We prove that the size #{ k(m-k) : 1≤ k≤ m/2}∩{ l(n-l) : 1≤ l≤ n/2} is (mn)^o(1), but can be arbitrarily large. This resolves Erdős problem # 443.
This study provides a new perspective on the relationship between circulant matrices and umbrella matrices from the viewpoint of Lie theory, working with real-valued matrices. For the first time, we prove that orthogonal circulant matrices whose characteristic value differs from -1 are precisely umbrella matrices, and we define the associated matrix group as the Circulant Umbrella Matrix (CUM) group. Furthermore, we show that the Lie algebra of this group consists of skew-symmetric circulant matrices and we establish a direct connection between the Lie group and its Lie algebra via the Cayley transformation. To visualize these theoretical findings, we construct a four-dimensional hypersurface by selecting an appropriate parametric orbit curve and analyzing its projections in three-dimensional subspaces. In particular, we emphasize how the combination of orthogonality, circulant structure, and zero-row-sum skew-symmetry yields rich geometric and algebraic properties. Finally, we evaluate circular convolution from an orthogonality viewpoint and demonstrate that the elements of the CUM Lie group offer an alternative solution to the Yang-Baxter equation (YBE) while supporting applications in signal processing, cryptography, and other fields that depend on specialized orthogonal transformations.
The purpose of this short note is to demonstrate that the Arithmetic-Logarithmic-Geometric (AM-LM-GM) inequality, Carlson's inequality, and the generalized form of the logarithmic mean (Stolarsky's mean) can be straightforwardly obtained from the Holder's inequality and its reverse.
We describe a new approach for calculating the winning probabilities in the game of Pass the Buck on arbitrary graphs. It is based on the Markov chain tree theorem, and reduces the problem to counting arborescences in directed graphs. We investigate the game on several classes of graphs, provide short derivations of existing results, and obtain several new ones.
We consider the number of ways in which k sides can be selected from a regular n-sided polygon so that, when the chosen sides are extended infinitely on both ends, they form a k-gon that contains the n-gon. We find formulas for three interpretations: the n sides are distinguishable, or rotations are indistinguishable, or both rotations and reflections are indistinguishable.
We introduce a simple structural property of bipartite graphs-edge-stability-which states that removing an edge whose endpoints lie in a minimum vertex cover does not change the vertex cover number. This property admits an elementary proof and serves as a bridge between several classical theorems in matching theory. We show that edge-stability follows directly from K & odblac;nig's Theorem and, conversely, can be used to rederive it. Moreover, we employ edge-stability to obtain short proofs of Berge's characterization of maximum independent sets and Hall's Marriage Theorem. Thus within the framework of Reichmeider's equivalence of Mengerian theorems, edge-stability, K & odblac;nig's, Berge's, and Hall's theorems are all equivalent. The approach highlights new conceptual and pedagogical connections between these cornerstone results in matching theory.
