On some solutions of a functional equation related to the partial sums of the Riemann zeta function

In this paper, we prove that infinite-dimensional vector spaces of α-dense curves are generated by means of the functional equations f(x)+f(2x)+⋯+f(nx)=0, with n≥2, which are related to the partial sums of the Riemann zeta function. These curves α-densify a large class of compact sets of the plane for arbitrary small α, extending the known result that this holds for the cases n=2,3. Finally, we prove the existence of a family of solutions of such functional equation which has the property of quadrature in the compact that densifies, that is, the product of the length of the curve by the nth power of the density approaches the Jordan content of the compact set which the curve densifies.

The Zeros of Riemann Zeta Partial Sums Yield Solutions to f(x) + f(2x) + · · · + f(nx) = 0

This paper proves that every zero of any n th , n ≥ 2, partial sum of the Riemann zeta function provides a vector space of basic solutions of the functional equation f(x)+f(2x)+⋯+f(nx)=0,x∈R . The continuity of the solutions depends on the sign of the real part of each zero.

A note on the real projection of the zeros of partial sums of Riemann zeta function

This paper proves that the real projection of each simple zero of any partial sum of the Riemann zeta function ζn(s):=∑nk=11ks,n>2 , is an accumulation point of the set {Res : ζ n (s) = 0}.

Computing the zeros of the partial sums of the Riemann zeta function

In this paper, we introduce a formula for the exact number of zeros of every partial sum of the Riemann zeta function inside infinitely many rectangles of the critical strips where they are situated.

Comparación de las macroalgas epizóicas de la madreperla Pinctada mazatlanica (Hanley 1856) con las de fondos rocosos en la bahía de La Paz, Baja California Sur, México

El presente estudio describe la comunidad de macroalgas epizóicas de Pinctada mazatlanica y la compara con la del substrato rocoso. Colectamos un total de 36 muestras de la comunidad de macroalgas, 18 muestras sobre ostras y otras tantas sobre substrato rocoso en la Península San Juan Nepomuceno, bahía de La Paz, México. Las algas fueron colectadas mediante buceo SCUBA raspando las distintas superficies (25 cm2 de substrato rocoso). El tamaño de muestra fue ajustado por curva de acumulación de especies y expresado con un modelo polinomial. Comparamos las comunidades de algas con análisis multivariantes de similitud basados en el índice de Bray-Curtis, entre substratos, diferentes alturas de ostras perleras y profundidades. Encontramos 27 especies de algas epizóicas (15.4% Clorophyta, 3.8% Phaeophyta y 80.8% Rhodophyta) con una disimilitud de 71.16% con respecto al substrato rocoso. El dendrograma mostró tres agrupaciones de macroalgas en P. mazatlanica. El primero caracterizado por Chondria, Jania, Herposiphonia tenella y Gracilaria. El segundo compuesto por Jania, Polysiphonia acuminata, P. decusata y Spyridia filamentosa. El tercero constituido por Polysiphonia sp., Jania, Herposiphonia tenella, Ceramium canouii y Amphiroa sp. Estas agrupaciones y los talos filamentosos de las algas epizóicas corresponden a estados iniciales de sucesión.

Formation of atomic-sized contacts controlled by electrochemical methods

Electrochemical methods have recently become an interesting tool for fabricating and characterizing nanostructures at room temperature. Simplicity, low cost and reversibility are some of the advantages of this technique that allows to work at the nanoscale without requiring sophisticated instrumentation. In our experimental setup, we measure the conductance across a nanocontact fabricated either by dissolving a macroscopic gold wire or by depositing gold in between two separated gold electrodes. We have achieved a high level of control on the electrochemical fabrication of atomic-sized contacts in gold. The use of electrochemistry as a reproducible technique to prepare nanocontacts will open several possibilities that are not feasible with other methodologies. It involves, also, the possibility of reproducing experiments that today are made by more expensive, complicated or irreversible methods. As example, we show here a comparison of the results when looking for shell effects in gold nanocontacts with those obtained by other techniques.

The Critical Strips of the Sums 1 + 2z + ... + nz

We give a partition of the critical strip, associated with each partial sum 1 + 2z + ... + nz of the Riemann zeta function for Re z < −1, formed by infinitely many rectangles for which a formula allows us to count the number of its zeros inside each of them with an error, at most, of two zeros. A generalization of this formula is also given to a large class of almost-periodic functions with bounded spectrum.

Specific and reversible immobilization of proteins tagged to the affinity polypeptide C-LytA on functionalized graphite electrodes

We have developed a general method for the specific and reversible immobilization of proteins fused to the choline-binding module C-LytA on functionalized graphite electrodes. Graphite electrode surfaces were modified by diazonium chemistry to introduce carboxylic groups that were subsequently used to anchor mixed self-assembled monolayers consisting of N,N-diethylethylenediamine groups, acting as choline analogs, and ethanolamine groups as spacers. The ability of the prepared electrodes to specifically bind C-LytA-tagged recombinant proteins was tested with a C-LytA-β-galactosidase fusion protein. The binding, activity and stability of the immobilized protein was evaluated by electrochemically monitoring the formation of an electroactive product in the enzymatic hydrolysis of the synthetic substrate 4-aminophenyl β-D-galactopyranoside. The hybrid protein was immobilized in an specific and reversible way, while retaining the catalytic activity. Moreover, these functionalized electrodes were shown to be highly stable and reusable. The method developed here can be envisaged as a general, immobilization procedure on the protein biosensor field.

Crystallographic orientation and electrode nature are key factors for electric current generation by Geobacter sulfurreducens

We have investigated the influence of electrode material and crystallographic structure on electron transfer and biofilm formation of Geobacter sulfurreducens. Single-crystal gold - Au(110), Au(111), Au(210) - and platinum - Pt(100), Pt(110), Pt(111), Pt(210) - electrodes were tested and compared to graphite rods. G. sulfurreducens electrochemically interacts with all these materials with different attachment kinetics and final current production, although redox species involved in the electron transfer to the anode are virtually the same in all cases. Initial bacterial colonization was fastest on graphite up to the monolayer level, whereas gold electrodes led to higher final current densities. Crystal geometry showed to have an important influence, with Au(210) sustaining a current density of up to 1442 (± 101) μA cm- 2 at the steady state, over Au(111) with 961 (± 94) μA cm- 2 and Au(110) with 944 (± 89) μA cm- 2. On the other hand, the platinum electrodes displayed the lowest performances, including Pt(210). Our results indicate that both crystal geometry and electrode material are key parameters for the efficient interaction of bacteria with the substrate and should be considered for the design of novel materials and microbial devices to optimize energy production.

Electrochemical features of Pt(S)[n(110) × (100)] surfaces in acidic media

Experiments have been carried out in sulfuric and perchloric acid solutions on Pt(S)[n(110) × (100)] electrodes. The comparison between the two different electrolytic media reveals an important influence of the anion in the voltammetric features. Total charge curves have been obtained with the CO charge displacement method in combination with voltammetric measurements. From these curves, the dependence of the pztc with the step density and the strength of the anion adsorption have been analyzed. The problem of the so-called third peak is treated for a series of electrodes that contain (110) terraces, revealing the requirement of (110) domains for occurrence of this adsorption state.

NO adsorption on Pt (111)/Bi surfaces

The adsorption of nitric oxide (NO) on a Pt (111) surface modified with irreversible adsorbed bismuth adatoms is reported. While the voltammetric results reveal a close interaction between the two co-adsorbed compounds. In-situ infrared spectroscopy and scanning tunnelling microscopy indicate the formation of segregated adlayers. Formation of compressed Bi adlayers with modified redox properties is proposed to reconcile both results. This agrees with the observation of Bi islands in the STM images when NO is coadsorbed, not observed in the absence of NO.

Nitrate reduction at Pt(1 0 0) single crystals and preferentially oriented nanoparticles in neutral media

The electroreduction of nitrate on Pt(1 0 0) electrodes in phosphate buffer neutral solution, pH 7.2, is reported. The sensitivity of the reaction to the crystallographic order of the surface is studied through the controlled introduction of defects by using stepped surfaces with (1 0 0) terraces of different length separated by monoatomic steps, either with (1 1 1) or (1 1 0) symmetry. The results of this study show that nitrate reduction occurs mainly on the well defined (1 0 0) terraces in the potential region where H adsorption starts to decrease, allowing the nitrate anion to access the surface. Adsorbed NO has been detected as a stable intermediate in this media. An oxidation process observed at 0.8 V has been identified as leading to the formation of adsorbed NO and being responsible for a secondary reduction process observed in the subsequent negative scan. Using in situ FTIRS, ammonium was found to be the main product of nitrate reduction. This species can be oxidized at high potentials resulting in adsorbed NO and nitrate (probably with nitrite as intermediate).

Study of dopamine reactivity on platinum single crystal electrode surfaces

Dopamine is the biological molecule responsible, among other functions, of the heart beat and blood pressure regulation. Its loss, in the human body, can result in serious diseases such as Parkinson's, schizophrenia or depression. Structurally, this molecule belongs to the group of catecholamines, together with epinephrine (adrenaline) and norepinephrine (noradrenaline). The hydroquinone moiety of the molecule can be easily oxidized to quinone, rendering the electrochemical methods a convenient approach for the development of dopamine biosensors. The reactivity of similar aromatic molecules, such as catechol and hydroquinone, at well-ordered platinum surfaces, has recently been investigated in our group. In this paper, we extend these studies to the structurally related molecule dopamine. The study has been performed in neutral pH, since this is closer to the natural conditions for these molecules in biological media. Cyclic voltammetry and in situ infra-red spectroscopy have been combined to extract information about the behavior of this molecule on well-defined platinum surfaces. Dopamine appears to be electrochemically active and reveals interesting adsorption phenomena at low potentials (0.15–0.25 V vs RHE), sensitive to the single crystal orientation. The adsorption of dopamine on these surfaces is very strong, taking place at much lower potentials than the electron transfer from solution species. Specifically, the voltammetry of Pt(1 1 1) and Pt(1 0 0) in dopamine solutions shows an oxidation peak at potentials close to the onset of hydrogen evolution, which is related to the desorption of hydrogen and the adsorption of dopamine. On the other hand, adsorption on Pt(1 1 0) is irreversible and the surface appears totally blocked. Spectroscopic results indicate that dopamine is adsorbed flat on the surface. At potentials higher than 0.6 V vs RHE the three basal planes show a common redox process. The initial formation of the quinone moiety is followed by a chemical step resulting in the formation of 5,6-dihydroxyindoline quinone as final product. This oxidation process has also been investigated by vibrational spectroscopy.

Size-Dependent and Step-Modulated Supramolecular Electrochemical Properties of Catechol-Derived Adlayers at Pt(hkl) Surfaces

The electrochemical reactivity of catechol-derived adlayers is reported at platinum (Pt) single-crystal electrodes. Pt(111) and stepped vicinal surfaces are used as model surfaces possessing well-ordered nanometer-sized Pt(111) terraces ranging from 0.4 to 12 nm. The electrochemical experiments were designed to probe how the control of monatomic step-density and of atomic-level step structure can be used to modulate molecule–molecule interactions during self-assembly of aromatic-derived organic monolayers at metallic single-crystal electrode surfaces. A hard sphere model of surfaces and a simplified band formation model are used as a theoretical framework for interpretation of experimental results. The experimental results reveal (i) that supramolecular electrochemical effects may be confined, propagated, or modulated by the choice of atomic level crystallographic features (i.e.monatomic steps), deliberately introduced at metallic substrate surfaces, suggesting (ii) that substrate-defect engineering may be used to tune the macroscopic electronic properties of aromatic molecular adlayers and of smaller molecular aggregates.

Probing the Electrocatalytic Oxygen Reduction Reaction Reactivity of Immobilized Multicopper Oxidase CueO

The bioelectrocatalytic (oxygen reduction reaction, ORR) properties of the multicopper oxidase CueO immobilized on gold electrodes were investigated. Macroscopic electrochemical techniques were combined with in situ scanning tunneling microscopy (STM) and surface-enhanced Raman spectroscopy at the ensemble and at the single-molecule level. Self-assembled monolayer of mercaptopropionic acid, cysteamine, and p-aminothiophenol were chosen as redox mediators. The highest ORR activity was observed for the protein attached to amino-terminated adlayers. In situ STM experiments revealed that the presence of oxygen causes distinct structure and electronic changes in the metallic centers of the enzyme, which determine the rate of intramolecular electron transfer and, consequently, affect the rate of electron tunneling through the protein. Complementary Raman spectroscopy experiments provided access for monitoring structural changes in the redox state of the type 1 copper center of the immobilized enzyme during the CueO-catalyzed oxygen reduction cycle. These results unequivocally demonstrate the existence of a direct electronic communication between the electrode substrate and the type 1 copper center.