* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).

If a Banach algebra has unit 1, then 1 cannot be a commutator ; i. e., for any x, y ∈ A.

If is a wrapped Cauchy distribution with the parameter representing the parameters of the corresponding " unwrapped " Cauchy distribution in the variable y where, then

If u ( t ) is the control signal sent to the system, y ( t ) is the measured output and r ( t ) is the desired output, and tracking error, a PID controller has the general form

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 the axes are named x, y, and z, then the x coordinate is the distance from the plane defined by the y and z axes.

If ( x, y ) are the Cartesian coordinates of a point, then (− x, y ) are the coordinates of its reflection across the second coordinate axis ( the Y axis ), as if that line were a mirror.

To make a figure larger or smaller is equivalent to multiplying the Cartesian coordinates of every point by the same positive number m. If ( x, y ) are the coordinates of a point on the original figure, the corresponding point on the scaled figure has coordinates

If the index finger of the right hand is pointed forward, the middle finger bent inward at a right angle to it, and the thumb placed at a right angle to both, the three fingers indicate the relative directions of the x -, y -, and z-axes in a right-handed system.

If S is an arbitrary set, then the set S < sup > N </ sup > of all sequences in S becomes a complete metric space if we define the distance between the sequences ( x < sub > n </ sub >) and ( y < sub > n </ sub >) to be, where N is the smallest index for which x < sub > N </ sub > is distinct from y < sub > N </ sub >, or 0 if there is no such index.

If x and y are real numbers, and if the graph of y is plotted against x, the derivative measures the slope of this graph at each point.

If the function f is not linear ( i. e. its graph is not a straight line ), however, then the change in y divided by the change in x varies: differentiation is a method to find an exact value for this rate of change at any given value of x.

If y = f ( x ) is differentiable at a, then f must also be continuous at a.

If X < sub > k </ sub > and Y < sub > k </ sub > are the DFTs of x < sub > n </ sub > and y < sub > n </ sub > respectively then the Plancherel theorem states:

If y is a point where the vector field v ( y ) ≠ 0, then there is a change of coordinates for a region around y where the vector field becomes a series of parallel vectors of the same magnitude.

If ~ is an equivalence relation on X, and P ( x ) is a property of elements of X, such that whenever x ~ y, P ( x ) is true if P ( y ) is true, then the property P is said to be well-defined or a class invariant under the relation ~.

If f: X → Y morphism of pointed spaces, then every loop in X with base point x < sub > 0 </ sub > can be composed with f to yield a loop in Y with base point y < sub > 0 </ sub >.

If the circumstances are faced frankly it is not reasonable to expect this to be true.

If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.

If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.

If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;

If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.

If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.

If it is an honest feeling, then why should she not yield to it??

If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.

If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.

If he is good, he may not be legal ; ;

If the man on the sidewalk is surprised at this question, it has served as an exclamation.

If the existent form is to be retained new factors that reinforce it must be introduced into the situation.

If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??

If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.

If he is a traditionalist, he is an eclectic traditionalist.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.

If your child is under 19 or is a student you may also claim an exemption for him if he qualifies as your dependent, even though he earns $600 or more.

If one component changes another component's vapor pressure, or if the volatility of a component is dependent on its percentage in the mixture, the law will fail.

If not, the null hypothesis is supported ( or, more correctly, not rejected ), meaning no effect of the independent variable ( s ) was observed on the dependent variable ( s ).

" If a dependent variable takes a while to fully absorb a shock.

If the under-twenty group and those above the age of sixty are regarded as a dependent group, the social, political, and economic implications for the 1990s and beyond are as grave for Ghana as they are for sub-Saharan Africa as a whole.

If an external time dependent force is present, the harmonic oscillator is described as a driven oscillator.

If too large a dose of opioid antagonist is given to a dependent patient, it will result in withdrawal symptoms ( possibly severe ).

If, however, a statute which did not itself contain any limitation is to be governed by another which is temporary only, the former will also be temporary and dependent upon the existence of the latter.

If generation of phenotypes is conditional, and dependent on external or environmental inputs, evolution can proceed by a " phenotype-first " route, with genetic change following, rather than initiating, the formation of morphological and other phenotypic novelties.

If the Battle of Solebay had not prevented it, they would have been able to force the Dutch population to surrender by starvation, as it was dependent for its survival on supplies of Baltic grain.

Bertrand Russell discussed the paradox briefly in § 38 of The Principles of Mathematics ( 1903 ), distinguishing between implication ( associated with the form " if p, then q "), which he held to be a relation between unasserted propositions, and inference ( associated with the form " p, therefore q "), which he held to be a relation between asserted propositions ; having made this distinction, Russell could deny that the Tortoise's attempt to treat inferring Z from A and B is equivalent to, or dependent on, agreeing to the hypothetical " If A and B are true, then Z is true.

If they are interest-relative, then they must be dependent on the existence of minds which are capable of having interested perspectives.

If minds are wholly dependent on brains, and brains on biochemistry, and biochemistry ( in the long run ) on the meaningless flux of the atoms, I cannot understand how the thought of those minds should have any more significance than the sound of the wind in the trees.

If, however, salutary acts can in virtue of the Divine justice give the right to an eternal reward, this is possible only because they themselves have their root in gratuitous grace, and consequently are of their very nature dependent ultimately on grace, as the Council of Trent emphatically declares ( Sess.

If an identically distributed sequence is independent, then the sequence is exchangeable ; however, the converse is false --- there exist exchangeable random variables that are statistically dependent, for example the Polya urn model.

If there are dependent sources in the circuit, another method must be used such as connecting a test source across A and B and calculating the voltage across or current through the test source.

If X is a domain of x and P ( x ) is a predicate dependent on x, then the universal proposition is expressed in Boolean algebra terms as

If the latter is chosen, then morality is no longer dependent on God, defeating the divine command theory.

If the material is destined for vinyl release, additional processing, such as dynamic range reduction, frequency dependent stereo – to – mono fold-down and equalization, may be applied to compensate for the limitations of that medium.

If the functions f < sub > i </ sub > are linearly dependent, then so are the columns of the Wronskian as differentiation is a linear operation, so the Wronskian vanishes.

For n functions of several variables, a generalized Wronskian is the determinant of an n by n matrix with entries D < sub > i </ sub >( f < sub > j </ sub >) ( with 0 ≤ i < n ), where each D < sub > i </ sub > is some constant coefficient linear partial differential operator of order i. If the functions are linearly dependent then all generalized Wronskians vanish.

* If you look for the self within the body, you can not find it there, since the body itself is dependent upon its parts.

* If you look for the self within the mind, you can not find it there, since the mind can only be said to exist in relation to external objects ; therefore the mind is also dependent upon causes and conditions outside of itself.

If the disease is introduced in agroforestry areas dependent upon Monterey Pine, such as New Zealand, it could have catastrophic effects in those countries as well.