Realization spaces of 4-polytopes are universal

Let $P\subset\R^d$ be a $d$-dimensional polytope. The {\em realization space} of~$P$ is the space of all polytopes $P'\subset\R^d$ that are combinatorially equivalent to~$P$, modulo affine transformations. We report on work by the first author, which shows that realization spaces of \mbox{4-dimensional} polytopes can be ``arbitrarily bad'': namely, for every primary semialgebraic set~$V$ defined over~$\Z$, there is a $4$-polytope $P(V)$ whose realization space is ``stably equivalent'' to~$V$. This implies that the realization space of a $4$-polytope can have the homotopy type of an arbitrary finite simplicial complex, and that all algebraic numbers are needed to realize all $4$- polytopes. The proof is constructive. These results sharply contrast the $3$-dimensional case, where realization spaces are contractible and all polytopes are realizable with integral coordinates (Steinitz's Theorem). No similar universality result was previously known in any fixed dimension.Comment: 10 page

Explicit determination of a 727-dimensional root space of the hyperbolic Lie algebra E10E_{10}

The 727-dimensional root space associated with the level-2 root \bLambda_1 of the hyperbolic Kac--Moody algebra $E_{10}$ is determined using a recently developed string theoretic approach to hyperbolic algebras. The explicit form of the basis reveals a complicated structure with transversal as well as longitudinal string states present.Comment: 12 pages, LaTeX 2

Missing Modules, the Gnome Lie Algebra, and E10E_{10}

We study the embedding of Kac-Moody algebras into Borcherds (or generalized Kac-Moody) algebras which can be explicitly realized as Lie algebras of physical states of some completely compactified bosonic string. The extra ``missing states'' can be decomposed into irreducible highest or lowest weight ``missing modules'' w.r.t. the relevant Kac-Moody subalgebra; the corresponding lowest weights are associated with imaginary simple roots whose multiplicities can be simply understood in terms of certain polarization states of the associated string. We analyse in detail two examples where the momentum lattice of the string is given by the unique even unimodular Lorentzian lattice $II_{1,1}$ or $II_{9,1}$, respectively. The former leads to the Borcherds algebra $g_{1,1}$, which we call ``gnome Lie algebra", with maximal Kac-Moody subalgebra $A_1$. By the use of the denominator formula a complete set of imaginary simple roots can be exhibited, whereas the DDF construction provides an explicit Lie algebra basis in terms of purely longitudinal states of the compactified string in two dimensions. The second example is the Borcherds algebra $g_{9,1}$, whose maximal Kac-Moody subalgebra is the hyperbolic algebra $E_{10}$. The imaginary simple roots at level 1, which give rise to irreducible lowest weight modules for $E_{10}$, can be completely characterized; furthermore, our explicit analysis of two non-trivial level-2 root spaces leads us to conjecture that these are in fact the only imaginary simple roots for $g_{9,1}$.Comment: 31 pages, LaTeX2e, AMS packages, PSTRICK

Vitamin E content of different animal products: Influence of animal nutrition

Zusammenfassung: In der vorliegenden Studie wurde der α-Tocopherolgehalt verschiedener Fleischstücke untersucht. Hähnchenschenkel hatte den höchsten α-Tocopherolgehalt, gefolgt von Hähnchenbrust und Schweineschulter (p Leber > Fettgewebe >Musculus longissimus dorsi. Die Nährstoffdichten betrugen 28.8, 7.3, 0.9 und 1.2 mg α-Tocopherol/MJ für Eigelb, Leber, Fettgewebe und Musculus longissimus dorsi der jeweiligen mit Vitamin E supplementierten Gruppe. Diese Ergebnisse zeigen, daß Fleisch, mit Ausnahme des Hähnchenschenkels, von Tieren mit supplementierten Diäten kein bedeutender Vitamin E-Lieferant ist. Hingegen wurde Eigelb durch fütterungsbedingte Modifikation zu einer guten Vitamin E-Quell

Hybrid CO₂-Ti:sapphire laser with tunable pulse duration for mid-infrared-pump terahertz-probe spectroscopy

We describe a mid-infrared pump – terahertz-probe setup based on a CO2 laser seeded with 10.6 μm wavelength pulses from an optical parametric amplifier, itself pumped by a Ti:Al2O3 laser. The output of the seeded CO2 laser produces high power pulses of nanosecond duration, which are synchronized to the femtosecond laser. These pulses can be tuned in pulse duration by slicing their front and back edges with semiconductor-plasma mirrors irradiated by replicas of the femtosecond seed laser pulses. Variable pulse lengths from 5 ps to 1.3 ns are achieved, and used in mid-infrared pump, terahertz-probe experiments with probe pulses generated and electro-optically sampled by the femtosecond laser

Catastrophic stress corrosion failure of Zr-base bulk metallic glass through hydrogen embrittlement

Zr-base bulk metallic glasses (BMG) are prone to pitting corrosion in halide containing solutions and also stress corrosion cracking (SCC) is often interpreted in this context. This work presents in situ SCC experiments on notched Zr52.5Cu17.9Ni14.6Al10Ti5 (at.%) BMG bars under 3-point bending in dilute NaCl solution. They show that pitting corrosion is only the initiating process. The pitted areas have a lower local corrosion potential and the reaction of Zr4+ to zirconyl ions in solution produces H+ that can be reduced and absorbed in the local acidic environment. So, hydrogen embrittlement causes the observed catastrophic failure and peculiar fracture surface characteristics. © 2019 The Author

BPS Saturation from Null Reduction

We show that any $d$-dimensional strictly stationary, asymptotically Minkowskian solution $(d\ge 4)$ of a null reduction of $d+1$-dimensional pure gravity must saturate the BPS bound provided that the KK vector field can be identified appropriately. We also argue that it is consistent with the field equations.Comment: 10 page

Damage-free single-mode transmission of deep-UV light in hollow-core PCF

Transmission of UV light with high beam quality and pointing stability is desirable for many experiments in atomic, molecular and optical physics. In particular, laser cooling and coherent manipulation of trapped ions with transitions in the UV require stable, single-mode light delivery. Transmitting even ~2 mW CW light at 280 nm through silica solid-core fibers has previously been found to cause transmission degradation after just a few hours due to optical damage. We show that photonic crystal fiber of the kagom\'e type can be used for effectively single-mode transmission with acceptable loss and bending sensitivity. No transmission degradation was observed even after >100 hours of operation with 15 mW CW input power. In addition it is shown that implementation of the fiber in a trapped ion experiment significantly increases the coherence times of the internal state transfer due to an increase in beam pointing stability

Effect of Build Orientation on the Microstructure, Mechanical and Corrosion Properties of a Biodegradable High Manganese Steel Processed by Laser Powder Bed Fusion

In the last decade, additive manufacturing technologies like laser powder bed fusion (LPBF) have emerged strongly. However, the process characteristics involving layer-wise build-up of the part and the occurring high, directional thermal gradient result in significant changes of the microstructure and the related properties compared to traditionally fabricated materials. This study presents the influence of the build direction (BD) on the microstructure and resulting properties of a novel austenitic Fe-30Mn-1C-0.02S alloy processed via LPBF. The fabricated samples display a {011} texture in BD which was detected by electron backscatter diffraction. Furthermore, isolated binding defects could be observed between the layers. Quasi-static tensile and compression tests displayed that the yield, ultimate tensile as well as the compressive yield strength are significantly higher for samples which were built with their longitudinal axis perpendicular to BD compared to their parallel counterparts. This was predominantly ascribed to the less severe effects of the sharp-edged binding defects loaded perpendicular to BD. Additionally, a change of the Young’s modulus in dependence of BD could be demonstrated, which is explained by the respective texture. Potentiodynamic polarization tests conducted in a simulated body fluid revealed only slight differences of the corrosion properties in dependence of the build design

Influence of isothermal omega precipitation aging on deformation mechanisms and mechanical properties of a β-type Ti-Nb alloy

In this study, the influence of ωiso precipitates on the active deformation mechanisms and the mechanical properties of the biomedical β-type Ti-40Nb alloy are revealed. Low temperature heat treatments (aging) at 573 K for durations up to 108.0 ks were carried out for a cold-rolled and recrystallized sample state. After an aging time of 3.6 ks the ωiso phase was determined by means of synchrotron XRD and the fraction and the crystallite size of ωiso increased progressively with increasing aging time. Due to the high intrinsic Young's modulus of the ωiso phase, the Young's modulus increased gradually with the aging time from 63 GPa, for the recrystallized reference condition, to values of 70 GPa (3.6 ks), 73 GPa (14.4 ks), 81 GPa (28.8 ks) and 96 GPa (108.0 ks). Depending on the aging time, also a change of the active deformation mechanisms occurred, resulting in significantly altered mechanical properties. For the single β-phase reference microstructure, stress-induced martensite (SIM) formation, {332} twinning and dislocation slip were observed under tensile loading, resulting in a low 0.2% proof stress of around 315 MPa but a high elongation at fracture of 26.2%. With increasing aging time, SIM formation and mechanical twinning are progressively hindered under tensile loading. SIM formation could not be detected for samples aged longer than 3.6 ks. The amount and thickness of deformation twins is clearly reduced with increasing aging time and for samples aged longer than 14.4 ks deformation twinning is completely suppressed. As a result of the changed deformation mechanisms and the increase of the critical stress for slip caused by ωiso, the 0.2% proof stress of the aged samples increased gradually from 410 MPa (3.6 ks) to around 910 MPa (108.0 ks). With regard to application as new bone implant material, a balanced ratio of a low Young's modulus of E = 73 GPa and higher 0.2% proof stress of 640 MPa was achieved after an aging time of 14.4 ks