Using Itō's lemma with f ( S )

Using Schur's lemma, this must be zero.

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.

Using Schur's lemma, all possible Hilbert space norms can only be a multiple of each other.

Using Schur's lemma, in an irreducible unitary representation, each of these Casimir invariants are multiples of the identity.

Using the lemma, one can " lift " a root r of the polynomial f mod p < sup > k </ sup > to a new root s mod p < sup > k + 1 </ sup > ( by taking m =

Using a Batista man to screen refugees represented a total misunderstanding of the democratic forces which alone can effectively oppose Castro.

Using visual imagery of a soroban, one can arrive at the answer in the same time as, or even faster than, is possible with a physical instrument.

Using this distance and other photographs, more information about the object, including its orbital elements, can be obtained .< ref >

Using these formulas, the area of any polygon can be found by dividing the polygon into triangles.

Using the " lock and key " metaphor, the antigen itself can be seen as a string of keys-any epitope being a " key "-each of which can match a different lock.

Using this binding mechanism, an antibody can tag a microbe or an infected cell for attack by other parts of the immune system, or can neutralize its target directly ( for example, by blocking a part of a microbe that is essential for its invasion and survival ).

Using these inhibitors it was discovered that cells can die while displaying a morphology similar to apoptosis without caspase activation.

Using modern full capacity firearms, a shooter can fire more shots with greater accuracy using a single gun in a two-handed grip than two guns with one-handed grips.

Using a solar roof, solar cells can provide electric power.

Using various modifiers, the activity of the enzyme can be regulated, enabling control of the biochemistry of the cell as a whole.

Using the above theorem it is easy to see that the original Borsuk Ulam statement is correct since if we take a map f: S < sup > n </ sup > → ℝ < sup > n </ sup > that does not equalize on any antipodes then we can construct a map g: S < sup > n </ sup > → S < sup > n-1 </ sup > by the formula

Using these tools, one can write a more complex text editor, and a simple compiler for a higher-level language and so on, until one can have a graphical IDE and an extremely high-level programming language.

Using pharmacogenomics, pharmaceutical companies can create drugs based on the proteins, enzymes and RNA molecules that are associated with specific genes and diseases.

Using braille translation software, a document can be embossed with relative ease, making braille production much more efficient and cost-effective.

Using summation notation, it can be written as

Using the previous relationships they can be expressed as:

Using the extension of the code, the encoded string 0011001011 can be grouped into codewords as 0 – 011 – 0 – 01 – 011, and these in turn can be decoded to the sequence of source symbols acabc.

Using the language of category theory, many areas of mathematical study can be cast into appropriate categories, such as the categories of all sets, groups, topologies, and so on.

Using the one-point compactification, one can also easily construct compact spaces which are not Hausdorff, by starting with a non-Hausdorff space.

Using Cartesian coordinates, it can be shown that

Using a hair dryer, some atheist groups have conducted tongue-in-cheek " debaptism " ceremonies, not intended to be taken seriously.

Using the techniques of modern biotechnology, one or two genes ( Smartstax from Monsanto in collaboration with Dow AgroSciences will use 8, starting in 2010 ) may be transferred to a highly developed crop variety to impart a new character that would increase its yield.

Using terms from formal language theory, the precise mathematical definition of this concept is as follows: Let S and T be two finite sets, called the source and target alphabets, respectively.

Using the relative yardstick favoured in many European countries, 27 % of Chileans would be poor, according to Juan Carlos Feres of the ECLAC.

Using the right-hand rule, we expect the curl to be into the page.

Using Cartesian coordinates, it can be shown that:

Using this essay as his guideline, Howard began plotting " The Tower of the Elephant ", a new Conan story that would be the first to truly integrate his new conception of the Hyborian world.

Using cathode ray tubes, Fred Williams would invent the Williams tube, which would be the first random access computer memory.

Using the Cartesian coordinate system, geometric shapes ( such as curves ) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape.

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

Using Unicode characters, the faces < big >&# x2680 ; &# x2681 ; &# x2682 ; &# x2683 ; &# x2684 ; &# x2685 ;</ big >, can be shown in text using the range U + 2680 – U + 2685 or using decimal.

Using the word apocrypha ( Greek: hidden away ) to describe texts, although not necessarily pejorative, implies to some people that the writings in question should not be included in the canon of the Bible.

Using some clever tricks, the merge operation can be done in time O ( n ), so the total running time is O ( n log n ).

Using small telescopes and the Lovell Telescope as an interferometer, they were shown to have a very small angular size.

Using the parallel axis theorem, the radius of gyration L of a rigid pendulum can be shown to be

Using immunoquantitative techniques, a decrease of Th protein staining level, number of locus coeruleus TH-expressing neurons and density of dendritic arborization surrounding the structure was shown in symptomatic Mecp2-deficient mice.

Using this premise, researches have shown significant reduction in the extent of vessel outgrowth using antibodies to sequester this growth factor ( i. e. bevacizumab ( Avastin ®)).

Using the Coupon collector's problem in probability theory, it can be shown that it takes, on average, 72 questions of the Magic 8 Ball for all 20 of its answers to appear at least once.

Using big-O notation, the performance of the interpolation algorithm on a data set of size N is O ( N ); however under the assumption of a uniform distribution of the data on the linear scale used for interpolation, the performance can be shown to be O ( log log N ).

Using the original X-Men as his agents to satisfy Lucifer's twisted sense of irony-and also due to his limited ability to create truly effective ionic constructs ; the duplicate Xavier was only his third construct shown and the other two lacked any kind of initiative-the duplicate Professor dispatched them to acquire three components from Quistalian outposts around the globe, intending to use them to alter the machine that had sent the Professor into the Nameless Dimension so that it could be used to allow Lucifer to return to Earth.

Using Lagrangian mechanics, it may be shown that the motion of a pendulum can be described by the dimensionless nonlinear equation

Using quantum mechanics it can be shown that for an electron in an external electric field E, the acceleration a < sub > ℓ </ sub > along coordinate direction ℓ is:

Using the free choice paradigm, Sharot and colleagues have shown that after making a choice, activity in the striatum changes to reflect the new evaluation of the choice object, increasing if the object was chosen and decreasing if it was rejected.

Using MRI scans, Dr. Murray Stein, of the University of California, San Diego, found that when people with the disorder are shown pictures of angry faces, their amygdala-the brain's fear center-lights up with more activity than it does in people without the condition.

Using scale analysis, it can be shown that the above equations of motion reduce within the boundary layer to become

Using an example that applies to instant-runoff voting ( IRV ) and to the Two-round system, it is shown, that these voting systems violate the mono-raise criterion.

Using this contextual paleogeographic information ( paleogeographic time series is shown in panels A-D ) the authors of this study proposed a null-hypothesis that assumes no spatial structure and two alternative hypothesis involving dispersal and other biogeographic constraints ( hypothesis are shown in panels E-G, listed as SMO, SM1, and SM2 ).

Using a head mounted display, the control of the camera can be facilitated by tracking the head as shown in the figure below.

: Using knowledge of San beliefs, researchers have shown that the art played a fundamental part in the religious lives of its San painters.

Using depth gauges, European Shags have been shown to dive to at least 45 metres.

Using single-cell recording techniques, Dr. Emilio Bizzi and his collaborators have shown the behavior of certain cells, known as " memory cells ," can undergo lasting alteration with practice.

Using a variety of experimental procedures, studies have shown that infants in their second year of life have an implicit understanding what other people see and what they know .. A popular paradigm used to study infants ' theory of mind is the violation of expectation procedure, which predicates on infants ' tendency to look longer at unexpected and surprising events compared to familiar and expected events.

Using standard antimicrobial susceptibility tests, Psilocybe semilanceata was shown to strongly inhibit the growth of the human pathogen methicillin-resistant Staphylococcus aureus ( MRSA ).