arXiv subjects
Publications and source records attributed to Jimmy Dillies.
We answer a question of Simental by providing a combinatorial interpretation of a formula which generalizes rational Catalan numbers and which appears in the study of Springer fibers. We provide an interpretation in terms of binary necklaces as well as anchored Dyck paths.
We consider some known and some new properties of the family of polynomials introduced by Ted Suffridge in 1969. We begin by giving a brief overview of their extremal properties in classic and more recent work. We also give a compact form for Suffridge polynomials which matches a general pattern discovered by Brandt. Our approach allows us to find the coefficients which Brandt's result was not giving explicitly. This new presentation provides us the tools to obtain an estimate of the rate of approximation of the generalized Koebe functions by univalent polynomials. Furthermore, we consider the presentation of Suffridge polynomials in Robertson's form and find the suiting Robertson measure. This suggests a new way to approximate step functions by continuous monotonic ones. We then study the lack of robustness of the univalency of these polynomials and suggest a new family of polynomials for which we conjecture the univalency of a subclass. Namely, we prove the quite surprising fact that by extending the family by letting the discrete argument in the polynomial coefficients become continuous one does not increase the set of univalent polynomials. Only the initial polynomials remain univalent. In this new one parameter family generalizing the Suffridge polynomials, it is remarkable that the Suffridge polynomials are already extremal as they correspond to the choice of the parameter set to 1; moreover the complex Fejér polynomials correspond the choice of the parameter set to 0, and the choice of the parameter set to -1 corresponds the polynomials $z+(z^N/N)$. Remarkably, computer simulations seem to clearly indicate that the image of the unit disc under these new polynomial mapping is a simply-connected region bounded by a simple curve. This justifies the conjectural univalency of these polynomials for the whole range of the parameters.
D. Dimitrov has posed the problem of finding polynomials that set the sharpness of the Koebe Quarter Theorem for polynomials and asked whether Suffridge polynomials are optimal. We disprove Dimitrov's conjecture for polynomials of degree 3, 4, 5 and 6. For polynomials of degree 1 and 2 the conjecture is obviously true. On the way we introduce a new family of polynomials that allows us to state a conjecture about the value of the Koebe radius for polynomials of a specific degree.
We give a short proof that the quartic polynomial $f(z)=\frac 1 6 z^{4} + \frac 2 3 z^{3} + \frac 7 6 z^{2} + z$ is univalent.
We propose an approach to engineer quartic metamaterials starting from the desired photonic states. We apply our method to the design of the high-k asymptotics of metamaterials, extreme non-reciprocity and complex bi-anisotropic media.
We exhibit an example of a K3 surface of Picard rank $14$ with a non-symplectic automorphism of order $16$ which fixes a rational curve and $10$ isolated points. This settles the existence problem for the last case of Al Tabbaa, Sarti and Taki's classification.
We construct a moduli space of four colorings on planar cubic graphs. More precisely, we introduce the notion of weak Hamiltonian, a generalization of Hamiltonian cycles, and relate it to 4-colorings. Weak Hamiltonians have a form of deformation, which we call mutation, which gives them a graph structure, the Weak Hamiltonian graph. This graph encodes the different colorings as 3 vertex cliques. Identifying vertices on these cliques, we obtain a new graph, the chromatic graph, whose vertices are exactly the colorings of the original graph. Also, this construction gives a heuristic argument on why 4 colors are sufficient to color planar maps.
C. Voisin and C. Borcea have constructed mirror pairs of families of Calabi-Yau threefolds by taking the quotient of the product of an elliptic curve with a K3 surface endowed with a non-symplectic involution. In this paper, we generalize the construction of Borcea and Voisin to any prime order and build three and four dimensional Calabi-Yau orbifolds. We classify the topological types that are obtained and show that, in dimension 4, orbifolds built with an involution admit a crepant resolution and come in topological mirror pairs. We show that for odd primes, there are generically no minimal resolutions and the mirror pairing is lost.
We classify primitive non-symplectic automorphisms of order 6 on K3 surfaces. We show how their study can be reduced to the study of non-symplectic automorphisms of order 3 and to a local analysis of the fixed loci. In particular, we determine the possible fixed loci and show that when the Picard lattice is fixed, K3 surfaces come in mirror pairs.
We classify orbifolds obtained by taking the quotient of a three tori by abelian extensions of Z/n x Z/n automorphisms, where each torus has a multiplicative Z/n action (n=3,4 or 6). This 'completes' the classification of orbifolds of the above type initiated by Donagi and Faraggi (hep-th/0403272) and, Donagi and Wendland in the cases n=2.