Congruences modulo 11 for broken 5-diamond partitions

The notion of broken k-diamond partitions was introduced by Andrews and Paule in 2007. For a fixed positive integer k, let \(\Delta _k(n)\) denote the number of broken k-diamond partitions of n. Recently, Paule and Radu conjectured two relations on \(\Delta _5(n)\) which were proved by Xiong and Jameson, respectively. In this paper, employing these relations, we prove that, for any prime p with \(p\equiv 1\ (\mathrm{mod}\ 4)\), there exists an integer \(\lambda (p)\in \{2,\ 3,\ 5,\ 6,\ 11\}\) such that, for \(n, \alpha \ge 0\), if \(p\not \mid (2n+1)\), then

$$\begin{aligned} \Delta _5\left( 11p^{\lambda (p)(\alpha +1)-1} n+\frac{11p^{\lambda (p)(\alpha +1)-1}+1}{2}\right) \equiv 0\ (\mathrm{mod}\ 11). \end{aligned}$$

Moreover, some non-standard congruences modulo 11 for \(\Delta _5(n)\) are deduced. For example, we prove that, for \(\alpha \ge 0\), \(\Delta _5\left( \frac{11\times 5^{5\alpha }+1}{2}\right) \equiv 7\ (\mathrm{mod}\ 11)\).

     

Arithmetic properties for a partition function related to the Ramanujan/Watson mock theta function $$\omega (q)$$

Recently, Andrews, Dixit, and Yee introduced a new partition function \(p_{\omega }(n)\) that denotes the number of partitions of n in which each odd part is less than twice the smallest part. The generating function of \(p_{\omega }(n)\) is associated with the third-order mock theta function \(\omega (q)\). Andrews, Passary, Sellers, and Yee proved three infinite families of congruences modulo 4 and 8 for \(p_{\omega }(n)\) and provided elementary proofs of congruences modulo 5 for \(p_{\omega }(n)\) which were first discovered by Waldherr. In this paper, we prove some new congruences modulo 5 and powers of 2 for \(p_{\omega }(n)\). In particular, we obtain some non-standard congruences for \(p_{\omega }(n)\). For example, we prove that for \(k\ge 0\), \( p_{\omega }\left( \frac{7\times 5^{2k+1}+1 }{3}\right) \equiv (-1)^k \ (\mathrm{mod}\ 5) \) and \( p_\omega \left( \frac{2^{2k+7}+1}{3}\right) \equiv 1251 \times (-1)^k \ (\mathrm{mod}\ 2^{11})\).     

On the “Hidden” Harmonics Associated to Best Approximants Due to Quasi-periodicity in Splitting Phenomena

The effects of quasi-periodicity on the splitting of invariant manifolds are examined. We have found that some harmonics that could be expected to be dominant in some ranges of the perturbation parameter actually are nondominant. It is proved that, under reasonable conditions, this is due to the arithmetic properties of the frequencies.     

Associative monotonic operations in fuzzy set theory

The properties of binary operations in a real interval are considered and used in the discussion of generalized operations on fuzzy sets, on fuzzy numbers and on fuzzy probabilistic sets.     

Are the known Δ2-cephems inactive as antibiotics because of an unfavourable steric orientation of their 4α-carboxylic group? Synthesis and biology of two Δ2-cephem-4β-carboxylic acids

Two representatives of the yet unknown type of Δ²-cephem-4β-carboxylic acids were prepared. Contrarily to the prediction based on the activity model of Cohen, both acids proved inactive as antibiotics. Possible reasons for this discrepancy are briefly discussed.     

Spectroscopic and electrochemical studies of transition-metal tetrasulfonated phthalocyanines: Part VI. The adsorption of iron tetrasulfonated phthalocyanine on single crystal silver electrodes

The influence of the underpotential deposition of Pb, Tl and Bi on the oxidation of ammonia and on the reduction and oxidation processes of hydroxylamine on platinum has been studied in alkaline solutions. Pronounced inhibition effects caused by underpotential metal adsorbates have been detected for either the oxidation of ammonia or the reduction of hydroxylamine. On the other hand, the same metal adsorbates markedly increase the electrocatalytic activity of the Pt electrode for hydroxylamine oxidation. The observed negative and positive catalytic effects have been interpreted in terms of the electronic and adsorption properties of the Pt substrate, which are strongly modified by the underpotential metal adsorbates.     

2,3,10,11‐Tetrathiatricyclo[10.4.0.04,9]hexadeca‐4,6,8,12,14,16‐hexaene (precession camera versus CCD)

The title compound, C₁₂H₈S₄, has crystallographic symmetry, the benzene groups thus being anti with respect to the plane of the four S atoms. The S—S and C—S bond lengths of the sulfur–carbon eight‐membered ring were found to be similar to those in other structures containing such sulfur–carbon rings. There is evidence for π–π interactions between the aromatic rings of neighboring molecules, linking them into sheets.     

2-Ketogluconic acid production by acetobacter pasteurianus

Production of 2-ketogluconic acid in batch fermentation was investigated.Acetobacter pasteurianus ATCC 6438, which produces selectively 2-ketogluconic acid only, was used. The optimal pH for glucose dehydrogenation to gluconate by resting cells was 5.0 and for gluconate dehydrogenation to 2-ketogluconate 4.25. When glucose medium was used, the 89% yield was achieved after 90 h. For the optimal productivity, medium containing glucose and gluconate with the molar glucose:gluconate ratio 7.4 was proposed, and the yield of 92% after 56 h was achieved. This composition of medium led to the elevation of dissolved oxygen concentration during fermentation. It consequently resulted in elevated gluconate dehydrogenase activity being discussed as the rate-limiting activity of the batch production.     

Bounds on fourth-generation fermion masses in the standard model

In the standard model, if the ratios of Yukawa couplings to gauge couplings are bounded functions of the renormalization point, then the fourth-generation fermion massesm _U , m _D , andm _L (in GeV) are constrained to lie within a domain approximately given by (m _U /213)^4.8+(m _D /209)^4.8+(m _L /161)^2.4<1.