Search Results (91)

We study the half-integer family $F(n,p)={}_3{F}_2\left(-\frac{2n+1}{2},1,\frac{2p-1}{2};\frac{2n+5}{2},\frac{2p+1}{2};1\right).$ Although this family belongs to the Karlsson–Minton class of ${}_3{F}_2\left(1\right)$ series with negative integral parameter differences, the most directly relevant symmetric formula of Shpot and Srivastava becomes singular at the free numerator parameter $a=1$(here a denotes the first numerator parameter of the Shpot–Srivastava reduction formula, the value of which controls the Beta factors in their evaluation). We show that this apparent singularity is removable and identify its finite value with the independently derived closed form. First, Euler’s integral representation and a recurrence for $J_{m,n}\left(x\right)={\int }_0^1{v}^{2m}{(1-x{v}^2)}^{-n}dv$ yield a closed form for ${}_2{F}_1\left(-\frac{2n+1}{2},1;\frac{2n+5}{2};x\right)$ as a polynomial in $(1-x)$ plus $arctanh\left(\sqrt{x}\right)/\sqrt{x}$. Rainville’s integral formula then reduces $F(n,p)$ to a rational part and a finite rational linear combination of odd harmonic sums $H_r^{\mathrm{odd}}={\sum }_{j=1}^r{(2j-1)}^{-1}$. For all integers $n\ge 0$ and $p\ge n+3$, this proves $F(n,p)\in \mathbb{Q}$. The threshold is sharp: at the adjacent boundary $p=n+2$ we obtain $F(n,n+2)=R_n+C_n{\pi }^2$ with $R_n,C_n\in \mathbb{Q}$ and $C_n\ne 0$, so rationality fails. We also prove, by analyticity and the independently derived formula, that the $a\to 1$ limit of the Shpot–Srivastava representation equals the present closed form. For computation, the exact assembly uses $O(n+p)$ arithmetic operations under the unit-cost model, while a direct fixed-precision evaluation is ill-conditioned for large n. A stable series/closed-form hybrid for the ${}_2{F}_1$ factor combined with adaptive positive-kernel Gauss–Jacobi quadrature gives relative errors at the machine-precision scale over the tested range up to $n=64$. Full article

We study the small-radius topology of the Euclidean tube generated by a smooth planar Gaussian path. Let $X={\left(X_t\right)}_{0\le t\le T}$ be a sufficiently regular Gaussian process with values in ${\mathbb{R}}^2$, and define its radius-$\epsilon$ tube by ${\mathcal{T}}_{\epsilon }=\{x\in {\mathbb{R}}^2:dist(x,X\left([0,T]\right))\le \epsilon \}$. When the sample path is a smooth immersed curve with finitely many transversal double points and no higher-order intersections, the tube is, for all sufficiently small radii, a regular neighbourhood of the finite planar graph traced by the curve. Consequently, its Euler characteristic is determined by the number N of self-intersections: $\chi \left({\mathcal{T}}_{\epsilon }\right)=1-N$ for all sufficiently small ε, almost surely. We combine this deterministic topological observation with the Kac–Rice formula applied to the two-parameter Gaussian difference field $F(s,t)=X_t-X_s$. This yields an exact integral expression for the expected limiting Euler characteristic $\mathbb{E}\left[{\chi }_0\right]$, where ${\chi }_0={\lim }_{\epsilon \downarrow 0}\chi \left({\mathcal{T}}_{\epsilon }\right)$. For stationary Gaussian coordinates with covariance r, the formula reduces to a one-dimensional integral depending only on r, $r^{\prime }$, and $r^{″}$. We then specialize to the squared-exponential covariance $r\left(u\right)=exp(-u^2/2{\mathsf{\ell }}^2)$, obtaining a fully explicit dimensionless quadrature depending only on $T/\mathsf{\ell }$. For the squared-exponential kernel, the expected number of limiting small-radius holes satisfies $\mathbb{E}\left[N\right]\sim {\displaystyle \frac{1}{8\pi }}{(T/\mathsf{\ell })}^2$ in the short-correlation regime. Finally, we add a constant linear drift and show that the expected number of self-intersections is modified by two competing mechanisms: a Gaussian density-damping term and a noncentral velocity-amplification term. Full article

Accurately predicting the terminal state of shaped charge jets (SCJs) is crucial for optimizing their penetration performance. The core challenge lies in a deep understanding of the complete physical chain from shock compression to unloading expansion. This paper presents a hybrid analytical–numerical model for SCJ formation that incorporates a shock-pressure-driven unloading term. Unlike classical PER theory, the proposed model explicitly introduces an unloading term and derives a quantitative expression for the momentum conversion factor $\left({\Pi }_{DMCF}\right)$ to quantitatively characterize the momentum redistribution during collapse. Our analysis finds that ${\Pi }_{DMCF}$ exhibits a typical S-shaped evolution law as the dimensionless Mach number $\left(Ma\right)$ varies. This study uses a logistic function with two characteristic parameters, $M{a}_0$ and $k$, to accurately fit the data. The research results indicate that the model parameters have clear physical connotations: $M{a}_0$ characterizes the critical condition for the material to transition from “strength-dominated” to “kinetic-energy-dominated” behavior, while k reflects the degree of abrupt transition. After calibrating the model parameters using high-fidelity numerical simulations, the jet morphology and velocity data obtained from X-ray flash photography experiments are compared and verified, confirming that the model can significantly improve the prediction accuracy. Especially for Ti₅₅Al₂₀V₅Zr₅Nb₁₅ HEA, the prediction error in the jet velocity is less than 4%, and the theoretically predicted shock pressure is highly correlated with the numerical results $\left({\mathrm{R}}^2=0.943\right)$. A further mechanistic analysis indicates that the proposed model successfully decodes the unique response of the HEA: its high dynamic strength results in a larger value, causing its momentum conversion efficiency to fall within a lower range under typical impact conditions. The theoretical framework constructed in this study provides a hybrid analytical–numerical and highly reliable theoretical tool for the accurate prediction of SCJs, as well as for the material selection and design of high-performance liners. Full article

We introduce a notion of curvature based on informational holonomy. Let $(M,g)$ be a smooth Riemannian manifold and let $\pi :\mathcal{P}\to M$ be a bundle of state spaces equipped fibrewise with a smooth divergence $D_x$ inducing an information metric $g^{{\mathcal{P}}_x}$. Assuming a connection on $\mathcal{P}$ compatible with this fibrewise information geometry, we measure the deviation of holonomy around small geodesic triangles by transporting a reference state ${\mu }_x$ and comparing it to its image via the induced informational distance $d_x=\sqrt{2D_x}$. Normalizing the resulting distance defect by the geometric area yields a continuous informational holonomy (sectional) curvature$K_{\mathrm{hol}}^{\mathrm{cont}}(x,\mathsf{\Pi })$. We prove that this limit exists for all $(x,\mathsf{\Pi })$ and equals the norm of a vector $W_x(\mathsf{\Pi };{\mu }_x)\in T_{{\mu }_x}{\mathcal{P}}_x$ depending linearly on the curvature of the connection along $\mathsf{\Pi }$. In geometric models induced from the Levi–Civita connection via an isometric representation, $K_{\mathrm{hol}}^{\mathrm{cont}}$ becomes a scalar invariant of $R^g{|}_{\mathsf{\Pi }}$ and, on spaces of constant sectional curvature, reduces to a constant multiple of $|{\sec }_g|$. On the discrete side, we consider quasi-uniform sampling graphs whose edges carry channels approximating parallel transport. Discrete triangle holonomies define a curvature estimator, and under explicit sampling, area-approximation, and channel-consistency assumptions, we establish a discrete-to-continuum convergence theorem with a quantitative error bound controlled by the sampling scale. Full article

The adsorption of aromatic hydrocarbons from liquid paraffin is essential because of their harmful nature, long-lasting presence, and detrimental effects on the quality of the product. In this study, we investigated the adsorption of naphthalene from liquid paraffin by using a nanoclay-based adsorbent prepared from boron enrichment process waste. The characterization of the prepared adsorbent was carried out by using X-ray Diffraction (XRD), Scanning Electron Microscopy (SEM), X-ray Photoelectron Spectroscopy (XPS) and N₂ adsorption–desorption techniques, which confirmed the development of a layered nanostructure containing boron that possesses a porous and high-surface-area format appropriate for the adsorption. The hydrothermal treatment significantly increased the BET surface area from 35.42 to 112.15 m²/g, indicating the successful formation of a porous nanostructure. The kinetic and isotherm parameters of the adsorption process were calculated from experimental data. The adsorption of naphthalene followed pseudo-second-order kinetics and the isotherm fit well to the Langmuir model. Adsorption experiments revealed that the optimum adsorption performance was achieved at pH 4.0, and equilibrium was reached within 90 min. The adsorption kinetics were best described by the pseudo-second-order model (R² > 0.99), while the equilibrium data showed excellent agreement with the Langmuir isotherm model (R² = 0.995), suggesting monolayer adsorption. The maximum adsorption capacity of BNC was determined as 365.20 mg/g, which was more than twice that of the raw BEW (247.59 mg/g). Thermodynamic analysis indicated that the adsorption process was spontaneous at lower temperatures and exothermic, with a ΔH° value of −15.42 kJ/mol for BNC. The results suggest that the adsorption occurs through a multi-step process, beginning with external film diffusion, followed by pore diffusion and surface interaction. Based on the kinetic, isotherm, and spectroscopic data, a supramolecular adsorption mechanism is suggested, which encompasses π-π interactions, van der Waals forces, and surface complexation between naphthalene and the nanoclay structure. These results indicate that boron enrichment process waste-derived nanoclay is a sustainable, economical, and efficient adsorbent for removing naphthalene from liquid paraffin. Full article

This paper presents a reinforcement learning (RL) framework based on the Koopman operator for high-dimensional nonlinear control. By leveraging nonlinear eigenvalue dynamics, the approach enables scalable and efficient policy optimization. We examined the challenge of controlling complex systems by embedding high-dimensional states $x\left(t\right)\in {\mathbb{R}}^n$ into a Koopman-invariant subspace $\varphi \left(x\right)\in {\mathbb{R}}^m$, where evolution becomes linear under the Koopman operator $\mathcal{K}$. By spectrally decomposing $\mathcal{K}=U\Lambda U^{-1}$, the eigenvalue dynamics are obtained, and $\mathcal{K}$ is reconstructed iteratively via dominant eigenpairs $\left(v_i,w_i\right)$. A policy network $\pi \left(a|s\right)$ selects actions $u\left(t\right)$, while a value function $V\left(s\right)$, expressed in Koopman eigenfunction coordinates, guides gradient-based policy updates. The framework integrates spectral stability constraints ($\rho \left(X\right)<1)$ and Lyapunov-based analysis to ensure convergence. We derive perturbation bounds for Koopman eigenvalues under policy updates and establish conditions for nonlinear mode interactions in the lifted space. The spectral policy gradient theorem for Koopman RL links eigenvalue dynamics to policy optimization, includes a constrained Bellman formulation in Koopman coordinates, and analyzes bifurcation of learning-induced eigenvalue shifts. Full article

4-Methyl- (1) and 4-ethyl-8-hydroxyquinoline (2) crystallize from a mixture of diethyl ether and chloroform in the triclinic space group P$\overline{1}$. X-ray analysis reveals that both compounds form discrete molecular dimers stabilized by intermolecular O-H∙∙∙N and C-H∙∙∙O hydrogen bonds, resulting in $R_2^2\left(5\right)$ cyclic synthons. This pattern of hydrogen bonds is further stabilized by intramolecular O-H∙∙∙N bonds so that the quinoline nitrogen atom acts as a bifurcated binding site. The dimers exhibit a planar geometry and arrange into layer-like structures held together by π∙∙∙π stacking and van der Waals forces. While the fundamental bonding motifs are similar, the increased steric demand of the ethyl group in compound 2 induces a shift in the crystallographic orientation of the layers and alters the degree of π-overlap compared to the methyl-substituted analogue 1. Full article

We study truncation-compatible families $F={\left(F_m\right)}_{m\ge 1}$ over $\mathbb{Q}\left[z\right]$ through an inverse-limit formalism, and we evaluate them at the punctured cyclotomic cosine points ${\alpha }_{k,n}=cos(2\pi k/n)$ with the specialization $z=n-1$. For symmetric families of uniformly bounded total x-degree $\le d$, we prove a stable-range rigidity theorem: for all $n\ge d+2$, the cosine-point evaluation factors through the finitely many punctured cosine power sums $P_1\left(n\right),\dots ,P_d\left(n\right)$. In the purely polynomial case, this implies eventual polynomiality in n. We then extend the framework to include fixed-product factors and package their cosine-point contribution in multiplicative invariants $M_Q\left(n\right)$. In the stable range, the bounded-degree symmetric part collapses as before; any remaining cyclotomic dependence occurs only through these explicit product terms. Finally, we show that coefficient extraction from such products produces further bounded-degree symmetric families, and we apply this to complete symmetric functions $h_r$ evaluated at cosine points. Full article

Let $(X,\mathfrak{A},P)$ and $(Y,\mathfrak{B},Q)$ be two probability spaces, R be their skew product on the product $\sigma$-algebra $\mathfrak{A}\otimes \mathfrak{B}$ and $\{({\mathfrak{A}}_y,S_y):y\in Y\}$ be a Q-disintegration of R. Then, let $\mathfrak{A}\divideontimes \mathfrak{B}$ be the $\sigma$-algebra generated $\mathfrak{A}\otimes \mathfrak{B}$ and by the family $\mathcal{M}:=\{E\subset X\times Y:\exists N\in {\mathfrak{B}}_0\forall y\notin N\widehat{S_y}\left(E^y\right)=0\}$ and $R_{\divideontimes }$ be the extension of R such that $\mathcal{M}$ becomes the family of $R_{*}$-zero sets ($\widehat{S_y}$ is the completion of $S_y$ and ${\mathfrak{B}}_0=\{B\in \mathfrak{B}:Q\left(B\right)=0\}$). We prove that there exists a lifting $\pi$ on ${\mathcal{L}}^{\infty }\left(R_{\divideontimes }\right)$ and liftings ${\sigma }_y$ on ${\mathcal{L}}^{\infty }\left(\widehat{S_y}\right)$, $y\in Y$, such that sections of $\pi$ determined by Y are lifting invariant (in particular, the sections are measurable), i.e., ${\left[\pi \left(f\right)\right]}^y={\sigma }_y\left({\left[\pi \left(f\right)\right]}^y\right) \mathrm{for} \mathrm{every} y\in Y \mathrm{and} \mathrm{every} f\in {\mathcal{L}}^{\infty }\left(R_{\divideontimes }\right).$ In general, if $\pi$ is an arbitrary lifting on the product, then some sections of $\pi \left(f\right)$ may be even nonmeasurable. The main novelty of my paper lies in expanding the domain of the measure in the product to $\mathfrak{A}\divideontimes \mathfrak{B}$ and constructing on such a much larger abstract space the suitable lifting. Such expansions used to be made only in case of topological spaces, where product of marginal Borel sets was replaced by the Borel subsets of the product space. However, several topological technics are then applied, not approachable in the abstract case. The main theorem is a generalization of earlier lifting results, where either separability of $\mathfrak{A}$ in the Frechet–Nikodým pseudometric was assumed or $R\ll P\times Q$. In case of a separable P and in the case when $R\ll P\times Q$, a characterization of stochastic processes possessing an equivalent measurable version is presented. The theorem is a strong generalization of earlier results (see the introduction) where it was proved only that the lifting modification of a measurable stochastic process (via the lifting constructed there) is again measurable. Full article

A promising direction for the creation of new biologically active derivatives of the alkaloid lupinine is the synthesis of “hybrid molecules” that combine a fragment of the alkaloid and the pharmacophore of 1,2,3-triazole in their structure. From a biological perspective, this work presents the first X-ray diffraction study of a single crystal of (1S,9aR)-1-({4-[4-(Benzyloxy)-3-methoxyphenyl]-1H-1,2,3-triazol-1-yl}methyl)octahydro-2H-quinolizine, a new, recently synthesized 1,2,3-triazole derivative of lupinine. A comparison of theoretically predicted and experimentally observed structural parameters was carried out. The FTIR spectroscopy study and vibrational properties calculations allowed us to interpret the FTIR absorption spectrum and localize specific vibrational modes in quinolizidine, 1,2,3-triazole, and benzene rings. Such information can be fruitful for further characterization of the synthesis process and products. The molecular docking of the compound was performed. It was shown that the studied molecules are capable of interacting with the Mpro binding site via non-covalent and hydrophobic interactions with subsites S3 (Met165, Glu166, Leu167, Pro168) and S5 (Gln189, Thr190, Gln192), which ensure the stabilization of the Mpro substrate. Blocking of the active site of the enzyme in the region of the oxyanion hole does not occur, but stable stacking interactions with the π-system of one of the catalytic amino acids, His41, are observed. Full article

Perimidine derivatives are versatile heterocycles with growing significance in medicinal chemistry and materials sciences. However, their structural variety remains limited. This study focused on the synthesis and crystal structure characterization of a new perimidine-based molecule. A bicyclic perimidine lactone, (1S,4R)-4,7,7-trimethyl-1-(1H-perimidin-2-yl)-2-oxabicyclo[2.2.1]heptan-3-one (1), was synthesized through an intramolecular dehydration of a monoamide intermediate formed from 1,8-diaminonaphthalene and (1S)-(–)-camphanic chloride under basic conditions. The product was purified and crystallized from acetone, giving single crystals suitable for X-ray diffraction. Structural analysis revealed two stereogenic centers and crystallization in the chiral tetragonal P4₃2₁2 space group, with stabilization through N—H···O and C—H···N hydrogen bonds as well as C—H···π interactions. Full article

The functional integration of chiral liquid crystals and π-conjugated compounds has great potential for creating novel exotic materials. A series of chiral donor–acceptor (D–A)-type fluorenone derivatives was synthesized to investigate the influence of molecular structure upon their liquid-crystalline phase-transition behavior, ferroelectricity, photophysical properties, and photoconductive properties. Polarizing optical microscopy (POM), differential scanning calorimetry (DSC), and X-ray diffraction (XRD) analyses revealed that several D–A-type fluorenone derivatives exhibited liquid crystal (LC) phases. These chiral LC fluorenone derivatives exhibited polarization hysteresis in the chiral smectic C (SmC*) phase. Among the four fluorenone-based ferroelectric liquid crystals (FLCs), (R,R)-2a exhibited the largest spontaneous polarization (over 3.0 × 10² nC cm⁻²). The formation of intramolecular charge-transfer (ICT) states in each compound was evidenced by the UV–vis absorption spectroscopy. Ambipolar carrier transport in the SmC* phases of the fluorenone-based FLCs was elucidated by the time-of-flight (TOF) method. The mobilities of holes and electrons in the SmC* phases were on the order of 10⁻⁵ cm² V⁻¹ s⁻¹, which is on par with the carrier mobilities of low-ordered smectic phases in conventional LC semiconductors. Full article

Eco-friendly silver nanoparticle systems are highly effective due to their large surface area and strong adsorption capacity. In this study, a novel silver (diethyldithiocarbamate)-decorated reduced graphene oxide nanocomposite (Ag(DDTC)@rGO) was synthesized via a simple green method, yielding a stable and monodispersed material. SEM and HRTEM analyses revealed uniform anchoring of the Ag(DDTC) complex on rGO, producing a coherent nanocomposite with strong physicochemical coupling. The Ag(DDTC)@rGO nanocomposite exhibited a high Brunauer–Emmett–Teller (BET) surface area (289 m² g⁻¹) with an average pore diameter of 45 nm, confirming the mesoporous nature of the composite. FTIR spectra showed characteristic bands of rGO and DDTC ligands, with new peaks at 620–640 cm⁻¹ confirming the successful anchoring of silver–diethyldithiocarbamate species onto rGO via Ag–S and Ag–O bond formation. Raman spectroscopy further confirmed the multilayered rGO structure after Ag(DDTC) incorporation. X-ray diffraction (XRD) identified a broad hybrid amorphous–crystalline pattern, favorable for catalytic and sensing functions. The superior malachite green adsorption capacity of Ag(DDTC)@rGO was attributed to synergistic electrostatic, π–π stacking, hydrogen bonding, and silver-mediated interactions. Furthermore, antibacterial assays demonstrated significant inhibition of P. aeruginosa ATCC 9027 and S. enterica ATCC 14028, further enhanced by mild heat activation (40–50 °C) that significantly improved the surface activation of silver nanoparticles. The multifunctional Ag(DDTC)@rGO nanocomposite exhibits strong adsorption and antibacterial properties, highlighting its potential for sustainable wastewater treatment and environmental remediation applications. Full article

Acetaminophen (APAP) is a widely used pharmaceutical increasingly detected as a contaminant in aquatic environments due to its persistent nature and incomplete removal by conventional wastewater treatment. This study investigates the adsorption performance and mechanisms of commercial activated carbon (M) and its hydrothermally modified form (MH) for APAP removal. Characterization via elemental analysis, X-ray photoelectron spectroscopy (XPS), and N₂ adsorption isotherms revealed that hydrothermal treatment reduced oxygen content and enhanced micro- and mesopore volumes, resulting in a more homogeneous and carbon-rich surface. Batch adsorption experiments conducted under varying pH (5–7) and temperature (30–40 °C) conditions showed that MH achieved up to 94.3% APAP removal, outperforming the untreated carbon by more than 15%. Kinetic modeling indicated that adsorption followed a pseudo-second-order mechanism (R² > 0.99), and isotherm data fitted best to the Langmuir model for MH and the Freundlich model for M, reflecting their differing surface properties. Adsorption was enhanced at lower pH and higher temperatures, consistent with an endothermic and pH-dependent mechanism. Complementary density functional theory (DFT) simulations confirmed that π–π stacking is the dominant interaction between APAP and the carbon surface. The most favorable configuration involved coplanar stacking with non-oxidized graphene (ΔG = −33 kJ/mol), while oxidized graphene models exhibited weaker interactions. Natural Bond Orbital (NBO) analysis further supported the prevalence of π–π interactions over dipole interactions. These findings suggest that surface deoxygenation and improved pore architecture achieved via hydrothermal treatment significantly enhance APAP adsorption, offering a scalable strategy for pharmaceutical pollutant removal in water treatment applications. Full article

Two single-crystal X-ray structure determinations of 2-aminopyrimidinium hydrogen tri oxofluorophosphate, (C₄H₆N₃)⁺·(HFO₃P)⁻, (I), and bis(2-aminopyrimidinium) trioxofluorophosphate, 2(C₄H₆N₃)⁺·(FO₃P)²⁻, (II), as well as their vibration spectra (FTIR on powder samples and the Raman spectra on unoriented single crystals) with a detailed assignment of vibrational modes are reported. The structure (I) consists of one independent 2-aminopyrimidinium cation and one hydrogen trioxofluorophosphate anion, while (II) consists of two symmetry independent 2-aminopyrimidinium cations and one trioxofluorophosphate anion. In (I), there is an O-H···O hydrogen bond of a moderate strength. A pair of these hydrogen bonds is situated about the symmetry centre and involved in the graph set motif R²₂(8). There are also N-H···O hydrogen bonds of a moderate strength, which are present in both structures while being involved in the graph set motifs R²₂(8), too. In addition, the N-H···O hydrogen bonds form R³₄(10) graph set motifs in (II). The latter motifs form ribbons which propagate parallel to the unit-cell axis a. In both structures, there are present π···π-electron ring interactions into which the primary amine groups are involved. In both structures, there are also present weak C-H···N hydrogen bonds with participation of the non-protonated ring N-atoms. The fluorine participates in the C-H···F hydrogen bonds in both title structures. The P-F distances are normal in both anions. The structure (I) differs from the known structure of 2-aminopyrimidinium hydrogen phosphite, the compositional isomer, though the main hydrogen bonds show similar geometry in both structures. The crystal of (I) was twinned. Full article