Let D be a (m, u, k, λ)-difference set in a group G relative to a subgroup U of G. We say D is symmetric if D^(−1) is also a (m, u, k, λ)-difference set. By a result of [7] D is symmetric if U is a normal subgroup of G. In general, D is non-symmetric when U is not normal in G. In this paper we study a condition under which D is symmetric and show that if D is semiregular then D is symmetric if and only if the dual of dev(D) is a divisible design. We also give a modification of Davis' product construction of relative difference sets and as an application we give a class of non-symmetric semiregular relative difference sets.

Let $V$ be a finite dimensional vector space over the complex number field $\mathbb{C}$. Suppose that, by the adjoint action, a reductive subgroup $\tilde G$ of $GL(V)$ acts on a subspace $\tilde L$ of End$(V)$ and a closed subgroup $G$ of $\tilde G$ acts on a subspace $L$ of $\tilde L$. In this paper, we give a sufficient condition on the inclusion $(G,L) \hookrightarrow (\tilde G, \tilde L)$ for which the orbits correspondence $L/G \to \tilde L/ \tilde G (O \mapsto \tilde O := $Ad$(\tilde G) \dot O)$ is injective. Moreover we show that the ring $\mathbb{C}[L]^G$ of $G$-invariants on $L$ is the integral closure of $\mathbb{C}[\tilde L]^{\tilde G}|_L$ in its quotient field. Then, if the ring $\mathbb{C}[\tilde L]^{\tilde G}|_L$ is normal, the restriction map rest : $\mathbb{C}[\tilde L]^{\tilde G} \to \mathbb{C}[L]^G (f \mapsto f|_L)$ is surjective. By using this, we give some examples for which $L/G \to \tilde L/ \tilde G$ is injective and rest $: \mathbb{C}[\tilde L]^{\tilde G} \to \mathbb{C}[L]^G$ is surjective.

In this paper, we give a combinatorial proof to the following new product formula: \[\Pi_{i=1}^m 2F0(-a_i, -b_i; z) =\Pi_{r=0}^n p(r) 2F0(-n, -r; z).\]

In this paper we prove two results on the Levi conditions for weakly hyperbolic systems with characteristics of constant multiplicities. A first result concerns scalar operators: we prove that Levi conditions defined by the second author in [29] are equivalent to the usual Levi conditions for scalar operator. A second result concerns systems whose principal symbol has a Jordan form made of a large number of $2 \times 2$ blocks. For these systems we express the first Levi condition via an invariant constructed from the sub-characteristic matrix. Moreover we show that this condition is necessary for the $C^\infty$ well-posedness.

Let $\mathfrak M$ be an invariant subspace of $L^2(\mathbb{T}^2)$. Considering the largest z-invariant (resp. $w$-invariant) subspace $\mathfrak {F}_z$ (resp. $\mathfrak {F}_w)$ in the wandering subspace $M \ominus zw{\mathfrak{M}}$ of $\mathfrak{M}$ with respect to the shift operator $zw$. If $\mathfrak{F}_w \ne \{0\}$ and $\mathfrak{F}_z \ne \{0\}$, then we consider the certain form of invariant subspaces $\mathfrak{M}$ of $L^2(T^2)$. Furthermore, we study certain classes of invariant subspaces of $L^2(T^2)$.

In this paper, we employ the notion of weighted sharing to consider the problem of uniqueness of meromorphic functions when they share two or three sets. Our results not only improve the results of Qiu and Fang [21], Lin and Yi [27], Lin and Yi [28], Fang and Lahiri [6] but also supplement the result of Lahiri [13] in a new direction and consequently provide an answer to the question of Gross [7].

We proved affine planes corresponding to quadratic planar functions over $F_{p^n}$ are semifield planes, and we determined affine planes corresponding to planar functions $f(x) = x^10 − αx^6 − α^2x^2$ by Ding and Yuan. Moreover we calculated explicit shapes of planar functions from the square mappings of almost all known finite commutative semifields.

It is shown that, for each inclusion of ergodic discrete measured equivalence relations, the commensurability can be characterized in terms of measure theoretical arguments. As an application, we also include a measure theoretical proof concerning a property of the commensurability groupoid which determines the commensurability in terms of operator algebras. It is proven that a family of typical elements in the commensurability groupoid is closed under the product operation. This proof supplements a gap in the proof of [2, Lemma 7.5].

Simple ends of maximal surfaces in $R^{2,1}$ naturally correspond to catenoidal or planar ends of minimal surfaces in $R^3$. We study some properties of flux of simple ends, which are different from those of catenoidal or planar ends. We also give a classification of maximal surfaces of genus zero with 3 simple ends.