The Structure of Total Reflection Orders in Affine Root Systems

Reflectable bases for affine reflection systems

Affine root systems and Dedekind's?-function

Previous article Next article Sign Variations of the Macdonald IdentitiesDennis StantonDennis Stantonhttps://doi.org/10.1137/0517103PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractSign variations are given for the Macdonald identities for root systems of small rank. Limiting cases of these identities give properties of the eta function. Two such formulas are explicitly given.[1] Richard Askey, Some basic hypergeometric extensions of integrals of Selberg and Andrews, SIAM J. Math. Anal., 11 (1980), 938–951 10.1137/0511084 82e:33002 0458.33002 LinkISIGoogle Scholar[2] F. Dyson, 1968, letter to L. Winquist, March Google Scholar[3] I. Macdonald, Affine root systems and Dedekind's $\eta$-function, Invent. Math., 15 (1972), 91–143 50:9996 0244.17005 CrossrefISIGoogle Scholar[4] I. G. Macdonald, Some conjectures for root systems, SIAM J. Math. Anal., 13 (1982), 988–1007 10.1137/0513070 84h:17006a 0498.17006 LinkISIGoogle Scholar[5] S. Milne, An elementary proof of the Macdonald identities for $A\sp {(1)}\sb l$, Adv. in Math., 57 (1985), 34–70 87e:17020 0586.33011 CrossrefISIGoogle Scholar[6] Lasse Winquist, An elementary proof of $p(11m+6)\equiv 0\,({\rm mod}\ 11)$, J. Combinatorial Theory, 6 (1969), 56–59 38:4434 0241.05006 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Leading coefficients of Morris type constant term identitiesAdvances in Applied Mathematics, Vol. 87 Cross Ref Logarithmic and Complex Constant Term Identities29 July 2013 Cross Ref A q-product Tutorial for a q-series Maple Package Cross Ref A New Proof of Winquist's IdentityJournal of Combinatorial Theory, Series A, Vol. 78, No. 2 Cross Ref Macdonald’s constant term conjectures for exceptional root systems1 January 1991 | Bulletin of the American Mathematical Society, Vol. 24, No. 2 Cross Ref A Stembridge-Stanton style elementary proof of the Habsieger-Kadell q-Morris identityDiscrete Mathematics, Vol. 79, No. 3 Cross Ref The crank of partitions mod 8,9 and 101 January 1990 | Transactions of the American Mathematical Society, Vol. 322, No. 1 Cross Ref An Elementary Approach to the Macdonald Identities Cross Ref A Unified Approach to Macdonald’s Root-System ConjecturesDoron Zeilberger17 July 2006 | SIAM Journal on Mathematical Analysis, Vol. 19, No. 4AbstractPDF (2886 KB)First layer formulas for characters of 𝑆𝐿(𝑛,𝐶)1 January 1987 | Transactions of the American Mathematical Society, Vol. 299, No. 1 Cross Ref Volume 17, Issue 6| 1986SIAM Journal on Mathematical Analysis History Submitted:17 December 1984Accepted:16 September 1985Published online:17 July 2006 InformationCopyright © 1986 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0517103Article page range:pp. 1454-1460ISSN (print):0036-1410ISSN (online):1095-7154Publisher:Society for Industrial and Applied Mathematics

The affine Toda field theory is studied as a 2+1-dimensional system. The third dimension appears as the discrete space dimension, corresponding to the simple roots in the $A_N$ affine root system, enumerated according to the cyclic order on the $A_N$ affine Dynkin diagram. We show that there exists a natural discretization of the affine Toda theory, where the equations of motion are invariant with respect to permutations of all discrete coordinates. The discrete evolution operator is constructed explicitly. The thermodynamic Bethe ansatz of the affine Toda system is studied in the limit $L,N\to\infty$. Some conjectures about the structure of the spectrum of the corresponding discrete models are stated.

Classification of concatenation measurement structures according to scale type

Short-range structural and chemical ordering in Si-rich chemicallydeposited a-Si1-xCx thin films have beeninvestigated via Raman scattering and the numerical modellingtechnique. Raman spectra have been presented over a wide frequencyrange including both Stokes and anti-Stokes scattering. Theinterpretation of the spectra is performed in terms of the wholedensity of vibrational states. In order to determine the latter,the structure of the a-Si1-xCx (x<0.5) system has beenmodelled and the corresponding dynamical properties have beencomputed in the harmonic approximation using the valence-force-fieldmodel. By integrating the Stokes and anti-Stokes, first-order, andmultiple-order processes, a fit of the experimental Raman spectrahas been achieved. As expected, our analysis shows a tendency tochemical ordering into a tetrahedrally coordinated network forSi-rich alloys. Nevertheless, a total chemical ordering is notachieved since homonuclear C-C bonds coexist with Si-Si and Si-Cones in these alloy compounds.

The Nekrasov-Okounkov formula gives an expression for the Fourier coefficients of the Euler functions as a sum of hook length products. This formula can be deduced from a specialization in a renormalization of the affine type $A$ Weyl denominator formula and the use of a polynomial argument. In this paper, we rephrase the renormalized Weyl-Kac denominator formula as a sum parametrized by affine Grassmannian elements. This naturally gives rise to the (dual) atomic length of the root system considered introduced by Chapelier-Laget and Gerber. We then provide an interpretation of this atomic length as the cardinality of some subsets of $n$-core partitions by using foldings of affine Dynkin diagrams. This interpretation does not permit the direct use of a polynomial argument for all affine root systems. We show that this obstruction can be overcome by computing the atomic length of certain families of integer partitions. Then we show how hook-length statistics on these partitions are connected with the Coxeter length on affine Grassmannian elements and Nekrasov-Okounkov type formulas.

A formal identity for affine root systems

The flow set with partial order is a mixed-integer set described by a budget on total flow and a partial order on the arcs that may carry positive flow. This set is a common substructure of resource allocation and scheduling problems with precedence constraints and robust network flow problems under demand/capacity uncertainty. We give a polyhedral analysis of the convex hull of the flow set with partial order. Unlike for the flow set without partial order, cover-type inequalities based on partial order structure are a function of a lifting sequence. We study the lifting sequences and describe structural results on the lifting coefficients for general and simpler special cases. We show that all lifting coefficients can be computed in polynomial time by solving maximum weight closure problems in general. For the special case of induced-minimal covers, we give a sequence-dependent characterization of the lifting coefficients. We prove, however, if the partial order is defined by an arborescence, then lifting is sequence-independent and all lifting coefficients can be computed in linear time. Moreover, if the partial order is defined by a path (total order), then the coefficients can be expressed explicitly. We also give a complete polyhedral description of the flow set with partial order for the polynomially-solvable total order case. We show that finding an optimal lifting order for a given induced-minimal cover and a given fractional solution is a submodular optimization problem, which is solved greedily. Finally, we present preliminary computational results with a cutting-plane algorithm based on the lifting and separation results.

In statistical decision theory involving a single decision-maker, one says that an information structure is better than another one if for any cost function involving a hidden state variable and an action variable which is restricted to be only a function of some measurement, the solution value under the former is not worse than the value under the latter. For finite probability spaces, Blackwell's celebrated theorem on comparison of information structures leads to a complete characterization on when one information structure is better than another. For stochastic games with incomplete information, due to the presence of competition among decision makers, in general such an ordering is not possible since additional information can lead to equilibria perturbations with positive or negative values to a player. However, for zero-sum games in a finite probability space, Peski introduced a complete characterization of ordering of information structures. In this paper, we obtain an infinite dimensional (standard Borel) generalization of Peski's result. A corollary of our analysis is that more information cannot hurt a decision maker taking part in a zero-sum game in standard Borel spaces. During our analysis, we establish two novel supporting results: (i) a refined existence result for equilibria in zero-sum games with incomplete information when compared with the prior literature and ii) a partial converse to Blackwell's ordering of information structures in the standard Borel space setup.

It was about 125 years ago that the light bulb was commercialized by Thomas Edison. No doubt a brilliant invention at the time, today its low power conversion efficiency is one of the reasons why lighting in the western world has such high energy consumption. Thus, the potential for saving energy is enormous in this area. The introduction of halogen, discharge and fluorescent lamps has lead to certain efficiency improvements, however more than half of the energy is still lost as heat. Light-emitting diodes (LEDs) are very promising candidates for high efficiency light sources, with modern devices showing internal quantum efficiencies of virtually 100 %. However, due to the high refractive index of the commonly used semiconductor materials it is very difficult to have a large extraction efficiency; in a standard cubic geometry most of the internally emitted light is trapped inside the device due to total internal reflection. Several methods have been developed in order to circumvent this problem, either by optimizing the device geometry in order to increase the escape cone or by incorporating a resonant structure in order to force the internal emission into the existing escape cone. The latter approach is called microcavity LED (MCLED) or resonant cavity LED (RCLED). In a MCLED the spontaneous internal emission is controlled by placing the emitter inside an optical cavity with a thickness of the order of its emitting wavelength. The resulting interference effects increase the part of the emission that can be extracted. Contrary to the other approaches this is possible without changing the device geometry and thus without additional costly back-end processing steps. The control of the farfield radiation pattern makes these devices particularly interesting for high brightness applications, which demand highly directional emitters, such as for printing, bar code reading, large area displays and optical communication. The extraction efficiency of a MCLED is inversely proportional to the effective cavity length. An ideal cavity, allowing an extraction efficiency close to unity, consists of a low refractive index material and has an optical length of λ/2. In contrast to this, to obtain high internal quantum efficiencies it is necessary to use high index cavities with an optical length of at least λ. It should be noted, that the large penetration depth of the optical field in the semiconductor-based distributed Bragg reflectors (DBRs) leads to a significant increase of the effective cavity length and thus further reduces the achievable extraction efficiencies. In this thesis novel concepts to reduce effective cavity lengths and therefore increase extraction efficiencies are implemented into standard MCLED structures. The phaseshift cavity principle whilst maintaining the electrical properties of a standard A cavity achieves optical properties approaching that of a λ/2 cavity. The use of AlOx instead of AlAs as the, low refractive index component in the DBRs leads to smaller penetration depths and a concomitant reduction of the effective cavity length. A similar effect can be obtained by combining a resonant cavity with a thin-film structure. Thanks to these design improvements, the external quantum efficiency of different types of MCLEDs was increased. Near infrared emitting InGaAs/GaAs MCLEDs including a phase-shift cavity were realized, as both bottom and top emitting structures. The external quantum efficiencies achieved for emission into air were 18 and 19 %, respectively. With the additional incorporation of an oxide based bottom DBR, the efficiency of top emitting near infrared MCLEDs was further increased to 28 %. Red emitting AlGaInP-based structures are not compatible with the phase-shift cavity principle. However the beneficial effect of the implementation of an oxide DBR is greater at their wavelength rather than in the near infrared. Thus with preliminary red emitting GaInP/AlGaInP MCLEDs containing a bottom oxide DBR external quantum efficiencies of 12% could be achieved. Unfortunately, the incorporation of an oxide DBR significantly complicates the device design and the device fabrication. These problems can be avoided by combining the resonant cavity with a thin-film structure instead. Initial non-optimized red emitting thin-film MCLEDs were realized by OSRAM Opto Semiconductors and characterized in this work. They show external quantum efficiencies of 23 % and 18 % with and without encapsulation, respectively. It is assumed that a significant fraction of the high external quantum efficiency is due to a strong photon recycling effect in these devices. Simulations presented in this thesis show that the theoretical limits for the MCLEDs discussed above are slightly higher than the values obtained, encouraging further device optimization. The thin-film MCLEDs seem to hold the biggest potential for high efficiency emission from MCLEDs, independent of the wavelength range of emission.

Decision making alternatives are sometimes incomparable with each other and violate the completeness axiom of the rational behavior axiom of von Neumann-Morgenstern. Lattice order theory has therefore been used to deal with the incomparability issue among alternatives. This paper proves that the incomparable relation of the set of alternatives in same level under lattice order structure is equal to equivalence relation. The alternatives set with lattice ordered structure is classified as subsets of alternatives in terms of their corresponding location layers. The subsets make up a chain structure under the preference relations of equivalence classes, which realizes the chain-forming of preference structure, and transforms the lattice order structure to total order or asymmetric weak order structure. A new decision making method based on lattice-ordered preference structure is proposed and the proposed method is efficient to make decisions for the incomparable paired alternatives in the decision making alternatives set.

The characterization of in-place material properties is important for quality control and condition assessment of the built infrastructure. Although various methods have been developed to characterize structural materials in situ, many suffer limitations and cannot provide complete or desired characterization, especially for inhomogeneous and complex materials such as concrete and rock. Recent advances in machine learning and artificial neural networks (ANN) can help address these limitations. In particular, physics-informed neural networks (PINN) portend notable advantages over traditional physics-based or purely data-driven approaches. PINN is a particular form of ANN, where physics-based equations are embedded within an ANN structure in order to regularize the outputs during the training process. This paper reviews the fundamentals of PINN, notes its differences from traditional ANN, and reviews applications of PINN for selected material characterization tasks. A specific application example is presented where mechanical wave propagation data are used to characterize in-place material properties. Ultrasonic data are obtained from experiments on long rod-shaped mortar and glass samples; PINN is applied to these data to extract inhomogeneous wave velocity data, which can indicate mechanical material property variations with respect to length.

Identical marker alleles in Podolic cattle ( Bos taurus) and Indian zebu ( Bos indicus)

A stable, convenient thermal pyrolysis process was demonstrated on a large scale pilot plant. The process successfully converted high viscosity copolymer atactic polypropylene to predominantly liquid fuels which could be burned in commercial burners. Energy yield of the process was very high - in excess of 93% including gas phase heating value. Design and operating data were obtained to permit design of a commercial size atactic conversion plant. Atactic polypropylene can be cracked at temperatures around 850/sup 0/F and residence time of 5 minutes. The viscosity of the cracked product increases with decrease in time/temperature. A majority of the pyrolysis was carried out at a pressure of 50 psig. Thermal cracking of atactic polypropylene is seen to result in sigificant coke formation (0.4% to 0.8% on a weight of feed basis) although the coke levels were of an order of magnitude lower than those obtained during catalytic cracking. The discrepancy between batch and continuous test data can be atrributed to lowered heat transfer and diffusion rates. Oxidative pyrolysis is not seen as a viable commercial alternative due to a significant amount of water formation. However, introduction of controlled quantities of oxygen at lower temperatures to affect change in feedstock viscosity could be considered. It is essential to have a complete characterization of the polymer composition and structure in order to obtain useful and duplicable data because the pyrolysis products and probably the pyrolysis kinetics are affected by introduction of abnormalities into the polymer structure during polymerization. The polymer products from continuous testing contained an olefinic content of 80% or higher. This suggests that the pyrolysis products be investigated for use as olefinic raw materials. Catalytic cracking does not seem to result in any advantage over the Thermal Cracking process in terms of reaction rates or temperature of operation.