• Honam Mathematical Journal
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• v.34 no.1
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• pp.63-76
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• 2012

In this paper, we present the convolution sum ${\sum}_{m<n/8}{\sigma}_1(2m){\sigma}_1(n-8m)$ evaluated for all $n{\in}\mathbb{N}$.

• Bulletin of the Korean Mathematical Society
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• v.54 no.3
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• pp.935-952
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• 2017

It is natural to try to find a kernel such that its convolution of integrable functions converges faster than that of the $Fej{\acute{e}}r$ kernel. In this paper, we introduce a weighted Fourier partial sums which are written as the convolution of signed good kernels and prove that the weighted Fourier partial sum converges in $L^2$ much faster than that of the $Ces{\grave{a}}ro$ means. In addition, we present two numerical experiments.

• Journal of applied mathematics & informatics
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• v.39 no.3_4
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• pp.359-380
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• 2021

The main purpose and motivation of this work is to investigate and provide some new results for coefficients derived from eta quotients related to 3. The result of this paper involve some restricted divisor numbers and their convolution sums. Also, our results give relation between the coefficients derived from infinite product, infinite sum and the convolution sum of restricted divisor functions.

• Journal of applied mathematics & informatics
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• v.41 no.3
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• pp.553-567
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• 2023

There are real life situations in our lives where the things are changing continuously or from time to time. It is a very important problem for one whether to continue the existing relationship or to form a new one after some occasions. That is, people, companies, cities, countries, etc. may change their opinion or position rapidly. In this work, we think of the problem of changing relationships from a mathematical point of view and think of an answer. In some sense, we comment these changes as power changes. Our number theoretical model will be based on this idea. Using the convolution sum of the restricted divisor function E, we obtain the answer to this problem.

• Honam Mathematical Journal
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• v.35 no.3
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• pp.351-372
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• 2013

In this paper, we study a distinction the two generating functions : ${\varphi}^k(q)=\sum_{n=0}^{\infty}r_k(n)q^n$ and ${\varphi}^{*,k}(q)={\varphi}^k(q)-{\varphi}^k(q^2)$ ($k$ = 2, 4, 6, 8, 10, 12, 16), where $r_k(n)$ is the number of representations of $n$ as the sum of $k$ squares. We also obtain some congruences of representation numbers and divisor function.

• Journal of applied mathematics & informatics
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• v.31 no.1_2
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• pp.45-54
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• 2013

Let $$S^{\pm}_{(n,k)}\;:=\{(a,b,x,y){\in}\mathbb{N}^4:ax+by=n,x{\equiv}{\pm}y\;(mod\;k)\}$$. From the formula $\sum_{(a,b,x,y){\in}S^{\pm}_{(n,k)}}\;ab=4\sum_{^{m{\in}\mathbb{N}}_{m<n/k}}\;{\sigma}_1(m){\sigma}_1(n-km)+\frac{1}{6}{\sigma}_3(n)-\frac{1}{6}{\sigma}_1(n)-{\sigma}_3(\frac{n}{k})+n{\sigma}_1(\frac{n}{k})$, we find the Diophantine solutions for modulo $2^{m^{\prime}}$ and $3^{m^{\prime}}$, where $m^{\prime}{\in}\mathbb{N}$.

• Journal of the Korean Mathematical Society
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• v.58 no.5
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• pp.1147-1180
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• 2021

By considering the polynomial function 𝜙_car(z) = 1 + z + z²/2, we define the class 𝓢^*_car consisting of normalized analytic functions f such that zf'/f is subordinate to 𝜙_car in the unit disk. The inclusion relations and various radii constants associated with the class 𝓢^*_car and its connection with several well-known subclasses of starlike functions is established. As an application, the obtained results are applied to derive the properties of the partial sums and convolution.

• Bulletin of the Korean Mathematical Society
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• v.39 no.1
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• pp.123-131
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• 2002

Let A(p, k) (p, k$\in$N) be the class of functions f(z) = $z^{p}$ + $a_{p+k}$ $z^{p+k}$+… analytic in the unit disk. We introduce a subclass H(p, k, λ, $\delta$, A, B) of A(p, k) by using the Ruscheweyh derivative. The object of the present paper is to show some properties of functions in the class H(p, k, λ, $\delta$, A, B). B).

• Journal of applied mathematics & informatics
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• v.8 no.3
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• pp.731-742
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• 2001

A Gaussian smoothing algorithm obtained from a cascade of convolutions with a seven-point kernel is described. We prove that the change of local sums after applying our algorithm to sinusoidal signals is reduced to about two thirds of the change by the binomial coefficients. Hence, our seven point kernel is better than the binomial coefficients when trend curves are needed to be generated. We also prove that if our Gaussian convolution is applied to sinusoidal functions, the amplitude of higher frequencies reduces faster than the lower frequencies and hence that it is a low pass filter.

• Journal of Korea Game Society
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• v.13 no.2
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• pp.49-58
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• 2013

In this study, a tree growth model is designed to represent a variety of trees consisting of a outdoor terrain of game efficiently and naturally. The proposed tree growth model is an integrated tree growth model, and is configured using the following approaches: (1) the tree modeling method based on growth volume and the convolution sums of divisor functions, which is used to model a variety kind of trees more intuitively and naturally; (2) a rendering method using a level of detail of branch based on instancing for real-time processing of numerous trees with complicated structures; and (3) a combination of the above methods to efficiently implement a game landscape. The natural and diverse growths of trees that emerged using the proposed tree growth model is evaluated through experimentation, along with the possibility of implementing the natural game landscape and the efficiency of real-time processing.