Hall’s theorem on differences of bijections characterizes the multisets \(\{a_1,\ldots,a_{|G|}\}\) in a finite abelian group \(G\) that can be written in the form \( a_i=b_i-c_i, \) where both \(b_1,\ldots,b_{|G|}\) and \(c_1,\ldots,c_{|G|}\) are enumerations of \(G\). The necessary and sufficient condition is the zero-sum condition \( a_1+\cdots+a_{|G|}=0. \) This paper studies the corresponding problem for finite nonabelian groups, with differences replaced by quotients. Thus we ask when a multiset \(A\) of cardinality \(|G|\) can be represented as \( A=\{b(i)c(i)^{-1}:1\le i\le |G|\}, \) where \(b\) and \(c\) are bijections onto \(G\). Passing to the abelianization gives a necessary condition, namely that the product of the images of the elements of \(A\) is trivial in \(G_{\rm ab}\). We show that this condition is not sufficient in general, even when the elements of \(A\) admit an ordering whose product is the identity in \(G\). The main structural result is a cycle-tiling criterion: quotient-realizability is equivalent to a decomposition of \(A\) into product-one words whose partial-product sets tile \(G\) by right translates. The use of permutation cycles is standard, but the criterion translates quotient-realizability into an exact tiling condition. We then use this criterion to construct a counterexample in \(S_3\), and we extend the same obstruction to infinitely many finite nonabelian groups.

We investigate diagonal equations \(ax^{m}+by^{m}-cz^{m}=1\) over finite fields \(F\) using combinatorial designs naturally associated with \(F\). Building on prior work that resolved the case \(a=b=c=1\), we obtain exact formulas for the solutions when \(a=1\) and \(b=c\), under circularity assumptions. For general coefficients, we present an algorithm that determines whether a given instance can be reduced to the settled cases, or else identifies it as requiring brute-force computation.

In this paper, we expand our interest in the 16th Hilbert’s problem to acquire a comprehensive understanding of the maximum number of crossing limit cycles in \(\mathbb{R}^3\), specifically within a class of three- dimensional discontinuous piecewise differential system generated by two arbitrary Euler systems separated by the unit sphere \(\mathbb{S}^2=\{ (x,y,z) \in\mathbb{R}^3; x^2 + y^2 + z^2 = 1\}\).

Let \(G\) be a graph with no isolated vertices. A \(k\)-coupon coloring of \(G\) is an assignment of colors from \([k]=\{1,2,\ldots,k\}\) to the vertices of \(G\) such that the neighborhood of every vertex contains all colors from \([k]\). The maximum integer \(k\) for which a \(k\)-coupon coloring exists is called the coupon coloring number of \(G\), and is denoted by \(\chi_c(G)\). In this paper, we investigate coupon coloring in inflated graphs arising from various classes of graphs. In addition, we introduce new graph operations based on inflation and study their effect on the existence and behavior of coupon colorings. Our results contribute to a deeper understanding of how inflation based graph operations influence coupon coloring.