scholarly journals A theta relation in genus 4

2001 ◽  
Vol 161 ◽  
pp. 69-83 ◽  
Author(s):  
Eberhard Freitag ◽  
Manabu Oura
Keyword(s):  
Linear Combination ◽  
Modular Group ◽  
Remarkable Property ◽  
Graded Ring ◽  
Defining Relation

The 2gtheta constants of second kind of genusggenerate a graded ring of dimensiong(g +1)/2. In the caseg ≥3 there must exist algebraic relations. In genusg =3 it is known that there is one defining relation. In this paper we give a relation in the caseg =4. It is of degree 24 and has the remarkable property that it is invariant under the full Siegel modular group and whose Φ-image is not zero. Our relation is obtained as a linear combination of code polynomials of the 9 self-dual doubly-even codes of length 24.

scholarly journals On Siegel modular forms part II

1995 ◽  
Vol 138 ◽  
pp. 179-197 ◽  
Author(s):  
Bernhard Runge
Keyword(s):  
Modular Forms ◽  
Modular Group ◽  
Congruence Subgroup ◽  
Siegel Modular Forms ◽  
Graded Ring

In this paper we compute dimension formulas for rings of Siegel modular forms of genus g = 3. Let denote the main congruence subgroup of level two, the Hecke subgroup of level two and the full modular group. We give the dimension formulas for genus g = 3 for the above mentioned groups and determine the graded ring of modular forms with respect to .

scholarly journals The vanishing of Poincaré series

1980 ◽  
Vol 23 (2) ◽  
pp. 151-161 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Positive Integer ◽  
Modular Form ◽  
Linear Combination ◽  
Cusp Form ◽  
Modular Group ◽  
Poincaré Series ◽  
Complex Field ◽  
Poincare Series ◽  
Holomorphic Modular Form ◽  
Image Position

Every holomorphic modular form of weight k > 2 is a sum of Poincaré series; see, for example, Chapter 5 of (5). In particular, every cusp form of even weight k ≧ 4 for the full modular group Γ(1) is a linear combination over the complex field C of the Poincaré series.Here mis any positive integer, z ∈ H ={z ∈ C: Im z>0} andThe summation is over all matriceswith different second rows in the (homogeneous) modular group, i.e. in SL(2, Z).The factor ½ is introducted for convenience.

2-ADIC PROPERTIES OF CERTAIN MODULAR FORMS AND THEIR APPLICATIONS TO ARITHMETIC FUNCTIONS

2005 ◽  
Vol 01 (01) ◽  
pp. 75-101 ◽  
Author(s):  
KEN ONO ◽  
YUICHIRO TAGUCHI
Keyword(s):  
Modular Forms ◽  
Quadratic Forms ◽  
Modular Group ◽  
Hecke Algebra ◽  
Arithmetic Functions ◽  
Partition Functions ◽  
Classifying Spaces ◽  
Graded Ring ◽  
Central Values

It is a classical observation of Serre that the Hecke algebra acts locally nilpotently on the graded ring of modular forms modulo 2 for the full modular group. Here we consider the problem of classifying spaces of modular forms for which this phenomenon continues to hold. We give a number of consequences of this investigation as they relate to quadratic forms, partition functions, and central values of twisted modular L-functions.

On generic complexity of the subset sum problem in monoids and groups of integer matrix of order two

2020 ◽  
Vol 25 (4) ◽  
pp. 10-15
Author(s):  
Alexander Nikolaevich Rybalov
Keyword(s):  
Linear Group ◽  
Modular Group ◽  
Special Linear Group ◽  
Second Order ◽  
Integer Matrix ◽  
Subset Sum ◽  
Algorithmic Problems ◽  
Almost All ◽  
Subset Sum Problems ◽  
Generic Complexity

Generic-case approach to algorithmic problems was suggested by A. Miasnikov, I. Kapovich, P. Schupp and V. Shpilrain in 2003. This approach studies behavior of an algo-rithm on typical (almost all) inputs and ignores the rest of inputs. In this paper, we prove that the subset sum problems for the monoid of integer positive unimodular matrices of the second order, the special linear group of the second order, and the modular group are generically solvable in polynomial time.

Recognition by Linear Combination of Models

1989 ◽  
Author(s):  
Shimon Ullman ◽  
Ronen Basri
Keyword(s):  
Linear Combination

scholarly journals An Accurate Reconstruction of CMB E Mode Signal over Large Angular Scales using Prior Information of CMB Covariance Matrix in ILC Algorithm

2020 ◽  
Author(s):  
Ujjal Purkayastha ◽  
Vipin Sudevan ◽  
Rajib Saha
Keyword(s):  
Power Spectrum ◽  
Linear Combination ◽  
Covariance Matrix ◽  
Large Scale ◽  
Prior Information ◽  
Future Generation ◽  
Temperature Anisotropy ◽  
Satellite Mission ◽  
Angular Power Spectrum ◽  
Accurate Reconstruction

Abstract Recently, the internal-linear-combination (ILC) method was investigated extensively in the context of reconstruction of Cosmic Microwave Background (CMB) temperature anisotropy signal using observations obtained by WMAP and Planck satellite missions. In this article, we, for the first time, apply the ILC method to reconstruct the large scale CMB E mode polarization signal, which could probe the ionization history, using simulated observations of 15 frequency CMB polarization maps of future generation Cosmic Origin Explorer (COrE) satellite mission. We find that the clean power spectra, from the usual ILC, are strongly biased due to non zero CMB-foregrounds chance correlations. In order to address the issues of bias and errors we extend and improve the usual ILC method for CMB E mode reconstruction by incorporating prior information of theoretical E mode angular power spectrum while estimating the weights for linear combination of input maps (Sudevan & Saha 2018b). Using the E mode covariance matrix effectively suppresses the CMB-foreground chance correlation power leading to an accurate reconstruction of cleaned CMB E mode map and its angular power spectrum. We compare the performance of the usual ILC and the new method over large angular scales and show that the later produces significantly statistically improved results than the former. The new E mode CMB angular power spectrum contains neither any significant negative bias at the low multipoles nor any positive foreground bias at relatively higher mutlipoles. The error estimates of the cleaned spectrum agree very well with the cosmic variance induced error.

scholarly journals Partially constrained internal linear combination: A method for low-noise CMB foreground mitigation

2021 ◽  
Vol 103 (10) ◽  
Author(s):  
Y. Sultan Abylkairov ◽  
Omar Darwish ◽  
J. Colin Hill ◽  
Blake D. Sherwin
Keyword(s):  
Linear Combination ◽  
Low Noise
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