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How do you use Lipschitz in a sentence? See 10+ example sentences showing how this word appears in different contexts, plus the exact meaning.

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Lipschitz in a sentence

Lipschitz meaning

  1. (Of a real-valued real function f) Such that there exists a constant K such that whenever x_1 and x_2 are in the domain of f,
  2. f(x_1)-f(x_2)
  3. x_1-x_2

Using Lipschitz

  • The main meaning on this page is: (Of a real-valued real function f) Such that there exists a constant K such that whenever x_1 and x_2 are in the domain of f, |f(x_1)-f(x_2)|≤K|x_1-x_2|.
  • In the example corpus, lipschitz often appears in combinations such as: lipschitz constant, lipschitz continuous, locally lipschitz.

Context around Lipschitz

  • Average sentence length in these examples: 23.5 words
  • Position in the sentence: 2 start, 9 middle, 2 end
  • Sentence types: 13 statements, 0 questions, 0 exclamations

Often before:

locally 3× injective 1× valued 1× large 1× sided 1× gita 1× bounded 1× best 1× unbounded 1×

Often after:

continuous 6× constant 5× function 1× danie 1× constants 1×

Sentence patterns with Lipschitz

These short patterns come directly from the example sentences and show how the word is often embedded:

an injective lipschitz function whose 1× called locally lipschitz continuous if 1× derivatives is lipschitz 1× for the lipschitz constant 1× gita lipschitz danie steele 1× have unbounded lipschitz constants however 1× is also lipschitz 1× is also lipschitz with the 1×

Sentence openings and endings

Common sentence openings:

a bilipschitz function 1× a function is 1× by the result 1× equivalently if x 1× for instance every 1× gita lipschitz danie 1×

Common sentence endings:

a sufficient condition 1× addiction awareness talk 1× an equicontinuous set 1× and short maps 1× bounded as well 1× derivative df exists 1×

Words around Lipschitz

Just before it in the word index:

lips 799 lippy 7 lipps 6 lipponen 8 lippold 2 lippo 4

Just after it in the word index:

lipschultz 2 lipschutz 2 lipscomb 70 lipscombe 5 lipset 2 lipsey 4

Similar in corpus frequency:

aanand 16 abcd 16 abdurrahman 16 aberdour 16 abergele 16 abg 16

Corpus analysis for Lipschitz

  • In this selection, "lipschitz" usually appears in the middle of the sentence. The average example has 23.5 words, and this corpus slice is mostly made up of statements.
  • Around the word, locally, injective, valued, continuous, constant and function stand out and add context to how "lipschitz" is used.
  • Recognizable usage signals include an injective lipschitz function whose and called locally lipschitz continuous if. That gives this page its own corpus information beyond isolated example sentences.
  • By corpus frequency, "lipschitz" sits close to words such as aanand, abcd and abdurrahman, which helps place it inside the broader word index.

Example types with lipschitz

The same corpus examples are grouped by length and sentence type, making it easier to see the contexts in which the word appears:

Short example sentences

For instance, every function that has bounded first derivatives is Lipschitz. (11 words)

More generally, a set of functions with bounded Lipschitz constant forms an equicontinuous set. (14 words)

Moreover, if K is the best Lipschitz constant of f, then whenever the total derivative Df exists. (17 words)

Longer example sentences

Equivalently, if X is a locally compact metric space, then f is locally Lipschitz if and only if it is Lipschitz continuous on every compact subset of X. In spaces that are not locally compact, this is a necessary but not a sufficient condition. (44 words)

However, the discrete metric space is free in the category of bounded metric spaces and Lipschitz continuous maps, and it is free in the category of metric spaces bounded by 1 and short maps. (34 words)

It is also true that every real-valued Lipschitz-continuous map defined on a subset of a metric space can be extended to a Lipschitz-continuous map on the whole space. (31 words)

Example sentences (13)

A bilipschitz function is the same thing as an injective Lipschitz function whose inverse function is also Lipschitz.

A function is called locally Lipschitz continuous if for every x in X there exists a neighborhood U of x such that f restricted to U is Lipschitz continuous.

By the result of the previous paragraph, the limit function is also Lipschitz, with the same bound for the Lipschitz constant.

Equivalently, if X is a locally compact metric space, then f is locally Lipschitz if and only if it is Lipschitz continuous on every compact subset of X. In spaces that are not locally compact, this is a necessary but not a sufficient condition.

It is also true that every real-valued Lipschitz-continuous map defined on a subset of a metric space can be extended to a Lipschitz-continuous map on the whole space.

It is possible that the function F could have a very large Lipschitz constant but a moderately sized, or even negative, one-sided Lipschitz constant.

Gita Lipschitz, Danie Steele, Nic Ingel, Paris Obel, Miya Ichikowitz and Evan Koton at the King David Victory Park addiction awareness talk.

For instance, every function that has bounded first derivatives is Lipschitz.

Advertentie

However, the discrete metric space is free in the category of bounded metric spaces and Lipschitz continuous maps, and it is free in the category of metric spaces bounded by 1 and short maps.

In particular, any continuously differentiable function is locally Lipschitz, as continuous functions are locally bounded so its gradient is locally bounded as well.

More generally, a set of functions with bounded Lipschitz constant forms an equicontinuous set.

Moreover, if K is the best Lipschitz constant of f, then whenever the total derivative Df exists.

This result does not hold for sequences in which the functions may have unbounded Lipschitz constants, however.

Advertentie

Common combinations with lipschitz

These word pairs occur most frequently in English texts:

Frequently asked questions

How do you use "lipschitz" in a sentence?
An example: "A bilipschitz function is the same thing as an injective Lipschitz function whose inverse function is also Lipschitz." This page contains 10+ example sentences with the word "lipschitz" from authentic English texts.
What does "lipschitz" mean?
Lipschitz means: (Of a real-valued real function f) Such that there exists a constant K such that whenever x_1 and x_2 are in the domain of f, |f(x_1)-f(x_2)|≤K|x_1-x_2|.
How many example sentences with "lipschitz" are there?
Voorbeeldzinnen.info contains at least 10+ example sentences with "lipschitz", drawn from a database of millions of English sentences.