There’s a lot of confusion about the correct definition of an Into Function.

It is sometimes defined as a function that [1]

…maps X into Y if and only if for each x in X, f(x) is in Y.

Or a function “…for which the range is a subset of the codomain” [2]. This seems to agree with the first definition. A quick Google search brings up this more formal definition [3], which also seems to be in agreement:

A function f is into Y iff the range of f is a subset of Y . i.e., R_f⊂Y.

However, other authors define an into function a little differently. For example, Kumar [4] sets the requirement that there must be at least one element in a set B which is not mapped by any element of set A. Cue Math [5] agrees with this definition:

“…for an into function, there should be at least one element in the output set B that should not be connected to the elements of the input set A.”

So, What’s the Correct Definition for an “Into Function”?

Perhaps the biggest clue in solving the puzzle is that entries for “Into function” are notably absent from Wolfram Mathworld and even Wikipedia.
So, why is that? It seems to indicate that a formal definition for Into Function does not exist.

That means there probably isn’t a “correct” definition at all. One thing is for sure, an into function is not a synonym for injective, as this post on Stack Exchange points out. The confusion may come from the fact that surjective functions are sometimes described as being “onto”. However, it doesn’t mean that an injective function is “into.”

This is one of those definitions that you have to take with a grain of salt. Treat it as a loose definition or a colloquialism. If a website is talking about an Into Function in the same breath as an Onto Function, then they are most likely (incorrectly) referring to an injective function. But if you are reading a good book on set theory, they are probably (but not always) using the definition pointed out in the Stack Exchange article (which cites [6]), which is

References

[1] Rodgers, N. (2011). Learning to Reason. An Introduction to Logic, Sets, and Relations. Wiley.
[2] Dybkær, R. (1968). APMIS Supplementum. Munksgaard.
[3] Tiwari, G. Set Theory, Functions and Real Numbers. Retrieved March 7, 2022 from: https://gauravtiwari.org/set-theory-functions-and-real-number-system/#into-function
[4] Kumar, (201). Theory of Automata. McGraw-Hill Education (India) Pvt Limited Language: English.
[5] Cue Math. Retrieved March 7, 2022 from https://www.cuemath.com/algebra/onto-function/
[6] Suppes, P. (1960). Axiomatic Set Theory. Dover.

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The Witch of Agnesi is a special case of a cubic hyperbola that is roughly bell-shaped.

The Witch of Agnesi (red) and rotated 90° (black) with a = 5.

xy² = a²(a – x).

Alternatively, the curve can be turned 90° so that it lies on the horizontal axis:

yx² = a²(a – y).

The parametric equations are [1]:

• x(t) = at
• y(t) = a/(1 + t)²

The curve is named after the 18th century mathematician Maria Agnesi (1718 to 1799). It has a few surprising real life applications including as an approximation for the spectral line distribution of optical lines and x-rays. It also approximates the amount of power dissipated in resonant circuits [2].

Finding Derivatives

The coordinates x₀ and y₀ satisfy the equation


We can take the derivative (with the chain rule) to get:

Construction of the Witch of Agnesi

Construction of the witch of Agnesi.

1. Choose a circle of diameter a.
2. Center the circle at (0, a/2)
3. Choose a point A on the line y = a; draw a line from A to the origin. Mark a new point B where this line crosses the circle.
4. If P is the point where the vertical line through P crosses the horizontal line through B, then the witch of Agnesi is the curve traced by P as A travels along the line y = a.

Why is it Called the Witch of Agnesi?

It turns out, it’s named the “witch” because of a mistake in translation [4]. Agnesi’s two-volume treatise on calculus included the curve, which she called versiera, which is Italian for “that-which-turns.” But when her work was translated into English by a Cambridge professor, he mistook the word for l’aversiera, which means “witch” in Italian.

References

[1] Fifty Famous Curves, Lots of Calculus Questions, And a Few Answers.
[2] Applications to the Witch. Retrieved March 7, 2022 from: https://cs.appstate.edu/~sjg/wmm/final/agnesifinal/applications.pdf
[3] The Witch of Agnesi. Retrieved March 6, 2022 from: https://mathwomen.agnesscott.org/women/witch.htm
[4] Lienhard, J. No. 1741: THE WITCH OF AGNESI. Retrieved March 7, 2022 from: https://www.uh.edu/engines/epi1741.htm

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What is a Positive Function?

A positive function has function values greater than zero (i.e., f(x) > 0). The domain (inputs) of the function can be negative, but the outputs (y-values) must be greater than zero. In other words, a positive function has values that are positive for all arguments of its domain.

A non-negative function is similar, except that it includes zero in its range.

Positive (red): values above the x-axis. Negative (blue): values below the x-axis.

Graph of f(x) = x³. The positive interval is shaded in red.

A caution: a positive function isn’t necessarily an increasing function (although it can be). The function f(x) = 4x² + 2, shown on the above graph, is completely above the x-axis, which means it is a positive function. However, notice that it is only increasing for function values on the right-hand side of the vertical axis; the function is decreasing for values to the left of the y-axis. In other words, positive functions can have derivatives that are negative or positive.

A couple of interesting properties:

• A positive function f(x) is log-convex if log f(x) is convex [1].
• A linear combination of positive functions is a positive function.

What is a Negative Function?

A negative function has values that are all negative (i.e., f(x) < 0). The domain (inputs) of the function can be positive, but every output (y-value) must be less than zero. In other words, a negative function has values that are negative for all arguments of its domain. Graphically, all output (y) values are below the horizontal axis.

This function is a negative function because all y-values are below the x-axis.

Positive Function and Integrals

The definite integral of a positive function represents area under the graph of the function from a to b.

Area under the curve of x² from [1, 5].

A positive function is integrable if it is a measurable function and if the integral is less than infinity [2].

References

Image created with Desmos.com.
[1] Ni, L. Additional Problems-Set 5. Retrieved March 6, 2021 from: https://mathweb.ucsd.edu/~lni/math220/Pre-pr5.pdf
[2] Hunter, J. Chapter 4: Integration. Retrieved March 6, 2022 from: https://www.math.ucdavis.edu/~hunter/measure_theory/measure_notes_ch4.pdf

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The Briggs logarithm (also called the Briggsian logarithm, common logarithm, or decadic logarithm) is a logarithm to base 10; the natural logarithm (to base e) are called Naperian logarithms [1].

Note though, that the Briggs logarithm is denoted as Log(N) in the literature; the natural log is denoted as ln(N) [2].

As an example of a Brigg’s logarithm, the logarithm of 100 to the base 10 is 2. This is written as:

log₁₀ 100 = 2.

Note the “10” on the base: this tells you it is a Brigg’s logarithm.

History of the Brigg’s Logarithm

Briggs logarithm was mentioned in Saunderson and de Moivre’s 1740 text The Elements of Algebra [3].

Briggs logarithm tables allowed efficient replacement of multiplications by additions. One notable addition to the tables was the future Euler’s number (e), which was hidden in the tables in the limit e^-1 = lim (1 – N^-1)^N, N → ∞ (Briggs never made it’s meaning explicit) [4].

The term “Briggs logarithm” was used in many early texts (1700s to 1800s). It fell out of common usage in the 1900s, replaced by the term “common logarithm.” However, you’ll still find the term used in mathematical texts, books on logarithms, and some specialized scientific fields.

Working with logarithms in general fell out of popular usage with the advent of powerful hand held calculators, which can handle every base. Instead of logarithms, calculators can solve exponential equations y = a^x with the inverse x = log(to base a)^y, which replaces the traditional “take the log of both sides.”

References

[1] Logarithm. Retrieved March 6, 2022 from: http://www2.cfcc.edu/faculty/cmoore/LogarithmInfo.htm
[2] PROPERTIES OF THE POSITIVE INTEGERS.
Retrieved March 6, 2022 from: https://mae.ufl.edu/~uhk/ALL-ABOUT-INTEGERS.pdf
[3] Saunderson, N., Saunderson, J. & de Moivre, A. (1740). The Elements of Algebra, in Ten Books. University Press.
[4] Dani, S. Papadopoulis, A. (Eds.) (2019). Geometry in History. Springer.

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The term broken function has several meanings in calculus.

1. The quotient of two functions.
   
   As this is a ratio, the denominator cannot be zero [1].
2. The term “broken function” may also be used less formally to describe ill-behaved functions like the Weierstrass function, which is nowhere differentiable [2].
3. The term is sometimes used as a synonym for piecewise functions (which are, literally, functions that are broken into pieces). An example is this article on satellite galaxies, where a broken function is characterized by a break splitting constant values with increasing linearity for a four-parameter function.
4. In partial fraction decomposition, functions are broken into pieces before integrating. This makes some complicated functions easier to integrate.
5. Colloquially speaking, a broken function may also refer to a function with holes.
   

   “Old Math also allowed grimy, dirty tampering with functions saying that y = x is a function, but also, saying that at x = 0 then y = 3, and y = x elsewhere. What I called a broken function” ~ AP [3].

In computer science, the term can be used to describe a programming function (i.e., a block of code that is intended to perform an action) that doesn’t work.

Broken Function Space

A broken function space (or polynomial space) is a special case of broken Sobolev spaces.
For S_h ∈ {K_h, P_h) and integer k &gr; 0 they are defined by [4]


Where P^k_d are the space of polynomial functions of, at most, degree k.

References

[1] Chapter 2: Functions of One Variable, Part 2. Retrieved March 6, 2022 from: https://www.studeersnel.nl/nl/document/maastricht-university/quantitative-methods-i/math-chapter-2-summary/5533756
[2] Lambert, F. Fractal zooms and infinite spaces: the Unbearable Quest for the Sublime. Retrieved March 6, 2022 from: https://www.academia.edu/12634007/Fractal_Zooms_and_Infinite_Spaces_the_Unbearable_Quest_for_the_Sublime
[3] Fibonacci sequence and now the AP subset sequence 0, 2, 5, 9, 14, 20, 27, 35, 44, 54, . . Retrieved March 6, 2022 from: https://groups.google.com/g/sci.math/c/iZi1o1dUksA?pli=1
[4] Pietro, D. & Lemaire, S. (2013). An extension of the Crouzeix–Raviart space to general mesheswith application to quasi-incompressible linear elasticity and Stokes flow. Mathematics of Computation.

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The folium of Descartes (or noeud de ruban in French) is a single loop with one node (ordinary double point) and two asymptotes at the ends.


It is an algebraic curve defined by the equation x³ + y³ – 3axy = 0.

Polar coordinates are

The curve can be graphed with parametric equations, as shown in this image:


However, the parametric equations are not defined at t = -1. Therefore, you have to create a piecewise function graph that excludes t = 1 (In other words, graph two parts, one either side of t = 1).

Folium of Descartes Properties

• The curve is symmetric about the line y = x.
• One ordinary double point at the origin.
• Area of loop interior = 3a²/2.
• Area between asymptote and curve “wing” = 3a²/2.
• Horizontal tangent lines at the origin and the point
  

Graph showing the two horizontal tangent lines (in green) for the folium of Descartes (created with Desmos.com).

The curve of Descartes is related to the trisectrix of Maclaurin by affine transformation.

Trisectrix of Maclaurin (purple) plotted with Folium of Descartes (blue). Graphed with Desmos.com.

Finding Tangent Lines

Tangent lines for the curves can be found using implicit differentiation.

Example question: Find the tangent line for the point (2, 4) on Descartes folium.

Solution:

Step 1: Differentiate both sides of the equation with respect to x:


Step 2: Insert the coordinates (2, 4) into the formula and solve:

History of the Folium of Descartes

The folium of Descartes is named after René Descartes (1596 to 1650), who was the first to discuss it. He discovered the folium thanks to a challenge he put out to Fermat — to find a tangent line for an arbitrary point. While Fermat won the challenge, Descartes discovered his folium during the process [1].

The curve provided a role in the early development of calculus and provides the proof for some parts of Fermat’s Last Theorem [2].

Albania issued a postage stamp depicting Descartes and his folium in 1966 [1].

References

[1] Amoroso, R. FE, FI, FO, FOLIUM: A DISCOURSE ON DESCARTES’ MATHEMATICAL CURIOSITY.
[2] Pricopie, S. & Udriste, C. Multiplicative group law on the folium of Descartes. Retrieved March 6, 2021 from: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.402.4317&rep=rep1&type=pdf

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A pedal curve is a new curve derived from a given curve and a point.

Positive Pedal Curve

Construction of a Limaçon as a pedal curve of a circle [1].

1. Choose a curve, with fixed point O (called the pedal origin), from which you want to create a pedal curve. For the above example, the choice is a circle centered at the origin.
2. Draw a tangent line at any point P on the curve. In the above image, point P is the black dot on the circumference (outer edge) of the circle; the tangent line is shown in blue.
3. Mark point Q on the tangent line (shown as a red dot in the above image) so that OQ and PQ are perpendicular (i.e., at right angles). If you look closely at the above image, you’ll see that the tangent line (blue) and perpendicular (green) are always at right angles with respect to each other.
4. Repeat steps 2 and 3 for more points P on the curve; Technically you could repeat these steps an infinite number of times (as in the image). However, if drawing by hand you’ll want to stop when you see the general shape of the new curve.

The locus* of point Q (on the tangent line) is called the pedal of the given curve with respect to point O. The curve that results from this procedure is sometimes called a primitive curve.

*A locus (plural loci) is the set of points with a location that satisfies or is determined by specified conditions. For example, the locus of the above image is the set of points Q that satisfy the conditions laid out in the above steps.

Negative Pedal Curve

A negative pedal curve of a Limaçonresults in a circle [1].

1. Choose a curve, with fixed point O, from which you want to create a pedal curve. The above image shows a Limaçon.
2. Draw a line from O to any point P on the curve. In this example, point O is the black point at the top of the Limaçon. Point P is the red dot; the line OP is shown in green.
3. Draw a line perpendicular to OP, passing point P. The perpendicular is shown in blue in the image above.
4. Repeat Steps 2 and 3 ad infinitum.

As you can tell from the image, the result is the circle. In other words, the negative pedal curve inverses (reverses) the pedal curve procedure and vice versa,

References

[1] Sam Derbyshire at the English Wikipedia, CC BY-SA 3.0 , via Wikimedia Commons
[2] Pedal Curve. Retrieved March 6, 2021 from:
http://xahlee.info/SpecialPlaneCurves_dir/Pedal_dir/pedal.html

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The serpentine curve, named because of its snakelike shape, is a cubic curve defined by the Cartesian equation [1]


or by the parametric equations

The serpentine is a subcase of one of the normal forms of the cubic curve f(x, y) = 0:
xy² + ey = ax³ + bx² + cx + d [1].

Graph of a serpentine curve.

Serpentine Curve Properties

• Local maximum at x = a and Local minimum at x = -1,

Where the first derivative is zero, i.e.,:



• Inflection points are at x = ± √(3a), where the second derivative equals zero, i.e.,[2]
  
  

History

The serpentine curve was first studied by L’Hôpital and Huygens in 1692. Later, it was studied by Newton, in 1701, who gave the curve its name [3]. It appears in the 1710 work Curves by Sir Isaac Newton in Lexicon Technicumby John Harris [1].

Applications of the Serpentine Curve

Snakes, perhaps not surprisingly, move in the pattern of a serpentine curve. According the Hirose [4], that’s because the curve has the “greatest amount of smoothness of contraction and relaxation of the motor muscles”. The model for snake movement is a little more challenging than the generic serpentine curve proposed by Newton:


Where:

• α is the winding angle,
• j_m is the mth order Bessel function.

Serpentine curves are also used in surveying, where they are also called S-curves. They are generally used to connect two railway lines or parallel roads intersect at a tiny angle [5].

In engineering, the trajectory of a Chaplygin sleigh with periodic actuation is a serpentine curve [6].

References

[1] Mactutor. Serpentine. Retrieved March 6, 2022 from: https://mathshistory.st-andrews.ac.uk/Curves/Serpentine/
[2] Weisstein, Eric W. “Serpentine Curve.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/SerpentineCurve.html
[3] Titu Maiorescu University. The International Conference Education and Creativity for a Knowledge based Society – Computer Science, 2012.
[4] S. Hirose. Biologically Inspired Robots: Snake-Like Locomotors and
Manipulators. Oxford University Press, 1993. Cited in Spranklin, B. DESIGN, ANALYSIS, AND FABRICATION OF A SNAKE-INSPIRED ROBOT WITH A RECTILINEAR GAIT. 2006.
[5] Types of Curves in Surveying. Retrieved March 6, 2022 from: https://dailycivil.com/types-of-curves-in-surveying/
[6] Fedonyuk, V. (2020). Dynamics and Control of Nonholonomic Systems with Internal Degrees of Freedom. Retrieved March 6, 2022 from: https://tigerprints.clemson.edu/cgi/viewcontent.cgi?article=3650&context=all_dissertations

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A natural equation of a curve specifies a curve independent of a choice of coordinate system or parameterization. It relates two properties of the curve: curvature and arc length.For every curve there is a dependence F(s, k) = 0 where s is the arc length and k is the curvature.

The curve is described implicitly, which might not make for easy calculations. However, the natural equation does show how curvature changes with arc length. In addition, the equation is invariant under rotations and translations [1].

The natural equation arose in the solution to the following space curve problem: Given two one-parameter functions, we want to find a space curve for which the functions are the curvature and torsion (i.e. how the curve twists).[2] Two curves that have the same natural equation also have the same curvature and torsion; they are congruent, only differing by location in the plane [3].

Natural Equations for Curves

The natural equation can also be expressed as F(κ2[α], s) = 0, where s is the arc length function of α. This table shows a few natural equations for curvature (κ2) using this notation [1]:

The natural equation will often be defined in terms of integrals; Euler was the first to represent the curve in this way to represent special curves — although he didn’t use the term “natural” to describe the equation [4].


Where:

• φ is the angle between the x-axis and the tangent line to the curve,
• κ is the curvature.

We can use this integral to solve the equations for curvature (κ = &kappa(s)) and torsion (τ = 0) with the parametric equations [5]

Natural Equation: Alternate Definitions

In quantum physics, the natural equation is “the pair of Hamilton canonical equations of the total energy function e_T” [6], which can be written as

Sometimes the term “natural equation” is used colloquially to mean “the most appropriate equation”. For example

“…we extended this result to a variable coefficient case and also to other situations, for example when we have lower order terms in the equation. The most natural equation from stochastic control is the Hamilton-Jacobi-Bellman equation.”[7]

An article on Princeton University’s website [8] references a natural equation in an article about “beautiful mathematics equations” including Bayes’ theorem and the Dirac equation. However, it seems that this entry, taken in context, may have nothing to do with the equations themselves but may be a synonym for a natural question.

References

[1] Abbena, E. et al. (2017). Modern Differential Geometry of Curves and Surfaces with Mathematica. CRC Press.
[2] Knill, O. & Chi, A. (2003). Entry Curves.
[3] Svetlana, N. et al. (2018). Biometric Inverse Problems. Taylor & Francis.
[4] Struik, D. (1961). Lectures on Classical Differential Geometry. Addison-Wesley.
[5] Weisstein, Eric W. “Natural Equation.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/NaturalEquation.html
[6] Crespin, D. The Quantum Captivity of Physics.
[7] Silvestre, L. (2010). Description of my current research.
[8] Higham, N. (2016). The Most Beautiful Equations in Applied Mathematics.

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The Quadrifolium (also called the Rosace a quatre jeulles or four-leaved clover) is a four-petaled rose curve with n = 2.

The quadrifolium has polar equation [1]

(x² + y²)³ = 4a²x²y²

corresponding algebraic equation

(x² + y²)³ = (x² – y²)²

and Cartesian equation

r = a sin(2θ).

Like all curves defined by polar coordinates as a function of trigonometric functions r = f(θ), there is an equivalent harmonic pattern function α_n = f(2πn/N) · {1} with θ → 2πn/N. For a quadrifolium with equation r = 2 cos(2θ), the harmonic pattern function is [2]

The area inside the curve is ½πa², which corresponds to half the area of the quadrifolium’s circumcircle (or the areas complement within the curve’s circumcircle). The area is derived from integration:

The arc length is s = 8 a E(½√3) = 9.6884…a
Where E(k) is a complete elliptic integral of the second kind [3].

Real Life Applications of the Quadrifolium

The curve is frequently used in modeling applications. For example, one interesting application of the quadrifolium is in simulations of hurricane flowfields. One hurricane data collection method involves flying in a figure-four pattern (i.e., a quadrifolium) through the center of a hurricane [4]. Quadrifoliums have been used in the estimation of flowfields in general, although they made not be as good as other models [5]. In modeling of a large-scale system of agents (like plane formations or mobile robot deployment), formation settles to a quadrifolium pattern [6]. Many plants take the moniker because of four leaves, like some varieties of clover and the purple grass Quadrifolium fuscum.

References

Graph created with Desmos.
[1] Giedre, S. Mathematics around us.
[2] Putnam, L. (2012). The Harmonic Pattern Function:
A Mathematical Model Integrating Synthesis of
Sound and Graphical Patterns. Retrieved March 5,2022 from: https://www.mat.ucsb.edu/Dissertations/The_Harmonic_Pattern_Function_Lance_Putnam.pdf
[3] Weisstein, Eric W. “Quadrifolium.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/Quadrifolium.html
[4] DeVries, L. & Paley, D. (2011). Multi-vehicle Control in a Strong Flowfield with Application to Hurricane Sampling.
[5] DeVries, L. (2014). OBSERVABILITY-BASED SAMPLING AND ESTIMATION OF FLOWFIELDS USING MULTI-SENSOR SYSTEMS.
[6] Qui, J. et al. (2015). Multi-Agent Deployment in 3-D via PDE Control. IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 60, NO. 4, APRIL.
Retrieved March 5, 2022 from: http://flyingv.ucsd.edu/papers/PDF/208.pdf

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