38,849 research outputs found
Expressing set-size equality
The word âequalityâ often requires disambiguation, which is provided by context or by an explicit modifier. For each sort of magnitude, there is at least one sense of âequalsâ with its correlated senses of âis greater thanâ and âis less thanâ. Given any two magnitudes of the same sortâtwo line segments, two plane figures, two solids, two time intervals, two temperature intervals, two amounts of money in a single currency, and the likeâthe one equals the other or the one is greater than the other or the one is greater than the other [sc. in appropriate correlated senses of âequalsâ, âis greater thanâ and âis less thanâ]. In case there are two or more appropriate senses of âequalsâ, the one intended is often indicated by an adverb. For example, one plane figure may be said to be equal in area to another and, in certain cases, one plane figure may be said to be equal in length to another.
Each sense of âequalityâ is tied to a specific domain and is therefore non-logical. Notice that in every cases âequalityâ is definable in terms of âis greater thanâ and also in terms of âis less thanâ both of which are routinely considered domain specific, non-logical.
The word âidentityâ in the logical sense does not require disambiguation. Moreover, it is not correlated âis greater thanâ and âis less thanâ. If it is not the case that a certain designated triangle is [sc. is identical to] an otherwise designated triangle, it is not necessary for the one to be greater than or less than the other.
Moreover, if two magnitudes are equal then a unit of measure can be chosen and, no matter what unit is chosen, each magnitude is the same multiple of the unit that the other is. But identity does not require units. In this regard, congruence is like identity and unlike equality.
In arithmetic, the logical concept of identity is coextensive with the arithmetic concept of equality. The logical concept of identity admits of an analytically adequate definition in terms of logical concepts: given any number x and any number y, x is y iff x has every property that y has.
The arithmetical concept of equality admits of an analytically adequate definition in terms of arithmetical concepts: given any number x and any number y, x equals y iff x is neither less than nor greater than y. As Aristotle told us and as Frege retold us, just because one relation is coextensive with another is no reason to conclude that they are one
Noncovalent Interactions by QMC: Speedup by One-Particle Basis-Set Size Reduction
While it is empirically accepted that the fixed-node diffusion Monte-Carlo
(FN-DMC) depends only weakly on the size of the one-particle basis sets used to
expand its guiding functions, limits of this observation are not settled yet.
Our recent work indicates that under the FN error cancellation conditions,
augmented triple zeta basis sets are sufficient to achieve a benchmark level of
0.1 kcal/mol in a number of small noncovalent complexes. Here we report on a
possibility of truncation of the one-particle basis sets used in FN-DMC guiding
functions that has no visible effect on the accuracy of the production FN-DMC
energy differences. The proposed scheme leads to no significant increase in the
local energy variance, indicating that the total CPU cost of large-scale
benchmark noncovalent interaction energy FN-DMC calculations may be reduced.Comment: ACS book chapter, accepte
The Power of an Example: Hidden Set Size Approximation Using Group Queries and Conditional Sampling
We study a basic problem of approximating the size of an unknown set in a
known universe . We consider two versions of the problem. In both versions
the algorithm can specify subsets . In the first version, which
we refer to as the group query or subset query version, the algorithm is told
whether is non-empty. In the second version, which we refer to as the
subset sampling version, if is non-empty, then the algorithm receives
a uniformly selected element from . We study the difference between
these two versions under different conditions on the subsets that the algorithm
may query/sample, and in both the case that the algorithm is adaptive and the
case where it is non-adaptive. In particular we focus on a natural family of
allowed subsets, which correspond to intervals, as well as variants of this
family
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