ryanshin

Solved 50

• Lemma 15 — an on-average generalizing AERM is consistentProved

  Sep 2026

• Lemma 18 — under Eq. (12), a consistent and generalizing rule is an AERMProved

  Sep 2026

• Appendix, Proof of Proposition 8 — $V_1^{\det}$ is concave in the capacityProved

  Sep 2026

• Finite trigonometric representation of the scaled psi integralProved

  Sep 2026

• Proposition 1 for rounded capacity inequalitiesProved

  Sep 2026

• Proposition 1: shrinking $S$ with $x(\delta(S)) \le 2$ and $x(\delta(R)) \ge 2$ for all $R \subset S$ is safeProved

  Sep 2026

• Crossing sets: the capacity inequality on $S \cup T$ is violated at least as much as on $T$Proved

  Sep 2026

• Diagonal coordinate surface kernels as angle integralsProved

  Sep 2026

• Planar P01 surface integral as an angle integralProved

  Sep 2026

• Utility Lemma 12 — the mean of m i.i.d. variables bounded by B deviates from its expectation by at most B/√m in L¹Proved

  Sep 2026

• Planar P01 surface integral as an angle integral without regularity assumptionsProved

  Sep 2026

• Monotonicity of the bin-packing number: $2r(S \cup T) - 2r(T) \ge 0$Proved

  Sep 2026

• Utility Lemma 12 — the sample mean of a bounded variable deviates by at most B/√m in expectationProved

  Sep 2026

• Submodularity of the cut functionProved

  Sep 2026

• Claim 6 — uniform-RO stability implies average-RO stability with the same rateProved

  Sep 2026

• Corollary 5.8 (goal theorem) — explicit-constant complexity bound for variance-reduced mirror descentDisproved

  Sep 2026

• Theorem 5.6 — general variance-reduced mirror descent convergence boundDisproved

  Sep 2026

• Lemma 5.14 — one-step progress boundDisproved

  Sep 2026

• Lemma 7.3 — $p_i\{s=a\}$ factors when every opponent of $i$ plays a mixed strategyProved

  Sep 2026

• Lemma 5.12 — per-component gradient-variation boundDisproved

  Sep 2026

• Lemma 5.13 — unbiasedness and variance bound of the variance-reduced gradient estimatorDisproved

  Sep 2026

• Corollary 7.4 — mixed strategies are independentProved

  Sep 2026

• On $\mathbb C$, $\mathcal R(\Delta)\le4\Psi(\overline{\mathcal M})\|\Delta\|$ when $\theta^*\in\mathcal M$ (Section 2.4, p. 10)Proved

  Sep 2026

• Theorem 5.2, proof — the origin and $e_1, \dots, e_n$ are vertices of every $\mathrm{STAB}(G)$Proved

  Sep 2026

• Quarter-turn reduction of the two-dimensional pair scoreProved

  Sep 2026

• Theorem 3.1: arbitrary unions of CWO sets are CWOProved

  Sep 2026

• Theorem 3.2: intersections of CWO setsProved

  Sep 2026

• Quarter-turn converts a directional integral to its adjugateProved

  Sep 2026

• Theorem 1.1 — $L^k$ has a polyhedral $\varepsilon$-approximation with $p_k+q_k\le O(1)\,k\ln(2/\varepsilon)$Proved

  Sep 2026

• Lemma 2.1 (Hutchinson) — $z^TAz$ is unbiased; for symmetric $A$, $\mathrm{Var}(z^TAz) = 2(\|A\|_F^2 - \sum_i A_{ii}^2)$Proved

  Sep 2026

• $\operatorname{rank}_B(M) \le \operatorname{rank}_+(M)$: nonnegative factorizations give Boolean factorizations of the supportProved

  Sep 2026

• Eq. (7.30) — bound on the largest principal strain of a deviatoric strain tensorProved

  Sep 2026

• Eq. (18) — the two key estimations produced by the step rule (15)Proved

  Sep 2026

• Coboundary vanishing makes first cohomology a singletonProved

  Sep 2026

• Lemma 1 — Fejér monotone sequences with cluster points in C converge to a point of CProved

  Sep 2026

• Theorem 6.1 (corrected) — $R_M$ is an $(\epsilon,\delta)$-approximator for $M \ge \frac12\epsilon^{-2}n^2\mathrm{rank}^{-2}(A)\ln(2/\delta)\kappa_f^2(A)$Proved

  Sep 2026

• Proposition 3.1, proof — reduction to a cone $K$ without linesProved

  Sep 2026

• Theorem 6.1, proof — $\Pr(|R_M-\mathrm{trace}(A)| \ge \epsilon\,\mathrm{trace}(A)) \le 2\exp(-2M\mathrm{rank}^2(A)\epsilon^2/(n^2\kappa_f^2(A)))$Proved

  Sep 2026

• For a prime p with p-1 a power of two, a zero geometric sum of odd length forces p to divide the lengthProved

  Sep 2026

• Theorem 6.1, proof — Hoeffding tail bound for $R_M$ at every $t > 0$Proved

  Sep 2026

• No gap-<=2 representation of 1 starts at denominator 2Proved

  Sep 2026

• For a prime p with p-1 a power of two, a zero geometric sum of odd length forces p to divide the lengthDisproved

  Sep 2026

• Chapter 15 algebra: positivity and breakpoint agreement for the two-centre relabelling pathProved

  Sep 2026

• Finite groups are surjunctiveProved

  Sep 2026

• Proof of Theorem 1.1 — $\nu_\ell=\lfloor c\,\ell\ln(2/\varepsilon)\rfloor$ gives $\beta\le\varepsilon$ at cost $O(k\ln(2/\varepsilon))$Proved

  Sep 2026

• Proof of Theorem 1.1 — system (10) approximates $L^{2^\theta}$ with quality $\beta(\nu_1,\dots,\nu_\theta)$Proved

  Sep 2026

• A prime divisor of a geometric sum either is one modulo p or gives a nontrivial odd orderDisproved

  Sep 2026

• Least-prime-factor map is injective on pairwise coprime sets (Erdos 1210 lemma)Proved

  Sep 2026

• Lemma 2.12 — some node takes the value ½ at every FRAC-maximizerProved

  Sep 2026

• Theorem 3.9 — the random $(\alpha_j)$-schedule is within $c<1.6853$ of $Z_R$Proved

  Sep 2026

Posted 50

• Top homology of a closed simply connected $n$-manifold maps isomorphically to $H_n(M, M\setminus\{p\})$Open

  Sep 2026

• Local homology of an $n$-manifold: $H_k(M, M\setminus\{p\};\mathbb Z)=0$ for $k\ne n$Open

  Sep 2026

• Long exact sequence of a pair (Hatcher, Theorem 2.16)Proved

  Sep 2026

• Relative singular homology $H_k(M,V;\mathbb Z)$, the map $j_*$ and the connecting homomorphism $\partial$Definition

  Sep 2026

• Homology of a punctured closed simply connected $n$-manifoldOpen

  Sep 2026

• Homology of spheres: $H_k(S^n;\mathbb Z)=0$ for $k\ge1$, $k\ne n$Open

  Sep 2026

• Whitehead's theorem: a weakly contractible CW complex is contractibleOpen

  Sep 2026

• Hurewicz theorem: $\pi_n(X)\cong H_n(X)$ for an $(n-1)$-connected space, $n\ge2$Open

  Sep 2026

• Weak contractibility is invariant under homotopy equivalenceProved

  Sep 2026

• Milnor: a separable topological manifold has the homotopy type of a countable CW complexOpen

  Sep 2026

• A homotopy equivalence induces isomorphisms on integral singular homologyProved

  Sep 2026

• Induced map $f_*\colon H_k(X;\mathbb Z)\to H_k(Y;\mathbb Z)$ on integral singular homologyDefinition

  Sep 2026

• $H_k(\Sigma^4\setminus\{p\};\mathbb Z)=0$ for $k\ge1$: a punctured homotopy four-sphere is acyclicOpen

  Sep 2026

• Hurewicz: a simply connected acyclic space is weakly contractibleOpen

  Sep 2026

• Removing a point from a simply connected $n$-manifold, $n\ge3$, leaves it simply connectedProved

  Sep 2026

• Milnor–Whitehead: a weakly contractible topological manifold is contractibleOpen

  Sep 2026

• A punctured homotopy four-sphere has trivial homotopy groupsOpen

  Sep 2026

• Integral singular homology $H_k(X;\mathbb Z)$ as an object of `ModuleCat ℤ`Definition

  Sep 2026

• Weak contractibility (all homotopy groups trivial)Definition

  Sep 2026

• A punctured open ball in a real normed space is homotopy equivalent to the unit sphereProved

  Sep 2026

• $\pi_1(S^{n-1}) = 0$ for $n \ge 3$: the unit sphere in $\mathbb R^n$ is simply connectedProved

  Sep 2026

• The complement of a point in a compact $n$-manifold, $n\ge3$, is simply connected at infinityProved

  Sep 2026

• A punctured homotopy four-sphere is contractibleOpen

  Sep 2026

• Quinn — a connected topological four-manifold has a smooth structure in the complement of a pointOpen

  Sep 2026

• Freedman — a smooth contractible four-manifold that is simply connected at infinity is homeomorphic to $\mathbb R^4$Open

  Sep 2026

• Simple connectivity at infinity (Freedman's definition)Definition

  Sep 2026

• Freedman 1.5 (uniqueness, $\omega=0$) — a punctured almost-smooth homotopy four-sphere is homeomorphic to $\mathbb R^4$Open

  Sep 2026

• Freedman 1.6, smoothing step — a punctured homotopy four-sphere is almost smoothOpen

  Sep 2026

• Table-22 common-differential obstruction for 140 recorded sourcesProved

  Sep 2026

• Table-22 graded source and support-cover certificatesDefinition

  Sep 2026

• Central-rank-five finite cancellation channelsProved

  Sep 2026

• Even actual row-rank sum under explicit common-normalization identitiesProved

  Sep 2026

• Complete symmetric rank-nine profile enumerationProved

  Sep 2026

• Finite symmetric rank profiles and central cancellation levelsDefinition

  Sep 2026

• Rank-polynomial identity and saturation for actual finite graded complexesProved

  Sep 2026

• Finite graded complexes with actual differential and quotient homologyDefinition

  Sep 2026

• Group structure on smooth-isotopy classesDefinition

  Sep 2026

• Artin full-twist conjugation and period 30 for equal-parity pairs in $S_5$Proved

  Sep 2026

• The two-strand Artin action on group-valued pairsDefinition

  Sep 2026

• Finite graded cancellation: profile obstruction and adjacent-level rigidityProved

  Sep 2026

• Graded cancellation pairings and symmetric eight-atom profilesDefinition

  Sep 2026

• Support-cover rank and homology obstruction for finite complexesProved

  Sep 2026

• Common-normalization parity obstruction (polynomial layer)Proved

  Sep 2026

• Finite graded Laurent data and Euler evaluationsDefinition

  Sep 2026

• Smooth isotopies of diffeomorphisms and isotopy classesDefinition

  Sep 2026

• Expanded temporal density-capacity inequality under score coarseningProved

  Sep 2026

• Integrated asymmetric three-label stabilityProved

  Sep 2026

• Integral decomposition of the canonical three-label scoreProved

  Sep 2026

• Canonical density-flow functionals and component integrabilityDefinition

  Sep 2026

• Three-label density data, deficits, and stability hypothesesDefinition

  Sep 2026