Anisotropic equations in weighted Sobolev spaces of higher order

We consider the strongly nonlinear boundary value problem,

Summation formulae for finite cotangent sums

Recently, a half-dozen remarkably general families of the finite trigonometric sums were summed in closed-form by choosing a particularly convenient integration contour and making use of the calculus of residues. In this sequel, we show that this procedure can be further extended and we find the summation formulae, in terms of the higher order Bernoulli polynomials and the ordinary Bernoulli polynomials, for four general families of the finite cotangent sums.     

Numerical solution of fuzzy differential equations under generalized differentiability

In this paper, we interpret a fuzzy differential equation by using the strongly generalized differentiability concept. Utilizing the Generalized Characterization Theorem, we investigate the problem of finding a numerical approximation of solutions. Then we show that any suitable numerical method for ODEs can be applied to solve numerically fuzzy differential equations under generalized differentiability. The generalized Euler approximation method is implemented and its error analysis, which guarantees pointwise convergence, is given. The method’s applicability is illustrated by solving a linear first-order fuzzy differential equation.     

Potentialities of two solventless extraction approaches—Stir bar sorptive extraction and headspace solid-phase microextraction for determination of higher alcohol acetates, isoamyl esters and ethyl esters in wines

A stir bar sorptive extraction with liquid desorption followed by large volume injection coupled to gas chromatography-quadrupole mass spectrometry (SBSE-LD/LVI-GC-qMS) was evaluated for the simultaneous determination of higher alcohol acetates (HAA), isoamyl esters (IsoE) and ethyl esters (EE) of fatty acids. The method performance was assessed and compared with other solventless technique, the solid-phase microextraction (SPME) in headspace mode (HS). For both techniques, influential experimental parameters were optimised to provide sensitive and robust methods. The SBSE-LD/LVI methodology was previously optimised in terms of extraction time, influence of ethanol in the matrix, liquid desorption (LD) conditions and instrumental settings. Higher extraction efficiency was obtained using 60 min of extraction time, 10% ethanol content, n-pentane as desorption solvent, 15 min for the back-extraction period, 10 mL min⁻¹ for the solvent vent flow rate and 10 °C for the inlet temperature. For HS-SPME, the fibre coated with 50/30 μm divinylbenzene/carboxen/polydimethylsiloxane (DVB/CAR/PDMS) afforded highest extraction efficiency, providing the best sensitivity for the target volatiles, particularly when the samples were extracted at 25 °C for 60 min under continuous stirring in the presence of sodium chloride (10% (w/v)). Both methodologies showed good linearity over the concentration range tested, with correlation coefficients higher than 0.984 for HS-SPME and 0.982 for SBES-LD approach, for all analytes. A good reproducibility was attained and low detection limits were achieved using both SBSE-LD (0.03-28.96 μg L⁻¹) and HS-SPME (0.02-20.29 μg L⁻¹) methodologies. The quantification limits for SBSE-LD approach ranging from 0.11 to 96.56 μg L⁻and from 0.06 to 67.63 μg L⁻¹ for HS-SPME. Using the HS-SPME approach an average recovery of about 70% was obtained whilst by using SBSE-LD obtained average recovery were close to 80%. The analytical and procedural advantages and disadvantages of these two methods have been compared.Both analytical methods were used to determine the HAA, IsoE and EE fatty acids content in “Terras Madeirenses” table wines. A total of 16 esters were identified and quantified from the wine extracts by HS-SPME whereas by SBSE-LD technique were found 25 esters which include 2 higher alcohol acetates, 4 isoamyl esters and 19 ethyl esters of fatty acids. Generally SBSE-LD provided higher sensitivity with decreased analysis time.     

广义度量方程的改进及其应用(I)

众所周知, 度量方程作为距离几何的基本内容和工具之一,在几何约束求解中扮演着主要的角色.改进了杨定华关于$n$维欧氏空间中两个等数量有限基本元素构成集合的广义度量方程, 建立了更为一般意义的、应用方便的广义度量方程,作为其初步应用,导出了两个单形之间的一些有趣的矩阵恒等式关系.特别地,将其两边取行列式,可以简洁得到关于联系两个单形的几何恒等式.     

Combined ℓ<Subscript>2</Subscript> data and gradient fitting in conjunction with ℓ<Subscript>1</Subscript> regularization

We are interested in minimizing functionals with ℓ₂ data and gradient fitting term and ℓ₁ regularization term with higher order derivatives in a discrete setting. We examine the structure of the solution in 1D by reformulating the original problem into a contact problem which can be solved by dual optimization techniques. The solution turns out to be a ’smooth’ discrete polynomial spline whose knots coincide with the contact points while its counterpart in the contact problem is a discrete version of a spline with higher defect and contact points as knots. In 2D we modify Chambolle’s algorithm to solve the minimization problem with the ℓ₁ norm of interacting second order partial derivatives as regularization term. We show that the algorithm can be implemented efficiently by applying the fast cosine transform. We demonstrate by numerical denoising examples that the ℓ₂ gradient fitting term can be used to avoid both edge blurring and staircasing effects.      

Mathematical solution for bending of short hybrid composite beams with variable fibers spacing

In this study, the static response is presented for a simply supported functionally graded hybrid beam subjected to a transverse uniform load. Material properties of the beam are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. By varying the fiber volume fraction within a symmetric laminated beam and combining two fiber types to create a hybrid functionally graded material (FGM) can offer desirable increases in axial and bending stiffness. The equations governing the hybrid FGM beams are determined using the principle of virtual work (PVW) arising from the higher order shear deformation theories. Numerical results on the transverse deflection, axial and shear stresses in a moderately thick hybrid FGM beam under uniform distributed load are discussed in depth. The effect of power-law exponent on the deflection and stresses are also commented.     

Polyanalytic Hardy decomposition of higher order Lipschitz functions

This paper is concerned with the problem of decomposing a higher order Lipschitz function on a closed Jordan curve Γ into a sum of two polyanalytic functions in each open domain defined by Γ. Our basic tools are the Hardy projections related to a singular integral operator arising in polyanalytic function theory, which, as it is proved here, represents an involution operator on the higher order Lipschitz classes. Our result generalizes the classical Hardy decomposition of Hölder continuous functions on the boundary of a domain.     

Rank one quotients of vector groups

ABSTRACT

In this paper we study the possible torsion in even-dimensional higher class groups Cl _2n (Λ)(n ≥ 1) of an order Λ in a semisimple algebra A over a number field F with a ring of integers 𝒪 _F . We show that for certain orders, called generalized Eichler orders, bip-torsion in Cl _2n (Λ) can only occur for primes p dividing prime ideals ? of 𝒪 _F , at which Λ is not maximal. In particular, the results apply to Eichler orders in quaternion algebras and to hereditary orders.     

Moments of discounted aggregate claim costs until ruin in a Sparre Andersen risk model with general interclaim times

In the context of a Sparre Andersen risk model with arbitrary interclaim time distribution, the moments of discounted aggregate claim costs until ruin are studied. Our analysis relies on a novel generalization of the so-called discounted density which further involves a moment-based component. More specifically, while the usual discounted density contains a discount factor with respect to the time of ruin, we propose to incorporate powers of the sum until ruin of the discounted (and possibly transformed) claims into the density. Probabilistic arguments are applied to derive defective renewal equations satisfied by the moments of discounted aggregate claim costs until ruin. Detailed examples concerning the discounted aggregate claims and the number of claims until ruin are studied upon assumption on the claim severities. Numerical illustrations are also given at the end.