Cohomological relation between Jacobi forms and skew‐holomorphic Jacobi forms

Eichler and Zagier developed a theory of Jacobi forms to understand and extend Maass' work on the Saito‐Kurokawa conjecture. Later Skoruppa introduced skew‐holomorphic Jacobi forms, which play an important role in understanding liftings of modular forms and Jacobi forms. In this paper, we explain a relation between Jacobi forms and skew‐holomorphic Jacobi forms in terms of a group cohomology. More precisely, we introduce an isomorphism from the direct sum of the space of Jacobi cusp forms on and the space of skew‐holomorphic Jacobi cusp forms on with the same half‐integral weight to the Eichler cohomology group of with a coefficient module coming from polynomials.     

Congruences involving the $$U_{\ell }$$Uℓ operator for weakly holomorphic modular forms

The Ramanujan Journal - Let $$\lambda $$ be an integer, and $$f(z)=\sum _{n\gg -\infty } a(n)q^n$$ be a weakly holomorphic modular form of weight$$\lambda +\frac{1}{2}$$ on $$\Gamma _0(4)$$ with...     

Eichler integrals and harmonic weak Maass forms

Recently, K. Bringmann, P. Guerzhoy, Z. Kent and K. Ono studied the connection between Eichler integrals and the holomorphic parts of harmonic weak Maass forms on the full modular group. In this article, we extend their result to more general groups, namely, H-groups by employing the theory of supplementary functions introduced and developed by M.I. Knopp and S.Y. Husseini. In particular, we show that the set of Eichler integrals, which have polynomial period functions, is the same as the set of holomorphic parts of harmonic weak Maass forms of which the non-holomorphic parts are certain period integrals of cusp forms. From this we deduce relations among period functions for harmonic weak Maass forms.     

The Eichler–Shimura cohomology theorem for Jacobi forms

Let \(\Gamma \) be a subgroup of finite index in \(\mathrm {SL}(2,\mathbb {Z})\). Eichler defined the first cohomology group of \(\Gamma \) with coefficients in a certain module of polynomials. Eichler and Shimura established that this group is isomorphic to the direct sum of two spaces of cusp forms on \(\Gamma \) with the same integral weight. These results were generalized by Knopp to cusp forms of real weights. In this paper, we define the first parabolic cohomology groups of Jacobi groups \(\Gamma ^{(1,j)}\) and prove that these are isomorphic to the spaces of (skew-holomorphic) Jacobi cusp forms of real weights. We also show that if \(j=1\) and the weights of Jacobi cusp forms are in \(\frac{1}{2}\mathbb {Z}-\mathbb {Z}\), then these isomorphisms can be written in terms of special values of partial L-functions of Jacobi cusp forms.     

Increasing the mass resolution in a TOF aerosol mass spectrometer using a nonlinear post focusing method

The mass resolution for the time of flight aerosol mass spectrometer for aerosol component analysis is dependent on the initial direction and energy of the ions. We have found that the shape of the optimum post focusing electric field is nonlinear. The maximum electric potential should be applied to the ions whose initial direction is 90°. To determine the post focusing effects, a laser ablation mass spectrometer was installed. By using this LA-MS, it was found that the average energy distribution of the laser ablated ions was 8 eV. To establish an optimum mass resolution, a time delay and a high voltage are needed. The study results showed that 1500 ns and 3.7 kV, respectively, were the optimum parameters for time delay and voltage for this system. Using these optimized parameters, good resolution between the isotope mass signals of copper was achieved.     

Optimization of critical factors affecting the performance of an allergen chip for the analysis of an allergen-specific human IgE in serum.

A sensitive and multiplexed assay of allergen-specific human immunoglobulin E (IgE) is of great significance in the precise diagnosis of allergies. We report on the optimization of critical factors for chip-based analysis of IgE in human serum with a high reliability. Extracts of two mite species were used as model allergens, and were spotted onto a glass slide for the construction of an allergen chip. Respective allergen-specific IgE in human serum was analyzed by using biotinylated anti-human IgE and a streptavidin-Cy3 conjugate. Factors affecting the performance of the allergen chip were investigated and optimized. Especially, the effect of additives, the concentrations of biotinylated anti-human IgE and the streptavidin-Cy3 conjugate, the serum dilution factor, and the concentration of allergen extract as a capturing agent were examined in detail. Under the optimized conditions, a chip-based analysis for sera from 43 patients revealed a reliable and reproducible diagnosis of respective allergies, showing a good correlation with a conventional CAP assay.     

Partitions weighted by the parity of the crank

The ‘crank’ is a partition statistic which originally arose to give combinatorial interpretations for Ramanujan's famous partition congruences. In this paper, we establish an asymptotic formula and a family of Ramanujan type congruences satisfied by the number of partitions of n with even crank Me(n) minus the number of partitions of n with odd crank Mo(n). We also discuss the combinatorial implications of q-series identities involving Me(n)−Mo(n). Finally, we determine the exact values of Me(n)−Mo(n) in the case of partitions into distinct parts. These values are at most two, and zero for infinitely many n.