Cubic symmetric polynomials yielding variations of the Erdős–Ginzburg–Ziv theorem

Let p be a prime and let $\varphi\in\mathbb{Z}_{p}[x_{1},x_{2},\ldots, x_{p}]$ be a symmetric polynomial, where  $\mathbb {Z}_{p}$ is the field of p elements. A sequence T in  $\mathbb {Z}_{p}$ of length p is called a φ-zero sequence if φ(T)=0; a sequence in $\mathbb {Z}_{p}$ is called a φ-zero free sequence if it does not contain any φ-zero subsequence. Motivated by the EGZ theorem for the prime p, we consider symmetric polynomials $\varphi\in \mathbb {Z}_{p}[x_{1},x_{2},\ldots, x_{p}]$ , which satisfy the following two conditions: (i) every sequence in  $\mathbb {Z}_{p}$ of length 2p?1 contains a φ-zero subsequence, and (ii) the φ-zero free sequences in  $\mathbb {Z}_{p}$ of maximal length are all those containing exactly two distinct elements, where each element appears p?1 times. In this paper, we determine all symmetric polynomials in $\mathbb {Z}_{p}[x_{1},x_{2},\ldots, x_{p}]$ of degree not exceeding 3 satisfying the conditions above.     

A dynamical approach to the large-time behavior of solutions to weakly coupled systems of Hamilton–Jacobi equations

We investigate the large-time behavior of the value functions of the optimal control problems on the n-dimensional torus which appear in the dynamic programming for the system whose states are governed by random changes. From the point of view of the study on partial differential equations, it is equivalent to consider viscosity solutions of quasi-monotone weakly coupled systems of Hamilton–Jacobi equations. The large-time behavior of viscosity solutions of this problem has been recently studied by the authors and Camilli, Ley, Loreti, and Nguyen for some special cases, independently, but the general cases remain widely open. We establish a convergence result to asymptotic solutions as time goes to infinity under rather general assumptions by using dynamical properties of value functions.     

Generalized Random Spectral Measures

In an attempt to examine the random version of the spectral theorem, the notion of random spectral measures and generalized random spectral measures are introduced and investigated. It is shown that each generalized random spectral measure on $(\mathbb C ,\mathcal{B}(\mathbb C ))$ admits a modification which is a random spectral measure.     

Optimality and duality in nonsmooth multiobjective fractional programming problem with constraints

In this paper, we establish some quotient calculus rules in terms of contingent derivatives for the two extended-real-valued functions defined on a Banach space and study a nonsmooth multiobjective fractional programming problem with set, generalized inequality and equality constraints. We define a new parametric problem associated with these problem and introduce some concepts for the (local) weak minimizers to such problems. Some primal and dual necessary optimality conditions in terms of contingent derivatives for the local weak minimizers are provided. Under suitable assumptions, sufficient optimality conditions for the local weak minimizers which are very close to necessary optimality conditions are obtained. An application of the result for establishing three parametric, Mond–Weir and Wolfe dual problems and several various duality theorems for the same is presented. Some examples are also given for our findings.

     

Facile synthesis of core‐surface crosslinked nanoparticles by interblock RAFT polymerization

A new, efficient method for synthesizing stable nanoparticles with poly(ethylene oxide) (PEO) functionalities on the core surface, in which the micellization and crosslinking reactions occur in one pot, has been developed. First, amphiphilic PEO‐b‐PS copolymers were synthesized by reversible addition fragmentation chain transfer (RAFT) radical polymerization of styrene using (PEO)‐based trithiocarbonate as a macro‐RAFT agent. The low molecular weight PEO‐b‐PS copolymer was dissolved in isopropyl alcohol where the block copolymer self‐assembled as core‐shell micelles, and then the core‐shell interface crosslink was performed using divinylbenzene as a crosslinking agent and 2,2′‐azobisisobutyronitrile as an initiator. The design of the amphiphilic RAFT agent is critical for the successful preparation of core‐shell interface crosslinked micellar nanoparticles, because of RAFT functional groups interconnect PEO and polystyrene blocks. The PEO functionality of the nanoparticles surface was confirmed by ¹H NMR and FTIR. The size and morphology of the nanoparticles was confirmed by scanning electron microscopy, transmission electron microscopy, and dynamic laser light scattering analysis. © 2010 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2010     

Comparison of three methods for fractionation and enrichment of low molecular weight proteins for SELDI-TOF-MS differential analysis

In most diseases, the clinical need for serum/plasma markers has never been so crucial, not only for diagnosis, but also for the selection of the most efficient therapies, as well as exclusion of ineffective or toxic treatment. Due to the high sample complexity, prefractionation is essential for exploring the deep proteome and finding specific markers.In this study, three different sample preparation methods (i.e., highly abundant protein precipitation, restricted access materials (RAM) combined with IMAC chromatography and peptide ligand affinity beads) were investigated in order to select the best fractionation step for further differential proteomic experiments focusing on the LMW proteome (MW inferior to 40,000 Da). Indeed, the aim was not to cover the entire plasma/serum proteome, but to enrich potentially interesting tissue leakage proteins. These three methods were evaluated on their reproducibility, on the SELDI-TOF-MS peptide/protein peaks generated after fractionation and on the information supplied.The studied methods appeared to give complementary information and presented good reproducibility (below 20%). Peptide ligand affinity beads were found to provide efficient depletion of HMW proteins and peak enrichment in protein/peptide profiles.     

Diagnostic di‐ and triphosphate cyclisation in the negative ion electrospray mass spectra of phosphoSer peptides

It has been shown previously that [M–H]^– anions of small peptides containing two phosphate residues undergo cyclisation of the phosphate groups, following collision‐induced dissociation (CID), to form a characteristic singly charged anion A (H₃P₂O₇^–, m/z 177). In the present study it is shown that the precursor anions derived from the diphosphopeptides of caerin 1.1 [GLLSVLGSVAKHVLPHVVPVIAEHL(NH₂)] and frenatin 3 [GLMSVLGHAVGNVLGGLFKPKS(OH)] also form the characteristic product anion A (m/z 177). Both of the precursor peptides show random structures in water, but partial helices in membrane‐mimicking solvents [e.g. in d₃‐trifluoroethanol/water (1:1)]. In both cases the diphosphopeptide precursor anions must have flexible conformations in order to allow approach of the phosphate groups with consequent formation of A: for example, the two pSer groups of 4,22‐diphosphofrenatin 3 are seventeen residues apart. Finally, CID tandem mass spectrometric (MS/MS) data from the [M–H]^– anion of the model triphosphoSer‐containing peptide GpSGLGpSGLGpSGL(OH) show the presence of both product anions A (m/z 177) and D (m/z 257, H₄P₃O₁₀^–). Ab initio calculations at the HF/6‐31+G(d)//AM1 level of theory suggest that cyclisation of the three phosphate groups occurs by a stepwise cascade mechanism in an energetically favourable reaction (ΔG = ?245 kJ mol^–1) with a maximum barrier of +123 kJ mol^–1. Copyright © 2011 John Wiley & Sons, Ltd.     

FRET detection of Octamer-4 on a protein nanoarray made by size-dependent self-assembly

An alternative approach for fabricating a protein array at nanoscale is suggested with a capability of characterization and/or localization of multiple components on a nanoarray. Fluorescent micro- and nanobeads each conjugated with different antibodies are assembled by size-dependent self-assembly (SDSA) onto nanometer wells that were created on a polymethyl methacrylate (PMMA) substrate by electron beam lithography (EBL). Antibody-conjugated beads of different diameters are added serially and electrostatically attached to corresponding wells through electrostatic attraction between the charged beads (confirmed by zeta potential analysis) and exposed p-doped silicon substrate underneath the PMMA layer. This SDSA method is enhanced by vibrated-wire-guide manipulation of droplets on the PMMA surface containing nanometer wells. Saturation rates of antibody-conjugated beads to the nanometer patterns are up to 97% under one component and 58–70% under two components nanoarrays. High-density arrays (up to 40,000 wells) could be fabricated, which can also be multi-component. Target detection utilizes fluorescence resonance energy transfer (FRET) from fluorescent beads to fluorescent-tagged secondary antibodies to Octamer-4 (Oct4), which eliminates the need for multiple steps of rinsing. The 100 nm green beads are covalently conjugated with anti-Oct4 to capture Oct4 peptides (39 kDa); where the secondary anti-Oct4 and F(ab)₂ fragment of anti-gIgG tagged with phycoerythrin are then added to function as an indicator of Oct4 detection. FRET signals are detected through confocal microscopes, and further confirmed by Fluorolog3 spectrofluorometer. The success rates of detecting Oct4 are 32% and 14% of the beads in right place under one and two component nanoarrays, respectively. Ratiometric FRET is used to quantify the amount of Oct4 peptides per each bead, which is estimated about 2 molecules per bead.