The main result of this note is that given any two time-symmetric cycles, one can find a time-symmetric extension of one by the other. This means that given a time-symmetric cycle, both time-symmetric doubles and square roots can be found. 1. Introduction. In [4], the idea of time-symmetric cycles was introduced. The basic idea underlying the definition of these cycles is that the dynamics asso...

1. Matrices, Vectors and their Basic Operations 1.1. Matrices 1.2. Vectors 1.3. Addition and Scalar Multiplication of Matrices 1.4. Multiplication of Matrices 2. Determinants 2.1. Square Matrices 2.2. Determinants 2.3. Cofactors and the Inverse Matrix 3. Systems of Linear Equations 3.1. Linear Equations 3.2. Cramer’s Rule 3.3. Eigenvalues of a Complex Square Matrix 3.4. Jordan Canonical Form 4....

We prove a new upper bound on the second moment of Maass form symmetric square L -functions defined over Gaussian integers. Combining this estimate with recent result Balog–Biro–Cherubini–Laaksonen, we improve error term in prime geodesic theorem for Picard manifold.

CONTENTS We use Zagier's method to compute the critical values of the 1. Introduction symmetric square L-functions of six cuspidal eigenforms of level 2. Calculating the Critical Values one with rational coefficients. According to the Bloch-Kato 3. Tables of Results conjecture, certain large primes dividing these critical values 4. An Observation must be the orders of elements in generalised Sh...

The PT-symmetric square well problem is considered in a SUSY framework. When the coupling strength Z lies below the critical value Z (crit) 0 where PT symmetry becomes spontaneously broken, we find a hierarchy of SUSY partner potentials, depicting an unbroken SUSY situation and reducing to the family of sec2-like potentials in the Z → 0 limit. For Z above Z 0 , there is a rich diversity of SUSY...