What do you mean by equicontinuous?

In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus sequences of functions.

How do you prove Equicontinuity?

Lemma 1 Let I be an interval in R and let fn : I → R for n ∈ N. Assume that f is differentiable at all interior points of I (if I is open, that’s all points of I). If there exists M ≥ 0 such that |fn(x)| ≤ M for all x ∈ I0, n ∈ N, then (fn) is equicontinuous. Proof.

What is relative compactness?

Relative compactness is another property of interest. Definition: A subset S of a topological space X is relative compact when the closure Cl(x) is compact. Note that relative compactness does not carry over to topological subspaces.

What do you mean by Stone Weierstrass Theorem?

In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval [a, b] can be uniformly approximated as closely as desired by a polynomial function.

Is Cos n x equicontinuous?

Since cos(n+x) has uniformly bounded derivatives, so they are equicontinuous as well. Hence {fn(x)} is equicontinuous. uniformly bounded, then (fn) is equicontinuous.

What is relative compaction used for?

Relative Compaction Test Set is used for laboratory determination of the maximum wet density of soils and aggregates by the California 216 Impact method. Relative compaction is the ratio of in-place wet density to test the maximum wet density of the same soil or aggregate.

What is locally compact?

In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which every point has a compact neighborhood.

Why is the Bolzano Weierstrass theorem important?

The Bolzano-Weierstrass Theorem says that no matter how “random” the sequence (xn) may be, as long as it is bounded then some part of it must converge. This is very useful when one has some process which produces a “random” sequence such as what we had in the idea of the alleged proof in Theorem 7.3.

Can a bounded sequence have no convergent subsequence?

Boundedness is simple, by your own comment. To show it has no convergent subsequences: Assume it has a convergent subsequence, (xnk), with limit K. Then nk≥n=xn.

What is the difference between relative compaction and relative density?

The concept of relative compaction is applicable to both cohesive and non-cohesive soil, while relative density (Dr) is more appropriate for end-product compaction specification for granular soils.

What is 95% relative compaction?

95% compaction means that the soil on the construction site has been compacted to 95% of the maximum density achieved in the lab. Remember the test in the lab uses a specific compaction effort so it is possible to achieve compaction above 100% in the field.

Is R Sigma compact?

Hence, by definition, R is σ-compact.

What does the Bolzano-Weierstrass theorem say?

What is Bolzano-Weierstrass theorem state?

The theorem states that each bounded sequence in Rn has a convergent subsequence. An equivalent formulation is that a subset of Rn is sequentially compact if and only if it is closed and bounded. The theorem is sometimes called the sequential compactness theorem.

Does every bounded sequence have a Cauchy subsequence?

e) TRUE Every bounded sequence has a Cauchy subsequence. We proved that every bounded sequence (sn) has a convergent subsequence (snk ), but all convergent sequences are Cauchy, so (snk ) is Cauchy.

Is every convergent sequence Cauchy?

Every convergent sequence is a Cauchy sequence. The converse statement is not true in general. However, in the metric space of complex or real numbers the converse is true.

What does relative compaction mean?

Relative compaction in this method is defined as the ratio of the in-place, wet density of a soil or aggregate to the test maximum wet density of the same soil or aggregate when compacted by a specific test method.

Why is relative compaction important?

3.1 Purpose of Compaction Compaction increases the shear strength of the soil. through soil. This is important if the soil is being used to retain water such as would be required for an earth dam. Compaction can prevent the build up of large water pressures that cause soil to liquefy during earthquakes.

What is MDD in compaction?

MDD – Maximum Dry Density. MWD – Maximum Wet Density. OMC – Optimum moisture Content.