Anna Schwarz

Matthias Gauly

Hansjörg Abel

Gürbüz Das

Julia Humburg

Th A Weiss

Gerhard Alexander

Silke Breves

Rautenschlein

Gürbüz Daş

Alexander A Th

Weiss

Gerhard Breves

Silke Rautenschlein
email: silke.rautenschlein@tiho-hannover.de

Pathobiology of Heterakis gallinarum mono-and co-infection with Histomonas meleagridis in layer chicken

Keywords: Heterakis gallinarum, Histomonas meleagridis, chicken, gut immunity

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Introduction

Due to changes in the legal requirements for housing of layer chickens a shift from cage to alternative production systems is occurring in European countries. This shift to more housing on litter and free-range production has led to the re-emergence of parasitic infections, such as infection with Heterakis gallinarum [START_REF] Permin | Prevalence of gastrointestinal helminths in different poultry production systems[END_REF]Maurer et al., 2009). H. gallinarum is one of the most frequently diagnosed nematode within the digestive tract of galliform birds [START_REF] Lund | Performance in chickens and turkeys of chicken-adapted Heterakis gallinarum[END_REF]. Infection with H. gallinarum is generally subclinical, but the parasite may also function as a vector for Histomonas meleagridis, which is known to induce severe pathological lesions in the gut and liver and leads to high mortality rates in susceptible hosts [START_REF] Gibbs | The occurrence of the protozoan parasite Histomonas meleagridis in the adults and eggs of the cecal worm Heterakis gallinae[END_REF]Springer et al., 1969;[START_REF] Lund | The reproductive potential of Heterakis gallinarum in various species of galliform birds: implications for survival of H. gallinarum[END_REF][START_REF] Esquenet | An outbreak of histomoniasis in free-range layer hens[END_REF]. In contrast to the turkey, histomoniasis in the chicken is known to show a high morbidity but low mortality (McDougald, 2005). Direct transmission of H. meleagridis has not been demonstrated for chickens in some studies [START_REF] Hu | Histomonas meleagridis in chickens: attempted transmission in the absence of vectors[END_REF] or may occur at lower rates than observed in turkeys [START_REF] Hess | Rapid transmission of the protozoan parasite Histomonas meleagridis in turkeys and specific pathogen free chickens following cloacal infection with a mono-eukaryotic culture[END_REF]. This emphasises the importance of H. gallinarum as a vector for H. meleagridis in chickens.

Although many studies have investigated the prevalence of H. gallinarum in the chicken [START_REF] Kurt | Cross-sectional survey on helminth infections of chickens in the Samsun region, Turkey[END_REF]Mungube et al., 2008;Maurer et al., 2009) and the induction of pathological lesions after H. gallinarum infection [START_REF] Kaushik | Studies on tissue responses in primary and subsequent infections with Heterakis gallinae in chickens and on the process of formation of caecal nodules[END_REF][START_REF] Riddell | Cecal and hepatic granulomas in chickens associated with Heterakis gallinarum infection[END_REF], no information is available on specific gut-associated immune reactions following the infection. A variety of studies in mammalian species demonstrated the importance of cell-mediated immune reactions in the clearance of nematode infection. The Th2 immune response is especially dominant following a nematode infection [START_REF] Dawson | Localized multigene expression patterns support an evolving Th1/Th2-like paradigm in response to infections with Toxoplasma gondii and Ascaris suum[END_REF][START_REF] Little | The characterization of intraepithelial lymphocytes, lamina propria leukocytes, and isolated lymphoid follicles in the large intestine of mice infected with the intestinal nematode parasite Trichuris muris[END_REF][START_REF] Scales | The role of IL-4, IL-13 and IL-4Rα in the development of protective and pathological responses to Trichinella spiralis[END_REF]. In mice it has been shown that local activation of T cells may play a role in the expulsion of the caecal nematode Trichuris muris. In resistant mice, which develop a Th2 response, the number of infiltrating CD4+ and CD8+ cells in the epithelium and lamina propria of the gut was highest at the time of worm expulsion [START_REF] Little | The characterization of intraepithelial lymphocytes, lamina propria leukocytes, and isolated lymphoid follicles in the large intestine of mice infected with the intestinal nematode parasite Trichuris muris[END_REF]. Increased numbers of T cells, particularly CD4+ and TCR γδ+ lymphocytes were observed in the abomasal mucosa of goats following primary infection with Haemonchus contortus [START_REF] Perez | Cellular phenotypes in the abomasal mucosa and abomasal lymph nodes of goats infected with Haemonchus contortus[END_REF]. The few studies on Ascaridia galli infection in chickens demonstrated the systemic and local increase in IL-4 and IL-13 mRNA expression in splenic and ileal tissues [START_REF] Degen | Th1/Th2 polarization by viral and helminth infection in birds[END_REF][START_REF] Kaiser | The avian immune genome -a glass half-full or half-empty?[END_REF] and induction of circulating IgG antibodies starting two to three weeks pi (Marcos-Atxutegi et al., 2009).

With regard to the caeca as the major site of H. gallinarum and H. meleagridis infection it may be hypothesized that the secretory response might also be affected by the infection. Pigs infected with Oesophagostomum dentatum showed, depending on the parasite stage, alterations in chloride secretion [START_REF] Leonhard-Marek | Electro-and transport-physiological changes in pig proximal colon during parasitic infection -neuronal and epithelial muscarinic responses[END_REF], which in turn is correlated to the water content of digesta [START_REF] Leonhard-Marek | Electrophysiological characterization of chloride secretion across the jejunum and colon of pigs as affected by age and weaning[END_REF]. Significant increase in Cl -secretion in response to histamine during the period of self-healing was also observed in Ascaris suum-infected pigs suggesting a net rise in fluid in the intestinal lumen at this stage [START_REF] Dawson | Localized multigene expression patterns support an evolving Th1/Th2-like paradigm in response to infections with Toxoplasma gondii and Ascaris suum[END_REF].

At present, the use of many effective antihelmintic drugs is prohibited or restricted due to consumer safety reasons. New ways to influence chicken health by feed additives are under investigation (Owens et al., 2008;Mountzouris et al., 2009;[START_REF] Solis De Los Santos | The natural feed additive caprylic acid decreases Campylobacter jejuni colonization in market-aged broiler chickens[END_REF].

It was shown in different mammalian models that non-starch polysaccharides (NSP) may have a beneficial effect on systemic and local immune functions and general performance of the animal [START_REF] Kelly-Quagliana | Dietary oligofructose and inulin modulate immune functions in mice[END_REF][START_REF] Roller | Prebiotic inulin enriched with oligofructose in combination with the probiotics Lactobacillus rhamnosus and Bifidobacterium lactis modulates intestinal immune functions in rats[END_REF][START_REF] Seifert | Inulin and oligofructose: review of experimental data on immune modulation[END_REF]. Inulin-fed rats showed an increased proportion of dendritic cells in the Peyer's patches and higher ex vivo secretion of IL-2, IL-10 and IFN-γ in spleen and mesenteric lymph node cell cultures [START_REF] Ryz | Long-chain inulin increases dendritic cells in the Peyer's patches and increases ex vivo cytokine secretion in the spleen and mesenteric lymph nodes of growing female rats, independent of zinc status[END_REF]. Furthermore, NSP influenced the course of intestinal nematode infection in mammalian species [START_REF] Petkevicius | The impact of diets varying in carbohydrates resistant to endogenous enzymes and lignin on populations of Ascaris suum and Oesophagostomum dentatum in pigs[END_REF][START_REF] Lund | The reproductive potential of Heterakis gallinarum in various species of galliform birds: implications for survival of H. gallinarum[END_REF][START_REF] Petkevicius | The effect of dietary carbohydrates with different digestibility on the populations of Oesophagostomum dentatum in the intestinal tract of pigs[END_REF]; The objective of this study was to investigate immunological and electrophysiological parameters of the intestine of layer chickens in response to experimental infection with the nematode H. gallinarum. In addition, we characterised the influence of a co-infection of H. gallinarum and H. meleagridis on these parameters, as well as the impact of NSP on the outcome of the infections.

Materials and Methods

Chickens. One day old female Lohmann Selected Leghorn (LSL) chickens were obtained from Lohmann Animal Breeding GmbH, Cuxhaven, Germany. They were housed in two isolation rooms; infected and non-infected groups were kept separately. They were divided randomly into three separate groups within each room and kept according to the regulations laid down for Animal Welfare. Water and feed was offered ad libitum. No vaccination programme was applied.

H. gallinarum.

The adult female worms were collected from infected chickens, which had been obtained from different farms. The collected worm eggs were positive for Histomonas meleagridis. After incubation of the eggs in 0.5% (w/v) formalin for 3 weeks at room temperature the embryonated eggs were stored at 4°C for one to eight months until inoculation.

Diets. Animals of the non-infected and infected groups were allotted to one of three different diets: control feed; control feed containing 9.1% of pea bran meal as a source of insoluble non-starch polysaccharides (NSP); control feed containing 9.1% of chicory root meal as a source of soluble NSP. The diets were offered in pelleted form. The percentages of the ingredients and the analysis of the diets are given in Table 1.

Haematoxylin & eosin staining and histopathology. Samples of distal caecum were fixed for 24 h in 4% phosphate-buffered formalin and then processed for histopathological examination by standard methods after haematoxylin and eosin (H&E) staining. The H&Estained tissue sections were examined by light microscopy for lesions such as epithelial erosion and ulceration, lymphocyte and heterophil infiltration and cell aggregation as well as accumulation of fibrinoous exudate in the lumen.

Flow cytometric analysis.

Single cell suspensions of spleen leukocytes were prepared using a slight modification to the method described by [START_REF] Liman | Induction of local and systemic immune reactions following infection of turkeys with avian Metapneumovirus (aMPV) subtypes A and B[END_REF]. The combination of the following antibodies was used to detect splenic T cells: mouse-antichicken-CD4 and -CD8α antibodies (Experiment 1 & 2) [START_REF] Chan | Identification of the avian homologues of mammalian CD4 and CD8 antigens[END_REF] and mouse-antichicken-TCRαβ (Vβ 1 ) and -TCRγδ antibodies (Experiment 2) [START_REF] Chen | Differential expression of two T cell receptors, TcR1 and TcR2, on chicken lymphocytes[END_REF], conjugated to phycoerythrin (R-PE) and to fluorescein (FITC), respectively (Southern Biotech, provided by Biozol, Eching, Germany). The antibodies were diluted in FACS buffer to a final concentration of 0.5 µg (anti-CD4), 0.8 µg (anti-CD8α), 2 µg (anti-TCRαβ (Vβ 1 )) and 5 µg (anti-TCRγδ) per ml. The percentage of stained cells was determined using a Beckman Coulter Epics XL © flow cytometer and EXPO 32 ADC software program (Beckman Coulter Company, Miami, Florida). The lymphocyte population was gated according to size and granularity, and 10,000 events per sample were analysed based on positive staining with FITC and R-PE. Immunohistochemistry. Cryostat sections of distal caeca (8-µm thick) were processed as described previously (Vervelde et al., 1996;[START_REF] Berndt | Chicken cecum immune response to Salmonella enterica serovars of different levels of invasiveness[END_REF]. Sections were stained with the following mouse-anti-chicken unlabeled monoclonal antibodies: anti-CD4, anti-CD8α, anti-TCRαβ (Vβ 1 ), anti-TCRγδ (at 0.5 µg/ml each) and anti-IgA (0.05 µg/ml) (Southern Biotech, provided by Biozol). The secondary anti-mouse IgG biotinylated antibody, ABC reagent (Vectastain ® Elite ® ABC Kit,Vector Laboratories Inc.) and the 3,3´-diaminobenzidine (DAB) peroxidase substrate Kit (Vector Laboratories Inc.) were used according to the manufacturer's instructions. The different lymphocyte populations in the caecal lamina propria were evaluated semi-quantitatively using the abundance score (Figure 3).

Real-time quantitative RT-PCR. Total RNA was isolated from distal caecum with 1000 µl TrifastGOLD (Peqlab, Erlangen, Germany) per sample according to the manufacturer's instructions.

Cytokine mRNA expression levels were quantified using TaqMan quantitative RT-PCR. Details of specific primers, cloning primers and probes are provided in Table 2 [START_REF] Rautenschlein | A comparison of immune responses to infection with virulent infectious bursal disease virus (IBDV) between specific-pathogen-free chickens infected at 12 and 28 days of age[END_REF][START_REF] Powell | The turkey, compared to the chicken, fails to mount an effective early immune response to Histomonas meleagridis in the gut[END_REF]. Real-time quantitative RT-PCR was performed using the Brilliant ® II QRT-PCR one-step master mix kit (STRATAGENE, Agilent Technologies Company, USA). Amplification and quantification of specific products was done using the Mx3005P TM thermal cycle system and Mx3005P TM Q PCR Software (STRATAGENE, Agilent Technologies Company). The following cycle profile was applied: one cycle at 50°C for 30 min and 95°C for 10 min, and 40 cycles at 95°C for 20 s and 60°C for 1 min. Results are expressed as x-fold change in mRNA expression levels in the tissues of inoculated birds compared to non-inoculated controls. The differences in template RNA levels of individual transcripts were normalised to 28S rRNA as previously described [START_REF] Powell | The turkey, compared to the chicken, fails to mount an effective early immune response to Histomonas meleagridis in the gut[END_REF]. Expression levels of 28S rRNA stayed constant in the tissues of inoculated and noninoculated animals showing the same threshold cycle values (C t ) throughout both experiments. GADPH was tested as an additional house-keeping gene [START_REF] Rautenschlein | A comparison of immune responses to infection with virulent infectious bursal disease virus (IBDV) between specific-pathogen-free chickens infected at 12 and 28 days of age[END_REF] but its expression levels in the caecal samples were not as stable as the expression level of 28S rRNA. In order to generate standard curves the target gene segment of chicken IFN-γ was cloned into the pCR3.1 vector (Invitrogen, Germany) [START_REF] Rautenschlein | A comparison of immune responses to infection with virulent infectious bursal disease virus (IBDV) between specific-pathogen-free chickens infected at 12 and 28 days of age[END_REF] and target gene segments of chicken IL-4 and IL-13 -into the pCR ® 4-TOPO ® vector (Invitrogen) following standard procedures.

Electrophysiological measurements. After removing the tunica serosa the segments of the distal caecum were mounted in Ussing chambers with the exposed area of 1 cm 2 . KG, Darmstadt, Germany. The buffer solutions had an osmolarity of 300 mosmol/l and pH of 7.45, when aerated with carbogen at 37 °C. To reduce endogenous production of prostaglandin the buffers were supplemented with Indomethacin (10 µmol/l). Short-circuit currents (Isc) and transepithelial tissue conductances (Gt) were measured using a computercontrolled voltage-clamp device (Mußler Ingenieurbüro für Mess-und Datentechnik, Aachen, Germany). Gt were determined by applying a current pulse of 100 µA for 200 ms every 6 s.

The following chemicals were added to the chambers with recovery intervals of 20-30 min: amiloride (0.1 mmol/l) in combination with tetraethylammonium (TEA) (5 mmol/l) and Ba 2+ (1mmol/l) mucosal to inhibit apical Na + and K + channels; Carbachol (0.1 mmol/l) and Forskolin (0.01 mmol/l) serosal as stimulators of the chloride secretion, 4,4´diisothiocyanostilbene-2,2´-disulfonic acid (DIDS) (0.2 mmol/l) mucosal to inhibit alternative chloride channels and 5-nitro-2-(3-phenylpropylamino) benzoate (NPPB) (0.5 mmol/l) serosal The substances, with the exception of Ba 2+ (Merck KG, Darmstadt, Germany), were obtained from Sigma-Aldrich Chemicals, St. Louis, MO, USA. Three caecal segments from each bird in Experiment 1 and two segments from each bird in Experiment 2 were used. The electrical responses were measured as a difference of an average of two to three basal values before and two to three values after reaching the maximal response to the application of the respective substance.

Experimental protocol. Experiment 1. One hundred and twenty animals were divided randomly into three groups of 40. The groups were kept on control diets or diets containing 9.1% of soluble or insoluble NSP. At 3 weeks of age 20 birds per group were selected randomly and inoculated orally with 200 embryonated H. gallinarum eggs that were positive for H. meleagridis. Throughout the experiments clinical examinations were carried out daily and of body weight gain was measured weekly. Necropsy was carried out 2, 3 and 5 weeks post infection (pi). Four to five birds chosen at random from each subgroup were necropsied and examined for pathological lesions. Samples of distal caecum were taken for histopathological and immunohistochemical examination and spleen samples for flow cytometric analysis. Distal caecal tissue of three birds per inoculated and non-inoculated group allotted to control diet was taken for quantification of cytokine mRNA expression levels by quantitative RT-PCR. On the last necropsy day parasite numbers from one caecum per animal were counted and the faeces of the birds were examined for the presence of H. gallinarum eggs.

For electrophysiological measurements four to five birds chosen at random per diet and infection group were necropsied at five weeks pi, and samples of distal caeca were analysed in the Ussing chamber. Experiment 2. The birds were treated prophylactically via the drinking water with dimetridazole (0.05% w/v) (Chevi-col, Chevita GmbH, Germany) against infection with H. meleagridis from two days before inoculation until day 7 post inoculation with H. gallinarum embryonated eggs. The experimental setup was identical to Experiment 1 except that caecal samples for immunohistochemical examination were only taken 2 and 5 weeks pi, and at 9 weeks pi, a total of 30 animals (five birds chosen at random from each subgroup) were necropsied for electrophysiological measurements.

Statistical methods. All data are expressed as mean per group ± standard deviation (SD).

The differences between groups were determined by paired t-test and Wilcoxon Rank Sum Test. The effects of diets and infection as two independent factors were investigated by twoway ANOVA. P values of <0.05 were considered as significant. The statistical analyses were performed using the SAS ® 9.1 programme.

Results

Influence of diet on performance, lesion development and immune parameters.

No difference was seen between the feeding groups with regard to clinical signs, pathological and histopathological lesions or local and systemic immune reactions. Therefore, for the evaluation of these parameters the three different feeding groups were treated as replicates and combined to one inoculated and non-inoculated animal group.

Clinical and post-mortem observations. In Experiment 1 birds in the dual-infected group showed mild clinical signs such as depression and ruffled feathers beginning at one week pi.

The mortality throughout this experiment was 0.8% -one bird died on day 13 pi. In Experiment 2 animals treated prophylactically with dimetridazol (0.05% w/v) against H. meleagridis co-infection showed no clinical signs or mortality. Dual-infected birds in Experiment 1 showed a significant reduction in body weight beginning at two weeks pi until the end of the trial (P <0.05), while in Experiment 2 a significant drop in body weight was only observed at 2 weeks pi (P <0.05) (data not shown).

Pathological examination of dual-infected birds in Experiment 1 showed formation of fibrinous to fibrino-hemorrhagic exudates in caecal lumen of 14 out of 15 inoculated chickens at 2 and 3 weeks pi (Table 3). One to three focal hepatic necrotic areas were found in six out of 15 birds 2 weeks pi. At 5 weeks pi only one bird out of 15 showed macroscopic lesions in the caeca. No pathological lesions were observed in H. gallinarum mono-infected birds in Experiment 2 (Table 3).

The infection rate for H. gallinarum (% of worm-positive birds out of inoculated birds) was 13% in the dual-infected group in Experiment 1 and 93% in the mono-infected group in Experiment 2. No faeces sample of inoculated birds in Experiment 1 and 33% of the faeces samples of inoculated birds in Experiment 2 were positive for H. gallinarum eggs at 5 weeks pi (Table 3). In Experiment 1 the average number of H. gallinarum worms per bird detected in caeca at 5 weeks pi was 0.33 (min 0 and max three worms) and in Experiment 2 was 12.1 (min 0 and max 37 worms).

Non-inoculated birds showed no clinical signs or pathological lesions and were wormnegative in both experiments. inoculated birds 2 weeks pi (Figure 1b, Experiment 1). Numerous histomonads and moderate numbers of bacterial colonies were found in the severely hyperplasic tunica muscularis. Three weeks pi, all inoculated birds still showed histopathological lesions. The structure of the mucosa showed reorganisation and re-epithelisation and still contained severe lymphocyte and heterophil infiltration in the lamina propria (Figure 1c,d). Five weeks pi, 93% (14 of 15) of inoculated birds showed moderate lymphocyte infiltration in the lamina propria and formation of lymphoid centers in cecal mucosa (Figure 1e).

Mono-infection with H. gallinarum (Experiment 2) resulted in mild to moderate lymphocyte infiltration of the lamina propria and formation of lymphoid aggregations. Bacterial colonies were not observed. Two, 3 and 5 weeks pi 93%, 86%, 80% of inoculated chickens showed the indicated microscopic lesions, respectively (Figure 1f), while the remaining birds were free of detectable histopathological changes.

No microscopic lesions were observed in the non-inoculated groups of either experiment (Figure 1a).

Changes in local lymphocyte populations. Different T cell subsets were investigated immunohistochemically in the lamina propria of the caeca of control and inoculated birds (Figure 2, 3). For the semi-quantitative evaluation we used an abundance score based on the incidence of positive stained cells in the caecal mucosa and on the changes in the tissue structure due to the infection (Figure 2). The changes in local T lymphocyte populations were comparable for the different T cell subtypes in each experiment (Figure 3). Dual-infected birds (Experiment 1) showed severe T lymphocyte infiltration in the lamina propria (Figure 2e,f). The increase in the abundance score of CD4+, CD8α+, TCRαβ (Vβ 1 )+ and TCRγδ+ T cells in the gut-mucosa 2 weeks pi was 5-to 6-fold compared to non-inoculated animals (Figure 3, P< 0.05). Mono-infected birds (Experiment 2) reacted with mild to moderate T lymphocyte infiltration in the caecal lamina propria 2 weeks pi (Figure 2b,c). The abundance Following H. gallinarum and H. meleagridis co-infection, no IgA+ B lymphocytes were detected in the destroyed caecal mucosa 2 weeks pi (Experiment 1). Five weeks pi, there were no differences in the abundance score of IgA+ cells between control and inoculated birds, when the mucosal structure was almost restored. No changes in IgA+ B cell populations were observed between mono-infected and parasite-free birds in Experiment 2 (data not shown).

Flow cytometric analysis of splenic lymphocyte populations. Co-infection of H.

gallinarum and H. meleagridis induced a significant decrease in the relative number of CD4+ splenic T lymphocytes at 2 weeks pi but not at 3 and 5 weeks pi in comparison to noninoculated layers (Experiment 1), (Figure 4, P< 0.05). In the case of dimetridazol treated birds no changes in the relative number of splenic CD4+ cell were detected compared to worm free controls (Experiment 2, Figure 4). No differences were seen in the percentages of splenic

Discussion

Mucosal immunity and the secretory response of the intestine play an important role in the control of parasitic intestinal infections [START_REF] Khan | Immune-mediated alteration in gut physiology and its role in host defence in nematode infection[END_REF]. In this study we investigated for the first time the gut-associated immune parameters and secretory responses of caeca in layer chickens in the light of different diets following mono-infection with H. gallinarum and dual infection with H. gallinarum and H. meleagridis While NSP showed immunomodulatory effects in the T cell compartment in mammalian species [START_REF] Lim | Dietary fibers modulate indices of intestinal immune function in rats[END_REF]Trushina et al., 2005;Watzl et al., 2005), we did not observe any NSP-influence on the splenic and caecal T cell numbers in worm-free or in nematode-infected birds at the points investigated. In broiler chickens fructo-oligosaccharide enhanced IgG antibody titres in plasma and reduced the proportion of B cells but did not affect the percentage of T cells in caecal tonsils [START_REF] Janardhana | Prebiotics modulate immune responses in the gut-associated lymphoid tissue of chickens[END_REF]. It may be speculated that splenic and gut T cell populations in the layer chicken are not influenced by NSP, but further studies investigating other immune parameters are needed to support our findings.

Previous studies on nematode infection in mammals showed that T cells are attracted locally to the site of infection [START_REF] Almeria | Isolation and phenotypic characterization of abomasal mucosal lymphocytes in the course of a primary Ostertagia ostertagi infection in calves[END_REF][START_REF] Bozic | Analysis of intestinal intraepithelial lymphocyte populations in experimental Trichinella spiralis infection of mice[END_REF]Balic et al., 2002;[START_REF] Perez | Cellular phenotypes in the abomasal mucosa and abomasal lymph nodes of goats infected with Haemonchus contortus[END_REF]. It was suggested that T lymphocytes, especially CD4+ cells, may play a significant role in resistance to worm infection [START_REF] Betts | Trichuris muris: CD4+ T cell-mediated protection in reconstituted SCID mice[END_REF][START_REF] Khan | Immune-mediated alteration in gut physiology and its role in host defence in nematode infection[END_REF][START_REF] Little | The characterization of intraepithelial lymphocytes, lamina propria leukocytes, and isolated lymphoid follicles in the large intestine of mice infected with the intestinal nematode parasite Trichuris muris[END_REF][START_REF] Rausch | Functional analysis of effector and regulatory T cells in a parasitic nematode infection[END_REF]Patel et al., 2009). In this study we demonstrated for the first time that, similar to mammals, H. gallinarum infection in chickens also leads to recruitment of T lymphocytes to the intestinal mucosa. In our experiments the local accumulation of immune cells in the caecal lamina propria was most pronounced 2 weeks pi for both mono-and dual-infected birds. The observed increase in CD4+, CD8α+, TCRαβ (Vβ 1 )+ and TCRγδ+ T cell subpopulation was similar for subsets within each experiment. Co- infection of H. gallinarum and H. meleagridis resulted in a significantly higher T cell infiltration rate compared to mono-infection with H. gallinarum, which may be due to a higher invasiveness of the protozoan parasite [START_REF] Brener | Prevalence and pathology of the nematode Heterakis gallinarum, the trematode Paratanaisia bragai, and the protozoan Histomonas meleagridis in the turkey, Meleagris gallopavo[END_REF].

Flow cytometric analysis of splenic T lymphocyte population revealed a significant decrease in CD4+ T cells in dual-infected birds at 2 weeks pi. This effect was not detected for splenic CD8α+ lymphocytes. This observation may suggest a significant role of CD4+ cells in H. meleagridis infection, as previously reported for other protozoan parasites (Vervelde et al., 1996;Yun et al., 2000) and provide evidence of an increased intestinal recruitment of CD4+cells to the caecal lamina propria (Taylor et al., 2009). Mono-infection with H. gallinarum did not cause any changes in the percentage of splenic T lymphocytes, which suggests that, in contrast to high invasive intestinal nematodes [START_REF] Dondji | Role for nitric oxide in hookworm-associated immune suppression[END_REF][START_REF] Dondji | CD4 T cells mediate mucosal and systemic immune responses to experimental hookworm infection[END_REF], the immune response following H. gallinarum infection is mainly localised in the gut.

Analysis of T helper (Th)1 and Th2 cytokine mRNA expression in caecal tissues demonstrated an increase in IFN-γ mRNA level in dually infected birds and an increase in IL-13 mRNA level in mono-infected birds compared to worm-free animals. Our data suggest that mono-infection with H. gallinarum elicit local Th2 type immune reactions in the caecal lamina propria as was shown for nematode infections in mammals [START_REF] Dawson | Localized multigene expression patterns support an evolving Th1/Th2-like paradigm in response to infections with Toxoplasma gondii and Ascaris suum[END_REF]Patel et al., 2009). Following H. gallinarum and H. meleagridis co-infection the local immunological reactions are dominated by H. meleagridis with induction of Th1 type cytokines comparable to the effect reported for other protozoan infections [START_REF] Lillehoj | Role of T lymphocytes and cytokines in coccidiosis[END_REF]Yun et al., 2000;[START_REF] Hong | Changes in immune-related gene expression and intestinal lymphocyte subpopulations following Eimeria maxima infection of chickens[END_REF].

Previous studies in mammals have demonstrated that nematode infection influences secretory functions of intestine (Madden et al., 2004;[START_REF] Dawson | Localized multigene expression patterns support an evolving Th1/Th2-like paradigm in response to infections with Toxoplasma gondii and Ascaris suum[END_REF]. To study electrogenic Cl -secretion in the caeca of chicken we used a modified protocol, which had been established for the diagnosis of cystic fibrosis in humans and for the characteristics of Cl -secretion in mice and pigs. Changes in short-circuit current can be attributed to changes in Cl -secretion after inhibition of the apical Na + and K + channels [START_REF] De Jonge | Ex vivo CF diagnosis by intestinal current measurements (ICM) in small aperture, circulating Ussing chambers[END_REF][START_REF] Bleich | Characterisation of chloride currents across the proximal colon in CftrTgH(neoim)1Hgu congenic mice[END_REF][START_REF] Leonhard-Marek | Electrophysiological characterization of chloride secretion across the jejunum and colon of pigs as affected by age and weaning[END_REF]. Major chloride secretion in intestine epithelium takes place via the CFTR channel [START_REF] Barrett | Chloride secretion by the intestinal epithelium: molecular basis and regulatory aspects[END_REF]. This channel can be stimulated in a Ca-dependent manner by carbachol or in a cAMP-dependent way by forskolin, and is inhibited by NPPB.

No obvious effects of the diet or infection were observed on electrogenic Cl -secretion.

Previous studies in rats showed significant decreases in Cl -ion transport in the proximal jejunum after dietary supplementation with cellulose or pectin [START_REF] Schwartz | Effects of dietary fiber on intestinal ion fluxes in rats[END_REF]. Our data did not reproduce this observation in caeca of chickens, where NSP fermentation takes place. Reduced caecal Cl -secretion of mono-infected birds fed soluble NSP instead of the control diet suggests that soluble NSP in combination with nematode infection may reduce intestinal fluid. Reduced Cl -secretion was also observed in H. gallinarum mono-infected animals fed with the insoluble NSP diet compared to non-inoculated birds. Experiments with pigs have shown that insoluble NSP benefit nematode infection [START_REF] Petkevicius | The impact of diets varying in carbohydrates resistant to endogenous enzymes and lignin on populations of Ascaris suum and Oesophagostomum dentatum in pigs[END_REF][START_REF] Petkevicius | The effect of dietary carbohydrates with different digestibility on the populations of Oesophagostomum dentatum in the intestinal tract of pigs[END_REF]. Reduction of intestinal fluid may be considered as a reaction of the host to a parasitic developmental stage. Pigs infected with Oesophagostomum dentatum showed higher Isc responses during the time of nematode penetration into the epithelium (high expulsion chances) and lower Isc responses at the histiotrophic phase (low expulsion chances) [START_REF] Leonhard-Marek | Electro-and transport-physiological changes in pig proximal colon during parasitic infection -neuronal and epithelial muscarinic responses[END_REF]. It is also possible that nematodes modulate the activation of Cl -channels by their excretory/secretory products to prevent expulsion.

The changes in electrogenic Cl -secretion showed a similar pattern for mono-and dualinfected birds despite the high grade of destruction of caecal tissue in the course of H. meleagridis co-infection. We suggest that the secretory response of the intestine was not significantly affected by the co-infection at 5 weeks pi, when the intestinal epithelia had already largely recovered from pathological lesions, as shown histologically. In our experiments we observed clinical signs and pathological changes only in H. gallinarum and H. meleagridis co-infected chickens. The birds had completely recovered from clinical signs and most histological lesions by 5 weeks pi. This recovery is consistent with previous studies in chickens [START_REF] Hess | Rapid transmission of the protozoan parasite Histomonas meleagridis in turkeys and specific pathogen free chickens following cloacal infection with a mono-eukaryotic culture[END_REF][START_REF] Powell | The turkey, compared to the chicken, fails to mount an effective early immune response to Histomonas meleagridis in the gut[END_REF]. Mono-infection with H. gallinarum induced only mild to moderate local histological reaction in the caecal lamina propria. Bacterial colonies, which were detected in the damaged mucosa of H. meleagridis infected birds, were absent. The absence of pathological lesions and detectable bacterial colonies in the caecal tissue may indicate the long-term adaptation of H. gallinarum in the chicken. Possibly, the dimetridazol treatment may also have affected the bacterial flora of the gut.

In our study H. meleagridis has clearly influenced the establishment of H. gallinarum infection in the caeca. Previously it had been demonstrated that chickens with pathological evidence of histomoniasis had lower numbers of H. gallinarum worms compared to those mono-infected with H. gallinarum [START_REF] Lund | Growth and development of Heterakis gallinae in turkeys and chickens infected with Histomonas meleagridis[END_REF][START_REF] Lund | Response of four breeds of chickens and one breed of turkeys to experimental Heterakis and Histomonas infections[END_REF]. The low infection rate and the absence of H. gallinarum eggs observed in our experiments suggest that the severe inflammation induced by H. meleagridis may provide unfavourable conditions for the establishment of H. gallinarum and decelerate the development of the nematode.

In summary, local T cell immune reactions in the caecal lamina propria with induction of the Th2-type cytokine IL-13 dominate the H. gallinarum immunopathogenesis in chickens.

NSP did not influence the gut-associated cellular immune response after H. gallinarum infection. Co-infection of H. gallinarum and H. meleagridis altered the local immune response and induced an increase in mRNA expression of the Th1-type cytokine IFN-γ, and elicited systemic immune reactions in the spleen. Furthermore, we did not observe obvious effects of the infection or different feed diets on the chloride secretion in intestinal epithelia.

With this study we have contributed to the understanding of immunological and electrophysiological responses of the chicken intestine following nematode infection.

(1969). Transmission of histomoniasis with male

Heterakis gallinarum (nematoda). Parasitology, 59, 401-405. Taylor, M.D., van der Werf, N., Harris, A., Graham, A.L., Bain, O., Allen, J.E. & Maizels, R. M. (2009). Early recruitment of natural CD4+ Foxp3+ Treg cells by infective larvae determines the outcome of filarial infection. European Journal of Immunology, 39, 192-206. Trushina, E.N., Martynova, E.A., Nikitiuk, D.B., Mustafina, O.K. & Baigarin, E.K. (2005).

[The influence of dietary inulin and oligofructose on the cell-mediated and humoral immunity in rats]. Voprosy pitaniia, 74, 22-27. Vervelde, L., Vermeulen, A.N. & Jeurissen, S.H. (1996). In situ characterization of leucocyte subpopulations after infection with Eimeria tenella in chickens. Parasite Immunology, 18, 247-256. Watzl, B., Girrbach, S. & Roller, M. (2005). Inulin, oligofructose and immunomodulation.

British

  on gut immunity, nematode infection and electrophysiological parameters in chickens.

  of the cystic fibrosis transmembrane conductance regulator (CFTR) channel.

Histopathology.

  The development and incidence of histopathological lesions after inoculation of the chickens with embryonated H. gallinarum eggs are shown in Table3. In the case of H. meleagridis co-infection severe interstitial lymphocyte, heterophil and macrophage infiltration, complete ulceration of intestinal epithelium and accumulation of fibrin exudates and detritus in the lumen of caeca was observed in 93% (14 of 15) of

  2-to 3-fold in infected birds compared to non-inoculated animals (Figure3, P< 0.05). Three to 5 weeks pi, dual-infected birds (Experiment 1) showed moderate to severe CD4+, CD8α+, TCRαβ (Vβ 1 )+ and TCRγδ+ T-cells infiltration (Figure2c, d, Figure3, P< 0.05). Five weeks pi, animals in Experiment 2 showed a small increase in the abundance score that was only significant for CD4+, CD8α+ and TCRγδ+ T-lymphocytes (Figure3, P< 0.05).

  CD8α+ (Experiment1 & 2) and TCRαβ (Vβ 1 )+ and TCRγδ+ (Experiment 2) T lymphocyte populations between infected and parasite-free birds. Detection of IFN-γ, IL-4 and IL-13 cytokine mRNA expression levels by real-time quantitative RT-PCR. Dual infection with H. gallinarum and H. meleagridis (Experiment 1) induced a statistically significant increase in IFN-γ mRNA expression in the caeca of inoculated birds in comparison to non-inoculated controls at 2 weeks pi (Figure 5a, P< 0.05).

  The changes in IL-4 and IL-13 cytokine mRNA expression were not significantly different compared to the control group in this experiment (P>0.05). Mono-infection with H. gallinarum (Experiment 2) resulted in a statistically significant increase in IL-13 mRNA expression 2 weeks pi in inoculated birds compared to worm-free controls (Figure5c, P< 0.05). The changes in IFN-γ and IL-4 mRNA expression in H. gallinarum mono-infected birds were not significantly different compared to the worm free control group.Electrophysiological measurements. The maximal changes in short-circuit currents in caeca of H. gallinarum and H. meleagridis co-infected birds (Experiment 1) and H. gallinarum mono-infected birds (Experiment 2) in response to the addition of carbachol, forskolin and NPPB are shown in Figure6. Mono-infected birds (Experiment 2) showed responses to carbachol and forskolin, whereas no significant effect of the infection or the diet was seen in dually infected birds (Experiment 1). Both substances induced a significantly lower increase in the maximal Isc response in mono-infected birds compared to non-inoculated birds (Experiment 2) when insoluble NSP were fed (Figure6a, b, P< 0.05). Forskolin also induced smaller maximal Isc responses in non-inoculated birds fed with soluble NSP than in noninoculated birds fed with insoluble NSP. Moreover, forskolin caused lower short-circuit current responses in inoculated birds fed with soluble NSP than in inoculated birds receiving control feed (Figure 6b, P< 0.05). NPPB addition resulted in decreases in Isc in all birds. Dual infection of H. gallinarum and H. meleagridis (Experiment 1) induced a significantly smaller decrease in the maximal Isc responses in inoculated compared to non-inoculated birds when fed insoluble NSP. Mono-infection with H. gallinarum (Experiment 2) led to smaller Isc decreases in tissues of inoculated birds fed soluble or insoluble NSP instead of control diet. Maximal Isc response was also less pronounced in non-inoculated birds fed with soluble NSP compared to non-inoculated birds receiving insoluble NSP (Figure 6c, P< 0.05). DIDS had no effect on Isc response in both experiments (data not shown).

  Histomonas meleagridis to recent times. International Journal for Parasitology, 4, 455-461. Madden, K.B., Yeung, K.A., Zhao, A., Gause, W.C., Finkelman, F.D., Katona, I.M., Urban, J.F., Jr. & Shea-Donohue, T. (2004). Enteric nematodes induce stereotypic STAT6dependent alterations in intestinal epithelial cell function. Journal of Immunology, 172, 5616-5621. Marcos-Atxutegi, C., Gandolfi, Aranguena, T., Sepulveda, R., Arevalo, M. & Simon, F. (2009). Antibody and inflammatory responses in laying hens with experimental primary infections of Ascaridia galli. Veterinary Parasitology, 161, 69-75. Maurer, V., Amsler, Z., Perler, E. & Heckendorn, F. (2009). Poultry litter as a source of gastrointestinal helminth infections. Veterinary Parasitology, 161, 255-260. McDougald, L.R. (2005). Blackhead disease (histomoniasis) in poultry: a critical review. Avian Diseases, 49, 462-476. Mountzouris, K.C., Balaskas, C., Xanthakos, I., Tzivinikou, A. & Fegeros, K. (2009). Effects of a multi-species probiotic on biomarkers of competitive exclusion efficacy in broilers challenged with Salmonella enteritidis. British Poultry Science, 50, 467-478. Mungube, E.O., Bauni, S.M., Tenhagen, B.A., Wamae, L.W., Nzioka, S.M., Muhammed, L. & Nginyi, J.M. (2008). Prevalence of parasites of the local scavenging chickens in a selected semi-arid zone of Eastern Kenya. Tropical Animal Health and Production, 40, 101-109. Owens, B., Tucker, L., Collins, M.A. & McCracken, K.J. (2008). Effects of different feed additives alone or in combination on broiler performance, gut microflora and ileal histology. British Poultry Science, 49, 202-212. Patel, N., Kreider, T., Urban, J.F., Jr. & Gause, W.C. (2009). Characterisation of effector mechanisms at the host:parasite interface during the immune response to tissue-dwelling intestinal nematode parasites. International Journal for Parasitology, 39, 13-21.

  .T., Johnson, J. & Reid, W.M.

  Journal of Nutrition, 93 Supplement 1, S49-55. Yun, C. H., Lillehoj, H. S. & Choi, K. D. (2000). Eimeria tenella infection induces local gamma interferon production and intestinal lymphocyte subpopulation changes.Infection andImmunity, 68, 1282Immunity, 68, -1288. .

Figure 1 .

 1 Figure 1. Histopathological lesions of the caecal wall of chickens inoculated with 200

Figure 2 .

 2 Figure 2. Abundance score of caecal T lymphocytes in the lamina propria of chickens orally

Figure 3 .Figure 4 .Figure 5 .Figure 6 .

 3456 Figure 3. Changes in local T lymphocyte populations in the caecal lamina propria following

  , F: forward primer, R: reverse primer, F C : cloning forward primer, R C : cloning reverse primer. a refers to genomic DNA sequence

  birds per inoculated group. The macroscopically observed lesions were thickening of caecal wall, haemorrhagic exudate and cheesy core in caecal lumen. Severity grades of microscopical lesions: 4.0 = severe hyperplasy of tunica muscularis, numerous histomonads, massive lymphocyte, heterophil and macrophage infiltration, coagulation necrosis; 3.0 = severe lymphocyte infiltration in the lamina propria and lymphoid centres in caecal tissue; 2.0 = moderate lymphocyte infiltration in the lamina propria and formation of lymphoid aggregations; 1.0 = mild lymphocyte infiltration in the lamina propria. Non-inoculated birds were negative for lesions, worms and eggs.

  Figure 1.

  Figure 4.

Table 1 .

 1 Ingredients and nutrient contents of experimental diets.

	Item		Diet 1
		C	I	S
	Component, %
	Barley	29.0	26.4	26.4
	Wheat	54.0	49.1	49.1
	Fishmeal	8.0	7.3	7.3
	Casein	4.5	4.1	4.1
	Soybean oil	2.0	1.8	1.8
	Premix 77046	1.0	0.9	0.9
	MCP 2	0.9	0.8	0.8
	CaCO 3	0.6	0.5	0.5
	Pea bran meal	-	9.1	-
	Chicory root meal	-	-	9.1
	Analyzed
	DM, g/kg	898	900	898
	Nutrients 3 , g/kg DM
	Crude ash	56	53	56
	Crude protein	219	203	204
	Crude lipids	40	35	37
	Sugar	33	30	69
	Starch	485	439	415
	ADF	32	88	38
	NDF	119	173	123
	Insoluble NSP	102	173	104
	Sol-NSP,g/kg	19	22	24
	Inulin,g/kg	-	-	71
	ME 4 , MJ/kg DM	13.29	12.06	12.25
	1 C = control diet, I = diet with insoluble NSP, S = diet with soluble NSP
	2 MCP = mono-calcium phosphate, 3 DM = dry matter, ADF = acid detergent fibre, NDF =
	neutral detergent fibre, NSP = non-starch polysaccharides 4 ME = metabolizable energy

Table 2 .

 2 Real-time quantitative RT-PCR primers and probes.

Table 3 .

 3 Infection rate and development of macroscopic and microscopic lesions in chickens inoculated with 200 embryonated eggs of Heterakis gallinarum (H. g.).

	Experiment/	H. g.	% faeces	% birds per group with caecal lesions at weeks pi
	/ treatment	infection	samples	Macroscopic	Microscopic
	with	rate in %	with H. g.		(severity grade)
	dimetridazol

E-mail: cavanagh@metronet.co.uk URL: http://mc.manuscriptcentral.com/cavp Avian Pathology


Oscar ; Valero
email: co.valero@uib.es

Salvador Romaguera
email: sromague@mat.upv.es

Pedro Tirado

Complete partial metric spaces have partially metrizable computational models-CMMSE2010

Keywords: computational model, complete partial metric, domain, formal ball, The Scott topology computational model, complete partial metric, domain, formal ball, the Scott topology AMS Subject Classification: MCS Codes: 68Q01, 68Q05, 54E50, 06A06, 06B35

come

Introduction

Motivated by the fact that metric spaces and domain theory constitute fundamental mathematical tools in computer science, several authors have investigated the problem of obtaining links between them and constructing, with this approach, suitable models in the theory of computation [2, 3, 5, 7, 8, 11-13, 15, 19, 23, etc]. In particular, Edalat and Heckmann [START_REF] Edalat | A computational model for metric spaces[END_REF] presented a very nice and explicit construction of a computational model for (complete) metric spaces by means of the notion of a formal ball. They proved, among other results, that the poset of formal balls of a metric space (X, d) is a domain if and only if (X, d) is complete and that this poset is an ω-domain if and only if (X, d) is complete and separable (previously, Lawson characterized in [START_REF] Lawson | Spaces of maximal points[END_REF] separable completely metrizable spaces in terms of ω-domains). Later on, Heckmann [START_REF] Heckmann | Approximation of metric spaces by partial metric spaces[END_REF] constructed a partial metric on the (continuous) poset of formal balls of a metric space (X, d) which extends the metric d and induces the Scott topology (similar results were obtained by Rutten [START_REF] Rutten | Weighted colimits and formal balls in generalized metric spaces[END_REF] with a different approach). Heckmann's construction is very interesting because the use of a distance function on the formal balls provides a "quantitative" computational model, which suggested, by one hand, the study of computational models from a quantitative point of view and, on the other hand, the study of partial metrics and related structures by means of domain theory (see [1, 9, 14, 16-18, 20-23, etc] for recent contributions in these directions).

Here, and improving a recent construction by Romaguera and Valero [START_REF] Romaguera | A quantitative computational model for complete partial metric spaces via formal balls[END_REF], we shall prove that the domain of formal balls of a complete partial metric space (X, p) can be endowed with a complete partial metric that, among other things, extends the partial metric p and induces the Scott topology. Motivated by this result we shall define the notion of a partially metrizable computational model and then we shall show that the topological spaces having a partially metrizable computational model are exactly those admitting a compatible complete partial metric.

Background

The letters N, R and R + will denote the set of positive integer numbers, the set of real numbers and the set of non-negative real numbers, respectively.

Our basic reference for domain theory is [START_REF] Gierz | Continuous Lattices and Domains[END_REF], for general topology is [START_REF] Engelking | Monografie Mat. Tom[END_REF] and for quasi-metric spaces it is [START_REF] Künzi | Nonsymmetric distances and their associated topologies: About the origins of basic REFERENCES ideas in the area of asymmetric topology[END_REF].

Let us recall that a partially ordered set, or poset for short, is a set L equipped with a partial order ; it will be denoted by (L, ) or simply by L if no confusion arises.

A subset D of a poset L is directed provided that it is nonempty and any pair of elements of D has an upper bound in D. The least upper bound of a subset D of L is denoted by D if it exists.

A poset L is said to be directed complete, and is called a dcpo, if every directed subset of L has a least upper bound.

An element x of L is said to be maximal if the condition x y implies x = y. The set of all maximal points of L will be denoted by Max((L, )) or simply by Max(L) if no confusion arises.

Let L be a poset and x, y ∈ L; we say that x is way below y, in symbols x y, if for each directed subset D of L for which D exists, the relation y D implies the existence of some z ∈ D with x z.

A poset L is called continuous if for each x ∈ L, the set ⇓ x = {u ∈ L : u x} is directed and x = (⇓ x).

A continuous poset which is also a dcpo is called a continuous domain or, simply, a domain.

A subset B of a poset L is a basis for L if for each x ∈ L, the set ⇓ x B = {u ∈ B : u x} is directed and x = (⇓ x B ). Recall that a poset has a basis if and only if it is continuous. Therefore, a dcpo has a basis if and only if it is a domain.

A dcpo having a countable basis is said to be an ω-continuous domain, or simply an ω-domain if no confusion arises. If (X, τ ) is a T 0 topological space, then the binary relation ≤ τ defined on X by x ≤ τ y ⇔ x ∈ cl{y}, is a partial order on X called the specialization order (see, for instance, p. 42 of [START_REF] Gierz | Continuous Lattices and Domains[END_REF]).

Following the modern terminology, by a quasi-metric on a set X we mean a function d : X × X → R + such that for all x, y, z ∈ X :

(i) x = y ⇔ d(x, y) = d(y, x) = 0; (ii) d(x, z) ≤ d(x, y) + d(y, z).
A quasi-metric space is a pair (X, d) such that X is a set and d is a quasi-metric on X.

Each quasi-metric d on X induces a T 0 topology τ d on X which has as a base the family of open balls

{B d (x, r) : x ∈ X, ε > 0}, where B d (x, ε) = {y ∈ X : d(x, y) < ε} for all x ∈ X and ε > 0.
Note that if (X, d) is a quasi-metric space, then the binary relation ≤ d defined on X by x ≤ d y ⇔ d(x, y) = 0, is a partial order on X, which coincides with the specialization order of (X, τ d ).

Given a quasi-metric d on X, then the function d -1 defined by d -1 (x, y) = d(y, x), is also a quasi-metric on X, called the conjugate of d, and the function d s defined by d s (x, y) = max{d(x, y), d -1 (x, y)} is a metric on X.

On the other hand, the notion of a partial metric space, and its equivalent weightable quasi-metric space, was introduced by Matthews in [START_REF] Matthews | Partial metric topology[END_REF] as a part of the study of denotational semantics of dataflow networks.

Let us recall that a partial metric on a set X is a function p :

X × X → R + such that for all x, y, z ∈ X : (i) x = y ⇔ p(x, x) = p(x, y) = p(y, y); (ii) p(x, x) ≤ p(x, y); (iii) p(x, y) = p(y, x); (iv) p(x, z) ≤ p(x, y) + p(y, z) -p(y, y).
If p is a partial metric on X we denote by ker p the subset of X consisting of all points x ∈ X such that p(x, x) = 0. Obviously, every metric d on X is a partial metric with ker d = X.

A partial metric space is a pair (X, p) such that X is a set and p is a partial metric on X.

Each partial metric p on X induces a T 0 -topology τ p on X which has as a base the family of open p-balls {B p (x, ε) : x ∈ X, ε > 0}, where B p (x, ε) = {y ∈ X : p(x, y) < ε + p(x, x)} for all x ∈ X and ε > 0.

We say that a topological space (X, τ ) admits a compatible partial metric if there is a partial metric p on X such that τ p = τ.

Matthews observed in [START_REF] Matthews | Partial metric topology[END_REF] that each partial metric p on a set X induces a quasimetric d p on X given by d p (x, y) = p(x, y)p(x, x) for all x, y ∈ X. Moreover τ p = τ dp .

If (X, p) is a partial metric space, then the binary relation ≤ p on X given by x ≤ p y ⇔ p(x, y) = p(x, x), is a partial order on X. Hence (X, ≤ p ) is a poset ( [START_REF] Heckmann | Approximation of metric spaces by partial metric spaces[END_REF][START_REF] Matthews | Partial metric topology[END_REF]). Note that in this case one has ≤ p =≤ dp .

In Definition 5.3 of [START_REF] Matthews | Partial metric topology[END_REF], Matthews also introduced the notion of a complete partial metric and stated, with a slight different terminology, that a partial metric p on a set X is complete if and only if the metric (d p ) s is complete on X.

We will say that a partial metric space (X, p) is complete if p is a complete partial metric on X. Recall [START_REF] Edalat | A computational model for metric spaces[END_REF] that if (X, d) is a metric space, then BX = X × R + is said to be the set of formal balls of (X, d), and the relation d defined on BX by (x, r) d (y, s) ⇔ d(x, y) ≤ rs, is a partial order on X. The pair (BX, d ) is called the poset of formal balls of (X, d).

Then, Edalat and Heckmann proved that for any metric space (X, d), the poset of formal balls is continuous, and the mapping i : X → BX, given by i(x) = (x, 0), is a homeomorphism between (X, τ d ) and (Max(BX), σ(BX) |Max(BX) ), and σ(BX) |Max(BX) = λ(BX) |Max(BX) .

As we mentioned above, Heckmann constructed in [START_REF] Heckmann | Approximation of metric spaces by partial metric spaces[END_REF] a partial metric on the set BX of formal balls of a metric space (X, d), that extends the metric d and such that both the partial order d and the Scott topology are induced by it.

The partial metric P on BX, given below, is a slight modification of Heckmann's original construction (see Section 4 of [START_REF] Romaguera | A quantitative computational model for complete partial metric spaces via formal balls[END_REF]): P ((x, r), (y, s)) = max {d(x, y), |r -s|} + r + s, for all (x, r), (y, s) ∈ BX.

In Theorem 1 below we extend Heckmann's construction to (complete) partial metric spaces which also provides an improvement of Theorem 4.1 of [START_REF] Romaguera | A quantitative computational model for complete partial metric spaces via formal balls[END_REF].

To this end, we first recall some pertinent concepts and facts. The poset of formal balls of a quasi-metric space (X, d) is the poset (BX, d ), where both BX and the partial order d are defined exactly as in the metric case (see [START_REF] Ali-Akbari | The space of formal balls and models of quasi-metric spaces[END_REF][START_REF] Romaguera | A quantitative computational model for complete partial metric spaces via formal balls[END_REF], where quasi-metric and partial metric versions of several results of Edalat and Heckmann were obtained. In particular, it was proved in Theorem 3.1 of [START_REF] Romaguera | A quantitative computational model for complete partial metric spaces via formal balls[END_REF] that a partial metric space (X, p) is complete if and only if (BX, d p ) is a domain ).

According to Waszkiewicz [START_REF] Waszkiewicz | Quantitative continuous domains[END_REF], a weak partial metric on a set X is a function p : X × X → R + such that for all x, y, z ∈ X : (i) x = y ⇔ p(x, x) = p(x, y) = p(y, y); (ii) p(x, y) = p(y, x); (iii) p(x, z) ≤ p(x, y) + p(y, z)p(y, y).

Heckmann [START_REF] Heckmann | Approximation of metric spaces by partial metric spaces[END_REF] showed that every weak partial metric p on a set X induces a partial metric p on X given by p (x, y) = max{p(x, y), p(x, x), p(y, y)} for all x, y ∈ X. Now let (X, p) be a partial metric space. Construct a function P : BX × BX → R + by P ((x, r), (y, s)) = p(x, y) + r + s, for all (x, r), (y, s) ∈ BX.

The proof of the following result is straightforward (compare with the first paragraph of 3.2 in [START_REF] Heckmann | Approximation of metric spaces by partial metric spaces[END_REF]). Proposition 1. For each partial metric space (X, p), the function P constructed above is a weak partial metric on BX.

Therefore, the function P : BX × BX → R + defined by P ((x, r), (y, s)) = max {P ((x, r), (y, s)), P ((x, r), (x, r)), P ((y, s), (y, s))} , for all (x, r), (y, s) ∈ BX, is a partial metric on BX. Note that, in fact, P ((x, r), (y, s)) = max{p(x, y) + r + s, p(x, x) + 2r, p(y, y) + 2s}, for all (x, r), (y, s) ∈ BX. Now we are in position of establishing our main result.

URL

Theorem 1. Let (X, p) be partial metric space and let P be the partial metric on BX defined above. Then (a) The mapping i : X → BX, given by i(x) = (x, 0), is an isometry between (X, p) and a τ (d P ) -1 -closed subspace of (BX, P ).

(b) ≤ P = dp , and i(Max(X,

≤ p )) = Max(BX). (c) If (X, p) is complete, then τ P = σ(BX). (d) If (X, p) is complete and ker p = Max(X, ≤ p ), then σ(BX) |Max(BX) = λ(BX) |Max(BX) .
(e) (BX, P ) is complete if and only if (X, p) is complete.

Proof. (a) Let x, y ∈ X. Then P (i(x), i(y)) = P ((x, 0), (y, 0)) = max{p(x, y), p(x, x), p(y, y)} = p(x, y).

So i is an isometry from (X, p) into (BX, P ). In order to show that i(X) is τ (d P ) -1 -closed in (BX, P ), let (y, s) ∈ BX such that there is a sequence (x n ) n∈N in X with d P ((x n , 0), (y, s)) < 1/n for all n ∈ N. It follows that p(x n , y)+s-p(x n , x n ) < 1/n, so s < p(x n , x n )-p(x n , y)+1/n ≤ 1/n for all n ∈ N, and hence s = 0. We conclude that (y, s) ∈ i(X) and consequently i(X) is τ (d P ) -1 -closed in (BX, P ).

(b) For each (x, r), (y, s) ∈ BX we have (x, r) ≤ P (y, s) ⇔ P ((x, r), (y, s)) = P ((x, r), (x, r)) ⇔ p(x, y) + r + s ≤ p(x, x) + 2r ⇔ p(x, y)p(x, x) ≤ rs ⇔ (x, r) dp (y, s).

So ≤ P = dp .
On the other hand, it is clear (compare Theorem 5.1 of [START_REF] Romaguera | A quantitative computational model for complete partial metric spaces via formal balls[END_REF]) that Max(BX) = {(x, 0) : x ∈ Max(X, ≤ p )}. Hence i(Max(X, ≤ p )) = Max(BX).

(c) If (X, p) is complete, we have that (BX, d p ) is a domain by Theorem 3.1 of [START_REF] Romaguera | A quantitative computational model for complete partial metric spaces via formal balls[END_REF], and hence the sets ⇑ (x, r), (x, r) ∈ BX, form a base for σ(BX). Moroever, by Corollary 3.1 of [START_REF] Romaguera | A quantitative computational model for complete partial metric spaces via formal balls[END_REF], we have that (x, r) (y, s) ⇔ d p (x, y) < rs, for all (x, r), (y, s) ∈ BX.

It is easy to check that for each (x, r) ∈ BX and each ε > 0, one obtains

⇑ (x, r + ε 2 ) ⊆ B P ((x, r), ε).
Consequently τ P ⊆ σ(BX). Now let (z, t) ∈ BX such that (x, r) ∈⇑ (z, t). Then, there is ε > 0 such that d p (z, x) < trε. Then, for (y, s) ∈ B P ((x, r), ε), we have and thus (y, s) ∈⇑ (z, t). Therefore σ(BX) ⊆ τ P .

d p (z, y) ≤ d p (z, x) + d p (x, y) < (t -r -ε) + (r -s + ε) = t -s,
(d) It suffices to show that ω(BX) |Max(BX) ⊆ σ(BX) |Max(BX) . Let (x, r) ∈ Max(BX) and (z, t) ∈ BX such that (x, r) ∈ BX\ ↑ (z, t). By (b), x ∈ Max(X, ≤ p ) and r = 0. Then p(x, x) = 0 by our hypothesis. Since d p (z, x) > t there exists ε > 0 such that d p (z, x) > t + ε, so, B P ((x, 0), ε) ∈ σ(BX) by (c). Then, for each (y, 0) ∈ B P ((x, 0), ε) ∩ Max(BX), we have p(y, y) = 0 and thus

t + ε < d p (z, x) ≤ d p (z, y) + d p (y, x) = d p (z, y) + d p (x, y) < d p (z, y) + ε.
Hence d p (z, y) > t, i.e., (y, 0) ∈ BX\ ↑ (z, t). We conclude that ω(BX) |Max(BX) ⊆ σ(BX) |Max(BX) , so σ(BX) |Max(BX) = λ(BX) |Max(BX) .

(e) The proof is similar to the one given in Theorem 4.1 (e) of [START_REF] Romaguera | A quantitative computational model for complete partial metric spaces via formal balls[END_REF], so we omit some details.

Suppose that (BX, P ) is complete and let (x n ) n∈N be a Cauchy sequence in the metric space (X, (d p ) s ). Then ((x n , 0)) n is a Cauchy sequence in the complete metric space (BX, (d P ) s ), so there is (z, t) ∈ BX such that ((x n , 0)) n converges to (z, t) in (BX, (d P ) s ). So, in particular, we obtain that for each ε > 0, p(z, z) + 2rp(z, z) < ε, and thus r = 0. Then, it immediately follows that (x n ) n converges to z in (X, (d p ) s ). We conclude that p is a complete partial metric on X.

Conversely, suppose that (X, p) is complete and let ((x n , r n )) n∈N be a Cauchy sequence in (BX, (d P ) s ). Then, for each ε > 0 there is n ε ∈ N such that p(x n , x m )p(x n , x n ) + r mr n < ε and p(x n , x m )p(x m , x m ) + r nr m < ε for all n, m ≥ n ε . Hence the sequence (r n ) n∈N is bounded, so there exist a subsequence (r nk ) k∈N of (r n ) n∈N and an r ∈ R + such that lim k r nk = r. It immediately follows that (x nk ) k∈N is a Cauchy sequence in (X, (d p ) s ). Then, there is x ∈ X such that (x nk ) k∈N converges to x in (X, (d p ) s ). Finally, it is easy to deduce that ((x n , r n )) n∈N converges to (x, r) in (BX, (d P ) s ). This concludes the proof. Remark 1. If (X, p) is partial metric space such that τ p is a T 1 -topology, then Max(X, ≤ p ) = X, so, by Theorem 1(b), i(X) = Max(BX) in this case.

The following example shows that condition "ker p = Max(X, ≤ p )" can not be omitted in Theorem 1 (d).

Example 1. Let X = {0, ∞} ∪ N\{1}, and let p : X × X → R + defined as p(0, 0) = 0, p(0, ∞) = 2, p(∞, ∞) = 0, p(n, n) = 1/n, p(0, n) = 1 + 1/n, and p(n, ∞) = 1 for all n ∈ N\{1}, and p(n, m) = 1 + 1/n + 1/m for all n, m ∈ N\{1} with n = m.

It is routine to check that p is a partial metric on X, and it is complete because every Cauchy sequence in (X, (d p ) s ) is eventually constant.

On the other hand, Max(X, ≤ p ) = X, so i(X) = Max(BX) by Theorem 1 (b). However ker p = {∞}.

We show that σ(BX) |Max(BX) = λ(BX) |Max(BX) . Indeed, we have (0, 0) ∈ BX\ ↑ (∞, 1) because d p (∞, 0) = p(∞, 0)p(∞, ∞) = 2. Nevertheless, for each n ∈ N we have (n + 1, 0) ∈ B P ((0, 0), 1/n)∩ ↑ (∞, 1), because P ((0, 0), (n + 1, 0)) = 1 + 1/(n + 1) < 1 + 1/n = P ((0, 0), (0, 0)) + 1/n, and d p (∞, n + 1) = 1 for all n ∈ N. Since by Theorem 1 (c), {B P ((0, 0), 1/n) : n ∈ N} is a σ(BX)-local base at (0, 0), we conclude that λ(BX) |Max(BX) is strictly finer than σ(BX) |Max(BX) .

URL

The computational model suggested by conditions (1)-(3) in p. 59 of [START_REF] Waszkiewicz | Quantitative continuous domains[END_REF] joint with Definition 5.1 of [START_REF] Romaguera | A quantitative computational model for complete partial metric spaces via formal balls[END_REF] and Theorem 1 above, motivate the following notions. Definition 1. A partially metrizable computational model is a triple (L, , P ) such that (L, ) is a domain and P is a complete partial metric on L that induces the Scott topology. Definition 2. A topological space (X, τ ) has a partially metrizable computational model if there is a partially metrizable computational model (L, , P ) and an homeomorphism φ between (X, τ ) and a τ (dP ) -1 -closed subset of (L, P ) such that φ(Max(X , ≤ τ )) = Max(L).

We conclude the paper with a characterization of those topological spaces that admit a compatible complete partial metric via the model proposed in Definition 1.

Theorem 2. A topological space has a partially metrizable computational model if and only if it admits a compatible complete partial metric.

Proof. Suppose that the topological space (X, τ ) has a partially metrizable computational model (L, , P ). Let φ an homeomorphism between (X, τ ) and a τ (dP ) -1closed subset of (L, P ) such that φ(Max(X , ≤ τ )) = Max(L). For each x, y ∈ X put p(x, y) = P (φ(x), φ(y)). Then p is a partial metric on X such that τ p = τ. Moreover p is complete: Indeed, let (x n ) n∈N be a Cauchy sequence in the metric space (X, (d p ) s ). Then (φ(x n )) n is a Cauchy sequence in the complete metric space (L, (d P ) s ), so there is z ∈ L such that (φ(x n )) n converges to z in (L, (d P ) s ), and hence in (L, (d P ) -1 ). Since φ(X) is τ (dP ) -1 -closed in L, it follows that z ∈ φ(X), and consequently (x n ) n converges to φ -1 (z) in (X, (d p ) s ). We conclude that p is a complete partial metric on X.

Conversely, if (X, τ ) admits a compatible complete partial metric p, then (BX, dp ) is a domain. So, by Theorem 1, (BX, dp , P ) is a partially metrizable computational model for (X, τ ).

The

  Scott topology σ(L) of a poset (L, ) is constructed as follows: A subset U of L is open with respect to σ(L) provided that: (i) U =↑ U, where ↑ U = {y ∈ L : x y for some x ∈ U }; (ii) for each directed subset D of L with D ∈ U, it follows that D ∩ U = ∅. If D is a subset of L, we denote by σ(L) |D the restriction of σ(L) to D. If (L, ) is a continuous poset, then the sets ⇑ x, x ∈ L, form an open base for σ(L), where ⇑ x = {y ∈ L : x y} (see [6, Proposition II-1.10]). The lower (or weak) topology of a poset L is the one that has as a subbase the collection of sets of the form L\ ↑ x, x ∈ L, and is denoted by ω(L). Let us recall topology of σ(L) and ω(L) is the Lawson topology of L, which is denoted by λ(L). If D is a subset of L, we denote by λ(L) |D the restriction of λ(L) to D.

3 .

 3 Partially metrizable computational models

  : http://mc.manuscriptcentral.com/gcom E-mail: ijcm@informa.

  : http://mc.manuscriptcentral.com/gcom E-mail: ijcm@informa.

URL: http://mc.manuscriptcentral.com/gcom E-mail: ijcm@informa.comInternational Journal of Computer Mathematics

The authors acknowledge the support of the Spanish Ministry of Science and Innovation, under grant MTM2009-12872-C02-01


V Arroyo

A Cordero

J R Torregrosa

M P Vassileva

Artificial Satellites Preliminary Orbit Determination by modified high-order Gauss method

Keywords: orbit determination, Gauss method, nonlinear systems, Newton method, iterative function, order of convergence, efficiency orbit determination, Gauss method, nonlinear systems, Newton's method, order of convergence, efficiency index AMS Subject Classification: 65H10; 65L20

In recent years, high-order methods have shown to be very useful in many practical applications, in which nonlinear systems arise. In this case, a classical method of positional astronomy have been modified in order to hold a nonlinear system in its establishments (that in the classical method is reduced to a single equation). At this point, high-order methods have been introduced in order to estimate the solutions of this system and, then, determine the orbit of the celestial body. We also have implemented a user friendly application, which will allow us to make a numerical and graphical comparison of the different methods with reference orbits, or user defined orbits.

Introduction

Orbit determination is an old problem with new applications: at the early XIX century, Gauss designed a method to predict the future positions of asteroids, as Ceres, or other celestial bodies of our solar system with elliptical orbits. Nowadays, orbit determination methods are an essential tool in order to, by example, correct the position of artificial satellites in their orbits. The first step in kind of methods is to determine preliminary orbits, as the motion analyzed is under the premises of the two bodies problem, not taking into account any other force than mutual gravitational attraction between both bodies. Thereafter, perturbations and other variables must be taken into account in order to refine the preliminary orbit.

Recently, several authors have focused this problem from other points of view: in [START_REF] Danchick | Gauss meets Newton again: How to make Gauss orbit determination from two position vectors more efficient and robust with Newton-Raphson iterations[END_REF], Danchick improves the original work of Gauss introducing Newton-Raphson's method (instead of fixed-point method in the original scheme) and modifying the iterative process in order to widen the region of convergence. Moreover, Gronchi in [START_REF] Gronchi | Multiple solutions in preliminary orbit determination from three observations[END_REF] introduces a generalization of the geometric interpretation that provides Charlier's theory on Laplace's method of preliminary orbit determination from three observations. This generalization can to take into account topocentric observations and is useful to understand when there are multiple solutions and where they are located. This new theory works for both Laplace's and Gauss's methods. From another point of view, Gronchi et al. in [START_REF] Gronchi | Orbit determination with the two-body integrals[END_REF] investigate a method to compute a finite set of preliminary orbits for solar system bodies using the first integral of the Kepler's problem, which is useful for the modern sets of astrometric observation.

The inertial system which the orbit is placed in is a geocentric system whose fundamental plane is the projection of the terrestrial equator to the celestial sphere, the perpendicular axis points to Celestial North and South Poles, and the fundamental direction point to the Vernal point γ, in Aries constellation.

If we focus on the orbital plane, it is possible to set a two-dimensional coordinate system, where the fundamental direction points to the perigee of the orbit, the closest point of the elliptical orbit to the focus and center of the system, the Earth. In order to place this orbit in the celestial sphere and determine completely the position of a body in the orbit, some elements (called orbital or keplerian elements) must be determined.

Then, the orbital elements are:

• Ω, (right ascension of the ascending node): defined as the equatorial angle between the Vernal point γ and the ascending node N ; it orients the orbit in the equatorial plane. • ω, (argument of the perigee): defined as the angle of the orbital plane, centered at the focus, between the ascending node N and the perigee of the orbit; it orients the orbit in its plane. • i, (inclination): Dihedral angle between the equatorial and the orbital planes.

• a, (semi-major axis): Which sets the size of the orbit.

• e, (eccentricity): Which gives the shape of the ellipse.

• T 0 , (perigee epoch): Time for the passing of the object over the perigee, to determine a reference origin in time. It can be denoted by a exact date, in Julian Days, or by the amount of time ago the object was over the perigee.

Different methods have been developed for this purpose (see [START_REF] Danby | Fundamentals of Celestial Mechanics[END_REF][START_REF]Methods of Orbit Determination[END_REF][START_REF] Sevilla | Mecánica Celeste Clásica[END_REF]), constituting a fundamental element in navigation control, tracking and supervision of artificial satellites. By using these methods, from position and velocity coordinates for a given time, it is possible to determine those orbital elements for the preliminary orbit, which should be refined with later observations from ground stations, whose geographic coordinates are already known. In order to get this aim, some angles (or anomalies) must be determined on the planar orbit. Firstly, the object position in the ellipse can be determined by an angle, the true anomaly (ν), with center on the focus of the ellipse, which is the covered angle by a position vector, from its last perigee epoch (ν = 0), to the observation instant. Another related angle with the previous one is the eccentric anomaly (E), whose center is on the center of the ellipse. This is the covered angle by a line from this center to the point where a circumference of radius a, the semi-major axis, is cut by a perpendicular line to X axis passing by the coordinates of the position vector, from its last Perigee epoch (E = 0) to the observation moment.

Using the Earth as the center of the coordinates system, it is useful to establish related units: the distance unit is the Earth radius (e.r.), approximately 6370 Km, and time unit is the minute, although some dates are described in Julian days (JD). Some fundamental constants are the Earth gravitational constant, k = 0.07436574(e.r.) 3 2 /min (see [START_REF]Methods of Orbit Determination[END_REF]), G, and the gravitational parameter µ = 1 mEarth (m Earth + m Object ) ≈ 1. Then, modified time variable is introduced as τ = k(t 2 -t 1 ), where t 1 is an initial arbitrary time and t 2 is the observation time. So, τ is considered here as a measure of time difference, which will simplify calculations. To estimate the velocity we can make use of the closed forms of the f and g series (see [START_REF]Methods of Orbit Determination[END_REF][START_REF] Sevilla | Mecánica Celeste Clásica[END_REF]),

F

f = 1-a | r1| [1-cos (E 2 -E 1 )] and g = τ - √ a 3 µ [(E 2 -E 1 )-sin (E 2 -E 1 )
], so we can express the rate respect two positions vectors and time as

˙ r 1 = r 2 -f • r 1 g . (1)
So, it is clear that, known two position vectors and its corresponding observational instants, the main objective of the different methods that determine preliminary orbits is the calculation of the semi-major axis, a, and the eccentric anomalies difference, E 2 -E 1 . When they have been calculated, it is possible to obtain by (1) the velocity vector corresponding to one of the known position vectors and, then, obtain the orbital elements.

Most of these methods have something in common: the need for finding the solution of a nonlinear equation or system, as in Gauss method. Usually, fixed point or secant methods are employed.

From the available input data, two position vectors and times for the observations, τ can be immediately deduced and other intermediate results as the difference in true anomalies, (ν 2 -ν 1 ), deduced by cos (ν

2 -ν 1 ) = r1• r2 | r 1 |•| r 2 | and sin (ν 2 -ν 1 ) = ± x 1 y 2 -x 2 y 1 |x 1 y 2 -x 2 y 1 | 1 -cos 2 (ν 2 -ν 1 )
, with positive sign for direct orbits, and negative for retrograde orbits.

Once the difference of true anomalies is obtained from the position vectors and times, the specific orbit determination method is used. In our particular case, we will introduce in the following section the classical Gauss method and, thereafter, we will modify it in order to estimate the value of the semi-major axis and eccentric anomalies by means of high-order iterative methods.

Let us also note that the inverse problem, it is the calculation of ephemeris (position an velocity in a given time) knowing the orbital elements, can be done easily, with direct operations that can be found in related bibliography (see [START_REF] Danby | Fundamentals of Celestial Mechanics[END_REF][START_REF]Methods of Orbit Determination[END_REF][START_REF] Sevilla | Mecánica Celeste Clásica[END_REF]). of the sector ABC and the triangle ABC, as Figure 1 illustrates, delimited by both position vectors, r 1 y r 2 . The ratio sector-triangle can be expressed as:

Gauss method of orbit determination

This method calculate a preliminary orbit of a celestial body by means of only two observations (position vectors). It is based on the relation between the areas

y = √ µp • τ r 2 r 1 sin (ν 2 -ν 1 ) = √ µ • τ 2 √ a √ r 2 r 1 sin ( E2-E1
2

) cos ( (with (ν 2 -ν 1 ) = π), and on the first

y 2 = m l + x (3)
and second

y 2 (y -1) = mX. ( 4
)
Gauss equations, where the constants of the problem (based on the data and the previously made calculations are

l = r 2 + r 1 4 √ r 2 r 1 cos ( ν2-ν1 2 ) - 1 2 and m = µτ 2 [2 √ r 2 r 1 cos ( ν2-ν1 2 )] 2 . ( 5
)
Moreover, also must be determined in the process the value of:

x = 1 2 1 -cos E 2 -E 1 2 = sin 2 E 2 -E 1 4 and X = E 2 -E 1 -sin (E 2 -E 1 ) sin 3 ( E 2 -E 1 2
) .

(6) With these equations we present two different schemes to solve the problem: the classical method, which reduces first and second Gauss equations to a unique nonlinear equation, solved by fixed point method, and the modified Gauss scheme, which solve directly the nonlinear system formed by both Gauss equations.

Classical Gauss method scheme

In the classical method, an only nonlinear equation is obtained by, substituting second Gauss equation (3) into first equation ( 4):

y = 1 + X(l + x). ( 7
)
Then a fixed-point scheme is used to estimate the solution of ( 7), making a first estimation of the ratio , y 0 = 1, and by using the first Gauss equation to get

x 0 = m y 2 0 -l.
From the definition of x in equation [START_REF] Gronchi | Orbit determination with the two-body integrals[END_REF], it is possible to calculate cosine and sine of the half difference of eccentric anomalies, which is known to be between 0 an π radians, determining uniquely the difference of eccentric anomalies:

cos E 2 -E 1 2 = 1 -2x 0 and sin E 2 -E 1 2 = + 4x 0 (1 -x 0 ). (8)
Then, with equation ( 6), an estimation of X, X 0 , can be calculated and used in the reduced nonlinear equation [START_REF] Jarrat | Some fourth order multipoint iterative methods for solving equations[END_REF] in order to get a better estimation of the ratio

y 1 = 1 + X 0 (l + x 0 ).
This iterative process gets new estimations of the ratio, until a given tolerance condition is satisfied. If there is convergence, the semi-major axis a, can be calculated by means of equation ( 2), from the last estimations of ratio and difference of eccentric anomalies, and the last phase of the process is then initiated, to determine velocity and orbital elements.

The Gauss method has some limitations, as the critical observation angles spread, in ν 2 -ν 1 = π, where the denominator of equation ( 2 Artificial Satellites Orbit Determination that this method is only convergent to a coherent solution if the observation angles spread is less than 70 o , where this method has order of convergence 1. The ratio y grows with spread, leading to an invalid solution, if it converges. So this method is suitable for small spreads in observations, that is, observations which are close to each other.

Modified Gauss schemes

It is possible to make a different approach to the problem, solving the nonlinear system formed by both Gauss equations with different higher order iterative methods, instead of solving a unique nonlinear equation, which have the ratio y as unknown.

Firstly, it is necessary to establish the nonlinear system to be solved. In this case we can use the ratio u = y and the difference of eccentric anomalies, v = E 2 -E 1 , as our unknowns, and substitute [START_REF] Gronchi | Orbit determination with the two-body integrals[END_REF] in first an second Gauss equations, (3) and ( 4), so the system F (u, v) = 0 becomes:

u 2 + u 2 2 1 -cos v 2 -m = 0, u 3 + u 2 -m v-sin v sin 3 v 2 = 0. ( 9
)
Let us note that l and m are constants, with the input data, calculated with (5). Moreover, it is easy to check that the jacobian matrix F (u, v) associated to this system is ill-conditioned, so the iterative methods used to estimate its solutions must be robust enough. General information about iterative methods to solve nonlinear equations and systems can be found in [START_REF] Traub | Iterative methods for the solution of equations[END_REF].

Firstly, we will use Newton's method. Then, new estimations of the solution can be deduced with the following iterative scheme:

x (k+1) = x (k) -F (x (k) ) -1 F (x (k) ), (10)
with convergence order up to 2. Also the well-known Jarrat's method (see [START_REF] Jarrat | Some fourth order multipoint iterative methods for solving equations[END_REF]) will be employed, with forth order of convergence and iterative expression:

x (k+1) = x (k) - 1 2 (3F (y (k) ) -F (x (k) )) -1 (3F (y (k) ) + F (x k ))F (x (k) )) -1 F (x (k) ) (11) where y (k) = x (k) -2 3 F (x (k) ) -1 F (x (k)
). Moreover, a new family of methods is introduced:

x (k+1) = y (k) -A -1 BF (x (k) ) -1 F (y (k) ), ( 12
)
where

y (k) = x (k) -cF (x (k) ) -1 F (x (k) ), A = a 1 F (x (k) ) + a 2 F (y (k) ) and B = b 1 F (x (k) )+b 2 F (y (k)
) which will be denoted by N 5 methods and whose convergence order will be proved to be five for some values of the parameters. In order to analyze the convergence of the new method we use Taylor expansions around the solution, whose notation was introduced in [START_REF] Cordero | A modified Newton-Jarratt's composition[END_REF].

Theorem 2.1 Let F : D ⊆ R n -→ R n be sufficiently differentiable at each point of an open neighborhood D of x ∈ R n ,
if c = 1, a 2 = 0, a 1 = -a2 5 , b 1 = 3a2
5 and b 2 = -a 1 and satisfies the error equation

e k+1 = -5C 4 2 -4C 2 2 C 3 + 3 2 C 2 C 3 C 2 - 3 2 C 3 C 2 + 6C 2 C 4 ) e 6 k + O(e 7 k )
, ). ( 14)

where e k = x(k) -x and C k = (1/k!)[F (x)] -1 F (k)(x), k = 2,
From ( 14), we have

F (x (k) ) -1 = I + X 2 e k + X 3 e 2 k + X 4 e 3 k + X 5 e 4 k [F (x)] -1 + O(e 5 k ), with X 2 = -2C 2 , X 3 = 4C 2 2 -3C 3 , X 4 = 6C 3 C 2 -8C 3 2 + 6C 2 C 3 -4C 4 and X 5 = 16C 4 2 -12C 3 C 2 2 -12C 2 C 3 c 2 + 8C 4 C 2 + 9C 2 3 -12C 2 2 C 3 + 8C 2 C 4 -5C 5 . Then, y (k) -x = x (k) -x -c F (x (k) ) -1 F (x (k) ) (15) = (1 -c)e k + cC 2 e 2 k -cM 0 e 3 k -cM 1 e 4 k + O(e 5 k ),
where

M 0 = C 3 + X 2 C 2 + X 3 and M 1 = C 4 + X 2 C 3 + X 3 C 2 + X 4 .
Again, by Taylor expansion:

F (y (k) ) = F (x) (1 -c)e k + Q 2 e 2 k + Q 3 e 3 k + Q 4 e 4 k + O(e 5 k ), (16)
where

Q 2 = (c + (1 -c) 2 )C 2 , Q 3 = -cM 0 + 2c(1 -c)C 2 2 + (1 -c) 3 C 3 and Q 4 = -cM 1 + c 2 C 3 2 -2c(1 -c)C 2 M 0 + 3c(1 -c) 2 C 3 C 2 + (1 -c) 4 C 4 . Moreover, F (y (k) ) = F (x) I + 2(1 -c)C 2 e k + T 2 e 2 k + T 3 e 3 k + T 4 e 4 k + O(e 5 k ), (17)
being

T 2 = 2cC 2 2 + 3(1 -c) 2 C 3 , T 3 = -2cC 2 M 0 + 6c(1 -c)C 3 C 2 + 4(1 -c) 5 C 4 and T 4 = -2cC 2 M 1 + 3c 2 C 3 C 2 2 -6c(1 -c)C 3 M 0 + 12c(1 -c) 2 C 4 C 2 + 5(1 -c) 4 C 5 .
So, the expansion of matrix A is obtained combining ( 14) and ( 17):

A = a 1 F (x (k) ) + a 2 F (y (k) ) = F (x) (a 1 + a 2 )I + L 1 e k + L 2 e 2 k + L 3 e 3 k + L 4 e 4 k + O(e 5 k ),
where

L 1 = 2(a 1 + a 2 (1 -c))C 2 , L 2 = 3a 1 C 3 + a 2 T 2 , L 3 = 4a 1 C 4 + a 2 T 3 and L 4 = 5a 1 C 5 + a 2 T 4
, and the inverse of A can be expressed as: Now, by ( 14) and ( 17), the Taylor expansion of matrix B is:

A -1 = 1 a 1 + a 2 I + Z 2 e k + Z 3 e 2 k + Z 4 e 3 k + Z 5 e 4 k [F (x)] -1 + O(e 5 k ), being Z 2 = -L 1 (a1+a2) 2 , Z 3 = -L 2 (a1+a2) 2 -Z 2 L 1 a 1 +a 2 , Z 4 = -L 3 (a1+a2) 2 -Z 2 L 2 +Z 3 L 1 a 1 +a 2 and Z 5 = -L 4 (a1+a2) 2 -Z 2 L 3 +Z 3 L 2 +Z 4 L
B = b 1 F (x (k) ) + b 2 F (y (k) ) = F (x) (b 1 + b 2 )I + P 1 e k + P 2 e 2 k + P 3 e 3 k + P 4 e 4 k + O(e 5 k ),
where

P 1 = 2(b 1 + b 2 (1 -c))C 2 , P 2 = 3b 1 C 3 + b 2 T 2 , P 3 = 4b 1 C 4 + b 2 T 3 and P 4 = 5b 1 C 5 + b 2 T 4 . Then, A -1 B = HI + S 1 e k + S 2 e 2 k + S 3 e 3 k + S 4 e 4 k + O[e k ], (18)
being H = b1+b2 a1+a2 and

S 1 = P1 a1+a2 +(b 1 +b 2 )Z 2 , S 2 = P2 a1+a2 +Z 2 P 1 +(b 1 +b 2 )Z 3 , S 3 = P 3 a 1 +a 2 +Z 2 P 2 +Z 3 P 1 +(b 1 +b 2 )Z 4 and S 4 = P 4 a 1 +a 2 +Z 2 P 3 +Z 3 P 2 +Z 4 P 1 +(b 1 +b 2 )Z 5
. Now, the Taylor expansion of the remaining product of the iterative expression (12) is:

F (x (k) ) -1 F (y (k) ) = (1 -c)e k + R 2 e 2 k + R 3 e 3 k + R 4 e 4 k + O(e 5 k ), ( 19
)
where

R 2 = Q 2 + (1 -c)X 2 , R 3 = Q 3 + X 2 Q 2 + (1 -c)X 3 and R 4 = Q 4 + X 2 Q 3 + X 3 Q 2 + (1 -c)X 4 .
And finally, combining (15), ( 18) and ( 19), we obtain:

e k+1 = ((1 -c) -H(1 -c))e k + (cC 2 -HR 2 -(1 -c)S 1 )e 2 k + (-cM 0 -HR 3 -S 1 R 2 -(1 -c)S 2 )e 3 k + (-cM 1 -HR 4 -S 1 R 3 -S 2 R 2 -(1 -c)S 3 )e 4 k + O(e 5 k ).
So, the solution of the following linear system will provide us the conditions of the parameters to have convergence order five.

(

-c) -H(1 -c) = 0 cC 2 -HR 2 -S 1 (1 -c) = 0 cM 0 + HR 3 + S 1 R 2 + S 2 (1 -c) = 0 cM 1 + HR 4 + S 1 R 3 + S 2 R 2 + S 3 (1 -c) = 0        1
From the two first equations is directly obtained that c = 1 and H = 1. Then, from the third equation, the following condition must be satisfied in order to obtain convergence order four: b1 a1+a2 -a 1 (b 1 +b 2 ) (a 1 +a 2 ) 2 = 1. Finally, the last equation gives us the condition to be satisfied for order five: 5a 1 +a 2 = 0. So, the values of the parameters that ensure fifth-order convergence are:

a 1 = -a 2 5 , b 1 = 3a 2
5 and b 2 = -a 1 and the resulting error equation is:

e k+1 = -5C 4 2 -4C 2 2 C 3 + 3 2 C 2 C 3 C 2 - 3 2 C 3 C 2 + 6C 2 C 4 ) e 6 k + O(e 7 k ).
In the last section, we will use a member of this family in order to compare the precision of the calculated orbit with the other methods. In particular, we will take a 2 = 5 and its iterative expression is:

y (k) = x (k) -F (x (k) ) -1 F (x (k) ) (20) x (k+1) = y (k) --F (x (k) ) + 5F (y (k) ) -1 3F (x (k) ) + F (y (k) ) F (x (k) ) -1 F (y (k) ).
Let us remark that this new uniparametric family of methods has better efficiency index than the well-known Jarrat's method and it is more efficient from the computational point of view, as it gets higher order of convergence with the same number of operations and only one more functional evaluation. In order to measure the efficiency of an iterative method, the efficiency index is defined as I = p 1/d (see [START_REF] Ostrowski | Solutions of equations and systems of equations[END_REF]), where p is the order of convergence and d is the total number of new functional evaluations (per iteration) required by the method. In the particular case of the modified Gauss method, the size of the nonlinear system involved is two; then, the respective efficiency indices are I N ewton = 1.1225, I Jarrat = 1.1487 and I N 5 = 1.1610. In spite of this, we will see in the next section that the behavior of the new method N 5 is better, because of the ill-conditioned system to be solved.

All this Newton's variants applied to the nonlinear system appearing in Gauss method, [START_REF] Sevilla | Mecánica Celeste Clásica[END_REF], are expected to be at least so accurate as the classical scheme, but to drastically reduce the number of iterations needed to find a solution to the problem, as it will be seen later.

Comparing Gauss method schemes

Now, tests are needed to analyze the previously described schemes. For that purpose a graphical application has been developed with Matlab GUIDE (Graphical User Interface Development Environment) to make graphical and numerical comparison. The schemes presented will work with 200 digits of mantissa as they use variable precision arithmetics, so we can set more restrictive tolerances.

The reference or test orbits we will use, found on [START_REF]Methods of Orbit Determination[END_REF], are:

• Test Orbit I: By using the first test positions vectors and times, we can first compare the number of iterations and estimated accuracy of classical (C), Newton (N ), Jarrat (J) and new fifth-order (N 5 ) schemes described in this paper with a 2 = 5, described in (20). As we can see in Table 1, with tolerance = 10 -100 , higher order methods reduce significantly the number of iterations, getting even more accuracy than the classical scheme. Due to limitations in number of digits and format in observations data, and to the last phase of calculations, some accuracy is lost, but it is hard to determine differences in errors in the presented schemes. As far as results can be represented, errors in the final results of the orbital elements, for classical Gauss method, are shown in Table 2, where the exact orbit elements are compared with the calculated ones by means of the classical method.

r 1 = [2.

Errors

C N J N 5 |a -a| 3.3032e-070 e.r. Now we can compare the new schemes with the classical, seeing in Table 2 the differences between the calculated orbital elements by the classical method and each one of the modified methods. It can be observed that Jarrat's and new fifthorder methods obtain almost the same estimation of the solution than Newton's method. Nevertheless, the reduction in the number of iterations needed justifies the use of high-order methods.

If we vary tolerance from 10 -100 up to 10 -498 , we can compare in Table 3 how number of iterations grows, making it clear that solving the nonlinear system, instead of reducing it to a nonlinear equation, does not increase number of iterations so fast as the classical scheme.

Finally, in Table 4, we can compare the number of iterations needed for different test orbits with different spreads in observations SP = ν 2 -ν 1 , to realize that the limitation of spread is still present, but overall process is made faster, not increasing iterations to find a solution in worse cases, that is, with bigger difference of true anomalies in observation.

Conclusion

A new approach to the problem of orbit determination is proposed, consisting in solving directly a nonlinear system formed by both Gauss equations, by means of well known iterative functions as Newton's and Jarrat's and a new method which have higher convergence order.

In the test of these variants of the Gauss methods, it is seen that they can reduce significantly the number of iterations, making the process faster, so it is possible to use more limiting tolerances to improve accuracy, without increasing much more the number of iterations. Some limitations of the classical scheme are still present in the alternatives introduced in this paper, such as spread limitation in observations, that is, the difference of true anomalies of observations. As the ratio y grows with spread, bigger spreads mean more iterations to find a solution, but in the proposed modified schemes this increment is very limited. If the difference is greater than 70 o , the process will probably lead to invalid solutions, which makes Gauss method suitable only for observations that are close enough.

Figure 1 .

 1 Figure 1. Ratio sector to triangle, in Gauss method.

  46080928705339, 2.04052290636432, 0.14381905768815] e.r. r 1 = [1.98804155574820, 2.50333354505224, 0.31455350605251] e.r. t 1 = 0JD t 2 = 0.01044412000000 JD Ω = 30 o ω = 10 o i = 15 o a = 4e.r. e = 0.2 T 0 = 0m • Test Orbit II: r 1 = [-1.75981065999937, 1.68112802634201, 1.16913429510899] e.r. r 2 = [-2.23077219993536, 0.77453561301361, 1.34602197883025] e.r. t 1 = 0 JD t 2 = 0.01527809000000 JD Ω = 80 o ω = 60 o i = 30 o a = 3 e.r. e = 0.1 T 0 = 0m • Test Orbit III: r 1 = [0.41136206679761, -1.66250000000000, 0.82272413359522] e.r. r 2 = [0.97756752977209, -1.64428006097667, -0.04236299091612] e.r. t 1 = 0 JD t 2 = 0.01316924000000 JD Ω = 120 o ω = 150 o i = 60 o a = 2 e.r. e = 0.05 T 0 = 0m, • Test Orbit IV: Determination r 1 = [0.65241964490697, 3.80258035509303, 2.22750000000000] e.r. r 2 = [-1.35626966531604, 2.95849708305651, 3.05100082701246] e.r. t 1 = 0 JD t 2 = 0.04622903000563 JD Ω = 45 o ω = 45 o i = 45 o a = 4.5 e.r. e = 0.01 T 0 = 0m

10 REFERENCES

 10 Scheme tol = 10 -100 tol = 10 -198 tol = 10 -498 C 54 106 172 N 8 9 10 J 5 5 6 N 5 4 5 5

  ) vanish. Moreover, it is known

	ecember MMSE2010˙OrbitDetIJCM˙R1 23, 2010 11:37	International	Journal	of	Computer	Mathematics
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  3, . . ..

	Proof: Taylor's expansion around x = x gives
	F (x (k) ) = F (x) e k + C 2 e 2 k + C 3 e 3 k + C 4 e 4 k + C 5 e 5 k + C 6 e 6 k + O(e 7 k ),	(13)
	F (x (k) ) = F (x) I + 2C 2 e k + 3C 3 e 2 k + 4C 4 e 3 k + 5C 5 e 4 k + 6C 6 e 5 k + O(e 6 k

Table 1 .

 1 Comparison of different Gauss method schemes for reference orbit I

	Scheme Iterations ||x (k+1) -x (k) ||
	C	54	1.8364e-101
	N	8	7.3258e-133
	J	5	7.0e-200
	N 5	4	1.0658e-108

Table 2 .

 2 Error in classical Gauss method for reference orbit I

	3.3032e-070 e.r.

Table 3 .

 3 Iterations if varying tolerances, for reference orbit I

		Test Orbit I Test Orbit II Test Orbit III Test Orbit IV
	Scheme SP = 12.23 o	SP = 22.06 o	SP = 31.46 o	SP = 30.29 o
	C	54	76	101	99
	N	8	8	8	8
	J	5	5	5	5
	N 5	4	4	5	5

Table 4 .

 4 Iterations needed for different spreads

URL: http://mc.manuscriptcentral.com/gcom E-mail: ijcm@informa.comInternational Journal of Computer Mathematics

Acknowledgement:

The authors would like to thank the referees for their valuable comments and suggestions to improve the readability of the paper.

This research was supported by Ministerio de Ciencia y Tecnología MTM2010-18539