One triangle has angles measuring A, β−α and 90. The other triangle has angles measuring B, α− β, and 90. The two right angles are placed back to back with the two triangles having the same height. Using algebra and substitution, we got something that is similar to the proof but a couple of variables are switched, meaning that there is more work to do. I wonder if this method can be used though...

Triangle algebras are equationally defined structures that are equivalent with certain residuated lattices on a set of intervals, which are called interval-valued residuated lattices (IVRLs). Triangle algebras have been used to construct Triangle Logic (TL), a formal fuzzy logic that is sound and complete w.r.t. the class of IVRLs. In this paper, we prove that the so-called pseudo-prelinear tri...

Abstract. Given any simple biorientable graph it is shown that there exists a weak *-Hopf algebra constructed on the vector space of graded endomorphisms of essential paths on the graph. This construction is based on a direct sum decomposition of the space of paths into orthogonal subspaces one of which is the space of essential paths. Two simple examples are worked out with certain detail, the...

Abstract Given a negatively graded Calabi-Yau algebra, we regard it as DG algebra with vanishing differentials and study its cluster category. We show that this is sign-twisted realise category triangulated hull of an orbit derived the singularity finite-dimensional Iwanaga-Gorenstein algebra. Along way, give two results stand on their own. First, coherent sheaves over has natural tilting subca...

Let ${\Bbbk }$ be an algebraically closed field, let Λ a finite dimensional -algebra, and $\widehat {\Lambda the repetitive algebra of Λ. For stable category finitely generated left -modules -mod, we prove that every Auslander-Reiten triangle in -mod is induced from sequence -modules. We use this fact to show irreducible morphisms fall into three canonical forms: (i) all component are split mon...

Abstract For a finite dimensional algebra A , the bounded homotopy category of projective -modules and derived are dual to each other via certain categories locally-finite cohomological functors. We prove that duality gives rise 2-categorical between strict 2-categories involving categories, respectively. apply study triangle autoequivalence groups.

Stature of the triangle): On the wisdom of the triangle and geometry and some rules on qualities and algebra. Jerusalem Such a really remarkable discovery. I wanted your opinion on it. You know the formula m over naught equals infinity, m being any positive number? [m/0 = ∞]. Well, why not reduce the equation to a simpler form by multiplying both sides by naught? In which case you have m equals...

The Gromov-hyperbolic δ or “fatness” of a hyperbolic geodesic triangle, defined to be the infimum of the perimeters of all inscribed triangles, is given an explicit analytical expression in term of the angle data of the triangle. By a hyperbolic extension of Fermat’s principle, the optimum inscribed triangle is easily constructed as the orthic triangle, that is, the triangle with its vertices a...

Using the most elementary methods and considerations, the solution of the star-triangle condition a 2+b2−c2 2ab = (a ′)2+(b′)2−(c′)2 2a′b′ is shown to be a necessary condition for the extension of the operator coalgebra of the six-vertex model to a bialgebra. A portion of the bialgebra acts as a spectrum-generating algebra for the algebraic Bethe ansatz, with which higher-dimensional representa...

We study connections among structures in commutative algebra, combinatorics, and discrete geometry, introducing an array of numbers, called Borel’s triangle, that arises in counting objects in each area. By defining natural combinatorial bijections between the sets, we prove that Borel’s triangle counts the Betti numbers of certain Borel-fixed ideals, the number of binary trees on a fixed numbe...