From a more advanced perspective, each of these definitions may be interpreted as giving the unique analytic continuation of e < sup > x </ sup > to the complex plane.

From the definitions above, it is evident that, and.

From a predicative viewpoint it is not valid to quantify over all real numbers in the process of generating a new real number, because this is believed to lead to a vicious-circle problem in the definitions.

By virtue of these interpretive connections, the network can function as a scientific theory: From certain observational data, we may ascend, via an interpretive string, to some point in the theoretical network, thence proceed, via definitions and hypotheses, to other points, from which another interpretive string permits a descent to the plane of observation.

From the definitions above, the sensitivity of this classifier is.

From these definitions we can say that ΔU is the energy added to the system, ΔF is the total work done on it, ΔG is the non-mechanical work done on it, and ΔH is the sum of non-mechanical work done on the system and the heat given to it.

From the definitions of Fresnel integrals, the infinitesimals dx and dy are thus:

From dictionary definitions, Frame ( 2002 ) insists that this is a stronger term than " inerrant ".

From 1964 until the opening of the line in 1995, Mansfield was, by some definitions, the largest town in Britain without a railway station, all the more remarkable because the town pioneered the railway in the East Midlands.

From the various definitions of high-fired ceramics, it is agreed that the earliest stoneware is encountered in the late Shang dynasty in China, with large quantities in production by the Han dynasty.

From current EASA definitions: Indicated airspeed means the speed of an aircraft as shown on its pitot static airspeed indicator calibrated to reflect standard atmosphere adiabatic compressible flow at sea level uncorrected for airspeed system errors.

From the above definitions, it is clear that there is a relationship between adaptedness and fitness ( a key population genetics concept ).

In Simon Prideaux's " From Organisational Theory to the New Communitarium of Amitai Etzioni ," he argues that Etzioni's communitarian methods are based upon earlier functionalist definitions of organisations.

From field experiments she conducted in the 1970s with the Dani people of Papua New Guinea, Rosch concluded that when categorizing an everyday object or experience, people rely less on abstract definitions of categories than on a comparison of the given object or experience with what they deem to be the object or experience best representing a category.

Suppose S is in a microstate indexed by m. From the above definitions, the total energy of the system S * is given by

* Human Intelligence: From Sleepers to Walk-ins, Thomas Patrick Carroll, Syllabus, 5 September 2006 – 24 October 2006 — many good definitions with historic examples and timely discussion of problems ; in outline form.

From the above definitions, some useful relationships can be derived by use of basic algebra.

From 2005 to late 2009, the Google search engine definitions feature, in the top-right corner of the site, was linked to Answers. com.

From dictionary definitions, Frame ( 2002 ) insists that " infalliblity " is a stronger term than " inerrancy.

From the above definitions it follows that is the first derivative of concentration with respect to time, i. e. the change in concentration with time.

From 6 – 10 years, children can understand meanings of words based on their definitions.

From the above definitions it follows that is the first derivative of concentration with respect to time, i. e. the change in concentration with time.

From the above definitions it follows that is the first derivative of concentration with respect to time, i. e. the change in concentration with time.

From the terraces -- eighteen in all -- broad flights of steps descend into the water or onto still more terraces barely above the level of the river.

From above one could only occasionally catch a glimpse of life on the floor of this green sea: a neighbor's gingham skirt flashing into sight for an instant on the path beneath her grape-arbor, or the movement of hands above a clothesline and the flutter of garments hung there, half-way down the block.

From above me and somewhere behind me, a rifle cracked.

From the early Middle Ages until after the Second Vatican Council the sacrament was administered, within the Latin Church, only when death was approaching and, in practice, bodily recovery was not ordinarily looked for, giving rise, as mentioned above to the name " Extreme Unction " ( i. e. final anointing ).

From the above, it is possible to reformulate the definition to either of the following:

From the above equation, it can be re-arranged to show that

From a geostructural perspective the Azores is located above an active triple junction between three of the world's large tectonic plates ( the North American Plate, the Eurasian Plate and the African Plate ), a condition that has translated into the existence of many faults and fractures in this region of the Atlantic.

From the above rule of calculation of heat with respect to volume, there follows one with respect to pressure.

From the Shockley ideal diode equation given above, it might appear that the voltage has a positive temperature coefficient ( at a constant current ), but usually the variation of the reverse saturation current term is more significant than the variation in the thermal voltage term.

From the above properties an important feature arises: Looking at two triangles ABD and BCD with the common edge BD ( see figures ), if the sum of the angles α and γ is less than or equal to 180 °, the triangles meet the Delaunay condition.

From another perspective, the farmer and the miller both reduce a risk and acquire a risk when they sign the futures contract: the farmer reduces the risk that the price of wheat will fall below the price specified in the contract and acquires the risk that the price of wheat will rise above the price specified in the contract ( thereby losing additional income that he could have earned ).

From the above equations, it can be shown that the spin singlet can also be written as

From the variables defined above, we find two other common expression for the value produced during a given period as:

From Muhammad IV ( 1859 – 1873 ) and Hassan I ( 1873 – 1894 ) the Alaouites tried to foster trading links, above all with European countries and the United States.

From his remarks on the importance of public, observable behavior ( as opposed to private experiences ), it may seem that Wittgenstein is simply a behaviorist — one who thinks that mental states are nothing over and above certain behavior.

From Persia, in medieval times polo spread to the Byzantines ( who called it tzykanion ), and after the Muslim conquests to the Ayyubid and Mameluke dynasties of Egypt and the Levant, whose elites favoured it above all other sports.

From above we can deduce that the eigenvalues of each σ < sub > i </ sub > are ± 1.

From a knowledge of the probabilities of each of these subprocesses – E ( A to C ) and P ( B to D ) – then we would expect to calculate the probability of both happening by multiplying them, using rule b ) above.

From the information above and the average cost per employee over time, or cost accrual ratio, a project manager can estimate:

* From the above property, one can deduce the following: If X is a separable space having an uncountable closed discrete subspace, then X cannot be normal.

From the above stated equations it can be expressed as follows:

From a variety of sites which were offered, Shrum recommended to the provincial government that the summit of Burnaby Mountain, 365 meters above sea level, be chosen for the new university.