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Born: Prince Adam Czartoryski, 1770.
Died: Edward Lord Bruce, 1610; Dr. John Boyse, translator of the Bible, 1643; Madame de Sevigné, 1696; Edmund Halley, astronomer, 1742; Dr. George Berkeley, Bishop of Cloyne, 1753.
Feast Day: Sts. Isaias and Sabbas, 273. St. Barbasceminus, 346. St. Hilary, B. 368. St. Felix.
ST. HILARY
St. Hilarius lived in the fourth century, and the active and influential part of his life was passed under the Emperor Constantius in the East, though he is included among the Fathers of the Western or Latin Church. He belonged to a family of distinction resident at Poitiers, in Gaul, and was brought up in paganism, but became a convert to Christianity, and in the year 354 was elected bishop of Poitiers. The first general council, held at Nice (Nicaea) in Bithynia, in 325, under the Emperor Constantine, had condemned the doctrine of Arius, but had not suppressed it; and Hilarius, about thirty years afterwards, when he had made himself acquainted with the arguments, became an opponent of the Arians, who were then numerous, and were patronised by the Emperor Constantius.
The council of Arles, held in 353, had condemned Athanasius and others, who were opponents of the Arian doctrine; and Hilarius, in the council of Beziers, held in 356, defended Athanasius, in opposition to Saturninus, bishop of Arles. He was in consequence deposed from his bishopric by the Arians, and banished by Constantius to Phrygia. There he remained about four years, occupied in composing his principal work, On the Trinity, in twelve books. Hilarius, besides his twelve books On the Trinity, wrote a work On Synods addressed to the bishops of Gaul and Britain, in which he gives an account of the various creeds adopted in the Eastern church subsequent to the council of Nice; and he addressed three books to the Emperor Constantius, of whose religious opinions he was always an energetic and fearless opponent. He continued, indeed, from the time when he became a bishop till the termination of his life in 368, to be zealously engaged in the Trinitarian controversy; and the final triumph of the Nicene creed over the Arian may be attributed in a great degree to his energetic exertions. After the death of Constantius, in 361, he was restored to his bishopric, and returned to Poitiers, where he died.
DR. JOHN BOYSE
A minute and interesting memoir of this eminent scholar, in Peck's Desiderata Curiosa makes us aware of his profound learning, his diligence in study, and his many excellences of character. Ultimately he was a prebendary of Ely; but when engaged in his task of translating the Bible, he was only rector of Boxworth. Boyse was one of a group of seven scholars at Cambridge to whom were committed the Apocryphal books; and when, after four years, this task was finished, he was one of two of that group sent to London to superintend the general revision. With other four learned men, Boyse was engaged for nine months at Stationers' Hall, in the business of revising the entire translation; and it is not unworthy of notice, as creditable to the trade of literature, that, while the task of translation passed unrewarded of the nation, that of revision was remunerated by the Company of Stationers sending each scholar thirty shillings a week. The idea of a guerdon for literary exertion was then a novelty-indeed a thing scarcely known in England.
Boyse was employed with Sir Henry Savile in that serious task of editing Chrysostom, which led to a celebrated witticism on the part of Sir Henry. Lady Savile, complaining one day to her husband of his being so abstracted from her society by his studies, expressed a wish that she were a book, as she might then receive some part of his attention. 'Then,' said Sir Henry, 'I should have you to be an almanack, that I might change you every year.' She threatened to burn Chrysostom, who seemed to be killing her husband; whereupon Dr. Boyse quietly remarked, 'That were a great pity, madam.' 'Why, who was Chrysostom?' inquired she. 'One of the sweetest preachers since the Apostles' times,' he calmly answered. 'Then,' said she, corrected by his manner and words, 'I would not burn him for the world.'
Boyse lived to eighty-two, though generally engaged eight hours a day in study. He seems to have been wise before his time as to the management of his physical system under intellectual labour, and his practice may even yet be described with advantage. 'He made but two meals, dinner and supper; betwixt which he never so much as drank, unless, upon trouble of flatulency, some small quantity of aqua-vitae and sugar. After meat he was careful, almost to curiosity, in picking and rubbing his teeth; esteeming that a special preservative of health; by which means he carried to his grave almost a Hebrew alphabet of teeth [twenty-two]. When that was done, he used to sit or walk an hour or more, to digest his meat, before he would go to his study. . . . He would never study at all, in later years, between supper and bed; which time, two hours at least, he would spend with his friends in discourse, hearing and telling harmless, delightful stories, whereof he was exceedingly full. . . . The posture of his body in studying was always standing, except when for ease he went upon his knees.' No modern physiologist could give a better set of rules than these for a studious life, excepting as far as absence of all reference to active exercise is concerned.
MADAME DE SEVIGNÉ
This celebrated woman, who has the glory of being fully as conspicuous in the graces of style as any writer of her age, died. after a few days' illness, at the town of Grignan. Her children were throughout life her chief object, and especially her daughter, to her affliction for whom we owe the greater part of that admirable collection of Letters upon which the fame of Madame de Sevigné is raised. La Harpe describes them as:
'the book of all hours, of the town, of the country, on travel. They are the conversations of a most agreeable woman. to which one need contribute nothing but one's own; which is a great charm to an idle person.'
Her Letters were not published till the eighteenth century, but they were written in the mid-day of the reign of Louis XIV.
'Their ease and freedom from affectation,' says Hallam, 'are more striking by contrast with the two epistolary styles which had been most admired in France-that of Balzac, which is laboriously tumid, and that of Voiture, which: becomes insipid by dint of affectation. Everyone perceives that in the letters of a Mother to her Daughter, the public, in a strict sense, is not thought of; and yet the habit of speaking and writing what men of wit and taste would desire to hear and read, gives a certain mannerism, I will not say air of effort, even to the letters of Madame de Sevigné. The abandonment of the heart to its casual impulses is not so genuine as in some that have since been published. It is at least clear that it is possible to become affected in copying her unaffected style; and some of Walpole's letters bear witness to this.
Her wit and talent of painting by single touches are very eminent; scarcely any collection of letters, which contain so little that can interest a distant age, are read with such pleasure. If they have any general fault, it is a little monotony and excess of affection towards her daughter, which is reported to have wearied its object, and, in contrast with this, a little want of sensibility towards all beyond her immediate friends, and a readiness to find something ludicrous in the dangers and sufferings of others.' Thus, in one letter she mentions that a lady of her acquaintance, having been bitten by a mail doe had gone to be dipped in the sea. and amuses herself by taking off the provincial accent with which. she will express herself on the first plunge. She makes a jest of La Voisin's execution, and thought that person was as little entitled to sympathy as any one; yet, when a woman is burned alive, it is not usual for another woman to turn it into drollery.-Literature of Europe.
Madame de Sevignés taste has been arraigned for slighting Racine; and she has been charged with the unfortunate prediction: 'Il passera comme café.' But it has been denied that these words can be found, though few like to give up so diverting a miscalculation of futurity.
BISHOP BERKELEY AND TARWATER
Berkeley was a poet, as well as a mathematician and philosopher; and his mind was not only well stored with professional and philosophical learning, but with information upon trade, agriculture, and the common arts of life. Having received benefit from the use of tar-water, when ill of the colic, he published a work on the Virtues of Tar-water, on which he said he had bestowed more pains than on any other of his productions. His last work, published but a few mouths before his death, was Further Thoughts on Tar-water; and it shows his enthusiastic character, that, when accused of fancying he had discovered a panacea in tar-water, he replied, that to speak out, he freely owns he suspects tar-water is a panacea.' Walpole has taken the trouble to preserve, from the newspapers of the day, the following epigram on Berkeley's tar-water:
Who dare deride what pious Cloyne has done?
The Church shall rise and vindicate her son;
She tells is all her bishops shepards are,
And Shepherds heal their rotten sheep with tar
In a letter written by Mr. John Whishaw, solicitor, May 25th, 1744, we find this account of Berkeley's panacea:
'The Bishop of Cloyne, in Ireland, has published a book, of two shillings price, upon the excellencies of tar-water, which is to keep ye bloud in due order, and a great remedy in many cases. His way of making it is to put, I think a gallon of water to a quart of tar, and after stirring it together. to let it stand forty-eight hours, and then pour off the clear and drink a glass of about half a pint in ye mornn, and as much at five in ye afternoon. So it's become common to call for a glass of tar-water in a coffee-house, as a dish of tea or coffee.'
GREAT FROSTS
On this day in 1205:
'began a frost which continued till the two and twentieth day of March, so that the ground could not be tilled; whereof it came to pass that, in summer following a quarter of wheat was sold for a mark of silver in many places of England, which for the more part in the days of King Henry the Second was sold for twelve pence; a quarter of beans or peas for half a mark; a quarter of oats for thirty pence, that were wont to be sold for fourpence. Also the money was so sore clipped that there was no remedy but to have it renewed.'
It has become customary in England to look to St. Herilary's Day as the coldest in the year; perhaps from its being a noted day about the middle of the noted coldest month. It is, however, just possible that the commencement of the extraordinary and fatal frost of 1205, on this day, may have had something to do with the notion; and it may be remarked, that in 1820 the 14th of January was the coldest day of the year, one gentleman's thermometer falling to four degrees Fahrenheit below zero. On a review of the greatest frosts in the English chronicles, it can only be observed that they have for the most part occurred throughout January, and only, in general, diverge a little into December on the one hand, and February on the other. Yet one of the most remarkable of modern frosts began quite at the end of January.
It was at that time in 1814 that London last saw the Thames begin to be so firmly frozen as to support a multitude of human beings on its surface. For a month following the 27th of the previous December, there had been a strong frost in England. A thaw took place on the 26th January, and the ice of the Thames came down in a huge 'pack,' which was suddenly arrested between the bridges by the renewal of the frost. On the 31st the ice pack was so firmly frozen in one mass, that people began to pass over it, and next day the footing appeared so safe, that thousands of persons ventured to cross.
Opposite to Queen-hithe, where the mass appeared most solid, upwards of thirty booths were erected, for the sale of liquors and viands, and for the playing of skittles. A sheep was set to a fire in a tent upon the ice, and sold in shilling slices, under the appellation of Lapland mutton. Musicians came, and dances were effected on the rough and slippery surface. What with the gay appearance of the booths, and the quantity of favourite popular amusements going on, the scene was singularly cheerful and exciting. On the ensuing day, faith in the ice having increased, there were vast multitudes upon it between the London and Blackfriars' Bridges; the tents for the sale of refreshments, and for games of hazard, had largely multiplied; swings and merry-go-rounds were added to skittles; in short, there were all the appearances of a Greenwich or Bartholomew Fair exhibited on this frail surface, and Frost Fair was a term in everybody's mouth.
Amongst those who strove to make a trade of the occasion, none were more active than the humbler class of printers. Their power of producing an article capable of preservation, as a memorial of the affair, brought them in great numbers to the scene. Their principal business consisted, accordingly, in the throwing off of little broadsides referring to Frost Fair, and stating the singular circumstances under which they were produced, in rather poor verses-such as the following:
Amidst the arts which on the Thames appear,
To tell the wonders of this icy year,
Printing claims prior place, which at one view
Erects a monument of THAT and You.
Another peculiarly active corps was the ancient fraternity of watermen, who, deserting their proper trade, contrived to render themselves serviceable by making convenient accesses from the landings, for which they charged a moderate toll. It was reported that some of these men realized as much as ten pounds a day by this kind of business.
All who remember the scene describe it as having been singular and picturesque. It was not merely a white icy plain, covered with flag-bearing booths and lively crowds. The peculiar circumstances under which this part of the river had finally been frozen, caused it to appear as a variegated ice country-hill and dale, and devious walk, all mixed together, with human beings thronging over every bit of accessible surface.
After Frost Fair had lasted with increasing activity for four days, a killing thaw came with the Saturday, and most of the traders who possessed any prudence struck their flags and departed. Many, reluctant to go while any customers remained, held on past the right time, and towards evening there was a strange medley of tents, and merry-go-rounds, and printing presses seen floating about on detached masses of ice, beyond recovery of their dismayed owners, who had themselves barely escaped with life. A large refreshment booth, belonging to one Lawrence, a publican of Queenhithe, which had been placed opposite Brook's Wharf, was floated off by the rising tide, at an early hour on Sunday morning, with nine men in the interior, and was borne with violence back towards Blackfriars' Bridge, catching fire as it went. Before the conflagration had gone far, the whole mass was dashed to pieces on one of the piers of the bridge, and the men with difficulty got to land. A vast number of persons suffered immersion both on this and previous days, and three men were drowned. By Monday nothing was to be seen where Frost Fair had been, but a number of ice-boards swinging lazily backwards and for-wards under the impulse of the tide.
There has been no recurrence of Frost Fair on the Thames from 1814 down to the present year (1861); but it is a phenomenon which, as a rule, appears to recur several times each century. The next previous occasion was in the winter of 1788-9; the next again in January 1740, when people dwelt in tents on the Thames for weeks. In 1715-16, the river was thickly frozen for several miles, and became the scene of a popular fete resembling that just described, with the additional feature of an ox roasted whole for the regalement of the people. The next previous instance was in January 1684. There was then a constant frost of seven weeks, producing ice eighteen inches thick. A contemporary, John Evelyn, who was an eyewitness of the scene, thus describes it:
'The frost continuing, more and more severe, the Thames, before London, was still planted with booths in formal streets, all sorts of trades and shops, furnished and full of commodities, even to a printing press, where the people and ladies took a fancy to have their names printed, and the day and the year set down when produced on the Thames: this humour took so universally, that it was estimated the printer gained five pounds a day, for printing a line only, at sixpence a name, besides what he got by ballads, &c. Coaches plied from Westminster to the Temple and from other stairs, to and fro, as in the streets; sheds, sliding with skates, or bull-baiting, horse and coach races, puppet-shows and interludes, cooks, tippling and other lewd places; so that it seemed to be a bacchanalian triumph or carnival on the water: while it was a severe judgment on the land, the trees not only splitting as if lightning-struck, but men and cattle perishing in divers places, and the very seas so locked up with ice, that no vessels could stir out or come in; the fowls, fish, and birds, and all our exotic plants and greens, universally perishing. Many parks of deer were destroyed; and all sorts of fuel so dear, that there were great contributions to keep the poor alive. Nor was this severe weather much less intense in most parts of Europe, even as far as Spain in the most southern tracts.
London, by reason of the excessive coldness of the air hindering the ascent of the smoke, was so filled with the fuliginous stream of the sea-coal, that hardly could any one see across the streets; and this filling of the lungs with the gross particles exceedingly obstructed the breath, so as one could scarcely breathe. There was no water to be had from the pipes or engines; nor could the brewers and divers other tradesmen work; and every moment was fall of disastrous accidents.'
Hollinshed describes a severe frost as occurring at the close of December 1564:
'On New Year's even,' he says, 'people went over and along the Thames on the ice from London Bridge to Westminster. Some played at the foot-ball as boldly there as if it had been on dry land. Divers of the court, being daily at Westminster, shot daily at pricks set upon the Thames; and the people, both men and women, went daily on the Thames in greater number than in any street of the city of London. On the 3rd day of January it began to thaw, and on the 5th day was no ice to be seen between London Bridge and Lambeth; which sudden thaw caused great floods and high waters, that bare down bridges and houses, and drowned many people, especially in Yorkshire.'
A protracted frost necessarily deranges the lower class of employments in such a city as London, and throws many poor persons into destitution. Just as sure as this is the fact, so sure is it that a vast horde of the class who systematically avoid regular work, preferring to live by their wits, simulate the characteristic appearances of distressed labourers, and try to excite the charity of the better class of citizens. Investing themselves in aprons, clutching an old spade, and hoisting as their signal of distress a turnip on the top of a pole or rake, they will wend their way through the west-end streets, proclaiming themselves in sepulchral tones as Frozen-out Gardeners, or simply calling, 'Hall frozen hout!' or chanting 'We've got no work to do The faces of the corps are duly dolorous; but one can nevertheless observe a sharp eye kept on the doors and windows they are passing, in order that if possible they may arrest some female gaze on which to practise their spell of pity. It is alleged on good grounds that the generality of these victims of the frost are impostors, and that their daily gatherings will often amount to double a skilled workman's wages. Nor do they usually discontinue the trade till long after the return of milder airs has liquidated even real claims upon the public sympathy.
When, like a sullen exile driven forth,
Southward, December drags his icy chain,
He graves fair pictures of drags native North
On the crisp window-pane.
So some pale captive blurs, with lips unshorn,
The latticed glass, and shapes rude outlines there,
With listless finger and a look forlorn,
Cheating his dull despair.
The fairy fragments of some Arctic scene
I see to-night; blank wastes of polar snow,
Ice-laden boughs, and feathery pines that lean
Over ravines below.
Black frozen lakes, and icy peaks blown bare,
Break the white surface of the crusted pane,
And spear-like leaves, long ferns, and blossoms fair
Linked in silvery chain.
Draw me, I pray thee, by this slender thread;
Fancy, thou sorceress, bending vision-wrought
O'er that dim well perpetually fed
By the clear springs of thought!
Northward I turn, and tread those dreary strands,
Lakes where the wild fowl breed, the swan abides;
Shores where the white fox, burrowing in the sands,
Harks to the droning tides.
And seas, where, drifting on a raft of ice,
The she-bear rears her young; and cliffs so high,
The dark-winged birds that emulate their rise
Melt through the pale blue sky.
There, all night long, with far diverging rays,
And stalking shades, the red Auroras glow;
From the keen heaven, meek suns with pallid blaze
Light up the Arctic snow.
Guide me, I pray, along those waves remote,
That deep unstartled from its primal rest;
Some errant sail, the fisher's lone light boat
Borne waif-like on its breast!
Lead me, I pray, where never shallop's keel
Brake the dull ripples throbbing to their caves;
Where the mailed glacier with his armed heel
Spurs the resisting waves!
Paint me, I pray, the phantom hosts that hold
Celestial tourneys when the midnight calls
On airy steeds, with lances bright and bold,
Storming her ancient halls.
Yet, while I look, the magic picture fades;
Melts the bright tracery from the frosted pane;
Trees, vales, and cliffs, in sparkling snows arrayed,
Dissolve in silvery rain.
Without, the day's pale glories sink and swell
Over the black rise of you wooded height;
The moon's thin crescent, like a stranded shell,
Left on the shores of night.
Hark how the north wind, with a hasty hand,
Rattling my casement, frames his mystic rhyme.
House thee, rude minstrel, chanting through the land,
Runes of the olden times.
INFERNAL MACHINES
The 14th of January 1858 was made memorable in France by an attempt at regicide, most diabolical in its character, and yet the project of a man who appears to have been by no means devoid of virtue and even benevolence. It was, however, the third time that what the French call an Infernal Machine was used in the streets of Paris, for regicidal purposes, within the present century.
The first was a Bourbonist contrivance directed against the life of the First Consul Bonaparte. This machine,' says Sir Walter Scott, in his Life of Napoleon, 'consisted of a barrel of gunpowder, placed on a cart, to which it was strongly secured, and charged with grape-shot, so disposed around the barrel as to be dispersed in every direction by the explosion. The fire was to be communicated by a slow match. It was the purpose of the conspirators, undeterred by the indiscriminate slaughter which such a discharge must occasion, to place the machine in the street, through which the First Consul must go to the opera; having contrived that it should explode exactly as his carriage should pass the spot.' Never, during all his eventful life, had Napoleon a narrower escape than on this occasion, on the 14th of December 1800. St. Regent applied the match, and an awful explosion took place. Several houses were damaged, twenty persons were killed on the spot, and fifty-three wounded, including St. Regent himself. Napoleon's carriage, however, had just got beyond the reach of harm. This atrocity led to the execution of St. Regent, Carbon, and other conspirators.
Fieschi's attempt at regicide in 1835 was more elaborate and scientific; there was something of the artillery officer in his mode of proceeding, although he was in truth nothing but a scamp. Fieschi hired a front room of a house in Paris, in a street through which royal cortéges were sometimes in the habit of passing; he proceeded to construct a weapon to be fired off through the open window, on some occasion when the king was expected to pass that way. He made a strong frame, supported by four legs. He obtained twenty-five musket barrels, which he ranged with their butt ends raised a little higher than the muzzles, in order that he might fire downwards, from a first floor window into the street. The barrels were not ranged quite parallel, but were spread out slightly like a fan; the muzzles were also not all at the same height; so that by this combined plan he obtained a sweep of fire, both in height and breadth, more extensive than he would otherwise have obtained.
Every year during Louis Philippe's reign there were certain days of rejoicing in July, in commemoration of the circumstances which placed him on the throne. On the 28th, the second day of the festival in 1835, a royal cortége was proceeding along this particular street, the Boulevard du Temple. Fieschi adjusted his machine, heavily loaded with ball (four to each barrel), and connected the touch-holes of all his twenty-five barrels with a train of gunpowder. He had a blind at his window, to screen his operations from view. Just as the cortége arrived, he raised his blind and fired, when a terrific scene was presented. Marshal Mortier, General de Verigny, the aide-de-camp of Marshal Maison, a colonel, several grenadiers of the Guard, and several bystanders, were killed, while the wounded raised the number of sufferers to nearly forty. In this, as in many similar instances, the person aimed at escaped. One ball grazed the king's arm, and another lodged in his horse's neck: but he and his sons were in other respects unhurt. Fieschi was executed; and his name obtained for some years that kind of notoriety which Madame Tussaud could give it.
We now come to the attempt of Orsini and his companions. A Birmingham manufacturer was commissioned to make six missiles according to a particular model. The missile was of oval shape, and had twenty-five nipples near one end, with percussion caps to fit them. The greatest thickness and weight of metal were at the nipple end, to ensure that it should come foremost to the ground. The inside was to be filled with detonating composition, such as fulminate of mercury; a concussion would explode the caps on the nipples, and communicate the explosion to the fulminate, which would burst the iron shell into innumerable fragments. A Frenchman residing in London bought alcohol, mercury, and nitric acid; made a detonating compound from these materials, and filled the shells with it. Then ensued a very complicated series of manoeuvres to get the conspirators and the shells to Paris, without exciting the suspicion of the authorities.
On the evening of the 14th of January 1858, the Emperor and Empress were to go to the opera; and Orsini and his confederates prepared for the occasion. At night, while the imperial carriage was passing, three explosions were heard. Several soldiers were wounded; the Emperor's hat was perforated; General Roquet was slightly wounded in the neck; two footmen were wounded while standing behind the Emperor's carriage; one horse was killed; the carriage was severely shattered; and the explosion extinguished most of the gas-lights near at hand. The Emperor, cool in the midst of danger, proceeded to the opera as if nothing had happened. When the police had sought out the cause of this atrocity, it was ascertained that Orsini, Pierri, Radio, and Gomez were all on the spot; three of the shell-grenades had been thrown by hand, and two more were found on Orsini and Pierri. The fragments of the three shells had inflicted the frightful number of more than five hundred wounds-Orsini himself had been struck by one of the pieces. Rudio and Gomez were condemned to the galleys; Orsini and Pierri were executed. Most readers will remember the exiting political events that followed this affair in England and France, nearly plunging the two countries into war.
THE FEAST OF THE ASS
Formerly, the Feast of the Ass was celebrated on this day, in commemoration of the 'Flight into Egypt.' Theatrical representations of Scripture history were originally intended to impress religious truths upon the minds of an illiterate people, at a period when books were not, and few could read. But the advantages resulting from this mode of instruction were counterbalanced by the numerous ridiculous ceremonies which they originated. Of these probably none exceeded in grossness of absurdity the Festival of the Ass, as annually performed on the 14th of January.
The escape of the Holy Family into Egypt was represented by a beautiful girl holding a child at her breast, and seated on an ass, splendidly decorated with trappings of gold-embroidered cloth. After having been led in solemn procession through the streets of the city in which the celebration was held, the ass, with its burden, was taken into the principal church, and placed near the high altar, while the various religious services were performed. In place, however, of the usual responses, the people on this occasion imitated the braying of an ass; and, at the conclusion of the service, the priest, instead of the usual benediction, brayed three times, and was answered by a general hee-hawing from the voices of the whole congregation. A hymn, as ridiculous as the ceremony, was sung by a double choir, the people joining in the chorus, and imitating the braying of an ass. Ducange has preserved this burlesque composition, a curious medley of French and mediæval Latin, which may be translated thus:
From the country of the East,
Came this strong and handsome beast:
This able ass, beyond compare,
Heavy loads and packs to bear.
Now, seignior ass, a noble bray,
Thy beauteous mouth at large display;
Abundant food our hay-lofts yield,
And oats abundant load the field.
Hee-haw! He-haw! He-haw!
True it is, his pace is slow,
Till he feels the quickening blow;
Till he feel the urging goad,
On his hinder part bestowed.
Now, seignior ass, &c.
He was born on Shechem's hill;
In Reuben's vales he fed his fill;
He drank of Jordan's sacred stream,
And gambolled in Bethlehem.
See that broad majestic ear!
Born he is the yoke to wear:
All his fellows he surpasses!
He's the very lord of asses!
In leaping he excels the fawn,
The deer, the colts upon the lawn;
Less swift the dromedaries ran,
Boasted of in Midian.
Gold from Araby the blest,
Seba myrrh, of myrrh the best,
To the church this ass did bring;
We his sturdy labours sing.
While he draws the loaded wain,
Or many a pack, he don't complain.
With his jaws, a noble pair,
He doth craunch his homely fare.
Now, seignior ass, &c.'
The bearded barley and its stem,
And thistles, yield his fill of them:
He assists to separate,
When it 's threshed, the chaff from wheat.
With your belly full of grain,
Bray, most honoured ass, Amen!
Bray out loudly, bray again,
Never mind the old Amen;
Without ceasing, bray again,
Amen! Amen! Amen! Amen!
The 'Festival of the Ass,' and other religious burlesques of a similar description, derive their origin from Constantinople; being instituted by the Patriarch Theophylact, with the design of weaning the people's minds from pagan ceremonies, particularly the Bacchanalian and calendary observances, by the substitution of Christian spectacles, partaking of a similar spirit of licentiousness,-a principle of accommodation to the manners and prejudices of an ignorant people, which led to a still further adoption of rites, more or less imitated from the pagans. According to the pagan mythology, an ass, by its braying, saved Vesta from brutal violence, and, in consequence, ' the coronation of the ass ' formed a part of the ceremonial feast of the chaste goddess.
An elaborate sculpture, representing a kneeling ass, in the church of St. Anthony at Padua, is said to commemorate a miracle that once took place in that city. It appears that one morning, as St. Anthony was carrying the sacrament to a dying person, some profane Jews refused to kneel as the sacred vessels were borne past them. But they were soon rebuked and put to contrition and shame, by seeing a pious ass kneel devoutly in honour of the host. The Jews, converted by this miracle, caused the sculpture to be erected in the church. It takes but little to make a miracle. The following anecdote, told by the Rev John Wesley, in his Journal, would, in other hands, have made a very good one:
'An odd circumstance,' says Mr. Wesley, 'happened at Rotherham during the morning preaching. It was well only serious persons were present. An ass walked gravely in at the gate, came up to the door of the house, lifted up his head, and stood stock still, in a posture of deep attention. Might not the dumb beast reprove many, who have far less decency, and not much more understanding?'
A somewhat similar asinine sensibility was differently displayed in the presence of King Henry IV of France-the ass, on this occasion, not exhibiting itself as a dumb animal. When passing through a small town, just as the King was getting tired of a long stupid speech de-livered by the mayor, an ass brayed out loudly; and Henry, with the greatest gravity and politeness of tone, said: 'Pray, gentlemen, speak one at a time, if you please.'
MALLARD DAY
The 14th of January is celebrated in All Souls College, Oxford, by a great merrymaking, in commemoration of the finding of an overgrown mallard in a drain, when they were digging a foundation for the college buildings, anno 1437. The following extract from a contemporary chronicle gives an account of the incident:
'Whenas Henrye Chichele, the late renowned archbishope of Cantorberye, had minded to founden a collidge in Oxenforde, for the hele of his soule and the smiles of all those who peryshed in the warres of Fraunce, fighteing valiantlye under our most gracious Henrye the fifthe, moche was he distraughten concerning the place he myghte choose for thilke purpose. Him thinkyth some whylest how he myghte place it withouten the eastern porte of the citie, both for the pleasauntnesse of the meadowes and the clere streamys therebye runninge. Agen him thinkyth odir whylest howe he mote builden it on the northe side for the heleful ayre there coming from the fieldes. Nowe while he doubteth thereon he dremt, and behold there appereth unto him one of righte godelye personage, sayinge and adviseing as howe he myghte platen his collidge in the highe strete of the citie, nere unto the chirche of our blessed ladie the Yirgine, and in witnesse that it was sowthe, and no vain and deceitful phantasie, welled him to laye the first stane of the foundation at the corner which turncth towards the Cattys Strete, where in delvinge he myghte of a suretye finde a scliwoppinge mallarde imprisoned in the sinke or sewere, wele yfattened and almost ybosten. Sure token of the thrivaunce of his future college.
' Moche doubteth he when he awoke on the nature of this vision, whethyr he mote give hede thereto or not. Then advisyth he there with monie docters and learnyd clerkys, who all seyde howe he oughte to maken trial upon it. Then comyth he to Oxenforde, and on a daye fixed, after masse seyde, proceedeth he in solemnee wyse, with spades and pickaxes for the nonce provided, to the place afore spoken of. But long they had not digged ere they herde, as it myghte seme, within the wam of the erthe, horrid strugglinges and flutteringes, and anon violent quaakinges of the distressyd mallarde. Then Chichele lyfteth up his hondes and seyth Bonedicite, &c. &c. Nowe when they broughte him forth, behold the size of his bodie was as that of a bustarde or an ostridge. And mock wonder was thereat; for the lycke had not been scene in this londe, ne in onie odir.'
We obtain no particulars of the merrymaking beyond a quaint song said to have been long sung on the occasion:
'THE MERRY OLD SONG OF THE ALL SOULS' MALLARD
Griffin, bustard, turkey, capon,
Let other hungry mortals gape on;
And on the bones their stomach fall hard,
But let All Souls' men have their MALLARD.
Oh! by the blood of King Edward,
It was a wopping, wopping MALLARD.
Because he saved, if some don't fool us,
The place that 's called th' head of Totes.
Oh! by the blood, &c.
The poets feign Jove turned a swan,
But let them prove it if they can;
As for our proof, 'tis not at all hard,
For it was a wopping, wopping MALLARD.
Therefore let us sing and dance a galliard,
To the remembrance of the MALLARD:
And as the MALLARD dives in pool,
Let us dabble, dive, and duck in bowl.
MISERRIMUS
In the north aisle of the cloister of Worcester Cathedral is a sepulchral slab, which bears only the word MISERRIMUS, expressing that a most miserable but unknown man reposes below. The most heedless visitor is arrested by this sad voice speaking, as it were, from the ground; and it is no wonder that the imaginations of poets and romancists have been awakened by it:
Miserrimus! 'and neither name nor date,
Prayer, text, or symbol, graven upon the stone;
Nought but that word assigned to the unknown,
That solitary word-to separate
From all, and cast a cloud around the fate
Of him who lies beneath. Most wretched one!
Who chose his epitaph?-Himself alone
Could thus have dared the grave to agitate,
And claim among the dead this awful crown;
Nor doubt that he marked also for his own,
Close to these cloistral steps, a burial-place,
That every foot might fall with heavier tread,
Trampling upon his vileness. Stranger, pass
Softly!-To save the contrite Jesus bled!
There has of course been much speculation regarding the identity of Miserrimus: even a novel has been written upon the idea, containing striking events and situations, and replete with pathos. It is alleged, however, that the actual person was no hero of strikingly unhappy story, but only a 'Rev Thomas Morris, who, at the Revolution refusing to acknowledge the king's supremacy [more probably refusing to take the oaths to the new monarch], was deprived of his preferment, and depended for the remainder of his life on the benevolence of different Jacobites.' At his death, viewing merely, we suppose, the extreme indigence to which he was reduced, and the humiliating way in which he got his living, he ordered that the only inscription on his tomb should be-MISERRIMUS!
Such freaks are not unexampled, and we cannot be always sure that there is a real correspondence between the inscription and the fact or instance, a Mr. Francis Cherry of Shottesbrooke, who died September 23rd, 1713, had his grave inscribed with no other words than Hic JACET PECCATORUM MAXIMUS (Here lies the Chief of Sinners), the truth being, if we are to believe his friend Hearne, that he was an upright and amiable man, of the most unexceptionable religious practice-in Hearne's own words, 'one of the most learned, modest, humble, and virtuous persons that I ever had the honour to be acquainted with.'
The writer can speak on good authority of a similar epitaph which a dying person of unhappy memory desired to be put upon his coffin. The person referred to was an Irish ecclesiastic who many years ago was obliged, in consequence of a dismal lapse, to become as one lost to the world. Fully twenty-five years after his wretched fall, an old and broken down man, living in an obscure lodging at Newington, a suburb of Edinburgh, sent for one of the Scottish Episcopalclergy, for the benefit of his ministrations as to a dying person. Mr. F- saw much in this aged man to interest him; he seemed borne down with sorrow and penitence. It was tolerably evident that he shunned society, and lived under a feigned name and character. Mr. F--- became convinced that he had been a criminal, but was not able to penetrate the mystery. The miserable man at length had to give some directions about his funeral-an evidently approaching event; and he desired that the only inscription on his coffin should be 'A CONTRITE SINNER.' He was in due time deposited without any further memorial in Warriston Cemetery, near Edinburgh.


text:
David Hickey (December 5, 1938 – November 12, 2021) was an American art critic who wrote for many American publications including Rolling Stone, ARTnews, Art in America, Artforum, Harper's Magazine, and Vanity Fair. He was nicknamed "The Bad Boy of Art Criticism" and "The Enfant Terrible of Art Criticism". He had been professor of English at the University of Nevada Las Vegas and distinguished professor of criticism for the MFA program in the Department of Art & Art History at the University of New Mexico.

Biography
Hickey graduated from Texas Christian University in 1961 and received his MA from the University of Texas two years later. In 1989, SMU Press published Prior Convictions, a volume of his short fiction. He was owner-director of A Clean Well-Lighted Place, an art gallery in Austin, Texas, and director of the Reese Palley Gallery in New York. He served as executive editor for Art in America magazine, as contributing editor to The Village Voice, as staff songwriter for Glaser Publications in Nashville, and as the arts editor for the Fort Worth Star-Telegram.

He wrote for most major cultural publications in the United States and abroad, including Rolling Stone, ARTnews, Art in America, Artforum, Interview, Harper's Magazine, Vanity Fair, Nest, The New York Times, and the Los Angeles Times. Hickey regularly published "Revisions", a monthly column for Art in America. He also wrote for European publications like The London Review of Books, frieze (London), Situation (Paris), and Parkett (Zurich).

He was known for his arguments against academicism and in favor of the effects of rough-and-tumble free markets on art. His critical essays have been published in two volumes: The Invisible Dragon: Four Essays on Beauty (1993) and Air Guitar: Essays on Art and Democracy (1997). In 2009, Hickey published a revised and updated version of The Invisible Dragon, adding an introduction that addressed changes in the art world since the book's original publication, as well as a new concluding essay.

"I write love songs for people who live in a democracy", he remarked.

Profiles of Hickey have appeared in Time magazine, The New York Times, the Los Angeles Times, Los Angeles Times Magazine, The Wall Street Journal, Newsweek, U.S. News & World Report, The Economist, and Flaunt, among other publications. Interviews with Hickey have been published in the Los Angeles Times, Bomb, the New Art Examiner, Public Events, A Gathering of the Tribes, The Art Newspaper, and other magazines. He has been interviewed several times on topics such as art and Las Vegas by the BBC, PBS, and NPR.

In 2014, Hickey began making posts on Facebook during an illness. Eighteen months later, art historian Julia Friedman suggested a project documenting his experience. Two books resulted from the collaboration: Wasted Words and Dust Bunnies, published in 2016. Both books appeared in a lengthy review published by The Times Literary Supplement.

In 2015, he wrote the essay "War Is Beautiful, They Say" for the book War is Beautiful: The New York Times Pictorial Guide to the Glamour of Armed Conflict by David Shields. This essay described the painterly influences and inspirations behind several war photographs published by The New York Times.

Hickey was married to art historian Libby Lumpkin. He died in Santa Fe, New Mexico, on November 12, 2021, at the age of 82.

Awards and accolades
In 1994, Hickey received the Frank Jewett Mather Award for art criticism from the College Art Association. In 2001, he became a recipient of a MacArthur Fellowship, the so-called "genius grant". In 2003, Hickey was inducted into the Nevada Writers Hall of Fame, sponsored by the Friends of the University of Nevada, Reno, Libraries. He won a Peabody Award for his 2006 documentary about Andy Warhol for the American Masters series, which aired on PBS in 2006.

Books
In 1997 Art Issues Press published Air Guitar: Essays on Art and Democracy, a memoir containing 23 essays or "love songs" addressing his experiences as a music critic and an art dealer. The Invisible Dragon was originally published in 1993 with a new revised and expanded edition published in 2012. It is a series of provocative essays that encourage readers to reconsider the role of beauty in art. Pirates and Farmers was published by Ridinghouse in 2013, and featured essays by Hickey from 1999 to 2013. This body of short essays looks at contemporary phenomena including super-collectors, the trope of the biennale and the loss of looking. 25 Women: Essays on Their Art was published by the University of Chicago Press in 2016, and featured essays by Hickey from the past twenty years. This body of short essays analyzes the work of Joan Mitchell, Bridget Riley, Fiona Rae, Lynda Benglis, Karen Carson, and many others. Wasted Words: The Essential Dave Hickey Online Compilation was published by PCP Press in 2016. This book features almost 3,000 digital comments on social media, prompting nearly 700,000 words in response from art lovers, acolytes, and skeptics between June 2014 and April 2015. These writings highlight the impact of digital technology on the author and his online-audience. Dust Bunnies: Dave Hickey's Online Aphorisms June 2014 - March 2015 was also published by PCP Press in 2016. This second volume was edited by art historian Julia Friedman. This book is a 124-page collection of fragments selected from Wasted Words, the vast digital discourse from Dave Hickey's social media pages during June 2014 and March 2015. These writings highlight verbal riffs of Hickey blasting away at digital natives and his online-audience. In 2017 the University of Chicago Press published the collection Perfect Wave: More Essays on Art and Democracy.

References

Further reading

External links
"Art Critic Dave Hickey On Facebook's 'Wasted Words' And The Lack Of Good Art" (8:32 min radio interview), KNPR Nevada Public Radio with Joe Schoenmann, March 3, 2016.
"Art Critic Dave Hickey speaking with Art Historian Julia Friedman" at SITE Santa Fe, April 15, 2016 -- YouTube Trailer
"Art Critic Dave Hickey speaking with Art Historian Julia Friedman" at UCLA Hammer Museum, May 11, 2016
"Dave Hickey presentation at the School of Visual Arts MFA Art Criticism and Writing Program: The God Ennui". September 17, 2009.
Article on Dave Hickey in Newsweek, March 14, 2009
Interview with Dave Hickey on The New York Times the Moment blog(dead link - archived version)
Interview with Dave Hickey in The Believer, November 2007, by Sheila Heti
Interview with Dave Hickey and Ilana Stanger on NYFA
"A Gathering of the Tribes", interview with Dave Hickey by Lee Klein
Essay on Dave Hickey featuring in Artforum, Summer 1995
Profile on the University of Nevada Las Vegas
Nevada Writers Hall of Fame
Dave Hickey's Facebook page

1938 births
2021 deaths
American art critics
University of Nevada, Las Vegas faculty
MacArthur Fellows
Frank Jewett Mather Award winners
Texas Christian University alumni
University of Texas alumni
University of New Mexico faculty
People from Fort Worth, Texas
Journalists from Texas
American male journalists
20th-century American writers
21st-century American non-fiction writers
20th-century American male writers
21st-century American male writers

redpajama_set_name: RedPajamaWikipedia


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\begin{document}

\title{\large\bf\sc Necessary and sufficient conditions for a subclass of $P$-tensor}

\author{
R. Deb$^{a,1}$ and A. K. Das$^{b,2}$\\
\emph{\small $^{a}$Jadavpur University, Kolkata , 700 032, India.}\\
\emph{\small $^{b}$Indian Statistical Institute, 203 B. T.
	Road, Kolkata, 700 108, India.}\\
\emph{\small $^{1}$Email: rony.knc.ju@gmail.com}\\
\emph{\small $^{2}$Email: akdas@isical.ac.in}\\
}

\date{}

\maketitle

\begin{abstract}
\noindent In this article, we introduce the class $B$-Nekrasov tensor in the context of tensor complementarity problem. We study some tensor theoretic properties. We show that the class of B-Nekrasov tensor contains the class of Nekrasov $Z$-tensor with positive diagonal entries. We present a necessary and sufficient condition for a $B$-Nekrasov tensor. We show that the class of $P$-tensor contains the class of $B$-Nekrasov tensor.\\

\noindent{\bf Keywords:} Tensor complementarity problem, $P$-tensors, Nekrasov tensors, Nekrasov $Z$-tensor, $B$-Nekrasov tensors, Nonsingular $H$-tensor, Diagonally dominant tensor.
\\

\noindent{\bf AMS subject classifications:}
\end{abstract}
\footnotetext[1]{Corresponding author}

\section{Introduction}

\noindent Given a positive integer $n,$ the set $\{1,2,...,n\}$ is denoted by $[n].$ A tensor of order $m$ and dimension $n$, $\mathcal{A}= (a_{i_1 i_2... i_m}) $ is a multidimensional array of entries $a_{i_1 i_2... i_m} \in \mathbb{C}$ where $i_l \in [n]$ with $l\in [m].$ The set of complex (real) tensors of order $m$ and dimension $n$ is denoted by $\mathbb{C}^{[m,n]} (\mathbb{R}^{[m,n]}).$ If under any permutation of the indices $i_1, i_2, ... ,i_m$ the entries $a_{i_1 i_2... i_m}$ are invariant then the tensor $\mathcal{A}= (a_{i_1 i_2... i_m}) \in \mathbb{C}^{[m,n]}$ is called a symmetric tensor. Tensors have applications in different areas of science and engineering. The common applications are in physics, quantum computing, spectral hypergraph theory, diffusion tensor imaging, image authenticity verification problem, optimization theory and many other areas. In optimization theory, the tensor complementarity problem (TCP) introduced by Song and Qi \cite{song2017properties}, is a subclass of the nonlinear complementarity problem, where the involved functions are special polynomials defined by a tensor.

\noindent Given a real tensor $\mathcal{A}$ of order $m$ and dimension $n$ and an real vector $q\in\mathbb{R}^n$ the tensor complementarity problem is to look for $v\in \mathbb{R}^n $ such that  satisfying
	\begin{equation}\label{tensor comp equation}
	v\geq 0,\;\;  \mathcal{A}v^{m-1}+q \geq 0,\;\;  v^T (\mathcal{A}v^{m-1}+q)  = 0.
	\end{equation}
This problem is denoted by TCP$(\mathcal{A}, q)$ and the solution set of TCP$(\mathcal{A}, q)$ is denoted by SOL$(\mathcal{A}, q).$
The tensor complementarity problem arises in optimization theory, game theory and in other areas. A class of multi-person noncooperative game \cite{huang2017formulating}, hypergraph clustering problem and traffic equilibrium problem \cite{huang2019tensor} can be formulated as the tensor complementarity problem.

\noindent The tensor complementarity problem can be viewed as one type of extension of the linear complementarity problem where the homogeneous polynomials involved in the function are constructed with the help of tensor.
\noindent Given a real square matrix $A$ of order $n$ and an vector $q\in \mathbb{R}^n$, the linear complementarity problem \cite{cottle2009linear} is to find $z$ satisfying
\begin{equation}\label{linear comp equation}
 z\geq 0,\;\;  Az + q \geq 0,\;\; z^T (Az + q) = 0.
\end{equation}
The problem is denoted by LCP$(q,A)$ and the solution set of LCP$(q,A)$ is denoted by SOL$(q,A).$ It is important that large number of formulations not only enrich the linear complementarity problem but also generate different matrix classes along with their computational methods.
 For details see \cite{das2019some}, \cite{neogy2006some}, \cite{dutta2021some} \cite{dutta2022on}, \cite{neogy2013weak}, \cite{jana2018semimonotone}, \cite{neogy2005almost}, \cite{neogy2011singular}, \cite{jana2019hidden}, \cite{jana2021more}, \cite{neogy2009modeling}, \cite{das2017finiteness}, \cite{das2018invex}. For details of game theory see \cite{mondal2016discounted}, \cite{neogy2008mathematical}, \cite{neogy2008mixture}, \cite{neogy2005linear}, \cite{neogy2016optimization}, \cite{das2016generalized}, \cite{neogy2011generalized} and for details of QMOP see \cite{mohan2004note}. Even matrix classes arise during the study of Lemke's algorithm as well as principal pivot transform. For details see \cite{mohan2001classes}, \cite{mohan2001more} \cite{neogy2005principal}, \cite{das2016properties}, \cite{neogy2012generalized}, \cite{jana2019hidden}, \cite{jana2021iterative}, \cite{jana2021more}, \cite{jana2018processability}.

\noindent Orea and Pe$\tilde{n}$a \cite{orera2019accurate} introduced Nekrasov $Z$-matrix and studied the methods to compute its inverse with high relative accuracy. Esnaola and Pe$\tilde{n}$a \cite{garcia2016b} introduced $B$-Nekrasov matrix and studied the error bounds for the linear complementarity problem involving a $B$-Nekrasov matrix.

\noindent In tensor complementarity theory, some special classes of matrices are extended to higher order structured tensors. In recent years many structured tensors are developed and studied well in context of TCP. The class of $P(P_0)$-tensor, $Z$-tensor, $M$-tensor, $H$-tensor, Nekrasov tensor and many other tensor classes are introduced and studied in the context of tensor complementarity problem. Zhang et al. \cite{zhang2014m} introduce $Z$-tensor as well as $M$-tensor in context of tensor complementarity problem and show that for a $Z$-tensor the tensor complementarity problem has least element property. $H$-tensor is introduced by Ding et al. \cite{ding2013m} and studied thoroughly in literature for its importance in tensor complementarity problem.
The Nekrasov tensors is introduced by Zhang and Bu \cite{zhang2018nekrasov}. They have proved that a Nekrasov tensor is a $H$-tensor. The positive definitene tensors have many important applications in multivariate network realizability analysis, automatic control, medical imaging and so on \cite{basser2002diffusion}, \cite{ni2008eigenvalue}, \cite{ qi2010higher}. Song and Qi \cite{song2014properties} introduce $P$-tensor. Severl properties of $P$-tensors are studied in context of tensor complementarity problem. For details see \cite{bai2016global}, \cite{ding2018p}, \cite{yuan2014some}. Qi and Luo \cite{qi2017tensor} has studied Several important results related to positive definite tensors. The positive definiteness of a homogeneous polynomial is equivalent to the positive definiteness of the symmetric tensor associated with it \cite{kannan2015some}. Since an even order real symmetric nonsingular $H$-tensor with positive diagonal entries is positive definite \cite{li2014criterions}, nonsingular H-tensors play an important role in checking the positive definiteness of even order real symmetric tensors.

\noindent In this paper, we introduce $B$-Nekrasov tensor. We discuss necessary and sufficient condition for $B$-Nekrasov tensor. We prove that a $B$-Nekrasov tensor is a $P$-tensor.

The paper is organised as follows. Section 2 contains some basic notations and results. In Section 3, we introduce $B$-Nekrasov tensor. We investigate the connection between Nekrasov $Z$-tensor and $B$-Nekrasov tensor. We propose a necessary and sufficient condition for $B$-Nekrasov tensor. We establish its connection with $P$-tensor.

\section{Preliminaries}

Here are some basic notations used in this paper. For any positive integer $n,$ let $\mathbb{R}^n$ denote the $n$-dimensional Euclidean space. Any vector $v\in \mathbb{C}^n$ is a column vector and $v^T$ denotes the row transpose of $v.$ Here $\mathbb{R}^n_+ =\{ v\in \mathbb{R}^n : v\geq 0 \}$, $\mathbb{R}^n_{+ +} =\{ v\in \mathbb{R}^n : v> 0 \}$. A diagonal matrix $D=[d_{ij}]_{n \times n}=diag(d_1, \; d_2,\; ..., \; d_n)$ is defined as $d_{ij}=\left \{ \begin{array}{ll}
	  d_i  &;\; \forall \; i=j, \\
	  0  &; \; \forall \; i \neq j.
	   \end{array}  \right.$

\begin{defn}\cite{cvetkovic2009new,bailey1969bounds,kolotilina2015some, garcia2014error}
A matrix $A=(a_{ij})\in \mathbb{C}^{[2,n]},$ is said to be a Nekrasov matrix if $|a_{ii}|> \Lambda_i(A), \; \forall\; i\in [n],$ where
\begin{center}
    $\Lambda_i(A) =\left\{\begin{array}{ll}
        \sum_{j=2}^n |a_{ij}| &, i=1,  \\
        \sum_{j=1}^{i-1} |a_{ij}| \frac{\Lambda_j(A)}{|a_{jj}|} + \sum_{j=i+1}^n |a_{ij}| &, i=2,3,...,n.
    \end{array} \right.$
\end{center}
\end{defn}

\begin{defn}\cite{mangasarian1976linear}
A matrix $A\in \mathbb{R}^{[2,n]},$ is said to be a $Z$-matrix if all its offdiagonal entries are nonpositive.
\end{defn}

\begin{defn}\cite{garcia2016b}
Given a real matrix $A(a_{ij})\in \mathbb{R}^{[2,n]},$ we can write $A$ as $A= B^+ + C,$ where
\begin{center}
    $B^+ = \left(\begin{array}{ccc}
        a_{11}-r_1^+ & ... & a_{1n}-r_1^+ \\
        \vdots & \vdots & \vdots \\
        a_{n1}-r_n^+ & ... & a_{nn}-r_n^+
    \end{array} \right)$ and
    $C = \left(\begin{array}{ccc}
        r_1^+ & ... & r_1^+ \\
        \vdots & \vdots & \vdots \\
        r_n^+ & ... & r_n^+
    \end{array} \right)$
\end{center}
with $r_i^+ = \max \{0, a_{i,j} | j\neq i\}.$ The matrix $A$ is said to be a $B$-Nekrasov matrix if $B^+$ is a Nekrasov $Z$-matrix whose diagonal entries are all positive.
\end{defn}

\noindent An identity tensor of order $m,$ $\mathcal{I}=(\delta_{i_1 i_2... i_m})\in \mathbb{C}^{[m,n]}$ is defined as follows:
$\delta_{i_1 i_2... i_m}= \left\{
\begin{array}{ll}
	  1  &:\; i_1= ...= i_m \\
	  0  &:\; else
	   \end{array}
 \right. .$
Let the zero tensor be denoted by $\mathcal{O},$ where each entry of $\mathcal{O}$ is zero. For $\mathcal{A}\in \mathbb{C}^{[m,n]}$ and $v\in \mathbb{C}^n,\; \mathcal{A}v^{m-1}\in \mathbb{C}^n $ is a vector defined by
\[ (\mathcal{A}v^{m-1})_i = \sum_{i_2, ..., i_m =1}^{n} a_{i i_2 ... i_r} v_{i_2}  \cdots v_{i_m} , \mbox{   for all } i \in [n] \]
and $\mathcal{A}v^m\in \mathbb{C} $ is a scalar defined by
\[ v^T\mathcal{A}v^{m-1}= \mathcal{A}v^m = \sum_{i_1, i_2, ..., i_m =1}^{n} a_{i_1 i_2 ...i_m} v_{i_1} v_{i_2} \cdots v_{i_m}. \]

\noindent The general product of tensors was introduced by Shao \cite{shao2013general}. Let $\mathcal{A}$ and $\mathcal{B}$ be two $n$ dimensional tensor of order $p \geq 2$ and $r \geq 1,$ respectively. The product $\mathcal{A} \mathcal{B}$ is an $n$ dimensional tensor $\mathcal{C}$ of order $((p-1)(r-1)) + 1$ with entries
\[c_{j \beta_1 \cdots \beta_{p-1} } =\sum_{j_2, \cdots ,j_p \in [n]} a_{j j_2 \cdots j_p} b_{j_2 \beta_1} \cdots b_{j_p \beta_{p-1}},\] where $j \in [n]$, $\beta_1, \cdots, \beta_{p-1} \in [n]^{r-1}$

\begin{defn}\cite{song2016properties}
Given $\mathcal{A} \in \mathbb{R}^{[m,n]} $ and $q\in \mathbb{R}^n$, a vector $v$ is said to be (strictly) feasible solution of TCP$(\mathcal{A},q),$ if $v \geq 0\; (>0)$ and $\mathcal{A}v^{m-1}+q \geq 0\; (>0)$.
\end{defn}

\begin{defn}\cite{song2016properties}
Given $\mathcal{A} \in \mathbb{R}^{[m,n]} $ and $q\in \mathbb{R}^n$, TCP$(\mathcal{A},q)$ is said to be (strictly) feasible if a (strictly) feasible vector exists.
\end{defn}

\begin{defn}\cite{song2016properties}
Given $\mathcal{A} \in \mathbb{R}^{[m,n]} $ and $q\in \mathbb{R}^n$, TCP$(\mathcal{A},q)$ is said to be solvable if there exists a feasible vector $v$ satisfying $v^{T}(\mathcal{A}v^{r-1}+q)=0$ and $v$ is said to be a solution of the TCP$(\mathcal{A},q)$.
\end{defn}

\begin{defn}
\cite{song2015properties} A tensor $\mathcal{A}\in \mathbb{R}^{[m,n]} $ is said to be a $P_0(P)$-tensor, if for each $v\in \mathbb{R}^n \backslash \{0\}$, there exists an index $j\in [n]$ such that $v_j \neq 0$ and $v_j (\mathcal{A}v^{m-1})_j \geq 0\; (>0)$.
\end{defn}

\noindent The row subtensors are defined in \cite{shao2016some}. Here for the sake of convenience we denote $i$-th rowsubtensor of $\mathcal{A}$ by $\mathcal{R}_i(\mathcal{A}).$
\begin{defn}\cite{shao2016some}
For each $i$ the $i$th row subtensor of $\mathcal{A}\in \mathbb{C}^{[m,n]}$ is denoted by $\mathcal{R}_i(\mathcal{A})$ and its entries are given as $(\mathcal{R}_i(\mathcal{A}))_{i_2 ... i_m}=(a_{i i_2... i_m})$, where $i_l\in [n]$ and $2\leq l \leq n.$
\end{defn}

\begin{defn}\cite{zhang2018nekrasov}
Let $\mathcal{A}\in \mathbb{C}^{[m,n]}$ such that $\mathcal{A}=(a_{i_1 i_2 ... i_m}).$ Let for all $i\in [n],$
\begin{equation*}
    R_i(\mathcal{A})= \sum_{(i_2 ... i_m) \neq (i ... i)} |a_{i i_2 ... i_m}|.
\end{equation*}
The tensor $\mathcal{A}$ is said to be diagonally dominant (strict diagonally dominant) if $|a_{i, ..., i}| \geq (>) R_i(\mathcal{A})$ for all $i\in [n].$
\end{defn}

\begin{defn}\cite{zhang2018nekrasov}
Let $\mathcal{A}\in \mathbb{C}^{[m,n]}$ is called quasi-diagonally dominant tensor if there exists a diagonal matrix $D=diag(d_1, d_2, \cdots, d_n)$ such that $\mathcal{A} D$ is strictly diagonally dominant tensor, i.e.,
\begin{equation*}
  |a_{i i \cdots i}|d_i^{m-1} > |a_{i i_2 \cdots i_m}|d_{i_2} \cdots d_{i_m}, \;\;\; \forall\; i\in [n].
\end{equation*}
\end{defn}

\noindent The concept of eigenvalues and eigenvectors for tensors was proposed by Qi \cite{qi2005eigenvalues} and Lim \cite{lim2005singular}. If a pair $(\lambda, v) \in \mathbb{C}\times (\mathbb{C}^n \backslash \ \{0\})$ satisfies the equation $\mathcal{A} v^{m-1} = \lambda v^{m-1},$ then $\lambda$ is said to be an eigenvalue of $\mathcal{A}$, and $v$ is said to be an eigenvector corresponding to $\lambda$ of $\mathcal{A}$, where $v^{[m−1]} = (v_1^{m−1},... , v_n^{m−1})^T.$ The quantity $\rho(\mathcal{A}) = \max\{|\lambda| \; : \; \lambda \text{ is an eigen value of } \mathcal{A}\}$ is said to be the spectral radius of $\mathcal{A}$.

\begin{defn}\cite{ding2013m}
A tensor is said to be $Z$ tensor if all its offdiagonal entries are nonpositive.
\end{defn}
\begin{defn}\cite{ding2013m}
A $Z$-tensor $\mathcal{A}$ is said to be an $M$-tensor if $\exists \; \mathcal{B}\; \geq \mathcal{O}$ and $s>0$ such that $\mathcal{A}= s \mathcal{I} - \mathcal{B}$, where $s\geq \rho(\mathcal{B})$. $\mathcal{A}$ is said to be a nonsingular $M$-tensor if $s>\rho(\mathcal{B}).$
\end{defn}

\begin{defn}\cite{ding2013m}
For a tensor $\mathcal{A}= a_{i_1 i_2 ... i_m}\in \mathbb{C}^{[m,n]},$ a tensor $\mathcal{C}(\mathcal{A})= c_{i_1 i_2 ... i_m}$ is said to be the comparison tensor of $\mathcal{A}$ if
\begin{center}
    $c_{i_1 i_2 ... i_m}=\left\{\begin{array}{ll}
       |a_{i_1 i_2 ... i_m}|  & , \text{ if } (i_1, i_2, ... ,i_m) =(i,i, ..., i), \\
       -|a_{i_1 i_2 ... i_m}|  & , \text{ if } (i_1, i_2, ... ,i_m) \neq (i,i, ..., i).
    \end{array} \right.$
\end{center}
\end{defn}

\begin{defn}\cite{ding2013m}
A tensor $\mathcal{A}$ is said to be an $H$-tensor, if its comparison tensor is an $M$-tensor, and it is said to be a nonsingular $H$-tensor, if its comparison tensor is a nonsingular $M$-tensor.
\end{defn}

\begin{defn}
A tensor $\mathcal{A}\in \mathbb{R}^{[m,n]}$ is said to be a Nekrasov $Z$-tensor if $\mathcal{A}$ is a Nekrasov tensor as well as a $Z$-tensor.
\end{defn}

\begin{defn}\cite{zhang2018nekrasov}
For $\mathcal{A}=(a_{i_1 i_2 ... i_m}) \in \mathbb{C}^{[m,n]}, \; a_{i i ... i}\neq 0, \; \forall \; i\in [n].$ Let
\begin{align*}
    \Lambda_1(\mathcal{A}) & =R_1(\mathcal{A}),\\
    \Lambda_i(\mathcal{A}) & =\sum_{i_2...i_m \in [i-1]^{m-1}} |a_{i i_2 ... i_m}| \left(\frac{\Lambda_{i_2} (\mathcal{A})}{|a_{i_2 i_2 ... i_2}|}\right)^{\frac{1}{m-1}} \cdots \left(\frac{\Lambda_{i_m} (\mathcal{A})}{|a_{i_m i_m ... i_m}|}\right)^{\frac{1}{m-1}}\\
    & \qquad + \sum_{i_2...i_m \notin [i-1]^{m-1}, (i_2 ... i_m)\neq (i ... i)} |a_{i i_2 ... i_m}|, \; i=2,3,...,n.
\end{align*}
 A tensor $\mathcal{A}=(a_{i_1 i_2 ... i_m})\in \mathbb{R}^{[m,n]}$ is said to be a Nekrasov tensor if
 \begin{equation}
     |a_{ii...i}| > \Lambda_i(\mathcal{A}), \; \forall\; i\in [n].
 \end{equation}
\end{defn}

\begin{theorem}\cite{ding2013m}\label{nonsingular H if and only if quasi-SDD}
A tensor $\mathcal{A}$ is a nonsingular $H$-tensor if and only if it is quasi-strictly diagonally dominant tensor.
\end{theorem}

\begin{theorem}\cite{zhang2018nekrasov}\label{nekrasov implies nonsingular H}
If a tensor $\mathcal{A}=(a_{i_1 i_2 ... i_m})\in \mathbb{C}^{[m,n]}$ is a Nekrasov tensor, then $\mathcal{A}$ is a nonsingular $H$-tensor.
\end{theorem}

\begin{theorem}\cite{ding2018p}\label{nonsingular H implies P}
A nonsingular $H$-tensor of even order with all positive diagonal entries is a $P$-tensor.
\end{theorem}

\begin{theorem}\cite{bai2016global}
For any $q\in \mathbb{R}^n$ and a $P$-tensor $\mathcal{A} \in \mathbb{R}^{[m,n]}$, the solution set of TCP$(\mathcal{A},q)$ is nonempty and compact.
\end{theorem}

\section{Main results}

We begin by introducing $B$-Nekrasov tensor. We want to define a class of tensors which will contain Nekrasov $Z$-tensors with positive diagonal entries. To do so we use a decomposition of tensors which will be useful in our discussion.\\

\noindent Given a tensor $\mathcal{A}=(a_{i_1 i_2 ... i_m})\in \mathbb{R}^{[m,n]},$ we can write $\mathcal{A}$ as
\begin{equation}\label{decomposition equation}
    \mathcal{A} = \mathcal{B}^+ + \mathcal{C},
\end{equation}
where $\forall \; i\in [n]$ the elements of rowsubtensors of $\mathcal{B}$ and $\mathcal{C}$ are given by,
\begin{equation}\label{law for B+}
(R_i(\mathcal{B}^+))_{i_2 \cdots i_m} = a_{i i_2 ... i_m} - r_i^+, \;\forall \; i_l\in [n], \; l\in [m]
\end{equation}
and
\begin{equation}\label{law for C}
(R_i(\mathcal{C}))_{i_2 \cdots i_m} =r_i^+ , \;\forall \; i_l\in [n], \; l\in [m]..
\end{equation}
with
\begin{equation}\label{law for r_i^+}
    r_i^+ =\max_{i_2, ..., i_m \in [n], \; ( i_2 \cdots i_m) \neq (i \cdots i)} \{ 0, a_{i i_2 ... i_m} \}
\end{equation}

\begin{prop}
Let $\mathcal{A}=(a_{i_1 i_2 ... i_m})\in \mathbb{R}^{[m,n]}.$ If $\mathcal{A}$ is decomposed as given in the equation (\ref{decomposition equation}), then $\mathcal{B}^+$ is a $Z$-tensor and $\mathcal{C}$ is nonnegative tensor.
\end{prop}

Here we define $B$-Nekrasov tensor.
\begin{defn}
A tensor $\mathcal{A}\in \mathbb{R}^{[m,n]}$ is said to be a $B$-Nekrasov tensor if it can be written in the form of (\ref{decomposition equation}) with $\mathcal{B}^+$ as a Nekrasov $Z$-tensor whose diagonal entries are all positive.
\end{defn}

Here is an example of a $B$-Nekrasov tensor which is not a Nekrasov tensor

\begin{examp}
Let $\mathcal{A}=(a_{i j k l})\in \mathbb{R}^{[4,4]}$ be such that $a_{1111}=10,\; a_{2222}=6.8,\; a_{3333}=3,\; a_{4444}=10,\; a_{1112}=a_{1211}=a_{1121}=1,\; a_{1444}=-1$ and $a_{1jkl}=2$ for all other entries of $j,\; k,\; l\in [4],$
$a_{2111}=2,\; a_{2322}=a_{2232}=a_{2223}=2,\;$ and $a_{2jkl}=3$ for all other entries of $j,\; k,\; l\in [4],$
$a_{3222}=-1$ and $a_{3jkl}=0$ for all other entries of $j,\; k,\; l\in [4]$ $a_{4441}=a_{4414}=a_{4144}=-3,\;$ and $a_{4jkl}=0$ for all other entries of $j,\; k,\; l\in [4].$

\noindent Then
\begin{equation}
    r_1^+ =\max \{0,1,-1,2\}=2,
\end{equation}
\begin{equation}
    r_2^+ =\max \{0,2,3\}=3,
\end{equation}
\begin{equation}
    r_3^+ =\max \{0,-1\}=0,
\end{equation}
\begin{equation}
    r_4+ =\max \{0,-3\}=0,
\end{equation}
Then the tensor can be decomposed into two tensor as $\mathcal{A}=\mathcal{B}^+ + \mathcal{C}$, where $(\mathcal{R}_1(\mathcal{C}))_{jkl}= r_1^+ =2$ forall $j,\; k,\; l\in [4],$ $(\mathcal{R}_2(\mathcal{C}))_{jkl}= r_2^+ =3$ forall $j,\; k,\; l\in [4]$ and all other entries of $\mathcal{C}$ are zeros. The tensor $\mathcal{B}^+$ is same as the tensor $\mathcal{B}$ of Example \ref{example of nekrasov Z tensor}, which is proved to be a Nekrasov $Z$-tensor. Also all the diagonal entries of $\mathcal{B}^+$ are $8,\; 3.8,\; 3,\; 10,$ which are positive. Therefore $\mathcal{B}^+$ is a Nekrasov $Z$-tensor with positive diagonal entries. Hence $\mathcal{A}$ is a $B$-Nekrasov tensor.

Now for $i=1$ we have $a_{1111}=10$ and $\Lambda_1(\mathcal{A})=\mathcal{R}_1(\mathcal{A}).$ Now
\begin{equation*}
    R_i(\mathcal{A})= \sum_{(i_2 ... i_m) \neq (i ... i)} |a_{i i_2 ... i_m}| = 122.
\end{equation*}
Clearly $10=a_{1111}< 122 = \Lambda_1(\mathcal{A}).$ Therefore $(\mathcal{A})$ is not a Nekrasov tensor.

\end{examp}

\begin{prop}
A Nekrasov $Z$-tensor with positive diagonal entries is a $B$-Nekrasov tensor.
\end{prop}
\begin{proof}
The result immediately follows from the fact that for a Nekrasov $Z$-tensor with positive diagonal entries $\mathcal{A},$ the tensor can be easily decomposed into $\mathcal{A}=\mathcal{B}^+ + \mathcal{C},$ where $\mathcal{B}^+ = \mathcal{A}$ and $\mathcal{C} =\mathcal{O}.$
\end{proof}

Here we present a necessary and sufficient condition for $B$-Nekrasov tensor.

\begin{theorem}
Let $\mathcal{A}\in \mathbb{R}^{[m,n]}.$ Then $\mathcal{A}$ is a $B$-Nekrasov tensor if and only if the following condition hold:\\

(a) $\sum_{i_2...i_m\in [n]^{m-1}} a_{1i_2...i_m} >0 $ and $\frac{1}{n^{m-1}}\left( \sum_{i_2...i_m\in [n]^{m-1}} a_{1i_2...i_m} \right) > a_{1 j_2 \cdots j_m},$ for $j_2 \cdots j_m \in [n]^{m-1}$ and $(j_2, \cdots ,j_m)\neq (1,... ,1).$

(b) $r_i^+ <\min \left \{ \frac{\sum_{i_2...i_m\in [n]} a_{ii_2...i_m} \alpha_{i_2}\cdots \alpha_{i_m} + \sum_{i_2, ..., i_m \notin [i-1]^{m-1}} a_{i i_2 \cdots i_m}}{\sum_{i_2, ..., i_m \in [i-1]^{m-1}} \alpha_{i_2} \cdots \alpha_{i_m} + n^{m-1} -(i-1)^{m-1}},  a_{i...i} \right\}$, where $i\in [n]-\{1\},$
where $\alpha_{i_l}=\left(\frac{\Lambda_{i_l} (\mathcal{B}^+)}{|b^+_{i_l i_l ... i_l}|}\right)^{\frac{1}{m-1}},\; \forall\; 1\leq l \leq m$

\end{theorem}
\begin{proof}
Let conditions (a) and (b) hold for a tensor $\mathcal{A}.$ Let $\mathcal{A} = \mathcal{B}^+ + \mathcal{C}$ where $\mathcal{B}^+ $ and $\mathcal{C}$ are given by (\ref{decomposition equation}). We prove that the tensor $\mathcal{B^+}$ is a Nekrasov tensor.  Now
\begin{align}
   (0\leq)\; \Lambda_1(\mathcal{B}^+) &= \sum_{(i_2,...,i_m) \neq (i, ..., i)} |b_{1 i_2...i_m}^+|\notag \\
    &= \sum_{(i_2,...,i_m) \neq (i, ..., i)} |a_{1 i_2...i_m} - r_1^+| \notag \\
    &= \sum_{(i_2,...,i_m) \neq (i, ..., i)} (r_1^+ - a_{1 i_2...i_m}) \notag \\
    &= (n^{m-1} -1) r_1^+ - \sum_{(i_2,...,i_m) \neq (i, ..., i)} a_{1 i_2...i_m} \notag \\
    &= \left( n^{m-1} r_1^+ - \sum_{i_2,...,i_m \in [n]} a_{1 i_2...i_m} \right) + (a_{1 1 \cdots 1} - r_1^+).\label{comparison for lambda1}
\end{align}
Now from (a) $\sum_{i_2...i_m\in [n]^{m-1}} a_{1 i_2...i_m} >0 $ and $\frac{1}{n^{m-1}}\left( \sum_{i_2...i_m\in [n]^{m-1}} a_{1i_2...i_m} \right) > a_{1 j_2 \cdots j_m},$ for $j_2, \cdots ,j_m \in [n].$ Which implies
\begin{equation}\label{comparison with max and sum}
   r_1^+ = \max\{ 0, a_{1 i_2...i_m}\} < \frac{1}{n^{m-1}}\left( \sum_{i_2...i_m\in [n]^{m-1}} a_{1i_2...i_m} \right)
\end{equation}
Therefore,
\begin{equation}\label{first equation for necessary part}
    \left( n^{m-1}  r_1^+ - \sum_{i_2,...,i_m \in [n]} a_{1 i_2...i_m} \right) < 0.
\end{equation}
Now from (\ref{comparison for lambda1}) and (\ref{first equation for necessary part}) we obtain,
\begin{equation}
    \Lambda_1(\mathcal{B}^+) < a_{1 1 \cdots 1} - r_1^+ = |a_{1 1...1} - r_1^+| = |b_{1 \cdots 1}^+|.
\end{equation}
Thus for $i=1,$ the tensor $\mathcal{B}^+$ satisfies the condition of Nekrasov tensor.

\noindent Now for $i= 2,3,..., n$, we have
\begin{align}
    0\leq \Lambda_i(\mathcal{B}^+) &=\sum_{i_2...i_m \in [i-1]^{m-1}} |b^+_{i i_2 ... i_m}| \left(\frac{\Lambda_{i_2} (\mathcal{B}^+)}{|b^+_{i_2 i_2 ... i_2}|}\right)^{\frac{1}{m-1}} \cdots \left(\frac{\Lambda_{i_m} (\mathcal{B}^+)}{|b^+_{i_m i_m ... i_m}|}\right)^{\frac{1}{m-1}} \notag \\
    & \qquad + \sum_{i_2...i_m \notin [i-1]^{m-1}, (i_2 ... i_m)\neq (i ... i)} |b^+_{i i_2 ... i_m}| \notag \\
    &= \sum_{i_2...i_m \in [i-1]^{m-1}} |a_{i i_2 \cdots i_m} - r_i^+| \left(\frac{\Lambda_{i_2} (\mathcal{B}^+)}{|b^+_{i_2 i_2 ... i_2}|}\right)^{\frac{1}{m-1}} \cdots \left(\frac{\Lambda_{i_m} (\mathcal{B}^+)}{|b^+_{i_m i_m ... i_m}|}\right)^{\frac{1}{m-1}} \notag \\
    & \qquad + \sum_{i_2...i_m \notin [i-1]^{m-1}, (i_2 ... i_m)\neq (i ... i)} |a_{i i_2 \cdots i_m} - r_i^+| \notag \\
    &= \sum_{i_2...i_m \in [i-1]^{m-1}} (r_i^+ - a_{i i_2 \cdots i_m}) \left(\frac{\Lambda_{i_2} (\mathcal{B}^+)}{|b^+_{i_2 i_2 ... i_2}|}\right)^{\frac{1}{m-1}} \cdots \left(\frac{\Lambda_{i_m} (\mathcal{B}^+)}{|b^+_{i_m i_m ... i_m}|}\right)^{\frac{1}{m-1}} \notag \\
    & \qquad + \sum_{i_2...i_m \notin [i-1]^{m-1}, (i_2 ... i_m)\neq (i ... i)} (r_i^+ - a_{i i_2 \cdots i_m}) \notag \\
    &= r_i^+ \sum_{i_2...i_m \in [i-1]^{m-1}} \alpha_{i_2} \cdots \alpha_{i_m} - \sum_{i_2...i_m \in [i-1]^{m-1}} a_{i i_2 \cdots i_m} \alpha_{i_2} \cdots \alpha_{i_m} \notag \\
    &\qquad + (n^{m-1} -(i-1)^{m-1} -1) r_i^+  - \sum_{i_2...i_m \notin [i-1]^{m-1}, (i_2 ... i_m)\neq (i ... i)} a_{i i_2 \cdots i_m} \notag \\
    &= \left[r_i^+ \left( \sum_{i_2...i_m \in [i-1]^{m-1}} \alpha_{i_2} \cdots \alpha_{i_m} + n^{m-1} -(i-1)^{m-1} \right)\right. \notag \\
    &\quad \left. - \sum_{i_2...i_m \in [i-1]^{m-1}} a_{i i_2 \cdots i_m} \alpha_{i_2} \cdots \alpha_{i_m} - \sum_{i_2...i_m \notin [i-1]^{m-1}} a_{i i_2 \cdots i_m} \right] \notag \\
    &\qquad \qquad - r_i^+ + a_{i \cdots i} \label{equation for necessary part of B Nekrasov}
\end{align}

\noindent Now by (b) we have  $r_i^+ <\min \left \{ \frac{\sum_{i_2...i_m\in [n]} a_{ii_2...i_m} \alpha_{i_2}\cdots \alpha_{i_m} + \sum_{i_2...i_m \notin [i-1]^{m-1}} a_{i i_2 \cdots i_m}}{\sum_{i_2...i_m \in [i-1]^{m-1}} \alpha_{i_2} \cdots \alpha_{i_m} + n^{m-1} -(i-1)^{m-1}},  a_{i...i} \right\}$, where $i=2,3,...,n.$ This implies
\begin{align}
    r_i^+ \left( \sum_{i_2...i_m \in [i-1]^{m-1}} \alpha_{i_2} \cdots \alpha_{i_m} + n^{m-1} -(i-1)^{m-1} \right) &< \sum_{i_2...i_m \in [i-1]^{m-1}} a_{i i_2 \cdots i_m} \alpha_{i_2} \cdots \alpha_{i_m} \notag \\
    &\qquad + \sum_{i_2...i_m \notin [i-1]^{m-1}} a_{i i_2 \cdots i_m} \label{second equation for necessary part}
\end{align}
Now by (\ref{equation for necessary part of B Nekrasov}) and (\ref{second equation for necessary part}) we conclude that
\begin{equation}
    \Lambda_i(\mathcal{B}^+) < a_{i \cdots i} - r_i^+ = |a_{i \cdots i} - r_i^+|= |b_{i \cdots i}^+|,
\end{equation}
Therefore $\mathcal{B}^+$ is a Nekrasov tensor. Note that by construction $\mathcal{B}^+$ is a $Z$-tensor and by the conditions (a) and (b), we have $b_{i...i}^+ =a_{i...i} -r_i^+>0\; \forall\; i\in [n].$ Therefore $\mathcal{B}^+$ is a Nekrasov $Z$-tensor with positive diagonal entries. Hence $\mathcal{A}$ is a $B$-Nekrasov tensor.

Conversely, let us assume that $\mathcal{A}$ is a $B$-Nekrasov tensor. We prove that (a) and (b) hold. Since $\mathcal{B}^+$ is a Nekrasov tensor with positive diagonal entries, we have
\begin{align*}
a_{1 1...1} - r_1^+ &= |a_{1 1...1} - r_1^+|\\
                    &> \sum_{(i_2,...,i_m) \neq (i, ..., i)} |a_{1 i_2...i_m} - r_1^+|\\
                    &= \sum_{(i_2,...,i_m) \neq (i, ..., i)} (r_1^+ - a_{1 i_2...i_m})\\
                    &= (n^{m-1} -1) r_1^+ - \sum_{(i_2,...,i_m) \neq (i, ..., i)} a_{1 i_2...i_m},
\end{align*}
and so,
\begin{equation}\label{comparison with max and sum}
   r_1^+ < \frac{1}{n^{m-1}}\left( \sum_{i_2...i_m\in [n]^{m-1}} a_{1i_2...i_m} \right).
\end{equation}
This implies $\sum_{i_2...i_m \in [n]^{m-1}} a_{1 i_2...i_m} >0$ and $\frac{1}{n^{m-1}}\left( \sum_{i_2...i_m\in [n]^{m-1}} a_{1i_2...i_m} \right) > a_{1 j_2 \cdots j_m},$ for $j_2 \cdots j_m \in [n]^{m-1}$ and $(j_2, \cdots ,j_m)\neq (1,... ,1).$ Hence condition (a) holds.

\noindent Again since $\mathcal{B}^+$ is a Nekrasov tensor with positive diagonal entries, $|a_{i i...i} -r_i^+|$ $= a_{i i...i} -r_i^+$ and we also have,

\begin{align*}
     a_{i i...i} -r_i^+ &= |a_{i i...i} -r_i^+|\\
                        &> \Lambda_i(\mathcal{B}^+)\\
                        & = \sum_{i_2 ... i_m \in [i-1]^{m-1}} |a_{i i_2 \cdots i_m} - r_i^+| \alpha_{i_2}\cdots \alpha_{i_m} \\
                        & \qquad + \sum_{i_2 ... i_m \notin [i-1]^{m-1}, (i_2 ... i_m)\neq (i ... i)} |a_{i i_2 \cdots i_m} - r_i^+|\\
                        &= \sum_{i_2 ... i_m \in [i-1]^{m-1}} (r_i^+ - a_{i i_2 \cdots i_m})\alpha_{i_2}\cdots \alpha_{i_m} \\
                        & \qquad + \sum_{i_2 ... i_m \notin [i-1]^{m-1}, (i_2 ... i_m)\neq (i ... i)} (r_i^+ - a_{i i_2 \cdots i_m})\\
                        &= r_i^+ \sum_{i_2 ... i_m \in [i-1]^{m-1}} \alpha_{i_2} \cdots \alpha_{i_m} - \sum_{i_2 ... i_m \in [i-1]^{m-1}} a_{i i_2 \cdots i_m} \alpha_{i_2} \cdots \alpha_{i_m} \\
                       &\qquad + \left(n^{m-1} -(i-1)^{m-1} -1\right) r_i^+  - \sum_{i_2 ... i_m \notin [i-1]^{m-1}, (i_2 ... i_m)\neq (i ... i)} a_{i i_2 \cdots i_m}.
\end{align*}
Then
\begin{align*}\label{second equation for sufficient part}
    r_i^+ \left( \sum_{i_2, ..., i_m \in [i-1]^{m-1}} \alpha_{i_2} \cdots \alpha_{i_m} + n^{m-1} -(i-1)^{m-1} \right) &< \sum_{i_2, ..., i_m \in [i-1]^{m-1}} a_{i i_2 \cdots i_m} \alpha_{i_2} \cdots \alpha_{i_m}\\
    &\qquad + \sum_{i_2, ..., i_m \notin [i-1]^{m-1}} a_{i i_2 \cdots i_m}.
\end{align*}
Therefore,
\begin{equation*}
    r_i^+ < \frac{\sum_{i_2...i_m\in [n]} a_{ii_2...i_m} \alpha_{i_2}\cdots \alpha_{i_m} + \sum_{i_2, ..., i_m \notin [i-1]^{m-1}} a_{i i_2 \cdots i_m}}{\sum_{i_2, ..., i_m \in [i-1]^{m-1}} \alpha_{i_2} \cdots \alpha_{i_m} + n^{m-1} -(i-1)^{m-1}}
\end{equation*}
and so (b) holds.
\end{proof}

Here we propose a weak restriction on a Nekrasov tensor to costruct a positive diagonal matrix $W$ which scales the Nekrasov tensor and transforms it to a strictly diagonally dominated tensor.

\begin{theorem}\label{theorem to convert into SDD}
Let for even $m,$ $\mathcal{B}=(b_{i_1 i_2 ... i_m})\in \mathbb{R}^{[m,n]}$ be a Nekrasov tensor such that, for each $i=1,2,..., n-1,\; b_{i i_2 ... i_m} \neq 0$ for some $i_2, ..., i_m > i.$ Then the matrix $W=diag(w_1, ..., w_n),$ with $w_i= \left( \frac{\Lambda_i(\mathcal{B})}{|b_{i...i}|} \right)^{\frac{1}{m-1}}$ for $i\in[n-1]$ and $w_n= \left( \frac{\Lambda_n(\mathcal{B})}{|b_{n...n}|} \right)^{\frac{1}{m-1}} + \epsilon,$ $\epsilon \in \left( 0, 1- \left( \frac{\Lambda_n(\mathcal{B})}{|b_{n...n}|} \right)^{\frac{1}{m-1}} \right),$ has positive diagonal entries and the tensor $\mathcal{A} = \mathcal{B} W$ is strictly diagonally dominated tensor.
\end{theorem}
\begin{proof}
Note that, by the hypothesis, for each $i\in [n-1],$ $b_{i i_2 ... i_m} \neq 0$ for some $i_2, ..., i_m > i.$ Then $\Lambda_i(\mathcal{B}) >0$ for $i\in [n-1]$ and so the diagonal entries of the matrix $W$ are all positive. Let $W=diag(w_1, ..., w_n).$ Now for $i\in [n-1]$ we have,
\begin{align}
    |b_{i...i}|w_i^{m-1} &=\Lambda_i(\mathcal{B}) \notag \\
                         &=\sum_{i_2...i_m \in [i-1]^{m-1}} |b_{i i_2 ... i_m}| \left(\frac{\Lambda_{i_2} (\mathcal{B})}{|b_{i_2 i_2 ... i_2}|}\right)^{\frac{1}{m-1}} \cdots \left(\frac{\Lambda_{i_m} (\mathcal{B})}{|b_{i_m i_m ... i_m}|}\right)^{\frac{1}{m-1}} \notag \\
                         & \qquad + \sum_{(i_2 ... i_m) \in [n-1]^{m-1} \backslash\ [i-1]^{m-1}, (i_2 ... i_m)\neq (i ... i)} |b_{i i_2 ... i_m}| \notag \\
                          &=\sum_{i_2...i_m \in [i-1]^{m-1}} |b_{i i_2 ... i_m}| \left(\frac{\Lambda_{i_2} (\mathcal{B})}{|b_{i_2 i_2 ... i_2}|}\right)^{\frac{1}{m-1}} \cdots \left(\frac{\Lambda_{i_m} (\mathcal{B})}{|b_{i_m i_m ... i_m}|}\right)^{\frac{1}{m-1}} \notag \\
                         & \qquad + \sum_{(i_2 ... i_m) \in [n-1]^{m-1} \backslash\ [i-1]^{m-1}, (i_2 ... i_m)\neq (i ... i)} |b_{i i_2 ... i_m}| \notag \\
                         & \qquad \qquad + \sum_{(i_2 ... i_m) \in [n]^{m-1} \backslash\ [n-1]^{m-1}}|b_{i i_2 ... i_m}| \label{main equation for scalling}
\end{align}

\noindent Since $\mathcal{B}$ is a Nekrasov tensor, $\Lambda_i(\mathcal{B}) < |b_{ii...i}|$ and so $w_i <1$ for all $i\in[n-1].$ Therefore
\begin{equation}\label{2nd main equation for scalling}
    |b_{i i_2...i_m} \geq |b_{i i_2 ... i_m}| w_{i_2}\cdots w_{i_m}, \; (i_2 ... i_m) \in ([n-1]^{m-1} \backslash\ [i-1]^{m-1}), (i_2 ... i_m)\neq (i ... i).
\end{equation}

\noindent Also by the assumption, $\epsilon < 1- \left( \frac{\Lambda_n(\mathcal{B})}{|b_{n...n}|} \right)^{\frac{1}{m-1}} $ an so $w_n <1.$ Therefore
\begin{equation}\label{3rd main equation for scalling}
    |b_{i i_2...i_m}| \geq |b_{i i_2 ... i_m}| w_{i_2}\cdots w_{i_m}, \text{ for } (i_2 ... i_m) \in [n]^{m-1} \backslash\ [n-1]^{m-1}.
\end{equation}
From (\ref{main equation for scalling}), (\ref{2nd main equation for scalling}) and (\ref{3rd main equation for scalling}) we deduce
\begin{equation}\label{4th main equation for scalling}
    |b_{i i ... i}| w_{i}\cdots w_{i} \geq \sum_{(i_2 ... i_m) \neq (i i ... i)} |b_{i i_2 ... i_m}| w_{i_2}\cdots w_{i_m}.
\end{equation}
Now, if either (\ref{2nd main equation for scalling}) or (\ref{3rd main equation for scalling}) are strict inequalities,  then (\ref{4th main equation for scalling}) is also strict.

\noindent If $b_{i i_2 ... i_m}\neq 0,$ for $(i_2 ... i_m) \in [n]^{m-1} \backslash\ [n-1]^{m-1}$, since $w_n <1,$ then $|b_{i i_2 ... i_m}| w_{i_2}\cdots w_{i_m} < |b_{i i_2 ... i_m}|$ and so (\ref{3rd main equation for scalling}) and (\ref{4th main equation for scalling}) are strict.

\noindent If $b_{i i_2 ... i_m}= 0,$ the by the assumption $b_{i i_2 ... i_m} \neq0$ for some $(i_2 ... i_m) \in ([n-1]^{m-1} \backslash\ [i-1]^{m-1}), (i_2 ... i_m)\neq (i ... i)$ Since $w_i <1,$ then $|b_{i i_2 ... i_m}| w_{i_2}\cdots w_{i_m} < |b_{i i_2 ... i_m}|$ and so $(\ref{2nd main equation for scalling})$ and (\ref{4th main equation for scalling}) are strict. Therefore the condition of strictly diagonally dominance for the tensor $\mathcal{A}W$ holds for $i\in [n-1].$

Finally, since $\epsilon |b_{n ...n}| >0,$ we have
\begin{align*}
    |b_{n ...n} w_n w_n \cdots w_n| &= \Lambda_n(\mathcal{B}) + \epsilon |b_{n ...n}|\\
    &= \sum_{i_2...i_m \in [n-1]^{m-1}} |b_{n i_2 ... i_m}| \left(\frac{\Lambda_{i_2} (\mathcal{B})}{|b_{i_2 i_2 ... i_2}|}\right)^{\frac{1}{m-1}} \cdots \left(\frac{\Lambda_{i_m} (\mathcal{B})}{|b_{i_m i_m ... i_m}|}\right)^{\frac{1}{m-1}}\\
    & \qquad +\sum_{(i_2 ... i_m)\in [n]^{m-1} \backslash\ [n-1]^{m-1}, (i_2 ... i_m)\neq (n...n)}|b_{n i_2 ... i_m}|  + \epsilon |b_{n ...n}|\\
    &\geq \sum_{(i_2 ... i_m) \neq (n...n)} |b_{n i_2 ... i_m}| w_{i_2}\cdots w_{i_m} + \epsilon |b_{n ...n}| \text{ using (\ref{3rd main equation for scalling}) }\\
    &> \epsilon |b_{n ...n}|
\end{align*}
and so the last condition of strictly diagonally dominance for the tensor $\mathcal{A}W$ holds for $i=n.$
\end{proof}

\begin{theorem}
Let for even $m,$ $\mathcal{A}=(a_{i_1 i_2 ... i_m})\in \mathbb{R}^{[m,n]},$ be a $B$-Nekrasov tensor such that for each $i=1,...,n-1,\; \exists\; i_2, ..., i_m >i$ with $a_{ii_2 ... i_m }< \max \{0, a_{i i_2 ... i_m}: (i_2 ... i_m) \neq (i ... i)\}= r_i^+.$ Let $\mathcal{B}^+$ be the tensor given by equation (\ref{decomposition equation}). Then the matrix $W=diag(w_1 ,\cdots w_n)$ with $w_i= \left(\frac{\Lambda_i(\mathcal{B}^+)}{a_{ii ... i} - r_i^+}\right)^{\frac{1}{m-1}},$ for $i\in [n-1]$ and $w_n = \left(\frac{\Lambda_n(\mathcal{B}^+)}{a_{nn ... n} - r_n^+}\right)^{\frac{1}{m-1}} + \epsilon,$ $\epsilon \in \left( 0, 1- \left(\frac{\Lambda_n(\mathcal{B}^+)}{a_{nn ... n} - r_n^+}\right)^{\frac{1}{m-1}} \right),$ has positive diagonal entries with each entry less than $1$ and the tensor $\mathcal{B}^+ W$ is a strictly diagonally dominant $Z$-tensor.
\end{theorem}
\begin{proof}
Note that $\mathcal{B}^+$ is a Nekrasov tensor by the definition of a $B$-Nekrasov tensor. Since $a_{ii_2 ... i_m }< r_i^+,\;\forall\; i\in [n],$ we conclude that $\mathcal{B}^+$ satisfies the conditions of the Theorem \ref{theorem to convert into SDD}. Therefore by Theorem \ref{theorem to convert into SDD}, we conclude that $\mathcal{B}^+ W$ is strictly diagonally dominant tensor. Also $\mathcal{B}^+$ is a $Z$-tensor since $B$ is a $B$-Nekrasov tensor. Therefore $\mathcal{B}^+W$ is a strictly diagonally dominant $Z$-tensor.
\end{proof}

\begin{prop}
Let $\mathcal{A}$ be a $B$-Nekrasov tensor of even order and $\mathcal{A}$ is decomposed as $\mathcal{B}^+ + \mathcal{C}.$ If $\mathcal{C}=O,$ then $\mathcal{A}$ is a nonsingular $H$-tensor.
\end{prop}
\begin{proof}
If $\mathcal{C}=O,$ then $\mathcal{A}=\mathcal{B}^+.$ Since $\mathcal{A}$ is a $B$-Nekrasov tensor, $\mathcal{B}^+$ is a Nekrasov $Z$-tensor with positive diagonal entries. This implies $\mathcal{B}^+$ is a Nekrasov tensor. Hence by Theorem \ref{nekrasov implies nonsingular H}, $\mathcal{B}^+=\mathcal{A}$ is a nonsingular $H$-tensor.
\end{proof}

\begin{lemma}
Let $\mathcal{B}, \; \mathcal{C}\in \mathbb{R}^{[m,n]},$ $m$ be even. If $\mathcal{B}$ is a $P$-tensor and $\mathcal{C}$ is a nonnegative tensor with constant row subtensors, then $\mathcal{B} +\mathcal{C}$ is a $P$-tensor.
\end{lemma}
\begin{proof}
Let $\mathcal{B},\; \mathcal{C}\in \mathcal{R}^{[m,n]}$ with the condition that $\mathcal{B}$ is a $P$-tensor and $\mathcal{C}$ is a nonnegative tensor with constant row subtensors. Then $\forall \;0\neq v \in \mathbb{R}^n \; \exists \; i\in [n]$ such that
\begin{equation}
    v_i (\mathcal{B}v^{m-1})_i >0.
\end{equation}
Let $\mathcal{C}= (c_{i_1 i_2 ... i_m})$ be a nonnegative tensor with costant row subtensors. Let the row subtensors of $\mathcal{C}$ are $\mathcal{R}_i(\mathcal{C})\; \forall \; i\in [n].$ Then for each $i\in [n],$ $(\mathcal{R}_i(\mathcal{C}))_{i_2 ... i_m} = c_i,\;\forall \; i_2, ..., i_m \in [n],$ for some $c_i \geq 0.$

\noindent If possible let $\mathcal{A}= \mathcal{B}+\mathcal{C}$ is not a $P$-tensor. Then $\exists \; 0\neq v\in \mathbb{R}^n$ such that $v_i (\mathcal{C} v^{m-1})_i \leq 0,\; \forall \; i\in [n].$ This implies for all $i\in [n],$
\begin{align}
   & v_i (\mathcal{B} v^{m-1})_i + v_i (\mathcal{C} v^{m-1})_i \leq 0 \notag \\
   \implies & v_i (\mathcal{B} v^{m-1})_i + v_i c_i (\sum_{i=1}^n v_i )^{m-1} \leq 0 \notag \\
   \implies & \frac{v_i^{m-2}}{(\tau^{m-1})^2}\left[ v_i (\mathcal{B} v^{m-1})_i + v_i c_i \tau^{m-1} \right] \leq 0 \text{ where, } \tau=(\sum_{i=1}^n v_i ) \neq0, \notag \\
   \implies & \left(\frac{v_i}{\tau}\right)^{m-1} \left(\mathcal{B} \left(\frac{v}{\tau}\right)^{m-1}\right)_i +  \left(\frac{v_i}{\tau}\right)^{m-1} c_i \leq 0 \notag \\
   \implies & y_i (\mathcal{A} y^{m-1})_i + y_i c_i \leq 0, \text{ where } y=\frac{1}{\tau} v\label{last equation to prove P tensor}.
\end{align}

\noindent Now we prove that for all $i\in [n],$ $y_i (\mathcal{A} y^{m-1})_i + y_i c_i \leq 0$ for arbitrary $c_i\geq0$ implies $y_i (\mathcal{A} y^{m-1})_i \leq 0.$
If not, then $\exists \; k\in [n]$ such that $y_k (\mathcal{A} y^{m-1})_k >0.$ Now if $y_k \geq 0$ then $y_k (\mathcal{A} y^{m-1})_k + y_k r_k >0.$ Again, if $y_k <0$ then for $\epsilon >0$ such that $y_k (\mathcal{A} y^{m-1})_k -\epsilon >0$ we can choose $r_k = \frac{y_k (\mathcal{A} y^{m-1})_k -\epsilon}{|y_k|} >0.$ Then $y_k (\mathcal{A} y^{m-1})_k + y_k r_k = \epsilon >0.$ This contradicts the assumption. Thus (\ref{last equation to prove P tensor}) implies $y_i (\mathcal{A} y^{m-1})_i \leq 0,\; \forall\; i\in [n].$ Therefore $\exists \; 0\neq y\in \mathbb{R}^n$ such that $y_i (\mathcal{A} y^{m-1})_i \leq 0,\; \forall\; i\in [n].$ This implies $\mathcal{A}$ is not a $P$-tensor.
\end{proof}

\begin{theorem}
If $\mathcal{A}$ is a $B$-Nekrasov tensor of even order, then $\mathcal{A}$ is a $P$-tensor.
\end{theorem}
\begin{proof}
By the definition of $B$-Nekrasov tensor $\mathcal{A}$ can be decomposed as $\mathcal{A}= \mathcal{B}^+ + \mathcal{C},$ where $\mathcal{B}^+$ is a Nekrasov $Z$-tensor whose all diagonal entries are positive and $\mathcal{C}$ is a nonnegative tensor with constant row subtensors. Now by Theorem \ref{1st theorem of Nekrasov Z tensor} we conclude that $\mathcal{B}^+$ is a $P$-tensor. Therefore the result follows from the fact that $\mathcal{A}$ is the sum of a $P$-tensor and a nonnegative tensor with constant row subtensors.
\end{proof}

\begin{remk}
The $P$-tensors play an important role in tensor complementarity theory. We know that the SOL$(\mathcal{A}, q)$ is nonempty and compact if the involved tensor $\mathcal{A}$ is a $P$-tensor. Here we prove that a $B$-Nekrasov tensor of even order $\mathcal{A}$ is a $P$-tensor. Hence we conclude that for a even order $B$-Nekrasov tensor $\mathcal{A},$ the solution set of TCP$(\mathcal{A}, q)$ is nonempty and compact.
\end{remk}

\section{Conclusion}
In  this article, we introduce $B$-Nekrasov tensor. We show that the class of $B$-Nekrasov tensor contains Nekrasov $Z$-tensor with positive diagonal entries. We present a necessary and sufficient condition for a $B$-Nekrasov tensor. We prove that the class of $B$-Nekrasov tensor is a subclass of $P$-tensor.

\section{Acknowledgment}
The author R. Deb is thankful to the Council of Scientific $\&$ Industrial Research (CSIR), India, Junior Research Fellowship scheme for financial support.

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


text:
Q: Restful web service hosted in IIS 7 not handling data contract members I have a requirement to create a simple HTTP C# restful web service hosted in IIS 7 that can support a POST from a callout and the format of the POST data does not seem to be handled properly. The requirement for the data member names is that they have a dash in them, such as custom-1. My problem is that the datamembers names who are custom-10 or higher are only giving me null for data. custom-1 thru custom-9 are fine.
If anyone can help, I appreciate it!!!
Here is the required format of the POST data:
 <person-search-request xmlns="someurl">
    <person>
        <custom-1 />
        <custom-2 />
        <custom-3 />
        <custom-4 />
        <custom-5 />
        <custom-6 />
        <custom-7 />
        <custom-8 />
        <custom-9 />
        <custom-10 />
        <custom-11 />
        <custom-12 />
        <custom-13 />
        <custom-14 />
        <custom-15 />
        <custom-16 />
        <custom-17 />
        <custom-18 />
        <custom-19 />
        <custom-20 />
      </person>
     </person-search-request>

My Web service data contract looks like this:
    [CollectionDataContract(Name = "person-search-request", Namespace="")]
    public class PersonsRequest : List<Person>
    { }

    [CollectionDataContract(Name = "person-search-response", Namespace="")]
    public class PersonsResponse : List<Person>
    { }

    [DataContract(Name = "person", Namespace = "")]
    public class Person
    {
        public Person()
        {
            Custom14 = String.Empty;
            Custom13 = String.Empty;
            Custom15 = String.Empty;
            Custom16 = String.Empty;
            Custom17 = String.Empty;
            Custom18 = String.Empty;
            Custom19 = String.Empty;
            Custom20 = String.Empty;
            Custom7 = String.Empty;
            Custom8 = String.Empty;
            Custom9 = String.Empty;
            Custom1 = String.Empty;
            Custom10 = String.Empty;
            Custom11 = String.Empty;
            Custom12 = String.Empty;
            Custom3 = String.Empty;
            Custom4 = String.Empty;
            Custom5 = String.Empty;
            Custom6 = String.Empty;
        }

        [DataMember(Name = "custom-14")]
        public string Custom14 { get; set; }
        [DataMember(Name = "custom-7")]
        public string Custom7 { get; set; }
        [DataMember(Name = "custom-8")]
        public string Custom8 { get; set; }
        [DataMember(Name = "custom-9")]
        public string Custom9 { get; set; }
        [DataMember(Name = "custom-13")]
        public string Custom13 { get; set; }
        [DataMember(Name = "custom-15")]
        public string Custom15 { get; set; }
        [DataMember(Name = "custom-16")]
        public string Custom16 { get; set; }
        [DataMember(Name = "custom-17")]
        public string Custom17 { get; set; }
        [DataMember(Name = "custom-18")]
        public string Custom18 { get; set; }
        [DataMember(Name = "custom-19")]
        public string Custom19 { get; set; }
        [DataMember(Name = "custom-20")]
        public string Custom20 { get; set; }
        [DataMember(Name = "custom-10")]
        public string Custom10 { get; set; }
        [DataMember(Name = "custom-11")]
        public string Custom11 { get; set; }
        [DataMember(Name = "custom-12")]
        public string Custom12 { get; set; }
        [DataMember(Name = "custom-1")]
        public string Custom1 { get; set; }
        [DataMember(Name = "custom-2")]
        public string Custom2 { get; set; }
        [DataMember(Name = "custom-3")]
        public string Custom3 { get; set; }
        [DataMember(Name = "custom-4")]
        public string Custom4 { get; set; }
        [DataMember(Name = "custom-5")]
        public string Custom5 { get; set; }
        [DataMember(Name = "custom-6")]
        public string Custom6 { get; set; }
        /// <summary>

    }

My Service endpoint is setup like this:
    [ServiceBehavior(IncludeExceptionDetailInFaults = true), AspNetCompatibilityRequirements(RequirementsMode = AspNetCompatibilityRequirementsMode.Allowed), ServiceContract]
public partial class Service
{

     [WebHelp(Comment = "Person POST")]
    [WebInvoke(UriTemplate = "person/v1.0/fetch", BodyStyle = WebMessageBodyStyle.Bare, ResponseFormat = WebMessageFormat.Xml, RequestFormat = WebMessageFormat.Xml)]
    [OperationContract]
    public PersonsResponse FetchPerson(PersonsRequest request)
    {
            ....
     }

}
The Post data looks like this (using Fiddler to test):
    <person-search-request>
    <person>
        <custom-1>1</custom-1>
        <custom-2>2</custom-2>
        <custom-3>3</custom-3>
        <custom-4>4</custom-4>
        <custom-5>5</custom-5>
        <custom-6>6</custom-6>
        <custom-7>7</custom-7>
        <custom-8>8</custom-8>
        <custom-9>9</custom-9>
        <custom-10>10</custom-10>
        <custom-11>11</custom-11>
        <custom-12>12</custom-12>
        <custom-13>13</custom-13>
        <custom-14>14</custom-14>
        <custom-15 />
        <custom-16 />
        <custom-17 />
        <custom-18 />
        <custom-19 />
        <custom-20 />
     </person>
</person-search-request>

The data comes into the endpoint with nulls in the 10-20 custom tags. Anyone know why??? :(
This is what I see when I debug from the request object passed into the method.
     Custom1    "1" string
    Custom10    null    string
    Custom11    null    string
    Custom12    null    string
    Custom13    null    string
    Custom14    null    string
    Custom15    null    string
    Custom16    null    string
    Custom17    null    string
    Custom18    null    string
    Custom19    null    string
    Custom2 "2" string
    Custom20  null  string
    Custom3 "3" string
    Custom4 "4" string
    Custom5 "5" string
    Custom6 "6" string
    Custom7 "7" string
    Custom8 "8" string
    Custom9 "9" string

A: I was able to fix this using the order attribute. Still acting unpredictable. If anyone else has a better explanation that would be appreciated!!!
Anyways, here is the fix (there are other not important members starting using 1 & 2 slot):
        [DataMember(Name = "custom-1", Order = 3)]
        public string Unused1 { get; set; }
        [DataMember(Name = "custom-2", Order = 4)]
        public string Unused2 { get; set; }
        [DataMember(Name = "custom-3", Order = 5)]
        public string Unused3 { get; set; }
        [DataMember(Name = "custom-4", Order = 6)]
        public string Unused4 { get; set; }
        [DataMember(Name = "custom-5", Order = 7)]
        public string Unused5 { get; set; }
        [DataMember(Name = "custom-6", Order = 9)]
        public string Unused6 { get; set; }
        [DataMember(Name = "custom-7", Order = 10)]
        public string License { get; set; }
        [DataMember(Name = "custom-8", Order = 11)]
        public string LicenseState { get; set; }
        [DataMember(Name = "custom-9", Order = 12)]
        public string Tax { get; set; }
        [DataMember(Name = "custom-10", Order = 13)]
        public string Unused10 { get; set; }
        [DataMember(Name = "custom-11", Order = 14)]
        public string Unused11 { get; set; }
        [DataMember(Name = "custom-12", Order = 15)]
        public string Unused12 { get; set; }
        [DataMember(Name = "custom-13", Order = 16)]
        public string ProfDesignation { get; set; }
        [DataMember(Name = @"custom-14", Order = 17)]
        public string NPI { get; set; }
        [DataMember(Name = "custom-15", Order = 18)]
        public string Address1 { get; set; }

        [DataMember(Name = "custom-16", Order = 19)]
        public string Address2 { get; set; }

        [DataMember(Name = "custom-17", Order = 20)]
        public string Address3 { get; set; }

        [DataMember(Name = "custom-18", Order = 21)]
        public string City { get; set; }

        [DataMember(Name = "custom-19", Order = 22)]
        public string State { get; set; }

        [DataMember(Name = "custom-20", Order = 23)]
        public string Zip { get; set; }

redpajama_set_name: RedPajamaStackExchange