scholarly journals Distribution of Modular Symbols in ℍ3

2020 ◽  
Author(s):  
Petru Constantinescu
Keyword(s):  
Method Of Moments ◽  
Cusp Form ◽  
General Setting ◽  
Kleinian Groups ◽  
Congruence Subgroups ◽  
Modular Symbols ◽  
Ring Of Integers ◽  
Second Moments ◽  
A New Technique ◽  
Imaginary Number

Abstract We introduce a new technique for the study of the distribution of modular symbols, which we apply to the congruence subgroups of Bianchi groups. We prove that if $K$ is a quadratic imaginary number field of class number one and $\mathcal{O}_K$ is its ring of integers, then for certain congruence subgroups of $\textrm{PSL}_2(\mathcal{O}_K)$, the periods of a cusp form of weight two obey asymptotically a normal distribution. These results are specialisations from the more general setting of quotient surfaces of cofinite Kleinian groups where our methods apply. We avoid the method of moments. Our new insight is to use the behaviour of the smallest eigenvalue of the Laplacian for spaces twisted by modular symbols. Our approach also recovers the first and second moments of the distribution.

scholarly journals Maqasid-based consumer preference index for Islamic home financing

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Hanudin Amin
Keyword(s):  
Consumer Choice ◽  
General Setting ◽  
Consumer Preference ◽  
Future Research ◽  
Building Society ◽  
Content Type ◽  
Customer Preference ◽  
Preference Index ◽  
A New Technique ◽  
Debt Policy

Purpose Using the maqasid-based consumer preference index (MCPi), this study aims to investigate customer preference for Islamic home financing. Design/methodology/approach The current study, based on valid 1,034 usable questionnaires and the MCPi, evaluates consumer choice for the supplied Islamic home finance products by 16 Islamic banks in Malaysia. Findings According to the findings, all banks have a moderate value of MCPi. Bank Islam Malaysia Berhad is at the top of the list, followed by Maybank Islamic, Commerce International Merchant Bankers Islamic and Malaysia Building Society Berhad. Research limitations/implications The MCPi is used in this study to test a new technique to measuring consumer preference. The contributions are confined to these particular variables – Educating Customer, Establishing Justice, Promoting Welfare and Fulfiling Islamic Debt Policy. The research also has limitations in terms of the facility’s general setting. Future research may shed light on these issues from new angles. Practical implications This research offers banks a new way to manage their products based on maqasid al-Sharīʿah. Originality/value In the context of Malaysia, this study introduces the MCPi, a new measure of consumer preference for home financing.

scholarly journals On the Distribution of Periods of Holomorphic Cusp Forms and Zeroes of Period Polynomials

2020 ◽  
Author(s):  
Asbjørn Christian Nordentoft
Keyword(s):  
Asymptotic Behavior ◽  
Method Of Moments ◽  
Cusp Form ◽  
Joint Distribution ◽  
Random Variables ◽  
Kloosterman Sums ◽  
Limiting Distribution ◽  
Independent Random Variables ◽  
Cusp Forms ◽  
Holomorphic Cusp Forms

Abstract In this paper, we determine the limiting distribution of the image of the Eichler–Shimura map or equivalently the limiting joint distribution of the coefficients of the period polynomials associated to a fixed cusp form. The limiting distribution is shown to be the distribution of a certain transformation of two independent random variables both of which are equidistributed on the circle $\mathbb{R}/\mathbb{Z}$, where the transformation is connected to the additive twist of the cuspidal $L$-function. Furthermore, we determine the asymptotic behavior of the zeroes of the period polynomials of a fixed cusp form. We use the method of moments and the main ingredients in the proofs are additive twists of $L$-functions and bounds for both individual and sums of Kloosterman sums.

scholarly journals Central values of additive twists of cuspidal L-functions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Asbjørn Christian Nordentoft
Keyword(s):  
Asymptotic Formula ◽  
Cusp Form ◽  
Dirichlet Character ◽  
Cusp Forms ◽  
Modular Symbols ◽  
Central Values ◽  
Arithmetic Distribution ◽  
Holomorphic Cusp Forms ◽  
Normally Distributed

Abstract Additive twists are important invariants associated to holomorphic cusp forms; they encode the Eichler–Shimura isomorphism and contain information about automorphic L-functions. In this paper we prove that central values of additive twists of the L-function associated to a holomorphic cusp form f of even weight k are asymptotically normally distributed. This generalizes (to k ≥ 4 {k\geq 4} ) a recent breakthrough of Petridis and Risager concerning the arithmetic distribution of modular symbols. Furthermore, we give as an application an asymptotic formula for the averages of certain “wide” families of automorphic L-functions consisting of central values of the form L ⁢ ( f ⊗ χ , 1 / 2 ) {L(f\otimes\chi,1/2)} with χ a Dirichlet character.

Poisson convergence and semi-induced properties of random graphs

1987 ◽  
Vol 101 (2) ◽  
pp. 291-300 ◽  
Author(s):  
Michał Karoński ◽  
Andrzej Ruciński
Keyword(s):  
Asymptotic Distribution ◽  
Random Graph ◽  
Random Graphs ◽  
Method Of Moments ◽  
Traditional Method ◽  
General Setting ◽  
Asymptotic Distributions ◽  
New Method ◽  
Isolated Trees ◽  
Ingenious Method

Barbour [l] invented an ingenious method of establishing the asymptotic distribution of the number X of specified subgraphs of a random graph. The novelty of his method relies on using the first two moments of X only, despite the traditional method of moments that involves all moments of X (compare [8, 10, 11, 14]). He also adjusted that new method for counting isolated trees of a given size in a random graph. (For further applications of Barbour's method see [4] and [10].) The main goal of this paper is to show how this method can be extended to a general setting that enables us to derive asymptotic distributions of subsets of vertices of a random graph with various properties.

Relative Galois module structure of rings of integers and elliptic functions

1983 ◽  
Vol 94 (3) ◽  
pp. 389-397 ◽  
Author(s):  
M. J. Taylor
Keyword(s):  
Number Field ◽  
Finite Extension ◽  
Elliptic Functions ◽  
Complex Numbers ◽  
Module Structure ◽  
Galois Module ◽  
Integral Ideal ◽  
Ring Of Integers ◽  
Rings Of Integers ◽  
Imaginary Number

Let K be a quadratic imaginary number field with discriminant less than −4. For N either a number field or a finite extension of the p-adic field p, we let N denote the ring of integers of N. Moreover, if N is a number field then we write for the integral closure of [½] in N. For an integral ideal & of K we denote the ray classfield of K with conductor & by K(&). Once and for all we fix a choice of embedding of K into the complex numbers .

scholarly journals Analysis of Arbitrary Reflector Antennas Applying the Geometrical Theory of Diffraction Together with the Master Points Technique

2013 ◽  
Vol 2013 ◽  
pp. 1-13
Author(s):  
María Jesús Algar ◽  
Jose-Ramón Almagro ◽  
Javier Moreno ◽  
Lorena Lozano ◽  
Felipe Cátedra
Keyword(s):  
Method Of Moments ◽  
Near Field ◽  
Complex Geometries ◽  
Geometrical Theory ◽  
Reflector Antennas ◽  
New Approach ◽  
Sampling Points ◽  
Time Requirements ◽  
Geometrical Theory Of Diffraction ◽  
A New Technique

An efficient approach for the analysis of surface conformed reflector antennas fed arbitrarily is presented. The near field in a large number of sampling points in the aperture of the reflector is obtained applying the Geometrical Theory of Diffraction (GTD). A new technique named Master Points has been developed to reduce the complexity of the ray-tracing computations. The combination of both GTD and Master Points reduces the time requirements of this kind of analysis. To validate the new approach, several reflectors and the effects on the radiation pattern caused by shifting the feed and introducing different obstacles have been considered concerning both simple and complex geometries. The results of these analyses have been compared with the Method of Moments (MoM) results.

Free quotients of subgroups of the Bianchi groups whose kernels contain many elementary matrices

1994 ◽  
Vol 116 (2) ◽  
pp. 253-273
Author(s):  
A. W. Mason ◽  
R. W. K. Odoni
Keyword(s):  
Normal Subgroup ◽  
Positive Integer ◽  
Commutator Subgroup ◽  
Quadratic Number ◽  
Congruence Subgroups ◽  
Quadratic Number Field ◽  
Ring Of Integers ◽  
Imaginary Quadratic Number ◽  
Low Dimensional ◽  
Fundamental Paper

AbstractLet d be a square-free positive integer and let be the ring of integers of the imaginary quadratic number field ℚ(√ − d) The Bianchi groups are the groups SL2() (or PSL2(). Let m be the order of index m in . In this paper we prove that for each d there exist infinitely many m for which SL2(m)/NE2(m) has a free, non-cyclic quotient, where NE2(m) is the normal subgroup of SL2(m) generated by the elementary matrices. When d is not a prime congruent to 3 (mod 4) this result is true for all but finitely many m. The proofs are based on the fundamental paper of Zimmert and its generalization due to Grunewald and Schwermer.The results are used to extend earlier work of Lubotzky on non-congruence subgroups of SL2(), which involves the concept of the ‘non-congruence crack’. In addition the results highlight a number of low-dimensional anomalies. For example, it is known that [SLn(m), SLnm)] = En(m), when n ≥ 3, where [SLn(m), SLn(m)] is the commutator subgroup of SL(m) and En(m) is the subgroup of SLn(m) generated by the elementary matrices. Our results show that this is not always true when n = 2.

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