1033 matching entries · Page 9 of 21
Independent random one-qubit complex Cliffords, or exact one-qubit 2-designs, form a depth-one n-qubit tensor-product ensemble with entangled anticoncentration error (2/3)^n and negligible distinctness for polynomially many queries.
For each h≥2, an h+3-state one-way nondeterministic automaton recognizes liveness of products of binary relations, while every equivalent two-way deterministic automaton with s states satisfies 2^⌊(h−2)/31⌋≤4(s+2)². The finite alphabet grows with h.
A real radial Schwartz function on the plane has nonnegative Fourier transform, is nonpositive beyond unit distance, and has normalized origin values making the Cohn–Elkies linear-programming bound attain the triangular-packing density π/(2√3).
A nonzero Schwartz function has twelve distinct time-frequency shifts, eleven in one irrational translate of a half-integer lattice plus the origin, that are linearly dependent. The authors also prove a finite counterexample without a fixed twelve-shift count.
In each of dimensions four and five, for every sufficiently small nonzero ε, an explicit smooth odd function on S^(d−1) cannot agree with a linear functional on any rotated normalized A_d root system. Through Kuperberg’s equivalence, these give counterexamples to Makeev’s universal-cover conjecture in those dimensions.
Xu constructs a fixed finite forbidden graph family with chromatic parameter p=2 and extremal surplus greater than c n^(3/2) for all sufficiently large n, yet with an extremal graph at each such order whose complement is connected.
For the spherical perceptron at margin −1 with quadratic penalty, after successive limits in system size, inverse temperature and density approaching the feasibility threshold from above, the gap and force laws have exponents γ=1/(2+θ), with 0.4126930<γ<0.4126934 and 0.4231063<θ<0.4231088. The same limits are claimed for a sufficiently small fixed Gaussian scale mixture.
For a loopless multigraph on 2m vertices with a perfect matching and feasible edge marginals x, the maximum entropy H(x) satisfies F(x)−(2−2/m)B(x)≤H(x)≤F(x), where F and B sum the binary-coordinate entropy terms. Optimizing gives the linear-error estimate proposed by Anari, Oveis Gharan and Vinzant.
For all positive integers m and n, the ordinary crossing number of the complete bipartite graph Kₘ,ₙ equals ⌊m/2⌋⌊(m−1)/2⌋⌊n/2⌋⌊(n−1)/2⌋ for plane drawings with continuous simple edge paths.
The rooted terminal-only Manhattan cost-radius spanning-tree decision problem is NP-complete under polynomial-time many-one reductions through a weak reduction from positive-integer Partition.
For every even k≥4 and prime p>k, a coloring of Z/(p^(k²/4))Z uses (k−1)^(k²/4−1)p colors and avoids nontrivial symmetrically colored k-term arithmetic progressions. At k=4 this gives 27p colors and an O(N^(1/4)) interval bound.
For a feasible d-dimensional Max-Cut SDP solution strengthened by triangle inequalities, a randomized polynomial-time rounding algorithm attains expected cut value at least (α_GW+2^(−O(d))) times the SDP value.
For a reduced planar absolute Mumford–Shah minimizer, each interior discontinuity point has a local C¹ arc, endpoint or 120-degree triple junction, with endpoints and junctions locally finite.
GPT-5.6 Sol proposed the Hodge-obstruction and rank-improvement strategy behind the bound Vol B_R(p)≤C(n)R^(n−2) on complete connected n-manifolds with Ric≥0 and Scal≥1.
Every four-partite four-uniform hypergraph with matching number two has vertex-cover number at most six.
Fourteen binary vertices form a simplicial seven-dimensional 0/1-polytope with the maximum 2d vertices but without central symmetry, countering the proposed cross-polytope characterization.
A polynomial-time algorithm finds an optimal size-k committee for any Thiele rule on voter-interval approval profiles, including voter-specific scoring sequences.
For the exponent-five Heisenberg group of order 125, the small unordered Davenport constant is twelve: x⁴y⁴v⁴ is product-one-free, and every thirteen-term sequence has a product-one subsequence in some ordering.
Every subset of R⁴ containing a unit segment in every direction has Hausdorff dimension four, without a measurability, compactness or stickiness condition.
The Paley graph P(17) and each join P(17)∨Kₜ satisfy χ(H)≤ω(H)(ω(H)+1)/2 for every induced subgraph H, yet are not perfectly divisible.
For every compact smooth positively curved surface Σ in R³, including smooth boundary, and every 3<p<∞, the Fourier extension operator maps Lᵖ(Σ) to Lᵖ(R³).
For fixed matrix dimension, whether the mean of i.i.d. binary random matrices is diagonalizable over R or C can be predicted with only finitely many errors almost surely. The argument uses eventual predictability of semialgebraic properties of bounded means.
Alpöge and Buckmaster construct smooth compactly supported initial velocity and forcing smooth through blowup time for 3D incompressible Euler on R³. A finite-energy classical solution develops unbounded vorticity in finite time, with divergent time-integrated vorticity sup norm and uniqueness in the stated preterminal class.
Under the stated balanced stochastic-block-model signal and community-growth assumptions, the authors give a high-probability polynomial-time pure node-private estimator attaining the prior exponential-time method’s minimax exact-recovery risk, with privacy parameter of order log n.
Lalonde gives a bipartite inner-product game with four and three inputs and six outputs. It has a perfect strategy using a nonmaximally entangled state, but no perfect finite-dimensional strategy using a maximally entangled state.
An explicit width-10 shifted template gives S(k+2)≥10S(k)+2. Using existing bounds for S(6) and S(11), the authors obtain S(8)≥5362 and S(13)≥2038282.
For the discrete-time switch–walk–switch walk on Z₂ wr Z, started with all lamps off and walker positions 0 or 2, the total-variation distance between the time-t laws is comparable to t^(−1/2), with constants independent of t.
For a one-parameter complex projective degeneration with simple-normal-crossing central fiber and smooth nearby fiber X_t, QED proves that H¹(X,Z) maps onto the monodromy-invariant subgroup H¹(X_t,Z)^T.
For every prime p, finite extension k/Q_p and smooth proper geometrically connected curve X/k of genus at least two, the manuscript claims a bijection between X(k) and sections of the full arithmetic étale fundamental-group sequence modulo conjugation by the full geometric group.
For p>2, a common-set/private-set construction shows Carbery’s proposed exponent c=2 fails. Grok also supplied the kernel inequality used to establish the optimal necessary constraint c≤p′.
Ajwani, Gajjala, Raman and Ray give 64 axis-parallel rectangles with packing number 16 and piercing number at least 32, disproving Wegner’s proposed bound 2ν−1. Their package construction yields piercing and clique-LP ratios approaching 5/2.
Under the paper’s pth-order smoothness and oracle assumptions, the final Halpern-based method for smooth monotone variational inequalities attains residual rate Õ(T^(−p)), or Õ(ε^(−1/p)) oracle complexity.
For every isotropic log-concave random vector X and symmetric matrix M, Var⟨MX,X⟩ ≤ 2 E|∇⟨MX,X⟩|². Letwin also derives a general KLS bound of order (log n)^(1/4).
At canonical point C of the DiVincenzo family, an explicit Schmidt-rank-two tight-frame witness proves one-copy undistillability but two-copy distillability for every local dimension d≥3.
For weighted k-matroid intersection with k≥2, the natural LP has integrality gap at most k−1+1/k. A deterministic LP-relative algorithm achieves that factor; the paper separately extends the bound to p-matchoids.
For each fixed prime power q, the greatest possible least degree D_q(n) omitted by a non-covering family of n congruence classes in F_q[x] satisfies D_q(n)=n/(q−1)+O_q(1). ChatGPT supplied a truncated CRT/Brun-sieve approach used in the upper bound.
In the IID request model, the stationary expected access cost of Transposition is at most OPT+1, where OPT orders items by decreasing request probability. GPT-5 Pro proposed the central inequality sⱼ≤pⱼ and a coefficient-nonnegativity route; Christian Coester supplied the final proof.
In every dimension d≥2, an IID uniformly elliptic nearest-neighbor random walk environment on Zᵈ with almost-sure transience in a fixed direction has an almost-sure annealed velocity that is deterministic and has positive projection in that direction.
For the zero-field Gaussian Sherrington–Kirkpatrick model at every fixed 0<β<1, rate-one-per-site heat-bath dynamics satisfy a dimension-independent Poincaré inequality for all observables with disorder probability tending to one. Uniformly chosen single-site updates have gap at least 1/(Cβ n).
For G=SmallGroup(128,859) over the algebraic closure of F₂, depth H*(G;k)=2 while every associated-prime quotient has dimension at least 3. The group-specific certificate enumerates 75 rank-two elementary abelian subgroups and checks their centralizer types.
For every M≥2 with N=4M−5, a nonempty open set of complex N×M measurement matrices has a noninjective phase-retrieval map. Hence an IID complex Gaussian matrix is noninjective with positive probability, so p_M<1.
A surface qdlt log Calabi–Yau pair with three big boundary components has a square dual complex, rather than the boundary of a two-dimensional standard simplex. This answers part (2) of Mauri and Moraga’s Question 9.9 negatively.
For n=4 and p=2, the AlphaEvolve-found cosine correlation matrix Σᵢⱼ=cos(π(i−j)/n) has a strictly smaller expected squared minimum absolute Gaussian coordinate than the regular-simplex correlation matrix.
For asymptotically strictly decreasing sequences, and for every index set Q with at least two members, the possible global product-intersection sets are exactly the subsets of positive integers containing 1. The paper also specifies the Q=0 and Q=1 cases.
At inverse temperature β=1, the quenched law of N^(1/3) times the two-replica overlap converges to an explicit random Airy₁-based measure for both Ising and spherical Sherrington–Kirkpatrick models. Its second moment gives the limit of N^(2/3) E⟨overlap²⟩, answering Talagrand’s Conjecture 11.7.5.
For every finite connected simple graph G=(V,E), a parallel chip-firing game with 2|E|−|V|<|σ|<2|E| has eventual minimal period two and activity one half.
Aletheia constructed two closed, infinite, countable compact subsets of the complex plane with the same zero transfinite diameter but different values of the polynomial sublevel-area quantity μ(F).
A finitely generated torsion-free discrete group has a scalar projection e in its reduced group C*-algebra whose canonical trace lies strictly between zero and one-half.
In Bucić’s alternative proof that the uniform Turán density of the three-uniform tetrahedron K₄³ is 1/2, ChatGPT 6 Pro found the finite sum-of-positive-quadratic-forms identity and proposed the tensor projection inequality used in the proof.
The retained AI-origin construction gives a C¹ circle diffeomorphism with irrational rotation number, a wandering interval and derivative modulus ω(t)=t log(1/t), the barrier case in Prasad’s Corollary 1.1.
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arXiv · algebra-geometry — last successful metadata check: 21/10/2026, 07:20:46 UTC
arXiv · algorithms-formal — last successful metadata check: 21/10/2026, 07:19:48 UTC
arXiv · analysis — last successful metadata check: 21/10/2026, 07:21:51 UTC
arXiv · combinatorics — last successful metadata check: 21/10/2026, 07:18:49 UTC
arXiv · geometry-topology — last successful metadata check: 21/10/2026, 07:22:23 UTC
arXiv · logic-foundations — last successful metadata check: 21/10/2026, 07:23:04 UTC
arXiv · optimization-information — last successful metadata check: 21/10/2026, 07:23:10 UTC
arXiv · probability-dynamics-physics — last successful metadata check: 21/10/2026, 07:22:36 UTC
arXiv · theoretical-physics — last successful metadata check: 21/10/2026, 07:22:50 UTC
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