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Poincare's invariance principle for Hamiltonian flows implies Kelvin's principle for solution to Incompressible Euler Equation. Iyer-Constantin Circulation Theorem offers a stochastic analog of Kelvin's principle for Navier-Stokes Equation.…
Minimal flow spaces of dimension 1 are among the most fundamental limit sets in dynamical systems. These invariant sets occur as the typical minimal sets in surface flows, the minimal sets of suspensions of subshifts (for example, in Lorenz…
Motivated by understanding the quality of tractable convex relaxations of intractable polytopes, Ko et al. gave a closed-form expression for the volume of a standard relaxation $\mathscr{Q}(G)$ of the boolean quadric polytope (also known as…
An extended formulation of a polytope is a linear description of this polytope using extra variables besides the variables in which the polytope is defined. The interest of extended formulations is due to the fact that many interesting…
If X is any connected Cayley graph on any finite abelian group, we determine precisely which flows on X can be written as a sum of hamiltonian cycles. (This answers a question of Brian Alspach.) In particular, if the degree of X is at least…
Symmetric edge polytopes of graphs and root polytopes of semi-balanced digraphs are two classes of lattice polytopes whose $h^*$-polynomials have interesting properties and generalize important graph polynomials. For both classes of…
Recently developed concept of dissipative measure-valued solution for compressible flows is a suitable tool to describe oscillations and singularities possibly developed in solutions of multidimensional Euler equations. In this paper we…
PQ-type adjacency polytopes $\nabla^{\rm PQ}_G$ are lattice polytopes arising from finite graphs $G$. There is a connection between $\nabla^{\rm PQ}_G$ and the engineering problem known as power-flow study, which models the balance of…
In this paper, we define extended trigonometric functions via series and employ the method of contour integration to investigate the parity of certain cyclotomic Euler sums and multiple polylogarithm function. We can provide the statement…
Partition functions, also known as homomorphism functions, form a rich family of graph invariants that contain combinatorial invariants such as the number of k-colourings or the number of independent sets of a graph and also the partition…
Gelfand-Tsetlin polytopes are prominent objects in algebraic combinatorics. The number of integer points of the Gelfand-Tsetlin polytope $\mathrm{GT}(\lambda)$ is equal to the dimension of the corresponding irreducible representation of…
Relative dimensions of isotypic components of N-th order tensor representations of the symmetric group on n letters give a Plancherel-type measure on the space of Young diagrams with n cells and at most N rows. It was conjectured by G.…
We introduce the notion of a generalized flow on a graph with coefficients in a R-representation and show that the module of flows is isomorphic to the first derived functor of the colimit. We generalize Kirchhoff's laws and build an exact…
Let K(X) be the collection of all non-zero finite dimensional subspaces of rational functions on an n-dimensional irreducible variety X. For any n-tuple L_1,..., L_n in K(X), we define an intersection index [L_1,..., L_n] as the number of…
A well known result by Lagarias and Ziegler states that there are finitely many equivalence classes of d-dimensional lattice polytopes having volume at most K, for fixed constants d and K. We describe an algorithm for the complete…
Structurally stable (rough) flows on surfaces have only finitely many singularities and finitely many closed orbits, all of which are hyperbolic, and they have no trajectories joining saddle points. The violation of the last property leads…
We will employ the method of contour integration to investigate the parity results of non-embedded cyclotomic multiple $t$-values, which we refer to as cyclotomic Euler $T$-sums. We can provide explicit parity formulas for the linear and…
We enumerate total cyclic orders on $\left\{1,\ldots,n\right\}$ where we prescribe the relative cyclic order of consecutive triples $(i,{i+1},{i+2})$, these integers being taken modulo $n$. In some cases, the problem reduces to the…
We study graphs of polynomial growth from the perspective of asymptotic geometry and descriptive set theory. The starting point of our investigation is a theorem of Krauthgamer and Lee who showed that every connected graph of polynomial…
The general position problem in graphs seeks the largest set of vertices such that no three vertices lie on a common geodesic. Its counting refinement, the general position polynomial $\psi(G)$, asks for all such possible sets. In this…