• Journal of the Korean Mathematical Society
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• v.60 no.1
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• pp.33-69
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• 2023

Let α ∈ (0, ∞), p ∈ (0, ∞) and q(·) : ℝⁿ → [1, ∞) satisfy the globally log-Hölder continuity condition. We introduce the weak Herz-type Hardy spaces with variable exponents via the radial grand maximal operator and to give its maximal characterizations, we establish a version of the boundedness of the Hardy-Littlewood maximal operator M and the Fefferman-Stein vector-valued inequality on the weak Herz spaces with variable exponents. We also obtain the atomic and the molecular decompositions of the weak Herz-type Hardy spaces with variable exponents. As an application of the atomic decomposition we provide various equivalent characterizations of our spaces by means of the Lusin area function, the Littlewood-Paley g-function and the Littlewood-Paley $g^*_{\lambda}$-function.

• Honam Mathematical Journal
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• v.40 no.4
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• pp.771-783
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• 2018

This paper, we investigate a strongly damped nonlinear Klein-Gordon equation with nonlinearities of variable exponent type $$u_{tt}-{\Delta}u-{\Delta}u_t+m^2u+{\mid}u_t{\mid}^{p(x)-2}u_t={\mid}u{\mid}^{q(x)-2}u$$ associated with initial and Dirichlet boundary conditions in a bounded domain. We obtain a nonexistence of solutions if variable exponents p (.), q (.) and initial data satisfy some conditions.

• Journal of the Korean Mathematical Society
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• v.55 no.4
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• pp.939-962
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• 2018

In this paper, we show that the commutator of the intrinsic square function with BMO symbols is bounded on the variable exponent Lebesgue spaces $L^{p({\cdot})}({\mathbb{R}}^n)$ applying a generalization of the classical Rubio de Francia extrapolation. As a consequence we further establish its boundedness on the variable exponent Morrey spaces $\mathcal{M_{p({\cdot}),u}$, Morrey-Herz spaces $M{\dot{K}}^{{\alpha}({\cdot}),{\lambda}}_{q,p({\cdot})}({\mathbb{R}}^n)$ and Herz type Hardy spaces $H{\dot{K}}^{{\alpha}({\cdot}),q}_{p({\cdot})}({\mathbb{R}}^n)$, where the exponents ${\alpha}({\cdot})$ and $p({\cdot})$ are variable. Observe that, even when ${\alpha}({\cdot}){\equiv}{\alpha}$ is constant, the corresponding main results are completely new.

• Bulletin of the Korean Mathematical Society
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• v.58 no.2
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• pp.365-384
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• 2021

Let A be an expansive dilation on ℝn, and p(·) : ℝn → (0, ∞) be a variable exponent function satisfying the globally log-Hölder continuous condition. Let Hp(·)A (ℝn) be the variable anisotropic Hardy space defined via the non-tangential grand maximal function. In this paper, the author obtains the boundedness of anisotropic convolutional ��-type Calderón-Zygmund operators from Hp(·)A (ℝn) to Lp(·) (ℝn) or from Hp(·)A (ℝn) to itself. In addition, the author also obtains the duality between Hp(·)A (ℝn) and the anisotropic Campanato spaces with variable exponents.

• Bulletin of the Korean Mathematical Society
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• v.61 no.2
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• pp.529-540
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• 2024

In this paper, using the Fourier transform, inverse Fourier transform and Littlewood-Paley decomposition technique, we prove the boundedness of bilinear pseudodifferential operators with symbols in the bilinear Hörmander class $BS^{m}_{1,1}$ in variable Triebel-Lizorkin spaces and variable Besov spaces.

• Communications of the Korean Mathematical Society
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• v.23 no.4
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• pp.479-485
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• 2008

In this note, we study arithmetic properties for the exponents of modular forms on ${\Gamma}_0(p)$ for primes p. Our aim is to refine the result of [4] by using the geometric property of the modular curve of ${\Gamma}_0(p)$.

• Journal of the Institute of Electronics and Information Engineers
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• v.49 no.9
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• pp.183-195
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• 2012

We propose a variable-latency Decimal Floating Point(DFP) adder which adopts the dual data path scheme. It is to speed addition and subtraction of operand that has identical exponents. The proposed DFP adder makes use of L. K. Wang's operand alignment algorithm, but operates through high speed data-path in guaranteed accuracy range. Synthesis results show that the area of the proposed DFP adder is increased by 8.26% compared to the L. K. Wang's DFP adder, though critical path delay is reduced by 10.54%. It also operates at 13.65% reduced path than critical path in case of an operation which has two DFP operands with identical exponents. We prove that the proposed DFP adder shows higher efficiency than L. K. Wang's DFP adder when the ratio of identical exponents is larger than 2%.

• Journal of applied mathematics & informatics
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• v.32 no.1_2
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• pp.137-152
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• 2014

In this paper, we prove, in the spirit of [3, 12, 20, 22, 23], the existence of infinitely many small solutions to the following quasilinear elliptic equation $-{\Delta}_{p(x)}u+{\mid}u{\mid}^{p(x)-2}u={\mid}u{\mid}^{q(x)-2}u+{\lambda}f(x,u)$ in a smooth bounded domain ${\Omega}$ of ${\mathbb{R}}^N$. We also assume that $\{q(x)=p^*(x)\}{\neq}{\emptyset}$, where $p^*(x)$ = Np(x)/(N - p(x)) is the critical Sobolev exponent for variable exponents. The proof is based on a new version of the symmetric mountainpass lemma due to Kajikiya [22], and property of these solutions are also obtained.

• Journal of Korea Water Resources Association
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• v.30 no.1
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• pp.3-13
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• 1997

Modifications of Prandtl's mixing length theory were used to obtain a power velocity distribution in which the coefficient and exponent are variable over a range from 1/4 to 1/7. A simple suspended-sediment concentration distribution was developed which can be associated with this modified velocity distribution. Using nominal values of ${\beta}$=1.0, $\kappa$=0.4 and visual accumulation tube values of fall velocity, the comparison between theory and field measurements by the USGS on the Rio Grande is fair. Doubling the value of the exponent results in a good comparison. Further research is needed to be able to better choose ${\beta}$, $\kappa$, and fall velocity values, but such research will not be able to account for the effects of large-scale turbulence and secondary flows. In a pragmatic sense, a special set of fairly detailed measurements can establish coefficients and exponents for any gaging site.

• Journal of the Korean earth science society
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• v.38 no.4
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• pp.283-292
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• 2017

In this study, the empirical models were established to estimate the concentrations of surface-level $PM_{2.5}$ over Seoul, Korea from 1 January 2012 to 31 December 2013. We used six different multiple linear regression models with aerosol optical thickness (AOT), ${\AA}ngstr{\ddot{o}}m$ exponents (AE) data from Moderate Resolution Imaging Spectroradiometer (MODIS) aboard Terra and Aqua satellites, meteorological data, and planetary boundary layer depth (PBLD) data. The results showed that $M_6$ was the best empirical model and AOT, AE, relative humidity (RH), wind speed, wind direction, PBLD, and air temperature data were used as input data. Statistical analysis showed that the result between the observed $PM_{2.5}$ and the estimated $PM_{2.5}$ concentrations using $M_6$ model were correlations (R=0.62) and root square mean error ($RMSE=10.70{\mu}gm^{-3}$). In addition, our study show that the relation strongly depends on the seasons due to seasonal observation characteristics of AOT, with a relatively better correlation in spring (R=0.66) and autumntime (R=0.75) than summer and wintertime (R was about 0.38 and 0.56). These results were due to cloud contamination of summertime and the influence of snow/ice surface of wintertime, compared with those of other seasons. Therefore, the empirical multiple linear regression model used in this study showed that the AOT data retrieved from the satellite was important a dominant variable and we will need to use additional weather variables to improve the results of $PM_{2.5}$. Also, the result calculated for $PM_{2.5}$ using empirical multi linear regression model will be useful as a method to enable monitoring of atmospheric environment from satellite and ground meteorological data.