Invariant (mathematics)

<h2 id="examples">Examples</h2>
A simple example of invariance is expressed in our ability to <a href="/facts/Counting/DsRhH9RT">count</a>. For a <a href="/facts/Finite_set/za1newlP">finite set</a> of objects of any kind, there is a number to which we always arrive, regardless of the <a href="/facts/Total_order/xnTtmuYJ">order</a> in which we count the objects in the <a href="/facts/Set_(mathematics)/BfucfHMq">set</a>. The quantity—a <a href="/facts/Cardinal_number/ccoKwU4R">cardinal number</a>—is associated with the set, and is invariant under the process of counting.
An <a href="/facts/List_of_mathematical_identities/TJsqTRa6">identity</a> is an equation that remains true for all values of its variables. There are also <a href="/facts/List_of_inequalities/NcG9HjhB">inequalities</a> that remain true when the values of their variables change.
The <a href="/facts/Distance/hbXlaEFk">distance</a> between two points on a <a href="/facts/Number_line/gFVyKQHr">number line</a> is not changed by <a href="/facts/Addition/apFWJBFB">adding</a> the same quantity to both numbers. On the other hand, <a href="/facts/Multiplication/VtcUjpNv">multiplication</a> does not have this same property, as distance is not invariant under multiplication.
<a href="/facts/Angle/gFUTEQYq">Angles</a> and <a href="/facts/Ratio/mFmhQt7T">ratios</a> of distances are invariant under <a href="/facts/Scaling_(geometry)/pFm6fvpu">scalings</a>, <a href="/facts/Rotation_(mathematics)/6gtkkcy0">rotations</a>, <a href="/facts/Translation_(geometry)/KlpWmnfX">translations</a> and <a href="/facts/Reflection_(mathematics)/5JHWnrql">reflections</a>. These transformations produce <a href="/facts/Similarity_(geometry)/Sbd6hEAN">similar</a> shapes, which is the basis of <a href="/facts/Trigonometry/K4hfaC6F">trigonometry</a>. In contrast, angles and ratios are not invariant under non-uniform scaling (such as stretching). The sum of a triangle's interior angles (180°) is invariant under all the above operations. As another example, all <a href="/facts/Circle/9fy5ggKn">circles</a> are similar: they can be transformed into each other and the ratio of the <a href="/facts/Circumference/r1rntU0f">circumference</a> to the <a href="/facts/Diameter/6uNn4VmU">diameter</a> is invariant (denoted by the Greek letter π (<a href="/facts/Pi/vC0UBj1q">pi</a>)).
Some more complicated examples:

<ul><li>The <a href="/facts/Real_part/2w7mqI7e">real part</a> and the <a href="/facts/Absolute_value/dwJBpEs5">absolute value</a> of a <a href="/facts/Complex_number/2w7mqI7e">complex number</a> are invariant under <a href="/facts/Complex_conjugation/7TEaoQDe">complex conjugation</a>.</li>
<li>The <a href="/facts/Tricolorability/QfTV00ln">tricolorability</a> of <a href="/facts/Knot_(mathematics)/UMtKJ36m">knots</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a></li>
<li>The <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree of a polynomial</a> is invariant under a linear change of variables.</li>
<li>The <a href="/facts/Topological_dimension/xdCJLVdw">dimension</a> and <a href="/facts/Homology_group/VFJmryEW">homology groups</a> of a topological object are invariant under <a href="/facts/Homeomorphism/jMfWg3Mn">homeomorphism</a>.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a></li>
<li>The number of <a href="/facts/Fixed_point_(mathematics)/vVVddTKc">fixed points</a> of a <a href="/facts/Dynamical_system/782gREl5">dynamical system</a> is invariant under many mathematical operations.</li>
<li>Euclidean distance is invariant under <a href="/facts/Orthogonal_transformation/s7LGBDFX">orthogonal transformations</a>.</li>
<li><a href="/facts/Area/KsNG1G9t">Area</a> is invariant under <a href="/facts/Linear_map/Q1PBKNVV">linear maps</a> which have <a href="/facts/Determinant/eGYqJ2TF">determinant</a> ±1 (see <a href="/facts/Equiareal_map/5MFDDkUr">Equiareal map § Linear transformations</a>).</li>
<li>Some invariants of <a href="/facts/Projective_transformation/wCkNzUjd">projective transformations</a> include <a href="/facts/Collinearity/SxbVBJYR">collinearity</a> of three or more points, <a href="/facts/Concurrent_lines/zPT6S4w9">concurrency</a> of three or more lines, <a href="/facts/Conic_section/teqgLptX">conic sections</a>, and the <a href="/facts/Cross-ratio/tj4DK3Vk">cross-ratio</a>.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a></li>
<li>The <a href="/facts/Determinant/eGYqJ2TF">determinant</a>, <a href="/facts/Trace_(linear_algebra)/dchFFvHt">trace</a>, <a href="/facts/Eigenvectors/8TjEoT8u">eigenvectors</a>, and <a href="/facts/Eigenvalues/8TjEoT8u">eigenvalues</a> of a <a href="/facts/Linear_endomorphism/Q1PBKNVV">linear endomorphism</a> are invariant under a <a href="/facts/Change_of_basis/Hnu1Spcc">change of basis</a>. In other words, the <a href="/facts/Spectrum_of_a_matrix/BIBnA4qf">spectrum of a matrix</a> is invariant under a change of basis.</li>
<li>The principal invariants of <a href="/facts/Tensors/pNjFo5tV">tensors</a> do not change with rotation of the coordinate system (see <a href="/facts/Invariants_of_tensors/fwpws1vj">Invariants of tensors</a>).</li>
<li>The <a href="/facts/Singular-value_decomposition/8t3v6wk3">singular values</a> of a <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> are invariant under orthogonal transformations.</li>
<li><a href="/facts/Lebesgue_measure/8us9cGoW">Lebesgue measure</a> is invariant under translations.</li>
<li>The <a href="/facts/Variance/ULBJKXD1">variance</a> of a <a href="/facts/Probability_distribution/EpsKKVRu">probability distribution</a> is invariant under translations of the <a href="/facts/Real_line/6eq42xX5">real line</a>. Hence the variance of a <a href="/facts/Random_variable/TwTBXnLT">random variable</a> is unchanged after the addition of a constant.</li>
<li>The <a href="/facts/Fixed_point_(mathematics)/vVVddTKc">fixed points</a> of a transformation are the elements in the <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> that are invariant under the transformation. They may, depending on the application, be called <a href="/facts/Symmetry/hpu7DK8p">symmetric</a> with respect to that transformation. For example, objects with <a href="/facts/Translational_symmetry/PYVMaqWb">translational symmetry</a> are invariant under certain translations.</li>
<li>The integral 
 
 
 
 
 ∫
 
 M
 
 
 K
 
 d
 μ
 
 
 {\textstyle \int _{M}K\,d\mu }
 
 of the Gaussian curvature 
 
 
 
 K
 
 
 {\displaystyle K}
 
 of a two-dimensional <a href="/facts/Riemannian_manifold/5KYnmEgF">Riemannian manifold</a> 
 
 
 
 (
 M
 ,
 g
 )
 
 
 {\displaystyle (M,g)}
 
 is invariant under changes of the <a href="/facts/Riemannian_metric/5KYnmEgF">Riemannian metric</a> 
 
 
 
 g
 
 
 {\displaystyle g}
 
. This is the <a href="/facts/Gauss%25E2%2580%2593Bonnet_theorem/pvTbtH91">Gauss–Bonnet theorem</a>.</li></ul>
<h3>MU puzzle</h3>
The <a href="/facts/MU_puzzle/kRLV40CS">MU puzzle</a><a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> is a good example of a logical problem where determining an invariant is of use for an <a href="/facts/Impossibility_proof/rAQEghH8">impossibility proof</a>. The puzzle asks one to start with the word MI and transform it into the word MU, using in each step one of the following transformation rules:

<ol><li>If a string ends with an I, a U may be appended (xI → xIU)</li>
<li>The string after the M may be completely duplicated (Mx → Mxx)</li>
<li>Any three consecutive I's (III) may be replaced with a single U (xIIIy → xUy)</li>
<li>Any two consecutive U's may be removed (xUUy → xy)</li></ol>
An example derivation (with superscripts indicating the applied rules) is

MI →2 MII →2 MIIII →3 MUI →2 MUIUI →1 MUIUIU →2 MUIUIUUIUIU →4 MUIUIIUIU → ...
In light of this, one might wonder whether it is possible to convert MI into MU, using only these four transformation rules. One could spend many hours applying these transformation rules to strings. However, it might be quicker to find a <a href="/facts/Predicate_(mathematical_logic)/5WazAvKF">property</a> that is invariant to all rules (that is, not changed by any of them), and that demonstrates that getting to MU is impossible. By looking at the puzzle from a logical standpoint, one might realize that the only way to get rid of any I's is to have three consecutive I's in the string. This makes the following invariant interesting to consider:

The number of I's in the string is not a multiple of 3.
This is an invariant to the problem, if for each of the transformation rules the following holds: if the invariant held before applying the rule, it will also hold after applying it. Looking at the net effect of applying the rules on the number of I's and U's, one can see this actually is the case for all rules:

<table><tbody><tr><th>Rule</th><th>#I's</th><th>#U's</th><th>Effect on invariant</th></tr><tr><td>1</td><td>+0</td><td>+1</td><td>Number of I's is unchanged. If the invariant held, it still does.</td></tr><tr><td>2</td><td>×2</td><td>×2</td><td>If n is not a multiple of 3, then 2×n is not either. The invariant still holds.</td></tr><tr><td>3</td><td>−3</td><td>+1</td><td>If n is not a multiple of 3, n−3 is not either. The invariant still holds.</td></tr><tr><td>4</td><td>+0</td><td>−2</td><td>Number of I's is unchanged. If the invariant held, it still does.</td></tr></tbody></table>
The table above shows clearly that the invariant holds for each of the possible transformation rules, which means that whichever rule one picks, at whatever state, if the number of I's was not a multiple of three before applying the rule, then it will not be afterwards either.
Given that there is a single I in the starting string MI, and one is not a multiple of three, one can then conclude that it is impossible to go from MI to MU (as the number of I's will never be a multiple of three).

<h2 id="invariant-set">Invariant set</h2>
A <a href="/facts/Subset/SJGERmbA">subset</a> S of the domain U of a mapping T: U → U is an invariant set under the mapping when 
 
 
 
 x
 ∈
 S
 
 ⟺
 
 T
 (
 x
 )
 ∈
 S
 .
 
 
 {\displaystyle x\in S\iff T(x)\in S.}
 
 The <a href="/facts/Element_(mathematics)/Q2mx1vHL">elements</a> of S are not necessarily <a href="/facts/Fixed_point_(mathematics)/vVVddTKc">fixed</a>, even though the set S is fixed in the <a href="/facts/Power_set/FVuf8PDD">power set</a> of U. (Some authors use the terminology setwise invariant,<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> vs. pointwise invariant,<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> to distinguish between these cases.)
For example, a circle is an invariant subset of the plane under a <a href="/facts/Rotation/yChkz49x">rotation</a> about the circle's center. Further, a <a href="/facts/Conical_surface/n6h4xmLu">conical surface</a> is invariant as a set under a <a href="/facts/Homothetic_transformation/WUmuRuzC">homothety</a> of space.
An invariant set of an operation T is also said to be stable under T. For example, the <a href="/facts/Normal_subgroup/aRDaGwLm">normal subgroups</a> that are so important in <a href="/facts/Group_theory/NfowcI5H">group theory</a> are those <a href="/facts/Subgroup/r91mBgpa">subgroups</a> that are stable under the <a href="/facts/Inner_automorphism/awJPzhDa">inner automorphisms</a> of the ambient <a href="/facts/Group_(mathematics)/IQTYqINt">group</a>.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a><a class="footnote-ref" id="fnref:13" href="#fn:13">13</a><a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>
In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, if a <a href="/facts/Linear_transformation/Q1PBKNVV">linear transformation</a> T has an <a href="/facts/Eigenvector/8TjEoT8u">eigenvector</a> v, then the line through 0 and v is an invariant set under T, in which case the eigenvectors span an <a href="/facts/Invariant_subspace/51Vm1W2X">invariant subspace</a> which is stable under T.
When T is a <a href="/facts/Screw_displacement/Cos1gnJb">screw displacement</a>, the <a href="/facts/Screw_axis/Cos1gnJb">screw axis</a> is an invariant line, though if the <a href="/facts/Pitch_(screw)/dqGzDfHv">pitch</a> is non-zero, T has no fixed points.
In <a href="/facts/Probability_theory/mFBn51rE">probability theory</a> and <a href="/facts/Ergodic_theory/rkqCbIy0">ergodic theory</a>, invariant sets are usually defined via the stronger property 
 
 
 
 x
 ∈
 S
 ⇔
 T
 (
 x
 )
 ∈
 S
 .
 
 
 {\displaystyle x\in S\Leftrightarrow T(x)\in S.}
 
<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a><a class="footnote-ref" id="fnref:16" href="#fn:16">16</a><a class="footnote-ref" id="fnref:17" href="#fn:17">17</a> When the map 
 
 
 
 T
 
 
 {\displaystyle T}
 
 is measurable, invariant sets form a <a href="/facts/Sigma-algebra/8LJINLNc">sigma-algebra</a>, the <a href="/facts/Invariant_sigma-algebra/Tzdt4gY7">invariant sigma-algebra</a>.

<h2 id="formal-statement">Formal statement</h2>

The notion of invariance is formalized in three different ways in mathematics: via <a href="/facts/Group_action/Vh9VtUUw">group actions</a>, presentations, and deformation.

<h3>Unchanged under group action</h3>
Firstly, if one has a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> G <a href="/facts/Group_action/Vh9VtUUw">acting</a> on a mathematical object (or set of objects) X, then one may ask which points x are unchanged, "invariant" under the group action, or under an element g of the group.
Frequently one will have a group acting on a set X, which leaves one to determine which objects in an associated set F(X) are invariant. For example, rotation in the plane about a point leaves the point about which it rotates invariant, while translation in the plane does not leave any points invariant, but does leave all lines parallel to the direction of translation invariant as lines. Formally, define the set of lines in the plane P as L(P); then a <a href="/facts/Rigid_motion/0KzWYQXl">rigid motion</a> of the plane takes lines to lines – the group of rigid motions acts on the set of lines – and one may ask which lines are unchanged by an action.
More importantly, one may define a function on a set, such as "radius of a circle in the plane", and then ask if this function is invariant under a group action, such as rigid motions.
Dual to the notion of invariants are <a href="/facts/Coinvariant/Vh9VtUUw">coinvariants</a>, also known as orbits, which formalizes the notion of <a href="/facts/Congruence_relation/TV3njpqF">congruence</a>: objects which can be taken to each other by a group action. For example, under the group of rigid motions of the plane, the <a href="/facts/Perimeter/yQv1AajA">perimeter</a> of a triangle is an invariant, while the set of triangles congruent to a given triangle is a coinvariant.
These are connected as follows: invariants are constant on coinvariants (for example, congruent triangles have the same perimeter), while two objects which agree in the value of one invariant may or may not be congruent (for example, two triangles with the same perimeter need not be congruent). In <a href="/facts/Classification_problem_(mathematics)/NWGgtNm6">classification problems</a>, one might seek to find a <a href="/facts/Complete_set_of_invariants/Ax3nQbRT">complete set of invariants</a>, such that if two objects have the same values for this set of invariants, then they are congruent.
For example, triangles such that all three sides are equal are congruent under rigid motions, via <a href="/facts/Congruence_(geometry)/2tKJ6AhB">SSS congruence</a>, and thus the lengths of all three sides form a complete set of invariants for triangles. The three angle measures of a triangle are also invariant under rigid motions, but do not form a complete set as incongruent triangles can share the same angle measures. However, if one allows scaling in addition to rigid motions, then the <a href="/facts/Similarity_(geometry)/Sbd6hEAN">AAA similarity criterion</a> shows that this is a complete set of invariants.

<h3>Independent of presentation</h3>
Secondly, a function may be defined in terms of some presentation or decomposition of a mathematical object; for instance, the <a href="/facts/Euler_characteristic/JCjzWUr2">Euler characteristic</a> of a <a href="/facts/Cell_complex/rN7buLMo">cell complex</a> is defined as the alternating sum of the number of cells in each dimension. One may forget the cell complex structure and look only at the underlying <a href="/facts/Topological_space/v2ut7CbK">topological space</a> (the <a href="/facts/Manifold/rXPERJtu">manifold</a>) – as different cell complexes give the same underlying manifold, one may ask if the function is independent of choice of presentation, in which case it is an intrinsically defined invariant. This is the case for the Euler characteristic, and a general method for defining and computing invariants is to define them for a given presentation, and then show that they are independent of the choice of presentation. Note that there is no notion of a group action in this sense.
The most common examples are:

<ul><li>The <a href="/facts/Differentiable_manifold/aSnbXKcV">presentation of a manifold</a> in terms of coordinate charts – invariants must be unchanged under <a href="/facts/Change_of_coordinates/Mtd0q4SP">change of coordinates</a>.</li>
<li>Various <a href="/facts/Manifold_decomposition/SVz7dIXz">manifold decompositions</a>, as discussed for Euler characteristic.</li>
<li>Invariants of a <a href="/facts/Presentation_of_a_group/f9emPUoI">presentation of a group</a>.</li></ul>
<h3>Unchanged under perturbation</h3>
Thirdly, if one is studying an object which varies in a family, as is common in <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a> and <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a>, one may ask if the property is unchanged under perturbation (for example, if an object is constant on families or invariant under change of metric).

<h2 id="invariants-in-computer-science">Invariants in computer science</h2>
For other uses of the word "invariant" in computer science, see <a href="/facts/Invariant_(disambiguation)/kaCoe2MA">invariant (disambiguation)</a>.
In <a href="/facts/Computer_science/lN3pEyPz">computer science</a>, an invariant is a <a href="/facts/Logical_assertion/CwJwURfm">logical assertion</a> that is always held to be true during a certain phase of execution of a <a href="/facts/Computer_program/e38jH9N9">computer program</a>. For example, a <a href="/facts/Loop_invariant/fIbNgcsI">loop invariant</a> is a condition that is true at the beginning and the end of every iteration of a loop.
Invariants are especially useful when reasoning about the <a href="/facts/Correctness_(computer_science)/m4yxw3gw">correctness of a computer program</a>. The theory of <a href="/facts/Optimizing_compiler/JXnv4LUd">optimizing compilers</a>, the methodology of <a href="/facts/Design_by_contract/PjacMRRd">design by contract</a>, and <a href="/facts/Formal_methods/tw4FCWIb">formal methods</a> for determining <a href="/facts/Program_correctness/m4yxw3gw">program correctness</a>, all rely heavily on invariants.
Programmers often use <a href="/facts/Assertion_(computing)/lxSfLdLY">assertions</a> in their code to make invariants explicit. Some <a href="/facts/Object_oriented/TxNMEcao">object oriented</a> <a href="/facts/Programming_language/cWTYbgWa">programming languages</a> have a special syntax for specifying <a href="/facts/Class_invariant/PqDFHgQy">class invariants</a>.

<h3>Automatic invariant detection in imperative programs</h3>
<a href="/facts/Abstract_interpretation/cj6wjzu1">Abstract interpretation</a> tools can compute simple invariants of given imperative computer programs. The kind of properties that can be found depend on the <a href="/facts/Abstract_interpretation/cj6wjzu1">abstract domains</a> used. Typical example properties are single integer variable ranges like 0<=x<1024, relations between several variables like 0<=i-j<2*n-1, and modulus information like y%4==0. Academic research prototypes also consider simple properties of pointer structures.<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a>
More sophisticated invariants generally have to be provided manually.
In particular, when verifying an imperative program using <a href="/facts/Hoare_logic/UoMg08wo">the Hoare calculus</a>,<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a> a loop invariant has to be provided manually for each loop in the program, which is one of the reasons that this approach is generally impractical for most programs.
In the context of the above <a href="/facts/MU_puzzle/kRLV40CS">MU puzzle</a> example, there is currently no general automated tool that can detect that a derivation from MI to MU is impossible using only the rules 1–4. However, once the abstraction from the string to the number of its "I"s has been made by hand, leading, for example, to the following C program, an abstract interpretation tool will be able to detect that ICount%3 cannot be 0, and hence the "while"-loop will never terminate.

void MUPuzzle(void) {
 volatile int RandomRule;
 int ICount = 1, UCount = 0;
 while (ICount % 3 != 0) // non-terminating loop
 switch(RandomRule) {
 case 1: UCount += 1; break;
 case 2: ICount *= 2; UCount *= 2; break;
 case 3: ICount -= 3; UCount += 1; break;
 case 4: UCount -= 2; break;
 } // computed invariant: ICount % 3 == 1 || ICount % 3 == 2
}

<ul><li><a href="/facts/Erlangen_program/HAVoCLHG">Erlangen program</a></li>
<li><a href="/facts/Graph_invariant/bjuIvgb1">Graph invariant</a></li>
<li><a href="/facts/Invariant_differential_operator/yoFWKPV6">Invariant differential operator</a></li>
<li><a href="/facts/Invariant_estimator/LQRCCnyM">Invariant estimator</a> in statistics</li>
<li><a href="/facts/Invariant_measure/X75B3PcM">Invariant measure</a></li>
<li><a href="/facts/Invariant_(physics)/UfHcehYs">Invariant (physics)</a></li>
<li><a href="/facts/Invariants_of_tensors/fwpws1vj">Invariants of tensors</a></li>
<li><a href="/facts/Invariant_theory/5fIUh9h3">Invariant theory</a></li>
<li><a href="/facts/Knot_invariant/BEvEEpHQ">Knot invariant</a></li>
<li><a href="/facts/Mathematical_constant/rzqTX307">Mathematical constant</a></li>
<li><a href="/facts/Mathematical_constants_and_functions/0q476laU">Mathematical constants and functions</a></li>
<li><a href="/facts/Scale_invariance/S96sxg2R">Scale invariance</a></li>
<li><a href="/facts/Symmetry_in_mathematics/KtLgoqyX">Symmetry in mathematics</a></li>
<li><a href="/facts/Topological_invariant/vbJR18Sx">Topological invariant</a></li>
<li><a href="/facts/Young%25E2%2580%2593Deruyts_development/3uUl8wc4">Young–Deruyts development</a></li></ul>

<h2 id="notes">Notes</h2>

<ul><li>Fraleigh, John B. (1976), A First Course In Abstract Algebra (2nd ed.), Reading: <a href="/facts/Addison-Wesley/A7UFATEG">Addison-Wesley</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-201-01984-1</li>
<li>Herstein, I. N. (1964), Topics In Algebra, Waltham: Blaisdell Publishing Company, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1114541016 {{citation}}: ISBN / Date incompatibility (help)</li>
<li>Kay, David C. (1969), College Geometry, New York: <a href="/facts/Holt%2c_Rinehart_and_Winston/Kkw5s1rc">Holt, Rinehart and Winston</a>, <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/69-12075">69-12075</a></li>
<li>McCoy, Neal H. (1968), Introduction To Modern Algebra, Revised Edition, Boston: <a href="/facts/Allyn_and_Bacon/RI2leS8r">Allyn and Bacon</a>, <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/68-15225">68-15225</a></li>
<li>J.D. Fokker, <a href="/facts/Hans_Zantema/Q2T3snqw">H. Zantema</a>, S.D. Swierstra (1991). "Iteratie en invariatie", Programmeren en Correctheid. Academic Service. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 90-6233-681-7.</li>
<li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/Invariant.html">"Invariant"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li>
<li><a href="/facts/Vladimir_L._Popov/U9VGpfRW">Popov, V.L.</a> (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Invariant">"Invariant"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li>
<li>Billingsley, Patrick (1995). <a href="https://www.wiley.com/en-gb/Probability+and+Measure%2C+Anniversary+Edition-p-9781118122372">Probability and Measure</a>. John Wiley & Sons. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-00710-2.</li>
<li>Douc, Randal; Moulines, Eric; Priouret, Pierre; Soulier, Philippe (2018). <a href="https://link.springer.com/book/10.1007/978-3-319-97704-1">Markov Chains</a>. Springer. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-319-97704-1">10.1007/978-3-319-97704-1</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-319-97703-4.</li>
<li>Klenke, Achim (2020). <a href="https://link.springer.com/book/10.1007/978-1-4471-5361-0">Probability Theory: A comprehensive course</a>. Universitext. Springer. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-1-4471-5361-0">10.1007/978-1-4471-5361-0</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-030-56401-8.</li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="http://www.u.arizona.edu/~wbraynen/software/VisualInvariants/index.html">"Applet: Visual Invariants in Sorting Algorithms"</a> <a href="https://web.archive.org/web/20220224135515/http://www.u.arizona.edu/~wbraynen/software/VisualInvariants/index.html">Archived</a> 2022-02-24 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a> by William Braynen in 1997</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">"Invariant Definition (Illustrated Mathematics Dictionary)". www.mathsisfun.com. Retrieved 2019-12-05. <a href="https://www.mathsisfun.com/definitions/invariant.html" target="_blank">https://www.mathsisfun.com/definitions/invariant.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Weisstein, Eric W. "Invariant". mathworld.wolfram.com. Retrieved 2019-12-05. <a href="http://mathworld.wolfram.com/Invariant.html" target="_blank">http://mathworld.wolfram.com/Invariant.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">"Invariant – Encyclopedia of Mathematics". www.encyclopediaofmath.org. Retrieved 2019-12-05. <a href="https://www.encyclopediaofmath.org/index.php/Invariant" target="_blank">https://www.encyclopediaofmath.org/index.php/Invariant</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Weisstein, Eric W. "Invariant". mathworld.wolfram.com. Retrieved 2019-12-05. <a href="http://mathworld.wolfram.com/Invariant.html" target="_blank">http://mathworld.wolfram.com/Invariant.html</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">"Invariant – Encyclopedia of Mathematics". www.encyclopediaofmath.org. Retrieved 2019-12-05. <a href="https://www.encyclopediaofmath.org/index.php/Invariant" target="_blank">https://www.encyclopediaofmath.org/index.php/Invariant</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Qiao, Xiaoyu (January 20, 2015). "Tricolorability.pdf" (PDF). Knot Theory Week 2: Tricolorability. Archived from the original (PDF) on May 25, 2024. Retrieved May 25, 2024. <a href="https://web.archive.org/web/20240525145629/https://web.math.ucsb.edu/~padraic/ucsb_2014_15/ccs_problem_solving_w2015/Tricolorability.pdf" target="_blank">https://web.archive.org/web/20240525145629/https://web.math.ucsb.edu/~padraic/ucsb_2014_15/ccs_problem_solving_w2015/Tricolorability.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Fraleigh (1976, pp. 166–167) - Fraleigh, John B. (1976), A First Course In Abstract Algebra (2nd ed.), Reading: Addison-Wesley, ISBN 0-201-01984-1 <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Kay (1969, pp. 219) - Kay, David C. (1969), College Geometry, New York: Holt, Rinehart and Winston, LCCN 69-12075 <a href="https://lccn.loc.gov/69-12075" target="_blank">https://lccn.loc.gov/69-12075</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Hofstadter, Douglas R. (1999) [1979], Gödel, Escher, Bach: An Eternal Golden Braid, Basic Books, ISBN 0-465-02656-7
Here: Chapter I. <a href="0-465-02656-7" target="_blank">0-465-02656-7</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Barry Simon. Representations of Finite and Compact Groups. American Mathematical Soc. p. 16. ISBN 978-0-8218-7196-6. <a href="978-0-8218-7196-6" target="_blank">978-0-8218-7196-6</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Judith Cederberg (1989). A Course in Modern Geometries. Springer. p. 174. ISBN 978-1-4757-3831-5. <a href="978-1-4757-3831-5" target="_blank">978-1-4757-3831-5</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Fraleigh (1976, p. 103) - Fraleigh, John B. (1976), A First Course In Abstract Algebra (2nd ed.), Reading: Addison-Wesley, ISBN 0-201-01984-1 <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Herstein (1964, p. 42) - Herstein, I. N. (1964), Topics In Algebra, Waltham: Blaisdell Publishing Company, ISBN 978-1114541016 <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">McCoy (1968, p. 183) - McCoy, Neal H. (1968), Introduction To Modern Algebra, Revised Edition, Boston: Allyn and Bacon, LCCN 68-15225 <a href="https://lccn.loc.gov/68-15225" target="_blank">https://lccn.loc.gov/68-15225</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Billingsley (1995), pp. 313–314 - Billingsley, Patrick (1995). Probability and Measure. John Wiley & Sons. ISBN 0-471-00710-2. <a href="https://www.wiley.com/en-gb/Probability+and+Measure%2C+Anniversary+Edition-p-9781118122372" target="_blank">https://www.wiley.com/en-gb/Probability+and+Measure%2C+Anniversary+Edition-p-9781118122372</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Douc et al. (2018), p. 99 - Douc, Randal; Moulines, Eric; Priouret, Pierre; Soulier, Philippe (2018). Markov Chains. Springer. doi:10.1007/978-3-319-97704-1. ISBN 978-3-319-97703-4. <a href="https://link.springer.com/book/10.1007/978-3-319-97704-1" target="_blank">https://link.springer.com/book/10.1007/978-3-319-97704-1</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Klenke (2020), p. 494-495 - Klenke, Achim (2020). Probability Theory: A comprehensive course. Universitext. Springer. doi:10.1007/978-1-4471-5361-0. ISBN 978-3-030-56401-8. <a href="https://link.springer.com/book/10.1007/978-1-4471-5361-0" target="_blank">https://link.springer.com/book/10.1007/978-1-4471-5361-0</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Bouajjani, A.; Drǎgoi, C.; Enea, C.; Rezine, A.; Sighireanu, M. (2010). "Invariant Synthesis for Programs Manipulating Lists with Unbounded Data" (PDF). Proc. CAV. doi:10.1007/978-3-642-14295-6_8. <a href="/wiki/Mihaela_Sighireanu" target="_blank">/wiki/Mihaela_Sighireanu</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Hoare, C. A. R. (October 1969). "An axiomatic basis for computer programming" (PDF). Communications of the ACM. 12 (10): 576–580. doi:10.1145/363235.363259. S2CID 207726175. Archived from the original (PDF) on 2016-03-04. <a href="/wiki/C.A.R._Hoare" target="_blank">/wiki/C.A.R._Hoare</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
</ol>

Invariant (mathematics) open-in-new

Invariant (mathematics)