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This paper applies the concepts of fuzzifying functions to the probability density function of a random variable and introduce a fuzzifying probability to better understand the probability arising from the uncertainties of the probability density function. Using the fuzzifying probability, we derive the fuzzifying expected value and the fuzzifying variance of a random variable with the fuzzifying probability density function. Additionally, we provide examples of a fuzzifying probability density function to validate that the proposed fuzzy concepts generalize crisp expected value and variance in probability theory.

In this article, we provide a comprehensive convergence and stability analysis of a semi-Lagrangian scheme for solving nonlinear Burgers' equations with a high-order spatial discretization. The analysis is for the iteration-free semi-Lagrangian scheme comprising the second-order backward finite difference formula (BDF2) for total derivative and the fourth-order central finite difference for diffusion term along the trajectory. The main highlight of the study is to thoroughly analyze the order of convergence of the discrete $ \ell_2 $-norm error $ \mathcal{O}(h^2+\triangle x^4+ \triangle x^{p+1}/h) $ by managing the relationship between the local truncation errors from each discretization procedure and the interpolation properties with a symmetric high-order discretization of the diffusion term. Furthermore, stability is established by the uniform boundedness of the numerical solution using the discrete Grönwall's Lemma. We provide numerical examples to support the validity of the theoretical convergence and stability analysis for the propounded backward semi-Lagrangian scheme.

The degenerate versions of special polynomials and numbers, initiated by Carlitz, have regained the attention of some mathematicians by replacing the usual exponential function in the generating function of special polynomials with the degenerate exponential function. To study the relations between degenerate special polynomials, $ \lambda $-umbral calculus, an analogue of umbral calculus, is intensively applied to obtain related formulas for expressing one $ \lambda $-Sheffer polynomial in terms of other $ \lambda $-Sheffer polynomials. In this paper, we study the connection between degenerate higher-order Daehee polynomials and other degenerate type of special polynomials. We present explicit formulas for representations of the polynomials using $ \lambda $-umbral calculus and confirm the presented formulas between the degenerate higher-order Daehee polynomials and the degenerate Bernoulli polynomials, for example. Additionally, we investigate the pattern of the root distribution of these polynomials.

This paper analyzes the stability of numerical solutions for a nonlinear Schrödinger equation that is widely used in several applications in quantum physics, optical business, etc. One of the most popular approaches to solving nonlinear problems is the application of a linearization scheme. In this paper, two linearization schemes—Newton and Picard methods were utilized to construct systems of linear equations and finite difference methods. Crank-Nicolson and backward Euler methods were used to establish numerical solutions to the corresponding linearized problems. We investigated the stability of each system when a finite difference discretization is applied, and the convergence of the suggested approximation was evaluated to verify theoretical analysis.

Research on the degenerate versions of special polynomials provides a new area, introducing the λ-analogue of special polynomials and numbers, such as λ-Sheffer polynomials. In this paper, we propose a new variant of type 2 Bernoulli polynomials and numbers by modifying a generating function. Then we derive explicit expressions and their representations that provide connections among existing λ-Sheffer polynomials. Also, we provide the explicit representations of the proposed polynomials in terms of the degenerate Lah-Bell polynomials and the higher-order degenerate derangement polynomials to confirm the presented identities.

We present a new type of degenerate poly-Bernoulli polynomials and numbers by modifying the polyexponential function in terms of the degenerate exponential functions and degenerate logarithm functions. Also, we introduce a new variation of the degenerate unipoly-Bernoulli polynomials by the similar modification. Based on these polynomials, we investigate some properties, new identities, and their relations to the known special functions and numbers such as the degenerate type 2-Bernoulli polynomials, the type 2 degenerate Euler polynomials, the degenerate Bernoulli polynomials and numbers, the degenerate Stirling numbers of the first kind, and λ-falling factorial sequence. In addition, we compute some of the proposed polynomials and present their zeros and behaviors for different variables in specific cases.

Abstract Background Epidermal growth factor receptor (EGFR) mutations in non–small-cell lung cancer predict sensitivity to EGFR tyrosine kinase inhibitors (TKIs). EGFR mutation types are associated with efficacy of EGFR TKIs. We investigated the clinical outcomes of afatinib, erlotinib, and gefitinib according to EGFR mutation type in patients with lung adenocarcinoma. Methods Between May 2010 and December 2018, we investigated 363 patients with advanced lung adenocarcinoma harboring EGFR mutations who received EGFR TKIs. Efficacies of EGFR TKIs such as response rate, progression-free survival (PFS), and overall survival (OS) were retrospectively evaluated according to exon 19 deletion (E19del), L858R point mutation (L858R) and uncommon mutations. Results The frequency of E19del was 48.2%, that of L858R was 42.4%, and that of uncommon mutations was 9.4%. E19del and L858R were associated with superior PFS and OS compared with uncommon mutations. Erlotinib showed significantly inferior OS than other TKIs (30.8 ± 3.3 in erlotinib vs. 39.1 ± 4.3 in afatinib vs. 48.4 ± 6.3 in gefitinib; p = 0.031) in patients with L858R. Gefitinib showed significantly inferior PFS (4.6 ± 1.1 in gefitinib vs. 11.6 ± 2.7 in afatinib vs. 10.6 ± 2.7 in erlotinib; p = 0.049) in patients with uncommon mutations. Conclusion Afatinib was significantly associated with a longer PFS, presenting constant effectiveness in all EGFR mutation types. Caution may be needed on the use of erlotinib for L858R and the use of gefitinib for uncommon EGFR mutations.

Among a remarkably large number of various extensions of polynomials and numbers, and diverse introductions of new polynomials and numbers, in this paper, we choose to introduce two new generalizations of some extended Bernoulli polynomials and numbers by using the Mittag–Leffler function and the confluent hypergeometric function. Then, we investigate certain properties and formulas of these newly introduced polynomials and numbers such as explicit representations, addition formulas, integral formulas, differential formulas, inequalities, and inequalities involving their integrals. Also, by using the theory of umbral calculus, five additional formulas regarding these new polynomials are provided. Furthermore, we propose to introduce four generalizations of the extended Euler and Genocchi polynomials. Finally, three natural problems are poised.

Photoswitchable nanoparticles can be used for selective imaging in biological systems but usually have only one color. Here the authors develop a two-color fluorescent emissive system that allows full on-off switching of one component color of the system while the other color is unaffected, which has implications for super-resolution imaging.

Abstract In this paper, we introduce the extended r-central factorial numbers of the second and first kinds and the extended r-central Bell polynomials, as extended versions and central analogues of some previously introduced numbers and polynomials. Then we study various properties and identities related to these numbers and polynomials and also their connections.

Abstract Recently, the central Fubini polynomials were introduced in connection with central factorial numbers of the second kind. In this paper, we consider two variable higher-order central Fubini polynomials as a ‘central analogue’ of two variable higher-order Fubini polynomials. We investigate some properties, identities, and recurrence relations for these polynomials by making use of generating functions and umbral calculus. In particular, we obtain various expressions for the two variable higher-order central Fubini polynomials and express them in terms of some families of special polynomials and vice versa.

Abstract The classical linearization problem concerns with determining the coefficients in the expansion of the product of two polynomials in terms of any given sequence of polynomials. As a generalization of this, we consider here sums of finite products of Chebyshev polynomials of the first, third, and fourth kinds, which are different from the ones previously studied. We represent each of them as linear combinations of Hermite, extended Laguerre, Legendre, Gegenbauer, and Jacobi polynomials. Here, the coefficients involve some terminating hypergeometric functions F12 ${}_{2}F_{1}$, F22 ${}_{2}F_{2}$, and F11 ${}_{1}F_{1}$. These representations are obtained by explicit computations.

In this study, the double hydrothermal method is proposed as a facile approach to the synthesis of ZnTe/ZnO core–shell nanorods. The coating thickness of the p-type ZnTe is varied to adjust the junction depth in the n-type ZnO nanorods, and the conductance measurements reveal the change in the conduction path in the heterojunction structures. Structural and chemical investigations conducted using X-ray diffraction, transmission electron microscopy, and X-ray photoelectron spectroscopy confirm the hetero-nanostructure formation of ZnTe/ZnO. The role of ZnTe in H2S-gas sensing by the ZnO nanorods is discussed. The enhanced sensing performance observed with a thin ZnTe coating confirms the importance of the base resistance of the nano-transducer in achieving high response characteristics. The composite structure also demonstrates a superior sensing performance of good repeatability, stability, linearity, and gas selectivity at temperatures greater than 200 °C.

Abstract We consider a system of acoustic wave equation possessing lower-order perturbation terms in a bounded domain in R2 $\mathbb{R}^{2}$. In this paper, we show the system is well-posed and stable with energy decays introducing a local discontinuous Galerkin (LDG) method. Also, we study an a priori L2 $L^{2}$-norm error estimate for the semi-discretized LDG method for the system under additional regularity assumptions. Further, numerical tests are presented to support the theoretical analysis.

Adenocarcinoma is the most common type of non-small cell lung cancer. Some causative genomic alterations in epidermal growth factor receptor (EGFR), including deletions in exon 19 (E19 dels) and a point mutation in E21, are known to have favourable prognoses due to sensitivity to tyrosine kinase inhibitors; however, the prognoses of other uncommon mutations are unclear. This study analysed the clinical significance of EGFR mutation types in lung adenocarcinoma. We retrospectively reviewed 1,020 subjects (mean age: 66.8 years, female: 41.7%) who were diagnosed with advanced lung adenocarcinoma, had EGFR mutation data, and did not undergo surgery from five medical institutes between 2010 and 2016. Subjects were classified according to EGFR mutation status, particularly for exon-specific mutations. EGFR positivity was defined as the presence of mutation and EGFR negativity was defined as wild-type EGFR. EGFR positivity was 38.0%, with the incidence of mutations in E18, E19, E20, and E21 was 3.6%, 51.0%, 3.4%, and 42.0%, respectively. The EGFR positive group survived significantly longer than the negative group (p

Tin sulfide (SnS) is known for its effective gas-detecting ability at low temperatures. However, the development of a portable and flexible SnS sensor is hindered by its high resistance, low response, and long recovery time. Like other chalcogenides, the electronic and gas-sensing properties of SnS strongly depend on its surface defects. Therefore, understanding the effects of its surface defects on its electronic and gas-sensing properties is a key factor in developing low-temperature SnS gas sensors. Herein, using thin SnS films annealed at different temperatures, we demonstrate that SnS exhibits n-type semiconducting behavior upon the appearance of S vacancies. Furthermore, the presence of S vacancies imparts the n-type SnS sensor with better sensing performance under UV illumination at room temperature (25 °C) than that of a p-type SnS sensor. These results are thoroughly investigated using various experimental analysis techniques and theoretical calculations using density functional theory. In addition, n-type SnS deposited on a polyimide substrate can be used to fabricate high-stability flexible sensors, which can be further developed for real applications.

We study some properties of Sheffer sequences for some special polynomials with complex Changhee and Daehee polynomials introducing their complex versions of the polynomials and splitting them into real and imaginary parts using trigonometric polynomial sequences. Moreover, considering their degenerate types of Sheffer sequences based on umbral composition, we present some useful expressions, properties, and examples about complex versions of the degenerate polynomials.

Nanocomposite structures, where the Fe, Fe2O3, or Ni2O3 nanoparticles with thin carbon layers are distributed among a single-wall carbon nanotube (SWCNT) network, are architectured using the co-arc discharge method. A synergistic effect between the nanoparticles and SWCNT is achieved with the composite structures, leading to the enhanced sensing response in ammonia detection. Thorough studies about the correlation between the electric properties and sensing performance confirm the independent operation of the receptor and transducer in the sensor structure by nanoparticles and SWCNT, respectively. Nanoparticles with a large specific surface area provide adsorption sites for the NH3 gas molecules, whereas hole carriers are supplied by the SWCNT to complete the chemisorption process. A new chemo-resistive sensor concept and its operating mechanism is proposed in our work. Furthermore, the separated receptor and transducer sensor scheme allows us more freedom in the design of sensor materials and structures, thereby enabling the design of high-performance gas sensors.

Two-dimensional (2D) nanomaterials have distinct optical and electrical properties owing to their unique structures. In this study, smooth 2D amorphous tin disulfide (SnS2) films were fabricated by atomic layer deposition (ALD), and applied for the first time to photoelectrochemical water splitting. The optimal stable photocurrent density of the 50-nm-thick amorphous SnS2 film fabricated at 140 °C was 51.5 µA/cm2 at an oxygen evolution reaction (0.8 V vs. saturated calomel electrode (SCE)). This value is better than those of most polycrystalline SnS2 films reported in recent years. These results are attributed mainly to adjustable optical band gap in the range of 2.80 to 2.52 eV, precise control of the film thickness at the nanoscale, and the close contact between the prepared SnS2 film and substrate. Subsequently, the photoelectron separation mechanisms of the amorphous, monocrystalline, and polycrystalline SnS2 films are discussed. Considering above advantages, the ALD amorphous SnS2 film can be designed and fabricated according to the application requirements.

In the classical connection problem, it is dealt with determining the coefficients in the expansion of the product of two polynomials with regard to any given sequence of polynomials. As a generalization of this problem, we will consider sums of finite products of Fubini polynomials and represent these in terms of orthogonal polynomials. Here, the involved orthogonal polynomials are Chebyshev polynomials of the first, second, third and fourth kinds, and Hermite, extended Laguerre, Legendre, Gegenbauer, and Jabcobi polynomials. These representations are obtained by explicit computations.