We will call the first the furnace and the second the refrigerator .” Carnot then explains how we can obtain motive power, i. e. “ work ”, by carrying a certain quantity of heat from body A to body B.

We thus obtain a functor from the category of pointed topological spaces to the category of groups.

We can factor this expression to obtain

We thus obtain the inequality in terms of dimensions of kernel, which can then be converted to the inequality in terms of ranks by the rank-nullity theorem.

We obtain the same variation in probability amplitudes by allowing the time at which the photon left the source to be indeterminate, and the time of the path now tells us when the photon would have left the source, and thus what the angle of its " arrow " would be.

' We lose time, we destroy trees to obtain paper necessary to print this word.

We can, however, seek to obtain some form of consensus, with others, of what is real.

There is a special case of Lagrange inversion theorem that is used in combinatorics and applies when and Take to obtain We have

We thus obtain the recursion:

We obtain the sequence of Fibonacci numbers which begins:

We are deprived of a leader than whom the whole world would scarcely obtain a greater, whether in knowledge of true religion or in integrity and innocence of life, or in thirst for study of the most holy things, or in exhausting labour in advancing piety, or in authority and fulness of teaching, or in anything that is praiseworthy and renowned.

We obtain a Kleene algebra A with 0 being the empty set and 1 being the set that only contains the empty string.

We start with the standard assumption of independence of the two sides, enabling us to obtain the joint probabilities of pairs of outcomes by multiplying the separate probabilities, for any selected value of the " hidden variable " λ. λ is assumed to be drawn from a fixed distribution of possible states of the source, the probability of the source being in the state λ for any particular trial being given by the density function ρ ( λ ), the integral of which over the complete hidden variable space is 1.

We now use the Pythagorean theorem, the fact that the slope of the curve is equal to the tangent of its angle, and some trigonometric identities to obtain ds in terms of dx:

We perform our ' simran ' or meditation by Guru's Grace only and open tenth door of our body to obtain " band-khallaassi " or the ' moksha ' ( liberation from bondage of birth and death and attain ultimate oneness with the Lord.

We obtain:

We obtain complete forgiveness ..." are from http :// aa-history. com / 12stephistory2. html

We can " cut " the plane along the real axis, from − 1 to 1, and obtain a sheet on which g ( z ) is a single-valued function.

We can obtain a formula for r by substituting estimates of the covariances and variances based on a sample into the formula above.

We then obtain an equation of polynomials whose left-hand side is simply p ( x ) and whose right-hand side has coefficients which are linear expressions of the constants A < sub > ir </ sub >, B < sub > ir </ sub >, and C < sub > ir </ sub >.

We may integrate the expression for to obtain the Helmholtz energy:

We obtain that

: We had to create our own journals because it was impossible to obtain entry into the official journals of psychiatry and medicine.

If T is a statistic that is approximately normally distributed under the null hypothesis, the next step in performing a Z-test is to estimate the expected value θ of T under the null hypothesis, and then obtain an estimate s of the standard deviation of T. We then calculate the standard score Z = ( T − θ ) / s, from which one-tailed and two-tailed p-values can be calculated as Φ (−| Z |) and 2Φ (−| Z |), respectively, where Φ is the standard normal cumulative distribution function.

We can therefore derive a semi-discrete numerical scheme for the above problem with cell centres indexed as, and with cell edge fluxes indexed as, by differentiating ( 6 ) with respect to time to obtain:

We define an equivalence relation ~ upon these pairs with the following rule:

We can embed the rational numbers into the reals by identifying the rational number r with the equivalence class of the sequence ( r, r, r, …).

We can define an even finer equivalence relation of oriented C < sup > r </ sup > curves by requiring φ to be φ ‘( t ) > 0.

We can consider operations such as equivalence ( whether two people have the same last name ), set membership ( whether a person has a name in a given list ), counting ( how many people have a given last name ), or finding the mode ( which name occurs most often ).

We simply take Ext ( A, B ) to be the set of equivalence classes of extensions of A by B, forming an abelian group under the Baer sum.

Let V be a vector space over a field K, and let N be a subspace of V. We define an equivalence relation ~ on V by stating that x ~ y if x − y ∈ N. That is, x is related to y if one can be obtained from the other by adding an element of N. From this definition, one can deduce that any element of N is related to the zero vector ; in other words all the vectors in N get mapped into the equivalence class of the zero vector.

We say that f is a weak homotopy equivalence if the homomorphisms f < sub >*</ sup > are all isomorphisms.

We say is Turing equivalent to and write if both and The equivalence classes of Turing equivalent sets are called Turing degrees.

* The most general precise definition is simply in terms of an equivalence relation R. We say that a is equivalent or congruent to b modulo R if aRb.

We can define a correspondent Kripke structure by taking ( i ) the same space S, ( ii ) accessibility relations that define the equivalence classes corresponding to the partitions, and ( iii ) a valuation function such that it yields value true to the primitive proposition p in all and only the states s such that, where is the event of the Aumann structure corresponding to the primitive proposition p. It is not difficult to see that the common knowledge accessibility function defined in the previous section corresponds to the finest common coarsening of the partitions for all, which is the finitary characterization of common knowledge also given by Aumann in the 1976 article.

We define an equivalence relation on this space by declaring that two functions f and g are equivalent to order k if f and g have the same value at p, and all of their partial derivatives agree at p up to ( and including ) their k-th order derivatives.

We now define the k-jet of a curve f through p to be the equivalence class of f under, denoted or.

Let p be a point of M. Consider the space consisting of smooth maps defined in some neighborhood of p. We define an equivalence relation on as follows.

We stressed the reciprocal relation of these systems with respect to the autonomic-somatic downward discharge as well as regarding the hypothalamic-cortical discharge.

We can say that that relation has being as well.

We axiomatize predicate calculus without equality, i. e. there are no special axioms expressing the properties of equality as a special relation symbol.

We can work the other way and start by choosing < as a transitive trichotomous binary relation ; then a total order ≤ can equivalently be defined in two ways:

Several alternative names have been proposed by various people: Generation We, Global Generation, Generation Next, the Net Generation, The name " Echo Boomers " refers to the size of the generation and its relation to the Baby Boomer generation.

We can also work backwards and find what the relation is from the number.

We define headers as finite subsets of C. A relational database schema is defined as a tuple S = ( D, R, h ) where D is the domain of atomic values ( see relational model for more on the notions of domain and atomic value ), R is a finite set of relation names, and

We note here that though Ritschl gives Jesus a unique and unapproachable position in His active relation to the kingdom, he declines to rise above this relative teaching.

We will derive the put-call parity relation by creating two portfolios with the same payoffs and invoking the above principle.

We also learn more about the work of the spirits of the light, as they, incarnated as human beings in many lives, have worked to give mankind greater skills in all matters, to make life on earth easier, and to give us greater knowledge about our true relation to God.

The images were " backed " by two advertising characters, the " T-Ladies ", who drink Tango ( in relation to " We Drink Tango Don't You Know ", the parent campaign of the 1998-1999 peroid ).

Next, the frame is extended to a model by specifying the truth-values of all propositions at each of the worlds in G. We do so by defining a relation ⊨ between possible worlds and propositional letters.

We can prove that these frames produce the same set of valid sentences as do any frames where all worlds can see all other worlds of W ( i. e., where R is a " total " relation ).

This is true for all sciences: the goal is to connect a “ natural phenomenon ” with its “ immediate cause .” We formulate hypotheses elucidating, as we see it, the relation of cause and effect for particular phenomena.

We believe that history and scripture prove that God deals with nations in relation to how they deal with Israel.

The title comes from Abraham Lincoln's " Gettysburg Address " and appears to have no relation to Ayn Rand's similarly titled We the Living ( published in 1936 ).

We can use the real process data of the productivity model ( above ) in order to show the logic of the growth accounting model and identify possible differences in relation to the productivity model.

We obtain the dispersion relation of:

Tolman remarked on this relation that " We have, moreover, of course the experimental verification of the expression in the case of moving electrons to which we shall call attention in § 29.

We just saw that the feed on relation is not transitive, but it still contains some transitivity: for instance: humans feed on rabbits, rabbits feed on carrots, and humans also feed on carrots.

We can use well-founded induction on any set with a well-founded relation, thus one is interested in when a quasi-order is well-founded.