The enveloping algebra of a covariant system

In an earlier work, Doplicher, Kastler and Robinson have examined a mathematical structure consisting of a pair (A, G), whereA is aC*-algebra andG is a locally compact automorphism group ofA. We call such a structure a covariant system. The enveloping von Neumann algebraA(A, G) of (A, G) is defined as a *-algebra of operator valued functions (called options) on the space of covariant representations of (A, G). The system (A, G) is canonically embedded in, and in fact generates, the von Neumann algebraA(A, G). Further we show there is a natural one-to-one correspondence between the normal *-representations ofA(A, G) and the proper covariant representations of (A, G). The relation ofA(A, G) to the covarainceC*-algebraC*(A, G) is also examined.     

At-column heating and a resistively heated, liquid-cooled thermal modulator for a low-resource bench-top GC x GC

A transportable GC x GC instrument is under development for on-site applications that would benefit from the enhanced resolution and powers of detection, which can be achieved by this method. In the present study, a low-resource GC x GC instrument using an electrically heated and liquid-cooled single-stage thermal modulator that requires no cryogenic materials is evaluated. The instrument also uses at-column heating, thus eliminating the need for a convection oven to house the two columns. The stainless-steel modulator tube is coated with PDMS, which can be heated to 350 degrees C for sample injection into the second-dimension column. The modulator is cooled to -30 degrees C by a 100 mL/min flow of PEG by means of a commercial liquid chiller and a small recirculating pump. Resistive heating of the modulator tube is provided by a programmable power supply, which uses a voltage program that results in increasing modulator temperature during an analysis. This, together with more rapid cooling by the use of a liquid cooling medium, results in reduced solute breakthrough following each heating cycle as the modulator cools to a temperature where quantitative trapping resumes. As a result, modulated peak widths at half-height of less than 40 ms are observed. Design and performance details are presented along with chromatograms of gasoline and an essential oil sample.     

Symbiotic symmetries of the two-Higgs-doublet model

The new phenomenon of symbiotic symmetries   is described in the context of the Two-Higgs-Doublet Model (THDM). The quartic potential has two or more separate sectors with unequal symmetries, but these unequal symmetries persist even though the different sectors are renormalized by one another. We discuss all such symmetries of the THDM, consistent with the SU(2)×U(1)$\mathit{SU}(2)\times U(1)$ gauge interactions, using the Pauli formalism.     

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 \(\alpha \ge 0\), \(\Delta _5\left( \frac{11\times 5^{5\alpha }+1}{2}\right) \equiv 7\ (\mathrm{mod}\ 11)\).

     

Analogues of Ramanujan’s partition identities

Ramanujan discovered that $$\sum_{n=0}^\infty p(5n+4)q^n=5 \prod_{j=1}^\infty \frac{(1-q^{5j})^5}{(1-q^j)^6}, $$ where p(n) is the number of partitions of n. Recently, H.-C. Chan and S. Cooper, and H.H. Chan and P.C. Toh established several analogues of Ramanujan’s partition identities by employing the theory of modular functions. Very recently, N.D. Baruah and K.K. Ojah studied the partition function $p_{[c^{l}d^{m}]}(n)$ which is defined by $$\sum_{n=0}^\infty p_{[c^ld^m]}(n)q^n= \frac{1}{\prod_{j=1}^\infty (1-q^{cj})^{l}(1-q^{dj})^m}. $$ They discovered some analogues of Ramanujan’s partition identities and deduced several interesting partition congruences. In this paper, we provide a uniform method to prove some of their results by utilizing an addition formula. In the process, we also establish some new analogues of Ramanujan’s partition identities and congruences for $p_{[c^{l}d^{m}]}(n)$ .     

A supramolecular structure of bis(1,3-diphenyl-1,3-propanedionato-O,O′) (1,10-phenanthroline-N,N′)cobalt(II) based on C–H…O, C–H…π and π…π interactions

The compound bis(1,3-diphenyl-1,3-propanedionato-O,O′) (1,10-phenanthroline-N, N′)cobalt(II) (Co(DBM)₂(1,10-phe)) was investigated as a potential host component for functional inclusion materials. The molecules of the complex in the crystal (a=21.015(5) ?, b=17.456(5) ?, c=9.503(3) ?, β=107.04(2)°, s.g. C 2/c) are centrosymmetric and assemble by way of C–H···O and C–H···π hydrogen bonds as well as π···π stacking interactions to provide a supramolecular ladder-like motif that is accompanied by a variant of the six phenyl-embrace synthon.     

Intramolecular N−H...O hydrogen bonding,quinoid effect,and partial π-electron delocalization in N-aryl Schiff bases of 2-hydroxy-1-naphthaldehyde: the crystal structures of planar N-(α-naphthyl)- and N-(β-naphthyl)-2-oxy-1-naphthaldimine

The structures of N-(-naphthyl)-2-oxy-1-naphthaldimine1 and N-(-naphthyl)-2-oxy-1-naphthaldimine2 have been investigated by X-ray analysis and by spectroscopic methods. Crystals of1 are monoclinic, space groupPn, with cell dimensionsa=10.823(3),b=5.826(2),c=11.899 (3) Å, and =99.66(3)°. Compound2 crystallizes in the orthorhombic space groupPca2₁ witha=17.564(3),b=6.314(2), andc=13.663(4) Å. The IR spectra exhibited neither N–H nor O–H stretching frequencies. The existence of theintramolecular hydrogen bonding of N–H...O type was predicted by spectroscopic experiment but unequivocally established by diffraction experiment in both cases1 and2. The molecules1 and2 are significantly planar with considerable quinoid effect at the 2-oxy-naphthaldimine moiety. Although essentially planar, both molecules1 and2 show the delocalization of -electrons only in the central part of the molecules including C=N imino group with pendent ring carbon atoms.Intermolecular attractions in the crystals belong to weak van der Waals interactions-between discrete planar molecules spatially arranged into the expectedherringbone motif in the solid state.