We extend categorical Morita equivalence to finite tensor categories gra\-ded by a finite group $G$. We show that two such categories are graded Morita equivalent if and only if their equivariant Drinfeld centers are equivalent as braided $G$-crossed tensor categories.

We define and study deductive systems in unital quantum-B algebras. Then we introduce the notion of filter in a unital quantum-B algebras with product, and show that filters coincide with deductive systems. We also introduce and characterize the commutative deductive systems in unital quantum-B algebras.

For any graph $G$, the Grundy (or First-Fit) chromatic number of $G$, denoted by $\Gamma(G)$ (also $\chi_{_{\sf FF}}(G)$), is defined as the maximum number of colors used by the First-Fit (greedy) coloring of the vertices of $G$. Determining the Grundy number is $NP$-complete, and obtaining bounds for $\Gamma(G)$ in terms of the known graph parameters is an active research topic. By a star partition of $G$ we mean any partition of $V(G)$ into say $V_1, \ldots, V_k$ such that each $G[V_i]$ contains a vertex adjacent to any other vertex in $V_i$. In this paper using the star partition of graphs we obtain the first upper bounds for the Grundy number in terms of the domination number. We also prove some bounds in terms of the domination number and girth of graphs.

The Suzuki algebra $A_{Nn}^{\mu \lambda}$, introduced by Suzuki Satoshi in 1998, is a class of cosemisimple Hopf algebras. It is not categorically Morita-equivalent to a group algebra in general. In this paper, the author gives a complete set of simple Yetter-Drinfeld modules over the Suzuki algebra $A_{N\,2n}^{\mu\lambda}$ and investigates the Nichols algebras over those simple Yetter-Drinfeld modules. The involved finite dimensional Nichols algebras of diagonal type are of Cartan type $A_1$, $A_1\times A_1$, $A_2$, $A_2\times A_2$, Super type ${\bf A}_{2}(q;\I_2)$ and the Nichols algebra $\ufo(8)$. There are $64$, $4m$ and $m^2$-dimensional Nichols algebras of non-diagonal type over $A_{N\,2n}^{\mu \lambda}$. The $64$-dimensional Nichols algebras are of dihedral rack type $\Bbb{D}_4$. The $4m$ and $m^2$-dimensional Nichols algebras $\mathfrak{B}(V_{abe})$ discovered first by Andruskiewitsch and Giraldi can be realized in the category of Yetter-Drinfeld modules over $A_{Nn}^{\mu \lambda}$. Using a result of Masuoka, we prove that $\dim\mathfrak{B}(V_{abe})=\infty$ under the condition $b^2=(ae)^{-1}$, $b\in\Bbb{G}_{m}$ for $m\geq 5$.

The existence of homoclinic solutions is obtained for the ordinary $p$-Laplacian systems $\frac{{\rm d} }{{\rm d}t}\left(|\dot{u}(t)|^{p-2}\dot{u}(t)\right)-\nabla K(t,u(t))+\nabla W(t,u(t))=0$, as the limit of the $2kT$-periodic solutions which are obtained by the Mountain Pass Theorem, where $K(t,x)$ satisfies some sub-$p$ linear or asymptotic-$p$ linear conditions and $W(t,x)$ satisfies some local super-$p$ linear conditions.

We prove that the symmetric group $S_n$ has a unique minimal cover $\mathcal{M}$ by maximal nilpotent subgroups, and we obtain an explicit and easily computed formula for the size of $\mathcal{M}$. In addition, we prove that the size of $\mathcal{M}$ is equal to the size of a maximal non-nilpotent subset of $S_n$. This cover $\mathcal{M}$ has attractive properties; for instance, it is a normal cover, and the number of conjugacy classes of subgroups in the cover is equal to the number of partitions of $n$ into distinct positive integers. We show that these results contrast with those for the alternating group $A_n$. In particular, we prove that, for all but finitely many values of $n$, no minimal cover of $A_n$ by maximal nilpotent subgroups is a normal cover and the size of a minimal cover of $A_n$ by maximal nilpotent subgroups is strictly greater than the size of a maximal non-nilpotent subset of $A_n$.

We discuss additivity and lineability of classes of functions generalizing the concept of injectivity. In particular, we prove that it is consistent with ZFC that the class of almost injective functions has maximal possible lineability, i.e., it contains (with the exception of the zero function) a vector space of cardinality $2^\cont$.

We prove the existence of a maximal dimensional vector space of real functions on the real plane all of whose nonzero members are bounded, entire, non-Lebesgue-integrable, and satisfy the equality of the two iterated integrals given in the conclusion of the Fubini theorem, with the additional property that these iterated integrals exist in the Riemann sense, but not in the Lebesgue one. Moreover, this vector space is dense in the space of smooth functions on the plane under its natural topology.