If competition beats you to it, this exciting new product era can have real headaches in store.

If the photographically realistic continuity of dreams, however bizarre their combinations, denies that it is purely a composition of the brain, it must be compounded from views of diverse realities, although some of them may never be encountered in what we are pleased to call the real life.

I kept saying, `` If I could just build up a reputation for myself, make some real money, get to be well known as an illustrator -- like Peter Askington, for instance -- then I could take some time off and paint ''.

If one quantizes a real scalar field, then one finds that there is only one kind of annihilation operator ; therefore, real scalar fields describe neutral bosons.

Existence is the principle that gives reality to an essence not the same in any way as the existence: " If things having essences are real, and it is not of their essence to be, then the reality of these things must be found in some principle other than ( really distinct from ) their essence.

If X is a Banach space and K is the underlying field ( either the real or the complex numbers ), then K is itself a Banach space ( using the absolute value as norm ) and we can define the continuous dual space as X ′ = B ( X, K ), the space of continuous linear maps into K.

If X is a real Banach space, then the polarization identity is

If A is a self-adjoint operator, then is always a real number ( not complex ).

If the Czech ethnic origin was to be stressed, combinations like " Bohemian of Bohemian language "( Čech českého jazyka ), " a real Bohemian " ( pravý Čech ) etc.

If the exponent r is even, then the inequality is valid for all real numbers x.

If the wave function is regarded as ontologically real, and collapse is entirely rejected, a many worlds theory results.

If wave function collapse is regarded as ontologically real as well, an objective collapse theory is obtained.

If the eigenvalues are all positive, then the frequencies are all real and the stationary point is a local minimum.

If zeal for orthodoxy caused him to overstep the limits of discretion, his real attitude towards Rome is sufficiently clear.

If a simply stable system response neither decays nor grows over time, and has no oscillations, it is marginally stable: in this case the system transfer function has non-repeated poles at complex plane origin ( i. e. their real and complex component is zero in the continuous time case ).

If this limit exists, then it may be computed by taking the limit as h → 0 along the real axis or imaginary axis ; in either case it should give the same result.

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 a DBMS system responses users ' request in a given time period, it can be regarded as a real time database.

If the matrix entries are real numbers, the matrix can be used to represent two linear mappings: one that maps the standard basis vectors to the rows of, and one that maps them to the columns of.

If this operation has to be done in real time video games there is an easy trick to boost performance.

* If we think of as the set of real numbers, then the direct product is precisely just the cartesian product,.

* If we think of as the group of real numbers under addition, then the direct product still consists of.

* If we think of as the ring of real numbers, then the direct product again consists of.

The best known example is the ring of quaternions H. If we allow only rational instead of real coefficients in the constructions of the quaternions, we obtain another division ring.

If we are discussing differentiable complex-valued functions, then Af and V are complex vector spaces, and Af may be any complex numbers.

If more than one eigenvalue is negative, then the stationary point is a more complex one, and is usually of little interest.

If X is Cauchy distributed with median μ and scale parameter γ, then the complex variable

If the weight of the cable and supporting wires are not negligible then the analysis is more complex.

If a stitch in a finished crocheted item breaks, the stitches above and below remain intact, and because of the complex looping of each stitch, the stitches on either side are unlikely to come loose unless heavily stressed.

If we look for solutions that have the form, where C is a constant, we discover the relationship, and thus must be one of the complex numbers or.

If < math > c ^ 2 < 4km </ math > there are two complex conjugate roots a ± ib, and the solution ( with the above boundary conditions ) will look like this:

If anything, the research has shown the situation is even more complex than initially imagined, and we believe not enough is known to structure the classification of psychiatric disorders according to etiology.

If the obstructing object provides multiple, closely spaced openings, a complex pattern of varying intensity can result.

If the ether is part of a more complex molecule, it is described as an alkoxy substituent, so-OCH < sub > 3 </ sub > would be considered a " methoxy -" group.

If the cell is synthesizing DNA each pore complex needs to transport about 100 histone molecules per minute.

If the cell is growing rapidly, each complex also needs to transport about 6 newly assembled large and small ribosomal subunits per minute from the nucleus to the cytosol, where they are used to synthesize proteins.

If it is unaffected, then it is a simple partial seizure ; otherwise it is a complex partial ( psychomotor ) seizure.

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 a complex function = is holomorphic, then u and v have first partial derivatives with respect to x and y, and satisfy the Cauchy – Riemann equations:

If the coordinates are chosen badly, the laws of motion may be more complex than necessary.

In The Origin of Species, he wrote, " If it could be demonstrated that any complex organ existed, which could not possibly have been formed by numerous, successive, slight modifications, my theory would absolutely break down.

If ( i ) or ( j ) have side effects or introduce run time errors, the subtle differences between ( l ) and ( l ') can be complex.

If, however, the set of allowed candidates is expanded to the complex numbers, every non-constant polynomial has at least one root ; this is the fundamental theorem of algebra.

* 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 ).