Let $a$ and $b$ be two distinct fixed positive integers such that $\min \{a,b\}> 1.$ First, we correct an oversight from [12]. Then, we show that the equation in the title with $b \equiv 3 \pmod 8$, $b$ prime and $a$ even has no solution in positive integers $n, x$. This generalizes a result of Szalay [10].

Functions of hyperbolic type encode representations on real or complex hyperbolic spaces, usually infinite-dimensional. These notes set up the complex case. As applications, we prove the existence of a non-trivial deformation family of representations of $\mathbf{SU}(1,n)$ and of its infinite-dimensional kin ${\rm Is}(\mathbf{H}_{\mathbf{C}}^{\infty})$. We further classify all the self-representations of ${\rm Is}(\mathbf{H}_{\mathbf{C}}^{\infty})$ that satisfy a compatibility condition for the subgroup ${\rm Is}(\mathbf{H}_{\mathbf{R}}^{\infty})$. It turns out in particular that translation lengths and Cartan arguments determine each other for these representations. In the real case, we revisit earlier results and propose some further constructions.

In this paper, we study the structure of attached prime ideals of generalized local cohomology modules. We show the Shifted Localization Principle, the Weak General Shifted Localization Principle and the Shifted Completion Principle for generalized local cohomology modules. We introduce a generalization of pseudo-support and pseudo-dimension using the generalized local cohomology modules and we show that pseudo-support is closed under certain conditions. As applications, we investigate the top non-vanishing and the associativity formula for multiplicities of generalized local cohomology modules.

Involving an extended 1-cocycle condition, we first define a coproduct on the space of bi-decorated planar rooted forests to equip it with a left counital bialgebraic structure. We introduce the concept of left counital $(\Omega, \alpha)$-cocyle bialgebras and show that the space of bi-decorated planar rooted forests is the free object in the category of left counital $(\Omega, \alpha)$-cocyle bialgebras. We then prove a generalized fact that a connected graded left counital bialgebra is a left counital right antipode Hopf algebra in the sense that the antipode is only right-sided. Having this fact in hand, a left counital Hopf algebraic structure on bi-decorated rooted forests is also established. Finally, we construct a Rota-Baxter system on the left counital Hopf algebra.

In this paper, we study tensor (or monoidal) categories of finite rank over an algebraically closed field $\mathbb F$. Given a tensor category $\mathcal{C}$, we have two structure invariants of $\mathcal{C}$: the Green ring (or the representation ring) $r(\mathcal{C})$ and the Auslander algebra $A(\mathcal{C})$ of $\mathcal{C}$. We show that a Krull-Schmit abelian tensor category $\mathcal{C}$ of finite rank is uniquely determined (up to tensor equivalences) by its two structure invariants and the associated associator system of $\mathcal{C}$. In fact, we can reconstruct the tensor category $\mathcal{C}$ from its two invariants and the associator system. More general, given a quadruple $(R, A, \phi, a)$ satisfying certain conditions, where $R$ is a $\mathbb{Z}_+$-ring of rank $n$, $A$ is a finite dimensional $\mathbb F$-algebra with a complete set of $n$ primitive orthogonal idempotents, $\phi$ is an algebra map from $A\otimes_{\mathbb F}A$ to an algebra $M(R, A)$ constructed from $A$ and $R$, and $a=\{a_{i,j,l}|1\leqslant i,j,l\leqslant n\}$ is a family of ``invertible" matrices over $A$, we can construct a Krull-Schmidt and abelian tensor category $\mathcal C$ over $\mathbb{F}$ such that $R$ is the Green ring of $\mathcal C$ and $A$ is the Auslander algebra of $\mathcal C$. In this case, $\mathcal C$ has finitely many indecomposable objects (up to isomorphisms) and finite dimensional Hom-spaces. Moreover, we will give a necessary and sufficient condition for such two tensor categories to be tensor equivalent.

We aim to construct a non-commutative algebraic geometry in the style of Chevalley by using generalised valuations. To this end, we introduce groupoid valuation rings and associate suitable value functions to them. We show that many results from classical valuation theory can be generalised in a natural way to this context and give several examples. In the final section, we give a very concrete example of what a non-commutative curve would look like in this new setting.

We study the large time behavior of solutions to the Neumann Problem of the porous medium equation. It is shown that the critical Fujita exponent is determined not only by the spatial dimension and the nonlinear exponent, but also by the coefficient $k$ of the first-order term.