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The paper is devoted to exhibiting a proof of an analytical extension of the well-known Loomis–Whitney inequality. Such a proof is completely independent of the original one and it is based on the technique of optimal transport, which leads also to fully characterize the equality case.

Waste activated sludge requires effective dewatering, high biological stability and retention of nutrients prior to disposal for agricultural application. The study was conducted to evaluate the impact of pressure-driven electro-dewatering (EDW) on improving sludge characteristics related to disposal in agriculture, including biological stability, pathogen availability, heavy metals concentrations and nutrients content. Thickened conditioned and mechanically dewatered sludge samples were collected from two wastewater treatment plants (WWTPs), characterized by different stabilization processes, and treated by a lab-scale device at 5, 15 and 25 V. EDW increased significantly the dry solid (DS) content, up to 43–45%, starting from 2 to 3% of raw sludge. The endogenous value of specific oxygen uptake rate (SOUR), monitored as indicator of biological stability, increased up to 56% and 39% after EDW tests for sludge from two WWTPs. On the other hand, the exogenous SOUR decreased, indicating a significant drop in the active bacterial population. Likewise, a 1–2 log unit reduction was observed for E. coli after EDW tests at 15 and 25 V. However, no remarkable removal of heavy metals, namely chromium, nickel, lead, copper and zinc, was achieved. Finally, the concentration of nutrients for soil, such as carbon, nitrogen, phosphorus and sulfur, was not affected by the EDW process. In conclusion, EDW exerts considerable effects on the biological characteristics of sludge, which should be considered in a proper design of sludge management to ensure safe and sustainable resource recovery.

The present paper deals with the problem of computing (or at least estimating) the $$\mathrm {LW}$$ -number $$\lambda (n)$$ , i.e., the supremum of all $$\gamma $$ such that for each convex body K in $${\mathbb {R}}^n$$ there exists an orthonormal basis $$\{u_1,\ldots ,u_n\}$$ such that $$\begin{aligned} {\text {vol}}_n(K)^{n-1} \ge \gamma \prod _{i=1}^n {\text {vol}}_{n-1} (K|u_i^{\perp }) , \end{aligned}$$ where $$K|u_i^{\perp }$$ denotes the orthogonal projection of K onto the hyperplane $$u_i^{\perp }$$ perpendicular to $$u_i$$ . Any such inequality can be regarded as a reverse to the well-known classical Loomis–Whitney inequality. We present various results on such reverse Loomis–Whitney inequalities. In particular, we prove some structural results, give bounds on $$\lambda (n)$$ and deal with the problem of actually computing the $$\mathrm {LW}$$ -constant of a rational polytope.

Incineration units for sewage sludge disposal require a 40% minimum of dry solid content to obtain an ignitable material for energy recovery. Electro-dewatering (EDW) process could be useful to accelerate and improve the removal of adsorbed/interstitial water from sludge, but this application must be as cheap as possible and, further, the use of noble metals as electrodes to avoid corrosion is not appropriate. With these constraints, the use of an organic conductive coating could be suitable to protect low carbon steel electrodes. In this work, we investigated the chemical, electrical and mechanical characteristics of coatings obtained with epoxy resin/multi-walled carbon nanotube (MWCNT) composites, with the aim to design a protective film. The filler/resin ratio is the main parameter to be considered because at high ratio the electrical conductivity increases and, on the reverse, the physical and mechanical performances worsen. Specimens with different concentrations of carbonaceous filler have been produced and their characteristics have been discussed in relation to the critical pigment volume concentration (CPVC) and to the percolation limit.

The concept of an $i$-symmetrization is introduced, which provides a convenient framework for most of the familiar symmetrization processes on convex sets. Various properties of $i$-symmetrizations are introduced and the relations between them investigated. New expressions are provided for the Steiner and Minkowski symmetrals of a compact convex set which exhibit a dual relationship between them. Characterizations of Steiner, Minkowski and central symmetrization, in terms of natural properties that they enjoy, are given and examples are provided to show that none of the assumptions made can be dropped or significantly weakened. Other familiar symmetrizations, such as Schwarz symmetrization, are discussed and several new ones introduced., Comment: A chacterization of central symmetrization has been added and several typos have been corrected. This version has been accepted for publication on Advances in Mathematics

Coatings for high temperatures (HT > 400 °C) are obtained from interpenetrating polymer network (IPN) binders formed by simultaneous polymerization of silicone and epoxide pre-polymers. A ceramic layer; mainly composed of silica and fillers; remains on the metal surface after a thermal treatment at 450 °C. The layer adhesion and the inorganic filler’s distribution have been investigated by, firstly, exchanging the organic substituents (methyl and phenyl) of the silicone chains and, secondly, by adding conductive graphene nanoplatelets with the aim to assure a uniform distribution of heat during the thermal treatment. The results are evidence that different substituent ratios affect the polymer initial layout. The adhesion tests of paint formulations are analysed and were related to instrumental analyses performed using glow discharge optical emission spectroscopy (GDOES); thermal analyses (TG/DTA and DSC); electron microscopy with energy dispersive X-ray analysis (SEM-EDX). A greater resistance to powdering using phenyl groups instead of methyl ones; and an improved distribution of fillers due to graphene nanoplatelet addition; is evidenced.

The sewage sludge dewatering produced by wastewater treatment plants (WWTP) is a multifaceted process due to the presence of colloid fractions. Electro-osmosis could be a suitable technique to reduce the water content of the final sludge. Electric fields of 10, 15, and 20 V/cm have been studied for electro-osmosis tests under the pressure of a static or rotating piston, obtaining a dry solids content up to 40–45%, with respect to 25–30% obtained by mechanical methods. In order to optimize the process, the corrosion behavior and the wear of the anodic material appear to be the main critical aspects, due to the high circulating current density and the use of a rotating electrode. We compared the efficiency and the corrosion resistance of dimensionally stable anodes (DSA) with respect to bare stainless steel (AISI 304) and stainless steel coated by PVD technique with TiN, AlTiN, and DLC. Characterization of the anode surfaces by SEM and potentiodynamic tests show that DSA is the most suitable materia...

The pressure-driven electro-dewatering (EDW) of sewage sludge was assessed using a lab device. The sludge samples were supplied from four different Wastewater Treatment Plants (WWTPs) around the metropolitan area of Milan (Italy), including both aerobically and anaerobically stabilised samples. The test results show that the EDW treatment enabled to dewater the sludge samples to a dry solids content of 18.4–31.1% (wt%), which means 2.0–12.7% improvement as compared to the conventional mechanical dewatering treatment used in these WWTPs. A follow-up test was carried out with the sample giving the best dewatering performance. A dry solids content of 39.3% (wt%) was achieved. Apart from the technological performance, the economic feasibility of EDW was evaluated. The energy consumption and sludge treatment cost associated with the EDW process were compared with the reference case (the corresponding WWTP currently operating with mechanical dewatering line). It was found that for the best performance case, technology upgrade from the conventional mechanical dewatering to the EDW dewatering will enable the WWTP to reduce its sludge management cost up to 35% per year.

The feasibility of pressure-driven electro-dewatering (EDW) on sludge samples taken after different biological processes, stabilisation methods or mechanical dewatering techniques was assessed. First, the influence of potential values on EDW of anaerobically and aerobically stabilised, mechanically dewatered, sludge samples was investigated. Preliminary tests carried out by applying a constant potential (10, 15 and 20 V) in a lab-scale device confirmed the possibility to reach a dry solid (DS) content of up to 42.9%, which corresponds to an increase of 15% of the dry content in dewatered sludge without the application of the electrical field. Dewatering increased with the applied potential but at the expense of a higher energy consumption. A potential equal to 15 V was chosen as the best compromise for EDW performance, in terms of DS content and energy consumption. Then, the influence of the mechanical dewatering was studied on aerobically stabilised sludge samples with a lower initial DS content: the higher initial water content led to a lower final DS content but with a considerable reduction of energy consumption. Finally, the biological process, studied by comparing sludge samples from conventional activated sludge and membrane bioreactor processes, didn’t evidence any influence on EDW. Experimental results shown that DS obtained after mechanical dewatering, volatile solids and conductivity are the main factors influencing EDW. Anaerobically digested sludge reached the highest DS content, thanks to lower organic fraction.

The cement hydration delay is due to comb-polymers (PCP), used as dispersant agent during the preparation of the cement paste. In order to evaluate the role of the PCP structure on hydration, the comb-polymer is separated in two main parts: the backbone (PAA) and the dispersing chain. A linear polymer made up of PEO, Mw = 1000, with a carboxylic head suitable to link the inorganic surface, was synthesized to simulate the effect of the dispersing-side chain. The hydration delay of C-S-H formation induced by comb-polymer on CEM I and C3S phase was analyzed by conduction calorimetry and the morphology of the crystalline structure in growth by SEM, specially studying the relation between structure and hydration time. The results show that the hydration delay is mainly ascribable to the comb-shaped structure as a whole, where PEO chain, as from considerations on energy, might assume conformations able to modify the salt concentration near the cement surface. The different hydration rates are tentatively related to the crystal growth and the surface texture observed by SEM.

Superhydrophobic state is one of the most interesting and attractive topic in these years due to the wide application possibilities. Roughness and surface tension are the main parameters to be controlled to obtain a surface with hydrophobic properties. The topic of this thesis is to investigate how substrate roughness and coating deposition parameters influence wettability properties. Different chemo-physical surface properties were performed by creating different substrate roughness and by applying different coatings. As substrate has been chosen mild carbon steel, use both as received and after sandblasting treatment. Over these substrates a thin organic film coating has been applied. The employed coatings were purchased or were synthesized and characterized in laboratory. Their deposition procedure was made of three main steps: cleaning, etching and post thermal treatment. Etching time and post thermal treatment has been widely surveyed to respectively improve active sites concentration on the metallic substrate and to improve the strength of anchoring bonding. For each kind of coating, three different alkyl chain lengths have been studied to understand their role in the wettability. Optical contact angle analysis was used to establish wetting properties, atomic force microscopy (AFM) was used to characterize samples mechanical properties (roughness and coatings adhesion force). Glow discharge optical emission spectroscopy (GDOES) characterization was used to quantify coatings thickness and to determine their chemical composition. Electrical samples properties were studied by electrochemical impedance (EI), while surface morphologies were detected by using scanning electron microscopy (SEM).

FTIR has been used to follow the evolution of a sol–gel preparation accomplished with two acid/acid steps. The reaction of methyltrimethoxysilane mixed with colloidal silica was taken as an example, aiming at proving the use of IR spectroscopy for the determination of reaction aspects like the kinetic behaviour that is essential for the scale-up of the application. The steps were carried out for 60 min each and samples were taken in each step at variable time intervals. For the spectroscopy analysis the dispersion samples were deposited on a spinning disc of KBr or ZnSe obtaining thin hybrid organic/inorganic films. The signals at 950 cm−1 and 1,000–1,100 cm−1 related to the hydrolysis and condensation reaction, respectively were recorded at variable time and their intensity normalised to the 1,273 cm−1band of Si-CH3 group not involved in the hydrolysis nor condensation reaction. The effects of pH and temperature have been investigated showing that reliable effective data on the reaction extent of SiO2-doped sol–gel dispersions can be obtained by the FTIR spectroscopy. The data have been supported and interpreted also by SEM observations. The reactions degree of the (poly)condensation sharply increases after the addition of more alkoxysilane. Short reaction times can be designed that could be compatible with industrial production.

The rheology of cement pastes can be controlled by polymeric dispersants such as branched polyelectrolytes that adsorb on the surfaces of silicate particles. In the present work, we analyze the adsorption behavior of ad hoc-prepared end-carboxylated poly(ethylene glycol), or PEG, on CaCO(3) particles as a model of cement in an early hydration stage. The experimental adsorption isotherms form the base of a theoretical study aimed at unraveling polymer conformational aspects of adsorption. The study was carried out with Monte Carlo simulations using a coarse-grained bead-and-spring model of linear end-charged polymer chains adsorbing on a flat, continuous, uniformly charged surface. The adsorption driving force was introduced by a Debye-Hückel electrostatic screened potential to describe the interaction between the negatively charged end group of PEG and the positively charged CaCO(3) surface empirically. With a suitable length-scale conversion between real PEG and the coarse-grained model, the calculated and experimental adsorption isotherms can be semiquantitatively compared. The theoretical results reproduce the fundamental aspects of polymer adsorption, in essential agreement with analytical approaches relating the isotherm shape to the polymer conformational properties. The conformational transition mushroom-brush of the adsorbed polymer is located on the isotherm and is related to the molecular shape. The solvent quality effect and the solution ionic strength are also considered, and their implications on the isotherms are discussed.

In this paper a study on the corrosion and wear protection of AM60B magnesium alloy, by means of multilayer coating, is described. The coating mainly consists of a porous oxide layer MgO, produced by oxidation in microarc regime, and a single or multilayer top coat of SiOx produced by sol–gel technique and deposited by either a dip or spray coating method. The anodic oxidation pretreatment improves the adhesion of the sol–gel layers, which seal the porosity of MgO and improve corrosion protection. Furthermore, the multilayer coating significantly improves AM60B wear resistance. Scanning electron microscopy and radiofrequency glow discharge optical emission spectroscopy were used to assess surface morphologies and in depth elemental profiles of the coatings. Electrochemical polarisation tests and wear tests were performed to evaluate the corrosion resistance behaviour and the wear resistance of the coated magnesium alloys.

A theorem due to Favard states that among all plane sets of given area and perimeter, the symmetric lens has maximum circumradius. This paper deals with the higher-dimensional problem of finding the convex body in ℝ 3 of given volume and mean width with the largest possible diameter. It is shown that the solution is the convex hull of a surface of revolution with constant Gauss curvature and a segment lying on the axis of revolution. Such a body is conjectured also to maximize the circumradius in the same class.

For any convex n-gon P we consider the polygons obtained by dropping a vertex or an edge of P. The area distance of P to such (n−1)-gons, divided by the area of P, is an affinely invariant functional on n-gons whose maximizers coincide with the affinely regular polygons. We provide a complete proof of this result. We extend these area functionals to planar convex bodies and we present connections with the affine isoperimetric inequality and parallel X-ray tomography.

The classical Minkowski sum of convex sets is defined by the sum of the corresponding support functions. The Lp-extension of such a definition makes use of the sum of the pth power of the support functions. An Lp-zonotope Zp is the p-sum of finitely many segments and is isometric to the unit ball of a subspace of lq, where 1/p + 1/q = 1. In this paper, a sharp upper estimate is given of the volume of Zp in terms of the volume of Z1, as well as a sharp lower estimate of the volume of the polar of Zp in terms of the same quantity. In particular, for p = 1, the latter result provides a new approach to Reisner's inequality for the Mahler conjecture in the class of zonoids.

The volume of the polar body of a symmetric convex set K of R d is investigated. It is shown that its reciprocal is a convex function of the time t along movements, in which every point of K moves with constant speed parallel to a fixed direction. This result is applied to find reverse forms of the L p -Blaschke-Santalo inequality for two-dimensional convex sets.

The Sylvester (d+2)-points problem deals with the probability S(K) that d + 2 random points taken from a convex compact subset K of are not the vertices of any convex polytope and asks for which sets S(K) is minimal or maximal. While it is known that ellipsoids are the only minimizers of S(K), the problem of the maximum is still open, unless d = 2. In this article we study generalizations of S(K), which include the Busemann functional – appearing in the formula for the volume of a convex set in terms of the areas of its central sections – and a functional introduced by Bourgain, Meyer, Milman and Pajor in connection with the local theory of Banach spaces. We also show that for these more general functionals ellipsoids are the only minimizers and, in the two-dimensional case, triangles (or parallelograms, in the symmetric case) are maximizers.

Basic properties of finite subsets of the integer lattice ℤn are investigated from the point of view of geometric tomography. Results obtained concern the Minkowski addition of convex lattice sets and polyominoes, discrete X-rays and the discrete and continuous covariogram, the determination of symmetric convex lattice sets from the cardinality of their projections on hyperplanes, and a discrete version of Meyer’s inequality on sections of convex bodies by coordinate hyperplanes.

It is known that given any convex bodyK ⊂ ℝ n there is a sequence of suitable iterated Steiner symmetrizations ofK that converges, in the Hausdorff metric, to a ball of the same volume. Hadwiger and, more recently, Bourgain, Lindenstrauss and Milman have given estimates from above of the numberN of symmetrizations necessary to transformK into a body whose distance from the equivalent ball is less than an arbitrary positive constant. In this paper we will exhibit some examples of convex bodies which are “hard to make spherical”. For instance, for any choice of positive integersn≥2 andm, we construct ann-dimensional convex body with the property that any sequence ofm symmetrizations does not decrease its distance from the ball. A consequence of these constructions are some lower bounds on the numberN.

SiO2 and Al2O3 supported Ni catalysts were synthesized in the form of xerogels: the SiO2 based materials were prepared starting from Ni propionate or glycolate salts and reacting them with tetraethoxysilane (TEOS) in propionic acid, Si(ethylene glycolate) or sodium silicate. The Al2O3 supported catalysts were prepared similarly from Ni propionate salts with Al iso-propoxide salts. Narrow metal particles and strong metal support interactions are observed in the sol-gel catalysts. The metal dispersion was higher for Al2O3 based materials than the SiO2 ones and it deeply depends on the Ni precursor for the silica supported Ni. Wet impregnated oxides with similar Ni loading have higher metal surface area than those from sol-gel processing. The influence of surface differences on the catalytic activity of the materials was studied following the CH4 and CO2 reaction in dry reforming conditions by pulse reaction tests.

The volume of the L p -centroid body of a convex body K ⊂ R d is a convex function of a time-like parameter when each chord of K parallel to a fixed direction moves with constant speed. This fact is used to study extrema of some affine invariant functionals involving the volume of the L p -centroid body and related to classical open problems like the slicing problem. Some variants of the L p -Busemann-Petty centroid inequality are established. The reverse form of these inequalities is proved in the two-dimensional case.

The reduction profile of several unpromoted and promoted metal molybdate catalysts was investigated correlating their reducibility with the reactivity in catalysis. Using the stoichiometric α- and β-nickel molybdate compounds it was observed that the reduction rate was significantly affected by the nature of the phase. The results show thatβ-NiMoO4 phase led to a significant increase in the reduction rate with respect to α phase. The increased resistance to reduction by hydrogen due to the structure of the catalytic system is reported. It was found that there is a relationship between the reducibility of the catalysts and selectivity to dehydrogenation products, indicating that the lattice oxygen plays an important role in the reaction. The effect of MoO3, TeO2 and Te2MoO7 added to NiMoO4 systems onthe reducibility of the catalyst and on the propylene oxidation were also studied. It wasobserved that the reduction rate was significantly affected by the nature of the doping element. The results show that NiMoO4–MoO3 combination led toa significant increase of the reduction resistance of the nickel molybdate while TeO2 or Te2MoO7 addition increases its oxygen depletion rate.Ni–Mo–O systems (Mo/Ni>1) were found to favour low COx selectivity, high selectivity to C3H4O and C3H4O2 and good propylene conversion. In presence of TeO2 and Te2MoO7 doped Ni–Mo–O system both acrolein and propylene conversion were increased with respect to the undoped system. Ni–Mo–Te–O catalysts have been found to have a reducibility trend which fits well with the acrolein and acrylic acid formation from propylene oxidation in presence of molecular oxygen.

Oxidative dehydrogenation (ODH) of propane with lattice oxygen of α-, α'- and β-NiMoO4 phases has been studied. Reduction tests of the different phases were performed by TG analysis under hydrogen at different temperatures. Under isothermal conditions lattice-oxygen-consumption rate was found to be higher on the β-phase with respect to the α- and α'-phases. TG analyses showed the existence of different sources of lattice oxygen in the catalyst, depending on the NiMoO4 phase. Catalytic activity was measured under steady state conditions, using a periodic flow reactor and it was then correlated to the reduction curve. It was found that the activation energy for ODH reaction over NiMoO4 of 54.3–62.7 kJ/mol is nearly identical to that obtained for the catalyst reduction under hydrogen (46–50.2 kJ/mol). The correlation between the activation energy of the reaction in a periodic operating system (POS) and that of solid reduction under hydrogen suggests that the reaction obeys a red–ox mechanism.

Ni–La/SiO2 catalysts were prepared by using a new metal organic precursor M(OC3H7)n, dissolved in organic solvent, hydrolysed and finally condensed to form inorganic polymers containing M–O–M or M–(µOH)–M linkages. An optimal distribution of both the active phase ‘Ni’ and the promoter ‘La2O3’ was ensured by addition of their corresponding salts, previously dissolved in propionic acid, to a silica solution prior to gelation. After drying under vacuum the precursor was submitted to thermal treatment in air at 600°C with a heating rate of 1°C min–1.

Increase in catalytic activity performance of a stoichiometric NiMoO4 system has been studied using a periodic flow reactor system. Aging results in an increase of selectivity, a decrease in BET surface area, a weak decrease in activity and a change in surface morphological properties. α-Nickel molybdate exposed to the reaction gases and temperature tends to transform progressively toward an α-aged phase more and more like to the most selective β-phase.

The propane oxydehydrogenation with monolayer lattice oxygen of undoped and K2O–, CaO– and P2O5–NiMoO4 was investigated by using a periodic‐flow reactor (PFR). The influence of the nature and the extent of the promoter has been emphasized relative to the doped catalysts with respect to pure NiMoO4 phases. It was observed that calcium and potassium promoters satisfactorily enhance propylene selectivity, and phosphorus promoter specifically increases the total activity while maintaining the propylene selectivity. Evidence found by thermogravimetric (TG) analyses (oxygen depletion rate) has shown a dependence on lattice oxygen mobility due to the presence of promoters. This dependence has been correlated to the propane conversion while the propylene selectivity was attributed to the acid–base properties.

Oxidative dehydrogenation of propane with nickel molybdate catalysts (αNiMoO4) has been investigated in flow reactors feeding the reactant gases continuously and periodically. The data obtained by the continuous operating system (COS) showed the determining role of lattice oxygen on the selectivity. Moreover, when only the lattice oxygen was consumed, very high propene selectivity was obtained.

We give a necessary condition for a convex body in $ \Bbb R^n $ to have minimal volume in the class of all convex sets with prescribed constant brightness.¶Such a condition permits us to prove that in the class of all convex bodies of revolution with given constant brightness the unique element of minimal volume is a body described by Blaschke as an example of a non-spherical solid of constant brightness.

Nella Neurochirurgia stereotassica, ed in quella «funzionale» in particolare, è basilare la massima accuratezza nel calcolo delle coordinate che identificano la posizione della struttura-bersaglio prescelta: la precisione necessaria è dell'ordine di 1 mm. Viene presentato un originale metodo matematico di calcolo delle coordinate di un target (T) posto ad una distanza nota da un sistema di riferimento. Quest'ultimo è costituito dal piano sagittale passante per la linea bicommissurale (CA-CP) ed il recesso infundibolare (I) del 3° ventricolo e risulta determinato in modo univoco da tre punti (CA, CP, I) le cui coordinate cartesiane sono ricavate direttamente su sezioni assiali TC. L'accuratezza di questo metodo, intrinsecamente valido, è stata valutata preliminarmente su fantoccio. Si tratta di una semplice e rigorosa procedura matematica che può essere adottata in Neurochirurgia funzionale, per il reperaggio di bersagli «invisibili» in interventi di tipo lesivo o di stimolazione su strutture cerebrali profonde. Gli autori hanno impiegato questo metodo di calcolo nel trattamento del morbo di Parkinson con stimolazione cerebrale profonda in cinque pazienti.

The catalytic activities of Rh and Ni metals supported on SiO2 or La2O3 were compared at CH4+CO2 reforming conditions and performing catalytic tests and thermal analyses in thermodynamic conditions favourable for carbon deposition. The deactivation of both the metals deeply depend on the support. An increased CO production is obtained with lanthana but the comparison between Rh and Ni on carbon deposition base suggests that a positive effect could be reached only at low methanation activity.

Coke formation in the dry reforming of methane was studied using a thermobalance (TG) and with a catalytic microreactor in the temperature range 800–950 K. Silica-supported and lanthana-supported nickel catalysts were examined. The effects of process variables such as temperature and gas composition (He dilution, CH4/CO2 ratio) on the coke formation rate were determined. The reactivity of H2 on several kinds of carbon was also investigated. The morphology of the coke was studied by scanning electron microscopy (SEM). The induction times for coke formation were significantly affected by temperature and by the CO content in the feed gas. The results of catalytic tests were consistent with the TG measurements. The behaviour of SiO2 and La2O3 supported Ni catalysts agree with a mechanism in which the lanthana support plays an important role in the carbon deposition.

We consider the class of convex bodies in ℝn with prescribed projection (n − 1)-volumes along finitely many fixed directions. We prove that in such a class there exists a unique body (up to translation) with maximumn-volume. The maximizer is a centrally symmetric polytope and the normal vectors to its facets depend only on the assigned directions.

Three Rh supported catalysts (Rh/SiO 2 , Rh/La 2 O 3 -SiO 2 , Rh/La 2 O 3 ) have been investigated by catalytic tests in the dry reforming reaction (CO 2 +CH 4 --> 2CO+2H 2 ). The La 2 O 3 shows a positive effect on the conversion and selectivity due to the CO 2 activation. The results lead to the supposition of a deep influence of the CH 4 partial pressure on the CO 2 conversion. The mechanism should involve the reaction of carbon dioxide on the support with methane adsorbed on the metal. Thermogravimetric analyses put in evidence the role of metal and seem to confirm a concerted reaction scheme in which carbon deposition and CO 2 conversion simultaneously occurr.

Variants of the brightness function of a convex body K in n-dimensional Euclidean are investigated. The Lambertian lightness function L(K; v , w ) gives the total reflected light resulting from illumination by a light source at infinity in the direction w that is visible when looking in the direction v . The partial brightness function R( K ; v , w ) gives the area of the projection orthogonal to v of the portion of the surface of K that is both illuminated by a light source from the direction w and visible when looking in the direction v . A class of functions called lightness functions is introduced that includes L(K;.) and R(K;.) as special cases. Much of the theory of the brightness function like uniqueness, stability, and the existence and properties of convex bodies of maximal and minimal volume with finitely many function values equal to those of a given convex body, is extended to lightness functions.

We initiate a systematic investigation into the nature of the function K(L; ) that gives the volume of the intersection of one convex body K in R n and a dilatate L of another convex body L in R n , as well as the function K(L; ) that gives the (n 1)- dimensional Hausdor measure of the intersection of K and the boundary @(L ) of L. The focus is on the concavity properties of K(L; ). Of particular interest is the case when K and L are symmetric with respect to the origin. In this situation, there is an interesting change in the concavity properties of K(L; ) between dimension 2 and dimensions 3 or higher. When L is the unit ball, an important special case with connections to E. Lutwak's dual Brunn- Minkowski theory, we prove that this change occurs between dimension 2 and dimensions 4 or higher, and conjecture that it occurs between dimension 3 and dimension 4. We also establish an isoperimetric inequality with equality condition for subsets of equatorial zones in the sphere S 2 , and apply this and the Brunn-Minkowski inequality in the sphere to obtain results related to this conjecture, as well as to the properties of a new type of symmetral of a convex body which we call the equatorial symmetral.

There are sequences of directions such that, given any compact set K in R^n, the sequence of iterated Steiner symmetrals of K in these directions converges to a ball. However examples show that Steiner symmetrization along a sequence of directions whose differences are square summable does not generally converge. (Note that this may happen even with sequences of directions which are dense in S^{n-1}.) Here we show that such sequences converge in shape. The limit need not be an ellipsoid or even a convex set. We also deal with uniformly distributed sequences of directions, and with a recent result of Klain on Steiner symmetrization along sequences chosen from a finite set of directions., 11 pages

The influence of dispersion on the interactions of hydrogen and carbon monoxide with Rh/V2O3 has been investigated (transmission electron microscopy and temperature-programmed desorption). The catalysts having a 0.2% Rh loading were more highly dispersed (8–12A˚) than the catalysts with 1.0% Rh (35–40A˚). Up to 523 K a higher selectivity towards ethanol than towards carbon dioxide is observed upon decreasing the metal dispersion. The metal particle size is being evaluated as a factor of importance for carbon monoxide dissociation.

In this paper we give two characterizations of miquelian Laguerre planes by transitivity properties of the groups ofL-translations andL-homologies respectively.

The following problem, arising from medical imaging, is addressed: Suppose that $T$ is a known tetrahedron in $\R^3$ with centroid at the origin. Also known is the orthogonal projection $U$ of the vertices of the image $\phi T$ of $T$ under an unknown rotation $\phi$ about the origin. Under what circumstances can $\phi$ be determined from $T$ and $U$?, Comment: 16 pages, 3 figures; minor revision based on reviewers' comments

Test data for catalytic oxidation of ethylene glycol to glyoxal performed with a laboratory continuous flow tubular reactor and successively with a pilot plant scale apparatus are reported using a Cu, Ag, P catalyst (94.5%, 4.5% and 1%, respectively). Reaction parameters (O2 concentration, temperature, space velocity) deeply affect the reactor thermal behavior. Relevant features of the complex reaction path can be hypothesized together to useful information on designing the reactor.

The projection body of order one Π1K of a convex body K in ℝn is the body whose support function is, up to a constant, the average mean width of the orthogonal projections of K onto hyperplanes through the origin.The paper contains an inequality for the support function of Π1K, which implies in particular that such a function is strictly convex, unless K has dimension one or two. Furthermore, an existence problem related to the reconstruction of a convex body is discussed to highlight the different behavior of the area measures of order one and of order n – 1.