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We solve the integration problem for generalized complex manifolds, obtaining as the natural integrating object a weakly holomorphic symplectic groupoid, which is a real symplectic groupoid with a compatible complex structure defined only…

A symmetry reduction of the Dirac equation is shown to yield the system of ordinary differential equations whose integrability by quadratures is closely connected to the stationary mKdV hierarchy.

Computing the real solutions to a system of polynomial equations is a challenging problem, particularly verifying that all solutions have been computed. We describe an approach that combines numerical algebraic geometry and sums of squares…

Let $M$ be a smooth manifold and $K\subset M$ be a simplicial complex of codimension at least 3. Functor calculus methods lead to a homotopical formula of $M\setminus K$ in terms of spaces $M\setminus T$ where $T$ is a finite subset of $K$.…

Transforming an asymmetric system into a symmetric system makes it possible to exploit the simplifying properties of symmetry in control problems. We define and characterize the family of symmetrizable systems, which can be transformed into…

Bilinear systems of equations are defined, motivated and analyzed for solvability. Elementary structure is mentioned and it is shown that all solutions may be obtained as rank one completions of a linear matrix polynomial derived from…

There is a canonical way to associate two simplicial complexes K, L to any relation $R\subset X\times Y$. Moreover, the geometric realizations of K and L are homotopy equivalent. This was studied in the fifties by C.H. Dowker. In this…

Using the notion of contiguity of simplicial maps, we adapt Farber's topological complexity to the realm of simplicial complexes. We show that, for a finite simplicial complex $K$, our discretized concept recovers the topological complexity…

The concepts of differentiation and integration for matrices are known. As far as each matrix is differentiable, it is not clear a priori whether a given matrix is integrable or not. Recently some progress was obtained for diagonalizable…

We show that under certain mild conditions, a metric simplicial complex which satisfies the Ptolemy inequality is a CAT(0) space. Ptolemy's inequality is closely related to inversions of metric spaces. For a large class of metric simplicial…

Let $n$ be a positive integer, and let $k$ be a field (of arbitrary characteristic) accessible to symbolic computation. We describe an algorithmic test for determining whether or not a finitely presented $k$-algebra $R$ has infinitely many…

We present an encoding of a polynomial system into vanishing and non-vanishing constraints on almost-principal minors of a symmetric, principally regular matrix, such that the solvability of the system over some field is equivalent to the…

We determine simplicity criteria in characteristics 0 and $p$ for a ubiquitous class of iterated skew polynomial rings in two indeterminates over a base ring. One obstruction to simplicity is the possible existence of a canonical normal…

For a finite abstract simplicial complex G with n sets, define the n x n matrix K(x,y) which is the number of subsimplices in the intersection of x and y. We call it the counting matrix of G. Similarly as the connection matrix L which is…

The (n,k)-hypersimplex is the convex hull of all 0/1-vectors of length n with coordinate sum k. We explicitly determine the extension complexity of all hypersimplices as well as of certain classes of combinatorial hypersimplices. To that…

For any two arbitrary positive integers `$n$' and `$m$', using the $m$--th KdV hierarchy and the $(n+m)$--th KdV hierarchy as building blocks, we are able to construct another integrable hierarchy (referred to as the $(n,m)$--th KdV…

In this article we establish an exponential lower bound on the Graver complexity of integer programs. This provides new type of evidence supporting the presumable intractability of integer programming. Specifically, we show that the Graver…

In this paper, isometric embedding results of Greene, Gromov and Rokhlin are extended to what are called "indefinite metric polyhedra". Two definitions for an indefinite metric polyhedron are given, an intuitive definition and a more useful…

We consider the algebraic Riccati equation for which the four coefficient matrices form an M-matrix K. When K is a nonsingular M-matrix or an irreducible singular M-matrix, the Riccati equation is known to have a minimal nonnegative…

All exactly integrable systems connected with the semisimple algebras of the second rank with an arbitrary choice of the grading in them are presented in explicit form. General solution of such systems are expressed in terms of the matrix…