A graph G is universal for a (finite) family H of graphs if every H is an element of H is a subgraph of G. For a given family 9-l, the goal is to determine the smallest number of edges an 9-l-universal graph can have. With the aim of unifying a number of recent results, we consider a family of graphs with bounded density. In particular, we construct a graph with O-d (n(2-1/(d+1))) edges which contains every n-vertex graph with density at most d is an element of Q (d >= 1), which is close to a Omega(n(2-1/d)) lower bound obtained by counting lifts of a carefully chosen (small) graph. When restricting the maximum degree of such graphs to be constant, we obtain near-optimal universality. If we further assume d is an element of N, we get an asymptotically optimal construction.
The Grassmannian & Gscr;(2,m), is the collection of all two-dimensional subspaces of a vector space of dimension m. It is one of the most widely studied objects in Algebraic Geometry and has interesting algebraic, geometric, and combinatorial properties. Since subspaces of dimension 2 are known as lines the Grassmannian & Gscr;(2,m), is known as the Grassmannian of lines. A class of linear codes, known as Grassmann codes, are used to understand the geometric and algebraic properties of the Grassmannian. Codes from Schubert subvarieties and polar subvarieties of the Grassmannian are known. For the case & ell; = 2, the parameters of Schubert codes have been established by Chen and the parameters of polar Grassmann codes have been established by Cardinali and Giuzzi. For general & ell;, the parameters for Schubert codes are known, but the parameters of polar Grassmann codes are not known for general & ell;. The case in which either a polarity condition or a Schubert condition is applied to the Grassmannian on their own is simpler. These conditions have been studied independently but not concurrently. In this work, we study linear codes derived from subvarieties of the Grassmannian defined from the intersection of Schubert and symplectic varieties. We determine the length, dimension and minimum distance for several cases of line Schubert symplectic Grassmann codes when both the Schubert conditions and some special symplectic polarity conditions are combined in different ways.
We present an extensive photometric and spectroscopic ultraviolet–optical–infrared campaign on the luminous fast blue optical transient (LFBOT) AT 2024wpp over the first ∼100 days. AT 2024wpp is the most luminous LFBOT discovered to date, with L _pk ≈ (2–4) × 10 ^45 erg s ^−1 (5–10 times that of the prototypical AT 2018cow). This extreme luminosity enabled the acquisition of the most detailed LFBOT UV light curve thus far. In the first ∼45 days, AT 2024wpp radiated >10 ^51 erg, surpassing AT 2018cow by an order of magnitude and requiring a power source beyond the radioactive ^56 Ni decay of traditional supernovae. Like AT 2018cow, the UV–optical spectrum of AT 2024wpp is dominated by a persistently blue thermal continuum throughout our monitoring, with blackbody parameters at a peak of T > 30,000 K and R _BB / t ≈ 0.2 c –0.3 c . We find evidence for cooling until ∼10 days; thereafter, T ≳ 20,000 K is maintained. We interpret the featureless spectra as a consequence of continuous energy injection from a central source of high-energy emission that maintains high ejecta ionization. After 35 days, faint (equivalent width (EW) ≲ 10 Å) H and He spectral features with kinematically separate velocity components centered at 0 and −6400 km s ^−1 emerge, implying spherical symmetry deviations. A near-infrared excess of emission above the optical blackbody emerges between 20 and 30 days, with a power-law spectrum F _ν _,NIR ∝ ν ^−0.3 at 30 days. We interpret this distinct emission component as either reprocessing of early UV emission in a dust echo or free–free emission in an extended medium above the optical photosphere. LFBOT asphericity and multiple outflow components (including mildly relativistic ejecta), together with the large radiated energy, are naturally realized by super-Eddington accretion disks around neutron stars or black holes and their outflows.
For a finite group G, the prime graph Γ(G) (also known as Gruenberg-Kegel graph) is defined to be the graph where the vertices are the primes that divide |G| such that two vertices p and q share an edge if and only if there is an element of order pq in G. The prime graphs of solvable groups have been classified. The prime graphs of groups whose noncyclic composition factors are isomorphic to a single nonabelian simple group T where |T| is divisible by three or four distinct primes have been classified except for the cases where T = PSL(2,q) for q≠ 2^5 and |PSL(2,q)| is divisible by exactly four primes. In this paper, we provide criteria for general classification results for certain classes of T, and then use them to classify the prime graphs of some T-solvable groups for T a suitably small PSL(2, q)-group. We also provide general results on the prime graphs of T-solvable groups where T is a member of the possibly infinite family of groups PSL(2, 2^f) such that f≥ 5, f is prime, and |PSL(2, 2^f)| is divisible by exactly four primes. This is the first paper to prove general results about the prime graphs of T-solvable groups where T belongs to a large (probably infinite) family of groups.
We analyze the origin of the large-scale bulk flow using the CosmicFlows-4 (CF4) peculiar-velocity catalog. We decompose the observed motions into internal components, generated by mass fluctuations within 200 Mpc/h, and external ones arising from structures beyond this volume. A weighted-average technique is developed to test the model's self-consistency while minimizing the impact of non-Gaussian distance errors. The CF4 velocities show excellent agreement with the predicted internal field, yielding beta = 0.31 pm 0.01. We also determine that the value of the Hubble constant that should be used for calculating peculiar velocities from the CF4 to be H0 = 75.0 pm 0.1 km/s/Mpc, consistent with CF4 calibrations. Using the minimum-variance formalism, we further separate the bulk flow into its internal and external contributions and find that the observed large-scale bulk flow is dominated by sources beyond 200 Mpc/h. The amplitude of this externally driven flow increases monotonically with scale, consistent with the influence of a distant, massive overdensity. These findings reinforce the reliability of the CF4 velocity field while calling into question the assumption of a spatially uniform flow generated by external sources. Our results challenge the commonly made hypothesis that the flow in our local volume due to external mass concentrations can be modeled as being spatially uniform.
