A Characterization Theorem for the K-functional Associated with the Algebraic Version of Trigonometric Jackson Integrals

2000 Mathematics Subject Classification: 41A25, 41A36.

A New Characterization of Weighted Peetre K-Functionals (II)

2000 Mathematics Subject Classification: 46B70, 41A25, 41A17, 26D10. ∗Part of the results were reported at the Conference “Pioneers of Bulgarian Mathematics”, Sofia, 2006.

Equivalence Between K-functionals Based on Continuous Linear Transforms

2000 Mathematics Subject Classification: 46B70, 41A10, 41A25, 41A27, 41A35, 41A36, 42A10.

Complete Pick positivity and unitary invariance

The characteristic function for a contraction is a classical complete unitary invariant devised by Sz.-Nagy and Foias. Just as a contraction is related to the Szego kernel k(S) (z, w) = (1 - z (w) over tilde)(-1) for |z|, |w| < 1, by means of (1/k(S))(T,T*) >= 0, we consider an arbitrary open connected domain Omega in C-n, a complete Pick kernel k on Omega and a tuple T = (T-1, ..., T-n) of commuting bounded operators on a complex separable Hilbert space H such that (1/k)(T,T*) >= 0. For a complete Pick kernel the 1/k functional calculus makes sense in a beautiful way. It turns out that the model theory works very well and a characteristic function can be associated with T. Moreover, the characteristic function is then a complete unitary invariant for a suitable class of tuples T.

On an Extremal Problem concerning Bernstein Operators

* The second author is supported by the Alexander-von-Humboldt Foundation. He is on leave from: Institute of Mathematics, Academia Sinica, Beijing 100080, People’s Republic of China.

Direct and Converse Theorems for Generalized Bernstein-Type Operators

2000 Mathematics Subject Classification: 41A25, 41A27, 41A36.

Теглови апроксимации с оператори от тип на Гудман–Шарма

P. E. Parvanov - The uniform weighted approximation errors of the Goodman–Sharma operators are characterized for functions.

Best Approximation and Moduli of Smoothness

2010 Mathematics Subject Classification: 41A25, 41A10.

On the approximation by convolution operators in homogeneous Banach spaces on R^d

AMS Subject Classification 2010: 41A25, 41A35, 41A40, 41A63, 41A65, 42A38, 42A85, 42B10, 42B20

The Lower Estimate for Bernstein Operator

MSC 2010: 41A10, 41A15, 41A25, 41A36

Weighted Approximation by a Class of Bernstein-Type Operators

AMS classification: 41A36, 41A10, 41A25, 41Al7.

On the Approximation by Convolution Operators in Homogeneous Banach Spaces of Periodic Functions

AMS Subject Classification 2010: 41A25, 41A27, 41A35, 41A36, 41A40, 42Al6, 42A85.

Functional analysis of missense variants in the TRESK (KCNK18) K+ channel

A loss of function mutation in the TRESK K2P potassium channel (KCNK18), has recently been linked with typical familial migraine with aura. We now report the functional characterisation of additional TRESK channel missense variants identified in unrelated patients. Several variants either had no apparent functional effect, or they caused a reduction in channel activity. However, the C110R variant was found to cause a complete loss of TRESK function, yet is present in both sporadic migraine and control cohorts, and no variation in KCNK18 copy number was found. Thus despite the previously identified association between loss of TRESK channel activity and migraine in a large multigenerational pedigree, this finding indicates that a single non-functional TRESK variant is not alone sufficient to cause typical migraine and highlights the genetic complexity of this disorder. Migraine is a common, disabling neurological disorder with a genetic, environmental and in some cases hormonal component. It is characterized by attacks of severe, usually unilateral and throbbing headache, can be accompanied by nausea, vomiting and photophobia and is clinically divided into two main subtypes, migraine with aura (MA) when a migraine is accompanied by transient and reversible focal neurological symptoms and migraine without aura (MO)1. The multifactorial and clinical heterogeneity of the disorder have considerably hindered the identification of common migraine susceptibility genes and most of our current understanding comes from the studies of familial hemiplegic migraine (FHM), a rare monogenic autosomal dominant form of MA2. So far, the three susceptibility genes that have been convincingly identified in FHM families all encode ion channels or transporters: CACNA1A encoding the α1 subunit of the Cav2.1 calcium channel3, SCN1A encoding the Nav1.1 sodium channel4 and ATP1A2 encoding the α2 subunit of the Na+/K+ pump5. It is believed that mutations in these genes may lead to increased efflux of glutamate and potassium in the synapse and thereby cause migraine by rendering the brain more susceptible to cortical spreading depression (CSD)6 which is thought to play a role in initiating a migraine attack7,8. However, these genes have not to date been implicated in common forms of migraine9. Nevertheless, current opinion suggests that typical migraine, like FHM, is also disorder of neuronal excitability, ion homeostasis and neurotransmitter release10,11,12. Mutations in the SLC4A4 gene encoding the sodium-bicarbonate cotransporter NBCe1, have recently been implicated in several different forms of migraine13, and a variety of genes involved in glutamate homeostasis (PGCP, MTDH14 and LRP115) and a cation channel (TRPM8)15 have also recently been implicated in migraine via genome-wide association studies. Ion channels are therefore highly likely to play an important role in the pathogenesis of typical migraine. TRESK (KCNK18), is a member of the two-pore domain (K2P) family of potassium channels involved in the control of cellular electrical excitability16. Regulation of TRESK activity by the calcium-dependent phosphatase calcineurin17, as well as its expression in dorsal root ganglia (DRG)18 and trigeminal ganglia (TG)19,20 has led to a proposed role for this channel in a variety of pain pathways. In a recent study, a frameshift mutation (F139Wfsx24) in TRESK was identified in a large multigenerational pedigree where it co-segregated perfectly with typical MA and a significant genome-wide linkage LOD score of 3.0. Furthermore, functional analysis revealed that this mutation caused a complete loss of TRESK function and that the truncated subunit was also capable of down regulating wild-type channel function. This therefore highlighted KCNK18 as potentially important candidate gene and suggested that TRESK dysfunction might play a possible role in the pathogenesis of familial migraine with visual aura20. Additional screening for KCNK18 mutations in unrelated sporadic migraine and control cohorts also identified a number of other missense variants; R10G, A34V, C110R, S231P and A233V20. The A233V variant was found only in the control cohort, whilst A34V was identified in a single Australian migraine proband for which family samples were not available, but it was not detected in controls. By contrast, the R10G, C110R, and S231P variants were found in both migraineurs and controls in both cohorts. In this study, we have investigated the functional effect of these variants to further probe the potential association of TRESK dysfunction with typical migraine.

An insight into alkali promotion: a density functional theory study of CO dissociation on K/Rh(111)

The important role of alkali additives in heterogeneous catalysis is, to a large extent, related to the high promotion effect they have on many fundamental reactions. The wide application of alkali additives in industry does not, however, reflect a thorough understanding of the mechanism of their promotional abilities. To investigate the physical origin of the alkali promotion effect, we have studied CO dissociation on clean Rh(111) and K-covered Rh(111) surfaces using density functional theory. By varying the position of potassium atoms relative to a dissociating CO, we have mapped out the importance of different K effects on the CO dissociation reactions. The K-induced changes in the reaction pathways and reaction barriers have been determined; in particular, a large reduction of the CO dissociation barrier has been identified. A thorough analysis of this promotion effect allows us to rationalize both the electronic and the geometrical factors that govern alkali promotion effect: (i) The extent of barrier reductions depends strongly on how close K is to the dissociating CO. (ii) Direct K-O bonding that is in a very short range plays a crucial role in reducing the barrier. (iii) K can have a rather long-range effect on the TS structure, which could reduce slightly the barriers.