Problems
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Ross introduced S-perfect numbers, integers expressible as 1 + ∑ λ_j d_j over their proper divisors with coefficients in S, and conjectured that they have the same density as the nondeficient numbers, plus a second conjecture relating odd…
For a simple 3-polytope with at least three faces of size at least 7, must p_6 ≥ 39/20 + p_3/2 - p_5/4 - ∑_k ≥ 7 p_k? Five minimal ten-face counterexamples refute the printed inequality.
Does there exist a single real-valued, divergence-free, time-independent Lipschitz velocity field u∈ W^1,∞( T^3; R^3), chosen independently of magnetic diffusivity, that is a fast dynamo for the kinematic induction equation on the flat…
Can one construct a plain-model, efficient, information-theoretically secure one-time unclonable-encryption scheme for one classical bit with exponentially small adversarial advantage?
Maz'ya and Shaposhnikova introduced a non-classical maximal operator M^diamond, the maximal convolution with the vector-valued signum kernel truncated to centered balls. One of Maz'ya's 75 open problems in analysis asks whether it can be…
What is the minimum asymptotic density δ_k of monochromatic k-term arithmetic progressions in every two-colouring of 1, …, n? The exact certificate gives δ_3 = 117/2192, matching the known 548-bead colouring.
For the adjacent-transposition chain on S_n with a regular parameter vector, Fill's spectral gap conjecture (recently resolved) leaves open the characterization of the equality cases. The paper settles them, constructing the additional…
Whether the real Kalton-Peck space Z_2 is isomorphic to its hyperplanes. It is not: no hyperplane of Z_2 is isomorphic to Z_2, proved through a rank parity theorem for symplectic spaces applied to the Kalton-Swanson symplectic structure.
The Howland-Kato conjecture that every nonzero positive commutator i[f(P),g(Q)] must arise from functions in appropriate Kato classes is false: i[arctan(P),arctan(Q)] is nonzero and nonnegative.
A question of Averkov, Hofscheier and Nill on whether the Ehrhart h^*-polynomial of a lattice polytope of large lattice width is real-rooted. Proved in fixed dimension for sufficiently large lattice width, giving strict log-concavity and…
A convex body is in Faber-Krahn position if it minimizes the first Dirichlet eigenvalue within its volume-preserving linear orbit. The paper proves this position is unique up to orthogonal transformations, answering a question of…
Is the depth of the mod-p cohomology ring of every finite group realized as the dimension of one of its associated primes? For G = SmallGroup(128, 859) over overlineF_2 the ring has depth 2 while every associated-prime quotient has…
Does the natural trace estimate hold for kinetic energy spaces in the unrestricted Gaussian velocity model on bounded domains (Question 1.8 of Albritton, Armstrong, Mourrat and Novack)? No: for each 1 ≤ p < 2 there are counterexamples on…
How many pairwise non-overlapping infinite circular cylinders of unit radius can simultaneously touch a unit ball? Kuperberg conjectured in 1990 that the maximum is six.
If a finite graph has girth at least five, must its minimum dual degree satisfy δ^*(G) ≤ -∂_n(G), where ∂_n(G) is the smallest eigenvalue of its distance matrix? The Hoffman-Singleton graph violates it: dual degree 7 against eigenvalue…
Zhu, Gyori, He, Lv, Salia and Xiao conjectured the maximum number of copies of a fixed cycle in an n-vertex graph of bounded circumference, attained by the join of a clique with an independent set. For every fixed s ≥ 3 and L ≥ 2s+2 and…
Erdős and Graham asked whether binomnk with 1 ≤ k ≤ n/2 must always have a divisor ≤ n that is close to n, meaning bigger than a fixed constant times n. Settled in both directions: true when k is large enough as a function of n, but false…
Every finite connected simple graph G satisfies α(G)≥ r(G)+ln(ρ(G)), where α(G) is the independence number, r(G) is the radius, and ρ(G) is the minimum number of pairwise vertex-disjoint paths whose vertices cover V(G).
Monical, Tokcan and Yong conjectured that Schubitopes, the generalized permutahedra arising as Newton polytopes of Schubert polynomials and of Demazure characters of GL_n, are Ehrhart positive. Disproved by an explicit Schubitope whose…
Among all nonconstant monic polynomials f whose roots lie in [-1, 1], determine inf_f |x ∈ R : |f(x)| < 1|.