Irregular Primes and Cyclotomic Invariants to 12 Million

Computations of irregular primes and associated cyclotomic invariants were extended to all primes up to 12 million using multisectioning/convolution methods and a novel approach which originated in the study of Stickelberger codes Shokrollahi (1996). The latter idea reduces the problem to that of finding zeros of a polynomial overF_pof degree  <  (p −  1) / 2 among the quadratic nonresidues mod p. Use of fast polynomial gcd-algorithms gives anO (p log²p loglog p)-algorithm for this task. A more efficient algorithm, with comparable asymptotic running time, can be obtained by using Schönhage–Strassen integer multiplication techniques and fast multiple polynomial evaluation algorithms; this approach is particularly efficient when run on primes p for whichp −  1 has small prime factors. We also give some improvements on previous implementations for verifying the Kummer–Vandiver conjecture and for computing the cyclotomic invariants of a prime.     

Stereoelectronic interaction effects on the conformational properties of hydrogen peroxide and its analogues containing S and Se atoms: An ab initio,hybrid-DFT study and NBO analysis

Ab initio molecular orbital (MP2/6-311+G**//MP2/6-31G+G**) and hybrid-density functional theory (B3LYP/6-311+G**//MP2/6-311+G**) methods and NBO analysis were used to study the stereoelectronic interaction effects on the conformational properties of hydrogen peroxide (1), hydrogen disulfide (2) and hydrogen diselenide (3). The results showed that the Gibbs free energy difference (G_T − G_S) values at 298.15 K and 1 atm between the skew (S) and trans (T) conformations (ΔG_T–S) increase from compound 1 to compound 2 but decrease from compound 2 to compound 3. The C conformations of compounds 1–3 are less stable than their S and T conformations. Based on these results, the racemization processes of the axial symmetrical (C₂ symmetry) conformations of compounds 1–3 take place via their T conformations. Based on the optimized ground state geometries using the MP2/6-311+G** level of theory, the NBO analysis of donor–acceptor (bond–antibond) interactions revealed that the stabilization (resonance) energy associated with LP₂M2 → σ*_M3-H4 electronic delocalization for the S conformations of compounds 1–3 are 1.35, 5.94 and 4.68 kcal mol⁻¹, respectively. There is excellent agreement between the variations of the calculated ΔG_T–S and stabilization (resonance) energies associated with LP₂M2→σ*_M3-H4 electronic delocalization for the S conformations of compounds 1–3. The correlations between resonance energies, orbital integrals, dipole moments, bond orders, structural parameters and conformational behaviors of compounds 1–3 have been investigated. Test were made of complete basis set methods (CBS-QB3, CBS-4 and CBS-Q), the first two gave results essentially indistinguishable from those we used, but the CBS-Q results were in disagreement with experimental and other theoretical results.     

Dielectric and piezoelectric properties of [0 0 1] and [0 1 1]-poled relaxor ferroelectric PZN–PT and PMN–PT single crystals

The properties of PZN–PT and PMN–PT single crystals of varying compositions and orientations have been investigated. Among the various compositions studied, [0 0 1]-optimally poled PZN-(6–7)%PT and PMN-30%PT exhibit superior dielectric and piezoelectric properties, with K^T ≈ 6800–8000, d₃₃ ≈ 2800 pC/N, d₃₁ ≈ −(1200–1800) pC/N for PZN-(6–7)%PT; and K^T = 7500–9000, d₃₃ = 2200–2500 pC/N and d₃₁ = −(1100–1400) pC/N for PMN-30%PT. These two compositions are also fairly resistant to over-poling. The [0 0 1]-poled electromechanical coupling factors (k₃₃, k₃₁ and k_t) are relatively insensitive to crystal composition. [0 1 1]-optimally poled PZN-7%PT single crystal also exhibits extremely high d₃₁ values of up to −4000 pC/N with k₃₁ ≈ 0.90–0.96. While [0 1 1]-poled PZN-7%PT single crystal becomes over-poled with much degraded properties when poled at and above 0.6 kV/mm, PZN-6%PT crystal shows no signs of over-poling even when poled to 2.0 kV/mm. The presence of a certain amount (i.e., 10–15%) of orthorhombic phase in a rhombohedral matrix has been found to be responsible for the superior transverse piezoelectric properties of [0 1 1]-optimally poled PZN-(6–7)%PT. The present work shows that flux-grown PZN–PT crystals exhibit superior and consistent properties and improved over-poling resistance to flux-grown PMN–PT crystals and that, for or a given crystal composition, flux-grown PMN–PT crystals exhibit superior over-poling resistance to their melt-grown counterparts.     

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.     

Modal analysis and modified cascade neural networks in identification of geometrical parameters of circular arches

A hybrid computational system, composed of the finite element method (FEM) and cascade neural network system (CNNs), is applied to the identification of three geometrical parameters of elastic arches, i.e. span l, height f and cross-sectional thickness h. FEM is used in the direct (forward) analysis, which corresponds to the mapping α = {l, f, h} → {ω_j}, where: α – vector of control parameters, ω_j – arch eigenfrequencies. The reverse analysis is related to the identification procedure in which the reverse mapping is performed {ω_j} → {α_i}. For the identification purposes a recurrent, three level CNNs of structure (D^k-H^k-1)_s was formulated, where: k – recurrence step, s = I, II, III-levels of cascade system. The Semi-Bayesian approach is introduced for the design of CNNs applying the MML Maximum Marginal Likelihood) criterion. The computation of hyperparameters is performed by means of the Bayesian procedure evidence. The numerical analysis proves a great numerical efficiency of the proposed hybrid approach for both the perfect (noiseless) values of eigenfrequencies and noisy ones simulated by an added artificial noise.     

Blue electroluminescent materials based on phenylanthracene-substituted fluorene derivatives for organic light-emitting diodes

Two phenylanthracene-substituted fluorene derivatives, 10-(9,9′-dimethyl-2-(10-phenylanthracen-9-yl)-9H-fluoren-7-yl)-phenylanthracene (1) and 2′,7′-di-(10-phenylanthracen-9-yl)-9,9′-spirobi[9H-fluorene] (2) have been designed, synthesized, and characterized. A device using compound 1 as an emitting material exhibited luminous efficiency, power efficiency, external quantum efficiency and CIE coordinates of 3.37 cd/A, 1.50 lm/W, 1.87% at 20 mA/cm² and (0.18, 0.25) at 7 V, respectively. Furthermore, by exploiting this efficient blue fluorescent material as a blue emitting material with the combination of red phosphorescent bis(2-phenylquinoline)acetylacetonate [(pq)₂Ir(acac)], an efficient white OLED (WOLED) with a external quantum efficiency of 1.70%, luminous efficiency of 1.38 cd/A, power efficiency of 0.94 lm/W at 20 mA/cm² and the color coordinates of (0.33, 0.36) at 14 V is demonstrated.     

On identifying codes that are robust against edge changes

Assume that G = (V, E) is an undirected graph, and C ⊆ V. For every v ∈ V, denote I_r(G; v) = {u ∈ C: d(u,v) ≤ r}, where d(u,v) denotes the number of edges on any shortest path from u to v in G. If all the sets I_r(G; v) for v ∈ V are pairwise different, and none of them is the empty set, the code C is called r-identifying. The motivation for identifying codes comes, for instance, from finding faulty processors in multiprocessor systems or from location detection in emergency sensor networks. The underlying architecture is modelled by a graph. We study various types of identifying codes that are robust against six natural changes in the graph; known or unknown edge deletions, additions or both. Our focus is on the radius r = 1. We show that in the infinite square grid the optimal density of a 1-identifying code that is robust against one unknown edge deletion is 1/2 and the optimal density of a 1-identifying code that is robust against one unknown edge addition equals 3/4 in the infinite hexagonal mesh. Moreover, although it is shown that all six problems are in general different, we prove that in the binary hypercube there are cases where five of the six problems coincide.     

An Efficient Algorithm for the k-Pairwise Disjoint Paths Problem in Hypercubes

A graph G(V, E) (|V|⩾2k) satisfies property A_k if, given k pairs of distinct nodes (s₁, t₁), …, (s_k, t_k) of V(G), there are k mutually node-disjoint paths, one connecting s_i and t_i for each i, 1⩽i⩽k. A necessary condition for any graph to satisfy A_k is that it is (2k−1)-connected. Hypercubes are important interconnection topologies for parallel computation and communication networks. It has been known that hypercubes of dimension n (which are n-connected) satisfy A_⌈n/2⌉. In this paper we give an algorithm which, given k=⌈n/2⌉ pairs of distinct nodes (s₁, t₁), …, (s_k, t_k) in the n-dimensional hypercube, finds the k disjoint paths of length at most n+⌈log n⌉+1 in O(n² log* n) time.     

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.     

Theoretical exploration to second-order nonlinear optical properties of new hybrid complexes via coordination interaction between (metallo)porphyrin and [MSiW11O39]3− (M = NbV or VV) polyoxometalates

The second-order nonlinear optical (NLO) properties of hybrid complexes via coordination interaction between porphyrin and Keggin-type polyoxometalates (POMs) α-[MSiW₁₁O₃₉]³⁻ (M = Nb^V or V^V) are investigated by time-dependent density functional theory (TDDFT). The calculated results show that this kind of organic–inorganic hybrid complexes possesses remarkably large molecular second-order NLO polarizability, especially for the ZnP₃P-CC-4-Py-[VSiW₁₁O₃₉]³⁻ (complex 4), which has a computed β₀ value of 261,410 a.u. and might be an excellent second-order NLO material. The effects of substituted metal atom (M), metalloporphyrin, and π-conjugation on NLO response are analyzed, the substituted metal atom (M) with a large electronegativity, the metalloporphyrin, and the lengthening of π-conjugation are helpful in enhancing the optical nonlinearity of these systems, which reveal the general rules to design the complexes with large optical nonlinearities. Furthermore, the solvent effect largely affects the first-order hyperpolarizability of the complex, it implies that the second-order polarizabilities increased with the increase of the solvent in polarity.     

Investigation of spectroscopic properties of Sm-Eu codoped phosphate glasses

Phosphate glasses with chemical compositions of 74.5NaH₂PO₄–20ZnO–5Li₂O–0.5Sm₂O₃ and 74NaH₂PO₄–20ZnO–5Li₂O–0.5Sm₂O₃–0.5Eu₂O₃ were synthetized by melt quenching method. We investigated the influence of Sm³⁺/Eu³⁺ doping on the optical properties of phosphate glasses. X-ray Diffraction indicates that the samples have an amorphous structure. DSC measurements show a good thermal stability of phosphate glasses. Using the absorption spectra, Judd–Ofelt analysis was applied to absorption bands of Sm³⁺ (4f⁵) to carry out the three phenomenological parameters of Judd–Ofelt (JO). According to the obtained values of Ω₂, Ω₄ and Ω₆, some radiative properties were theoretically determined. We report both the photoluminescence (PL) and the PL lifetime measurements of a prominent emission transition ⁴G_5/2 → ⁶H_5/2 (604 nm) of Sm³⁺ both in absence and in presence of Eu³⁺. It is shown that Eu³⁺ ions act as sensitizers for Sm³⁺ ions and contribute largely to the improvement of the radiative properties of phosphate glasses. An improvement of the PL lifetime value after adding Eu³⁺ ions (4.58 ms) is reported. The predicted lifetime (τ_rad) calculated by Judd–Ofelt theory and the experimental lifetime (τ_meas) for the prepared phosphate glasses were compared with those of other works. Photoluminescence (PL) intensity of ⁴G_5/2 → ⁶H_5/2 (604 nm), ⁴G_5/2 → ⁶H_7/2 (567 nm), ⁴G_5/2 → ⁶H_9/2 (650 nm) and ⁴G_5/2 → ⁶H_11/2 (706 nm) and the quantum efficiency (η) for the excited ⁴G_5/2 level were enhanced after adding Eu³⁺. The radiative properties obtained for (Sm, Eu) codoped phosphate glasses suggest that the present material can be a potential candidate for the development of color display devices.     

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.     

Theoretical characterization and design of highly efficient iridium (III) complexes bearing guanidinate ancillary ligand

A density functional theory/time-depended density functional theory was used to investigate the synthesized guanidinate-based iridium(III) complex [(ppy)₂Ir{(NⁱPr)₂C(NPh₂)}] (1) and two designed derivatives (2 and 3) to determine the influences of different cyclometalated ligands on photophysical properties. Except the conventional discussions on geometric relaxations, absorption and emission properties, many relevant parameters, including spin-orbital coupling (SOC) matrix elements, zero-field-splitting parameters, radiative rate constants (k_r) and so on were quantitatively evaluated. The results reveal that the replacement of the pyridine ring in the 2-phenylpyridine ligand with different diazole rings cannot only enlarge the frontier molecular orbital energy gaps, resulting in a blue-shift of the absorption spectra for 2 and 3, but also enhance the absorption intensity of 3 in the lower-energy region. Furthermore, it is intriguing to note that the photoluminescence quantum efficiency (Φ_PL) of 3 is significantly higher than that of 1. This can be explained by its large SOC value <T₁|H_SO|S_n> (n = 3–4) and large transition electric dipole moment (μ_S3), which could significantly contribute to a larger k_r. Besides, compared with 1, the higher emitting energy (E_T1) and smaller <S₀|H_SO|T₁>² value for 3 may lead to a smaller non-radiative decay rate. Additionally, the detailed results also indicate that compared to 1 with pyridine ring, 3 with imidazole ring performs a better hole injection ability. Therefore, the designed complex 3 can be expected as a promising candidate for highly efficient guanidinate-based phosphorescence emitter for OLEDs applications.     

Tuning the push–pull configuration for efficient second-order nonlinear optical properties in some chalcone derivatives

Using the density functional theory methods, we effectively tune the second-order nonlinear optical (NLO) properties in some chalcone derivatives. Various unique push–pull configurations are used to efficiently enhance the intramolecular charge transfer process over the designed derivatives, which result in significantly larger amplitudes of the first hyperpolarizability as compared to their parent molecule. The ground state molecular geometries have been optimized using B3LYP/6-311G** level of theory. A variety of methods including B3LYP, CAM-B3LYP, PBE0, M06, BHandHLYP and MP2 are tested with 6-311G** basis set to calculate the first hyperpolarizability of parent system 1. The results of M06 are found closer to highly correlated MP2 method, which has been selected to calculate static and frequency dependent first hyperpolarizability amplitudes of all selected systems. At M06/6-311G** level of theory, the permanent electronic dipole moment (μ_tot), polarizability (α₀) and static first hyperpolarizability (β_tot) amplitudes for parent system 1 are found to be 5.139 Debye, 274 a. u. and 24.22 × 10⁻³⁰ esu, respectively. These amplitudes have been significantly enhanced in designed derivatives 2 and 3. More importantly, the (β_tot) amplitudes of systems 2 and 3 mount to 75.78 × 10⁻³⁰ and 128.51 × 10⁻³⁰ esu, respectively, which are about 3 times and 5 times larger than that of their parent system 1. Additionally, we have extended the structure-NLO property relationship to several newly synthesized chalcone derivatives. Interestingly, the amplitudes of dynamic frequency dependent hyperpolarizability μβ_ω (SHG) are also significantly larger having values of 366.72 × 10⁻⁴⁸, 856.32 × 10⁻⁴⁸ and 1913.46 × 10⁻⁴⁸ esu for systems 1–3, respectively, at 1400 nm of incident laser wavelength. The dispersion behavior over a wide range of change in wavelength has also been studied adopting a range of wavelength from 1907 to 544 nm. Thus, the present work realizes the potential of designed derivatives as efficient NLO-phores for modern NLO applications.     

Factoring Polynomials Over Local Fields

We describe an efficient new algorithm for factoring a polynomial Φ(x) over a fieldkthat is complete with respect to a discrete prime divisor. For every irreducible factorϕ (x) of Φ(x) this algorithm returns an integral basis fork[ x ] / ϕ(x)k[ x ] overk.     

A DFT study on the physical adsorption of cyclophosphamide derivatives on the surface of fullerene C60 nanocage

The physical sorption of a series of cyclophosphamide drug derivatives with formula

where x = 2, X = F (2), Cl (3), Br (4); x = 3, X = F (5), Cl (6), Br (7) and x = 4, X = F (8), Cl (9), Br (10) on the surface of fullerene C₆₀ was studied using density functional theory (DFT) at B3LYP and B3PW91 levels. The most negative binding energies obtained using the B3LYP approach and corrected for geometrical BSSE and dispersion energies (gCP-D3-ΔE_binding) were measured for compounds 8 (among isolated drugs) and 13 (among complexes). The dipole moments of isolated drugs were obtained close to those of their complexes with C₆₀ (∼4.0–5.5 Debye) indicating their hydrophililic nature that is an appropriate property appealing for drug delivery in biological media. The adsorption of all drugs on the surface of fullerene was endergonic with all of the ΔG_adsorption > 0. The ΔH_adsorption values at B3PW91 level were only negative for complexes 14–16 indicating their exothermic adsorption nature. The HOMO–LUMO band gaps of complexes 11–19 were about 2.7 eV and are comparable with the gap in C₆₀ but are much smaller than the gaps in isolated drugs 2–10 (6.5–8.0 eV) reflecting the increase in electrical conductivities upon complexation. The QTAIM data supported the covalent character of the C–O, C–N and N–H bonds, the intermediate character of PO, P–O and P–N bonds while the electrostatic nature of PO…C(fullerene) interactions. According to the gCP-D3-ΔE_binding binding energies and ΔH_adsorption values at B3LYP level, it seems that the complexes 12 and 13 can be the most promising prodrug + carrier delivery systems.

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Transition-aware DVS algorithm for real-time systems using tree structure analysis

Dynamic voltage scaling (DVS) is a key technique for embedded real-time systems to reduce energy consumption by lowering the supply voltage and operating frequency. Many existing DVS algorithms have to generate the canonical schedules or estimate the lengths of slack time in advance for generating the voltage scaling decisions. Therefore, these methods have to compute the schedules with exponential time complexities in general. In this paper, we consider a set of jitter-controlled, independent, periodic, hard real-time tasks scheduled according to preemptive pinwheel model. Our approach constructs a tree structure corresponding to a schedule and maintains the data structure at each early-completion point. Our approach consists of off-line and on-line algorithms which consider the effects of transition time and energy. The off-line and on-line algorithm takes O(k + n log n) and O(k + (p_max/p_min)) time complexity, respectively, where n, k, p_max and p_min denotes the number of tasks, jobs, longest and shortest task period, respectively. Experimental results show that the proposed approach is effective in reducing computational complexity, transition time and energy overhead.     

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.