If numerical iterative methods have to be employed, the aim is to iterate until full machine accuracy is obtained ( the best that is possible with a finite word length on the computer, and within the mathematical and / or physical approximations made ).

If we consider the thesis and its converse as definition, then the hypothesis is an hypothesis about the application of the mathematical theory developed from the definition.

If a mathematical function is used to represent the production function, and if that production function is homogeneous, returns to scale are represented by the degree of homogeneity of the function.

If one agrees that set theory is an appealing foundation of mathematics, then all mathematical objects must be defined as sets of some sort.

If a mathematical surface can be intersected by a ray, it can be rendered using ray casting.

If T is a theory whose objects of discourse can be interpreted as natural numbers, we say T is arithmetically sound if all theorems of T are actually true about the standard mathematical integers.

" If we mistakenly confuse precision with accuracy, then we might be misled into thinking that an explanation expressed in precise mathematical or graphical terms is somehow more rigorous or useful than one that takes into account particulars of history, institutions or business strategy.

If the two loose ends of an overhand knot are joined together ( without creating additional crossings ), this becomes equivalent to the trefoil knot of mathematical knot theory.

If mathematics is a part of logic, then questions about mathematical objects reduce to questions about logical objects.

If propositions are sets of possible worlds, however, then all mathematical truths are the same set ( the set of all possible worlds ).

If the names of the academic papers are reversed, they describe real mathematical events.

If modern mathematical practices are what distinguish modern professional mathematicians from older ideas of folk mathematics.

If mathematics has been informally used throughout history, in numerous cultures and continents, then it could be argued that " mathematical practice " is the practice, or use, of mathematics in everyday life.

If the first block has index 1, the mathematical formula for CBC encryption is

Describing this situation after many decades, Ambartsumian said, " If an astronomer publishes an article with a mathematical content in a physics journal, then the most likely thing that will happen to it is oblivion.

If Göttingen had been the “ Mecca ” of mathematics in the 1920s and early ‘ 30s, Princeton, following the decimation of German mathematics under the Nazis, had become the center of the mathematical world in the 1940s.

If one inserts a more realistic barrier model into the simplest form of the Schrödinger equation, then an awkward mathematical problem arises over the resulting differential equation: it is known to be mathematically impossible in principle to solve this equation exactly in terms of the usual functions of mathematical physics, or in any simple way.

If more exact values for v ( f ) are required, then provides formulae that give values for v ( f ) to an absolute mathematical accuracy of better than 8 × 10 < sup >− 10 </ sup >.

# If a human m is a sufficiently skillful mathematical logician ( equipped with a sufficiently powerful computer if necessary ) then if m is given L ( m ), he or she can construct T ( L ( m )) and

If an ideal, mathematical point source of light is placed on-axis in the input plane of the first lens, then there will be a uniform, collimated field produced in the output plane of the first lens.

If any mathematical methods can aid in the growth of such wisdom, then catastrophe theory will be part of them.

If we define a decimal expansion such that some digit is dependent on some yet unsolved mathematical problem, we know beforehand that we cannot tell if this number is the same as some other decimal expansion which is independent of this problem.

If one is to use Gödel's technique to prove the proposition that T cannot prove, one must first prove ( the mathematical statement representing ) the consistency of T, a daunting and perhaps impossible task.

If you could see all your opponents ' cards, you would always be able to calculate the correct decision with mathematical certainty.

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 you drive greater distances than that, you'll just be skimming the surface and will never discover the enchantment, fascination and beauty which lured you in the first place to explore the hinterlands.

If, however, a layer of paper remains stuck to the surface, the adhesive has not failed.

If there is refraction at a collective spherical surface, or through a thin positive lens, O ' 2 will lie in front of O ' 1 so long as the angle u2 is greater than u1 ( under correction ); and conversely with a dispersive surface or lenses ( over correction ).

If the above errors be eliminated, the two astigmatic surfaces united, and a sharp image obtained with a wide aperture — there remains the necessity to correct the curvature of the image surface, especially when the image is to be received upon a plane surface, e. g. in photography.

If rotation is arrested, the energy stored in fashioning the concave surface must be dissipated, for example through friction, before an equilibrium flat surface is restored.

If a player is unsure of whether a bet is a single or multi-roll bet, it can be noted that all single-roll bets will be displayed on the playing surface in one color ( usually red ), while all multi-roll bets will be displayed in a different color ( usually yellow ).

If the ball has a rough surface, the fluid flowing past it will make it rotate.

where Q is the electric charge transferred through the surface over a time t. If Q and t are measured in coulombs and seconds respectively, I is in amperes.

If the missile starts to roll, the gyroscopic force of the disk drives the control surface into the airflow, cancelling the motion.

If a particle of the beam crosses this surface, there will be some kind of interaction.

If an organism ( such as bacteria ) attaches to a valve surface and forms a vegetation, the host immune response is blunted.

If we choose the shape of the control volume such that all flow in or out occurs perpendicular to its surface, then the flow of matter into the system performs work as if it were a piston of fluid pushing mass into the system, and the system performs work on the flow of matter out as if it were driving a piston of fluid.

If a previous fresco was being painted over, the surface would be roughened to give a key.

If a solid surface is chilled below the dew point of the surrounding air and the surface itself is colder than freezing, frost will form on the surface.

If the solid surface is rough, and the liquid is in intimate contact with the solid asperities, the droplet is in the Wenzel state.

If the imaging surface is cleaned of contaminants and residual moisture before the second image is taken, then no image of the missing section would appear.

* If creatures similar to plesiosaurs lived in the waters of the Loch Ness, they would be seen very frequently as they would have to surface several times a day to breathe.

If the fluid being measured is significantly dense, hydrostatic corrections may have to be made for the height between the moving surface of the manometer working fluid and the location where the pressure measurement is desired except when measuring differential pressure of a fluid ( for example across an orifice plate or venturi ), in which case the density ρ should be corrected by subtracting the density of the fluid being measured.

If the universe is static and populated by an infinite number of stars, any sight line from Earth must end at the ( very bright ) surface of a star, so the night sky should be completely bright.