Fast Constructive Recognition of a Black Box Group Isomorphic toSnorAnusing Goldbach’s Conjecture

The well-known Goldbach Conjecture (GC) states that any sufficiently large even number can be represented as a sum of two odd primes. Although not yet demonstrated, it has been checked for integers up to 10¹⁴. Using two stronger versions of the conjecture, we offer a simple and fast method for recognition of a gray box group G known to be isomorphic to S_n(or A_n) with knownn ≥  20, i.e. for construction1of an isomorphism from G toS_n (or A_n). Correctness and rigorous worst case complexity estimates rely heavily on the conjectures, and yield times of O([ρ + ν + μ ] n log²n) or O([ ρ + ν + μ ] n logn / loglog n) depending on which of the stronger versions of the GC is assumed to hold. Here,ρ is the complexity of generating a uniform random element of G, ν is the complexity of finding the order of a group element in G, and μ is the time necessary for group multiplication in G. Rigorous lower bound and probabilistic approach to the time complexity of the algorithm are discussed in the Appendix.     

Approximate Solutions of Polynomial Equations

In this paper, we introduce “approximate solutions" to solve the following problem: given a polynomial F(x, y) over Q, where x represents an n -tuple of variables, can we find all the polynomials G(x) such that F(x, G(x)) is identically equal to a constant c in Q ? We have the following: let F(x, y) be a polynomial over Q and the degree of y in F(x, y) be n. Either there is a unique polynomial g(x)  ∈ Q [ x ], with its constant term equal to 0, such that F(x, y)  = ∑_j = 0ⁿc_j(y − g(x))^jfor some rational numbers c_j, hence, F(x, g(x)  + a)  ∈ Q for all a ∈ Q, or there are at most t distinct polynomials g₁(x),⋯ , g_t(x), t ≤ n, such that F(x, g_i(x))  ∈ Q for 1  ≤ i ≤ t. Suppose that F(x, y) is a polynomial of two variables. The polynomial g(x) for the first case, or g₁(x),⋯ , g_t(x) for the second case, are approximate solutions of F(x, y), respectively. There is also a polynomial time algorithm to find all of these approximate solutions. We then use Kronecker’s substitution to solve the case of F(x, y).     

Bounds for the Roots of Lacunary Polynomials

A polynomial P(X)  = X^d + a_d − 1X^d − 1 + ⋯ is called lacunary when a_d − 1 =  0. We give bounds for the roots of such polynomials with complex coefficients. These bounds are much smaller than for general polynomials.     

The STO problem is NP-complete

We prove that the problem STO of deciding whether or not a finite setEof term equations is subject to occur-check is in NP.Eis subject to occur-check if the execution of the Martelli–Montanari unification algorithm gives for inputEa setE^′  ∪ {x = t}, wheret ≠ xandxappears int. Aptet al. (1994) proved that STO is NP-hard leaving the problem of NP-completeness open.     

Giant-grain silicon (GGS) and its application to stable thin-film transistor

We developed a giant-grain silicon (GGS) by Ni-mediated crystallization of amorphous silicon (a-Si) with a silicon-nitride (SiN_x) cap layer. Ni particles were sputtered onto the SiN_x/a-Si layer and then it was annealed at around 600 °C. The Ni diffuses through a SiN_x cap and then forms NiSi₂ crystallites in a-Si, which is able to induce crystallization. The grain size can be controlled from a few to 100 μm. The grain size can be increased with increasing the cap layer thickness or by decreasing the Ni density on the SiN_x. The p-channel GGS poly-Si TFT exhibited a field-effect mobility of 101 cm²/Vs and a threshold voltage of −3.6 V and is very stable under gate or hot carrier bias-stress. These superior performances may be due to the smooth surface of GGS poly-Si and solid-phase crystallization of a-Si.     

Tetrel bonds between PySiX3 and some nitrogenated bases: Hybridization,substitution, and cooperativity

Ab initio calculations have been performed to study the influence of hybridization, substitution, and cooperativity on the tetrel bond in the complexes of PySiX₃ (Py = pyridine and X = halogen). The tetrel bond becomes stronger in the order of p-PySiF₃⋯NCH(sp) < p-PySiF₃⋯NHCH₂(sp²) < p-PySiF₃⋯NH₃(sp³). The electron-donating group in the electron donor strengthens the tetrel bond, and this effect is mainly achieved through electrostatic and polarization interactions. Tetrel bonding displays cooperative effects with triel bonding and chalcogen bonding, characterized by shorter binding distances and greater electron densities. The cooperative effects between triel/chalcogen bond and tetrel bond have been analyzed by molecular electrostatic potentials and charge transfer. Energy decomposition indicates that many-body effects are mainly caused by polarization energy. The geometries of Si⋯N interaction and its applications in crystal materials have been characterized and evidenced by a CSD research.     

Cationic P⋯N interaction in XH3P+⋯NCY complexes (X = H,F, CN,NH2, OH; Y = H,Li, F,Cl) and its cooperativity with hydrogen/lithium/halogen bond

The geometries, interaction energies and bonding properties of cationic pnicogen bond (CPB) interactions are studied in binary XH₃P⁺⋯NCY (X = H, F, CN, NH₂, OH; Y = H, Li, F, Cl) complexes by means of MP2/aug-cc-pVTZ calculations. Interaction energies of these binary complexes span a large range, from −16.36 kcal/mol in (NH₂)H₃P⁺⋯NCF to −71.36 kcal/mol in FH₃P⁺⋯NCLi complex. The spin–spin coupling constant across P⋯N interaction depends considerably on the nature of X and Y substituents. The characteristic of CPB interactions is analyzed in terms of parameters derived from quantum theory of atoms in molecules (QTAIM) and natural bond orbital (NBO) analyses. The charge transfer from the nitrogen base to the cationic acid stabilizes these pnicogen–bonded complexes. For a given XH₃P⁺, the net charge transfer value increases as the interaction energy of the complex becomes more negative, i.e., NCLi > NCCl > NCH > NCF. Moreover, mutual influence between the CPB and hydrogen/halogen/lithium bond is studied in the ternary XH₃P⁺⋯NCY⋯NCH complexes. The results indicate that the formation of a Y⋯N interaction tends to strengthen CPB in the ternary systems.     

Some Applications of Magma in Designs and Codes: Oval Designs,Hermitian Unitals and Generalized Reed–Muller Codes

We describe three applications of Magma to problems in the area of designs and the associated codes:  •  Steiner systems, Hadamard designs and symmetric designs arising from an oval in an even-order plane, leading in the classical case to bent functions and difference-set designs;  •  the Hermitian unital as a 2-(q³ +  1, q +  1, 1) design, and the code overF_p where p divides q +  1;  •  a basis of minimum-weight vectors for the code over F_pof the design of points and hyperplanes of the affine geometry AG_d(F_p), where p is a prime.     

Modeling of the energies and splitting of the Qx and Qy bands in positional isomers of zinc pyridinoporphyrazines by TDDFT approach: Can TDDFT help distinguishing the structural isomers?

Electronic structures, energies and splitting of the Q_x and Q_y bands for positional isomers of zinc mono-, di-, tri-, and tetra pyridinoporphyrazines as well as parent zinc phthalocyanine were investigated using density functional theory (DFT) and time-dependent (TD) DFT approaches. The influence of the Hartree–Fock exchange on excited state energies and Q_x and Q_y bands splitting were studied using GGA BP86 and hybrid B3LYP and PBE1PBE exchange-correlation functionals. Solvent effects were estimated using the polarized continuum model (PCM) approach and cyclohexane, toluene, or DMSO as solvents. It was found that general trends in the Q_x and Q_y band energies and splitting correlate very well with the available experimental data on pyridinoporphyrazines and follow the trends in HOMO–LUMO and HOMO–LUMO + 1 energy gaps as well as LUMO–LUMO + 1 splitting. TDDFT trends allow estimation of the Q_x and Q_y band energies and splitting in unknown tripyridinoporphyrazines and in individual positional isomers of tetrapyridinoporphyrazines.     

Graph Isomorphism is in SPP

We show that Graph Isomorphism is in the complexity class SPP, and hence it is in ⊕P (in fact, in Mod_kP for each k ⩾ 2). These inclusions for Graph Isomorphism were not known prior to membership in SPP. We derive this result as a corollary of a more general result: we show that a generic problem FIND-GROUP has an FP^SPP algorithm. This general result has other consequences: for example, it follows that the hidden subgroup problem for permutation groups, studied in the context of quantum algorithms, has an FP^SPP algorithm. Also, some other algorithmic problems over permutation groups known to be at least as hard as Graph Isomorphism (e.g., coset intersection) are in SPP, and thus in Mod_kP for each k ⩾ 2.     

Ferroelectric properties of direct-patterned half-micron thick PZT film

Ferroelectric properties of direct-patterned PZT(PbZr_0.52Ti_0.48O₃) films with 460 μm × 460 μm size and 510 nm thick were analyzed for applying to micro-detecting devices. A photosensitive solution containing ortho-nitrobenzaldehyde was used for the preparation of direct-patterned PZT film. PZT solution was coated on Pt(1 1 1)/Ti/SiO₂/Si(1 0 0) substrate for three times to obtain half-micron thick film and three times of direct-patterning process were repeated to define a pattern on multi-layer PZT film. Through intermediate and final anneal procedure of direct-patterned PZT film, any shrinkage along horizontal direction was not observed within this experimental condition, i.e., the size of the pattern was preserved after annealing, only a thickness reduction was observed after each annealing treatment. Ferroelectric properties of direct-patterned PZT film with 460 μm × 460 μm size and 510 nm thick were compared with those of un-patterned conventional PZT film and shown to be almost the same. Through this work, the high potentiality of direct-patternable PZT film for applying to micro-devices without the introduction of physical damages from dry-etching could be confirmed.     

Conditional diagnosability of hypercubes under the comparison diagnosis model

Processor fault diagnosis plays an important role in multiprocessor systems for reliable computing, and the diagnosability of many well-known networks has been explored. Lai et al. proposed a novel measure of diagnosability, called conditional diagnosability, by adding an additional condition that any faulty set cannot contain all the neighbors of any vertex in a system. We make a contribution to the evaluation of diagnosability for hypercube networks under the comparison model and prove that the conditional diagnosability of n-dimensional Hypercube Q_n is 3(n ? 2) + 1 for n ? 5. The conditional diagnosability of Q_n is about three times larger than the classical diagnosability of Q_n.     

Designing of potential inhibitors against Staphylococcus aureus sortase A: Combined analogue and structure based approach with in vitro validation

Staphylococcus aureus sortase A is an attractive target of Gram-positive bacteria that plays a crucial role in anchoring of surface proteins to peptidoglycan present in bacterial cell wall. Inhibiting sortase A is an elementary and essential effort in preventing the pathogenesis. In this context, in silico virtual screening of in-house database was performed using ligand based pharmacophore model as a filter. The developed pharmacophore model AAHR 11 consists of two acceptors, one hydrophobic and one ring aromatic feature. Top ranked molecule KKR1 was docked into the active site of the target. After profound analysis, it was analyzed and optimized based on the observations from its binding pose orientation. Upgraded version of KKR1 was KKR2 and has improved docking score, binding interactions and best fit in the binding pocket. KKR1 along with KKR2 were further validated using 100 ns molecular dynamic studies. Both KKR1 and KKR2 contain Indole-thiazolidine moiety and were synthesized. The disk diffusion assay has good initial results (ZI of KKR1, KKR2 were 24, 38 mm at 10 μg/mL and ZI of Ampicillin was 22 at 10 μg/mL) and calculated MICs of the molecules (KKR1 5.56 ± 0.28 μg/mL, KKR2 1.32 ± 0.12 μg/mL, Ampicillin 8 ± 1.1 μg/mL) were in good agreement with standard drug Ampicillin. KKR1 has shown IC₅₀ of 1.23 ± 0.14 μM whereas the optimized lead molecule KKR2 show IC₅₀ of 0.008 ± 0.07 μM. Results from in silico were validated by in vitro studies and proved that indole-thiazolidine molecules would be useful for future development as lead molecules against S. aureus sortase A.     

Curves of Genus 2 with (N,N) Decomposable Jacobians

Let C be a curve of genus 2 and ψ₁: C  −  → E ₁  a map of degree n, from C to an elliptic curveE₁ , both curves defined over C. This map induces a degree n map φ₁:P¹  −  → P ¹  which we call a Frey–Kani covering. We determine all possible ramifications for φ₁. If ψ₁:C  −  → E ₁  is maximal then there exists a maximal map ψ₂: C  −  → E ₂  , of degree n, to some elliptic curveE₂ such that there is an isogeny of degree n²from the JacobianJ_C to E₁ × E₂. We say thatJ_C is (n, n)-decomposable. If the degree n is odd the pair (ψ₂, E₂) is canonically determined. For n =  3, 5, and 7, we give arithmetic examples of curves whose Jacobians are (n, n)-decomposable.     

Aviation emission inventory development and analysis

An up to date and accurate aviation emission inventory is a prerequisite for any detailed analysis of aviation emission impact on greenhouse gases and local air quality around airports. In this paper we present an aviation emission inventory using real time air traffic trajectory data. The reported inventory is in the form of a 4D database which provides resolution of 1° × 1° × 1000 ft for temporal and spatial emission analysis. The inventory is for an ongoing period of six months starting from October 2008 for Australian Airspace.In this study we show 6 months of data, with 492,936 flights (inbound, outbound and over flying). These flights used about 2515.83 kt of fuel and emitted 114.59 kt of HC, 200.95 kt of CO, 45.92 kt of NO_x, 7929.89 kt of CO₂, and 2.11 kt of SO_x. From the spatial analysis of emissions data, we found that the CO₂ concentration in some parts of Australia is much higher than other parts, especially in some major cities. The emission results also show that NO_x emission of aviation may have a significant impact on the ozone layer in the upper troposphere, but not in the stratosphere.It is expected that with the availability of this real time aviation emission database, environmental analysts and aviation experts will have an indispensable source of information for making timely decisions regarding expansion of runways, building new airports, applying route charges based on environmentally congested airways, and restructuring air traffic flow to achieve sustainable air traffic growth.     

A study of nonlinear dispersive equations with solitary-wave solutions having compact support

With the use of Adomian decomposition method, the prototypical, genuinely nonlinear K(m,n) equation, u_t+(u^m)_x+(uⁿ)_xxx=0, which exhibits compactons — solitons with finite wavelength — is solved exactly. Two numerical illustrations, K(2,2) and K(3,3), are investigated to illustrate the pertinent features of the proposed scheme. The technique is presented in a general way so that it can be used in nonlinear dispersive equations.     

Red emitting MTiO3 (M = Ca or Sr) phosphors doped with Eu3+ or Pr3+ with some cations as co-dopants

Ca (or Sr)TiO₃:Eu³⁺, M (Li⁺ or Na⁺ or K⁺) and CaTiO₃:Pr³⁺, M (Li⁺ or Na⁺ or Ag⁺ or K⁺ or Gd³⁺ or La³⁺) powders were prepared by combustion synthesis method and the samples were further heated to ～1000 °C to improve the crystallinity. The structure and morphology of materials were examined by X-ray diffraction (XRD) and a scanning electron microscopy (SEM). The morphologies of SrTiO₃:Eu³⁺, CaTiO₃:Eu³⁺ or CaTiO₃:Pr³⁺ powders co-doped with other metal ions were very similar. Small and coagulated particles of nearly cubical shapes with small size distribution having smooth and regular surface were formed. Photo-luminescence spectra of CaTiO₃:Pr³⁺ and co-doped either with Li⁺, Na⁺, K⁺, Ag⁺, La³⁺ or Gd³⁺ ions showed red emissions at 613 nm due to the ¹D₂ → ³H₄ transition of Pr³⁺. The variation of intensity of emission peak with different co-doping follows the order: K⁺ > Ag⁺ > Na⁺ > Li⁺ > La³⁺ > Gd³⁺. The characteristic emissions of CaTiO₃:Eu³⁺ lattices had strong emission at 614 and 620 nm for ⁵D₀ → ⁷F₂ with other weak transitions observed at 580, 592, 654, 705 nm for ⁵D₀ → ⁷F_n transitions where n = 0, 1, 3, 4 respectively in all host lattices. Photoluminescence intensity in SrTiO₃:Eu³⁺ is more than CaTiO₃:Eu³⁺ lattices. A remarkable increase of photoluminescence intensity (in ⁵D₀ → ⁷F₂ transition) was observed if co-doped with Li⁺ ions in CaTiO₃:Eu³⁺ and SrTiO3:Eu³⁺.     

Evolution of structure,stability, and nonlinear optical properties of the heterodinuclear CNLin (n = 1–10) clusters

The lowest-energy structures and stabilities of the heterodinuclear clusters, CNLi_n (n = 1–10) and relevant CNLi_n⁺ (n = 1–10) cations, are studied using the density functional theory with the 6-311 + G(3df) basis set. The CNLi₆ and CNLi₅⁺ clusters are the first three-dimensional ones in the CNLi_n^0/+ series, respectively, and the CN group always caps the Li_n^0/+ moiety in the CNLi_n^0/+ (n = 1–9) configurations. The CN triple bond is found to be completely cleaved in the CNLi₁₀^0/+ clusters where the C and N atoms are bridged by two Li atoms. The CNLi_n (n = 2–10) clusters are hyperlithiated molecules with delocalized valence electrons and consequently possess low VIP values of 3.780–5.674 eV. Especially, the CNLi₈ and CNLi₁₀ molecules exhibit lower VIPs than that of Cs atom and can be regarded as heterobinuclear superalkali species. Furthermore, these two superalkali clusters show extraordinarily large first hyperpolarizabilities of 19,423 and 42,658 au, respectively. For the CNLi_n⁺ cationic species, the evolution of the energetic and electronic properties with the cluster size shows a special stability for CNLi₂⁺.     

Liquid phase separation,solidification and phase transformations of Gd–Ti and Gd–Ti–Al–Cu alloys

Phase equilibria of the quaternary Gd–Ti–Al–Cu system have been studied with particular respect to solidification and phase separation phenomena in metallic glasses. Along the section Gd_55−xTi_xAl₂₅Cu₂₀ the primary solidifying phase changes from Gd₂CuAl (x=0) toward α-Ti (x≥10) with rising Ti-fraction x. This is accompanied by an upturn of the liquidus temperature from T_L=745 °C to T_L>1100 °C. The miscibility gap predicted from thermodynamic calculations for Gd_55−xTi_xAl₂₅Cu₂₀ melts at intermediate Ti-fractions was not verified experimentally. Unlike binary Gd–Ti melts, levitated Gd–Ti–Al–Cu droplets do not exhibit liquid phase separation features after quenching from different holding temperatures, even at high melt undercooling up to 200 K prior to solidification.