Inductive freeness of Ziegler’s canonical multiderivations for restrictions of reflection arrangements

Supersolvable restrictions of reflection arrangements

The aim of this note is a classification of all nice and all inductively factored reflection arrangements. It turns out that apart from the supersolvable instances only the monomial groups $G(r,r,3)$ for $r \ge 3$ give rise to nice reflection arrangements. As a consequence of this and of the classification of all inductively free reflection arrangements from Hoge and Röhrle (2015) we deduce that the class of all inductively factored reflection arrangements coincides with the class of all supersolvable reflection arrangements. Moreover, we extend these classifications to hereditarily factored and hereditarily inductively factored reflection arrangements.

Suppose that $W$ is a finite, unitary, reflection group acting on the complex vector space $V$. Let ${\mathcal A} = {\mathcal A}(W)$ be the associated hyperplane arrangement of $W$. Terao has shown that each such reflection arrangement ${\mathcal A}$ is free. Let $L({\mathcal A})$ be the intersection lattice of ${\mathcal A}$. For a subspace $X$ in $L({\mathcal A})$ we have the restricted arrangement ${\mathcal A}^X$ in $X$ by means of restricting hyperplanes from ${\mathcal A}$ to $X$. In 1992, Orlik and Terao conjectured that each such restriction is again free. In this note we settle the outstanding cases confirming the conjecture. In 1992, Orlik and Terao also conjectured that every reflection arrangement is hereditarily inductively free. In contrast, this stronger conjecture is false however; we give two counterexamples.

Recursively free reflection arrangements

Based on reflection arrangement, two antenna polarization measurement techniques, the absolute polarization measurement method and the polarization‐transfer measurement method are presented with experimental results. In the absolute polarization measurement method or the system calibration, the polarization characteristics of transmitting and receiving dual‐polarization antennas used in the measurement system are determined using the two polarization‐isolated scattering calibrators. In the polarization‐transfer measurement method, the antenna being tested operates in either transmitting mode or receiving mode. Since the transmitting and receiving antennas are located side by side in the measurement system, the phase reference signal between two antennas can be implemented with a short cable.

On inductively free restrictions of reflection arrangements

The collection of reflecting hyperplanes of a finite Coxeter group is called a reflection arrangement and it appears in many subareas of combinatorics and representation theory. We focus on the problem of counting regions of reflection arrangements and their deformations. Inspired by the recent work of Bernardi, we show that the notion of moves and sketches can be used to provide a uniform and explicit bijection between regions of (the Catalan deformation of) a reflection arrangement and certain non-nesting partitions. We then use the exponential formula to describe a statistic on these partitions such that distribution is given by the coefficients of the characteristic polynomial. Finally, we consider a sub-arrangement of type C arrangement called the threshold arrangement and its Catalan and Shi deformations.

Suppose that W is a finite, unitary reflection group acting on the complex vector space V. Let 𝒜 = 𝒜(W) be the associated hyperplane arrangement of W. Terao [J. Fac. Sci. Univ. Tokyo 27 (1980), 293–320] has shown that each such reflection arrangement 𝒜 is free. There is the stronger notion of an inductively free arrangement. In 1992, Orlik and Terao [Arrangements of hyperplanes, Springer-Verlag, Berlin 1992, Conjecture 6.91] conjectured that each reflection arrangement is inductively free. It has been known for quite some time that the braid arrangement as well as the Coxeter arrangements of type B ℓ and type D ℓ are inductively free. Barakat and Cuntz [Adv. Math. 229 (2012), 691–709] completed this list only recently by showing that every Coxeter arrangement is inductively free. Nevertheless, Orlik and Terao's conjecture is false in general. In a paper which will appear in Tôhoku Math. J., we already gave two counterexamples to this conjecture among the exceptional complex reflection groups. In this paper we classify all inductively free reflection arrangements. In addition, we show that the notions of inductive freeness and that of hereditary inductive freeness coincide for reflection arrangements. As a consequence of our classification, we get an easy, purely combinatorial characterization of inductively free reflection arrangements 𝒜 in terms of exponents of the restrictions to any hyperplane of 𝒜.

A finite (pseudo-)reflection group G naturally gives rise to a hyperplane arrangement, i.e., its reflection arrangement. We show that G is reducible if and only if its reflection arrangement is reducible.

Inductive freeness of Ziegler's canonical multiderivations for reflection arrangements

AlGaN-Based flip-chip Ultraviolet Light-Emitting Diodes at 365 nm with epitaxial ITO transparent ohmic contact layers and Al reflective were fabricated. The epitaxial ITO thin film exhibits higher transmittance than that of sputter ITO at 365nm, which is 93.6% and 85%, respectively. The epitaxial ITO thin film is more suitable for 365nm UV-LED. And the reflectance of the ITO/Al layers is 81.2% at 365nm, much higher than that of the ITO/Ag layers, which is only 53.2% at 365nm. When the current injection is 350mA, the forward voltages are 3.43V and 4.05V for flip-chip UV-LED and conventional UV-LED, respectively. The forward voltages of flip-chip UV-LED is much lower than that of conventional UV-LED, because the series resistance (Rs) of the flip-chip UV-LED is 0.73 Ω, much lower than 2.98 Ω of conventional UV-LED. The flip-chip UV-LED with epitaxial ITO/Al reflective mirror and Symmetry Electrode arrangement is more suitable for high power application.

Differential cross sections for elastic scattering of the 59.54 keV γ-rays in elements with 22 ≤Z ≤92 have been measured over the angular range 10○–160○corresponding to the momentum transfer 0.4 ≤x ≤4.7 A-1. The measurements at forward and backward angles were performed using the 241Am radioactive point-source, target and the Ge detectors in the transmission and reflection arrangements, respectively. The measured differential scattering cross sections are compared with those based on the form-factor (FF) formalism and state-of-the-art S-matrix calculations to differentiate between their relative efficacies and to check angular-dependence of the anomalous scattering factors (ASF) incorporated as correction to the modified form-factor (MF). The S-matrix values exhibit agreement with the measured data at backward angles and differences ∼10% at forward angles. The scattering cross sections based on the MF including ASF’s are in general lower at various angles by up to 20% for medium- and high-Z elements. The observed deviations being higher at the forward angles infer possibility of angular-dependence of ASF’s.

The reflection arrangement of a Coxeter group is a well-known instance of a free hyperplane arrangement. In 2002, Terao showed that equipped with a constant multiplicity each such reflection arrangement gives rise to a free multiarrangement. In this note we show that this multiarrangment satisfies the stronger property of inductive freeness in case the Coxeter group is of type A.

Optical methods of stress analysis including photoelasticity, moirr, holographic and moire interferometry and caustics have extensively been used in fracture mechanics problems. Among them, the optical method of caustics is gaining ground in the determination of stress intensity factors in crack problems under static and dynamic loading. In this method [ 1,2], the area in the vicinity of the crack tip is illuminated by a collimated light and the reflected or transmitted rays from an envelope in space. When this envelope is cut by a reference plane, a highly illuminated curve, the caustic, is formed. By measuring a diameter of the caustic the stress intensity factor is determined. The caustic is the image of the circumference of a circle on the specimen centered at the crack tip, which is called the initial curve. In the application of the method care should be taken to ensure that the initial curve lies in the plane-stress region, so that the plane-stress assumption made in developing the evaluation equations of caustic is valid. The implication of this assumption regarding the selection of material, load level, specimen size and optical arrangement has been discussed by Konsta-Gdoutos and Gdoutos [3-5]. In the beginning, the method of caustics was applied to transparent specimens and later on its use was extended to metals by Theorcaris and Gdoutos [6,7]. The optical quality of the specimens used in the method of caustics should meet high standards. Specimens used in transmission arrangements must have constant thickness and density. On the other hand, specimens used in reflection arrangements must have an optically planar surface of high reflectivity at least in the neighborhood of the crack tip where the caustic is generated. In metal specimens mirrored surfaces are prepared by grinding, lapping and polishing their front surface. Preparation of mirror-like surfaces of specimens used in the method of caustics constitutes not only an elaborate task but difficulties arise in a number of cases. A characteristic example is bone, which due to its spongy nature cannot be adequately polished to the point that Snell's reflection law applies. Analogous problems are encountered in fiber or particulate composite materials. Another class of problems in which the method of caustics cannot be

In this chapter we collect the basic definitions and results involving the intersection lattice of a hyperplane arrangement. Then we explain the key induction technique called deletion-restriction and apply it to deduce the main properties of the characteristic polynomial and of the Poincare polynomial of an arrangement. These polynomials enter into Zaslavsky’s Theorem expressing the number of regions (resp. bounded regions) of the complement of a real arrangement. In this chapter we also introduce several important classes of hyperplane arrangements: the supersolvable arrangements, the graphic arrangements and the reflection arrangements.