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(i) both commutative and associative;
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(ii) neither commutative nor associative;
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(iii) commutative but not associative;
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(iv) associative but not commutative.
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Some results on the Cohen–Macaulayness of the canonical module;
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We study the S 2-fication of rings which are quotients by lattices ideals;
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Given a simplicial lattice ideal of codimension two I, its Macaulayfication is given explicitly from a system of generators of I.
Given an infinite set X, the Stone space S(X) is ultrafilter compact.
For every infinite set X, every countable filterbase of X extends to an ultra-filter i? for every infinite set X, S(X) is countably compact.
ω has a free ultrafilter i? every countable, ultrafilter compact space is countably compact.
We also show the following:There are a permutation model 𝒩 and a set X ∈ 𝒩 such that X has no free ultrafilters and S(X) is not compact but S(X) is countably compact and every countable filterbase of X extends to an ultrafilter.
It is relatively consistent with ZF that every countable filterbase of ω extends to an ultrafilter but there exists a countable filterbase of ? which does not extend to an ultrafilter. Hence, it is relatively consistent with ZF that ? has free ultrafilters but there exists a countable filterbase of ? which does not extend to an ultrafilter.
We give conditions under which there exist unique solutions of such equations.
Then we prove both a sufficient maximum principle (a verification theorem) and a necessary maximum principle via Hida-Malliavin calculus.
As an application we solve a problem of optimal consumption from a cash flow modelled by an SVIE.
A contains a central element;
A satisfies the additional identity (x, x3, x) = 0.
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bottom‐up
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top‐down
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To provide students with efficient and simple criteria to decide whether a continuous function is also uniformly continuous;
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To provide students with skill to recognize graphically significant classes of both uniformly and nonuniformly continuous functions.
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illustrate the qualitative behaviour of the solutions of the Rayleigh problem by a phase plane analysis;
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describe how the method of multiple scales provides a framework within which previous analyses can be seen;
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describe an improvement to the method of multiple scales, which may also prove useful in other applications.
Every locally finite family of open sets of X is finite iff every pairwise disjoint, locally finite family of open sets is finite.
Every locally finite family of subsets of X is finite iff every pairwise disjoint, locally finite family of subsets of X is finite iff every locally finite family of closed subsets of X is finite.
The statement “every locally finite family of closed sets of X is finite” implies the proposition “every locally finite family of open sets of X is finite”. The converse holds true in case X is T4 and the countable axiom of choice holds true.
We also show:It is relatively consistent with ZF the existence of a non countably compact T1 space such that every pairwise disjoint locally finite family of closed subsets is finite but some locally finite family of subsets is infinite.
It is relatively consistent with ZF the existence of a countably compact T4 space including an infinite pairwise disjoint locally finite family of open (resp. closed) sets.
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If X has strong measure zero aid if Y is contained in an F σ, set of measure zero, then X + Y has measure zero (Proposition 9).
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If X is a measure zero set with property s 0 and Y is a Sierpinski set, then X + Y has property s 0 (Theorem 12).
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If X is a meager set with property s 0 and Y is a Lusin set, then X + Y has property s 0 (Theorem 17).
- i)What is the meaning of swing when more than two parties fight an election?
- ii)How can the distribution of seats at an election be determined from a prediction of national swing?
- iii)How can swing analysis be extended to help determine an electoral strategy for political parties?
- iv)What explanation can be provided for the swing in Scotland?
- v)What explanation can be provided for variations in the English swing?
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if h: UA → UB and g: VA → VB are morphisms with Gh = Tg, there exists a morphism f: A → B such that Uf = h and Vf = g;
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V carries U-initial monosources into T-initial mono-sources.