Relative (p,q)-φ order and relative (p,q)-φ type based on some growth properties of composite p-adic entire functions

2020 ◽  
Vol 29 (1) ◽  
pp. 09-16
Author(s):  
Biswas Tanmay
Keyword(s):  
Weak Type ◽  
Entire Functions ◽  
Closed Field ◽  
Unbounded Function ◽  
Algebraically Closed Field ◽  
Growth Properties ◽  
Positive Integers

Let K be a complete ultrametric algebraically closed field and A (K) be the K-algebra of entire functions on K. For any p adic entire functions f ∈ A (K) and r > 0, we denote by |f| (r) the number sup {|f (x) | : |x| = r} where |·| (r) is a multiplicative norm on A (K) . In this paper we study some growth properties of composite p-adic entire functions on the basis of their relative (p, q)-ϕ order, relative (p, q)-ϕ type and relative (p, q)-ϕ weak type where p, q are any two positive integers and ϕ (r) : [0, +∞) → (0, +∞) is a non-decreasing unbounded function of r.

Some growth properties of composite p-adic entire functions on the basis of their relative order and relative lower order

2019 ◽  
Vol 12 (03) ◽  
pp. 1950044
Author(s):  
Tanmay Biswas
Keyword(s):  
Growth Rates ◽  
Lower Order ◽  
Entire Functions ◽  
Closed Field ◽  
Relative Order ◽  
Algebraically Closed Field ◽  
Growth Properties

Let [Formula: see text] be a complete ultrametric algebraically closed field and [Formula: see text] be the [Formula: see text]-algebra of entire functions on [Formula: see text]. For [Formula: see text], [Formula: see text], we wish to introduce the notions of relative order and relative lower order of [Formula: see text] with respect to [Formula: see text]. Hence, after proving some basic results, in this paper, we estimate some growth rates of composite p-adic entire functions on the basis of their relative orders and relative lower orders.

scholarly journals $(p,q)$th order oriented growth measurement of composite $p$-adic entire functions

2018 ◽  
Vol 10 (2) ◽  
pp. 248-272
Author(s):  
T. Biswas
Keyword(s):  
Lower Order ◽  
Complex Analysis ◽  
Entire Functions ◽  
Log P ◽  
Closed Field ◽  
Order Of Growth ◽  
Oriented Growth ◽  
Growth Measurement ◽  
Growth Properties ◽  
Positive Integers

Let $\mathbb{K}$ be a complete ultrametric algebraically closed field and let $\mathcal{A}\left(\mathbb{K}\right)$ be the $\mathbb{K}$-algebra of entire functions on $\mathbb{K}$. For any $p$-adic entire function $f\in \mathcal{A}\left( \mathbb{K}\right) $ and $r>0$, we denote by $|f|\left(r\right)$ the number $\sup \left\{ |f\left( x\right) |:|x|=r\right\}$, where $\left\vert \cdot \right\vert (r)$ is a multiplicative norm on $\mathcal{A}\left( \mathbb{K}\right)$. For any two entire functions $f\in \mathcal{A}\left(\mathbb{K}\right)$ and $g\in \mathcal{A}\left(\mathbb{K}\right)$ the ratio $\frac{|f|(r)}{|g|(r)}$ as $r\rightarrow \infty $ is called the comparative growth of $f$ with respect to $g$ in terms of their multiplicative norms. Likewise to complex analysis, in this paper we define the concept of $(p,q)$th order (respectively $(p,q)$th lower order) of growth as $\rho ^{\left( p,q\right) }\left( f\right) =\underset{r\rightarrow +\infty }{\lim \sup } \frac{\log ^{[p]}|f|\left( r\right) }{\log ^{\left[ q\right] }r}$ (respectively $\lambda ^{\left( p,q\right) }\left( f\right) =\underset{ r\rightarrow +\infty }{\lim \inf }\frac{\log ^{[p]}|f|\left( r\right) }{\log ^{\left[ q\right] }r}$), where $p$ and $q$ are any two positive integers. We study some growth properties of composite $p$-adic entire functions on the basis of their $\left(p,q\right)$th order and $(p,q)$th lower order.

scholarly journals On the(p,q)th Relative Order Oriented Growth Properties of Entire Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Luis Manuel Sánchez Ruiz ◽  
Sanjib Kumar Datta ◽  
Tanmay Biswas ◽  
Golok Kumar Mondal
Keyword(s):  
Quantitative Assessment ◽  
Entire Functions ◽  
Relative Order ◽  
Order Of Growth ◽  
Alternative Definition ◽  
Oriented Growth ◽  
Growth Properties ◽  
Self Similar ◽  
Definition Of ◽  
Positive Integers

The relative order of growth gives a quantitative assessment of how different functions scale each other and to what extent they are self-similar in growth. In this paper for any two positive integerspandq, we wish to introduce an alternative definition of relative(p,q)th order which improves the earlier definition of relative(p,q)th order as introduced by Lahiri and Banerjee (2005). Also in this paper we discuss some growth rates of entire functions on the basis of the improved definition of relative(p,q)th order with respect to another entire function and extend some earlier concepts as given by Lahiri and Banerjee (2005), providing some examples of entire functions whose growth rate can accordingly be studied.

scholarly journals Growth Estimates of Composite Entire Functions in the Light of Slowly Changing Functions Based Relative Order, Relative Type and Relative Weak Type

2015 ◽  
Vol 54 (1) ◽  
pp. 59-74
Author(s):  
S. K. Datta ◽  
T. Biswas ◽  
S. Bhattacharyya
Keyword(s):  
Weak Type ◽  
Entire Functions ◽  
Maximum Modulus ◽  
Relative Order ◽  
Maximum Term ◽  
Growth Properties ◽  
Relative Type

Abstract In the paper we prove some growth properties of maximum term and maximum modulus of composition of entire functions on the basis of relative L*-order, relative L*-type and relative L*-weak type.

scholarly journals Algebraic Barth-Lefschetz theorems

1996 ◽  
Vol 142 ◽  
pp. 17-38 ◽  
Author(s):  
Lucian Bădescu
Keyword(s):  
Positive Integer ◽  
Projective Space ◽  
Algebraic Variety ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Arbitrary Characteristic ◽  
Basic Properties ◽  
Positive Integers ◽  
Lefschetz Theorems ◽  
Algebraic Scheme

We shall work over a fixed algebraically closed field k of arbitrary characteristic. By an algebraic variety over k we shall mean a reduced algebraic scheme over k. Fix a positive integer n and e = (e0, el,…, en) a system of n + 1 weights (i.e. n + 1 positive integers e0, el,…, en). If k[T0, Tl,…, Tn] is the polynomial k-algebra in n + 1 variables, graded by the conditions deg(Ti) = ei i = 0, 1,…, n, denote by Pn(e) = Proj(k[T0, T1,…, Tn]) the n-dimensional weighted projective space over k of weights e. We refer the reader to [3] for the basic properties of weighted projective spaces.

INTERPOLATION FOR P-ADIC ENTIRE FUNCTIONS

2010 ◽  
Vol 03 (02) ◽  
pp. 251-262
Author(s):  
Kamal Boussaf
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Interpolation Functions

Let K be a complete ultrametric algebraically closed field. This paper is aimed at studying problems of interpolation for entire functions on K. We prove an analogue of M . Lazard's divisors theorem for entire functions and then we show that sequences of interpolation are the injective sequences (ai)i∈ N in K such that lim i→∞ |ai| = +∞. This result does not assume K to be spherically complete. In order to construct concrete entire interpolation functions, we introduce Lagrange entire functions and define L-sequences. We show that a sequence (ai)i∈ N in K defines a Lagrange entire function if and only if (ai)i∈ N is an L-sequence.

scholarly journals ON THE MEASUREMENT OF GROWTH PROPERTIES OF ENTIRE AND MEROMORPHIC FUNCTIONS FOCUSING THEIR RELATIVE TYPE AND RELATIVE WEAK TYPE

2017 ◽  
pp. 1011
Author(s):  
Sanjib Kumar Datta ◽  
Tanmay Biswas
Keyword(s):  
Meromorphic Functions ◽  
Weak Type ◽  
Entire Functions ◽  
Order Type ◽  
Relative Order ◽  
Relative Growth ◽  
Growth Properties ◽  
Relative Type ◽  
Growth Indicators

The concepts of relative growth indicators such as relative order, relative type, relative weak type, etc. have widely been used to avoid comparing growths of entire and meromorphic functions just with exp functions. Using the notions of several relative growth indicators as mentioned earlier, in this paper we would like to find out the limits in terms of classical growth indicators (i.e. order, type, weak type etc.) in which the relative type, relative weak type, etc. of meromorphic functions with respect to entire functions should lie.

scholarly journals On Some Growth Properties of Entire Functions Using Their Maximum Moduli Focusingp,qth Relative Order

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Luis Manuel Sanchez Ruiz ◽  
Sanjib Kumar Datta ◽  
Tanmay Biswas ◽  
Golok Kumar Mondal
Keyword(s):  
Entire Function ◽  
Growth Rates ◽  
Lower Order ◽  
Entire Functions ◽  
Relative Order ◽  
Growth Properties ◽  
Definition Of ◽  
Positive Integers

We discuss some growth rates of composite entire functions on the basis of the definition of relativep,qth order (relativep,qth lower order) with respect to another entire function which improve some earlier results of Roy (2010) wherepandqare any two positive integers.

scholarly journals Generalized Relative Type and Generalized Weak Type of Entire Functions

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
Sanjib Kumar Datta ◽  
Tanmay Biswas ◽  
Debasmita Dutta
Keyword(s):  
Entire Function ◽  
Weak Type ◽  
Entire Functions ◽  
Relative Growth ◽  
Growth Properties ◽  
Relative Type

We study some relative growth properties of entire functions with respect to another entire function on the basis of generalized relative type and generalized relative weak type.

-MINIMUM SPANNING LENGTHS AND AN EXTENSION TO BURNSIDE’S THEOREM ON IRREDUCIBILITY

2020 ◽  
pp. 1-16
Author(s):  
W. E. LONGSTAFF
Keyword(s):  
Closed Field ◽  
Algebraically Closed Field ◽  
Complete Calculation ◽  
Jordan Matrix ◽  
Invertible Matrices ◽  
Positive Integers

Abstract We introduce the $\textbf{h}$ -minimum spanning length of a family ${\mathcal A}$ of $n\times n$ matrices over a field $\mathbb F$ , where $\textbf{h}$ is a p-tuple of positive integers, each no more than n. For an algebraically closed field $\mathbb F$ , Burnside’s theorem on irreducibility is essentially that the $(n,n,\ldots ,n)$ -minimum spanning length of ${\mathcal A}$ exists if ${\mathcal A}$ is irreducible. We show that the $\textbf{h}$ -minimum spanning length of ${\mathcal A}$ exists for every $\textbf{h}=(h_1,h_2,\ldots , h_p)$ if ${\mathcal A}$ is an irreducible family of invertible matrices with at least three elements. The $(1,1, \ldots ,1)$ -minimum spanning length is at most $4n\log _{2} 2n+8n-3$ . Several examples are given, including one giving a complete calculation of the $(p,q)$ -minimum spanning length of the ordered pair $(J^*,J)$ , where J is the Jordan matrix.

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