Let $$E/{\mathbb {Q}}$$ be an elliptic curve and p odd prime where E has good reduction, assume that admits a rational p-isogeny. In this paper we study the anticyclotomic Iwasawa theory of over imaginary quadratic field in which splits, relate to characters by variation method Greenberg–Vatsal. As result our obtain proofs (under relatively mild hypotheses) Perrin-Riou’s Heegner point main conj...

In this study, we consider the stochastic Konno–Oono system to investigate soliton solutions under multiplicative sense. The noise is considered firstly in Stratonovich sense and secondly Ito^ Applications of include current-fed strings interacting with an external magnetic field. F-expansion method used find different types form dark, singular, complex combo, solitary, periodic, mixed rational...

This paper is concerned with α– and D–negative imaginary systems. The definition of α–negative imaginary transfer functions is firstly introduced. Then, the relationship between negative imaginary and α–negative imaginary transfer functions is studied. By means of the generalized inverse, an α–negative imaginary lemma is proposed to test the α–negative imaginary property of transfer functions. ...

It is doubtful whether I shall be able to meet the high standards set by previous speakers. I shall not prove deep theorems. I shall not present astonishing new results. Instead of this I shall try to catch your attention by a fictitious dialogue. I shall employ the help of imaginary discussants like the “Bayesian” or the “experimentalist.” A “chairman” will determine who speaks next, but he sh...

An imaginary cube is a three dimensional object which has square projections in three orthogonal directions just as a cube has. In this paper, we study imaginary cubes and present sculptures based on imaginary cubes. We show that there are 16 kinds of minimal convex imaginary cubes. The list of 16 representative minimal convex imaginary cubes contains, as well as a regular tetrahedron and a cub...