Bollob\'as proved that for every $k$ and $\ell$ such that $k\mathbb{Z}+\ell$ contains an even number, an $n$-vertex graph containing no cycle of length $\ell \bmod k$ can contain at most a linear number of edges. The precise (or asymptotic) value of the maximum number of edges in such a graph is known for very few pairs $\ell$ and $k$. In this work we precisely determine the maximum number of edges in a graph containing no cycle of length $0 \bmod 4$.
The evolution of cooperation in networked systems helps to understand the dynamics in social networks, multiagent systems, and biological species. The self-persistence of individual strategies is common in real-world decision making. The self-replacement of strategies in evolutionary dynamics forms a selection amplifier, allows an agent to insist on its autologous strategy, and helps the networked system to avoid full defection. In this paper, we study the self-interaction learning in the networked evolutionary dynamics. We propose a self-interaction landscape to capture the strength of an agent's self-loop to reproduce the strategy based on local topology. We find that proper self-interaction can reduce the condition for cooperation and help cooperators to prevail in the system. For a system that favors the evolution of spite, the self-interaction can save cooperative agents from being harmed. Our results on random networks further suggest that an appropriate self-interaction landscape can significantly reduce the critical condition for advantageous mutants, especially for large-degree networks
We prove Kantorovich duality for a linearized version of a recently proposed non-quadratic quantum optimal transport problem, where quantum channels realize the transport. As an application, we determine optimal solutions of both the primal and the dual problem using this duality in the case of quantum bits and distinguished cost operators, with certain restrictions on the states involved. Finally, keeping the same restrictions regarding the states involved, we use this information on optimal solutions to give an analytical proof of the triangle inequality even for the square of the induced quantum Wasserstein divergences.
BACKGROUND:A number of novel imidazole compounds with a triaryl were created, manufactured, and tested for their insecticidal and acaricidal effects in an effort to find new agrochemicals with notable pesticidal qualities. RESULTS:One-pot four-component reactions of benzyl, aromatic aldehydes, ammonium acetate, pentylamine and/or allylamine in a 1:1:1:1 m ratio were developed to synthesize a series of 1-pentyl-2,4,5-trisubstututed-1H-imidazoles (5a-e) and 1-allyl-2,4,5-triaryl-1H-imidazole derivatives (5f-j). In addition, this process used a catalytic quantity of cheap and easily accessible p-toluenesulfonic acid, demonstrating its affordability and suitability for environmentally friendly synthesis. The insecticidal effectiveness of the target synthesized compounds against Spodoptera littorals species was assessed. CONCLUSION:With median lethal concentrations (LC50) of 4.926 mg/L for fourth-instar larvae and 7.241 mg/L for second-instar larvae, comparable with the commercial insecticide fipronil, bioassays demonstrated that compound 5i had a strong insecticidal effect. In addition, some biochemical characteristics of the produced molecules were examined, including total protein, total lipids, and glutathione S-transferase and acetylcholinesterase activity. © 2026 Society of Chemical Industry.
The ability to anticipate rhythmic and melodic structures in music is considered a fundamental human trait, present across all cultures and predating linguistic comprehension in human development. Yet, it remains unclear the extent to which this ability is already developed at birth. Here, we used temporal response functions to assess rhythmic and melodic neural encoding in newborns (N = 49) exposed to classical monophonic musical pieces (real condition) and control stimuli with shuffled tones and inter-onset intervals (shuffled condition). We computationally quantified context-based rhythmic and melodic expectations and dissociated these high-level processes from low-level acoustic tracking, such as local changes in timing and pitch. We observed encoding of probabilistic rhythmic expectations only in response to real but not shuffled music. This proves newborns' ability to rely on rhythmic statistical regularities to generate musical expectations. We found no evidence for the tracking of melodic information, demonstrating a downweighting of this dimension compared to the rhythmic one. This study provides neurophysiological evidence that the capacity to track statistical regularities in music is present at birth and driven by rhythm. Melodic tracking, in contrast, may receive more weight through development with exposure to signals relevant to communication, such as speech and music.
