minimal solution of

USAGE SUMMARY

The phrase "minimal solution of" is correct and usable in written English.
It can be used in contexts such as mathematics, optimization, or problem-solving where you are referring to the simplest or least complex solution to a problem. Example: "In this optimization problem, we are looking for the minimal solution of the equation to achieve the best results."

✓ Grammatically correct

Science

Human-verified examples from authoritative sources

Exact Expressions

35 human-written examples

Since is a weakly minimal solution of,.

Journal of Inequalities and Applications

Then is a weakly minimal solution of.

Journal of Inequalities and Applications

We now show that is a weakly minimal solution of.

Journal of Inequalities and Applications

Assume that is not a weakly minimal solution of.

Journal of Inequalities and Applications

If is a weakly minimal solution of, then (4.13).

Journal of Inequalities and Applications

and is the minimal solution of this equation.

Journal of Inequalities and Applications

Human-verified similar examples from authoritative sources

Similar Expressions

25 human-written examples

( S P − ⪯ )  ⪯-minimize F ( x ), subject to x ∈ X, where minimal solutions of ( S P − ⪯ ) are defined in the following way: Definition 4 (Minimal solutions of ( S P − ⪯ ) w.r.t. the preorder ⪯).

Fixed Point Theory and Applications

The maximal and minimal solutions of problem (1.1) are called its extremal solutions.

Advances in Difference Equations

Rong and Wu [9] introduced ε-weakly minimal solutions of vector optimization problems.

Journal of Inequalities and Applications

This implies that (x^) and (y^) are the maximal and minimal solutions of system (1.1), respectively.

Boundary Value Problems

Next, we formulate a comparison statement for minimal solutions of two Riccati equations.

Advances in Difference Equations

Expert writing Tips

Best practice

When seeking the "minimal solution of" a problem, clearly define the criteria for minimality. Are you minimizing cost, complexity, or another factor? Specificity enhances clarity.

Common error

Avoid using "minimal solution of" when a trivial solution (e.g., zero) exists but is not useful or relevant. Ensure the solution meets all necessary conditions, not just the fewest possible.

Linguistic Context

The phrase "minimal solution of" functions as a noun phrase, typically used to identify the solution that satisfies a given problem or condition with the least magnitude or complexity. As Ludwig AI states, the phrase is grammatically correct and commonly used.

Expression frequency: Common

Frequent in

Science

100%

Less common in

News & Media

0%

Formal & Business

0%

Ludwig's WRAP-UP

The phrase "minimal solution of" is a grammatically sound and commonly employed term, particularly within scientific and mathematical domains. As Ludwig AI confirms, it is used to describe the solution that satisfies a given problem with the least magnitude or complexity. While alternatives like "least solution of" or "simplest solution of" exist, it's crucial to differentiate "minimal" from "trivial" and to clearly define the criteria for minimality. Usage is concentrated in scientific contexts, with authoritative sources such as the Journal of Inequalities and Applications frequently employing the phrase.

More alternative expressions

Phrases that express similar concepts, ordered by semantic similarity:

least solution of

Replaces "minimal" with "least", emphasizing the smallest or lowest value.

smallest solution of

Substitutes "minimal" with "smallest", highlighting the size or magnitude of the solution.

simplest solution of

Uses "simplest" instead of "minimal", focusing on the ease of understanding or implementation of the solution.

optimal solution of

Uses "optimal" instead of "minimal", shifting the focus to the best or most favorable solution.

weakest solution of

Employs "weakest" to suggest a solution that barely satisfies the conditions.

reduced solution of

Uses "reduced" instead of "minimal", emphasizing a solution that has been simplified or made smaller.

basic solution of

Replaces "minimal" with "basic", emphasizing a fundamental or elementary solution.

elementary solution of

Employs "elementary" to suggest a solution that is fundamental and straightforward.

most economical solution of

Substitutes "minimal" with "most economical", implying a solution that minimizes cost or resources.

lower bound solution of

Specifies "lower bound" instead of "minimal", highlighting a solution at the lowest possible limit.

FAQs

How to use "minimal solution of" in a sentence?

You can use "minimal solution of" to describe the smallest or least complex answer to a problem. For example: "The algorithm aims to find the "minimal solution of" the equation."

What is an alternative to "minimal solution of"?

Alternatives include "least solution of", "simplest solution of", or "smallest solution of" depending on the context.

When is it appropriate to use "minimal solution of"?

Use "minimal solution of" when you specifically want to emphasize that the solution is the smallest, least complex, or requires the fewest resources compared to other possible solutions.

Is there a difference between "minimal solution of" and "optimal solution of"?

Yes. While both describe desirable outcomes, "minimal solution of" focuses on reducing something (like complexity or cost), while "optimal solution of" refers to the best possible solution considering all factors, not just minimizing one aspect.