Inequalities for the gamma function

Some inequalities for the gamma function are given. These results refine the classical Stirling approximation and its many recent improvements. Received: 2 May 2008, Revised: 2 September 2008     

Trace results on domains with self-similar fractal boundaries

This work deals with trace theorems for a family of ramified bidimensional domains Ω with a self-similar fractal boundary Γ^∞. The fractal boundary Γ^∞ is supplied with a probability measure μ called the self-similar measure. Emphasis is put on the case when the domain is not a −δ domain and the fractal is not post-critically finite, for which classical results cannot be used. It is proven that the trace of a function in H¹(Ω) belongs to for all real numbers p1. A counterexample shows that the trace of a function in H¹(Ω) may not belong to BMO(μ) (and therefore may not belong to ). Finally, it is proven that the traces of the functions in H¹(Ω) belong to H^s(Γ^∞) for all real numbers s such that 0s<d_H/4, where d_H is the Hausdorff dimension of Γ^∞. Examples of functions whose traces do not belong to H^s(Γ^∞) for all s>d_H/4 are supplied.There is an important contrast with the case when Γ^∞ is post-critically finite, for which the functions in H¹(Ω) have their traces in H^s(Γ^∞) for all s such that 0s<d_H/2.     

Limitations on the size of semigroups of matrices

If a nonzero linear functional has finite, countable, or bounded range when restricted to an irreducible semigroup of complex matrices, it is shown that itself has the same property. Similar results are proven under the hypothesis that a nontrivial ideal of is finite, countable, or bounded.     

Integrals of equivariant forms and a Gauss-Bonnet theorem for constructible sheaves

The Berline-Vergne integral localization formula for equivariantly closed forms ([N. Berline, M. Vergne, Classes caractéristiques équivariantes. Formules de localisation en cohomologie équivariante, C. R. Acad. Sci. Paris 295 (1982) 539-541], Theorem 7.11 in [N. Berline, E. Getzler, M. Vergne, Heat Kernels and Dirac Operators, Springer-Verlag, 1992]) is well-known and requires the acting Lie group to be compact. In this article, we extend this result to real reductive Lie groups G_R.As an application of this generalization, we prove an analogue of the Gauss-Bonnet theorem for constructible sheaves. If F is a G_R-equivariant sheaf on a complex projective manifold M, then the Euler characteristic of M with respect to F

     

A remark on Andrews-Askey integral

In this paper, we use the Andrews-Askey integral and the q-Chu-Vandermonde formula to derive a more general integral formula. Applications of the new integral formula are also given, which include to derive the q-Pfaff-Saalschütz formula and the terminating Sears's transformation formula.     

Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation

This paper discusses a randomized non-autonomous logistic equation , where B(t) is a 1-dimensional standard Brownian motion. In [D.Q. Jiang, N.Z. Shi, A note on non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl. 303 (2005) 164-172], the authors show that E[1/N(t)] has a unique positive T-periodic solution E[1/Np(t)] provided a(t), b(t) and α(t) are continuous T-periodic functions, a(t)>0, b(t)>0 and . We show that this equation is stochastically permanent and the solution Np(t) is globally attractive provided a(t), b(t) and α(t) are continuous T-periodic functions, a(t)>0, b(t)>0 and mint∈[0,T]a(t)>maxt∈[0,T]α²(t). By the way, the similar results of a generalized non-autonomous logistic equation with random perturbation are yielded.     

Determination of lanthanum and cerium in cmsx-4 nickel-based superalloys by inductively coupled plasma mass spectrometryEI北大核心CSCD

CMSX-4 is the second-generation nickel-based single crystal superalloy used widely in the world. The oxidation resistance and corrosion resistance of CMSX-4 alloy can be improved by adding trace lanthanum (La), cerium (Ce) and other rare earth elements. A method for the simultaneous determination of La and Ce in CMSX-4 nickel-based superalloy by wet dissolution-inductively coupled plasma mass spectrometry was established. The sample was heated and dissolved under normal pressure by aqua regia and hydrofluoric acid, and the interference of fluorine ion was eliminated by using perchloric acid. The amount of dissolved acid and the digestion conditions were optimized. The limits of detection were 0.23 μg/g for La and 0.85 μg/g for Ce under optimized conditions. The spiked recoveries were 95.0%–98.9% with the relative standard deviations of 1.3%–3.9%, which can meet the requirements of accurate and rapid determination of La and Ce in CMSX-4 nickel-based superalloy. © 2023, Youke Publishing Co.,Ltd. All rights reserved.     

A Humidity-Induced Large Electronic Conductivity Change of 107 on a Metal-Organic Framework for Highly Sensitive Water Detection

The electronic conductivity (EC) of metal–organic frameworks (MOFs) is sensitive to strongly oxidizing guest molecules. Water is a relatively mild species, however, the effect of H₂O on the EC of MOFs is rarely reported. We explored the effect of H₂O on the EC in the MOFs (NH₂)₂-MIL-125 and its derivatives with experimental and theoretical investigations. Unexpectedly, a large EC increase of 10⁷ on H₂SO₄@(NH₂)₂-MIL-125 by H₂O was observed. Brønsted acid–base pairs formed with the −NH₂ groups, and H₂SO₄ played an important role in promoting the charge transfer from H₂O to the MOF. Based on H₂SO₄@(NH₂)₂-MIL-125, a high-performance chemiresistive humidity sensor was developed with the highest sensitivity, broadest detection range, and lowest limit of detection amongst all reported sensing materials to date. This work not only demonstrated that H₂O can remarkably influence the EC of MOFs, but it also revealed that post-modification of the structure of MOFs could enhance the influence of the guest molecule on their EC to design high-performance sensing materials.     

The Influence of the Electrodeposition Conditions on the Electroanalytical Performance of the Bismuth Film Electrode for Lead Determination

The bismuth film is a great promise as a suitable material to replace the mercury electrodes due to its low toxicity and good cathodic potential range. This work studies the influence of the electrodeposition conditions in the morphology and electroanalytical performance of the bismuth film electrodeposited onto copper electrode. The bismuth films were obtained in nitric or hydrochloric acid solutions with and without the presence of sodium citrate. The films were characterized by field emission scanning electron microscopy (FE‐SEM) and scanning electron microscopy with energy dispersive X‐ray spectrometry (SEM‐EDX). The microscopic analysis of the bismuth film obtained in HCl solution with sodium citrate (BIFE‐Cit) showed more homogeneous structure with higher content of bismuth than the film obtained in HCl only (BiFE‐HCl). The BiFE‐Cit exhibited a better analytical performance for lead with good adherence to the copper substrate.     

Iridium Oxide Film Electrodes for Anodic Stripping Voltammetry

The new iridium oxide film electrode, applied for the determination of lead(II), cadmium(II) and copper(II) traces using differential pulse anodic stripping voltammetry (DP ASV) is presented. The electrode display an interesting stripping voltammetric performance which compares with electrodes commonly used in voltammetry. The deposited film is known as anodically electrodeposited iridium oxide film (AEIROF). The AEIROF electrode is characterized by long‐term stability (more than 40 days) and very good reproducibility of the analytical signals in this time (≤12% for 0.5 μM of lead). The regeneration of iridium film is very simple in a time shorter than 60 seconds. The effects of various factors such as: thickness of AEIROF film, preconcentration potential and time, supporting electrolyte composition, potential interferences are optimized. The detection limit for AEIROF film electrode based on glassy carbon for an accumulation time of 30 s is as low as 7 nM for lead(II). The repeatability of the method at a concentration level of the lead(II) as low as 0.5 μM, expressed as RSD is 2.5% (n=10). The proposed method was successfully applied and validated by studying certified reference material CTA‐OTL‐1. Such an attractive use of ‘mercury–free’ ‐ environmentally friendly electrodes offers great promise to measure trace metals.