Signs in the 𝑐𝑑-index of Eulerian partially ordered sets

Bayer, Margaret M.

doi:10.1090/s0002-9939-00-05831-7

Cited by 4 publications

(3 citation statements)

References 13 publications

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“…However, the thermal expansion coefficients of YSZ and Sc 2 O 3 are both in the range of 10 −5 K −1 [58, 59]. A temperature change of 500 °C to the annealing temperature will only change the strain state in the films (additional to the mismatch induced strain) by less than 1%.…”

Section: Formal Considerationsmentioning

confidence: 99%

¹⁸O-tracer diffusion along nanoscaled Sc₂O₃/yttria stabilized zirconia (YSZ) multilayers: on the influence of strain

Aydin

Korte

Janek

2013

Science and Technology of Advanced Materials

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The oxygen tracer diffusion coefficient describing transport along nano-/microscaled YSZ/Sc2O3 multilayers as a function of the thickness of the ion-conducting YSZ layers has been measured by isotope exchange depth profiling (IEDP), using secondary ion mass spectrometry (SIMS). The multilayer samples were prepared by pulsed laser deposition (PLD) on (0001) Al2O3 single crystalline substrates. The values for the oxygen tracer diffusion coefficient were analyzed as a combination of contributions from bulk and interface contributions and compared with results from YSZ/Y2O3-multilayers with similar microstructure. Using the Nernst–Einstein equation as the relation between diffusivity and electrical conductivity we find very good agreement between conductivity and diffusion data, and we exclude substantial electronic conductivity in the multilayers. The effect of hetero-interface transport can be well explained by a simple interface strain model. As the multilayer samples consist of columnar film crystallites with a defined interface structure and texture, we also discuss the influence of this particular microstructure on the interfacial strain.

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Section: Formal Considerationsmentioning

confidence: 99%

¹⁸O-tracer diffusion along nanoscaled Sc₂O₃/yttria stabilized zirconia (YSZ) multilayers: on the influence of strain

Aydin

Korte

Janek

2013

Science and Technology of Advanced Materials

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“…Stanley [76] noted that it is sometimes useful to write the cd-index as a polynomial in c and e 2 , where e = a − b and thus e 2 = c 2 − 2d. Purtill [67] showed that if P is a convex polytope, then Φ P (c, d) can be written as a polynomial in the noncommuting variables c, d, and −e 2 = 2d − c 2 with nonnegative coefficients.…”

Section: Conjecture 23mentioning

confidence: 99%

“…In the most general case, Bayer [2] determined which cd coefficients are bounded for all Eulerian posets. Theorem 3.4…”

Section: Nonnegativitymentioning

confidence: 99%

The $cd$-Index: A Survey

Bayer

2019

Preprint

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This is a survey of the cd-index of Eulerian partially ordered sets. The cd-index is an encoding of the numbers of chains, specified by ranks, in the poset. It is the most efficient such encoding, incorporating all the affine relations on the flag numbers of Eulerian posets. Eulerian posets include the face posets of regular CW spheres (in particular, of convex polytopes), intervals in the Bruhat order on Coxeter groups, and the lattices of regions of oriented matroids. The paper discusses inequalities on the cd-index, connections with other combinatorial parameters, computation, and algebraic approaches.

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Eulerian abstract polytopes

Hartley¹

2011

Aequat. Math.

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The classical theory of regular convex polytopes has inspired many combinatorial analogues. In this article, we examine two of them, the eulerian posets and the abstract regular polytopes, and see what the overlap between the concepts is. It is shown that a section regular polytope is eulerian if and only if it is spherical, or it has even rank and is locally spherical. Equivelar polytopes of rank less than 4 are eulerian, and some progress is made towards a characterisation of equivelar eulerian posets in higher rank. In particular, necessary conditions are given for an equivelar quotient of a cube or a torus to be eulerian. Mathematics Subject Classification (2000). 51M20, 20F65, 52B15.

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Signs in the 𝑐𝑑-index of Eulerian partially ordered sets

Cited by 4 publications

References 13 publications

¹⁸O-tracer diffusion along nanoscaled Sc₂O₃/yttria stabilized zirconia (YSZ) multilayers: on the influence of strain

¹⁸O-tracer diffusion along nanoscaled Sc₂O₃/yttria stabilized zirconia (YSZ) multilayers: on the influence of strain

The $cd$-Index: A Survey

Eulerian abstract polytopes

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Signs in the 𝑐𝑑-index of Eulerian partially ordered sets

Cited by 4 publications

References 13 publications

18O-tracer diffusion along nanoscaled Sc2O3/yttria stabilized zirconia (YSZ) multilayers: on the influence of strain

18O-tracer diffusion along nanoscaled Sc2O3/yttria stabilized zirconia (YSZ) multilayers: on the influence of strain

The $cd$-Index: A Survey

Eulerian abstract polytopes

Contact Info

Product

Resources

About

¹⁸O-tracer diffusion along nanoscaled Sc₂O₃/yttria stabilized zirconia (YSZ) multilayers: on the influence of strain

¹⁸O-tracer diffusion along nanoscaled Sc₂O₃/yttria stabilized zirconia (YSZ) multilayers: on the influence of strain