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Using and mathematical

Using programs or proofs of bounded lengths, it is possible to construct an analogue of the Berry expression in a formal mathematical language, as has been done by Gregory Chaitin.

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 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 mathematical rules and procedures based on properties of reducible configurations, Appel and Haken found an unavoidable set of reducible configurations, thus proving that a minimal counterexample to the four-color conjecture could not exist.

Using RPN, one could get the result of the mathematical expression this way:

Using mathematical methods he experimented with various designs until he arrived at a shape cast from a single piece of iron, the Scots plough.

Using a mathematical approach, specifically geometric models, Gottfried Leibniz and Descartes discussed the infinite divisibility of extension.

# Using a mathematical expression, such as a polynomial or a trigonometric function, and a single point on the corresponding curve instead of storing or transmitting the entire graphic curve or a series of points on it.

Using floating-point variables and mathematical functions ( sin (), cos (), atan2 (), log (), exp (), sqrt () are the most popular ones ) of double precision as opposed to single precision comes at execution cost: the operations with double precision are usually slower.

Using late-model compilers, the machine attained a standard mathematical operations rate of 500 kilo-FLOPS, but handcrafted computer assembler delivered about 1. 0 mega-FLOPS.

Using his mathematical and analytical training, he concluded that dissimilar metals have a tendency to adhere to each other when subjected to very high pressure.

Using a mathematical model, the researchers then determined that two major tributaries, the Stony Brook and Muddy River, are the predominant sources of E. coli in the lower Charles River.

# Using graphical displays to suggest underlying mathematical principles.

Using his mathematical and medical expertise, he was able to develop a scale that would allow doctors to quantify the potency of their medication.

Using this notation and some knowledge about mathematical functions and probability theory we formalise different versions of the binding and hiding properties of commitments.

Using the chip, Max has Euclid analyze mathematical patterns in the Torah.

Using the mathematical identities

Using notable sociological and anthropological field studies as the grounding for such general ideas, the book makes a persuasive case for treating groups as social systems that can be analyzed in terms of a verbal analogue of the mathematical method of studying equilibrium and stability of systems.

Using that mathematical tool Hambardzumyan resolved a number of nonlinear problems of the light diffusion.

Using mathematical filters, the researchers decoded the signals to generate movies of what the cats saw and were able to reconstruct recognizable scenes and moving objects.

Using mathematical formulations of quantum mechanics, such as the Schrödinger equation, the wave function can be solved.

Using fractal geometry to describe natural forms such as coastlines, mathematical physicist Mitchell Feigenbaum developed software capable reconfiguring coastlines, borders, and mountain ranges to fit a multitude of map scales and projections.

* Using a mathematical algorithm to generate a new password based on the previous password ( OTPs are effectively a chain and must be used in a predefined order ).

* Using a mathematical algorithm where the new password is based on a challenge ( e. g., a random number chosen by the authentication server or transaction details ) and / or a counter.

Using and formulation

Using a different formulation of Gödel's theorems, namely, that of Raymond Smullyan and Emil Post, Webb shows one can derive convincing arguments for oneself of both the truth and falsity of p. He furthermore argues that all arguments about the philosophical implications of Gödel's theorems are really arguments about whether the Church-Turing thesis is true.

* Using the formulation " Let us bless our God, from whose wealth we have eaten ," in preparing for Grace after meals.

Using Bowlby's early formulation, she conducted observational research on infant-parent pairs ( or dyads ) during the child's first year, combining extensive home visits with the study of behaviours in particular situations.

* Using the Griffiths inequality in the formulation of Ginibre, Aizenman and Simon proved that the two point spin correlation of the ferromagnetics XY model in dimension, coupling and inverse temperature is dominated by ( i. e. has upper bound given by ) the two point correlation of the ferromagnetic Ising model in dimension, coupling and inverse temperature

Using the definition of the integral flux Φ and the average cross section σ, the same formulation as before is found:

Using this formulation, silicone grease is a translucent white viscous paste, with exact properties dependent on the type and proportion of the components ..

Using and let

Using the avante-garde magazine Playboy as his model, Arledge convinced his superiors at WRCA to let him film a pilot of a show he called " For Men Only.

Using the bottleneck technique most of the time, Elmore really let his guitar sound as I had never heard a guitar sound before.

Using the Dirac notation, let the system initially be in the state where

Using the close correspondence between divisors and holomorphic line bundles on a Riemann surface, the theorem can also be stated in a different, yet equivalent way: let L be a holomorphic line bundle on X.

* Using indicator variables, let x < sub > i </ sub > equal 1 if x is in A < sub > i </ sub > and 0 otherwise.

Using the standard formalism of probability, let and be two random variables defined on probability spaces and.

Using the abstract index notation, let a, b, c, ... refer to and μ, ν, ... refer to the tangent bundle.

Using the aforementioned glitch with the Oak's Parcel will let you warp to the map with the number of steps you've walked until the door you walk in.

Using the parts from all the captured Autobots she creates a giant Autobot with herself in control, striking a deal with them that if they would co-operate she would let them go.

* Using " rent " and " let " interchangeably.

Using the parts from all the captured Autobots she created a giant Autobot with herself in control, striking a deal with them that if they would co-operate she would let them go.

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