The phrase "are equivalent by" is not correct in standard written English.
It is typically used incorrectly; a more appropriate phrase would be "are equivalent to" when comparing two things.
Example: "The two equations are equivalent to each other in terms of their solutions."
Alternatives: "are equal to" or "are the same as".
Note that the two random dynamical systems are equivalent by (3.19).
Now a-Weyl's theorem and generalized a-Weyl's theorem hold for T (resp. for d A, B ) and are equivalent by Theorem 3.3 ii).
As a matter of fact estimates (31) and (106) are equivalent by noticing that if v solves then so does v - k, whenever k is a real number.
Shi [50] and Robinson [48] proved that the variational inequalities and the Wiener-Hopf equations are equivalent by using the projection technique.
In fact, they are equivalent by noting and For (2.9), pre- and postmultiplying by and respectively, we get the inequality (2.10); similarly, for (2.10), pre- and postmultiplying by respectively, we get the inequality (2.11).
Notice that all permutations of (left {frac {N}{Q}, frac {N}{Q}-1, frac {N}{Q}+ 1 right }) are equivalent by symmetry and lead to an identical power consumption; therefore, we provide only one of these permutations in the Table.
Every $1000 of monthly pension income is equivalent, by some estimates, to $170,000 to $240,000 in investments.
Original and new results were equivalent by about age 14 yr in both genders.
Then by defining, we obtain a new sequence (g n ) whose basin is equivalent (by the biholomorphic map U 0) to the basin of the sequence (f n ).
Both samples were equivalent by mean age and semester as well as by gender, areas of study, and public versus private university distributions.
OCls recoveries were equivalent by UAE-MSC and Soxhlet methods UAE-MSC optimized extraction conditions reduced 30 times less solvent and decreased the extraction time from several hours to ten minutes, respect to Soxhlet.
