We give a domain-theoretic analogue of the classical BanachAlaoglu theorem, showing that the patch topology on the weak* topology is compact. Various theorems follow concerning the stable compactness of spaces of valuations on a topological space. We conclude with reformulations of the patch topology in terms of polar sets or Minkowski functionals, showing, in particular, that the ‘sandwich set...

Both the theory of differential subordination and its dual, superordination, introduced by Professors Miller Mocanu are based on reinterpreting certain inequalities for real-valued functions case complex-valued functions. Studying superordination properties using different types operators is a technique that still widely used, some studies resulting in sandwich-type theorems as present paper. T...

We show thatm points and n smooth algebraic surfaces of bounded degree in R satisfying suitable nondegeneracy conditions can have at most O(m 2k 3k−1 n 3k−3 3k−1 +m+ n) incidences, provided that any collection of k points have at most O(1) surfaces passing through all of them, for some k ≥ 3. In the case where the surfaces are spheres and no three spheres meet in a common circle, this implies t...

We use recent extensions of the Borsuk–Ulam theorem for Stiefel manifolds to generalize ham sandwich mass assignments. A k-dimensional assignment continuously imposes a measure on each affine subspace $${\mathbb {R}}^d$$ . Given finite collection assignments different dimensions, one may ask if there is some sequence subspaces $$S_{k-1} \subset S_k \ldots S_{d-1} {\mathbb such that $$S_i$$ bise...

If a convex body K ? ?n is covered by the union of bodies C1, …, CN, multiple subadditivity questions can be asked. Two classical results regard width (the smallest distance between two parallel hyperplanes that sandwich K) and inradius largest radius ball contained in K): sum widths Ci at least (this plank theorem Thøger Bang), inradii due to Vladimir Kadets). We adapt existing proofs these pr...

Following ideas of Iriyeh and Shibata we give a short proof the three-dimensional Mahler conjecture for symmetric convex bodies. Our contributions include, in particular, simple self-contained proofs their two key statements. The first these is an equipartition (ham sandwich type) theorem which refines celebrated result Hadwiger and, as usual, can be proved using from equivariant topology. seco...

We consider a nonstandard odd reduction of supermatrices (as compared with the standard even one) which arises in connection with possible extension of manifold structure group reductions. The study was initiated by consideration of the generalized noninvertible superconformal-like transformations. The features of evenand oddreduced supermatrices are investigated on a par. They can be unified i...

A k-weighted graph is a directed graph where each vertex has k associated weights. Partitioning such a graph so that each weight is roughly evenly divided into each piece is useful for balancing a parallel task with k distinct components. Such a goal arises in crash simulation, in which there there are 2 components. We present a 2-weighted graph partitioning algorithm, based on the Ham Sandwich...