Open registry federation
639 source-grounded records. Verification labels remain separate from source authentication and publication status.
Imported from VibeMathed under CC BY 4.0. Each record links to its registry entry and named primary source. Registry verification is preserved verbatim.
Englert and Rehacek conjectured which measurement is globally information-optimal for an ensemble of equiangular equiprobable pure states. Their conjecture holds, via the remaining entropy inequalities of Holevo and Utkin.
a lower bound for Tokuda's sequence; the general Shellsort complexity question stays open
Is the depth of the mod- cohomology ring of every finite group realized as the dimension of one of its associated primes? For over the ring has depth while every associated-prime quotient has dimension at least .
up to polylogarithmic factors
Baker asked, as recorded by Poonen, whether a fixed smooth quasiprojective variety over a finite field must acquire a smooth rational hyperplane section after every sufficiently high-dimensional linearly nondegenerate embedding. Poonen predicted no for every positive-dimensional variety, and that prediction is correct.
After normalization, stable phase retrieval holds over the -spans of independent real-valued centered random variables exactly when all but possibly one coordinate satisfies a uniform two-sided bound. This confirms the characterization conjectured by Calderbank, Daubechies, Freeman and Freeman.
part (1) of the conjecture
Bounds the weighted sum taken over primitive sets of integers (sets where no element divides another).
An asymptotic formula for , the limiting probability that a random permutation has an invariant set of size : it is asymptotically times a smooth positive function, sharpening a line of estimates running through Łuczak-Pyber and Eberhard-Ford-Green.
Partial: this settles n=16 only. Combined with Pang's n>=17 the conjecture holds for all n>=16, leaving the small cases open.
Is computing a Kemeny-optimal aggregate ranking NP-hard when the input consists of exactly three complete rankings? Hardness was known for every even ; three voters was the minimal open case, and is polynomial-time solvable.
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.
Both bounds move, and the gap stays enormous: the lower bound rises from to and the upper falls from to , so is still undetermined between an exponent of and one of . 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,…
Spahn and Zeilberger's third challenge asks whether the restricted permutation counts and are holonomic for all . Answered affirmatively.
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.
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…
Record lower bound only. The sub-2 ceiling is the codimension-two case and does not bound the record ladder (k=17 has complement mass 11). 4/3 and 2 are conjectures; the proved gap is [1.28249, 2].
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.
density-one set of n; the conjecture itself remains open
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.
Whether perfectly complete quantum key agreement can be built from quantumly secure one-way functions in a black-box way. It cannot: for any protocol where Alice and Bob exchange only classical messages, make at most and quantum queries to a Boolean random oracle and agree on a key with certainty, an eavesdropper given the classical messages recovers the key with certainty in classical…
For an even cycle of size and depth with , is the optimal QAOA approximation ratio for MaxCut exactly , as Farhi, Goldstone and Gutmann conjectured?
improved bounds; the exact Hadwiger-Debrunner numbers remain open
a batch of problems from published lists, resolved in one paper
Monical, Tokcan and Yong conjectured that Schubitopes, the generalized permutahedra arising as Newton polytopes of Schubert polynomials and of Demazure characters of , are Ehrhart positive. Disproved by an explicit Schubitope whose Ehrhart polynomial has a negative coefficient in its monomial expansion.
disproved at d = 4; the conjecture for smaller d is untouched
Can a finite set of lattice points determine many rectangles but few isosceles triangles? Both parts of the governing question have negative answers, quantified by explicit blowup rates, and the resulting configurations give obstructions in the Mizohata-Takeuchi circle of problems.
After Chen-He-Ye-Yuan's counterexample to direct three-block ADMM, the subclass in which the third constraint block is the identity matrix remained unresolved: the literature contained neither a convergence proof nor a counterexample. Disproved: an explicit rational counterexample exists in which the first two blocks are strongly convex quadratics and direct three-block ADMM produces a bounded nonconvergent orbit of…
The interchange graph has the -matrices with row sums and column sums as vertices, adjacent when they differ by a single interchange. Brualdi asked whether is always Hamiltonian. It satisfies more: it is maximally Hamiltonian, Hamilton-laceable when bipartite and Hamilton-connected when not.
Does the Hodge bundle over the moduli stack of genus curves contain any nontrivial sub-bundles? Posed by Dawei Chen around 2015; the answer is no.
Let be the least size forcing a set to contain distinct with , and all in . The upper bound matches the standard construction , so .
Sampling a nearly uniform Eulerian tour of a directed Eulerian multigraph was stuck at -type running times coming from arborescence sampling. A randomized algorithm achieves worst case, breaking that barrier on sparse graphs.
Zhao's Generalized Vanishing Conjecture asks whether, for a differential operator with constant coefficients, for all large forces for all large . Refuted by an explicit five-variable counterexample.
the locally noetherian case; whether such a category can fail to admit a projective generator is left open as Problem 1.3
The Hessian conjecture asks whether every polynomial with has a polynomial gradient inverse. It is known for , false for , and open exactly in dimension four, where it implies the plane Jacobian conjecture. Proved for every quartic polynomial in dimension four: the quartic case reduces to wi…
Hamaker and Reiner conjectured that the order complex of an open interval in the ASM weak order is contractible unless is the long element of a standard parabolic subgroup, in which case it is homotopy equivalent to a sphere. False: there is an interval in the ASM weak order on whose order complex is not contractible even though has no such form, detected by a nonzero Mobius function value.