Explore the vast range of titles available.

We consider -direct sums (1 ) and -direct sums of countably many normed spaces and find the dual of these spaces. We characterize the support functionals of arbitrary elements in these spaces to characterize smoothness and approximate smoothness, both locally and globally. These results let us answer the Chmieliński, Khurana, and Sain question raised in [4] on the existence of a non-approximately smooth normed space whose every element is smooth. We also characterize Birkhoff-James orthogonality and its pointwise symmetry in these spaces.

In this article, by making use of a q-analogue of the familiar Borel distribution, we introduce two new subclasses: S symmetric α , λ , q ( b , A , B ) and S conjugate α , λ , q ( b , A , B ) of starlike and convex functions in the open unit disk Δ with respect to symmetric and conjugate points. We obtain some properties including the Taylor-Maclaurin coefficient estimates for functions in each of these subclasses and deduce various corollaries and consequences of the main results. We also indicate relevant connections of each of these subclasses S symmetric α , λ , q ( b , A , B ) and S conjugate α , λ , q ( b , A , B ) with the function classes which were investigated in several earlier works. Finally, in the concluding section, we choose to comment on the recent usages, especially in Geometric Function Theory of Complex Analysis, of the basic (or q-) calculus and also of its trivial and inconsequential (p, q)-variation involving an obviously redundant (or superfluous) parameter p.

The aim of the present paper is to find the necessary and sufficient conditions for subclasses of starlike functions with respect to symmetric points, starlike functions with respect to conjugate points, starlike functions with respect to symmetric conjugate points associated with Pascal distribution series and inclusion relations for such subclasses in the open unit disk U. Further, we consider an integral operator related to Pascal distribution series. Keywords: Analytic functions, Starlike functions with respect to symmetric points, Starlike functions with respect to conjugate points, Starlike functions with respect to symmetric conjugate points, Pascal distribution series. AMS Subject Classification: 30C45.

Accurate and consistent annual runoff prediction in a region is a hot topic in management, optimization, and monitoring of water resources. A novel prediction model (ESMD-SE-WPD-LSTM) is presented in this study. Firstly, extreme-point symmetric mode decomposition (ESMD) is used to produce several intrinsic mode functions (IMF) and a residual (Res) by decomposing the original runoff series. Secondly, sample entropy (SE) method is employed to measure the complexity of each IMF. Thirdly, wavelet packet decomposition (WPD) is adopted to further decompose the IMF with the maximum SE into several appropriate components. Then long short-term memory (LSTM) model, a deep learning algorithm based recurrent approach, is employed to predict all components. Finally, forecasting results of all components are aggregated to generate the final prediction. The proposed model, which is applied to seven annual series from different areas in China, is evaluated based on four evaluation indexes (R, MAE, MAPE and RMSE). Results indicate that ESMD-SE-WPD-LSTM outperforms other benchmark models in terms of four evaluation indexes. Hence the proposed model can provide higher accuracy and consistency for annual runoff prediction, rendering it an efficient instrument for scientific management and planning of water resources.

In this paper, we introduce new classes of functions defined within the open unit disk by integrating the concepts of (λ,γ)-symmetrical functions, generalized Janowski functions, and quantum calculus. We derive a structural formula and a representation theorem for the class Sqλ,γ(x,y,z). Utilizing convolution techniques and quantum calculus, we investigate convolution conditions supported by examples and corollary, establishing sufficient conditions. Additionally, we derive properties related to coefficient estimates, which further elucidate the characteristics of the defined function classes.

Drought is one of the severe natural disasters to impact human society and occurs widely and frequently in China, causing considerable damage to the living environment of humans. The Yellow River basin (YRB) of China shows great vulnerability to drought in the major basins; thus, drought monitoring in the YRB is particularly important. Based on monthly data of 124 meteorological stations from 1961 to 2015, the Standardized Precipitation Evapotranspiration Index (SPEI) was used to explore the temporal and spatial patterns of drought in the YRB. The periods and trends of drought were identified by Extreme-point Symmetric Mode Decomposition (ESMD), and the research stages were determined by Bernaola-Galvan Segmentation Algorithm (BGSA). The annual and seasonal variation, frequency and intensity of drought were studied in the YRB. The results indicated that (1) for the past 55 years, the drought in the YRB has increased significantly with a tendency rate of −0.148 (10 a) −1 , in which the area Lanzhou to Hekou was the most vulnerable affected (−0.214 (10 a) −1 ); (2) the drought periods (2.9, 5, 10.2 and 18.3 years) and stages (1961–1996, 1997–2002 and 2003–2015) were characterized and detected by ESMD and BGSA; (3) the sequence of drought frequency was summer, spring, autumn and winter with mean values of 71.0%, 47.2%, 10.2% and 6.9%, respectively; and (4) the sequence of drought intensity was summer, spring, winter and autumn with mean values of 0.93, 0.40, 0.05 and 0.04, respectively.

In this paper, we investigate the sharp bounds of the second Hankel determinant of Logarithmic coefficients for the starlike and convex functions with respect to symmetric points in the open unit disk.

In our current study, we apply differential subordination and quantum calculus to introduce and investigate a new class of analytic functions associated with the q-differential operator and the symmetric balloon-shaped domain. We obtain sharp results concerning the Maclaurin coefficients the second and third-order Hankel determinants, the Zalcman conjecture, and its generalized conjecture for this newly defined class of q-starlike functions with respect to symmetric points.

Let Ss*(ez) denote the class of starlike functions with respect to symmetric points subordinate to the exponential function, i.e., the functions which satisfy in the unit disk U the condition 2zf′(z)f(z)−f(−z)≺ez(z∈U). We obtained the sharp estimate of the second-order Hankel determinants H2,3(f) and improved the estimate of the third-order H3,1(f) for this functions class Ss*(ez).