Union And Intersection Quotes

While I am interested both in economics and in philosophy, the union of my interests in the two fields far exceeds their intersection

There will be times in which things appear hopeless. You will begin to doubt everything around you. You will even begin to doubt yourself. You will think things will never look up and you may be in the deepest, darkest, loneliest place in the world. Everything which had once been infused with wonder may appear disappointing and harsh. You may grow cynical and come to believe that this is simply the way the world is...that one must bear with the unforgiving realities of the world and only hope that it doesn’t get worse. You might grow suspicious of others, as adults tend to do, and close yourself off from the rest of the world. You might just look to the past and reminisce about better days...or you might just dwell in one place for a little too long and become nostalgic for the future. Just remember—regardless of where you are, what experiences you have, and who you have become—that there will always be those who have loved you. Those whom you may have taken for granted, but have nonetheless, always had you in their hearts and in their hopes and wishes. Lives that you have touched: whether you realize it or not. To separation you may venture, but indissolubly in union shall you drift...you will always be at the whims of forces, both great and small, and far beyond your capacity to control. That’s how all our stories go. Innumerable arcs intersect and scatter into a vast indefinite sea.

We face so many overlapping and intersecting crises that we can't afford to fix them one at a time. We need integrated solutions, solutions that radically bring down emissions while creating huge numbers of good, unionized jobs and delivering meaningful justice to those who have been most abused and excluded under the current extractive economy.

Innumerable arcs intersect and scatter into a vast indefinite sea.

Here are the words that have brought me to a new understanding. Here are the words that will bind us forever. From this day forward, I will speak your name with gratitude, knowing it is the mantra of my soul. I will let you go, knowing we are eternal. We were born to walk this world in intersecting lines. We are circles and signposts and parallels. I have left markers for you at every turn. Look for me in everything that catches your breath. Let the simple miracle of your own presence overwhelm you. For you are beautiful, in ways that can't be described. And we are love at its most inexplicable. With these words, I am one with divinity. With these words, I am one with you.

Assume that P is a second-species infinite point-set. Cantor shows that P's first derived set, P', can be "decomposed" or broken down into the union of two different subsets, Q and R, where Q is the set of all points belonging to first-species derived sets of P', and R is the set of all points that are contained in every single derived set of P', meaning R is the set of just those points that all the derived sets of P' have in common. Why not take a second and read that last sentence over again. R is the important part, and it's actually how Cantor first defines 'intersection' for sets, here via the infinite sequence of derived sets P', P'', P''',...(the sequence being infinite because P is a second-species-set).

Finally, Europe’s post-war history is a story shadowed by silences; by absence. The continent of Europe was once an intricate, interwoven tapestry of overlapping languages, religions, communities and nations. Many of its cities—particularly the smaller ones at the intersection of old and new imperial boundaries, such as Trieste, Sarajevo, Salonika, Cernovitz, Odessa or Vilna—were truly multicultural societies avant le mot, where Catholics, Orthodox, Muslims, Jews and others lived in familiar juxtaposition. We should not idealise this old Europe. What the Polish writer Tadeusz Borowski called ‘the incredible, almost comical melting-pot of peoples and nationalities sizzling dangerously in the very heart of Europe’ was periodically rent with riots, massacres and pogroms—but it was real, and it survived into living memory. Between 1914 and 1945, however, that Europe was smashed into the dust. The tidier Europe that emerged, blinking, into the second half of the twentieth century had fewer loose ends. Thanks to war, occupation, boundary adjustments, expulsions and genocide, almost everybody now lived in their own country, among their own people. For forty years after World War Two Europeans in both halves of Europe lived in hermetic national enclaves where surviving religious or ethnic minorities the Jews in France, for example—represented a tiny percentage of the population at large and were thoroughly integrated into its cultural and political mainstream. Only Yugoslavia and the Soviet Union—an empire, not a country and anyway only part-European, as already noted—stood aside from this new, serially homogenous Europe. But since the 1980s, and above all since the fall of the Soviet Union and the enlargement of the EU, Europe is facing a multicultural future. Between them refugees; guest-workers; the denizens of Europe’s former colonies drawn back to the imperial metropole by the prospect of jobs and freedom; and the voluntary and involuntary migrants from failed or repressive states at Europe’s expanded margins have turned London, Paris, Antwerp, Amsterdam, Berlin, Milan and a dozen other places into cosmopolitan world cities whether they like it or not.

A peculiar use is made of the cross symbol by the Muyscas Indians, of Peru; two ropes are stretched crosswise over the surface of the water (pool or stream), and fruits, oil, and precious stones are thrown in as a sacrifice at the point of intersection.145 Here the divinity is evidently the water, not the cross, which only signifies the place of sacrifice. The symbolism is somewhat obscure. Water, and particularly deep water, usually has a maternal significance, roughly corresponding to “womb.” The point of intersection of the two ropes is the point of union where the “crossing” takes place. (Note the double meaning of this word! According to all the analogies, the aim of fertility magic is to bring about the increase of the things marked for sacrifice.)

On April 1, 1865, in Virginia, Pickett was defending an intersection known as Five Forks, six miles south of the Appomattox River and a good bit closer to the Southside Railroad, the last remaining supply line to Richmond. While thirty thousand Union troops led by Little Phil Sheridan approached from the southeast, Pickett’s twelve thousand, spread two miles wide behind fences and in ditches, braced to meet them. Pickett’s supreme commander, Robert E. Lee, was headquartered ten miles away, near Petersburg. Should Pickett fall to Sheridan, Lee would be forced from Petersburg, the Federals would capture Richmond, and the Confederate cause would be lost. Someone mentioned shad. The spring spawning run was in full penetration of the continent. The fish were in the rivers. Tom Rosser, another Confederate general, had caught some, and on the morning of April 1st ordered them baked for his midday dinner, near Hatcher’s Run, several miles from Five Forks. He invited Pickett and Major General Fitzhugh Lee, nephew of Robert E. Lee, to join him. Pickett readily accepted, and rode off from his battle station with Lee. The historian Shelby Foote continues the narrative (“The Civil War,” vol. 3, p. 870): “Neither told any subordinate where he was going or why, perhaps to keep from dividing the succulent fish too many ways; with the result that when the attack exploded—damped from their hearing, as it was, by a heavy stand of pines along Hatcher’s Run—no one knew where to find them. Pickett only made it back to his division after half its members had been shot or captured, a sad last act for a man who gave his name to the most famous charge in a war whose end was hastened by his threehour absence at a shad bake.

Braid groups have many important practical applications. For example, they are used to construct efficient and robust public key encryption algorithms.7 Another promising direction is designing quantum computers based on creating complex braids of quantum particles known as anyons. Their trajectories weave around each other, and their overlaps are used to build “logic gates” of the quantum computer.8 There are also applications in biology. Given a braid with n threads, we can number the nails on the two plates from 1 to n from left to right. Then, connect the ends of the threads attached to the nails with the same number on the two plates. This will create what mathematicians call a “link”: a union of loops weaving around each other. In the example shown on this picture, there is only one loop. Mathematicians’ name for it is “knot.” In general, there will be several closed threads. The mathematical theory of links and knots is used in biology: for example, to study bindings of DNA and enzymes.9 We view a DNA molecule as one thread, and the enzyme molecule as another thread. It turns out that when they bind together, highly non-trivial knotting between them may occur, which may alter the DNA. The way they entangle is therefore of great importance. It turns out that the mathematical study of the resulting links sheds new light on the mechanisms of recombination of DNA. In mathematics, braids are also important because of their geometric interpretation. To explain it, consider all possible collections of n points on the plane. We will assume that the points are distinct; that is, for any two points, their positions on the plane must be different. Let’s choose one such collection; namely, n points arranged on a straight line, with the same distance between neighboring points. Think of each point as a little bug. As we turn on the music, these bugs come alive and start moving on the plane. If we view the time as the vertical direction, then the trajectory of each bug will look like a thread. If the positions of the bugs on the plane are distinct at all times – that is, if we assume that the bugs don’t collide – then these threads will never intersect. While the music is playing, they can move around each other, just like the threads of a braid. However, we demand that when we stop the music after a fixed period of time, the bugs must align on a straight line in the same way as at the beginning, but each bug is allowed to end up in a position initially occupied by another bug. Then their collective path will look like a braid with n threads. Thus, braids with n threads may be viewed as paths in the space of collections of n distinct points on the plane.10

The Banach-Tarski Theorem is an astonishing result. We have decomposed a ball into finitely many pieces, moved around the pieces without changing their size or shape, and then reassembled them into two balls of the same size as the original. I think the theorem teaches us something important about the notion of volume. As noted earlier, it is an immediate consequence of the theorem that some of the Banach-Tarski pieces must lack definite volumes and, therefore, that not every subset of the unit ball can have a well-defined volume. A little more precisely, the theorem teaches us that there is no way of assigning volumes to the Banach-Tarski pieces while preserving three-dimensional versions of the principles we called Uniformity and (finite) Additivity in chapter 7. (Proof: Suppose that each of the (finitely many) Banach-Tarski pieces has a definite finite volume. Since the pieces are disjoint, and since their union is the original ball, Additivity entails that the sum of the volumes of the pieces must equal the volume of the original ball. But Uniformity ensures that the volume of each piece is unchanged as we move it around. Since the reassembled pieces are disjoint, and since their union is two balls, Additivity entails that the sum of their volumes must be twice the volume of the original ball. But since the volume of the original ball is finite and greater than zero, it is impossible for the sum of the pieces to equal both the volume of the original ball and twice the volume of the original ball.) If I were to assign the Banach-Tarski Theorem a paradoxicality grade of the kind we used in chapter 3, I would assign it an 8. The theorem teaches us that although the notion of volume is well-behaved when we focus on ordinary objects, there are limits to how far it can be extended when we consider certain extraordinary objects - objects that can only be shown to exist by assuming the Axiom of Choice.

The Times ad marked a seminal intersection in the history of cancer. With it, cancer declared its final emergence from the shadowy interiors of medicine into the full glare of public scrutiny, morphing into an illness of national and international prominence. This was a generation that no longer whispered about cancer. There was cancer in newspapers and cancer in books, cancer in theater and in films: in 450 articles in the New York Times in 1971; in Aleksandr Solzhenitsyn's Cancer Ward, a blistering account of a cancer hospital in the Soviet Union; in Love Story, a 1970 film about a twenty-four-year-old woman who dies of leukemia; in Bang the Drum Slowly, a 1973 release about a baseball catcher diagnosed with Hodgkin's disease; in Brian's Song, the story of the Chicago Bears star Brian Piccolo, who died of testicular cancer. A torrent of op-ed pieces and letters appeared in newspapers and magazines.

A person gathers all their resources to compose a foursquare philosophy for surviving each day, an engagement driving at a union of seemly inapposite associations to spotlight an androgyny of inspiration for living better. Combating self-alienation, roving after dusk without a map, unsure of the topography that lies ahead, a sincere pathfinder tentatively picks their way by using penetrating low beams and flashing wide-angle high beams. Only by continuing on the bewildering path, can we find what we seek. The writer peers into the encasement of gloom seeking out a deferential of lightness and darkness in the midst of the incongruous elements that foreshadow a person’s peripatetic quest to steer a meaningful life. By displaying the coexistence and intersection of blackened sequential realism overlaid on a snowy field of internalized temporal legend, the narrator assiduously lumbers to shed a ban of moonlight on the battered pages of their brash secular existence.

Chapel Parking Lot Access: From Denver, drive I-70 west 80 miles to exit 195. Turn right into Copper Mountain Resort. Continue approximately 1 mile, then turn left into the chapel parking lot. Walk south and west to the Village Center Plaza in the area of the American Eagle chairlift. To intersect the CT, go about 150 yards diagonally southeast between the condos on the left and the American Eagle lift on the right, Segment 8, mile 1.6. Union Creek Ski Area Access: Instead of parking at the chapel lot, continue through Copper Mountain Village to the Union Creek drop-off parking area. (Parking is permitted here during off-ski-season months.) Cross over the covered bridge, go past the ticket office and under the elevated walkway, turn right on a gravel road, pass under a ski lift, and go about 200 yards on the road. Turn left up the road, around a green security gate, and follow the road east uphill about 400 yards to a wide area in the road. Pick up the single-track of the CT to the right, by a painted white rock, mile 2.1 of CT Segment 8. Tennessee Pass Trailhead Access: See Segment 9 on page 128.

Non-monopolists exaggerate their distinction by defining their market as the intersection of various smaller markets: British food ∩ restaurant ∩ Palo Alto Rap star ∩ hackers ∩ sharks Monopolists, by contrast, disguise their monopoly by framing their market as the union of several large markets: search engine ∪ mobile phones ∪ wearable computers ∪ self-driving cars

My short time in Pretoria made me realize that it can best be described as that place where the brushstrokes of life blend the old with the new in a way that helps to create a story of a place that will forever be deeply tucked into the breathing spaces of my heart, as a place of fondness. A reminder that even when the lessons doesn’t go according to plan, there are always chances to be like the statue of President Nelson Mandela, open arms – embracing the future and using the past, especially the most difficult chapters, to help to infuse new life through the wisdom gained by being like the middle part of the Union Buildings, a space of collaboration. In the words of South African British poet William Polmer, “Creativity is the power to connect the seemingly unconnected.” And when the connection is made, that place is simply called Pretoria. And if one should look a little deeper at the connection, you’ll understand that Pretoria is simply a word with a Latin origin, Praetor, that means Leader, a perfect place to house the Union Buildings, the place where our difference becomes one, and that knowledge becomes the spectrum of where the old and the new intersect, and we call that… Pretoria…Leader within.

The intersection of faith and human rights provided Carter with some of his most gratifying moments in office. In 1979 Carter and Brezhnev completed what Carter described as a “highly emotional” prisoner swap, trading two Soviet spies being held in the United States for four dissidents in the USSR, including three Jewish refuseniks and Georgi Vins, a Russian Baptist pastor jailed in 1974 for conducting an underground ministry in the Soviet Union. A mere four days after he was transported from prison in a Siberian cattle car, Vins joined the president in Plains for church. “Up early to prepare my Sunday school lesson, from 1 Kings 21—about Ahab, Jezebel, and Naboth,” Carter wrote in his diary. “Since Georgi Vins is with me, the parallel between this lesson and his persecution in the Soviet Union was remarkable.” Vins, sitting next to Rosalynn in the pew, pulled off his shoe, lifted the inner sole, and showed her a small, wrinkled photograph of Jimmy Carter he had kept in prison.

Bush went off to a tour of “the caves” beneath the city that had been arranged for him by Deng Xiaoping. It was a memorable Christmas Day. “Dig tunnels deep,” Mao had ordered his people, “store grain everywhere.” Mao’s mission: to create a means by which his people might survive a nuclear assault from the Soviet Union. The result: a vast project to create enormous underground shelters across China. Bush was met at an intersection and taken to a clothing store, where his tour guide pressed a concealed button that operated a trapdoor. They climbed down into the tunnels. Bush walked from room to room. There was enough space, he thought, for thousands in this subterranean kingdom.