Adler’s lattice equation has acquired the status of a master equation among 2D discrete integrable systems. In this paper we derive what we believe are the first explicit solutions of this equation. In particular it turns out necessary to establish a non-trivial seed solution from which soliton solutions can subsequently be constructed using the Bäcklund transformation. As a corollary we find t...

New static regular axially symmetric solutions of SU (2) Euclidean Yang-Mills theory are constructed numerically. They represent calorons having trivial Polyakov loop at spacial infinity. The solutions are labeled by two integers m, n. It is shown that besides known, charge one self-dual periodic instanton solution, there are other non-self dual solutions of the Yang-Mills equations naturally c...

The Regular Post Embedding Problem is a variant of Post’s Correspondence Problem where one compares strings with the subword relation and imposes additional regular constraints on admissible solutions. It is known that this problem is decidable, albeit with very high complexity. We consider and solve variant problems where the set of solutions is compared to regular constraint sets and where on...

‎three infinite families of finite abelian groups will be‎ ‎described such that each member of these families has‎ ‎the r'edei $k$-property for many non-trivial values of $k$‎.

We describe a general framework for studying duality between different phase spaces which share the same symmetry group H. Solutions corresponding to collective dynamics become dual in the sense that they are generated by the same curve in H. Explicit examples of phase spaces which are dual with respect to a common non trivial coadjoint orbit Oc,0 (α, 1) ⊂ h ∗ are constructed on the cotangent b...

We present a reformulation of SU(2) Yang-Mills theory in the maximal Abelian gauge, where the non-Abelian gauge field components are exactly integrated out at the expense of a new Abelian tensor field. The latter can be treated in a semiclassical approximation and the corresponding saddle point equation is derived. Besides the non-trivial solutions, which are presumably related to non-perturbat...

We investigate the infra-red stable fixed points of the Yukawa couplings in the minimal version of the supersymmetric standard model with R-parity violation. Retaining only the R-parity violating couplings of higher generations, we analytically study the solutions of the renormalization group equations of these couplings together with the topand b-quark Yukawa couplings. We show that only the B...

Platformer games let players solve real-time, physics-based puzzles by jumping and moving around to reach different goals. Designing levels for this context is a non-trivial task; the placement of well-timed jumps, moving platforms, interesting traps, etc., has a complex relationship to in-game challenge and the existence of possible solutions. In this work, we describe three different search a...

The roots of -1 in the set of biquaternions (quaternions with complex components, or complex numbers with quaternion real and imaginary parts) are studied and it is shown that there is an infinite number of non-trivial complexified quaternion roots (and two degenerate solutions which are the complex imaginary operator and the set of unit pure real quaternions). The non-trivial roots are shown t...