Using the disintegration theorem we can define a restriction of Lebesgue measure to the-dimensional affine subspace of where the Gaussian distribution is supported, i. e..

For the function field even to be defined, V here must be an irreducible algebraic set ; in which case the function field ( for an affine variety ) is just the field of fractions of the coordinate ring of V. Using polynomial equations, it is easy to define sets that have ' mixed dimension ': a union of a curve and a plane in space, for example.

Using the above matrix representation of the affine group on the plane, the matrix M is a 2 × 2 real matrix.

Using the dictionary between ( commutative ) rings and affine schemes local properties are thus properties of the rings A < sub > i </ sub >.

Using the classification of hermitian symmetric spaces, the first family gives the following complex affine holonomy groups:

Using control points ( automatically or manually entered ), the computer performs transformations on one image to make major features align with a second image.

Using the natural isomorphism between and the space of linear transformations from V to V ,< ref name =" natural iso "> Let L ( V, V ) be the space of linear transformations from V to V. Then the natural map

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 frequency transformations and impedance scaling, the normalized low-pass filter may be transformed into high-pass, band-pass, and band-stop filters of any desired cutoff frequency or bandwidth.

Using natural transformations as morphisms, one can form the category of all representations of G in C. This is just the functor category C < sup > G </ sup >.

Using transformations, XForms can be automatically created from XML schemas, and XForms can be converted to XHTML forms.

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

Using Flynn's taxonomy, these two schemes of dealing with data are generally referred to as SIMD ( single instruction, multiple data ) and SISD ( single instruction, single data ), respectively.

Using Copper or CPU register manipulations, each sprite ' channel ' can be reused multiple times in a single frame to increase the total sprites per frame.

Using a phone that supports 3G and Bluetooth 2. 0, multiple Bluetooth-capable laptops can be connected to the Internet.

Using a Game Boy Advance link cable, up to four Game Boy Advance units can be linked together and the game can be played with multiple players using only one copy of the game.

Using multiple components with load balancing, instead of a single component, may increase reliability through redundancy.

* Using a faster " flick " motion, the rider can spin the board multiple revolutions before landing.

Using multiple methods helps the navigator detect errors, and simplifies procedures.

Using GDI, it is very easy to draw on multiple devices, such as a screen and a printer, and expect proper reproduction in each case.

Using editing, multiple exposures, jump cutting, superimposition, slow-motion and other camera techniques to her fullest advantage, Deren creates continued motion through discontinued space, while abandoning the established notions of physical space and time, with the ability to turn her vision into a stream of consciousness.

Using multiple carburetors ( two or three ) always results in one carburetor feeding more cylinders than the other, which also produces running and tuning problems.

Using multiple coils speeds up acquisition time in exact proportion to the coils used.

Using the original Q-Link software as a D-64 file, it can be accessed using either the VICE Commodore 64 emulator ( available on multiple platforms, including Windows and Linux ), or by using authentic Commodore hardware connected to the Internet by way of a serial cable connected to a PC with internet access.

* Using larger or multiple carburetors, to create a more controllable air / fuel mixture to burn, and to get it into the engine more smoothly.

Using multiple valves per cylinder results in the same effect – it is often more difficult to fit several small valves than to have larger single valves due to the valve gear required.

* Using multiple forms of contraceptive simultaneously, usually one by the female, and one by the male.

Using this style, he appeared on multiple Davis albums, including Black Beauty: Live at the Fillmore West and Miles Davis at Fillmore: Live at the Fillmore East.

Using multiple linear regression analyses, after adjusting for potential confounders, they claim betel consumption was statistically significantly associated with obesity.

Using a globally distributed, partially replicated database, StreetTalk could meld multiple widely separated networks into a single network that allowed seamless resource-sharing.

Using the LCD ( or any multiple of it, such as the product of the denominators ) as a denominator enables addition, subtraction or comparison of fractions:

Using multiple dental impressions in a lineup may enable forensic odontologists to significantly decrease the current bias in matching bite marks to the teeth of a suspect.

Using a mask, multiple bits in a byte, nibble, word ( etc.

Using a sophisticated passing attack, the Chargers and quarterback Dan Fouts set multiple offensive records during Gibbs ' two seasons there.

Using multiple spot beam antennas and advanced on-board switching and processing systems, ACTS pioneered new initiatives in communications satellite technology.

Using David's telepathic abilities, he reintegrated the multiple personalities into David's core personality.

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 the same prefixes to mean two different things within the same industry has caused some confusion.

Using different working liquids ( oils, salt solutions, etc.

Using the principle of equivalence, Einstein concluded that the same thing holds in any gravitational field, that the rate of clocks R at different heights was altered according to the gravitational field g. When g is slowly varying, it gives the fractional rate of change of the ticking rate.

Using different copolymers most of the remaining codons were then determined.

Using a modified Quake engine, it featured single-player and multiplayer game modes, as well as four character classes to choose from, each with different abilities.

Using two sets of rollers that travelled at different speeds, it was later used in the first cotton spinning mill.

Using needles of varying shape and thickness as well as different varieties of yarn can also change the effect.

Using them its simple to send individually controlled copies of your film to various recipients with different expiry dates.

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

Using his theory, he discovered homologies showing that slightly changed body parts served different functions to meet new conditions, and he found an intermediate stage in the evolution of distinct sexes.

Using this way of framing the issue, Cousin stridently argued that different competing philosophical ideologies all had some claim on truth, as they all had arisen in defense of some truth.

Using a different style, Rabbi Nathan b. Jechiel created a lexicon called the Arukh in the 11th century to help translate difficult words.

Using a dissimilar metal to complete the circuit creates a circuit in which the two legs generate different voltages, leaving a small difference in voltage available for measurement.

Using the same dough that was used for the firm ’ s crispbread, Vitawheat, he experimented with different shapes and flavourings, and the final product was launched in time for the Christmas market in 1929.

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 Fourier analysis, wave packets can be analyzed into infinite sums ( or integrals ) of sinusoidal waves of different wavenumbers or wavelengths.

Using Fregean terms: whenever their presuppositions hold, the two hypotheses have the same referent ( truth-value ), but different senses ; that is, they express two different ways to determine that truth-value.

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 his modern telescope he succeeded in subdividing the nebula into different stars.

Using six different climate model simulations, a large multi-university study of ocean ecosystems predicts, by 2050 A. D., " a global increase in primary production of 0. 7 % at the low end to 8. 1 % at the high end ," although with " very large regional differences " including " a contraction of the highly productive marginal sea ice biome by 42 % in the Northern Hemisphere and 17 % in the Southern Hemisphere.

Using different algorithms on a single data set can sometimes yield different " best " cladograms, because each algorithm may have a unique definition of what is " best ".