It is possible, although highly doubtful, that he killed none at all but merely let his reputation work for him by privately claiming every unsolved murder in the state.

Yet long before the scheduled time for return, Donovan would be watching for every speck in the sky.

The Nazis knew this, of course, and while their chief quarry was the industrial centers, they let a few drop every time they went over, hoping for a lucky hit.

Opposite every gate was a hitching post or a stone carriage-step, set with a rusty iron ring for tying a horse.

In town after town my companion pointed out the Negro school and the White school, and in every instance the former made a better appearance ( it was newer, for one thing ).

As shown in Figure 1, there is a connection for communication between every pair of points.

Just as in the case of every prodigy child, we must watch for the efficacy of my teaching to show up in the future -- if he should master all the strenuous exercises I inflicted on him.

The steady purpose of our society is to assure justice, before God, for every individual.

After we got a script and the spots for the songs were blocked out, we'd get together for an hour or so every day.

The White House had chewing gum until it could chew no more, and every Christmas, Mr. Wrigley sent the President a check for $100, to be divided among all the help.

Probably every visitor has a favorite time for his first sight of it.

When we went for our walks Lilly's brother would come along every once in a while.

Several times in my youth I voted the Socialist ticket, but less because I was Socialist than because I was not either a Republican or a Democrat, and I voted for Franklin Roosevelt every time he was a candidate.

It would present a forum for them in almost every community.

This sum spent for foreign economic aid, the peace corps, food for peace, or any other program to solve the problems of the underdeveloped countries would be an investment that would pay off in world peace, increased world trade, and prosperity for every country on the globe.

If the railroads, for example, regularly slaughtered 25,000 passengers each year, the high priests of the cult would have cause to tremble for their personal safety, for such a holocaust would excite demands for the hanging of every railroad president in the United States.

The highway system is an agency of government, and when it grinds up 40,000 Americans every year the government is destroying its own taxpayers, which is obviously a silly thing for any government to do.

Even apart from the fact that now at the age of 31 my personal life is being totally disrupted for the second time for no very compelling reason -- I cannot help looking around at the black leather jacket brigades standing idly on the street corners and in the taverns of every American city and asking myself if our society has gone mad.

Some authors require in addition that μ ( C ) < ∞ for every compact set C. If a Borel measure μ is both inner regular and outer regular, it is called a regular Borel measure.

Note that a locally finite Borel measure automatically satisfies μ ( C ) < ∞ for every compact set C.

In this case, is the smallest σ-algebra that contains the open intervals of R. While there are many Borel measures μ, the choice of Borel measure which assigns for every interval is sometimes called " the " Borel measure on R. In practice, even " the " Borel measure is not the most useful measure defined on the σ-algebra of Borel sets ; indeed, the Lebesgue measure is an extension of " the " Borel measure which possesses the crucial property that it is a complete measure ( unlike the Borel measure ).

To clarify, when one says that the Lebesgue measure is an extension of the Borel measure, it means that every Borel measurable set E is also a Lebesgue measurable set, and the Borel measure and the Lebesgue measure coincide on the Borel sets ( i. e., for every Borel measurable set ).

That this form of compactness holds for closed and bounded subsets of Euclidean space is known as the Heine – Borel theorem.

Weyl brought the early period of the development of the theory of Lie groups to fruition, for not only did he classify irreducible representations of semisimple Lie groups and connect the theory of groups with quantum mechanics, but he also put Lie's theory itself on firmer footing by clearly enunciating the distinction between Lie's infinitesimal groups ( i. e., Lie algebras ) and the Lie groups proper, and began investigations of topology of Lie groups ( Borel ( 2001 ), ).

Since there is a one-to-one correspondence between Borel regular measures in the interval and functions of bounded variation ( that assigns to each function of bounded variation the corresponding Lebesgue-Stieltjes measure, and the integral with respect to the Lebesgue-Stieltjes measure agrees with the Riemann-Stieltjes integral for continuous functions ), the above stated theorem generalizes the original statement of F. Riesz.

A measure μ on the Borel subsets of G is called left-translation-invariant if for all Borel subsets S of G and all a in G one has

* μ ( gE ) = μ ( E ) for any g in G and Borel set E ( left-translation-invariance ).

The left Haar measure satisfies the inner regularity condition for all σ-finite Borel sets, but may not be inner regular for all Borel sets.

This guarantees for any partition of a set X the existence of a subset C of X containing exactly one element from each part of the partition.

: For any set A there is a function f such that for any non-empty subset B of A, f ( B ) lies in B.

In this view, Christianity is seen as a religion in its own right, rather than a subset of Judaism, if one makes the common assumption that Judaism is not universal, however see Noahide Laws and Christianity and Judaism for details.

In analytic geometry, any equation involving the coordinates specifies a subset of the plane, namely the solution set for the equation.

Sometimes the name Sporozoa is taken as a synonym for the Apicomplexa, or occasionally as a subset.

This extension is needed for, among other things, modeling the concepts of " is an element of " or " is a subset of " in set theory, without running into logical inconsistencies such as Russell's paradox.

Conversely, a subset R defines a binary function if and only if, for any x in X and y in Y, there exists a unique z in Z such that ( x, y, z ) belongs to R.

An ideal of the Boolean algebra A is a subset I such that for all x, y in I we have x ∨ y in I and for all a in A we have a ∧ x in I.

For example, in the mid-Atlantic US, the X-Acto name is likelier to evoke only a specific subset of these knives ( the pencil-shaped hobby knife ), which may explain why the " utility knife " name, with its specificity, is more common there for the larger type.

Basic English, also known as Simple English, is an English-based controlled language created ( in essence as a simplified subset of English ) by linguist and philosopher Charles Kay Ogden as an international auxiliary language, and as an aid for teaching English as a Second Language.

), along with a 50-word list from a more specialised subset of that general field, to make a basic 1000 words vocabulary for everyday work and life.

* Convex cone, a subset C of a vector space V is a convex cone if αx + βy belongs to C, for any positive scalars α, β, and any x, y in C

In two dimensions, closed disks are compact since for any infinite number of points sampled from a disk, some subset of those points must get arbitrarily close either to a point within the disc, or to a point on the boundary.

It is also possible for a proper subset of an infinite set to have the same cardinality as the original set, something that cannot happen with proper subsets of finite sets.

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.

Even outside the Acme :: hierarchy, some modules are still written largely for amusement ; one example is Lingua :: Romana :: Perligata, which can be used to write Perl programs in a subset of Latin.

Others, especially words belonging to the large subset of collective nouns known as terms of venery ( words for groups of animals ), are specific to one kind of constituent object.

Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.

for some given functions α ( x, y ) and β ( x, y ) defined in an open subset of R < sup > 2 </ sup >.

However, modern x86 processors also ( typically ) decode and split instructions into dynamic sequences of internally buffered micro-operations, which not only helps execute a larger subset of instructions in a pipelined ( overlapping ) fashion, but also facilitates more advanced extraction of parallelism out of the code stream, for even higher performance.

Programs for reading documents may not be required to read the external subset.

This use allows notations to be defined only in a DTD stored as an external entity and referenced only as the external subset of documents, and allows these documents to remain compatible with validating XML or SGML parsers that have no direct support for notations.

Most XML schema languages are only replacements for element declarations and attribute list declarations, in such a way that it becomes possible to parse XML documents with non-validating XML parsers ( if the only purpose of the external DTD subset was to define the schema ).