A note on multivalued groups

Buchstaber (Mosc Math J 6(1):57?C84, 2006) defined multivalued groups. In this paper we will show that the first isomorphism theorem and Lagrange theorem dose not hold for multivalued groups. Finally we define stabilizer of an action and we show that orbit-stabilizer theorem is not true for multivalued-groups.     

EXPERIMENTAL AND THEORETICAL INVESTIGATION OF THE FIRST FOLD CREATION IN THIN WALLED COLUMNS

In this paper, a theoretical formula is presented to predict the instantaneous folding force of the first fold creation in square and rectangular columns under axial loading.The rectangular column is a small and single model of honeycombs with square or rectangular cells.Calculations are based on the analysis of the Basic Folding Mechanism （BFM）.For this purpose, sum of the dissipated energy rate under bending around horizontal and inclined hinge lines and the dissipated energy rate under extensional deformations was equal to the work rate of the external force on the structure.The final formula obtained in this research, reasonably predicts the instantaneous folding force of single-cell square and rectangular honeycombs instead of the average value.Finally, according to the calculated theoretical relation, the instantaneous folding force of the first fold creation in a square column was sketched versus folding distance and compared to the experimental results, which showed a good correlation.     

A note on <Emphasis Type="Italic">p</Emphasis>-groups of order ≤ <Emphasis Type="Italic">p</Emphasis><Superscript>4</Superscript>

In [1], we defined c(G), q(G) and p(G). In this paper we will show that if G is a p-group, where p is an odd prime and |G| ≤ p ⁴, then c(G) = q(G) = p(G). However, the question of whether or not there is a p-group G with strict inequality c(G) = q(G) < p(G) is still open.     

Surfactant-Free Hydrothermal Synthesis of Mesoporous Niobia Samples and Their Photoinduced Decomposition of Terephthalic Acid (TPA)

This study introduces a low temperature surfactant-free hydrothermal method to synthesize mesoporous Nb₂O₅ photocatalysts using NbCl₅ and H₂O₂ as precursors that are subsequently calcinated at 300, 400 and 450 °C and are assigned as mNb₂O₅-300, mNb₂O₅-400 and mNb₂O₅-450, respectively. Commercial niobia sample was used as reference sample for comparison purpose. All of materials were characterized by XRD, SEM, UV–Vis DRS, FTIR, TG/DTG and BET techniques. The synthesized Nb₂O₅ particles especially mNb₂O₅-300 sample shows a high surface area (240 m²/g), a large pore volume (0.21 cm³/g) and an identifying morphology of these features. Photocatalytic decomposition of terephthalic acid was evaluated using UV–Vis spectrophotometer. The photocatalytic reactions followed pseudo-first-order kinetics with an apparent rate constant of k = 105 × 10^?3 min^?1 for mNb₂O₅-300 sample with the highest activity among all samples at natural pH (pH = 6). Meanwhile, it was observed that optimum pH of 4 resulted in fast photocatalytic reaction for mNb₂O₅-300 sample.     

Effects of nonionic micelles on dephosphorylation and aromatic nucleophilic substitution

Terminal hydroxyl groups of micellized dodecyl (10) and (23) polyoxyethylene glycol (C12E10 and C12E23, respectively) are deprotonated by OH-, and the alkoxide ions react with 2,4-dinitrochlorobenzene (DNCB) giving ethers, but do not react with p-nitrophenyl dimethyl and diethyl phosphate (pNPDMP and pNPDEP, respectively), which react wholly with OH-. These substrates have similar hydrophobicities and should locate similarly in the micellar interfacial regions. These differences in reactivities are due to differences in relative nucleophilicities of OH- and alkoxide ions in aromatic nucleophilic substitution and dephosphorylation in micelles. Reactivities of DNCB and the related ethers indicate that local concentrations of OH- are similar in the aqueous and micellar interfacial regions but those of the alkoxide ions are higher in C12E10 than in C12E23 micelles due to differences in the molar volumes of the interfacial regions. With time the initially formed ethers react at similar rates with OH- generating 2,4-dinitrophenoxide ion.     

Influence of population and radius of cavity bubbles on size of sodium chloride crystals in ultrasound ablation

Crystal size distribution as a function of power in ultrasound ablation is simulated by using a theoretical method based on population and radius of bubbles. In this method, population and radius of bubbles are related to ultrasound power and Debye temperature of crystals. The best power for production of crystals with uniform size is obtained from the theory. In addition, it is shown that the population of bubbles effects the radius of the productions and the radius of bubbles influences the population of products. Results from the theory are supported by experimental results.     

QUASI-PERMUTATION REPRESENTATIONS OF ALTERNATING AND SYMMETRIC GROUPS

The quantities c(G), q(G) and p(G) for finite groups were defined by H. Behravesh. In this article, these quantities for the alternating group An and the symmetric group Sn are calculated. It is shown that c(G) = q(G)=p(G) = n, when G = An or Sn.