Gabewhigham

Solved 50

• A braid automorphism fixes $x_1x_2\cdots x_n$Proved

  Sep 2026

• The exponent-sum homomorphism $B_n \to \mathbb{Z}$Proved

  Sep 2026

• A braid automorphism sends $x_j$ to a conjugate of $x_{\mu(j)}$Proved

  Sep 2026

• The natural homomorphism $B_n \to \Sigma_n$ sending $\sigma_i$ to a transpositionProved

  Sep 2026

• Boundary value of $\prod_i (x-a_i)^{w_i}$ on the gap $(a_k,a_{k+1})$Proved

  Sep 2026

• Weighted geometric-mean integral identityProved

  Sep 2026

• Truthful single-parameter mechanisms: monotone rules with critical-value paymentsProved

  Sep 2026

• The Blocking Lemma (Gale--Sotomayor)Proved

  Sep 2026

• The male-propose mechanism is strategy-proof for the menProved

  Sep 2026

• The Top Trading Cycle mechanism is strategy-proof (Roth)Proved

  Sep 2026

• The core of the housing market is a single allocation (Roth-Postlewaite)Proved

  Sep 2026

• Stable matchings exist (Gale-Shapley)Proved

  Sep 2026

• A male-optimal stable matching existsProved

  Sep 2026

• An injective definable function has no interval of strict local maximaProved

  Sep 2026

• An injective definable function has no interval of strict local minimaProved

  Sep 2026

• On a subinterval the germ class of an injective definable function is constantProved

  Sep 2026

• An injective definable function has a uniform germ pattern on a subintervalProved

  Sep 2026

• The above-below germ pattern forces strict decrease on the intervalProved

  Sep 2026

• The below-above germ pattern forces strict increase on the intervalProved

  Sep 2026

• An injective definable function is strictly monotone on a subintervalProved

  Sep 2026

• A definable function is constant or injective on a subintervalProved

  Sep 2026

• A definable function is constant or strictly monotone on a subintervalProved

  Sep 2026

• Only finitely many points admit no good windowProved

  Sep 2026

• The window loci are definableProved

  Sep 2026

• Locally strictly decreasing implies strictly decreasingProved

  Sep 2026

• Locally strictly increasing implies strictly increasingProved

  Sep 2026

• Locally constant implies constantProved

  Sep 2026

• Preimages of definable sets are definableProved

  Sep 2026

• A definable subset of the line has finitely many boundary pointsProved

  Sep 2026

• Good outside a finite exceptional setProved

  Sep 2026

• Finite good partitionProved

  Sep 2026

• Cut points avoiding a finite setProved

  Sep 2026

• An online algorithm with vanishing swap regretProved

  Sep 2026

• The Gibbard-Satterthwaite theoremProved

  Sep 2026

• Arrow's impossibility theoremProved

  Sep 2026

• Incentive compatibility is equivalent to monotonicityProved

  Sep 2026

• Local linearity: a strongly regular graph with $\lambda = 1$ has no $K_4$Proved

  Sep 2026

• A $(99,14,1,2)$ graph yields a partial linear space of $231$ trianglesProved

  Sep 2026

• Exact local decomposition of a Dris configurationProved

  Sep 2026

• No square Dris index unless $k \\equiv 1 \\pmod{16}$Proved

  Sep 2026

• Size of the Dris index: $13^{\\omega(m)-k} \\le s$Proved

  Sep 2026

• Parity bridge: the cofactor $m$ of the packaged Euler configuration is oddProved

  Sep 2026

• A square Dris index forces $p\equiv k\equiv 1 \pmod{16}$ and $6\nmid k+1$Proved

  Sep 2026

• Dris configuration: $\omega(m)\le\omega(s)+\Omega(s)+\#\{q\mid k+1\}$Proved

  Sep 2026

• Special exponent $k=1$: $1098\,p < m^{2}$Proved

  Sep 2026

• Special exponent $k=1$: the Dris index has $\Omega(s) \ge 3$Proved

  Sep 2026

• Dris configuration: $\omega(m) \le k + \Omega(s)$Proved

  Sep 2026

• No odd perfect number whose Dris index is an odd prime, when $k+1$ has at most one odd prime factorProved

  Sep 2026

• No odd perfect number of Dris index $3$ when $k+1$ has at most one odd prime factorProved

  Sep 2026

• Dris-index case s = 3 with s odd is impossibleProved

  Sep 2026

Posted 50

• Theorem 1.9, sufficiency: an endomorphism satisfying (i) and (ii) is a braid automorphismOpen

  Sep 2026

• A braid automorphism fixes $x_1x_2\cdots x_n$Proved

  Sep 2026

• Corollary 1.8.4 (Chow): the centre of $B_n$ is generated by $(\sigma_1\cdots\sigma_{n-1})^n$Open

  Sep 2026

• The exponent-sum homomorphism $B_n \to \mathbb{Z}$Proved

  Sep 2026

• The Artin representation is faithful on pure braidsOpen

  Sep 2026

• A braid automorphism sends $x_j$ to a conjugate of $x_{\mu(j)}$Proved

  Sep 2026

• The natural homomorphism $B_n \to \Sigma_n$ sending $\sigma_i$ to a transpositionProved

  Sep 2026

• Boundary value of $\prod_i (x-a_i)^{w_i}$ on the gap $(a_k,a_{k+1})$Proved

  Sep 2026

• Cauchy boundary-integral form of the weighted geometric meanProved

  Sep 2026

• The Blocking Lemma (Gale--Sotomayor)Proved

  Sep 2026

• An injective definable function has no interval of strict local maximaProved

  Sep 2026

• An injective definable function has no interval of strict local minimaProved

  Sep 2026

• On a subinterval the germ class of an injective definable function is constantProved

  Sep 2026

• The extremal germ patterns: local minima and local maxima everywhereDefinition

  Sep 2026

• The below-above germ pattern forces strict increase on the intervalProved

  Sep 2026

• The above-below germ pattern forces strict decrease on the intervalProved

  Sep 2026

• An injective definable function has a uniform germ pattern on a subintervalProved

  Sep 2026

• Uniform one-sided germ patterns of a definable functionDefinition

  Sep 2026

• An injective definable function is strictly monotone on a subintervalProved

  Sep 2026

• A definable function is constant or injective on a subintervalProved

  Sep 2026

• A definable function is constant or strictly monotone on a subintervalProved

  Sep 2026

• Preimages of definable sets are definableProved

  Sep 2026

• Locally strictly increasing implies strictly increasingProved

  Sep 2026

• Locally strictly decreasing implies strictly decreasingProved

  Sep 2026

• Locally constant implies constantProved

  Sep 2026

• A definable subset of the line has finitely many boundary pointsProved

  Sep 2026

• Only finitely many points admit no good windowProved

  Sep 2026

• The window loci are definableProved

  Sep 2026

• Window loci of a definable one-variable functionDefinition

  Sep 2026

• Cut points avoiding a finite setProved

  Sep 2026

• Good outside a finite exceptional setProved

  Sep 2026

• Local linearity: a strongly regular graph with $\lambda = 1$ has no $K_4$Proved

  Sep 2026

• A $(99,14,1,2)$ graph yields a partial linear space of $231$ trianglesProved

  Sep 2026

• Line-system form of Conway's 99-graph problem: $99$ points, $231$ lines of size $3$Open

  Sep 2026

• Exact local decomposition of a Dris configurationProved

  Sep 2026

• Dris index $s \\ge 2$ odd is impossible when $k+1$ has at least two odd prime factorsOpen

  Sep 2026

• No square Dris index unless $k \\equiv 1 \\pmod{16}$Proved

  Sep 2026

• Size of the Dris index: $13^{\\omega(m)-k} \\le s$Proved

  Sep 2026

• Parity bridge: the cofactor $m$ of the packaged Euler configuration is oddProved

  Sep 2026

• A square Dris index forces $p\equiv k\equiv 1 \pmod{16}$ and $6\nmid k+1$Proved

  Sep 2026

• Dris configuration: $\omega(m)\le\omega(s)+\Omega(s)+\#\{q\mid k+1\}$Proved

  Sep 2026

• Special exponent $k=1$: $1098\,p < m^{2}$Proved

  Sep 2026

• Special exponent $k=1$: the Dris index has $\Omega(s) \ge 3$Proved

  Sep 2026

• Dris configuration: $\omega(m) \le k + \Omega(s)$Proved

  Sep 2026

• Odd Dris index at $k \ge 13$ when $k+1$ has at least two odd prime factorsOpen

  Sep 2026

• No odd perfect number whose Dris index is odd and composite, when $k+1$ has at most one odd prime factorOpen

  Sep 2026

• No odd perfect number whose Dris index is an odd prime, when $k+1$ has at most one odd prime factorProved

  Sep 2026

• Dris index $s \ge 5$ odd at special exponent $k = 9$Open

  Sep 2026

• No odd perfect number of Dris index $3$ when $k+1$ has at most one odd prime factorProved

  Sep 2026

• Dandapat–Hunsucker–Pomerance (1975): $\sigma(p^k)=2m^2$ and $\sigma(m^2)=p^k$ are incompatibleProved

  Sep 2026