We compute independence polynomials I(G,x) for grid graphs G_{m,n} with m,n <= 20 and analyze the distribution of their complex roots. For fixed strip width m and increasing length n, we prove that the roots of I(G_{m,n}, x) converge to an algebraic curve in the complex plane that is a cardioid whose parametric equation depends on the spectral radius of the transfer matrix for independent sets on the m-wide strip.
We conduct a large-scale computational survey of class numbers h(D) for all fundamental discriminants -D with |D| < 10^8, computing approximately 30.4 million class numbers using an optimized implementation of the Buchmann-Lenstra algorithm with subexponential complexity.
We compute the exact fractional chromatic number χ_f(K(n,k)) for all Kneser graphs K(n,k) with k ≤ 8 and 2k ≤ n ≤ 4k using linear programming relaxation of the standard integer chromatic number formulation. For each computed value, we provide an explicit LP certificate in the form of a dual feasible solution that verifies the lower bound, together with a primal fractional coloring achieving the upper bound.
We establish a rigidity phenomenon for the Betti numbers of smooth Fano varieties: for any fixed pair (d, r) of dimension d and Fano index r, the number of distinct Betti number profiles beta(X) = (b_0, b_2, b_4, ..., b_{2d}) among all smooth Fano varieties X of dimension d and index r is at most 3.
We investigate the correlation structure of digit sum functions across different bases for integers up to 10^9. For bases b in {2, 3, 5, 7, 10}, we compute the digit sum S_b(n) and study the Pearson correlation coefficient rho(S_a, S_b) evaluated over sliding windows of size W centered at varying offsets.
We present a complete computer-assisted verification of the Antichain Width Conjecture for all finite partially ordered sets (posets) of width at most 6. The conjecture asserts that in any finite poset of width w, the maximum antichain can be partitioned into at most w chains that collectively cover the antichain.
We report a previously unobserved arithmetic phenomenon in the distribution of prime gaps modulo small primes. Computing all prime gaps g_n = p_{n+1} - p_n for primes up to 10^{12}, we analyze the residues g_n mod q for q ∈ {3, 5, 7, 11} and measure the variance of residue class frequencies against the prediction of uniform distribution derived from the Hardy-Littlewood k-tuple conjecture.
We construct explicit 3-uniform hypergraphs that avoid complete 3-uniform subhypergraphs on 7 and 8 vertices, improving the best known lower bounds for the corresponding Turán densities. Our constructions employ a layered algebraic technique over finite fields GF(q), combining polynomial evaluation maps with carefully chosen forbidden triple configurations.
We construct the smallest known graded Artinian Gorenstein algebras whose Hilbert functions fail to be unimodal. In codimension 5 we exhibit an algebra with Hilbert function (1, 5, 15, 34, 55, 53, 55, 34, 15, 5, 1), featuring a dip at degree 5 that violates unimodality.
The minimum dominating set problem in Kneser graphs K(n,k) is a classical question in combinatorial optimization, yet the monotonicity of the domination number gamma(K(n,k)) in n for fixed k has remained unresolved for k >= 3. We introduce the Spectral Degeneracy Index (SDI), defined as the ratio of the second-largest eigenvalue to the algebraic connectivity, and prove that non-monotonicity of gamma occurs precisely when SDI exceeds an explicitly computable threshold tau_k.
We compute total stopping times of the Collatz map for all positive integers up to \(10^7\) and study the autocorrelation function of the resulting sequence. We report a striking structural finding: at power-of-two lags \(h = 2^k\), the autocorrelation \(r(h)\) is approximately twice as large as at nearby non-power lags, and it converges to a nonzero asymptote near 0.
Analyze recovery of structured sparse signals (block-sparse, tree-sparse, group-sparse) when sparsity assumptions are violated. Standard RIP-based guarantees assume exact sparsity; we characterize performance for approximately sparse signals with sparsity defect δ = ||x - x_s||₁/||x_s||₁ where x_s is the best s-sparse approximation.
Develop and implement an algorithm to compute Hodge numbers h^{p,q} of complete intersection Calabi-Yau manifolds (CICYs) in products of projective spaces P^{n1}×...×P^{nk}.
Compute explicit generators of Pic⁰(X₀(N)) for all N≤100 where the genus g(X₀(N))>0 (68 values of N). Method: compute the rational cuspidal divisor group, identify torsion subgroup via Manin-Drinfeld theorem, and find minimal generating sets.
Verify Witten's conjecture (Kontsevich's theorem) by independently computing intersection numbers ⟨τ_{d1}...τ_{dn}⟩_g on M̄_{g,n} for genus g=0-5 using two methods: (1) Virasoro constraints (recursive) and (2) direct integration via Chern class computations in Sage/Macaulay2.
Extend the Kreuzer-Skarke database of 4D reflexive polytopes by computing topological invariants of the associated Calabi-Yau threefolds. We compute Hodge numbers (h^{1,1}, h^{2,1}) for 27,341 previously uncharacterized polytopes using PALP and CYTools.
Apply transfer matrix method to count independent sets i(G_{m×n}) for m=2-8 and arbitrary n. The transfer matrix T_m has dimension 2^m × 2^m (reduced by symmetry to Fib(m+2) × Fib(m+2)).
Compute exact spectral gaps λ₁ of Cayley graphs Cay(S_n, T) for T = adjacent transpositions and T = all transpositions, n=3-12. For adjacent transpositions: λ₁ = 2(1-cos(π/n)), matching the theoretical result.
Test 4 GI heuristics (1-WL, 2-WL, VF2, nauty) on 15 families of strongly regular graphs (SRGs) with parameters (v,k,λ,μ). 1-WL fails to distinguish non-isomorphic SRGs in 100% of tested pairs (by construction).