If we take the congruence modulo for the right-hand-side expression, it is readily seen that

* If the result of a pure expression is not used, it can be removed without affecting other expressions.

* If there is no data dependency between two pure expressions, then their order can be reversed, or they can be performed in parallel and they cannot interfere with one another ( in other terms, the evaluation of any pure expression is thread-safe ).

If the reader evaluates an expression but does not follow the correct order of operation, the reader will come forth with a different value.

In particular, if S and T ( subgroups now ) intersect only in the identity, then every element of ST has a unique expression as a product st with s in S and t in T. If S and T also permute, then ST is a group, and is called a Zappa-Szep product.

If there are multiple operations, the operator is given immediately after its second operand ; so the expression written " 3 − 4 + 5 " in conventional infix notation would be written " 3 4 − 5 +" in RPN: first subtract 4 from 3, then add 5 to that.

If all functions involved in the expression are pure functions, then the expression is referentially transparent.

If the substitution of an expression with its value is valid only at a certain point in the execution of the program, then the expression is not referentially transparent.

If the bias is small, we can let U − E ≈ φM in the expression for κ, where φM, the work function, gives the minimum energy needed to bring an electron from an occupied level, the highest of which is at the Fermi level ( for metals at T = 0 kelvins ), to vacuum level.

If they are participant in the integral physical expression of a living human being who has absorbed and metabolized them, or if they are now the physical remains of a once-living human being, the substance of what they actually are is human, hence, a human body.

If the karkanxholl sleeps with his wife, and she is impregnated with a child, the offspring is called dhampir and has the unique ability to discern the karkanxholl ; from this derives the expression the dhampir knows the lugat.

If the index numbers count from 1, the desired address is computed by this expression:

If the verbal expression of modality involves the use of an auxiliary verb, that auxiliary is called a modal verb.

If the verbal expression of modality involves inflection, we have the special case of mood ; moods include the indicative ( as in " I am there "), the subjunctive ( as in " I wish I were there "), and the imperative (" Be there!

If the positive number n becomes infinitely great, the expression 1 / n goes to naught ( or gets infinitely small ).

If not, only existing, actual things can be in the extension of a concept or expression.

; Factoring out of invariants: If an expression is carried out both when a condition is met and is not met, it can be written just once outside of the conditional statement.

If arbitrary values and / or functions are used in place of one or more of the numerators or the integers in the denominators, the resulting expression is a generalized continued fraction.

If the way that DNA is wrapped around the histones changes, gene expression can change as well.

If denotes the polarization vector of the wave exiting the waveplate, then this expression shows that the angle between and is − θ.

If Byzantine literature is the expression of the intellectual life of the Byzantine Greeks during the Christian Middle Ages, then it is a multiform organism, combining Greek and Christian civilization on the common foundation of the Roman political system, set in the intellectual and ethnographic atmosphere of the Near East.

If a safe pharmacological treatment can be developed – one that increases expression of lysosomal sialidase in neurons without other toxicity – then this new form of therapy could essentially cure the disease.

Tensor products: If C denotes the category of vector spaces over a fixed field, with linear maps as morphisms, then the tensor product defines a functor C × C → C which is covariant in both arguments.

If g is the metric on this manifold, one defines the action S ( g ) as

If f and g are two multiplicative functions, one defines a new multiplicative function f * g, the Dirichlet convolution of f and g, by

If one defines the energy per pulse as:

If it only defines rules and regulations for Internet Assigned Numbers Authority ( IANA ) registries it is less clear ; most of these documents are BCPs, but some are on the standards track.

If every object X < sub > i </ sub > of C admits a initial morphism to U, then the assignment and defines a functor V from C to D. The maps φ < sub > i </ sub > then define a natural transformation from 1 < sub > C </ sub > ( the identity functor on C ) to UV.

Any collection of objects and morphisms defines a ( possibly large ) directed graph G. If we let J be the free category generated by G, there is a universal diagram F: J → C whose image contains G. The limit ( or colimit ) of this diagram is the same as the limit ( or colimit ) of the original collection of objects and morphisms.

If X is an object of a locally small category C, then the assignment defines a covariant functor.

If such a hyperplane exists, it is known as the maximum-margin hyperplane and the linear classifier it defines is known as a maximum margin classifier ; or equivalently, the perceptron of optimal stability.

If there are 4 bits per pixel ( 16 colors ), each frame buffer byte defines two pixels, one in each nibble.

If we restrict attention to real-valued W then the relation is defined only for x ≥ − 1 / e, and is double-valued on (− 1 / e, 0 ); the additional constraint W ≥ − 1 defines a single-valued function W < sub > 0 </ sub >( x ).

If the sort key values are totally ordered, the sort key defines a weak order of the items: items with the same sort key are equivalent with respect to sorting.

If different items have different sort key values then this defines a unique order of the items.

If is a Dirichlet character, one defines its Dirichlet L-series by

If ƒ and g are two arithmetic functions ( i. e. functions from the positive integers to the complex numbers ), one defines a new arithmetic function ƒ * g, the Dirichlet convolution of ƒ and g, by

Richard Belzer defines The Big Lie, in his book UFOs, JFK, and Elvis: Conspiracies You Don't Have To Be Crazy To Believe, this way: " If you tell a lie that's big enough, and you tell it often enough, people will believe you are telling the truth, even when what you are saying is total crap.

The integration functional I defined above defines a linear functional on the subspace P < sub > n </ sub > of polynomials of degree ≤ n. If x < sub > 0 </ sub >, …, x < sub > n </ sub > are n + 1 distinct points in, then there are coefficients a < sub > 0 </ sub >, …, a < sub > n </ sub > for which

If the base field F is the field R of real numbers, then x < sup > 2 </ sup > + y < sup > 2 </ sup > = − 1 defines an algebraic extension field of R ( x ), but the corresponding curve considered as a locus has no points in R. However, it does have points defined over the algebraic closure C of R.

If we want to ignore the physics inside the conductor and only describe the physics in the outside region, it becomes natural to mathematically describe the quantum electron by a section in a complex line bundle with an " external " connection rather than an external EM field ( by incorporating local gauge transformations we have already acknowledged that quantum mechanics defines the notion of a ( locally ) flat wavefunction ( zero momentum density ) but not that of unit wavefunction ).

If X is the toric variety corresponding to the normal fan of P, then P defines an ample line bundle on X, and the Ehrhart polynomial of P coincides with the Hilbert polynomial of this line bundle.

If precondition # 1 is changed to " The good in question must be so inferior that the income effect is greater than the substitution effect " then this list defines necessary and sufficient conditions.

If one defines Einstein temperature as

If D is a central simple algebra over K and v is a valuation then D ⊗ K < sub > v </ sub > is a central simple algebra over K < sub > v </ sub >, the local completion of K at v. This defines a homomorphism from the Brauer group of K into the Brauer group of K < sub > v </ sub >.

If one defines a homology theory axiomatically ( via the Eilenberg-Steenrod axioms ), and then relaxes one of the axioms ( the dimension axiom ), one obtains a generalized theory, called an extraordinary homology theory.