With the vertex at Af in the C-plane we assume that Af is the parametric location on C of an ordinary intersection Q between C and Af.

The fact that there can not be any limit points of the set except in closed intervals follows from the argument used in Lemma 1, namely, that near any tangent point in the C-plane the curves C and Af are analytic, and therefore the difference between them must be a monotone function in some neighborhood on either side of the tangent point.

A tangent point Q in the C-plane occurs when C and Af are tangent to one another.

An ordinary point will be any point of intersection A such that in every neighborhood of A in the C-plane, Af meets both the interior and the exterior of C.

We know that in the C-plane both C and Af are analytic.

We will refer to the plane of C and Af as the C-plane and to the plane of the graph as the Aj.

We must now show that on some component of the graph there exist two points for which the corresponding diagonal points in the C-plane are on opposite sides of C.

Still, we must remember that we cannot construct and justify generalizations of this sort unless we are ready to consider many special instances of influence moving between such areas as theology, philosophy, political thought, and literature.

On C, from the point P at Af to the point Q at Af, we construct the chord, and upon the chord as a side erect a square in such a way that as S approaches zero the square is inside C.

Using the above theorem it is easy to see that the original Borsuk Ulam statement is correct since if we take a map f: S < sup > n </ sup > → ℝ < sup > n </ sup > that does not equalize on any antipodes then we can construct a map g: S < sup > n </ sup > → S < sup > n-1 </ sup > by the formula

Since all problems in NP can be reduced to this problem it follows that for all problems in NP we can construct a non-deterministic Turing machine that decides the complement of the problem in polynomial time, i. e., NP is a subset of co-NP.

Whereas objectivism is concerned with the " object of our knowledge ", constructivism emphasises " how we construct knowledge ".

We can indeed enumerate algorithms to construct a function T, about which we initially assume that it is a function from the natural numbers onto the reals.

: When we set out to construct a given discipline, we distinguish, first of all, a certain small group of expressions of this discipline that seem to us to be immediately understandable ; the expressions in this group we call PRIMITIVE TERMS or UNDEFINED TERMS, and we employ them without explaining their meanings.

: To construct U is to write down the definition of a general-recursive function U ( n, x ) that correctly interprets the number n and computes the appropriate function of x. to construct U directly would involve essentially the same amount of effort, and essentially the same ideas, as we have invested in constructing the universal Turing machine.

The coproduct is given by the disjoint union: given sets A < sub > i </ sub > where i ranges over some index set I, we construct the coproduct as the union of A < sub > i </ sub >×

Given any vector space V over K we can construct the tensor algebra T ( V ) of V. The tensor algebra is characterized by the fact:

In a way, by working with these particular schemata building blocks, we have reduced the complexity of our problem ; instead of building high-performance strings by trying every conceivable combination, we construct better and better strings from the best partial solutions of past samplings.

For every dyadic fraction r ∈ ( 0, 1 ), we are going to construct an open subset U ( r ) of X such that:

" " if we have prior knowledge about a topic in a text, we construct meaning based on our experience, and we can adjust and change those plans as we go along.

Montero had set up a strong position, using every bale and box we had in addition to barricades of logs and brush.

And here again we hear the same refrain mentioned above: `` the paramount goal of the United States set long ago was to guard the rights of the individual, ensure his development, enlarge his opportunity ''.

We feel uncomfortable at being bossed by a corporation or a union or a television set, but until we have some knowledge about these phenomena and what they are doing to us, we can hardly learn to control them.

So when textbooks, like that of Baker set out drawings of the ' Ptolemaic System ', complete with earth in the center and the seven heavenly bodies epicyclically arranged on their several deferents, we have nothing but a misleading 20th-century idea of what never existed historically.

Criticism is as old as literary art and we can set the stage for our study of three moderns if we see how certain critics in the past have dealt with the ethical aspects of literature.

In entering this union we will be surrendering most, if not all, of our economic autonomy to international bodies such as the Atlantic Institute ( recently set up ) or the O.E.C.D., I.M.F. and others.

During the war, we set up schools for the teaching of psychological warfare, which included the teaching of propaganda, both black and white and the various shades of grey in between.

They set bouncing betties to jump and explode at testicle level while we more mercifully had them go off at the head.

The extent to which we can persuade the less developed countries to appraise their own resources, to set targets toward which they should be working, to establish in the light of this forward perspective the most urgent priorities for their immediate attention, and to do the other things which they must do to help themselves, all on a realistic long-term basis, will depend importantly on the incentives we place before them.

Personally, I think we ought to set up an immediate naval blockade of Cuba.

`` We, the Subscribers, do agree, that as soon as a convenient Number of Persons have subscribed to this, or a similar Writing, We will present a petition to the Hon'ble General Court of the Commonwealth of Massachusetts, praying for an Act incorporating into a Body politic the subscribers to such Writing with Liberty to build such a Bridge, and a Right to demand a Toll equal to that received at Malden Bridge, and on like Terms, and if such an Act shall be obtained, then we severally agree each with the others, that we will hold in the said Bridge the several shares set against our respective Names, the whole into two hundred shares being divided, and that we will pay such sums of Money at such Times and in such Manners, as by the said proposed Corporation, shall be directed and required ''.

With each vertex we associate certain numerical values, namely the set of positive differences in the parameter T between the vertex and its corresponding forward corner points.

When we look at countries like Iran, Iraq, Pakistan, and Burma, where substantial progress has been made in creating a minimum supply of modern men and of social overhead capital, and where institutions of centralized government exist, we find a second category of countries with a different set of problems and hence different priorities for policy.

When an evaluative situation is set up, and no concern is with the details that lead to an over-all estimate, we say that roleplaying is used for evaluation.

Without such uncertainty we are left with a set of dogmas and myths.

In our own practice, to have the last `` intonaco '' plaster coat thick enough to match, and at the same time to avoid fine cracks in drying, we found that it had to be put on in two layers, letting the first set awhile before applying the second.

This is the principle which we will invoke in every case to set up a functional equation.

In practical situations there will be restrictions on the admissible operating conditions, and we regard the vectors as belonging to a fixed and bounded set S.

Within the map plane, we have rectangular coordinates and.

It is easy enough to " translate " between polar and rectangular coordinates in the plane: let, as above, direction and distance be and respectively, then we have

Thus for the rectangular guide the Laplacian is evaluated, and we arrive at

The plans of all the Assyrian buildings are rectangular, and we know that long ago, as now, the Eastern architects used this outline almost invariably, and upon it reared some of the most lovely and varied forms ever devised.

First we pack the pennies in a rectangular grid.

For example, in ( early ) tensor analysis elasticity, namely Young's Modulus, should not change simply because we measure it with tensile testing of a rectangular object ( a bar ) or cylindrical one ( a rod ).

The general form of rejection sampling assumes that the board is not necessarily rectangular but is shaped according to some distribution that we know how to sample from ( e. g. using inversion sampling ), and which is at least as high at every point as the distribution we want to sample from, so that the former completely encloses the latter.

But if we apply the fractional Fourier transform to the rectangular signal, the transformation output will be in the domain between time and frequency.

First we pack the pennies in a rectangular grid.

Clockwise from top left, we see a hinge-mounted stamp, stamp about to be hinge-mounted, stamp damaged by a hinge, and loose stamp hingesMultiple hinge remnants are visible on the back of this stamp. In philately stamp hinges or mounts are small, folded, transparent, rectangular pieces of paper coated with a mild gum.

From Alambra and Marki in central Cyprus we know that the houses were rectangular with many rooms, with lanes allowing people to move freely in the community.

Especially when using rectangular grid finite-difference models ( e. g. MODFLOW, made by the USGS ), we deal with Cartesian coordinates.

The form of the squares becomes more rectangular the longer north we come.

For cylindrical or rectangular necks, we have