any given compact

USAGE SUMMARY

The phrase "any given compact" is correct and usable in written English.
It can be used in mathematical contexts, particularly in topology or analysis, to refer to any specific compact set being discussed. Example: "In any given compact space, every open cover has a finite subcover."

✓ Grammatically correct

Science

Human-verified examples from authoritative sources

Exact Expressions

5 human-written examples

Indeed, for any given compact subset, let.

Fixed Point Theory and Applications

Thus we have lim n → ∞ ∥ K f n ∥ LB = 0. for any given compact operator K on LB.

Journal of Inequalities and Applications

It follows from (3.4) that, for any given compact set E in the interior of (R^{3}_), there is a positive orbit of (2.2) leaving ultimately E, which will not return to E afterwards.

Advances in Difference Equations

In this case, for any given compact set E in the interior of (R^{3}_), we can take sufficiently large C such that bigl{ (x,y,z vert V x,y,z) = C bigr} cap E = emptyset, which implies the non-permanence of system (1.5).

Advances in Difference Equations

Since by a well-known theorem, for any compact operator (widehat{K}:Xto Y), where X and Y are Banach spaces, the weak convergence (x_{n}stackrel{w}{to}x_{0}) implies the norm convergence (widehat{K}x_{n}towidehat{K}x_{0}) [31], we have lim_{jrightarrowinfty}|Kq_{j}|_{H^{infty}_{mu}}=0, (17) for any given compact operator K from (mathcal{B}^{alpha}_{0}) to (H^{infty}_{mu}).

Journal of Inequalities and Applications

Human-verified similar examples from authoritative sources

Similar Expressions

54 human-written examples

The paper is devoted to a design of control laws with time-invariant parameters, which insure the desired quality (stability or accurate output tracking) of any system with unknown parameters from a given compact set of minimum-phase systems with given relative degree.

IFAC Proceedings Volumes

Let be a given compact set in.

Advances in Difference Equations

Let J ⊂ R be a given compact interval.

Boundary Value Problems

We assume that the parameters lie in a given compact set and optimise the worst case over this set.

Automatica

The systems under consideration contain uncertain parameters whose values are known only to within a given compact bounding set.

IFAC Proceedings Volumes

(1) For a given compact set (bar{mathcal{D}}subsetmathcal {R}^{m+n}), let (d=sup{|w-w^{0}||winbar{mathcal{D}}}), where (w^{0}) is the initial iterate.

Journal of Inequalities and Applications

Expert writing Tips

Best practice

When using "any given compact", ensure the context clearly establishes the nature and relevance of the compact set within your argument or proof. This phrase is most effective when the properties of compact sets are central to the discussion.

Common error

Avoid using "any given compact" without a solid understanding of what compactness implies (e.g., closedness, boundedness in Euclidean space). Misapplying the concept can lead to incorrect assumptions in mathematical reasoning.

Linguistic Context

The phrase "any given compact" functions as a determiner phrase modifying the noun "compact". It introduces a compact set in a general sense, allowing for the application of properties or theorems that hold true for all such sets, as shown in Ludwig's examples.

Expression frequency: Rare

Frequent in

Science

100%

Less common in

News & Media

0%

Formal & Business

0%

Encyclopedias

0%

Ludwig's WRAP-UP

In summary, the phrase "any given compact" is a formal and precise term primarily used within mathematical and scientific contexts. As Ludwig AI explained, it serves to introduce a compact set in a general manner, ensuring that any subsequent statements or proofs apply universally to all compact sets. While grammatically correct, its usage is relatively rare and concentrated within academic and scientific writing. When using the phrase, ensure a clear understanding of compactness and its implications to avoid conceptual errors. Alternative phrases like "any specific compact set" or "any particular compact set" may offer similar meanings with slight variations in emphasis.

More alternative expressions

Phrases that express similar concepts, ordered by semantic similarity:

any specific compact set

Replaces "given" with "specific" to emphasize the particularity of the compact set.

any particular compact set

Substitutes "given" with "particular" to highlight the distinctiveness of the compact set under consideration.

any compact set under consideration

Rephrases to emphasize that the compact set is part of the current discussion or analysis.

any assumed compact set

Indicates that the compact set is being taken as a premise or condition.

any defined compact set

Highlights that the compact set has been previously defined or specified.

any compact set in question

Focuses on a compact set that is currently being investigated or analyzed.

any compact subset provided

Emphasizes that the compact set is being supplied or offered as part of a larger context.

any hypothetical compact set

Introduces a compact set that is theoretical or for the sake of argument.

any pre-defined compact set

Highlights that the compact set was established or determined beforehand.

an arbitrarily chosen compact set

Emphasizes the random or unconstrained selection of the compact set.

FAQs

How is "any given compact" used in mathematical proofs?

In mathematical proofs, "any given compact" typically introduces a general compact set to which certain theorems or properties are then applied. It indicates that the subsequent argument holds for all compact sets, not just specific examples.

What distinguishes "any given compact" from "a compact set"?

"Any given compact" emphasizes the generality of the statement, meaning it applies to every compact set. "A compact set" can refer to a specific instance or example, without necessarily implying a universal application.

Can I use other adjectives instead of "given" in "any given compact"?

Yes, you can use alternatives such as "any specific compact set" or "any particular compact set" to add emphasis or clarify the context. The choice depends on the nuance you want to convey.

Is "any given compact set" more formal than "a compact set"?

While both phrases are acceptable in formal writing, "any given compact set" tends to be slightly more formal due to its explicit emphasis on generality. The phrase "a compact set" is commonly used as well and is typically interchangeable depending on the context.