In this extended abstract we announce a proof that, in a Coxeter group of rank 3, low elements are in bijection with small inversion sets. This gives a partial confirmation of Conjecture 2 in Dyer, ...Hohlweg '16. That same article provides the main ingredient: the bipodality of the set of small roots is used to propagate information on the vertices of inversion polytopes.
A famous conjecture of Stanley states that his chromatic symmetric function distinguishes trees. As a quasisymmetric analogue, we conjecture that the chromatic quasisymmetric function of Shareshian ...and Wachs and of Ellzey distinguishes directed trees. This latter conjecture would be implied by an affirmative answer to a question of Hasebe and Tsujie about the $P$-partition enumerator distinguishing posets whose Hasse diagrams are trees. They proved the case of rooted trees and our results include a generalization of their result.
Let X ⊆ {0, 1} n. Then the daisy cube Q n (X) is introduced as the sub-graph of Q n induced by the intersection of the intervals I(x, 0 n) over all x ∈ X. Daisy cubes are partial cubes that include ...Fibonacci cubes, Lucas cubes, and bipartite wheels. If u is a vertex of a graph G, then the distance cube polynomial D G,u (x, y) is introduced as the bivariate polynomial that counts the number of induced subgraphs isomorphic to Q k at a given distance from the vertex u. It is proved that if G is a daisy cube, then D G,0 n (x, y) = C G (x + y − 1), where C G (x) is the previously investigated cube polynomial of G. It is also proved that if G is a daisy cube, then D G,u (x, −x) = 1 holds for every vertex u in G.
This thesis deals with problems related to the structure of the solutions to some specific polynomial equations. A brief introduction to the type of problems we are interested in is given in Chapter ...1. In Chapter 2 we recall some standard results in number theory and additive combinatorics. In Chapter 3 we look at partition regularity of equations of the form xa + yb = zc over Z/pZ. In particular we look at the equation x+y= z2. In Chapter 4 we prove that any 2-colouring of N has infinitely many monochromatic solutions to the equation x + y = z². This work is joint with Ben Green. In Chapter 5 we use the same methods as in Chapter 4 to prove partition regularity of the equation x-y=z2. In Chapter 6 we show that a linear combination of kth powers is partition regular if and only if the corresponding linear equation is partition regular, provided the number of variables is large enough. This is based on joint work with Sam Chow and Sean Prendiville. In Chapter 7 we look at Heath-Brown's method of counting the zeros of a quadratic form in four variables, and in particular how the error term in this count is affected by the weight function used. In Chapter 8 we try to count the number of zeros of a quadratic form in four variables that lie in a fixed congruence class.
A central theme of this thesis is the quantization and generalization of objects from areas of mathematics such as set theory and combinatorics with application to quantum information. We represent ...mathematical objects such as Latin squares or functions as string diagrams over the category of finite-dimensional Hilbert spaces obeying various diagrammatic axioms. This leads to a direct understanding of how these objects arise in quantum mechanics and gives insight into how quantum analogues can be defined. In Part I of this thesis we introduce quantum Latin squares (QLS), quantum objects which generalize the classical Latin squares from combinatorics. We present a new method for constructing a unitary error basis (UEB) from a quantum Latin square equipped with extra data, which we show simultaneously generalizes the existing shift-and-multiply and Hadamard methods. We introduce two different notions of orthogonality for QLS which we use to construct families of mutually unbiased bases and perfect tensors respectively. We also introduce a further generalization of Latin squares called quantum Latin isometry squares. We use these to produce quantum error codes and to give a new way of characterizing UEBs. In Part II we show that maximal families of mutually unbiased bases (MUBs) are characterized in all dimensions by partitioned UEBs, up to a choice of a family of Hadamards. Furthermore, we give a new construction of partitioned UEBs, and thus maximal families of MUBs, from a finite field, which is simpler and more direct than previous proposals. We introduce new tensor diagrammatic characterizations of maximal families of MUBs, partitioned UEBs, and finite fields as algebraic structures defined over Hilbert spaces. In Part III we introduce quantum functions and quantum sets, which quantize the classical notions. We show that these structures form a 2-category 'QSet'. We extend this framework to introduce quantum graphs and quantum homomorphisms which form a 2-category 'QGraph'. We show that these 2-categories capture several different notions of quantum morphism and noncommutative graph from various previous papers in noncommutative topology, quantum non-local games and quantum information. We later use the correspondence between our quantum graph morphisms and those from quantum non-local games to show that pairs of quantum isomorphic graphs with multiple connected components are built up of quantum isomorphisms on those components.
The enumeration of generalized Tamari intervals PRÉVILLE-RATELLE, LOUIS-FRANÇOIS; VIENNOT, XAVIER
Transactions of the American Mathematical Society,
07/2017, Letnik:
369, Številka:
7
Journal Article
Recenzirano
Odprti dostop
Let \overleftarrow {v} be the path obtained from v by reading the unit steps of v in reverse order, replacing the east steps by north steps and vice versa. We show that the poset Tam (v) is ...isomorphic to the dual of the poset Tam (\overleftarrow {v}). We do so by showing bijectively that the poset Tam (v) is isomorphic to the poset based on rotation of full binary trees with the fixed canopy v, from which the duality follows easily. This also shows that Tam (v) is a lattice for any path v. We also obtain as a corollary of this bijection that the usual Tamari lattice, based on Dyck paths of height n, can be partitioned into the (smaller) lattices Tam (v), where the v are all the paths on the square grid that consist of n-1 unit steps.
In this paper, we study b-Smarandache m1m2 curves of biharmonic new type b-slant helix in the Sol3. We characterize the b-Smarandache m1m2 curves in terms of their Bishop curvatures. Finally, we find ...out their explicit parametric equations in the Sol3. PUBLICATION ABSTRACT
In this paper, making use of the author's method appeared in 1 , we define nonnull inclined curve in L^sup 5^. We also give some new characterizations of these curves in Lorentzian 5-space L^sup 5^. ...PUBLICATION ABSTRACT
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