text:
# Open Period Integrals of Elliptically Fibered K3 surfaces

Let M be the period domain for elliptic K3 surfaces $(X,\Omega)$ with a holomorphic two-form. Denote the fiber class $f$. Then $$M=\{\Omega\in f^\perp\otimes \mathbb{C}\,:\, \Omega\cdot \Omega=0, \,\Omega\cdot \overline{\Omega}>0\}$$ where $\cdot$ is the extension of the intersection product on the K3 lattice to $\mathbb{C}$. There is an elliptic fibration $X\rightarrow \mathbb{P}^1$ whose fiber class is $f$. Now consider a path connecting two singular fibers of $X\rightarrow \mathbb{P}^1$ and let $C$ be a cylinder lying over this path. Then the integral $$\int_C\Omega$$ is locally a well-defined function $f_C\,:\,M\rightarrow \mathbb{C}$ because any two choices of cylinder differ by a homology lying within a fiber of the elliptic fibration, and $\Omega$ restricts to zero on such a fiber. My question is the following: Is $f_C$ a linear function (in the sense that it is a restriction of a linear function on the vector space containing the quadric $M$)? This would make sense, because whenever a collection of cylinders join to form an actual homology class, the sum of these integral is linear by definition.


text:
Paola Lucchesi on Sat, 11 Dec 1999 14:24:20 +0100

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 Syndicate: AIM: AN AMERICAN OASIS IN KOSOVO

>aim\pri\trae\rm
>
>
>*** BONNSTIL - AN AMERICAN OASIS IN KOSOVO
>
>
>        AIM Pristina, November 30, 1999
>
>
>        The Americans are building a large military base
>with an enormous airport. For weeks and months now, Kosovo
>and partly the world, have been abuzz with rumours about
>civil-engineering works that have started in mid this summer
>at a plateau in the south-eastern part of Kosovo.
>This location is near the village of Sojevo, some 3-4
>kilometres from Urosevac on the road to Gnjilane. Among
>these speculations the most frequently mentioned was a story
>about the construction of a mammoth-size military airport
>which, if necessary, could even replace the one in Aviano,
>Italy. The story that went around was that the interest of
>the West and NATO was shifting further to the East, former
>satellite countries and territories of former Soviet Union,
>towards Near and Middle East. In the public eye such
>speculations seem and sound very intriguing and evoke
>associations.
>
>
>        But they yet have to be confirmed. The only thing
>one can see there is this huge muddy building site where
>people are working and building round the clock. Currently,
>they are working on a 4.5 km long and 4 km wide strip.
>However, according to official data, the entire facility, or
>camp Bonnstil as the Americans call it, should cover the
>area of 755 acres, i.e. over 305 hectares.
>
>
>        Only experts can evaluate the military importance of
>this area, i.e. what this American military camp can mean
>for Kosovo and this part of the Balkans. Even laymen could
>tell you that no one invests so much money in a facility to
>be used only by some 5 thousand American soldiers and the
>American control sector in Kosovo, as the Americans claim to
>be the case. At this moment, the view from the camp itself
>does not lead to such a conclusion. The installed equipment and
>materiel mostly include  military "Havis" jeeps and tents,
>as well as a great quantity of satellite communication
>equipment. Those who had a chance to visit the camp several
>times could tell you that people living there have no
>problem communicating with the States, but cannot establish
>contacts with Kosovo. One could see an occasional tank, but,
>naturally, a great number of soldiers. The guards, as well
>as those on watchtowers were armed.
>
>
>        According to what we could see it was obvious that
>all stories about an airport are unfounded. There were
>nearly 30 military helicopters on the heliport, including
>several of the "Apache" type and a number with Red Cross
>signs. There was enough space for an even greater number of
>such aircraft. Captain Patrick Sweeney, who is in charge of
>the press, told us that the people have probably confused
>the heliport with the airport and that is how the rumour
>about military airport started. It is all easy to see.
>Airplanes cannot take off from nor land here and there are
>no overflights in these parts. "Had there been any, everyone
>would have seen that we had no secrets", explained the
>American officer.
>
>
>        Most of the activities are organised under tents.
>Judging by their intensity and the pace of the works it
>could be concluded that the planned target of accommodating
>all soldiers in prefabricated houses, which the Americans
>call SEA Huts, has not been achieved yet. The plan is to put
>up 160 of such houses for soldiers and 26 for officers and
>non-commissioned officers, with all the accompanying
>infrastructure and facilities, including services. Green
>areas have also been envisaged, as well as recreational
>grounds, cultural centre, work-out gyms and even a small
>hospital for soldiers. Civilians will be allowed to use it
>in case of emergency or medical analyses for which the most
>sophisticated medical equipment is required. The construction
>of a large storehouse for all types of ammunition is
>underway and probably much more than could meet the eye.
>
>
>        The prison will be isolated in several tents
>surrounded by barbed wire. The number of inmates changes
>daily, told us Captain Sweeney, but they usually number some
>hundred inmates accused of all kinds of crimes, from grave
>crimes to traffic offences. He explained how judges from
>Pristina come here for the purpose of investigation and how
>they cannot wait to get rid of this kind of obligation.
>Nearby, also surrounded with barbed wire was a kind of dump
>with wreckages of some 30 tanks and other armoured carriers
>and heavy vehicles which once belonged to the Serbian army
>and police. Sweeney told us that several weeks ago, in
>agreement with Belgrade, these wrecks were loaded onto
>trucks and handed over at the North Kosovo border.
>
>
>        While we were leaving we had the impression that
>this was a huge construction site in a forming stage.
>Nothing looked finished. The facilities we have mentioned,
>as well as 24 administrative and other official premises,
>yet have to be built. Until now, including the costs for a
>small local camp in Gnjilane, over USD 300 million have been
>already invested. About 5 thousand locals are engaged on
>different tasks. Inside this base, the Americans should live
>as if they were in the States. Question is whether something
>like that is possible since, after all, this is Kosovo.
>Captain Sweeney told us that he comes from North Carolina,
>that his family lives in Chicago. Smiling he also told us
>that his wife is momentarily in Germany, while he is
>stationed in Kosovo...
>
>
>AIM Pristina
>
>
># Fehim REXHEPI
>
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text:
## Friday, July 19, 2013

### Orthogonal Polynomials: Mathematica Script

A family of polynomials $\{ \phi_{0}, \phi_1, ..., \phi_{n} \}$ (the subscript indicates the degree) are orthogonal with respect to a weighting function $w(x)$, over an interval $[a,b]$, if $\langle\phi_n, \phi_m\rangle = \int_{a}^{b} w(x) \phi_{n}(x) \phi_{m}(x) dx = \delta_{nm} c(n)$
In other words, given a $w(x)$ and an interval $[a, b]$ an orthogonal sequence of polynomials can be found via Gram-Schmidt orthogonalization of monomials.

This is done by a straightforward Mathematica script (link at the end). A possible use is:

P = OrthoPoly[Exp[-x], 0, 10, 3] // FullSimplify

This spits out the first three members of the family of orthogonal polynomials with $w(x)=\exp(-x)$, over the interval [0,10].

One can use the zeros of these polynomials to obtain corresponding Gauss quadrature points. For example:

Solve[P[[3]] == 0, x] // N

yields {{x -> 0.180784}, {x -> 0.752396}}.

The code is available here.


text:
# Beginner help with Arrays & Complexity

My goal is to create a method that gets an array that only has 1's and 0's in it, and then turns every number into the amount of 'steps' it is away from the closest 0, left or right.

So for example [1] - [1] - [0] - [1] - [1] - [1] - [1] - [1] - [0] - [1] would be [2] - [1] - [0] - [1] - [2] - [3] - [2] - [1] - [0] - [1]

I also need to do it as simple as possible in terms of time and space complexity.

I'm not sure how to make it count how far a certain spot is from a 0 if it's from the right of the 0, or if there's another 0 nearby I'm not sure how to make it count which one is closest.

I'm assuming two for loops are needed?

Made a sketch: no idea if it makes sense.

 for (int i=0; i<a.length; i++)
{

for (int j=i; j<a.length; j++)
{
if (a[j]==0)
{
if (i<j)
{
a[i]=j-i;
}
if(j<i)
{
a[i]=i-j;
}
}
}
}

• Welcome to Computer Science! The title you have chosen is not well suited to representing your question. Please take some time to improve it; we have collected some advice here. Thank you! – Raphael Feb 4 '17 at 10:55
• @Raphael It's actually O(n). two loops are not nested. (And I didn't want to give this info because it seems like OP is already almost on to it.) – Billiska Feb 4 '17 at 11:11
• All I see are some loops. Try to explain in words how you think you should proceed. The steps are: Understanding the problem. Figuring out how to solve it. Writing code. You left out the most important middle step. – gnasher729 Feb 4 '17 at 11:45
• And two loops in the way you are using them are not needed. The problem can be solved easily in linear time. – gnasher729 Feb 4 '17 at 11:47
• Since j starts at i then j cannot be < i – paparazzo Feb 4 '17 at 12:11

Let's try to solve a simpler problem first:

Given an array that only has 1's and 0's in it, the method turns every number into the amount of 'steps' it is away from the closest 0 to the left.

Now this problem is easy. A simple solution is: traverse the array from left to right, keeping track of the distance from the last zero encountered and updating each element with the distance.

A simple code:

dist = INF
for(i=0;i<a.length;i++){
if(a[i] == 0)
dist = 0;
else
dist++;
b[i] = dist;
}

Now once you have solved the simpler problem for closest 0 to the left, you can similarly solve the problem for closest 0 to the right. With these two solutions, you should be able to easily get a solution for the original problem (by taking a min of the two solutions).

The solution evidently has O(n) time complexity, and can be easily coded to use only O(1) space by carefully manipulating the updates in the original array only.

Just some hints that will tell you how to solve it efficiently:

If you know where the first zero is, then you know what numbers to write down at all positions before and up to that zero. For example, if a [4] is the first zero, then you know the numbers for positions 0, 1, 2, 3 and 4 (4, 3, 2, 1, 0). You don't know the number for position 5 at this point.

If you know where two consecutive zeroes are, then you know what numbers to write down at all positions from the first to the second of these two zeroes. For example, if there is a zero at a [4] and the next zero is at a [9] then you know all the numbers for positions 4 to 9 (0, 1, 2, 2, 1, 0).

And if you know the position of the last zero, then you know what numbers to write down at all positions from that zero to the end of the array.

• I think that's an overly complicated idea. – Raphael Feb 4 '17 at 13:18

Just iterate in each direction and track the distance from the last zero

You could get fancier with some back tracking to only go one direction but would not be much faster

2n is still O(n)

lastZero is the distance from the last zero

int[] result  //this store the answer / calculation
int lastZero = null
for(i = 0; i < len; i++)
{
if(input[i] = 0)
{
lastZero = 0
result[i] = 0
}
else if (lastZero != null)
{
lastZero++
result[i] = lastZero
}
}
//now go from right to left
lastZero = null
for(i = len - 1; i <= 0; i--)
{
if(input[i] = 0)
{
lastZero = 0
}
else if (lastZero != null)
{
lastZero++
if(result[i] > lastZero or result[i] = null)
result[i] = lastZero
}
}

• Once again, we're looking for explanation, not code dumps. – David Richerby Feb 4 '17 at 12:44
• @BDriz Input is your input array. Result is the array of results (calculation). – paparazzo Feb 4 '17 at 13:25
• @Paparazzi No, I never claimed that it didn't work. That's now twice that you've mischaracterized what I said about that answer. – David Richerby Feb 4 '17 at 15:29
• @Paparazzi A set of truths does not make a good answer. A single block of code certainly does not. – Raphael Feb 4 '17 at 16:15
• It's okay if you write a block of code that solves it and there's an explanation attached. I'm just trying to learn and now I'm going through the logic of it. – B Driz Feb 4 '17 at 16:41


text:
# The New International Encyclopædia/Maine (United States)

MAINE (called ‘The Province or Countie of Mayne’ in the charter granted by Charles I. in 1639, because regarded as part of ‘the Mayne land of New England’). A North Atlantic State of the United States, belonging to the New England group. It is the most northeasterly State of the Union, and lies between latitudes 43° 4’ and 47° 28’ N.. and between longitudes 66° 57’ and 71° 7’ W. It is bounded on the northwest by the Canadian Province of Quebec, on the northeast by the Province of New Brunswick, on the southeast by the Atlantic Ocean, and on the west by the State of New Hampshire. Its extreme length is a little over 300 miles, its extreme width 185 miles, and its total area 33,040 square miles, the land amounting to 29,895 square miles. Maine is nearly as large as all the rest of the New England States combined, and thirty-fifth in size among the States of the Union.

Topography. The surface of the State is in general moderately hilly, becoming mountainous in the west and north. The elevation rises from the coast northward and from east to west toward the interior. Along the coast in the southwest corner the surface falls into low, flat, and even marshy land, and the country maintains its low and generally level character as far as the Kennebec River. East of the Kennebec the coast region rises to abrupt and bold elevations of 1000 to 1500 feet. The general slope of the State is determined by the main plateau which crosses it in a northeasterly direction and is the continuation of the Appalachians. This system appears here only in somewhat isolated heights which, while preserving the continuity of the Appalachians, can scarcely be called a mountain range. The height of this divide varies from 2000 feet above sea level on the west, where it enters the State, to 600 feet at its eastern extremity, where it dies away in the Aroostook region on the border of New Brunswick. The most prominent peak is Mount Katahdin, 5200 feet high. Nearer the western border stand Mount Abraham, 3388 feet, and Mount Bigelow, 3600 feet.

The fringe-like coast, with its tasseled edge of bold promontories and rocky islands, has given rise to the appropriate appellation of ‘hundred harbored Maine.’ The fiord-like harbors are the result of the drowning of the river valleys scored out by glacial action, and extending seaward far beyond the present shore line. This has provided Maine with excellent natural harbors, perfectly protected by the islands, but suffering from a serious disadvantage in the great rise and fall of the tides and the resulting swift currents. At Portland the tide rises 11 feet, increasing northward. Yet in spite of this disadvantage the coast between Portland and Eastport contains proportionately the greatest number of good harbors to be found anywhere along the coast of the United States.

There are upward of 600 lakes and ponds, aggregating a total area of over 2300 square miles, or one-fourteenth the total area of the State. They lie for the most part in the elevated plateau region, and are of great natural beauty. The most noted are: Moosehead Lake, 120 square miles in extent, the largest inland body of water in New England, and the source of the Kennebec River; the Rangeley Lakes, with an area of 90 square miles, the headwaters of the Androscoggin River; Chamberlin Lake, supplying the Saint John; and Chesuncook Lake, connected with the Penobscot River.

The largest rivers of Maine are naturally those of the southern slope. The chief of these are the Saco, Androscoggin, Kennebec, Penobscot, and Saint Croix, the last named on the eastern boundary. They are of little value for commerce, being navigable only a few miles inland—the Kennebec 26 miles to Augusta; the Penobscot 27 miles to Bangor. They rise at high elevations, and their precipitous character offers the explanation of their value as sources of water power. The power available for industrial purposes is enormous, the Adroscoggin alone having developed over 70,000 horse-power. Besides, the lakes in connection with these rivers furnish reservoirs for the storage of water, and thus make possible a uniform and constant supply of power through all the year. It is estimated that about 2,000,000 horse-power is available. This is utilized especially at the falls which characterize most of the rivers, and which are due to the unyielding granite beds that cross their path. The Saint John River traverses the northwest corner of the State, and forms part of the boundary with New Brunswick.

AREA AND POPULATION OF MAINE BY COUNTIES.

 County Map Index. County Seat. Area in squaremiles. Population. 1890. 1900. Androscoggin C 7 Auburn 480 48,968 54,242 Aroostook F 2 Houlton 6,408 49,589 60,744 Cumberland C 8 Portland 1,014 90,949 100,689 Franklin B 5 Farmington 1,764 17,053 18,444 Hancock G 6 Ellsworth 1,390 37,312 37,241 Kennebec D 6 Augusta 880 57,012 59,117 Knox E 7 Rockland 327 31,473 30,406 Lincoln D 7 Wiscasset 520 21,996 19,669 Oxford B 5 Paris 1,981 30,586 32,238 Penobscot F 5 Bangor 3,254 72,865 76,346 Piscataquis E 5 Dover 3,626 16,134 16,949 Sagadahoc D 7 Bath 259 19,452 20,330 Somerset D 3 Skowhegan 3,831 32,627 33,849 Waldo E 6 Belfast 748 27,759 24,185 Washington H 6 Machias 2,456 44,482 45,232 York B 8 Alfred 957 62,829 64,885

Climate. Maine lies in the heart of the temperate zone, yet in climate it ranks as boreal in the northwestern part, and the southeastern part is only a region of transition into the austral zone—a region of overlapping floras and faunas. The winter is very severe, except in a narrow strip along the coast. The summers are never hot. The mean temperature for January is 20° F. and 10° in the north. In July the mean temperature is 65° F. in the north and 70° in the south. There is a wide range of temperatures through the year, the average maximum shade temperature of 90° F. occurring all along the coast to Eastport. The anticyclones and north winds of winter bring an average minimum of 20° F. below zero to Eastport and Augusta, while the vicinity of Mount Katahdin records an average minimum of 30° below zero. The weather is subject to sudden falls in temperature in short periods because of the frequent passage of cyclonic and anticyclonic centres and the subjection of the State to alternate oceanic and continental influences. The southern counties have a growing season of less than six months, and the northern counties of only about five months. The average annual rainfall for the whole State is 40 inches, very evenly distributed throughout the year, though there is a maximum precipitation in the late summer and autumn. There is an average annual snowfall at the coast of 60 inches, increasing rapidly to the north and west to over 110 inches. The relative humidity for the whole State is above 70 per cent., though the absolute humidity is rather low, on account of low temperatures. The normal wind direction for January is northwest, and for July is southwest. The State lies in the zone of maximum cyclonic frequency. The cool summers, the lakes, the forests and the rocky seaside attract multitudes of summer tourists, and the coast and lakes are lined with summer cottages. Some of the resorts, notably Bar Harbor, on the eastern end of Mount Desert (q.v.), are among the most popular of the American summer resorts.

For Flora and Fauna, see these paragraphs under United States.

Geology and Soils. Maine has had a very complex geological history. In pre-Cambrian time the State was crossed by two great mountain ranges. One extended along the northwestern boundary through the White Mountains to Long Island Sound east of the Connecticut River; the other extended northeast, along the coast. These mountains were much folded, largely gneissic and schistose, and they were worn down to base level in later Paleozoic time. Through the centre of the State, between these two ranges, a long gulf extended from Gaspé Peninsula to the southwest. By the end of Devonian time this trough was filled with sediments from the aging mountains, only to be uplifted, crumpled, and somewhat metamorphosed, again worn to base level and depressed, allowing carboniferous deposits to be laid down unconformably upon all the older beds. It was uplifted and again worn to base level in Cretaceous time, since which the region has been broadly lifted into a plateau of low elevation and again dissected. The Pleistocene ice sheet covered the State entirely, discharging its marginal ice into the sea. The effect of glaciation was to denude the higher lands, accentuate the river valleys, create many lake basins, and leave the surface strewn with a coating of till. Since Pleistocene time there has been a considerable subsidence, resulting in the submergence of the coastal lowlands, and converting the higher hills and ridges into a fringe of islands, and the drowned valleys into fiords. The latest crustal movement has been a very slight uplift along the coast, uncovering small plains of marine clays which interlock with the rocky headlands. This same slight uplift has revived the rivers, furnishing very valuable water power at the bay heads.

The soils of Maine, with the exception of the Pleistocene marine clays along the coast, are almost wholly glacial. Along some of the rivers and lakes and in many old lake bottoms, long since filled up or drained out, there are alluvial plains of great fertility. The higher outcrops of the crystalline old land are largely denuded of all soil, while the whole surface of the State is more or less strewn with glacial débris. The drift is in places arable, though in wide areas it is unsuited for agriculture. The character of the soil and the large supply and even distribution of rainfall have been determining factors in making the State a rich forest region.

Mineral Resources and Mining. The fact that the surface is largely made up of the bases of worn-down old mountains explains at once the nature of the mineral resources of Maine. Granite rock outcrops all along the coast and for many miles inland. The fiords are incised in it, making it possible often to load the blocks on to the boats direct from the quarry. This fact of convenience and cheap transportation is a large factor, explaining the rank of Maine as second of all the States in granite production, the output in 1889 being valued at $1,321,182. The granite occurs in great veins or in eruptive masses. Some of these were cooled very slowly, evidently at great depth, the crystalline structure being so extremely coarse that the quartz and feldspar constituents can be separated easily and marketed pure. Maine is one of the leading sources of feldspar and silica, used in the manufacture of earthenware, porcelain, glass, flour and wall tile, wood-filler, sandpaper, and scouring soap. The output of silica in 1899 was valued at$50,336, the supply occurring as vein quartz in the crystalline rocks. Along the coast in occasional places and in Aroostook County and other places in the interior, are found beds of crystalline limestone, often a fine quality of marble. The value of the limestones, including the marble, was in 1899 $1,028,375, giving the State the sixth place in the production of this commodity. In the central part of the State valuable slate quarries are worked for roofing, tables, and blackboards; the output in 1899, valued at$181,766, ranked the State fourth in this resource. The total value of quarry products in 1899 was $2,531,223. The crystalline rocks of Maine produce many rare minerals, one of which, tourmalin, is obtained in Oxford County in larger and more beautiful crystals than anywhere else in the world. Copper, silver, gold, iron, tin, and manganese occur in small amounts, the iron and manganese being used only in local smelters. The State reported 26 mineral springs in 1899 as of commercial importance, 10 of these being in Androscoggin County. In this year 1,850,132 gallons were marketed with a value of$179,450. Potable waters of the highest quality are found in great abundance everywhere, the glacial drift being the usual source of supply for domestic use.

Forests. Maine is one of the leading States in the Union in the extent of wooded area and in the annual value received from the forest products. The woodland is estimated at 23,700 square miles, or 79 per cent. of the total land area. The forests cover largely the interior and northern portions of the State, the coast and lower valleys being generally cleared. The land rapidly reforests, and since most of the land is not well suited to cultivation, reforestation is extensive. Large areas at one time under cultivation have been allowed to go back to woodland. The primeval forests of pine are all gone, and the second growth is being used to some extent. The spruce forests are the most extensive and most heavily drawn upon at present. Authorities estimated in 1890 that the State had 21,239,000,000 feet of standing spruce. Of this nearly a third was in the region drained by the Saint John River, and nearly a fourth in the region drained by the Penobscot. The densest spruce forests—the largest, finest trees—are found in the upper basin of the Androscoggin River. The basin of this river has contributed about 44 per cent. of the supply used in the manufacture of pulp and paper. The Kennebec and Penobscot basins have contributed the remainder. The table on the following page shows that the paper and wood pulp industry is largely a development of the last decade of the nineteenth century. The pulp mills are located along the three rivers above mentioned. A belt of white birch timber extends entirely across the State, and this species figures prominently in the lumber industry. From it spools to the value of a million dollars are manufactured annually. Spool timber is shipped extensively to Scotland. Poplar is cut most extensively in the Kennebec region, where it is used in paper manufacture. In the Saint John and Penobscot basins there are extensive forests of cedar. Much has been done in the way of constructing dams, canals, and sluices to facilitate the transportation of logs.

Fisheries. Maine ranks second among the New England States in the value of its fisheries and first in the number of men engaged in the fishing industry. In 1898 the total number of persons so employed was 16,954, of whom 8237 were shoresmen and 6770 were employed in shore or boat fisheries. The total value of the product for the same year was $2,654,919, of which$992,855 represented the lobster fisheries, this being more important in Maine than in all the other New England States combined. Clam fishing ranks next, being also more important than in any other New England State. The salmon fishery on the Penobscot River, though small, is the largest on the Atlantic Coast. The canning industry is very prominent in connection with the shore fisheries, particularly the preparation of small herring as sardines. Clams also figure prominently in the canning products. The State law of 1895 resulted in the discontinuance of lobster-canning, which was formerly very important. The preparation of smoked herring is of considerable moment.

Agriculture and Live Stock. Like that of the other New England States, the surface is too rugged and broken, also the soil for the most part too sterile, to admit of the extensive development of agriculture. In the river valleys, however, the soil is very fertile. The Aroostook Valley, in the northeastern part of Maine, is the largest district of fertile farming land in New England. Only 32.9 per cent. of the total land area is included in the farms, and of this acreage only 37.9 per cent. is improved. But these areas increased steadily until 1880. Since then both have decreased—the latter very considerably. During the period from 1850 to 1900 the average acreage per farm remained nearly constant, the average for the latter year—106.2 acres—being the largest for the entire period. The average size of the farms varies greatly in the different counties. The State is remarkable for the small percentage of rented farms—less than 5 per cent. of the total.

As elsewhere in New England, the influence of the Western competition has obliged the Maine farmers to give up cereal farming in part, especially wheat. Oats is now the principal cereal, although this too decreased in the last two decades of the century. Buckwheat, the next most important cereal, has more than held its own, but corn and wheat decreased from 1880 to 1900. The area devoted to cereals is rapidly increasing in Aroostook County, constituting in 1900 one-half of the total for the State. The potato crop, though having a less acreage than oats, produces a greater income than all the cereal crops combined. In Aroostook County an average of 6.6 acres per farm is devoted to potatoes. Intensive methods of cultivation are used, and a very high per acre product is obtained for potatoes, as also for most other crops. The acreage of hay and forage is over seven times as great as that of the cereals. Increased attention is being given to dairying and market gardening, the poorer soils being turned into permanent pastures and the most fertile being appropriated for the raising of market products. Sweet corn is the most important of these crops. Much attention is given to the raising of apples, the total number of trees in 1900 exceeding 4,100,000—an increase of 39.3 per cent. for the decade ending in that year. Other orchard and small fruits receive little attention.

Stock-raising is not a leading industry. Since 1880 the number of sheep has decreased over one-half, and the number of neat cattle had decreased rapidly after 1860 until the last decade of the century. The number of horses more than doubled during the last half of the century. Likewise the growing interest in dairying has resulted in the number of milch cows increasing considerably during the same period. In 1900, 29.9 per cent. of the farms of the State derived their principal income from dairy produce. The production of milk increased 71.8 per cent. during the decade ending in that year. Over $5,600,000 was realized from the sale of dairy products, about two-fifths being the returns for milk, two-fifths for butter, and one-fifth for cream. The following tables show the importance of the leading kinds of crops and farm animals for the census years 1890 and 1900:  1900 1890 Hay and forage, acres 1,270,254 1,300,302 Oats, acres 108,661 121,612 Buckwheat, acres 25,292 22,395 Corn, acres 16,856 10,891 Barley, acres 8,809 11,972 Potatoes, acres 71,765  1900 1890 Dairy cows 173,592 157,278 Other cattle 165,255 141,832 Horses 106,299 109,156 Mules and asses 401 278 Sheep 252,213 370,484 Swine 79,018 91,297 Manufactures. Manufacturing is of much importance, 74,800 wage-earners, or 10.8 per cent. of the population, being engaged in this line in 1900. The percentage of the population thus occupied has increased continuously in the last half century. It was only 4.8 in 1850. The total value of the product in 1900 was$127,361,000. The development of the manufacturing industry is due to the extensive water power afforded by the numerous rivers of the State and also to the excellent commercial advantages offered by the harbors, and, in later years, by the railways. The factories are located, almost without exception, along the rivers or on the coast, and most of them are run by water power. The manufacture of textiles—cotton and woolen goods—leads in importance. Both cotton and woolen mills were in operation during the early years of the nineteenth century. The cotton mills of to-day are all west of the Kennebec River. Lewiston is the chief centre. The woolen mills are more widely distributed, but also derive their power from the streams. The manufactures of cotton show a slight decrease in value during the decade 1890-1900. This is probably due in part to the increasing competition of the Southern States.

Shipbuilding formerly depended upon the forests for its supplies, but with the increased use of steel in vessel construction, the shipbuilding industry of Maine has declined. The industry is one of the oldest in the State, a vessel having been built as early as 1608. For a long time Maine held first rank in the industry, and indeed constructed more than half of all the sea-going vessels of the nation. At Bath, the principal shipbuilding centre, the construction of steel vessels has attained considerable importance. The leather industry also dates from an early period. It was one of the State's leading industries between the years 1861 and 1870. The bark of the hemlock was used in tanning, and the industry is declining as the supply of this bark becomes scarcer. Other noteworthy industries are represented by the foundries and machine shops and printing and publishing houses. The following table gives a comparison of the chief industries for the years indicated:

INDUSTRIES  Year  Number of
establishments
Wage-earners—
average
number
Value of products,
including custom
work and repairing

 Total for selected industries for State ${\displaystyle \scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}}$
1900  1,689  43,730  $73,368,312 1890 1,580 39,651 54,631,124 Increase 1890 to 1900 ...... 109 4,079$18,737,188
Per cent. of increase ...... 6.9  10.3  34.3

 Per cent. of total of all industries in State ${\displaystyle \scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}}$
1900 25.2  58.5  57.6
1890 31.5  56.3  57.1

 Cotton goods ${\displaystyle \scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}}$
1900 15  13,723  $14,631,086 1890 23 13,912 15,316,909  Fish, canning and preserving ${\displaystyle \scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}}$ 1900 117 5,567 4,779,733 1890 35 2,342 1,660,881  Flouring and grist-mill products ${\displaystyle \scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}}$ 1900 227 192 3,399,832 1890 210 262 3,254,690  Foundry and machine-shop products ${\displaystyle \scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}}$ 1900 112 2,143 3,298,706 1890 82 1,768 2,628,572  Leather, tanned, curried, and finished ${\displaystyle \scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}}$ 1900 31 587 2,451,713 1890 51 852 3,363,672  Lumber and timber products ${\displaystyle \scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}}$ 1900 838 6,834 13,489,401 1890 894 11,540 11,849,654  Paper and wood pulp ${\displaystyle \scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}}$ 1900 35 4,851 13,223,275 1890 17 1,509 3,281,051  Printing and publishing, newspapers and periodicals ${\displaystyle \scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}}$ 1900 120 1,309 2,190,017 1890 105 949 1,719,477  Ship and boat building, wooden ${\displaystyle \scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}}$ 1900 115 1,369 2,491,765 1890 85 1,450 2,818,565  Wool manufactures ${\displaystyle \scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}}$ 1900 79 7,155 13,412,784 1890 78 5,067 8,737,653 Transportation and Commerce. Maine is the only one of tlie New England States in which there was a large railroad construction in the last decade of the nineteenth century. This mileage increased from 1377 miles in 1800 to 1928 in 1900. Prior to this decade railway construction had been confined mainly to the central and southwestern parts of the State. During that decade the Bangor and Aroostook line was built into the northeast counties, and its influenoe was largely responsilile for the rapid development of the forest and farming industries which has taken place in that section. The Canadian Pacific crosses the State from east to west. The construction of electric railways, including interurban lines, is increasing rapidly. The electric railway mileage in 1901 aggregated 280 miles. Among the many fine harbors, that of Portland (Casco Bay), especially, is easy of access, deep, large, and well protected, and is often unobstructed by ice when harbors farther west and south are frozen over. Lines of steamers ply regularly between the largest cities of the State and Boston; also between Portland and New York, Saint John, N. B., and Halifax. For the year ending June 30, 1901, the exports from the Portland and Falmouth customs district amounted to$12,403,958 and the imports to $633,114. The chief imports are coal, fish, sugar, iron, molasses, and wool; the chief exports are cotton goods, canned vegetables, boots, shoes, lumber, bacon, hams, etc. Banks. In 1902 there were 86 national banks with loans aggregating$27,837,000; cash, etc., $2,027,000; capital,$10,531,000; and deposits, $26,263,000. In 1901 there was$69,533,058 deposited in savings banks, the total number of depositors being 196,583.

Finance. Receipts of the State Treasury for the two years ending December 31, 1900, were $3,953,647; disbursements,$3,754,768; balance on hand, December 31, 1900, $198,879. The principal sources of State revenue during 1899 and 1900 were: From cities, towns, etc.,$1,815,902; savings banks, $851,566; railroads,$323,052; insurance companies, $131,405; new corporations,$71,565. The total revenue for the State during the two years was $1,783,021. The bonded debt at the close of 1900 was$2,103,000, a reduction of $516,300 since 1890. Outstanding temporary loans against the State aggregated$350,000.

Government. The present Constitution was adopted by the people in town meetings, in December, 1819. To secure an amendment a two-thirds vote of both Houses and a majority vote of the people at their next biennial meeting are necessary. A three months' residence in the State and the ability to read English are required of voters. The suffrage is withheld from paupers, Indians not taxed, and persons under guardianship. Soldiers from the State may vote when serving outside the State.

Legislature. The Legislature, composed of a Senate of 31 members and a House of Representatives of 151 members, elected on the second Monday of September, biennially, meets on the first Wednesday in January. The Representatives are elected from the towns, the Senators from districts bounded by county lines. The Governor's veto is overcome by a two-thirds vote of each House. Revenue bills originate in the Lower House. The power of impeachment rests with the House, the trial of impeachments with the Senate. The State has four representatives in the Lower House of Congress.

Executive. The Governor is elected biennially (a plurality being necessary to election). He has the advice of a council of seven members elected biennially by the Legislature on joint ballot, but only one councilor can be elected from any district prescribed for the election of Senators. The Governor and Council have power to remit penalties and grant reprieves, pardons, and commutations. The president of the Senate and Speaker of the House are respectively in the line of succession to the Governorship in case of vacancy in that office. A Secretary of State and a Treasurer are elected biennially on joint ballot.

Judiciary. The Supreme Court, composed of eight judges, is appointed by the Governor and Council for a term of seven years. The County of Cumberland, embracing the city of Portland, has a Superior Court of one judge appointed in the same way. Probate judges are elected by the people of each county for terms of four years. Judges of inferior courts are appointed by the Governor and Council for terms of seven years. Judges of municipal and police courts are similarly appointed, and hold for four years. An Attorney-General is chosen biennially on joint ballot.

Local Government. A sheriff is elected biennially in each county. The indebtedness of a municipality is limited to 5 per cent. of its valuation.

Other Constitutional or Statutory Provisions. In 1846 Maine passed the first prohibitory liquor law, and in 1884 inserted a prohibition clause in the Constitution. The laws against the manufacture of and the traffic in intoxicating liquors are very strict and supported by severe penalties. Liquors for medicinal, mechanical, and manufacturing purposes are sold in towns and cities under the supervision of State agents. Husbands are not liable for debts contracted by their wives in their own name, but the latter may be sued for them. A wife may hold real and personal estate separately from her husband, and may convey or devise the same by will. The legal rate of interest is 6 per cent., and any rate is allowed on contract.

Militia. In 1900 there were 142,175 males of militia age. In the State militia force there were 93 commissioned officers and 1165 enlisted men.

Population. The following figures indicate the growth of the population: 1790, 96,540; 1810, 228,705; 1830, 399,455; 1850, 583,169; 1860, 628,279; 1870, 626,915; 1880, 648,936; 1890, 661,086; 1900, 694,466. In 1900 Maine ranked in respect to population thirtieth among the States of the Union. The West has drawn heavily upon the population here. The increase in the last decade amounted to 5 per cent., as compared with 20.7 for the United States. The largest increase for the decade, as also for the last half of the century, was in Aroostook County. The foreign population in 1900—mostly Canadian—numbered 93,330. The French element is prominent in the southwestern counties. In 1900 there was an average of 23.2 inhabitants to the square mile. In the same year twenty-five cities had each a population exceeding 4000, and together constituted 36.2 per cent. of the total population.

Cities. The following figures give the population for the largest towns in 1900: Portland, 50,145; Lewiston, 23,761; Bangor, 21,850; Biddeford, 16,145. Augusta is the capital.

Religion. The immigration into the State during recent years, including the large number of French Canadians, has increased the Catholic population until the Catholic communicants number over one-third of the total Church membership. The strongest Protestant denominations in their order are the Baptists, Methodists, and Congregationalists.

Education. In the year 1900, 5.1 per cent. of the population ten years of age and over was illiterate. For the native white population alone it was 2.4 per cent. As in the other New England States, much attention was given here to education from the earliest colonial period. In 1821, soon after the establishment of Statehood, a law was passed requiring every town and plantation to raise and expend not less than forty cents annually per inhabitant for school purposes. A law of 1828 set apart twenty townships of public land for the establishment of a public school fund. This was augmented by certain moneys received from Massachusetts on account of claims against the United States for services in the War of 1812. The district plan of school supervision has been abolished and town management has taken its place. There is a strictly enforced compulsory attendance law which covers the age period from seven to fifteen inclusive. The whole number of scholars in attendance during the school year of 1901 was 132,802, of whom 97,038 were in average daily attendance. In 1901 there were 1349 graded and 2896 ungraded schools. The average length of the school term in 1900 was 141 days. Secondary education was very generally provided for, even before public aid was extended to it, by a large number of private or denominational academies. In 1873 provision was made for State aid to towns which maintained schools fur free academic instruction. This proved a death blow to many of the old academies. In many instances the old academy buildings were occupied by the free high school. In 1889 towns were authorized to contract with any academy or high school for the tuition of their scholars, and to receive the same aid from the State as if they had established a free high school within the town. A large number of academies now receive State aid. In 1901 there were 211 free high schools receiving aid from the State, with a total attendance of 13,183. In 1900 the State employed 6445 teachers, of whom 5360 were women. State normal schools are maintained at Farmington, Castine, and Gorham. In 1895 the State authorized the State superintendent to hold three or more summer schools, each of two weeks' duration, for the benefit of teachers. In 1901, 1408 of the State's teachers had graduated from normal schools. The State has charge of the examination and certification of teachers. The total expenditure for schools in 1900 was $1,712,795, of which$1,229,004 was paid as salaries to teachers and superintendents. Higher education is provided for at the State University at Orono, and by the following denominational colleges: Bowdoin (Congregational) at Brunswick, Bates (Free Baptist) at Lewiston, and Colby (Baptist) at Waterville.

Charitable and Penal Institutions. There are insane hospitals at Augusta and Bangor, an orphan asylum in Bangor, the military and naval orphans' asylum at Bath, and the Maine General Hospital, a school for the deaf, and an eye and ear infirmary, all at Portland. The National Government maintains a soldiers' home at Togus and a marine hospital at Portland. The State prison is at Thomaston, the reform school is near Portland, and the industrial school for girls is at Hallowell. Convicts at the State prison are employed in the manufacture of carriages, harness, furniture, and brooms. At some of the county jails the prisoners make heels and inner soles for boots and shoes. In a few counties the jails are leased, together with the labor of the convicts. Farm labor and certain other lines of work are engaged in at the reform school.

The following have been the Governors of the State since its admission into the Union:

 William King Democrat 1820-21 William D. Williamson (acting) Democrat 1821 Albion K. Parris Democrat 1822-27 Enoch Lincoln “ 1827-29 Nathan Cutter (acting) “ 1829-30 Jonathan D. Hunton “ 1830-31 Samuel E. Smith “ 1831-34 Robert P. Dunlap “ 1834-38 Edward Kent Whig 1838-39 John Fairfield Democrat 1839-40 Edward Kent Whig 1840-41 John Fairfield Democrat 1841-43 Edward Kavanagh “ 1843-44 Hugh J. Anderson “ 1844-47 John W. Dana “ 1847-50 John Hubbard “ 1850-53 William G. Crosby Whig and Free-Soil 1853-55 Anson P. Morrill Republican 1855-56 Samuel Wells Democrat 1856-57 Hannibal Hamlin Republican 1857 Joseph H. Williams (acting) “ 1857-58 Lot M. Morrill “ 1858-60 Israel Washburn “ 1861-63 Abner Coburn “ 1863-64 Samuel Cony “ 1864-67 Joshua L. Chamberlain “ 1867-71 Sidney Perham “ 1871-74 Nelson Dingley “ 1874-76 Selden Connor “ 1876-79 Alonzo Garcelon Democrat-Greenback 1879-80 Samuel F. Davis Republican 1880-81 Harris M. Plaistead Democrat-Greenback 1881-83 Frederick Robie Republican 1883-87 Joseph K. Bodwell “ 1887 Sebastian S. Marble (acting) “ 1887-89 Edwin C. Burleigh “ 1889-93 Henry B. Cleaves “ 1893-97 Llewellyn Powers “ 1897-1901 John F. Hill “ 1901—

Bibliography. Thoreau, The Maine Woods (Boston, 1881); Boardman, “The Climate, Soil, Physical Resources and Agricultural Capabilities of the State of Maine,” in United States Department of Agriculture Miscellaneous Special Report iv. (Washington, 1884); Drake, The Pine Tree Coast (Boston, 1891); Hubbard, Woods and Lakes of Maine (Boston, 1891); McDonald, The Government of Maine, Its History and Administration (New York, 1902); Little, “One Hundred Books on Maine,” in Bowdoin College Library Bulletin (Brunswick, 1891); Hall, “Reference List on Maine Local History,” in New York State Library Bulletin Bibliography, vol. ii. (Albany, 1891); Maine Historical Society Collections (Portland, 1831 et seq.); De Costa, The Northmen in Maine (Albany, 1870); Chamberlain, Maine, Her Place in History (Augusta, 1877); Varney, Brief History of Maine (Portland, 1888); Abbott and Elwell, History of Maine (Portland, 1893).


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# CMLJ Contributor Guidelines

### Scope and Aims of the Journal

CMLJ is the leading international journal for capital markets lawyers and academics. It is a think-tank for this area of law and practice based in the UK (in recognition of the UK’s leading position in this field) but includes material from, and relevant to, all of the major financial centres around the world.

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### CMLJ:

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J Griffiths, ‘Copyright in English Literature: Denying the Public Domain’ [2000] EIPR 150, 151

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text:
Faculty of Physics
M.V.Lomonosov Moscow State University
Regular Article

# The effect of the beam cross-sectional velocity variation on the start conditions of the oscillator backward wave

## D. I. Trubetskov$^1$, G.M. Vdovina$^2$

### Memoirs of the Faculty of Physics 2016. N 5.

• Article
Annotation

Wave propagation in a drifting low-voltage electron beam has been studied. The causes of such a flow instabilities and features of its interaction with the electromagnetic wave have been discussed. The effect of beam cross–sectional velocity variation on the starting conditions BWO has been considered.

Rissian citation:
Д.И. Трубецков, Г.М. Вдовина.
Учен. зап. физ. фак-та Моск. ун-та. 2016. № 5. 165502.
Authors
D. I. Trubetskov$^1$, G.M. Vdovina$^2$
$^1$Department of Electronics, Vibrations, and Waves, Faculty of Nonlinear Processes, National Research Saratov State University Saratov 410012, Russia
$^2$National Research Nuclear University MEPhI (Moscow Engineering Physics Institute) Moscow 115409, Russia

### Science News of the Faculty of Physics, Lomonosov Moscow State University

This new information publication, which is intended to convey to the staff, students and graduate students, faculty colleagues and partners of the main achievements of scientists and scientific information on the events in the life of university physicists.


text:
A priori bounds for the vorticity of axially symmetric solutions to the Navier-Stokes equations

Abstract

We obtain a pointwise, a priori bound for the vorticity of axially symmetric solutions to the three-dimensional Navier-Stokes equations. The bound is in the form of a reciprocal of a power of the distance to the axis of symmetry. This seems to be the first general pointwise upper bound established for the axially symmetric Navier-Stokes equations.

Article information

Source
Adv. Differential Equations Volume 15, Number 5/6 (2010), 531-560.

Dates
First available in Project Euclid: 18 December 2012


text:
# zbMATH — the first resource for mathematics

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Oscillation of third order nonlinear functional dynamic equations on time scales. (English) Zbl 1213.34116

The purpose of the paper is to give oscillation criteria for the third order nonlinear functional dynamic equation

${\left(a\left(t\right){\left[{\left(r\left(t\right){x}^{{\Delta }}\left(t\right)\right)}^{{\Delta }}\right]}^{\gamma }\right)}^{{\Delta }}+f\left(t,x\left(g\left(t\right)\right)\right)=0$

on a time scale $𝕋$, where $\gamma$ is a quotient of odd positive integers, $a$ and $r$ are positive rd-continuous functions on $𝕋$, and the function $g:𝕋\to 𝕋$ satisfies ${lim}_{t\to \infty }g\left(t\right)=\infty$ and $f\in C\left(𝕋×ℝ,ℝ\right)$. Some examples are given to illustrate the main results.

##### MSC:
 34N05 Dynamic equations on time scales or measure chains 34K11 Oscillation theory of functional-differential equations
##### Keywords:
oscillation theory; dynamic equations on time scales
##### References:
 [1] Agarwal R., Bohner M. and Saker S. H., Oscillation of second order delay dynamic equations, Canad. Appl. Math. Quart., 13, 1–17, (2005) [2] Bohner M. and Peterson A., Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, (2001) [3] Bohner M. and Peterson A., Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, (2003) [4] Bohner M. and Saker S. H., Oscillation of second order half-linear dynamic equations on discrete time scales, Internat. J. Difference Equs., 1, 208–218, (2006) [5] Gera M., Graef J. R. and Gregus M., On oscillatory and asymptotic properties of solutions of certain nonlinear third order differential equations, Nonlinear Anal., 32, 417–425, (1998) · Zbl 0945.34021 · doi:10.1016/S0362-546X(97)00483-5 [6] Došlý O. and Hilger E., A necessary and sufficient condition for oscillation of the Sturm-Liouville dynamic equation on time scales, Special Issue on Dynamic Equations on Time Scales (Agarwal P. P., Bohner M. and O’Regan D., eds.), J. Comp. Appl. Math., 141(1–2), 571–585, (2002) [7] Elabbasy E. M. and Hassan T. S., Oscillation of third order nonlinear functional differential equations, Diff. Eq. Appl., submitted [8] Erbe L., Hassan T. S. and Peterson A., Oscillation criteria for nonlinear damped dynamic equations on time scales, Appl. Math. Comp., 203, 343–357, (2008) · Zbl 1162.39005 · doi:10.1016/j.amc.2008.04.038 [9] Erbe L., Hassan T. S. and Peterson A., Oscillation criteria for nonlinear functional neutral dynamic equations on time scales, J. Diff. Eq. Appl., 15, 1097–1115, (2009) · Zbl 1193.34135 · doi:10.1080/10236190902785199 [10] Erbe L., Hassan T. S., Peterson A. and Saker S. H., Oscillation criteria for half-linear delay dynamic equations on time scales, Nonlinear Dynam. Sys. Th., 9, 51–68, (2009) [11] Erbe L., Hassan T. S., Peterson A. and Saker S. H., Oscillation criteria for sublinear half-linear delay dynamic equations on time scales, Int. J. Diff. Equ., 3, 227–245, (2008) [12] Erbe L., Peterson A. and Saker S. H., Asymptotic behavior of solutions of a third-order nonlinear dynamic equation on time scales, J. Comp. Appl. Math., 181, 92–102, (2005) · Zbl 1075.39010 · doi:10.1016/j.cam.2004.11.021 [13] Erbe L., Peterson A. and Saker S. H., Hille and Nehari type criteria for third order dynamic equations, J. Math. Anal. Appl., 329, 112–131, (2007) · Zbl 1128.39009 · doi:10.1016/j.jmaa.2006.06.033 [14] Erbe L., Peterson A. and Saker S. H., Oscillation and asymptotic behavior a third-order nonlinear dynamic equation, Canad. Quart. Appl. Math., 14, 2, (2006) [15] Hardy G. H., Littlewood J. E. and Polya G., Inequalities, second ed., Cambridge University Press, Cambridge, (1988) [16] Hassan T. S., Oscillation criteria for half-linear dynamic equations on time scales, J. Math. Anal. Appl., 345, 176–185, (2008) · Zbl 1156.34022 · doi:10.1016/j.jmaa.2008.04.019 [17] Hilger S., Analysis on measure chains – a unified approach to continuous and discrete calculus, Results Math., 18, 18–56, (1990) [18] Kac V. and Cheung P., Quantum Calculus, Universitext, (2002) [19] Zhang B. G. and Deng X., Oscillation of delay differential equations on time scales, Math. Comp. Mod., 36, 1307–1318, (2002) · Zbl 1034.34080 · doi:10.1016/S0895-7177(02)00278-9 [20] Şahiner Y. and Stavroulakis I. S., Oscillation of first order delay dynamic equations, Dynam. Systems Appl., 15, 645–655, (2006) [21] Wu H., Zhuang R. and Mathsen R. M., Oscillation criteria of second-order nonlinear neutral variable delay dynamic equations, Appl. Math. Comp., 178, 321–331, (2006)


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This is the second part of the in-depth sequence analysis. In the previous post, we processed data to the required format, plotted a Sankey diagram, and did some distribution, frequency, time lapse and entropy analysis with visualization. For dessert, clustering! Clustering is an exploratory data analysis method aimed at finding automatically homogeneous groups or clusters in the data. It simplifies the large number of distinct sequences in a few types of trajectories.

Let’s assume that we want to identify four segments of customers based on their behavior (purchase sequences). We will use the hierarchical clustering method Ward for clustering our customers with the following code:

# CLUSTERING
library(cluster)
df.om <- seqdist(df.seq, method='OM', indel=1, sm='TRATE', with.missing=TRUE) # computing the optimal matching distances
clusterward <- agnes(df.om, diss=TRUE, method="ward") # building a Ward hierarchical clustering
df.cl4 <- cutree(clusterward, k=4) # cut the tree for creating 4 clusters
cl4.lab <- factor(df.cl4, labels=paste("Cluster", 1:4)) # creating label with the number of cluster for each customer

Once we have identified clusters, we can plot three types of graphics we are familiar with from the previous post. These grafics can help us to identify the typical patterns that characterize the clusters. We will start with a distribution analysis for each cluster which shows the state distribution at each time point (the columns of the sequence object), continue with a frequency plot, and finish with a mean time spent in each state plot:

# distribution chart
seqdplot(df.seq, group=cl4.lab, border=NA)
# frequence chart
seqfplot(df.seq, group=cl4.lab, pbarw=T, border=NA)
# mean time plot
seqmtplot(df.seq, group=cl4.lab, border=NA)

It is also possible an advanced approach of clustering. The command below finds and plots the representative set that, with a neighborhood radius of 10% (default tsim value), covers at least 35% (trep parameter) of the sequences in each of the four cl4.lab groups:

seqrplot(df.seq, group=cl4.lab, dist.matrix=df.om, trep=0.35, border=NA)

In the resulting plot the selected representative sequences are plotted bottom-up according to their representativeness score with a bar width proportional to the number of sequences assigned to them. At the top of the plot, two parallel series of symbols standing each for a representative sequence are displayed horizontally on a scale ranging from 0 to the maximal theoretical distance Dmax. The location of the symbol associated with the representative sequence indicates on axis A the discrepancy within the subset of sequences and on axis B the mean distance to the representative sequence.
We learn from the plots that nine, three, one and three representatives, respectively, are necessary for each of the four groups to achieve the 35% coverage and that the actual coverage is 36.5%, 36.4%, 38.3% and 43.6%, respectively.

So, what is the main point of preceding analysis? We can use it for:

• Defining customers who left our company using solutions I mentioned in the previous post and intending to recover them;
• Both predicting customer’s sequences and intending to affect them by recommending the right product at the right time during the customer’s life-time. The right product best matches another one and leads to increase customer’s life-time value;
• Defining unsuccessful combination of products/carts;
• Clustering customers based on their sequences and working with homogeneous segments;

and so on.


text:
# Absolutely Continuous Function is Continuous

 It has been suggested that this article or section be renamed: indicate that function is real One may discuss this suggestion on the talk page.

## Theorem

Let $I$ be an interval of $\R$.

If a function $f: I \to \R$ is absolutely continuous, then it is continuous on $I$.

## Proof

Take any $x \in I$. We will prove that $f$ is continuous at $x$.

In order to do this we will use the $\epsilon$-$\delta$ definition of continuity.

Take any $\epsilon > 0$, and choose the corresponding $\delta > 0$ obtained from the definition of absolute continuity for $f$.

If $y \in I$ is any point such that $\size {y - x} < \delta$, then:

$\size {\map f y - \map f x} < \epsilon$

as $f$ is absolutely continuous.

This is the particular case of just one interval in the definition of absolute continuity.

Hence, $f$ is continuous at $x$.

As $x$ was arbitrary, this proves that $f$ is continuous on all of $I$.

$\blacksquare$


text:
# A mercury manometer (ρ = 13,600 kg/m 3 ) is connected to an air duct to measure the pressure inside.... 1 answer below »

A mercury manometer (ρ = 13,600 kg/m3) is connected to an air duct to measure the pressure inside. The difference

Difference in the manometer levels is 15 mm, and the atmospheric pressure is 100 kPa. (a) Judging from Fig. P1–62, determine if the pressure in the duct is above or below the atmospheric pressure. (b) Determine the absolute pressure in the duct.

FIGURE P1–62

Parimal P
Given data: $$P_atm=100KPa$$ h = 15 mm = 0.015 m ρ = 13,600 kg/m^3 Calculation: a) From the given fig. It's clearly visible that the fluid column in the duct is at the lower level. So, the pressure in the duct is above the atmospheric pressure....

Looking for Something Else? Ask a Similar Question


text:
# Measure theoretic definition of curl

Is there a good measure theoretic definition of curl?

To give an idea of the sort of equation that I'm looking for, here's now I define grad and div. For the gradient, say we are given a Fréchet differentiable function $f:X\rightarrow\mathbb{R}$, then we can define $\nabla f(x)$ to be the element in $X$ such that $$\langle \nabla f(x),\eta \rangle = f^\prime(x)\eta.$$ Hence, $\nabla f(x)$ is the Riesz representative of the Fréchet derivative (Note, we've assumed a Hilbert space.) For the divergence, we have $f:X\rightarrow X$ and can define $$\nabla \cdot f(x) = \lim_{\Omega\rightarrow \{x\}}\frac{1}{\mu(\Omega)} \int_{\Omega} \nabla\cdot f(x) d\mu =\lim_{\Omega\rightarrow \{x\}}\frac{1}{\mu(\Omega)} \int_{\partial \Omega} f(x)\cdot n d\mu.$$ Here, $\lim_{\Omega\rightarrow \{x\}} 1/\mu(\Omega)$ is shorthand for $\lim_{k\rightarrow \infty} 1/\mu(\Omega_k)$ where $\Omega_{k+1}\subseteq\Omega_k$ and $\lim_{k\rightarrow\infty}\Omega_k = \{x\}$. In addition, $\mu$ denotes some measure. In any case, the first equality follows from the Lebesgue differentiation theorem and the second follows from integration by parts. In this way, we require that $X$ be finite dimensional.

Now, I'd like to get something along these lines for curl. More specifically, I'm interested in a definition of curl that does not use differential forms or the exterior calculus.

I haven't checked this closely, but if it helps, I believe that for $f:X\rightarrow X$, we have that $$(\nabla \times f(x))\times \delta x = f^\prime(x)\delta x-f^\prime(x)^*\delta x$$ where $f^\prime(x)^*$ denotes the adjoint of the Fréchet derivative of $f$ at $x$. In other words, the cross product between the curl of $f$ and $\delta x$ is the antisymmetric part of the Fréchet derivative.

-
Not sure if this might be useful (probably not), but you should get the curl if you replace $\cdot$ with $\times$ in the definition of divergence. This holds in classical vector calculus (cfr. Schey, Div grad curl and all of that, Ex. III-29), and should extend to general measure-theoretic setting. –  Giuseppe Negro Jan 7 '13 at 2:16
Maybe this answer helps: mathoverflow.net/questions/52829/… –  Christian Blatter Sep 10 '13 at 15:44

Let $v=(v_i)$ be a vector field. If $c=(c_i)$ is the curl of $v$, as you did with the divergence, you can write:

$$c_i(x) = \lim_{A_i\to \{x\}}\frac{1}{\mu(A_i)} \int_{A_i}(\nabla\times v)_i\,d\mu = \lim_{A_i\to \{x\}}\frac{1}{\mu(A_i)} \int_{\partial A_i} (v\times dx)_i,$$

where: $$(v\times dx)_i = \sum_{j,k}\epsilon_{ijk} v_j\,dx_k$$

are the components of the usual cross product, and $A_i$ are flat surfaces with smooth boundary, orthogonal to $dx_i$. For example, $A_1$ is parallel to the $yz$ plane, and oriented in that way.

Note: integration by parts (componentwise) is exactly equivalent to exterior differentiation and Stokes theorem.

-
I accidentally downvoted this and cannot now change my vote unless you edit your answer. If you do that, just let me know and I'll change my vote. Apologies. –  Quinn Culver Jan 7 '14 at 23:27
Done! Thank you. –  geodude Feb 3 '14 at 16:27


text:
# I'm not electrified by the existence of the blanket [electrical] and [electricity] tags, let's remove them, or at least blacklist them

Both these tags are typically used when the poster isn't intending to dig for an actually descriptive tag. Neither of these tags should be existing, as they apply to every single question on this network.

Both tags are indicative of a bad question:

I don't mean to imply the tag makes the question bad, but tagging something with is a pretty clear indication that the question hasn't seen the required amount of TLE, and the site would, in my humble opinion, be better off if we enforced not tagging with this pseudo-tag.

• I did my best to add a play on words into the title, as is the requirement for any good tag-deletion/burnination question, at least on StackOverflow. Please don't PUNish me if it's not up to standards. Jun 1 '20 at 12:55
• Electrocution would be a fitting end for those tags Jun 1 '20 at 13:50
• But, but... almost all of my EE questions involve electricity! :) Jun 2 '20 at 11:21
• I was thinking about that the other day. It is pretty ridiculous that those tags exist. It's almost like if the Physics SE had a science tag or if Stack Overflow had a code tag LOL.
– user103380
Jun 3 '20 at 3:29
• Soemone has added "status review" tag so it should automagically be being escalated.
– Russell McMahon Mod
Jun 8 '20 at 11:14
• There is also the electric tag. Jun 9 '20 at 9:08
• @Lundin: electrical electricity or electronical electricity? Jun 11 '20 at 19:34
• This is in the CM team backlog, and we'll get back to you when we have updates.
– JNat StaffMod
Sep 1 '20 at 14:22
• @JNat um, I feel bad for asking/pointing this out, but these two tags aren't blacklisted for use, so there's new daily additions, what's the state of this? Sep 14 '20 at 7:14
• I still don't have a date for you, @MarcusMüller, and will update y'all as soon as I have it. The CM team is working on this initiative currently, and I'll see if I can squeeze this in this week (no promises, though!).
– JNat StaffMod
Sep 14 '20 at 9:41
• @JNat no pressure! The question was more whether we can simply blacklist the tag now so that we have less to care about later. Sep 14 '20 at 12:12
• This issue is now in Dev and will be worked on as a part of that queue.
– Juan M StaffMod
Sep 29 '20 at 13:31
• @JuanM thank you, that's good news! Sep 29 '20 at 14:00

I agree with you, but as I'm currently the only "active" mod (not active, really), I'll leave it open to collect some votes at least. And I hope that fresh new mod(s) will be willing to work on these issues, too.

These tags look like a candidate for intrinsic tags.

Intrinsic tags are

term[s] that can apply to all questions on the site [...]

Hopefully, this meta post can also function as a formal request to the Community Manager to blacklist them, as it is also stated,

post a request on your meta and we'll mark it as intrinsic.

(a moderator may need to edit and add to this question, as per What posts should be escalated to staff using [status-review], and how do I escalate them?)

• I don't know the inner workings of the mod tools, but I believe mods nowadays are able to "burninate" tags without involving CMs. Some mods over at SO have been very active in tag clean-up in the past few years. At any rate, widely used tags like these at least need moderator support to get rid of. Jun 4 '20 at 6:41
• @Lundin you're correct that burninating can be done by community and mods, but blacklisting need CM. I made an edit to make it more clear it's about blacklist. Jun 4 '20 at 7:08

These tags are intrinsic, they need to go away

In general, all adjective-only tags are bad: they change the meaning depending on which noun they are applied to. "Electrical" is obviously meaningless without a noun, or at best equivalent to "Electricity".

"Electricity" is essentially a negative tag - it can be applied to any electromagnetic phenomenon in which the magnetic part is insignificant. This is also not useful, similarly to a "non-java" tag on Stack Overflow or "non-windows" tag on Super User.

• Would something like "electrical-field" be more suitable for questions where the magnetic part is insignificant? And btw I now also found the "field" tag... for all our farming questions. Jun 9 '20 at 9:07

These tags are now blocked

two intrinsic tags