mathematics

Symmetric decreasing rearrangement

Symmetric decreasing rearrangement is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Symmetric decreasing rearrangement rather than just read about it. In short: In mathematics, the symmetric decreasing rearrangement of a function is a function which is symmetric and decreasing, and whose level sets are of the same size as those of the original function. Definition for sets Given a measurable set, A , {\displaystyle A,} in R n , {\displaystyle \mathbb {R} ^{n},} one defines the symmetric rearrangement of A , {\displaystyle A,} called A ∗ , {\displaystyle A^{*},} as the ball…

Key takeaways

Symmetric decreasing rearrangement belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Symmetric decreasing rearrangement to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Symmetric decreasing rearrangement from memory before moving on to harder problems.

Reference excerpt

In mathematics, the symmetric decreasing rearrangement of a function is a function which is symmetric and decreasing, and whose level sets are of the same size as those of the original function.

Definition for sets Given a measurable set, A , {\displaystyle A,} in R n , {\displaystyle \mathbb {R} ^{n},} one defines the symmetric rearrangement of A , {\displaystyle A,} called A ∗ , {\displaystyle A^{*},} as the ball centered at the origin, whose volume (Lebesgue measure) is the same as that of the set A . {\displaystyle A.} An equivalent definition is

A ∗ = { x ∈ R n : ω n ⋅ | x | n < | A | } , {\displaystyle A^{*}=\left\{x\in \mathbb {R} ^{n}:\,\omega _{n}\cdot |x|^{n}<|A|\right\},}

where ω n {\displaystyle \omega _{n}} is the volume of the unit ball and where | A | {\displaystyle |A|} is the volume of A . {\displaystyle A.}

Definition for functions The rearrangement of a non-negative, measurable real-valued function f {\displaystyle f} whose level sets f − 1 ( y ) {\displaystyle f^{-1}(y)} (for y ∈ R ≥ 0 {\displaystyle y\in \mathbb {R} _{\geq 0}} ) have finite measure is

f ∗ ( x ) = ∫ 0 ∞ I { y : f ( y ) > t } ∗ ( x ) d t , {\displaystyle f^{*}(x)=\int _{0}^{\infty }\mathbb {I} _{\{y:f(y)>t\}^{*}}(x)\,dt,}

where I A {\displaystyle \mathbb {I} _{A}} denotes the indicator function of the set A . {\displaystyle A.} In words, the value of f ∗ ( x ) {\displaystyle f^{*}(x)} gives the height t {\displaystyle t} for which the radius of the symmetric rearrangement of { y : f ( y ) > t } {\displaystyle \{y:f(y)>t\}} is equal to x . {\displaystyle x.} We have the following motivation for this definition. Because the identity

g ( x ) = ∫ 0 ∞ I { y : g ( y ) > t } ( x ) d t , {\displaystyle g(x)=\int _{0}^{\infty }\mathbb {I} _{\{y:g(y)>t\}}(x)\,dt,}

holds for any non-negative function g , {\displaystyle g,} the above definition is the unique definition that forces the identity I A ∗ = I A ∗ {\displaystyle \mathbb {I} _{A}^{*}=\mathbb {I} _{A^{*}}} to hold.

Properties

The function f ∗ {\displaystyle f^{*}} is a symmetric and decreasing function whose level sets have the same measure as the level sets of f , {\displaystyle f,} that is,

| { x : f ∗ ( x ) > t } | = | { x : f ( x ) > t } | . {\displaystyle |\{x:f^{*}(x)>t\}|=|\{x:f(x)>t\}|.}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Symmetric decreasing rearrangement

Start with the simplest possible case. Write down what Symmetric decreasing rearrangement claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Symmetric decreasing rearrangement before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Symmetric decreasing rearrangement ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Symmetric decreasing rearrangement

In research: Symmetric decreasing rearrangement appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Symmetric decreasing rearrangement in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Symmetric decreasing rearrangement is common in secondary-school and first-year university syllabi. It links to neighbouring topics Measure theory, Multivariable calculus, Real analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Symmetric decreasing rearrangement outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Symmetric decreasing rearrangement in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Symmetric decreasing rearrangement means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Symmetric decreasing rearrangement out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Symmetric decreasing rearrangement in simple terms?

In mathematics, the symmetric decreasing rearrangement of a function is a function which is symmetric and decreasing, and whose level sets are of the same size as those of the original function. Definition for sets Given a measurable set, A , {\displaystyle A,} in R n , {\displaystyle \mathbb {R} ^…

Why does Symmetric decreasing rearrangement matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Symmetric decreasing rearrangement?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Symmetric decreasing rearrangement.