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The study of topological invariants of finite topological spaces is relevant because they can be used as models of a wide class of topological spaces, including regular CW-complexes. In this work, we present a new module for the Kenzo system that allows the computation of homology groups with generators of finite topological spaces in different situations. Our algorithms combine new constructive versions of well-known results about topological spaces with combinatorial methods used on finite spaces. In the particular case of h-regular spaces, effective and reasonably efficient methods are implemented and the technique of discrete vector fields is applied in order to improve the previous algorithms.

In this work we present an interface between SageMath and Kenzo, together with an optional Kenzo package. Our work makes it possible to communicate both computer algebra programs and enhances the SageMath system with new capabilities in algebraic topology, dealing in particular with simplicial objects of infinite nature.

Spain is the leading producer of tropical fruit crops in Europe. Within Spain, papaya cultivation is concentrated in the Canary Islands, where 350 ha are planted and 16,000 t produced annually. Recently, papaya has also been introduced into the south-east of Spain. More than 100 ha of papaya orchards under protected cultivation have now planted in south-east Spain. This region has a subtropical, semiarid, Mediterranean climate with a hot and dry summer and a mild winter, with high temperatures in summer (above 40°C) and low ones in winter (sometimes below 10°C) being the main climate challenges. Papaya cultivation in south-east Spain takes place very often in low-technology parral greenhouses, 3.5-4 m tall, covered with plastic and only provided with natural ventilation. In order to improve the climate inside greenhouses, different systems of nebulization and heating are being evaluated. ‘Intenzza’, a ‘Formosa’ hybrid from Mexico, is so far the most planted cultivar. Other cultivars under cultivation are ‘Siluet’, ‘Sweet Sense’, ‘Tainung’, ‘BH-65’ and ‘Caballero’. Papaya multiplication is by seed, but the use of molecular markers to determine papaya sex allows rapid selection of hermaphrodite plants in the nursery. This technique makes crop management easier. Plant density ranges from 1600 to 2700 plants ha(‑1), transplanted more frequently in spring (March-April), although transplanting at other times of year is also possible. Harvest begins 7 months after transplanting. The yield reaches 200 t ha(‑1) and shows two peaks, one after winter and the other after summer. The main pest so far observed is the red mite, while the most important diseases are powdery mildew and botrytis. There is no record of Papaya ring spot virus (PRSV) incidence in our fields. Given the higher cost of production of papaya in Spain, we believe that the key for the successful introduction of Spanish papaya in European markets is to allow fruit ripening on the tree, and to produce sweet fruits of excellent quality.

In countries new to producing ‘Manzanillo’ olive trees (Olea europaea), free cross-pollination is often insufficient to obtain high levels of fruit set. An appropriate pollination design is therefore essential to ensure a timely, abundant, and compatible pollen supply. With a view to determining whether a pollination deficit exists in a nontraditional olive area such as the northern Mexico, pollination experiments were carried out in two consecutive seasons in both a monovarietal and a multivarietal Manzanillo orchard, where Sevillano, Barouni, Picual, Pendolino, Mission, Nevadillo, and Frantoio trees were growing nearby. The pollination treatments were self-, open, and cross-pollination with ‘Barouni’ and ‘Sevillano’ pollen, the latter only in the multivarietal orchard. The results confirmed the full self-incompatible condition of ‘Manzanillo’. Open-pollination did not improve fruit set in the monovarietal orchard, but it did so significantly in the multivarietal plot, where fruit set levels under open-pollination matched those of cross-pollination. Lower pollen adhesion, as well as occasional decreased germination, and reduced and delayed pollen tube growth were observed under self-pollination, highlighting self-incompatibility reactions. The reduction in fertilization rates led to low fruit set under self-pollination. Positive effects of open- and cross-pollination treatments were also noted on fruit weight (despite higher crop loads) and pulp-to-pit ratios. A strategic plantation design, including appropriate pollinizers in the right number and position, is therefore suggested for increasing ‘Manzanillo’ fruit quality and yield in Mexico. Both ‘Barouni’ and ‘Sevillano’ served as efficient pollinizers for ‘Manzanillo’, although we recommend ‘Barouni’ as a more efficient because the bloom periods of them matched that of ‘Manzanillo’

Papaya (Carica papaya L.) is one of the main horticultural crops of many tropical and subtropical regions. The fruit is sold either as a fresh product or processed into drinks, jams, candies, dried and crystallized fruit, while the enzyme papain is used for medicinal purposes. Papaya fruits have high vitamins A and C contents, as well as are good sources of calcium. Brazil is one of the most important producers and exporters of papaya in the world; in 2016 c.a., 1,424,650 tons of papaya was produced in 30,372 hectares of the territory. Optimum light absorption and utilization by the canopy are important factors for maximizing papaya crop growth and productivity. Thus, knowing how papaya responds to light is important to develop management strategies to optimize fruit yield and quality. This short review aims to present the current research knowledge related to the effects of light intensity on the photosynthetic processes and growth of papaya. We demonstrate that photosynthetically active radiation (PAR) greatly affects the physiology of papaya. Understanding the interaction between light and physiological processes is extremely important for a sustainable profitable production under either greenhouse or field conditions. By using improved light science-based management, growers may optimize photosynthetic carbon assimilation and increase papaya yield and fruit quality

Papaya is a tropical fruit crop that produces large-sized berries of high nutritional value. Due to climatic restrains, papaya cultivation in Europe is restricted to limited areas of South East Spain where, even so, it must be grown under protected cultivation in greenhouses to be profitable. The knowledge of papaya fruit growth dynamic and its thermal requirements measured in Growing Degree Days (GDD) from bloom to ripening are essential to predict fruit size and harvest dates. Having this in mind, the objective of this work was to establish the best model describing papaya fruit growth and GDD required for fruit ripening in four cultivars selected for their contrasting vigor and fruit size. The results show that papaya fruits grow according to a simple sigmoid curve. For the sake of simplicity and the good fitting obtained, we recommend Gompertz equation for describing papaya fruit growth. Seasonal changes in the time required from flowering to harvest were detected. Fruits setting in autumn required 8 months to mature, while the fruits setting in spring reached ripening after only 5 months. Heat units required for fruit ripening ranged between 1293 GDD for ‘Siluet’ and 1488 GDD for ‘Red Lady’. Its calculation was revealed as a more useful tool to predict harvest dates than counting calendar days. Initial fruit size was found to be a clear determinant of fruit size at harvest; thus, small fruitlets remained proportionally small as ripening, while large fruitlets reached premium sizes at harvest.

Manzanillo’ olive (Olea europaea L.) cultivar suffers from pollination deficits that make it difficult for growers to achieve high productivity. This variety has been designated as completely self-incompatible by some authors, while others label it as partially self-incompatible. The aim of this study was to determine the degree of self-incompatibility and identify obstacles to self-fertilization in ‘Manzanillo’. For this, in a multivarietal orchard in El Ejido-Almeria, Spain (36° 50’ NL, 2° 24’‘ WL), a completely randomized experiment was established with four pollination treatments (self-pollination, free-pollination, and cross-pollination with pollen from ‘Picual’ and ‘Sevillano’). Pollen adhesion and germination, pollen tube growth and fertilization every three days after pollination were evaluated. Analyses showed no differences among treatments in pollen adhesion. The effect of pollination on pollen germination was erratic, because although free-pollination showed more germination than self-pollination at the beginning, the differences disappeared at nine days after anthesis. The clearest self-incompatibility reaction was detected in the growth of the pollen tube, with little and late growth of the pollen tube under self-pollination. The lowest fertilization rate was observed with self-pollination, and the highest with free-pollination

Papaya is a tropical crop increasingly cultivated in the greenhouses of subtropical regions such as South East Spain, where the determination of the best planting season is important to ensure a stable fruit production and quality during the year. In this work, we studied plant growth, yield, and fruit quality, comparing spring and autumn planting seasons in ‘Intenzza’ cultivar. The results showed that planting in spring favors plant growth, leading to an earlier entry into production. Total yield and fruit quality were similar in both planting seasons, although the spring cycle provided higher profits due to greater commercial yield and lower discards. Our results confirm that adverse environmental conditions affect the crop in a similar way regardless of the planting season, so different growing cycles are not very useful for filling the production gaps and fighting against seasonality in our region, unless unfavorable climate conditions are avoided inside the greenhouse.

Greenhouse cultivation of table grapes is a challenge due to difficulties imposed by their perennial habit and chilling requirements. Despite difficulties, greenhouse cultivation allows ripening long before that in the open field. Nonetheless, for harvesting “Flame Seedless” in the most profitable periods, a cultural practices timetable has to be established. In this context, an estimation of development rate as a function of temperature becomes essential. This work puts forward a procedure to determine “Flame Seedless” threshold temperatures and heat requirements from bud break to ripening. “Flame Seedless” required an average of 1633 growing degree days (GDD) in the open field with a base temperature of 5 °C and an upper threshold temperature of 30 °C. Strikingly, only 1542 GDD were required within the greenhouse. This procedure forecast “Flame Seedless” ripening with an accuracy of three and six days in the open field and greenhouse, improving predictions based on the average number of days between bud break and ripening. The procedure to predict oncoming harvest date was found satisfactory, just four days earlier than the real date. If we used the typical meteorological year instead of the average year, then the prediction was greatly improved since harvest was forecast just one day before its occurrence.

In this work we present an ongoing project on the improving of a previous symboliccomputation algorithm computing discrete vector fields on finite topological spaces. To this aim, we consider different strategies to choose each one of the possible vectors at each step of the algorithm and we apply some reinforcement learning techniques.

This chapter provides information on the various factors affecting the fruit set, fruit development, maturity and ripening of pawpaws.

Lack of color in the skin of red table grape varieties is a serious problem in areas of warm climate. This problem is often addressed by the application of ethylene release products such as ethephon. Strict regulation in the use of this product in EU forces European grape producers to look for suitable alternatives. With the aim to increase red skin color, we applied regulated deficit irrigation (RDI) strategies from veraison until harvest on &ldquo, Flame Seedless&rdquo, table grape vines cultivated under nets and under a plastic greenhouse in South East Spain, and compared yield and fruit quality with vines fully irrigated under the same net and plastic greenhouses. Our results show a modest improvement in the percentage of commercial clusters with better skin color, probably because the short duration of the deficit irrigation period only caused a slight decrease in soil water content and a mild water stress in RDI vines. Larger differences were observed under the more limiting conditions of the plastic greenhouse for light environment, especially when berry skin color was measured by CIRG (color index of red grape). More noticeable effect of RDI was noted on fruit earliness. Water savings were also remarkable. Negative effects of RDI on berry size or total soluble solid content were not perceived. Our results suggest that RDI is a suitable strategy to save irrigation water without substantial negative effects on yield and berry size. However, the effects on skin color were insufficient in the trial conditions.

In the species of the Cucurbitaceae family, the occurrence of separate male and female flowers in the same plant (monoecy) is controlled by an ethylene biosynthesis ACS gene, which specifically suppresses the development of stamen in the female flower. In watermelon, a mutation of loss of function in CitACS4 promotes the conversion of female into hermaphrodite flowers, and of monoecious into andromonoecious plants. We have studied whether the ethylene produced by CitACS4 enzyme could also be involved in other ethylene-regulated traits, including pistillate flowering transition and the number of female flowers per plant, the development of floral organs other than stamens, as well as fruit and seed set, and fruit development. A linkage analysis approach was performed in three independent F2 populations segregating for the two alleles of the gene (M, monoecious; m, andromonoecious), and the different traits under study. The CitACS4m allele not only cosegregated with andromonoecy, but also with earlier pistillate transition, an increased number of pistillate flowers per plant, and a slower growth and maturation of petals and carpels, which delayed anthesis time in hermaphrodite flowers. The m allele was also found to be linked to a reduced fruit set, which was not caused by a deficiency in pollination or fertilization. The gene also affected the longitudinal and transverse growth rates of the ovary and fruit, which means that fruits from andromonoecious plants (mm) were rounder than those from monoecious (MM) ones. Taken together, these data indicate that the locus defined by the ethylene biosynthesis and sex-determining gene CitACS4 acts as a pleiotropic regulator of the complete development of the pistillate flower and the earlier development of the fruit.

Relationships between plant water status and gas exchange parameters at increasing levels of water stress were determined in Algerie loquats which grown in 50 I pots. Changes in soil water content and stem water potential and their effects on stomatal conductance (Gs) and net photosynthesis (Pn) rate were followed in control plants and in plants without irrigation until the latter reached near permanent wilting point and some leaf abscission took place. Then, the irrigation was restarted and the comparison repeated. Soil water content and stem water potential gradually diminished in response to drought reaching the minimum values of 0.9 mm and −5.0 MPa, respectively, 9 days after watering suspension. Compromised plant water status had drastic effects on Gs values that dropped by 97% in the last day of the drought period. Pn was diminished by 80% at the end of the drought period. The increasing levels of water stress did not cause a steady increase in leaf temperature in non-irrigated plants. Non-irrigated plants wilted and lost some leaves due to the severity of the water stress. However, all non-irrigated plants survived and reached similar Pn than control plants just a week after the irrigation was restarted, confirming drought tolerance of loquat and suggesting that photosynthesis machinery remained intact.

Olive (Olea europaea L.) is a wind-pollinated, preferentially allogamous species. However, inconsistent results are often reported for the classification of some cultivars as either self-compatible or self-incompatible, probably because of the influence of management and environmental conditions during bloom. Cross- pollination significantly enhances fertilization and fruit set in most olive cultivars. However, this improvement partly depends on the compatibility relationships between recipient and pollinizer trees in the species, where a few cases of cross- incompatibility have been reported recently. Different methods have been used to evaluate the success and inter-compatibility in pollination assays. The most common method for the determination of inter-compatibility is based on the observation of pollen-pistil interactions in cross-pollination trials. This method requires the usage of pollination bags to prevent the access of unwanted pollen grains to the stigmas of recipient flowers and the subsequent observation of pollen-pistil interaction features (pollen adhesion, pollen germination, pollen tube growth in the pistil and fertilization) and/or initial and final fruit set in response to experimental crosses. Lately, simple sequence repeats (SSRs) or microsatellites are increasingly becoming the markers of choice for paternity analysis of seeds to measure the success of crosses, especially when potential pollen donors are multiple and not applied by hand. SSRs offer a reliable way to test the paternity of seeds by checking the presence of parental alleles in the progenies. Knowledge of self-compatibility and cross-compatibility relationships of olive cultivars is a vital factor in the success of olive crosses and, therefore, of breeding programs. The advantages and drawbacks of the different methods to evaluate compatibility relationships within and between olive cultivars are presented in this work.

Papaya is an alternative crop of increasing interest for diversifying the crops cultivated in greenhouses of southeast Spain. The proximity to European markets allows offering consumers with high eating quality papaya and competing with other exporting countries from South America. It is important to harvest fruits at the optimum maturity for export. In this investigation, we studied the effect of the maturity stage at harvest on fruit quality as well as the quality of fruits harvested at different times of the year. Fruits of 'BH-65' cultivated in greenhouse were harvested at three different maturity stages defined by their skin color (stage 1: fruit with up to 30% yellow color; stage 2: 50-60% yellow color; stage 3: 90-100% yellow color) and in three different periods of the year (March, June and September). Physico-chemical quality parameters (skin and pulp color, weight, pulp firmness, soluble solids (TSS) and titrable acidity) were determined at harvest and at 5, 10 and 15 days after storage at 12°C and 85% HR. Sensorial attributes (external appearance, odor, flavor and texture) were also evaluated in fruits that had reached edible firmness. Our results indicate that the maturity stages at harvest influenced fruit firmness and color but not TSS content and acidity. TSS was slightly higher in stage 2 and 3 fruits than in stage 1, although the difference was only significant in some dates. Regardless of the maturity stage at harvest, TSS did not vary during storage. Fruits harvested at stage 2 were best rated in sensory evaluations in both, external appearance and the overall evaluation of flavor and texture. Harvest time influenced fruit quality: September fruits tend to be lighter and with slightly higher TSS content than fruits harvested in other periods. We suggest that the fruits should be harvested at stage 1 for export and at stage 2 for domestic market.

The objective of this study was to determine a suitable plot size for field experiments with papaya genotypes. Two experiments were carried out using a randomized complete block design with 11 and 12 papaya genotypes, respectively. In both experiments, plots consisted of one row, with 10 plants each. Spacing between rows was 3.5 m, with 1.5 m between plants. The characteristic evaluated was fruit production in t ha-1 in first year of cultivation, and the basic unit used was one plant. Suitable plot size was estimated using Lin and Binns, and Hatheway’s methods. These methods are complementary and should be used together in the determination of the optimum plot size. The results of these tests showed that the optimum plot size for the evaluation of yield in papaya was four plants by plot with four replications each assuming 30% of the precision for establishing differences among the means of two genotypes. Key words: Breeding, Carica papaya, intrablock correlation.

‘Arbequina’ is the olive cultivar preferred by growers for the design of the trendy high-density orchards due to its low vigor, high yield, and extraordinary oil quality. Although, ‘Arbequina’ is considered by many authors as self-compatible, increasing evidence questions its full self-compatibility. Our results clearly illustrate that, contrary to earlier assumptions, ‘Arbequina’ olive behaves as a self-incompatible variety. Self-pollen rejection in the style, low levels of self-fertilization and decreased initial and final fruit set after self-pollination in comparison to cross-pollination treatments clearly indicate that ’Arbequina’ is self-incompatible. Furthermore, even seed and fruit size were depressed in response to self-pollination, despite the lighter crop produced in self-pollinated shoots. This was observed in both ‘on’ and ‘off’ ‘Arbequina’ trees. Both pollinizers tested, ‘Manzanillo’ and ‘Picual’, matched results obtained under free open-pollination and significantly increased the yield obtained under self-pollination. Our results suggest the convenience of establishing pollinizers for large, high-density ’Arbequina’ orchards, especially in new areas of olive cultivation, where wind pollination may be not enough to ensure beneficial cross pollen flow.

The cultivation of papaya under greenhouse has become recently a profitable alternative for horticultural crops in different regions of the world, among them South East Spain. The objective of this work was to determine the sample size to evaluate fruit quality features in mature fruits of ‘BH-65’ papaya cultivated under greenhouse in Almería (Spain). With this aim, fruits were harvested at stage 2 in July 2013, and at the end of the production cycle in March 2014. On each occasion and when fruits reached maturation phenological stage 5, 26 fruits were evaluated for physical characteristics (weight, length, diameter, cavity width), total soluble solids content, and color attributes of skin and pulp. The optimal sample size was calculated using a deterministic method. The physical parameters and the skin and pulp color attributes of fruits of ‘BH-65’ papaya cultivar present different experimental accuracy among themselves and between harvest seasons, requiring different sample sizes. Higher sample size is required for evaluating fruit weight. 25 fruits were necessary at harvest performed in July, but only 7 fruits in March, considering an error of 15% around the average.

Golden genotype of papaya (Carica papaya), named for its yellowish leaves, produces fruits very much appreciated by consumers worldwide. However, its growth and yield are considerably lower than those of other genotypes, such as ‘Sunrise Solo’, which has intensely green leaves. We undertook an investigation with the goal of evaluating key physiological traits that can affect biomass accumulation of both Golden and Sunrise Solo genotypes. Papaya seeds from two different genotypes with contrasting leaf colour ‘Sunrise Solo’ and Golden were grown in greenhouse conditions. Plant growth (plant height, leaf number, stem diameter, leaf area, plant dry weight), leaf gas exchanges, leaf carbon balance, RuBisCO oxygenation and carboxylation rates, nitrogen, as well as chlorophyll concentrations and fluorescence variables were assessed. Although no significant differences were observed for photosynthetic rates between genotypes, the accumulation of small differences in photosynthesis, day after day, over a long period, might contribute to some extend to a higher C-budget in Sunrise Solo, higher leaf area and, thus, to higher productivity. Additionally, we consider that physiological processes other than photosynthesis and leaf respiration can be as well involved in lower growth and yield of Golden. One of these aspects could be related to the higher rates of photorespiration observed in Sunrise Solo, which could improve the rate of N assimilation into organic compounds, such as amino acids, thus contributing to the higher biomass production in Sunrise Solo relative to Golden. Further experiments to evaluate the effects of N metabolism on physiology and growth of Golden are required as it has the potential to limit its yield., This study provides the first complete picture of leaf carbon balance in two economically important genotypes of papaya and demonstrates that neither stomatal effects nor reduced photochemical and carboxylation capacities of the Golden genotype affected CO2 assimilation through photosynthesis. We consider that physiological processes other than photosynthesis/leaf respiration (LCB) can be as well involved in the worst growth and yield of Golden. For example, higher rates of photorespiration were observed in Sunrise Solo, which could improve the rate of N assimilation into organic compounds, thus contributing to the higher biomass production in Sunrise Solo relative to Golden.

Poor skin color in red table grape cultivars is one of the most troublesome difficulty producers face in the management of this crop. Recent EU limitations of ethylene release products left growers with poor prospects to improve the color of the berry. In this work, we have explored ‘Crimson Seedless’ response to deficit irrigation from veraison to the end of harvest and we observed an improvement in the development of the berry skin color. Earlier yield was also obtained in plants under deficit irrigation, with a higher percentage of yield in the first picking operations, due to earlier acquisition of color by the berries. Negative effects on berry size or on total yield were not observed. Therefore, we may recommend deficit irrigation from veraison to harvest as a mean to increase the quality of the berry and to reduce water consumption.