The generating function for the Dirichlet series $L_{m}(s)$

Chen, William Y. C.; Fan, Neil J. Y.; Jia, Jeffrey Y. T.

doi:10.1090/s0025-5718-2011-02520-2

Cited by 2 publications

(2 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…But excessive amount of Cu 2 + ion not only imbalance the cellular process that lead to Wilson's disease, Parkinson's disease but also causes many problems such as gastro intestinal disturbance, liver and kidney damage. [1][2][3][4][5] According to World Health Organization (WHO) maximum permissible limit is 20.5 μM in drinking water. [6] Moreover, copper ions are released from metallurgical plants and mixed with water, dust and fume that increase environmental pollution.…”

Section: Introductionmentioning

confidence: 99%

Nano Gold Hybrid Polyvinyl Alcohol Films for Sensing of Cu²⁺ ions

et al. 2019

View full text Add to dashboard Cite

An efficient and eco‐friendly approach is adopted for the synthesis of highly stable L‐alanine assisted nano gold based polyvinyl alcohol nanocomposite hybrid films (Au/PVA).The –COOH groups of L‐alanine and oxygeneous groups of polyvinyl alcohol are utilized for the synthesis of size controlled gold nanoparticles. The as‐synthesized nanocomposites show rapid and selective detection of Cu2+ ion in aqueous medium. With addition Cu2+ ion to as‐synthesized nanocomposites, the SPR peaks of Au NPs (538 nm) decrease with occurrence of a red shift as well as a new peak is observed at higher wavelength which is due to the aggregation of gold nanoparticles in the nanocomposites hybrid films. Scanning electron microscope (SEM) micrograph and energy dispersive spectroscopy (EDS) reveal the aggregation of gold nanoparticles in the nanocomposite hybrid films. The limit of detection and limit of quantification for Cu2+ ion are 40 nM and 133 nM respectively. This technique for detection of Cu2+ ion in waste water samples is inexpensive and eco‐friendly since no any toxic chemicals are used. In the presence of Cu2+ ion, the colour of the hybrid film is changed hence, this method also involves visual detection of Cu2+ ion without use of electrolyte or transducer. Further, many reported methods represent the in situ monitoring of Cu2+ ion, but the present hybrid film can be used for in situ as well as ex situ monitoring of Cu2+ ion.

show abstract

Section: Introductionmentioning

confidence: 99%

Nano Gold Hybrid Polyvinyl Alcohol Films for Sensing of Cu²⁺ ions

et al. 2019

View full text Add to dashboard Cite

show abstract

“…Hoffman [5] showed that the exponential generating function for the number of snakes of type B n also equals the right hand side of (1.1), that is, the generating function of the Springer numbers. Recently, Chen, Fan and Jia [3] obtained a formula for the generating function of s m,n for arbitrary m, which in principle leads to a combinatorial interpretation of the numbers s m,n in terms of alternating augmented m-signed permutations. Note that for m = 2, alternating augmented 2-signed permutations are exactly snakes of type B n .…”

Section: Introductionmentioning

confidence: 99%

Labeled Ballot Paths and the Springer Numbers

Chen¹,

Fan

Jia

2011

SIAM J. Discrete Math.

Self Cite

View full text Add to dashboard Cite

The Springer numbers are defined in connection with the irreducible root systems of type B n , which also arise as the generalized Euler and class numbers introduced by Shanks. Combinatorial interpretations of the Springer numbers have been found by Purtill in terms of André signed permutations, and by Arnol'd in terms of snakes of type B n . We introduce the inversion code of a snake of type B n and establish a bijection between labeled ballot paths of length n and snakes of type B n . Moreover, we obtain the bivariate generating function for the number B(n, k) of labeled ballot paths starting at (0, 0) and ending at (n, k). Using our bijection, we find a statistic α such that the number of snakes π of type B n with α(π) = k equals B(n, k). We also show that our bijection specializes to a bijection between labeled Dyck paths of length 2n and alternating permutations on [2n].

show abstract

The generating function for the Dirichlet series $L_{m}(s)$

Cited by 2 publications

References 24 publications

Nano Gold Hybrid Polyvinyl Alcohol Films for Sensing of Cu²⁺ ions

Nano Gold Hybrid Polyvinyl Alcohol Films for Sensing of Cu²⁺ ions

Labeled Ballot Paths and the Springer Numbers

Contact Info

Product

Resources

About

The generating function for the Dirichlet series $L_{m}(s)$

Cited by 2 publications

References 24 publications

Nano Gold Hybrid Polyvinyl Alcohol Films for Sensing of Cu2+ ions

Nano Gold Hybrid Polyvinyl Alcohol Films for Sensing of Cu2+ ions

Labeled Ballot Paths and the Springer Numbers

Contact Info

Product

Resources

About

Nano Gold Hybrid Polyvinyl Alcohol Films for Sensing of Cu²⁺ ions

Nano Gold Hybrid Polyvinyl Alcohol Films for Sensing of Cu²⁺ ions