Used and loved by millions
Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com
with an upper bound of
Grammar usage guide and real-world examplesTable of contents
Usage summary
Human-verified examples
Expert writing tips
Linguistic context
Ludwig's wrap-up
Alternative expressions
FAQs
Human-verified examples from authoritative sources
Exact Expressions
15 human-written examples
Adequate condition are obtained in terms of linear matrix inequalities (LMIs) which are independent of time delays such that the resultant collection of discrete time-delay systems are stable with an upper bound of the quadratic performance index.
From data available in the literature, the mean contractile force (Fc) generated by individual dermal fibroblasts within the collagen-glycosaminoglycan scaffold was calculated to range between 11 and 41 nN (Fc = 26 ± 13 nN, mean ± SD), with an upper bound of cell contractility estimated at 450 nN.
Science
It is 3.6°C, with an upper bound of 4.9°C.
Science & Research
Sozio and Gionis [28] also study the problem of querying the optimal community with an upper bound of the size.
Science
In addition, in Figure 7b, we compare the performance of IBRC with an upper bound of the optimal solution to assess its efficiency.
In the case of a 5000 VM deployment Varanus achieves a mean latency of 1.64 s with an upper bound of 2.04 s.
Science
Human-verified similar examples from authoritative sources
Similar Expressions
45 human-written examples
Ruiz and Allahverdi [12] minimized the bi-criteria of makespan and maximum tardiness with an upper bound on maximum tardiness of the flowshop scheduling problem.
For networks with fixed controlled nodes, an upper bound of the objective function is obtained which is shown to be convex and independent of the time-horizon.
Science
The aim of the filtering problem is to design a least squares filter for the formulated nonlinear system with uncertain polynomials, and an upper bound of the filtering error covariance is found and subsequently minimized at each time step.
Science
Consequently, based on the analysis of those three approaches mix in this study, we suggest the alternative interval of carbon price benchmark in the national unified carbon trading market with a lower bound of 30 yuan/toO2 to an upper bound of 50 yuan/tCO2.
Science
Each surface describes the end-of-life SRH prevalence of a birth cohort for the range of lifespan permitted by the survey, but with a lower bound of 70 and an upper bound of 100.
Science
Expert writing Tips
Best practice
Ensure that the value following the phrase is the absolute maximum the variable can reach under the described conditions.
Common error
Avoid using redundant phrases like "with a maximum upper bound of", as an upper bound by definition is a maximum limit. Additionally, do not use this phrase when you actually mean an average or a typical value; it must strictly refer to a constraint.
Source & Trust
85%
Authority and reliability
5/5
Expert rating
Real-world application tested
Linguistic Context
The phrase "with an upper bound of" functions as a complex prepositional phrase that introduces a numerical or conceptual limit. In the context of Ludwig AI, it typically modifies a noun phrase (like "temperature increase" or "quadratic performance index") to establish a constraint.
Frequent in
Science
75%
Academia
20%
Formal & Business
5%
Less common in
News & Media
3%
Encyclopedias
1%
Social Media
1%
Ludwig's WRAP-UP
The phrase "with an upper bound of" is an essential tool for high-level technical writing. Ludwig AI identifies it as a highly formal expression primarily utilized within the scientific community to denote strict limits or statistical caps. It is most frequently used to qualify estimates, such as p-values, temperature projections or mechanical forces, by providing the maximum possible threshold. When using this phrase, writers should ensure they are describing a definitive limit rather than a general range. For less technical contexts, alternatives like "up to a maximum of" may be more accessible, but for peer-reviewed research, the original phrase remains the gold standard for clarity and precision.
More alternative expressions(10)
Phrases that express similar concepts, ordered by semantic similarity:
bounded above by
Uses a more formal mathematical verb to describe the same constraint.
with a maximum limit of
Replaces technical terminology with a more general synonym for bound.
subject to a ceiling of
Employs a common metaphorical term for a maximum limit.
capped at
A more concise and slightly less formal way to state a limit.
not exceeding
Focuses on the negative constraint rather than the boundary itself.
limited to a maximum of
Adds emphasis on the restrictive nature of the boundary.
up to a maximum of
Common in non-technical contexts to describe range limits.
within an upper limit of
Uses limit instead of bound, which is slightly more accessible.
restricted to an upper bound of
Explicitly mentions the restriction applied to the boundary.
possessing a peak value of
Focuses on the highest point reached rather than the theoretical limit.
FAQs
How do I use "with an upper bound of" in a sentence?
You can use it to define a limit for a variable, for example: "The study estimated the temperature increase at 3.6°C "with an upper bound of" 4.9°C."
What is the difference between "with an upper bound of" and "up to"?
While both indicate a limit, "with an upper bound of" is more formal and used in statistical or mathematical contexts, whereas "up to" is common in everyday English.
Can I say "capped at" instead?
Yes, "capped at" is a suitable alternative in professional or business writing, though "with an upper bound of" remains preferred in pure science.
Is "with an upper bound of" considered formal?
Yes, it is highly formal and most frequently found in academic journals, scientific research papers and technical documentation.
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Table of contents
Usage summary
Human-verified examples
Expert writing tips
Linguistic context
Ludwig's wrap-up
Alternative expressions
FAQs
Source & Trust
85%
Authority and reliability
5/5
Expert rating
Real-world application tested