Château Gombert

We investigate the redshift evolution and group richness dependence of the quiescent galaxy fraction and red sequence (RS) parameters in COSMOS galaxy groups, spanning a wide redshift range, from z=0 to z=3.7. We analyzed the deep and well-characterized sample of groups recently detected with the AMICO algorithm in the COSMOS(-Web) field. Our study of the quiescent galaxy population is based on a machine-learning classification tool based on rest-frame magnitudes. The algorithm learns from several traditional methods to estimate the probability of a galaxy being quiescent, achieving high precision and recall. Starting from this classification, we computed quiescent galaxy fractions within groups via two methods: one based on the membership probabilities provided by AMICO, which rely on an analytical model, and another using a model-independent technique. We then detected the RS by estimating the ridgeline position using probability-weighted photometric data, followed by σ clipping to remove outliers. This analysis was performed using both rest-frame magnitudes and observer-frame magnitudes with rest-frame matching. We compared the results from both approaches and investigated the redshift and richness dependence of the RS parameters. We found that the quiescent galaxy population in groups builds up steadily from z = 1.5-2 across all richnesses, with faster and earlier growth in the richest groups. The first galaxies settle onto the RS ridgeline by z ∼ 2, consistent with current evolutionary scenarios. Notably, we reported a rare protocluster core hosting quiescent galaxies at z = 3.4, potentially one of the most distant early RSs observed. Extending our study to X-ray properties, we found that X-ray faint groups have, on average, lower quiescent fractions than X-ray bright ones, likely reflecting their typical location in filaments where pre-processing is lower. Leveraging the broad wavelength coverage of COSMOS2025, we traced RS evolution using observed and rest-frame colors over ∼ 12 Gyr, finding no significant trends in either the slope or the scatter of the ridgeline.

Our main result is a succinct counterpoint to Courcelle's meta-theorem as follows: every cw-nontrivial monadic second-order (MSO) property is either NP-hard or coNP-hard over graphs given by succinct representations. Succint representations are Boolean circuits computing the adjacency relation. Cw-nontrivial properties are those which have infinitely many models and infinitely many countermodels with bounded cliquewidth. Moreover, we explore what happens when the cw-nontriviality condition is dropped and show that, under a reasonable complexity assumption, the previous dichotomy fails, even for questions expressible in first-order logic.

Let B \mathfrak {B} be the set of odd integers that are sums of two coprime squares. We prove that the trigonometric polynomial S ( α ; N ) = ∑ b ∈ B , b ≤ N e ( b α ) S(\alpha ;N) = \sum _{b\in \mathfrak {B},b\le N}e(b\alpha ) satisfies S ( α ; N ) N / log ⁡ N ≪ A , A ′ 1 φ ( q ) + q N ( log ⁡ N ) 7 + 1 ( log ⁡ N ) A \begin{equation*} \frac {S(\alpha ; N)}{N/\sqrt {\log N}}\ll _{A,A’} \frac {1}{\varphi (q)} + \sqrt {\frac {q}{N}}(\log N)^{7} +\frac {1}{(\log N)^A} \end{equation*} for any A , A ′ ≥ 0 A,A’\ge 0 and when ( a , q ) = 1 (a,q)=1 and | q α − a | ≤ ( log ⁡ N ) A ′ / N |q\alpha -a|\le (\log N)^{A’}/N . We use this estimate together with a variant of the circle method influenced by Green and Tao’s Transference Principle to obtain the number of representations of a large enough odd integer N N as a sum b + b 1 + b 2 b+b_1+b_2 , where b ∈ B b\in \mathfrak {B} while b 1 b_1 (resp. b 2 b_2 ) belongs to a general subset B 1 \mathfrak {B}_1 (resp. B 2 \mathfrak {B}_2 ) of B \mathfrak {B} of relative positive density. We further show that the above bound is effective when 0 ≤ A > 1 / 2 0\le A>1/2 .

We study two positional games played on hypergraphs, whose edges may be interpreted as winning sets. Two players take turns picking a previously unpicked vertex of the hypergraph. We say a player fills an edge if that player has picked all the vertices of that edge. In the Maker-Maker convention, whoever first fills an edge wins, or we get a draw if no edge is filled. In the Maker-Breaker convention, the first player aims at filling an edge while the second player aims at preventing the first player from filling an edge. Our main result is that, for both games, deciding whether the first player has a winning strategy is a PSPACE-complete problem even when restricted to 4-uniform hypergraphs (of bounded maximum degree). For the Maker-Maker convention, this improves on the known PSPACE-completeness result for hypergraphs of rank 4. For the Maker-Breaker convention, this improves on the known PSPACE-completeness result for 5-uniform hypergraphs, and closes the complexity gap since the problem for hypergraphs of rank 3 is known to be solvable in polynomial time. As a corollary of our construction, we actually get a stronger result: deciding whether the first player has a winning strategy for the vertex-$C_4$-game played on arbitrary graphs, where the winning sets are the vertex sets of 4-cycles, is a PSPACE-complete problem for both conventions.

International audience