Using the Fibonacci sequence would keep the number of grains within any one field at minimum.

Using this method, scientists and farmers hope to keep the number of resistant genes very low, and use genetic drift to ensure any resistance that does emerge does not spread.

Using the number 2 would give the symbol for singular " You "; adding the plural indicator ( a small cross at the top ) would produce the pronouns " We " and plural " You ".

Using a three-digit Julian day number saves one byte of computer storage over a two-digit month plus two-digit day, for example, " January 17 " is 017 in Julian versus 0117 in month-day format.

Using the diagonal argument, it is possible to define a real number x, which is not equal to G ( n ) for any n. This means that there is a language L ' that defines x, which is undefinable in L.

Using its authority under Section 5 of the FTC Act, which prohibits unfair or deceptive practices, the FTC has brought a number of cases to enforce the promises in corporate privacy statements, including promises about the security of consumers ’ personal information.

Using base-10 ( the familiar decimal notation ) as an example, the number 152853. 5047, which has ten decimal digits of precision, is represented as the significand 1528535047 together with an exponent of 5 ( if the implied position of the radix point is after the first most significant digit, here 1 ).

Using this sequence, he constructs nested intervals whose intersection contains a real number not in the sequence.

Using their data on the time occupied by travel to work, Table 2 below shows the absolute number of people who responded with travel times " at least 15 but less than 20 minutes " is higher than the numbers for the categories above and below it.

Using that value, perform a right-shift operation by that number of bits on the register holding the intermediate result

Using a statistical description for data, information theory quantifies the number of bits needed to describe the data, which is the information entropy of the source.

Using the Seagram as a prototype, Mies ' office designed a number of modern high-rise office towers, notably the Chicago Federal Center, which includes the Dirksen and Kluczynski Federal Buildings and Post Office ( 1959 ) and the IBM Plaza in Chicago ; the Westmount Square in Montreal, and the Toronto-Dominion Centre in 1967.

Using the overall SADC figures as a comparative benchmark, an even smaller number came to Malawi – 219, 000 in 1998, 254, 000 in 1999 and 222, 526 in 2000.

Using the number in your space, jump that number of blocks in a straight line to a new space.

Using this material, planning a single turn may take a number of hours.

Using statistics collected from tens of thousands of coincidence events, a set of simultaneous equations for the total activity of each parcel of tissue along many LORs can be solved by a number of techniques, and, thus, a map of radioactivities as a function of location for parcels or bits of tissue ( also called voxels ) may be constructed and plotted.

Using RPGs to initiate the attack, they blew through the wall and killed a number of Salvadorean soldiers.

Using this technique, a small number of triodes could be made to do the work that formerly required dozens of triodes.

Using this study as a baseline, a number of different mission profiles were looked at over the next six months.

Using number of dice to set a task difficulty approximates fixed target numbers without needing to recall those numbers, thus streamlining actions requiring fast play ( combat ).

Using the explicit continued fraction expansion of e, one can show that e is not a Liouville number ( although the partial quotients in its continued fraction expansion are unbounded ).

Using M551 hulls, the NTC created a number of mock vehicles for training exercises resembling common Soviet / Warsaw pact types.

Using a little linear algebra, one can fairly easily construct approximations, which sample an arbitrary number of points to the left and a ( possibly different ) number of points to the right of the center point, for any order of derivative.

Using affine transformations multiple different euclidean transformations including translation can be combined by simply multiplying the corresponding matrices.

Using matrices, dual numbers can be represented as

Using the notation that is sometimes used to concisely describe diagonal matrices, we can write:

Using that, it follows that addition and composition of matrices obey the usual rules for matrix addition and matrix multiplication.

Using the linearity of matrices,

Using this logic, Dayhoff derived matrices as high as PAM250.

Using NumPy in Python gives functionality comparable to MATLAB since they are both interpreted, and they both allow the user to write fast programs as long as most operations work on arrays or matrices instead of scalars.

Using a coarser notion of equivalence that also allows transposition, there are 4 inequivalent matrices of order 16, 3 of order 20, 36 of order 24, and 294 of order 28.

Using Robert Steinberg's work on universal central extensions of classical algebraic groups, John Milnor defined the group K < sub > 2 </ sub >( A ) of a ring A as the center, isomorphic to H < sub > 2 </ sub >( E ( A ), Z ), of the universal central extension of the group E ( A ) of infinite elementary matrices over A.

Using transformation matrices containing homogeneous coordinates, translations can be seamlessly intermixed with all other types of transformations.

Using the anticommutators of the gamma matrices, one can show that for any and,

Using ConA-couple matrices, such enzymes may be immobilized in high quantities without a concurrent loss of activity and / or stability.

Using the action principle, a Lagrangian can be constructed from the trace of a polynomial function of these matrices, leading to Hamiltonian equations of motion.

Using these new tactics and the support of authors, the Company petitioned Parliament again in both 1707 and 1709 to introduce a bill providing for copyright.

Using osmotic shock, it is possible temporarily open vesicles ( filling in them with the required solution ) and then centrifugate down again and resuspend in a different solution.

Using this, we can get the relation, again choosing the right sign:

Using this power to great advantage, the Mule conquers planets simply by visiting them in force, with his own army, instilling the inhabitants with great fear, then again with great loyalty to himself.

Using PB GRAPHIC statements, a GRAPHIC ( WINDOW, BITMAP or control ) is first selected as the current GRAPHIC target, then operations are done on it without requiring it to be identified again.

Using a hangover cure discovered on Blood Island, he becomes an adult again and gets on the Rollercoaster of Death to confront LeChuck.

Using the fan makes Alice shrink again to a size small enough to crawl into one of the cracks, which takes her diving in her pool of tears.

Using the Stirling approximation again for large N, ln ( N!

Using parallel thinking ( not the Yoga version ) in this way will again protect the individuality of each of the pupils.

Using a gem stone that carries Raven's pure essence within it, the Titans manage to free Raven of her father's evil, although there will always be the possibility of it happening again.

Using again just the syntactic transformations available in this formalism, one can obtain so called fixed point combinators ( the best-known of which is the Y combinator ); these, by definition, have the property that f ( Y ( f )) = Y ( f ) for all functions f.

Using the fake Emerald, Sonic manages to perform Chaos Control and escape before it explodes, but Tails, thinking Sonic is dead, fights Eggman again.

Using a motorbike found in David's car, Jack manages to get himself and the semi-conscious Charley to the safety of the airlock before he falls unconscious again.

Using the chain rule, the reciprocal rule, and the fact that the derivative of the exponential function is again the exponential function, we see that the formula is correct for the first derivative of f for all x > 0 and that p < sub > 1 </ sub >( x ) is a polynomial of degree 0.

Using a robotic decoy in suit to distract Crocker, Jimmy repairs the Big Wand so that the fairies ' magic works again and Crocker would be defeated.

Using 3 AA batteries and the replica Key to Vector Sigma accessory ( placed on the backside of the display stand ), with the Matrix placed on it, it can change colors from blue to purple, to green, to red, and back again.

Using his Amahl Farouk body again, Shadow King appears as a member of Daken's Brotherhood of Mutants ( consisting of Sabretooth, Mystique, Skinless Man, the Blob from Earth-295, and the Omega Clan ) in a plot to attack the members of X-Force.

Using the danger of radicalism as an excuse, the same three powers abrogated the constitution and in 1793 again stripped Poland of territory.

where e is the identity of G. Though the f ″ defined a couple of paragraphs back is only defined for G-irreducible representations, and though A ⊗ B is not a G-irreducible representation in general, we claim this argument could be made correct since A ⊗ B is simply the direct sum of copies of Bs, and we have shown that each copy all maps to the same G × G-irreducible subrepresentation of C, we have just showed that as an irreducible G × G-subrepresentation of C is contained in A ⊗ B as another irreducible G × G-subrepresentation of C. Using Schur's lemma again, this means both irreducible representations are the same.

You can check your own contributions to refresh your memory about which pages you have worked on ( and to easily access these again ), and also to find out whether there have been any subsequent edits ( see item 18 in the Using section below ).

Using the cosmic treadmill, the two Reverse-Flashes intended to force Wally to watch the aftermath of his first battle with Zoom ( when his wife Linda Park miscarried ) over and over again, but the intervention of Barry Allen, from a time shortly before his death in Crisis on Infinite Earths, transferred Professor Zoom back to his present, allowing Wally to not only defeat the new Zoom, but also to undo Linda's miscarriage.

Using this biplane model, Chatterjee was able to calculate possible methods of gliding, and determined that Microraptor most likely employed a phugoid style of gliding: launching itself from a perch, the animal would have swooped downward in a deep U-shaped curve and then lifted again to land on another tree.

Using this program, SysOps can once again offer their BBSes via direct modem-to-modem connections.

Using representability once again, the cohomology operation is given by an element of.