638 research outputs found
Postnikov invariants of crossed complexes
No full textAbstractWe determine the Postnikov tower and Postnikov invariants of a crossed complex in a purely algebraic way. Using the fact that crossed complexes are homotopy types for filtered spaces, we use the above “algebraically defined” Postnikov tower and Postnikov invariants to obtain from them those of filtered spaces. We argue that a similar “purely algebraic” approach to Postnikov invariants may also be used in other categories of spaces
Postnikov--Stanley polynomials are Lorentzian
No full textPostnikov--Stanley polynomials are a generalization of skew dual Schubert polynomials to the setting of arbitrary Weyl groups. We prove that Postnikov--Stanley polynomials are Lorentzian by showing that they are degree polynomials of Richardson varieties. Our result yields an interesting class of Lorentzian polynomials related to the geometry of Richardson varieties, generalizes the result that dual Schubert polynomials are Lorentzian (Huh--Matherne--Mészáros--St. Dizier 2022), and resolves the conjecture that Postnikov--Stanley polynomials have M-convex support (An--Tung--Zhang 2024).10 pages, 1 figur
Self-homotopy equivalences of Postnikov conjugates
No full textThe purpose of this note is to generalize a result of Wilkerson [4] and show that the self homotopy equivalences of the Postnikov approximations of a space X determine, in a rather simple manner, simultaneously the self homotopy equivalences of X and the self homotopy equivalences of the Postnikov conjugates of X (i.e. the spaces with the same Postnikov approximations as X).</p
Postnikov invariants of H-spaces
No full textIt is known that the order of all Postnikov k-invariants of an H-space of finite type is finite. This paper establishes the finiteness of the order of the k-invariants of X in dimensions m ≤ 2n if X is an (n-1)-connected H-space which is not necessarily of finite type (n ≥ 1). Similar results hold more generally for higher k-invariants if X is an iterated loop space. Moreover, we provide in all cases explicit universal upper bounds for the order of the k-invariants of X
Diaryliodonium Tetracyanidometallates Self-Assemble into Halogen-Bonded Square-Like Arrays
No full textTwo diphenyliodonium tetracyanidometallates, [Ph2I](2)[M(CN)(4)] (M = Ni and Pd), were prepared through anion metathesis. Their X-ray structural analyses show that the structure-defining contact for both crystals is the charge-assisted I center dot center dot center dot N halogen bond (HaB) formed between the I atom of the iodonium cations and the N atoms of the CN- ligands. These HaBs assemble the bidentate and 90 degrees-orienting HaB donor Ph2I+ and the tetradentate, square planar, and 90/180 degrees-orienting HaB acceptors [M(CN)(4)](2-) into supramolecular rectangles, which further assemble into infinite chains by sharing the vertexes occupied by the [M(CN)(4)](2-) anions. The noncovalent nature of these contacts was confirmed by density functional theory calculations (M06/def2-TZVP) followed by combined topological analysis of the electron density distribution in the quantum theory of the atoms-in-molecules approach and noncovalent interaction analysis. The philicities of the HaB partners were further verified by the analysis of electron localization function projections, electron density/electrostatic potential profiles along the I center dot center dot center dot N bond paths, natural bond orbital analysis, and the natural population analysis or atoms-in-molecules charge sums in model systems
On the synthesis of fixed order stabilizing controllers
No full textIn this dissertation, we consider two problems concerning the synthesis of fixed order
controllers for Single Input, Single Output systems. The first problem deals with the
synthesis of absolutely stabilizing fixed order controllers for Lure-Postnikov systems. The
second problem deals with the synthesis of fixed order stabilizing controllers directly from
the empirical frequency response data and from some coarse information of the plant.
Lure-Postnikov systems are frequently encountered in mechanical engineering applications.
Analytical tools for synthesizing stabilizing fixed structure controllers, such as the
PID controllers examining the absolute stability of Lure-Postnikov systems, have recently
been studied in the literature. However, tools for synthesizing controllers of arbitrary order
have not been studied yet. We propose a systematic method for synthesizing absolutely
stabilizing controllers of arbitrary order for the Lure-Postnikov systems. Our approach is
based on recent results in the literature on approximation of the set of stabilizing controller
parameters that render a family of real and complex polynomials Hurwitz. We provide an
example of a robotic system to illustrate the procedure developed.
Exact analytical models of plants may not be readily available for controller design.
The current approach is to synthesize controllers through the identification of the analytical
model of the plant from empirical frequency response data. In this dissertation, we
depart from this conventional approach. We seek to synthesize controllers directly (i.e.
without resort to identification) from the empirical frequency response data of the plant and coarse information about it. The coarse information required is the number of nonminimum
phase zeros of the plant(or the number of poles of the plant with positive real
parts) and the frequency range beyond which the phase response of the LTI plant does not
change appreciably and the amplitude response goes to zero. We also assume that the LTI
plant does not have purely imaginary zeros or poles. The method of synthesizing stabilizing
controllers involves the use of generalized Hermite-Biehler theorem for counting the
roots of rational functions and the use of recently developed Sum-of-Squares techniques
for checking the nonnegativity of a polynomial in an interval through the Markov-Lucaks
theorem. The method does not require an explicit analytical model of the plant that must
be stabilized or the order of the plant, rather, it only requires the empirical frequency response
data of the plant. The method also allows for measurement errors in the frequency
response of the plant. We illustrate the developed procedure with an example. Finally, we
extended the technique to the synthesis of controllers of arbitrary order that also guarantee
performance specifications such as the phase margin and gain margin
Nonparametric study of the evolution of the cosmological equation of state with SNeIa, BAO, and high-redshift GRBs
No full textWe study the dark energy equation of state as a function of redshift in a nonparametric way, without imposing any a priori w (z) (ratio of pressure over energy density) functional form. As a check of the method, we test our scheme through the use of synthetic data sets produced from different input cosmological models that have the same relative errors and redshift distribution as the real data. Using the luminosity-time correlation for gamma-ray burst (GRB) X-ray afterglows (the Dainotti et al. correlation), we are able to utilize GRB samples from the Swift satellite as probes of the expansion history of the universe out to z 10. Within the assumption of a flat Friedmann-Lemaître-Robertson-Walker universe and combining supernovae type Ia (SNeIa) data with baryonic acoustic oscillation constraints, the resulting maximum likelihood solutions are close to a constant w = –1. If one imposes the restriction of a constant w , we obtain w = -0.99 0.06 (consistent with a cosmological constant) with the present-day Hubble constant as and density parameter as , while nonparametric w (z) solutions give us a probability map that is centered at and . Our chosen GRB data sample with a full correlation matrix allows us to estimate the amount, as well as quality (errors), of data needed to constrain w (z) in the redshift range extending an order of magnitude beyond the farthest SNeIa measured
Postnikov factorizations at infinity
No full textAbstractWe have developed Postnikov sections for Brown–Grossman homotopy groups and for Steenrod homotopy groups in the category of exterior spaces, which is an extension of the proper category. The homotopy fibre of a fibration in the factorization associated with Brown–Grossman groups is an Eilenberg–Mac Lane exterior space for this type of groups and it has two non-trivial consecutive Steenrod homotopy groups. For a space which is first countable at infinity, one of these groups is given by the inverse limit of the homotopy groups of the neighbourhoods at infinity, the other group is isomorphic to the first derived of the inverse limit of this system of groups. In the factorization associated with Steenrod groups the homotopy fibre is an Eilenberg–Mac Lane exterior space for this type of groups and it has two non-trivial consecutive Brown–Grossman homotopy groups. We also obtain a mix factorization containing both kinds of previous factorizations and having homotopy fibres which are Eilenberg–Mac Lane exterior spaces for both kinds of groups.Given a compact metric space embedded in the Hilbert cube, its open neighbourhoods provide the Hilbert cube the structure of an exterior space and the homotopy fibres of the factorizations above are Eilenberg–Mac Lane exterior spaces with respect to inward (or approaching) Quigley groups
Exchange interactions and magnetic anisotropy in the ``Ni<span class='mathrm'><sub>4</sub></span>'' magnetic molecule
No full textPostnikov A, Brüger M, Schnack J. Exchange interactions and magnetic anisotropy in the ``Ni<sub>4</sub>'' magnetic molecule. Phase Transitions. 2005;78(1-3):47-59
(k,m)-Catalan numbers and hook length polynomials for plane trees
No full textAbstractMotivated by a formula of A. Postnikov relating binary trees, we define the hook length polynomials for m-ary trees and plane forests, and show that these polynomials have a simple binomial expression. An integer value of this expression is Ck,m(n)=1mn+1(mn+1)kn, which we call the (k,m)-Catalan number. For proving the hook length formulas, we also introduce a combinatorial family, (k,m)-ary trees, which are counted by the (k,m)-Catalan numbers
- …
