The many aspects of counting lattice points in polytopes

A wide variety of topics in pure and applied mathematics involve the problem of counting the number of lattice points inside a convex bounded polyhedron, for short called a polytope. Applications range from the very pure (number theory, toric Hilbert functions, Kostant’s partition function in representation theory) to the most applied (cryptography, integer programming, contingency tables). This paper is a survey of this problem and its applications. We review the basic structure theorems about this type of counting problem. Perhaps the most famous special case is the theory of Ehrhart polynomials, introduced in the 1960s by Eugène Ehrhart. These polynomials count the number of lattice points in the different integral dilations of an integral convex polytope. We discuss recent algorithmic solutions to this problem and conclude with a look at what happens when trying to count lattice points in more complicated regions of space.     

Postsynthetic Improvement of the Physical Properties in a Metal–Organic Framework through a Single Crystal to Single Crystal Transmetallation

A single crystal to single crystal transmetallation process takes place in the three‐dimensional (3D) metal–organic framework (MOF) of formula Mg^II₂{Mg^II₄[Cu^II₂(Me₃mpba)₂]₃}?45 H₂O ( 1 ; Me₃mpba^4?=N,N′‐2,4,6‐trimethyl‐1,3‐phenylenebis(oxamate)). After complete replacement of the Mg^II ions within the coordination network and those hosted in the channels by either Co^II or Ni^II ions, 1 is transmetallated to yield two novel MOFs of formulae Co₂^II{Co^II₄[Cu^II₂(Me₃mpba)₂]₃}?56 H₂O ( 2 ) and Ni₂^II{Ni^II₄[Cu^II₂(Me₃mpba)₂]₃}? 54 H₂O ( 3 ). This unique postsynthetic metal substitution affords materials with higher structural stability leading to enhanced gas sorption and magnetic properties.     

Catalytic Borylation using an Air‐Stable Zinc Boryl Reagent: Systematic Access to Elusive Acylboranes

The use of borylzinc reagents in palladium‐catalyzed borylation chemistry is described (i.e. a boron analogue of the Negishi coupling), including a one‐pot bench‐top protocol using an air‐ and moisture‐stable bis(boryl)zinc reagent. The steric/electronic properties of the boryl fragment employed enable a systematic method for accessing acylboranes, a rare class of organoboron species with great potential in chemical synthesis. The reactions proceed under mild conditions, use inexpensive commercial sources of palladium, and demonstrate a remarkable functional‐group tolerance.     

Reactivity with Amines of Bis(cyanamide) and Bis(cyanoguanidine) Complexes of the Iron Triad

Treatment of bis(cyanamide) [M(N≡CNEt₂)₂L₄](BPh₄)₂ and bis(cyanoguanidine) [M{N≡CN(H)C(NH₂)=NH}₂L₄](BPh₄)₂ complexes [M = Fe, Ru, Os; L = P(OEt)₃] with an excess of amine RNH₂ (R = nPr, iPr) affords mixed‐ligand complexes with cyanamide and amine [M(NH₂R)(N≡CNEt₂)L₄](BPh₄)₂ ( 1a – 5a ) and [M(NH₂R){N≡CN(H)C(NH₂)=NH}L₄](BPh₄)₂ ( 1b , 2b ). The complexes were characterized by spectroscopy and X‐ray crystal structure determination of [M(NH₂iPr)(N≡CNEt₂){P(OEt)₃}₄](BPh₄)₂ [M = Ru ( 3a ), Os ( 5a )].     

Bound state solutions of D‐dimensional schrödinger equation with exponential‐type potentials

In this article, using an exactly‐solvable multiparameter exponential‐type potential we propose a unified treatment of the analytical bound—state solutions of the Schrödinger equation for exponential‐type potentials in D‐dimensions. Our proposal accepts different approximations to the centrifugal term; however, its usefulness is exemplified in the frame of the Green and Aldrich approach. This fact enables us to compare our results with specific potentials found in the literature and that are obtained here as particular cases of our proposal. That is, instead of solving a specific exponential‐type potential, by resorting each time to a specialized method, the energy spectra and wavefunctions are derived straightforward from the proposed approach. Furthermore, our proposal can be used as an alternative way in the search of solutions to new exponential‐type potentials besides that one can study different approximations to the term . © 2014 Wiley Periodicals, Inc.     

Micelle‐Triggered β‐Hairpin to α‐Helix Transition in a 14‐Residue Peptide from a Choline‐Binding Repeat of the Pneumococcal Autolysin LytA

Choline‐binding modules (CBMs) have a ββ‐solenoid structure composed of choline‐binding repeats (CBR), which consist of a β‐hairpin followed by a short linker. To find minimal peptides that are able to maintain the CBR native structure and to evaluate their remaining choline‐binding ability, we have analysed the third β‐hairpin of the CBM from the pneumococcal LytA autolysin. Circular dichroism and NMR data reveal that this peptide forms a highly stable native‐like β‐hairpin both in aqueous solution and in the presence of trifluoroethanol, but, strikingly, the peptide structure is a stable amphipathic α‐helix in both zwitterionic (dodecylphosphocholine) and anionic (sodium dodecylsulfate) detergent micelles, as well as in small unilamellar vesicles. This β‐hairpin to α‐helix conversion is reversible. Given that the β‐hairpin and α‐helix differ greatly in the distribution of hydrophobic and hydrophilic side chains, we propose that the amphipathicity is a requirement for a peptide structure to interact and to be stable in micelles or lipid vesicles. To our knowledge, this “chameleonic” behaviour is the only described case of a micelle‐induced structural transition between two ordered peptide structures.     

Surface‐Functionalized Nanoparticles by Olefin Metathesis: A Chemoselective Approach for In Vivo Characterization of Atherosclerosis Plaque

The use of click chemistry reactions for the functionalization of nanoparticles is particularly useful to modify the surface in a well‐defined manner and to enhance the targeting properties, thus facilitating clinical translation. Here it is demonstrated that olefin metathesis can be used for the chemoselective functionalization of iron oxide nanoparticles with three different examples. This approach enables, in one step, the synthesis and functionalization of different water‐stable magnetite‐based particles from oleic acid‐coated counterparts. The surface of the nanoparticles was completely characterized showing how the metathesis approach introduces a large number of hydrophilic molecules on their coating layer. As an example of the possible applications of these new nanocomposites, a focus was taken on atherosclerosis plaques. It is also demonstrated how the in vitro properties of one of the probes, particularly its Ca²⁺‐binding properties, mediate their final in vivo use; that is, the selective accumulation in atherosclerotic plaques. This opens promising new applications to detect possible microcalcifications associated with plaque vulnerability. The accumulation of the new imaging tracers is demonstrated by in vivo magnetic resonance imaging of carotids and aorta in the ApoE^?/? mouse model and the results were confirmed by histology.     

Oxidation and β‐Alkylation of Alcohols Catalysed by Iridium(I) Complexes with Functionalised N‐Heterocyclic Carbene Ligands

The borrowing hydrogen methodology allows for the use of alcohols as alkylating agents for C?C bond forming processes offering significant environmental benefits over traditional approaches. Iridium(I)‐cyclooctadiene complexes having a NHC ligand with a O‐ or N‐functionalised wingtip efficiently catalysed the oxidation and β‐alkylation of secondary alcohols with primary alcohols in the presence of a base. The cationic complex [Ir(NCCH₃)(cod)(MeIm(2‐ methoxybenzyl))][BF₄] (cod=1,5‐cyclooctadiene, MeIm=1‐methylimidazolyl) having a rigid O‐functionalised wingtip, shows the best catalyst performance in the dehydrogenation of benzyl alcohol in acetone, with an initial turnover frequency (TOF₀) of 1283 h^?1, and also in the β‐alkylation of 2‐propanol with butan‐1‐ol, which gives a conversion of 94 % in 10 h with a selectivity of 99 % for heptan‐2‐ol. We have investigated the full reaction mechanism including the dehydrogenation, the cross‐aldol condensation and the hydrogenation step by DFT calculations. Interestingly, these studies revealed the participation of the iridium catalyst in the key step leading to the formation of the new C?C bond that involves the reaction of an O‐bound enolate generated in the basic medium with the electrophilic aldehyde.     

General Variable Neighborhood Search for computing graph separators

Computing graph separators in networks has a wide range of real-world applications. For instance, in telecommunication networks, a separator determines the capacity and brittleness of the network. In the field of graph algorithms, the computation of balanced small-sized separators is very useful, especially for divide-and-conquer algorithms. In bioinformatics and computational biology, separators are required in grid graphs providing a simplified representation of proteins. This papers presents a new heuristic algorithm based on the Variable Neighborhood Search methodology for computing vertex separators. We compare our procedure with the state-of-the-art methods. Computational results show that our procedure obtains the optimum solution in all of the small and medium instances, and high-quality results in large instances.