text:
houseprlondon
~ PR Agency in London
Tag Archives: The Secret Life of Pets
Donkey from Shrek tops House PR/Empire Cinemas' poll of cinemagoers' favourite movie animal character
Posted by houseprlondon in Media Brands & Entertainment PR, Media Brands & Entertainment public relations, Uncategorized
Empire Cinemas, house pr, pr agency london, The Secret Life of Pets
By House PR – Media Brands & Entertainment public relations
In a unique poll organised by House PR and Empire Cinemas, the cheeky Donkey character from the hit Shrek movies scored the highest score of cinemagoers' favourite animal character.
The research was carried out ahead of the release at the beginning of July of the new animated blockbuster, The Secret Life of Pets.
Empire Cinemas is the UK's biggest independent chain of cinema theatres and the research was yet another sign of our unique relationship with our pets and other animals. And the producers of The Secret Life of Pets – Universal Pictures and Illumination Entertainment – are banking on this. The fact is, don't we all really want to know what our pets are doing once we scuttle off to our day jobs and they are left home alone?
The animated comedy production, directed by Chris "Despicable Me 1 and 2" Renaud, is the fifth collaboration of the two companies and stars the voices of some of the biggest names in comedy – Kevin Hart, Louis C.K. and Eric Stonestreet, as well as Albert Brooks, Ellie Kemper, Hannibal Buress and Bobby Moynihan.
Jon Nutton is the Head of Marketing at Empire Cinemas and he says that animals come up time and again as moviegoers' favourite characters, both in animated films such as The Jungle Book and real-life productions like Howard Bound. He also says that he expects The Secret Life of Pets to become a firm favourite that the whole family will enjoy.
The research into moviegoers' tastes also highlighted the enduring appeal of other famous movie animal characters.
Simba from The Lion King came in at second place attracting over a quarter of all votes. However, this result is just an average of all respondents to the survey. For young adults, Simba scored a whopping 33% of votes cast.
The forgetful Dory of Finding Nemo and the soon to be released Finding Dory fame came in third position of favourite movie animal characters.
The recent film remake of Rudyard Kipling's classic novel The Jungle Book also put Baloo the bear in the top five ranking, reminding us of the astonishing impact of the loveable character, an appeal that one can now say stretches over several decades.
The fascinating research also threw up differences in preferences based on the gender of respondents. 23% of female cinemagoers voted for Eeyore from Winnie The Pooh, 9% more than male cinemagoers. Female cinemagoers also voted more than twice as much compared to male cinemagoers for Marley from the hit film Marley and Me.
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Ginny Paton
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Tacking Point Lighthouse. Built in 1879 – the 3rd oldest lighthouse in Australia.
but there's a first for everything. Not sure the local Hastings Council would approve of me smashing plates against the lighthouse either. Spoil sports.
This entry was posted in Man-made and tagged Man-made by stevemanning1976. Bookmark the permalink.
Can I tell you how much I enjoyed your fabulous photos and the descriptions!!!!! Do you do this for a living?? If not, you should.
Thanks so much for your kinds words Carol.


Network Monitoring – When it comes to network monitoring, the thin line between legal monitoring and privacy bleaching gets more and more vague. However, if you are managing a network, monitoring is one thing you want to be doing at any time.
Network monitoring is like policing what is being accessed on the network and what network traffic is being transmitted across the network. This is something you want to be doing at any one time. Well-managed networks that have less downtime are ones that are regularly under surveillance. In a busy network environment, it can go a long way in determining the productivity of the team.
This entry was posted in Products and tagged Monitoring, Security, SelfActivated, System on May 20, 2015 by Andrew S.


text:
Камнелазы () — род стафилинид из подсемейства Paederinae.

Описание
Глаза равные по длине вискам. Переднегрудь без эпимер, со свободными дыхальцами. Голова личинок между верхними челюстями с восемью зубцами

Систематика
К роду относятся:
 вид:
 вид:
 вид:

Примечания

Paederinae
Роды жесткокрылых

redpajama_set_name: RedPajamaWikipedia


This is a vital step to take to answer the 'Should I Franchise My Business?' question.
Remember, whether your business can be franchised and whether it should be franchised, are of course, two different questions.
To reach a decision about whether you should franchise your business, you need to know what the outcome of doing so would be.
In working with clients, this is the first vital piece of work we do together.
The franchise development plan will lay out for you everything involved in franchising your business, how much you will have to invest in total and what the return will be.
The plan will give you a crystal clear picture of what you will have to do and what the outcome of franchising your business will be, enabling you to make a fully informed decision on whether to continue down the franchising route.
What's Included in a Franchise Development Plan?
Many would-be franchisors make the mistake of compromising this stage of the process by trying to do things 'on the cheap'. There simply is no shortcut at this stage.
What Are The Costs of a Having a Good Franchise Development Plan Done?
Before answering that, I take you back to analogy of buying a property…Would you just buy a building when you weren't quite sure about how solid it was? I'd hope the answer would be 'no', and that you'd at least commit to a structural survey, in that case, to ensure the foundations are what they should be.


text:
Biologically active ethanamine of benzothiazolone
C07D277/68 - with hetero atoms or with carbon atoms having three bonds to hetero atoms with at the most one bond to halogen, e.g. ester or nitrile radicals, directly attached in position 2
A61P11/06 - Antiasthmatics
A61K31/428 -
(57) Abstract:
The invention relates to a new derived benzothiazolone formulas I and II, where X denotes naphthyl and its pharmaceutically acceptable salts. The method of obtaining the compounds of formula I includes a selective reduction of the compounds of formula II. The compounds of formula I can be used for getting medicines for the treatment of respiratory diseases such as asthma and chronic bronchitis. The technical result - getting a new derived benzothiazolone. 3 s and 5 C.p. f-crystals, 1 PL.
< / BR>
The invention relates to new ethanaminium of benzothiazolone, methods for their preparation and the pharmaceutical compositions and methods of treatment using them. New connections are shown for use as agonists dopamine-DA2-and receptor agonists2-adrenergic receptors.
Derivatives benzothiazolone known in the art. For example, in applications for international patent publications numbers W0 92/08708 and W0 93/23385 reveal biologically active amines and among them biologically active aminoacyl derivatives benzothiazolone, which are agonists2-adrenergic receptors and the diseases of the respiratory tract.
Publication number W0 93/24473 discloses 7-(2-amino-ethyl)-benzothiazolinone the compounds of formula:
where X and Y represent independently from each other, - S(O)nor-O-; n takes on the values 0, 1 or 2; p, q and r are independently of each value of 2 or 3; Z is a phenyl, optionally substituted with halogen, OR1, NO2or NR2R3; or Z represents a five - or six-membered heterocycle containing N, O or S; and R1, R2and R3represent independently from each other hydrogen or C1-6alkyl. The compounds are agonists2-adrenergic receptors and agonists dopamine-DA2-receptor and can be used to treat reversible obstructive diseases of the respiratory tract.
Found a new group of ethanamine of benzothiazolone may be used as agonists2-adrenergic receptors and agonists dopamine-DA2-receptor.
The present invention is the creation of new compounds of formula I,
where X denotes a naphthyl, optionally substituted by alkyl or halogen;
p, q and r independently of one another represent 2 or 3; and their physiologically acceptable salts.
Sceptor and K2-adrenergic receptors. However, they do not show or show a small agonism to1-adrenergic receptors. These compounds have a high duration of action and coefficient DA2/2.
In the above formula I preferably p is 3; q is preferably equal to 2; g is preferably equal to 2.
X preferably represents an unsubstituted naphthyl.
When X represents an alkyl-substituted naphthyl, a Deputy may be selected from C1-6the alkyl linear or branched chain, e.g. methyl, ethyl, n-propyl, isopropyl, n-butyl, isobutyl, pentila straight or branched chain or hexyl straight or branched chain. Preferred halogen substituents are F, Cl and Br. When the alkyl substituent is chiral, can be formed optical isomers, and all of them are also included in the scope of the present invention. In the scope of the invention also includes compounds of formula I in tautomeric form.
Pharmaceutically acceptable salts of compounds of formula I include additive salts of inorganic and organic acids, such as hydrochloride, hydrobromide, sulfates, phosphates, maleate, t is ursuleti, naphthalenesulfonate, methanesulfonate, sulfamate, ascorbates, salicylates, acetates, diphenylacetate, trimeniaceae, adipate, fumarate, succinate, lactates, glutarate, gluconate, hydroxyethylmethacrylate, for example 1-hydroxy or 3-hydroxy-2-naphthaleneacetic or oleates. Compounds can also form salts with suitable bases. The compounds of formula I can be obtained in the form of salts, which represents a pharmaceutically acceptable salt. When this is desirable, such salts can be converted into the free base using conventional methods. Pharmaceutically acceptable salts can be obtained by the reaction of compounds of formula I with an appropriate acid or base in the presence of a suitable solvent.
The present invention also relates to a method for producing compounds of formula I, comprising a selective reduction of the compounds of formula II,
in which p, q, g and X are defined above.
Suitable reducing agents include electrophilic reducing agents such as DIBORANE and Alan (aluminum hydride), or nucleophilic reducing agents, for example a complex metal hydride, such as sodium bis(2-methoxyethoxy) aluminerie. Preferred aprotic solvents such as tetrahydrofuran, diethyl ether or 1,2-dimethoxyethane. The reaction may be conducted at a temperature of from about 0oC to about 100oC, preferably at the boiling point under reflux.
The compounds of formula II can be obtained through the interaction of the amine of formula III:
with the appropriate acid of the formula IV
HOOC-(CH2)p-1-SO2-(CH2)d-O-(CH2)r-X IV
or with the appropriate acid chloride using traditional methods. For example, the interaction may be carried out in the presence of dicyclohexylcarbodiimide using the method of Sheehan and Hess (Sheehan and Hess, J. Am.Chem.Soc., 1955, 77, 1067), or in the presence of 1,1'-carbonyldiimidazole as described Stabm (Staab, Angew. Chem. Int. Ed. Engl., 1962, 1, 351), or in the presence of hexaflurophosphate postreproductive in solvent type DMF. Acid necessary for the above-mentioned process may be obtained from the corresponding esters by hydrolysis in aqueous methanol in the presence of lithium hydroxide. The acid chlorides can be obtained from acids, for example, through reaction with oxalylamino or thionyl chloride in toluene at a temperature is obtained by the known methods for example, using a method described in literature (J. Med.Chem., 1987, 30, 1116).
The compounds of formula IV can be obtained, as described in the above examples, and methods used for this easily reproducible by any technician with a medium level of knowledge in this area and can be adapted for a list of compounds of the formula IV.
In the above methods may be necessary to protect certain functional groups, for example hydroxy - or amino-groups present in the original compounds. Suitable protecting groups and methods for their removal are described, for example, in the following activities ("Protective Groups in Organic Synthesis" by T. W. Green and P. G. M. Wuts, John Wiley and Sons Inc., 1991).
Hydroxy-group can be, for example, protected with allotropic groups, such as phenylmethyl, diphenylmethyl or triphenylmethyl or such as derivatives tetrahydropyranyl. Suitable protective groups for amino groups include arylmethyl groups such as benzyl, (R,S)--phenylethyl, diphenylmethyl or triphenylmethyl, and acyl groups such as acetyl, trichloroacetyl or TRIFLUOROACETYL. For removal of the protective groups can be used traditional methods. For example, arylmethylidene palladium on charcoal. Tetrahydropyranyl groups can be cleaved by hydrolysis under acidic conditions. Acyl groups can be removed by hydrolysis with a base such as sodium hydroxide or potassium carbonate, or grouping, such as trichloroacetyl, you can delete the recovery, for example, zinc and acetic acid.
Intermediate compounds of formula II are new compounds and are also included in the scope of the present invention.
The compounds of formula I and their salts are agonists dopamine-DA2-receptor. Linking the ability of the tested compounds against the binding site of the receptor DA2on the membranes of bovine pituitary gland can be determined by studying the substitution of [3H]-N-n-propylnorapomorphine and [3H] -spiperone, respectively in the absence and in the presence of neytralinogo analogue of GTP (D. R. Sibley, A. DeLean and I. Creese, Anterior Pituitary Dopamine Receptors, Demonstration of Interconvertible High and Low Affinity States of the D-2 Dopamine Receptor, J. Biol.Chem., 1982, 257(11), 6351-6361). The activity of DA2the receptor can also be demonstrated in functional screening test on an isolated ear artery of the rabbit, as described by brown and O'connor (Brown and O'connor, Br. J. Pharmacol., 1981, 73, 189P). Join vlauterbach Guinea pigs, as described by Dougallm, Harper, Jackson and Leff (I. G. Dougall, D. Harper, D. M. Jackson and P. Leff, Br.J.Pharmacol., 1991, 104, 1057).
Activity1the receptor can be analyzed using isolated ear artery of the rabbit under the test described in the examples.
The duration can be investigated using the method of Coleman and co-authors (Coleman et al., in "Novel and versatile superfusion system: its use in the evaluation of some spasmogenic and spasmolytic agents using guinea-pig isolated tracheal smooth muscle". Journal of Pharmacological Methods 21, 71-86 (1989)).
The compounds of formula I and their salts can be used in the treatment of a wide range of respiratory diseases, including conditions such as asthma, including bronchial asthma, allergic asthma, endogenous asthma (e.g. late asthma and hypersensitivity of the respiratory tract), as well as bronchitis and others (see, for example, UK Patent N 2022078 and Br. J. Pharmacol., 1987, 24, 4983).
The compounds of formula I and their salts are also used in the treatment of glaucoma and various other conditions, such as inflammatory and allergic skin diseases, and cancer, such as small-cell lung cancer, and congestive heart failure.
In the context of the present invention the term "treatment" clucosamine is the creation of compounds of the formula I, their pharmaceutically acceptable salts, used in therapy.
In addition, the invention relates to the use of compounds of the formula I or their pharmaceutically acceptable salts for obtaining medicines used in the treatment of respiratory diseases, in particular, in the treatment of asthma or chronic bronchitis.
Another objective of the present invention is a method of treatment of respiratory diseases, this method includes the introduction of a therapeutically effective amount of the compounds of formula I or its pharmaceutically acceptable salt to a patient suffering from such condition or are sensitive to it.
Another objective of the present invention is to provide a method for treatment or prevention of glaucoma, this method includes the introduction of a therapeutically effective amount of the compounds of formula I or its pharmaceutically acceptable salt to a patient suffering from such condition or tend to him.
To solve the aforementioned problems of injected dose will, of course, vary depending on the method of administration and the treatment. However, in General, satisfactory results can be obtained with the introduction of the connection tradeline this dose is 1-4 servings per day, or in the form of slow release. For a person's total daily dose ranges from about 70 mg to 1400 mg, and standard forms suitable for injection include from 20 mg to 1400 mg of the compound in a mixture with a solid or liquid pharmaceutical diluent or carrier.
The compound of the formula I can be used by itself or in the form of appropriate pharmaceutical compositions for local, enteral or parenteral administration.
For example, ingaliruyet device with a measured dose can be used for introducing connections, dispersible in a suitable propellant in the presence or in the absence of fillers, such as ethanol, surfactants, lubricating and stabilizing agents.
Suitable propellants include hydrocarbons, harperperennial and hydrofluroalkane propellants or mixtures of any of such propellants. Especially preferred propellants are P134a and P227, each of which can be used by itself or in combination with other propellants and/or surfactants and/or other fillers, for example, in combination with each other.
Can also be sprayed aqueous suspension or, p is the only dosage forms.
Dry powder inhalers can be used for introducing connections, single or in combination with pharmaceutically acceptable carrier, while in the latter case is either finely ground powder, or an ordered mixture. A dry powder inhaler may be a single dose or mnogochasovym and may use dry powder or capsules containing the powder.
Inhaler with measured dose dispenser and ingaliruyet device for spraying a dry powder is well known in the art, and many such devices is available.
Tablets and gelatin capsules, which may be optionally coated, containing the active substance may include one or more ingredients selected from diluents, carriers, binders, lubricants, stabilizers and other commonly used ingredients.
Injectable solutions of the active ingredient may also contain, for example, one or more ingredients selected from preservatives, stabilizers, means regulating the viscosity, emulsifying agents, buffers and other commonly used ingredients.
The present invention relates also to pharmaceutical is rmula I or its pharmaceutically acceptable salt in a mixture with a pharmaceutically acceptable diluent or carrier.
The present invention is illustrated without in any way limiting its scope to the following examples in which all temperatures are in degrees Celsius.
4-hydroxy-7-[2-[3-[2-[2-(1- naphthalenyl)ethoxy]ethylsulfanyl]propylamino]ethyl]-1,3-benzothiazol-2(3H)-she hydrochloride
a) 1-[2-(adenylate)ethyl]naphthalene
A mixture of freshly etilenovogo ether (60 ml), 2-(1-naphthalenyl)ethanol (8,61 g) and acetate mercury (0.4 g) is heated at boiling temperature under reflux for 20 hours. Volatile material is removed under vacuum (water pump) and the residue is distilled in Kugeler (Kugelrohr) to give the compound indicated in the title, in the form of oil (8,15 g), which was contaminated with a small amount of 2-(1-naphthalenyl)ethanol.
So Kip. 160-170oC at 0.04 mm Hg;
Mass spectrum: gc/ms 198 (M), 155 (100).
b) Methyl 3-[2-[2-(1-naphthalenyl)ethoxy]ethylthio]propanoate
The product obtained in stage a), (8,15 g), methyl 3-mercaptopropanol (a 4.86 g) and 2,2'-azobis(2-methylpropionitrile) (0.2 g) is heated at a temperature of 50oC for 1 hour, after this time TLC indicates that some amount of starting material still remains in the mixture. In the mixture contribute more to the Crude reaction mixture was purified using flash chromatography (petrol:diethyl ether 5:1 as eluent) to give the compounds specified in the header, in the form of oil (11,07 g).
Mass spectrum: gc/ms 318 (M), 154 (100);
1H NMR (360 MHz, CDCl3) : 2,61 (2H, T.), of 2.72 (2H, T.), of 2.81 (2H, etc), to 3.38 (2H, etc), to 3.64 (2H, etc), 3,70 (3H, s), 3,80 (2H, etc), 7,40 (2H, m), and 7.5 (2H, m), 7,73 (1H, d), a 7.85 (1H, d), of 8.06 (1H, d).
C) Methyl 3-[2-[2-(1-naphthalenyl)ethoxy]ethylsulfanyl]propanoate
A solution of OXONETM(18,4 g) in water (70 ml) is added dropwise at 10oC to a solution of the product obtained at stage b) (3,18 g) in methanol (70 ml). The suspension is stirred at room temperature for 5 hours. The reaction mixture was diluted with water (200 ml) and extracted with dichloromethane (3 x 50 ml). The combined organic extracts washed with water and a solution of sodium sulfite, and then dried to obtain the connection specified in the header (3,39 g), which is used in the next stage without further purification.
1H NMR (360 MHz, CDCl3) 2,70 (2H, etc), 3,06 (2H, etc), 3,17 (2H, etc), 3,26 (2H, etc ), 3,71 (3H, s), 3,85 (4H, m), 7,40 (2H, m), and 7.5 (2H, m), 7,73 (1H, d), a 7.85 (1H, d), of 8.06 (1H, d).
g) 3-[2-[2-(1-naphthalenyl)ethoxy]ethylsulfanyl]propanoic acid
The solution of the monohydrate of lithium hydroxide (2,02 g) in water (30 ml) are added to a suspension of the product obtained at stage b) (3,35 g) in methanol (40 ml). The mixture of the PE the IDF spent. A large part of the methanol is removed under vacuum and the residue diluted with water (100 ml). The basic solution was washed with ether. The aqueous layer is acidified with hydrochloric acid and extracted with ether. The ether is washed with water, brine, then dried (MgSO4) and remove the solvent under vacuum to obtain the compound indicated in the title (is 3.08 g). It then purified using flash chromatography (chloroform:acetic acid 9:1 as eluent) to obtain the compound indicated in the heading, after trituration with hexane as a white solid (0,687 g).
So pl. 105-108oC;
Mass spectrum: gc/ms 336 (M),141 (100);
1H NMR (360 MHz, CDCl3) : 2,73(2H, etc), 3,15-up 3.22 (4H, m), 3.33 and-3,37 (2H, etc ), 3,79-3,86 (4H, m), 7,33-rate of 7.54 (4H, m), 7,72 (1H, d), to 7.84 (1H, d), of 8.00 (1H, d).
Data analysis: found: C - 60,90; H - 6,34; S - 9,48%
Calculated data: 60,70; H - 5,99; S - AT 9.53%
d) N-[2-(4-hydroxy-2-oxo-3H-1,3-benzothiazol-7-yl)ethyl] -3-[2-[2-(1-naphthalenyl)ethoxy]ethylsulfanyl]propanamide
The product obtained in stage g) (0.625 g), dissolved in dimethylformamide (10 ml) under nitrogen atmosphere. To this stirred solution was added 1,1'-carbonyldiimidazole (0,301 g) and the mixture is stirred for 2 hours at room temperature. Nam (0,259 ml). The mixture is stirred overnight, after which the volatile products are removed under vacuum. The residue is transferred into ethyl acetate and washed with diluted hydrochloric acid and then brine. The organic layer is dried (MgSO4) and remove the solvent under vacuum to obtain the connection specified in the header (1,30 g), in the form of oil, which is then used without further purification.
Mass spectrum FAB 529 (M+H).
e) 4-hydroxy-7-[2-[3-[2-[2-(1-naphthalenyl)ethoxy] ethylsulfanyl] propylamino]ethyl]-1,3-benzothiazol-2(3H)-she hydrochloride
Borane-tertrahydrofuran ring solution (1.0 M in THF to 8.6 ml) is added dropwise to a stirred solution of the product obtained in stage d) (1,30 g) in dry tetrahydrofuran (100 ml). The reaction mixture is heated at the boiling point under reflux in an inert atmosphere for 1 hour. HPLC treatment phase using acetonitrile and 0.1% triperoxonane acid as eluent shows the presence of a large amount of starting material, and therefore contribute an additional amount of borane (8.6 ml) and then heated the mixture at the boiling point under reflux for 1 hour. The reaction mixture is cooled and carefully add metabasic concentrated hydrochloric acid (beats. the density of 1.18; 0.75 ml). This solution is heated at the boiling point under reflux for 20 minutes and then remove the solvent under vacuum. The residue was subjected to purification by using preparative HPLC with a treatment phase using acetonitrile and 0.1% triperoxonane acid as eluent. Eventually, after receiving cleaners containing hydrochloride salt and recrystallization from ethanol obtain the target product as a white powder (0,38 g).
Mass spectrum: FAB 515(M+H);
1H NMR (360 MHz, D6-DMSO) : 2,00-of 2.08 (2H, m), 2,86 - only 2.91 (2H, m), 2,96 (2H, m),3,06 (2H, m) and 3.15 (2H, etc), to 3.33 (2H, etc), 3,38 is 3.40 (2H, m ), 3,76-3,82 (4H, m ), to 6.80 (1H, d), 6.89 in (1H, d), 7,42-7,58 (4H, m), 7,80 (1H, d), a 7.92 (1H, d), of 8.09 (1H, D), which 9.22 (2H, sh.C.), and 10.20 (1H, s ), 11,82 (1H, s).
Data analysis:
found: 56,16; H is 5.98; N - 5,40; S - 11,77; Cl - 6,00; H2ON - 1,20%
designed for 0,28 mol of water:
With - 56,16; H - of 5.89; N - 5,04; S - To 11.52; Cl - 6,39; H2O - 0,91%
The activity of DA2-receptor has been demonstrated in functional screening test on an isolated ear artery of the rabbit, as described by brown and O'connor (Brown and O'connor, Br.J.Pharmacol., 1981, 73, 189R) (see table 1).
Activity2-adrenergic receptors was demonstrated on isolated trachea of the Guinea pig, CT>Activity1the receptor was analyzed by means of a screening test using an isolated ear artery of the rabbit, as described below (see table 1).
Isolated ear artery of the rabbit
Rabbits male NZW (2.5 to 3.0 kg) killed by intravenous injection pentobarbitone sodium (60 mg/kg). Remove the ears, cut out the proximal part of the artery in the middle ear and kanyoro it using polypropylene cannula (outer diameter 0.75 mm). After removal of artery cleaned of residues of connective tissue and cook 6 rings width of 5 mm, containing part of the vascular smooth muscle. Tissue is placed on a thin hooks made of tungsten wire (0.25 mm in diameter) in the bath for bodies with a volume of 20 ml containing Krebs solution of the following composition (mm): NaCl - 117,56; NaHCO3- 25,00; KCl And 5.36; NaH2PO4- 0,89; MgSO4- 1,18; glucose - 11,10 and CaCl2is 2.55. In the Krebs solution is injected cocaine (30 μm) and propranolol (1 μm) to block the absorption of neurons and receptors, respectively. Add ascorbate (100 μm) to prevent oxidation of the catecholamine. This solution support at a temperature of 37oC and continuously bubbled through him, a mixture of 95% O2and 5% CO2. Top Provolone stationary substrate in the bath. Changes in isometric load recorded on flat recorder Advance Bryans AB500.
General description of the experiment
At the beginning of each experiment, each tissue sample is subjected to the action of the load in 1.0, This load is restored two or three times during the stabilization period of about 60 minutes, until, until it becomes constant. At that time, as there is a stabilization load, bath wash. Curves the concentration of agonist-effect (E/[A]) construct the cumulative addition of agonist increments of 0.5 Log10. Reaction (reduction) is recorded as a percentage of the maximum response to the standard agonist.
Quantitative expression of agonism
As standard agonist phenylephrine was used. First build the curve E/[A] to phenylephrine. Then phenylephrine washed and build a curve E/[a] for the tested compounds. Answers to compounds that cause agonism, expressed as a percentage of the maximum response to phenylephrine. The value of the asymptote to the curve of the tested compounds, related to the corresponding value on the curve for phenylephrine, shows the specific activity of these compounds (Specifices the East. It represents the negative logarithm of the concentration of agonist that causes a response equal to half of the maximum response. For compounds with specific activity is significantly below 1, for example, of 0.8, it is possible to calculate the efficiency ( ) and the value of the affinity (pKAusing the comparative analysis method. This analysis assumes that phenylephrine acts in this system as a full agonist, and thus uses it to define the parameters of the operating model Emand n (ref. Leff et al., "Estimation of agonist affinity and efficacy by direct and operational model fitting", J. Pharmacol. Methods. , 1989, 23, 225-237). These parameters can then be used for comparative analysis of the tested compounds. Affinity is expressed as pKA(the negative logarithm of the concentration of the agonist, which involved half of the receptors).
Quantitative expression of antagonism
Compounds that do not exhibit agonism, were tested as antagonists by incubating tissue in such a high concentration of the compound, to the extent possible, with the subsequent construction of curves E/[A] to phenylephrine. Degree offset to the right of these curves to phenylephrine in CPA the evaluation of the affinity shown as the values of pA2(the negative logarithm of the concentration of antagonist that causes a 2-fold shift to the right from the reference curve E/[A]).
Confirmation1-mediated agonism
As a standard antagonist1was used prazosin. If the tested compound showed agonism when the level of the asymptotes on the curve E/[a] for the tested compounds were added prazosin (1 μm) in order to determine if the response is reversible. If1antagonist leads to a reversion of the response of the tested compounds, it confirms that agonism in this case is 1-mediated.
1. Derived benzothiazolone formula I
where X denotes naphthyl,
and its pharmaceutically acceptable salts.
2. Derived under item 1, characterized in that it is a salt, such as hydrochloride, hydrobromide, sulfate, phosphate, maleate, tartrate, citrate, benzoate, 4-methoxybenzoate, 2 - or 4-hydroxybenzoate, 4-chlorobenzoate, bansilalpet, p-toluensulfonate, naphthalenesulfonate, methanesulfonate, sulpham, ascorbate, salicylate, acetate, diphenylacetate, trimeniaceae, adipate, fumarate, succinate, lactate, glutarate, gluconate, 1-hydroxy-2-on is characterized in that it is a salt, such as hydrochloride.
4. Derivative of the formula I on p. 1, characterized in that it is suitable for the production of medicines for the treatment of respiratory diseases.
5. Derivative of the formula I on p. 4, wherein the respiratory disease is asthma.
6. Derivative of the formula I on p. 4, wherein the respiratory disease is chronic bronchitis.
7. The method of obtaining the derived benzothiazolone formula I, comprising a selective reduction of the compounds of formula II
in which X is defined in paragraph 1.
8. Derived benzothiazolone formula II
where X is defined in paragraph 1.
Connection benzothiazolone, methods and intermediate compounds for their production, as well as containing pharmaceutical compositions // 2172313
The invention relates to new derivatives of benzothiazolone General formula I where X is-SO2NH - or-NНSО2-; p, q and r independently of one another represent 2 or 3; Y represents thienyl, optionally substituted C1-6by alkyl or halogen, or phenylthio or phenyl, optionally substituted C1-6by alkyl or halogen; each R independently represents H or C1-6alkyl; its optical isomers and pharmaceutically acceptable salts
2-adrenoreceptor" target="_blank">
Derivatives of 7-(2-amino-ethyl)benzothiazolone or their pharmaceutically acceptable salts, methods for their preparation, derivatives of n-[2-(4-hydroxy-2-oxo-3h-1,3-benzothiazol-7-yl) ethylamide]the pharmaceutical composition exhibiting agonistic activity against 2-adrenoreceptor // 2114108
The invention relates to new compounds, methods for their preparation and pharmaceutical preparations, in which they enter and treatment options, based on their use
The method of splitting alkoxygroup in the synthesis of 2-cyano-6 - hydroxybenzothiazole // 2059628
The invention relates to organic chemistry, in particular to the synthesis of substituted 6-hydroxybenzothiazole containing labile fragments
Hydrazine derivatives or their salts // 2051902
Bicyclic carboxamide and their pharmaceutically acceptable salts and pharmaceutical compositions on their basis // 2176642
The invention relates to new bicyclic to carboxamide formula (i) in which (1) X represents N and (a) Z is =CR1-CR2and Y is N, Z is =CR1and Y represents O, S or NR4or (C) Z is = CR1-N= and Y represents CR2or (2), X represents NR4Z represents CR1= and Y is N, Q is O, R1and R2are СОR6, C(= NOR6R13, alkyl-C(=NOR6R13, NR8R9, CF3or R6, R3is1-6alkoxygroup, R4represents H or alkyl, R5is heteroaryl, optionally substituted with halogen, alkyl, CONR11R12, CF3or CN, aryl, substituted with halogen; R6represents H, alkyl, cycloalkyl, aryl, heteroaryl, heterocycle, arylalkyl, heteroaromatic or heteroseksualci, R7represents alkyl, hydroxy, OR10, NR8R9CN, CO2H, CO2R10, CONR11R12, R8and R9represent H or alkyl, or NR8R9represents a heterocyclic ring, optionally substituted by R14, R10represents an alkyl, heterocycle, R11and R12represent H or alkyl, and the salts
Treatment of severe broncho-obstructive syndrome in children of early age // 2174838
The invention relates to medicine, in particular to Pediatrics
Diazepinones inhibitors of phosphodiesterase-iv // 2174517
The invention relates to new, containing diazepambuy group of compounds with biological activity, and more particularly to a derivative of diazepinone, their racemic forms, their isomeric configurations defined by the carbon in position 3 diazepinone-4-about the kernel and their pharmacologically acceptable salts, intermediate compounds and pharmaceutical compositions inhibiting phosphodiesterase IV activity
Ascomycin, the method of production thereof and pharmaceutical composition based on them // 2173688
The invention relates to new ascomycin General formula I, where Y represents a phenylene; Z is selected from carboxyl and physiologically hydrolyzable of oxycarbonyl or alkyl, CNS, alkylamino or dialkylamino bearing from 1 to 4 carboxyl or physiologically hydrolyzable oxycarbonyl group; Q is O or S; R1Is H, alkyl or aryl; R2is hydrogen or hydroxyl; R3is methyl, ethyl, propyl or allyl; R4is hydroxyl or alkoxyl; R5-oxoprop or (H, OH), R6- oxoprop, H, HE H, alkoxyl); n is an integer 1 or 2, in free form or in the form of a pharmaceutically acceptable salt
The immunophilin-specific ligands as anti-asthma and immunodepressant agents and method of production thereof, pharmaceutically ready form, the drug and its production method // 2172743
Bromide l-(-)-enantiomer(endo,syn)-(-)-3-(3-hydroxy-1-oxo-2 - phenylpropoxy)-8-methyl - 8-(1-methylethyl)-8-asone - bicyclo[3.2.1]octane with a purity of 90-100% of an inhaled drug for the inhibition of the bronchoconstriction induced by acetylcholine // 2171258
The compound 7-(2-imidazolidinone) quinoline, suitable as agonists alpha-2-adrenergic receptors // 2161967
The invention relates to the field of organic chemistry and medicine and relates to new quinoline derivatives of General formula I, where R represents unsubstituted C1-C3-alkyl or C2-C3alkenyl; R1is cyano, which are agonists alpha-2-adrenergic receptors and can be used when obtaining drugs suitable for the treatment of nasal congestion, glaucoma, diarrhea, asthma
Drugs fluticasone propionate // 2161485
Xantina derivatives, their stereoisomers and the physiologically tolerated salts, pharmaceutical composition having reducing pathological hyperactivity eosinophilic granules activity // 2159617
The invention relates to chemistry, medicine and pharmacology, new nitrogen-containing heterocyclic compounds possessing biological activity, and more particularly to a derivative of xanthine f-ly I, their stereoisomers and the physiologically tolerated salts, and pharmaceutical compositions having reducing pathological hyperactivity eosinophilic granules activity
Derivatives benzazepin, benzoxazepin or benzodiazepin-n - acetic acid, method of their production and the drug containing // 2159768
Pharmaceutical composition with antidepressant activity // 2157685
The invention relates to medicine
The use of riluzole for the treatment of neuro-aids and a method of obtaining drugs // 2157205
Benzothiazole derivatives // 2251419
FIELD: medicine, organic chemistry.
SUBSTANCE: the present innovation deals with new benzothiazole derivatives and medicinal preparation containing these derivatives for treating diseases mediated by adenosine receptor A2.A.. The present innovation provides efficient treatment of the above-mentioned diseases.
EFFECT: higher efficiency of therapy.
14 cl, 354 ex
Indeno-, naphtho-, and benzocycloheptadihydrothiazole derivatives, production thereof and uses as anorexia pharmaceuticals // 2252219
FIELD: organic chemistry, pharmacology.
SUBSTANCE: invention relates to polycyclic dihydrothiazoles of formula I , containing in substituted alkyl residues in 2-posiiton, as well as physiologically accepted salts thereos, having anorexia action. In formula Y is direct bond; X is CH2; R1 and R1' are independently H, Cl; R2 and R3 are H; R4 is (C8-C16-cycloalkyl, (CH2)n-A-R8, wherein n = 1-6, excepted group of formula -CH2-O-CH2-phenyl with unsubstituted phenyl; A is O, S; R8 is methyl or (CH2)m-aryl, where in m = 0-6; and aryl may represent phenyl, wherein aryl group may be optionally substituted with one or two substituents, selected from Cl, O-(C1-C6)-alkyl or (C1-C6)-alkyl. Also disclosed is method for production thereof.
EFFECT: new anorexia pharmaceuticals.
5 cl, 4 ex, 2 tbl
Derivatives of 8,8a-dihydroindeno[1,2-d]thiazole substituted at 8a position, method for their preparing and their applying as medicinal agents, for example, as anorectics // 2261860
FIELD: organic chemistry, medicine.
SUBSTANCE: invention relates to polycyclic dihydrothiazoles and their physiologically acceptable salts and physiologically functional derivatives. Invention describes compounds of the formula (I): wherein r1 and R1' mean independently of one another atoms H, F, Cl, Br and J; R2 means hydrogen atom (H); R3 means chlorine (Cl), bromine (Br) atom; R4 means phenyl, and a method for their preparing. Compounds can be used, for example, as anorectics for prophylaxis or obesity treatment.
EFFECT: improved preparing method, valuable medicinal properties of compounds.
5 cl, 2 tbl, 1 ex

redpajama_set_name: RedPajamaCommonCrawl


We serve authentic, upscale Mexican food in a comfortable setting. Couples, families, and singles will feel at home both in our Dining Room, and in our Tequila Lounge. There is a Children's Menu available.
Monday through Friday 4 -7 PM in our Tequila Lounge.
Special prefix dinner menu for $23.95 offered Sunday through Tuesday.
"I'm loving the vibe of this restaurant. They have a brunch, lunch, and dinner menu. Also, a gluten free menu. My waiter was welcoming and friendly. Guacamole is made fresh in front of you. Salsa and chips are given once you sit down. And once you run low on chips, they automatically refill it without even having to ask!
I had the chicken tacos and it tasted amazing and fresh. I was full. I definitely want to explore more of this menu, especially the coconut margarita."
"If your looking for a great Mexican restaurant for an intimate dinner or dinner party, this is your place. The food was amazing. I got as an appetizer the chicken and beef empanadas and they are the second best empanadas I ever had (my grandmas being the first best). As my main course I had a skirt steak with plantains on the side that was delicious and cooked the way I asked. They also have nachos and dip on the table. With the meals for everyone they bring out rice and beans. The service was really great, they explained to us any questions we had about the menu and informed us that some courses had nuts if anyone was allergic. I can't wait to go back to the place again and try more food!"
"First time there- Thursday around 5:30ish. Few people there so it was nice and quiet for a good drink and dinner. $6 margaritas! I ordered the taco de pollo- came with 4 small tacos but it definitely filled me up! It was good for the most part. A tad spicy for me. The guacamole was delicious- I couldn't stop eating it, very fresh.
The service was very good. Always made sure everything was to your liking.
I will be back but I think I will try something else for the main course."


1 New England Historical and Genealogical Register, (New England Historic and Genealogical Society), 1994, Vol. 148, Pages 255-258.
2 The American Genealogist, New Haven, Connecticut: D. L. Jacobus (issues newer than 5 years not online), 1932, Vol. 9, p. 212, New England Historic Genealogical Society (Online Database).
3 Browning, Charles H., ed., Americans Of Royal Descent , Philadelphia: Porter and Coates (1883), 287, Google Books (Digital Library).
4 Burke, John and John Bernard Burke, The Royal Families of England, Scotland, and Wales, with their Descendants, Sovereigns and Subjects, Vol. II, London: J. Billing, Printer and Stereotyper (1851), Pedigree xv: Frevile Lambton Burne, Esq., Google Books (Digital Library).
5 Foster, Joseph, comp., The Royal Lineage of our Noble and Gentle Families. Together with their Paternal Ancestry , Aylesbury: Privately Printed by Hazell, Watson and Viney, Limited (1884), 84, Heritage Quest (Digital Library).
6 Fraser, ed. Antonia, The Lives of the Kings and Queens of England , New York: Alfred A. Knopf (1975), 70.
7 Luther, George A., editor and compiler, The Luther Genealogy: A History of the Descendants of Captain John Luther, Who Arrived in the Massachusetts Bay Colony, 1630-1635 , Penobscot Press (2001), 81.
8 Pitcairn, Constance, The History of the Fife Pitcairns with Transcripts from Old Charters , Edinburgh and London: William Blackwood and Sons (1905), 419, Internet Archive (Digital Library).
9 Roberts, Gary Boyd, Notable Kin, Volume 1 , Santa Clarita, California: Carl Boyer, 3rd (1998), 62.
10 Roberts, Gary Boyd, The Royal Descents of 600 Immigrants to the American Colonies or the United States , Baltimore: Genealogical Publishing Co. (2008), 270-370.
11 Roberts, Gary Boyd, The Royal Descents of 900 Immigrants to the American Colonies, Quebec, or the United States (2 vols.) , Baltimore: Genealogical Publishing Company (2018), 500, 558.
12 Weis, Frederick Lewis, et. al., The Magna Charta Sureties, 1215: The Barons Named in the Magna Charta, 1215 and Some of their . . ., 5th edition , Baltimore, Maryland: Genealogical Publishing Co., Inc. (Third Printing, 2006), 25.
13 Wikipedia, "Eleanor of Castile", (accessed 02/21/2010).


It is a big step to decide to seek legal assistance for your employment matter. Here at Bryan Schwartz Law, we have represented hundreds of people who have faced discrimination, harassment, wage theft, retaliation, and numerous other workplace wrongs. We understand that deciding to contact a lawyer can be daunting. We hope this blog post can help clarify the process and help you better understand what to expect when you contact a workers' rights attorney. If you are considering contacting an attorney for help with your employment problem, here are some tips on how to present your best case and what to expect from the initial consultation process.
Telling your potential employment lawyers that you feel you were wrongfully terminated is not going to help him or her very much in deciding whether to take your case. Be prepared to flesh out the story of why you feel you were wronged with a 3-D presentation—Deadlines, Documents, and Dates. As a general practice at Bryan Schwartz Law, we ask all potential clients to initially provide us with a brief summary of the workplace issue they are contacting us about and for a detailed timeline of events. This information is meant to help us understand and evaluate your case as quickly as possible. We recognize that your employment matter is very important to you and that you need to receive information from us in a timely manner. If you are prepared from the beginning to provide us with your written summary and timeline, as well as documents related to your case, we can get back to you as quickly as possible. Below are a few more details about the 3-D Presentation.
The attorney you contact will most likely ask you early on for documents supporting your claims. Depending on the type of case that you have, examples of these types of documents could include: termination letters; disciplinary documents; administrative charge documents with, for example, the Department of Fair Employment and Housing or the Equal Employment Opportunity Commission; paystubs; W-2s; and emails supporting your claims. When you are asked for documents, we suggest you go through your documents carefully and select only the documents you think are the most helpful to your case. If you submit hundreds of pages to your potential attorney and most of the pages do not help your case, it will take longer for the lawyer to evaluate the case. Furthermore, you may inadvertently send the message that you are not interested in putting a lot of effort into your own case. As with many things, a good rule to go by is quality, not quantity.
If possible, it is a great idea to be prepared with electronic scans (or extra copies) of the documents that you think support your case before you start contacting attorneys. It is also a good idea to prepare a summary and timeline in advance—your potential attorney will admire the fact that you are so prepared, and it might even help you to organize your thoughts and get a clearer picture of your case from the beginning.
Our friends at Workplace Fairness—a terrific website with many helpful articles about employment issues that affect workers—have a very helpful article about building the best case you can. In section 2 of the article, Workplace Fairness lists the types of documents you should gather in preparation for your case. This list is a great starting point, though some documents may not be available or necessary in your particular situation.
Please bear in mind that every firm that represents employees operates their intake process a little bit differently. Here at Bryan Schwartz Law, your case will be assessed by an attorney right from the start. Like many firms that represent workers, our firm is small. It is always of the utmost importance to us that you are treated with respect and that your case is evaluated in a timely manner. We understand how stressful it is to try to find an attorney, and we understand that you are eager to hear back from us. However, it is the nature of our work that we may have out-of-town travel, depositions, court dates, or brief filing deadlines that come up at any given time. Therefore, it may take us slightly more time to review your case during one of those weeks than it would during a week with fewer pressing deadlines, etc.
If you have contacted an attorney and have not heard back within a few days (or shorter if your matter is particularly urgent), you will not be faulted if you call to follow up. Bear in mind, however, that acting like too much of a "squeaky wheel" may not benefit your case in the end. Lawyers must assess whether your case would be a good "fit" for their firm, and if a potential client has unrealistic expectations, or is simply rude, this can be a red flag for the case down the road. Why? Lawyers are constantly thinking ahead—about things like, how would this person hold up if he or she had to testify on the stand? How would he or she do in a deposition? Will he or she cooperate in the litigation process even if it gets frustrating at times? Will he or she be reasonable in settlement negotiations?
On a related note, you should make sure to honor the commitments that you make to your potential attorney. For example, if you have not already provided documents to the firm, they will most likely request supporting documents for your case. If you say you will email documents by a certain date, hold yourself to that commitment. Just as it is the attorney's responsibility to follow up with you, you have a responsibility to follow up with your potential attorney. If you have a delay in sending documents or otherwise responding, a short "heads-up" email will do the trick.
When we get a potential client call at Bryan Schwartz Law, it is always our goal to treat you with dignity and respect. It is also our job and our responsibility to candidly assess whether our firm will take your case. When you contact an attorney about your case, bear in mind that it is possible that the lawyer will tell you news that you will not like to hear. Employment cases often involve painful experiences and can bring up strong emotions. Remember that, unfortunately, there are many situations in which employees face treatment that is unfair, disrespectful, or even downright cruel, but that do not rise to the level of a legal violation.
Another part of our job is to assess your case for issues you may not have considered. Sometimes clients initially believe they have one kind of case, and end up hiring an attorney to represent them in entirely different employment matters of which they learn from counsel. It is an employment lawyer's job to spot workplace violations, so don't be surprised if the lawyer with whom you are speaking asks you a question that you may not have expected—e.g., you may come in thinking you have a discrimination case, and the attorney will ask if you were paid overtime.
We recommend against exaggerating about the strength of the merits of your case. Attorneys will see through such pronouncements, and they may even reflect poorly on your credibility. If your case truly is a "slam dunk," let your documents show that. Clients who spend extensive time repeating that their case is a winner often do themselves a disservice.
4. What if the Firm Does Not Take Your Case?
Many considerations go into a law firm's decision in whether to take a case—these reasons may have to do with the firm's assessment of the merits of your case, or they may have to do with the firm's workload, variety of cases at a given time, geographical considerations, conflicts of interest, or other factors. If your case is turned down, please do not be angry. Many winning cases have been turned down by multiple law firms before the clients have found the right fit. That being said, be willing to hear and understand vulnerabilities in your case if an attorney shares them with you. You may disagree, another lawyer may see the case differently, but we urge you to be realistic about your case. Employment cases are challenging to win, and you are not doing yourself a favor if you refuse to take a realistic look at yours. We wish you the best of luck in your case and commend your bravery for standing up for your rights.
If you feel you have been wronged in the workplace, contact Bryan Schwartz Law today.
 Please note that every firm handles potential client inquiries in its own way. If you are inquiring about a firm's legal services and have questions about the process, it is always a good idea to ask them directly what you can expect.
 A prime example of this issue is the lack of protection for employees who are terminated in retaliation for taking bereavement leave for the death of a close loved one. Currently, the law affords no protection for this situation, though efforts continue to change this. See, e.g., Lange v. Showbiz Pizza Time, Inc. (D. Kan. 1998) 12 Fed.Supp.2d 1150, 1154-1155. AB 2340, a bill currently pending in the California Assembly, would provide up to three days of job-protected time off for bereavement of the loss of a spouse, child, parent, sibling, grandparent, grandchild, or domestic partner (available at: http://leginfo.ca.gov/pub/09-10/bill/asm/ab_2301-2350/ab_2340_bill_20100506_amended_asm_v98.pdf), and our firm strongly supports this legislation. Governor Schwarzenegger vetoed a similar piece of legislation in 2007. (Veto statement available at: http://info.sen.ca.gov/pub/07-08/bill/sen/sb_0501-0550/sb_549_vt_20071013.html).


I deleted the old logo and just want text for now. I went to header and delete image and in the preview it look ok but when i go to the page it looks funny.
I love to help out you out but please read this first Have you tried to install in a root domain? Have you tried to Resave permalinks? Have you tried to disable all plug in? Have you tried the default theme of Wordpress?
Center Header Logo in Vantage?


text:
Subnational population projections
Subnational population projections (national parks): 2018 based
The results from the 2018-based projections for national parks.
This is not the latest release in the series: Subnational population projections
2018 based
These projections provide an indication of the possible size and structure of national parks' population for the period 2018 to 2043. However, this release focusses on the first 10 years of the projection period, as projections tend to become increasingly uncertain in the longer-term, as trends may change over that timescale.
The projections are not forecasts. They do not attempt to predict the impact of government policies, changing economic circumstances, or other factors (like the coronavirus pandemic) on future population.
This release provides analysis of the latest national park population projections, including comparisons with previous projections.
Main points for 2018 to 2028
Projected population change
The populations of the Brecon Beacons and Pembrokeshire Coast are projected to increase by 3.5% (1,190) and 2.5% (550) respectively.
The population of Snowdonia is projected to decline by 1.9% (480).
Projections by age
All three national parks are projected to see a large percentage increase in people aged 65 or over, and a larger increase in people aged 75 or over.
The number of people of working age (aged 16 to 64 years old) and children (aged 0 to 15 years old) are projected to decrease in Pembrokeshire Coast and Snowdonia.
The number of people in Brecon Beacons is projected to see a smaller decrease in the number of people of working age and an increase in the number of children.
Components of change
Negative natural change is projected to increase in all three national parks.
Positive net migration is projected to increase in all three national parks.
Positive net migration is projected to be larger than the negative natural change in both Brecon Beacons and Pembrokeshire Coast, leading to overall projected population increases.
Positive net migration is projected to be smaller than the negative natural change in Snowdonia, leading to overall projected population decreases.
Subnational population projections (national parks), 2018 based , file type: PDF, file size: 2 MB
StatsWales website
Office for National Statistics website
Stephanie Harries
Email: stats.popcensus@gov.wales


text:
\section{Introduction}
\label{intro}

In this paper we study the sum rules obtained summing the inverse powers of the eigenvalues
of the Helmholtz equation on a heterogeneous sphere. For the special case of constant density
(to which we can arbitrarily assign the value $\Sigma=1$), one obtains that the eigenfunctions
are the spherical harmonics, $Y_{lm}(\theta,\phi)$, with  $|m| \leq l$ and $l=0,1,2,\dots$,
and the corresponding eigenvalues, $l (l+1)$, are $2l+1$ degenerate.

The sum rule of order $p$ is thus defined in terms of the eigenvalues as~\footnote{Note that to obtain a finite result,
	the zero mode, corresponding to $l=0$,  has to be excluded from the series.}
\begin{equation}
Z_p = \sum_{l=1}^\infty \frac{2l+1}{(l (l+1))^p}
\end{equation}
with $p>1$.

For the case of an arbitrary density, $\Sigma(\theta,\phi)>0$ over the sphere, however,
the approach outlined above cannot be adopted, since it requires to calculate exactly
{\sl each} of the eigenvalues of the Helmholtz equation. A similar situation occurrs
in the calculation of the sum rules for quantum billiards on finite domains in the plane,
since the eigenvalues are known exactly only for a limited number of shapes (rectangle, circle, ellipse and
symmetric circular annulus, among others).
In particular, Itzykson, Moussa and Luck~\cite{Itzykson86} were able to obtain explicit integral
expressions for the sum rules of inverse powers of the Dirichlet eigenvalues of the Laplacian
on arbitrary domains in two dimensions using a conformal transformation from the domain to the unit
disk, without having to know the eigenvalues exactly. Berry~\cite{Berry86}
applied the method of Ref.~\cite{Itzykson86} to Aharonov-Bohm quantum billiards, obtaining explicit
expressions for different shapes. Steiner \cite{Steiner87} also discussed the sum rule for Aharonov-Bohm quantum billiard
of circular shape, extending an approach previously developed in \cite{Steiner85} for confinement potentials.

More recently Kvitsinky has considered the spectral sum rules for nearly circular domains, particularly a $N$-sided regular polygons~\cite{Kvitsinsky96}; Dittmar~\cite{Dittmar02} has obtained the sum rules for fixed and free membrane problems for simply connected domains of the plane, conformally transforming the domain into the unit disk. Sum rules for specific domains are obtained in Ref.~\cite{Dittmar11}. Dostani\'c~\cite{Dostanic11} has obtained the regularized trace of the inverse Dirichlet laplacian on
a bounded convex domain.

In a series of papers, Refs.~\cite{Amore13A, Amore13B, Amore14, Amore18}, we have derived
general integral expressions for the spectral sum rules of  inhomogenous strings and membranes,
for different boundary conditions; the case of Neumann or periodic boundary conditions, discussed
in Ref.~\cite{Amore14}, requires a careful handling of the traces, which are in principle ill defined due to the  singular contribution stemming from the zero mode. Ref.~\cite{Amore18}, finally, introduces a  "regularized" sum rule, which is obtained
exploting the symmetries of a problem or different boundary conditions.
The purpose of this paper is to extend the approach outlined in Ref.~\cite{Amore14} to the case of the heterogenous sphere.

The paper is organized as follows: in section \ref{general} we describe the general approach and
define the sum rules in terms of the appropriate traces; in section \ref{pt} we obtain the
perturbative corrections to the energy of the lowest mode ("zero-mode"); in section \ref{order}
we derive the general integral expressions for the sum rules of order two and three, explicitly proving that
all divergent contributions cancel out, and  we apply these results to a non-trivial example.
Finally, in section \ref{conclusions}, we draw our conclusions and discuss future work.
The expressions for the perturbative corrections to the energy of the fundamental mode
and for the integrals appearing in the sum rules of order two and tree for an arbitrary density
are reported in  the Appendices \ref{appA} and \ref{appB} respectively.

\section{Exact sum rules: general expressions}
\label{general}

Our starting point is the Helmholtz equation on a unit 2-sphere, in presence of a
variable density
\begin{equation}
-\Delta \psi_n(\theta,\phi)  = E_n \Sigma(\theta,\phi) \psi_n(\theta,\phi) \label{eq_Helmoltz_dens}
\end{equation}
where
\begin{equation}
\Delta \equiv \frac{1}{\sin \theta} \frac{\partial}{\partial \theta} {\sin \theta} \frac{\partial}{\partial \theta} +  \frac{1}{\sin^2 \theta} \frac{\partial^2}{\partial \phi^2}
\nonumber
\end{equation}
is the angular part of the spherical Laplacian operator.

As discussed in Ref.~\cite{Amore10}, one can define $\Phi_n=\sqrt{\Sigma} \psi_n $
and cast this equation into the equivalent form
\begin{equation}
\frac{1}{\sqrt{\Sigma}} (-\Delta) \frac{1}{\sqrt{\Sigma}} \Phi_n(\theta,\phi) = E_n \Phi_n(\theta,\phi)
\label{eq_Helmoltz_dens2}
\end{equation}
in terms  of the hermitian operator $\hat{O} \equiv \frac{1}{\sqrt{\Sigma}} (-\Delta) \frac{1}{\sqrt{\Sigma}}$.

Since the lowest eigenvalue of $\hat{O}$ vanishes, it is convenient to introduce
the modified operator, following Ref.~\cite{Amore14},
\begin{equation}
\hat{O}_\gamma \equiv \frac{1}{\sqrt{\Sigma}} (-\Delta +\gamma) \frac{1}{\sqrt{\Sigma}}
\end{equation}
where $\gamma$ is a constant parameter which will be eventually sent to zero.

Our ultimate goal is to obtain the Green's function associated with $\hat{O}_\gamma$ on the unit sphere; the first
step in this direction is to write the Green's function associated with the operator $(-\Delta+ \gamma)$ on the unit sphere,
which obeys the  spectral decomposition
\begin{equation}
G_\gamma(\theta,\phi,\theta',\phi')  = \frac{1}{4\pi \gamma} + \sum_{l=1}^\infty \sum_{m=-l}^l
\frac{Y_{lm}(\theta,\phi)Y_{lm}^\star(\theta',\phi')}{l (l+1)+\gamma}
\end{equation}

In particular, for $\gamma \rightarrow 0$ one can write
\begin{equation}
G_\gamma(\theta,\phi,\theta',\phi')  = \frac{1}{4\pi \gamma} + \sum_{q=0}^\infty (-\gamma)^q
G^{(q)}(\theta,\phi,\theta',\phi')
\end{equation}
where
\begin{equation}
G^{(q)}(\theta,\phi,\theta',\phi')  \equiv
\sum_{l=1}^\infty \sum_{m=-l}^l  \frac{Y_{lm}(\theta,\phi)Y_{lm}^\star(\theta',\phi')}{(l (l+1))^{q+1}}
\end{equation}

These functions obey the properties:
\begin{equation}
\begin{split}
-\Delta G^{(0)}(\theta,\phi,\theta',\phi') &= \frac{\delta(\phi-\phi') \delta(\theta-\theta')}{\sin\theta} - \frac{1}{4\pi} \nonumber \\
-\Delta G^{(q)}(\theta,\phi,\theta',\phi') &= G^{(q-1)}_\gamma(\theta,\phi,\theta',\phi') \hspace{1cm} , \hspace{1cm} q=1,2,\dots \nonumber
\end{split}
\end{equation}
and
\begin{equation}
G^{(q+1)}(\theta,\phi,\theta',\phi')  =
\int d\Omega''  G^{(0)}(\theta,\phi,\theta'',\phi'') G^{(q)}(\theta'',\phi'',\theta',\phi') \nonumber
\end{equation}

Notice that $G^{(q)}(\theta,\phi,\theta',\phi')$ ($q=0,1,\dots$) are {\sl finite} since they do not contain
contributions from the mode $l=0$ (in Refs.~\cite{Amore13A,Amore13B,Amore14} we actually referred to
$G^{(0)}(\theta,\phi,\theta',\phi')$ as to a "regularized" Green's function).

Using the property
\begin{equation}
\sum_{m=-l}^l Y_{lm}(\theta,\phi) Y_{lm}^\star(\theta',\phi') = \frac{2 l+1}{4\pi} P_l(x(\theta,\phi,\theta',\phi'))
\end{equation}
where $x(\theta,\phi,\theta',\phi') \equiv \hat{e}(\theta,\phi) \cdot \hat{e}(\theta',\phi')$ and  $\hat{e}(\theta,\phi) \equiv \sin\theta \cos \phi \ \hat{i}
+ \sin\theta \sin \phi \ \hat{j} + \cos\theta \ \hat{k}$, we can cast the Green's functions in the form
\begin{equation}
G^{(q)}(\theta,\phi,\theta',\phi')  = \frac{1}{4\pi}
\sum_{l=1}^\infty   \frac{2l+1}{(l (l+1))^{q+1}} P_l(x(\theta,\phi,\theta',\phi'))
\end{equation}

It is worth noticing that the Green's function $G^{(0)}(\Omega,\Omega')$ is a special case of the generalized
Green's function discussed in Ref.~\cite{Szmytkowski06}:
\begin{equation}
\bar{G}_L(\Omega,\Omega') \equiv \sum_{\begin{array}{c}
l=0 \\
l \neq L \\
\end{array}} \sum_{m=-l}^l \frac{Y_{lm}(\Omega) Y_{lm}^\star (\Omega')}{L (L+1)-l (l+1)}
\end{equation}
and
\begin{equation}
G^{(0)}(\Omega,\Omega') =  - \bar{G}_0(\Omega,\Omega')
\end{equation}

The explicit expression for $G^{(0)}(\Omega,\Omega')$ is well-known and it can be found
in Refs.~\cite{Szmytkowski06, Freeden80, Englis98,Gustavsson01}
\begin{equation}
G^{(0)}(x)  = \frac{1}{4\pi} \left[\log 2 - 1- \log(1-x) \right] \nonumber \\
\end{equation}

The formulas for the Green's functions of order one and two can be found in Refs.~\cite{Englis98,Gustavsson01}
\begin{equation}
\begin{split}
G^{(1)}(x)  &= \frac{1}{4\pi} \left[\log\left(\frac{1-x}{1+x}\right) \log\left(\frac{2}{1+x}\right) - \frac{1}{2} \log^2 \left(\frac{2}{1+x}\right)  +  {\rm Li}_2\left(-\frac{1-x}{1+x}\right) +1 \right] \nonumber \\
G^{(2)}(x)  &= \frac{1}{4\pi} \left[ \frac{\pi^2}{6} -2 +2 \zeta(3) + \log\left(\frac{1-x}{2}\right) {\rm Li}_2\left(\frac{1-x}{2}\right)  - {\rm Li}_2\left(\frac{1+x}{2}\right) - 2 {\rm Li}_3\left(\frac{1-x}{2}\right) \right] \nonumber
\end{split}
\end{equation}
where ${\rm Li}_\nu(z) \equiv \frac{z}{\Gamma(\nu )}\int_0^\infty \frac{t^{\nu -1}}{\left(e^t-z\right)} dt$
is the polylogarithm of order $\nu$ ($\nu >0$).

The Green's function associated with $\hat{O}_\gamma$ can be now expressed  as
\begin{equation}
G_{\hat{O}_\gamma}(\theta,\phi,\theta',\phi') = \sqrt{\Sigma (\theta,\phi)}  G_\gamma (\theta,\phi,\theta',\phi')  \sqrt{\Sigma (\theta',\phi')}
\end{equation}
since
\begin{equation}
\begin{split}
\hat{O}_\gamma G_{\hat{O}_\gamma} &= \frac{1}{\sqrt{\Sigma (\theta,\phi)}}  (-\Delta+\gamma) G_\gamma (\theta,\phi,\theta',\phi')  \sqrt{\Sigma (\theta',\phi')} \nonumber \\
=& \frac{1}{\sqrt{\Sigma (\theta,\phi)}} \frac{\delta(\theta-\theta') \delta(\phi-\phi')}{\sin\theta}  \sqrt{\Sigma (\theta',\phi')}  \nonumber \\
=&  \delta (\Omega-\Omega')
\end{split}
\end{equation}

Exploiting the invariance of the trace with respect to unitary transformations and using the completeness of the basis of the homogeneous
problem we can write the sum rule~\cite{Amore13A,Amore13B,Amore14}
\begin{equation}
Z_p(\gamma) \equiv \sum_{n=0}^\infty \frac{1}{E_n(\gamma)^p}
\end{equation}
as
\begin{equation}
Z_p(\gamma) = \int \ G_{\hat{O}_\gamma}(\Omega_1,\Omega_2) \ \dots \  G_{\hat{O}_\gamma}(\Omega_p,\Omega_1) \  d\Omega_1 \dots d\Omega_p
\label{eq_trace}
\end{equation}

Unfortunately, eq.~(\ref{eq_trace}) is not very useful since it diverges as $\gamma \rightarrow 0$, due to
the singular behavior of $G_{\hat{O}_\gamma}$ in this limit. For this reason
it is then convenient to introduce the regularized sum rule
\begin{equation}
\tilde{Z}_p(\gamma) = \sum_{n=1}^\infty \frac{1}{E_n^p(\gamma)} = Z_p(\gamma) - \frac{1}{E_0(\gamma)^p}
\label{eq_Zp}
\end{equation}
by taking out the contributions stemming from the zero mode (we will discuss soon the calculation of
$E_0(\gamma)$ using perturbation theory for $|\gamma| \ll 1$).

Since $\tilde{Z}_p(\gamma)$ is now well behaved for $\gamma \rightarrow 0$, we conclude that
\begin{equation}
\sum_{n=1}^\infty \frac{1}{E_n^p} = \lim_{\gamma \rightarrow 0} \left[ Z_p(\gamma) - \frac{1}{E_0(\gamma)^p}\right]
\end{equation}

Some remarks:
\begin{itemize}
\item For $\gamma \rightarrow 0$, $Z_p(\gamma)$ and $1/E_0(\gamma)^p$ can be Laurent expanded  around $\gamma=0$:
\begin{equation}
\begin{split}
Z_p(\gamma) &= z_{-p} \gamma^{-p} +z_{-p+1} \gamma^{-p+1} + \dots + z_0 + z_1 \gamma + \dots \nonumber \\
\frac{1}{E_0(\gamma)^p} &= \epsilon_{-p} \gamma^{-p} +\epsilon_{-p+1} \gamma^{-p+1} + \dots + \epsilon_0 + \epsilon_1 \gamma + \dots \nonumber
\end{split}
\end{equation}
\item The finiteness of $\lim_{\gamma\rightarrow 0} \tilde{Z}_p(\gamma)$ requires that
\begin{equation}
z_{-p} = \epsilon_{-p} \hspace{.5cm} , \hspace{0.5cm} z_{-p+1} = \epsilon_{-p+1}
 \hspace{.5cm} , \hspace{0.5cm}  \dots  \hspace{.5cm} , \hspace{0.5cm}  z_{-1} = \epsilon_{-1} \nonumber
\end{equation}
\item  The singular (for $\gamma \rightarrow 0$) part of the heterogeneous Green's functions appearing in eq.~(\ref{eq_Zp})   may contribute to $z_0$ as long as it combines with suitable contributions from the remaining Green's functions, that are vanishing
with the appropriate strength (of course this is not the case if the spectrum does not contain a zero mode). It is easy
to check that $Z_p(\gamma)$ contains at most Green's functions of order $p+1$, $G^{(p+1)}$;

\item The calculation of $\tilde{Z}_p$ requires calculating the lowest eigenvalue using perturbation theory up to order $p+1$:
\begin{equation}
E_0 = E_0^{(1)} \gamma + E_0^{(2)} \gamma^2 + \dots \nonumber
\end{equation}
from which
\begin{equation}
\begin{split}
\frac{1}{E_0(\gamma)^p} &= \frac{1}{\gamma^p (E_0^{(1)})^p} - p \frac{E_0^{(2)}}{\gamma^{p-1} (E_0^{(1)})^{p+1} }  \nonumber \\
+&
\frac{p}{2}\frac{1}{\gamma^{p-2} (E_0^{(1)})^{p+2}}
\left((p+1) (E_0^{(2)})^2-2 E_0^{(1)} E_0^{(3)}\right) + \dots \nonumber
\end{split}
\end{equation}

Specifically, for $p=2$ and $p=3$ one has
\begin{equation}
\begin{split}
\frac{1}{E_0(\gamma)^2} =& \frac{1}{\gamma ^2 \left[E_0^{(1)}\right]^2}-\frac{2 E_0^{(2)}}{\gamma  \left[E_0^{(1)}\right]^3} +\frac{3 \left[ E_0^{(2)}\right]^2-2 E_0^{(1)} E_0^{(3)}}{\left[E_0^{(1)}\right]^4} + O(\gamma) \nonumber \\
\frac{1}{E_0(\gamma)^3} =&  \frac{1}{\gamma ^3 \left[E_0^{(1)}\right]^3}
-\frac{3 E_0^{(2)}}{\gamma ^2 \left[E_0^{(1)}\right]^4}
-\frac{3 \left(-2 \left[ E_0^{(2)}\right]^2+E_0^{(1)} E_0^{(3)}\right)}{\gamma \left[E_0^{(1)}\right]^5}\nonumber \\
+& \frac{-10 \left[ E_0^{(2)}\right]^3+12 E_0^{(1)} E_0^{(2)} E_0^{(3)}-3 \left[E_0^{(1)}\right]^2 E_0^{(4)}}{\left[ E_0^{(1)} \right]^6} + O(\gamma) \nonumber
\end{split}
\end{equation}

\item The explicit expressions for the exact sum rules of order $2$ and $3$ are
\begin{equation}
\begin{split}
\tilde{Z}_2 &= \int G^{(0)}(\Omega_1,\Omega_2) \Sigma(\Omega_2)  G^{(0)}(\Omega_2,\Omega_1) \Sigma(\Omega_1) d\Omega_1 d\Omega_2 \nonumber \\
-& \frac{1}{2\pi} \int \Sigma(\Omega_1) G^{(1)}(\Omega_1,\Omega_2) \Sigma(\Omega_2) d\Omega_1 d\Omega_2
\nonumber \\
-& \frac{3 \left[ E_0^{(2)}\right]^2-2 E_0^{(1)} E_0^{(3)}}{\left[E_0^{(1)}\right]^4}  \nonumber \\
\tilde{Z}_3 =& \int G^{(0)}(\Omega_1,\Omega_2) \Sigma(\Omega_2)  G^{(0)}(\Omega_2,\Omega_3) \Sigma(\Omega_3) G^{(0)}(\Omega_3,\Omega_1) \Sigma(\Omega_1)  d\Omega_1 d\Omega_2 d\Omega_3 \nonumber \\
-& \frac{3}{2\pi} \int \Sigma(\Omega_1)  G^{(1)}(\Omega_1,\Omega_2) \Sigma(\Omega_2)   G^{(0)}(\Omega_2,\Omega_3)  \Sigma(\Omega_3)   d\Omega_1 d\Omega_2 d\Omega_3
\nonumber \\
+& \frac{3}{(4\pi)^2} \left( \int \Sigma(\Omega_1)  G^{(2)}(\Omega_1,\Omega_2)  \Sigma(\Omega_2)  d\Omega_1 d\Omega_2 \right) \ \left( \int \Sigma(\Omega_3) d\Omega_3 \right)
\nonumber \\
-& \frac{-10 \left[ E_0^{(2)}\right]^3+12 E_0^{(1)} E_0^{(2)} E_0^{(3)}-3 \left[E_0^{(1)}\right]^2 E_0^{(4)}}{\left[ E_0^{(1)} \right]^6}  \nonumber
\end{split}
\end{equation}

The expressions for higher order sum rules can be worked out in a completely similar way.

Before being able to cast $\tilde{Z}_2$ and $\tilde{Z}_3$ into a simpler form, we need to apply perturbation theory to derive
the explicit expression for $E_0(\gamma)$ up to a given order. This is done in the next section.

\end{itemize}

\section{Perturbation theory for the zero mode}
\label{pt}

Consider the eigenvalue equation for the lowest mode
\begin{equation}
(-\Delta + \gamma) \psi_0(\Omega) = E_0 \Sigma(\Omega) \psi_0(\Omega)
\label{Helmoltz_eq}
\end{equation}
and assume $\gamma \rightarrow 0$ and
\begin{subequations}
\begin{align}
\label{eqPT1}
E_0 &=  \sum_{k=1}^\infty E_0^{(k)} \gamma^k  \\
\label{eqPT2}
\psi_0(\Omega) =& Y_{00}(\Omega) + \sum_{k=1}^\infty \psi_0^{(k)}(\Omega) \gamma^k
\end{align}
\end{subequations}

By inserting these expressions inside the Helmholtz equation (\ref{Helmoltz_eq})
one obtains a system of equations, one for each order in $\gamma$.

Starting to zero order, one has the equation
\begin{equation}
-\Delta \psi_0^{(0)} = 0 \label{eq_PT0}
\end{equation}
from which we obtain the leading contributions to the eigenvalue and to the wave function (normalized over
the total solid angle)
\begin{subequations}
\begin{align}
E_0^{(0)} &= 0 \\
\psi_0^{(0)}(\Omega) &= Y_{00}(\Omega)
\end{align}
\end{subequations}

To first order one needs  to solve the equation
\begin{equation}
-\Delta \psi_0^{(1)} + \psi_0^{(0)} = E_0^{(1)} \Sigma \psi_0^{(0)} \label{eq_PT1}
\end{equation}

Using eq.(\ref{eq_PT0}), we can project equation (\ref{eq_PT1}) over the  zero mode, obtaining
\begin{equation}
E_0^{(1)} = \frac{4\pi}{\int \Sigma(\theta,\phi) d\Omega} \ .
\end{equation}

We now write the first order correction to  the wave function as
\begin{equation}
\psi_0^{(1)}(\theta,\phi) = \sum_{l=1}^\infty \sum_{m=-l}^l c_{lm}^{(1)} Y_{lm}(\theta,\phi)
\end{equation}
and substitute inside  equation (\ref{eq_PT1}).

With straightforward algebra we obtain
\begin{equation}
\psi_0^{(1)}(\theta,\phi) = \frac{E_0^{(1)}}{\sqrt{4\pi}} \int G^{(0)}(\Omega,\Omega') \Sigma(\Omega') d\Omega'
\end{equation}
where $G^{(0)}$ is the regularized Green's function introduced earlier.

To order $k$ ($k \geq 2$) one obtains the equation
\begin{equation}
(-\Delta )\psi_0^{(k)}(\theta,\phi) + \psi_0^{(k-1)}(\theta,\phi) = \Sigma(\theta,\phi) \sum_{j=1}^{k-1} E_0^{(j)} \psi_0^{(k-j)}(\theta,\phi)
\end{equation}

The corrections of order $k$ to the eigenvalue and to the eigenfunction are obtained as done to order $1$ and they read
\begin{subequations}
\begin{align}
E_0^{(k)} &=  - \frac{\sum_{j=1}^{k-1} E_0^{(j)} \langle \psi_0^{(0)} | \Sigma | \psi_0^{(k-j)} \rangle}{\langle \psi_0^{(0)} |  \Sigma | \psi_0^{(0)} \rangle} \label{eq_EN_recur}\\
\psi_0^{(k)}(\Omega) &=  \sum_{j=1}^k E_0^{(j)} \int d\Omega' G^{(0)}(\Omega,\Omega') \Sigma(\Omega') \psi_0^{(k-j)}(\Omega') \nonumber \\
-& \int d\Omega'  G^{(0)}(\Omega,\Omega') \psi_0^{(k-1)}(\Omega') \label{eq_WF_recur}
\end{align}
\end{subequations}

The expressions for the perturbative corrections to the energy of the zero mode
up to fourth order, obtained solving recursively  eqs.~(\ref{eq_EN_recur})
and (\ref{eq_WF_recur}), are reported in  Appendix \ref{appA}.

\section{Exact sum rules of given order}
\label{order}

The final expressions for the sum rules can now be worked out, using the explicit expressions for the perturbative corrections to
$E_0$ obtained in the previous section. We will concentrate only on the sum rules of order two and three, although similar expressions
can be obtained also for sum rules of higher order.

In particular, for $\gamma \rightarrow 0$, we find that
\begin{equation}
\begin{split}
Z_2(\gamma) &\approx \left(\frac{\int \Sigma(\Omega)d\Omega}{4\pi} \right)^2 \frac{1}{\gamma^2}
+ \frac{\int \Sigma(\Omega) G^{(0)}(\Omega,\Omega') \Sigma(\Omega') d\Omega d\Omega'}{2\pi} \frac{1}{\gamma} \nonumber \\
+& \left[ \int  G^{(0)}(\Omega,\Omega') \Sigma(\Omega') G^{(0)}(\Omega',\Omega) \Sigma(\Omega) d\Omega d\Omega' \right. \nonumber \\
-& \left. \frac{ \int  \Sigma(\Omega) G^{(1)}(\Omega,\Omega') \Sigma(\Omega') d\Omega d\Omega' }{2\pi} \right] + O(\gamma)
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}
\frac{1}{E_0^2(\gamma)} &\approx \left(\frac{\int \Sigma(\Omega)d\Omega}{4\pi} \right)^2 \frac{1}{\gamma^2}
+ \frac{\int \Sigma(\Omega) G^{(0)}(\Omega,\Omega') \Sigma(\Omega') d\Omega d\Omega'}{2\pi} \frac{1}{\gamma} \nonumber \\
+ & \left[ 2 \frac{\int   \Sigma(\Omega)  G^{(0)}(\Omega,\Omega') \Sigma(\Omega') G^{(0)}(\Omega',\Omega'')  \Sigma(\Omega'') d\Omega d\Omega' d\Omega''}{\int \Sigma(\Omega) d\Omega} \nonumber \right. \\
- & \left. \left(\frac{ \int  \Sigma(\Omega) G^{(0)}(\Omega,\Omega') \Sigma(\Omega') d\Omega d\Omega' }{\int \Sigma(\Omega) d\Omega}\right)^2 - \frac{ \int  \Sigma(\Omega) G^{(1)}(\Omega,\Omega') \Sigma(\Omega') d\Omega d\Omega' }{2\pi} \right] \nonumber \\
+ & O(\gamma)
\end{split}
\end{equation}

As a result, we see that the singularities in $\tilde{Z}_2(\gamma)$  cancel identically for $\gamma \rightarrow 0$, as anticipated,  and
the sum rule of order two is therefore
\begin{equation}
\begin{split}
\sum_{n=1}^\infty \frac{1}{E_n^2} &=
\int G^{(0)}(\Omega,\Omega')\Sigma(\Omega') G^{(0)}(\Omega',\Omega)\Sigma(\Omega) d\Omega d\Omega' \nonumber \\
-& 2 \ \frac{\int \Sigma(\Omega) G^{(0)}(\Omega,\Omega') \Sigma(\Omega') G^{(0)}(\Omega',\Omega'') \Sigma(\Omega'')
d\Omega d\Omega' d\Omega''}{\int \Sigma(\Omega) d\Omega}\nonumber \\
+& \left( \frac{\int \Sigma(\Omega) G^{(0)}(\Omega,\Omega')\Sigma(\Omega') d\Omega d\Omega'}{\int \Sigma(\Omega) d\Omega} \right)^2
\end{split}
\end{equation}

In the case of the sum rule of order three we also obtain that the singularities for $\gamma \rightarrow 0$ cancel out
identically inside $\tilde{Z}_3(\gamma)$ and the sum rule reads
\begin{equation}
\begin{split}
\sum_{n=1}^\infty \frac{1}{E_n^3} &=
\int G^{(0)}(\Omega,\Omega')\Sigma(\Omega') G^{(0)}(\Omega',\Omega)\Sigma(\Omega) G^{(0)}(\Omega',\Omega'')\Sigma(\Omega'')
d\Omega d\Omega' d\Omega''\nonumber \\
- & 3 \ \frac{\int \Sigma(\Omega) G^{(0)}(\Omega,\Omega') \Sigma(\Omega') G^{(0)}(\Omega',\Omega'') \Sigma(\Omega'')
G^{(0)}(\Omega'',\Omega''') \Sigma(\Omega''')  d\Omega d\Omega' d\Omega'' d\Omega'''}{\int \Sigma(\Omega) d\Omega}\nonumber \\
+ & 3 \ \frac{\left(\int \Sigma(\Omega) G^{(0)}(\Omega,\Omega')\Sigma(\Omega') d\Omega d\Omega' \right)}{\left(\int \Sigma(\Omega) d\Omega\right)^2} \nonumber \\
\cdot &  \left(\int \Sigma(\Omega) G^{(0)}(\Omega,\Omega')\Sigma(\Omega') G^{(0)}(\Omega',\Omega'')\Sigma(\Omega'') d\Omega d\Omega' d\Omega''\right) \nonumber \\
-& \left( \frac{\int \Sigma(\Omega) G^{(0)}(\Omega,\Omega')\Sigma(\Omega') d\Omega d\Omega'}{\int \Sigma(\Omega) d\Omega} \right)^3
\end{split}
\end{equation}

Using the definitions in Appendix \ref{appB} one has
\begin{equation}
\begin{split}
\sum_{n=1}^\infty \frac{1}{E_n^2} &=  \mathcal{J}_1^{(0,0)} - \frac{1}{2\pi} \mathcal{I}_2^{(0,0)}  + \left(\frac{ \mathcal{I}_1^{(0)} }{4\pi}\right)^2 \\
\sum_{n=1}^\infty \frac{1}{E_n^3} &= \mathcal{J}_2^{(0,0,0)} - \frac{3}{4\pi} \mathcal{I}_3^{(0,0,0)} + \frac{3}{16 \pi^2} \mathcal{I}_1^{(0)} \mathcal{I}_2^{(0,0)} -  \left(\frac{ \mathcal{I}_1^{(0)} }{4\pi}\right)^3 \label{exactsr}
\end{split}
\end{equation}

As an application we consider the density
\begin{equation}
\Sigma(\theta,\phi) = 1 + \kappa \ Y_{10}(\theta,\phi) = 1 + \frac{1}{2} \sqrt{\frac{3}{\pi }} \kappa \cos (\theta )
\nonumber
\end{equation}
where the requirement $\Sigma(\Omega)>0$ on the sphere implies the condition $|\kappa| < 2 \sqrt{\pi/3} \approx 2.04665$.

We have calculated explicitly the integrals appearing in the sum rules of order two and three:
\begin{equation}
\begin{split}
\mathcal{I}_1^{(0)}     &=  \frac{\kappa^2}{2}  \nonumber \\
\mathcal{I}_2^{(0,0)}   &=  \frac{\kappa^2}{4}  \nonumber \\
\mathcal{I}_3^{(0,0,0)} &=  \frac{\kappa^2}{8} + \frac{\kappa^4}{120 \pi} \nonumber \\
\mathcal{J}_1^{(0,0)}   &= 1 + \frac{\kappa^2}{8 \pi} \nonumber \\
\mathcal{J}_2^{(0,0,0)} &= 2 ( \zeta(3)-1) + \frac{3 \kappa^2}{32 \pi} \nonumber
\end{split}
\end{equation}

The explicit expressions for the sum rules (\ref{exactsr}) are
\begin{equation}
\begin{split}
\sum_{n=1}^\infty \frac{1}{E_n^2} &= 1 + \frac{\kappa^4}{64 \pi^2}  \\
\sum_{n=1}^\infty \frac{1}{E_n^3} &= 2 (\zeta (3)-1) +\frac{11 \kappa^4}{640 \pi ^2}-\frac{\kappa ^6}{512 \pi ^3}
\end{split}
\label{sumruleexact}
\end{equation}

\begin{figure}
\begin{center}
\bigskip\bigskip\bigskip
\includegraphics[width=12cm]{figure1.eps} \ \ \  \
\caption{$\left| \sum_{n=1}^\infty \frac{1}{E_n^2} - 1 \right|$ as a function of $\kappa$. The solid line is the exact result $\frac{\kappa^4}{64\pi^2}$, while the dotted, dashed and dot-dashed lines are the numerical results obtained approximating the  eigenvalues with the Rayleigh-Ritz method with $l_{max}=30$,$60$ and $90$ respectively.}
\label{Fig_1}
\end{center}
\end{figure}

\begin{figure}
	\begin{center}
		\bigskip\bigskip\bigskip
		\includegraphics[width=12cm]{figure2.eps}  \ \ \ \
		\caption{$\left| \sum_{n=1}^\infty \frac{1}{E_n^3} - 2 (\zeta(3)-1) \right|$ as a function of $\kappa$. The solid line is the exact result $\frac{\kappa^4}{64\pi^2}$, while the dotted, dashed and dot-dashed lines are the numerical results obtained approximating the  eigenvalues with the Rayleigh-Ritz method with $l_{max}=30$,$60$ and $90$ respectively.}
		\label{Fig_2}
	\end{center}
\end{figure}

In Fig.~\ref{Fig_1} we have compared the exact result for $\left| \sum_{n=1}^\infty \frac{1}{E_n^2} - 1  \right| = \frac{\kappa^4}{64\pi^2}$, with the approximate sum rule obtained calculating the eigenvalues numerically with the Rayleigh-Ritz method, using the states with $1 \leq l \leq l_{max}$ and $|m|\leq l$, respectively with $l_{max}=30$ (dotted curve), $l_{max}=60$ (dashed curve) and $l_{max} = 90$ (dot-dashed curve).

The numerical sum rule is calculated using the lowest $N$ numerical eigenvalues ($N=320$, $1240$ and $2760$, respectively) and completing the series using the asymptotic behavior predicted by Weyl's law, $E_n^{(Weyl)} \approx n$, for $n \rightarrow \infty$:
\begin{eqnarray}
\sum_{n=1}^\infty \frac{1}{E_n^2}  \approx  \sum_{n=1}^{N} \frac{1}{\left(E_n^{(RR)}\right)^2} + \sum_{n=N+1}^\infty \frac{1}{\left(E_n^{(Weyl)}\right)^2}
\end{eqnarray}

A similar result for the case of the sum rule of order three is displayed in Fig.~\ref{Fig_2}.

\section{Conclusions}
\label{conclusions}

We have used the method of Ref.~\cite{Amore14} to derive general integral formulas for the sums of
inverse powers of the eigenvalues of the Laplacian on a heterogeneous sphere with arbitrary density.
Due to the presence of a zero mode, i.e. of a mode with vanishing eigenvalue, the spectral sum rules
need to be "renormalized", by taking out the singular contribution of the fundamental mode: this
is achieved by performing an infinitesimal shift $\gamma$ on the Laplacian, thus rendering all the eigenvalues
finite and then subtracting the contributions stemming from the lowest eigenvalue, for a finite infinitesimal
shift (calculated using perturbation theory). The resulting sum rule is now analytical at $\gamma=0$ and
it corresponds to the sum over the non-vanishing eigenvalues.

We have applied our general formulas to a non--trivial problem, corresponding to the variable density
$\Sigma(\theta,\phi) = 1 +\frac{1}{2} \sqrt{\frac{3}{\pi}} \kappa \cos\theta$, with $|\kappa | < 2 \sqrt{\frac{\pi }{3}}$,
obtaining the exact expressions for the sum rules of order two and three as functions of $\kappa$.
These results have been verified numerically using the Rayleigh-Ritz method to calculate numerically the eigenvalues.

\section*{Acknowledgements}
The research of P.A. was supported by the Sistema Nacional de Investigadores (M\'exico).

redpajama_set_name: RedPajamaArXiv


text:
namespace Bloggable.Data.Models
{
    using System.Collections.Generic;
    using System.ComponentModel.DataAnnotations;
    using System.ComponentModel.DataAnnotations.Schema;

    using Bloggable.Common.Constants;
    using Bloggable.Data.Models.Base;

    public class Tag : IdentifiableAuditInfo<int>
    {
        private ICollection<Post> posts;

        public Tag()
        {
            this.posts = new HashSet<Post>();
        }

        [Required]
        [Index(IsUnique = true)]
        [MinLength(TagValidationConstants.TagNameMinLength)]
        [MaxLength(TagValidationConstants.TagNameMaxLength)]
        public string Name { get; set; }

        public virtual ICollection<Post> Posts
        {
            get { return this.posts; }
            set { this.posts = value; }
        }
    }
}

redpajama_set_name: RedPajamaGithub


text:
Josep Martí i Brillas († 5 de febrer de 1967) va ser alcalde d'Esplugues de Llobregat en dues èpoques diferents. La primera, entre el juliol de 1926 i l'abril de 1927, i la segona, entre l'octubre de 1945 i l'abril de 1946. En tots dos casos va haver d'abandonar l'alcaldia per motius de salut.

Trajectòria política
En la primera etapa va succeir Fèlix Brillas durant el règim de Primo de Rivera. Durant aquest mandat es va crear una escola de pàrvuls i es van fer algunes obres, com l'eixamplament de la carretera de Cornellà a Fogàs. A causa d'una malaltia va dimitir i va traspassar el càrrec a Pere Astals i Vila, fins aleshores primer tinent d'alcalde.

La segona etapa va ser encara més breu: va prendre possessió el 20 d'octubre de 1945, en un any de molts canvis al capdavant de l'alcaldia; el 30 de novembre del mateix any se'n va apartar, també per malaltia, i va ser d'alcalde accidental Josep Maria de Pouplana i Carot fins al 17 d'abril següent, data en què Gaietà Faura i Rodon en va prendre el relleu. Durant el 1945 Alemanya va ser derrotada en la Segona Guerra Mundial i el règim de Franco es va haver d'adaptar a la nova situació internacional.

Referències

Marti


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People read books at Shenzhen Book City CBD store on the fourth day of the Chinese New Year, the Year of the Ox, on February 15, 2021 in Shenzhen, Guangdong Province of China.
Zou Bixiong-VCG
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Walsh is the author of The Subplot: What China Is Reading and Why It Matters published by Columbia Global Reports on February 8
A grandmother, desperate to protect her family from a poisonous, airborne disease, locks everyone up in a hermetically-sealed flat. Years go by and the family survive the pandemic, only to discover that they lack the extra nostrils and yellow puss developed by those exposed to the illness which are needed to re-enter the real world. This story, written in 2015 by Tang Fei, one of China's young SF authors, is ironically called "The Path to Freedom."
It may be exactly what it appears to be, a work of speculative science fiction. Or perhaps it was written as an allegory about stagnation within a closed society. Either way, Tang Fei's simple tale reads like an unnervingly accurate prediction of China's radical response to COVID-19. And it reminds us that Chinese authors can be political, critical and topical, as long as they're writing fiction. In China fiction, as opposed to nonfiction, offers space for authors to distance themselves from what they are writing, and for readers to interpret the meanings themselves, to read between the lines. Of course, fiction works this way in many countries but in a repressive country like China is emerges as less of a choice, than a necessity.
For a brief moment, non-fiction flourished in the very earliest days of the outbreak in Wuhan. As people hunkered down in their apartments writers, sensing widespread anger and suffering, dared to speak plainly and bravely. Poems flooded the internet, expressing fear of the unknown, theories about government cover-ups and commemorating the unburied dead, including the whistle-blower doctor Li Wenliang who died from the virus not long after being penalized for "making false comments." Most famous of all was Wuhan author and "battlefield diarist" Fang Fang, whose daily Weibo posts documented the individual heartbreak and hardship of those at the frontline and, therefore, the reality behind all the depersonalized—and often absurd—official statistics pumped out by the Chinese Communist Party's jittery propaganda department.
The CCP's statistics often feel fictional. From impressive economic milestones pinned to important dates celebrating the Chinese Communist Party, to bombastic claims that, since 2002, there are no unsolved murders in China. Or even absurdist graphs showing precipitous spikes in coronavirus cases, followed by sheer drops every time a new official or government policy was put in place, uncannily tidy bookkeeping is a soft target for Western journalists looking to pick holes in the official narrative.
Right now, the official narrators are riding high. Thanks to the comparative chaos, high death tolls and fraught vaccine debates in the West, and to a tightly-programmed year of propaganda and crackdowns to mark the CCP's centenary, patriotism and pride in the government's authoritarian tactics are soaring. And, as 1.2 million cubic meters of fake snow are pumped out over Beijing for the 2022 Winter Olympics, the absolutism and efficacy of the CCP's zero-COVID-19 strategy, not to mention its ability to manufacture and harness the weather itself (Beijing is too dry for snow) is tribute to the government's ability to insist upon its own desired reality.
Prior to the pandemic, Fang Fang's award-winning fiction—including the banned novel Soft Burial about the traumas of land reform—was widely praised for its "Wuhan flavour" and for her realistic focus on the lives of ordinary people. Yet, just as Lu Xiaobo was not arrested for his poetry, but for his part in the Charter 08 manifesto, so Fang Fang, once hailed the "conscience" of Wuhan, has been left out in the cold since writing her personal testimony. She can no longer find anyone to publish her work in China and nationalist cyberbullies have bombarded her with tens of thousands of messages, including death threats, for agreeing to publish her diary abroad—proof in their eyes that she was a traitor pandering to Western audiences.
For writers instinctively tuned to the hidden influence of any master narrative, this era of economic ascension, authoritarianism and patriotism has been difficult. The ill-fated protagonist Winston Smith in George Orwell's Nineteen Eighty-Four said it "might be true that the average human being was better off now than he had been before… The only evidence to the contrary was the mute protest in your own bones." So too, many writers in China are attuned to something being amiss, even if the numbers do add up. When visiting his family in rural Henan, Yan Lianke, who started out as an army propaganda writer and has become one of China's finest authors, wrote about the despair he felt when his brother suggested he should write stories that won't upset the government. "Our lives are good," said his brother, "isn't that enough?"
Three years later in 2015 Yan published his novel The Day the Sun Died about an entire village in the grip of a terrifying sleepwalking epidemic, a sickness that homogenises and atomises people, leaving them numb to themselves, and others. Yan publicly voiced his support for Fang Fang's diary and online he implored his creative writing students to not let individual memories of hardship be drowned out by songs of victory: "Let's not talk of collective, national, ethnic memories," he said, "but our own."
The reason authors and readers are drawn to magical realism, according to Salman Rushdie, comes from the realization that there is no agreed reality. It is no surprise then that many Chinese writers, including literary giants Can Xue, Mo Yan and Yu Hua not mention a raft of next generation surrealists, channel their personal experiences and anxieties into fiction, and do so in the tradition of magical realism. Yan Lianke has developed his own genre called "mythorealism" to grapple with what he calls contemporary China's "unrealistic reality." He has characterized both China's metamorphosis from agrarian socialist to urban capitalist, as well as the government's handling of SARS and COVID-19 as nothing short of miraculous; no different to the smelting campaigns of the Great Leap Forward, in which people were expected to believe that sand could be turned into steel. Worlds shaped by this kind of magical, alchemical thinking underwrite most of his novels.
Naturally, most authoritarian governments long for the day when everyone is on the same page as them. In China, this pressure has varied in severity over the last three decades, with a dynamic period of artistic output in the lead up to Tiananmen in 1989, and then a sharp tightening of the reins. Under President Xi Jinping, a self-confessed bibliophile, art must now contribute to a "harmonious" and "healthy" society, the bedrock of political legitimacy through stability, known as weiwen or 维稳. He has instructed writers and artists in China to take "patriotism as a theme", for their work to "serve the people and serve socialism." He states that it "is necessary to broadly unite and gather patriotic and dedicated literature and art workers under the leadership of the party." Online writing platforms are now given a "socialist rating," and those that fall short must be rectified. While the state-backed China Writers Association has a tendency to celebrate the (politically correct) writer, rather than the writing itself.
And yet, this is not dissimilar to the way Western readers think of Chinese writers. We demand the mirror image: opposition and defiance. The false dichotomy of patriot or traitor surrounding Fang Fang, while indicative of the fickle and temperamental political climate she's had to weather, is a way to understand how we in the West engage with Chinese literature itself. Fang Fang certainly did not write her diary as a way to appeal to Western publishers and readers. But our eagerness to embrace it is as willful dissent (which, in large part, it was not: it was a plaintive and honest attempt to tell the uncensored truth) is also a reminder that, like the Chinese Communist Party itself (or the online nationalists), we tend to favor art that appears to adhere to our own political doctrines of free speech and anti-censorship. As a result Western readers have an undeniable penchant for anything that is controversial or banned in China, as well as a suspicion of writers, including Nobel Literature Prize-winning author Mo Yan, who thrive there. And it's almost certainly why most people have not read, let alone heard about, much Chinese literature.
But the sheer scale and diversity of Chinese fiction—from a vast online fantasy fiction universe, to a migrant worker poetry movement, underground comics, peculiar crime capers and a golden age of science fiction—is both a tribute to the ways in which cultural strictures have both warped and inadvertently created works of incredible innovation and illumination, which reflect neither what Western readers nor Chinese censors want. The majority are not, in fact, writing about politics at all, but simply reclaiming—or escaping through fantasy—the messiness of personal and social experience. But even that risks being a political act when so many personal experiences, memories and desires contradict or clash with the Party's authorized story and collective ideology. Much like the people who evolve in Tang Fei's story to live with inhospitable air, Chinese fiction has developed some fascinating evolutionary quirks to survive China's political storms—and we have a lot to learn from it.
Since the start of the Trump Administration, Brexit in the U.K., and the rise of right-wing populism in countries such as Austria, Poland and France, much of the West has suffered its own crisis of reality. People on both sides of the increasingly polarized political spectrum believe that the other side pushes fake news, delusional data and dogged biases in order to control the narrative. We are in danger of finding ourselves trapped in increasingly small, sealed rooms. For Winston Smith freedom was the ability to say that 2 + 2 = 4. For Yan Lianke, literature offers an opportunity to say that 1 + 1 does not equal 2. I predict that, as our own narratives are being rewritten, as we discover that we have not been on the same page for a long time, our writers of fiction will increasingly look to the mesmeric, non-linear, fragmented worlds of Chinese writers who learnt to sublimate this crisis a long time ago.
TIME Ideas hosts the world's leading voices, providing commentary on events in news, society, and culture. We welcome outside contributions. Opinions expressed do not necessarily reflect the views of TIME editors.
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