Optimal software-implemented Itoh–Tsujii inversion for $$\mathbb {F}_{2^{m}}$$

Field inversion in \(\mathbb {F}_{2^{m}}\) dominates the cost of modern software implementations of certain elliptic curve cryptographic operations, such as point encoding/hashing into elliptic curves (Brown et al. in: Submission to NIST, 2008; Brown in: IACR Cryptology ePrint Archive 2008:12, 2008; Aranha et al. in: Cryptology ePrint Archive, Report 2014/486, 2014) Itoh–Tsujii inversion using a polynomial basis and precomputed table-based multi-squaring has been demonstrated to be highly effective for software implementations (Taverne et al. in: CHES 2011, 2011; Oliveira et al. in: J Cryptogr Eng 4(1):3–17, 2014; Aranha et al. in: Cryptology ePrint Archive, Report 2014/486, 2014), but the performance and memory use depend critically on the choice of addition chain and multi-squaring tables, which in prior work have been determined only by suboptimal ad-hoc methods and manual selection. We thoroughly investigated the performance/memory tradeoff for table-based linear transforms used for efficient multi-squaring. Based upon the results of that investigation, we devised a comprehensive cost model for Itoh–Tsujii inversion and a corresponding optimization procedure that is empirically fast and provably finds globally-optimal solutions. We tested this method on eight binary fields commonly used for elliptic curve cryptography; our method found lower-cost solutions than the ad-hoc methods used previously, and for the first time enables a principled exploration of the time/memory tradeoff of inversion implementations.     

Balanced numerical semigroups

We investigate the Frobenius number, genus, type, and minimal presentation of a class of numerical semigroups of embedding dimension 4 of the form \(S = \langle a_1, a_2, a_3, a_4 \rangle \) such that \(a_1 + a_4 = a_2 + a_3\). The investigation focuses on determining the Apery set of S with respect to the multiplicity.     

Characteristics of Second Graders' Mathematical Writing

This study compared the characteristics of second graders' mathematical writing between an intervention and comparison group. Two six‐week Project M2 units were implemented with students in the intervention group. The units position students to communicate in ways similar to mathematicians, including engaging in verbal discourse where they themselves make sense of the mathematics through discussion and debate, writing about their reasoning on an ongoing basis, and utilizing mathematical vocabulary while communicating in any medium. Students in the comparison group learned from the regular school curriculum. Students in both the intervention and comparison groups conveyed high and low levels of content knowledge as indicated in archived data from an open‐response end‐of‐the‐year assessment. A multivariate analysis of variance indicated several differences favoring the intervention group. Both the high‐ and low‐level intervention subgroups outperformed the comparison group in their ability to (a) provide reasoning, (b) attempt to use formal mathematical vocabulary, and (c) correctly use formal mathematical vocabulary in their writing. The low‐level intervention subgroup also outperformed the respective comparison subgroup in their use of (a) complete sentences and (b) linking words. There were no differences between groups in their attempt at writing and attempts at and usage of informal mathematical vocabulary.     

Preparation of a Series of 5-Methyl-3-(substituted)-[1,2,4]triazines

Procedures for the synthesis of thirty-six 5-methyl-3-(substituted)-[1,2,4]triazines have been described. These compounds were evaluated for antagonism at metabotropic glutamate receptor subtype 5 (mGluR5). Two compounds, 5b and 3c, were determined to be low micromolar inhibitors of mGluR5.     

Bile UPLC-MS fingerprinting and bile acid fluxes during human liver transplantation

Bile flow restoration is a crucial step in the recovery process post transplantation of the liver. Here, metabolic trajectories based on changes in bile secretion - a known marker of functionality - have been utilised as an approach for discovering bile fluxes during transplantation. A total of ten liver transplants were monitored and from these 68 bile samples from both donors and recipients were collected and analysed using ultra-performance LC-MS in combination with multivariate statistical analysis. Based on the principal component scores constructed from the total bile fingerprint, differentiation of the bile acid concentrations before and after transplantation was detected. A trend was also observed, by constructing metabolic trajectories, whereby the post-transplant profiles approached the position of pre-transplant profiles within 30-60?min of the restoration of bile secretion function. The ten major conjugated bile acid salts were measured and a significant increase in concentrations of taurocholic acid and taurochenodeoxycholic acid were seen after transplantation. In addition, the ratios of secondary bile acids detected in gall bladder and hepatic bile were measured before and after transplantation. This study suggests that bile acid ratios in the donor liver at the pre-transplant and post-transplant stage may be important and that profiling of secreted bile after transplantation may aid clinical assessment and progress post-transplantation.     

Fabrication of spiropyran-containing thin film sensors used for the simultaneous identification of multiple metal ions

In this article, a methacrylate-based spiropyran-containing copolymer was used as a colorimetric sensor to identify multiple metal ions simultaneously. Through UV-vis absorption spectroscopy, the relative binding affinity of merocyanine to each metal ion was investigated by displacement studies of a bound metal ion with a second metal ion of a higher binding affinity. We also show that because each metal ion gives rise to a distinct spectral response, partial least-squares discriminant analysis (PLS-DA) can be used to analyze the UV-vis absorbance spectra to identify the two metal ions that are present in solution at varying concentrations simply by dipping a coated polymer substrate into solution after irradiation. Partial least-squares regression analysis (PLS) was used to determine the metal ions in solution for several binary mixtures quantitatively. We also demonstrate that the quantitative determination depends on the relative binding preference of merocyanine to each metal ion.     

ToF-SIMS imaging and depth profiling of HeLa cells treated with bromodeoxyuridine

Time of flight secondary ion mass spectrometry 2D images and molecular depth profiles of human HeLa cells treated with bromodeoxyuridine (BrdU) were acquired in the dual beam mode (Bi(3) (+) analysis beam, C(60) (+) etching beam). Several preparation protocols were investigated and were compared to a simple wash-and-dry method. The feasibility of using C(60) to clean the samples prior to imaging with Bi was also investigated quantitatively by calibrating full depth profiles of the cells using atomic force microscopy. BrdU was used as a marker for the cell nucleus, facilitating identification and localization of sub-cellular features during depth profiling. Results show that C(60) can be used to remove the surface contamination and to access different layers within the cells for 2D imaging. For a 1 nA, 10 keV C(60) (+) beam incident at 45° and rastered over a 500 × 500 μm(2) area, ~1 nm of biological material was sputtered every second. Our results show that HeLa cells were completely removed after etching with 1.3×10(15) C(60) (+) ions per cm(2), giving an average etching rate of 3.9 nm for every 10(13) C(60) per cm(2) at 10 keV and 45° incidence.     

Structural and electric field effects of ions in aqueous nanodrops

Ensemble infrared photodissociation (IRPD) spectra in the hydrogen stretch region (～2950-3800 cm(-1)) are reported for M(H(2)O)(35-37), with M = I(-), Cl(-), HCO(3)(-), OH(-), tetrabutyl-, tetrapropyl-, and tetramethylammonium, Cs(+), Na(+), Li(+), H(+), Ba(2+), Ca(2+), Co(2+), Mg(2+), La(3+), and Tm(3+), at 133 K. A single, broad feature is observed in the bonded-OH region of the spectra that indicates that the water network in these clusters is bulk-like and likely resembles liquid water more strongly than ice. The free-OH region for all of these clusters is dominated by peaks corresponding to water molecules that accept two and donate one hydrogen bond (AAD water molecules), indicating that AAD water molecules are more abundant at the surface of these ions than AD water molecules. A-only water molecules are present in significant abundance only for the trivalent metal cations. The frequency of the AAD free-OH stretch band shifts nearly linearly with the charge state of the ion, consistent with a Stark shift attributable to the ion's electric field. From these data, a frequency range of 3704.9-3709.7 cm(-1) is extrapolated for the free-OH of AAD water molecules at the (uncharged) bulk liquid water surface, consistent with sum-frequency generation spectroscopy experiments. Differences in both the bonded- and the free-OH regions of the spectra for these ions are attributable to ion-induced patterning of the water network that extends to the surface of the clusters, which includes water molecules in the third and fourth solvation shells; that is, these ions pattern water molecules at long distance to various extents. These spectra are simulated using two different electrostatic models previously used to calculate OH-stretch spectra of bulk water and aqueous solutions and parametrized for bonded-OH frequencies. These models qualitatively reproduce a number of features in the experimental spectra, although it is evident that more sophisticated treatment of water molecule and ion polarizability and vibrational coupling is necessary for more quantitative comparisons.     

Activation of CO2 by a heterobimetallic Zr/Co complex

At room temperature, the early/late heterobimetallic complex Co((i)Pr(2)PNMes)(3)Zr(THF) has been shown to oxidatively add CO(2), generating (OC)Co((i)Pr(2)PNMes)(2)(μ-O)Zr((i)Pr(2)PNMes). This compound can be further reduced under varying conditions to generate either the Zr oxoanion (THF)(3)Na-O-Zr(MesNP(i)Pr(2))(3)Co(CO) or the Zr carbonate complex (THF)(4)Na(2)(CO(3))-Zr(MesNP(i)Pr(2))(3)Co(CO). Additionally, reactivity of the CO(2)-derived product has been observed with PhSiH(3) to generate the Co-hydride/Zr-siloxide product (OC)(H)Co((i)Pr(2)PNMes)(3)ZrOSiH(2)Ph.