The Jacobson radical of inverse Laurent power series of rings

2018 ◽  
Vol 17 (04) ◽  
pp. 1850061 ◽  
Author(s):  
S. Karimi ◽  
Sh. Sahebi ◽  
M. Habibi
Keyword(s):  
Differential Operator ◽  
Power Series ◽  
Large Class ◽  
Jacobson Radical ◽  
Pseudo Differential Operator ◽  
Operator Ring ◽  
Derivation Formula ◽  
Differential Operator Ring ◽  
Text Condition

For a ring [Formula: see text] with a derivation [Formula: see text], we determine the Jacobson radical of the pseudo-differential operator ring [Formula: see text] for a large class of rings with a [Formula: see text]-condition. Various types of examples of these rings are provided.

On Annihilator Ideals of Pseudo-differential Operator Rings

2015 ◽  
Vol 22 (04) ◽  
pp. 607-620 ◽  
Author(s):  
R. Manaviyat ◽  
A. Moussavi
Keyword(s):  
Differential Operator ◽  
Countable Subset ◽  
Pseudo Differential Operator ◽  
Rigid Ring ◽  
Baer Ring ◽  
Operator Ring ◽  
Operator Type ◽  
Differential Operator Ring ◽  
Zip Ring ◽  
Armendariz Ring

Let R be a ring with a derivation δ and R((x-1; δ)) denote the pseudo-differential operator ring over R. We study the relations between the set of annihilators in R and the set of annihilators in R((x-1; δ)). Among applications, it is shown that for an Armendariz ring R of pseudo-differential operator type, the ring R((x-1; δ)) is Baer (resp., quasi-Baer, PP, right zip) if and only if R is a Baer (resp., quasi-Baer, PP, right zip) ring. For a δ-weakly rigid ring R, R((x-1; δ)) is a left p.q.-Baer ring if and only if R is left p.q.-Baer and every countable subset of left semicentral idempotents of R has a generalized countable join in R.

The Mehler-Fock-Clifford transform and pseudo-differential operator on function spaces

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2457-2469
Author(s):  
Akhilesh Prasad ◽  
S.K. Verma
Keyword(s):  
Differential Operator ◽  
Function Spaces ◽  
Pseudo Differential Operator ◽  
Test Function ◽  
Basic Properties ◽  
Cone Function ◽  
Translation Operators

In this article, weintroduce a new index transform associated with the cone function Pi ??-1/2 (2?x), named as Mehler-Fock-Clifford transform and study its some basic properties. Convolution and translation operators are defined and obtained their estimates under Lp(I, x-1/2 dx) norm. The test function spaces G? and F? are introduced and discussed the continuity of the differential operator and MFC-transform on these spaces. Moreover, the pseudo-differential operator (p.d.o.) involving MFC-transform is defined and studied its continuity between G? and F?.

scholarly journals Some properties of generalized hypergeometric Appell polynomials

2020 ◽  
Vol 12 (1) ◽  
pp. 129-137 ◽  
Author(s):  
L. Bedratyuk ◽  
N. Luno
Keyword(s):  
Differential Operator ◽  
Power Series ◽  
Hypergeometric Function ◽  
Explicit Formula ◽  
Series Representation ◽  
Generalized Hypergeometric Function ◽  
Appell Polynomials ◽  
Pochhammer Symbol ◽  
Formal Power ◽  
Polynomial Family

Let $x^{(n)}$ denotes the Pochhammer symbol (rising factorial) defined by the formulas $x^{(0)}=1$ and $x^{(n)}=x(x+1)(x+2)\cdots (x+n-1)$ for $n\geq 1$. In this paper, we present a new real-valued Appell-type polynomial family $A_n^{(k)}(m,x)$, $n, m \in {\mathbb{N}}_0$, $k \in {\mathbb{N}},$ every member of which is expressed by mean of the generalized hypergeometric function ${}_{p} F_q \begin{bmatrix} \begin{matrix} a_1, a_2, \ldots, a_p \:\\ b_1, b_2, \ldots, b_q \end{matrix} \: \Bigg| \:z \end{bmatrix}= \sum\limits_{k=0}^{\infty} \frac{a_1^{(k)} a_2^{(k)} \ldots a_p^{(k)}}{b_1^{(k)} b_2^{(k)} \ldots b_q^{(k)}} \frac{z^k}{k!}$ as follows $$ A_n^{(k)}(m,x)= x^n{}_{k+p} F_q \begin{bmatrix} \begin{matrix} {a_1}, {a_2}, {\ldots}, {a_p}, {\displaystyle -\frac{n}{k}}, {\displaystyle -\frac{n-1}{k}}, {\ldots}, {\displaystyle-\frac{n-k+1}{k}}\:\\ {b_1}, {b_2}, {\ldots}, {b_q} \end{matrix} \: \Bigg| \: \displaystyle \frac{m}{x^k} \end{bmatrix} $$ and, at the same time, the polynomials from this family are Appell-type polynomials. The generating exponential function of this type of polynomials is firstly discovered and the proof that they are of Appell-type ones is given. We present the differential operator formal power series representation as well as an explicit formula over the standard basis, and establish a new identity for the generalized hypergeometric function. Besides, we derive the addition, the multiplication and some other formulas for this polynomial family.

scholarly journals Regular Bohr-Sommerfeld quantization rules for a h-pseudo-differential operator. The method of positive commutators

2016 ◽  
Vol Volume 23 - 2016 - Special... ◽  
Author(s):  
Abdelwaheb Ifa ◽  
Michel Rouleux
Keyword(s):  
Differential Operator ◽  
Gram Matrix ◽  
Pseudo Differential Operator ◽  
Quantization Rule ◽  
International Audience ◽  
Scalar Products ◽  
Quantization Rules

International audience We revisit in this Note the well known Bohr-Sommerfeld quantization rule (BS) for a 1-D Pseudo-differential self-adjoint Hamiltonian within the algebraic and microlocal framework of Helffer and Sjöstrand; BS holds precisely when the Gram matrix consisting of scalar products of some WKB solutions with respect to the " flux norm " is not invertible. Dans le cadre algébrique et microlocal élaboré par Helffer et Sjöstrand, on propose une ré-écriture de la règle de quantification de Bohr-Sommerfeld pour un opérateur auto-adjoint h-Pseudo-différentiel 1-D; elle s'exprime par la non-inversibilité de la matrice de Gram d'un couple de solutions WKB dans une base convenable, pour le produit scalaire associé à la " norme de flux " .

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