Some new equivalences of Anderson's paving conjectures

Anderson's paving conjectures are known to be equivalent to the Kadison-Singer problem. We prove some new equivalences of Anderson's conjectures that require the paving of smaller sets of matrices. We prove that if the strictly upper triangular operators are paveable, then every 0 diagonal operator is paveable. This result follows from a new paving condition for positive operators. In addition, we prove that if the upper triangular Toeplitz operators are paveable, then all Toeplitz operators are paveable.

     

Adding Remnant Magnetization and Anisotropic Exchange to Propeller‐like Single‐Molecule Magnets through Chemical Design

The selective replacement of the central iron(III) ion with vanadium(III) in a tetrairon(III) propeller‐shaped single‐molecule magnet has allowed us to increase the ground spin state from S=5 to S=13/2. As a consequence of the pronounced anisotropy of vanadium(III), the blocking temperature for the magnetization has doubled. Moreover, a significant remnant magnetization, practically absent in the parent homometallic molecule, has been achieved owing to the suppression of zero‐field tunneling of the magnetization for the half‐integer molecular spin. Interestingly, the contribution of vanadium(III) to the magnetic anisotropy barrier occurs through the anisotropic exchange interaction with iron(III) spins and not through single ion anisotropy as in most single‐molecule magnets.     

Remotely detected MRI velocimetry in microporous bead packs

Many NMR and MRI methods probe fluid dynamics within macro- and mesoporous materials, but with few exceptions, they report on its macroscopically averaged properties. MRI methods are generally unable to localize microscopic features of flow within macroscopic samples because the fraction of the enclosing detector volume occupied by these features is so small. We have recently overcome this problem using remotely detected MRI velocimetry, a technique in which spatial, chemical, and velocity information about elements of the flow is encoded with a conventional NMR coil and detected sensitively at the sample outflow by a volume-matched microdetector. Here, we apply this method to microporous model systems, recording MRI images that correlate local velocity, spin relaxation, and time-of-flight in microscopic resolution and three spatial dimensions. Our results illustrate that remotely detected MRI is an effective approach to elucidate flow dynamics in porous materials including bead pack microreactors and chromatography columns.     

Domain-wall spin dynamics in kagome antiferromagnets

We report magnetization and neutron scattering measurements down to 60 mK on a new family of Fe based kagome antiferromagnets, in which a strong local spin anisotropy combined with a low exchange path network connectivity lead to domain walls intersecting the kagome planes through strings of free spins. These produce unfamiliar slow spin dynamics in the ordered phase, evolving from exchange-released spin flips towards a cooperative behavior on decreasing the temperature, probably due to the onset of long-range dipolar interaction. A domain structure of independent magnetic grains is obtained that could be generic to other frustrated magnets.     

Rational enantioselective design of chiral heterobimetallic single-chain magnets: synthesis, crystal structures and magnetic properties of oxamato-bridged M(II)Cu(II) chains (M=Mn, Co)

A new series of neutral oxamato-bridged M(II)Cu(II) chiral chains of general formula [MCuL(x)(S)(m)(H(2)O)(n)]·aS·bH(2)O [L(1)=(M)-1,1'-binaphthalene-2,2'-bis(oxamate) with M=Mn (1a) and Co (1b); L(2)=(P)-1,1'-binaphthalene-2,2'-bis(oxamate) with M=Mn (2a) and Co (2b)] and the analogous racemic chains of formula [MCuL(3)(S)(m)(H(2)O)(n)]·aS·bH(2)O [L(3)=1,1'-binaphthalene-2,2'-bis(oxamate) with M=Mn (3a) and Co (3b)] have been prepared by reaction of the corresponding dianionic oxamatocopper(II) complex [Cu(L(x))](2-) with Mn(2+) or Co(2+) cations in either dimethylformamide (DMF) or dimethyl sulfoxide (DMSO). Solid circular dichroism (CD) spectra of the bimetallic chain compounds were recorded to establish their chiral and enantiomeric nature. They exhibit maximum positive and negative Cotton effects, each pair of enantiomeric chains being non-superimposable mirror images. The crystal structures of the Mn(II)Cu(II) (1a-3a) and the Co(II)Cu(II) (1b and 2b) chain compounds were solved by single-crystal X-ray diffraction methods. Our attempts to obtain X-ray quality crystals of 3b were unsuccessful. The values of the shortest interchain Mn···Mn and Co···Co distances are indicative of a good isolation of neighbouring chains in the crystal lattice, which is caused by the bulky aromatic ligand. Although all the Mn(II)Cu(II) and Co(II)Cu(II) chains exhibit ferrimagnetic behaviour (-J(MnCu)=18.9-26.6 cm(-1) and -J(CoCu)=19.5-32.5 cm(-1)), only the enantiopure Co(II)Cu(II) chains (1b and 2b) show slow magnetic relaxation at low temperatures (T(B)=0.6-1.8 K), which is a characteristic of single-chain magnets (SCMs) and is related to the magnetic anisotropy of the high-spin Co(II) ion. Analysis of the SCM behaviour of 1b and 2b, based on Glauber's theory for an Ising one-dimensional system, shows a thermally activated mechanism for the magnetic relaxation (Arrhenius law dependence). The energy barriers (E(a)) to reverse the magnetisation direction are 8.2 (1b) and 8.1cm(-1) (2b), whereas the pre-exponential factor (τ(0)) is 1.9×10(-8) (1b) and 6.0×10(-9) s (2b). Interestingly, the racemic Co(II)Cu(II) chain analogue, 3b, showed no evidence of SCM behaviour.     

Operator algebras of functions

We present some general theorems about operator algebras that are algebras of functions on sets, including theories of local algebras, residually finite-dimensional operator algebras and algebras that can be represented as the scalar multipliers of a vector-valued reproducing kernel Hilbert space. We use these to further develop a quantized function theory for various domains that extends and unifies Agler's theory of commuting contractions and the Arveson-Drury-Popescu theory of commuting row contractions. We obtain analogous factorization theorems, prove that the algebras that we obtain are dual operator algebras and show that for many domains, supremums over all commuting tuples of operators satisfying certain inequalities are obtained over all commuting tuples of matrices.