The partial KKM principle for an abstract convex space is an abstract form of the classical KKM theorem. In this paper, we derive generalized forms of the Ky Fan minimax inequality, the von Neumann-Sion minimax theorem, the von Neumann-Fan intersection theorem, the Fan-type analytic alternative, and the Nash equilibrium theorem for abstract convex spaces satisfying the partial KKM principle. Th...

Let R be a locally finitely generated algebra over a discrete valuation ring V of mixed characteristic. For any of the homological properties, the Direct Summand Theorem, the Monomial Theorem, the Improved New Intersection Theorem, the Vanishing of Maps of Tors and the Hochster-Roberts Theorem, we show that it holds for R and possibly some other data defined over R, provided the residual charac...

The Helly number of a family of sets with empty intersection is the size of its largest inclusionwise minimal sub-family with empty intersection. Let F be a finite family of open subsets of an arbitrary locally arc-wise connected topological space Γ. Assume that for every sub-family G ⊆ F the intersection of the elements of G has at most r connected components, each of which is a Q-homology cel...

The Blaschke–Leichtweiss theorem (Leichtweiss in Abh Math Semin Univ Hambg 75:257–284, 2005) states that the smallest area convex domain of constant width w 2-dimensional spherical space $${\mathbb {S}}^2$$ is Reuleaux triangle for all $$0<w\le \frac{\pi }{2}$$ . In this paper we extend result to family wide r-disk domains , where $$0<r\le Here a an intersection disks radius r with centers cont...

Cohen, Goresky and Ji showed that there is a Künneth theorem relating the intersection homology groups I H∗(X × Y ) to I H∗(X) and I H∗(Y ), provided that the perversity p̄ satisfies rather strict conditions. We consider biperversities and prove that there is a Künneth theorem relating I H∗(X × Y ) to I H∗(X) and I H∗(Y ) for all choices of p̄ and q̄. Furthermore, we prove that the Künneth theorem...

We express the beta invariant of a loopless matroid as tropical self-intersection number diagonal its fan (a “local” Poincaré-Hopf theorem). This provides another example uncovering “geometry” matroids by expressing their invariants in terms tropicalised geometric constructions. also prove global theorem and initiate study more general Lefschetz-Hopf trace formula proving two special cases curv...

Morley's trisector theorem states that " The points of intersection of the adjacent trisectors of the angles of any triangle are the vertices of an equilateral triangle " [10]. The notation and terminology used in this paper have been introduced in the

Based on Weitzenböck’s theorem and Nagata’s counterexample for Hilbert’s fourteenth problem we construct two finitely generated invariant rings R,S ⊂ K[x1, x2, . . . , xn] s.t. the intersection R ∩ S is not finitely generated as a K-algebra.