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Does exhaustive AdaBoost always converge to a finite cycle of weak classifiers and weight vectors on every finite training set? A finite instance whose orbit never becomes periodic answers no.
Does every globally asymptotically stable homogeneous polynomial vector field admit a homogeneous polynomial Lyapunov function? No. A planar homogeneous cubic vector field with integer coefficients is globally asymptotically stable yet admits no positive definite homogeneous polynomial with nonpositive Lie derivative, and no real-analytic Lyapunov function even locally, though it does have exponential and rational s…
This is the MODIFIED conjecture, not the original Lyons-Sidorova one, and it is proved for continuous bounded-variation paths. Prior work had a line-image result under the stronger assumption of infinite radius on every subinterval; this removes that assumption.
In the circle of Kummer's regular primes and Vandiver's conjecture, the paper proves that almost all primes are partially regular, yielding a partial Vandiver theorem for a density-one set of primes, with consequences for Kubota-Leopoldt p-adic L-functions, Eisenstein congruences and K-theory torsion.
The total Chern class of Symd(Cn) as a torus representation is a symmetric polynomial whose coefficients were conjectured positive, with a binomial log-concavity refinement. Both are established.
For a differential poset P, must the weighted 2-multichain series MP,2(q) be a rational multiple of FP(q)2, the square of its rank generating series?
Both bounds move, and the gap stays enormous: the lower bound rises from exp(Ω(log2k)) to exp(Ω(k1/3)) and the upper falls from exp(O(k4)) to exp(O(k2)), so A(k) is still undetermined between an exponent of k1/3 and one of k2. The paper's own closing discussion argues its lower-bound construction is near the limit of the method and that beating it needs additional randomness,…
For every n≥2 the paper exhibits an n-dimensional K-polystable toric Q-Fano variety whose alpha invariant is exactly 2n+12, answering a question of Liu and Zhuang on whether a K-semistable example exists with alpha invariant between n+11 and n1.
Pak and Slonim conjectured that stretched Schubert structure constants are eventually polynomial. They are. Monomial coefficients in affine families of key and Schubert polynomials are eventually polynomial, and the Schubert duality of Watanabe carries this to the structure constants. The same result settles the polynomiality half of a conjecture of Alexandersson and Alhajjar for key polynomials.
Iterates of a firmly nonexpansive operator converge weakly but not strongly, by Genel and Lindenstrauss. Whether their Cesaro means converge strongly was open. They need not: an explicit curve gives a counterexample.
The Elton–Odell theorem gives, in every infinite-dimensional normed space, a unit-sphere sequence with mutual distances at least 1+ε. Over C, identifying vectors differing by a unimodular scalar gives a toroidal distance. Does every infinite-dimensional complex normed space admit such a uniformly separated sequence for that distance? Yes.
Does quantum memory give a query-complexity advantage for learning an unknown quantum channel, when protocols without it must measure after each channel use and keep only a classical transcript? It does, and the paper also determines how little coherent memory suffices for the advantage to appear.
Reading computed the order dimension of the poset of regions for most finite Coxeter arrangements, observed that an exceptional type whose dimension exceeds its rank would be the first known simplicial arrangement with that property, and recorded the general guess that every simplicial region poset has dimension equal to its rank (Problem 9.3 of his 2016 chapter); Segovia later asked the analogous question for orien…
Must every r-differential poset have at least as many elements in each rank as Yr, the r-th Cartesian power of Young's lattice? For r=3 the new construction has fourth-rank size 50 against 51 for Y3.
Online Shadow Tomography with logm dependence, while retaining poly(log(d)/ϵ) dependence. Also, matching the best classical bounds for Adaptive Data Analysis
The honest reading, which the paper gives itself: the reduction is implicit in earlier work of Aboulker, Oijid, Petit, Rocton and Simon, and the model itself surfaced that reference when asked about originality. So this establishes the conjecture and supplies a polynomial-time algorithm, while the underlying idea is a rediscovery rather than a first. It is a striking record of a model producing an argument and then…
Three immediate consequences follow for undirected unweighted planar graphs: better compression of the Okamura-Seymour metric, less space for constant-time exact distance oracles, and a faster distributed algorithm.
How large can a measurable A⊆[0,R]2 be while avoiding the vertices of upward-oriented axis-aligned right triangles of area 1/2? At most Oc(R2/(logR)c), with a matching-shaped lower bound construction.
Conditional on the randomized exact-volume Small-Set Expansion Hypothesis, and stated for least-squares objectives rather than sparse convex optimization in general.
Determine the leading asymptotic of the largest eigenvalue of the N-Majorana quartic SYK Hamiltonian as N→∞. The preprint proves λ1/N→4∫0∞g0(t)4dt≈0.32504 almost surely, via the limiting free energy at every fixed positive temperature.
If a chromatic symmetric function is Schur positive, must every finite-variable specialization XG(x1,…,xk) have a saturated Newton polytope? A 12-vertex bipartite graph realizes weights (6,6,0) and (8,2,2) but omits their midpoint (7,4,1).
Monical, Tokcan and Yong conjectured that every fixed positive power of the Vandermonde determinant fails to have saturated Newton polytope in sufficiently many variables. For every even power k≥4 there is an explicit lattice point of the Newton polytope of aδkk with vanishing coefficient, obtained from a Dyson constant-term identity; the odd case follows by alternation, proving the conjecture.
Let nk be the least n>2k such that (n−k)(n−k+1)⋯(n−1) has no prime factor in (k,2k). Erdos conjectured a superpolynomial lower bound; for all large k, nk>elog2k/(20loglogk).