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OtherUPDATE ON SUGAR SENSING
Plant Sugar-Response Pathways. Part of a Complex Regulatory Web
Susan I. Gibson
Published December 2000. DOI: https://doi.org/10.1104/pp.124.4.1532
This article has a correction. Please see:
Correction - April 01, 2001
Copyright © 2000 American Society of Plant Physiologists
In addition to playing a central role in metabolism, soluble sugars such as Glc and Suc help regulate many developmental and physiological processes in plants (for review, see Koch, 1996; Smeekens, 1998; Sheen et al., 1999; Yu, 1999). For example, sugar levels have been postulated to play an important role in determining the time at which some plant species flower. Treatments that induce flowering can also lead to increased transport of carbohydrates from leaves to shoot apical meristems (Corbesier et al., 1998). This increased sugar transport takes place prior to the rise in metabolic activity that occurs during the transition to flowering, suggesting that sugar levels do not simply rise in response to greater metabolic demand and that sugars may be acting to signal the transition to flowering (for review, see Bernier et al., 1993). Recent findings that Arabidopsis can be induced to flower under conditions of complete darkness by supplying the aerial portions of the plant with exogenous Suc also suggest a role for sugar levels in regulating floral induction (Roldán et al., 1999). A role for sugars in seed germination is suggested by reports that exogenous sugars allow wild-type seeds to germinate in the presence of abscisic acid (ABA; Garciarrubio et al., 1997; Finkelstein and Lynch, 2000). Other developmental processes affected by soluble sugar levels include tuber formation by potatoes (Müller-Röber et al., 1992) and control of root to shoot ratios in a variety of plant species (for review, see Wilson 1988). Sugars are also thought to help control key metabolic processes such as photosynthesis (Krapp et al., 1993) and starch synthesis and breakdown (for review, see Koch, 1996). Strong evidence for the importance of sugars in controlling plant processes is also provided by reports that sugars help regulate the expression of a significant number of plant genes (for review, see Koch, 1996).
Whereas sugars have been implicated in control of many plant processes, the molecular mechanisms by which sugars act remain largely unknown. In contrast, sugar-response pathways have been relatively well characterized in the yeast Saccharomyces cerevisiae (for review, see Johnston, 1999). Information obtained from studies on yeast may be used to make predictions regarding plant sugar-response pathways. For example, as yeast mediate sugar responses via multiple signal transduction pathways, plants may also be expected to utilize several sugar-response pathways. In addition, plant sugar-response pathways may employ homologs of some of the components of yeast sugar-response pathways. The available evidence suggests that some of the most important factors thought to act in yeast sugar-response pathways, such as hexokinase (Hohmann et al., 1999) and SNF1 protein kinase (for review, see Hardie et al., 1998), also play important roles in plant sugar-response pathways. However, although plant and yeast sugar-response pathways are likely to share many features, plants utilize some factors not involved in yeast sugar-response pathways. For example, a calcium-dependent protein kinase acts in sugar-regulated gene expression in plants, but is not known to play a similar role in yeast (Ohto and Nakamura, 1995).
Despite the importance of soluble sugars in regulating plant development and physiology, many fundamental questions regarding plant sugar responses have barely begun to be addressed. For example, precisely which plant processes are sugar regulated remains controversial. In addition, which plant processes are affected by sugars acting in metabolism, as opposed to in signaling, has yet to be determined in most cases. In fact the identities of the molecule(s) that trigger sugar-response pathways remain in question. Although Glc or Suc may act directly as signaling molecules in some sugar-response pathways, other pathways may sense the level of a different sugar or sugar metabolite. In addition, some sugar-response pathways may sense the rate of flux through a particular metabolic pathway, rather than the absolute levels of sugars or sugar metabolites (Krapp et al., 1993). The pathways by which plants respond to sugars as signaling molecules remain to be elucidated. Characterization of these pathways is complicated by the fact that they form part of a complex regulatory web that also includes phytohormone and environmental-response pathways. Several approaches are being employed to address these issues. The remainder of this review focuses on a description of these approaches and some of the results obtained using them.
USE OF SUGAR ANALOGS TO CHARACTERIZE PLANT SUGAR RESPONSES AND RESPONSE PATHWAYS
Experiments aimed at testing the effects of sugar concentration on plant developmental or physiological processes often employ treatments designed to alter endogenous sugar concentrations. However, these experiments typically do not allow a distinction to be made between the role of sugars as metabolites as opposed to as signaling molecules.
How then can the role of sugars as signaling molecules be distinguished from their role as metabolites? One possibility is to use non-metabolizable sugar analogs. In theory, demonstration that a non-metabolizable sugar analog can trigger a particular sugar response should provide a good indication that the response is the result of sugars acting as signaling molecules, rather than as metabolites. In addition, it should be possible to use different sugar analogs to obtain information about sugar-response pathways. For example, to determine whether uptake via a specific sugar-transport system is necessary to trigger a particular sugar response, the effects of sugar analogs that are, or are not, substrates for that transport system can be determined. In a similar manner, whether phosphorylation by hexokinase is an essential step in a sugar-response pathway can be assessed by comparing the effects of sugar analogs that are hexokinase substrates versus those that are not. This type of analysis has led to the hypothesis that phosphorylation of sugars by hexokinase, but not further metabolism, plays a key role in triggering sugar-regulated expression of certain genes (Graham et al., 1994; Jang and Sheen, 1994).
WHICH SUGAR ANALOGS ARE REALLY "NON-METABOLIZABLE"?
Experiments of the type described above require the availability of appropriate sugar analogs. Experiments to test the role of a particular Glc transporter in sugar responses, for example, ideally would compare the effects of a Glc analog that is transported as efficiently as Glc with the effects of a Glc analog that is not transported at all. Figure 1 presents an idealized model of the transport and metabolism of some of the most commonly used sugar analogs. Although this idealized model may be fairly accurate for some plant systems, unfortunately it is quite inaccurate for others. For example, celery suspension cultures have similar growth rates when fed Man, Suc, or mannitol, suggesting that contrary to the idealized model, Man and mannitol are efficiently metabolized by these cells (Stoop and Pharr, 1993). InChenopodium rubrum cell-suspension cultures, the primary metabolic products of exogenously-supplied Man and 2-deoxy-Glc are not Man-6-P and 2-deoxy-Glc-6-P (Klein and Stitt, 1998). Also, 3-O-methyl-Glc is a poor substrate for some Glc-uptake systems (Komor et al., 1985). Therefore, the extent to which a particular sugar analog is transported and metabolized varies tremendously among plant species. As a result, unequivocal interpretation of sugar-analog experiments requires that transport and metabolism of the analogs be characterized in the species used in the experiments. This characterization, unfortunately, has yet to be reported for some popular model organisms, such as Arabidopsis.
Idealized model of sugar-analog metabolism. Arrows indicate different steps in transport and metabolism of sugar analogs. Crosses through arrows indicate that, according to the idealized model (shown in blue), the indicated compounds do not progress through that step. Selected exceptions to the idealized model are shown in red and italicized. "1" refers to reference by Stoop and Pharr (1993). "2" refers to reference by Komor et al. (1985).
DO HEXOKINASES ACT AS SENSORS?
The question of how plants sense sugar is fundamental to our understanding of plant sugar-response pathways. In yeast, hexokinases have been postulated to have dual functions in Glc sensing and phosphorylation. According to this model the conformational alteration induced in hexokinase PII during binding and/or phosphorylation of Glc may allow hexokinase PII to interact with some as yet unidentified factor(s), thereby triggering sugar responses (Entian and Fröhlich, 1984). Hexokinases have also been suggested to act as hexose sensors in plants (Graham et al., 1994; Jang and Sheen, 1994). However, models postulating that hexokinases play a role in Glc sensing that is separable from their role in Glc phosphorylation remain controversial and require further testing (Halford et al., 1999). Progress toward testing these models is provided by the recent identification of mutations that have differential effects on the hexose-sensing and phosphorylation activities of yeast hexokinase PII (Hohmann et al., 1999). In addition, transgenic Arabidopsis expressing a yeast hexokinase gene exhibit increased hexokinase catalytic activity, but are actually less sensitive to exogenous Glc (Jang et al., 1997). These findings suggest that it is possible to separate the Glc sensing and phosphorylation activities of hexokinases. Further identification and characterization of hexokinase mutants should help clarify the role of hexokinases in hexose sensing.
Although the debate over whether hexokinases act as sensors for some sugar-response pathways is likely to continue, the available evidence indicates that hexokinases are not involved in all plant sugar-response pathways. For example, expression of several genes is known to be regulated by Suc, but not by Glc, suggesting that hexokinases are not involved in sugar-regulated expression of these genes (Chiou and Bush, 1998; Rook et al., 1998; Loreti et al., 2000; Müller et al., 2000).
SNF1-RELATED PROTEIN KINASES ACT AS "METABOLIC SENSORS"
The SNF1-related protein kinases represent a large family of proteins that are conserved among animals, fungi, and plants. Many organisms encode several SNF1-related protein kinases. In mammalian systems, SNF1-related protein kinases were first identified as key components of environmental stress-response pathways. In S. cerevisiae, SNF1 protein kinases are activated by Glc deprivation and play a central role in response to nutritional stress. Both environmental and nutritional stresses result in increased AMP:ATP ratios, suggesting that SNF1-related protein kinases may function as "metabolic sensors" that enable organisms to regulate metabolism and gene expression in response to changes in cellular energy status. Although less is known about the role(s) of SNF1-related protein kinases in plants, they are likely to be involved in response to nutritional and/or environmental stress (for review, see Hardie et al., 1998). A role for these proteins in plant sugar-response pathways is suggested by experiments in which antisense expression of a SNF1-related protein kinase gene in potato resulted in loss of sugar-inducible expression of Suc synthase (Purcell et al., 1998). A role for plant SNF1-related protein kinases in stress responses is similarly suggested by the finding that mutations in the ArabidopsisSOS2 gene, which encodes a protein that is similar in sequence to the yeast SNF1 protein kinase, lead to an osmo-sensitive phenotype (Liu et al., 2000). As plant SNF1-related protein kinases may function in both environmental stress and sugar responses, some apparent sugar responses may actually result from stimulation of an SNF1-mediated stress-response pathway, rather than induction of an SNF1-mediated sugar-response pathway.
Understanding how plant SNF1-related protein kinases act as metabolic sensors will require increased information regarding the regulation of these proteins and the identities of their substrates. Progress in this area has been rapid, with recent reports indicating Suc deprivation, cytokinin, and light increase transcript levels of a SNF1-related protein kinase from wheat (Ikeda et al., 1999). In addition, an SNF1-related protein kinase is up-regulated by ABA and inhibits gibberellin-induced expression of certain genes (Gómez-Cadenas et al., 1999). These findings suggest that phytohormones play an important role in some SNF1-mediated responses. Several SNF1-related protein kinase substrates have been identified. Proteins shown to be phosphorylated, and consequently inactivated, by SNF1-related protein kinases include 3-hydroxy-3-methylglutaryl coenzyme A reductase, nitrate reductase, and Suc phosphate synthase. As these enzymes play key roles in isoprenoid biosynthesis, nitrogen metabolism, and Suc synthesis, respectively, their inactivation by SNF1-related protein kinases provides a means by which these kinases may help regulate several major metabolic pathways (Sugden et al., 1999).
IDENTIFYING SUGAR-RESPONSE MUTANTS
A powerful method for characterizing plant sugar responses is to identify and analyze mutants that are defective in one or more sugar responses. Sugar-response mutants can also serve as invaluable tools for identifying components of sugar-response pathways. In recent years, several laboratories have conducted genetic screens to identify plant sugar-response mutants (Table I). These screens have used Arabidopsis as a model system and have generally employed one of two approaches. One approach has been to screen for mutants that are defective in the expression of a particular sugar-regulated gene. Mutants isolated using this approach include thelba and hba mutants, which are defective in β-amylase expression (Mita et al., 1997a, 1997b), the rsrmutants, which are defective in patatin expression (Martin et al., 1997), and the sun mutants, which are defective in plastocyanin expression (Dijkwel et al., 1997).
Sugar-response mutants and corresponding loci
A second approach used to identify sugar-response mutants is to screen for altered sensitivity to the inhibitory effects of high concentrations of exogenous Glc or Suc on early seedling development. As shown in Figure 2, wild-type Arabidopsis seeds sown on media containing high (e.g. 0.3m) concentrations of Suc germinate, but the majority of seedlings fail to develop green, expanded cotyledons or true leaves. High concentrations of exogenous Glc exert a similar effect. It is interesting that germinating seeds/seedlings are only sensitive to high sugar concentrations during the first approximately 48 h after the start of imbibition. This finding suggests that completion of some critical, but as yet unidentified, developmental or metabolic transition results in loss of susceptibility to high sugar concentrations (S. Gibson, R. Laby, and D. Kim, unpublished data).
Effects of high concentrations of exogenous sugar on seedling development. Wild-type ArabidopsIs seedlings were grown on the indicated media for 16 d prior to photographing. The red bars indicate 2 mm.
The sugar-insensitive, or sis (Laby et al., 2000) and Glc-insensitive, or gin (Zhou et al., 1998;Arenas-Huertero et al., 2000), mutants exhibit reduced sensitivity to high sugar concentrations. As high concentrations of Suc and Glc inhibit wild-type seedling development, one of the first questions that arises is whether these mutants exhibit altered sensitivity to both sugars, or just to the one used in their selection. To date, there have been no reports regarding the response of gin mutants to Suc. However, the sis mutants have altered responses to both sugars (Laby et al., 2000; S. Gibson, D. Kim, L. Hoot, and R. Laby, unpublished data). These results suggest that Suc inhibits wild-type seedling development by being metabolized to Glc, thereby triggering a hexose-response pathway. Besides being insensitive to high concentrations of Glc and Suc, at least most of the sis andgin mutants also exhibit decreased sensitivity to the inhibitory effects of exogenous Man on early seedling development (Zhou et al., 1998; Laby et al., 2000).
In contrast to the sis and gin mutants, thepleiotropic regulatory locus 1 (prl1) mutant shows increased sensitivity to Glc and Suc (Németh et al., 1998). The PRL1 gene encodes a protein with sequences characteristic of WD-40 repeat proteins (Németh et al., 1998). It is interesting that the PRL1 protein interacts with the yeast SNF1 protein, as well as with two SNF1-related proteins from Arabidopsis, in a yeast two-hybrid system (Bhalerao et al., 1999). These results suggest that PRL1 may play an important role in one or more sugar-response pathways. However, as SNF1 and related proteins may act in stress responses, as well as in sugar responses, the possibility that PRL1 primarily functions in stress responses must also be considered (Gibson and Graham, 1999).
MANY SUGAR-RESPONSE MUTANTS ARE OSMO-TOLERANT
As high concentrations of exogenous sugars were used to identify the prl1, sis, and gin mutants, an important question is whether these mutants are defective in their response to osmotic stress. When grown on concentrations of either mannitol or sorbitol that are equimolar to the Glc or Suc concentrations used in their selection (0.3–0.33 m), thegin1 (Zhou et al., 1998), sis1, 2, 4, and5 (Laby et al., 2000; S. Gibson, D. Kim, L. Hoot, and R. Laby, unpublished data) mutants appear similar to wild-type plants. However, when grown on higher concentrations of sorbitol, thesis1, 2, 4, and 5 mutants display osmo-tolerant phenotypes during early seedling development (Laby et al., 2000; S. Gibson, D. Kim, L. Hoot, and R. Laby, unpublished data). Testing of the gin1 mutant at higher concentrations of sorbitol or mannitol has not been reported. The molecular basis for the osmo-tolerant phenotype of these mutants currently remains to be determined. One possibility is that mutants defective in the ability to sense and/or respond to sugar may accumulate unusually high concentrations of endogenous sugars, which could have an osmo-protectant effect.
Although most, and possibly all, mutants that are resistant to the inhibitory effects of high concentrations of exogenous sugars on early seedling development are also resistant to osmotic stress during the same developmental time period, several lines of evidence suggest that the mutants' sugar-response phenotypes are not simply a result of osmotic-stress tolerance. First, osmo-tolerant and sugar-response phenotypes are genetically separable, as some mutants, such as theabi2-1 mutant, that are osmo-tolerant exhibit normal sugar responses (Laby et al., 2000). In addition, at least most of the sugar-response mutants characterized to date are also resistant to the inhibitory effects of concentrations of Man (e.g. 1–4 mm) that are too low to exert an osmotic stress. Man is a Glc analog that has been postulated to inhibit seed germination via a hexokinase-mediated sugar-response pathway (Pego et al., 1999). However, Man metabolism has yet to be analyzed in Arabidopsis. Therefore, the possibility remains that hexokinase-mediated inhibition of Arabidopsis seed germination by Man is the result of Man-6-P being metabolized to form a toxic product, rather than of induction of a sugar-response pathway. As a result, the possibility that some sugar-response mutants may have relatively broad defects in stress responses, rather than being specifically defective in sugar responses, cannot be ruled out at this time.
MANY SUGAR-RESPONSE MUTANTS ARE ALSO DEFECTIVE IN PHYTOHORMONE RESPONSE OR METABOLISM
Characterization of plant sugar-response mutants reveals that many of them also exhibit defects in phytohormone response or metabolism. For example, the prl1 mutant shows increased sensitivity not only to sugars, but also to ABA, ethylene, cytokinin, and auxin (Németh et al., 1998). In addition, the sis5 (Laby et al., 2000), sun6 (Huijser et al., 2000), and gin6(Arenas-Huertero et al., 2000) mutants are allelic to the ABA-insensitive mutant abi4, and the sis4 (Laby et al., 2000) and gin1 (J. Sheen, personal communication) mutants are allelic to the ABA-biosynthesis mutant aba2(Table I). It is interesting that mutations in the ABI5gene, but not mutations in the ABI1, ABI2, orABI3 genes also confer a weak sugar-insensitive phenotype (Arenas-Huertero et al., 2000; Huijser et al., 2000; Laby et al., 2000). These results are significant because they show that only mutations in specific ABI genes cause a sugar-insensitive phenotype. Only certain paclobutrazol-resistant mutants also display a sugar-insensitive phenotype. For example, the sis2 mutant is resistant to high concentrations of sugar and to paclobutrazol, an inhibitor of gibberellin biosynthesis. In contrast, the spy3mutant is resistant to paclobutrazol, but not to high concentrations of sugar (D. Kim, L. Hoot, and S. Gibson, unpublished data). Mutants that over-produce ethylene or that exhibit a constitutive ethylene response also have sugar-insensitive phenotypes (Zhou et al., 1998; S. Gibson, R. Laby, and D. Kim, unpublished data).
The above results raise the question of how certain mutations might affect both sugar responses and phytohormone responses or metabolism. For example, how might mutations in the ABI4 gene lead to ABA and sugar-insensitive phenotypes? As shown in Figure3, several different types of mechanisms are possible. Distinguishing between these mechanisms will require more information about ABA and sugar-response pathways than is currently available. One possibility is that ABA and Glc act in the same pathway, with either acting first (Fig. 3, A and B). For example, ABA might act via a signal transduction pathway that requires ABI4, but not ABI1, 2, or 3 to induce expression of a Glc sensor. Wild-type plants would then be expected to produce relatively low levels of the Glc sensor in response to endogenous ABA. Endogenous Glc might then act via the Glc sensor to cause low level activation of a pathway that slows early seedling development. Exogenous ABA or Glc might then lead to over-stimulation of this pathway, strongly inhibiting early seedling development. Plants carrying mutations in ABI4 would be ABA insensitive as a result of being unable to increase Glc sensor levels in response to ABA. These plants would also be Glc insensitive as a result of producing lower levels of the Glc sensor in response to endogenous ABA.
Models for inhibition of early seedling development by Glc and ABA. Glc and ABA may inhibit seed germination and early seedling development via one of the following mechanisms, or by a combination of two or more of them. A, Glc and ABA may act in the same pathway, with ABA acting first. B, Glc and ABA may act in the same pathway, with Glc acting first. C, Glc and ABA may act via converging pathways. D, Glc and ABA may act via completely independent pathways.
Glc and ABA, alternatively, might act via pathways that are initially independent, but then converge (Fig. 3C). For example, Glc and ABA might act via independent pathways to induce expression of ABI4, thereby activating a pathway that slows early seedling development. Mutations in ABI4 would then lead to ABA- and Glc-insensitive phenotypes. The relationship between the converging pathways might also be less direct. For example, they might have complementary effects on second messenger levels. Glc and ABA might also act via completely independent pathways (Fig. 3D). For example, ABA (acting via a pathway that requires ABI4, but not ABI1, 2 or 3) and Glc might inhibit expression of different lipase genes involved in seed storage lipid mobilization. A mutation in ABI4 might then result in increased levels of the ABA-inhibited lipase, which could lead to a Glc-insensitive phenotype by compensating for decreased levels of the Glc-inhibited lipase in response to exogenous Glc.
The mechanism(s) by which phytohormones and sugars inhibit early seedling development are likely to be even more complex than suggested by the above models. First, these models describe responses to only a single phytohormone (ABA) and Glc, whereas several phytohormones, Glc, light, and other factors are involved in early seedling development. Second, more than one of the above models may be required to describe the mechanism(s) by which just one phytohormone and Glc affect early seedling development. In other words, ABA and Glc might each affect early seedling development via several response pathways. Some of these pathways might be involved in response to ABA and Glc, whereas others might be involved in response to only one of these factors. Precedent for this type of "combinatorial" model is provided by studies on the effects of ABA and gibberellin on seed germination. ABA and gibberellin are postulated to regulate seed germination via several pathways, some of which are responsive to both phytohormones and others of which are responsive to just one phytohormone. For example, ABA and gibberellin have antagonistic effects on Ca2+levels, whereas only gibberellin acts via effects on cGMP levels (for review, see Lovegrove and Hooley, 2000).
Plant development, physiology, and metabolism are regulated by input from a number of signaling/response pathways. These pathways include those involved in response to phytohormones, environmental stimuli, and metabolites such as sugars and nitrogen. In recent years it has become increasingly clear that the idea that plants respond to each of these stimuli via separate, linear pathways is over-simplified. Instead, many researchers feel it may be more useful to consider plant response pathways as forming an interconnected web. A signal that affects one part of the web can then affect other parts of the web, more or less strongly and directly. As a result, determining whether a mutation affects response to a particular stimulus by altering a factor that acts directly or indirectly in a response pathway for that stimulus, or by exerting a compensatory effect on a response pathway for a different stimulus, may be difficult.
Elucidation of sugar-response pathways, as well as characterization of the relationships between these pathways and other response pathways, will require much more extensive knowledge regarding the components of these pathways than is currently available. Isolation of more genes identified by sugar-response mutations represents an important step toward this goal. The availability of these genes will then make possible, for example, experiments designed to detect direct, physical interactions between the factors encoded by these genes, as well as regulation of one gene product by another. Use of high-density cDNA microarrays and similar technologies to identify genes regulated in response to a variety of stimuli will also help provide information critical to constructing models of plant response webs.
Dr. Ruth Finkelstein is thanked for several stimulating conversations. The many helpful suggestions made regarding this manuscript by Dr. Bonnie Bartel, Dr. Janet Braam, Lydia Hoot, and Donna Pattison are also gratefully acknowledged.
↵1 This work was supported by the U.S. Department of Energy, Energy Biosciences Program (grant no. DE–FG03–00ER15061).
↵* E-mail sig{at}bioc.rice.edu; fax 713–348–5154.
Received August 14, 2000.
Accepted September 20, 2000.
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(1995) Sugar-induced increase of calcium-dependent protein kinases associated with the plasma membrane in leaf tissues of tobacco. Plant Physiol 109:973–981.
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(1999) Mannose inhibits Arabidopsis germination via a hexokinase-mediated step. Plant Physiol 119:1017–1023.
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Halford NG
(1998) Antisense expression of a sucrose non-fermenting-1-related protein kinase sequence in potato results in decreased expression of sucrose synthase in tubers and loss of sucrose-inducibility of sucrose synthase transcripts in leaves. Plant J 14:195–202.
Roldán M,
Gómez-Mena C,
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(1999) Sucrose availability on the aerial part of the plant promotes morphogenesis and flowering of Arabidopsis in the dark. Plant J 20:581–590.
Rook F,
Gerrits N,
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(1998) Sucrose-specific signalling represses translation of the Arabidopsis ATB2 bZIP transcription factor gene. Plant J 15:253–263.
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(1999) Sugars as signaling molecules. Curr Opin Plant Biol 2:410–418.
(1998) Sugar regulation of gene expression in plants. Curr Opin Plant Biol 1:230–234.
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You are going to email the following Plant Sugar-Response Pathways. Part of a Complex Regulatory Web
Plant Physiology Dec 2000, 124 (4) 1532-1539; DOI: 10.1104/pp.124.4.1532


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text:
Sunrise Airways to Bring More Tourism to Haiti
Cap Haitien is about to get an uptick in tourism this summer. Sunrise Airways, a new airliner, has submitted an application to Turk and Caicos Islands (TCI) Air Transport Licensing Authority (ATLA) to ferry passengers between Providenciales and Cap Haitien International Airports (IA).
Sunrise Airways' application must be made public to permit competing air-passenger carriers to make objections. If objections are made, a hearing will be held to determine their validity. The outcome of the hearing will be the deciding factor to either approve or refuse the air carrier an operating license. If no one raises objections, the application will proceed through the review process and sign-offs more quickly. TCI ATLA Director, Tom Swann, has reported to the news media he is not aware of any objections to Sunrise Airways' planned service route.
Sunrise is planning six flights each week, from Monday through Saturday between Providenciales and recently re-named Cap Haitien IA, now Hugo Chavez IA. It is the biggest airport after Port-au-Prince IA in Haiti. The aircraft Sunrise will use for its flights to Haiti is a Jetstream 19-passenger seat turbine-engine airliner.
Once Sunrise's application has been approved by the licensing authority, it must go to the Civil Aviation Authority (CAA) to get an Air Operators Certificate from them. CAA will vet pilot education and experience in flying turbine-engine aircraft. CAA will also review the Sunrise business plan to ensure they have the industry knowledge necessary to operate an air-carrier company.
Read more: Cap-Haitian, Transportation, Turks and Caicos Islands, tourism, airline, Travel Expenses, Cap-Haitian Airport, Sunrise Airways, Providenciales, Travel and Tourism
« Spicy Haitian Marinades, Haitian Food | Main | Haiti and Dominican Republic Partner for Reforestation Campaign »
Subject: Sunrise Airways to Bring More Tourism to Haiti edit
Gary Bodeau wants a peace of the pie in the new Government
Port-Au-Prince, One of the Largest Cities in the World without a Sewage System
Immigrant child crying as she is separated from parents
Radio Caraibes and Patrick Moussignac, Ranmasse, Jean Monard Metellus
Archives Nationales d'Haiti
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why are there so many Black Owned Mega Churches but no Mega...
Thomassique, Haiti
Verrettes, Haiti

redpajama_set_name: RedPajamaCommonCrawl


text:
Summary of Leadership Lessons from West Point
Doug Crandall
Jossey-Bass, 2006
"In business, if you make bad decisions, people lose money and perhaps jobs," according to an Army captain who graduated from West Point. "In the military, if you make bad decisions...people can die." In that light, these leadership lessons from the U.S. Military Academy at West Point seem highly credible. After all, most fields of endeavor do not involve life-and-death consequences, but being in the military is not an ordinary profession. This grim truth not only requires but demands "competent" military leaders, as author Jim Collins discusses in his foreword to this compilation edited by Major Doug Crandall. getAbstract appreciates these valuable leadership lessons and their source. The Leader to Leader Institute, formerly the Peter F. Drucker Foundation for Nonprofit Management, publishes a comprehensive series of leadership books, 21 of which have been printed in 28 languages. So, when the Institute describes this book as "one of the most important...published in our 16 years," it merits a respectful look.
Book editor Major Doug Crandall serves at the U.S. Military Academy as executive officer to the dean of the Academic Board. Twenty past or present West Point teachers, graduates or serving officers contributed to this book. Crandall, who taught a variety of leadership courses, received an Excellence in Teaching Award at West Point.
"Be, Know, Do"
West Point has been recognized as America's premier leadership training institution for more than 200 years. From its earliest days, West Point has turned inexperienced young men – and, more recently, young women – into outstanding leaders with exemplary character, courage, skills and knowledge. The U.S. Army's pivotal leadership principle, "Be, Know, Do," reflects this:
"Be" – Incorporate the honorable values and attributes that characterize an officer.
"Know" – Master the needed technical and leadership skills.
"Do" – Apply knowledge, skills and leadership, to achieve a mission.
As it trains the Army's future leaders, West Point employs a guiding philosophy based on two vital precepts: 1) The future – and thus war in the future – is unknowable; and 2) In battle, it is best to have imaginative and resourceful leaders who can successfully deal with unforeseen circumstances. Such leaders will be able to "adapt" to all kinds of situations and crises. West Point is to leadership what Wal-Mart is to low prices. Indeed, if you want to learn about leadership, this is the right place. The leadership concepts it uses to "educate...
Flat Army
Work Is Love Made Visible
Welcome to Management
From We Will to At Will
Pakistan Under Siege


text:
Tag Archives: plagiarism
If the Daily Mail 'steals' your visualisation, they're giving you publishing permission on their site
Nathan Yau has written about the Daily Mail using his data visualisation without permission. It's not the first time this has happened, nor even the second.
One of my former Telegraph trainees Raziye Akkoc had the same experience when her world map of immigration was embedded in a Daily Mail article.
Hi Daily Mail. I love that you love my stuff, but please say Telegraph or Raziye Akkoc/link http://t.co/cVnjdIqxcl pic.twitter.com/JL0XL7R8Yl
— Raziye Akkoc (@RazAkkoc) April 17, 2015
This entry was posted in online journalism and tagged daily mail, data visualisation, datavis, huffington post, Nathan Yau, Oatmeal, plagiarism, Raziye Akkoc, Stuff.co.nz on February 21, 2016 by Paul Bradshaw.
The DailyMail.com content farm, pride vs cynicism, and 3 things that could happen next
Image by Yew Wei Tan
James King's account of a year "ripping off the web" at DailyMail.com is the latest in an ongoing drip-drip of uncomfortable revelations about how publishers and broadcasters do their work.
The media may have been the Fourth Estate; but blogs have been performing their role as 'Estate 4.5′ (as Jane Singer put it) for some time now, opening up publishers, journalists' and editors' working practices to public scrutiny on a regular basis.
Two things strike me about King's account, however.
The Brits are coming
The first is to wonder whether a young UK journalist would ever have written the story that King did. Or, perhaps, why none ever has. Continue reading →
This entry was posted in online journalism and tagged DailyMail.com, ethics, gawker, James King, MailOnline, plagiarism, pride, scepticism on March 6, 2015 by Paul Bradshaw.
Plagiarists should at least be *competent* plagiarists – Media Ooops 002
This is a shorter version of an article appearing on the Wardman Wire.
Plagiarism is an interesting game.
You can either rewrite the piece, find a bit more information, leave other bits out, and – if you're the Daily Mail – reduce the reading age by a year or three.
Or you can acknowledge that the story came from somewhere else, and give a hat-tip for a nugget, or a small fee for an article.
Or you can try and ride both horses and end up sitting on your backside in the middle.
So, we have Exhibit A, from Dizzy Thinks:
The Department for Culture, Media and Sport has a dedicated civil servant working on the Queen's Diamond Jubilee celebrations in 2012? Not particular shocking really, but there is an oddity.
According to an FoI release, one of the roles of this civil servant is the development of equalities impact assessment for the Queen's celebratory bash. Why does a celebration for one person need an equalities impact assessment?
Mind you, as an eagle-eyed reader put to to me. Perhaps it's because she's (a) a woman, (b) a pensioner, (c) dependent on state benefits, and (d) married to an immigrant?
and Exhibit B, from the Daily Mail:
The Department for Culture, Media and Sport has a dedicated civil servant working on the Queen's Diamond Jubilee celebrations in 2012. One of the roles of the civil servant is the development of an 'equalities impact assessment'. Why does a celebration for one person need an equalities impact assessment? Is it because she's a woman, a pensioner, relies on the state for handouts — and is married to a foreigner?
The two are nearly the same, and it's only an item in a Diary column, for heaven's sake. A tip would cost about twenty pounds or a gift voucher, and an acknowledgement would cost nothing.
(Hat-tip: Dizzy).
[Update: re-edited]
This entry was posted in newspapers and tagged daily mail, dizzy thinks, humour, knockabout, plagiarism on December 13, 2010 by Paul Bradshaw.
Cooks Source anger moves on to Dairy Goat Journal's Dave Belanger
UPDATE 2 – from Cathy in the comments (Nov 11): Dave Belanger has now paid the fee.
UPDATE – thanks to Vicki in the comments (Nov 11): Dave Belanger has responded to Suzanne, reinstating the image on their website with a credit and link, and offering to pay. However, he has refused to pay the amount requested by Suzanne, and Suzanne is now planning to take the magazine to court. Her reasoning is admirable, and it's fair to say that contributions of commenters have helped her to make a well-informed stance:
"Countryside Publications is a five million dollar company. He accused me of being opportunistic by asking for an increased fee for the unauthorized and uncredited use.
"This is not about money. I may never see the $2100. If I do, it will be a long time from now. If I wanted to make a quick buck, I'd take the $500 [offered]. (I could use it.) But if I let him not only steal the photo but pay no penalty for it, there's no reason for him to not steal again. After all, what did it cost him? He can steal photos all he wants and only pay for them (at a price he sets) if he's caught. Just who is opportunistic? He published my photo without authorization or credit then says, here, take $500 or NOTHING."
There's also some detail about the possible impact on the publishers from Internet users:
"P.S. He mentioned receiving phone calls and emails from my readers and said he was not concerned about it. He admitted there had also been some subscription cancellations, but that people cancelled subscriptions and started subscriptions every day and that he had no reason to believe any subscription cancellations were related to his treatment of my work."
The original post:
Oh dear. It appears another magazine editor is about to feel the force of a thousand emails following a blogger's complaint of breach of copyright and – more importantly – said editor's response to their request for fair payment and acknowledgement of authorship.
The editor in question is Dave Belanger who – apparently – hung up on Suzanne McMinn when she called to ask that her photo – used in Dairy Goat Journal – was properly credited.
With 80 comments already – many of them saying they have called and written to the magazine – and the case also being discussed on the fake Cooks Source Facebook page – you can only hope Dave looks at the Cooks Source and reacts quickly.
*All about this that I can find looks credible, but I'm extra cautious of this being an opportunistic hoax.
via Ulrike in the comments.
This entry was posted in magazines, online journalism, regulation, social media and tagged cooks source, copyright, dairy goat journal, dave belanger, facebook, plagiarism on November 10, 2010 by Paul Bradshaw.
Cooks Source magazine gets Facebook backlash for copying material without permission
UPDATE 7: The official Cooks Source webpage now features a rather confusing statement on the saga, apologising to Monica Gaudio and saying they have made the donation asked for. The page claims that their Facebook page was "cancelled" and "since hacked". It's not clear what they mean by these terms as the original Facebook page is still up and, clearly, could not be hacked if it had been "cancelled". They may be referring to the duplicate Facebook page which also claims (falsely) the original was "hacked". In addition the statement says they have "cancelled" their website – but as the statement is published on their website it may be that by "cancelled" they mean all previous content has been removed. This discussion thread picks out further inconsistencies and omissions.
UPDATE 5: The magazine's Facebook page has now been updated with a message from editor Judith saying she "did apologise" but "apparently it wasn't enough for her", shown below:
UPDATE 2: Reddit users have been digging further into the magazine's use of copyrighted content. They've also identified a planned sister magazine, whose Facebook page has also been the recipient of a few comments.
UPDATE 6: Edward Champion has chased down the copyright holders of both text and images found in Cooks Source which appear to have been used without permission.
UPDATE 4: A list of mainstream media reports on the story is also being maintained on the magazine's Facebook page.
***ORIGINAL BLOG POST STARTS HERE***
For much of today people have been tweeting and blogging about the magazine editor with 30 years' experience demonstrating a by now familiar misunderstanding of copyright law and the 'public domain'.
To the writer whose material they used without permission she apparently responded that "the web is considered "public domain" and you should be happy we just didn't "lift" your whole article and put someone else's name on it!"
What makes this of particular interest is how the affair has blown up not just across Twitter and Reddit but on the magazine's own Facebook page, demonstrating how this sort of mistake can impact very directly on your own readers – and stockists and advertisers:
Meanwhile, others were suggesting investigating the magazine further:
It all adds up to a perfect lesson for magazine editors – not just in copyright, but in PR and community management.
UPDATE 1: It seems that users are going through the latest issue and suggesting where the content may have been taken from.
UPDATE 3 On a separate topics page on the Facebook page the details are being collated.
This entry was posted in magazines, online journalism and tagged cooks source, copyright, edward champion, facebook, judith griggs, plagiarism, reddit on November 4, 2010 by Paul Bradshaw.
The biggest moments in Indian blogging history (guest post)
Pramit Singh gives a comprehensive overview of blogging history – and the blogging scene – in India.
During the Mumbai Terror attacks, a blog started by Dina Mehta was perhaps the first place to provide useful links and phone numbers. During the unprecedented Bihar Floods in August 2008, a blog was the first site providing useful information. During the Tsunami in December 2004, another blog came to the rescue. I can go on.
My point is: Indian blogs have proven themselves time and time again when it comes to providing timely information before anyone else. Continue reading →
This entry was posted in online journalism and tagged aamir khan, Amit Agarwal, amit kapoor, Amit Varma, amitabh bachhan, anil ambani, anuraq kashyap, asian tsunami, bihar floods, blog monkeys, Coca Cola water controversy, comment spamming, dainik bhaskar, Desipundit, dina mehta, divanation, Gaurav Sabnis, history, iipm, India, indian express, indiauncut.com, instablogs, Kamla Bhatt, karan johar, labnol, Mamata Banerji, Mayank Austen Soofi, Mediaah, MediaNet, Meenakshi Reddy Madhavan, mumbai attacks, Nandigram, ndtv, Outlook, p sainath, plagiarism, Pradyuman Maheshwari, Pramit Singh, pratibha patil, Rajesh Jain, Rashmi Bansal, ravish kumar, sakaal times, Sanjiv Bikchandani, seo monkeys, SEZ, Shanmugam Manjunath, shekhar kapoor, Shivam Vjj, Star Syndrome, Sunita Narain, Tarun Tejpal, Tehelka, the hindu, Times of India., TR Vivek on February 4, 2009 by Paul Bradshaw.


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text:
\section{Introduction}

The Painlev\'e differential equations were discovered more than a hundred years ago and since the eighties have appeared in many branches on mathematics and physics, including several of Dubrovin's seminal papers on Frobenius manifolds.

The reason behind the ubiquitous appearance of these equations is that they are innately linked to the Toda hierarchy. In \cite{DZ1}, Dubrovin and Zhang proved that the tau function of a generic solution to the extended Toda hierarchy is annihilated by some combinations of the Virasoro operators. It is such Virasoro constraints that regulate the correlation functions of many systems in random matrix theory, in string theory and topological field theory.  For example in \cite{DZ}, expressions for the genus $g\geq 1$ total Gromov--Witten potential were obtained via the genus zero quantities derived from the Virasoro constraints.

The link between Toda--type systems and Painlev\'e equations becomes explicit when the latter are re-formulated in the so called {\it sigma form}  introduced in \cite{JMMS} as the equation satisfied by the logarithmic derivative of the isomonodromic $\tau$--function. An other approach to obtain the sigma form was proposed by Okamoto who developed the Hamiltonian theory of the Painlev\'e differential equations and showed that  all  B\"acklund transformations can be obtained as natural affine Weyl groups actions on the sigma form (\cite{okam1}, \cite{okam2}, \cite{okam3}, \cite{okam4}).

In this paper, we present the sigma form for the second Painlev\'e hierarchy, an infinite sequence of non linear ODEs containing
$$
{\rm P}_{\rm{\scriptstyle II}} :\quad\quad w_{zz} = 2 w^3+z\,w+\alpha_1,
$$
as its simplest equation.The
 $n$-th element  is of order $2n$, and depends on $n$ parameters denoted by ${t}_1,\dots,{t}_{n-1}$ and $\alpha_n$:
\begin{equation}\label{PIIhier}
    {\rm P}^{(n)}_{\rm{\scriptstyle II}} :\,
    \left(
    \od{z}
    + 2 w
    \right)
    \mathcal{L}_{n} \left[w_{z}-w^{2}\right]
    + \sum_{l=1}^{n-1} {t}_{l} \left( \od{z}
    + 2 w \right) \mathcal{L}_{l}\left[w_{z}-w^{2}\right]
    = z w
    + \alpha_{n},
    \, n\ge 1,
\end{equation}
where $\mathcal{L}_{n}$ is the
operator defined by the recursion relation
\begin{equation}\label{i:lenard}
    \od{z}
    {\mathcal L}_{n+1}
    = \left(
    \frac{{\rm d^3}}{{\rm d}z^3}
    + 4 \left( w_z
    - w^2 \right)
    \od{z}
    + 2 \left( w_z
    - w^2 \right)_z
    \right){{\mathcal L}}_n,
    \quad
    {\mathcal L}_0
    \left[w_z - w^2 \right]
    = {\textstyle\frac12},
\end{equation}
with boundary condition
\begin{equation}\label{eq:L0}
    \mathcal L_{n}[0]
    :=0,
    \quad
    \forall n \geq 1.
\end{equation}

The Hamiltonian form of the second Painlev\'e hierarchy was produced in \cite{MM} where the authors gave canonical coordinates $P_1, \dots, P_n, Q_1, \dots, Q_n$ and a Hamiltonian function ${\mathcal H}^{(n)}$ such that ${\rm P}^{(n)}_{\rm{\scriptstyle II}}$ is equivalent to
\begin{equation}\label{i:hpq}
    \frac{\partial Q_i}{\partial z}
     =\frac{\partial {\mathcal H}^{(n)}}{\partial P_i},
    \qquad
    \frac{\partial P_i}{\partial z}
    = - \frac{\partial {\mathcal H}^{(n)}}{\partial Q_i},
    \quad
    i = 1, \dots, n.
\end{equation}
In particular $ {\mathcal H}^{(n)}$ is a polynomial in
$P_1,\dots,P_n,Q_1,\dots,Q_n$ and that the Hamiltonian equations satisfy the Painlev\'e property.

The sigma function is by definition the evaluation of the Hamiltonian on solutions, namely
\begin{equation}\label{eq-s-def}
    \sigma_n(z)
    := {\mathcal H}^{(n)}
    \left(
    P_1(z), \dots, P_n(z), Q_1(z), \dots, Q_n(z)
    \right).
\end{equation}
Our main result in this paper is the following

\begin{thm} \label{SFthm}
Consider the Lenard operators $\widehat{\mathcal L}_k$ defined by
\begin{align}\label{i:lenards}
    \od{z}
    \widehat{\mathcal L}_{k+1} \left[\sigma_n'-\frac{t_{n-1}}{2}\right]
    &= \left(
    \frac{{\rm d^3}}{{\rm d}z^3}
    + 2 \left(2\sigma_n'- t_{n-1}\right)
    \od{z}
    + 2 \sigma_n''
    \right)
    \widehat{\mathcal L}_k\left[\sigma_n'-\frac{t_{n-1}}{2}\right],
    \\
    \notag
    \widehat{\mathcal L}_0 \left[\sigma_n'-\frac{t_{n-1}}{2}\right]
    &= {\textstyle\frac12},
    \quad
    t_0 = - z,
\end{align}
with the boundary condition
\begin{equation}\label{eq:L0s}
    \widehat{\mathcal L}_{k}[0]
    := \left(
    - \frac{ t_{n-1}}{2}
    \right)^k
    \frac{1}{k}
    \left(
    \begin{array}{c} 2k\\ k\\ \end{array}
    \right),
\end{equation}
and define
\begin{equation}\label{eq:def-fn}
    f_n
    = \displaystyle \sum_{l = 1}^{n} t_l \widehat{\mathcal{L}}_l \LieBrackets{ \sigma_n^{\prime} \brackets{z} - \dfrac{t_{n - 1}}{2}},
    \quad
    t_n = 1.
\end{equation}
Then, for $ n > 1$, the $n$-th element of the second Painlev\'e hierarchy \eqref{PIIhier} is equivalent to
\begin{equation}\label{SF1}
    - f_n^{\prime}
    + \brackets{f_n^{\prime}}^2
    + \brackets{z - 2 f_n}
    \brackets{
    \sum_{l=1}^n t_l
    \left( \widehat{\mathcal L}_l^{\prime\prime} - \widehat{\mathcal L}_{l+1} \right)
    + 2 f_1 f_n
    + \sigma_n \brackets{z}
    - \frac{1}{2} t_{n-1} z
    }
    = \alpha_n \brackets{\alpha_n - 1}.
\end{equation}
\end{thm}

\begin{rem}
We note that an other equation of order $2n+1$ for the sigma function of the $n$-the element of the second Painlev\'e hierarchy was produced by Stuart Andrew, a former master student of the second author in integral form \cite{SA}. The formula \eqref{SF1} in this paper is an explicit ODE of order $2 n$ as expected (see Lemma \ref{lm:order} in Section \ref{se:proof}).
\end{rem}

\textbf{Acknowledgements.}
The authors would like to express their gratitude to Volodya Rubtsov for introducing them to each other.
The authors are also grateful to Vladimir Poberezhnyi, who initiated I.B. to the Painlev{\'e}  equations theory and constantly supported her during her scientific work.
The research of I.B. is a part of her PhD program studies
at the Higher School of Economics (HSE).  I. B. would like to thank to Faculty of Mathematics for giving her such opportunity. The research of M.M. is supported by the EPSRC Research Grant $EP/P021913/1$. The research of I.B. was partially supported by the RFBR Grant 18-01-00461 A. We dedicate this paper in lasting memory of Boris A. Dubrovin.

\section{Hamiltonian structure of the second Painlev\'e hierarchy}

In this section we resume some results in \cite{MM} that turn out to be useful in our proof of Theorem \ref{SFthm}.

Let us consider the isomonodromic deformation problem for the $\PIIn{n}$ hierarchy
\begin{align*}
\dfrac{\partial \Psi}{\partial z}
&= \mathcal{B} \Psi
=
	\begin{pmatrix}
    -\lambda & w \\
    w & \lambda
    \end{pmatrix}
    \Psi
,
\\
\dfrac{\partial \Psi}{\partial \lambda}
&= \mathcal{A}^{(n)} \Psi
= \dfrac{1}{\lambda}
\left[
\begin{pmatrix}
- \lambda z & - \alpha_n
\\
- \alpha_n & \lambda z
\end{pmatrix}
+ M^{(n)}
+ \sum_{l = 1}^{n - 1} t_l M^{(l)}
\right]
,
\\
\brackets{2 k + 1} \dfrac{\partial \Psi}{\partial t_k}
&=
\left(
M^{(k)}
-
\begin{pmatrix}
0 & \brackets{\partial_z + 2 w} \mathcal{L}_k \LieBrackets{w^{\prime} - w^2}
\\
\brackets{\partial_z + 2 w} \mathcal{L}_k \LieBrackets{w^{\prime} - w^2} & 0
\end{pmatrix}
\right)
\Psi,
\end{align*}
where the matrix $M^{(l)}$ is defined as
\begin{equation*}
M^{(l)}
=
\begin{pmatrix}
\displaystyle \sum_{j = 1}^{2 l + 1} A_j^{(l)} \lambda^j
&
\displaystyle \sum_{j = 1}^{2l} B_j^{(l)} \lambda^j
\\
\displaystyle \sum_{j = 1}^{2l} C_j^{(l)} \lambda^j
&
- \displaystyle \sum_{j = 1}^{2 l + 1} A_j^{(l)} \lambda^j
\end{pmatrix},
\end{equation*}
with
\begin{align}
	\label{entrMn}
    A_{2l+1}^{\brackets{l}}
    &= 4^l;
    \qquad
    A_{2k}^{\brackets{l}}
    = 0,
    \quad
    k
    = 0, \dots, l;
    \\
    \notag
	A_{2k+1}^{(l)}
	&= \dfrac{4^{k+1}}{2}
	\left\{ {\mathcal{L}}_{l-k} \left[ w^{\prime} - w^2 \right]
	- \od{z} \left( \od{z} + 2 w \right) {\mathcal{L}}_{l-k-1} \left[ w^{\prime} - w^2 \right]
	\right\},
    \quad
    k = 0, \dots l -1;
    \\
    \notag
	B_{2k+1}^{(l)}
	&= \dfrac{4^{k+1}}{2} \od{z} \left( \od{z} + 2 w \right)
	{\mathcal{L}}_{l-k-1} \left[ w^{\prime} - w^2 \right],
    \quad
    k = 0, \dots, l - 1;
    \\
    \notag
	B_{2k}^{(l)}
	&= - 4^k \left( \od{z} + 2 w \right)
	{\mathcal{L}}_{l-k} \left[ w^{\prime} - w^2 \right],
    \quad
    k = 1, \dots, l.
\end{align}

The compatibility condition
\begin{equation*}
    \dfrac{\partial \mathcal{A}^{(n)}}{\partial z}
    - \dfrac{\partial \mathcal{B}}{\partial \lambda}
    = \LieBrackets{\mathcal{B}, \mathcal{A}^{(n)}}
\end{equation*}
gives the $n$-th member of the second Painlev{\'e}  hierarchy \eqref{PIIhier}.

It is convenient to introduce new notations to define the matrix $\mathcal{A}^{(n)}$. Let us set
\begin{align} \label{entrAn}
	a_{2k+1}^{(n)}
	&= \sum_{l=1}^n t_l A_{2k+1}^{(l)},
	\quad
	k = 1, \dots, n;
	&
	a_1^{(n)}
	&= \sum_{l=1}^{n} t_l A_1^{(l)} - z;
	\\
	\notag
	b_{2k+1}^{(n)}
	&= \sum_{l=1}^{n} t_l B_{2k+1}^{(l)},
	\quad
	k = 1, \dots, n - 1;
	\\
	\notag
	b_{2k}^{(n)}
	&= \sum_{l=1}^{n} t_l B_{2k}^{(l)},
	\quad
	k = 1, \dots, n;
	&
	b_0^{(n)}
	&= - \alpha_n,
	\quad t_n = 1.
\end{align}

Therefore, $\mathcal{A}^{(n)}$ can be represented in the following form
\begin{equation*}
    \mathcal{A}^{(n)}
    =
    \begin{pmatrix}
    \displaystyle \sum_{k=0}^{n} a_{2k+1}^{(n)} \lambda^{2k} &
    \displaystyle \sum_{k=0}^{2n} b_{k}^{(n)} \lambda^{k-1} \\
    \displaystyle \sum_{k=0}^{2n} \brackets{-1}^k b_{k}^{(n)} \lambda^{k-1} 	&
    - \displaystyle \sum_{k=0}^{n} a_{2k+1}^{(n)} \lambda^{2k} \\
    \end{pmatrix}.
\end{equation*}

The canonical coordinates are given by relations
\begin{gather*}
    P_k
    = \Pi_{2k}
    = \dfrac{a_{2(n-k)+1}^{(n)} + b_{2(n-k)+1}^{(n)}}{a_{2n+1}^{(n)}},
    \quad
    Q_k
    = \sum_{j = 1}^n \dfrac{1}{2j} b_{2j}^{(n)} \dfrac{\partial S_{2j}}{\partial \Pi_{2k}},
    \quad
    k
    = 1, \dots, n;
    \\
    S_k = \sum_{j=1}^{2n} q_j^k,
    \quad
    k = 1, \dots, 2n;
    \\
    \Pi_1
    = q_1 + \dots + q_{2n},
    \quad
    \Pi_2
    = \sum_{1 \leq j \leq 2n} q_j q_k,
    \quad
    \dots,
    \quad
    \Pi_{2n}
    = q_1 q_2 \dots q_{2n},
\end{gather*}
where $q_j$ are solutions of the following equation
\begin{equation*}
    \sum_{k=0}^{n-1} \brackets{b_{2k+1}^{(n)} + a_{2k+1}^{(n)}} q_j^{2k} + a_{2n+1}^{(n)} q_j^{2n} = 0,
\end{equation*}
and $p_j = \displaystyle \sum_{k=0}^{n} b_{2k}^{(n)} q_j^{2k-1}$.

The coordinates $P_1, \dots, P_n$, $Q_1, \dots, Q_n$  are canonical with the Poisson structure
\begin{equation*}
	\PoissonBrackets{P_i, P_j}
	= \PoissonBrackets{Q_i, Q_j}
	= 0,
	\quad
	\PoissonBrackets{P_i, Q_j}
	= \delta_{ij},
	\quad
	i, j
	= 1, \dots, n.
\end{equation*}

The corresponding Hamiltonian in terms of the canonical coordinates is
\begin{multline} \label{HIIn}
	{\mathcal H}^{(n)}
	\brackets{P_1, \dots, P_n, Q_1, \dots, Q_n, z}
	= - \dfrac{1}{4^n}
	\left(
	\displaystyle \sum_{l = 0}^{n - 1}
	a_{2 l + 1}^{(n)} a_{2 (n - l) - 1}^{(n)}
	- \displaystyle \sum_{l = 0}^{n - 1}
	b_{2 l + 1}^{(n)} b_{2 (n - l) - 1}^{(n)}
	\right.
	\\
	\left.
	+ \displaystyle \sum_{l = 0}^n
	b_{2l}^{(n)} b_{2 (n - l)}^{(n)}
	\right)
	+ \dfrac{Q_n}{4^n},
\end{multline}
where the coefficients of $\mathcal{A}^{(n)}$ can be expressed as polynomials in the canonical coordinates by theorem 6.1 in \cite{MM}.

We conclude this section be reminding a useful formula valid both for the Lenard operators $\mathcal L_n$ and $\widehat{\mathcal L}_n$ \cite{AFord}:
\begin{align} \label{eq:lzz}
    {\mathcal L}_{n+1}
    =
    \partial_z^2{\mathcal L}_{n}
    &+ 3{\mathcal L}_{n}{\mathcal L}_{1}
    + \displaystyle \sum_{j = 1}^{l - 1}
    \brackets{{\mathcal{L}}_{l - j}
    \brackets{4 {\mathcal{L}}_1    {\mathcal{L}}_j
    - {\mathcal{L}}_{j + 1}
    + 2 {\mathcal{L}}_{j}^{\prime\prime}
    }
    - {\mathcal{L}}_j^{\prime}
    {\mathcal{L}}_{l - j}^{\prime}
    },
    \quad
    n > 1.
\end{align}

\section{Proof of main theorem}\label{se:proof}

Firstly, the following correlation between the sigma coordinates and the solution of the ${\rm P}^{(n)}_{\rm{\scriptstyle II}}$ equation was proved in \cite{SA}:

\begin{lem} \label{sigmaWcorrel} The sigma function $ \sigma_n$ of definition \eqref{eq-s-def} is related to the solution $w_n$ of \eqref{PIIhier} by the following
$$
w_n^{\prime} - w_n^2 = \sigma_n^{\prime} - \dfrac{t_{n - 1}}{2},
$$
where $t_0 = - z$.
\end{lem}

\begin{proof}
By definition \eqref{eq-s-def} of the sigma-coordinates, we have
\begin{equation*}
    \sigma_n(z)
    := {\mathcal H}^{(n)}
    \left(
    P_1(z), \dots, P_n(z), Q_1(z), \dots, Q_n(z)
    \right).
\end{equation*}
where $P_1, \dots, P_n$, $Q_1, \dots, Q_n$ are canonical coordinates. Its first derivative is
\begin{equation*}
    \sigma_n^{\prime} \brackets{z}
    = \PoissonBrackets{{\mathcal H}^{(n)}, {\mathcal H}^{(n)}}
    + \dfrac{\partial {\mathcal H}^{(n)}}{\partial z}
    = \dfrac{\partial {\mathcal H}^{(n)}}{\partial z}
    .
\end{equation*}
By formula \eqref{HIIn}, the only term in ${\mathcal H}^{(n)}$ that depends explicitly on $z$ is
\begin{equation*}
    - \dfrac{1}{2^{2n - 1}} a_1^{(n)} a_{2n - 1}^{(n)}.
\end{equation*}
Hence, using formulas \eqref{entrMn} and \eqref{entrAn}, $\sigma_n^{\prime} (z)$ is calculated as
\begin{align*}
    \sigma_n^{\prime} \brackets{z}
    &= \dfrac{\partial {\mathcal H}^{(n)}}{\partial z}
    = - \dfrac{1}{2^{2n - 1}} \partial_z
    \brackets{a_1^{(n)} a_{2n - 1}^{(n)}}
    \\
    &= - \dfrac{1}{2^{2n - 1}} \partial_z
    \brackets{
    \brackets{\displaystyle \sum_{l = 1}^n t_l A_1^{(l)} - z}
    \displaystyle \sum_{l = 1}^n t_l A_{2n - 1}^{(l)}
    }
    \\
    &= \dfrac{1}{2^{2n - 1}}
    \displaystyle \sum_{l = 1}^n t_l A_{2n - 1}^{(l)}
    = \dfrac{1}{2^{2n - 1}} \brackets{
    t_n A_{2n - 1}^{(n)}
    + t_{n - 1} A_{2n - 1}^{(n - 1)}
    }
    \\
    &= \mathcal{L}_1 \LieBrackets{w_n^{\prime} - w_n^2}
    + \dfrac{t_{n - 1}}{2}
    = w_n^{\prime} - w_n^2
    + \dfrac{t_{n - 1}}{2}.
\end{align*}
\end{proof}

Note that thanks to Lemma \ref{sigmaWcorrel}, we can define all Lenard operators in terms of $\sigma_n (z)$ rather than $w (z)$. However, we need to be careful in the integration step to extract $\mathcal L_{n+1} \LieBrackets{w^{\prime} - w^2}$ from formula \eqref{i:lenard}. The fact that the integrand is exact was proved in \cite{Lax}, and the choice of the integration constant depends on the condition we impose on $\mathcal L_{n}[0]$.

\begin{lem}
For the $n$-th member of the second Pailev\'e hierarchy, $n > 1$, the Lenard operators defined by \eqref{i:lenard}, \eqref{eq:L0}
coincide with the operators defined recursively by relations \eqref{i:lenards} with boundary conditions \eqref{eq:L0s}, or in other words
$$
    \mathcal L_k\left[w_n^{\prime} - w_n^2\right]
    = \widehat{\mathcal L}_k \left[\sigma_n'-\frac{t_{n-1}}{2}\right],
$$
where $w_n$ denotes the solution of \eqref{PIIhier}.
\end{lem}

\begin{proof}
We prove this statement by induction. We know that
$$
    {\mathcal L}_1 \left[ w_n^{\prime} - w_n^2 \right]
    = w_n^{\prime}- w_n^2.
$$
On the other side, using \eqref{i:lenards} we have
$$
    \od{z}
    \widehat{\mathcal L}_{1} \left[\sigma_n'-\frac{t_{n-1}}{2}\right]
    =  \sigma_n'',
$$
so that
$$
    \widehat{\mathcal L}_{1} \left[\sigma_n'-\frac{t_{n-1}}{2}\right]
    =  \sigma_n'
    + \const,
$$
and by imposing the boundary condition \eqref{eq:L0s}, we obtain
$$
    \widehat{\mathcal L}_{1} \left[\sigma_n'-\frac{t_{n-1}}{2}\right]
    =  \sigma_n'
    - \frac{ t_{n-1}}{ 2},
$$
that  due to Lemma \ref{sigmaWcorrel} gives $\mathcal L_1 \left[ w^{\prime} - w^2 \right] = \widehat{\mathcal L}_1 \left[ \sigma_n'-\frac{t_{n-1}}{2} \right]$.

Let us now assume that the statement is true for $k = l$ and prove it for $k= l + 1$.

Let us call $c_{l + 1}$ the constant term of $\widehat{\mathcal{L}}_{l + 1}  \LieBrackets{ \sigma_n'-\frac{t_{n-1}}{2} }$. By formula \eqref{eq:lzz}, we have that $c_{l + 1}$ is defined as
\begin{align*}
    c_{l + 1}
    &= 3 c_1 c_l
    + \displaystyle \sum_{j = 1}^{l - 1}
    c_{l - j}
    \brackets{4 c_1 c_j
    - c_{j + 1}
    }
    .
\end{align*}
This discrete equation is solved by $c_k=\left(- \frac{ t_{n-1}}{ 2}\right)^k\frac{1}{k}\left(\begin{array}{c} 2k\\ k\\ \end{array}\right)$ as we wanted to prove.
\end{proof}

\begin{rem}
When $n = 1$, $t_{n-1}=t_0=-z$ is no longer constant. In this there is no change of boundary condition and the operators $\widehat{\mathcal L}_k$ are simply  the standard operators ${\mathcal L}_k$ defined by \eqref{i:lenard}, \eqref{eq:L0}, applied to $\sigma_1'+ z/2$. In this proof of theorem \ref{SFthm} we consider $n\geq 1$ and prove \eqref{SF1} as well as the following (valid for $n=1$)
\begin{equation}\label{SF1n1}
    - f_1^{\prime} + \brackets{f_1^{\prime}}^2
    + \brackets{z - 2 f_1}
    \brackets{
    \left(  \widehat{\mathcal L}_1^{\prime\prime}
    - \widehat{\mathcal L}_{2} \right)
    + 2 f_1^2
    + \sigma_1 \brackets{z}
    + \frac{z^2}{4}}
    = \alpha_1 \brackets{\alpha_1 - 1},
\end{equation}
\end{rem}

Now we can proceed to the proof of theorem \ref{SFthm}.

\begin{proof}[Proof of theorem \ref{SFthm}]
Suppose that $z - 2 f_n \neq 0$, i.e. $\alpha_n \notin \frac{1}{2} \mathbb{Z}$. Then $w\brackets{z}$ is expressed from \eqref{PIIhier} as
\begin{equation} \label{wExpr}
    w = \dfrac{f_n^{\prime} - \alpha_n}{z - 2 f_n}.
\end{equation}

From lemma \ref{sigmaWcorrel} and \eqref{wExpr} we obtain
\begin{equation} \label{gnExpr}
    - f_n^{\prime}
    + \brackets{f_n^{\prime}}^2
    + \brackets{z - 2 f_n} f_n^{\prime\prime}
    - \undertext{\widehat{\mathcal{L}}_1 \LieBrackets{\sigma_n^{\prime} - \frac{t_{n-1}}{2}}}{\left( \sigma_n^{\prime} - \frac{t_{n-1}}{2} \right)}
    \brackets{z - 2 f_n}^2
    = \alpha_n \brackets{\alpha_n - 1}.
\end{equation}
This equation involves derivatives of $\sigma_n \brackets{z}$ of order $2n + 1$. Since we are looking for an ODE of order $2n$, we need to remove these. To this aim,
we differentiate \eqref{gnExpr} obtaining
\begin{equation}\label{dgnExpr}
    \brackets{z - 2 f_n}
    \brackets{f_n^{\prime\prime\prime}
    + 4 \widehat{\mathcal{L}}_1 f_n^{\prime}
    + 2 \widehat{\mathcal{L}}_1^{\prime} f_n
    - 2 \widehat{\mathcal{L}}_1  - z \widehat{\mathcal{L}}_1^{\prime}}
    = 0,
\end{equation}
and use the Lenard recursion relation \eqref{i:lenard} obtaining
\begin{equation} \label{dgnExpr2}
    \brackets{z - 2 f_n}
    \dfrac{d}{dz} \brackets{
    \sum_{l = 1}^n t_l \widehat{\mathcal{L}}_{l + 1}
    - z\widehat{\mathcal{L}}_1- \sigma_n \brackets{z} + h_n \brackets{z}
    + \const
    }
    = 0,
\end{equation}
where we understand that $\widehat{\mathcal{L}}_k$ is applied to $\sigma_n'-\frac{t_{n-1}}{2}$  and
\begin{equation*}
   h_n(z)=\dfrac{1}{2} z
    \brackets{t_{n - 1} \brackets{1 - \delta_{n, 1}} - \dfrac{1}{2} z \, \delta_{n, 1}}.
\end{equation*}
By our assumption, $z - 2 f_n \neq 0$. Thus, \eqref{dgnExpr2} becomes
\begin{equation} \label{dgnExpr4}
    \sum_{l = 1}^n t_l \widehat{\mathcal{L}}_{l + 1}
    - z \widehat{\mathcal{L}}_1
    - \sigma_n \brackets{z}
    + h_n \brackets{z}
    = 0,
\end{equation}
where we have absorbed the integration constant in sigma (constant shifts in sigma do not change the dynamics).

So we have two equations, \eqref{gnExpr} and \eqref{dgnExpr4} that both contain derivatives of $\sigma_n \brackets{z}$ of order $2n+1$. We can replace $f_n''$ in
 \eqref{gnExpr} by $f_n'' -\left( \sum_{l = 1}^n t_l \widehat{\mathcal{L}}_{l + 1}
    - z \widehat{\mathcal{L}}_1
    - \sigma_n \brackets{z}
    + h_n \brackets{z}\right)$ thus obtaining \eqref{SF1}.
\end{proof}

\begin{lem}\label{lm:order} The sigma form \eqref{SF1} is an ODE of order $2n$.
\end{lem}

\proof By using formula \eqref{eq:lzz} we see immediately that \eqref{SF1} is equivalent to
\begin{multline*}
    - f_n^{\prime}
    + \brackets{f_n^{\prime}}^2
    + \brackets{z - 2 f_n}
    \left(
    \sigma_n \brackets{z}
    - z \frac{t_{n-1}}{2}
    - \brackets{ \sigma_n^{\prime} \brackets{z}
    - \dfrac{t_{n - 1}}{2}}
    f_n

    \phantom{\displaystyle \sum_{j = 1}^{l - 1}\widehat{\mathcal{L}}}
    \right.
    \\
    \left.
    - \left(
    \displaystyle \sum_{l = 1}^{n} t_l
    \displaystyle \sum_{j = 1}^{l - 1}
    \brackets{\widehat{\mathcal{L}}_{l - j}
    \brackets{4 \widehat{\mathcal{L}}_1  \widehat{\mathcal{L}}_j
    - \widehat{\mathcal{L}}_{j + 1}
    + 2 \widehat{\mathcal{L}}_{j}^{\prime\prime}
    }
    - \widehat{\mathcal{L}}_j^{\prime}
    \widehat{\mathcal{L}}_{l - j}^{\prime}
    }
    \right)
    \right)
    = \alpha_n \brackets{\alpha_n - 1},
\end{multline*}
which is an ODE of order $2n$.
\endproof

To demonstrate how our theorem \ref{SFthm} works, we give two examples for cases $n = 1$ and $n = 2$.

\begin{exmp}
For $n = 1$ we use \eqref{SF1n1} to obtain
\begin{align*}
    - \brackets{\sigma_1^{\prime\prime} + \dfrac{1}{2}}
    + \brackets{\sigma_1^{\prime\prime} + \dfrac{1}{2}}^2
    - 2 \sigma_1^{\prime}
    \brackets{\sigma_1 + \dfrac{1}{4} z^2 - \brackets{\sigma_1^{\prime} + \dfrac{z}{2}}^2}
    &= \alpha_1 \brackets{\alpha_1 - 1},
    \\
    \brackets{\sigma_1^{\prime\prime}}^2
    - 2 \sigma_1 \sigma_1^{\prime}
    + 2 z \brackets{\sigma_1^{\prime}}^2
    + 2 \brackets{\sigma_1^{\prime}}^3
    &= \brackets{\alpha_1 - \dfrac{1}{2}}^2.
\end{align*}
\begin{rem}
If we consider the following map of the Okamoto Hamiltonian in \cite{okam3}
\begin{equation*}
    \text{H}_{\text{II}} \brackets{p, q}
    = \dfrac{1}{2} p \brackets{p - 2 q^2 - z}
    - \brackets{\alpha + \dfrac{1}{2}} q
    \quad
    \mapsto
    \quad
    2 \text{H}_{\text{II}} \brackets{2 p, \frac{1}{4} q}
    + \frac{1}{2} q,
\end{equation*}
the Okamoto sigma form for the second Painlev{\'e}  equation in \cite{okam3} coincides with our sigma form.
\end{rem}
\end{exmp}

\begin{exmp}
Set $n = 2$ in \eqref{SF1}:
\begin{gather*}
    - f_2^{\prime}
    + \brackets{f_2^{\prime}}^2
    + \brackets{z - 2 f_2}
    \brackets{
    t_1 \left(\widehat{\mathcal L}_1''
    - \widehat{\mathcal L}_{2}\right)
    + \left(\widehat{\mathcal L}_2''
    - \widehat{\mathcal L}_{3}\right)
    + 2 f_1 f_2 + \sigma_2 \brackets{z}
    - \frac{1}{2} t_{1} z
    }
    = \alpha_2 \brackets{\alpha_2 - 1},
\end{gather*}
where
\begin{gather*}
    f_1
    = \widehat{\mathcal{L}}_1 \LieBrackets{\sigma_2^{\prime} - \dfrac{t_1}{2}},
    \quad
    f_2
    = t_1 \widehat{\mathcal{L}}_1 \LieBrackets{\sigma_2^{\prime} - \dfrac{t_1}{2}}
    + \widehat{\mathcal{L}}_2 \LieBrackets{\sigma_2^{\prime} - \dfrac{t_1}{2}},
\end{gather*}
with the Lenard operators
\begin{gather*}
    \widehat{\mathcal{L}}_1
    \LieBrackets{\sigma_2^{\prime} - \dfrac{t_1}{2}}
    = \sigma_2^{\prime} - \dfrac{t_1}{2},
    \qquad
    \widehat{\mathcal{L}}_2
    \LieBrackets{\sigma_2^{\prime} - \dfrac{t_1}{2}}
    = \sigma_2^{\prime\prime\prime}
    + 3 \brackets{\sigma_2^{\prime}}^2
    - 3 t_1 \brackets{
    \sigma_2^{\prime}
    - \dfrac{1}{4} t_1},
\end{gather*}
\begin{align*}
    \widehat{\mathcal{L}}_3
    \LieBrackets{\sigma_2^{\prime} - \dfrac{t_1}{2}}
    = \sigma_2^{(\text{v})}
    &+ 10 \sigma_2^{\prime} \sigma_2^{\prime\prime\prime}
    + 5 \brackets{\sigma_2^{\prime\prime}}^2
    + 10 \brackets{\sigma_2^{\prime}}^3
    - 5 t_1 \brackets{\sigma_2^{\prime\prime\prime}
    + 3 \brackets{\sigma_2^{\prime}}^2}
    + \dfrac{5}{2} t_1^2 \brackets{
    3 \sigma_2^{\prime}
    - \dfrac{1}{2} t_1}.
\end{align*}

Then the sigma form for $\PIIn{2}$ is
\begin{align*}
    &
    \brackets{\sigma_2^{(\text{iv})}}^2
    + 12 \sigma_2^{\prime} \sigma_2^{\prime\prime} \sigma_2^{(\text{iv})}
    - 4 t_1 \sigma_2^{\prime\prime} \sigma_2^{(\text{iv})}
    - \sigma_2^{(\text{iv})}
    + 4 \sigma_2^{\prime} \sigma_2^{\prime\prime\prime}
    - 2 t_1 \brackets{\sigma_2^{\prime\prime\prime}}^2
    \\
    & \quad
    - 2 \brackets{\sigma_2^{\prime\prime}}^2 \sigma_2^{\prime\prime\prime}
    + 20 \brackets{\sigma_2^{\prime}}^3 \sigma_2^{\prime\prime\prime}
    - 24 t_1 \brackets{\sigma_2^{\prime}}^2 \sigma_2^{\prime\prime\prime}
    + 9 t_1^2 \sigma_2^{\prime} \sigma_2^{\prime\prime\prime}
    - 2 z \sigma_2^{\prime} \sigma_2^{\prime\prime\prime}
    \\
    & \quad \quad
    - t_1^3 \sigma_2^{\prime\prime\prime}
    + 2 z t_1 \sigma_2^{\prime\prime\prime}
    - 2 \sigma_2 \sigma_2^{\prime\prime\prime}
    + 30 \brackets{\sigma_2^{\prime}}^2 \brackets{\sigma_2^{\prime\prime}}^2
    - 20 t_1 \sigma_2^{\prime} \brackets{\sigma_2^{\prime\prime}}^2
    + \dfrac{7}{2} t_1^2 \brackets{\sigma_2^{\prime\prime}}^2
    \\
    & \quad \quad \quad
    + z \brackets{\sigma_2^{\prime\prime}}^2
    - 6 \sigma_2^{\prime} \sigma_2^{\prime\prime}
    + 2 t_1 \sigma_2^{\prime\prime}
    + 24 \brackets{\sigma_2^{\prime}}^5
    - 46 t_1 \brackets{\sigma_2^{\prime}}^4
    + 34 t_1^2 \brackets{\sigma_2^{\prime}}^3
    - 4 z \brackets{\sigma_2^{\prime}}^3
    \\
    & \quad \quad \quad \quad
    - 12 t_1^3 \brackets{\sigma_2^{\prime}}^2
    + 8 z t_1 \brackets{\sigma_2^{\prime}}^2
    - 6 \sigma_2 \brackets{\sigma_2^{\prime}}^2
    + 2 t_1^4 \sigma_2^{\prime}
    - 4 z t_1^2 \sigma_2^{\prime}
    + 4 t_1 \sigma_2 \sigma_2^{\prime}
    \\
    & \quad \quad \quad \quad \quad
    - \dfrac{1}{8} t_1^5
    + \dfrac{1}{2} t_1^3 z
    - \dfrac{1}{2} t_1^2 \sigma_2
    - \dfrac{1}{2} t_1 z^2
    + z \sigma_2
    = \alpha_2 \brackets{\alpha_2 - 1}.
\end{align*}
\end{exmp}

\section{B{\"a}cklund  transformations}

The B{\"a}cklund  transformations of \eqref{PIIhier} have two generators  \cite{okam3}
\begin{align*}
    {s}:
    &
    \qquad
     \Tilde{w}_n \brackets{z, {t}; \tilde{\alpha}_n}
        = w_n \brackets{z, {t}; \alpha_n}
        - \dfrac{2 \alpha_n - 1}{2 \displaystyle \sum_{l = 0}^{n} t_l \mathcal{L}_l \LieBrackets{w_n^{\prime} - w_n^2}},
    \quad
    \tilde{\alpha}_n
    = 1 - \alpha_n
    \\
    {r}:
    &
    \qquad
     {w_n} \brackets{z, {t}; -{\alpha}_n}
        = - w_n\brackets{z, {t}; {\alpha}_n}.
\end{align*}

\begin{thm}
The B{\"a}cklund  transformations of the sigma forms \eqref{SF1}, \eqref{SF1n1} act on the sigma function
\begin{align*}
    {s} \brackets{\sigma_n}
    &= \sigma_n,
    &
    {r} \brackets{\sigma_n}
    &= \sigma_n-2 w_n.
\end{align*}
\end{thm}
\begin{proof}
Since the right hand side of the sigma form \eqref{SF1}, \eqref{SF1n1} is invariant under the $s$-action, $\sigma_n$ is also invariant under this action.
    Regarding the $r$-action, we have
    \begin{equation*}
        {r} \brackets{w_n^{\prime} - w^2}
        = - w_n^{\prime} - w_n^2
        = \sigma_n^{\prime} - 2 w_n^{\prime}.
    \end{equation*}
    After integration w.r.t. $z$ we obtain the final formula.

\end{proof}

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


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Greenacre Recycling Ltd are based in Newhaven in a central position in Sussex, skirting both East Sussex, Brighton and West Sussex.
Greenacre Recycling Ltd have been providing domestic and commercial skip hire and grab truck hire services in East Sussex county since 1996. Greenacre Recycling is proud to be working with East Sussex County Council, Freedom Leisure and Blind Veterans UK as well as serving domestic clients locally throughout East Sussex, helping both domestic and commercial customers to be more environmentally responsible with their waste by recycling 95% of collected waste at their plant in Newhaven. Our ethos of recycling 95% of waste is as relevant and important to the environment today, as it has ever been.
Since 1996, Greenacre Recycling have been proudly providing domestic and commercial skip hire and grab lorry services to West Sussex county in the South East of England, providing the local region with key ethical and environmental benefits such as 95% recycled waste, aggregate deliveries and waste removal. Key clients in West Sussex include Horsham District Council, Crawley Borough Council, Hargreaves Group and Graves Son & Pilcher Properties. Greenacre Recycling continue to help local authorities, contractors and sub-contractors, and local building firms and businesses meet superior recycling targets, as well as providing local domestic residences with ethical and green waste management services.
Working with Brighton's Green party and city council, Greenacre Recycling have been utilising their wealth of experience to safely clear household and commercial waste in a green and environmentally friendly manner in the most efficient way. Whether the job is removing bulky furniture after a room renovation, clearing building waste like bricks and glass, or removing soil and hardcore, Greenacre Recycling have a proud history of working with Brighton & Hove to recycle up to 95% of all waste at their recycling plant in Newhaven.
Greenacre Recycling Ltd have been providing Seaford with domestic and commercial waste management services since 1996. Our process of recycling 95% of waste in a safe and green manner has been a core value shared by Seaford Town Council, and one that has deepened our relationship with public services to ensure they achieve their environmental goals for waste and recycling. Our friendly and experienced staff are always happy to answer any questions and can advise on the best way of clearing any waste.
We work very closely with Eastbourne Borough Council, alongside East Sussex County Council, and have undertaken a number of waste recycling projects for them. From domestic, public sector and commercial contractor and sub-contractor work, Greenacre Recycling can help with any waste removal and recycling requirements, whether the job is to remove bulky furniture after a room renovation, or clearing building waste like bricks and glass from sites, or if you need to remove soil and hardcore in large quantities.
For over 22 years, Greenacre Recycling Ltd have been removing waste and recycling materials for Lewes, the county town of East Sussex and formerly all of Sussex. Recycling projects include domestic and commercial waste removal in a safe and green manner, working for Lewes local government and East Sussex County Council as well as local businesses, contractors and sub-contractors on domestic or building site schemes. This has resulted in us being approved under the Buy With Confidence scheme. We make sure that at least 95% of all the waste we remove is recycled. All the materials we recover are brought back to our recycling centre in Newhaven where they are separated and then recycled.
Our headquarters are based in Newhaven, all the magic happens at our recycling centre on New Road. It's where 95% of all the waste we remove from building sites or domestic properties is recycled. We've developed a strong relationship with the local community and Newhaven Town Council, sharing our environmental goals to recycle as much as possible for a greener, safer place to live.
Saltdean
Saltdean is a small coastal village situated between Lewes, Newhaven and Brighton, sharing unitary authorities with both Lewes and Brighton & Hove. The work we do in Saltdean is largely serving the local communities domestic and property development projects, alongside any public works for either Brighton and Hove Council or Lewes' local government. As with all our waste management solutions, we seek to recycle 95% of all the waste we remove from building sites or domestic properties.
Greenacre Recycling have completed a number of waste removal and recycling project in Uckfield and the wider Wealden District, working with Uckfield Town Council and East Sussex County Council. Our green and environmentally friendly ethos for reclycling at least 95% of all waste we remove from building sites, domestic properties, or commercial. Our most popular services are all types of skip hire sizes for small domestic projects up to large building skips, and our grab truck hire for large removal of soil and hardcore, and the delivery of aggregates.
Situated in Mid Sussex, Haywards Heath is a central location within the South East region on the way to London. Known as a commuter town, there are nevertheless waste removal works required, particularly for road works, property development, domestic or commercial, and public building works around Sandrocks and Hurstwood Farm. For all waste removal projects, we ensure that at least 95% of all waste clearance is recycled at our recylcing plant in nearby Newhaven.
Greenacre Recylcing Ltd are proud to have been serving Horsham District Council and West Sussex County Council since 1996, ensuring that all waste clearances and building site soil and hardcore removal is at least 95% recycled at our recycling centre in nearby Newhaven. We provide Horsham with skip hire, grab truck hire, and man with a van services to help get any job done, no matter the size of the waste.
Our ethos of environmentally friendly waste removal at affordable budgets, ensuring that at least 95% of all waste removed is recycled is a shared value of the local community, domestic residences and with Adur & Worthing Borough Council. We've been proudly serving Worthing, and the wider West Sussex county for over 22 years, since 1996 - providing skip hire, grab truck hire, aggregates and man with a van services for any waste removal project or building site clearance.
Greenacre Recycling Ltd are proud to have been working with Crawley Borough Council and West Sussex County Council for over 20 years, supporting both domestic and commercial clients to recycle over 95% of their waste. Utilising our skip hire or grab truck hire services, Greenacre Recycling are uniquely positioned to provide building site clearances and waste management services, extracting waste and recycling over 95% of it at our recycling plant in Newhaven.
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Greenacre Recycling Limited
BN9 0HE
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Greenacre Recycling Ltd, registered as a limited company in England and Wales under company number: 04182562.
Registered Company Address: Greenacre Recycling Ltd, c/o JB Wood Accountancy Services Ltd, Greenacre Recycling Centre, New Road, Newhaven BN9 0HE
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Follow the 'Fishing' link under 'Recreation Links' for seasonal fishing tips.
On April 22nd, 2019 at 11 a.m. the headwater elevation was 1015.40 feet above mean sea level.
The surface water temperature on April 22nd, 2019 at Laurel River Lake's Natural Resource Management Office was 67.4° at the surface.
Laurel River Lake does not have a set generation schedule. However, generation often occurs during hours of peak electricity usage. Be safe and listen for horns that indicate when generation will begin.
Be safe and wear your lifejacket!


text:
[[Aboard the Physaric ship, the astral projections of Trudy and Sharon continue their awkward conversation with "Nega-Nick". The former emperor has just declared that he has no interest in answering their questions.]]
Sharon: [Scowling] Oh, please. Like we'll believe THAT. Your ego is so massive you can't HELP but gloat about your nefarious schemes...
Nega-Nick: [Giving them a mischievous smile] You'd like to think that, huh? Well, you see...
Nega-Nick: I DID the "megalomaniacal emperor" thing. I've done the "skulking about impersonating my goody two-shoes twin" thing. And you know what? I've learned a thing or two since then.
[[He kneels down to pick something up off the floor as Trudy and Sharon watch.]]
Nega-Nick: Patience is a virtue, and silence can be your ally. I don't NEED to gloat anymore, as I have far bigger ambitions than you can imagine. You dweebs no longer matter.
[[He begins absentmindedly tossing the small pebble he just picked up in his hand.]]
Nega-Nick: [Smirking] Sure, revenge is sweet and all, but that's for petty little dorks who can't see the big picture. And believe me, THIS picture is bigger than you can possibly imagine...
GPF HOME SEARCH ARCHIVE ARCHIVES BY DATE STORY INDEX STORYLINES: Year One: Nov. 1998 to Oct. 1999Year Two: Nov. 1999 to Oct. 2000Year Three: Nov. 2000 to Oct. 2001Surreptitious MachinationsYear Five: Nov. 2002 to Oct. 2003Year Six: Nov. 2003 to Oct. 2004Year Seven: Nov. 2004 to Oct. 2005To Thine Own Self...Year Nine: Mar. 2008 to Oct. 2008Year Ten: Nov. 2008 to Oct. 2009Year Eleven: Nov. 2009 to Oct. 2010Year Twelve: Nov. 2010 to Oct. 2011Year Thirteen: Nov. 2011 to Oct. 2012Year Fourteen: Nov. 2012 to Oct. 2013Scylla and CharybdisYear Seventeen: Nov. 2015 to Oct. 2016Year Eighteen: Nov. 2016 to Oct. 2017Year Nineteen: Nov. 2017 to Oct. 2018Year Twenty: Nov. 2018 to Oct. 2019Year Twenty-One: Nov. 2019 to Oct. 2020 A Change of Perspective Pouch Critters War Council Puppet Mastered Telepresence of Mind More Perforated Than Emmental Principio ad FinemYear Twenty-Two: Nov. 2020 to Oct. 2021Year Twenty-Three: Nov. 2021 to Oct. 2022
JUN July 2020 AUG


text:
Spondylitis is an inflammation of the vertebrae. It is a form of spondylopathy. In many cases, spondylitis involves one or more vertebral joints, as well, which itself is called spondylarthritis.

Types
Pott disease is a tuberculous disease of the vertebrae marked by stiffness of the vertebral column, pain on motion, tenderness on pressure, prominence of certain vertebral spines, and occasionally abdominal pain, abscess formation, and paralysis.

Ankylosing spondylitis (pronounced ank-kih-low-sing spon-dill-eye-tiss),  or AS, is a form of arthritis that primarily affects the spine, although other joints can become involved. It causes inflammation of the spinal joints (vertebrae) that can lead to severe, chronic pain and discomfort. In more advanced cases this inflammation can lead to ankylosis, new bone formation in the spine causing sections of the spine to fuse in a fixed, immobile position.

A combination of spondylitis and inflammation of the intervertebral disc space is termed a spondylodiscitis.

References

 National Institutes of Health

External links
 National Institutes of Health

Vertebral column disorders
Orthopedic problems

redpajama_set_name: RedPajamaWikipedia


text:
Battersby Joins Belper Town On Short Term Loan
Grimsby Town Football Club can confirm that goalkeeper Ollie Battersby has joined Belper Town on a one-month loan deal.
Ollie has joined up with the Northern Premier League Midlands Division side until 11th October 2021 and made his debut for the club on Saturday 11th September in a 2-1 victory at home to Cambridge City.
Progressing through the GTFC Academy system, Battersby penned his first professional contract with the Mariners in December 2018 and made his professional debut for his boyhood club in November 2020 during a 3-1 defeat at home to Leicester City U21's in the EFL Trophy.
Battersby was sent on loan to Gateshead in December 2020 however, his spell at the club was short-lived after being recalled just five days later due to James McKeown being forced to serve a period of self-isolation.
We would like to wish Ollie all the best on his short spell away from Blundell Park.
Good luck Ollie!
Ollie Battersby


According to Stein Kj0lberg, marketing manager for marine products at Jotun Paints, "The launch of SeaQuantum heralds a breakthrough in tin-free antifouling systems: Jotun's new coating has undergone extensive testing and offers up to five years of effective protection against fouling on ship bottoms." Developed in conjunction with Japanese chemicals group Nippon Oil and Fats (NOF), Jotun has reportedly uncov- ered a practical tin-free solution suitable for virtually any application.
Although tin-free alternatives have been on the market for quite some time, none have proven to go the distance in meeting the shipping sector's need for five-year systems. Lasting only three years instead of the required five, most of these substitute systems still cannot offer the same self-polishing effect posted by tin-based coatings. Contrastingly, the self-polishing aspect is present in SeaQuantum — holding properties that bear chemical and physical properties - that are almost uncanny to tin-based antifouling coatings. In translation, the product blends protection against fouling with a hydrolysis reaction, making the vessel's bottom increasingly smooth as the coating wears — creating a more efficient vessel. The company recommends the product be applied to both newbuilds and full bottom upgrades.
Undergoing a little less than seven years of trials, the new coating from Jotun. according to Kj0lberg is "the most tin-free marine antifouling system on the market today." Jotun has basked in a product and research venture with NOF for many years. With Jotun developing the formulation for the actual coating, NOF spearheaded a silyl polymer that can replace the environmentally-dangerous tin compounds used in today's self-polishing antifouling coatings.


5 COMPATIBILITY REPORT FOR MARY AND BEN toward Ben. Ben could feel stifled, or let down, but also may sense the practical and supportive nature of Mary, who is helpful in keeping partnership progress toward shared goals and objectives on track. You may be put off by a feeling that your partner is too closed, or a feeling of emotional restriction, but you do well when you focus on shared objectives. The sign and house position of the Moon and Saturn in each partner's chart will more closely indicate the area of shared interaction. It will be helpful if you try to move beyond any fears you may have regarding intimacy, which can leave you or your partner feeling left out, lonely or undesirable, and do some work on yourselves. You may accomplish much regarding your own inner development through these interactions with each other. Ben's Uranus in strong square (within 2.5 degrees) with Mary's Moon The planetary energies conflict; internal and creative tensions bring rich rewards through effort over time. This aspect, of the Moon in square with partner's Uranus, indicates a partnership based on mutual interests, friendship, and unique appreciation of individuality, although there also can be emotional upsets, unexpected turns of feeling, or disregard for the other's point of view. The unconventional, indiscreet or inconsiderate actions of Ben may go against the grain of Mary, or may grow into emotional disconnects between you. Change definitely comes about in your life as a result of this partnership. Adaptability is required, and also the awareness of your separate interests. In a romantic context, you may find an instantaneous and almost magnetic attraction between you, a strong pull towards getting together quickly and intimately, but this aspect may also indicate some exasperating distancing, or aloof feelings between you, or imbalance of emotion. You may feel not at all the same, or perhaps disregard the other's feelings, in what may turn out to be an unusual romance in terms of environment or styles. Domestic life between you may be unsettled and outside influences or the friction of career direction versus home life may separate you. You can withstand these pressures when an allowing and mature understanding is present between you. The at times irascible nature of your interaction may need some constructive realignment, or perhaps one partner's moodiness may need to be given more freedom. Refreshing personality traits and good humor between you provides a helpful source of resolution, serving to enliven you with a sense of joy in your feelings for each other. Ben's Neptune in quintile (within 0.7 degrees) with Mary's Moon and Ben's Moon in strong sextile (within 2.0 degrees) with Mary's Neptune.


In the year 2000, it was alleged that some 300,000 children were serving as soldiers in armed conflicts in more than thirty countries around the world. Nearly 120,000 of these were allegedly engaged in various conflicts on the African continent.
EZRA is a factional tale inspired by the Sierra Leonean conflict. It is centred on one event: a drug fuelled atrocious attack on a village by rebel soldiers. The jigsaw puzzle of what occurred that night is reconstructed through the testimonies of three witnesses: Ezra, an ex-combatant, his sister Onitcha, a mute, and Cynthia, Ezra's fellow ex-soldier.
What is supposed to be reconciliation soon becomes a trial, as Onitcha chooses this as the arena to reveal a secret she has kept from her brother.