what is canonical representation

• Stack Overflow Public questions & answers
• Stack Overflow for Teams Where developers & technologists share private knowledge with coworkers
• Talent Build your employer brand
• Advertising Reach developers & technologists worldwide
• Labs The future of collective knowledge sharing
• About the company

Collectives™ on Stack Overflow

Find centralized, trusted content and collaborate around the technologies you use most.

Q&A for work

Connect and share knowledge within a single location that is structured and easy to search.

Get early access and see previews of new features.

What does the term "canonical form" or "canonical representation" in Java mean?

I have often heard this term being used, but I have never really understood it.

What does it mean, and can anyone give some examples/point me to some links?

EDIT: Thanks to everyone for the replies. Can you also tell me how the canonical representation is useful in equals() performance, as stated in Effective Java?

12 Answers 12

I believe there are two related uses of canonical: forms and instances.

A canonical form means that values of a particular type of resource can be described or represented in multiple ways, and one of those ways is chosen as the favored canonical form. (That form is canonized , like books that made it into the bible, and the other forms are not.) A classic example of a canonical form is paths in a hierarchical file system, where a single file can be referenced in a number of ways:

The classic definition of the canonical representation of that file would be the last path. With local or relative paths you cannot globally identify the resource without contextual information. With absolute paths you can identify the resource, but cannot tell if two paths refer to the same entity. With two or more paths converted to their canonical forms, you can do all the above, plus determine if two resources are the same or not, if that is important to your application (solve the aliasing problem ).

Note that the canonical form of a resource is not a quality of that particular form itself; there can be multiple possible canonical forms for a given type like file paths (say, lexicographically first of all possible absolute paths). One form is just selected as the canonical form for a particular application reason, or maybe arbitrarily so that everyone speaks the same language.

Forcing objects into their canonical instances is the same basic idea, but instead of determining one "best" representation of a resource, it arbitrarily chooses one instance of a class of instances with the same "content" as the canonical reference, then converts all references to equivalent objects to use the one canonical instance.

This can be used as a technique for optimizing both time and space. If there are multiple instances of equivalent objects in an application, then by forcing them all to be resolved as the single canonical instance of a particular value, you can eliminate all but one of each value, saving space and possibly time since you can now compare those values with reference identity (==) as opposed to object equivalence ( equals() method).

A classic example of optimizing performance with canonical instances is collapsing strings with the same content. Calling String.intern() on two strings with the same character sequence is guaranteed to return the same canonical String object for that text. If you pass all your strings through that canonicalizer, you know equivalent strings are actually identical object references, i.e., aliases

The enum types in Java 5.0+ force all instances of a particular enum value to use the same canonical instance within a VM, even if the value is serialized and deserialized. That is why you can use if (day == Days.SUNDAY) with impunity in java if Days is an enum type. Doing this for your own classes is certainly possible, but takes care. Read Effective Java by Josh Bloch for details and advice.

Wikipedia points to the term Canonicalization .

A process for converting data that has more than one possible representation into a "standard" canonical representation. This can be done to compare different representations for equivalence, to count the number of distinct data structures, to improve the efficiency of various algorithms by eliminating repeated calculations, or to make it possible to impose a meaningful sorting order.

The Unicode example made the most sense to me:

Variable-length encodings in the Unicode standard, in particular UTF-8, have more than one possible encoding for most common characters. This makes string validation more complicated, since every possible encoding of each string character must be considered. A software implementation which does not consider all character encodings runs the risk of accepting strings considered invalid in the application design, which could cause bugs or allow attacks. The solution is to allow a single encoding for each character. Canonicalization is then the process of translating every string character to its single allowed encoding. An alternative is for software to determine whether a string is canonicalized, and then reject it if it is not. In this case, in a client/server context, the canonicalization would be the responsibility of the client.

In summary, a standard form of representation for data. From this form you can then convert to any representation you may need.

A good example for understanding "canonical form/representation" is to look at the XML schema datatype definition of "boolean":

• the "lexical representation" of boolean can be one of: {true, false, 1, 0} whereas
• the "canonical representation" can only be one of {true, false}

This, in essence, means that

• "true" and "1" get mapped to the canonical repr. "true" and
• "false" and "0" get mapped to the canoncial repr. "false"

see the w3 XML schema datatype definition for boolean

The word "canonical" is just a synonym for "standard" or "usual". It doesn`t have any Java-specific meaning.

• 3 canonical has a richer meaning than standard or usual IMO. –  squid Nov 12, 2015 at 15:20

reduced to the simplest and most significant form without losing generality

An easy way to remember it is the way "canonical" is used in theological circles, canonical truth is the real truth so if two people find it they have found the same truth. Same with canonical instance. If you think you have found two of them (i.e. a.equals(b) ) you really only have one (i.e. a == b ). So equality implies identity in the case of canonical object.

Now for the comparison. You now have the choice of using a==b or a.equals(b) , since they will produce the same answer in the case of canonical instance but a==b is comparison of the reference (the JVM can compare two numbers extremely rapidly as they are just two 32 bit patterns compared to a.equals(b) which is a method call and involves more overhead.

Another good example might be: you have a class that supports the use of cartesian (x, y, z), spherical (r, theta, phi) and cylindrical coordinates (r, phi, z). For purposes of establishing equality (equals method), you would probably want to convert all representations to one "canonical" representation of your choosing, e.g. spherical coordinates. (Or maybe you would want to do this in general - i.e. use one internal representation.) I am not an expert, but this did occur to me as maybe a good concrete example.

A canonical form means a naturally unique representation of the element

canonical representation means view the character in different style for example if I write a letter A means another person may write the letter A in different style:)

This is according to OPTICAL CHARACTER RECOGNITION FIELD

The OP's questions about canonical form and how it can improve performance of the equals method can both be answered by extending the example provided in Effective Java.

Consider the following class:

The equals method in this example has added cost by using String 's equalsIgnoreCase method. As mentioned in the text

you may want to store a canonical form of the field so the equals method can do a cheap exact comparison on canonical forms rather than a more costly nonstandard comparison.

What does Joshua Bloch mean when he says canonical form ? Well, I think Dónal's concise answer is very appropriate. We can store the underlying String field in the CaseInsensitiveString example in a standard way, perhaps the uppercase form of the String . Now, you can reference this canonical form of the CaseInsensitiveString , its uppercase variant, and perform cheap evaluations in your equals and hashcode methods.

Canonical Data in RDBMS, Graph Data; Think as "Normalization" or "Normal form" of a data in a RDBMS. Same data exists in different tables, represented with a unique identifier and mapped it in different tables. or Think a single form of a data in Graph Database that represented in many triples.

Major benefit of it is to make Dml (Data manipulation) more efficient since you can upsert (insert/update) only one value instead of many.

In TypeScript, when defining reusable types for functions, it's common to have two related types: a canonical form for return types and a looser form for parameters. Here's what these terms typically mean:

Canonical Form (Return Type): The canonical form, in the context of return types, represents the strictest and most specific type that describes what a function returns. It's often used to define the precise shape of the data that a function is expected to return. This form aims to capture the exact structure and type of the returned value as accurately as possible.

Looser Form (Parameters): The looser form, typically applied to parameters, is a type that is more permissive or general. It allows for various input values, which may include a wider range of possibilities. This form is less specific than the canonical form and is designed to accept different input values and types.

This approach provides flexibility in your code. By having a canonical form for return types, you can ensure that functions produce specific types of results, which is especially useful for type checking and understanding what functions return. On the other hand, using a looser form for parameters allows functions to accept a broader range of input values, making them more versatile.

Here's a simple example:

In this example, the multiply function has a canonical form that specifies it returns a number. The add function has a looser form for parameters, allowing it to accept both numbers and strings as inputs and returning a type that can be either a number or a string.

Using these canonical and looser forms in your type definitions can help you strike a balance between type safety and flexibility in your code.

• the question is (only) marked with the java tag... (and 15 years old [in 9 days]) –  user85421 Nov 2, 2023 at 14:47

Your Answer

Reminder: Answers generated by artificial intelligence tools are not allowed on Stack Overflow. Learn more

Sign up or log in

Post as a guest.

Required, but never shown

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy .

Not the answer you're looking for? Browse other questions tagged java or ask your own question .

• Featured on Meta
• The 2024 Developer Survey Is Live
• The return of Staging Ground to Stack Overflow
• The [tax] tag is being burninated
• Policy: Generative AI (e.g., ChatGPT) is banned

Hot Network Questions

• The symmetry of multiplications with the number 9
• Is it allowed to use patents for new inventions?
• Is it rational for heterosexuals to be proud that they were born heterosexual?
• How can I use a transistor to control the segments on this 7-segment display?
• Am I seeing double? What kind of helicopter is this, and how many blades does it actually have?
• Are there any jobs that are forbidden by law to convicted felons?
• Why are ETFs so bad at tracking Japanese indices?
• What is the origin of the idiom "say the word"?
• How do Authenticators work?
• What is the meaning of the 'ride out the clock'?
• Why is array access not an infix operator?
• Find characters common among all strings
• What is the point of triggering of a national snap election immediately after losing the EU elections?
• Word for a country declaring independence from an empire
• Symbolic solution of trigonometric equation
• Have I ruined my AC by running it with the outside cover on?
• A short story in French about furniture that leaves a mansion by itself, and comes back some time later
• Mismatching Euler characteristic of the Torus
• How to convert units when calculating a dimensionless quantity?
• Windows Server 2022 Support for Intel X520-DA2
• Geometric Brownian Motion as the limit of a Binomial Tree?
• How to implement a quantum multiplier in Qiskit 1.0
• An application of the (100/e)% rule applied to postdocs: moving on from an academic career, perhaps
• What does "far right tilt" actually mean in the context of the EU in 2024?

Rational Canonical Form: A Summary

The basic idea.

This post is intended to be a hopefully-not-too-intimidating summary of the rational canonical form (RCF) of a linear transformation. Of course, anything which involves the word "canonical" is probably intimidating no matter what . But even so, I've attempted to write a distilled version of the material found in (the first half of) section 12.2 from Dummit and Foote's Abstract Algebra .

In sum, the RCF is important because it allows us to classify linear transformations on a vector space up to conjugation. Below we'll set up some background, then define the rational canonical form, and close by discussing why the RCF looks the way it does. Next week we'll go through an explicit example to see exactly how the RCF can be used to classify linear transformations.        

From English to Math            

The background.

Let $F$ be a field and let $V$ be a finite dimensional $F$-vector space. Given a linear transformation $T:V\to V$, we can simultaneously view $V$ as an $F[x]$-module by defining the action   $$x\cdot v := Tv$$ for any vector $v\in V$. Then for any polynomial $p(x)\in F[x]$, we know what the action $p(x)\cdot v$ "looks like," namely if $p(x)=a_nx^n+\cdots+a_1x+a_0$ with $a_i\in F$, then     $$  \begin{align} p(x)\cdot v &= (a_nT^n+\cdots+a_1T+a_0I)v\\ &= a_nT^nv + \cdots + a_1Tv+a_0v \end{align} $$   where $I:V\to V$ is the identity. Notice that the last line is indeed another vector in $V$ and so this action makes sense. (And one can easily check that the module axioms are satisfied.)

One key observation is that $V$ is a torsion * $F[x]$-module. In particular, if $m_T(x)$ denotes the minimal polynomial of $T$, then for any $v\in V$ we have $m_T(x)v=0$ (since $m_T(x)v=m_T(T)v$ and $m_T(T)=0$ is the zero-linear transformation by definition). What's more, $V$ is finitely generated (it has a finite number of basis vectors) by assumption. So since $F[x]$ is a PID (as $F$ is a field), we may conclude   $$V\cong F[x]/(a_1(x))\oplus\cdots\oplus F[x]/(a_d(x))$$   where $a_1(x)\mid\cdots\mid a_d(x)$ are the invariant factors of $V$ and where $a_d(x)=m_T(x)$. (This is the Fundamental Theorem of Finitely Generated Modules over a PID, the subject of last week's post .) These invariant factors play a key role in defining the rational canonical form.

The Rational Canonical Form

Let $a(x)=x^k+b_{n-1}x^{k-1}+\cdots+b_1x+b_0$ be any monic polynomial in $F[x]$. Construct a $k\times k$ matrix by placing $1's$ along the subdiagonal and the negative of all the coefficients of $a(x)$ - except the leading term - along the last column. We call this the companion matrix of $a(x)$:   $$ \mathscr{C}_{a(x)}:= \begin{pmatrix}0 & 0 & \cdots  &\cdots& \cdots & -b_0\\ 1 & 0 & \cdots  &\cdots& \cdots &-b_1\\ 0 & 1 & \cdots  &\cdots& \cdots &-b_2\\ 0 & 0 &\ddots  & & &\vdots\\ \vdots & \vdots & & \ddots & &\vdots\\ 0 & 0 &\ldots & \ldots & 1 & -b_{k-1} \end{pmatrix}.$$

(For example if $a(x)=x^2+x+1\in \mathbb{Q}(x)$, then $\mathscr{C}_{a(x)}=\bigl( \begin{smallmatrix}  0 & -1\\ 1 & -1 \end{smallmatrix} \bigr).$ )

Letting $n=\text{dim}(V)$, we define the rational canonical form of $T:V\to V$ to be the $n\times n$ (block-diagonal) matrix $$ \begin{pmatrix} \mathscr{C}_{a_1(x)} & & & &\\ & \mathscr{C}_{a_2(x)} & & &\\ & & \ddots &&\\ &&& & \mathscr{C}_{a_d(x)} \end{pmatrix}$$   where the polynomials $a_1(x)\mid a_2(x)\mid\cdots \mid a_d(x)$ are the invariant factors of the representation of $V$ from above.

Recall that our vector space $V$ can be written as the direct sum $$V\cong F[x]/(a_1(x))\oplus\cdots\oplus F[x]/(a_d(x)).$$ For the moment, let's focus our attention on just one of the factors $$F[x]/(a(x))$$ (I've taken off the index for simplicity). Just like $V$, this quotient space wears two hats: it is both an $F[x]$-module and it is an $F$-vector space! As an $F$-vector space, it has the basis $\mathscr{B}=\{\bar{1},\bar{x},\ldots,\overline{x}^{k-1}\}$ where $\bar{x}:=x+(a(x))$ indicates a coset (this is the example from Dummit and Foote, section 11.1, following Proposition 1).  And just like $V$, this quotient space gets "acted on" by the linear transformation $T$ via $$Tv=x\cdot v$$ where here $v$ is a vector (i.e. a coset!) in $F[x]/(a(x))$.

A natural question to ask is, "What is the matrix representation of $T:F[x]/(a(x))\to F[x]/(a(x)) $ with respect to the basis $\mathcal{B}$?" From undergrad linear algebra we know the $i$th column of this matrix is simply the coefficients (with respect to $\mathcal{B}$) of the image of the $i$th basis vector under $T$. Explicitly:

 The last line follows because  $$  \begin{align}  \overline{x}^k=x^k+(a(x))&= a(x)-(b_{k-1}x^{k-1}+\cdots+b_0)+(a(x))\\  &= -b_{k-1}x^{k-1}-\cdots-b_0+(a(x))\\  &=-b_{k-1}\overline{x}^{k-1}-\cdots-b_0\overline{1}.  \end{align}  $$

Notice this matrix representation for $T$ is the precisely the companion matrix of $a(x)$! But keep in mind, we obtained it by restricting the action of $T$ to just one of the factors $F[x]/(a(x))$ in the direct sum above. To obtain the matrix representation of $T$ as it acts on the entire space $V$, we need to take the direct sum** of all the companion matrices. This corresponds to writing down the matrix of $T:V\to V$ with respect to the basis $\mathscr{B}=\mathscr{B}_1\cup \cdots\cup \mathscr{B}_m$ where $\mathscr{B}_i$ is the basis for $F[x]/(a_i(x))$. And this is exactly how we defined the rational canonical form of $T$ above.

As a final remark, any two similar matrices (or equivalently, similar linear transformations) share the same rational canonical form. (See Dummit and Foote, section 12.2 Theorem 15.) This means that the RCF acts like a "name tag" because it allows us to find representatives of the distinct conjugacy classes of linear transformations of a given order.

Next week we'll illustrate this fact with an explicit example.   

* Recall: to say an $R$-module $M$ is torsion means for every $m\in M$ there exists a nonzero $r\in R$ such that $rm=0$.

 ** By definition, the direct sum of matrices $A_1,\ldots, A_d$ is the block diagonal matrix with the $A_i$'s down the diagonal and zeros elsewhere.

Operator Norm, Intuitively

One unspoken rule of algebra, a first look at quantum probability, part 1, constructing the tensor product of modules.

Login Get started

Canonical Representation of a Number

• SchoolTutoring Academy
• August 31, 2012
• No Comments

Prime and composite numbers:

An integer is called prime if it is divisible only by 1 and itself. i.e. a prime number cannot be divisible by any other number than 1 and itself. A prime number has exactly 2 positive divisors. A number which has more than 2 positive divisors is said to be a composite number. i.e. a composite number contains at least one divisor other than 1 and itself. Here we need to make a note that 1 is neither prime nor composite.

Canonical representation of a number:

Every integer can be expressed as the product of primes and this factoring of the integer into primes is unique except the order of primes.

So, any integer n can be expressed as, n = p 1 k1 xp 2 k2 x …x p m km , where p 1 ,p 2 ,…,p m are primes where 1<p 1 <p 2 <…<p k and k 1 ,k 2 ,…,k m are positive integers.

Number of all positive divisors of a number:

If we take a number 6, we can say that there are only 4 divisors for 6 which are 1,2,3,6. Similarly if we take a number 24 we say that there are 8 divisors namely 1,2,3,4,6,8,12,24. But what if we have to find the number of divisors of a big number such as 2000. There is a formula for finding the number of divisors of a given number.

If n = p 1 k1 xp 2 k2 x …x p m km is the canonical representation of a number n then the number of positive divisors of n is given by (1+k 1 )(1+k 2 )…(1+k m ).

2000 = 2 4 x 5 3

Number of positive divisors of 2000 = (1+4)(1+3) = 5×4 = 20.

Sum of all positive divisors of a number:

As we discussed earlier, when it is difficult to find the number of divisors, it is obviously more difficult to find the sum of divisors. There is a formula for finding the sum of divisors of a given number.

If n = p 1 k1 xp 2 k2 x …x p m km is the canonical representation of a number n then the sum of positive divisors of n is given by

Sum of positive divisors of 2000 =[2 5 -1/2-1][5 4 -1/5-1] = [31/1] x [624/4] = 31 x 156 = 4836. SchoolTutoring Academy is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Patterson visit: Tutoring in Patterson.

Priming and Decision-Making

Associative, distributive and commutative properties.

• Engineering Mathematics
• Discrete Mathematics
• Operating System
• Computer Networks
• Digital Logic and Design
• C Programming
• Data Structures
• Theory of Computation
• Compiler Design
• Computer Org and Architecture
• Digital Electronics and Logic Design Tutorials

Number Systems

• Number System and Base Conversions
• 1's and 2's complement of a Binary Number
• BCD or Binary Coded Decimal
• Error Detection Codes: Parity Bit Method

Boolean Algebra and Logic Gates

• Logic Gates - Definition, Types, Uses
• Basic Conversion of Logic Gates
• Realization of Logic Gate Using Universal gates

Canonical and Standard Form

• Types of Integrated Circuits

Minimization Techniques

• Minimization of Boolean Functions
• Introduction of K-Map (Karnaugh Map)
• 5 variable K-Map in Digital Logic
• Various Implicants in K-Map
• Don't Care (X) Conditions in K-Maps
• Quine McCluskey Method
• Two Level Implementation of Logic Gates
• Combinational Circuits
• Half Adder in Digital Logic
• Full Adder in Digital Logic
• Half Subtractor in Digital Logic
• Full Subtractor in Digital Logic
• Parallel Adder and Parallel Subtractor
• Sequential Binary Multiplier
• Multiplexers in Digital Logic
• What is a demultiplexer ?
• Binary Decoder in Digital Logic
• Encoder in Digital Logic
• Code Converters - Binary to/from Gray Code
• Magnitude Comparator in Digital Logic

Sequential Circuits

• Introduction of Sequential Circuits
• Difference between combinational and sequential circuit
• Latches in Digital Logic
• Flip-Flop types, their Conversion and Applications

Conversion of Flip-Flop

• Conversion of S-R Flip-Flop into D Flip-Flop
• Conversion of S-R Flip-Flop into T Flip-Flop
• Conversion of J-K Flip-Flop into T Flip-Flop
• Conversion of J-K Flip-Flop into D Flip-Flop

Register, Counter, and Memory Unit

• Counters in Digital Logic
• Ripple Counter in Digital Logic
• Ring Counter in Digital Logic
• General Purpose Registers
• Shift Registers in Digital Logic
• Computer Memory
• Random Access Memory (RAM)
• Read Only Memory (ROM)

LMNs and GATE PYQs

• LMN - Digital Electronics
• Digital Logic and Design - GATE CSE Previous Year Questions

Practice Questions - Digital Logic & Design

• Logic functions and Minimization
• Sequential circuits
• Number Representation

Canonical Form – In Boolean algebra, the Boolean function can be expressed as Canonical Disjunctive Normal Form known as minterm and some are expressed as Canonical Conjunctive Normal Form known as maxterm .  In Minterm, we look for the functions where the output results in “1” while in Maxterm we look for functions where the output results in “0”.  We perform the Sum of minterm also known as the Sum of products (SOP).  We perform Product of Maxterm also known as Product of sum (POS).  Boolean functions expressed as a sum of minterms or product of maxterms are said to be in canonical form. 

Standard Form – A Boolean variable can be expressed in either true or complementary forms. In standard form Boolean function will contain all the variables in either true form or complemented form while in canonical number of variables depends on the output of SOP or POS. 

A Boolean function can be expressed algebraically from a given truth table by forming a : 

• minterm for each combination of the variables that produces a 1 in the function and then takes the OR of all those terms.
• maxterm for each combination of the variables that produces a 0 in the function and then takes the AND of all those terms.

Truth table representing minterm and maxterm –  

Sum of minterms –   The minterms whose sum defines the Boolean function are those which give the 1’s of the function in a truth table. Since the function can be either 1 or 0 for each minterm, and since there are 2^n minterms, one can calculate all the functions that can be formed with n variables to be (2^(2^n)). It is sometimes convenient to express a Boolean function in its sum of minterm form.   

Product of maxterms –

When dealing with Boolean algebra, the product of maxterms is a handy way to express how combinations of inputs lead to a result of 0. Maxterms basically tell us which combinations of inputs won’t give us a 1 as an output. They are the opposite of minterms, which tell us when we get a 1.

• Solution –   A = A(B + B’) = AB + AB’  This function is still missing one variable, so  A = AB(C + C’) + AB'(C + C’) = ABC + ABC’+ AB’C + AB’C’  The second term B’C is missing one variable; hence,  B’C = B’C(A + A’) = AB’C + A’B’C  Combining all terms, we have  F = A + B’C = ABC + ABC’ + AB’C + AB’C’ + AB’C + A’B’C  But AB’C appears twice, and  according to theorem 1 (x + x = x), it is possible to remove one of those occurrences. Rearranging the minterms in ascending order, we finally obtain  F = A’B’C + AB’C’ + AB’C + ABC’ + ABC  = m1 + m4 + m5 + m6 + m7  SOP is represented as Sigma(1, 4, 5, 6, 7)
• Solution –   F = xy + x’z = (xy + x’)(xy + z) = (x + x’)(y + x’)(x + z)(y + z) = (x’ + y)(x + z)(y + z)  x’ + y = x’ + y + zz’ = (x’+ y + z)(x’ + y + z’) x + z = x + z + yy’ = (x + y + z)(x + y’ + z) y + z = y + z + xx’ = (x + y + z)(x’ + y + z)  F = (x + y + z)(x + y’ + z)(x’ + y + z)(x’ + y + z’)  = M0*M2*M4*M5  POS is represented as Pi(0, 2, 4, 5)
• Solution – F(A, B, C) = Sigma(1, 4, 5, 6, 7)  F'(A, B, C) = Sigma(0, 2, 3) = m0 + m2 + m3  Now, if we take the complement of F’ by DeMorgan’s theorem, we obtain F in a different form:  F = (m0 + m2 + m3)’  = m0’m2’m3′  = M0*M2*M3  = PI(0, 2, 3) 
• Solution – F = (x+x’)y'(z+z’)+x(y+y’)z’ +xyz  F = xy’z+ xy’z’+x’y’z+x’y’z’+ xyz’+xy’z’+xyz 

 Advantages of Canonical Form:

Uniqueness: The canonical form of a boolean function is unique, which means that there is only one possible canonical form for a given function. Clarity: The canonical form of a boolean function provides a clear and unambiguous representation of the function. Completeness: The canonical form of a boolean function can represent any possible boolean function, regardless of its complexity.  

Disadvantages of Canonical Form:

Complexity: The canonical form of a boolean function can be complex, especially for functions with many variables. Computation: Computing the canonical form of a boolean function can be computationally expensive, especially for large functions. Redundancy: The canonical form of a boolean function can be redundant, which means that it can contain unnecessary terms or variables that do not affect the function.  

Advantages of Standard Form:

Simplicity: The standard form of a boolean function is simpler than the canonical form, making it easier to understand and work with. Efficiency: The standard form of a boolean function can be implemented using fewer logic gates than the canonical form, which makes it more efficient in terms of hardware and computation. Flexibility: The standard form of a boolean function can be easily modified and combined with other functions to create new functions that meet specific design requirements.  

Disadvantages of Standard Form:

Non-uniqueness: The standard form of a boolean function is not unique, which means that there can be multiple possible standard forms for a given function. Incompleteness: The standard form of a boolean function may not be able to represent some complex boolean functions. Ambiguity: The standard form of a boolean function can be ambiguous, especially if it contains multiple equivalent expressions.  

Please Login to comment...

Similar reads.

• Digital Logic

Improve your Coding Skills with Practice

What kind of Experience do you want to share?

Canonical representation

Continue reading

Also in this chapter

• Fourier transform
• Z-transform
• Delay: Minimum, mixed, and maximum
• Two-length wavelets
• Illustrations of spectra
• Delay in general
• Autocorrelation
• Zero-phase wavelets
• Symmetric wavelets
• Ricker wavelet
• Appendix G: Exercises

External links

• Digital Imaging and Deconvolution
• Publications
• Wavelets - Chapter 7

Navigation menu

• Java.lang Package classes
• Java.lang - Home
• Java.lang - Boolean
• Java.lang - Byte
• Java.lang - Character
• Java.lang - Character.Subset
• Java.lang - Character.UnicodeBlock
• Java.lang - Class
• Java.lang - ClassLoader
• Java.lang - Compiler
• Java.lang - Double
• Java.lang - Enum
• Java.lang - Float
• Java.lang - InheritableThreadLocal
• Java.lang - Integer
• Java.lang - Long
• Java.lang - Math
• Java.lang - Number
• Java.lang - Object
• Java.lang - Package
• Java.lang - Process
• Java.lang - ProcessBuilder
• Java.lang - Runtime
• Java.lang - RuntimePermission
• Java.lang - SecurityManager
• Java.lang - Short
• Java.lang - StackTraceElement
• Java.lang - StrictMath
• Java.lang - String
• Java.lang - StringBuffer
• Java.lang - StringBuilder
• Java.lang - System
• Java.lang - Thread
• Java.lang - ThreadGroup
• Java.lang - ThreadLocal
• Java.lang - Throwable
• Java.lang - Void
• Java.lang Package extras
• Java.lang - Interfaces
• Java.lang - Errors
• Java.lang - Exceptions
• Java.lang Package Useful Resources
• Java.lang - Useful Resources
• Java.lang - Discussion
• Selected Reading
• UPSC IAS Exams Notes
• Developer's Best Practices
• Questions and Answers
• Effective Resume Writing
• HR Interview Questions
• Computer Glossary

Java - String intern() Method

Description.

The Java String intern() method is used to retrieve the canonical representation for the current string object. In short, the intern() method is used to make an exact copy of a String in heap memory and store it in the String constant pool. A pool is a special storage space in Java heap memory where the string literals can be stored.

A canonical representation indicates that values of a specific type of resource can be described or represented in multiple ways, with one of those ways are chosen as the preferred canonical form.

Note − The equality of two objects can easily be tested by testing the equality of their canonical forms.

Following is the syntax of the Java String intern() method −

It does not accept any parameter.

Return Value

This method returns a canonical representation for the string object.

Creating Copy of Current String Example

If the given string is not null , the intern() method creates the same copy of the current string object in the heap memory.

In the following program, we are instantiating the string class with the value "TutorialsPoint" . Using the intern() method, we are trying to create an exact copy of the current string in the heap memory.

On executing the above program, it will produce the following result −

Getting Null Pointer While Creating Copy of Current String Example

If the given string has a null value , the intern() method throws the NullPointerException .

In the following example, we are creating a string literal with the null value . Using the intern() method, we are trying to retrieve the canonical representation of the current string object.

Following is the output of the above program −

Checking Equality while Creating Copy of Current String Example

Using the intern() method we can check the equality of the string objects based on their conical forms.

In this program, we are creating an object of the string class with the value "Hello" . Then, using the intern() method, we are trying to retrieve the canonical representation of the string object, and compare them using the equals() method.

The above program, produces the following output −

To Continue Learning Please Login

www.springer.com The European Mathematical Society

• StatProb Collection
• Recent changes
• Current events
• Random page
• Project talk
• Request account
• What links here
• Related changes
• Special pages
• Printable version
• Permanent link
• Page information
• View source

Lévy canonical representation

2020 Mathematics Subject Classification: Primary: 60E07 Secondary: 60G51 [ MSN ][ ZBL ]

A formula for the logarithm $ \mathop{\rm ln} \phi ( \lambda ) $ of the characteristic function of an infinitely-divisible distribution :

$$ \mathop{\rm ln} \phi ( \lambda ) = i \gamma \lambda - \frac{\sigma ^ {2} \lambda ^ {2} }{2} + \int\limits _ {- \infty } ^ { 0 } \left ( e ^ {i \lambda x } - 1 - \frac{i \lambda x }{1 + x ^ {2} } \right ) \ d M ( x) + $$

$$ + \int\limits _ { 0 } ^ \infty \left ( e ^ {i \lambda x } - 1 - \frac{i \lambda x }{1 + x ^ {2} } \right ) d N ( x) , $$

where the characteristics of the Lévy canonical representation, $ \gamma $, $ \sigma ^ {2} $, $ M $, and $ N $, satisfy the following conditions: $ - \infty < \gamma < \infty $, $ \sigma ^ {2} \geq 0 $, and $ M ( x) $ and $ N ( x) $ are non-decreasing left-continuous functions on $ ( - \infty , 0 ) $ and $ ( 0 , \infty ) $, respectively, such that

$$ \lim\limits _ {x \rightarrow \infty } \ N ( x) = \lim\limits _ {x \rightarrow - \infty } \ M ( x) = 0 $$

$$ \int\limits _ { - 1} ^ { 0 } x ^ {2} d M ( x) < \infty ,\ \ \int\limits _ { 0 } ^ { 1 } x ^ {2} d N ( x) < \infty . $$

To every infinitely-divisible distribution there corresponds a unique system of characteristics $ \gamma $, $ \sigma ^ {2} $, $ M $, $ N $ in the Lévy canonical representation, and conversely, under the above conditions on $ \gamma $, $ \sigma ^ {2} $, $ M $, and $ N $ the Lévy canonical representation with respect to such a system determines the logarithm of the characteristic function of some infinitely-divisible distribution.

Thus, for the normal distribution with mean $ a $ and variance $ \sigma ^ {2} $:

$$ \gamma = a ,\ \sigma ^ {2} = \sigma ^ {2} ,\ \ N ( x) \equiv 0 ,\ M ( x) \equiv 0 . $$

For the Poisson distribution with parameter $ \lambda $:

$$ \gamma = \frac \lambda {2} ,\ \ \sigma ^ {2} = 0 ,\ \ M ( x) \equiv 0 ,\ \ N ( x) = \left \{ \begin{array}{rl} - \lambda & \textrm{ for } x \leq 1 , \\ 0 & \textrm{ for } x > 1 . \\ \end{array} \right .$$

To the stable distribution with exponent $ \alpha $, $ 0 < \alpha < 2 $, corresponds the Lévy representation with

$$ \sigma ^ {2} = 0 ,\ \ \textrm{ any } \ \gamma ,\ M ( x) = \frac{c _ {1} }{| x | ^ \alpha } ,\ \ N ( x) = - \frac{c _ {2} }{x ^ \alpha } , $$

where $ c _ {i} \geq 0 $, $ i = 1 , 2 $, are constants $ ( c _ {1} + c _ {2} > 0 ) $. The Lévy canonical representation of an infinitely-divisible distribution was proposed by P. Lévy in 1934. It is a generalization of a formula found by A.N. Kolmogorov in 1932 for the case when the infinitely-divisible distribution has finite variance. For $ \mathop{\rm ln} \phi ( \lambda ) $ there is a formula equivalent to the Lévy canonical representation, proposed in 1937 by A.Ya. Khinchin and called the Lévy–Khinchin canonical representation . The probabilistic meaning of the functions $ N $ and $ M $ and the range of applicability of the Lévy canonical representation are defined as follows: To every infinitely-divisible distribution function $ F $ corresponds a stochastically-continuous process with stationary independent increments

$$ X = \{ {X ( t) } : {0 \leq t < \infty } \} ,\ X ( 0) = 0 , $$

$$ F ( X) = {\mathsf P} \{ X ( 1) < x \} . $$

In turn, a separable process $ X $ of the type mentioned has with probability 1 sample trajectories without discontinuities of the second kind; hence for $ b > a > 0 $ the random variable $ Y ( [ a , b ) ) $ equal to the number of elements in the set

$$ \left \{ {t } : {a \leq \lim\limits _ {\tau \downarrow 0 } \ X ( t + \tau ) - \lim\limits _ {\tau \downarrow 0 } \ X ( t - \tau ) < b , 0 \leq t \leq 1 } \right \} , $$

i.e. to the number of jumps with heights in $ [ a , b ) $ on the interval $ [ 0 , 1 ] $, exists. In this notation, one has for the function $ N $ corresponding to $ F $,

$$ {\mathsf E} \{ Y ( [ a , b ) ) \} = N ( b) - N ( a) . $$

A similar relation holds for the function $ M $.

Many properties of the behaviour of the sample trajectories of a separable process $ X $ can be expressed in terms of the characteristics of the Lévy canonical representation of the distribution function $ {\mathsf P} \{ X ( 1) < x \} $. In particular, if $ \sigma ^ {2} = 0 $,

$$ \lim\limits _ {x \rightarrow 0 } N ( x) > - \infty ,\ \ \lim\limits _ {x \rightarrow 0 } M ( x) < \infty , $$

$$ \gamma = \int\limits _ {- \infty } ^ { 0 } \frac{x}{1 + x ^ {2} } d M ( x) + \int\limits _ { 0 } ^ \infty \frac{x}{1 + x ^ {2} } d N ( x) , $$

then almost-all the sample functions of $ X $ are with probability 1 step functions with finitely many jumps on any finite interval. If $ \sigma ^ {2} = 0 $ and if

$$ \int\limits _ { - 1} ^ { 0 } | x | d M ( x) + \int\limits _ { 0 } ^ { 1 } x d N ( x) < \infty , $$

then with probability 1 the sample trajectories of $ X $ have bounded variation on any finite interval. Directly in terms of the characteristics of the Lévy canonical representation one can calculated the infinitesimal operator corresponding to the process $ X $, regarded as a Markov random function. Many analytical properties of an infinitely-divisible distribution function can be expressed directly in terms of the characteristics of its Lévy canonical representation.

There are analogues of the Lévy canonical representation for infinitely-divisible distributions given on a wide class of algebraic structures.

• Probability and statistics
• Probability theory and stochastic processes
• Distribution theory
• Stochastic processes
• This page was last edited on 19 January 2022, at 01:14.
• Privacy policy
• About Encyclopedia of Mathematics
• Disclaimers
• Impressum-Legal

Stack Exchange Network

Stack Exchange network consists of 183 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Q&A for work

Connect and share knowledge within a single location that is structured and easy to search.

What is a canonical representation?

We had our Digital Systems class today, and our professor kept throwing around the word 'canonical' around a whole lot, but I'm very confused as to what this is.

What I understood is that it's the way of representing an expression uniquely (i.e., two non-equivalent functions cannot have the same canonical form) Is my interpretation correct.

So would Sum of Minterms be canonical or not? If we simplify the SoM form, would it still be canonical?

• digital-logic
• boolean-algebra

2 Answers 2

Canonical form basically holds every variable in its group.

So if you have three variables named A, B and C, your SoM could be A~BC+~ABC+AB~C = Y .

Now you can simplify this to reduce the number of variables in the equation. This simplification is easy for us to solve manually. But for a computer, it needs to know the A, B and C values as a group ( 101+011+110 from the example above). Because of this, canonical form holds significance.

If you simplify, it just becomes a normal Boolean expression and not a canonical form

So would Sum of Minterms be canonical or not?

When Sum of Products is in its canonical form, it is called 'Sum of Minterms'. Similarly, Product of Sums in its canonical form is called 'Product of Max terms'.

So yes, SoM is canonical.

For a Boolean equation to be in canonical form means that all the terms in it contain all the variables, irrespective of whether a variable in a term is inverted or not.

For example, you have 3 variables (p, q, r) and a function f = p’qr + pq’r + pqr’ + pqr

Here the equation is in canonical form and its simplified form (standard form), after carrying out simplification is, f = pq + qr + pr

Your Answer

Sign up or log in, post as a guest.

Required, but never shown

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy .

Not the answer you're looking for? Browse other questions tagged digital-logic boolean-algebra or ask your own question .

Hot network questions.

• What's the maximum amount of material that a puzzle with unique solution can have?
• British child with Italian mother travelling to Italy
• Group with a translation invariant ultrafilter
• Why are ETFs so bad at tracking Japanese indices?
• Selecting an opamp for a voltage follower circuit using a LTspice simulation
• What does "far right tilt" actually mean in the context of the EU in 2024?
• Why a truly uninformative prior does not exist?
• Find characters common among all strings
• How do Authenticators work?
• What is with the different pronounciation of "janai" that consistently happens in "Jojo's Bizarre Adventure"?
• Wifi deteriorates over time; cronjob to restart network manager
• How to negotiate such toxic competitiveness during my masters studies
• Was it known in ancient Rome and Greece that boiling water made it safe to drink and if so, what was the theory behind this?
• How was damno derived from damnum?
• How to make Bash remove quotes after parameter expansion?
• A question about cohomology of the classifying spaces of compact groups
• Customary learning areas for Shavuos?
• Strict toposes as a finite limit theory
• Anime about a girl who can see spirits that nobody else can
• Ubuntu Terminal with alternating colours for each line
• How can I use a router without gateway?
• In beamer, can I have a caption only in article mode, but some of its content in all modes?
• How to unplug heating element connector in tumble dryer
• How to implement a quantum multiplier in Qiskit 1.0

• Previous : Literals In Queries
• Next : JDBC and HADB Date and Time Operations

Canonical String Format

Each type in a 5800 system has a canonical representation as a string value. The canonical string representation of each type is shown in Table 4–1 .

This canonical string encoding is used in the following places:

When exposing the field as a directory component or a filename component in a virtual view

When converting a typed value to a string as the result of the getAsString operation on a NameValueRecord or a QueryResultSet operation

When parsing a literal value as described in Literals for 5800 System Data Types to create a typed query value from a string representation of that value.

The Canonical String Decode Operation

The inverse of the canonical string encoding is used in the following places:

It is always allowed to store a string value into any metadata field, no matter what the type of the field is. The actual data stored is the result of applying the canonical string decode operation to the incoming string value.

On a virtual view lookup operation, the canonical string decode operation is used on the supplied filename to derive the actual metadata values to look up, given their string representations in the filename.

The decode operation is allowed to accept incoming string values that would never be a legal output for an encode operation. Some examples of this include:

decodeBinary of an odd number of hex digits. The convention is to left-justify the supplied digits in the binary value. For example, the string "b0a" corresponds to the binary literal [b0a0] .

decodeDate of a non-standard date format.

A double value encoded with a non-canonical number of digits. For example, .00145E20 instead of 1.45E17 .

Example 4–1 Virtual View Lookup Operation

If you take a value V and encode it into a string S , and then perform the canonical decode operation on S to get a new value V’ . Does V always equal V’ ? The answer is yes in most cases, but not always.

What is actually guaranteed is the weaker statement that if encode(V) = S and if decode(S)=V’ , then encode(V’) is also equal to S .

• © 2010, Oracle Corporation and/or its affiliates

Canonical representations

• First Online: 01 January 2006

Cite this chapter

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 152))

1876 Accesses

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Institutional subscriptions

Unable to display preview.  Download preview PDF.

Editor information

Rights and permissions.

Reprints and permissions

Copyright information

© 1991 Springer-Verlag

About this chapter

(1991). Canonical representations. In: Aplevich, J.D. (eds) Implicit Linear Systems. Lecture Notes in Control and Information Sciences, vol 152. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0044368

Download citation

DOI : https://doi.org/10.1007/BFb0044368

Published : 20 January 2006

Publisher Name : Springer, Berlin, Heidelberg

Print ISBN : 978-3-540-53537-9

Online ISBN : 978-3-540-46759-5

eBook Packages : Springer Book Archive

Share this chapter

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

• Publish with us

Policies and ethics

• Find a journal
• Track your research

Stack Exchange Network

Stack Exchange network consists of 183 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Q&A for work

Connect and share knowledge within a single location that is structured and easy to search.

matrix for a "canonical" rotation

Let $p$ be an arbitrary point on the unit sphere $$ S = \{(x,y,z) \mid x^2+y^2+z^2 = 1 \},$$ other than the north or south poles $(0,0,\pm 1)$.

There is a one-dimensional family of rotations which take $n = (0,0,1)$ to $p$, but one rotation is canonical: the one that keeps $n$, $p$, and $0$ in a plane. In other words the cross product $u = n \times p$ should be an axis of rotation.

Is there a nice way to write down the matrix $M \in SO(3)$ for this canonical rotation?

I observe that we have three equations: (1) $M^T M = I$, (2) $M n = p$, and (3) $M u = u $.

Since I want $M \in SO(3)$ rather than simply $M \in O(3)$ we also have $\det M = 1$.

This should be enough to determine $M$, but I still don't see an obvious way to write down a formula.

• linear-algebra

Let the point $p$ be given in cartesian coordinates by $(\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\phi)$, so that $\theta$ and $\phi$ are the usual spherical coordinates on $S^2$. Then the matrix that rotates $n=(1,0,0)$ to $p$, and also preserves the plane passing through both of them and the origin, is given by

$M=\left(\begin{array}{ccc}\sin^2\phi + \cos\theta\cos^2\phi & (-1+\cos\theta)\sin\phi\cos\phi & \sin\theta\cos\phi\\ (-1+\cos\theta)\sin\phi\cos\phi & \cos^2\phi+\cos\theta\sin^2\phi & \sin\theta\sin\phi\\ -\sin\theta\cos\phi & -\sin\theta\sin\phi & \cos\theta \end{array}\right).$

It's straightforward but tedious to check that this is indeed an orthogonal matrix with determinant $1$ that sends $n$ to $p$ and fixes $u=n\times p=(-\sin\theta\sin\phi,\sin\theta\cos\phi,0)$, and hence the desired transformation.

I produced this matrix by considering that, for each fixed $\phi$, what you desire is a one-parameter subgroup of $SO(3)$: the group of rotations that fix $u$. It therefore is the family of exponentials of an infinitesimal transformation, i.e. an element of the Lie algebra $\frak{so}\mathrm{(3)}$. For example, a tiny rotation tilting the $z$-axis toward the $x$-axis differs from the identity by $\left(\begin{array}{ccc} & & \epsilon\\ & & \\ -\epsilon & & \end{array}\right)$, and $\exp\left(\begin{array}{ccc} & & \theta\\ & & \\ -\theta & & \end{array}\right) = \left(\begin{array}{ccc} \cos\theta & & \sin\theta\\ & 1 & \\ -\sin\theta & & \cos\theta \end{array}\right).$

Similarly, the Lie algebra element corresponding to an infinitesimal rotation of the $z$-axis in the direction of the $y$-axis is given by $\left(\begin{array}{ccc} & & \\ & & \epsilon\\ & -\epsilon& \end{array}\right)$, and $\exp\left(\begin{array}{ccc} & & \\ & & \theta\\ & -\theta & \end{array}\right) = \left(\begin{array}{ccc} 1 & & \\ & \cos\theta & \sin\theta\\ & -\sin\theta & \cos\theta \end{array}\right).$

We want the Lie algebra element which has components $\cos\phi$ toward the $x$-axis and $\sin\phi$ toward the $y$-axis; thus we desire the exponential $M=\exp\left(\begin{array}{ccc} & & \theta\cos\phi\\ & & \theta\sin\phi\\ -\theta\cos\phi & -\theta\sin\phi & \end{array}\right)=\exp(A).$

This matrix $A$ is diagonalizable, so its exponential $M$ can be computed by hand, which is how I arrived at my answer above.

You must log in to answer this question.

Not the answer you're looking for browse other questions tagged linear-algebra rotations ., hot network questions.

• British child with Italian mother travelling to Italy
• Draw Memory Map/Layout/Region in TikZ
• Can we combine a laser with a gauss rifle to get a cinematic 'laser rifle'?
• Average value of lattice parameters from cell trajectory
• What do humans do uniquely, that computers apparently will not be able to?
• How to unplug heating element connector in tumble dryer
• Find characters common among all strings
• is_decimal Function Implementation in C++
• Customary learning areas for Shavuos?
• Why do I get different results for the products of two identical expressions?
• How can I use a router without gateway?
• I am international anyway
• Why is "second" an adverb in "came a close second"?
• Geometric Brownian Motion as the limit of a Binomial Tree?
• Was it known in ancient Rome and Greece that boiling water made it safe to drink and if so, what was the theory behind this?
• Strict toposes as a finite limit theory
• Can someone explain the damage distrubution on this aircraft that flew through a hailstorm?
• My son (infant) has received a Schengen tourist visa (Switzerland). However, his name is not printed as per passport. Please help
• Movie I saw in the 80s where a substance oozed off of movie stairs leaving a wet cat behind
• Resolving conflicts
• A question about cohomology of the classifying spaces of compact groups
• Have I ruined my AC by running it with the outside cover on?
• Is it allowed to use patents for new inventions?
• Should I ask for authorship or ignore?

Help | Advanced Search

Mathematics > Quantum Algebra

Title: canonicalizing zeta generators: genus zero and genus one.

Abstract: Zeta generators are derivations associated with odd Riemann zeta values that act freely on the Lie algebra of the fundamental group of Riemann surfaces with marked points. The genus-zero incarnation of zeta generators are Ihara derivations of certain Lie polynomials in two generators that can be obtained from the Drinfeld associator. We characterize a canonical choice of these polynomials, together with their non-Lie counterparts at even degrees $w\geq 2$, through the action of the dual space of formal and motivic multizeta values. Based on these canonical polynomials, we propose a canonical isomorphism that maps motivic multizeta values into the $f$-alphabet. The canonical Lie polynomials from the genus-zero setup determine canonical zeta generators in genus one that act on the two generators of Enriquez' elliptic associators. Up to a single contribution at fixed degree, the zeta generators in genus one are systematically expanded in terms of Tsunogai's geometric derivations dual to holomorphic Eisenstein series, leading to a wealth of explicit high-order computations. Earlier ambiguities in defining the non-geometric part of genus-one zeta generators are resolved by imposing a new representation-theoretic condition. The tight interplay between zeta generators in genus zero and genus one unravelled in this work connects the construction of single-valued multiple polylogarithms on the sphere with iterated-Eisenstein-integral representations of modular graph forms.

Submission history

Access paper:.

• Other Formats

Ancillary files ( details ) :

• ArithmeticDerivations_z3,z5,z7.nb
• Polyswt12.txt

References & Citations

• INSPIRE HEP
• Google Scholar
• Semantic Scholar

BibTeX formatted citation

Bibliographic and Citation Tools

Code, data and media associated with this article, recommenders and search tools.

• Institution

arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs .

Deep multimodal fusion for 3D mineral prospectivity modeling: Integration of geological models and simulation data via canonical-correlated joint fusion networks

• Zheng, Yang
• Wu, Jingjie
• Xie, Shaofeng
• Chen, Yudong
• Xiao, Keyan
• Pfeifer, Norbert
• Mao, Xiancheng

Data-driven three-dimensional (3D) mineral prospectivity modeling (MPM) employs diverse 3D exploration indicators to express geological architecture and associated characteristics in ore systems. The integration of 3D geological models with 3D computational simulation data enhances the effectiveness of 3D MPM in representing the geological architecture and its coupled geodynamic processes that govern mineralization. Despite variations in modality (i.e., data source, representation, and information abstraction levels) between geological models and simulation data, the cross-modal gap between these two types of data remains underexplored in 3D MPM. This paper presents a novel 3D MPM approach that robustly fuses multimodal information from geological models and simulation data. Acknowledging the coupled and correlated nature of geological architectures and geodynamic processes, a joint fusion strategy is employed, aligning information from both modalities by enforcing their correlation. A joint fusion neural network is devised to extract maximally correlated features from geological models and simulation data, fusing them in a cross-modality feature space. Specifically, correlation analysis (CCA) regularization is utilized to maximize the correlation between features of the two modalities, guiding the network to learn coordinated and joint fused features associated with mineralization. This results in a more effective 3D mineral prospectivity model that harnesses the strengths from both modalities for mineral exploration targeting. The proposed method is evaluated in a case study of the world-class Jiaojia gold deposit, NE China. Extensive experiments were carried out to compare the proposed method with state-of-the-art methods, methods using unimodal data, and variants without CCA regularization. Results demonstrate the superior performance of the proposed method in terms of prediction accuracy and targeting efficacy, highlighting the importance of CCA regularization in enhancing predictive power in 3D MPM.

• Mineral prospectivity modeling;
• Multimodal fusion;
• 3D geological models;
• Geodynamic simulation data;
• Canonical correlation analysis