Related papers: Counting joints in vector spaces over arbitrary fi…
Virtual knots are associated with knot diagrams, which are not obligatory planar. The recently suggested generalization from N=2 to arbitrary N of the Kauffman-Khovanov calculus of cycles in resolved diagrams can be straightforwardly…
In this paper we study possibilities of efficient reasoning in combinations of theories over possibly non-disjoint signatures. We first present a class of theory extensions (called local extensions) in which hierarchical reasoning is…
Projective spaces for finite-dimensional vector spaces over general fields are considered. The geometry of these spaces and the theory of line bundles over these spaces is presented. Particularly, the space of global regular sections of…
By proving a connected sum formula for the Legendrian invariant $\lambda_+$ in knot Floer homology we exhibit infinitely many transversely non simple knots.
A new proof for adjoint systems of linear equations is presented. The argument is built on the principles of Algorithmic Differentiation. Application to scalar multiplication sets the base line. Generalization yields adjoint inner vector,…
We consider an algebraic variety and its foliation, both defined over a number field. We prove upper bounds for the geometric complexity of the intersection between a leaf of the foliation and a subvariety of complementary dimension (also…
Let $L$ be a set of $n$ lines in $\reals^d$, for $d\ge 3$. A {\em joint} of $L$ is a point incident to at least $d$ lines of $L$, not all in a common hyperplane. Using a very simple algebraic proof technique, we show that the maximum…
In this paper a version of Knaster-Kuratowski-Mazurkiewicz theorem for products of simplices is formulated. Some corollaries for measure partition in the plane and cutting families of sets in the plane by lines are given.
The micro-local Gevrey regularity of a class of "sums of squares" with real analytic coefficients is studied in detail. Some partial regularity result is also given.
We study the restriction of the Fourier transform to quadratic surfaces in vector spaces over finite fields. In two dimensions, we obtain the sharp result by considering the sums of arbitrary two elements in the subset of quadratic surfaces…
Relative index theorems, which deal with what happens with the index of elliptic operators when cutting and pasting, are abundant in the literature. It is desirable to obtain similar theorems for other stable homotopy invariants, not the…
We apply Heegaard-Floer homology theory to establish generalized slicing Bennequin inequalities closely related to a recent result of T. Mrowka and Y. Rollin proved using Seiberg-Witten monopoles.
A vector topology on a vector space over a topological field is a (not necessarily Hausdorff) topology by which the addition and scalar multiplication are continuous. We prove that, if an isomorphism between the lattice of topologies of two…
We characterize the global hypoellipticity, almost hypoellipticity and solvability for a class of systems of real vector fields on the (n + 1)-dimensional torus as well as the same properties about the sum of squares associated to the…
Iosevich and Senger (2008) showed that if a subset of the d-dimensional vector space over a finite field is large enough, then it contains many k-tuples of mutually orthogonal vectors. In this note, we provide a graph theoretic proof of…
In their paper Scholze and Weinstein show that a certain diagram of perfectoid spaces is Cartesian. In this paper, we generalize their result. This generalization will be used in a forthcoming paper of ours to compute certain non-trivial…
We extend the diagrammatic calculus of syllogisms introduced in our previous paper to the general case of n-term syllogisms, showing that the valid ones are exactly those whose conclusion follows by calculation. Moreover, by pointing out…
It is shown that one can count $k$-edge paths in an $n$-vertex graph and $m$-set $k$-packings on an $n$-element universe, respectively, in time ${n \choose k/2}$ and ${n \choose mk/2}$, up to a factor polynomial in $n$, $k$, and $m$; in…
The Szemer\'edi-Trotter theorem gives a bound on the maximum number of incidences between points and lines on the Euclidean plane. In particular it says that $n$ lines and $n$ points determine $O(n^{4/3})$ incidences. Let us suppose that an…
In this paper we set up the family Seiberg-Witten theory. It can be applied to the counting of nodal pseudo-holomorphic curves in a symplectic 4-manifold (especially a Kahler surface). A new feature in this theory is that the chamber…