On asymptotic distributions of exceedance statistics

In this study, two independent samples X₁,X₂,…,X_n and Y₁,Y₂,…,Y_m with respective distribution functions F and Q are considered. The joint asymptotic distributions of exceedance statistics defined as the number of Y observations falling into a random interval of order statistics constructed from the X sample is investigated. The results can be used in the context of a two-sample problem.     

Stability of Equilibria for the $\mathfrak{so}(4)$ Free Rigid Body

The stability for all generic equilibria of the Lie–Poisson dynamics of the \mathfrakso(4)\mathfrak{so}(4) rigid body dynamics is completely determined. It is shown that for the generalized rigid body certain Cartan subalgebras (called of coordinate type) of \mathfrakso(n)\mathfrak{so}(n) are equilibrium points for the rigid body dynamics. In the case of \mathfrakso(4)\mathfrak{so}(4) there are three coordinate type Cartan subalgebras whose intersection with a regular adjoint orbit gives three Weyl group orbits of equilibria. These coordinate type Cartan subalgebras are the analogues of the three axes of equilibria for the classical rigid body in \mathfrakso(3)\mathfrak{so}(3). In addition to these coordinate type Cartan equilibria there are others that come in curves.     

2‐[Benzoyl­(hydroxy­imino)­methyl­amino]­benzoic acid

The structure of the title compound, C₁₅H₁₂N₂O₄, consists of a polymeric arrangement, around inversion centres, of mol­ecules linked through O—H⋯N and O—H⋯O hydrogen bonds; there are also intramolecular hydrogen bonds. All these hydrogen‐bond interactions result in the formation of infinite chains parallel to the [010] direction. The oxime group has an E conformation.     

Boundedness of Sublinear Operators and Commutators on Generalized Morrey Spaces

In this paper the authors study the boundedness for a large class of sublinear operators \({T_{\alpha}, \alpha \in [0,n)}\) generated by Calderón–Zygmund operators (α = 0) and generated by Riesz potential operator (α > 0) on generalized Morrey spaces \({M_{p,\varphi}}\) . As an application of the above result, the boundeness of the commutator of sublinear operators \({T_{b,\alpha}, \alpha \in [0,n)}\) on generalized Morrey spaces is also obtained. In the case \({b \in BMO}\) and T _b,α is a sublinear operator, we find the sufficient conditions on the pair \({(\varphi_1,\varphi_2)}\) which ensures the boundedness of the operators \({T_{b,\alpha}, \alpha \in [0,n)}\) from one generalized Morrey space \({M_{p,\varphi_1}}\) to another \({M_{q,\varphi_2}}\) with 1/p ? 1/q = α/n. In all the cases the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on \({(\varphi_1,\varphi_2)}\) , which do not assume any assumption on monotonicity of \({\varphi_1, \, \varphi_2}\) in r. Conditions of these theorems are satisfied by many important operators in analysis, in particular, Littlewood–Paley operator, Marcinkiewicz operator and Bochner–Riesz operator.     

Ultrasound-assisted rapid synthesis of <Emphasis Type="Italic">β</Emphasis>-aminoketones with direct-type catalytic Mannich reaction using bismuth(III) triflate in aqueous media at room temperature

An innovative, powerful, efficient and relatively rapid method was developed to synthesise various β-aminoketone derivatives from cyclohexanone, substituted aromatic amines and aromatic or hetero-aromatic aldehydes via ultrasound-assisted direct-type catalytic Mannich reaction using bismuth(III) triflate in water. Good yields of the desired β-aminoketones were obtained at room temperature by ultrasound-assisted reaction within 1–2 h. The major advantages of the proposed method are undemanding conditions, easy operation, low toxicity, shorter reaction time, anti selectivity and higher yields in comparison with conventional methods.     

Existence Results for Periodic Solutions of Nonautonomous Second-Order Differential Systems with $(q,p)$-Laplacian

In this paper, we consider the existence for periodic solutions of nonautonomous second-order differential systems with $(q, p)$-Laplacian by using the least action principle and the minimax methods.     

A numerical approach for a crack problem by Gauss–Chebyshev quadrature

In this paper, a problem of a crack in an orthotropic strip is studied under plane strain conditions. It is assumed that normal displacements and shear stresses do not act on neither of the boundaries of the strip. Cauchy-type singular integral equation for the crack problem is derived by using the theory of plane elasticity and the Fourier transformation technique. A quadrature collocation approach is adopted for the numerical solutions of the singular integral equation. The effect of relative thickness and mechanical properties of strip on Mode I stress intensity factors (SIFs) are examined under different loading conditions. Some sample results are given for SIFs; also, material orthotropy and geometrical effects are discussed in detail.     

4‐Bromo‐2‐{[4‐(3‐mesityl‐3‐methyl­cyclo­butyl)thia­zol‐2‐yl]hydrazono­meth­yl}phenol,with N—H⋯N,C—H⋯π and π–π inter­actions

The title compound, C₂₄H₂₆BrN₃OS, crystallizes in the triclinic space group P, with two independent mol­ecules in the asymmetric unit. The mol­ecules adopt an E geometry about the azomethine C=N double bond. The structure is stabilized as dimers by N—H⋯N hydrogen bonding. C—H⋯π and π–π inter­actions are also effective in the crystal packing.     

N1‐(4‐Bromo­phen­yl)‐N2‐hydr­oxy‐2‐oxo‐2‐phenyl­acetamidine

In the title compound, C₁₄H₁₁BrN₂O₂, which has the oxime group in an E conformation, molecules are linked by strong O—H⋯O and N—H⋯O hydrogen bonds into chains of edge‐fused rings, unlike closely related compounds.