If A is a directed set, we often write a net from A to X in the form ( x < sub > α </ sub >), which expresses the fact that the element α in A is mapped to the element x < sub > α </ sub > in X.

Haupt's critical work is distinguished by a combination of the most painstaking investigation with bold conjecture ; his oft-cited dictum that " If the sense requires it, I am prepared to write Constantinopolitanus where the MSS have the monosyllabic interjection o " well expresses this boldness.

If the and are real numbers, a more elegant proof is available: the identity expresses the fact that the absolute value of the product of two quaternions is equal to the product of their absolute values, in the same way that the Brahmagupta – Fibonacci two-square identity does for complex numbers.

If the domain is the set of all real numbers, the following second-order sentence expresses the least upper bound property:

: If a solid body is enclosed on all sides by another solid body, of the two bodies that one first became hard which, in the mutual contact, expresses on its own surface the properties of the other surface.

If one expresses oneself in a particular fashion, and someone says " that's true ", is there anything useful at all that can be said in general terms about the relationship between these two acts?

As she expresses it, “ If you don ’ t like the way the world is, you have an obligation to change it.

If expresses on other sectors are included, there was a period in the early 1990s when British Rail operated more 100 mph services per day than any other country.

If symbolic, it is probably taking up the Biblical language found in Amos 5: 27, " therefore I shall take you into exile beyond Damascus "; Damascus was part of Israel under King David, and the Damascus Document expresses an eschatalogical hope of the restoration of a Davidic monarchy.

" If Not Now When " expresses anger against those claim they want positive change but are too apathetic to act.

If the disfellowshipped member expresses repentance and abides by the conditions imposed upon him or her, disfellowshipment usually lasts approximately one year.

If, in the trustee's judgment, `` reasonable market conditions '' did not prevail during any given year, he was to be allowed to petition the court for an extension of time within the ten-year period.

If it is assumed that the formula given by Lodge of cosec Af applies, the pressure difference along the major axes can be calculated from the angle of inclination of the major axis, and from this the interfacial tension can be calculated.

If one assumes that the average flux did not change between measurements, a mass-distribution curve is obtained which relates the flux of particles larger than a given radius to the inverse 7/2 power of the radius.

If, however, we are seeking the optimal R-stage policy for a given feed state, only one search for a maximum is required at the final step.

If the latter knows nothing about the absent sitter except his name ( given by the experimenter ), he cannot possibly give any clues, conscious or unconscious, far less ask leading questions.

`` If you substitute ' atom ' for ' angel ', the problem is not insoluble, given the metallic content of the pin in question ''.

If a defendant has been convicted and can prove that his lawyer did not adequately handle his case and that there is a reasonable probability that the result of the trial would have been different had the lawyer given competent representation, he is entitled to a new trial.

** Trichotomy: If two sets are given, then either they have the same cardinality, or one has a smaller cardinality than the other.

** 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 condition is thought to be cellulitis rather than abscess, consideration should be given to possibility of strep species as cause that are still sensitive to traditional anti-staphylococcus agents such as dicloxacillin or cephalexin in patients able to tolerate penicillin ).

If we use a cylindrical coordinate system with basis vectors, then the gradient of and the divergence of are given by

If the resulting four kilobyte table size is too large for a given target platform, the table lookup operation can be performed with a single 256-entry 32-bit ( i. e. 1 kilobyte ) table by the use of circular rotates.

* If the balance factor of P is-2 then the right subtree outweighs the left subtree of the given node, and the balance factor of the right child ( R ) must be checked.

* If the balance factor of P is 2, then the left subtree outweighs the right subtree of the given node, and the balance factor of the left child ( L ) must be checked.

If the path of the particle or centre of mass of the rigid body passes through the given origin, its angular momentum is zero.

If two equivalent amplifiers are being compared, the amplifier with higher gain settings would be more sensitive as it would take less input signal to produce a given amount of power.

If the norm of a Banach space satisfies this identity, the associated inner product which makes it into a Hilbert space is given by the polarization identity.

If BCG is accidentally given subcutaneously, then a local abscess may form ( a BCG-oma ) that can sometimes ulcerate, and may require treatment with antibiotics.

If given subcutaneously, it may induce local infection and spread to the regional lymph nodes, causing either suppurative and nonsuppurative lymphadenitis.

If BCG is accidentally given to an immunocompromised patient ( e. g., an infant with SCID ), it can cause disseminated or life-threatening infection.

If players cannot move either die in a roll, given the position of their checkers then that turn is over and the turn passes to the opponent.

If a source emits a known luminous intensity I < sub > v </ sub > ( in candelas ) in a well-defined cone, the total luminous flux Φ < sub > v </ sub > in lumens is given by

He explained that this is one of the most important analects: " If the people be led by laws, and uniformity sought to be given them by punishments, they will try to avoid the punishment, but have no sense of shame.

If they be led by virtue, and uniformity sought to be given them by the rules of propriety, they will have the sense of the shame, and moreover will become good.

* 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.

If the vertex is at Af, and if the interior of C is on the left as one moves in the direction of increasing t, then every such corner can be found from the curve obtained by rotating C clockwise through 90-degrees about the vertex.

If not, the curve is subdivided parametrically into two segments, 0 ≤ t ≤ 0. 5 and 0. 5 ≤ t ≤ 1, and the same procedure is applied recursively to each half.

If the substituents are numbered from 1 ( highest priority ) to 4 ( lowest priority ), then the sense of rotation of a curve passing through 1, 2 and 3 distinguishes the stereoisomers.

If the cable is heavy then the resulting curve is between a catenary and a parabola.

If the density of the chain is variable then the analysis above can be adapted to produce equations for the curve given the density, or given the curve to find the density.

If f is a continuous function, meaning that its graph is an unbroken curve with no gaps, then Q is a continuous function away from.

If production of one good increases along the curve, production of the other good decreases, an inverse relationship.

If the positions of the two synchronized stations are known, then the position of the receiver can be determined as being somewhere on a particular hyperbolic curve where the time difference between the received signals is constant.

If the demand curve shifted the marginal revenue curve would shift as well and a new equilibrium and supply " point " would be established.

If there is a downward-sloping demand curve then by necessity there is a distinct marginal revenue curve.

If the piston is slowly lowered, the system will trace a curve of increasing temperature and pressure within the gas region of the phase diagram.

* If R denotes the ring CY of polynomials in two variables with complex coefficients, then the ideal generated by the polynomial Y < sup > 2 </ sup > − X < sup > 3 </ sup > − X − 1 is a prime ideal ( see elliptic curve ).

If there were really two different memory stores operating in this time frame, one could expect a discontinuity in this curve.

If we let Δx and Δy be the distances ( along the x and y axes, respectively ) between two points on a curve, then the slope given by the above definition,

This is because each point on the supply curve is the answer to the question " If this firm is faced with this potential price, how much output will it be able to and willing to sell?

This is because each point on the demand curve is the answer to the question " If this buyer is faced with this potential price, how much of the product will it purchase?

If the demand decreases, then the opposite happens: a shift of the curve to the left.

If the supply curve starts at < tt > S2 </ tt >, and shifts leftward to < tt > S1 </ tt >, the equilibrium price will increase and the equilibrium quantity will decrease as consumers move along the demand curve to the new higher price and associated lower quantity demanded.

If the ball is spinning counterclockwise, it will curve right from the hitter's point of view and curve left if spinning clockwise.