Gallai’s path decomposition conjecture states that for a connected graph G on n vertices, there exists a path decomposition of size ⌈n/2⌉ . The Levi graph of order one, denoted by L_1(m,k) , is a bipartite graph with vertex partition (A, B), where A is the collection of all (k-1) -element subsets of [m], and B is the collection of all k-element subsets of [m]. In this graph, a (k-1) -element subset is adjacent to a k-element subset if and only if it is properly contained within the k-element subset. The path number of a graph G is the minimum size of its path decomposition. Gallai’s conjecture can be seen as a conjecture on the upper bound of the path number of a connected graph. In this work, we prove the conjecture for L_1(m,k) for all m ≥ 2 and 2 ≤ k ≤ m . Moreover, we determine the path number of L_1(m,2) for all m.
We study nonlinear Volterra integral equations with two-variable non-convolution kernels. Under mild structural hypotheses and a Carathéodory nonlinearity satisfying linear growth and monotonicity on an interval generated by an ordered pair of functions acting as bounds, we adapt the method of upper and lower solutions and the monotone iteration scheme, and we obtain the existence of solutions confined to a prescribed ordered interval, which play the role of minimal and maximal solutions in the above-mentioned ordered interval. A compactness mechanism follows from equicontinuity provided by diagonal integrability and uniform boundedness provided by the order interval, via the Arzelà- Ascoli theorem. On the other hand, the positivity of the kernel ensures order preservation. We provide quantitative sufficient conditions that make the construction of ordered bounds explicit. Illustrative examples—power-law, positivity-restricted Bessel-type, and a genuinely non-convolution kernel—demonstrate the scope of the theory. Finally, a worked non-convolution example admitting a closed form offers a practical benchmark for numerical methods.
Data science has promise to be a useful tool in the prevention of genocide and related atrocity crimes. The generally agreed upon definition from the United Nations’ 1948 Genocide Convention has significant weight behind it, but attempts to operationalize it has proven to have mixed success. In this paper, we introduce our program toward genocide prevention which, in part, requires quantitative approaches in addition to more traditional methods in genocide studies. We apply data-scientific approaches to operationalize this definition of genocide, in particular, to predict the onset of genocide, and identify preventive policy options. Toward forecasting, we reaffirm some of the risk metrics and methodologies of the Early Warning Project, attaining accuracy scores in retrospective analysis of the forecasting problem, applying a bootstrapping methodology to achieve an aggregate median test-set accuracy of 80
Social choice theory research demonstrates that single transferable voting (STV) results in more proportionally representative legislative bodies. We aim to understand how using multi-member districts and ranked ballots with STV would affect the representation of political parties in the Colorado House of Representatives. We investigated this objective by producing 10,000 multi-member districting plans of Colorado, generating ranked ballots for each of these plans using returns from the 2022 Colorado attorney general race, and simulating STV using these ballots. Our simulated STV elections for the Colorado House of Representatives gave more proportional representation for Democrats and Republicans than the current first-past-the-post system.
Today, Bernard Bolzano is well known to both mathematicians and historians of mathematics, at the very least because of the function that bears his name and the theorem named after him and Karl Weierstrass, or because, according to the usual narrative, his mathematical work was practically unknown to his contemporaries and remained in oblivion until Weierstrass and some of his students began to spread the word about it from the 1860s. This article provides an overview of Bolzano’s contributions to mathematics on two fronts. Firstly, it addresses the reception of the mathematical works that he published during his lifetime among his contemporaries, showing that, although they were not widely known, they were more so than is generally believed. On the other hand, it offers some insights into the mathematical writings that are part of his written estate, both published to date and yet unpublished, showing the diversity and richness of his contributions.
