provided.