We study modular forms of some congruence subgroups. In this paper, we treat the cases level is 2-power, 3-power or 5. Structures of graded rings and many identities of infinite sum or infinite product are given. Theory of rational (1/3, 1/4, 1/5 etc...
In mathematics, a modular form is a holomorphic function on the complex upper half-plane, H {\displaystyle {\mathcal {H}}} , that roughly satisfies a functional
We study differential equations satisfied by modular forms associated to $Γ_1\timesΓ_2$, where $Γ_i (i=1,2)$ are genus zero subgroups of $SL_2(\mathbf R)$ commensurable with $SL_2(\mathbf Z)$, e.g., $Γ_0(N)$ or $Γ_0(N)^*$. In some examples, th...
The leitmotif of these Notes is the idea of a vertex operator algebra (VOA) and the relationship between VOAs and elliptic functions and modular forms. This is to some extent analogous to the relationship between a finite group and its irreducible...
Given two parties performing experiments in separate laboratories, we provide a diagrammatic formulation of what it means for the joint statistics of their experiments to satisfy local realism. In particular, we show that the principles of locality a...
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Formal techniques have been shown to be useful in the development of correct software. But the level of expertise required of practitioners of these techniques prohibits their widespread adoption. Formal techniques need to be tailored to the commer...
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Infrared (IR) luminosity is fundamental to understanding the cosmic star formation history and AGN evolution, since their most intense stages are often obscured by dust. Japanese infrared satellite, AKARI, provided unique data sets to probe these bot...
? Check how to display country flags (multiple selections) in SharePoint Online lists or Microsoft Lists using modern experience JSON formatting ? ? Using multiple selection choice col...
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Effective stellar feedback is used in models of galaxy formation to drive realistic galaxy evolution. Models typically include energy injection from supernovae as the dominant form of stellar feedback, often in some form of sub-grid recipe. However,...
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In the context of signed line graphs, this article introduces a modified inflation technique to study strong Gram congruence of non-negative (integral quadratic) unit forms, and uses it to show that weak and strong Gram congruence coincide among posi...
This manual accompanies the release of the particle data for 24 simulations of the EAGLE suite of cosmological hydrodynamical simulations of galaxy formation by the virgo consortium. It describes how to download these snapshots and how to extract dat...
Star clusters stand at the crossroads between galaxies and single stars. Resolving the formation of star clusters in cosmological simulations represents an ambitious and challenging goal, since modelling their internal properties requires very high r...
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We prove an explicit formula for a projection of singular vectors in the Verma module over a rank 2 Kac-Moody Lie algebra onto the universal enveloping algebra of the Heisenberg Lie algebra and of $sl_{2}$ (Theorem 3). The formula is derived from a...