Structural and Computational Existence Results for Multidimensional Subshifts

Symbolic dynamics is a branch of mathematics that studies the structure of infinite sequences of symbols, or in the multidimensional case, infinite grids of symbols. Classes of such sequences and grids defined by collections of forbidden patterns are called subshifts, and subshifts of finite type are defined by finitely many forbidden patterns. The simplest examples of multidimensional subshifts are sets of Wang tilings, infinite arrangements of square tiles with colored edges, where adjacent edges must have the same color. Multidimensional symbolic dynamics has strong connections to computability theory, since most of the basic properties of subshifts cannot be recognized by computer programs, but are instead characterized by some higher-level notion of computability. This dissertation focuses on the structure of multidimensional subshifts, and the ways in which it relates to their computational properties. In the first part, we study the subpattern posets and Cantor-Bendixson ranks of countable subshifts of finite type, which can be seen as measures of their structural complexity. We show, by explicitly constructing subshifts with the desired properties, that both notions are essentially restricted only by computability conditions. In the second part of the dissertation, we study different methods of defining (classes of ) multidimensional subshifts, and how they relate to each other and existing methods. We present definitions that use monadic second-order logic, a more restricted kind of logical quantification called quantifier extension, and multi-headed finite state machines. Two of the definitions give rise to hierarchies of subshift classes, which are a priori infinite, but which we show to collapse into finitely many levels. The quantifier extension provides insight to the somewhat mysterious class of multidimensional sofic subshifts, since we prove a characterization for the class of subshifts that can extend a sofic subshift into a nonsofic one.

The effects of water and sucrose contents on the physicochemical properties of non-directly expanded rice flour extrudates

Rice flour was processed by extrusion cooking in the presence of variable contents of water and sucrose. The process was carried out in a twin-screw extruder under the conditions given by a centre rotational experimental design of second order. The effects of the independent variables, water content (27.9 to 42.1%), and sucrose content (0.1 to 19.9%) on the physicochemical properties of the extrudates were investigated. The water absorption index (WAI), water solubility index (WSI), volumetric expansion index (VEI), and bulk density (BD) were determined as dependent variables. BD was determined for samples before and after frying. An increase in water contents resulted in higher WAI and VEI, and lower WSI and BD for extrudates before and after frying. Higher sucrose levels led to increased values of WAI and VEI and to reduced values of WSI and BD. Both independent variables had significant influence on the physicochemical properties of rice flour extrudates. However, the sucrose content was the most significant. The interaction between these two independent variables and their quadratic effect were also important for the responses studied.

Effects of heat treatment and storage temperature on the use of açaí drink by nutraceutical and beverage industries

This study assesses the storage temperature effect on the anthocyanins of pasteurized and unpasteurized açaí pulp. The data was obtained using a pasteurized and lyophilized pulp (PLP) to evaluate the temperature effect (0, 25, and 40 °C). Part of non-pasteurized frozen pulp (NPP) was pasteurized (NPP-P) at 90 °C for 30 seconds; both pulps were stored at 40 °C. The anthocyanin content reduction in the drink was evaluated from the half-life time (t1/2), activation energy (Ea), temperature quotient (Q10), and the reaction rate constant (k). The t1/2 of the PLP anthocyanins stored at 40 °C was 1.8 times less than that stored at 25 °C and 15 times less than that stored at 0 °C; therefore, the higher temperatures decreased the stability of anthocyanins. The pasteurization increased the t1/2 by 6.6 times (10.14 hours for NPP and 67.28 hours for NPP-P). The anthocyanin degradation on NPP-P followed a first order kinetic, while NPP followed a second order kinetic; thus it can be said that the pasteurization process can improve the preservation of anthocyanins in the pulp.

Optimization and functionality of millet supplemented pasta

Abstract Millets are having superior nutritional qualities and health benefits; hence they can be used for supplementation of pasta. Pasta was prepared using composite flour (CF) of durum wheat semolina (96%) and carrot pomace (4%) supplemented with finger millet flour (FMF, 0-20g), pearl millet flour (PMF, 0-30g) and carboxy methyl cellulose (CMC, 2-4g). Second order polynomial described the effect of FMF, PMF and CMC on lightness, firmness, gruel loss and overall acceptability of extruded pasta products. Results indicate that an increasing proportion of finger and pearl millet flour had signed (p≤0.05) negative effect on lightness, firmness, gruel loss and overall acceptability. However, CMC addition showed significant (p≤0. 05) positive effect on firmness, overall acceptability and negative effect on gruel loss of cooked pasta samples. Numeric optimization results showed that optimum values for extruded pasta were 20g FMF, 12g PMF and 4g CMC per 100g of CF and 34ml water with 0.981 desirability. The pasta developed is nutritionally rich as it contains protein (10.16g), fat (6g), dietary fiber (16.71g), calcium (4.23mg), iron (3.99mg) and zinc (1.682mg) per 100g.

A regional arrangement proposal for the UNASUR

The article analyses the current process of economic integration in South America. Thus, concentrating our attention on the UNASUR regional integration process, two questions arise: First, is UNASUR the most viable institution to achieve a consistent economic integration process in South America? Second, what model of economic integration should be adopted in the case of UNASUR, which would ensure macroeconomic stability and avoid financial and exchange rate crises in the South America? To answer these questions, the article proposes, based on the Keynes (1944/1980)'s revolutionary analysis presented in his International Clearing Union, during the Bretton Woods Conference in 1944, a regional arrangement to UNASUR.

Rationality in children: the first steps

Not all categorization is conceptual. Many of the experimental findings concerning infant and animal categorization invite the hypothesis that the subjects form abstract perceptual representations, mental models or cognitive maps that are not composed of concepts. The paper is a reflection upon the idea that conceptual categorization involves the ability to make categorical judgements under the guidance of norms of rationality. These include a norm of truth-seeking and a norm of good evidence. Acceptance of these norms implies willingness to defer to cognitive authorities, unwillingness to commit oneself to contradictions, and knowledge of how to reorganize one's representational system upon discovering that one has made a mistake. It is proposed that the cognitive architecture required for basic rationality is similar to that which underlies pretend-play. The representational system must be able to make room for separate 'mental spaces' in which alternatives to the actual world are entertained. The same feature underlies the ability to understand modalities, time, the appearance-reality distinction, other minds, and ethics. Each area of understanding admits of degrees, and mastery (up to normal adult level) takes years. But rational concept-management, at least in its most rudimentary form, does not require a capacity to form second-order representations. It requires knowledge of how to operate upon, and compare, the contents of different mental spaces.

Kinetic and mechanistic studies for the molybdenum- catalyzed oxidation of organic sulfides and olefins by t- DuO2H

This research was focussed on the effects of light, solvent and substituents in the molybdenum-catalyzed oxidation of phenylmethyl sulfides with t-Bu02H and on the effect of light in the molybdenum-catalyzed epoxidation of l-octene with t-Bu02H. It was shown that the Mo(CO)6-catalyzed oxidation of phenylmethyl sulfide with t-Bu02H~ at 35°C, proceeds 278 times faster underUV light than under laboratory lighting, whereas the Mo02(acac)2-catalyzed oxidation proceeds only 1.7 times faster under UV light than under normal laboratory lighting. The difference between the activities of both catalysts was explained by the formation of the catalytically active species, Mo(VI). The formation of the Mo(VI) species, from Mo(CO)6 was observed from the IR spectrum of Mo(CO)6 in the carbonyl region. The Mo(CO)6-catalyzed epoxidation of l-octene with t-Bu02H showed that the reaction proceeded 4.6 times faster under UV light than in the dark or under normal laboratory lighting; the rates of epoxidations were found to be the same in the dark and under normal laboratory lighting. The kinetics of the epoxidations of l-octene with t-Bu02H, catalyzed by Mo02(acac)2 were found to be complicated; after fast initial rates, the epoxidation rates decreased with time. The effect of phenylmethyl sulfide on the Mo(CO)6-catalyzed epoxidation of l-octene waS studied. It was shown that instead of phenylmethyl sulfide, phenylmethyl sulfone, which formed rapidly at 85°C, lowered the reaction rate. The epoxidation of l-octene was found to be 2.5 times faster in benzene than in ethanol. The substituent effect on the Mo02(acac)2-catalyzed oxidations of p-OH, p-CHgO, P-CH3' p-H, p-Cl, p-Br, p-CHgCO, p-HCO and P-N02 substituted phenylmethyl sulfides were studied. The oxidations followed second order kinetics for each case; first order dependency on catalyst concentration was also observed in the oxidation of p-CHgOPhSMeand PhSMe. It was found that electron-donating groups on the para position of phenylmethyl sulfide increased the rate of reaction, while electronwithdrawing groups caused the reaction rate to decrease. The reaction constants 0 were determined by using 0, 0- and 0* constants. The rate effects were paralleled by the activation energies for oxidation. The decomposition of t-Bu02H in the presence of M.o (CO)6, Mo02 (acac)2 and VO(acac)2 was studied. The rates of decomposition were found to be very small compared to the oxidation rates at high concentration of catalysis. The relative rates of the Mo02(acac)2-catalyzed oxidation of p-N02PhSMe by t-Bu02H in the presence of either p-CH30PhSMe or PhSMe clearly show that PhSMe and p-CHgOPhSMe act as co-catalysts in the oxidation of p-N02PhSMe. Benzene, mesity1ene and cyclohexane were used to determine the effect of solvent in the Mo02 (acac)2 and Mo(CO)6-catalyzed oxidation of phenylmethyl sulfide. The results showed that in the absence of hydroxylic solvent, a second molecule of t-Bu02H was involved in the transition state. The complexation of the solvent with the catalyst could not be explained.The oxidations of diphenyl sulfoxide catalyzed by VO(acac)2, Mo(CO)6 and Mo02(acac)2 showed that VO(acac)2 catalyzed the oxidation faster than Mo(CO)6 and Mo02 (acac)2_ Moreover, the Mo(CO)6-catalyzed oxidation of diphenyl sulfoxide proceeded under UV light at 35°C.

Solute effects on the production and decay of primary species in the flash photolysis of indole

Maximum production rates ofs and decay kinetics for the hydrated electron, the indolyl neutral radical and the indole triplet state have been obtained in the microsecond, broadband (X > 260 nm) flash photolysis of helium-saturated, neutral aqueous solutions of indole, in the absence and in the presence of the solutes NaBr, BaCl2*2H20 and CdSCV Fluorescence spectra and fluorescence lifetimes have also been obtained in the absence and in the presence of the above solutes, The hydrated electron is produced monophotonically and biphotonically at an apparent maximum rate which is increased by BaCl2*2H20 and decreased by NaBr and CdSOif. The neutral indolyl radical may be produced monophotonically and biphotonically or strictly monophotonically at an apparent maximum rate which is increased by NaBr and CdSO^ and is unaffected by BaCl2*2H20. The indole triplet state is produced monophotonically at a maximum rate which is increased by all solutes. The hydrated electron decays by pseudo first order processes, the neutral indolyl radical decays by second order recombination and the indole triplet state decays by combined first and second order processes. Hydrated electrons are shown to react with H , H2O, indole, Na and Cd"*""1"". No evidence has been found for the reaction of hydrated electrons with Ba . The specific rate of second order neutral indolyl radical recombination is unaffected by NaBr and BaCl2*2H20, and is increased by CdSO^. Specific rates for both first and second order triplet state decay processes are increased by all solutes. While NaBr greatly reduced the fluorescence lifetime and emission band intensity, BaCl2*2H20 and CdSO^ had no effect on these parameters. It is suggested that in solute-free solutions and in those containing BaCl2*2H20 and CdSO^, direct excitation occurs to CTTS states as well as to first excited singlet states. It is further suggested that in solutions containing NaBr, direct excitation to first excited singlet states predominates. This difference serves to explain increased indole triplet state production (by ISC from CTTS states) and unchanged fluorescence lifetimes and emission band intensities in the presence of BaCl2*2H20 and CdSOt^., and increased indole triplet state production (by ISC from S^ states) and decreased fluorescence lifetime and emission band intensity in the presence of NaBr. Evidence is presented for (a) very rapid (tx ^ 1 us) processes involving reactions of the hydrated electron with Na and Cd which compete with the reformation of indole by hydrated electron-indole radical cation recombination, and (b) first and second order indole triplet decay processes involving the conversion of first excited triplet states to vibrationally excited ground singlet states.

Dinocyst biochronology and palynofacies : inferred systems tract character of Miocene sequences from New Jersey mid-Atlantic transect (ODP legs 174A and 174AX) /

One of the main objectives of the mid-Atlantic transect is to improve dating resolution of sequences and unconfonnity surfaces. Dinoflagellate cysts from two Ocean Drilling Program boreholes, the onshore Leg 174AX Ocean View Site and Leg 174A continental shelf Site 1071, are used to provide age estimates for sequences and unconfonnities fonned on the New Jersey continental margin during the Miocene epoch. Despite the occasional lack of dinocysts in barren and oxidized sections, dinocyst biochronology still offers greater age control than that provided by other microfossils in marginal marine environments. An early Miocene to late Miocene chronology based on ages detennined for the two study sites is presented. In addition, .palynofacies are used to unravel the systems tract character of the Miocene sequences and provide insight into the effects of taphonomy and preservation of palynomorphs in marginal marine and shelf environments under different ~ea level conditions. More precise placement of maximum flooding surfaces is possible through the identification of condensed sections and palynofacies shifts can also reveal subaerially exposed sections and surfaces not apparent in seismic or lithological analyses. The problems with the application of the pollen record in the interpretation of Miocene climate are also discussed. Palynomorphs provide evidence for a second-order lowering of sea level during the Miocene, onto which higher order sea level fluctuations are super-imposed. Correlation of sequences and unconfonnities is attempted between onshore boreholes and from the onshore Ocean View borehole to offshore Site 1071.

Kinetic and product studies for the oxidtion of organic sulfides and sulfoxides in the presence of transition metal complexes

Rates and products of the oxidation of diphenyl sulfide, phenyl methyl sulfide, p-chlorophenyl methyl sulfide and diphenyl sulfoxide have been determined. Oxidants included t-Bu02H alone, t-Bu02H plus molybdenum or vanadium catalysts and the molybdenum peroxo complex Mo0(02)2*HMPT. Reactions were chiefly carried out in ethanol at temperatures ranging from 20° to 65°C. Oxidation of diphenyl sulfide by t-Bu02H in absolute ethanol at 65°C followed second-order kinetics with k2 = 5.61 x 10 G M~1s"1, and yielded only diphenyl sulfoxide. The Mo(C0)g-catalyzed reaction gave both the sulfoxide and the sulfone with consecutive third-order kinetics. Rate = k3[Mo][t-Bu02H][Ph2S] + k^[Mo][t-Bu02H][Ph2S0], where log k3 = 12.62 - 18500/RT, and log k^ = 10.73 - 17400/RT. In the absence of diphenyl sulfide, diphenyl sulfoxide did not react with t-Bu02H plus molybdenum catalysts, but was oxidized by t-Bu02H-V0(acac)2. The uncatalyzed oxidation of phenyl methyl sulfide by t-Bu02H in absolute ethanol at 65°C gave a second-order rate constant, k = 3.48 x 10~"5 M^s""1. With added Mo(C0)g, the product was mainly phenyl methyl sulfoxide; Rate = k3[Mo][t-Bu02H][PhSCH3] where log k3 = 22.0 - 44500/RT. Both diphenyl sulfide and diphenyl sulfoxide react readily with the molybdenum peroxy complex, Mo0(02)2'HMPT in absolute ethanol at 35°C, yielding diphenyl sulfone. The observed features are mainly in agreement with the literature on metal ion-catalyzed oxidations of organic compounds by hydroperoxides. These indicate the formation of an active catalyst and the complexation of t-Bu02H with the catalyst. However, the relatively large difference between the activation energies for diphenyl sulfide and phenyl methyl sulfide, and the non-reactivity of diphenyl sulfoxide suggest the involvement of sulfide in the production of an active species.

Learning styles and achievement in postsecondary classrooms

The purpose of this study was to investigate the relationship between learning styles and academic achievement in postsecondary education. It was the intent of the study to establish if there was a relationship between student learning style, teacher style, learner/teacher matching and/or mismatching, student gender and age, to the academic grades of students. This study was basically a replication of a study completed by Mary J. Thompson and Terrance P. O'Brien in 1991 on two campuses of a southeast community college in the United States. In the present study, 243 students and 18 teachers from two different campuses of a community college in the Province of Ontario participated in the research. All participants were administered the Gregorc Style Delineator and students identified by program, age and gender. Data were tested by two analysis of variance (ANOVA) models. In the first ANOVA model considered in this study, significant main effects were manifested in regard to the teaching style, age group and gender. With the exception of gender, these findings were very similiar to those of the original study. Duncan's multiple range test revealed that Concrete Sequential (CS) teachers assigned significantly lower grades than did teachers dominant in any of the other three learning styles. Post hoc testing revealed that students 25 years of age and older received significantly higher grades than did younger students. Female students also received significantly higher grades than did male students. In the second ANOVA model student/teacher learning style match/mismatch did emerge as a significant main effect. However, Duncan's multiple range test and Chi square analysis did not substantiate the relationship. Forty-eight references are cited.

The Effects of Annealing URu2Si2 on the Resistivity and Meissner Effect

The enigmatic heavy fermion URu2Si2, which is the subject of this thesis, has attracted intensive theoretical and experimental research since 1984 when it was firstly reported by Schlabitz et al. at a conference [1]. The previous bulk property measurements clearly showed that one second order phase transition occurs at the Hidden Order temperature THO ≈ 17.5 K and another second order phase transition, the superconducting transition, occurs at Tc ≈ 1 K. Though twenty eight years have passed, the mechanisms behind these two phase transitions are still not clear to researchers. Perfect crystals do not exist. Different kinds of crystal defects can have considerable effects on the crystalline properties. Some of these defects can be eliminated, and hence the crystalline quality improved, by annealing. Previous publications showed that some bulk properties of URu2Si2 exhibited significant differences between as-grown samples and annealed samples. The present study shows that the annealing of URu2Si2 has some considerable effects on the resistivity and the DC magnetization. The effects of annealing on the resistivity are characterized by examining how the Residual Resistivity Ratio (RRR), the fitting parameters to an expression for the temperature dependence of the resistivity, the temperatures of the local maximum and local minimum of the resistivity at the Hidden Order phase transition and the Hidden Order Transition Width ∆THO change after annealing. The plots of one key fitting parameter, the onset temperature of the Hidden Order transition and ∆THO vs RRR are compared with those of Matsuda et al. [2]. Different media used to mount samples have some impact on how effectively the samples are cooled because the media have different thermal conductivity. The DC magnetization around the superconducting transition is presented for one unannealed sample under fields of 25 Oe and 50 Oe and one annealed sample under fields of 0 Oe and 25 Oe. The DC field dependent magnetization of the annealed Sample1-1 shows a typical field dependence of a Type-II superconductor. The lower critical field Hc1 is relatively high, which may be due to flux pinning by the crystal defects.

Reduced power output in skeletal muscles devoid of skMLCK: RLC phosphorylation contributes to peak performance

Regulatory light chain (RLC) phosphorylation in fast twitch muscle is catalyzed by skeletal myosin light chain kinase (skMLCK), a reaction known to increase muscle force, work, and power. The purpose of this study was to explore the contribution of RLC phosphorylation on the power of mouse fast muscle during high frequency (100 Hz) concentric contractions. To determine peak power shortening ramps (1.05 to 0.90 Lo) were applied to Wildtype (WT) and skMLCK knockout (skMLCK-/-) EDL muscles at a range of shortening velocities between 0.05-0.65 of maximal shortening velocity (Vmax), before and after a conditioning stimulus (CS). As a result, mean power was increased to 1.28 ± 0.05 and 1.11 ± .05 of pre-CS values, when collapsed for shortening velocity in WT and skMLCK-/-, respectively (n = 10). In addition, fitting each data set to a second order polynomial revealed that WT mice had significantly higher peak power output (27.67 ± 1.12 W/ kg-1) than skMLCK-/- (25.97 ± 1.02 W/ kg-1), (p < .05). No significant differences in optimal velocity for peak power were found between conditions and genotypes (p > .05). Analysis with Urea Glycerol PAGE determined that RLC phosphate content had been elevated in WT muscles from 8 to 63 % while minimal changes were observed in skMLCK-/- muscles: 3 and 8 %, respectively. Therefore, the lack of stimulation induced increase in RLC phosphate content resulted in a ~40 % smaller enhancement of mean power in skMLCK-/-. The increase in power output in WT mice suggests that RLC phosphorylation is a major potentiating component required for achieving peak muscle performance during brief high frequency concentric contractions.

L’intégration de la perception visuelle du mouvement

La perception visuelle du mouvement est essentielle à l’exécution de déplacements sécuritaires ainsi qu’à l’interaction efficace avec notre environnement. C’est pourquoi il est nécessaire de comprendre la nature des mécanismes responsables de l’analyse de l’information sur le mouvement, ainsi que l’effet du vieillissement sur la réponse de ces mécanismes. Deux études seront présentées. La première avait pour but l’analyse des mécanismes responsables de la perception du mouvement de rotation fractale, nouveau stimulus introduit par Benton, O’Brien & Curran (2007). Ce type de stimulus a été créé afin d’isoler les mécanismes sensibles à la forme. Plusieurs auteurs ont suggéré que les mécanismes sensibles au mouvement de deuxième ordre utiliseraient les indices de position afin d’extraire l’information sur le mouvement (Seiffert & Cavanagh, 1998). Ainsi, la présente étude visait à déterminer si la rotation fractale est analysée par de tels mécanismes. Les résultats obtenus suggèrent que les mécanismes sensibles à la rotation fractale seraient basés sur l’orientation; tandis que ceux sensibles à la rotation de premier ordre, basés sur l’énergie. De plus, une certaine dissociation des mécanismes responsables du traitement de la rotation fractale et de premier ordre serait présente. La deuxième étude avait pour but, quant à elle, d’établir l’effet du vieillissement sur l’intégration du mouvement de premier et deuxième ordre. Les résultats indiquent que les mécanismes sensibles au mouvement de deuxième ordre seraient davantage affectés, comparativement à ceux de premier ordre. Ainsi, les fonctions visuelles requérant une intégration corticale de plus haut niveau seraient davantage affectées par l’effet du vieillissement.

Théorèmes d'existence pour des systèmes d'équations différentielles et d'équations aux échelles de temps.

Nous présentons dans cette thèse des théorèmes d’existence pour des systèmes d’équations différentielles non-linéaires d’ordre trois, pour des systèmes d’équa- tions et d’inclusions aux échelles de temps non-linéaires d’ordre un et pour des systèmes d’équations aux échelles de temps non-linéaires d’ordre deux sous cer- taines conditions aux limites. Dans le chapitre trois, nous introduirons une notion de tube-solution pour obtenir des théorèmes d’existence pour des systèmes d’équations différentielles du troisième ordre. Cette nouvelle notion généralise aux systèmes les notions de sous- et sur-solutions pour le problème aux limites de l’équation différentielle du troisième ordre étudiée dans [34]. Dans la dernière section de ce chapitre, nous traitons les systèmes d’ordre trois lorsque f est soumise à une condition de crois- sance de type Wintner-Nagumo. Pour admettre l’existence de solutions d’un tel système, nous aurons recours à la théorie des inclusions différentielles. Ce résultat d’existence généralise de diverses façons un théorème de Grossinho et Minhós [34]. Le chapitre suivant porte sur l’existence de solutions pour deux types de sys- tèmes d’équations aux échelles de temps du premier ordre. Les résultats d’exis- tence pour ces deux problèmes ont été obtenus grâce à des notions de tube-solution adaptées à ces systèmes. Le premier théorème généralise entre autre aux systèmes et à une échelle de temps quelconque, un résultat obtenu pour des équations aux différences finies par Mawhin et Bereanu [9]. Ce résultat permet également d’obte- nir l’existence de solutions pour de nouveaux systèmes dont on ne pouvait obtenir l’existence en utilisant le résultat de Dai et Tisdell [17]. Le deuxième théorème de ce chapitre généralise quant à lui, sous certaines conditions, des résultats de [60]. Le chapitre cinq aborde un nouveau théorème d’existence pour un système d’in- clusions aux échelles de temps du premier ordre. Selon nos recherches, aucun résultat avant celui-ci ne traitait de l’existence de solutions pour des systèmes d’inclusions de ce type. Ainsi, ce chapitre ouvre de nouvelles possibilités dans le domaine des inclusions aux échelles de temps. Notre résultat a été obtenu encore une fois à l’aide d’une hypothèse de tube-solution adaptée au problème. Au chapitre six, nous traitons l’existence de solutions pour des systèmes d’équations aux échelles de temps d’ordre deux. Le premier théorème d’existence que nous obtenons généralise les résultats de [36] étant donné que l’hypothèse que ces auteurs utilisent pour faire la majoration a priori est un cas particulier de notre hypothèse de tube-solution pour ce type de systèmes. Notons également que notre définition de tube-solution généralise aux systèmes les notions de sous- et sur-solutions introduites pour les équations d’ordre deux par [4] et [55]. Ainsi, nous généralisons également des résultats obtenus pour des équations aux échelles de temps d’ordre deux. Finalement, nous proposons un nouveau résultat d’exis- tence pour un système dont le membre droit des équations dépend de la ∆-dérivée de la fonction.