left sum

USAGE SUMMARY

The phrase "left sum" is correct and usable in written English.
It can be used in mathematical contexts, particularly in calculus or numerical analysis, to refer to the sum of the areas of rectangles under a curve, using the left endpoints of the intervals. Example: "To approximate the area under the curve, we will calculate the left sum using the specified intervals."

✓ Grammatically correct

Science

Human-verified examples from authoritative sources

Exact Expressions

19 human-written examples

Accounting for the use of EC as VM, the VM funding needs are reduced from ( left (sum _{i} J^{i} P^{i} -text {CA}right)^ ) to (left (sum _{i} J^{i} P^{i} -{text {EC}} (L -text {CA}right)^ ) in (15).

Probability, Uncertainty and Quantitative Risk

Proceeding as in Crépey et al. (2017), one could show that, accounting for the use of economic capital as variation margin, Proposition 4.1 is still valid, provided one replaces ( left (sum _{i} J^{i} P^{i} -text {CA}right)^ ) with (left (sum _{i} J^{i} P^{i} -{text {EC}} (L -text {CA}right)^) in aL -textulas.

Probability, Uncertainty and Quantitative Risk

In conclusion, the objectives of our research are left{begin{array}{c}max left({E}_Uright)=max left[left({sum}_{1le ile n}{varepsilon}_iright)/left({sum}_{1le ile n}{E}_O^iright)right] max left({T}_Nright)=max left({min}_{1le ile n}left(frac{E_O^i}{varepsilon_i}right)right) operatorname{Min}left {Y}_{E2E}^{v_i}right)=min left(sum {y}_{v_j,{v}_k}right)end{array}right.

EURASIP Journal on Wireless Communications and Networking

Variants of TAL TA L) L = T + A The time already lived and the time still left sum up to the total lifespan.

Genus

standard exponential random variates and set ({mathbf {u}}=(E_{1},ldots,E_{d})/ left (sum _{i=1}^{d} E_{i} right)).

Journal of Statistical Distributions and Applications

There are ADD libraries to efficiently compute operations such as addition, multiplication, and marginalization (left (sum _{x_{i} in X_{i}}right)).

Journal of the Brazilian Computer Society

Human-verified similar examples from authoritative sources

Similar Expressions

41 human-written examples

We also have further examples originating from geometric applications, given by functions, begin{aligned} F_{k,-alpha }[r]:= F_{k,-alpha }(lambda (r)) = left{ sum limits _{i_1<cdots <i_k}left[ <span class="lh">sum _{s=1}^k lambda _{i_s}(r)right] ^{-alpha }right} ^{-frac{1}{alpha }}, quad alpha > 0, end{aligned} (4.13)also defined in the cone ({mathcal {P}}_k) for (k=1,ldots,n).

Bulletin of Mathematical Sciences

To see this, let (A := left | sum _{underline {i} in Lambda _{X}^{prime }}c_{underline {i}}underline {a}^{underline {i}}d^{w_{X} - sum underline {i}} right |).

Pacific Journal of Mathematics for Industry

Here, (max _{i,j; ineq j} left | sum _{l=0}^{-j2pie^{-j2pi (lambda _{m} - lambda _{n})l / N} right |) is the fitness function which should be optimized.

EURASIP Journal on Advances in Signal Processing

r = rxy = frac{{nleft( {sum {xy} } right) - left( {sum x } right)left( {sum y } right)}}{{left( {nsqrt {sum {x^{2} - left( x right)^{2} } } } right)left( {nsqrt {sum {y^{2} - left( {sum y } right)^{2} } } } right)}}where: r = correlation coefficient; x = concentration; y = absorbance; n = no. of observations.

Chemistry Central Journal

1. Choose a finite subset (Lambda subset (mathbb {Z}_{geq 0})^{n}) such that (# left {sum underline {i} mid underline {i} in Lambda right } = # Lambda ).   2.

Pacific Journal of Mathematics for Industry

Expert writing Tips

Best practice

When using "left sum" in mathematical writing, clearly define the interval and function being approximated. Provide a visual representation to aid understanding.

Common error

Avoid using "left sum" interchangeably with other numerical integration methods like the right sum or midpoint rule. Each method yields different approximations, so specify the method used for clarity.

Linguistic Context

The phrase "left sum" functions as a technical noun phrase in mathematics, particularly in the context of numerical integration and calculus. It refers to a specific method of approximating the area under a curve using rectangles. Ludwig AI confirms its usability in mathematical contexts.

Expression frequency: Rare

Frequent in

Science

100%

Less common in

News & Media

0%

Formal & Business

0%

Academia

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Ludwig's WRAP-UP

The term "left sum" is a technical term used in mathematics, specifically in numerical integration, to describe a method of approximating the definite integral of a function. Ludwig AI confirms that the phrase is grammatically correct and usable in written English. The usage is primarily found within scientific and academic contexts, making it a formal term. While alternatives like "left Riemann sum" or "left endpoint sum" exist, "left sum" is the more concise option. When using this term, ensure clarity by defining the function and interval being approximated to avoid confusion with other approximation methods.

More alternative expressions

Phrases that express similar concepts, ordered by semantic similarity:

left-hand sum

Variant spelling using a hyphen.

left Riemann sum

More specific mathematical term.

left endpoint sum

More descriptive mathematical term.

lower Riemann sum

Similar, but considers the lowest value within each interval.

numerical approximation using left endpoints

Describes the procedure without using the concise term.

area approximation using left rectangles

Emphasizes the geometric interpretation.

sum of areas of rectangles at the left

Rephrases the definition of the concept.

approximation from the left

Shorter, more general term.

estimating area with left-side values

Informal description of the calculation.

underestimation using rectangles

Highlights that left sums often underestimate area.

FAQs

What is a "left sum" used for?

A "left sum", also known as a left Riemann sum or left endpoint sum, is a method for approximating the definite integral of a function. It involves dividing the area under the curve into rectangles and summing their areas using the left endpoint of each interval to determine the height of the rectangle.

How does a "left sum" differ from a right sum?

While both are methods for approximating definite integrals, the "left sum" uses the left endpoint of each interval to determine the height of the rectangle, whereas the right sum uses the right endpoint. This can result in different approximations, especially when the function is not monotonic.

Is a "left sum" always an underestimation of the area?

No, a "left sum" is not always an underestimation. If the function is increasing over the interval, the left sum will underestimate the area. However, if the function is decreasing, the "left sum" will overestimate the area. Whether it's an under or overestimation depends on the function's behavior.

What are some alternatives to using a "left sum" for approximating integrals?

Besides the right sum, other alternatives include the midpoint rule, which uses the midpoint of each interval, and the trapezoidal rule, which averages the left and right endpoints. More advanced techniques like Simpson's rule offer even more accurate approximations. You could also say "left Riemann sum" or "left endpoint sum".