If you have taken this stroll in the morning, and you have the time and inclination, walk to the right along the crowded Corso for half a dozen blocks to visit the fine private collection of paintings -- mainly of the sixteenth and seventeenth centuries -- in the Palazzo Doria ( open Sunday, Tuesday, and Thursday, 10:00 to 1:00 ).

* The Lusternik – Schnirelmann theorem: If the sphere S < sup > n </ sup > is covered by n + 1 open sets, then one of these sets contains a pair ( x, − x ) of antipodal points.

* If the metric space X is compact and an open cover of X is given, then there exists a number such that every subset of X of diameter < δ is contained in some member of the cover.

But now, intermarriage is often the result of living in an open society .... If our children end up marrying non-Jews, we should not reject them.

If a user added more than one drive to a system the user had to open the case and cut a trace in the circuit board to permanently change the drive's number, or hand-wire an external switch to allow it to be changed externally.

If a rotating frame is chosen so that just the angular position of an object is held fixed, more complicated motion, such as elliptical and open orbits, appears because the centripetal and centrifugal forces will not balance.

If the content or presentation of a course is inconsistent with preconceived information, the student will learn if he or she is open to varying concepts.

If is continuous in an open set Ω and the partial derivatives of ƒ with respect to x and y exist in Ω, and satisfies the Cauchy – Riemann equations throughout Ω, then ƒ is holomorphic ( and thus analytic ).

* If ƒ ( z ) is locally integrable in an open domain Ω ⊂ C, and satisfies the Cauchy – Riemann equations weakly, then ƒ agrees almost everywhere with an analytic function in Ω.

If the door opens inward and there is a fire, there can be a crush of people who run for the door and they will not be able to open it.

If we allow also the system boundary to move ( e. g. due to moving pistons ) we get a rather general form of the first law for open systems.

If people are also free to move between the countries, in addition to FTA, it would also be considered an open border.

Presheaves: If X is a topological space, then the open sets in X form a partially ordered set Open ( X ) under inclusion.

If the receiving partner has wounds on his genitals, or if the giving partner has wounds or open sores on or in his or her mouth, or bleeding gums, this poses an increased risk of STD transmission.

If these conditions can change, they are better represented by two predicates open ( t ) and on ( t ) that depend on time ; such predicates are called fluents.

If the open string is in tune, but sharp or flat when frets are pressed, the bridge saddle position can be adjusted with a screwdriver or hex key to remedy the problem.

If not blocked or countered, the open row of three will be extended to an open row of four, which threatens to win in two ways.

( i ) If A is open,

If ƒ is complex differentiable at every point z < sub > 0 </ sub > in an open set U, we say that ƒ is holomorphic on U. We say that ƒ is holomorphic at the point z < sub > 0 </ sub > if it is holomorphic on some neighborhood of z < sub > 0 </ sub >.

If an open verdict is returned, the inquest can be reopened if new evidence is found and presented to the coroner.

If there is a crystallographic structure to the target, and especially in semiconductor substrates where the crystal structure is more open, particular crystallographic directions offer much lower stopping than other directions.

# If A is an open or closed subset of R < sup > n </ sup > ( or even Borel set, see metric space ), then A is Lebesgue measurable.

# If A is a Lebesgue measurable set, then it is " approximately open " and " approximately closed " in the sense of Lebesgue measure ( see the regularity theorem for Lebesgue measure ).

If other parts of the program need memory ( a part assigned to open and close the door, for example ), then someone may be trapped inside, since the software cannot open the door.

If Mahzeer was planning to set up the prime minister for Muller he would have to do it in the next few minutes.

If the defendant pleads not guilty, a date is set for a preliminary hearing or a trial.

If they don't then for the algorithm to be effective it must provide a set of rules for extracting a square root.

If we cannot make explicit choices, how do we know that our set exists?

If we try to choose an element from each set, then, because X is infinite, our choice procedure will never come to an end, and consequently, we will never be able to produce a choice function for all of X.

** If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem ; see the section " Weaker forms " below.

** If the set A is infinite, then there exists an injection from the natural numbers N to A ( see Dedekind infinite ).

If the automorphisms of an object X form a set ( instead of a proper class ), then they form a group under composition of morphisms.

* If a set S is in M and S is congruent to T then T is also in M and a ( S )

* Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i. e. S ⊆ Q ⊆ T. If there is a unique number c such that a ( S ) ≤ c ≤ a ( T ) for all such step regions S and T, then a ( Q )

* If M is some set and S denotes the set of all functions from M to M, then the operation of functional composition on S is associative:

* If you want to be happy, set a goal that commands your thoughts, liberates your energy, and inspires your hopes.

If a is algebraic over K, then K, the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. In the special case where K = Q is the field of rational numbers, Q is an example of an algebraic number field.

If X and Y are Banach spaces over the same ground field K, the set of all continuous K-linear maps T: X → Y is denoted by B ( X, Y ).

If the score is tied the set is halved.

If each player wins a set, or both sets end tied, there is a 3-end tiebreaker to determine a winner.

If a node has fewer than two children, some of the child pointers may be set to a special null value, or to a special sentinel node.

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.

In the words of Geoffrey Crowther, then editor of The Economist, " If the economic relationships between nations are not, by one means or another, brought fairly close to balance, then there is no set of financial arrangements that can rescue the world from the impoverishing results of chaos.

If, for instance, an addition operation was requested, the arithmetic logic unit ( ALU ) will be connected to a set of inputs and a set of outputs.

If the addition operation produces a result too large for the CPU to handle, an arithmetic overflow flag in a flags register may also be set.