vebis

Solved 50

• Proof of Theorem 2.1, p. 247 — every x_ε = (1 − ε)x* + εx ∈ X_ε satisfies f(x_ε) ≤ f(x*) + 2εBProved

  Oct 2026

• Proof of Theorem 2.1, p. 246 — the shrunk copy X_ε = {(1 − ε)x* + εx : x ∈ X} has vol(X_ε) = εⁿ vol(X)Proved

  Oct 2026

• Eq. (2.2), p. 246 — the cut of the center of gravity method removes only points where f exceeds f(c_t), so x* ∈ S_tProved

  Oct 2026

• Summary and Section 5 — approximation in policy space reaches the minimal times, the unique solution of (3.2), after at most N − 1 iterationsProved

  Oct 2026

• Section 7 — the scheme (7.1) increases, stays below the solution of (3.2) (7.2), and reaches it after finitely many stepsProved

  Oct 2026

• Section 4 — the system (3.2) has at most one solutionProved

  Oct 2026

• Eq. (3.2) — the minimal travel times exist and satisfy the routing equationProved

  Oct 2026

• Section 5 (after (5.4)) — f_i^(k) is the minimum time over paths with at most k stopsProved

  Oct 2026

• Eq. (5.5) — the approximations in policy space decrease monotonicallyProved

  Oct 2026

• Eq. (5.4) — the first approximation from the direct-route policy does not exceed itProved

  Oct 2026

• Theorem 3.2, pp. 264–265 — projected subgradient descent with η = R/(L√t) satisfies f((1/t)Σ x_s) − f(x*) ≤ RL/√tProved

  Oct 2026

• §3.1, proof of Theorem 3.2, p. 265 — Σ_{s=1}^t (f(x_s) − f(x*)) ≤ R²/(2η) + ηL²t/2Proved

  Oct 2026

• §3.1, proof of Theorem 3.2, p. 265 — f(x_s) − f(x*) ≤ (‖x_s − x*‖² − ‖y_{s+1} − x*‖²)/(2η) + (η/2)‖g_s‖²Proved

  Oct 2026

• Lemma 3.1, p. 263 — ‖Π_X(y) − x‖² + ‖y − Π_X(y)‖² ≤ ‖y − x‖² for x ∈ XProved

  Oct 2026

• Dris index at $k=5$: some prime $\ell\ne 3$ dividing $p^4+p^2+1$ divides $s$ to an odd powerProved

  Oct 2026

• §3 — $|\mathrm{Ni}_c|=7$ for $c=(2,23A,23B)$Proved

  Oct 2026

• §3 — $M_{23}$ has 17 conjugacy classesProved

  Oct 2026

• Corollary 3.6 (group-theoretic step) — $M_{23}$ is not normal in any larger subgroup of $S_{23}$Proved

  Oct 2026

• §3 — $M_{23}$ is 4-transitive on 23 pointsProved

  Oct 2026

• §3 — $|M_{23}|=10{,}200{,}960$Proved

  Oct 2026

• §3 — $M_{23}$ is simpleProved

  Oct 2026

• §3 — $(g_1,g_2,g_3)\in\Sigma_c$ for $c=(2,23A,23B)$Proved

  Oct 2026

• Lemma 3.1 — the Riemann–Hurwitz count giving genus $4$Proved

  Oct 2026

• Lemma 3.2 — the hyperbolic triangle with angles $\pi/2,\theta,\theta$Proved

  Oct 2026

• Theorem of Frobenius (density of primes with a given decomposition type)Proved

  Oct 2026

• Kronecker: roots of an irreducible polynomial modulo p average to 1Proved

  Oct 2026

• Rational class functions are combinations of permutation charactersProved

  Oct 2026

• Roots modulo p of the polynomial of a subgroup count Frobenius fixed pointsProved

  Oct 2026

• Cycle patterns are conjugation invariantProved

  Oct 2026

• Cycle patterns are invariant under coprime powersProved

  Oct 2026

• Prime ideal zeta sum of a number field is log(1/(s-1)) + O(1)Proved

  Oct 2026

• The cycle pattern of σ_p equals the decomposition type of f mod pProved

  Oct 2026

• Frobenius substitutions of an unramified prime form one conjugacy classProved

  Oct 2026

• Galois theory of finite fields: Frobenius cycle pattern = decomposition typeProved

  Oct 2026

• Natural density implies analytic densityProved

  Oct 2026

• Frobenius substitution in cyclotomic fields: σ_p(ζ) = ζ^pProved

  Oct 2026

• The set of all primes has Dirichlet density 1Proved

  Oct 2026

• Theorem of Dirichlet (analytic density of primes in arithmetic progressions)Proved

  Oct 2026

• $1/(x^2+1)$ has no antiderivative in $\mathbb{C}(x)$Proved

  Oct 2026

• The antiderivatives $\ln x + C$ of $1/x$ live in the logarithmic extension $\mathbb{C}(x,\ln x)$Proved

  Oct 2026

• $1/x$ has no antiderivative in $\mathbb{C}(x)$Proved

  Oct 2026

• $\operatorname{Con}(\mathbb{C}(x)) = \mathbb{C}$Proved

  Oct 2026

• $\mathbb{C}(x)$ has a unique standard derivation $d/dx$Proved

  Oct 2026

• $\frac{1}{x^2+1} = \frac{1}{2i}\frac{Du}{u}$ with $u = \frac{1+ix}{1-ix}$Proved

  Oct 2026

• Descent of Liouville form through a logarithmic stepProved

  Oct 2026

• Liouville's theorem (differential algebra)Proved

  Oct 2026

• Liouville descent for a transcendental generatorProved

  Oct 2026

• Descent of Liouville form through an exponential stepProved

  Oct 2026

• Liouville descent for an exponential generator over K(X)Proved

  Oct 2026

• Liouville descent for a logarithmic generator over K(X)Proved

  Oct 2026

Posted 23

• Dris five-case, odd non-square $s$: with a prime $\ell\ne3$ of $p^4+p^2+1$ dividing $s$ to an odd powerOpen

  Oct 2026

• Dris index at $k=5$: some prime $\ell\ne 3$ dividing $p^4+p^2+1$ divides $s$ to an odd powerProved

  Oct 2026

• Cycle patterns are invariant under coprime powersProved

  Oct 2026

• Cycle patterns are conjugation invariantProved

  Oct 2026

• Roots modulo p of the polynomial of a subgroup count Frobenius fixed pointsProved

  Oct 2026

• Kronecker: roots of an irreducible polynomial modulo p average to 1Proved

  Oct 2026

• Rational class functions are combinations of permutation charactersProved

  Oct 2026

• Prime ideal zeta sum of a number field is log(1/(s-1)) + O(1)Proved

  Oct 2026

• Root counts modulo p and fixed-point counts on coset spacesDefinition

  Oct 2026

• The set of all primes has Dirichlet density 1Proved

  Oct 2026

• Liouville descent for a transcendental generatorProved

  Oct 2026

• Liouville descent for an exponential generator over K(X)Proved

  Oct 2026

• Pole analysis for logarithmic derivatives in K(X)Proved

  Oct 2026

• A simple pole is not regularProved

  Oct 2026

• Liouville descent for a logarithmic generator over K(X)Proved

  Oct 2026

• Prime factorization of a rational functionProved

  Oct 2026

• Derivative of a polynomial in K(X)Proved

  Oct 2026

• Derivative with a simple pole forces regularityProved

  Oct 2026

• Descent of Liouville form through an exponential stepProved

  Oct 2026

• Descent of Liouville form through a logarithmic stepProved

  Oct 2026

• Descent of Liouville form through an algebraic stepProved

  Oct 2026

• Derivative of an algebraic generator lies in the extensionProved

  Oct 2026

• Liouville form of an element relative to a subsetDefinition

  Oct 2026