meaning of unique representation

Unique Representation in Polynomial Forms/General Result

Let $f$ be a polynomial form in the indeterminates $\set {X_j: j \in J}$ such that $f: \mathbf X^k \mapsto a_k$.

For $r \in \R$, $\mathbf X^k \in M$, let $r \mathbf X^k$ denote the polynomial form that takes the value $r$ on $\mathbf X^k$ and zero on all other monomials .

Let $Z$ denote the set of all multiindices indexed by $J$.

Then the sum representation:

has only finitely many non-zero terms .

Moreover it is everywhere equal to $f$, and is the unique such sum.

Dropping the zero terms from the sum we can write the polynomial $f$ as

for some $a_{k_i}\in R$, $i = 1, \ldots, r$.

Suppose that the sum has infinitely many non-zero terms.

Then infinitely many $a_k$ are non-zero, which contradicts the definition of a polynomial.

Therefore the sum consists of finitely many non-zero terms.

Let $\mathbf X^m \in M$ be arbitrary.

So $\hat f = f$.

Finally suppose that:

is another such representation with $b_m \ne a_m$ for some $m \in Z$.

Therefore $\hat f$ as defined above is the only such representation.

$\blacksquare$

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representation

Definition of representation

Examples of representation in a sentence.

These examples are programmatically compiled from various online sources to illustrate current usage of the word 'representation.' Any opinions expressed in the examples do not represent those of Merriam-Webster or its editors. Send us feedback about these examples.

Word History

15th century, in the meaning defined at sense 1

Phrases Containing representation

• proportional representation
• self - representation

Dictionary Entries Near representation

representant

representationalism

Cite this Entry

“Representation.” Merriam-Webster.com Dictionary , Merriam-Webster, https://www.merriam-webster.com/dictionary/representation. Accessed 15 Aug. 2024.

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Kids definition of representation, legal definition, legal definition of representation, more from merriam-webster on representation.

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Color Meanings – Discover the Power and Symbolism Behind Every Hue

Colors play an exceptionally prominent role in our lives. They influence our thinking, inspire our decision-making, and impact our moods. From causing changes to changing reactions, colors are more powerful than we think. Depending on our interpretations, they can be used for both good and evil. Fortunately, we have the power to welcome some colors and shun others. This guide will take you on a journey through the exciting universe of color meanings.

It’s important to note that not all colors were created equal. Though some shades are similar, that doesn’t mean they’ll awaken the same emotions. To truly grasp the meaning of color, we must learn to accept this. It’s also imperative to relinquish some control. Try as we might, we won’t always have a handle on our emotions. Colors have such a substantial effect on the mind and body, and until we make our peace with this reality, we’ll remain forever perplexed by inexplicable and random reactions.

You can read about each color meaning by clicking the links below:

Recognizing Your Responses

Have you ever noticed a shift in your attitude while surrounded by a particular color? Perhaps you’ve been overwhelmed with nostalgia while staring at a specific shade. You’ve likely written these occurrences off as a common human response, but the cause is far more complex than that. Color psychology teaches us that much of human behavior is dictated by color. When we open our minds to this possibility, we begin to understand the scope of our perceptions.

Personal Cognizance

Once we acknowledge that colors influence doings, it makes way for personal interpretations. In other words, we level the playing field by regaining some control of how we react to colors. You may realize that the color red triggers anger. To avoid unnecessary bouts of rage, limit your encounters with this hue. On the other hand, yellow might make you feel cheery and upbeat. If so, introduce more of this color into your life. With a healthy dose of awareness, you can somewhat manage the moods and emotions that colors produce.

Uncovering New Meanings

The psychology behind color is fascinating and ever-expanding. With that said, new meanings are always coming to light. The more knowledge we acquire, the more informed we become. This prospect promises great hope for future research.

What’s more, our minds are in a perpetual state of evolvement. In essence, though some colors have fixed explanations, these findings can always shift. This fact is both unsettling and reassuring. As researchers continue to delve into the many facets of the color wheel, new information is continually revealing itself.

Diversify Your Understandings

While reading these articles, you’ll develop a deeper understanding of what colors mean, how they make an impression on us, and the ways we perceive them. The notion that colors are so dynamic may seem absurd to some, but there’s no refuting this truth. This concept often eludes us because we don’t realize how frequently colors control us. Certain hues can evoke fond memories. Others may give rise to dreadful feelings.

An Overview of Color Meanings

To help you discover the celebrated phenomenon of colors, here’s an overview of each color’s meaning, what it symbolizes, and how it affects human emotion and behavior.

1. Red Color Meaning

The color of passion and energy. Red draws attention like no other color and radiates a strong and powerful energy that motivates us to take action. It is also linked to sexuality and stimulates deep and intimate passion. Red is ubiquitously used to warn and signal caution and danger.

Read more about the color red

2. Orange Color Meaning

The color of enthusiasm and emotion. Orange exudes warmth and joy and is considered a fun color that provides emotional strength. It is optimistic and upliftning, adds spontaneity and positivity to life and it encourages social communication and creativity. It is a youthful and energetic color.

Read more about the color orange

3. Yellow Color Meaning

The color of happiness and optimism. Yellow is a cheerful and energetic color that brings fun and joy to the world. It makes learning easier as it affects the logical part of the brain, stimulating mentality and perception. It inspires thought and curiosity and boosts enthusiasm and confidence.

Read more about the color yellow

4. Green Color Meaning

The color of harmony and health. Green is a generous, relaxing color that revitalizes our body and mind. It balances our emotions and leaves us feeling safe and secure. It also gives us hope, with promises of growth and prosperity, and it provides a little bit of luck to help us along the way.

Read more about the color green

5. Turquoise Color Meaning

The color of calmness and clarity. Turquoise stabilizes emotions and increases empathy and compassion. It emits a cool calming peace, gives us a boost of positive mental energy that improves concentration and clarifies our mind, and creates a balance that clears the path to spiritual growth.

Read more about the color turquoise

6. Blue Color Meaning

The color of trust and loyalty. Blue has a calming and relaxing effect on our psyche, that gives us peace and makes us feel confident and secure. It dislikes confrontation and too much attention, but it is an honest, reliable and responsible color and you can always count on its support.

Read more about the color blue

7. Purple Color Meaning

The color of spirituality and imagination. Purple inspires us to divulge our innermost thoughts, which enlightens us with wisdom of who we are and encourages spiritual growth. It is often associated with royalty and luxury, and its mystery and magic sparks creative fantasies.

Read more about the color purple

8. Pink Color Meaning

The color of love and compassion. Pink is kind and comforting, full of sympathy and compassion, and makes us feel accepted. Its friendly, playful spirit calms and nurtures us, bringing joy and warmth into our lives. Pink is also a feminine and intuitive color that is bursting with pure romance.

Read more about the color pink

9. Brown Color Meaning

The color of stability and reliability. Brown is dependable and comforting. A great counselor and friend full of wisdom. You can count on its help if you need an honest opinion, support and protection. It stabilizes us, helps us stay grounded and inspires us to appreciate the simple things in life.

Read more about the color brown

10. Black Color Meaning

The color of power and sophistication. Black is an incredibly strong and intimidating color that exudes authority and makes us feel secure and protected. Often seen at formal and prestigious events, this mysterious marvel arouses and seduces our senses with its elegance and sexiness.

Read more about the color black

11. Gray Color Meaning

The color of compromise and control. Gray is neutral, conservative and unemotional. It is practically solid as a rock, making it incredibly stable, reliable and calming. It has a peaceful, relaxing and soothing presence. Gray avoids attention but offers mature, insightful advice to anyone who asks.

Read more about the color gray

12. White Color Meaning

The color of purity and innocence. White is a true balance of all colors and is associated with cleanliness, simplicity and perfection. It loves to make others feel good and provides hope and clarity by refreshing and purifying the mind. It also promotes open-mindedness and self-reflection.

Read more about the color white

Explore the World of Colors

Colors are an incredibly detailed subject and an endless source of fascination. If you want to learn more about color meanings and how they affect us every day, check out this color symbolism chart . If you’re interested in learning how to pick the best colors for your designs, begin reading about color theory . If you need to find some color codes for a design, here’s a list of colors with color names .

Thanks for visiting – enjoy your stay!

Colors, like features, follow the changes of the emotions. – Pablo Picasso

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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 Commented 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.

A canonical form means a naturally unique representation of the element

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.

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 Commented Nov 2, 2023 at 14:47

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Stonehenge’s ‘altar stone’ originally…

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Stonehenge’s ‘altar stone’ originally came from scotland and not wales, new research shows.

FILE – The world heritage site of Stonehenge is seen in Wiltshire, England on Dec. 17, 2013. (AP Photo/Alastair Grant, File)

In this photo provided by researchers in August 2024, Stonehenge’s Altar Stone lies underneath two Sarsen stones in Wiltshire, England. (Nick Pearce/Aberystwyth University via AP)

WASHINGTON (AP) — The ancient ritual meaning of Stonehenge is still a mystery, but researchers are one step closer to understanding how the famous stone circle was created.

The unique stone lying flat at the center of the monument was brought to the site in southern England from near the tip of northeast Scotland, researchers reported Wednesday in the journal Nature. It’s not clear whether the 16-foot (5-meter) stone was carried by boat or across land — a journey of more than 460 miles (740 kilometers).

“It’s a surprise that it’s come from so far away,” said University of Exeter archaeologist Susan Greaney, who was not involved in the study.

For more than a hundred years, scientists believed that Stonehenge’s central sandstone slab — long called the “altar stone” — came from much closer Wales. But a study last year by some of the same researchers showed that the stone didn’t match the geology of Wales’ sandstone formations. The actual source of the stone remained unknown until now.

For the study, the team was not permitted to chip away rocks at the site, but instead analyzed minerals in bits of rock that had been collected in previous digs, some dating back to the 1840s. They found a match in the sandstone formations of Orcadian Basin in northeast Scotland, a region that includes parts of the tip of the Scottish peninsula as well as the Orkney Islands.

“That geological ‘fingerprint’ isn’t repeated in any other area of sediment in the U.K.,” said Aberystwyth University geologist Nick Pearce, a study co-author.

Greaney said the difficult logistics of moving the stone such a long distance show a high level of coordination and cultural connection between these two regions of ancient Britain.

Stonehenge was constructed around 5,000 years ago, with stones forming different circles brought to the site at different times. The placement of stones allows for the sun to rise through a stone “window” during summer solstice. The ancient purpose of the altar stone — which lies flat at the heart of Stonehenge, now beneath other rocks — remains a mystery.

“Stonehenge isn’t a settlement site, but a place of ceremony or ritual,” said Heather Sebire, senior curator at English Heritage, who was not involved in the study. She said that past archaeological excavations had not uncovered evidence of feasting or daily living at the site.

Previous research has shown cultural connections — such as similarities in pottery styles — between the area around Stonehenge and Scotland’s Orkney Islands. Other stones at Stonehenge came from western Wales.

While Britain is dotted with other Neolithic stone circles, “the thing that’s unique about Stonehenge is the distance from which the stones have been sourced,” said Aberystwyth University’s Richard Bevins, a study co-author.

The Associated Press Health and Science Department receives support from the Howard Hughes Medical Institute’s Science and Educational Media Group. The AP is solely responsible for all content.

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I have not been the biggest fan of Kamala Harris, but to my surprise, the candidate who underwhelmed in 2020 is gone. Since she became the presumptive Democratic nominee, I have watched all of candidate Harris’s public appearances for a sense of how she intends to run and possibly govern. The audiences have been vastly different — among them, the annual conclave of Zeta Phi Beta sorority , a National Federation of Teachers convention and the Philadelphia rally where Tim Walz ticket made his first official appearance as her running mate.

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Uniqueness of binary representation

how to prove that binary number is always represent unique decimal number? (uniqueness of binary system) i.e every binary number determines unique decimal number i.e there is one-to-one relationship between binary and decimal.

• number-systems

• $\begingroup$ Before you state there is a one-to-one correspondence, please be aware of the points like 10.0000000 and 1.11111111. Both of them are 2 in decimal rep. So one should not expect a completely 1-1 correspondence. $\endgroup$ –  Zheng Liu Commented Sep 7, 2015 at 5:10
• $\begingroup$ Not for decimal place value... $\endgroup$ –  LOKENDRA SINGH RATHORE Commented Sep 7, 2015 at 5:12
• $\begingroup$ Yes.The definition of the real number system is usually done without any reference to a number-base.Every real number has a representation in any number-base, base 2 or base ten ,or any other base.It's the same set : the real numbers. $\endgroup$ –  DanielWainfleet Commented Sep 7, 2015 at 5:13
• $\begingroup$ Possible duplicate of Binary expansion Unique $\endgroup$ –  David K Commented Jan 19, 2016 at 22:08

To prove that binary numbers represent unique decimal numbers, simply prove that two unequal sequences of bits will always produce different decimal numbers. This can be done with inequalities.

Define two numbers $b \ne c$ and give their binary representations via:

$b = b_1b_2\cdots b_n$

$c = c_1c_2\cdots c_n$

Define $k$ such that $k$ is the least number such that $b_k \ne c_k$

Let $d = (b_1b_2\cdots b_{k-1}) \times 2^{n-k+2} = (c_1c_2\cdots c_{k-1}) \times 2^{n-k+2}$.

Then define $b' = b - d$, $c' = c - d$

WLOG, say $b_k = 1, c_k = 0$.

The number $c_{k+1}\cdots c_n$ is at most $2^{n-k+1}-1$, while the value $b_k = 2^{n-k+1}$.

This shows that $b' > c' \Rightarrow b' \ne c'$ for any difference in the sequence of bits. They will always yield unique decimal numbers.

Thus, $b \ne c$ since $b' + d \ne c' + d$.

The proof of decimal numbers yielding unique bit sequences uses the same logic.

EDIT: In light of Zheng's comment, this is good for a finite decimal expansion. If the decimal expansion is infinite, it can end up not being a one-to-one.

• $\begingroup$ Any Reference.......or suggestion of reading $\endgroup$ –  LOKENDRA SINGH RATHORE Commented Sep 7, 2015 at 5:22

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