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In computer science, a sorting algorithm is an algorithm that puts elements of a list into an order. The most frequently used orders are numerical order and lexicographical order, and either ascending or descending. Efficient sorting is important for optimizing the efficiency of other algorithms (such as search and merge algorithms) that require input data to be in sorted lists. Sorting is also often useful for canonicalizing data and for producing human-readable output.
Formally, the output of any sorting algorithm must satisfy two conditions:
The output is in monotonic order (each element is no smaller/larger than the previous element, according to the required order).
The output is a permutation (a reordering, yet retaining all of the original elements) of the input.
Although some algorithms are designed for sequential access, the highest-performing algorithms assume data is stored in a data structure which allows random access.
== History and concepts ==
From the beginning of computing, the sorting problem has attracted a great deal of research, perhaps due to the complexity of solving it efficiently despite its simple, familiar statement. Among the authors of early sorting algorithms around 1951 was Betty Holberton, who worked on ENIAC and UNIVAC. Bubble sort was analyzed as early as 1956. Asymptotically optimal algorithms have been known since the mid-20th century – new algorithms are still being invented, with the widely used Timsort dating to 2002, and the library sort being first published in 2006.
Comparison sorting algorithms have a fundamental requirement of Ω(n log n) comparisons (some input sequences will require a multiple of n log n comparisons, where n is the number of elements in the array to be sorted). Algorithms not based on comparisons, such as counting sort, can have better performance.
Sorting algorithms are prevalent in introductory computer science classes, where the abundance of algorithms for the problem provides a gentle introduction to a variety of core algorithm concepts, such as big O notation, divide-and-conquer algorithms, data structures such as heaps and binary trees, randomized algorithms, best, worst and average case analysis, time–space tradeoffs, and upper and lower bounds.
Sorting small arrays optimally (in the fewest comparisons and swaps) or fast (i.e. taking into account machine-specific details) is still an open research problem, with solutions only known for very small arrays (<20 elements). Similarly optimal (by various definitions) sorting on a parallel machine is an open research topic.
== Classification ==
Sorting algorithms can be classified by:
Computational complexity
Best, worst and average case behavior in terms of the size of the list. For typical serial sorting algorithms, good behavior is O(n log n), with parallel sort in O(log2 n), and bad behavior is O(n2). Ideal behavior for a serial sort is O(n), but this is not possible in the average case. Optimal parallel sorting is O(log n).
Swaps for "in-place" algorithms.
Memory usage (and use of other computer resources). In particular, some sorting algorithms are "in-place". Strictly, an in-place sort needs only O(1) memory beyond the items being sorted; sometimes O(log n) additional memory is considered "in-place".
Recursion: Some algorithms are either recursive or non-recursive, while others may be both (e.g., merge sort).
Stability: stable sorting algorithms maintain the relative order of records with equal keys (i.e., values).
Whether or not they are a comparison sort. A comparison sort examines the data only by comparing two elements with a comparison operator.
General method: insertion, exchange, selection, merging, etc. Exchange sorts include bubble sort and quicksort. Selection sorts include cycle sort and heapsort.
Whether the algorithm is serial or parallel. The remainder of this discussion almost exclusively concentrates on serial algorithms and assumes serial operation.
Adaptability: Whether or not the presortedness of the input affects the running time. Algorithms that take this into account are known to be adaptive.
Online: An algorithm such as Insertion Sort that is online can sort a constant stream of input.
=== Stability ===
Stable sort algorithms sort equal elements in the same order that they appear in the input. For example, in the card sorting example to the right, the cards are being sorted by their rank, and their suit is being ignored. This allows the possibility of multiple different correctly sorted versions of the original list. Stable sorting algorithms choose one of these, according to the following rule: if two items compare as equal (like the two 5 cards), then their relative order will be preserved, i.e. if one comes before the other in the input, it will come before the other in the output.
Stability is important to preserve order over multiple sorts on the same data set. For example, say that student records consisting of name and class section are sorted dynamically, first by name, then by class section. If a stable sorting algorithm is used in both cases, the sort-by-class-section operation will not change the name order; with an unstable sort, it could be that sorting by section shuffles the name order, resulting in a nonalphabetical list of students.
More formally, the data being sorted can be represented as a record or tuple of values, and the part of the data that is used for sorting is called the key. In the card example, cards are represented as a record (rank, suit), and the key is the rank. A sorting algorithm is stable if whenever there are two records R and S with the same key, and R appears before S in the original list, then R will always appear before S in the sorted list.
When equal elements are indistinguishable, such as with integers, or more generally, any data where the entire element is the key, stability is not an issue. Stability is also not an issue if all keys are different.
Unstable sorting algorithms can be specially implemented to be stable. One way of doing this is to artificially extend the key comparison so that comparisons between two objects with otherwise equal keys are decided using the order of the entries in the original input list as a tie-breaker. Remembering this order, however, may require additional time and space.
One application for stable sorting algorithms is sorting a list using a primary and secondary key. For example, suppose we wish to sort a hand of cards such that the suits are in the order clubs (♣), diamonds (♦), hearts (♥), spades (♠), and within each suit, the cards are sorted by rank. This can be done by first sorting the cards by rank (using any sort), and then doing a stable sort by suit:
Within each suit, the stable sort preserves the ordering by rank that was already done. This idea can be extended to any number of keys and is utilised by radix sort. The same effect can be achieved with an unstable sort by using a lexicographic key comparison, which, e.g., compares first by suit, and then compares by rank if the suits are the same.
== Comparison of algorithms ==
This analysis assumes that the length of each key is constant and that all comparisons, swaps and other operations can proceed in constant time.
Legend:
n is the number of records to be sorted.
Comparison column has the following ranking classifications: "Best", "Average" and "Worst" if the time complexity is given for each case.
"Memory" denotes the amount of additional storage required by the algorithm.
The run times and the memory requirements listed are inside big O notation, hence the base of the logarithms does not matter.
The notation log2 n means (log n)2.
=== Comparison sorts ===
Below is a table of comparison sorts. Mathematical analysis demonstrates a comparison sort cannot perform better than O(n log n) on average.
=== Non-comparison sorts ===
The following table describes integer sorting algorithms and other sorting algorithms that are not comparison sorts. These algorithms are not limited to Ω(n log n) unless meet unit-cost random-access machine model as described below.
Complexities below assume n items to be sorted, with keys of size k, digit size d, and r the range of numbers to be sorted.
Many of them are based on the assumption that the key size is large enough that all entries have unique key values, and hence that n ≪ 2k, where ≪ means "much less than".
In the unit-cost random-access machine model, algorithms with running time of
n
⋅
k
d
{\displaystyle \scriptstyle n\cdot {\frac {k}{d}}}
, such as radix sort, still take time proportional to Θ(n log n), because n is limited to be not more than
2
k
d
{\displaystyle 2^{\frac {k}{d}}}
, and a larger number of elements to sort would require a bigger k in order to store them in the memory.
Samplesort can be used to parallelize any of the non-comparison sorts, by efficiently distributing data into several buckets and then passing down sorting to several processors, with no need to merge as buckets are already sorted between each other.
=== Others ===
Some algorithms are slow compared to those discussed above, such as the bogosort with unbounded run time and the stooge sort which has O(n2.7) run time. These sorts are usually described for educational purposes to demonstrate how the run time of algorithms is estimated. The following table describes some sorting algorithms that are impractical for real-life use in traditional software contexts due to extremely poor performance or specialized hardware requirements.
Theoretical computer scientists have detailed other sorting algorithms that provide better than O(n log n) time complexity assuming additional constraints, including:
Thorup's algorithm, a randomized algorithm for sorting keys from a domain of finite size, taking O(n log log n) time and O(n) space.
A randomized integer sorting algorithm taking
O
(
n
log
log
n
)
{\displaystyle O\left(n{\sqrt {\log \log n}}\right)}
expected time and O(n) space.
One of the authors of the previously mentioned algorithm also claims to have discovered an algorithm taking
O
(
n
log
n
)
{\displaystyle O\left(n{\sqrt {\log n}}\right)}
time and O(n) space, sorting real numbers, and further claims that, without any added assumptions on the input, it can be modified to achieve
O
(
n
log
n
/
log
log
n
)
{\displaystyle O\left(n\log n/{\sqrt {\log \log n}}\right)}
time and O(n) space.
== Popular sorting algorithms ==
While there are a large number of sorting algorithms, in practical implementations a few algorithms predominate. Insertion sort is widely used for small data sets, while for large data sets an asymptotically efficient sort is used, primarily heapsort, merge sort, or quicksort. Efficient implementations generally use a hybrid algorithm, combining an asymptotically efficient algorithm for the overall sort with insertion sort for small lists at the bottom of a recursion. Highly tuned implementations use more sophisticated variants, such as Timsort (merge sort, insertion sort, and additional logic), used in Android, Java, and Python, and introsort (quicksort and heapsort), used (in variant forms) in some C++ sort implementations and in .NET.
For more restricted data, such as numbers in a fixed interval, distribution sorts such as counting sort or radix sort are widely used. Bubble sort and variants are rarely used in practice, but are commonly found in teaching and theoretical discussions.
When physically sorting objects (such as alphabetizing papers, tests or books) people intuitively generally use insertion sorts for small sets. For larger sets, people often first bucket, such as by initial letter, and multiple bucketing allows practical sorting of very large sets. Often space is relatively cheap, such as by spreading objects out on the floor or over a large area, but operations are expensive, particularly moving an object a large distance – locality of reference is important. Merge sorts are also practical for physical objects, particularly as two hands can be used, one for each list to merge, while other algorithms, such as heapsort or quicksort, are poorly suited for human use. Other algorithms, such as library sort, a variant of insertion sort that leaves spaces, are also practical for physical use.
=== Simple sorts ===
Two of the simplest sorts are insertion sort and selection sort, both of which are efficient on small data, due to low overhead, but not efficient on large data. Insertion sort is generally faster than selection sort in practice, due to fewer comparisons and good performance on almost-sorted data, and thus is preferred in practice, but selection sort uses fewer writes, and thus is used when write performance is a limiting factor.
==== Insertion sort ====
Insertion sort is a simple sorting algorithm that is relatively efficient for small lists and mostly sorted lists, and is often used as part of more sophisticated algorithms. It works by taking elements from the list one by one and inserting them in their correct position into a new sorted list similar to how one puts money in their wallet. In arrays, the new list and the remaining elements can share the array's space, but insertion is expensive, requiring shifting all following elements over by one. Shellsort is a variant of insertion sort that is more efficient for larger lists.
==== Selection sort ====
Selection sort is an in-place comparison sort. It has O(n2) complexity, making it inefficient on large lists, and generally performs worse than the similar insertion sort. Selection sort is noted for its simplicity and also has performance advantages over more complicated algorithms in certain situations.
The algorithm finds the minimum value, swaps it with the value in the first position, and repeats these steps for the remainder of the list. It does no more than n swaps and thus is useful where swapping is very expensive.
=== Efficient sorts ===
Practical general sorting algorithms are almost always based on an algorithm with average time complexity (and generally worst-case complexity) O(n log n), of which the most common are heapsort, merge sort, and quicksort. Each has advantages and drawbacks, with the most significant being that simple implementation of merge sort uses O(n) additional space, and simple implementation of quicksort has O(n2) worst-case complexity. These problems can be solved or ameliorated at the cost of a more complex algorithm.
While these algorithms are asymptotically efficient on random data, for practical efficiency on real-world data various modifications are used. First, the overhead of these algorithms becomes significant on smaller data, so often a hybrid algorithm is used, commonly switching to insertion sort once the data is small enough. Second, the algorithms often perform poorly on already sorted data or almost sorted data – these are common in real-world data and can be sorted in O(n) time by appropriate algorithms. Finally, they may also be unstable, and stability is often a desirable property in a sort. Thus more sophisticated algorithms are often employed, such as Timsort (based on merge sort) or introsort (based on quicksort, falling back to heapsort).
==== Merge sort ====
Merge sort takes advantage of the ease of merging already sorted lists into a new sorted list. It starts by comparing every two elements (i.e., 1 with 2, then 3 with 4...) and swapping them if the first should come after the second. It then merges each of the resulting lists of two into lists of four, then merges those lists of four, and so on; until at last two lists are merged into the final sorted list. Of the algorithms described here, this is the first that scales well to very large lists, because its worst-case running time is O(n log n). It is also easily applied to lists, not only arrays, as it only requires sequential access, not random access. However, it has additional O(n) space complexity and involves a large number of copies in simple implementations.
Merge sort has seen a relatively recent surge in popularity for practical implementations, due to its use in the sophisticated algorithm Timsort, which is used for the standard sort routine in the programming languages Python and Java (as of JDK7). Merge sort itself is the standard routine in Perl, among others, and has been used in Java at least since 2000 in JDK1.3.
==== Heapsort ====
Heapsort is a much more efficient version of selection sort. It also works by determining the largest (or smallest) element of the list, placing that at the end (or beginning) of the list, then continuing with the rest of the list, but accomplishes this task efficiently by using a data structure called a heap, a special type of binary tree. Once the data list has been made into a heap, the root node is guaranteed to be the largest (or smallest) element. When it is removed and placed at the end of the list, the heap is rearranged so the largest element remaining moves to the root. Using the heap, finding the next largest element takes O(log n) time, instead of O(n) for a linear scan as in simple selection sort. This allows Heapsort to run in O(n log n) time, and this is also the worst-case complexity.
==== Recombinant sort ====
Recombinant sort is a non-comparison-based sorting algorithm developed by Peeyush Kumar et al in 2020. The algorithm combines bucket sort, counting sort, radix sort, hashing, and dynamic programming techniques. It employs an n-dimensional Cartesian space mapping approach consisting of two primary phases: a Hashing cycle that maps elements to a multidimensional array using a special hash function, and an Extraction cycle that retrieves elements in sorted order. Recombinant Sort achieves O(n) time complexity for best, average, and worst cases, and can process both numerical and string data types, including mixed decimal and non-decimal numbers.
==== Quicksort ====
Quicksort is a divide-and-conquer algorithm which relies on a partition operation: to partition an array, an element called a pivot is selected. All elements smaller than the pivot are moved before it and all greater elements are moved after it. This can be done efficiently in linear time and in-place. The lesser and greater sublists are then recursively sorted. This yields an average time complexity of O(n log n), with low overhead, and thus this is a popular algorithm. Efficient implementations of quicksort (with in-place partitioning) are typically unstable sorts and somewhat complex but are among the fastest sorting algorithms in practice. Together with its modest O(log n) space usage, quicksort is one of the most popular sorting algorithms and is available in many standard programming libraries.
The important caveat about quicksort is that its worst-case performance is O(n2); while this is rare, in naive implementations (choosing the first or last element as pivot) this occurs for sorted data, which is a common case. The most complex issue in quicksort is thus choosing a good pivot element, as consistently poor choices of pivots can result in drastically slower O(n2) performance, but good choice of pivots yields O(n log n) performance, which is asymptotically optimal. For example, if at each step the median is chosen as the pivot then the algorithm works in O(n log n). Finding the median, such as by the median of medians selection algorithm is however an O(n) operation on unsorted lists and therefore exacts significant overhead with sorting. In practice choosing a random pivot almost certainly yields O(n log n) performance.
If a guarantee of O(n log n) performance is important, there is a simple modification to achieve that. The idea, due to Musser, is to set a limit on the maximum depth of recursion. If that limit is exceeded, then sorting is continued using the heapsort algorithm. Musser proposed that the limit should be
1
+
2
⌊
log
2
(
n
)
⌋
{\displaystyle 1+2\lfloor \log _{2}(n)\rfloor }
, which is approximately twice the maximum recursion depth one would expect on average with a randomly ordered array.
==== Shellsort ====
Shellsort was invented by Donald Shell in 1959. It improves upon insertion sort by moving out of order elements more than one position at a time. The concept behind Shellsort is that insertion sort performs in
O
(
k
n
)
{\displaystyle O(kn)}
time, where k is the greatest distance between two out-of-place elements. This means that generally, they perform in O(n2), but for data that is mostly sorted, with only a few elements out of place, they perform faster. So, by first sorting elements far away, and progressively shrinking the gap between the elements to sort, the final sort computes much faster. One implementation can be described as arranging the data sequence in a two-dimensional array and then sorting the columns of the array using insertion sort.
The worst-case time complexity of Shellsort is an open problem and depends on the gap sequence used, with known complexities ranging from O(n2) to O(n4/3) and Θ(n log2 n). This, combined with the fact that Shellsort is in-place, only needs a relatively small amount of code, and does not require use of the call stack, makes it is useful in situations where memory is at a premium, such as in embedded systems and operating system kernels.
=== Bubble sort and variants ===
Bubble sort, and variants such as the Comb sort and cocktail sort, are simple, highly inefficient sorting algorithms. They are frequently seen in introductory texts due to ease of analysis, but they are rarely used in practice.
==== Bubble sort ====
Bubble sort is a simple sorting algorithm. The algorithm starts at the beginning of the data set. It compares the first two elements, and if the first is greater than the second, it swaps them. It continues doing this for each pair of adjacent elements to the end of the data set. It then starts again with the first two elements, repeating until no swaps have occurred on the last pass. This algorithm's average time and worst-case performance is O(n2), so it is rarely used to sort large, unordered data sets. Bubble sort can be used to sort a small number of items (where its asymptotic inefficiency is not a high penalty). Bubble sort can also be used efficiently on a list of any length that is nearly sorted (that is, the elements are not significantly out of place). For example, if any number of elements are out of place by only one position (e.g. 0123546789 and 1032547698), bubble sort's exchange will get them in order on the first pass, the second pass will find all elements in order, so the sort will take only 2n time.
==== Comb sort ====
Comb sort is a relatively simple sorting algorithm based on bubble sort and originally designed by Włodzimierz Dobosiewicz in 1980. It was later rediscovered and popularized by Stephen Lacey and Richard Box with a Byte Magazine article published in April 1991. The basic idea is to eliminate turtles, or small values near the end of the list, since in a bubble sort these slow the sorting down tremendously. (Rabbits, large values around the beginning of the list, do not pose a problem in bubble sort) It accomplishes this by initially swapping elements that are a certain distance from one another in the array, rather than only swapping elements if they are adjacent to one another, and then shrinking the chosen distance until it is operating as a normal bubble sort. Thus, if Shellsort can be thought of as a generalized version of insertion sort that swaps elements spaced a certain distance away from one another, comb sort can be thought of as the same generalization applied to bubble sort.
==== Exchange sort ====
Exchange sort is sometimes confused with bubble sort, although the algorithms are in fact distinct. Exchange sort works by comparing the first element with all elements above it, swapping where needed, thereby guaranteeing that the first element is correct for the final sort order; it then proceeds to do the same for the second element, and so on. It lacks the advantage that bubble sort has of detecting in one pass if the list is already sorted, but it can be faster than bubble sort by a constant factor (one less pass over the data to be sorted; half as many total comparisons) in worst-case situations. Like any simple O(n2) sort it can be reasonably fast over very small data sets, though in general insertion sort will be faster.
=== Distribution sorts ===
Distribution sort refers to any sorting algorithm where data is distributed from their input to multiple intermediate structures which are then gathered and placed on the output. For example, both bucket sort and flashsort are distribution-based sorting algorithms. Distribution sorting algorithms can be used on a single processor, or they can be a distributed algorithm, where individual subsets are separately sorted on different processors, then combined. This allows external sorting of data too large to fit into a single computer's memory.
==== Counting sort ====
Counting sort is applicable when each input is known to belong to a particular set, S, of possibilities. The algorithm runs in O(|S| + n) time and O(|S|) memory where n is the length of the input. It works by creating an integer array of size |S| and using the ith bin to count the occurrences of the ith member of S in the input. Each input is then counted by incrementing the value of its corresponding bin. Afterward, the counting array is looped through to arrange all of the inputs in order. This sorting algorithm often cannot be used because S needs to be reasonably small for the algorithm to be efficient, but it is extremely fast and demonstrates great asymptotic behavior as n increases. It also can be modified to provide stable behavior.
==== Bucket sort ====
Bucket sort is a divide-and-conquer sorting algorithm that generalizes counting sort by partitioning an array into a finite number of buckets. Each bucket is then sorted individually, either using a different sorting algorithm or by recursively applying the bucket sorting algorithm.
A bucket sort works best when the elements of the data set are evenly distributed across all buckets.
==== Radix sort ====
Radix sort is an algorithm that sorts numbers by processing individual digits. n numbers consisting of k digits each are sorted in O(n · k) time. Radix sort can process digits of each number either starting from the least significant digit (LSD) or starting from the most significant digit (MSD). The LSD algorithm first sorts the list by the least significant digit while preserving their relative order using a stable sort. Then it sorts them by the next digit, and so on from the least significant to the most significant, ending up with a sorted list. While the LSD radix sort requires the use of a stable sort, the MSD radix sort algorithm does not (unless stable sorting is desired). In-place MSD radix sort is not stable. It is common for the counting sort algorithm to be used internally by the radix sort. A hybrid sorting approach, such as using insertion sort for small bins, improves performance of radix sort significantly.
== Memory usage patterns and index sorting ==
When the size of the array to be sorted approaches or exceeds the available primary memory, so that (much slower) disk or swap space must be employed, the memory usage pattern of a sorting algorithm becomes important, and an algorithm that might have been fairly efficient when the array fit easily in RAM may become impractical. In this scenario, the total number of comparisons becomes (relatively) less important, and the number of times sections of memory must be copied or swapped to and from the disk can dominate the performance characteristics of an algorithm. Thus, the number of passes and the localization of comparisons can be more important than the raw number of comparisons, since comparisons of nearby elements to one another happen at system bus speed (or, with caching, even at CPU speed), which, compared to disk speed, is virtually instantaneous.
For example, the popular recursive quicksort algorithm provides quite reasonable performance with adequate RAM, but due to the recursive way that it copies portions of the array it becomes much less practical when the array does not fit in RAM, because it may cause a number of slow copy or move operations to and from disk. In that scenario, another algorithm may be preferable even if it requires more total comparisons.
One way to work around this problem, which works well when complex records (such as in a relational database) are being sorted by a relatively small key field, is to create an index into the array and then sort the index, rather than the entire array. (A sorted version of the entire array can then be produced with one pass, reading from the index, but often even that is unnecessary, as having the sorted index is adequate.) Because the index is much smaller than the entire array, it may fit easily in memory where the entire array would not, effectively eliminating the disk-swapping problem. This procedure is sometimes called "tag sort".
Another technique for overcoming the memory-size problem is using external sorting, for example, one of the ways is to combine two algorithms in a way that takes advantage of the strength of each to improve overall performance. For instance, the array might be subdivided into chunks of a size that will fit in RAM, the contents of each chunk sorted using an efficient algorithm (such as quicksort), and the results merged using a k-way merge similar to that used in merge sort. This is faster than performing either merge sort or quicksort over the entire list.
Techniques can also be combined. For sorting very large sets of data that vastly exceed system memory, even the index may need to be sorted using an algorithm or combination of algorithms designed to perform reasonably with virtual memory, i.e., to reduce the amount of swapping required.
== Related algorithms ==
Related problems include approximate sorting (sorting a sequence to within a certain amount of the correct order), partial sorting (sorting only the k smallest elements of a list, or finding the k smallest elements, but unordered) and selection (computing the kth smallest element). These can be solved inefficiently by a total sort, but more efficient algorithms exist, often derived by generalizing a sorting algorithm. The most notable example is quickselect, which is related to quicksort. Conversely, some sorting algorithms can be derived by repeated application of a selection algorithm; quicksort and quickselect can be seen as the same pivoting move, differing only in whether one recurses on both sides (quicksort, divide-and-conquer) or one side (quickselect, decrease-and-conquer).
A kind of opposite of a sorting algorithm is a shuffling algorithm. These are fundamentally different because they require a source of random numbers. Shuffling can also be implemented by a sorting algorithm, namely by a random sort: assigning a random number to each element of the list and then sorting based on the random numbers. This is generally not done in practice, however, and there is a well-known simple and efficient algorithm for shuffling: the Fisher–Yates shuffle.
Sorting algorithms are ineffective for finding an order in many situations. Usually, when elements have no reliable comparison function (crowdsourced preferences like voting systems), comparisons are very costly (sports), or when it would be impossible to pairwise compare all elements for all criteria (search engines). In these cases, the problem is usually referred to as ranking and the goal is to find the "best" result for some criteria according to probabilities inferred from comparisons or rankings. A common example is in chess, where players are ranked with the Elo rating system, and rankings are determined by a tournament system instead of a sorting algorithm.
== See also ==
Collation – Assembly of written information into a standard order
K-sorted sequence
Schwartzian transform – Programming idiom for efficiently sorting a list by a computed key
Search algorithm – Any algorithm which solves the search problem
Quantum sort – Sorting algorithms for quantum computers
== References ==
== Further reading ==
Knuth, Donald E. (1998), Sorting and Searching, The Art of Computer Programming, vol. 3 (2nd ed.), Boston: Addison-Wesley, ISBN 0-201-89685-0
Sedgewick, Robert (1980), "Efficient Sorting by Computer: An Introduction", Computational Probability, New York: Academic Press, pp. 101–130, ISBN 0-12-394680-8
== External links ==
Sorting Algorithm Animations at the Wayback Machine (archived 3 March 2015).
Sequential and parallel sorting algorithms – Explanations and analyses of many sorting algorithms.
Dictionary of Algorithms, Data Structures, and Problems – Dictionary of algorithms, techniques, common functions, and problems.
Slightly Skeptical View on Sorting Algorithms – Discusses several classic algorithms and promotes alternatives to the quicksort algorithm.
15 Sorting Algorithms in 6 Minutes (Youtube) – Visualization and "audibilization" of 15 Sorting Algorithms in 6 Minutes.
A036604 sequence in OEIS database titled "Sorting numbers: minimal number of comparisons needed to sort n elements" – Performed by Ford–Johnson algorithm.
XiSort – External merge sort with symbolic key transformation – A variant of merge sort applied to large datasets using symbolic techniques.
Sorting Algorithms Used on Famous Paintings (Youtube) – Visualization of Sorting Algorithms on Many Famous Paintings.
A Comparison of Sorting Algorithms – Runs a series of tests of 9 of the main sorting algorithms using Python timeit and Google Colab. | Wikipedia/Sorting_algorithm |
Algorithmic entities refer to autonomous algorithms that operate without human control or interference. Recently, attention is being given to the idea of algorithmic entities being granted (partial or full) legal personhood. Professor Shawn Bayern and Professor Lynn M. LoPucki popularized through their papers the idea of having algorithmic entities that obtain legal personhood and the accompanying rights and obligations.
== Legal algorithmic entities ==
Academics and politicians have been discussing over the last few years whether it is possible to have a legal algorithmic entity, meaning that an algorithm or AI is granted legal personhood. In most countries, the law only recognizes natural or real persons and legal persons. The main argument is that behind every legal person (or layers of legal persons), there is eventually a natural person.
In some countries there have been made some exceptions to this in the form of the granting of an environmental personhood to rivers, waterfalls, forests and mountains. In the past, some form of personhood also existed for certain religious constructions such as churches and temples.
Certain countries – albeit for publicity purposes – have shown willingness to grant (some form of) legal personhood to robots. On the 27th of October 2017, Saudi Arabia became to first country in the world to grant citizenship to a robot when it gave “Sophia” a passport. In the same year, official residency status was granted to a chatbot named “Shibuya Mirai” in Tokyo, Japan.
The general consensus is that AI in any case cannot be regarded as a natural or real person and that granting AI (legal) personhood at this stage is unwanted from a societal point of view. However, the academic and public discussions continue as AI software becomes more sophisticated and companies are increasingly implementing artificial intelligence to assist in all aspects of business and society. This leads to some scholars to wonder whether AI should be granted legal personhood as it is not unthinkable to one day have a sophisticated algorithm capable of managing a firm completely independent of human interventions.
Brown argues that the question of whether legal personhood for AI may be granted is tied directly to the issue of whether AI can or should even be allowed to legally own property. Brown “concludes that that legal personhood is the best approach for AI to own personal property.” This is an especially important inquiry since many scholars already recognize AI as having possession and control of some digital assets or even data. AI can also create written text, photo, art, and even algorithms, though ownership of these works is not currently granted to AI in any country because it is not recognized as a legal person.
=== United States ===
Bayern (2016) argues that this is already possible currently under US law. He states, that, in the United States, creating an AI controlled firm without human interference or ownership is already possible under current legislation by creating a “zero member LLC”:
(1) an individual member creates a member-managed LLC, filing the appropriate paperwork with the state; (2) the individual (along, possibly, with the LLC, which is controlled by the sole member) enters into an operating agreement governing the conduct of the LLC; (3) the operating agreement specifies that the LLC will take actions as determined by an autonomous system, specifying terms or conditions as appropriate to achieve the autonomous system’s legal goals; (4) the sole member withdraws from the LLC, leaving the LLC without any members. The result is potentially a perpetual LLC—a new legal person—that requires no ongoing intervention from any preexisting legal person in order to maintain its status.
Sherer (2018) argues – after conducting an analysis on New York's (and other states’) LLC law(s), the Revised Uniform Limited Liability Company Act (RULLCA) and US case law on fundamentals of legal personhood – that this option is not viable, but agrees with Bayern on the existence of a ‘loophole’ whereby an AI system could “effectively control a LLC and thereby have the functional equivalent of legal personhood”. Bayern's loophole of “entity cross-ownership” would work as follows:
(1) Existing person P establishes member-managed LLCs A and B, with identical operating agreements both providing that the entity is controlled by an autonomous system that is not a preexisting legal person; (2) P causes A to be admitted as a member of B and B to be admitted as a member of A; (3) P withdraws from both entities.
Unlike the zero member LLC, the entity cross-ownership would not trigger a response by the law for having a memberless entity as what remains are two entities each having one member. In corporations, this sort of situations is often prevented by formal provisions in the statutes (predominantly for voting rights for shares), however, such limitations do not seem to be in place for LLCs as they are more flexible in arranging control and organization.
=== Europe ===
In Europe, certain academics from different countries have started to look at the possibilities in their respective jurisdictions. Bayern et al. (2017) compared the UK, Germany and Switzerland to the findings of Bayern (2016) earlier for the US to see whether such “loopholes” in the law exist there as well to set up an algorithmic entity.
Some smaller jurisdiction are going further and adapting their laws for the 21st century technological changes. Guernsey has granted (limited) rights to electronic agents and Malta is currently busy creating a robot citizenship test.
While it is unlikely the EU would allow for AI to receive legal personality at this moment, the European Parliament did however request the European Commission in a February 2017 resolution to “creating a specific legal status for robots in the long run, so that at least the most sophisticated autonomous robots could be established as having the status of electronic persons responsible for making good any damage they may cause, and possibly applying electronic personality to cases where robots make autonomous decisions or otherwise interact with third parties independently”.
Not all parts of the supranational European bodies agreed as the European Economic and Social Committee gave in its own initiative an opposing opinion given May 2017: “The EESC is opposed to any form of legal status for robots or AI (systems), as this entails an unacceptable risk of moral hazard. Liability law is based on a preventive, behavior-correcting function, which may disappear as soon as the maker no longer bears the liability risk since this is transferred to the robot (or the AI system). There is also a risk of inappropriate use and abuse of this kind of legal status.”
In reaction to the European Parliament's request, the European Commission set up a High Level Expert Group to tackle issues and take initiative in a number of subjects relating to automation, robotics and AI. The High Level Expert Group released a draft document for AI ethical guidelines and a document defining AI in December 2018. The document on ethical guidelines was opened for consultation and received extensive feedback. The European Commission is taking a careful approach legislating AI by emphasizing on ethics, but at the same time – as the EU is behind in AI research to the United States and China – focusing on how to narrow the gap with competitors by creating a more inviting regulatory framework for AI research and development. Giving (limited) legal personality to AI or even allow certain forms of algorithmic entities might create an extra edge.
== References == | Wikipedia/Algorithmic_entities |
In mathematics, a system of bilinear equations is a special sort of system of polynomial equations, where each equation equates a bilinear form with a constant (possibly zero). More precisely, given two sets of variables represented as coordinate vectors x and y, then each equation of the system can be written
y
T
A
i
x
=
g
i
,
{\displaystyle y^{T}A_{i}x=g_{i},}
where, i is an integer whose value ranges from 1 to the number of equations, each
A
i
{\displaystyle A_{i}}
is a matrix, and each
g
i
{\displaystyle g_{i}}
is a real number. Systems of bilinear equations arise in many subjects including engineering, biology, and statistics.
== See also ==
Systems of linear equations
== References ==
Charles R. Johnson, Joshua A. Link 'Solution theory for complete bilinear systems of equations' - http://onlinelibrary.wiley.com/doi/10.1002/nla.676/abstract
Vinh, Le Anh 'On the solvability of systems of bilinear equations in finite fields' - https://arxiv.org/abs/0903.1156
Yang Dian 'Solution theory for system of bilinear equations' - https://digitalarchive.wm.edu/handle/10288/13726
Scott Cohen and Carlo Tomasi. 'Systems of bilinear equations'. Technical report, Stanford, CA, USA, 1997.- ftp://reports.stanford.edu/public_html/cstr/reports/cs/tr/97/1588/CS-TR-97-1588.pdf | Wikipedia/System_of_bilinear_equations |
A matrix difference equation is a difference equation in which the value of a vector (or sometimes, a matrix) of variables at one point in time is related to its own value at one or more previous points in time, using matrices. The order of the equation is the maximum time gap between any two indicated values of the variable vector. For example,
x
t
=
A
x
t
−
1
+
B
x
t
−
2
{\displaystyle \mathbf {x} _{t}=\mathbf {Ax} _{t-1}+\mathbf {Bx} _{t-2}}
is an example of a second-order matrix difference equation, in which x is an n × 1 vector of variables and A and B are n × n matrices. This equation is homogeneous because there is no vector constant term added to the end of the equation. The same equation might also be written as
x
t
+
2
=
A
x
t
+
1
+
B
x
t
{\displaystyle \mathbf {x} _{t+2}=\mathbf {Ax} _{t+1}+\mathbf {Bx} _{t}}
or as
x
n
=
A
x
n
−
1
+
B
x
n
−
2
{\displaystyle \mathbf {x} _{n}=\mathbf {Ax} _{n-1}+\mathbf {Bx} _{n-2}}
The most commonly encountered matrix difference equations are first-order.
== Nonhomogeneous first-order case and the steady state ==
An example of a nonhomogeneous first-order matrix difference equation is
x
t
=
A
x
t
−
1
+
b
{\displaystyle \mathbf {x} _{t}=\mathbf {Ax} _{t-1}+\mathbf {b} }
with additive constant vector b. The steady state of this system is a value x* of the vector x which, if reached, would not be deviated from subsequently. x* is found by setting xt = xt−1 = x* in the difference equation and solving for x* to obtain
x
∗
=
[
I
−
A
]
−
1
b
{\displaystyle \mathbf {x} ^{*}=[\mathbf {I} -\mathbf {A} ]^{-1}\mathbf {b} }
where I is the n × n identity matrix, and where it is assumed that [I − A] is invertible. Then the nonhomogeneous equation can be rewritten in homogeneous form in terms of deviations from the steady state:
[
x
t
−
x
∗
]
=
A
[
x
t
−
1
−
x
∗
]
{\displaystyle \left[\mathbf {x} _{t}-\mathbf {x} ^{*}\right]=\mathbf {A} \left[\mathbf {x} _{t-1}-\mathbf {x} ^{*}\right]}
== Stability of the first-order case ==
The first-order matrix difference equation [xt − x*] = A[xt−1 − x*] is stable—that is, xt converges asymptotically to the steady state x*—if and only if all eigenvalues of the transition matrix A (whether real or complex) have an absolute value which is less than 1.
== Solution of the first-order case ==
Assume that the equation has been put in the homogeneous form yt = Ayt−1. Then we can iterate and substitute repeatedly from the initial condition y0, which is the initial value of the vector y and which must be known in order to find the solution:
y
1
=
A
y
0
y
2
=
A
y
1
=
A
2
y
0
y
3
=
A
y
2
=
A
3
y
0
{\displaystyle {\begin{aligned}\mathbf {y} _{1}&=\mathbf {Ay} _{0}\\\mathbf {y} _{2}&=\mathbf {Ay} _{1}=\mathbf {A} ^{2}\mathbf {y} _{0}\\\mathbf {y} _{3}&=\mathbf {Ay} _{2}=\mathbf {A} ^{3}\mathbf {y} _{0}\end{aligned}}}
and so forth, so that by mathematical induction the solution in terms of t is
y
t
=
A
t
y
0
{\displaystyle \mathbf {y} _{t}=\mathbf {A} ^{t}\mathbf {y} _{0}}
Further, if A is diagonalizable, we can rewrite A in terms of its eigenvalues and eigenvectors, giving the solution as
y
t
=
P
D
t
P
−
1
y
0
,
{\displaystyle \mathbf {y} _{t}=\mathbf {PD} ^{t}\mathbf {P} ^{-1}\mathbf {y} _{0},}
where P is an n × n matrix whose columns are the eigenvectors of A (assuming the eigenvalues are all distinct) and D is an n × n diagonal matrix whose diagonal elements are the eigenvalues of A. This solution motivates the above stability result: At shrinks to the zero matrix over time if and only if the eigenvalues of A are all less than unity in absolute value.
== Extracting the dynamics of a single scalar variable from a first-order matrix system ==
Starting from the n-dimensional system yt = Ayt−1, we can extract the dynamics of one of the state variables, say y1. The above solution equation for yt shows that the solution for y1,t is in terms of the n eigenvalues of A. Therefore the equation describing the evolution of y1 by itself must have a solution involving those same eigenvalues. This description intuitively motivates the equation of evolution of y1, which is
y
1
,
t
=
a
1
y
1
,
t
−
1
+
a
2
y
1
,
t
−
2
+
⋯
+
a
n
y
1
,
t
−
n
{\displaystyle y_{1,t}=a_{1}y_{1,t-1}+a_{2}y_{1,t-2}+\dots +a_{n}y_{1,t-n}}
where the parameters ai are from the characteristic equation of the matrix A:
λ
n
−
a
1
λ
n
−
1
−
a
2
λ
n
−
2
−
⋯
−
a
n
λ
0
=
0.
{\displaystyle \lambda ^{n}-a_{1}\lambda ^{n-1}-a_{2}\lambda ^{n-2}-\dots -a_{n}\lambda ^{0}=0.}
Thus each individual scalar variable of an n-dimensional first-order linear system evolves according to a univariate nth-degree difference equation, which has the same stability property (stable or unstable) as does the matrix difference equation.
== Solution and stability of higher-order cases ==
Matrix difference equations of higher order—that is, with a time lag longer than one period—can be solved, and their stability analyzed, by converting them into first-order form using a block matrix (matrix of matrices). For example, suppose we have the second-order equation
x
t
=
A
x
t
−
1
+
B
x
t
−
2
{\displaystyle \mathbf {x} _{t}=\mathbf {Ax} _{t-1}+\mathbf {Bx} _{t-2}}
with the variable vector x being n × 1 and A and B being n × n. This can be stacked in the form
[
x
t
x
t
−
1
]
=
[
A
B
I
0
]
[
x
t
−
1
x
t
−
2
]
{\displaystyle {\begin{bmatrix}\mathbf {x} _{t}\\\mathbf {x} _{t-1}\\\end{bmatrix}}={\begin{bmatrix}\mathbf {A} &\mathbf {B} \\\mathbf {I} &\mathbf {0} \\\end{bmatrix}}{\begin{bmatrix}\mathbf {x} _{t-1}\\\mathbf {x} _{t-2}\end{bmatrix}}}
where I is the n × n identity matrix and 0 is the n × n zero matrix. Then denoting the 2n × 1 stacked vector of current and once-lagged variables as zt and the 2n × 2n block matrix as L, we have as before the solution
z
t
=
L
t
z
0
{\displaystyle \mathbf {z} _{t}=\mathbf {L} ^{t}\mathbf {z} _{0}}
Also as before, this stacked equation, and thus the original second-order equation, are stable if and only if all eigenvalues of the matrix L are smaller than unity in absolute value.
== Nonlinear matrix difference equations: Riccati equations ==
In linear-quadratic-Gaussian control, there arises a nonlinear matrix equation for the reverse evolution of a current-and-future-cost matrix, denoted below as H. This equation is called a discrete dynamic Riccati equation, and it arises when a variable vector evolving according to a linear matrix difference equation is controlled by manipulating an exogenous vector in order to optimize a quadratic cost function. This Riccati equation assumes the following, or a similar, form:
H
t
−
1
=
K
+
A
′
H
t
A
−
A
′
H
t
C
[
C
′
H
t
C
+
R
]
−
1
C
′
H
t
A
{\displaystyle \mathbf {H} _{t-1}=\mathbf {K} +\mathbf {A} '\mathbf {H} _{t}\mathbf {A} -\mathbf {A} '\mathbf {H} _{t}\mathbf {C} \left[\mathbf {C} '\mathbf {H} _{t}\mathbf {C} +\mathbf {R} \right]^{-1}\mathbf {C} '\mathbf {H} _{t}\mathbf {A} }
where H, K, and A are n × n, C is n × k, R is k × k, n is the number of elements in the vector to be controlled, and k is the number of elements in the control vector. The parameter matrices A and C are from the linear equation, and the parameter matrices K and R are from the quadratic cost function. See here for details.
In general this equation cannot be solved analytically for Ht in terms of t; rather, the sequence of values for Ht is found by iterating the Riccati equation. However, it has been shown that this Riccati equation can be solved analytically if R = 0 and n = k + 1, by reducing it to a scalar rational difference equation; moreover, for any k and n if the transition matrix A is nonsingular then the Riccati equation can be solved analytically in terms of the eigenvalues of a matrix, although these may need to be found numerically.
In most contexts the evolution of H backwards through time is stable, meaning that H converges to a particular fixed matrix H* which may be irrational even if all the other matrices are rational. See also Stochastic control § Discrete time.
A related Riccati equation is
X
t
+
1
=
−
[
E
+
B
X
t
]
[
C
+
A
X
t
]
−
1
{\displaystyle \mathbf {X} _{t+1}=-\left[\mathbf {E} +\mathbf {B} \mathbf {X} _{t}\right]\left[\mathbf {C} +\mathbf {A} \mathbf {X} _{t}\right]^{-1}}
in which the matrices X, A, B, C, E are all n × n. This equation can be solved explicitly. Suppose
X
t
=
N
t
D
t
−
1
,
{\displaystyle \mathbf {X} _{t}=\mathbf {N} _{t}\mathbf {D} _{t}^{-1},}
which certainly holds for t = 0 with N0 = X0 and with D0 = I. Then using this in the difference equation yields
X
t
+
1
=
−
[
E
+
B
N
t
D
t
−
1
]
D
t
D
t
−
1
[
C
+
A
N
t
D
t
−
1
]
−
1
=
−
[
E
D
t
+
B
N
t
]
[
[
C
+
A
N
t
D
t
−
1
]
D
t
]
−
1
=
−
[
E
D
t
+
B
N
t
]
[
C
D
t
+
A
N
t
]
−
1
=
N
t
+
1
D
t
+
1
−
1
{\displaystyle {\begin{aligned}\mathbf {X} _{t+1}&=-\left[\mathbf {E} +\mathbf {BN} _{t}\mathbf {D} _{t}^{-1}\right]\mathbf {D} _{t}\mathbf {D} _{t}^{-1}\left[\mathbf {C} +\mathbf {AN} _{t}\mathbf {D} _{t}^{-1}\right]^{-1}\\&=-\left[\mathbf {ED} _{t}+\mathbf {BN} _{t}\right]\left[\left[\mathbf {C} +\mathbf {AN} _{t}\mathbf {D} _{t}^{-1}\right]\mathbf {D} _{t}\right]^{-1}\\&=-\left[\mathbf {ED} _{t}+\mathbf {BN} _{t}\right]\left[\mathbf {CD} _{t}+\mathbf {AN} _{t}\right]^{-1}\\&=\mathbf {N} _{t+1}\mathbf {D} _{t+1}^{-1}\end{aligned}}}
so by induction the form
X
t
=
N
t
D
t
−
1
{\displaystyle \mathbf {X} _{t}=\mathbf {N} _{t}\mathbf {D} _{t}^{-1}}
holds for all t. Then the evolution of N and D can be written as
[
N
t
+
1
D
t
+
1
]
=
[
−
B
−
E
A
C
]
[
N
t
D
t
]
≡
J
[
N
t
D
t
]
{\displaystyle {\begin{bmatrix}\mathbf {N} _{t+1}\\\mathbf {D} _{t+1}\end{bmatrix}}={\begin{bmatrix}-\mathbf {B} &-\mathbf {E} \\\mathbf {A} &\mathbf {C} \end{bmatrix}}{\begin{bmatrix}\mathbf {N} _{t}\\\mathbf {D} _{t}\end{bmatrix}}\equiv \mathbf {J} {\begin{bmatrix}\mathbf {N} _{t}\\\mathbf {D} _{t}\end{bmatrix}}}
Thus by induction
[
N
t
D
t
]
=
J
t
[
N
0
D
0
]
{\displaystyle {\begin{bmatrix}\mathbf {N} _{t}\\\mathbf {D} _{t}\end{bmatrix}}=\mathbf {J} ^{t}{\begin{bmatrix}\mathbf {N} _{0}\\\mathbf {D} _{0}\end{bmatrix}}}
== See also ==
Matrix differential equation
Difference equation
Linear difference equation
Dynamical system
Algebraic Riccati equation
== References == | Wikipedia/Matrix_difference_equation |
In number theory, the Erdős–Moser equation is
1
k
+
2
k
+
⋯
+
(
m
−
1
)
k
=
m
k
,
{\displaystyle 1^{k}+2^{k}+\cdots +(m-1)^{k}=m^{k},}
where m and k are restricted to the positive integers—that is, it is considered as a Diophantine equation. The only known solution is 11 + 21 = 31, and Paul Erdős conjectured that no further solutions exist. Any further solutions must have m > 10109.
Throughout this article, p refers exclusively to prime numbers.
== Constraints on solutions ==
In 1953, Leo Moser proved that, in any further solutions,
k is even,
p | (m − 1) implies (p − 1) | k,
p | (m + 1) implies (p − 1) | k,
p | (2m + 1) implies (p − 1) | k,
m − 1 is squarefree,
m + 1 is either squarefree or 4 times an odd squarefree number,
2m − 1 is squarefree,
2m + 1 is squarefree,
∑
p
|
(
m
−
1
)
m
−
1
p
≡
−
1
(
mod
m
−
1
)
,
{\displaystyle \sum _{p|(m-1)}{\frac {m-1}{p}}\equiv -1{\pmod {m-1}},}
∑
p
|
(
m
+
1
)
m
+
1
p
≡
−
2
(
mod
m
+
1
)
(if
m
+
1
is even)
,
{\displaystyle \sum _{p|(m+1)}{\frac {m+1}{p}}\equiv -2{\pmod {m+1}}\quad {\text{(if }}m+1{\text{ is even)}},}
∑
p
|
(
2
m
−
1
)
2
m
−
1
p
≡
−
2
(
mod
2
m
−
1
)
,
{\displaystyle \sum _{p|(2m-1)}{\frac {2m-1}{p}}\equiv -2{\pmod {2m-1}},}
∑
p
|
(
2
m
+
1
)
2
m
+
1
p
≡
−
4
(
mod
2
m
+
1
)
,
{\displaystyle \sum _{p|(2m+1)}{\frac {2m+1}{p}}\equiv -4{\pmod {2m+1}},}
and
m > 10106.
In 1966, it was additionally shown that
6 ≤ k + 2 < m < 3 (k + 1) / 2, and
m − 1 cannot be prime.
In 1994, it was shown that
lcm(1,2,…,200) divides k,
m ≡ 3 (mod 2ord2(k) + 3), where ord2(n) is the 2-adic valuation of n; equivalently, ord2(m − 3) = ord2(k) + 3,
for any odd prime p divding m, we have k ≢ 0, 2 (mod p − 1),
any prime factor of m must be irregular and > 10000.
In 1999, Moser's method was refined to show that m > 1.485 × 109,321,155.
In 2002, it was shown: §4 that k must be a multiple of 23 · 3# · 5# · 7# · 19# · 1000#, where the symbol # indicates the primorial; that is, n# is the product of all prime numbers ≤ n. This number exceeds 5.7462 × 10427.
In 2009, it was shown that 2k / (2m − 3) must be a convergent of ln(2); in what the authors of that paper call "one of very few instances where a large scale computation of a numerical constant has an application", it was then determined that m > 2.7139 × 101,667,658,416.
In 2010, it was shown that
m ≡ 3 (mod 8) and m ≡ ±1 (mod 3), and
(m2 − 1) (4m2 − 1) / 12 has at least 4,990,906 prime factors, none of which are repeated.
The number 4,990,906 arises from the fact that ∑4990905n=1 1/pn < 19/6 < ∑4990906n=1 1/pn, where pn is the nth prime number.
== Moser's method ==
First, let p be a prime factor of m − 1. Leo Moser showed that this implies that p − 1 divides k and that
which upon multiplying by p yields
p
+
m
−
1
≡
0
(
mod
p
2
)
.
{\displaystyle p+m-1\equiv 0{\pmod {p^{2}}}.}
This in turn implies that m − 1 must be squarefree. Furthermore, since nontrivial solutions have m − 1 > 2 and since all squarefree numbers in this range must have at least one odd prime factor, the assumption that p − 1 divides k implies that k must be even.
One congruence of the form (1) exists for each prime factor p of m − 1. Multiplying all of them together yields
∏
p
|
(
m
−
1
)
(
1
+
m
−
1
p
)
≡
0
(
mod
m
−
1
)
.
{\displaystyle \prod _{p|(m-1)}\left(1+{\frac {m-1}{p}}\right)\equiv 0{\pmod {m-1}}.}
Expanding out the product yields
1
+
∑
p
|
(
m
−
1
)
m
−
1
p
+
(
higher-order terms
)
≡
0
(
mod
m
−
1
)
,
{\displaystyle 1+\sum _{p|(m-1)}{\frac {m-1}{p}}+({\text{higher-order terms}})\equiv 0{\pmod {m-1}},}
where the higher-order terms are products of multiple factors of the form (m − 1) / p, with different values of p in each factor. These terms are all divisible by m − 1, so they all drop out of the congruence, yielding
1
+
∑
p
|
(
m
−
1
)
m
−
1
p
≡
0
(
mod
m
−
1
)
.
{\displaystyle 1+\sum _{p|(m-1)}{\frac {m-1}{p}}\equiv 0{\pmod {m-1}}.}
Dividing out the modulus yields
Similar reasoning yields the congruences
The congruences (2), (3), (4), and (5) are quite restrictive; for example, the only values of m < 1000 which satisfy (2) are 3, 7, and 43, and these are ruled out by (4).
We now split into two cases: either m + 1 is even, or it is odd.
In the case that m + 1 is even, adding the left-hand sides of the congruences (2), (3), (4), and (5) must yield an integer, and this integer must be at least 4. Furthermore, the Euclidean algorithm shows that no prime p > 3 can divide more than one of the numbers in the set {m − 1, m + 1, 2m − 1, 2m + 1}, and that 2 and 3 can divide at most two of these numbers. Letting M = (m − 1) (m + 1) (2m − 1) (2m + 1), we then have
Since there are no nontrivial solutions with m < 1000, the part of the LHS of (6) outside the sigma cannot exceed 0.006; we therefore have
∑
p
|
M
1
p
>
3.16.
{\displaystyle \sum _{p|M}{\frac {1}{p}}>3.16.}
Therefore, if
∑
p
≤
x
1
p
<
3.16
{\displaystyle \sum _{p\leq x}{\frac {1}{p}}<3.16}
, then
M
>
∏
p
≤
x
p
{\displaystyle M>\prod _{p\leq x}p}
. In Moser's original paper, bounds on the prime-counting function are used to observe that
∑
p
≤
10
7
1
p
<
3.16.
{\displaystyle \sum _{p\leq 10^{7}}{\frac {1}{p}}<3.16.}
Therefore, M must exceed the product of the first 10,000,000 primes. This in turn implies that m > 10106 in this case.
In the case that m + 1 is odd, we cannot use (3), so instead of (6) we obtain
1
m
−
1
+
2
2
m
−
1
+
4
2
m
+
1
+
∑
p
|
N
1
p
≥
3
−
1
3
=
2.666...
,
{\displaystyle {\frac {1}{m-1}}+{\frac {2}{2m-1}}+{\frac {4}{2m+1}}+\sum _{p|N}{\frac {1}{p}}\geq 3-{\frac {1}{3}}=2.666...,}
where N = (m − 1) (2m − 1) (2m + 1). On the surface, this appears to be a weaker condition than (6), but since m + 1 is odd, the prime 2 cannot appear on the greater side of this inequality, and it turns out to be a stronger restriction on m than the other case.
Therefore any nontrivial solutions have m > 10106.
In 1999, this method was refined by using computers to replace the prime-counting estimates with exact computations; this yielded the bound m > 1.485 × 109,321,155.: Thm 2
== Bounding the ratio m / (k + 1) ==
Let Sk(m) = 1k + 2k + ⋯ + (m − 1)k. Then the Erdős–Moser equation becomes Sk(m) = mk.
=== Method 1: Integral comparisons ===
By comparing the sum Sk(m) to definite integrals of the function xk, one can obtain the bounds 1 < m / (k + 1) < 3.: §1¶2
The sum Sk(m) = 1k + 2k + ⋯ + (m − 1)k is the upper Riemann sum corresponding to the integral
∫
0
m
−
1
x
k
d
x
{\textstyle \int _{0}^{m-1}x^{k}\,\mathrm {d} x}
in which the interval has been partitioned on the integer values of x, so we have
S
k
(
m
)
>
∫
0
m
−
1
x
k
d
x
.
{\displaystyle S_{k}(m)>\int _{0}^{m-1}x^{k}\;\mathrm {d} x.}
By hypothesis, Sk(m) = mk, so
m
k
>
(
m
−
1
)
k
+
1
k
+
1
,
{\displaystyle m^{k}>{\frac {(m-1)^{k+1}}{k+1}},}
which leads to
Similarly, Sk(m) is the lower Riemann sum corresponding to the integral
∫
1
m
x
k
d
x
{\textstyle \int _{1}^{m}x^{k}\,\mathrm {d} x}
in which the interval has been partitioned on the integer values of x, so we have
S
k
(
m
)
≤
∫
1
m
x
k
d
x
.
{\displaystyle S_{k}(m)\leq \int _{1}^{m}x^{k}\;\mathrm {d} x.}
By hypothesis, Sk(m) = mk, so
m
k
≤
m
k
+
1
−
1
k
+
1
<
m
k
+
1
k
+
1
,
{\displaystyle m^{k}\leq {\frac {m^{k+1}-1}{k+1}}<{\frac {m^{k+1}}{k+1}},}
and so
Applying this to (7) yields
m
k
+
1
<
(
1
+
1
m
−
1
)
m
=
(
1
+
1
m
−
1
)
m
−
1
⋅
(
m
m
−
1
)
<
e
⋅
m
m
−
1
.
{\displaystyle {\frac {m}{k+1}}<\left(1+{\frac {1}{m-1}}\right)^{m}=\left(1+{\frac {1}{m-1}}\right)^{m-1}\cdot \left({\frac {m}{m-1}}\right)<e\cdot {\frac {m}{m-1}}.}
Computation shows that there are no nontrivial solutions with m ≤ 10, so we have
m
k
+
1
<
e
⋅
11
11
−
1
<
3.
{\displaystyle {\frac {m}{k+1}}<e\cdot {\frac {11}{11-1}}<3.}
Combining this with (8) yields 1 < m / (k + 1) < 3, as desired.
=== Method 2: Algebraic manipulations ===
The upper bound m / (k + 1) < 3 can be reduced to m / (k + 1) < 3/2 using an algebraic method:: Lemat 4
Let r be a positive integer. Then the binomial theorem yields
(
ℓ
+
1
)
r
+
1
=
∑
i
=
0
r
+
1
(
r
+
1
i
)
ℓ
r
+
1
−
i
.
{\displaystyle (\ell +1)^{r+1}=\sum _{i=0}^{r+1}{\binom {r+1}{i}}\ell ^{r+1-i}.}
Summing over ℓ yields
∑
ℓ
=
1
m
−
1
(
ℓ
+
1
)
r
+
1
=
∑
ℓ
=
1
m
−
1
(
∑
i
=
0
r
+
1
(
r
+
1
i
)
ℓ
r
+
1
−
i
)
.
{\displaystyle \sum _{\ell =1}^{m-1}(\ell +1)^{r+1}=\sum _{\ell =1}^{m-1}\left(\sum _{i=0}^{r+1}{\binom {r+1}{i}}\ell ^{r+1-i}\right).}
Reindexing on the left and rearranging on the right yields
∑
ℓ
=
2
m
ℓ
r
+
1
=
∑
i
=
0
r
+
1
(
r
+
1
i
)
∑
ℓ
=
1
m
−
1
ℓ
r
+
1
−
i
{\displaystyle \sum _{\ell =2}^{m}\ell ^{r+1}=\sum _{i=0}^{r+1}{\binom {r+1}{i}}\sum _{\ell =1}^{m-1}\ell ^{r+1-i}}
∑
ℓ
=
1
m
ℓ
r
+
1
=
1
+
∑
i
=
0
r
+
1
(
r
+
1
i
)
S
r
+
1
−
i
(
m
)
{\displaystyle \sum _{\ell =1}^{m}\ell ^{r+1}=1+\sum _{i=0}^{r+1}{\binom {r+1}{i}}S_{r+1-i}(m)}
S
r
+
1
(
m
+
1
)
−
S
r
+
1
(
m
)
=
1
+
(
r
+
1
)
S
r
(
m
)
+
∑
i
=
2
r
+
1
(
r
+
1
i
)
S
r
+
1
−
i
(
m
)
{\displaystyle S_{r+1}(m+1)-S_{r+1}(m)=1+(r+1)S_{r}(m)+\sum _{i=2}^{r+1}{\binom {r+1}{i}}S_{r+1-i}(m)}
Taking r = k yields
m
k
+
1
=
1
+
(
k
+
1
)
S
k
(
m
)
+
∑
i
=
2
k
+
1
(
k
+
1
i
)
S
k
+
1
−
i
(
m
)
.
{\displaystyle m^{k+1}=1+(k+1)S_{k}(m)+\sum _{i=2}^{k+1}{\binom {k+1}{i}}S_{k+1-i}(m).}
By hypothesis, Sk = mk, so
m
k
+
1
=
1
+
(
k
+
1
)
m
k
+
∑
i
=
2
k
+
1
(
k
+
1
i
)
S
k
+
1
−
i
(
m
)
{\displaystyle m^{k+1}=1+(k+1)m^{k}+\sum _{i=2}^{k+1}{\binom {k+1}{i}}S_{k+1-i}(m)}
Since the RHS is positive, we must therefore have
Returning to (9) and taking r = k − 1 yields
m
k
=
1
+
k
⋅
S
k
−
1
(
m
)
+
∑
i
=
2
k
(
k
i
)
S
k
−
i
(
m
)
{\displaystyle m^{k}=1+k\cdot S_{k-1}(m)+\sum _{i=2}^{k}{\binom {k}{i}}S_{k-i}(m)}
m
k
=
1
+
∑
s
=
1
k
(
k
s
)
S
k
−
s
(
m
)
.
{\displaystyle m^{k}=1+\sum _{s=1}^{k}{\binom {k}{s}}S_{k-s}(m).}
Substituting this into (10) to eliminate mk yields
(
1
+
∑
s
=
1
k
(
k
s
)
S
k
−
s
(
m
)
)
(
m
−
(
k
+
1
)
)
=
1
+
∑
i
=
2
k
+
1
(
k
+
1
i
)
S
k
+
1
−
i
(
m
)
.
{\displaystyle \left(1+\sum _{s=1}^{k}{\binom {k}{s}}S_{k-s}(m)\right)(m-(k+1))=1+\sum _{i=2}^{k+1}{\binom {k+1}{i}}S_{k+1-i}(m).}
Reindexing the sum on the right with the substitution i = s + 1 yields
(
1
+
∑
s
=
1
k
(
k
s
)
S
k
−
s
(
m
)
)
(
m
−
(
k
+
1
)
)
=
1
+
∑
s
=
1
k
(
k
+
1
s
+
1
)
S
k
−
s
(
m
)
{\displaystyle \left(1+\sum _{s=1}^{k}{\binom {k}{s}}S_{k-s}(m)\right)(m-(k+1))=1+\sum _{s=1}^{k}{\binom {k+1}{s+1}}S_{k-s}(m)}
m
−
(
k
+
1
)
+
(
m
−
(
k
+
1
)
)
∑
s
=
1
k
(
k
s
)
S
k
−
s
(
m
)
=
1
+
∑
s
=
1
k
k
+
1
s
+
1
(
k
s
)
S
k
−
s
(
m
)
{\displaystyle m-(k+1)+(m-(k+1))\sum _{s=1}^{k}{\binom {k}{s}}S_{k-s}(m)=1+\sum _{s=1}^{k}{\frac {k+1}{s+1}}{\binom {k}{s}}S_{k-s}(m)}
We already know from (11) that k + 1 < m. This leaves open the possibility that m = k + 2; however, substituting this into (12) yields
0
=
∑
s
=
1
k
(
k
+
1
s
+
1
−
1
)
(
k
s
)
S
k
−
s
(
k
+
2
)
{\displaystyle 0=\sum _{s=1}^{k}\left({\frac {k+1}{s+1}}-1\right){\binom {k}{s}}S_{k-s}(k+2)}
0
=
∑
s
=
1
k
k
−
s
s
+
1
(
k
s
)
S
k
−
s
(
k
+
2
)
{\displaystyle 0=\sum _{s=1}^{k}{\frac {k-s}{s+1}}{\binom {k}{s}}S_{k-s}(k+2)}
0
=
k
−
k
k
+
1
(
k
k
)
S
k
−
k
(
k
+
2
)
+
∑
s
=
1
k
−
1
k
−
s
s
+
1
(
k
s
)
S
k
−
s
(
k
+
2
)
{\displaystyle 0={\frac {k-k}{k+1}}{\binom {k}{k}}S_{k-k}(k+2)+\sum _{s=1}^{k-1}{\frac {k-s}{s+1}}{\binom {k}{s}}S_{k-s}(k+2)}
0
=
0
+
∑
s
=
1
k
−
1
k
−
s
s
+
1
(
k
s
)
S
k
−
s
(
k
+
2
)
,
{\displaystyle 0=0+\sum _{s=1}^{k-1}{\frac {k-s}{s+1}}{\binom {k}{s}}S_{k-s}(k+2),}
which is impossible for k > 1, since the sum contains only positive terms. Therefore any nontrivial solutions must have m ≠ k + 2; combining this with (11) yields
k
+
2
<
m
.
{\displaystyle k+2<m.}
We therefore observe that the left-hand side of (12) is positive, so
Since k > 1, the sequence
{
(
k
+
1
)
/
(
s
+
1
)
−
m
+
(
k
+
1
)
}
s
=
1
∞
{\displaystyle \left\{(k+1)/(s+1)-m+(k+1)\right\}_{s=1}^{\infty }}
is decreasing. This and (13) together imply that its first term (the term with s = 1) must be positive: if it were not, then every term in the sum would be nonpositive, and therefore so would the sum itself. Thus,
0
<
k
+
1
1
+
1
−
m
+
(
k
+
1
)
,
{\displaystyle 0<{\frac {k+1}{1+1}}-m+(k+1),}
which yields
m
<
3
2
⋅
(
k
+
1
)
{\displaystyle m<{\frac {3}{2}}\cdot (k+1)}
and therefore
m
k
+
1
<
3
2
,
{\displaystyle {\frac {m}{k+1}}<{\frac {3}{2}},}
as desired.
== Continued fractions ==
Any potential solutions to the equation must arise from the continued fraction of the natural logarithm of 2: specifically, 2k / (2m − 3) must be a convergent of that number.
By expanding the Taylor series of (1 − y)k eky about y = 0, one finds
(
1
−
y
)
k
=
e
−
k
y
(
1
−
k
2
y
2
−
k
3
y
3
+
k
(
k
−
2
)
8
y
4
+
k
(
5
k
−
6
)
30
y
5
+
O
(
y
6
)
)
as
y
→
0.
{\displaystyle (1-y)^{k}=e^{-ky}\left(1-{\frac {k}{2}}y^{2}-{\frac {k}{3}}y^{3}+{\frac {k(k-2)}{8}}y^{4}+{\frac {k(5k-6)}{30}}y^{5}+O(y^{6})\right)\quad {\text{as }}y\rightarrow 0.}
More elaborate analysis sharpens this to
for k > 8 and 0 < y < 1.
The Erdős–Moser equation is equivalent to
1
=
∑
j
=
1
m
−
1
(
1
−
j
m
)
k
.
{\displaystyle 1=\sum _{j=1}^{m-1}\left(1-{\frac {j}{m}}\right)^{k}.}
Applying (14) to each term in this sum yields
T
0
−
k
2
m
2
T
2
−
k
3
m
3
T
3
+
k
(
k
−
2
)
8
m
4
T
4
+
k
(
5
k
−
6
)
30
m
5
T
5
−
k
3
6
m
6
T
6
<
∑
j
=
1
m
−
1
(
1
−
j
m
)
k
<
T
0
−
k
2
m
2
T
2
−
k
3
m
3
T
3
+
k
(
k
−
2
)
8
m
4
T
4
+
k
2
2
m
5
T
5
,
{\displaystyle {\begin{aligned}T_{0}-{\frac {k}{2m^{2}}}T_{2}-{\frac {k}{3m^{3}}}T_{3}+{\frac {k(k-2)}{8m^{4}}}T_{4}+{\frac {k(5k-6)}{30m^{5}}}T_{5}-{\frac {k^{3}}{6m^{6}}}T_{6}\qquad \qquad \\<\sum _{j=1}^{m-1}\left(1-{\frac {j}{m}}\right)^{k}<T_{0}-{\frac {k}{2m^{2}}}T_{2}-{\frac {k}{3m^{3}}}T_{3}+{\frac {k(k-2)}{8m^{4}}}T_{4}+{\frac {k^{2}}{2m^{5}}}T_{5},\end{aligned}}}
where
T
n
=
∑
j
=
1
m
−
1
j
n
z
j
{\displaystyle T_{n}=\sum _{j=1}^{m-1}j^{n}z^{j}}
and z = e−k/m. Further manipulation eventually yields
We already know that k/m is bounded as m → ∞; making the ansatz k/m = c + O(1/m), and therefore z = e−c + O(1/m), and substituting it into (15) yields
1
−
1
e
c
−
1
=
O
(
1
m
)
as
m
→
∞
;
{\displaystyle 1-{\frac {1}{e^{c}-1}}=O\left({\frac {1}{m}}\right)\quad {\text{as }}m\rightarrow \infty ;}
therefore c = ln(2). We therefore have
and so
1
z
=
e
k
/
m
=
2
+
2
a
m
+
a
2
+
2
b
m
2
+
O
(
1
m
3
)
as
m
→
∞
.
{\displaystyle {\frac {1}{z}}=e^{k/m}=2+{\frac {2a}{m}}+{\frac {a^{2}+2b}{m^{2}}}+O\left({\frac {1}{m^{3}}}\right)\quad {\text{as }}m\rightarrow \infty .}
Substituting these formulas into (15) yields a = −3 ln(2) / 2 and b = (3 ln(2) − 25/12) ln(2). Putting these into (16) yields
k
m
=
ln
(
2
)
(
1
−
3
2
m
−
25
12
−
3
ln
(
2
)
m
2
+
O
(
1
m
3
)
)
as
m
→
∞
.
{\displaystyle {\frac {k}{m}}=\ln(2)\left(1-{\frac {3}{2m}}-{\frac {{\frac {25}{12}}-3\ln(2)}{m^{2}}}+O\left({\frac {1}{m^{3}}}\right)\right)\quad {\text{as }}m\rightarrow \infty .}
The term O(1/m3) must be bounded effectively. To that end, we define the function
F
(
x
,
λ
)
=
(
1
−
1
t
−
1
+
x
λ
2
t
+
t
2
(
t
−
1
)
3
+
x
2
λ
3
t
+
4
t
2
+
t
3
(
t
−
1
)
4
−
x
2
λ
(
λ
−
2
x
)
8
t
+
11
t
2
+
11
t
3
+
t
4
(
t
−
1
)
5
)
|
t
=
e
λ
.
{\displaystyle F(x,\lambda )=\left.\left(1-{\frac {1}{t-1}}+{\frac {x\lambda }{2}}{\frac {t+t^{2}}{(t-1)^{3}}}+{\frac {x^{2}\lambda }{3}}{\frac {t+4t^{2}+t^{3}}{(t-1)^{4}}}-{\frac {x^{2}\lambda (\lambda -2x)}{8}}{\frac {t+11t^{2}+11t^{3}+t^{4}}{(t-1)^{5}}}\right)\right\vert _{t=e^{\lambda }}.}
The inequality (15) then takes the form
and we further have
F
(
x
,
ln
(
2
)
(
1
−
3
2
x
−
0.004
x
2
)
)
>
−
0.005
15
x
2
−
100
x
3
and
F
(
x
,
ln
(
2
)
(
1
−
3
2
x
−
0.004
x
2
)
)
<
−
0.00015
x
2
+
100
x
3
{\displaystyle {\begin{aligned}F(x,\ln(2)(1-{\tfrac {3}{2}}x\,{\phantom {-\;0.004x^{2}}}))&>{\phantom {-}}0.005{\phantom {15}}x^{2}-100x^{3}\quad {\text{and}}\\F(x,\ln(2)(1-{\tfrac {3}{2}}x-0.004x^{2}))&<-0.00015x^{2}+100x^{3}\end{aligned}}}
for x ≤ 0.01. Therefore
F
(
1
m
,
ln
(
2
)
(
1
−
3
2
m
−
0.004
m
2
)
)
>
−
110000
m
3
for
m
>
2202
⋅
10
4
and
F
(
1
m
,
ln
(
2
)
(
1
−
3
2
m
−
0.004
m
2
)
)
<
−
110000
m
3
for
m
>
0
734
⋅
10
6
.
{\displaystyle {\begin{aligned}F\left({\frac {1}{m}},\ln(2)\left(1-{\frac {3}{2m}}\,{\phantom {\;-{\frac {0.004}{m^{2}}}}}\right)\right)&>{\phantom {-}}{\frac {110000}{m^{3}}}\quad {\text{for }}m>2202\cdot 10^{4}\quad {\text{and}}\\F\left({\frac {1}{m}},\ln(2)\left(1-{\frac {3}{2m}}-{\frac {0.004}{m^{2}}}\right)\right)&<-{\frac {110000}{m^{3}}}\quad {\text{for }}m>{\phantom {0}}734\cdot 10^{6}.\end{aligned}}}
Comparing these with (17) then shows that, for m > 109, we have
ln
(
2
)
(
1
−
3
2
m
−
0.004
m
2
)
<
k
m
<
ln
(
2
)
(
1
−
3
2
m
)
,
{\displaystyle \ln(2)\left(1-{\frac {3}{2m}}-{\frac {0.004}{m^{2}}}\right)<{\frac {k}{m}}<\ln(2)\left(1-{\frac {3}{2m}}\right),}
and therefore
0
<
ln
(
2
)
−
2
k
2
m
−
3
<
0.0111
(
2
m
−
3
)
2
.
{\displaystyle 0<\ln(2)-{\frac {2k}{2m-3}}<{\frac {0.0111}{(2m-3)^{2}}}.}
Recalling that Moser showed that indeed m > 109, and then invoking Legendre's theorem on continued fractions, finally proves that 2k / (2m − 3) must be a convergent to ln(2). Leveraging this result, 31 billion decimal digits of ln(2) can be used to exclude any nontrivial solutions below 10109.
== See also ==
List of conjectures by Paul Erdős
List of things named after Paul Erdős
List of unsolved problems in mathematics
Sums of powers
Faulhaber's formula
== References == | Wikipedia/Erdős–Moser_equation |
In number theory, the Fermat–Catalan conjecture is a generalization of Fermat's Last Theorem and of Catalan's conjecture. The conjecture states that the equation
has only finitely many solutions (a, b, c, m, n, k) with distinct triplets of values (am, bn, ck) where a, b, c are positive coprime integers and m, n, k are positive integers satisfying
The inequality on m, n, and k is a necessary part of the conjecture. Without the inequality there would be infinitely many solutions, for instance with k = 1 (for any a, b, m, and n and with c = am + bn), with m=n=k=2 (for the infinitely many Pythagorean triples), and e.g.
7
5
+
393
3
=
7792
2
{\displaystyle 7^{5}+393^{3}=7792^{2}}
.
== Known solutions ==
As of 2015 the following ten solutions to equation (1) which meet the criteria of equation (2) are known:
1
m
+
2
3
=
3
2
{\displaystyle 1^{m}+2^{3}=3^{2}\;}
(for
m
>
6
{\displaystyle m>6}
to satisfy Eq. 2)
2
5
+
7
2
=
3
4
{\displaystyle 2^{5}+7^{2}=3^{4}\;}
7
3
+
13
2
=
2
9
{\displaystyle 7^{3}+13^{2}=2^{9}\;}
2
7
+
17
3
=
71
2
{\displaystyle 2^{7}+17^{3}=71^{2}\;}
3
5
+
11
4
=
122
2
{\displaystyle 3^{5}+11^{4}=122^{2}\;}
33
8
+
1549034
2
=
15613
3
{\displaystyle 33^{8}+1549034^{2}=15613^{3}\;}
1414
3
+
2213459
2
=
65
7
{\displaystyle 1414^{3}+2213459^{2}=65^{7}\;}
9262
3
+
15312283
2
=
113
7
{\displaystyle 9262^{3}+15312283^{2}=113^{7}\;}
17
7
+
76271
3
=
21063928
2
{\displaystyle 17^{7}+76271^{3}=21063928^{2}\;}
43
8
+
96222
3
=
30042907
2
{\displaystyle 43^{8}+96222^{3}=30042907^{2}\;}
The first of these (1m + 23 = 32) is the only solution where one of a, b or c is 1, according to the Catalan conjecture, proven in 2002 by Preda Mihăilescu. While this case leads to infinitely many solutions of (1) (since one can pick any m for m > 6), these solutions only give a single triplet of values (am, bn, ck).
== Partial results ==
It is known by the Darmon–Granville theorem, which uses Faltings's theorem, that for any fixed choice of positive integers m, n and k satisfying (2), only finitely many coprime triples (a, b, c) solving (1) exist.: p. 64 However, the full Fermat–Catalan conjecture is stronger as it allows for the exponents m, n and k to vary.
The abc conjecture implies the Fermat–Catalan conjecture.
For a list of results for impossible combinations of exponents, see Beal conjecture#Partial results. Beal's conjecture is true if and only if all Fermat–Catalan solutions have m = 2, n = 2, or k = 2.
== See also ==
Sums of powers, a list of related conjectures and theorems
== References ==
== External links ==
Perfect Powers: Pillai's works and their developments. Waldschmidt, M.
Sloane, N. J. A. (ed.). "Sequence A214618 (Perfect powers z^r that can be written in the form x^p + y^q, where x, y, z are positive coprime integers and p, q, r are positive integers satisfying 1/p + 1/q + 1/r < 1)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. | Wikipedia/Fermat–Catalan_conjecture |
In number theory, the Ramanujan–Nagell equation is an equation between a square number and a number that is seven less than a power of two. It is an example of an exponential Diophantine equation, an equation to be solved in integers where one of the variables appears as an exponent.
The equation is named after Srinivasa Ramanujan, who conjectured that it has only five integer solutions, and after Trygve Nagell, who proved the conjecture. It implies non-existence of perfect binary codes with the minimum Hamming distance 5 or 6.
== Equation and solution ==
The equation is
2
n
−
7
=
x
2
{\displaystyle 2^{n}-7=x^{2}\,}
and solutions in natural numbers n and x exist just when n = 3, 4, 5, 7 and 15 (sequence A060728 in the OEIS).
This was conjectured in 1913 by Indian mathematician Srinivasa Ramanujan, proposed independently in 1943 by the Norwegian mathematician Wilhelm Ljunggren, and proved in 1948 by the Norwegian mathematician Trygve Nagell. The values of n correspond to the values of x as:-
x = 1, 3, 5, 11 and 181 (sequence A038198 in the OEIS).
== Triangular Mersenne numbers ==
The problem of finding all numbers of the form 2b − 1 (Mersenne numbers) which are triangular is equivalent:
2
b
−
1
=
y
(
y
+
1
)
2
⟺
8
(
2
b
−
1
)
=
4
y
(
y
+
1
)
⟺
2
b
+
3
−
8
=
4
y
2
+
4
y
⟺
2
b
+
3
−
7
=
4
y
2
+
4
y
+
1
⟺
2
b
+
3
−
7
=
(
2
y
+
1
)
2
{\displaystyle {\begin{aligned}&\ 2^{b}-1={\frac {y(y+1)}{2}}\\[2pt]\Longleftrightarrow &\ 8(2^{b}-1)=4y(y+1)\\\Longleftrightarrow &\ 2^{b+3}-8=4y^{2}+4y\\\Longleftrightarrow &\ 2^{b+3}-7=4y^{2}+4y+1\\\Longleftrightarrow &\ 2^{b+3}-7=(2y+1)^{2}\end{aligned}}}
The values of b are just those of n − 3, and the corresponding triangular Mersenne numbers (also known as Ramanujan–Nagell numbers) are:
y
(
y
+
1
)
2
=
(
x
−
1
)
(
x
+
1
)
8
{\displaystyle {\frac {y(y+1)}{2}}={\frac {(x-1)(x+1)}{8}}}
for x = 1, 3, 5, 11 and 181, giving 0, 1, 3, 15, 4095 and no more (sequence A076046 in the OEIS).
== Equations of Ramanujan–Nagell type ==
An equation of the form
x
2
+
D
=
A
B
n
{\displaystyle x^{2}+D=AB^{n}}
for fixed D, A, B and variable x, n is said to be of Ramanujan–Nagell type. The result of Siegel implies that the number of solutions in each case is finite. By representing
n
=
3
m
+
r
{\displaystyle n=3m+r}
with
r
∈
{
0
,
1
,
2
}
{\displaystyle r\in \{0,1,2\}}
and
B
n
=
B
r
y
3
{\displaystyle B^{n}=B^{r}y^{3}}
with
y
=
B
m
{\displaystyle y=B^{m}}
, the equation of Ramanujan–Nagell type is reduced to three Mordell curves (indexed by
r
{\displaystyle r}
), each of which has a finite number of integer solutions:
r
=
0
:
(
A
x
)
2
=
(
A
y
)
3
−
A
2
D
{\displaystyle r=0:\qquad (Ax)^{2}=(Ay)^{3}-A^{2}D}
,
r
=
1
:
(
A
B
x
)
2
=
(
A
B
y
)
3
−
A
2
B
2
D
{\displaystyle r=1:\qquad (ABx)^{2}=(ABy)^{3}-A^{2}B^{2}D}
,
r
=
2
:
(
A
B
2
x
)
2
=
(
A
B
2
y
)
3
−
A
2
B
4
D
{\displaystyle r=2:\qquad (AB^{2}x)^{2}=(AB^{2}y)^{3}-A^{2}B^{4}D}
.
The equation with
A
=
1
,
B
=
2
,
D
>
0
{\displaystyle A=1,\ B=2,\ D>0}
has at most two solutions, except in the case
D
=
7
{\displaystyle D=7}
corresponding to the Ramanujan–Nagell equation. This does not hold for
D
<
0
{\displaystyle D<0}
, such as
D
=
−
17
{\displaystyle D=-17}
, where
x
2
−
17
=
2
n
{\displaystyle x^{2}-17=2^{n}}
has the four solutions
(
x
,
n
)
=
(
5
,
3
)
,
(
7
,
5
)
,
(
9
,
6
)
,
(
23
,
9
)
{\displaystyle (x,n)=(5,3),(7,5),(9,6),(23,9)}
. In general, if
D
=
−
(
4
k
−
3
⋅
2
k
+
1
+
1
)
{\displaystyle D=-(4^{k}-3\cdot 2^{k+1}+1)}
for an integer
k
⩾
3
{\displaystyle k\geqslant 3}
there are at least the four solutions
(
x
,
n
)
=
{
(
2
k
−
3
,
3
)
(
2
k
−
1
,
k
+
2
)
(
2
k
+
1
,
k
+
3
)
(
3
⋅
2
k
−
1
,
2
k
+
3
)
{\displaystyle (x,n)={\begin{cases}(2^{k}-3,3)\\(2^{k}-1,k+2)\\(2^{k}+1,k+3)\\(3\cdot 2^{k}-1,2k+3)\end{cases}}}
and these are the only four if
D
>
−
10
12
{\displaystyle D>-10^{12}}
. There are infinitely many values of D for which there are exactly two solutions, including
D
=
2
m
−
1
{\displaystyle D=2^{m}-1}
.
== Equations of Lebesgue–Nagell type ==
An equation of the form
x
2
+
D
=
A
y
n
{\displaystyle x^{2}+D=Ay^{n}}
for fixed D, A and variable x, y, n is said to be of Lebesgue–Nagell type. This is named after Victor-Amédée Lebesgue, who proved that the equation
x
2
+
1
=
y
n
{\displaystyle x^{2}+1=y^{n}}
has no nontrivial solutions.
Results of Shorey and Tijdeman imply that the number of solutions in each case is finite. Bugeaud, Mignotte and Siksek solved equations of this type with A = 1 and 1 ≤ D ≤ 100. In particular, the following generalization of the Ramanujan–Nagell equation:
y
n
−
7
=
x
2
{\displaystyle y^{n}-7=x^{2}\,}
has positive integer solutions only when x = 1, 3, 5, 11, or 181.
== See also ==
Pillai's conjecture
Scientific equations named after people
== Notes ==
== References ==
Beukers, F. (1981). "On the generalized Ramanujan-Nagell equation I" (PDF). Acta Arithmetica. 38 (4): 401–403. doi:10.4064/aa-38-4-389-410.
Bugeaud, Y.; Mignotte, M.; Siksek, S. (2006). "Classical and modular approaches to exponential Diophantine equations II. The Lebesgue–Nagell equation". Compositio Mathematica. 142: 31–62. arXiv:math/0405220. doi:10.1112/S0010437X05001739. S2CID 18534268.
Lebesgue (1850). "Sur l'impossibilité, en nombres entiers, de l'équation xm = y2 + 1". Nouv. Ann. Math. Série 1. 9: 178–181.
Ljunggren, W. (1943). "Oppgave nr 2". Norsk Mat. Tidsskr. 25: 29.
Nagell, T. (1948). "Løsning till oppgave nr 2". Norsk Mat. Tidsskr. 30: 62–64.
Nagell, T. (1961). "The Diophantine equation x2 + 7 = 2n". Ark. Mat. 30 (2–3): 185–187. Bibcode:1961ArM.....4..185N. doi:10.1007/BF02592006.
Ramanujan, S. (1913). "Question 464". J. Indian Math. Soc. 5: 130.
Saradha, N.; Srinivasan, Anitha (2008). "Generalized Lebesgue–Ramanujan–Nagell equations". In Saradha, N. (ed.). Diophantine Equations. Narosa. pp. 207–223. ISBN 978-81-7319-898-4.
Shorey, T. N.; Tijdeman, R. (1986). Exponential Diophantine equations. Cambridge Tracts in Mathematics. Vol. 87. Cambridge University Press. pp. 137–138. ISBN 0-521-26826-5. Zbl 0606.10011.
Siegel, C. L. (1929). "Uber einige Anwendungen Diophantischer Approximationen". Abh. Preuss. Akad. Wiss. Phys. Math. Kl. 1: 41–69.
== External links ==
"Values of X corresponding to N in the Ramanujan–Nagell Equation". Wolfram MathWorld. Retrieved 2012-05-08.
Can N2 + N + 2 Be A Power Of 2?, Math Forum discussion | Wikipedia/Ramanujan–Nagell_equation |
Diophantus and Diophantine Equations is a book in the history of mathematics, on the history of Diophantine equations and their solution by Diophantus of Alexandria. It was originally written in Russian by Isabella Bashmakova, and published by Nauka in 1972 under the title Диофант и диофантовы уравнения. It was translated into German by Ludwig Boll as Diophant und diophantische Gleichungen (Birkhäuser, 1974) and into English by Abe Shenitzer as Diophantus and Diophantine Equations (Dolciani Mathematical Expositions 20, Mathematical Association of America, 1997).
== Topics ==
In the sense considered in the book, a Diophantine equation is an equation written using polynomials whose coefficients are rational numbers. These equations are to be solved by finding rational-number values for the variables that, when plugged into the equation, make it become true. Although there is also a well-developed theory of integer (rather than rational) solutions to polynomial equations, it is not included in this book.
Diophantus of Alexandria studied equations of this type in the second century AD. Scholarly opinion has generally held that Diophantus only found solutions to specific equations, and had no methods for solving general families of equations. For instance, Hermann Hankel has written of the works of Diophantus that "not the slightest trace of a general, comprehensive method is discernible; each problem calls for some special method which refuses to work even for the most closely related problems". In contrast, the thesis of Bashmakova's book is that Diophantus indeed had general methods, which can be inferred from the surviving record of his solutions to these problems.
The opening chapter of the books tells what is known of Diophantus and his contemporaries, and surveys the problems published by Diophantus. The second chapter reviews the mathematics known to Diophantus, including his development of negative numbers, rational numbers, and powers of numbers, and his philosophy of mathematics treating numbers as dimensionless quantities, a necessary preliminary to the use of inhomogeneous polynomials. The third chapter brings in more modern concepts of algebraic geometry including the degree and genus of an algebraic curve, and rational mappings and birational equivalences between curves.
Chapters four and five concern conic sections, and the theorem that when a conic has at least one rational point it has infinitely many. Chapter six covers the use of secant lines to generate infinitely many points on a cubic plane curve, considered in modern mathematics as an example of the group law of elliptic curves. Chapter seven concerns Fermat's theorem on sums of two squares, and the possibility that Diophantus may have known of some form of this theorem. The remaining four chapters trace the influence of Diophantus and his works through Hypatia and into 19th-century Europe, particularly concentrating on the development of the theory of elliptic curves and their group law.
The German edition adds supplementary material including a report by Joseph H. Silverman on progress towards a proof of Fermat's Last Theorem. An updated version of the same material was included in the English translation.
== Audience and reception ==
Very little mathematical background is needed to read this book.
Despite "qualms about Bashmakova's historical claims", reviewer David Graves writes that "a wealth of material, both mathematical and historical, is crammed into this remarkable little book", and he recommends it to any number theorist or scholar of the history of mathematics. Reviewer Alan Osborne is also positive, writing that it is "well-crafted, ... offering considerable historical information while inviting the reader to explore a great deal of mathematics."
== References == | Wikipedia/Diophantus_and_Diophantine_Equations |
Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu at Paderborn University. The integers 23 and 32 are two perfect powers (that is, powers of exponent higher than one) of natural numbers whose values (8 and 9, respectively) are consecutive. The theorem states that this is the only case of two consecutive perfect powers. That is to say, that
== History ==
The history of the problem dates back at least to Gersonides, who proved a special case of the conjecture in 1343 where (x, y) was restricted to be (2, 3) or (3, 2). The first significant progress after Catalan made his conjecture came in 1850 when Victor-Amédée Lebesgue dealt with the case b = 2.
In 1976, Robert Tijdeman applied Baker's method in transcendence theory to establish a bound on a,b and used existing results bounding x,y in terms of a, b to give an effective upper bound for x,y,a,b. Michel Langevin computed a value of
exp
exp
exp
exp
730
≈
10
10
10
10
317
{\displaystyle \exp \exp \exp \exp 730\approx 10^{10^{10^{10^{317}}}}}
for the bound, resolving Catalan's conjecture for all but a finite number of cases.
Catalan's conjecture was proven by Preda Mihăilescu in April 2002. The proof was published in the Journal für die reine und angewandte Mathematik, 2004. It makes extensive use of the theory of cyclotomic fields and Galois modules. An exposition of the proof was given by Yuri Bilu in the Séminaire Bourbaki. In 2005, Mihăilescu published a simplified proof.
== Pillai's conjecture ==
Pillai's conjecture concerns a general difference of perfect powers (sequence A001597 in the OEIS): it is an open problem initially proposed by S. S. Pillai, who conjectured that the gaps in the sequence of perfect powers tend to infinity. This is equivalent to saying that each positive integer occurs only finitely many times as a difference of perfect powers: more generally, in 1931 Pillai conjectured that for fixed positive integers A, B, C the equation
A
x
n
−
B
y
m
=
C
{\displaystyle Ax^{n}-By^{m}=C}
has only finitely many solutions (x, y, m, n) with (m, n) ≠ (2, 2). Pillai proved that for fixed A, B, x, y, and for any λ less than 1, we have
|
A
x
n
−
B
y
m
|
≫
x
λ
n
{\displaystyle |Ax^{n}-By^{m}|\gg x^{\lambda n}}
uniformly in m and n.
The general conjecture would follow from the ABC conjecture.
Pillai's conjecture means that for every natural number n, there are only finitely many pairs of perfect powers with difference n. The list below shows, for n ≤ 64, all solutions for perfect powers less than 1018, such that the exponent of both powers is greater than 1. The number of such solutions for each n is listed at OEIS: A076427. See also OEIS: A103953 for the smallest solution (> 0).
== See also ==
== Notes ==
== References ==
Bilu, Yuri (2004), "Catalan's conjecture (after Mihăilescu)", Astérisque, 294: vii, 1–26, MR 2111637
Catalan, Eugene (1844), "Note extraite d'une lettre adressée à l'éditeur", J. Reine Angew. Math. (in French), 27: 192, doi:10.1515/crll.1844.27.192, MR 1578392
Cohen, Henri (2005). Démonstration de la conjecture de Catalan [A proof of the Catalan conjecture]. Théorie algorithmique des nombres et équations diophantiennes (in French). Palaiseau: Éditions de l'École Polytechnique. pp. 1–83. ISBN 2-7302-1293-0. MR 0222434.
Metsänkylä, Tauno (2004), "Catalan's conjecture: another old Diophantine problem solved" (PDF), Bulletin of the American Mathematical Society, 41 (1): 43–57, doi:10.1090/S0273-0979-03-00993-5, MR 2015449
Mihăilescu, Preda (2004), "Primary Cyclotomic Units and a Proof of Catalan's Conjecture", J. Reine Angew. Math., 2004 (572): 167–195, doi:10.1515/crll.2004.048, MR 2076124
Mihăilescu, Preda (2005), "Reflection, Bernoulli numbers and the proof of Catalan's conjecture" (PDF), European Congress of Mathematics, Zurich: Eur. Math. Soc.: 325–340, MR 2185753, archived from the original (PDF) on 2022-06-26
Ribenboim, Paulo (1994), Catalan's Conjecture, Boston, MA: Academic Press, Inc., ISBN 0-12-587170-8, MR 1259738 Predates Mihăilescu's proof.
Tijdeman, Robert (1976), "On the equation of Catalan" (PDF), Acta Arith., 29 (2): 197–209, doi:10.4064/aa-29-2-197-209, MR 0404137
== External links ==
Weisstein, Eric W. "Catalan's conjecture". MathWorld.
Ivars Peterson's MathTrek
On difference of perfect powers
Jeanine Daems: A Cyclotomic Proof of Catalan's Conjecture | Wikipedia/Catalan's_conjecture |
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. Theta functions are parametrized by points in a tube domain inside a complex Lagrangian Grassmannian, namely the Siegel upper half space.
The most common form of theta function is that occurring in the theory of elliptic functions. With respect to one of the complex variables (conventionally called z), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions, making it a quasiperiodic function. In the abstract theory this quasiperiodicity comes from the cohomology class of a line bundle on a complex torus, a condition of descent.
One interpretation of theta functions when dealing with the heat equation is that "a theta function is a special function that describes the evolution of temperature on a segment domain subject to certain boundary conditions".
Throughout this article,
(
e
π
i
τ
)
α
{\displaystyle (e^{\pi i\tau })^{\alpha }}
should be interpreted as
e
α
π
i
τ
{\displaystyle e^{\alpha \pi i\tau }}
(in order to resolve issues of choice of branch).
== Jacobi theta function ==
There are several closely related functions called Jacobi theta functions, and many different and incompatible systems of notation for them.
One Jacobi theta function (named after Carl Gustav Jacob Jacobi) is a function defined for two complex variables z and τ, where z can be any complex number and τ is the half-period ratio, confined to the upper half-plane, which means it has a positive imaginary part. It is given by the formula
ϑ
(
z
;
τ
)
=
∑
n
=
−
∞
∞
exp
(
π
i
n
2
τ
+
2
π
i
n
z
)
=
1
+
2
∑
n
=
1
∞
q
n
2
cos
(
2
π
n
z
)
=
∑
n
=
−
∞
∞
q
n
2
η
n
{\displaystyle {\begin{aligned}\vartheta (z;\tau )&=\sum _{n=-\infty }^{\infty }\exp \left(\pi in^{2}\tau +2\pi inz\right)\\&=1+2\sum _{n=1}^{\infty }q^{n^{2}}\cos(2\pi nz)\\&=\sum _{n=-\infty }^{\infty }q^{n^{2}}\eta ^{n}\end{aligned}}}
where q = exp(πiτ) is the nome and η = exp(2πiz). It is a Jacobi form. The restriction ensures that it is an absolutely convergent series. At fixed τ, this is a Fourier series for a 1-periodic entire function of z. Accordingly, the theta function is 1-periodic in z:
ϑ
(
z
+
1
;
τ
)
=
ϑ
(
z
;
τ
)
.
{\displaystyle \vartheta (z+1;\tau )=\vartheta (z;\tau ).}
By completing the square, it is also τ-quasiperiodic in z, with
ϑ
(
z
+
τ
;
τ
)
=
exp
(
−
π
i
(
τ
+
2
z
)
)
ϑ
(
z
;
τ
)
.
{\displaystyle \vartheta (z+\tau ;\tau )=\exp {\bigl (}-\pi i(\tau +2z){\bigr )}\vartheta (z;\tau ).}
Thus, in general,
ϑ
(
z
+
a
+
b
τ
;
τ
)
=
exp
(
−
π
i
b
2
τ
−
2
π
i
b
z
)
ϑ
(
z
;
τ
)
{\displaystyle \vartheta (z+a+b\tau ;\tau )=\exp \left(-\pi ib^{2}\tau -2\pi ibz\right)\vartheta (z;\tau )}
for any integers a and b.
For any fixed
τ
{\displaystyle \tau }
, the function is an entire function on the complex plane, so by Liouville's theorem, it cannot be doubly periodic in
1
,
τ
{\displaystyle 1,\tau }
unless it is constant, and so the best we can do is to make it periodic in
1
{\displaystyle 1}
and quasi-periodic in
τ
{\displaystyle \tau }
. Indeed, since
|
ϑ
(
z
+
a
+
b
τ
;
τ
)
ϑ
(
z
;
τ
)
|
=
exp
(
π
(
b
2
ℑ
(
τ
)
+
2
b
ℑ
(
z
)
)
)
{\displaystyle \left|{\frac {\vartheta (z+a+b\tau ;\tau )}{\vartheta (z;\tau )}}\right|=\exp \left(\pi (b^{2}\Im (\tau )+2b\Im (z))\right)}
and
ℑ
(
τ
)
>
0
{\displaystyle \Im (\tau )>0}
, the function
ϑ
(
z
,
τ
)
{\displaystyle \vartheta (z,\tau )}
is unbounded, as required by Liouville's theorem.
It is in fact the most general entire function with 2 quasi-periods, in the following sense:
== Auxiliary functions ==
The Jacobi theta function defined above is sometimes considered along with three auxiliary theta functions, in which case it is written with a double 0 subscript:
ϑ
00
(
z
;
τ
)
=
ϑ
(
z
;
τ
)
{\displaystyle \vartheta _{00}(z;\tau )=\vartheta (z;\tau )}
The auxiliary (or half-period) functions are defined by
ϑ
01
(
z
;
τ
)
=
ϑ
(
z
+
1
2
;
τ
)
ϑ
10
(
z
;
τ
)
=
exp
(
1
4
π
i
τ
+
π
i
z
)
ϑ
(
z
+
1
2
τ
;
τ
)
ϑ
11
(
z
;
τ
)
=
exp
(
1
4
π
i
τ
+
π
i
(
z
+
1
2
)
)
ϑ
(
z
+
1
2
τ
+
1
2
;
τ
)
.
{\displaystyle {\begin{aligned}\vartheta _{01}(z;\tau )&=\vartheta \left(z+{\tfrac {1}{2}};\tau \right)\\[3pt]\vartheta _{10}(z;\tau )&=\exp \left({\tfrac {1}{4}}\pi i\tau +\pi iz\right)\vartheta \left(z+{\tfrac {1}{2}}\tau ;\tau \right)\\[3pt]\vartheta _{11}(z;\tau )&=\exp \left({\tfrac {1}{4}}\pi i\tau +\pi i\left(z+{\tfrac {1}{2}}\right)\right)\vartheta \left(z+{\tfrac {1}{2}}\tau +{\tfrac {1}{2}};\tau \right).\end{aligned}}}
This notation follows Riemann and Mumford; Jacobi's original formulation was in terms of the nome q = eiπτ rather than τ. In Jacobi's notation the θ-functions are written:
θ
1
(
z
;
q
)
=
θ
1
(
π
z
,
q
)
=
−
ϑ
11
(
z
;
τ
)
θ
2
(
z
;
q
)
=
θ
2
(
π
z
,
q
)
=
ϑ
10
(
z
;
τ
)
θ
3
(
z
;
q
)
=
θ
3
(
π
z
,
q
)
=
ϑ
00
(
z
;
τ
)
θ
4
(
z
;
q
)
=
θ
4
(
π
z
,
q
)
=
ϑ
01
(
z
;
τ
)
{\displaystyle {\begin{aligned}\theta _{1}(z;q)&=\theta _{1}(\pi z,q)=-\vartheta _{11}(z;\tau )\\\theta _{2}(z;q)&=\theta _{2}(\pi z,q)=\vartheta _{10}(z;\tau )\\\theta _{3}(z;q)&=\theta _{3}(\pi z,q)=\vartheta _{00}(z;\tau )\\\theta _{4}(z;q)&=\theta _{4}(\pi z,q)=\vartheta _{01}(z;\tau )\end{aligned}}}
The above definitions of the Jacobi theta functions are by no means unique. See Jacobi theta functions (notational variations) for further discussion.
If we set z = 0 in the above theta functions, we obtain four functions of τ only, defined on the upper half-plane. These functions are called Theta Nullwert functions, based on the German term for zero value because of the annullation of the left entry in the theta function expression. Alternatively, we obtain four functions of q only, defined on the unit disk
|
q
|
<
1
{\displaystyle |q|<1}
. They are sometimes called theta constants:
ϑ
11
(
0
;
τ
)
=
−
θ
1
(
q
)
=
−
∑
n
=
−
∞
∞
(
−
1
)
n
−
1
/
2
q
(
n
+
1
/
2
)
2
ϑ
10
(
0
;
τ
)
=
θ
2
(
q
)
=
∑
n
=
−
∞
∞
q
(
n
+
1
/
2
)
2
ϑ
00
(
0
;
τ
)
=
θ
3
(
q
)
=
∑
n
=
−
∞
∞
q
n
2
ϑ
01
(
0
;
τ
)
=
θ
4
(
q
)
=
∑
n
=
−
∞
∞
(
−
1
)
n
q
n
2
{\displaystyle {\begin{aligned}\vartheta _{11}(0;\tau )&=-\theta _{1}(q)=-\sum _{n=-\infty }^{\infty }(-1)^{n-1/2}q^{(n+1/2)^{2}}\\\vartheta _{10}(0;\tau )&=\theta _{2}(q)=\sum _{n=-\infty }^{\infty }q^{(n+1/2)^{2}}\\\vartheta _{00}(0;\tau )&=\theta _{3}(q)=\sum _{n=-\infty }^{\infty }q^{n^{2}}\\\vartheta _{01}(0;\tau )&=\theta _{4}(q)=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{n^{2}}\end{aligned}}}
with the nome q = eiπτ.
Observe that
θ
1
(
q
)
=
0
{\displaystyle \theta _{1}(q)=0}
.
These can be used to define a variety of modular forms, and to parametrize certain curves; in particular, the Jacobi identity is
θ
2
(
q
)
4
+
θ
4
(
q
)
4
=
θ
3
(
q
)
4
{\displaystyle \theta _{2}(q)^{4}+\theta _{4}(q)^{4}=\theta _{3}(q)^{4}}
or equivalently,
ϑ
01
(
0
;
τ
)
4
+
ϑ
10
(
0
;
τ
)
4
=
ϑ
00
(
0
;
τ
)
4
{\displaystyle \vartheta _{01}(0;\tau )^{4}+\vartheta _{10}(0;\tau )^{4}=\vartheta _{00}(0;\tau )^{4}}
which is the Fermat curve of degree four.
== Jacobi identities ==
Jacobi's identities describe how theta functions transform under the modular group, which is generated by τ ↦ τ + 1 and τ ↦ −1/τ. Equations for the first transform are easily found since adding one to τ in the exponent has the same effect as adding 1/2 to z (n ≡ n2 mod 2). For the second, let
α
=
(
−
i
τ
)
1
2
exp
(
π
τ
i
z
2
)
.
{\displaystyle \alpha =(-i\tau )^{\frac {1}{2}}\exp \left({\frac {\pi }{\tau }}iz^{2}\right).}
Then
ϑ
00
(
z
τ
;
−
1
τ
)
=
α
ϑ
00
(
z
;
τ
)
ϑ
01
(
z
τ
;
−
1
τ
)
=
α
ϑ
10
(
z
;
τ
)
ϑ
10
(
z
τ
;
−
1
τ
)
=
α
ϑ
01
(
z
;
τ
)
ϑ
11
(
z
τ
;
−
1
τ
)
=
−
i
α
ϑ
11
(
z
;
τ
)
.
{\displaystyle {\begin{aligned}\vartheta _{00}\!\left({\frac {z}{\tau }};{\frac {-1}{\tau }}\right)&=\alpha \,\vartheta _{00}(z;\tau )\quad &\vartheta _{01}\!\left({\frac {z}{\tau }};{\frac {-1}{\tau }}\right)&=\alpha \,\vartheta _{10}(z;\tau )\\[3pt]\vartheta _{10}\!\left({\frac {z}{\tau }};{\frac {-1}{\tau }}\right)&=\alpha \,\vartheta _{01}(z;\tau )\quad &\vartheta _{11}\!\left({\frac {z}{\tau }};{\frac {-1}{\tau }}\right)&=-i\alpha \,\vartheta _{11}(z;\tau ).\end{aligned}}}
== Theta functions in terms of the nome ==
Instead of expressing the Theta functions in terms of z and τ, we may express them in terms of arguments w and the nome q, where w = eπiz and q = eπiτ. In this form, the functions become
ϑ
00
(
w
,
q
)
=
∑
n
=
−
∞
∞
(
w
2
)
n
q
n
2
ϑ
01
(
w
,
q
)
=
∑
n
=
−
∞
∞
(
−
1
)
n
(
w
2
)
n
q
n
2
ϑ
10
(
w
,
q
)
=
∑
n
=
−
∞
∞
(
w
2
)
n
+
1
2
q
(
n
+
1
2
)
2
ϑ
11
(
w
,
q
)
=
i
∑
n
=
−
∞
∞
(
−
1
)
n
(
w
2
)
n
+
1
2
q
(
n
+
1
2
)
2
.
{\displaystyle {\begin{aligned}\vartheta _{00}(w,q)&=\sum _{n=-\infty }^{\infty }\left(w^{2}\right)^{n}q^{n^{2}}\quad &\vartheta _{01}(w,q)&=\sum _{n=-\infty }^{\infty }(-1)^{n}\left(w^{2}\right)^{n}q^{n^{2}}\\[3pt]\vartheta _{10}(w,q)&=\sum _{n=-\infty }^{\infty }\left(w^{2}\right)^{n+{\frac {1}{2}}}q^{\left(n+{\frac {1}{2}}\right)^{2}}\quad &\vartheta _{11}(w,q)&=i\sum _{n=-\infty }^{\infty }(-1)^{n}\left(w^{2}\right)^{n+{\frac {1}{2}}}q^{\left(n+{\frac {1}{2}}\right)^{2}}.\end{aligned}}}
We see that the theta functions can also be defined in terms of w and q, without a direct reference to the exponential function. These formulas can, therefore, be used to define the Theta functions over other fields where the exponential function might not be everywhere defined, such as fields of p-adic numbers.
== Product representations ==
The Jacobi triple product (a special case of the Macdonald identities) tells us that for complex numbers w and q with |q| < 1 and w ≠ 0 we have
∏
m
=
1
∞
(
1
−
q
2
m
)
(
1
+
w
2
q
2
m
−
1
)
(
1
+
w
−
2
q
2
m
−
1
)
=
∑
n
=
−
∞
∞
w
2
n
q
n
2
.
{\displaystyle \prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1+w^{2}q^{2m-1}\right)\left(1+w^{-2}q^{2m-1}\right)=\sum _{n=-\infty }^{\infty }w^{2n}q^{n^{2}}.}
It can be proven by elementary means, as for instance in Hardy and Wright's An Introduction to the Theory of Numbers.
If we express the theta function in terms of the nome q = eπiτ (noting some authors instead set q = e2πiτ) and take w = eπiz then
ϑ
(
z
;
τ
)
=
∑
n
=
−
∞
∞
exp
(
π
i
τ
n
2
)
exp
(
2
π
i
z
n
)
=
∑
n
=
−
∞
∞
w
2
n
q
n
2
.
{\displaystyle \vartheta (z;\tau )=\sum _{n=-\infty }^{\infty }\exp(\pi i\tau n^{2})\exp(2\pi izn)=\sum _{n=-\infty }^{\infty }w^{2n}q^{n^{2}}.}
We therefore obtain a product formula for the theta function in the form
ϑ
(
z
;
τ
)
=
∏
m
=
1
∞
(
1
−
exp
(
2
m
π
i
τ
)
)
(
1
+
exp
(
(
2
m
−
1
)
π
i
τ
+
2
π
i
z
)
)
(
1
+
exp
(
(
2
m
−
1
)
π
i
τ
−
2
π
i
z
)
)
.
{\displaystyle \vartheta (z;\tau )=\prod _{m=1}^{\infty }{\big (}1-\exp(2m\pi i\tau ){\big )}{\Big (}1+\exp {\big (}(2m-1)\pi i\tau +2\pi iz{\big )}{\Big )}{\Big (}1+\exp {\big (}(2m-1)\pi i\tau -2\pi iz{\big )}{\Big )}.}
In terms of w and q:
ϑ
(
z
;
τ
)
=
∏
m
=
1
∞
(
1
−
q
2
m
)
(
1
+
q
2
m
−
1
w
2
)
(
1
+
q
2
m
−
1
w
2
)
=
(
q
2
;
q
2
)
∞
(
−
w
2
q
;
q
2
)
∞
(
−
q
w
2
;
q
2
)
∞
=
(
q
2
;
q
2
)
∞
θ
(
−
w
2
q
;
q
2
)
{\displaystyle {\begin{aligned}\vartheta (z;\tau )&=\prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1+q^{2m-1}w^{2}\right)\left(1+{\frac {q^{2m-1}}{w^{2}}}\right)\\&=\left(q^{2};q^{2}\right)_{\infty }\,\left(-w^{2}q;q^{2}\right)_{\infty }\,\left(-{\frac {q}{w^{2}}};q^{2}\right)_{\infty }\\&=\left(q^{2};q^{2}\right)_{\infty }\,\theta \left(-w^{2}q;q^{2}\right)\end{aligned}}}
where ( ; )∞ is the q-Pochhammer symbol and θ( ; ) is the q-theta function. Expanding terms out, the Jacobi triple product can also be written
∏
m
=
1
∞
(
1
−
q
2
m
)
(
1
+
(
w
2
+
w
−
2
)
q
2
m
−
1
+
q
4
m
−
2
)
,
{\displaystyle \prod _{m=1}^{\infty }\left(1-q^{2m}\right){\Big (}1+\left(w^{2}+w^{-2}\right)q^{2m-1}+q^{4m-2}{\Big )},}
which we may also write as
ϑ
(
z
∣
q
)
=
∏
m
=
1
∞
(
1
−
q
2
m
)
(
1
+
2
cos
(
2
π
z
)
q
2
m
−
1
+
q
4
m
−
2
)
.
{\displaystyle \vartheta (z\mid q)=\prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1+2\cos(2\pi z)q^{2m-1}+q^{4m-2}\right).}
This form is valid in general but clearly is of particular interest when z is real. Similar product formulas for the auxiliary theta functions are
ϑ
01
(
z
∣
q
)
=
∏
m
=
1
∞
(
1
−
q
2
m
)
(
1
−
2
cos
(
2
π
z
)
q
2
m
−
1
+
q
4
m
−
2
)
,
ϑ
10
(
z
∣
q
)
=
2
q
1
4
cos
(
π
z
)
∏
m
=
1
∞
(
1
−
q
2
m
)
(
1
+
2
cos
(
2
π
z
)
q
2
m
+
q
4
m
)
,
ϑ
11
(
z
∣
q
)
=
−
2
q
1
4
sin
(
π
z
)
∏
m
=
1
∞
(
1
−
q
2
m
)
(
1
−
2
cos
(
2
π
z
)
q
2
m
+
q
4
m
)
.
{\displaystyle {\begin{aligned}\vartheta _{01}(z\mid q)&=\prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1-2\cos(2\pi z)q^{2m-1}+q^{4m-2}\right),\\[3pt]\vartheta _{10}(z\mid q)&=2q^{\frac {1}{4}}\cos(\pi z)\prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1+2\cos(2\pi z)q^{2m}+q^{4m}\right),\\[3pt]\vartheta _{11}(z\mid q)&=-2q^{\frac {1}{4}}\sin(\pi z)\prod _{m=1}^{\infty }\left(1-q^{2m}\right)\left(1-2\cos(2\pi z)q^{2m}+q^{4m}\right).\end{aligned}}}
In particular,
lim
q
→
0
ϑ
10
(
z
∣
q
)
2
q
1
4
=
cos
(
π
z
)
,
lim
q
→
0
−
ϑ
11
(
z
∣
q
)
2
q
−
1
4
=
sin
(
π
z
)
{\displaystyle \lim _{q\to 0}{\frac {\vartheta _{10}(z\mid q)}{2q^{\frac {1}{4}}}}=\cos(\pi z),\quad \lim _{q\to 0}{\frac {-\vartheta _{11}(z\mid q)}{2q^{-{\frac {1}{4}}}}}=\sin(\pi z)}
so we may interpret them as one-parameter deformations of the periodic functions
sin
,
cos
{\displaystyle \sin ,\cos }
, again validating the interpretation of the theta function as the most general 2 quasi-period function.
== Integral representations ==
The Jacobi theta functions have the following integral representations:
ϑ
00
(
z
;
τ
)
=
−
i
∫
i
−
∞
i
+
∞
e
i
π
τ
u
2
cos
(
2
π
u
z
+
π
u
)
sin
(
π
u
)
d
u
;
ϑ
01
(
z
;
τ
)
=
−
i
∫
i
−
∞
i
+
∞
e
i
π
τ
u
2
cos
(
2
π
u
z
)
sin
(
π
u
)
d
u
;
ϑ
10
(
z
;
τ
)
=
−
i
e
i
π
z
+
1
4
i
π
τ
∫
i
−
∞
i
+
∞
e
i
π
τ
u
2
cos
(
2
π
u
z
+
π
u
+
π
τ
u
)
sin
(
π
u
)
d
u
;
ϑ
11
(
z
;
τ
)
=
e
i
π
z
+
1
4
i
π
τ
∫
i
−
∞
i
+
∞
e
i
π
τ
u
2
cos
(
2
π
u
z
+
π
τ
u
)
sin
(
π
u
)
d
u
.
{\displaystyle {\begin{aligned}\vartheta _{00}(z;\tau )&=-i\int _{i-\infty }^{i+\infty }e^{i\pi \tau u^{2}}{\frac {\cos(2\pi uz+\pi u)}{\sin(\pi u)}}\mathrm {d} u;\\[6pt]\vartheta _{01}(z;\tau )&=-i\int _{i-\infty }^{i+\infty }e^{i\pi \tau u^{2}}{\frac {\cos(2\pi uz)}{\sin(\pi u)}}\mathrm {d} u;\\[6pt]\vartheta _{10}(z;\tau )&=-ie^{i\pi z+{\frac {1}{4}}i\pi \tau }\int _{i-\infty }^{i+\infty }e^{i\pi \tau u^{2}}{\frac {\cos(2\pi uz+\pi u+\pi \tau u)}{\sin(\pi u)}}\mathrm {d} u;\\[6pt]\vartheta _{11}(z;\tau )&=e^{i\pi z+{\frac {1}{4}}i\pi \tau }\int _{i-\infty }^{i+\infty }e^{i\pi \tau u^{2}}{\frac {\cos(2\pi uz+\pi \tau u)}{\sin(\pi u)}}\mathrm {d} u.\end{aligned}}}
The Theta Nullwert function
θ
3
(
q
)
{\displaystyle \theta _{3}(q)}
as this integral identity:
θ
3
(
q
)
=
1
+
4
q
ln
(
1
/
q
)
π
∫
0
∞
exp
[
−
ln
(
1
/
q
)
x
2
]
{
1
−
q
2
cos
[
2
ln
(
1
/
q
)
x
]
}
1
−
2
q
2
cos
[
2
ln
(
1
/
q
)
x
]
+
q
4
d
x
{\displaystyle \theta _{3}(q)=1+{\frac {4q{\sqrt {\ln(1/q)}}}{\sqrt {\pi }}}\int _{0}^{\infty }{\frac {\exp[-\ln(1/q)\,x^{2}]\{1-q^{2}\cos[2\ln(1/q)\,x]\}}{1-2q^{2}\cos[2\ln(1/q)\,x]+q^{4}}}\,\mathrm {d} x}
This formula was discussed in the essay Square series generating function transformations by the mathematician Maxie Schmidt from Georgia in Atlanta.
Based on this formula following three eminent examples are given:
[
2
π
K
(
1
2
2
)
]
1
/
2
=
θ
3
[
exp
(
−
π
)
]
=
1
+
4
exp
(
−
π
)
∫
0
∞
exp
(
−
π
x
2
)
[
1
−
exp
(
−
2
π
)
cos
(
2
π
x
)
]
1
−
2
exp
(
−
2
π
)
cos
(
2
π
x
)
+
exp
(
−
4
π
)
d
x
{\displaystyle {\biggl [}{\frac {2}{\pi }}K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}{\biggr ]}^{1/2}=\theta _{3}{\bigl [}\exp(-\pi ){\bigr ]}=1+4\exp(-\pi )\int _{0}^{\infty }{\frac {\exp(-\pi x^{2})[1-\exp(-2\pi )\cos(2\pi x)]}{1-2\exp(-2\pi )\cos(2\pi x)+\exp(-4\pi )}}\,\mathrm {d} x}
[
2
π
K
(
2
−
1
)
]
1
/
2
=
θ
3
[
exp
(
−
2
π
)
]
=
1
+
4
2
4
exp
(
−
2
π
)
∫
0
∞
exp
(
−
2
π
x
2
)
[
1
−
exp
(
−
2
2
π
)
cos
(
2
2
π
x
)
]
1
−
2
exp
(
−
2
2
π
)
cos
(
2
2
π
x
)
+
exp
(
−
4
2
π
)
d
x
{\displaystyle {\biggl [}{\frac {2}{\pi }}K({\sqrt {2}}-1){\biggr ]}^{1/2}=\theta _{3}{\bigl [}\exp(-{\sqrt {2}}\,\pi ){\bigr ]}=1+4\,{\sqrt[{4}]{2}}\exp(-{\sqrt {2}}\,\pi )\int _{0}^{\infty }{\frac {\exp(-{\sqrt {2}}\,\pi x^{2})[1-\exp(-2{\sqrt {2}}\,\pi )\cos(2{\sqrt {2}}\,\pi x)]}{1-2\exp(-2{\sqrt {2}}\,\pi )\cos(2{\sqrt {2}}\,\pi x)+\exp(-4{\sqrt {2}}\,\pi )}}\,\mathrm {d} x}
{
2
π
K
[
sin
(
π
12
)
]
}
1
/
2
=
θ
3
[
exp
(
−
3
π
)
]
=
1
+
4
3
4
exp
(
−
3
π
)
∫
0
∞
exp
(
−
3
π
x
2
)
[
1
−
exp
(
−
2
3
π
)
cos
(
2
3
π
x
)
]
1
−
2
exp
(
−
2
3
π
)
cos
(
2
3
π
x
)
+
exp
(
−
4
3
π
)
d
x
{\displaystyle {\biggl \{}{\frac {2}{\pi }}K{\bigl [}\sin {\bigl (}{\frac {\pi }{12}}{\bigr )}{\bigr ]}{\biggr \}}^{1/2}=\theta _{3}{\bigl [}\exp(-{\sqrt {3}}\,\pi ){\bigr ]}=1+4\,{\sqrt[{4}]{3}}\exp(-{\sqrt {3}}\,\pi )\int _{0}^{\infty }{\frac {\exp(-{\sqrt {3}}\,\pi x^{2})[1-\exp(-2{\sqrt {3}}\,\pi )\cos(2{\sqrt {3}}\,\pi x)]}{1-2\exp(-2{\sqrt {3}}\,\pi )\cos(2{\sqrt {3}}\,\pi x)+\exp(-4{\sqrt {3}}\,\pi )}}\,\mathrm {d} x}
Furthermore, the theta examples
θ
3
(
1
2
)
{\displaystyle \theta _{3}({\tfrac {1}{2}})}
and
θ
3
(
1
3
)
{\displaystyle \theta _{3}({\tfrac {1}{3}})}
shall be displayed:
θ
3
(
1
2
)
=
1
+
2
∑
n
=
1
∞
1
2
n
2
=
1
+
2
π
−
1
/
2
ln
(
2
)
∫
0
∞
exp
[
−
ln
(
2
)
x
2
]
{
16
−
4
cos
[
2
ln
(
2
)
x
]
}
17
−
8
cos
[
2
ln
(
2
)
x
]
d
x
{\displaystyle \theta _{3}\left({\frac {1}{2}}\right)=1+2\sum _{n=1}^{\infty }{\frac {1}{2^{n^{2}}}}=1+2\pi ^{-1/2}{\sqrt {\ln(2)}}\int _{0}^{\infty }{\frac {\exp[-\ln(2)\,x^{2}]\{16-4\cos[2\ln(2)\,x]\}}{17-8\cos[2\ln(2)\,x]}}\,\mathrm {d} x}
θ
3
(
1
2
)
=
2.128936827211877158669
…
{\displaystyle \theta _{3}\left({\frac {1}{2}}\right)=2.128936827211877158669\ldots }
θ
3
(
1
3
)
=
1
+
2
∑
n
=
1
∞
1
3
n
2
=
1
+
4
3
π
−
1
/
2
ln
(
3
)
∫
0
∞
exp
[
−
ln
(
3
)
x
2
]
{
81
−
9
cos
[
2
ln
(
3
)
x
]
}
82
−
18
cos
[
2
ln
(
3
)
x
]
d
x
{\displaystyle \theta _{3}\left({\frac {1}{3}}\right)=1+2\sum _{n=1}^{\infty }{\frac {1}{3^{n^{2}}}}=1+{\frac {4}{3}}\pi ^{-1/2}{\sqrt {\ln(3)}}\int _{0}^{\infty }{\frac {\exp[-\ln(3)\,x^{2}]\{81-9\cos[2\ln(3)\,x]\}}{82-18\cos[2\ln(3)\,x]}}\,\mathrm {d} x}
θ
3
(
1
3
)
=
1.691459681681715341348
…
{\displaystyle \theta _{3}\left({\frac {1}{3}}\right)=1.691459681681715341348\ldots }
== Some interesting relations ==
If
|
q
|
<
1
{\displaystyle |q|<1}
and
a
>
0
{\displaystyle a>0}
, then the following theta functions
θ
3
(
a
,
b
;
q
)
=
∑
n
=
−
∞
∞
q
a
n
2
+
b
n
{\displaystyle \theta _{3}(a,b;q)=\sum _{n=-\infty }^{\infty }q^{an^{2}+bn}}
θ
4
(
a
,
b
;
q
)
=
∑
n
=
−
∞
∞
(
−
1
)
n
q
a
n
2
+
b
n
{\displaystyle \theta _{4}(a,b;q)=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{an^{2}+bn}}
have interesting arithmetical and modular properties. When
a
,
b
,
p
{\displaystyle a,b,p}
are positive integers, then
log
(
θ
3
(
p
2
,
p
2
−
a
;
q
)
θ
3
(
p
2
,
p
2
−
b
;
q
)
)
=
−
∑
n
=
1
∞
q
n
(
∑
d
|
n
n
/
d
≡
±
a
(
p
)
(
−
1
)
d
d
−
∑
d
|
n
n
/
d
≡
±
b
(
p
)
(
−
1
)
d
d
)
{\displaystyle \log \left({\frac {\theta _{3}\left({\frac {p}{2}},{\frac {p}{2}}-a;q\right)}{\theta _{3}\left({\frac {p}{2}},{\frac {p}{2}}-b;q\right)}}\right)=-\sum _{n=1}^{\infty }q^{n}\left(\sum _{\begin{array}{cc}d|n\\n/d\equiv \pm a(p)\end{array}}{\frac {(-1)^{d}}{d}}-\sum _{\begin{array}{cc}d|n\\n/d\equiv \pm b(p)\end{array}}{\frac {(-1)^{d}}{d}}\right)}
log
(
θ
4
(
p
2
,
p
2
−
a
;
q
)
θ
4
(
p
2
,
p
2
−
b
;
q
)
)
=
−
∑
n
=
1
∞
q
n
(
∑
d
|
n
n
/
d
≡
±
a
(
p
)
1
d
−
∑
d
|
n
n
/
d
≡
±
b
(
p
)
1
d
)
{\displaystyle \log \left({\frac {\theta _{4}\left({\frac {p}{2}},{\frac {p}{2}}-a;q\right)}{\theta _{4}\left({\frac {p}{2}},{\frac {p}{2}}-b;q\right)}}\right)=-\sum _{n=1}^{\infty }q^{n}\left(\sum _{\begin{array}{cc}d|n\\n/d\equiv \pm a(p)\end{array}}{\frac {1}{d}}-\sum _{\begin{array}{cc}d|n\\n/d\equiv \pm b(p)\end{array}}{\frac {1}{d}}\right)}
Also if
q
=
e
π
i
z
{\displaystyle q=e^{\pi iz}}
,
I
m
(
z
)
>
0
{\displaystyle Im(z)>0}
, the functions with :
ϑ
+
(
z
)
=
θ
+
(
a
,
p
;
z
)
=
q
p
/
8
+
a
2
/
(
2
p
)
−
a
/
2
θ
3
(
p
2
,
p
2
−
a
;
q
)
{\displaystyle \vartheta _{+}(z)=\theta _{+}(a,p;z)=q^{p/8+a^{2}/(2p)-a/2}\theta _{3}\left({\frac {p}{2}},{\frac {p}{2}}-a;q\right)}
and
ϑ
−
(
z
)
=
θ
−
(
a
,
p
;
z
)
=
q
p
/
8
+
a
2
/
(
2
p
)
−
a
/
2
θ
4
(
p
2
,
p
2
−
a
;
q
)
{\displaystyle \vartheta _{-}(z)=\theta _{-}(a,p;z)=q^{p/8+a^{2}/(2p)-a/2}\theta _{4}\left({\frac {p}{2}},{\frac {p}{2}}-a;q\right)}
are modular forms with weight
1
/
2
{\displaystyle 1/2}
in
Γ
(
2
p
)
{\displaystyle \Gamma (2p)}
i.e. If
a
1
,
b
1
,
c
1
,
d
1
{\displaystyle a_{1},b_{1},c_{1},d_{1}}
are integers such that
a
1
,
d
1
≡
1
(
2
p
)
{\displaystyle a_{1},d_{1}\equiv 1(2p)}
,
b
1
,
c
1
≡
0
(
2
p
)
{\displaystyle b_{1},c_{1}\equiv 0(2p)}
and
a
1
d
1
−
b
1
c
1
=
1
{\displaystyle a_{1}d_{1}-b_{1}c_{1}=1}
there exists
ϵ
±
=
ϵ
±
(
a
1
,
b
1
,
c
1
,
d
1
)
{\displaystyle \epsilon _{\pm }=\epsilon _{\pm }(a_{1},b_{1},c_{1},d_{1})}
,
(
ϵ
±
)
24
=
1
{\displaystyle (\epsilon _{\pm })^{24}=1}
, such that for all complex numbers
z
{\displaystyle z}
with
I
m
(
z
)
>
0
{\displaystyle Im(z)>0}
, we have
ϑ
±
(
a
1
z
+
b
1
c
1
z
+
d
1
)
=
ϵ
±
c
1
z
+
d
1
ϑ
±
(
z
)
{\displaystyle \vartheta _{\pm }\left({\frac {a_{1}z+b_{1}}{c_{1}z+d_{1}}}\right)=\epsilon _{\pm }{\sqrt {c_{1}z+d_{1}}}\vartheta _{\pm }(z)}
== Explicit values ==
=== Lemniscatic values ===
Proper credit for most of these results goes to Ramanujan. See Ramanujan's lost notebook and a relevant reference at Euler function. The Ramanujan results quoted at Euler function plus a few elementary operations give the results below, so they are either in Ramanujan's lost notebook or follow immediately from it. See also Yi (2004). Define,
φ
(
q
)
=
ϑ
00
(
0
;
τ
)
=
θ
3
(
0
;
q
)
=
∑
n
=
−
∞
∞
q
n
2
{\displaystyle \quad \varphi (q)=\vartheta _{00}(0;\tau )=\theta _{3}(0;q)=\sum _{n=-\infty }^{\infty }q^{n^{2}}}
with the nome
q
=
e
π
i
τ
,
{\displaystyle q=e^{\pi i\tau },}
τ
=
n
−
1
,
{\displaystyle \tau =n{\sqrt {-1}},}
and Dedekind eta function
η
(
τ
)
.
{\displaystyle \eta (\tau ).}
Then for
n
=
1
,
2
,
3
,
…
{\displaystyle n=1,2,3,\dots }
φ
(
e
−
π
)
=
π
4
Γ
(
3
4
)
=
2
η
(
−
1
)
φ
(
e
−
2
π
)
=
π
4
Γ
(
3
4
)
2
+
2
2
φ
(
e
−
3
π
)
=
π
4
Γ
(
3
4
)
1
+
3
108
8
φ
(
e
−
4
π
)
=
π
4
Γ
(
3
4
)
2
+
8
4
4
φ
(
e
−
5
π
)
=
π
4
Γ
(
3
4
)
2
+
5
5
φ
(
e
−
6
π
)
=
π
4
Γ
(
3
4
)
1
4
+
3
4
+
4
4
+
9
4
12
3
8
φ
(
e
−
7
π
)
=
π
4
Γ
(
3
4
)
13
+
7
+
7
+
3
7
14
3
8
⋅
7
16
φ
(
e
−
8
π
)
=
π
4
Γ
(
3
4
)
2
+
2
+
128
8
4
φ
(
e
−
9
π
)
=
π
4
Γ
(
3
4
)
1
+
2
+
2
3
3
3
φ
(
e
−
10
π
)
=
π
4
Γ
(
3
4
)
64
4
+
80
4
+
81
4
+
100
4
200
4
φ
(
e
−
11
π
)
=
π
4
Γ
(
3
4
)
11
+
11
+
(
5
+
3
3
+
11
+
33
)
−
44
+
33
3
3
+
(
−
5
+
3
3
−
11
+
33
)
44
+
33
3
3
52180524
8
φ
(
e
−
12
π
)
=
π
4
Γ
(
3
4
)
1
4
+
2
4
+
3
4
+
4
4
+
9
4
+
18
4
+
24
4
2
108
8
φ
(
e
−
13
π
)
=
π
4
Γ
(
3
4
)
13
+
8
13
+
(
11
−
6
3
+
13
)
143
+
78
3
3
+
(
11
+
6
3
+
13
)
143
−
78
3
3
19773
4
φ
(
e
−
14
π
)
=
π
4
Γ
(
3
4
)
13
+
7
+
7
+
3
7
+
10
+
2
7
+
28
8
4
+
7
28
7
16
φ
(
e
−
15
π
)
=
π
4
Γ
(
3
4
)
7
+
3
3
+
5
+
15
+
60
4
+
1500
4
12
3
8
⋅
5
2
φ
(
e
−
16
π
)
=
φ
(
e
−
4
π
)
+
π
4
Γ
(
3
4
)
1
+
2
4
128
16
φ
(
e
−
17
π
)
=
π
4
Γ
(
3
4
)
2
(
1
+
17
4
)
+
17
8
5
+
17
17
+
17
17
2
φ
(
e
−
20
π
)
=
φ
(
e
−
5
π
)
+
π
4
Γ
(
3
4
)
3
+
2
5
4
5
2
6
φ
(
e
−
36
π
)
=
3
φ
(
e
−
9
π
)
+
2
φ
(
e
−
4
π
)
−
φ
(
e
−
π
)
+
π
4
Γ
(
3
4
)
2
4
+
18
4
+
216
4
3
{\displaystyle {\begin{aligned}\varphi \left(e^{-\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}={\sqrt {2}}\,\eta \left({\sqrt {-1}}\right)\\\varphi \left(e^{-2\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {2+{\sqrt {2}}}}{2}}\\\varphi \left(e^{-3\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {1+{\sqrt {3}}}}{\sqrt[{8}]{108}}}\\\varphi \left(e^{-4\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {2+{\sqrt[{4}]{8}}}{4}}\\\varphi \left(e^{-5\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\sqrt {\frac {2+{\sqrt {5}}}{5}}}\\\varphi \left(e^{-6\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {{\sqrt[{4}]{1}}+{\sqrt[{4}]{3}}+{\sqrt[{4}]{4}}+{\sqrt[{4}]{9}}}}{\sqrt[{8}]{12^{3}}}}\\\varphi \left(e^{-7\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {{\sqrt {13+{\sqrt {7}}}}+{\sqrt {7+3{\sqrt {7}}}}}}{{\sqrt[{8}]{14^{3}}}\cdot {\sqrt[{16}]{7}}}}\\\varphi \left(e^{-8\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {{\sqrt {2+{\sqrt {2}}}}+{\sqrt[{8}]{128}}}{4}}\\\varphi \left(e^{-9\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {1+{\sqrt[{3}]{2+2{\sqrt {3}}}}}{3}}\\\varphi \left(e^{-10\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {{\sqrt[{4}]{64}}+{\sqrt[{4}]{80}}+{\sqrt[{4}]{81}}+{\sqrt[{4}]{100}}}}{\sqrt[{4}]{200}}}\\\varphi \left(e^{-11\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {11+{\sqrt {11}}+(5+3{\sqrt {3}}+{\sqrt {11}}+{\sqrt {33}}){\sqrt[{3}]{-44+33{\sqrt {3}}}}+(-5+3{\sqrt {3}}-{\sqrt {11}}+{\sqrt {33}}){\sqrt[{3}]{44+33{\sqrt {3}}}}}}{\sqrt[{8}]{52180524}}}\\\varphi \left(e^{-12\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {{\sqrt[{4}]{1}}+{\sqrt[{4}]{2}}+{\sqrt[{4}]{3}}+{\sqrt[{4}]{4}}+{\sqrt[{4}]{9}}+{\sqrt[{4}]{18}}+{\sqrt[{4}]{24}}}}{2{\sqrt[{8}]{108}}}}\\\varphi \left(e^{-13\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {13+8{\sqrt {13}}+(11-6{\sqrt {3}}+{\sqrt {13}}){\sqrt[{3}]{143+78{\sqrt {3}}}}+(11+6{\sqrt {3}}+{\sqrt {13}}){\sqrt[{3}]{143-78{\sqrt {3}}}}}}{\sqrt[{4}]{19773}}}\\\varphi \left(e^{-14\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {{\sqrt {13+{\sqrt {7}}}}+{\sqrt {7+3{\sqrt {7}}}}+{\sqrt {10+2{\sqrt {7}}}}+{\sqrt[{8}]{28}}{\sqrt {4+{\sqrt {7}}}}}}{\sqrt[{16}]{28^{7}}}}\\\varphi \left(e^{-15\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt {7+3{\sqrt {3}}+{\sqrt {5}}+{\sqrt {15}}+{\sqrt[{4}]{60}}+{\sqrt[{4}]{1500}}}}{{\sqrt[{8}]{12^{3}}}\cdot {\sqrt {5}}}}\\2\varphi \left(e^{-16\pi }\right)&=\varphi \left(e^{-4\pi }\right)+{\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {\sqrt[{4}]{1+{\sqrt {2}}}}{\sqrt[{16}]{128}}}\\\varphi \left(e^{-17\pi }\right)&={\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\frac {{\sqrt {2}}(1+{\sqrt[{4}]{17}})+{\sqrt[{8}]{17}}{\sqrt {5+{\sqrt {17}}}}}{\sqrt {17+17{\sqrt {17}}}}}\\2\varphi \left(e^{-20\pi }\right)&=\varphi \left(e^{-5\pi }\right)+{\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\sqrt {\frac {3+2{\sqrt[{4}]{5}}}{5{\sqrt {2}}}}}\\6\varphi \left(e^{-36\pi }\right)&=3\varphi \left(e^{-9\pi }\right)+2\varphi \left(e^{-4\pi }\right)-\varphi \left(e^{-\pi }\right)+{\frac {\sqrt[{4}]{\pi }}{\Gamma \left({\frac {3}{4}}\right)}}{\sqrt[{3}]{{\sqrt[{4}]{2}}+{\sqrt[{4}]{18}}+{\sqrt[{4}]{216}}}}\end{aligned}}}
If the reciprocal of the Gelfond constant is raised to the power of the reciprocal of an odd number, then the corresponding
ϑ
00
{\displaystyle \vartheta _{00}}
values or
ϕ
{\displaystyle \phi }
values can be represented in a simplified way by using the hyperbolic lemniscatic sine:
φ
[
exp
(
−
1
5
π
)
]
=
π
4
Γ
(
3
4
)
−
1
slh
(
1
5
2
ϖ
)
slh
(
2
5
2
ϖ
)
{\displaystyle \varphi {\bigl [}\exp(-{\tfrac {1}{5}}\pi ){\bigr ]}={\sqrt[{4}]{\pi }}\,{\Gamma \left({\tfrac {3}{4}}\right)}^{-1}\operatorname {slh} {\bigl (}{\tfrac {1}{5}}{\sqrt {2}}\,\varpi {\bigr )}\operatorname {slh} {\bigl (}{\tfrac {2}{5}}{\sqrt {2}}\,\varpi {\bigr )}}
φ
[
exp
(
−
1
7
π
)
]
=
π
4
Γ
(
3
4
)
−
1
slh
(
1
7
2
ϖ
)
slh
(
2
7
2
ϖ
)
slh
(
3
7
2
ϖ
)
{\displaystyle \varphi {\bigl [}\exp(-{\tfrac {1}{7}}\pi ){\bigr ]}={\sqrt[{4}]{\pi }}\,{\Gamma \left({\tfrac {3}{4}}\right)}^{-1}\operatorname {slh} {\bigl (}{\tfrac {1}{7}}{\sqrt {2}}\,\varpi {\bigr )}\operatorname {slh} {\bigl (}{\tfrac {2}{7}}{\sqrt {2}}\,\varpi {\bigr )}\operatorname {slh} {\bigl (}{\tfrac {3}{7}}{\sqrt {2}}\,\varpi {\bigr )}}
φ
[
exp
(
−
1
9
π
)
]
=
π
4
Γ
(
3
4
)
−
1
slh
(
1
9
2
ϖ
)
slh
(
2
9
2
ϖ
)
slh
(
3
9
2
ϖ
)
slh
(
4
9
2
ϖ
)
{\displaystyle \varphi {\bigl [}\exp(-{\tfrac {1}{9}}\pi ){\bigr ]}={\sqrt[{4}]{\pi }}\,{\Gamma \left({\tfrac {3}{4}}\right)}^{-1}\operatorname {slh} {\bigl (}{\tfrac {1}{9}}{\sqrt {2}}\,\varpi {\bigr )}\operatorname {slh} {\bigl (}{\tfrac {2}{9}}{\sqrt {2}}\,\varpi {\bigr )}\operatorname {slh} {\bigl (}{\tfrac {3}{9}}{\sqrt {2}}\,\varpi {\bigr )}\operatorname {slh} {\bigl (}{\tfrac {4}{9}}{\sqrt {2}}\,\varpi {\bigr )}}
φ
[
exp
(
−
1
11
π
)
]
=
π
4
Γ
(
3
4
)
−
1
slh
(
1
11
2
ϖ
)
slh
(
2
11
2
ϖ
)
slh
(
3
11
2
ϖ
)
slh
(
4
11
2
ϖ
)
slh
(
5
11
2
ϖ
)
{\displaystyle \varphi {\bigl [}\exp(-{\tfrac {1}{11}}\pi ){\bigr ]}={\sqrt[{4}]{\pi }}\,{\Gamma \left({\tfrac {3}{4}}\right)}^{-1}\operatorname {slh} {\bigl (}{\tfrac {1}{11}}{\sqrt {2}}\,\varpi {\bigr )}\operatorname {slh} {\bigl (}{\tfrac {2}{11}}{\sqrt {2}}\,\varpi {\bigr )}\operatorname {slh} {\bigl (}{\tfrac {3}{11}}{\sqrt {2}}\,\varpi {\bigr )}\operatorname {slh} {\bigl (}{\tfrac {4}{11}}{\sqrt {2}}\,\varpi {\bigr )}\operatorname {slh} {\bigl (}{\tfrac {5}{11}}{\sqrt {2}}\,\varpi {\bigr )}}
With the letter
ϖ
{\displaystyle \varpi }
the Lemniscate constant is represented.
Note that the following modular identities hold:
2
φ
(
q
4
)
=
φ
(
q
)
+
2
φ
2
(
q
2
)
−
φ
2
(
q
)
3
φ
(
q
9
)
=
φ
(
q
)
+
9
φ
4
(
q
3
)
φ
(
q
)
−
φ
3
(
q
)
3
5
φ
(
q
25
)
=
φ
(
q
5
)
cot
(
1
2
arctan
(
2
5
φ
(
q
)
φ
(
q
5
)
φ
2
(
q
)
−
φ
2
(
q
5
)
1
+
s
(
q
)
−
s
2
(
q
)
s
(
q
)
)
)
{\displaystyle {\begin{aligned}2\varphi \left(q^{4}\right)&=\varphi (q)+{\sqrt {2\varphi ^{2}\left(q^{2}\right)-\varphi ^{2}(q)}}\\3\varphi \left(q^{9}\right)&=\varphi (q)+{\sqrt[{3}]{9{\frac {\varphi ^{4}\left(q^{3}\right)}{\varphi (q)}}-\varphi ^{3}(q)}}\\{\sqrt {5}}\varphi \left(q^{25}\right)&=\varphi \left(q^{5}\right)\cot \left({\frac {1}{2}}\arctan \left({\frac {2}{\sqrt {5}}}{\frac {\varphi (q)\varphi \left(q^{5}\right)}{\varphi ^{2}(q)-\varphi ^{2}\left(q^{5}\right)}}{\frac {1+s(q)-s^{2}(q)}{s(q)}}\right)\right)\end{aligned}}}
where
s
(
q
)
=
s
(
e
π
i
τ
)
=
−
R
(
−
e
−
π
i
/
(
5
τ
)
)
{\displaystyle s(q)=s\left(e^{\pi i\tau }\right)=-R\left(-e^{-\pi i/(5\tau )}\right)}
is the Rogers–Ramanujan continued fraction:
s
(
q
)
=
tan
(
1
2
arctan
(
5
2
φ
2
(
q
5
)
φ
2
(
q
)
−
1
2
)
)
cot
2
(
1
2
arccot
(
5
2
φ
2
(
q
5
)
φ
2
(
q
)
−
1
2
)
)
5
=
e
−
π
i
/
(
25
τ
)
1
−
e
−
π
i
/
(
5
τ
)
1
+
e
−
2
π
i
/
(
5
τ
)
1
−
⋱
{\displaystyle {\begin{aligned}s(q)&={\sqrt[{5}]{\tan \left({\frac {1}{2}}\arctan \left({\frac {5}{2}}{\frac {\varphi ^{2}\left(q^{5}\right)}{\varphi ^{2}(q)}}-{\frac {1}{2}}\right)\right)\cot ^{2}\left({\frac {1}{2}}\operatorname {arccot} \left({\frac {5}{2}}{\frac {\varphi ^{2}\left(q^{5}\right)}{\varphi ^{2}(q)}}-{\frac {1}{2}}\right)\right)}}\\&={\cfrac {e^{-\pi i/(25\tau )}}{1-{\cfrac {e^{-\pi i/(5\tau )}}{1+{\cfrac {e^{-2\pi i/(5\tau )}}{1-\ddots }}}}}}\end{aligned}}}
=== Equianharmonic values ===
The mathematician Bruce Berndt found out further values of the theta function:
φ
(
exp
(
−
3
π
)
)
=
π
−
1
Γ
(
4
3
)
3
/
2
2
−
2
/
3
3
13
/
8
φ
(
exp
(
−
2
3
π
)
)
=
π
−
1
Γ
(
4
3
)
3
/
2
2
−
2
/
3
3
13
/
8
cos
(
1
24
π
)
φ
(
exp
(
−
3
3
π
)
)
=
π
−
1
Γ
(
4
3
)
3
/
2
2
−
2
/
3
3
7
/
8
(
2
3
+
1
)
φ
(
exp
(
−
4
3
π
)
)
=
π
−
1
Γ
(
4
3
)
3
/
2
2
−
5
/
3
3
13
/
8
(
1
+
cos
(
1
12
π
)
)
φ
(
exp
(
−
5
3
π
)
)
=
π
−
1
Γ
(
4
3
)
3
/
2
2
−
2
/
3
3
5
/
8
sin
(
1
5
π
)
(
2
5
100
3
+
2
5
10
3
+
3
5
5
+
1
)
{\displaystyle {\begin{array}{lll}\varphi \left(\exp(-{\sqrt {3}}\,\pi )\right)&=&\pi ^{-1}{\Gamma \left({\tfrac {4}{3}}\right)}^{3/2}2^{-2/3}3^{13/8}\\\varphi \left(\exp(-2{\sqrt {3}}\,\pi )\right)&=&\pi ^{-1}{\Gamma \left({\tfrac {4}{3}}\right)}^{3/2}2^{-2/3}3^{13/8}\cos({\tfrac {1}{24}}\pi )\\\varphi \left(\exp(-3{\sqrt {3}}\,\pi )\right)&=&\pi ^{-1}{\Gamma \left({\tfrac {4}{3}}\right)}^{3/2}2^{-2/3}3^{7/8}({\sqrt[{3}]{2}}+1)\\\varphi \left(\exp(-4{\sqrt {3}}\,\pi )\right)&=&\pi ^{-1}{\Gamma \left({\tfrac {4}{3}}\right)}^{3/2}2^{-5/3}3^{13/8}{\Bigl (}1+{\sqrt {\cos({\tfrac {1}{12}}\pi )}}{\Bigr )}\\\varphi \left(\exp(-5{\sqrt {3}}\,\pi )\right)&=&\pi ^{-1}{\Gamma \left({\tfrac {4}{3}}\right)}^{3/2}2^{-2/3}3^{5/8}\sin({\tfrac {1}{5}}\pi )({\tfrac {2}{5}}{\sqrt[{3}]{100}}+{\tfrac {2}{5}}{\sqrt[{3}]{10}}+{\tfrac {3}{5}}{\sqrt {5}}+1)\end{array}}}
=== Further values ===
Many values of the theta function and especially of the shown phi function can be represented in terms of the gamma function:
φ
(
exp
(
−
2
π
)
)
=
π
−
1
/
2
Γ
(
9
8
)
Γ
(
5
4
)
−
1
/
2
2
7
/
8
φ
(
exp
(
−
2
2
π
)
)
=
π
−
1
/
2
Γ
(
9
8
)
Γ
(
5
4
)
−
1
/
2
2
1
/
8
(
1
+
2
−
1
)
φ
(
exp
(
−
3
2
π
)
)
=
π
−
1
/
2
Γ
(
9
8
)
Γ
(
5
4
)
−
1
/
2
2
3
/
8
3
−
1
/
2
(
3
+
1
)
tan
(
5
24
π
)
φ
(
exp
(
−
4
2
π
)
)
=
π
−
1
/
2
Γ
(
9
8
)
Γ
(
5
4
)
−
1
/
2
2
−
1
/
8
(
1
+
2
2
−
2
4
)
φ
(
exp
(
−
5
2
π
)
)
=
π
−
1
/
2
Γ
(
9
8
)
Γ
(
5
4
)
−
1
/
2
1
15
2
3
/
8
×
×
[
5
3
10
+
2
5
(
5
+
2
+
3
3
3
+
5
+
2
−
3
3
3
)
−
(
2
−
2
)
25
−
10
5
]
φ
(
exp
(
−
6
π
)
)
=
π
−
1
/
2
Γ
(
5
24
)
Γ
(
5
12
)
−
1
/
2
2
−
13
/
24
3
−
1
/
8
sin
(
5
12
π
)
φ
(
exp
(
−
1
2
6
π
)
)
=
π
−
1
/
2
Γ
(
5
24
)
Γ
(
5
12
)
−
1
/
2
2
5
/
24
3
−
1
/
8
sin
(
5
24
π
)
{\displaystyle {\begin{array}{lll}\varphi \left(\exp(-{\sqrt {2}}\,\pi )\right)&=&\pi ^{-1/2}\Gamma \left({\tfrac {9}{8}}\right){\Gamma \left({\tfrac {5}{4}}\right)}^{-1/2}2^{7/8}\\\varphi \left(\exp(-2{\sqrt {2}}\,\pi )\right)&=&\pi ^{-1/2}\Gamma \left({\tfrac {9}{8}}\right){\Gamma \left({\tfrac {5}{4}}\right)}^{-1/2}2^{1/8}{\Bigl (}1+{\sqrt {{\sqrt {2}}-1}}{\Bigr )}\\\varphi \left(\exp(-3{\sqrt {2}}\,\pi )\right)&=&\pi ^{-1/2}\Gamma \left({\tfrac {9}{8}}\right){\Gamma \left({\tfrac {5}{4}}\right)}^{-1/2}2^{3/8}3^{-1/2}({\sqrt {3}}+1){\sqrt {\tan({\tfrac {5}{24}}\pi )}}\\\varphi \left(\exp(-4{\sqrt {2}}\,\pi )\right)&=&\pi ^{-1/2}\Gamma \left({\tfrac {9}{8}}\right){\Gamma \left({\tfrac {5}{4}}\right)}^{-1/2}2^{-1/8}{\Bigl (}1+{\sqrt[{4}]{2{\sqrt {2}}-2}}{\Bigr )}\\\varphi \left(\exp(-5{\sqrt {2}}\,\pi )\right)&=&\pi ^{-1/2}\Gamma \left({\tfrac {9}{8}}\right){\Gamma \left({\tfrac {5}{4}}\right)}^{-1/2}{\frac {1}{15}}\,2^{3/8}\times \\&&\times {\biggl [}{\sqrt[{3}]{5}}\,{\sqrt {10+2{\sqrt {5}}}}{\biggl (}{\sqrt[{3}]{5+{\sqrt {2}}+3{\sqrt {3}}}}+{\sqrt[{3}]{5+{\sqrt {2}}-3{\sqrt {3}}}}\,{\biggr )}-{\bigl (}2-{\sqrt {2}}\,{\bigr )}{\sqrt {25-10{\sqrt {5}}}}\,{\biggr ]}\\\varphi \left(\exp(-{\sqrt {6}}\,\pi )\right)&=&\pi ^{-1/2}\Gamma \left({\tfrac {5}{24}}\right){\Gamma \left({\tfrac {5}{12}}\right)}^{-1/2}2^{-13/24}3^{-1/8}{\sqrt {\sin({\tfrac {5}{12}}\pi )}}\\\varphi \left(\exp(-{\tfrac {1}{2}}{\sqrt {6}}\,\pi )\right)&=&\pi ^{-1/2}\Gamma \left({\tfrac {5}{24}}\right){\Gamma \left({\tfrac {5}{12}}\right)}^{-1/2}2^{5/24}3^{-1/8}\sin({\tfrac {5}{24}}\pi )\end{array}}}
== Nome power theorems ==
=== Direct power theorems ===
For the transformation of the nome in the theta functions these formulas can be used:
θ
2
(
q
2
)
=
1
2
2
[
θ
3
(
q
)
2
−
θ
4
(
q
)
2
]
{\displaystyle \theta _{2}(q^{2})={\tfrac {1}{2}}{\sqrt {2[\theta _{3}(q)^{2}-\theta _{4}(q)^{2}]}}}
θ
3
(
q
2
)
=
1
2
2
[
θ
3
(
q
)
2
+
θ
4
(
q
)
2
]
{\displaystyle \theta _{3}(q^{2})={\tfrac {1}{2}}{\sqrt {2[\theta _{3}(q)^{2}+\theta _{4}(q)^{2}]}}}
θ
4
(
q
2
)
=
θ
4
(
q
)
θ
3
(
q
)
{\displaystyle \theta _{4}(q^{2})={\sqrt {\theta _{4}(q)\theta _{3}(q)}}}
The squares of the three theta zero-value functions with the square function as the inner function are also formed in the pattern of the Pythagorean triples according to the Jacobi Identity. Furthermore, those transformations are valid:
θ
3
(
q
4
)
=
1
2
θ
3
(
q
)
+
1
2
θ
4
(
q
)
{\displaystyle \theta _{3}(q^{4})={\tfrac {1}{2}}\theta _{3}(q)+{\tfrac {1}{2}}\theta _{4}(q)}
These formulas can be used to compute the theta values of the cube of the nome:
27
θ
3
(
q
3
)
8
−
18
θ
3
(
q
3
)
4
θ
3
(
q
)
4
−
θ
3
(
q
)
8
=
8
θ
3
(
q
3
)
2
θ
3
(
q
)
2
[
2
θ
4
(
q
)
4
−
θ
3
(
q
)
4
]
{\displaystyle 27\,\theta _{3}(q^{3})^{8}-18\,\theta _{3}(q^{3})^{4}\theta _{3}(q)^{4}-\,\theta _{3}(q)^{8}=8\,\theta _{3}(q^{3})^{2}\theta _{3}(q)^{2}[2\,\theta _{4}(q)^{4}-\theta _{3}(q)^{4}]}
27
θ
4
(
q
3
)
8
−
18
θ
4
(
q
3
)
4
θ
4
(
q
)
4
−
θ
4
(
q
)
8
=
8
θ
4
(
q
3
)
2
θ
4
(
q
)
2
[
2
θ
3
(
q
)
4
−
θ
4
(
q
)
4
]
{\displaystyle 27\,\theta _{4}(q^{3})^{8}-18\,\theta _{4}(q^{3})^{4}\theta _{4}(q)^{4}-\,\theta _{4}(q)^{8}=8\,\theta _{4}(q^{3})^{2}\theta _{4}(q)^{2}[2\,\theta _{3}(q)^{4}-\theta _{4}(q)^{4}]}
And the following formulas can be used to compute the theta values of the fifth power of the nome:
[
θ
3
(
q
)
2
−
θ
3
(
q
5
)
2
]
[
5
θ
3
(
q
5
)
2
−
θ
3
(
q
)
2
]
5
=
256
θ
3
(
q
5
)
2
θ
3
(
q
)
2
θ
4
(
q
)
4
[
θ
3
(
q
)
4
−
θ
4
(
q
)
4
]
{\displaystyle [\theta _{3}(q)^{2}-\theta _{3}(q^{5})^{2}][5\,\theta _{3}(q^{5})^{2}-\theta _{3}(q)^{2}]^{5}=256\,\theta _{3}(q^{5})^{2}\theta _{3}(q)^{2}\theta _{4}(q)^{4}[\theta _{3}(q)^{4}-\theta _{4}(q)^{4}]}
[
θ
4
(
q
5
)
2
−
θ
4
(
q
)
2
]
[
5
θ
4
(
q
5
)
2
−
θ
4
(
q
)
2
]
5
=
256
θ
4
(
q
5
)
2
θ
4
(
q
)
2
θ
3
(
q
)
4
[
θ
3
(
q
)
4
−
θ
4
(
q
)
4
]
{\displaystyle [\theta _{4}(q^{5})^{2}-\theta _{4}(q)^{2}][5\,\theta _{4}(q^{5})^{2}-\theta _{4}(q)^{2}]^{5}=256\,\theta _{4}(q^{5})^{2}\theta _{4}(q)^{2}\theta _{3}(q)^{4}[\theta _{3}(q)^{4}-\theta _{4}(q)^{4}]}
=== Transformation at the cube root of the nome ===
The formulas for the theta Nullwert function values from the cube root of the elliptic nome are obtained by contrasting the two real solutions of the corresponding quartic equations:
[
θ
3
(
q
1
/
3
)
2
θ
3
(
q
)
2
−
3
θ
3
(
q
3
)
2
θ
3
(
q
)
2
]
2
=
4
−
4
[
2
θ
2
(
q
)
2
θ
4
(
q
)
2
θ
3
(
q
)
4
]
2
/
3
{\displaystyle {\biggl [}{\frac {\theta _{3}(q^{1/3})^{2}}{\theta _{3}(q)^{2}}}-{\frac {3\,\theta _{3}(q^{3})^{2}}{\theta _{3}(q)^{2}}}{\biggr ]}^{2}=4-4{\biggl [}{\frac {2\,\theta _{2}(q)^{2}\theta _{4}(q)^{2}}{\theta _{3}(q)^{4}}}{\biggr ]}^{2/3}}
[
3
θ
4
(
q
3
)
2
θ
4
(
q
)
2
−
θ
4
(
q
1
/
3
)
2
θ
4
(
q
)
2
]
2
=
4
+
4
[
2
θ
2
(
q
)
2
θ
3
(
q
)
2
θ
4
(
q
)
4
]
2
/
3
{\displaystyle {\biggl [}{\frac {3\,\theta _{4}(q^{3})^{2}}{\theta _{4}(q)^{2}}}-{\frac {\theta _{4}(q^{1/3})^{2}}{\theta _{4}(q)^{2}}}{\biggr ]}^{2}=4+4{\biggl [}{\frac {2\,\theta _{2}(q)^{2}\theta _{3}(q)^{2}}{\theta _{4}(q)^{4}}}{\biggr ]}^{2/3}}
=== Transformation at the fifth root of the nome ===
The Rogers-Ramanujan continued fraction can be defined in terms of the Jacobi theta function in the following way:
R
(
q
)
=
tan
{
1
2
arctan
[
1
2
−
θ
4
(
q
)
2
2
θ
4
(
q
5
)
2
]
}
1
/
5
tan
{
1
2
arccot
[
1
2
−
θ
4
(
q
)
2
2
θ
4
(
q
5
)
2
]
}
2
/
5
{\displaystyle R(q)=\tan {\biggl \{}{\frac {1}{2}}\arctan {\biggl [}{\frac {1}{2}}-{\frac {\theta _{4}(q)^{2}}{2\,\theta _{4}(q^{5})^{2}}}{\biggr ]}{\biggr \}}^{1/5}\tan {\biggl \{}{\frac {1}{2}}\operatorname {arccot} {\biggl [}{\frac {1}{2}}-{\frac {\theta _{4}(q)^{2}}{2\,\theta _{4}(q^{5})^{2}}}{\biggr ]}{\biggr \}}^{2/5}}
R
(
q
2
)
=
tan
{
1
2
arctan
[
1
2
−
θ
4
(
q
)
2
2
θ
4
(
q
5
)
2
]
}
2
/
5
cot
{
1
2
arccot
[
1
2
−
θ
4
(
q
)
2
2
θ
4
(
q
5
)
2
]
}
1
/
5
{\displaystyle R(q^{2})=\tan {\biggl \{}{\frac {1}{2}}\arctan {\biggl [}{\frac {1}{2}}-{\frac {\theta _{4}(q)^{2}}{2\,\theta _{4}(q^{5})^{2}}}{\biggr ]}{\biggr \}}^{2/5}\cot {\biggl \{}{\frac {1}{2}}\operatorname {arccot} {\biggl [}{\frac {1}{2}}-{\frac {\theta _{4}(q)^{2}}{2\,\theta _{4}(q^{5})^{2}}}{\biggr ]}{\biggr \}}^{1/5}}
R
(
q
2
)
=
tan
{
1
2
arctan
[
θ
3
(
q
)
2
2
θ
3
(
q
5
)
2
−
1
2
]
}
2
/
5
tan
{
1
2
arccot
[
θ
3
(
q
)
2
2
θ
3
(
q
5
)
2
−
1
2
]
}
1
/
5
{\displaystyle R(q^{2})=\tan {\biggl \{}{\frac {1}{2}}\arctan {\biggl [}{\frac {\theta _{3}(q)^{2}}{2\,\theta _{3}(q^{5})^{2}}}-{\frac {1}{2}}{\biggr ]}{\biggr \}}^{2/5}\tan {\biggl \{}{\frac {1}{2}}\operatorname {arccot} {\biggl [}{\frac {\theta _{3}(q)^{2}}{2\,\theta _{3}(q^{5})^{2}}}-{\frac {1}{2}}{\biggr ]}{\biggr \}}^{1/5}}
The alternating Rogers-Ramanujan continued fraction function S(q) has the following two identities:
S
(
q
)
=
R
(
q
4
)
R
(
q
2
)
R
(
q
)
=
tan
{
1
2
arctan
[
θ
3
(
q
)
2
2
θ
3
(
q
5
)
2
−
1
2
]
}
1
/
5
cot
{
1
2
arccot
[
θ
3
(
q
)
2
2
θ
3
(
q
5
)
2
−
1
2
]
}
2
/
5
{\displaystyle S(q)={\frac {R(q^{4})}{R(q^{2})R(q)}}=\tan {\biggl \{}{\frac {1}{2}}\arctan {\biggl [}{\frac {\theta _{3}(q)^{2}}{2\,\theta _{3}(q^{5})^{2}}}-{\frac {1}{2}}{\biggr ]}{\biggr \}}^{1/5}\cot {\biggl \{}{\frac {1}{2}}\operatorname {arccot} {\biggl [}{\frac {\theta _{3}(q)^{2}}{2\,\theta _{3}(q^{5})^{2}}}-{\frac {1}{2}}{\biggr ]}{\biggr \}}^{2/5}}
The theta function values from the fifth root of the nome can be represented as a rational combination of the continued fractions R and S and the theta function values from the fifth power of the nome and the nome itself. The following four equations are valid for all values q between 0 and 1:
θ
3
(
q
1
/
5
)
θ
3
(
q
5
)
−
1
=
1
S
(
q
)
[
S
(
q
)
2
+
R
(
q
2
)
]
[
1
+
R
(
q
2
)
S
(
q
)
]
{\displaystyle {\frac {\theta _{3}(q^{1/5})}{\theta _{3}(q^{5})}}-1={\frac {1}{S(q)}}{\bigl [}S(q)^{2}+R(q^{2}){\bigr ]}{\bigl [}1+R(q^{2})S(q){\bigr ]}}
1
−
θ
4
(
q
1
/
5
)
θ
4
(
q
5
)
=
1
R
(
q
)
[
R
(
q
2
)
+
R
(
q
)
2
]
[
1
−
R
(
q
2
)
R
(
q
)
]
{\displaystyle 1-{\frac {\theta _{4}(q^{1/5})}{\theta _{4}(q^{5})}}={\frac {1}{R(q)}}{\bigl [}R(q^{2})+R(q)^{2}{\bigr ]}{\bigl [}1-R(q^{2})R(q){\bigr ]}}
θ
3
(
q
1
/
5
)
2
−
θ
3
(
q
)
2
=
[
θ
3
(
q
)
2
−
θ
3
(
q
5
)
2
]
[
1
+
1
R
(
q
2
)
S
(
q
)
+
R
(
q
2
)
S
(
q
)
+
1
R
(
q
2
)
2
+
R
(
q
2
)
2
+
1
S
(
q
)
−
S
(
q
)
]
{\displaystyle \theta _{3}(q^{1/5})^{2}-\theta _{3}(q)^{2}={\bigl [}\theta _{3}(q)^{2}-\theta _{3}(q^{5})^{2}{\bigr ]}{\biggl [}1+{\frac {1}{R(q^{2})S(q)}}+R(q^{2})S(q)+{\frac {1}{R(q^{2})^{2}}}+R(q^{2})^{2}+{\frac {1}{S(q)}}-S(q){\biggr ]}}
θ
4
(
q
)
2
−
θ
4
(
q
1
/
5
)
2
=
[
θ
4
(
q
5
)
2
−
θ
4
(
q
)
2
]
[
1
−
1
R
(
q
2
)
R
(
q
)
−
R
(
q
2
)
R
(
q
)
+
1
R
(
q
2
)
2
+
R
(
q
2
)
2
−
1
R
(
q
)
+
R
(
q
)
]
{\displaystyle \theta _{4}(q)^{2}-\theta _{4}(q^{1/5})^{2}={\bigl [}\theta _{4}(q^{5})^{2}-\theta _{4}(q)^{2}{\bigr ]}{\biggl [}1-{\frac {1}{R(q^{2})R(q)}}-R(q^{2})R(q)+{\frac {1}{R(q^{2})^{2}}}+R(q^{2})^{2}-{\frac {1}{R(q)}}+R(q){\biggr ]}}
=== Modulus dependent theorems ===
In combination with the elliptic modulus, the following formulas can be displayed:
These are the formulas for the square of the elliptic nome:
θ
4
[
q
(
k
)
]
=
θ
4
[
q
(
k
)
2
]
1
−
k
2
8
{\displaystyle \theta _{4}[q(k)]=\theta _{4}[q(k)^{2}]{\sqrt[{8}]{1-k^{2}}}}
θ
4
[
q
(
k
)
2
]
=
θ
3
[
q
(
k
)
]
1
−
k
2
8
{\displaystyle \theta _{4}[q(k)^{2}]=\theta _{3}[q(k)]{\sqrt[{8}]{1-k^{2}}}}
θ
3
[
q
(
k
)
2
]
=
θ
3
[
q
(
k
)
]
cos
[
1
2
arcsin
(
k
)
]
{\displaystyle \theta _{3}[q(k)^{2}]=\theta _{3}[q(k)]\cos[{\tfrac {1}{2}}\arcsin(k)]}
And this is an efficient formula for the cube of the nome:
θ
4
⟨
q
{
tan
[
1
2
arctan
(
t
3
)
]
}
3
⟩
=
θ
4
⟨
q
{
tan
[
1
2
arctan
(
t
3
)
]
}
⟩
3
−
1
/
2
(
2
t
4
−
t
2
+
1
−
t
2
+
2
+
t
2
+
1
)
1
/
2
{\displaystyle \theta _{4}{\biggl \langle }q{\bigl \{}\tan {\bigl [}{\tfrac {1}{2}}\arctan(t^{3}){\bigr ]}{\bigr \}}^{3}{\biggr \rangle }=\theta _{4}{\biggl \langle }q{\bigl \{}\tan {\bigl [}{\tfrac {1}{2}}\arctan(t^{3}){\bigr ]}{\bigr \}}{\biggr \rangle }\,3^{-1/2}{\bigl (}{\sqrt {2{\sqrt {t^{4}-t^{2}+1}}-t^{2}+2}}+{\sqrt {t^{2}+1}}\,{\bigr )}^{1/2}}
For all real values
t
∈
R
{\displaystyle t\in \mathbb {R} }
the now mentioned formula is valid.
And for this formula two examples shall be given:
First calculation example with the value
t
=
1
{\displaystyle t=1}
inserted:
Second calculation example with the value
t
=
Φ
−
2
{\displaystyle t=\Phi ^{-2}}
inserted:
The constant
Φ
{\displaystyle \Phi }
represents the Golden ratio number
Φ
=
1
2
(
5
+
1
)
{\displaystyle \Phi ={\tfrac {1}{2}}({\sqrt {5}}+1)}
exactly.
== Some series identities ==
=== Sums with theta function in the result ===
The infinite sum of the reciprocals of Fibonacci numbers with odd indices has the identity:
∑
n
=
1
∞
1
F
2
n
−
1
=
5
2
∑
n
=
1
∞
2
(
Φ
−
2
)
n
−
1
/
2
1
+
(
Φ
−
2
)
2
n
−
1
=
5
4
∑
a
=
−
∞
∞
2
(
Φ
−
2
)
a
−
1
/
2
1
+
(
Φ
−
2
)
2
a
−
1
=
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{F_{2n-1}}}={\frac {\sqrt {5}}{2}}\,\sum _{n=1}^{\infty }{\frac {2(\Phi ^{-2})^{n-1/2}}{1+(\Phi ^{-2})^{2n-1}}}={\frac {\sqrt {5}}{4}}\sum _{a=-\infty }^{\infty }{\frac {2(\Phi ^{-2})^{a-1/2}}{1+(\Phi ^{-2})^{2a-1}}}=}
=
5
4
θ
2
(
Φ
−
2
)
2
=
5
8
[
θ
3
(
Φ
−
1
)
2
−
θ
4
(
Φ
−
1
)
2
]
{\displaystyle ={\frac {\sqrt {5}}{4}}\,\theta _{2}(\Phi ^{-2})^{2}={\frac {\sqrt {5}}{8}}{\bigl [}\theta _{3}(\Phi ^{-1})^{2}-\theta _{4}(\Phi ^{-1})^{2}{\bigr ]}}
By not using the theta function expression, following identity between two sums can be formulated:
∑
n
=
1
∞
1
F
2
n
−
1
=
5
4
[
∑
n
=
1
∞
2
Φ
−
(
2
n
−
1
)
2
/
2
]
2
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{F_{2n-1}}}={\frac {\sqrt {5}}{4}}\,{\biggl [}\sum _{n=1}^{\infty }2\,\Phi ^{-(2n-1)^{2}/2}{\biggr ]}^{2}}
∑
n
=
1
∞
1
F
2
n
−
1
=
1.82451515740692456814215840626732817332
…
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{F_{2n-1}}}=1.82451515740692456814215840626732817332\ldots }
Also in this case
Φ
=
1
2
(
5
+
1
)
{\displaystyle \Phi ={\tfrac {1}{2}}({\sqrt {5}}+1)}
is Golden ratio number again.
Infinite sum of the reciprocals of the Fibonacci number squares:
∑
n
=
1
∞
1
F
n
2
=
5
24
[
2
θ
2
(
Φ
−
2
)
4
−
θ
3
(
Φ
−
2
)
4
+
1
]
=
5
24
[
θ
3
(
Φ
−
2
)
4
−
2
θ
4
(
Φ
−
2
)
4
+
1
]
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{F_{n}^{2}}}={\frac {5}{24}}{\bigl [}2\,\theta _{2}(\Phi ^{-2})^{4}-\theta _{3}(\Phi ^{-2})^{4}+1{\bigr ]}={\frac {5}{24}}{\bigl [}\theta _{3}(\Phi ^{-2})^{4}-2\,\theta _{4}(\Phi ^{-2})^{4}+1{\bigr ]}}
Infinite sum of the reciprocals of the Pell numbers with odd indices:
∑
n
=
1
∞
1
P
2
n
−
1
=
1
2
θ
2
[
(
2
−
1
)
2
]
2
=
1
2
2
[
θ
3
(
2
−
1
)
2
−
θ
4
(
2
−
1
)
2
]
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{P_{2n-1}}}={\frac {1}{\sqrt {2}}}\,\theta _{2}{\bigl [}({\sqrt {2}}-1)^{2}{\bigr ]}^{2}={\frac {1}{2{\sqrt {2}}}}{\bigl [}\theta _{3}({\sqrt {2}}-1)^{2}-\theta _{4}({\sqrt {2}}-1)^{2}{\bigr ]}}
=== Sums with theta function in the summand ===
The next two series identities were proved by István Mező:
θ
4
2
(
q
)
=
i
q
1
4
∑
k
=
−
∞
∞
q
2
k
2
−
k
θ
1
(
2
k
−
1
2
i
ln
q
,
q
)
,
θ
4
2
(
q
)
=
∑
k
=
−
∞
∞
q
2
k
2
θ
4
(
k
ln
q
i
,
q
)
.
{\displaystyle {\begin{aligned}\theta _{4}^{2}(q)&=iq^{\frac {1}{4}}\sum _{k=-\infty }^{\infty }q^{2k^{2}-k}\theta _{1}\left({\frac {2k-1}{2i}}\ln q,q\right),\\[6pt]\theta _{4}^{2}(q)&=\sum _{k=-\infty }^{\infty }q^{2k^{2}}\theta _{4}\left({\frac {k\ln q}{i}},q\right).\end{aligned}}}
These relations hold for all 0 < q < 1. Specializing the values of q, we have the next parameter free sums
π
e
π
2
⋅
1
Γ
2
(
3
4
)
=
i
∑
k
=
−
∞
∞
e
π
(
k
−
2
k
2
)
θ
1
(
i
π
2
(
2
k
−
1
)
,
e
−
π
)
{\displaystyle {\sqrt {\frac {\pi {\sqrt {e^{\pi }}}}{2}}}\cdot {\frac {1}{\Gamma ^{2}\left({\frac {3}{4}}\right)}}=i\sum _{k=-\infty }^{\infty }e^{\pi \left(k-2k^{2}\right)}\theta _{1}\left({\frac {i\pi }{2}}(2k-1),e^{-\pi }\right)}
π
2
⋅
1
Γ
2
(
3
4
)
=
∑
k
=
−
∞
∞
θ
4
(
i
k
π
,
e
−
π
)
e
2
π
k
2
{\displaystyle {\sqrt {\frac {\pi }{2}}}\cdot {\frac {1}{\Gamma ^{2}\left({\frac {3}{4}}\right)}}=\sum _{k=-\infty }^{\infty }{\frac {\theta _{4}\left(ik\pi ,e^{-\pi }\right)}{e^{2\pi k^{2}}}}}
== Zeros of the Jacobi theta functions ==
All zeros of the Jacobi theta functions are simple zeros and are given by the following:
ϑ
(
z
;
τ
)
=
ϑ
00
(
z
;
τ
)
=
0
⟺
z
=
m
+
n
τ
+
1
2
+
τ
2
ϑ
11
(
z
;
τ
)
=
0
⟺
z
=
m
+
n
τ
ϑ
10
(
z
;
τ
)
=
0
⟺
z
=
m
+
n
τ
+
1
2
ϑ
01
(
z
;
τ
)
=
0
⟺
z
=
m
+
n
τ
+
τ
2
{\displaystyle {\begin{aligned}\vartheta (z;\tau )=\vartheta _{00}(z;\tau )&=0\quad &\Longleftrightarrow &&\quad z&=m+n\tau +{\frac {1}{2}}+{\frac {\tau }{2}}\\[3pt]\vartheta _{11}(z;\tau )&=0\quad &\Longleftrightarrow &&\quad z&=m+n\tau \\[3pt]\vartheta _{10}(z;\tau )&=0\quad &\Longleftrightarrow &&\quad z&=m+n\tau +{\frac {1}{2}}\\[3pt]\vartheta _{01}(z;\tau )&=0\quad &\Longleftrightarrow &&\quad z&=m+n\tau +{\frac {\tau }{2}}\end{aligned}}}
where m, n are arbitrary integers.
== Relation to the Riemann zeta function ==
The relation
ϑ
(
0
;
−
1
τ
)
=
(
−
i
τ
)
1
2
ϑ
(
0
;
τ
)
{\displaystyle \vartheta \left(0;-{\frac {1}{\tau }}\right)=\left(-i\tau \right)^{\frac {1}{2}}\vartheta (0;\tau )}
was used by Riemann to prove the functional equation for the Riemann zeta function, by means of the Mellin transform
Γ
(
s
2
)
π
−
s
2
ζ
(
s
)
=
1
2
∫
0
∞
(
ϑ
(
0
;
i
t
)
−
1
)
t
s
2
d
t
t
{\displaystyle \Gamma \left({\frac {s}{2}}\right)\pi ^{-{\frac {s}{2}}}\zeta (s)={\frac {1}{2}}\int _{0}^{\infty }{\bigl (}\vartheta (0;it)-1{\bigr )}t^{\frac {s}{2}}{\frac {\mathrm {d} t}{t}}}
which can be shown to be invariant under substitution of s by 1 − s. The corresponding integral for z ≠ 0 is given in the article on the Hurwitz zeta function.
== Relation to the Weierstrass elliptic function ==
The theta function was used by Jacobi to construct (in a form adapted to easy calculation) his elliptic functions as the quotients of the above four theta functions, and could have been used by him to construct Weierstrass's elliptic functions also, since
℘
(
z
;
τ
)
=
−
(
log
ϑ
11
(
z
;
τ
)
)
″
+
c
{\displaystyle \wp (z;\tau )=-{\big (}\log \vartheta _{11}(z;\tau ){\big )}''+c}
where the second derivative is with respect to z and the constant c is defined so that the Laurent expansion of ℘(z) at z = 0 has zero constant term.
== Relation to the q-gamma function ==
The fourth theta function – and thus the others too – is intimately connected to the Jackson q-gamma function via the relation
(
Γ
q
2
(
x
)
Γ
q
2
(
1
−
x
)
)
−
1
=
q
2
x
(
1
−
x
)
(
q
−
2
;
q
−
2
)
∞
3
(
q
2
−
1
)
θ
4
(
1
2
i
(
1
−
2
x
)
log
q
,
1
q
)
.
{\displaystyle \left(\Gamma _{q^{2}}(x)\Gamma _{q^{2}}(1-x)\right)^{-1}={\frac {q^{2x(1-x)}}{\left(q^{-2};q^{-2}\right)_{\infty }^{3}\left(q^{2}-1\right)}}\theta _{4}\left({\frac {1}{2i}}(1-2x)\log q,{\frac {1}{q}}\right).}
== Relations to Dedekind eta function ==
Let η(τ) be the Dedekind eta function, and the argument of the theta function as the nome q = eπiτ. Then,
θ
2
(
q
)
=
ϑ
10
(
0
;
τ
)
=
2
η
2
(
2
τ
)
η
(
τ
)
,
θ
3
(
q
)
=
ϑ
00
(
0
;
τ
)
=
η
5
(
τ
)
η
2
(
1
2
τ
)
η
2
(
2
τ
)
=
η
2
(
1
2
(
τ
+
1
)
)
η
(
τ
+
1
)
,
θ
4
(
q
)
=
ϑ
01
(
0
;
τ
)
=
η
2
(
1
2
τ
)
η
(
τ
)
,
{\displaystyle {\begin{aligned}\theta _{2}(q)=\vartheta _{10}(0;\tau )&={\frac {2\eta ^{2}(2\tau )}{\eta (\tau )}},\\[3pt]\theta _{3}(q)=\vartheta _{00}(0;\tau )&={\frac {\eta ^{5}(\tau )}{\eta ^{2}\left({\frac {1}{2}}\tau \right)\eta ^{2}(2\tau )}}={\frac {\eta ^{2}\left({\frac {1}{2}}(\tau +1)\right)}{\eta (\tau +1)}},\\[3pt]\theta _{4}(q)=\vartheta _{01}(0;\tau )&={\frac {\eta ^{2}\left({\frac {1}{2}}\tau \right)}{\eta (\tau )}},\end{aligned}}}
and,
θ
2
(
q
)
θ
3
(
q
)
θ
4
(
q
)
=
2
η
3
(
τ
)
.
{\displaystyle \theta _{2}(q)\,\theta _{3}(q)\,\theta _{4}(q)=2\eta ^{3}(\tau ).}
See also the Weber modular functions.
== Elliptic modulus ==
The elliptic modulus is
k
(
τ
)
=
ϑ
10
(
0
;
τ
)
2
ϑ
00
(
0
;
τ
)
2
{\displaystyle k(\tau )={\frac {\vartheta _{10}(0;\tau )^{2}}{\vartheta _{00}(0;\tau )^{2}}}}
and the complementary elliptic modulus is
k
′
(
τ
)
=
ϑ
01
(
0
;
τ
)
2
ϑ
00
(
0
;
τ
)
2
{\displaystyle k'(\tau )={\frac {\vartheta _{01}(0;\tau )^{2}}{\vartheta _{00}(0;\tau )^{2}}}}
== Derivatives of theta functions ==
These are two identical definitions of the complete elliptic integral of the second kind:
E
(
k
)
=
∫
0
π
/
2
1
−
k
2
sin
(
φ
)
2
d
φ
{\displaystyle E(k)=\int _{0}^{\pi /2}{\sqrt {1-k^{2}\sin(\varphi )^{2}}}d\varphi }
E
(
k
)
=
π
2
∑
a
=
0
∞
[
(
2
a
)
!
]
2
(
1
−
2
a
)
16
a
(
a
!
)
4
k
2
a
{\displaystyle E(k)={\frac {\pi }{2}}\sum _{a=0}^{\infty }{\frac {[(2a)!]^{2}}{(1-2a)16^{a}(a!)^{4}}}k^{2a}}
The derivatives of the Theta Nullwert functions have these MacLaurin series:
θ
2
′
(
x
)
=
d
d
x
θ
2
(
x
)
=
1
2
x
−
3
/
4
+
∑
n
=
1
∞
1
2
(
2
n
+
1
)
2
x
(
2
n
−
1
)
(
2
n
+
3
)
/
4
{\displaystyle \theta _{2}'(x)={\frac {\mathrm {d} }{\mathrm {d} x}}\,\theta _{2}(x)={\frac {1}{2}}x^{-3/4}+\sum _{n=1}^{\infty }{\frac {1}{2}}(2n+1)^{2}x^{(2n-1)(2n+3)/4}}
θ
3
′
(
x
)
=
d
d
x
θ
3
(
x
)
=
2
+
∑
n
=
1
∞
2
(
n
+
1
)
2
x
n
(
n
+
2
)
{\displaystyle \theta _{3}'(x)={\frac {\mathrm {d} }{\mathrm {d} x}}\,\theta _{3}(x)=2+\sum _{n=1}^{\infty }2(n+1)^{2}x^{n(n+2)}}
θ
4
′
(
x
)
=
d
d
x
θ
4
(
x
)
=
−
2
+
∑
n
=
1
∞
2
(
n
+
1
)
2
(
−
1
)
n
+
1
x
n
(
n
+
2
)
{\displaystyle \theta _{4}'(x)={\frac {\mathrm {d} }{\mathrm {d} x}}\,\theta _{4}(x)=-2+\sum _{n=1}^{\infty }2(n+1)^{2}(-1)^{n+1}x^{n(n+2)}}
The derivatives of theta zero-value functions are as follows:
θ
2
′
(
x
)
=
d
d
x
θ
2
(
x
)
=
1
2
π
x
θ
2
(
x
)
θ
3
(
x
)
2
E
[
θ
2
(
x
)
2
θ
3
(
x
)
2
]
{\displaystyle \theta _{2}'(x)={\frac {\mathrm {d} }{\mathrm {d} x}}\,\theta _{2}(x)={\frac {1}{2\pi x}}\theta _{2}(x)\theta _{3}(x)^{2}E{\biggl [}{\frac {\theta _{2}(x)^{2}}{\theta _{3}(x)^{2}}}{\biggr ]}}
θ
3
′
(
x
)
=
d
d
x
θ
3
(
x
)
=
θ
3
(
x
)
[
θ
3
(
x
)
2
+
θ
4
(
x
)
2
]
{
1
2
π
x
E
[
θ
3
(
x
)
2
−
θ
4
(
x
)
2
θ
3
(
x
)
2
+
θ
4
(
x
)
2
]
−
θ
4
(
x
)
2
4
x
}
{\displaystyle \theta _{3}'(x)={\frac {\mathrm {d} }{\mathrm {d} x}}\,\theta _{3}(x)=\theta _{3}(x){\bigl [}\theta _{3}(x)^{2}+\theta _{4}(x)^{2}{\bigr ]}{\biggl \{}{\frac {1}{2\pi x}}E{\biggl [}{\frac {\theta _{3}(x)^{2}-\theta _{4}(x)^{2}}{\theta _{3}(x)^{2}+\theta _{4}(x)^{2}}}{\biggr ]}-{\frac {\theta _{4}(x)^{2}}{4\,x}}{\biggr \}}}
θ
4
′
(
x
)
=
d
d
x
θ
4
(
x
)
=
θ
4
(
x
)
[
θ
3
(
x
)
2
+
θ
4
(
x
)
2
]
{
1
2
π
x
E
[
θ
3
(
x
)
2
−
θ
4
(
x
)
2
θ
3
(
x
)
2
+
θ
4
(
x
)
2
]
−
θ
3
(
x
)
2
4
x
}
{\displaystyle \theta _{4}'(x)={\frac {\mathrm {d} }{\mathrm {d} x}}\,\theta _{4}(x)=\theta _{4}(x){\bigl [}\theta _{3}(x)^{2}+\theta _{4}(x)^{2}{\bigr ]}{\biggl \{}{\frac {1}{2\pi x}}E{\biggl [}{\frac {\theta _{3}(x)^{2}-\theta _{4}(x)^{2}}{\theta _{3}(x)^{2}+\theta _{4}(x)^{2}}}{\biggr ]}-{\frac {\theta _{3}(x)^{2}}{4\,x}}{\biggr \}}}
The two last mentioned formulas are valid for all real numbers of the real definition interval:
−
1
<
x
<
1
∩
x
∈
R
{\displaystyle -1<x<1\,\cap \,x\in \mathbb {R} }
And these two last named theta derivative functions are related to each other in this way:
ϑ
4
(
x
)
[
d
d
x
ϑ
3
(
x
)
]
−
ϑ
3
(
x
)
[
d
d
x
θ
4
(
x
)
]
=
1
4
x
θ
3
(
x
)
θ
4
(
x
)
[
θ
3
(
x
)
4
−
θ
4
(
x
)
4
]
{\displaystyle \vartheta _{4}(x){\biggl [}{\frac {\mathrm {d} }{\mathrm {d} x}}\,\vartheta _{3}(x){\biggr ]}-\vartheta _{3}(x){\biggl [}{\frac {\mathrm {d} }{\mathrm {d} x}}\,\theta _{4}(x){\biggr ]}={\frac {1}{4\,x}}\,\theta _{3}(x)\,\theta _{4}(x){\bigl [}\theta _{3}(x)^{4}-\theta _{4}(x)^{4}{\bigr ]}}
The derivatives of the quotients from two of the three theta functions mentioned here always have a rational relationship to those three functions:
d
d
x
θ
2
(
x
)
θ
3
(
x
)
=
θ
2
(
x
)
θ
4
(
x
)
4
4
x
θ
3
(
x
)
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\,{\frac {\theta _{2}(x)}{\theta _{3}(x)}}={\frac {\theta _{2}(x)\,\theta _{4}(x)^{4}}{4\,x\,\theta _{3}(x)}}}
d
d
x
θ
2
(
x
)
θ
4
(
x
)
=
θ
2
(
x
)
θ
3
(
x
)
4
4
x
θ
4
(
x
)
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\,{\frac {\theta _{2}(x)}{\theta _{4}(x)}}={\frac {\theta _{2}(x)\,\theta _{3}(x)^{4}}{4\,x\,\theta _{4}(x)}}}
d
d
x
θ
3
(
x
)
θ
4
(
x
)
=
θ
3
(
x
)
5
−
θ
3
(
x
)
θ
4
(
x
)
4
4
x
θ
4
(
x
)
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\,{\frac {\theta _{3}(x)}{\theta _{4}(x)}}={\frac {\theta _{3}(x)^{5}-\theta _{3}(x)\,\theta _{4}(x)^{4}}{4\,x\,\theta _{4}(x)}}}
For the derivation of these derivation formulas see the articles Nome (mathematics) and Modular lambda function!
== Integrals of theta functions ==
For the theta functions these integrals are valid:
∫
0
1
θ
2
(
x
)
d
x
=
∑
k
=
−
∞
∞
4
(
2
k
+
1
)
2
+
4
=
π
tanh
(
π
)
≈
3.129881
{\displaystyle \int _{0}^{1}\theta _{2}(x)\,\mathrm {d} x=\sum _{k=-\infty }^{\infty }{\frac {4}{(2k+1)^{2}+4}}=\pi \tanh(\pi )\approx 3.129881}
∫
0
1
θ
3
(
x
)
d
x
=
∑
k
=
−
∞
∞
1
k
2
+
1
=
π
coth
(
π
)
≈
3.153348
{\displaystyle \int _{0}^{1}\theta _{3}(x)\,\mathrm {d} x=\sum _{k=-\infty }^{\infty }{\frac {1}{k^{2}+1}}=\pi \coth(\pi )\approx 3.153348}
∫
0
1
θ
4
(
x
)
d
x
=
∑
k
=
−
∞
∞
(
−
1
)
k
k
2
+
1
=
π
csch
(
π
)
≈
0.272029
{\displaystyle \int _{0}^{1}\theta _{4}(x)\,\mathrm {d} x=\sum _{k=-\infty }^{\infty }{\frac {(-1)^{k}}{k^{2}+1}}=\pi \,\operatorname {csch} (\pi )\approx 0.272029}
The final results now shown are based on the general Cauchy sum formulas.
== A solution to the heat equation ==
The Jacobi theta function is the fundamental solution of the one-dimensional heat equation with spatially periodic boundary conditions. Taking z = x to be real and τ = it with t real and positive, we can write
ϑ
(
x
;
i
t
)
=
1
+
2
∑
n
=
1
∞
exp
(
−
π
n
2
t
)
cos
(
2
π
n
x
)
{\displaystyle \vartheta (x;it)=1+2\sum _{n=1}^{\infty }\exp \left(-\pi n^{2}t\right)\cos(2\pi nx)}
which solves the heat equation
∂
∂
t
ϑ
(
x
;
i
t
)
=
1
4
π
∂
2
∂
x
2
ϑ
(
x
;
i
t
)
.
{\displaystyle {\frac {\partial }{\partial t}}\vartheta (x;it)={\frac {1}{4\pi }}{\frac {\partial ^{2}}{\partial x^{2}}}\vartheta (x;it).}
This theta-function solution is 1-periodic in x, and as t → 0 it approaches the periodic delta function, or Dirac comb, in the sense of distributions
lim
t
→
0
ϑ
(
x
;
i
t
)
=
∑
n
=
−
∞
∞
δ
(
x
−
n
)
{\displaystyle \lim _{t\to 0}\vartheta (x;it)=\sum _{n=-\infty }^{\infty }\delta (x-n)}
.
General solutions of the spatially periodic initial value problem for the heat equation may be obtained by convolving the initial data at t = 0 with the theta function.
== Relation to the Heisenberg group ==
The Jacobi theta function is invariant under the action of a discrete subgroup of the Heisenberg group. This invariance is presented in the article on the theta representation of the Heisenberg group.
== Generalizations ==
If F is a quadratic form in n variables, then the theta function associated with F is
θ
F
(
z
)
=
∑
m
∈
Z
n
e
2
π
i
z
F
(
m
)
{\displaystyle \theta _{F}(z)=\sum _{m\in \mathbb {Z} ^{n}}e^{2\pi izF(m)}}
with the sum extending over the lattice of integers
Z
n
{\displaystyle \mathbb {Z} ^{n}}
. This theta function is a modular form of weight n/2 (on an appropriately defined subgroup) of the modular group. In the Fourier expansion,
θ
^
F
(
z
)
=
∑
k
=
0
∞
R
F
(
k
)
e
2
π
i
k
z
,
{\displaystyle {\hat {\theta }}_{F}(z)=\sum _{k=0}^{\infty }R_{F}(k)e^{2\pi ikz},}
the numbers RF(k) are called the representation numbers of the form.
=== Theta series of a Dirichlet character ===
For χ a primitive Dirichlet character modulo q and ν = 1 − χ(−1)/2 then
θ
χ
(
z
)
=
1
2
∑
n
=
−
∞
∞
χ
(
n
)
n
ν
e
2
i
π
n
2
z
{\displaystyle \theta _{\chi }(z)={\frac {1}{2}}\sum _{n=-\infty }^{\infty }\chi (n)n^{\nu }e^{2i\pi n^{2}z}}
is a weight 1/2 + ν modular form of level 4q2 and character
χ
(
d
)
(
−
1
d
)
ν
,
{\displaystyle \chi (d)\left({\frac {-1}{d}}\right)^{\nu },}
which means
θ
χ
(
a
z
+
b
c
z
+
d
)
=
χ
(
d
)
(
−
1
d
)
ν
(
θ
1
(
a
z
+
b
c
z
+
d
)
θ
1
(
z
)
)
1
+
2
ν
θ
χ
(
z
)
{\displaystyle \theta _{\chi }\left({\frac {az+b}{cz+d}}\right)=\chi (d)\left({\frac {-1}{d}}\right)^{\nu }\left({\frac {\theta _{1}\left({\frac {az+b}{cz+d}}\right)}{\theta _{1}(z)}}\right)^{1+2\nu }\theta _{\chi }(z)}
whenever
a
,
b
,
c
,
d
∈
Z
4
,
a
d
−
b
c
=
1
,
c
≡
0
mod
4
q
2
.
{\displaystyle a,b,c,d\in \mathbb {Z} ^{4},ad-bc=1,c\equiv 0{\bmod {4}}q^{2}.}
=== Ramanujan theta function ===
=== Riemann theta function ===
Let
H
n
=
{
F
∈
M
(
n
,
C
)
|
F
=
F
T
,
Im
F
>
0
}
{\displaystyle \mathbb {H} _{n}=\left\{F\in M(n,\mathbb {C} )\,{\big |}\,F=F^{\mathsf {T}}\,,\,\operatorname {Im} F>0\right\}}
be the set of symmetric square matrices whose imaginary part is positive definite.
H
n
{\displaystyle \mathbb {H} _{n}}
is called the Siegel upper half-space and is the multi-dimensional analog of the upper half-plane. The n-dimensional analogue of the modular group is the symplectic group Sp(2n,
Z
{\displaystyle \mathbb {Z} }
); for n = 1, Sp(2,
Z
{\displaystyle \mathbb {Z} }
) = SL(2,
Z
{\displaystyle \mathbb {Z} }
). The n-dimensional analogue of the congruence subgroups is played by
ker
{
Sp
(
2
n
,
Z
)
→
Sp
(
2
n
,
Z
/
k
Z
)
}
.
{\displaystyle \ker {\big \{}\operatorname {Sp} (2n,\mathbb {Z} )\to \operatorname {Sp} (2n,\mathbb {Z} /k\mathbb {Z} ){\big \}}.}
Then, given τ ∈
H
n
{\displaystyle \mathbb {H} _{n}}
, the Riemann theta function is defined as
θ
(
z
,
τ
)
=
∑
m
∈
Z
n
exp
(
2
π
i
(
1
2
m
T
τ
m
+
m
T
z
)
)
.
{\displaystyle \theta (z,\tau )=\sum _{m\in \mathbb {Z} ^{n}}\exp \left(2\pi i\left({\tfrac {1}{2}}m^{\mathsf {T}}\tau m+m^{\mathsf {T}}z\right)\right).}
Here, z ∈
C
n
{\displaystyle \mathbb {C} ^{n}}
is an n-dimensional complex vector, and the superscript T denotes the transpose. The Jacobi theta function is then a special case, with n = 1 and τ ∈
H
{\displaystyle \mathbb {H} }
where
H
{\displaystyle \mathbb {H} }
is the upper half-plane. One major application of the Riemann theta function is that it allows one to give explicit formulas for meromorphic functions on compact Riemann surfaces, as well as other auxiliary objects that figure prominently in their function theory, by taking τ to be the period matrix with respect to a canonical basis for its first homology group.
The Riemann theta converges absolutely and uniformly on compact subsets of
C
n
×
H
n
{\displaystyle \mathbb {C} ^{n}\times \mathbb {H} _{n}}
.
The functional equation is
θ
(
z
+
a
+
τ
b
,
τ
)
=
exp
(
2
π
i
(
−
b
T
z
−
1
2
b
T
τ
b
)
)
θ
(
z
,
τ
)
{\displaystyle \theta (z+a+\tau b,\tau )=\exp \left(2\pi i\left(-b^{\mathsf {T}}z-{\tfrac {1}{2}}b^{\mathsf {T}}\tau b\right)\right)\theta (z,\tau )}
which holds for all vectors a, b ∈
Z
n
{\displaystyle \mathbb {Z} ^{n}}
, and for all z ∈
C
n
{\displaystyle \mathbb {C} ^{n}}
and τ ∈
H
n
{\displaystyle \mathbb {H} _{n}}
.
=== Poincaré series ===
The Poincaré series generalizes the theta series to automorphic forms with respect to arbitrary Fuchsian groups.
== Derivation of the theta values ==
=== Identity of the Euler beta function ===
In the following, three important theta function values are to be derived as examples:
This is how the Euler beta function is defined in its reduced form:
β
(
x
)
=
Γ
(
x
)
2
Γ
(
2
x
)
{\displaystyle \beta (x)={\frac {\Gamma (x)^{2}}{\Gamma (2x)}}}
In general, for all natural numbers
n
∈
N
{\displaystyle n\in \mathbb {N} }
this formula of the Euler beta function is valid:
4
−
1
/
(
n
+
2
)
n
+
2
csc
(
π
n
+
2
)
β
[
n
2
(
n
+
2
)
]
=
∫
0
∞
1
x
n
+
2
+
1
d
x
{\displaystyle {\frac {4^{-1/(n+2)}}{n+2}}\csc {\bigl (}{\frac {\pi }{n+2}}{\bigr )}\beta {\biggl [}{\frac {n}{2(n+2)}}{\biggr ]}=\int _{0}^{\infty }{\frac {1}{\sqrt {x^{n+2}+1}}}\,\mathrm {d} x}
=== Exemplary elliptic integrals ===
In the following some Elliptic Integral Singular Values are derived:
=== Combination of the integral identities with the nome ===
The elliptic nome function has these important values:
q
(
1
2
2
)
=
exp
(
−
π
)
{\displaystyle q({\tfrac {1}{2}}{\sqrt {2}})=\exp(-\pi )}
q
[
1
4
(
6
−
2
)
]
=
exp
(
−
3
π
)
{\displaystyle q[{\tfrac {1}{4}}({\sqrt {6}}-{\sqrt {2}})]=\exp(-{\sqrt {3}}\,\pi )}
q
(
2
−
1
)
=
exp
(
−
2
π
)
{\displaystyle q({\sqrt {2}}-1)=\exp(-{\sqrt {2}}\,\pi )}
For the proof of the correctness of these nome values, see the article Nome (mathematics)!
On the basis of these integral identities and the above-mentioned Definition and identities to the theta functions in the same section of this article, exemplary theta zero values shall be determined now:
θ
3
[
exp
(
−
π
)
]
=
θ
3
[
q
(
1
2
2
)
]
=
2
π
−
1
K
(
1
2
2
)
=
2
−
1
/
2
π
−
1
/
2
β
(
1
4
)
1
/
2
=
2
−
1
/
4
π
4
Γ
(
3
4
)
−
1
{\displaystyle \theta _{3}[\exp(-\pi )]=\theta _{3}[q({\tfrac {1}{2}}{\sqrt {2}})]={\sqrt {2\pi ^{-1}K({\tfrac {1}{2}}{\sqrt {2}})}}=2^{-1/2}\pi ^{-1/2}\beta ({\tfrac {1}{4}})^{1/2}=2^{-1/4}{\sqrt[{4}]{\pi }}\,{\Gamma {\bigl (}{\tfrac {3}{4}}{\bigr )}}^{-1}}
θ
3
[
exp
(
−
3
π
)
]
=
θ
3
{
q
[
1
4
(
6
−
2
)
]
}
=
2
π
−
1
K
[
1
4
(
6
−
2
)
]
=
2
−
1
/
6
3
−
1
/
8
π
−
1
/
2
β
(
1
3
)
1
/
2
{\displaystyle \theta _{3}[\exp(-{\sqrt {3}}\,\pi )]=\theta _{3}{\bigl \{}q{\bigl [}{\tfrac {1}{4}}({\sqrt {6}}-{\sqrt {2}}){\bigr ]}{\bigr \}}={\sqrt {2\pi ^{-1}K{\bigl [}{\tfrac {1}{4}}({\sqrt {6}}-{\sqrt {2}}){\bigr ]}}}=2^{-1/6}3^{-1/8}\pi ^{-1/2}\beta ({\tfrac {1}{3}})^{1/2}}
θ
3
[
exp
(
−
2
π
)
]
=
θ
3
[
q
(
2
−
1
)
]
=
2
π
−
1
K
(
2
−
1
)
=
2
−
1
/
8
cos
(
1
8
π
)
π
−
1
/
2
β
(
3
8
)
1
/
2
{\displaystyle \theta _{3}[\exp(-{\sqrt {2}}\,\pi )]=\theta _{3}[q({\sqrt {2}}-1)]={\sqrt {2\pi ^{-1}K({\sqrt {2}}-1)}}=2^{-1/8}\cos({\tfrac {1}{8}}\pi )\,\pi ^{-1/2}\beta ({\tfrac {3}{8}})^{1/2}}
θ
4
[
exp
(
−
2
π
)
]
=
θ
4
[
q
(
2
−
1
)
]
=
2
2
−
2
4
2
π
−
1
K
(
2
−
1
)
=
2
−
1
/
4
cos
(
1
8
π
)
1
/
2
π
−
1
/
2
β
(
3
8
)
1
/
2
{\displaystyle \theta _{4}[\exp(-{\sqrt {2}}\,\pi )]=\theta _{4}[q({\sqrt {2}}-1)]={\sqrt[{4}]{2{\sqrt {2}}-2}}\,{\sqrt {2\pi ^{-1}K({\sqrt {2}}-1)}}=2^{-1/4}\cos({\tfrac {1}{8}}\pi )^{1/2}\,\pi ^{-1/2}\beta ({\tfrac {3}{8}})^{1/2}}
== Partition sequences and Pochhammer products ==
=== Regular partition number sequence ===
The regular partition sequence
P
(
n
)
{\displaystyle P(n)}
itself indicates the number of ways in which a positive integer number
n
{\displaystyle n}
can be split into positive integer summands. For the numbers
n
=
1
{\displaystyle n=1}
to
n
=
5
{\displaystyle n=5}
, the associated partition numbers
P
{\displaystyle P}
with all associated number partitions are listed in the following table:
The generating function of the regular partition number sequence can be represented via Pochhammer product in the following way:
∑
k
=
0
∞
P
(
k
)
x
k
=
1
(
x
;
x
)
∞
=
θ
3
(
x
)
−
1
/
6
θ
4
(
x
)
−
2
/
3
[
θ
3
(
x
)
4
−
θ
4
(
x
)
4
16
x
]
−
1
/
24
{\displaystyle \sum _{k=0}^{\infty }P(k)x^{k}={\frac {1}{(x;x)_{\infty }}}=\theta _{3}(x)^{-1/6}\theta _{4}(x)^{-2/3}{\biggl [}{\frac {\theta _{3}(x)^{4}-\theta _{4}(x)^{4}}{16\,x}}{\biggr ]}^{-1/24}}
The summandization of the now mentioned Pochhammer product is described by the Pentagonal number theorem in this way:
(
x
;
x
)
∞
=
1
+
∑
n
=
1
∞
[
−
x
Fn
(
2
n
−
1
)
−
x
Kr
(
2
n
−
1
)
+
x
Fn
(
2
n
)
+
x
Kr
(
2
n
)
]
{\displaystyle (x;x)_{\infty }=1+\sum _{n=1}^{\infty }{\bigl [}-x^{{\text{Fn}}(2n-1)}-x^{{\text{Kr}}(2n-1)}+x^{{\text{Fn}}(2n)}+x^{{\text{Kr}}(2n)}{\bigr ]}}
The following basic definitions apply to the pentagonal numbers and the card house numbers:
Fn
(
z
)
=
1
2
z
(
3
z
−
1
)
{\displaystyle {\text{Fn}}(z)={\tfrac {1}{2}}z(3z-1)}
Kr
(
z
)
=
1
2
z
(
3
z
+
1
)
{\displaystyle {\text{Kr}}(z)={\tfrac {1}{2}}z(3z+1)}
As a further application one obtains a formula for the third power of the Euler product:
(
x
;
x
)
3
=
∏
n
=
1
∞
(
1
−
x
n
)
3
=
∑
m
=
0
∞
(
−
1
)
m
(
2
m
+
1
)
x
m
(
m
+
1
)
/
2
{\displaystyle (x;x)^{3}=\prod _{n=1}^{\infty }(1-x^{n})^{3}=\sum _{m=0}^{\infty }(-1)^{m}(2m+1)x^{m(m+1)/2}}
=== Strict partition number sequence ===
And the strict partition sequence
Q
(
n
)
{\displaystyle Q(n)}
indicates the number of ways in which such a positive integer number
n
{\displaystyle n}
can be splitted into positive integer summands such that each summand appears at most once and no summand value occurs repeatedly. Exactly the same sequence is also generated if in the partition only odd summands are included, but these odd summands may occur more than once. Both representations for the strict partition number sequence are compared in the following table:
The generating function of the strict partition number sequence can be represented using Pochhammer's product:
∑
k
=
0
∞
Q
(
k
)
x
k
=
1
(
x
;
x
2
)
∞
=
θ
3
(
x
)
1
/
6
θ
4
(
x
)
−
1
/
3
[
θ
3
(
x
)
4
−
θ
4
(
x
)
4
16
x
]
1
/
24
{\displaystyle \sum _{k=0}^{\infty }Q(k)x^{k}={\frac {1}{(x;x^{2})_{\infty }}}=\theta _{3}(x)^{1/6}\theta _{4}(x)^{-1/3}{\biggl [}{\frac {\theta _{3}(x)^{4}-\theta _{4}(x)^{4}}{16\,x}}{\biggr ]}^{1/24}}
=== Overpartition number sequence ===
The Maclaurin series for the reciprocal of the function ϑ01 has the numbers of over partition sequence as coefficients with a positive sign:
1
θ
4
(
x
)
=
∏
n
=
1
∞
1
+
x
n
1
−
x
n
=
∑
k
=
0
∞
P
¯
(
k
)
x
k
{\displaystyle {\frac {1}{\theta _{4}(x)}}=\prod _{n=1}^{\infty }{\frac {1+x^{n}}{1-x^{n}}}=\sum _{k=0}^{\infty }{\overline {P}}(k)x^{k}}
1
θ
4
(
x
)
=
1
+
2
x
+
4
x
2
+
8
x
3
+
14
x
4
+
24
x
5
+
40
x
6
+
64
x
7
+
100
x
8
+
154
x
9
+
232
x
10
+
…
{\displaystyle {\frac {1}{\theta _{4}(x)}}=1+2x+4x^{2}+8x^{3}+14x^{4}+24x^{5}+40x^{6}+64x^{7}+100x^{8}+154x^{9}+232x^{10}+\dots }
If, for a given number
k
{\displaystyle k}
, all partitions are set up in such a way that the summand size never increases, and all those summands that do not have a summand of the same size to the left of themselves can be marked for each partition of this type, then it will be the resulting number of the marked partitions depending on
k
{\displaystyle k}
by the overpartition function
P
¯
(
k
)
{\displaystyle {\overline {P}}(k)}
.
First example:
P
¯
(
4
)
=
14
{\displaystyle {\overline {P}}(4)=14}
These 14 possibilities of partition markings exist for the sum 4:
Second example:
P
¯
(
5
)
=
24
{\displaystyle {\overline {P}}(5)=24}
These 24 possibilities of partition markings exist for the sum 5:
=== Relations of the partition number sequences to each other ===
In the Online Encyclopedia of Integer Sequences (OEIS), the sequence of regular partition numbers
P
(
n
)
{\displaystyle P(n)}
is under the code A000041, the sequence of strict partitions is
Q
(
n
)
{\displaystyle Q(n)}
under the code A000009 and the sequence of superpartitions
P
¯
(
n
)
{\displaystyle {\overline {P}}(n)}
under the code A015128. All parent partitions from index
n
=
1
{\displaystyle n=1}
are even.
The sequence of superpartitions
P
¯
(
n
)
{\displaystyle {\overline {P}}(n)}
can be written with the regular partition sequence P and the strict partition sequence Q can be generated like this:
P
¯
(
n
)
=
∑
k
=
0
n
P
(
n
−
k
)
Q
(
k
)
{\displaystyle {\overline {P}}(n)=\sum _{k=0}^{n}P(n-k)Q(k)}
In the following table of sequences of numbers, this formula should be used as an example:
Related to this property, the following combination of two series of sums can also be set up via the function ϑ01:
θ
4
(
x
)
=
[
∑
k
=
0
∞
P
(
k
)
x
k
]
−
1
[
∑
k
=
0
∞
Q
(
k
)
x
k
]
−
1
{\displaystyle \theta _{4}(x)={\biggl [}\sum _{k=0}^{\infty }P(k)x^{k}{\biggr ]}^{-1}{\biggl [}\sum _{k=0}^{\infty }Q(k)x^{k}{\biggr ]}^{-1}}
== Notes ==
== References ==
Abramowitz, Milton; Stegun, Irene A. (1964). Handbook of Mathematical Functions. New York: Dover Publications. sec. 16.27ff. ISBN 978-0-486-61272-0. {{cite book}}: ISBN / Date incompatibility (help)
Akhiezer, Naum Illyich (1990) [1970]. Elements of the Theory of Elliptic Functions. AMS Translations of Mathematical Monographs. Vol. 79. Providence, RI: AMS. ISBN 978-0-8218-4532-5.
Farkas, Hershel M.; Kra, Irwin (1980). Riemann Surfaces. New York: Springer-Verlag. ch. 6. ISBN 978-0-387-90465-8.. (for treatment of the Riemann theta)
Hardy, G. H.; Wright, E. M. (1959). An Introduction to the Theory of Numbers (4th ed.). Oxford: Clarendon Press.
Mumford, David (1983). Tata Lectures on Theta I. Boston: Birkhauser. ISBN 978-3-7643-3109-2.
Pierpont, James (1959). Functions of a Complex Variable. New York: Dover Publications.
Rauch, Harry E.; Farkas, Hershel M. (1974). Theta Functions with Applications to Riemann Surfaces. Baltimore: Williams & Wilkins. ISBN 978-0-683-07196-2.
Reinhardt, William P.; Walker, Peter L. (2010), "Theta Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
Whittaker, E. T.; Watson, G. N. (1927). A Course in Modern Analysis (4th ed.). Cambridge: Cambridge University Press. ch. 21. (history of Jacobi's θ functions)
== Further reading ==
Farkas, Hershel M. (2008). "Theta functions in complex analysis and number theory". In Alladi, Krishnaswami (ed.). Surveys in Number Theory. Developments in Mathematics. Vol. 17. Springer-Verlag. pp. 57–87. ISBN 978-0-387-78509-7. Zbl 1206.11055.
Schoeneberg, Bruno (1974). "IX. Theta series". Elliptic modular functions. Die Grundlehren der mathematischen Wissenschaften. Vol. 203. Springer-Verlag. pp. 203–226. ISBN 978-3-540-06382-7.
Ackerman, Michael (1 February 1979). "On the generating functions of certain Eisenstein series". Mathematische Annalen. 244 (1): 75–81. doi:10.1007/BF01420339. S2CID 120045753.
Harry Rauch with Hershel M. Farkas: Theta functions with applications to Riemann Surfaces, Williams and Wilkins, Baltimore MD 1974, ISBN 0-683-07196-3.
Charles Hermite: Sur la résolution de l'Équation du cinquiéme degré Comptes rendus, C. R. Acad. Sci. Paris, Nr. 11, March 1858.
== External links ==
Moiseev Igor. "Elliptic functions for Matlab and Octave".
This article incorporates material from Integral representations of Jacobi theta functions on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. | Wikipedia/Theta_functions |
The Erdős–Straus conjecture is an unproven statement in number theory. The conjecture is that, for every integer
n
{\displaystyle n}
that is greater than or equal to 2, there exist positive integers
x
{\displaystyle x}
,
y
{\displaystyle y}
, and
z
{\displaystyle z}
for which
4
n
=
1
x
+
1
y
+
1
z
.
{\displaystyle {\frac {4}{n}}={\frac {1}{x}}+{\frac {1}{y}}+{\frac {1}{z}}.}
In other words, the number
4
/
n
{\displaystyle 4/n}
can be written as a sum of three positive unit fractions.
The conjecture is named after Paul Erdős and Ernst G. Straus, who formulated it in 1948, but it is connected to much more ancient mathematics; sums of unit fractions, like the one in this problem, are known as Egyptian fractions, because of their use in ancient Egyptian mathematics. The Erdős–Straus conjecture is one of many conjectures by Erdős, and one of many unsolved problems in mathematics concerning Diophantine equations.
Although a solution is not known for all values of n, infinitely many values in certain infinite arithmetic progressions have simple formulas for their solution, and skipping these known values can speed up searches for counterexamples. Additionally, these searches need only consider values of
n
{\displaystyle n}
that are prime numbers, because any composite counterexample would have a smaller counterexample among its prime factors. Computer searches have verified the truth of the conjecture up to
n
≤
10
17
{\displaystyle n\leq 10^{17}}
.
If the conjecture is reframed to allow negative unit fractions, then it is known to be true. Generalizations of the conjecture to fractions with numerator 5 or larger have also been studied.
== Background and history ==
When a rational number is expanded into a sum of unit fractions, the expansion is called an Egyptian fraction. This way of writing fractions dates to the mathematics of ancient Egypt, in which fractions were written this way instead of in the more modern vulgar fraction form
a
b
{\displaystyle {\tfrac {a}{b}}}
with a numerator
a
{\displaystyle a}
and denominator
b
{\displaystyle b}
. The Egyptians produced tables of Egyptian fractions for unit fractions multiplied by two, the numbers that in modern notation would be written
2
n
{\displaystyle {\tfrac {2}{n}}}
, such as the Rhind Mathematical Papyrus table; in these tables, most of these expansions use either two or three terms. These tables were needed, because the obvious expansion
2
n
=
1
n
+
1
n
{\displaystyle {\tfrac {2}{n}}={\tfrac {1}{n}}+{\tfrac {1}{n}}}
was not allowed: the Egyptians required all of the fractions in an Egyptian fraction to be different from each other. This same requirement, that all fractions be different, is sometimes imposed in the Erdős–Straus conjecture, but it makes no significant difference to the problem, because for
n
>
2
{\displaystyle n>2}
any solution to
4
n
=
1
x
+
1
y
+
1
z
{\displaystyle {\tfrac {4}{n}}={\tfrac {1}{x}}+{\tfrac {1}{y}}+{\tfrac {1}{z}}}
where the unit fractions are not distinct can be converted into a solution where they are all distinct; see below.
Although the Egyptians did not always find expansions using as few terms as possible, later mathematicians have been interested in the question of how few terms are needed. Every fraction
a
b
{\displaystyle {\tfrac {a}{b}}}
has an expansion of at most
a
{\displaystyle a}
terms, so in particular
2
n
{\displaystyle {\tfrac {2}{n}}}
needs at most two terms,
3
n
{\displaystyle {\tfrac {3}{n}}}
needs at most three terms, and
4
n
{\displaystyle {\tfrac {4}{n}}}
needs at most four terms. For
2
n
{\displaystyle {\tfrac {2}{n}}}
, two terms are always needed, and for
3
n
{\displaystyle {\tfrac {3}{n}}}
, three terms are sometimes needed, so for both of these numerators, the maximum number of terms that might be needed is known. However, for
4
n
{\displaystyle {\tfrac {4}{n}}}
, it is unknown whether four terms are sometimes needed, or whether it is possible to express all fractions of the form
4
n
{\displaystyle {\tfrac {4}{n}}}
using only three unit fractions; this is the Erdős–Straus conjecture. Thus, the conjecture covers the first unknown case of a more general question, the problem of finding for all
a
{\displaystyle a}
the maximum number of terms needed in expansions for fractions
a
b
{\displaystyle {\tfrac {a}{b}}}
.
One way to find short (but not always shortest) expansions uses the greedy algorithm for Egyptian fractions, first described in 1202 by Fibonacci in his book Liber Abaci. This method chooses one unit fraction at a time, at each step choosing the largest possible unit fraction that would not cause the expanded sum to exceed the target number. After each step, the numerator of the fraction that still remains to be expanded decreases, so the total number of steps can never exceed the starting numerator, but sometimes it is smaller. For example, when it is applied to
3
n
{\displaystyle {\tfrac {3}{n}}}
, the greedy algorithm will use two terms whenever
n
{\displaystyle n}
is 2 modulo 3, but there exists a two-term expansion whenever
n
{\displaystyle n}
has a factor that is 2 modulo 3, a weaker condition. For numbers of the form
4
n
{\displaystyle {\tfrac {4}{n}}}
, the greedy algorithm will produce a four-term expansion whenever
n
{\displaystyle n}
is 1 modulo 4, and an expansion with fewer terms otherwise. Thus, another way of rephrasing the Erdős–Straus conjecture asks whether there exists another method for producing Egyptian fractions, using a smaller maximum number of terms for the numbers
4
n
{\displaystyle {\tfrac {4}{n}}}
.
The Erdős–Straus conjecture was formulated in 1948 by Paul Erdős and Ernst G. Straus, and published by Erdős (1950). Richard Obláth also published an early work on the conjecture, a paper written in 1948 and published in 1950, in which he extended earlier calculations of Straus and Harold N. Shapiro in order to verify the conjecture for all
n
≤
10
5
{\displaystyle n\leq 10^{5}}
.
== Formulation ==
The conjecture states that, for every integer
n
≥
2
{\displaystyle n\geq 2}
, there exist positive integers
x
{\displaystyle x}
,
y
{\displaystyle y}
, and
z
{\displaystyle z}
such that
4
n
=
1
x
+
1
y
+
1
z
.
{\displaystyle {\frac {4}{n}}={\frac {1}{x}}+{\frac {1}{y}}+{\frac {1}{z}}.}
For instance, for
n
=
5
{\displaystyle n=5}
, there are two solutions:
4
5
=
1
2
+
1
4
+
1
20
=
1
2
+
1
5
+
1
10
.
{\displaystyle {\frac {4}{5}}={\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{20}}={\frac {1}{2}}+{\frac {1}{5}}+{\frac {1}{10}}.}
Multiplying both sides of the equation
4
n
=
1
x
+
1
y
+
1
z
{\displaystyle {\tfrac {4}{n}}={\tfrac {1}{x}}+{\tfrac {1}{y}}+{\tfrac {1}{z}}}
by
n
x
y
z
{\displaystyle nxyz}
leads to an equivalent polynomial form
4
x
y
z
=
n
(
x
y
+
x
z
+
y
z
)
{\displaystyle 4xyz=n(xy+xz+yz)}
for the problem.
=== Distinct unit fractions ===
Some researchers additionally require that the integers
x
{\displaystyle x}
,
y
{\displaystyle y}
, and
z
{\displaystyle z}
be distinct from each other, as the Egyptians would have, while others allow them to be equal. For
n
≥
3
{\displaystyle n\geq 3}
, it does not matter whether they are required to be distinct: if there exists a solution with any three integers, then there exists a solution with distinct integers. This is because two identical unit fractions can be replaced through one of the following two expansions:
1
2
r
+
1
2
r
⇒
1
r
+
1
+
1
r
(
r
+
1
)
1
2
r
+
1
+
1
2
r
+
1
⇒
1
r
+
1
+
1
(
r
+
1
)
(
2
r
+
1
)
{\displaystyle {\begin{aligned}{\frac {1}{2r}}+{\frac {1}{2r}}&\Rightarrow {\frac {1}{r+1}}+{\frac {1}{r(r+1)}}\\{\frac {1}{2r+1}}+{\frac {1}{2r+1}}&\Rightarrow {\frac {1}{r+1}}+{\frac {1}{(r+1)(2r+1)}}\\\end{aligned}}}
(according to whether the repeated fraction has an even or odd denominator) and this replacement can be repeated until no duplicate fractions remain. For
n
=
2
{\displaystyle n=2}
, however, the only solutions are permutations of
4
2
=
1
2
+
1
2
+
1
1
{\displaystyle {\tfrac {4}{2}}={\tfrac {1}{2}}+{\tfrac {1}{2}}+{\tfrac {1}{1}}}
.
=== Negative-number solutions ===
The Erdős–Straus conjecture requires that all three of
x
{\displaystyle x}
,
y
{\displaystyle y}
, and
z
{\displaystyle z}
be positive. This requirement is essential to the difficulty of the problem. Even without this relaxation, the Erdős–Straus conjecture is difficult only for odd values of
n
{\displaystyle n}
, and if negative values were allowed then the problem could be solved for every odd
n
{\displaystyle n}
by the following formula:
4
n
=
1
(
n
−
1
)
/
2
+
1
(
n
+
1
)
/
2
−
1
n
(
n
−
1
)
(
n
+
1
)
/
4
.
{\displaystyle {\frac {4}{n}}={\frac {1}{(n-1)/2}}+{\frac {1}{(n+1)/2}}-{\frac {1}{n(n-1)(n+1)/4}}.}
== Computational results ==
If the conjecture is false, it could be proven false simply by finding a number
4
n
{\displaystyle {\tfrac {4}{n}}}
that has no three-term representation. In order to check this, various authors have performed brute-force searches for counterexamples to the conjecture. Searches of this type have confirmed that the conjecture is true for all
n
{\displaystyle n}
up to
10
17
{\displaystyle 10^{17}}
.
In such searches, it is only necessary to look for expansions for numbers
4
n
{\displaystyle {\tfrac {4}{n}}}
where
n
{\displaystyle n}
is a prime number. This is because, whenever
4
n
{\displaystyle {\tfrac {4}{n}}}
has a three-term expansion, so does
4
m
n
{\displaystyle {\tfrac {4}{mn}}}
for all positive integers
m
{\displaystyle m}
. To find a solution for
4
m
n
{\displaystyle {\tfrac {4}{mn}}}
, just divide all of the unit fractions in the solution for
4
n
{\displaystyle {\tfrac {4}{n}}}
by
m
{\displaystyle m}
:
4
n
=
1
x
+
1
y
+
1
z
⇒
4
m
n
=
1
m
x
+
1
m
y
+
1
m
z
.
{\displaystyle {\frac {4}{n}}={\frac {1}{x}}+{\frac {1}{y}}+{\frac {1}{z}}\ \Rightarrow \ {\frac {4}{mn}}={\frac {1}{mx}}+{\frac {1}{my}}+{\frac {1}{mz}}.}
If
4
n
{\displaystyle {\tfrac {4}{n}}}
were a counterexample to the conjecture, for a composite number
n
{\displaystyle n}
, every prime factor
p
{\displaystyle p}
of
n
{\displaystyle n}
would also provide a counterexample
4
p
{\displaystyle {\tfrac {4}{p}}}
that would have been found earlier by the brute-force search. Therefore, checking the existence of a solution for composite numbers is redundant, and can be skipped by the search. Additionally, the known modular identities for the conjecture (see below) can speed these searches by skipping over other values known to have a solution. For instance, the greedy algorithm finds an expansion with three or fewer terms for every number
4
n
{\displaystyle {\tfrac {4}{n}}}
where
n
{\displaystyle n}
is not 1 modulo 4, so the searches only need to test values that are 1 modulo 4. One way to make progress on this problem is to collect more modular identities, allowing computer searches to reach higher limits with fewer tests.
The number of distinct solutions to the
4
n
{\displaystyle {\tfrac {4}{n}}}
problem, as a function of
n
{\displaystyle n}
, has also been found by computer searches for small
n
{\displaystyle n}
and appears to grow somewhat irregularly with
n
{\displaystyle n}
. Starting with
n
=
3
{\displaystyle n=3}
, the numbers of distinct solutions with distinct denominators are
Even for larger
n
{\displaystyle n}
there can sometimes be relatively few solutions; for instance there are only seven distinct solutions for
n
=
73
{\displaystyle n=73}
.
== Theoretical results ==
In the form
4
x
y
z
=
n
(
x
y
+
x
z
+
y
z
)
{\displaystyle 4xyz=n(xy+xz+yz)}
, a polynomial equation with integer variables, the Erdős–Straus conjecture is an example of a Diophantine equation. The Hasse principle for Diophantine equations suggests that these equations should be studied using modular arithmetic. If a polynomial equation has a solution in the integers, then taking this solution modulo
q
{\displaystyle q}
, for any integer
q
{\displaystyle q}
, provides a solution in modulo-
q
{\displaystyle q}
arithmetic. In the other direction, if an equation has a solution modulo
q
{\displaystyle q}
for every prime power
q
{\displaystyle q}
, then in some cases it is possible to piece together these modular solutions, using methods related to the Chinese remainder theorem, to get a solution in the integers. The power of the Hasse principle to solve some problems is limited by the Manin obstruction, but for the Erdős–Straus conjecture this obstruction does not exist.
On the face of it this principle makes little sense for the Erdős–Straus conjecture. For every
n
{\displaystyle n}
, the equation
4
x
y
z
=
n
(
x
y
+
x
z
+
y
z
)
{\displaystyle 4xyz=n(xy+xz+yz)}
is easily solvable modulo any prime, or prime power, but there appears to be no way to piece those solutions together to get a positive integer solution to the equation. Nevertheless, modular arithmetic, and identities based on modular arithmetic, have proven a very important tool in the study of the conjecture.
=== Modular identities ===
For values of
n
{\displaystyle n}
satisfying certain congruence relations, one can find an expansion for
4
n
{\displaystyle {\tfrac {4}{n}}}
automatically as an instance of a polynomial identity. For instance, whenever
n
{\displaystyle n}
is 2 modulo 3,
4
n
{\displaystyle {\tfrac {4}{n}}}
has the expansion
4
n
=
1
n
+
1
(
n
+
1
)
/
3
+
1
n
(
n
+
1
)
/
3
.
{\displaystyle {\frac {4}{n}}={\frac {1}{n}}+{\frac {1}{(n+1)/3}}+{\frac {1}{n(n+1)/3}}.}
Here each of the three denominators
n
{\displaystyle n}
,
(
n
+
1
)
/
3
{\displaystyle (n+1)/3}
, and
n
(
n
+
1
)
/
3
{\displaystyle n(n+1)/3}
is a polynomial of
n
{\displaystyle n}
, and each is an integer whenever
n
{\displaystyle n}
is 2 modulo 3. The greedy algorithm for Egyptian fractions finds a solution in three or fewer terms whenever
n
{\displaystyle n}
is not 1 or 17 mod 24, and the 17 mod 24 case is covered by the 2 mod 3 relation, so the only values of
n
{\displaystyle n}
for which these two methods do not find expansions in three or fewer terms are those congruent to 1 mod 24.
Polynomial identities listed by Mordell (1967) provide three-term Egyptian fractions for
4
n
{\displaystyle {\tfrac {4}{n}}}
whenever
n
{\displaystyle n}
is one of:
2 mod 3 (above),
3 mod 4,
2 or 3 mod 5,
3, 5, or 6 mod 7, or
5 mod 8.
Combinations of Mordell's identities can be used to expand
4
n
{\displaystyle {\tfrac {4}{n}}}
for all
n
{\displaystyle n}
except possibly those that are 1, 121, 169, 289, 361, or 529 mod 840. The smallest prime that these identities do not cover is 1009. By combining larger classes of modular identities, Webb and others showed that the natural density of potential counterexamples to the conjecture is zero: as a parameter
N
{\displaystyle N}
goes to infinity, the fraction of values in the interval
[
1
,
N
]
{\displaystyle [1,N]}
. that could be counterexamples tends to zero in the limit.
=== Nonexistence of identities ===
If it were possible to find solutions such as the ones above for enough different moduli, forming a complete covering system of congruences, the problem would be solved. However, as Mordell (1967) showed, a polynomial identity that provides a solution for values of
n
{\displaystyle n}
congruent to
r
{\displaystyle r}
mod
p
{\displaystyle p}
can exist only when
r
{\displaystyle r}
is not congruent to a square modulo
p
{\displaystyle p}
. (More formally, this kind of identity can exist only when
r
{\displaystyle r}
is not a quadratic residue modulo
p
{\displaystyle p}
.) For instance, 2 is a non-square mod 3, so Mordell's result allows the existence of an identity for
n
{\displaystyle n}
congruent to 2 mod 3. However, 1 is a square mod 3 (equal to the square of both 1 and 2 mod 3), so there can be no similar identity for all values of
n
{\displaystyle n}
that are congruent to 1 mod 3. More generally, as 1 is a square mod
n
{\displaystyle n}
for all
n
>
1
{\displaystyle n>1}
, there can be no complete covering system of modular identities for all
n
{\displaystyle n}
, because 1 will always be uncovered.
Despite Mordell's result limiting the form of modular identities for this problem, there is still some hope of using modular identities to prove the Erdős–Straus conjecture. No prime number can be a square, so by the Hasse–Minkowski theorem, whenever
p
{\displaystyle p}
is prime, there exists a larger prime
q
{\displaystyle q}
such that
p
{\displaystyle p}
is not a quadratic residue modulo
q
{\displaystyle q}
. One possible approach to proving the conjecture would be
to find for each prime
p
{\displaystyle p}
a larger prime
q
{\displaystyle q}
and a congruence solving the
4
n
{\displaystyle {\tfrac {4}{n}}}
problem for
n
{\displaystyle n}
congruent to
p
{\displaystyle p}
mod
q
{\displaystyle q}
. If this could be done, no prime
p
{\displaystyle p}
could be a counterexample to the conjecture and the conjecture would be true.
=== The number of solutions ===
Elsholtz & Tao (2013) showed that the average number of solutions to the
4
n
{\displaystyle {\tfrac {4}{n}}}
problem (averaged over the prime numbers up to
n
{\displaystyle n}
) is upper bounded polylogarithmically in
n
{\displaystyle n}
. For some other Diophantine problems, the existence of a solution can be demonstrated through asymptotic lower bounds on the number of solutions, but this works best when the number of solutions grows at least polynomially, so the slower growth rate of Elsholtz and Tao's result makes a proof of this type less likely. Elsholtz and Tao classify solutions according to whether one or two of
x
{\displaystyle x}
,
y
{\displaystyle y}
, or
z
{\displaystyle z}
is divisible by
n
{\displaystyle n}
; for prime
n
{\displaystyle n}
, these are the only possibilities, although (on average) most solutions for composite
n
{\displaystyle n}
are of other types. Their proof uses the Bombieri–Vinogradov theorem, the Brun–Titchmarsh theorem, and a system of modular identities, valid when
n
{\displaystyle n}
is congruent to
−
c
{\displaystyle -c}
or
−
1
c
{\displaystyle -{\tfrac {1}{c}}}
modulo
4
a
b
{\displaystyle 4ab}
, where
a
{\displaystyle a}
and
b
{\displaystyle b}
are any two coprime positive integers and
c
{\displaystyle c}
is any odd factor of
a
+
b
{\displaystyle a+b}
. For instance, setting
a
=
b
=
1
{\displaystyle a=b=1}
gives one of Mordell's identities, valid when
n
{\displaystyle n}
is 3 mod 4.
== Generalizations ==
As with fractions of the form
4
n
{\displaystyle {\tfrac {4}{n}}}
, it has been conjectured that every fraction
5
n
{\displaystyle {\tfrac {5}{n}}}
(for
n
>
1
{\displaystyle n>1}
) can be expressed as a sum of three positive unit fractions. A generalized version of the conjecture states that, for any positive
k
{\displaystyle k}
, all but finitely many fractions
k
n
{\displaystyle {\tfrac {k}{n}}}
can be expressed as a sum of three positive unit fractions. The conjecture for fractions
5
n
{\displaystyle {\tfrac {5}{n}}}
was made by Wacław Sierpiński in a 1956 paper, which went on to credit the full conjecture to Sierpiński's student Andrzej Schinzel.
Even if the generalized conjecture is false for any fixed value of
k
{\displaystyle k}
, then the number of fractions
k
n
{\displaystyle {\tfrac {k}{n}}}
with
n
{\displaystyle n}
in the range from 1 to
N
{\displaystyle N}
that do not have three-term expansions must grow only sublinearly as a function of
N
{\displaystyle N}
. In particular, if the Erdős–Straus conjecture itself (the case
k
=
4
{\displaystyle k=4}
) is false, then the number of counterexamples grows only sublinearly. Even more strongly, for any fixed
k
{\displaystyle k}
, only a sublinear number of values of
n
{\displaystyle n}
need more than two terms in their Egyptian fraction expansions. The generalized version of the conjecture is equivalent to the statement that the number of unexpandable fractions is not just sublinear but finite.
When
n
{\displaystyle n}
is an odd number, by analogy to the problem of odd greedy expansions for Egyptian fractions, one may ask for solutions to
k
n
=
1
x
+
1
y
+
1
z
{\displaystyle {\tfrac {k}{n}}={\tfrac {1}{x}}+{\tfrac {1}{y}}+{\tfrac {1}{z}}}
in which
x
{\displaystyle x}
,
y
{\displaystyle y}
, and
z
{\displaystyle z}
are distinct positive odd numbers. Solutions to this equation are known to always exist for the case in which k = 3.
== See also ==
List of sums of reciprocals
== Notes ==
== References == | Wikipedia/Erdős–Straus_conjecture |
The Beal conjecture is the following conjecture in number theory:
If
A
x
+
B
y
=
C
z
{\displaystyle A^{x}+B^{y}=C^{z}}
,
where A, B, C, x, y, and z are positive integers with x, y, z > 2, then A, B, and C have a common prime factor.
Equivalently,
The equation
A
x
+
B
y
=
C
z
{\displaystyle A^{x}+B^{y}=C^{z}}
has no solutions in positive integers and pairwise coprime integers A, B, C if x, y, z > 2.
The conjecture was formulated in 1993 by Andrew Beal, a banker and amateur mathematician, while investigating generalizations of Fermat's Last Theorem. Since 1997, Beal has offered a monetary prize for a peer-reviewed proof of this conjecture or a counterexample. The value of the prize has increased several times and is currently $1 million.
In some publications, this conjecture has occasionally been referred to as a generalized Fermat equation, the Mauldin conjecture, and the Tijdeman-Zagier conjecture.
== Related examples ==
To illustrate, the solution
3
3
+
6
3
=
3
5
{\displaystyle 3^{3}+6^{3}=3^{5}}
has bases with a common factor of 3, the solution
7
3
+
7
4
=
14
3
{\displaystyle 7^{3}+7^{4}=14^{3}}
has bases with a common factor of 7, and
2
n
+
2
n
=
2
n
+
1
{\displaystyle 2^{n}+2^{n}=2^{n+1}}
has bases with a common factor of 2. Indeed the equation has infinitely many solutions where the bases share a common factor, including generalizations of the above three examples, respectively
3
3
n
+
[
2
(
3
n
)
]
3
=
3
3
n
+
2
,
n
≥
1
;
{\displaystyle 3^{3n}+[2(3^{n})]^{3}=3^{3n+2},\quad \quad n\geq 1;}
[
b
(
a
n
−
b
n
)
k
]
n
+
(
a
n
−
b
n
)
k
n
+
1
=
[
a
(
a
n
−
b
n
)
k
]
n
,
a
>
b
,
b
≥
1
,
k
≥
1
,
n
≥
3
;
{\displaystyle [b(a^{n}-b^{n})^{k}]^{n}+(a^{n}-b^{n})^{kn+1}=[a(a^{n}-b^{n})^{k}]^{n},\quad \quad a>b,\quad b\geq 1,\quad k\geq 1,\quad n\geq 3;}
and
[
a
(
a
n
+
b
n
)
k
]
n
+
[
b
(
a
n
+
b
n
)
k
]
n
=
(
a
n
+
b
n
)
k
n
+
1
,
a
≥
1
,
b
≥
1
,
k
≥
1
,
n
≥
3.
{\displaystyle [a(a^{n}+b^{n})^{k}]^{n}+[b(a^{n}+b^{n})^{k}]^{n}=(a^{n}+b^{n})^{kn+1},\quad \quad a\geq 1,\quad b\geq 1,\quad k\geq 1,\quad n\geq 3.}
Furthermore, for each solution (with or without coprime bases), there are infinitely many solutions with the same set of exponents and an increasing set of non-coprime bases. That is, for solution
A
1
x
+
B
1
y
=
C
1
z
{\displaystyle A_{1}^{x}+B_{1}^{y}=C_{1}^{z}}
we additionally have
A
n
x
+
B
n
y
=
C
n
z
;
{\displaystyle A_{n}^{x}+B_{n}^{y}=C_{n}^{z};}
n
≥
2
{\displaystyle n\geq 2}
where
A
n
=
(
A
n
−
1
y
z
+
1
)
(
B
n
−
1
y
z
)
(
C
n
−
1
y
z
)
{\displaystyle A_{n}=(A_{n-1}^{yz+1})(B_{n-1}^{yz})(C_{n-1}^{yz})}
B
n
=
(
A
n
−
1
x
z
)
(
B
n
−
1
x
z
+
1
)
(
C
n
−
1
x
z
)
{\displaystyle B_{n}=(A_{n-1}^{xz})(B_{n-1}^{xz+1})(C_{n-1}^{xz})}
C
n
=
(
A
n
−
1
x
y
)
(
B
n
−
1
x
y
)
(
C
n
−
1
x
y
+
1
)
{\displaystyle C_{n}=(A_{n-1}^{xy})(B_{n-1}^{xy})(C_{n-1}^{xy+1})}
Any solutions to the Beal conjecture will necessarily involve three terms all of which are 3-powerful numbers, i.e. numbers where the exponent of every prime factor is at least three. It is known that there are an infinite number of such sums involving coprime 3-powerful numbers; however, such sums are rare. The smallest two examples are:
271
3
+
2
3
3
5
73
3
=
919
3
=
776,151,559
3
4
29
3
89
3
+
7
3
11
3
167
3
=
2
7
5
4
353
3
=
3,518,958,160,000
{\displaystyle {\begin{aligned}271^{3}+2^{3}\ 3^{5}\ 73^{3}=919^{3}&=776{,}151{,}559\\3^{4}\ 29^{3}\ 89^{3}+7^{3}\ 11^{3}\ 167^{3}=2^{7}\ 5^{4}\ 353^{3}&=3{,}518{,}958{,}160{,}000\\\end{aligned}}}
What distinguishes Beal's conjecture is that it requires each of the three terms to be expressible as a single power.
== Relation to other conjectures ==
Fermat's Last Theorem established that
A
n
+
B
n
=
C
n
{\displaystyle A^{n}+B^{n}=C^{n}}
has no solutions for n > 2 for positive integers A, B, and C. If any solutions had existed to Fermat's Last Theorem, then by dividing out every common factor, there would also exist solutions with A, B, and C coprime. Hence, Fermat's Last Theorem can be seen as a special case of the Beal conjecture restricted to x = y = z.
The Fermat–Catalan conjecture is that
A
x
+
B
y
=
C
z
{\displaystyle A^{x}+B^{y}=C^{z}}
has only finitely many solutions with A, B, and C being positive integers with no common prime factor and x, y, and z being positive integers satisfying
1
x
+
1
y
+
1
z
<
1
{\textstyle {\frac {1}{x}}+{\frac {1}{y}}+{\frac {1}{z}}<1}
. Beal's conjecture can be restated as "All Fermat–Catalan conjecture solutions will use 2 as an exponent".
The abc conjecture would imply that there are at most finitely many counterexamples to Beal's conjecture.
== Partial results ==
In the cases below where n is an exponent, multiples of n are also proven, since a kn-th power is also an n-th power. Where solutions involving a second power are alluded to below, they can be found specifically at Fermat–Catalan conjecture#Known solutions. All cases of the form (2, 3, n) or (2, n, 3) have the solution 23 + 1n = 32 which is referred below as the Catalan solution.
The case x = y = z ≥ 3 is Fermat's Last Theorem, proven to have no solutions by Andrew Wiles in 1994.
The case (x, y, z) = (2, 3, 7) and all its permutations were proven to have only four non-Catalan solutions, none of them contradicting Beal conjecture, by Bjorn Poonen, Edward F. Schaefer, and Michael Stoll in 2005.
The case (x, y, z) = (2, 3, 8) and all its permutations were proven to have only two non-Catalan solutions, which doesn't contradict Beal conjecture, by Nils Bruin in 2003.
The case (x, y, z) = (2, 3, 9) and all its permutations are known to have only one non-Catalan solution, which doesn't contradict Beal conjecture, by Nils Bruin in 2003.
The case (x, y, z) = (2, 3, 10) and all its permutations were proven by David Zureick-Brown in 2009 to have only the Catalan solution.
The case (x, y, z) = (2, 3, 11) and all its permutations were proven by Freitas, Naskręcki and Stoll to have only the Catalan solution.
The case (x, y, z) = (2, 3, 15) and all its permutations were proven by Samir Siksek and Michael Stoll in 2013 to have only the Catalan solution.
The case (x, y, z) = (2, 4, 4) and all its permutations were proven to have no solutions by combined work of Pierre de Fermat in the 1640s and Euler in 1738. (See one proof here and another here)
The case (x, y, z) = (2, 4, 5) and all its permutations are known to have only two non-Catalan solutions, which doesn't contradict Beal conjecture, by Nils Bruin in 2003.
The case (x, y, z) = (2, 4, n) and all its permutations were proven for n ≥ 6 by Michael Bennett, Jordan Ellenberg, and Nathan Ng in 2009.
The case (x, y, z) = (2, 6, n) and all its permutations were proven for n ≥ 3 by Michael Bennett and Imin Chen in 2011 and by Bennett, Chen, Dahmen and Yazdani in 2014.
The case (x, y, z) = (2, 2n, 3) was proven for 3 ≤ n ≤ 107 except n = 7 and various modulo congruences when n is prime to have no non-Catalan solution by Bennett, Chen, Dahmen and Yazdani.
The cases (x, y, z) = (2, 2n, 9), (2, 2n, 10), (2, 2n, 15) and all their permutations were proven for n ≥ 2 by Bennett, Chen, Dahmen and Yazdani in 2014.
The case (x, y, z) = (3, 3, n) and all its permutations have been proven for 3 ≤ n ≤ 109 and various modulo congruences when n is prime.
The case (x, y, z) = (3, 4, 5) and all its permutations were proven by Siksek and Stoll in 2011.
The case (x, y, z) = (3, 5, 5) and all its permutations were proven by Bjorn Poonen in 1998.
The case (x, y, z) = (3, 6, n) and all its permutations were proven for n ≥ 3 by Bennett, Chen, Dahmen and Yazdani in 2014.
The case (x, y, z) = (2n, 3, 4) and all its permutations were proven for n ≥ 2 by Bennett, Chen, Dahmen and Yazdani in 2014.
The cases (5, 5, 7), (5, 5, 19), (7, 7, 5) and all their permutations were proven by Sander R. Dahmen and Samir Siksek in 2013.
The cases (x, y, z) = (n, n, 2) and all its permutations were proven for n ≥ 4 by Darmon and Merel in 1995 following work from Euler and Poonen.
The cases (x, y, z) = (n, n, 3) and all its permutations were proven for n ≥ 3 by Édouard Lucas, Bjorn Poonen, and Darmon and Merel.
The case (x, y, z) = (2n, 2n, 5) and all its permutations were proven for n ≥ 2 by Bennett in 2006.
The case (x, y, z) = (2l, 2m, n) and all its permutations were proven for l, m ≥ 5 primes and n = 3, 5, 7, 11 by Anni and Siksek.
The case (x, y, z) = (2l, 2m, 13) and all its permutations were proven for l, m ≥ 5 primes by Billerey, Chen, Dembélé, Dieulefait, Freitas.
The case (x, y, z) = (3l, 3m, n) is direct for l, m ≥ 2 and n ≥ 3 from work by Kraus.
The Darmon–Granville theorem uses Faltings's theorem to show that for every specific choice of exponents (x, y, z), there are at most finitely many coprime solutions for (A, B, C).: p. 64
The impossibility of the case A = 1 or B = 1 is implied by Catalan's conjecture, proven in 2002 by Preda Mihăilescu. (Notice C cannot be 1, or one of A and B must be 0, which is not permitted.)
A potential class of solutions to the equation, namely those with A, B, C also forming a Pythagorean triple, were considered by L. Jesmanowicz in the 1950s. J. Jozefiak proved that there are an infinite number of primitive Pythagorean triples that cannot satisfy the Beal equation. Further results are due to Chao Ko.
Peter Norvig, Director of Research at Google, reported having conducted a series of numerical searches for counterexamples to Beal's conjecture. Among his results, he excluded all possible solutions having each of x, y, z ≤ 7 and each of A, B, C ≤ 250,000, as well as possible solutions having each of x, y, z ≤ 100 and each of A, B, C ≤ 10,000.
If A, B are odd and x, y are even, Beal's conjecture has no counterexample.
By assuming the validity of Beal's conjecture, there exists an upper bound for any common divisor of x, y and z in the expression
a
x
m
+
b
y
n
=
z
r
{\displaystyle ax^{m}+by^{n}=z^{r}}
.
== Prize ==
For a published proof or counterexample, banker Andrew Beal initially offered a prize of US $5,000 in 1997, raising it to $50,000 over ten years, but has since raised it to US $1,000,000.
The American Mathematical Society (AMS) holds the $1 million prize in a trust until the Beal conjecture is solved. It is supervised by the Beal Prize Committee (BPC), which is appointed by the AMS president.
== Variants ==
The counterexamples
3
5
+
10
2
=
7
3
{\displaystyle 3^{5}+10^{2}=7^{3}}
,
7
3
+
13
2
=
2
9
{\displaystyle 7^{3}+13^{2}=2^{9}}
, and
1
m
+
2
3
=
3
2
{\displaystyle 1^{m}+2^{3}=3^{2}}
show that the conjecture would be false if one of the exponents were allowed to be 2. The Fermat–Catalan conjecture is an open conjecture dealing with such cases (the condition of this conjecture is that the sum of the reciprocals is less than 1). If we allow at most one of the exponents to be 2, then there may be only finitely many solutions (except the case
1
m
+
2
3
=
3
2
{\displaystyle 1^{m}+2^{3}=3^{2}}
).
A variation of the conjecture asserting that x, y, z (instead of A, B, C) must have a common prime factor is not true. A counterexample is
27
4
+
162
3
=
9
7
,
{\displaystyle 27^{4}+162^{3}=9^{7},}
in which 4, 3, and 7 have no common prime factor. (In fact, the maximum common prime factor of the exponents that is valid is 2; a common factor greater than 2 would be a counterexample to Fermat's Last Theorem.)
The conjecture is not valid over the larger domain of Gaussian integers. After a prize of $50 was offered for a counterexample, Fred W. Helenius provided
(
−
2
+
i
)
3
+
(
−
2
−
i
)
3
=
(
1
+
i
)
4
{\displaystyle (-2+i)^{3}+(-2-i)^{3}=(1+i)^{4}}
.
== See also ==
ABC conjecture
Euler's sum of powers conjecture
Jacobi–Madden equation
Prouhet–Tarry–Escott problem
Taxicab number
Pythagorean quadruple
Sums of powers, a list of related conjectures and theorems
Distributed computing
BOINC
== References ==
== External links ==
The Beal Prize office page
Bealconjecture.com
Math.unt.edu
Beal Conjecture at PlanetMath.
Mathoverflow.net discussion about the name and date of origin of the theorem | Wikipedia/Beal's_conjecture |
"The Equation" is the eighth episode of the first season of the American science fiction drama television series Fringe. The episode follows the Fringe team's investigation into the kidnapping of a young musical prodigy (Charlie Tahan) who has become obsessed with finishing one piece of music. Dr. Walter Bishop (John Noble) returns to St. Claire's Hospital in an effort to find the boy's whereabouts.
The episode was written by supervising producer J. R. Orci and co-executive producer David H. Goodman, and was directed by Gwyneth Horder-Payton. Actress Gillian Jacobs guest starred as the boy's kidnapper. The episode featured her character in a "pretty violent and quite messy" fight with Olivia Dunham (Anna Torv), Torv's first for the series. The two actresses spent several weeks practicing the fight's choreography.
"The Equation" first aired in the United States on November 18, 2008, garnering an estimated 9.175 million viewers. The episode was the Fox network's fifth ranked show for the week, and helped Fox win the night among adults aged 18 to 49. Critical reception to "The Equation" ranged from mixed to positive, with most reviewers praising the asylum storyline.
== Plot ==
While helping fix a woman's car engine on the side of the road in Middletown, Connecticut, Andrew Stockston (Adam Grupper) sees a sequence of red and green flashing lights and is hypnotized into a suggestive state. Upon 'waking up', he does not have any memory of what happened while hypnotized, but sees that the woman and his son Ben (Charlie Tahan), a young musical prodigy, are missing. Phillip Broyles (Lance Reddick) reveals that similar cases have ended with the victims being returned, but left insane from the trauma of the incident. All the victims were academics and accomplished in their respective fields.
When interviewing Andrew, Olivia Dunham (Anna Torv) learns that nine months previously, Ben survived a car accident with a new, extraordinary ability to play the piano, despite never taking lessons. Dr. Walter Bishop (John Noble) recalls memories of red and green lights, but he's unable to remember more. While trying to dredge up the old memories, Walter recounts a previous unsuccessful mind control experiment he had worked on for an advertising agency, who wished to compel customers to buy their products using flashing lights. He deduces that someone succeeded in producing the lights using wavelengths, and these caused Andrew to sustain a "hypnagogic trance" that allowed his son to be abducted. He successfully tests an experiment on Peter Bishop (Joshua Jackson).
Andrew's sketch leads to the identification of the kidnapper as Joanne Ostler (Gillian Jacobs), a MIT neurologist who was previously believed deceased. Joanne tricks Ben into helping her complete an unfinished equation by using the image of his mother, who died in the car accident. Meanwhile, Walter suddenly remembers that he heard about the lights from former mathematician Dashiell Kim (Randall Duk Kim), an old bunkmate at St. Claire's Hospital who disappeared under similar circumstances. To discover the child's whereabouts, Olivia encourages Walter to return to St. Claire's. The visit does not go well, and Walter is held by hospital administrator Dr. Bruce Sumner (William Sadler), who remains unconvinced of Walter's sanity.
Peter figures out Joanne's assumed name using an FBI database, while Walter manages to convince Kim into giving up a vague description of Joanne's whereabouts by telling him there is a little boy who needs their help. Kim says he was kept in "a dungeon in a red castle." Olivia and Peter use the information to find the boy once they arrange for Walter's release. However, Joanne escapes with the completed formula, which she gives to Mitchell Loeb (Chance Kelly). Loeb kills her, but not before using the equation to allow him to pass through solid matter.
== Production ==
"The Equation" was written by supervising producer J. R. Orci and co-executive producer David H. Goodman. Both would go on to separately write other first season episodes, including Orci's "The Transformation" and Goodman's "Safe", which resolved the fate of the eponymous equation. "The Equation" was directed by filmmaker Gwyneth Horder-Payton, her first and only credit for the series to date.
The episode featured guest actress Gillian Jacobs as the kidnapper Joanne Ostler. Jacobs explained her character's motivations, "I'm a very mysterious figure, and at the beginning of the episode you see that I have taken this boy, have kidnapped him. I have taken him to this room and have him hooked him up to these EKG machines. I'm trying to get him to finish writing this piano piece which I need to help solve an equation... It's very important to me."
Horder-Payton considered the confrontation between Olivia and Joanne to be the former's "first big fight scene." She added, "It's been well choreographed and they've been practicing for several weeks." According to Torv, she and Jacobs broke the moves down "really simply" and then "put them together into really small little bits." The director called it a "pretty violent and quite messy" fight, to which first assistant director Colin MacLellan added, "It's a pretty phenomenal brawl actually to have two women kick the crap out of each other."
The first season DVD includes several scenes that were omitted from the final cut of the episode. The first centers on Walter waking up in the middle of the night, explaining to a disgruntled Peter that he's attempting to "shift my circadian rhythm to the nocturnal cycle." The other scene shows Olivia and Peter playing poker, which leads her to realize an important part of the episode's case.
== Themes and analysis ==
Flashing multicolored lights, specifically red and green, are a consistent theme of the first season, with Into the Looking Glass: Exploring the Worlds of Fringe author Sarah Clarke Stuart calling them "noteworthy recurring images." They can also be seen in the computer graphics of the Observer's binoculars, as well as during the second and third seasons as a way to distinguish the two universes.
The character of Walter goes through much development during the first season. Stuart believed that he changes the most out of the main cast, and his return to St. Claire's reflects this progression. Actor John Noble explained his character's evolution in a November 2008 interview, "We see Walter from a different angle, very vulnerable. He goes back to the asylum again, and we see the very, very fearful man return for a while. Although he does have some wonderful moments earlier in the episode, when he goes back inside he turns into this incredibly fearful, stuttering fellow that we saw when we first met him. It's a very interesting journey we see Walter go through."
In the episode, Walter sees himself several times while staying in the mental institution. Many reviewers expressed curiosity about this "Visitor" or "Second Walter". After looking through "Walter's Lab Notes," released by Fox after each episode, Ramsey Isler of IGN speculated that Walter suffered from a multiple personality disorder, while AOL TV's Jane Boursaw thought it was either an hallucination or his alternate universe counterpart.
== Reception ==
=== Ratings ===
"The Equation" first broadcast on the Fox network in the United States on November 18, 2008. It was watched by an estimated 9.18 million viewers, earning a 4.1/10 ratings share among adults aged 18 to 49, meaning that it was seen by 4.4 percent of all 18- to 49-year-olds, and 11 percent of all 18- to 49-year-olds watching television at the time of broadcast. The episode also received a 5.6/8 ratings share among all households. Fringe and its lead-in show, House M.D., helped Fox win the night in the adult demographic, as it was Fringe's highest rating since the season's second episode. It was Fox's fifth ranked show for the week.
=== Reviews ===
The episode received mixed to positive reviews from television critics. Fearnet columnist Alyse Wax called it a "pretty good episode", and believed it to be "far more enjoyable than last week's", as it lacked that episode's "conspiracy nonsense" and John Scott storyline. However, Wax continued that "The Equation" "seemed rather pedestrian" because "nothing too freaky" happened, and wished that Walter and Peter had been used more. Jane Boursaw of AOL TV considered Walter's return to the asylum "heartbreaking".
IGN's Travis Fickett rated "The Equation" 7.5/10, and called it a "solid episode" despite a few perceived plotholes. He liked Walter and Peter's actions in the asylum, and concluded "At this point, whether a solid single episode is enough to keep you watching Fringe likely has to do with your overall patience with the series and whatever its ultimate goals might be." Erin Dougherty of Cinema Blend called it the best episode since "The Same Old Story" since it contained "suspense and drama and a minimal amount of conspiracy theories", making her feel "seriously giddy". While still calling it "entertaining", she disliked Walter seeing himself in the asylum, believing it "was really strange and didn’t go with the flow of the story".
The A.V. Club writer Noel Murray graded the episode with a B+, explaining that he believed it to be mainly an original story; what kept him from promoting it to the "elusive 'A' level–something no Fringe episode has yet done for me" was the ending, which was "like something out of dozens of mediocre cop shows". Despite this, Murray found it and the asylum storyline to be "compelling". Conversely, UGO Networks was critical of the episode, writing "Fringe continues to wobble in story quality. Last night's episode was a perfectly good way to waste an hour, but far off the track of last week's episodes. The episode featured some plot conveniences that were a bit hard to swallow".
== References ==
Works cited
Stuart, Sarah Clarke (2011). Into the Looking Glass: Exploring the Worlds of Fringe. ECW Press. ISBN 978-1-77041-051-0.
== External links ==
"The Equation" at Fox
"The Equation" at IMDb | Wikipedia/The_Equation |
The equation of time describes the discrepancy between two kinds of solar time. The two times that differ are the apparent solar time, which directly tracks the diurnal motion of the Sun, and mean solar time, which tracks a theoretical mean Sun with uniform motion along the celestial equator. Apparent solar time can be obtained by measurement of the current position (hour angle) of the Sun, as indicated (with limited accuracy) by a sundial. Mean solar time, for the same place, would be the time indicated by a steady clock set so that over the year its differences from apparent solar time would have a mean of zero.
The equation of time is the east or west component of the analemma, a curve representing the angular offset of the Sun from its mean position on the celestial sphere as viewed from Earth. The equation of time values for each day of the year, compiled by astronomical observatories, were widely listed in almanacs and ephemerides.: 14
The equation of time can be approximated by a sum of two sine waves:
Δ
t
e
y
=
−
7.659
sin
(
D
)
+
9.863
sin
(
2
D
+
3.5932
)
{\displaystyle \Delta t_{ey}=-7.659\sin(D)+9.863\sin \left(2D+3.5932\right)}
[minutes]
where:
D
=
6.240
040
77
+
0.017
201
97
(
365.25
(
y
−
2000
)
+
d
)
{\displaystyle D=6.240\,040\,77+0.017\,201\,97(365.25(y-2000)+d)}
where
d
{\displaystyle d}
represents the number of days since 1 January of the current year,
y
{\displaystyle y}
.
== Concept ==
During a year the equation of time varies as shown on the graph; its change from one year to the next is slight. Apparent time, and the sundial, can be ahead (fast) by as much as 16 min 33 s (around 3 November), or behind (slow) by as much as 14 min 6 s (around 11 February). The equation of time has zeros near 15 April, 13 June, 1 September, and 25 December. Ignoring very slow changes in the Earth's orbit and rotation, these events are repeated at the same times every tropical year. However, due to the non-integral number of days in a year, these dates can vary by a day or so from year to year. As an example of the inexactness of the dates, according to the U.S. Naval Observatory's Multiyear Interactive Computer Almanac the equation of time was zero at 02:00 UT1 on 16 April 2011.: 277
The graph of the equation of time is closely approximated by the sum of two sine curves, one with a period of a year and one with a period of half a year. The curves reflect two astronomical effects, each causing a different non-uniformity in the apparent daily motion of the Sun relative to the stars:
the obliquity of the ecliptic (the plane of the Earth's annual orbital motion around the Sun), which is inclined by about 23.44 degrees relative to the plane of the Earth's equator; and
the eccentricity of the Earth's orbit around the Sun, which is about 0.0167.
The equation of time vanishes only for a planet with zero axial tilt and zero orbital eccentricity. Two examples of planets with large equations of time are Mars and Uranus. On Mars the difference between sundial time and clock time can be as much as 50 minutes, due to the considerably greater eccentricity of its orbit. The planet Uranus, which has an extremely large axial tilt, has an equation of time that makes its days start and finish several hours earlier or later depending on where it is in its orbit.
== Notation ==
The United States Naval Observatory states "the Equation of Time is the difference apparent solar time minus mean solar time", i.e. if the sun is ahead of the clock the sign is positive, and if the clock is ahead of the sun the sign is negative. The equation of time is shown in the upper graph above for a period of slightly more than a year. The lower graph (which covers exactly one calendar year) has the same absolute values but the sign is reversed as it shows how far the clock is ahead of the sun. Publications may use either format: in the English-speaking world, the former usage is the more common, but is not always followed. Anyone who makes use of a published table or graph should first check its sign usage. Often, there is a note or caption which explains it. Otherwise, the usage can be determined by knowing that, during the first three months of each year, the clock is ahead of the sundial. The mnemonic "NYSS" (pronounced "nice"), for "new year, sundial slow", can be useful. Some published tables avoid the ambiguity by not using signs, but by showing phrases such as "sundial fast" or "sundial slow" instead.
== History ==
The phrase "equation of time" is derived from the medieval Latin aequātiō diērum, meaning "equation of days" or "difference of days". The word equation is used in the medieval sense of "reconciliation of a difference". The word aequātiō (and Middle English equation) was used in medieval astronomy to tabulate the difference between an observed value and the expected value (as in the equation of the centre, the equation of the equinoxes, the equation of the epicycle). Gerald J. Toomer uses the medieval term "equation", from the Latin aequātiō (equalization or adjustment), for Ptolemy's difference between the mean solar time and the apparent solar time. Johannes Kepler's definition of the equation is "the difference between the number of degrees and minutes of the mean anomaly and the degrees and minutes of the corrected anomaly.": 155
The difference between apparent solar time and mean time was recognized by astronomers since antiquity, but prior to the invention of accurate mechanical clocks in the mid-17th century, sundials were the only reliable timepieces, and apparent solar time was the generally accepted standard. Mean time did not supplant apparent time in national almanacs and ephemerides until the early 19th century.
=== Early astronomy ===
The irregular daily movement of the Sun was known to the Babylonians.
Book III of Ptolemy's Almagest (2nd century) is primarily concerned with the Sun's anomaly, and he tabulated the equation of time in his Handy Tables. Ptolemy discusses the correction needed to convert the meridian crossing of the Sun to mean solar time and takes into consideration the nonuniform motion of the Sun along the ecliptic and the meridian correction for the Sun's ecliptic longitude. He states the maximum correction is 8+1⁄3 time-degrees or 5⁄9 of an hour (Book III, chapter 9). However he did not consider the effect to be relevant for most calculations since it was negligible for the slow-moving luminaries and only applied it for the fastest-moving luminary, the Moon.
Based on Ptolemy's discussion in the Almagest, values for the equation of time (Arabic taʿdīl al-ayyām bi layālayhā) were standard for the tables (zij) in the works of medieval Islamic astronomy.
=== Early modern period ===
A description of apparent and mean time was given by Nevil Maskelyne in the Nautical Almanac for 1767: "Apparent Time is that deduced immediately from the Sun, whether from the Observation of his passing the Meridian, or from his observed Rising or Setting. This Time is different from that shewn by Clocks and Watches well regulated at Land, which is called equated or mean Time." He went on to say that, at sea, the apparent time found from observation of the Sun must be corrected by the equation of time, if the observer requires the mean time.
The right time was originally considered to be that which was shown by a sundial. When good mechanical clocks were introduced, they agreed with sundials only near four dates each year, so the equation of time was used to "correct" their readings to obtain sundial time. Some clocks, called equation clocks, included an internal mechanism to perform this "correction". Later, as clocks became the dominant good timepieces, uncorrected clock time, i.e., "mean time", became the accepted standard. The readings of sundials, when they were used, were then, and often still are, corrected with the equation of time, used in the reverse direction from previously, to obtain clock time. Many sundials, therefore, have tables or graphs of the equation of time engraved on them to allow the user to make this correction.: 123
The equation of time was used historically to set clocks. Between the invention of accurate clocks in 1656 and the advent of commercial time distribution services around 1900, there were several common land-based ways to set clocks. A sundial was read and corrected with the table or graph of the equation of time.
If a transit instrument was available or accuracy was important, the sun's transit across the meridian (the moment the sun appears to be due south or north of the observer, known as its culmination) was noted; the clock was then set to noon and offset by the number of minutes given by the equation of time for that date. A third method did not use the equation of time; instead, it used stellar observations to give sidereal time, exploiting the relationship between sidereal time and mean solar time.: 57–58 The more accurate methods were also precursors to finding the observer's longitude in relation to a prime meridian, such as in geodesy on land and celestial navigation on the sea.
The first tables to give the equation of time in an essentially correct way were published in 1665 by Christiaan Huygens. Huygens, following the tradition of Ptolemy and medieval astronomers in general, set his values for the equation of time so as to make all values positive throughout the year. This meant that any clock being set to mean time by Huygens's tables was consistently about 15 minutes slow compared to today's mean time.
Another set of tables was published in 1672–73 by John Flamsteed, who later became the first Astronomer Royal of the new Royal Greenwich Observatory. These appear to have been the first essentially correct tables that gave today's meaning of Mean Time (previously, as noted above, the sign of the equation was always positive and it was set at zero when the apparent time of sunrise was earliest relative to the clock time of sunrise). Flamsteed adopted the convention of tabulating and naming the correction in the sense that it was to be applied to the apparent time to give mean time.
The equation of time, correctly based on the two major components of the Sun's irregularity of apparent motion, was not generally adopted until after Flamsteed's tables of 1672–73, published with the posthumous edition of the works of Jeremiah Horrocks.: 49
Robert Hooke (1635–1703), who mathematically analyzed the universal joint, was the first to note that the geometry and mathematical description of the (non-secular) equation of time and the universal joint were identical, and proposed the use of a universal joint in the construction of a "mechanical sundial".: 219
=== 18th and early 19th centuries ===
The corrections in Flamsteed's tables of 1672–1673 and 1680 gave mean time computed essentially correctly and without need for further offset. But the numerical values in tables of the equation of time have somewhat changed since then, owing to three factors:
General improvements in accuracy that came from refinements in astronomical measurement techniques,
Slow intrinsic changes in the equation of time, occurring as a result of small long-term changes in the Earth's obliquity and eccentricity (affecting, for instance, the distance and dates of perihelion), and
The inclusion of small sources of additional variation in the apparent motion of the Sun, unknown in the 17th century but discovered from the 18th century onwards, including the effects of the Moon (See barycentre), Venus and Jupiter.
From 1767 to 1833, the British Nautical Almanac and Astronomical Ephemeris tabulated the equation of time in the sense 'add or subtract (as directed) the number of minutes and seconds stated to or from the apparent time to obtain the mean time'. Times in the Almanac were in apparent solar time, because time aboard ship was most often determined by observing the Sun. This operation would be performed in the unusual case that the mean solar time of an observation was needed. In the issues since 1834, all times have been in mean solar time, because by then the time aboard ship was increasingly often determined by marine chronometers. The instructions were consequently to add or subtract (as directed) the number of minutes stated to or from the mean time to obtain the apparent time. So now addition corresponded to the equation being positive and subtraction corresponded to it being negative.
As the apparent daily movement of the Sun is one revolution per day, that is 360° every 24 hours, and the Sun itself appears as a disc of about 0.5° in the sky, simple sundials can be read to a maximum accuracy of about one minute. Since the equation of time has a range of about 33 minutes, the difference between sundial time and clock time cannot be ignored. In addition to the equation of time, one also has to apply corrections due to one's distance from the local time zone meridian and summer time, if any.
The tiny increase of the mean solar day due to the slowing down of the Earth's rotation, by about 2 ms per day per century, which currently accumulates up to about 1 second every year, is not taken into account in traditional definitions of the equation of time, as it is imperceptible at the accuracy level of sundials.
== Major components ==
=== Eccentricity of the Earth's orbit ===
The Earth revolves around the Sun. As seen from Earth, the Sun appears to revolve once around the Earth through the background stars in one year. If the Earth orbited the Sun with a constant speed, in a circular orbit in a plane perpendicular to the Earth's axis, then the Sun would culminate every day at exactly the same time, and be a perfect time keeper (except for the very small effect of the slowing rotation of the Earth). But the orbit of the Earth is an ellipse not centered on the Sun, and its speed varies between 30.287 and 29.291 km/s, according to Kepler's laws of planetary motion, and its angular speed also varies, and thus the Sun appears to move faster (relative to the background stars) at perihelion (currently around 3 January) and slower at aphelion a half year later.
At these extreme points, this effect varies the apparent solar day by 7.9 s/day from its mean. Consequently, the smaller daily differences on other days in speed are cumulative until these points, reflecting how the planet accelerates and decelerates compared to the mean.
As a result, the eccentricity of the Earth's orbit contributes a periodic variation which is (in the first-order approximation) a sine wave with:
amplitude: 7.66 minutes
period: one year
zero points: perihelion (at the beginning of January) and aphelion (beginning of July)
extreme values: early April (negative) and early October (positive)
This component of the EoT is represented by aforementioned factor a:
a
=
−
7.659
sin
(
6.240
040
77
+
0.017
201
97
(
365
(
y
−
2000
)
+
d
)
)
{\displaystyle a=-7.659\sin(6.240\,040\,77+0.017\,201\,97(365(y-2000)+d))}
=== Obliquity of the ecliptic ===
Even if the Earth's orbit were circular, the perceived motion of the Sun along our celestial equator would still not be uniform. This is a consequence of the tilt of the Earth's rotational axis with respect to the plane of its orbit, or equivalently, the tilt of the ecliptic (the path the Sun appears to take in the celestial sphere) with respect to the celestial equator. The projection of this motion onto our celestial equator, along which "clock time" is measured, is a maximum at the solstices, when the yearly movement of the Sun is parallel to the equator (causing amplification of perceived speed) and yields mainly a change in right ascension. It is a minimum at the equinoxes, when the Sun's apparent motion is more sloped and yields more change in declination, leaving less for the component in right ascension, which is the only component that affects the duration of the solar day. A practical illustration of obliquity is that the daily shift of the shadow cast by the Sun in a sundial even on the equator is smaller close to the solstices and greater close to the equinoxes. If this effect operated alone, then days would be up to 24 hours and 20.3 seconds long (measured solar noon to solar noon) near the solstices, and as much as 20.3 seconds shorter than 24 hours near the equinoxes.
In the figure on the right, we can see the monthly variation of the apparent slope of the plane of the ecliptic at solar midday as seen from Earth. This variation is due to the apparent precession of the rotating Earth through the year, as seen from the Sun at solar midday.
In terms of the equation of time, the inclination of the ecliptic results in the contribution of a sine wave variation with:
amplitude: 9.87 minutes
period: 1/2 year
zero points: equinoxes and solstices
extreme values: beginning of February and August (negative) and beginning of May and November (positive).
This component of the EoT is represented by the aforementioned factor "b":
b
=
9.863
sin
(
2
(
6.240
040
77
+
0.017
201
97
(
365
(
y
−
2000
)
+
d
)
)
+
3.5932
)
{\displaystyle b=9.863\sin \left(2(6.240\,040\,77+0.017\,201\,97(365(y-2000)+d))+3.5932\right)}
== Secular effects ==
The two above mentioned factors have different wavelengths, amplitudes and phases, so their combined contribution is an irregular wave. At epoch 2000 these are the values (in minutes and seconds with UT dates):
On shorter timescales (thousands of years) the shifts in the dates of equinox and perihelion will be more important. The former is caused by precession, and shifts the equinox backwards compared to the stars. But it can be ignored in the current discussion as our Gregorian calendar is constructed in such a way as to keep the vernal equinox date at 20 March (at least at sufficient accuracy for our aim here). The shift of the perihelion is forwards, about 1.7 days every century. In 1246 the perihelion occurred on 22 December, the day of the solstice, so the two contributing waves had common zero points and the equation of time curve was symmetrical: in Astronomical Algorithms Meeus gives February and November extrema of 15 m 39 s and May and July ones of 4 m 58 s. Before then the February minimum was larger than the November maximum, and the May maximum larger than the July minimum. In fact, in years before −1900 (1901 BCE) the May maximum was larger than the November maximum. In the year −2000 (2001 BCE) the May maximum was +12 minutes and a couple seconds while the November maximum was just less than 10 minutes. The secular change is evident when one compares a current graph of the equation of time (see below) with one from 2000 years ago, e.g., one constructed from the data of Ptolemy.
== Practical use ==
If the gnomon (the shadow-casting object) is not an edge but a point (e.g., a hole in a plate), the shadow (or spot of light) will trace out a curve during the course of a day. If the shadow is cast on a plane surface, this curve will be a conic section (usually a hyperbola), since the circle of the Sun's motion together with the gnomon point define a cone. At the spring and autumnal equinoxes, the cone degenerates into a plane and the hyperbola into a line. With a different hyperbola for each day, hour marks can be put on each hyperbola which include any necessary corrections. Unfortunately, each hyperbola corresponds to two different days, one in each half of the year, and these two days will require different corrections. A convenient compromise is to draw the line for the "mean time" and add a curve showing the exact position of the shadow points at noon during the course of the year. This curve will take the form of a figure eight and is known as an analemma. By comparing the analemma to the mean noon line, the amount of correction to be applied generally on that day can be determined.
The equation of time is used not only in connection with sundials and similar devices, but also for many applications of solar energy. Machines such as solar trackers and heliostats have to move in ways that are influenced by the equation of time.
Civil time is the local mean time for a meridian that often passes near the center of the time zone, and may possibly be further altered by daylight saving time. When the apparent solar time that corresponds to a given civil time is to be found, the difference in longitude between the site of interest and the time zone meridian, daylight saving time, and the equation of time must all be considered.
== Calculation ==
The equation of time is obtained from a published table, or a graph. For dates in the past such tables are produced from historical measurements, or by calculation; for future dates, of course, tables can only be calculated. In devices such as computer-controlled heliostats the computer is often programmed to calculate the equation of time. The calculation can be numerical or analytical. The former are based on numerical integration of the differential equations of motion, including all significant gravitational and relativistic effects. The results are accurate to better than 1 second and are the basis for modern almanac data. The latter are based on a solution that includes only the gravitational interaction between the Sun and Earth, simpler than but not as accurate as the former. Its accuracy can be improved by including small corrections.
The following discussion describes a reasonably accurate (agreeing with almanac data to within 3 seconds over a wide range of years) algorithm for the equation of time that is well known to astronomers.: 89 It also shows how to obtain a simple approximate formula (accurate to within 1 minute over a large time interval), that can be easily evaluated with a calculator and provides the simple explanation of the phenomenon that was used previously in this article.
=== Mathematical description ===
The precise definition of the equation of time is:: 1529
E
O
T
=
G
H
A
−
G
M
H
A
{\displaystyle \mathrm {EOT} =\mathrm {GHA} -\mathrm {GMHA} }
The quantities occurring in this equation are:
EOT, the time difference between apparent solar time and mean solar time;
GHA, the Greenwich Hour Angle of the apparent (actual) Sun;
GMHA = Universal Time − Offset, the Greenwich Mean Hour Angle of the mean (fictitious) Sun.
Here time and angle are quantities that are related by factors such as: 2π radians = 360° = 1 day = 24 hours. The difference, EOT, is measurable since GHA is an angle that can be measured and Universal Time, UT, is a scale for the measurement of time. The offset by π = 180° = 12 hours from UT is needed because UT is zero at mean midnight while GMHA = 0 at mean noon. Universal Time is discontinuous at mean midnight so another quantity day number N, an integer, is required in order to form the continuous quantity time t: t = N + UT/24 hr days. Both GHA and GMHA, like all physical angles, have a mathematical, but not a physical discontinuity at their respective (apparent and mean) noon. Despite the mathematical discontinuities of its components, EOT is defined as a continuous function by adding (or subtracting) 24 hours in the small time interval between the discontinuities in GHA and GMHA.
According to the definitions of the angles on the celestial sphere GHA = GAST − α (see hour angle) where:
GAST is the Greenwich apparent sidereal time (the angle between the apparent vernal equinox and the meridian in the plane of the equator). This is a known function of UT.
α is the right ascension of the apparent Sun (the angle between the apparent vernal equinox and the actual Sun in the plane of the equator).
On substituting into the equation of time, it is
E
O
T
=
G
A
S
T
−
α
−
U
T
+
o
f
f
s
e
t
{\displaystyle \mathrm {EOT} =\mathrm {GAST} -\alpha -\mathrm {UT} +\mathrm {offset} }
Like the formula for GHA above, one can write GMHA = GAST − αM, where the last term is the right ascension of the mean Sun. The equation is often written in these terms as: 275 : 45
E
O
T
=
α
M
−
α
{\displaystyle \mathrm {EOT} =\alpha _{M}-\alpha }
where αM = GAST − UT + offset. In this formulation a measurement or calculation of EOT at a certain value of time depends on a measurement or calculation of α at that time. Both α and αM vary from 0 to 24 hours during the course of a year. The former has a discontinuity at a time that depends on the value of UT, while the latter has its at a slightly later time. As a consequence, when calculated this way EOT has two, artificial, discontinuities. They can both be removed by subtracting 24 hours from the value of EOT in the small time interval after the discontinuity in α and before the one in αM. The resulting EOT is a continuous function of time.
Another definition, denoted E to distinguish it from EOT, is
E
=
G
M
S
T
−
α
−
U
T
+
o
f
f
s
e
t
{\displaystyle E=\mathrm {GMST} -\alpha -\mathrm {UT} +\mathrm {offset} }
Here GMST = GAST − eqeq, is the Greenwich mean sidereal time (the angle between the mean vernal equinox and the mean Sun in the plane of the equator). Therefore, GMST is an approximation to GAST (and E is an approximation to EOT); eqeq is called the equation of the equinoxes and is due to the wobbling, or nutation of the Earth's axis of rotation about its precessional motion. Since the amplitude of the nutational motion is only about 1.2 s (18″ of longitude) the difference between EOT and E can be ignored unless one is interested in subsecond accuracy.
A third definition, denoted Δt to distinguish it from EOT and E, and now called the Equation of Ephemeris Time: 1532 (prior to the distinction that is now made between EOT, E, and Δt the latter was known as the equation of time) is
Δ
t
=
Λ
−
α
{\displaystyle \Delta t=\Lambda -\alpha }
here Λ is the ecliptic longitude of the mean Sun (the angle from the mean vernal equinox to the mean Sun in the plane of the ecliptic).
The difference Λ − (GMST − UT + offset) is 1.3 s from 1960 to 2040. Therefore, over this restricted range of years Δt is an approximation to EOT whose error is in the range 0.1 to 2.5 s depending on the longitude correction in the equation of the equinoxes; for many purposes, for example correcting a sundial, this accuracy is more than good enough.
=== Right ascension calculation ===
The right ascension, and hence the equation of time, can be calculated from Newton's two-body theory of celestial motion, in which the bodies (Earth and Sun) describe elliptical orbits about their common mass center. Using this theory, the equation of time becomes:
Δ
t
=
M
+
λ
p
−
α
{\displaystyle \Delta t=M+\lambda _{p}-\alpha }
where the new angles that appear are:
M = 2π(t − tp)/tY, is the mean anomaly, the angle from the periapsis of the elliptical orbit to the mean Sun; its range is from 0 to 2π as t increases from tp to tp + tY;
tY = 365.2596358 days is the length of time in an anomalistic year: the time interval between two successive passages of the periapsis;
λp = Λ − M, is the ecliptic longitude of the periapsis;
t is dynamical time, the independent variable in the theory. Here it is taken to be identical with the continuous time based on UT (see above), but in more precise calculations (of E or EOT) the small difference between them must be accounted for: 1530 as well as the distinction between UT1 and UTC.
tp is the value of t at the periapsis.
To complete the calculation three additional angles are required:
E, the Sun's eccentric anomaly (note that this is different from M);
ν, the Sun's true anomaly;
λ = ν + λp, the Sun's true longitude on the ecliptic.
All these angles are shown in the figure on the right, which shows the celestial sphere and the Sun's elliptical orbit seen from the Earth (the same as the Earth's orbit seen from the Sun). In this figure ε is the obliquity, while e = √1 − (b/a)2 is the eccentricity of the ellipse.
Now given a value of 0 ≤ M ≤ 2π, one can calculate α(M) by means of the following well-known procedure:: 89
First, given M, calculate E from Kepler's equation:: 159
M
=
E
−
e
sin
E
{\displaystyle M=E-e\sin {E}}
Although this equation cannot be solved exactly in closed form, values of E(M) can be obtained from infinite (power or trigonometric) series, graphical, or numerical methods. Alternatively, note that for e = 0, E = M, and by iteration:: 2
E
≈
M
+
e
sin
M
{\displaystyle E\approx M+e\sin {M}}
This approximation can be improved, for small e, by iterating again:
E
≈
M
+
e
sin
M
+
1
2
e
2
sin
2
M
{\displaystyle E\approx M+e\sin {M}+{\frac {1}{2}}e^{2}\sin {2M}}
,
and continued iteration produces successively higher order terms of the power series expansion in e. For small values of e (much less than 1) two or three terms of the series give a good approximation for E; the smaller e, the better the approximation.
Next, knowing E, calculate the true anomaly ν from an elliptical orbit relation: 165
ν
=
2
arctan
(
1
+
e
1
−
e
tan
1
2
E
)
{\displaystyle \nu =2\arctan \left({\sqrt {\frac {1+e}{1-e}}}\tan {\tfrac {1}{2}}E\right)}
The correct branch of the multiple valued function arctan x to use is the one that makes ν a continuous function of E(M) starting from νE=0 = 0. Thus for 0 ≤ E < π use arctan x = arctan x, and for π < E ≤ 2π use arctan x = arctan x + π. At the specific value E = π for which the argument of tan is infinite, use ν = E. Here arctan x is the principal branch, |arctan x| < π/2; the function that is returned by calculators and computer applications. Alternatively, this function can be expressed in terms of its Taylor series in e, the first three terms of which are:
ν
≈
E
+
e
sin
E
+
1
4
e
2
sin
2
E
{\displaystyle \nu \approx E+e\sin {E}+{\frac {1}{4}}e^{2}\sin {2E}}
.
For small e this approximation (or even just the first two terms) is a good one. Combining the approximation for E(M) with this one for ν(E) produces:
ν
≈
M
+
2
e
sin
M
+
5
4
e
2
sin
2
M
{\displaystyle \nu \approx M+2e\sin {M}+{\frac {5}{4}}e^{2}\sin {2M}}
.
The relation ν(M) is called the equation of the center; the expression written here is a second-order approximation in e. For the small value of e that characterises the Earth's orbit this gives a very good approximation for ν(M).
Next, knowing ν, calculate λ from its definition:
λ
=
ν
+
λ
p
{\displaystyle \lambda =\nu +\lambda _{p}}
The value of λ varies non-linearly with M because the orbit is elliptical and not circular. From the approximation for ν:
λ
≈
M
+
λ
p
+
2
e
sin
M
+
5
4
e
2
sin
2
M
{\displaystyle \lambda \approx M+\lambda _{p}+2e\sin {M}+{\frac {5}{4}}e^{2}\sin {2M}}
.
Finally, knowing λ calculate α from a relation for the right triangle on the celestial sphere shown above: 22
α
=
arctan
(
cos
ε
tan
λ
)
{\displaystyle \alpha =\arctan \left(\cos {\varepsilon }\tan {\lambda }\right)}
Note that the quadrant of α is the same as that of λ, therefore reduce λ to the range 0 to 2π and write
α
=
arctan
(
cos
ε
tan
λ
+
k
π
)
{\displaystyle \alpha =\arctan \left(\cos {\varepsilon }\tan {\lambda }+k\pi \right)}
,
where k is 0 if λ is in quadrant 1, it is 1 if λ is in quadrants 2 or 3 and it is 2 if λ is in quadrant 4. For the values at which tan is infinite, α = λ.
Although approximate values for α can be obtained from truncated Taylor series like those for ν,: 32 it is more efficacious to use the equation: 374
α
=
λ
−
arcsin
(
y
sin
(
α
+
λ
)
)
{\displaystyle \alpha =\lambda -\arcsin \left(y\sin \left(\alpha +\lambda \right)\right)}
where y = tan2(ε/2). Note that for ε = y = 0, α = λ and iterating twice:
α
≈
λ
−
y
sin
2
λ
+
1
2
y
2
sin
4
λ
{\displaystyle \alpha \approx \lambda -y\sin {2\lambda }+{\frac {1}{2}}y^{2}\sin {4\lambda }}
.
=== Final calculation ===
The equation of time is obtained by substituting the result of the right ascension calculation into an equation of time formula. Here Δt(M) = M + λp − α[λ(M)] is used; in part because small corrections (of the order of 1 second), that would justify using E, are not included, and in part because the goal is to obtain a simple analytical expression. Using two-term approximations for λ(M) and α(λ) allows Δt to be written as an explicit expression of two terms, which is designated Δtey because it is a first order approximation in e and in y.
1)
Δ
t
e
y
=
−
2
e
sin
M
+
y
sin
(
2
M
+
2
λ
p
)
=
−
7.659
sin
M
+
9.863
sin
(
2
M
+
3.5932
)
{\displaystyle \Delta t_{ey}=-2e\sin {M}+y\sin \left(2M+2\lambda _{p}\right)=-7.659\sin {M}+9.863\sin \left(2M+3.5932\right)}
minutes
This equation was first derived by Milne,: 375 who wrote it in terms of λ = M + λp. The numerical values written here result from using the orbital parameter values, e = 0.016709, ε = 23.4393° = 0.409093 radians, and λp = 282.9381° = 4.938201 radians that correspond to the epoch 1 January 2000 at 12 noon UT1. When evaluating the numerical expression for Δtey as given above, a calculator must be in radian mode to obtain correct values because the value of 2λp − 2π in the argument of the second term is written there in radians. Higher order approximations can also be written,: Eqs (45) and (46) but they necessarily have more terms. For example, the second order approximation in both e and y consists of five terms: 1535
2)
Δ
t
e
2
y
2
=
Δ
t
e
y
−
5
4
e
2
sin
2
M
+
4
e
y
sin
M
cos
(
2
M
+
2
λ
p
)
−
1
2
y
2
sin
(
4
M
+
4
λ
p
)
{\displaystyle \Delta t_{e^{2}y^{2}}=\Delta t_{ey}-{\frac {5}{4}}e^{2}\sin {2M}+4ey\sin {M}\cos \left(2M+2\lambda _{p}\right)-{\frac {1}{2}}y^{2}\sin \left(4M+4\lambda _{p}\right)}
This approximation has the potential for high accuracy, however, in order to achieve it over a wide range of years, the parameters e, ε, and λp must be allowed to vary with time.: 86 : 1531,1535 This creates additional calculational complications. Other approximations have been proposed, for example, Δte: 86 which uses the first order equation of the center but no other approximation to determine α, and Δte2 which uses the second order equation of the center.
The time variable, M, can be written either in terms of n, the number of days past perihelion, or D, the number of days past a specific date and time (epoch):
3)
M
=
2
π
t
Y
n
{\displaystyle M={\frac {2\pi }{t_{Y}}}n}
days
=
M
D
+
2
π
t
Y
D
{\displaystyle =M_{D}+{\frac {2\pi }{t_{Y}}}D}
days
=
6.240
040
77
+
0.017
201
97
D
{\displaystyle =6.240\,040\,77+0.017\,201\,97D}
4)
M
=
6.240
040
77
+
0.017
201
97
D
{\displaystyle M=6.240\,040\,77+0.017\,201\,97D}
Here MD is the value of M at the chosen date and time. For the values given here, in radians, MD is that measured for the actual Sun at the epoch, 1 January 2000 at 12 noon UT1, and D is the number of days past that epoch. At periapsis M = 2π, so solving gives D = Dp = 2.508109. This puts the periapsis on 4 January 2000 at 00:11:41 while the actual periapsis is, according to results from the Multiyear Interactive Computer Almanac (abbreviated as MICA), on 3 January 2000 at 05:17:30. This large discrepancy happens because the difference between the orbital radius at the two locations is only 1 part in a million; in other words, radius is a very weak function of time near periapsis. As a practical matter this means that one cannot get a highly accurate result for the equation of time by using n and adding the actual periapsis date for a given year. However, high accuracy can be achieved by using the formulation in terms of D.
When D > Dp, M is greater than 2π and one must subtract a multiple of 2π (that depends on the year) from it to bring it into the range 0 to 2π. Likewise for years prior to 2000 one must add multiples of 2π. For example, for the year 2010, D varies from 3653 on 1 January at noon to 4017 on 31 December at noon; the corresponding M values are 69.0789468 and 75.3404748 and are reduced to the range 0 to 2π by subtracting 10 and 11 times 2π respectively.
One can always write:
5) D = nY + d
where:
nY = number of days from the epoch to noon on 1 January of the desired year
0 ≤ d ≤ 364 (365 if the calculation is for a leap year).
The resulting equation for years after 2000, written as a sum of two terms, given 1), 4) and 5), is:
a
=
−
7.659
sin
(
6.240
040
77
+
0.017
201
97
(
365.25
(
y
−
2000
)
+
d
)
)
{\displaystyle a=-7.659\sin(6.240\,040\,77+0.017\,201\,97(365.25(y-2000)+d))}
b
=
9.863
sin
(
2
(
6.240
040
77
+
0.017
201
97
(
365.25
(
y
−
2000
)
+
d
)
)
+
3.5932
)
{\displaystyle b=9.863\sin \left(2(6.240\,040\,77+0.017\,201\,97(365.25(y-2000)+d))+3.5932\right)}
6)
Δ
t
e
y
=
a
+
b
{\displaystyle \Delta t_{ey}=a+b}
[minutes]
In plain text format:
7) EoT = -7.659sin(6.24004077 + 0.01720197(365*(y-2000) + d)) + 9.863sin( 2 (6.24004077 + 0.01720197 (365*(y-2000) + d)) + 3.5932 ) [minutes]
Term "a" represents the contribution of eccentricity, term "b" represents contribution of obliquity.
The result of the computations is usually given as either a set of tabular values, or a graph of the equation of time as a function of d. A comparison of plots of Δt, Δtey, and results from MICA all for the year 2000 is shown in the figure. The plot of Δtey is seen to be close to the results produced by MICA, the absolute error, Err = |Δtey − MICA2000|, is less than 1 minute throughout the year; its largest value is 43.2 seconds and occurs on day 276 (3 October). The plot of Δt is indistinguishable from the results of MICA, the largest absolute error between the two is 2.46 s on day 324 (20 November).
==== Continuity ====
For the choice of the appropriate branch of the arctan relation with respect to function continuity a modified version of the arctangent function is helpful. It brings in previous knowledge about the expected value by a parameter. The modified arctangent function is defined as:
arctan
η
x
=
arctan
x
+
π
round
(
η
−
arctan
x
π
)
{\displaystyle \arctan _{\eta }x=\arctan x+\pi \operatorname {round} {\left({\frac {\eta -\arctan x}{\pi }}\right)}}
.
It produces a value that is as close to η as possible. The function round rounds to the nearest integer.
Applying this yields:
Δ
t
(
M
)
=
M
+
λ
p
−
arctan
M
+
λ
p
(
cos
ε
tan
λ
)
{\displaystyle \Delta t(M)=M+\lambda _{p}-\arctan _{M+\lambda _{p}}\left(\cos {\varepsilon }\tan {\lambda }\right)}
.
The parameter M + λp arranges here to set Δt to the zero nearest value which is the desired one.
=== Secular change ===
The difference between the MICA and Δt results was checked every 5 years over the range from 1960 to 2040. In every instance the maximum absolute error was less than 3 s; the largest difference, 2.91 s, occurred on 22 May 1965 (day 141). However, in order to achieve this level of accuracy over this range of years it is necessary to account for the secular change in the orbital parameters with time. The equations that describe this variation are:: 86 : 1531,1535
e
=
1.6709
×
10
−
2
−
4.193
×
10
−
5
(
D
36
525
)
−
1.26
×
10
−
7
(
D
36525
)
2
ε
=
23.4393
−
0.013
(
D
36
525
)
−
2
×
10
−
7
(
D
36
525
)
2
+
5
×
10
−
7
(
D
36
525
)
3
degrees
λ
p
=
282.938
07
+
1.7195
(
D
36
525
)
+
3.025
×
10
−
4
(
D
36
525
)
2
degrees
{\displaystyle {\begin{aligned}e&=1.6709\times 10^{-2}-4.193\times 10^{-5}\left({\frac {D}{36\,525}}\right)-1.26\times 10^{-7}\left({\frac {D}{36525}}\right)^{2}\\\varepsilon &=23.4393-0.013\left({\frac {D}{36\,525}}\right)-2\times 10^{-7}\left({\frac {D}{36\,525}}\right)^{2}+5\times 10^{-7}\left({\frac {D}{36\,525}}\right)^{3}{\mbox{ degrees}}\\\lambda _{\mathrm {p} }&=282.938\,07+1.7195\left({\frac {D}{36\,525}}\right)+3.025\times 10^{-4}\left({\frac {D}{36\,525}}\right)^{2}{\mbox{ degrees}}\end{aligned}}}
According to these relations, in 100 years (D = 36525), λp increases by about 0.5% (1.7°), e decreases by about 0.25%, and ε decreases by about 0.05%.
As a result, the number of calculations required for any of the higher-order approximations of the equation of time requires a computer to complete them, if one wants to achieve their inherent accuracy over a wide range of time. In this event it is no more difficult to evaluate Δt using a computer than any of its approximations.
In all this note that Δtey as written above is easy to evaluate, even with a calculator, is accurate enough (better than 1 minute over the 80-year range) for correcting sundials, and has the nice physical explanation as the sum of two terms, one due to obliquity and the other to eccentricity that was used previously in the article. This is not true either for Δt considered as a function of M or for any of its higher-order approximations.
=== Alternative calculation ===
Another procedure for calculating the equation of time can be done as follows. Angles are in degrees; the conventional order of operations applies.
n = 360°/365.24 days,
where n is the Earth's mean angular orbital velocity in degrees per day, a.k.a. "the mean daily motion".
A
=
(
D
+
9
)
n
{\displaystyle A=\left(D+9\right)n}
where D is the date, counted in days starting at 1 on 1 January (i.e. the days part of the ordinal date in the year). 9 is the approximate number of days from the December solstice to 31 December. A is the angle the Earth would move on its orbit at its average speed from the December solstice to date D.
B
=
A
+
0.0167
⋅
360
∘
π
sin
(
(
D
−
3
)
n
)
{\displaystyle B=A+0.0167\cdot {\frac {360^{\circ }}{\pi }}\sin \left(\left(D-3\right)n\right)}
B is the angle the Earth moves from the solstice to date D, including a first-order correction for the Earth's orbital eccentricity, 0.0167 . The number 3 is the approximate number of days from 31 December to the current date of the Earth's perihelion. This expression for B can be simplified by combining constants to:
B
=
A
+
1.914
∘
⋅
sin
(
(
D
−
3
)
n
)
{\displaystyle B=A+1.914^{\circ }\cdot \sin \left(\left(D-3\right)n\right)}
.
C
=
A
−
arctan
tan
B
cos
23.44
∘
180
∘
{\displaystyle C={\frac {A-\arctan {\frac {\tan B}{\cos 23.44^{\circ }}}}{180^{\circ }}}}
Here, C is the difference between the angle moved at mean speed, and at the angle at the corrected speed projected onto the equatorial plane, and divided by 180° to get the difference in "half-turns". The value 23.44° is the tilt of the Earth's axis ("obliquity"). The subtraction gives the conventional sign to the equation of time. For any given value of x, arctan x (sometimes written as tan−1 x) has multiple values, differing from each other by integer numbers of half turns. The value generated by a calculator or computer may not be the appropriate one for this calculation. This may cause C to be wrong by an integer number of half-turns. The excess half-turns are removed in the next step of the calculation to give the equation of time:
E
O
T
=
720
(
C
−
nint
C
)
{\displaystyle \mathrm {EOT} =720\left(C-\operatorname {nint} {C}\right)}
minutes
The expression nint(C) means the nearest integer to C. On a computer, it can be programmed, for example, as INT(C + 0.5). Its value is 0, 1, or 2 at different times of the year. Subtracting it leaves a small positive or negative fractional number of half turns, which is multiplied by 720, the number of minutes (12 hours) that the Earth takes to rotate one half turn relative to the Sun, to get the equation of time.
Compared with published values, this calculation has a root mean square error of only 3.7 s. The greatest error is 6.0 s. This is much more accurate than the approximation described above, but not as accurate as the elaborate calculation.
==== Solar declination ====
The value of B in the above calculation is an accurate value for the Sun's ecliptic longitude (shifted by 90°), so the solar declination δ becomes readily available:
δ
=
−
arcsin
(
sin
23.44
∘
⋅
cos
B
)
{\displaystyle \delta =-\arcsin \left(\sin 23.44^{\circ }\cdot \cos B\right)}
which is accurate to within a fraction of a degree.
== See also ==
Azimuth – Horizontal angle from north or other reference cardinal direction
Milankovitch cycles – Global climate cycles
== Notes ==
== References ==
== External links ==
NOAA Solar Calculator
"USNO data services". Archived from the original on 29 May 2014. (include rise/set/transit times of the Sun and other celestial objects)
The equation of time described on the Royal Greenwich Observatory web site
The Equation of Time and the Analemma, by Kieron Taylor
An article by Brian Tung containing a link to a C program using a more accurate formula than most (particularly at high inclinations and eccentricities). The program can calculate solar declination, Equation of Time, or Analemma
Doing calculations using Ptolemy's geocentric planetary models with a discussion of his E.T. graph
Equation of Time Longcase Clock by John Topping C.1720
The equation of time correction-table A page describing how to correct a clock to a sundial
Solar tempometer: Calculate your solar time, including the equation of time | Wikipedia/Equation_of_time |
Equation were a British, young Devon-based folk supergroup formed in 1995, which combined the core talents of the Lakeman Brothers with Kathryn Roberts and Kate Rusby, later replaced for a spell by Cara Dillon. The name of the band refers to the initials of the band members' names, KR2 + SL3.
Their first single "He Loves Me" was originally released in 1996 on the Blanco y Negro-WEA label and was followed by four studio albums.
The Times reviewed their first album release Hazy Daze. in 1998, scoring 7/10 and giving a favourable comparison to Fairport Convention.
== Discography ==
=== Albums ===
Hazy Daze (1998, Blanco Y Negro-WEA) - (Putumayo US)
The Lucky Few (1998, Black Burst Records - Rough Trade) - (Putumayo US)
First Name Terms (2002, IScream Music)
Return to Me (Recorded 1995/1996 - Released 2003, Rough Trade Records)
=== Singles and EPs ===
"In Session" (1995)
"He Loves Me" (1996)
The Dark Ages EP (2000)
== References == | Wikipedia/Equation_(band) |
The Equation Group, also known in China as APT-C-40, is a highly sophisticated threat actor suspected of being tied to the Tailored Access Operations (TAO) unit of the United States National Security Agency (NSA). Kaspersky Labs describes them as one of the most sophisticated advanced persistent threats in the world and "the most advanced (...) we have seen", operating alongside the creators of Stuxnet and Flame. Most of their targets have been in Iran, Russia, Pakistan, Afghanistan, India, Syria and Mali.
The name originated from the group's extensive use of encryption. By 2015, Kaspersky documented 500 malware infections by the group in at least 42 countries, while acknowledging that the actual number could be in the tens of thousands due to its self-terminating protocol.
In 2017, WikiLeaks published a discussion held within the CIA on how it had been possible to identify the group. One commenter wrote that "the Equation Group as labeled in the report does not relate to a specific group but rather a collection of tools" used for hacking.
== Discovery ==
At the Kaspersky Security Analysts Summit held in Mexico on February 16, 2015, Kaspersky Lab announced its discovery of the Equation Group. According to Kaspersky Lab's report, the group has been active since at least 2001, with more than 60 actors. The malware used in their operations, dubbed EquationDrug and GrayFish, is found to be capable of reprogramming hard disk drive firmware. Because of the advanced techniques involved and high degree of covertness, the group is suspected of ties to the NSA, but Kaspersky Lab has not identified the actors behind the group.
== Probable links to Stuxnet and the NSA ==
In 2015 Kaspersky's research findings on the Equation Group noted that its loader, "GrayFish", had similarities to a previously discovered loader, "Gauss",[repository] from another attack series, and separately noted that the Equation Group used two zero-day attacks later used in Stuxnet; the researchers concluded that "the similar type of usage of both exploits together in different computer worms, at around the same time, indicates that the EQUATION group and the Stuxnet developers are either the same or working closely together".: 13
=== Firmware ===
They also identified that the platform had at times been spread by interdiction (interception of legitimate CDs sent by a scientific conference organizer by mail),: 15 and that the platform had the "unprecedented" ability to infect and be transmitted through the hard drive firmware of several major hard drive manufacturers, and create and use hidden disk areas and virtual disk systems for its purposes, a feat which would require access to the manufacturer's source code to achieve,: 16–18 and that the tool was designed for surgical precision, going so far as to exclude specific countries by IP and allow targeting of specific usernames on discussion forums.: 23–26
=== Codewords and timestamps ===
The NSA codewords "STRAITACID" and "STRAITSHOOTER" have been found inside the malware. In addition, timestamps in the malware seem to indicate that the programmers worked overwhelmingly Monday–Friday in what would correspond to a 08:00–17:00 (8:00 AM - 5:00 PM) workday in an Eastern United States time zone.
=== The LNK exploit ===
Kaspersky's global research and analysis team, otherwise known as GReAT, claimed to have found a piece of malware that contained Stuxnet's "privLib" in 2008. Specifically it contained the LNK exploit found in Stuxnet in 2010. Fanny is classified as a worm that affects certain Windows operating systems and attempts to spread laterally via network connection or USB storage.[repository] Kaspersky stated that they suspect that the Equation Group has been around longer than Stuxnet, based on the recorded compile time of Fanny.
=== Link to IRATEMONK ===
F-Secure claims that the Equation Group's malicious hard drive firmware is TAO program "IRATEMONK", one of the items from the NSA ANT catalog exposed in a 2013 Der Spiegel article. IRATEMONK provides the attacker with an ability to have their software application persistently installed on desktop and laptop computers, despite the disk being formatted, its data erased or the operating system re-installed. It infects the hard drive firmware, which in turn adds instructions to the disk's master boot record that causes the software to install each time the computer is booted up. It is capable of infecting certain hard drives from Seagate, Maxtor, Western Digital, Samsung, IBM, Micron Technology and Toshiba.
== 2016 breach of the Equation Group ==
In August 2016, a hacking group calling itself "The Shadow Brokers" announced that it had stolen malware code from the Equation Group. Kaspersky Lab noticed similarities between the stolen code and earlier known code from the Equation Group malware samples it had in its possession including quirks unique to the Equation Group's way of implementing the RC6 encryption algorithm, and therefore concluded that this announcement is legitimate. The most recent dates of the stolen files are from June 2013, thus prompting Edward Snowden to speculate that a likely lockdown resulting from his leak of the NSA's global and domestic surveillance efforts stopped The Shadow Brokers' breach of the Equation Group. Exploits against Cisco Adaptive Security Appliances and Fortinet's firewalls were featured in some malware samples released by The Shadow Brokers. EXTRABACON, a Simple Network Management Protocol exploit against Cisco's ASA software, was a zero-day exploit as of the time of the announcement. Juniper also confirmed that its NetScreen firewalls were affected. The EternalBlue exploit was used to conduct the damaging worldwide WannaCry ransomware attack.
== 2022 alleged Northwestern Polytechnical University hack ==
In 2022, an investigation conducted by the Chinese National Computer Virus Emergency Response Center (CVERC) and computer security firm Qihoo 360 attributed an extensive cyber attack on China's Northwestern Polytechnical University (NPU) to the NSA's Office of Tailored Access Operations (TAO), compromising tens of thousands of network devices in China over the years and exfiltrating over 140GB of high-value data.
The CVERC alleged that the attack involved a "longer period of preparatory work", setting up an anonymized attack infrastructure by leveraging SunOS zero-days to compromise institutions with large network traffic in 17 countries, 70% of which neighbored China. Those compromised machines were used as "springboards" to gain access into the NPU by leveraging man-in-the-middle and spear-phishing attacks against students and teachers. The report also claims the NSA had used two cover companies, "Jackson Smith Consultants" and "Mueller Diversified Systems" to purchase US-based IP addresses that would later be used in the FOXACID platform to launch attacks on the Northwestern.
CVERC and 360 identified 41 different tools and malware samples during forensic analysis, many of which were similar or consistent with TAO weapons exposed in the Shadow Brokers leak. Investigators also attributed the attack to the Equation Group due to a mixture of attack times, human errors and American English keyboard inputs. Forensic analysis on one of the tools, called "NOPEN", which required human input, indicated that 98% of all attacks occurred during U.S. working hours, with no cyber-attacks being logged during weekends or during American holidays such as Memorial Day and Independence Day.
== See also ==
Global surveillance disclosures (2013–present)
United States intelligence operations abroad
Firmware hacking
== References ==
== External links ==
Equation Group: Questions and Answers by Kaspersky Lab, Version: 1.5, February 2015
A Fanny Equation: "I am your father, Stuxnet" by Kaspersky Lab, February 2015
fanny.bmp source - at GitHub, November 30, 2020
Technical Write-up - at GitHub, February 10, 2021 | Wikipedia/Equation_Group |
A chemical equation is the symbolic representation of a chemical reaction in the form of symbols and chemical formulas. The reactant entities are given on the left-hand side and the product entities are on the right-hand side with a plus sign between the entities in both the reactants and the products, and an arrow that points towards the products to show the direction of the reaction. The chemical formulas may be symbolic, structural (pictorial diagrams), or intermixed. The coefficients next to the symbols and formulas of entities are the absolute values of the stoichiometric numbers. The first chemical equation was diagrammed by Jean Beguin in 1615.
== Structure ==
A chemical equation (see an example below) consists of a list of reactants (the starting substances) on the left-hand side, an arrow symbol, and a list of products (substances formed in the chemical reaction) on the right-hand side. Each substance is specified by its chemical formula, optionally preceded by a number called stoichiometric coefficient. The coefficient specifies how many entities (e.g. molecules) of that substance are involved in the reaction on a molecular basis. If not written explicitly, the coefficient is equal to 1. Multiple substances on any side of the equation are separated from each other by a plus sign.
As an example, the equation for the reaction of hydrochloric acid with sodium can be denoted:
2HCl + 2Na → 2NaCl + H2
Given the formulas are fairly simple, this equation could be read as "two H-C-L plus two N-A yields two N-A-C-L and H two." Alternately, and in general for equations involving complex chemicals, the chemical formulas are read using IUPAC nomenclature, which could verbalise this equation as "two hydrochloric acid molecules and two sodium atoms react to form two formula units of sodium chloride and a hydrogen gas molecule."
=== Reaction types ===
Different variants of the arrow symbol are used to denote the type of a reaction:
=== State of matter ===
To indicate physical state of a chemical, a symbol in parentheses may be appended to its formula: (s) for a solid, (l) for a liquid, (g) for a gas, and (aq) for an aqueous solution. This is especially done when one wishes to emphasize the states or changes thereof. For example, the reaction of aqueous hydrochloric acid with solid (metallic) sodium to form aqueous sodium chloride and hydrogen gas would be written like this:
2HCl(aq) + 2Na(s) → 2NaCl(aq) + H2(g)
That reaction would have different thermodynamic and kinetic properties if gaseous hydrogen chloride were to replace the hydrochloric acid as a reactant:
2HCl(g) + 2Na(s) → 2NaCl(s) + H2(g)
Alternately, an arrow without parentheses is used in some cases to indicate formation of a gas ↑ or precipitate ↓. This is especially useful if only one such species is formed. Here is an example indicating that hydrogen gas is formed:
2HCl + 2Na → 2 NaCl + H2 ↑
=== Catalysis and other conditions ===
If the reaction requires energy, it is indicated above the arrow. A capital Greek letter delta (Δ) or a triangle (△) is put on the reaction arrow to show that energy in the form of heat is added to the reaction. The expression hν is used as a symbol for the addition of energy in the form of light. Other symbols are used for other specific types of energy or radiation.
Similarly, if a reaction requires a certain medium with certain specific characteristics, then the name of the acid or base that is used as a medium may be placed on top of the arrow. If no specific acid or base is required, another way of denoting the use of an acidic or basic medium is to write H+ or OH− (or even "acid" or "base") on top of the arrow. Specific conditions of the temperature and pressure, as well as the presence of catalysts, may be indicated in the same way.
=== Notation variants ===
The standard notation for chemical equations only permits all reactants on one side, all products on the other, and all stoichiometric coefficients positive. For example, the usual form of the equation for dehydration of methanol to dimethylether is:
2 CH3OH → CH3OCH3 + H2O
Sometimes an extension is used, where some substances with their stoichiometric coefficients are moved above or below the arrow, preceded by a plus sign or nothing for a reactant, and by a minus sign for a product. Then the same equation can look like this:
2
CH
3
OH
→
−
H
2
O
CH
3
OCH
3
{\displaystyle {\ce {2CH3OH->[{\overset {}{\ce {-H2O}}}]CH3OCH3}}}
Such notation serves to hide less important substances from the sides of the equation, to make the type of reaction at hand more obvious, and to facilitate chaining of chemical equations. This is very useful in illustrating multi-step reaction mechanisms. Note that the substances above or below the arrows are not catalysts in this case, because they are consumed or produced in the reaction like ordinary reactants or products.
Another extension used in reaction mechanisms moves some substances to branches of the arrow. Both extensions are used in the example illustration of a mechanism.
Use of negative stoichiometric coefficients at either side of the equation (like in the example below) is not widely adopted and is often discouraged.
2
CH
3
OH
−
H
2
O
⟶
CH
3
OCH
3
{\displaystyle {\ce {2 CH3OH \;-\; H2O -> CH3OCH3}}}
== Balancing chemical equations ==
Because no nuclear reactions take place in a chemical reaction, the chemical elements pass through the reaction unchanged. Thus, each side of the chemical equation must represent the same number of atoms of any particular element (or nuclide, if different isotopes are taken into account). The same holds for the total electric charge, as stated by the charge conservation law. An equation adhering to these requirements is said to be balanced.
A chemical equation is balanced by assigning suitable values to the stoichiometric coefficients. Simple equations can be balanced by inspection, that is, by trial and error. Another technique involves solving a system of linear equations.
Balanced equations are usually written with smallest natural-number coefficients. Yet sometimes it may be advantageous to accept a fractional coefficient, if it simplifies the other coefficients. The introductory example can thus be rewritten as
HCl
+
Na
⟶
NaCl
+
1
2
H
2
{\displaystyle {\ce {HCl + Na -> NaCl + 1/2 H2}}}
In some circumstances the fractional coefficients are even inevitable. For example, the reaction corresponding to the standard enthalpy of formation must be written such that one molecule of a single product is formed. This will often require that some reactant coefficients be fractional, as is the case with the formation of lithium fluoride:
Li
(
s
)
+
1
2
F
2
(
g
)
⟶
LiF
(
s
)
{\displaystyle {\ce {Li(s) + 1/2F2(g) -> LiF(s)}}}
=== Inspection method ===
The method of inspection can be outlined as setting the most complex substance's stoichiometric coefficient to 1 and assigning values to other coefficients step by step such that both sides of the equation end up with the same number of atoms for each element. If any fractional coefficients arise during this process, the presence of fractions may be eliminated (at any time) by multiplying all coefficients by their lowest common denominator.
Example
Balancing of the chemical equation for the complete combustion of methane
?
CH
4
+
?
O
2
⟶
?
CO
2
+
?
H
2
O
{\displaystyle {\ce {{\mathord {?}}\,{CH4}+{\mathord {?}}\,{O2}->{\mathord {?}}\,{CO2}+{\mathord {?}}\,{H2O}}}}
is achieved as follows:
A coefficient of 1 is placed in front of the most complex formula (CH4):
1
CH
4
+
?
O
2
⟶
?
CO
2
+
?
H
2
O
{\displaystyle {\ce {1{CH4}+{\mathord {?}}\,{O2}->{\mathord {?}}\,{CO2}+{\mathord {?}}\,{H2O}}}}
The left-hand side has 1 carbon atom, so 1 molecule of CO2 will balance it. The left-hand side also has 4 hydrogen atoms, which will be balanced by 2 molecules of H2O:
1
CH
4
+
?
O
2
⟶
1
CO
2
+
2
H
2
O
{\displaystyle {\ce {1{CH4}+{\mathord {?}}\,{O2}->1{CO2}+2H2O}}}
Balancing the 4 oxygen atoms of the right-hand side by 2 molecules of O2 yields the equation
1
CH
4
+
2
O
2
⟶
1
CO
2
+
2
H
2
O
{\displaystyle {\ce {1 CH4 + 2 O2 -> 1 CO2 + 2 H2O}}}
The coefficients equal to 1 are omitted, as they do not need to be specified explicitly:
CH
4
+
2
O
2
⟶
CO
2
+
2
H
2
O
{\displaystyle {\ce {CH4 + 2 O2 -> CO2 + 2 H2O}}}
It is wise to check that the final equation is balanced, i.e. that for each element there is the same number of atoms on the left- and right-hand side: 1 carbon, 4 hydrogen, and 4 oxygen.
=== System of linear equations ===
For each chemical element (or nuclide or unchanged moiety or charge) i, its conservation requirement can be expressed by the mathematical equation
∑
j
∈
reactants
a
i
j
s
j
=
∑
j
∈
products
a
i
j
s
j
{\displaystyle \sum _{j\,\in \,{\text{reactants}}}\!\!\!\!\!a_{ij}s_{j}\ =\!\!\!\!\!\sum _{j\,\in \,{\text{products}}}\!\!\!\!\!a_{ij}s_{j}}
where
aij is the number of atoms of element i in a molecule of substance j (per formula in the chemical equation), and
sj is the stoichiometric coefficient for the substance j.
This results in a homogeneous system of linear equations, which are readily solved using mathematical methods. Such system always has the all-zeros trivial solution, which we are not interested in, but if there are any additional solutions, there will be infinite number of them. Any non-trivial solution will balance the chemical equation. A "preferred" solution is one with whole-number, mostly positive stoichiometric coefficients sj with greatest common divisor equal to one.
==== Example ====
Let us assign variables to stoichiometric coefficients of the chemical equation from the previous section and write the corresponding linear equations:
s
1
CH
4
+
s
2
O
2
⟶
s
3
CO
2
+
s
4
H
2
O
{\displaystyle {\ce {{\mathit {s}}_{1}{CH4}+{\mathit {s}}_{2}{O2}->{\mathit {s}}_{3}{CO2}+{\mathit {s}}_{4}{H2O}}}}
C:
s
1
=
s
3
H:
4
s
1
=
2
s
4
O:
2
s
2
=
2
s
3
+
s
4
{\displaystyle \quad \;\;\;{\begin{aligned}{\text{C:}}&&s_{1}&=s_{3}\\{\text{H:}}&&4s_{1}&=2s_{4}\\{\text{O:}}&&2s_{2}&=2s_{3}+s_{4}\end{aligned}}}
All solutions to this system of linear equations are of the following form, where r is any real number:
s
1
=
r
s
2
=
2
r
s
3
=
r
s
4
=
2
r
{\displaystyle {\begin{aligned}s_{1}&=r\\s_{2}&=2r\\s_{3}&=r\\s_{4}&=2r\end{aligned}}}
The choice of r = 1 yields the preferred solution,
s
1
=
1
s
2
=
2
s
3
=
1
s
4
=
2
{\displaystyle {\begin{aligned}s_{1}&=1\\s_{2}&=2\\s_{3}&=1\\s_{4}&=2\end{aligned}}}
which corresponds to the balanced chemical equation:
CH
4
+
2
O
2
⟶
CO
2
+
2
H
2
O
{\displaystyle {\ce {CH4 + 2 O2 -> CO2 + 2 H2O}}}
=== Matrix method ===
The system of linear equations introduced in the previous section can also be written using an efficient matrix formalism. First, to unify the reactant and product stoichiometric coefficients sj, let us introduce the quantity
ν
j
=
{
−
s
j
for a reactant
+
s
j
for a product
{\displaystyle \nu _{j}={\begin{cases}-s_{j}&{\text{for a reactant}}\\+s_{j}&{\text{for a product}}\end{cases}}}
called stoichiometric number, which simplifies the linear equations to
∑
j
=
1
J
a
i
j
ν
j
=
0
{\displaystyle \sum _{j=1}^{J}a_{ij}\nu _{j}=0}
where J is the total number of reactant and product substances (formulas) in the chemical equation.
Placement of the values aij at row i and column j of the composition matrix
A =
[
a
1
,
1
a
1
,
2
⋯
a
1
,
J
a
2
,
1
a
2
,
2
⋯
a
2
,
J
⋮
⋮
⋱
⋮
]
{\displaystyle {\begin{bmatrix}a_{1,1}&a_{1,2}&\cdots &a_{1,J}\\a_{2,1}&a_{2,2}&\cdots &a_{2,J}\\\vdots &\vdots &\ddots &\vdots \end{bmatrix}}}
and arrangement of the stoichiometric numbers into the stoichiometric vector
ν =
[
ν
1
ν
2
⋮
ν
J
]
{\displaystyle {\begin{bmatrix}\nu _{1}\\\nu _{2}\\\vdots \\\nu _{J}\end{bmatrix}}}
allows the system of equations to be expressed as a single matrix equation:
Aν = 0
Like previously, any nonzero stoichiometric vector ν, which solves the matrix equation, will balance the chemical equation.
The set of solutions to the matrix equation is a linear space called the kernel of the matrix A. For this space to contain nonzero vectors ν, i.e. to have a positive dimension JN, the columns of the composition matrix A must not be linearly independent. The problem of balancing a chemical equation then becomes the problem of determining the JN-dimensional kernel of the composition matrix. It is important to note that only for JN = 1 will there be a unique preferred solution to the balancing problem. For JN > 1 there will be an infinite number of preferred solutions with JN of them linearly independent. If JN = 0, there will be only the unusable trivial solution, the zero vector.
Techniques have been developed to quickly calculate a set of JN independent solutions to the balancing problem, which are superior to the inspection and algebraic method in that they are determinative and yield all solutions to the balancing problem.
Example
Using the same chemical equation again, write the corresponding matrix equation:
s
1
CH
4
+
s
2
O
2
⟶
s
3
CO
2
+
s
4
H
2
O
{\displaystyle {\ce {{\mathit {s}}_{1}{CH4}+{\mathit {s}}_{2}{O2}->{\mathit {s}}_{3}{CO2}+{\mathit {s}}_{4}{H2O}}}}
C:
H:
O:
[
1
0
1
0
4
0
0
2
0
2
2
1
]
[
ν
1
ν
2
ν
3
ν
4
]
=
0
{\displaystyle {\begin{matrix}{\text{C:}}\\{\text{H:}}\\{\text{O:}}\end{matrix}}\quad {\begin{bmatrix}1&0&1&0\\4&0&0&2\\0&2&2&1\end{bmatrix}}{\begin{bmatrix}\nu _{1}\\\nu _{2}\\\nu _{3}\\\nu _{4}\end{bmatrix}}=\mathbf {0} }
Its solutions are of the following form, where r is any real number:
[
ν
1
ν
2
ν
3
ν
4
]
=
[
−
s
1
−
s
2
s
3
s
4
]
=
r
[
−
1
−
2
1
2
]
{\displaystyle {\begin{bmatrix}\nu _{1}\\\nu _{2}\\\nu _{3}\\\nu _{4}\end{bmatrix}}={\begin{bmatrix}-s_{1}\\-s_{2}\\s_{3}\\s_{4}\end{bmatrix}}=r{\begin{bmatrix}-1\\-2\\1\\2\end{bmatrix}}}
The choice of r = 1 and a sign-flip of the first two rows yields the preferred solution to the balancing problem:
[
−
ν
1
−
ν
2
ν
3
ν
4
]
=
[
s
1
s
2
s
3
s
4
]
=
[
1
2
1
2
]
{\displaystyle {\begin{bmatrix}-\nu _{1}\\-\nu _{2}\\\nu _{3}\\\nu _{4}\end{bmatrix}}={\begin{bmatrix}s_{1}\\s_{2}\\s_{3}\\s_{4}\end{bmatrix}}={\begin{bmatrix}1\\2\\1\\2\end{bmatrix}}}
== Ionic equations ==
An ionic equation is a chemical equation in which electrolytes are written as dissociated ions. Ionic equations are used for single and double displacement reactions that occur in aqueous solutions.
For example, in the following precipitation reaction:
CaCl
2
+
2
AgNO
3
⟶
Ca
(
NO
3
)
2
+
2
AgCl
↓
{\displaystyle {\ce {CaCl2 + 2AgNO3 -> Ca(NO3)2 + 2 AgCl(v)}}}
the full ionic equation is:
Ca
2
+
+
2
Cl
−
+
2
Ag
+
+
2
NO
3
−
⟶
Ca
2
+
+
2
NO
3
−
+
2
AgCl
↓
{\displaystyle {\ce {Ca^2+ + 2Cl^- + 2Ag+ + 2NO3^- -> Ca^2+ + 2NO3^- + 2AgCl(v)}}}
or, with all physical states included:
Ca
2
+
(
aq
)
+
2
Cl
−
(
aq
)
+
2
Ag
+
(
aq
)
+
2
NO
3
−
(
aq
)
⟶
Ca
2
+
(
aq
)
+
2
NO
3
−
(
aq
)
+
2
AgCl
↓
{\displaystyle {\ce {Ca^2+(aq) + 2Cl^{-}(aq) + 2Ag+(aq) + 2NO3^{-}(aq) -> Ca^2+(aq) + 2NO3^{-}(aq) + 2AgCl(v)}}}
In this reaction, the Ca2+ and the NO3− ions remain in solution and are not part of the reaction. That is, these ions are identical on both the reactant and product side of the chemical equation. Because such ions do not participate in the reaction, they are called spectator ions. A net ionic equation is the full ionic equation from which the spectator ions have been removed. The net ionic equation of the proceeding reactions is:
2
Cl
−
+
2
Ag
+
⟶
2
AgCl
↓
{\displaystyle {\ce {2Cl^- + 2Ag+ -> 2AgCl(v)}}}
or, in reduced balanced form,
Ag
+
+
Cl
−
⟶
AgCl
↓
{\displaystyle {\ce {Ag+ + Cl^- -> AgCl(v)}}}
In a neutralization or acid/base reaction, the net ionic equation will usually be:
H
+
(
aq
)
+
OH
−
(
aq
)
⟶
H
2
O
(
l
)
{\displaystyle {\ce {H+ (aq) + OH^{-}(aq) -> H2O(l)}}}
There are a few acid/base reactions that produce a precipitate in addition to the water molecule shown above. An example is the reaction of barium hydroxide with phosphoric acid, which produces not only water but also the insoluble salt barium phosphate. In this reaction, there are no spectator ions, so the net ionic equation is the same as the full ionic equation.
3
Ba
(
OH
)
2
+
2
H
3
PO
4
⟶
6
H
2
O
+
Ba
3
(
PO
4
)
2
↓
{\displaystyle {\ce {3Ba(OH)2 + 2H3PO4 -> 6H2O + Ba3(PO4)2(v)}}}
3
Ba
2
+
+
6
OH
−
+
6
H
+
+
2
PO
4
3
−
⏟
phosphate
⟶
6
H
2
O
+
Ba
3
(
PO
4
)
2
↓
⏟
barium
phosphate
{\displaystyle {\ce {{3Ba^{2}+}+{6OH^{-}}+{6H+}}}+\underbrace {\ce {2PO4^{3}-}} _{\ce {phosphate}}{\ce {->{6H2O}+\underbrace {Ba3(PO4)2(v)} _{barium~phosphate}}}}
Double displacement reactions that feature a carbonate reacting with an acid have the net ionic equation:
2
H
+
+
CO
3
2
−
⏟
carbonate
⟶
H
2
O
+
CO
2
↑
{\displaystyle {\ce {2H+}}+\underbrace {{\ce {CO3^2-}}} _{{\ce {carbonate}}}{\ce {-> H2O + CO2 (^)}}}
If every ion is a "spectator ion" then there was no reaction, and the net ionic equation is null.
Generally, if zj is the multiple of elementary charge on the j-th molecule, charge neutrality may be written as:
∑
j
=
1
J
z
j
ν
j
=
0
{\displaystyle \sum _{j=1}^{J}z_{j}\nu _{j}=0}
where the νj are the stoichiometric coefficients described above. The zj may be incorporated
as an additional row in the aij matrix described above, and a properly balanced ionic equation will then also obey:
∑
j
=
1
J
a
i
j
ν
j
=
0
{\displaystyle \sum _{j=1}^{J}a_{ij}\nu _{j}=0}
== History ==
== Typesetting ==
== See also ==
Mathematical notation
Comparison of TeX editors
TeX extentions for science and chemistry notation
Chemistry notation in TeX
== Notes ==
== References == | Wikipedia/Chemical_equation |
An equation clock is a mechanical clock which includes a mechanism that simulates the equation of time, so that the user can read or calculate solar time, as would be shown by a sundial. The first accurate clocks, controlled by pendulums, were patented by Christiaan Huyghens in 1657. For the next few decades, people were still accustomed to using sundials, and wanted to be able to use clocks to find solar time. Equation clocks were invented to fill this need.
Early equation clocks have a pointer that moves to show the equation of time on a dial or scale. The clock itself runs at constant speed. The user calculates solar time by adding the equation of time to the clock reading. Later equation clocks, made in the 18th century, perform the compensation automatically, so the clock directly shows solar time. Some of them also show mean time, which is often called "clock time".
== Simulation mechanisms ==
All equation clocks include a mechanism that simulates the equation of time, so a lever moves, or a shaft rotates, in a way that represents the variations of the equation of time as the year progresses. There are two frequently-used types of mechanism:
=== Cam and lever mechanism ===
In this type of mechanism, a shaft is driven by the clock so it rotates once a year, at constant speed. The shaft carries a cam, which is approximately "kidney shaped" such that its radius is essentially a graph of the annual variation of the equation of time. A follower and lever rest against the cam, so that as the cam rotates the lever moves in a way that represents the changing equation of time. This lever drives other components in the clock.
=== Double shaft mechanism ===
To a close approximation, the equation of time can be represented as the sum of two sine waves, one with a period of one year and the other with a period of six months, with the relative phase varying very slowly (marginally noticeable over the course of a century). See the explanation in Equation of time for more detail.
The double shaft mechanism has two shafts rotating at constant speeds: one turns once a year, and the other twice a year. Cranks or pins attached to the two shafts move the two ends of a combining lever (sometimes referred to as a whippletree) sinusoidally; if the dimensions are chosen correctly, the midpoint of the rod moves in a way that simulates the equation of time.
== Types of equation clock ==
During the period when equation clocks were made and used, all clocks were made by hand. No two are exactly alike. Many equation clocks also have other features, such as displays of the phase of the moon or the times of sunrise and sunset. Leaving aside such additions, there are four different ways in which the clocks function. The following paragraphs are intended, not as detailed descriptions of individual clocks, but as illustrations of the general principles of these four different types of equation clock. The basic workings of particular clocks resemble these, but details vary. Pictures and descriptions of various equation clocks, which still exist in museums, can be accessed through the External links listed below.
=== Clocks without solar time displays ===
Many equation clocks, especially early ones, have a normal clock mechanism, showing mean time, and also a display that shows the equation of time. An equation of time simulation mechanism drives the pointer on this display. The user has to add the equation of time to clock time to calculate solar time.
=== Clocks that directly display solar time ===
Most later equation clocks, made in the 18th century, directly display solar time. Many of them also display mean time and the equation of time, but the user does not have to perform addition. Three types exist:
==== Clocks with movable minute markings ====
Clocks have been constructed in which the minute markings are on a circular plate that can be turned around the same axis as the hands. The axis passes through a hole in the centre of the plate, and the hands are in front of the plate. The minutes part of the time shown by the clock is given by the position of the minute hand relative to the markings on the plate. The hand is driven clockwise at constant speed by the clock mechanism, and the plate is turned by the mechanism that simulates the equation of time, rotating anticlockwise as the equation of time increases, and clockwise when it decreases. If the gear ratios are correct, the clock shows solar time. Mean time can also be shown by a separate, stationary set of minute markings on the dial, outside the edge of the plate. The hour display is not adjusted for the equation of time, so the hour reading is slightly approximate. This has no practical effect, since it is always easy to see which hour is correct. These clocks are mechanically simpler than the other types described below, but they have disadvantages: Solar time is difficult to read without looking closely at the minute markings, and the clock cannot be made to strike the hour in solar time.
==== Clocks with variable pendula ====
These clocks include a device at the top of the pendulum that slightly changes its effective length, so the speed of the clock varies. This device is driven by a simulation mechanism which moves to simulate the rate of change of the equation of time, rather than its actual value. For example, during the months of December and January, when the equation of time is decreasing so a sundial runs slower than usual, the mechanism makes the pendulum effectively longer than usual, so the clock runs slower and keeps pace with sundial time. At other times of the year, the pendulum is shortened, so the clock runs faster, again keeping pace with sundial time. This type of mechanism shows only solar (sundial) time. Clocks using it cannot easily be made to show mean time unless a separate clock mechanism, with its own pendulum, is included. There are some equation clocks in which this is done, but it requires the clock case to be very sturdy, to avoid coupling between the pendula. Another disadvantage of variable pendulum clocks is that the equation of time cannot be easily displayed.
==== Clocks that do mechanical addition ====
In some later equation clocks, a pendulum swings at a constant frequency, controlling a normal clock mechanism. Often, this mechanism drives a display showing mean (clock) time. However, there are additional components: an equation of time simulation mechanism as described above, and a device that automatically adds the equation of time to clock time, and drives a display that shows solar time. The addition is done by an analogue method, using a differential gear. This type of equation clock mechanism is the most versatile. Both solar and mean times can be easily and clearly displayed, as can the equation of time. Striking the hours in both kinds of time is also easy to arrange. After its invention in 1720, this mechanism became the standard one, and was used for much of the 18th century, until the demand for equation clocks ceased.
== Slow changes in the equation of time ==
Slow changes in the motions of the Earth cause gradual changes in the annual variation of the equation of time. The graph at the top of this article shows the annual variation as it is at present, around the year 2000. Many centuries in the past or future, the shape of the graph would be very different. Most equation clocks were constructed some three centuries ago, since when the change in the annual variation of the equation of time has been small, but appreciable. The clocks embody the annual variation as it was when they were made. They do not compensate for the slow changes, which were then unknown, so they are slightly less accurate now than they were when new. The greatest error from this cause is currently about one minute. Centuries in the future, if these clocks survive, the errors will be larger.
== Similar modern devices ==
Equation clocks, as such, are no longer widely used. However, components functionally the same as those in equation clocks are still used in, for example, solar trackers, which move so as to follow the movements of the Sun in the sky. Many of them do not sense the position of the Sun. Instead, they have a mechanism which rotates about a polar axis at a constant speed of 15 degrees per hour, keeping pace with the average speed of the Earth's rotation relative to the Sun. Sometimes, a digital representation of this rotation is generated, rather than physical rotation of a component. The equation of time is then added to this constant rotation, producing a rotation of the tracker that keeps pace with the apparent motion of the Sun. Generally, these machines use modern technology, involving electronics and computers, instead of the mechanical devices that were used in historic equation clocks, but the function is the same.
== See also ==
Equation of time
Sundial
Clock
Differential (mechanical device)
== References and footnotes ==
== External links ==
Note: In some of these historical materials, clock time is called "equal time", and sundial time is called "apparent time" or "true solar time".
Variable pendulum clock in the British Museum.
British Museum equation clocks, with descriptions.
Pocket watch that works as an equation clock. The description suggests that it does mechanical addition.
Clock with movable minute markings on a ring.
Letter from Joseph Williamson written c.1715, claiming the invention of clocks showing solar time, working by movable minute marks or variable pendula.
The Equation of Time gives illustrated examples of the development of equation clocks | Wikipedia/Equation_clock |
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt to find approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences like economics, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living cells in medicine and biology.
Before modern computers, numerical methods often relied on hand interpolation formulas, using data from large printed tables. Since the mid-20th century, computers calculate the required functions instead, but many of the same formulas continue to be used in software algorithms.
The numerical point of view goes back to the earliest mathematical writings. A tablet from the Yale Babylonian Collection (YBC 7289), gives a sexagesimal numerical approximation of the square root of 2, the length of the diagonal in a unit square.
Numerical analysis continues this long tradition: rather than giving exact symbolic answers translated into digits and applicable only to real-world measurements, approximate solutions within specified error bounds are used.
== Applications ==
The overall goal of the field of numerical analysis is the design and analysis of techniques to give approximate but accurate solutions to a wide variety of hard problems, many of which are infeasible to solve symbolically:
Advanced numerical methods are essential in making numerical weather prediction feasible.
Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of ordinary differential equations.
Car companies can improve the crash safety of their vehicles by using computer simulations of car crashes. Such simulations essentially consist of solving partial differential equations numerically.
In the financial field, (private investment funds) and other financial institutions use quantitative finance tools from numerical analysis to attempt to calculate the value of stocks and derivatives more precisely than other market participants.
Airlines use sophisticated optimization algorithms to decide ticket prices, airplane and crew assignments and fuel needs. Historically, such algorithms were developed within the overlapping field of operations research.
Insurance companies use numerical programs for actuarial analysis.
== History ==
The field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was already in use more than 2000 years ago. Many great mathematicians of the past were preoccupied by numerical analysis, as is obvious from the names of important algorithms like Newton's method, Lagrange interpolation polynomial, Gaussian elimination, or Euler's method. The origins of modern numerical analysis are often linked to a 1947 paper by John von Neumann and Herman Goldstine,
but others consider modern numerical analysis to go back to work by E. T. Whittaker in 1912.
To facilitate computations by hand, large books were produced with formulas and tables of data such as interpolation points and function coefficients. Using these tables, often calculated out to 16 decimal places or more for some functions, one could look up values to plug into the formulas given and achieve very good numerical estimates of some functions. The canonical work in the field is the NIST publication edited by Abramowitz and Stegun, a 1000-plus page book of a very large number of commonly used formulas and functions and their values at many points. The function values are no longer very useful when a computer is available, but the large listing of formulas can still be very handy.
The mechanical calculator was also developed as a tool for hand computation. These calculators evolved into electronic computers in the 1940s, and it was then found that these computers were also useful for administrative purposes. But the invention of the computer also influenced the field of numerical analysis, since now longer and more complicated calculations could be done.
The Leslie Fox Prize for Numerical Analysis was initiated in 1985 by the Institute of Mathematics and its Applications.
== Key concepts ==
=== Direct and iterative methods ===
Direct methods compute the solution to a problem in a finite number of steps. These methods would give the precise answer if they were performed in infinite precision arithmetic. Examples include Gaussian elimination, the QR factorization method for solving systems of linear equations, and the simplex method of linear programming. In practice, finite precision is used and the result is an approximation of the true solution (assuming stability).
In contrast to direct methods, iterative methods are not expected to terminate in a finite number of steps, even if infinite precision were possible. Starting from an initial guess, iterative methods form successive approximations that converge to the exact solution only in the limit. A convergence test, often involving the residual, is specified in order to decide when a sufficiently accurate solution has (hopefully) been found. Even using infinite precision arithmetic these methods would not reach the solution within a finite number of steps (in general). Examples include Newton's method, the bisection method, and Jacobi iteration. In computational matrix algebra, iterative methods are generally needed for large problems.
Iterative methods are more common than direct methods in numerical analysis. Some methods are direct in principle but are usually used as though they were not, e.g. GMRES and the conjugate gradient method. For these methods the number of steps needed to obtain the exact solution is so large that an approximation is accepted in the same manner as for an iterative method.
As an example, consider the problem of solving
3x3 + 4 = 28
for the unknown quantity x.
For the iterative method, apply the bisection method to f(x) = 3x3 − 24. The initial values are a = 0, b = 3, f(a) = −24, f(b) = 57.
From this table it can be concluded that the solution is between 1.875 and 2.0625. The algorithm might return any number in that range with an error less than 0.2.
=== Conditioning ===
Ill-conditioned problem: Take the function f(x) = 1/(x − 1). Note that f(1.1) = 10 and f(1.001) = 1000: a change in x of less than 0.1 turns into a change in f(x) of nearly 1000. Evaluating f(x) near x = 1 is an ill-conditioned problem.
Well-conditioned problem: By contrast, evaluating the same function f(x) = 1/(x − 1) near x = 10 is a well-conditioned problem. For instance, f(10) = 1/9 ≈ 0.111 and f(11) = 0.1: a modest change in x leads to a modest change in f(x).
=== Discretization ===
Furthermore, continuous problems must sometimes be replaced by a discrete problem whose solution is known to approximate that of the continuous problem; this process is called 'discretization'. For example, the solution of a differential equation is a function. This function must be represented by a finite amount of data, for instance by its value at a finite number of points at its domain, even though this domain is a continuum.
== Generation and propagation of errors ==
The study of errors forms an important part of numerical analysis. There are several ways in which error can be introduced in the solution of the problem.
=== Round-off ===
Round-off errors arise because it is impossible to represent all real numbers exactly on a machine with finite memory (which is what all practical digital computers are).
=== Truncation and discretization error ===
Truncation errors are committed when an iterative method is terminated or a mathematical procedure is approximated and the approximate solution differs from the exact solution. Similarly, discretization induces a discretization error because the solution of the discrete problem does not coincide with the solution of the continuous problem. In the example above to compute the solution of
3
x
3
+
4
=
28
{\displaystyle 3x^{3}+4=28}
, after ten iterations, the calculated root is roughly 1.99. Therefore, the truncation error is roughly 0.01.
Once an error is generated, it propagates through the calculation. For example, the operation + on a computer is inexact. A calculation of the type
a
+
b
+
c
+
d
+
e
{\displaystyle a+b+c+d+e}
is even more inexact.
A truncation error is created when a mathematical procedure is approximated. To integrate a function exactly, an infinite sum of regions must be found, but numerically only a finite sum of regions can be found, and hence the approximation of the exact solution. Similarly, to differentiate a function, the differential element approaches zero, but numerically only a nonzero value of the differential element can be chosen.
=== Numerical stability and well-posed problems ===
An algorithm is called numerically stable if an error, whatever its cause, does not grow to be much larger during the calculation. This happens if the problem is well-conditioned, meaning that the solution changes by only a small amount if the problem data are changed by a small amount. To the contrary, if a problem is 'ill-conditioned', then any small error in the data will grow to be a large error.
Both the original problem and the algorithm used to solve that problem can be well-conditioned or ill-conditioned, and any combination is possible.
So an algorithm that solves a well-conditioned problem may be either numerically stable or numerically unstable. An art of numerical analysis is to find a stable algorithm for solving a well-posed mathematical problem.
== Areas of study ==
The field of numerical analysis includes many sub-disciplines. Some of the major ones are:
=== Computing values of functions ===
One of the simplest problems is the evaluation of a function at a given point. The most straightforward approach, of just plugging in the number in the formula is sometimes not very efficient. For polynomials, a better approach is using the Horner scheme, since it reduces the necessary number of multiplications and additions. Generally, it is important to estimate and control round-off errors arising from the use of floating-point arithmetic.
=== Interpolation, extrapolation, and regression ===
Interpolation solves the following problem: given the value of some unknown function at a number of points, what value does that function have at some other point between the given points?
Extrapolation is very similar to interpolation, except that now the value of the unknown function at a point which is outside the given points must be found.
Regression is also similar, but it takes into account that the data are imprecise. Given some points, and a measurement of the value of some function at these points (with an error), the unknown function can be found. The least squares-method is one way to achieve this.
=== Solving equations and systems of equations ===
Another fundamental problem is computing the solution of some given equation. Two cases are commonly distinguished, depending on whether the equation is linear or not. For instance, the equation
2
x
+
5
=
3
{\displaystyle 2x+5=3}
is linear while
2
x
2
+
5
=
3
{\displaystyle 2x^{2}+5=3}
is not.
Much effort has been put in the development of methods for solving systems of linear equations. Standard direct methods, i.e., methods that use some matrix decomposition are Gaussian elimination, LU decomposition, Cholesky decomposition for symmetric (or hermitian) and positive-definite matrix, and QR decomposition for non-square matrices. Iterative methods such as the Jacobi method, Gauss–Seidel method, successive over-relaxation and conjugate gradient method are usually preferred for large systems. General iterative methods can be developed using a matrix splitting.
Root-finding algorithms are used to solve nonlinear equations (they are so named since a root of a function is an argument for which the function yields zero). If the function is differentiable and the derivative is known, then Newton's method is a popular choice. Linearization is another technique for solving nonlinear equations.
=== Solving eigenvalue or singular value problems ===
Several important problems can be phrased in terms of eigenvalue decompositions or singular value decompositions. For instance, the spectral image compression algorithm is based on the singular value decomposition. The corresponding tool in statistics is called principal component analysis.
=== Optimization ===
Optimization problems ask for the point at which a given function is maximized (or minimized). Often, the point also has to satisfy some constraints.
The field of optimization is further split in several subfields, depending on the form of the objective function and the constraint. For instance, linear programming deals with the case that both the objective function and the constraints are linear. A famous method in linear programming is the simplex method.
The method of Lagrange multipliers can be used to reduce optimization problems with constraints to unconstrained optimization problems.
=== Evaluating integrals ===
Numerical integration, in some instances also known as numerical quadrature, asks for the value of a definite integral. Popular methods use one of the Newton–Cotes formulas (like the midpoint rule or Simpson's rule) or Gaussian quadrature. These methods rely on a "divide and conquer" strategy, whereby an integral on a relatively large set is broken down into integrals on smaller sets. In higher dimensions, where these methods become prohibitively expensive in terms of computational effort, one may use Monte Carlo or quasi-Monte Carlo methods (see Monte Carlo integration), or, in modestly large dimensions, the method of sparse grids.
=== Differential equations ===
Numerical analysis is also concerned with computing (in an approximate way) the solution of differential equations, both ordinary differential equations and partial differential equations.
Partial differential equations are solved by first discretizing the equation, bringing it into a finite-dimensional subspace. This can be done by a finite element method, a finite difference method, or (particularly in engineering) a finite volume method. The theoretical justification of these methods often involves theorems from functional analysis. This reduces the problem to the solution of an algebraic equation.
== Software ==
Since the late twentieth century, most algorithms are implemented in a variety of programming languages. The Netlib repository contains various collections of software routines for numerical problems, mostly in Fortran and C. Commercial products implementing many different numerical algorithms include the IMSL and NAG libraries; a free-software alternative is the GNU Scientific Library.
Over the years the Royal Statistical Society published numerous algorithms in its Applied Statistics (code for these "AS" functions is here);
ACM similarly, in its Transactions on Mathematical Software ("TOMS" code is here).
The Naval Surface Warfare Center several times published its Library of Mathematics Subroutines (code here).
There are several popular numerical computing applications such as MATLAB, TK Solver, S-PLUS, and IDL as well as free and open-source alternatives such as FreeMat, Scilab, GNU Octave (similar to Matlab), and IT++ (a C++ library). There are also programming languages such as R (similar to S-PLUS), Julia, and Python with libraries such as NumPy, SciPy and SymPy. Performance varies widely: while vector and matrix operations are usually fast, scalar loops may vary in speed by more than an order of magnitude.
Many computer algebra systems such as Mathematica also benefit from the availability of arbitrary-precision arithmetic which can provide more accurate results.
Also, any spreadsheet software can be used to solve simple problems relating to numerical analysis.
Excel, for example, has hundreds of available functions, including for matrices, which may be used in conjunction with its built in "solver".
== See also ==
== Notes ==
== References ==
=== Citations ===
=== Sources ===
== External links ==
=== Journals ===
Numerische Mathematik, volumes 1–..., Springer, 1959–
volumes 1–66, 1959–1994 (searchable; pages are images). (in English and German)
Journal on Numerical Analysis (SINUM), volumes 1–..., SIAM, 1964–
=== Online texts ===
"Numerical analysis", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Numerical Recipes, William H. Press (free, downloadable previous editions)
First Steps in Numerical Analysis (archived), R.J.Hosking, S.Joe, D.C.Joyce, and J.C.Turner
CSEP (Computational Science Education Project), U.S. Department of Energy (archived 2017-08-01)
Numerical Methods, ch 3. in the Digital Library of Mathematical Functions
Numerical Interpolation, Differentiation and Integration, ch 25. in the Handbook of Mathematical Functions (Abramowitz and Stegun)
Tobin A. Driscoll and Richard J. Braun: Fundamentals of Numerical Computation (free online version)
=== Online course material ===
Numerical Methods (Archived 28 July 2009 at the Wayback Machine), Stuart Dalziel University of Cambridge
Lectures on Numerical Analysis, Dennis Deturck and Herbert S. Wilf University of Pennsylvania
Numerical methods, John D. Fenton University of Karlsruhe
Numerical Methods for Physicists, Anthony O’Hare Oxford University
Lectures in Numerical Analysis (archived), R. Radok Mahidol University
Introduction to Numerical Analysis for Engineering, Henrik Schmidt Massachusetts Institute of Technology
Numerical Analysis for Engineering, D. W. Harder University of Waterloo
Introduction to Numerical Analysis, Doron Levy University of Maryland
Numerical Analysis - Numerical Methods (archived), John H. Mathews California State University Fullerton | Wikipedia/Numerical_solution |
In field theory, a branch of mathematics, the minimal polynomial of an element α of an extension field of a field is, roughly speaking, the polynomial of lowest degree having coefficients in the smaller field, such that α is a root of the polynomial. If the minimal polynomial of α exists, it is unique. The coefficient of the highest-degree term in the polynomial is required to be 1.
More formally, a minimal polynomial is defined relative to a field extension E/F and an element of the extension field E/F. The minimal polynomial of an element, if it exists, is a member of F[x], the ring of polynomials in the variable x with coefficients in F. Given an element α of E, let Jα be the set of all polynomials f(x) in F[x] such that f(α) = 0. The element α is called a root or zero of each polynomial in Jα
More specifically, Jα is the kernel of the ring homomorphism from F[x] to E which sends polynomials g to their value g(α) at the element α. Because it is the kernel of a ring homomorphism, Jα is an ideal of the polynomial ring F[x]: it is closed under polynomial addition and subtraction (hence containing the zero polynomial), as well as under multiplication by elements of F (which is scalar multiplication if F[x] is regarded as a vector space over F).
The zero polynomial, all of whose coefficients are 0, is in every Jα since 0αi = 0 for all α and i. This makes the zero polynomial useless for classifying different values of α into types, so it is excepted. If there are any non-zero polynomials in Jα, i.e. if the latter is not the zero ideal, then α is called an algebraic element over F, and there exists a monic polynomial of least degree in Jα. This is the minimal polynomial of α with respect to E/F. It is unique and irreducible over F. If the zero polynomial is the only member of Jα, then α is called a transcendental element over F and has no minimal polynomial with respect to E/F.
Minimal polynomials are useful for constructing and analyzing field extensions. When α is algebraic with minimal polynomial f(x), the smallest field that contains both F and α is isomorphic to the quotient ring F[x]/⟨f(x)⟩, where ⟨f(x)⟩ is the ideal of F[x] generated by f(x). Minimal polynomials are also used to define conjugate elements.
== Definition ==
Let E/F be a field extension, α an element of E, and F[x] the ring of polynomials in x over F. The element α has a minimal polynomial when α is algebraic over F, that is, when f(α) = 0 for some non-zero polynomial f(x) in F[x]. Then the minimal polynomial of α is defined as the monic polynomial of least degree among all polynomials in F[x] having α as a root.
== Properties ==
Throughout this section, let E/F be a field extension over F as above, let α ∈ E be an algebraic element over F and let Jα be the ideal of polynomials vanishing on α.
=== Uniqueness ===
The minimal polynomial f of α is unique.
To prove this, suppose that f and g are monic polynomials in Jα of minimal degree n > 0. We have that r := f−g ∈ Jα (because the latter is closed under addition/subtraction) and that m := deg(r) < n (because the polynomials are monic of the same degree). If r is not zero, then r / cm (writing cm ∈ F for the non-zero coefficient of highest degree in r) is a monic polynomial of degree m < n such that r / cm ∈ Jα (because the latter is closed under multiplication/division by non-zero elements of F), which contradicts our original assumption of minimality for n. We conclude that 0 = r = f − g, i.e. that f = g.
=== Irreducibility ===
The minimal polynomial f of α is irreducible, i.e. it cannot be factorized as f = gh for two polynomials g and h of strictly lower degree.
To prove this, first observe that any factorization f = gh implies that either g(α) = 0 or h(α) = 0, because f(α) = 0 and F is a field (hence also an integral domain). Choosing both g and h to be of degree strictly lower than f would then contradict the minimality requirement on f, so f must be irreducible.
=== Minimal polynomial generates Jα ===
The minimal polynomial f of α generates the ideal Jα, i.e. every g in Jα can be factorized as g=fh for some h' in F[x].
To prove this, it suffices to observe that F[x] is a principal ideal domain, because F is a field: this means that every ideal I in F[x], Jα amongst them, is generated by a single element f. With the exception of the zero ideal I = {0}, the generator f must be non-zero and it must be the unique polynomial of minimal degree, up to a factor in F (because the degree of fg is strictly larger than that of f whenever g is of degree greater than zero). In particular, there is a unique monic generator f, and all generators must be irreducible. When I is chosen to be Jα, for α algebraic over F, then the monic generator f is the minimal polynomial of α.
== Examples ==
=== Minimal polynomial of a Galois field extension ===
Given a Galois field extension
L
/
K
{\displaystyle L/K}
the minimal polynomial of any
α
∈
L
{\displaystyle \alpha \in L}
not in
K
{\displaystyle K}
can be computed as
f
(
x
)
=
∏
σ
∈
Gal
(
L
/
K
)
(
x
−
σ
(
α
)
)
{\displaystyle f(x)=\prod _{\sigma \in {\text{Gal}}(L/K)}(x-\sigma (\alpha ))}
if
α
{\displaystyle \alpha }
has no stabilizers in the Galois action. Since it is irreducible, which can be deduced by looking at the roots of
f
′
{\displaystyle f'}
, it is the minimal polynomial. Note that the same kind of formula can be found by replacing
G
=
Gal
(
L
/
K
)
{\displaystyle G={\text{Gal}}(L/K)}
with
G
/
N
{\displaystyle G/N}
where
N
=
Stab
(
α
)
{\displaystyle N={\text{Stab}}(\alpha )}
is the stabilizer group of
α
{\displaystyle \alpha }
. For example, if
α
∈
K
{\displaystyle \alpha \in K}
then its stabilizer is
G
{\displaystyle G}
, hence
(
x
−
α
)
{\displaystyle (x-\alpha )}
is its minimal polynomial.
=== Quadratic field extensions ===
==== Q(√2) ====
If F = Q, E = R, α = √2, then the minimal polynomial for α is a(x) = x2 − 2. The base field F is important as it determines the possibilities for the coefficients of a(x). For instance, if we take F = R, then the minimal polynomial for α = √2 is a(x) = x − √2.
==== Q(√d) ====
In general, for the quadratic extension given by a square-free
d
{\displaystyle d}
, computing the minimal polynomial of an element
a
+
b
d
{\displaystyle a+b{\sqrt {d\,}}}
can be found using Galois theory. Then
f
(
x
)
=
(
x
−
(
a
+
b
d
)
)
(
x
−
(
a
−
b
d
)
)
=
x
2
−
2
a
x
+
(
a
2
−
b
2
d
)
{\displaystyle {\begin{aligned}f(x)&=(x-(a+b{\sqrt {d\,}}))(x-(a-b{\sqrt {d\,}}))\\&=x^{2}-2ax+(a^{2}-b^{2}d)\end{aligned}}}
in particular, this implies
2
a
∈
Z
{\displaystyle 2a\in \mathbb {Z} }
and
a
2
−
b
2
d
∈
Z
{\displaystyle a^{2}-b^{2}d\in \mathbb {Z} }
. This can be used to determine
O
Q
(
d
)
{\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {d\,}}\!\!\!\;\;)}}
through a series of relations using modular arithmetic.
=== Biquadratic field extensions ===
If α = √2 + √3, then the minimal polynomial in Q[x] is a(x) = x4 − 10x2 + 1 = (x − √2 − √3)(x + √2 − √3)(x − √2 + √3)(x + √2 + √3).
Notice if
α
=
2
{\displaystyle \alpha ={\sqrt {2}}}
then the Galois action on
3
{\displaystyle {\sqrt {3}}}
stabilizes
α
{\displaystyle \alpha }
. Hence the minimal polynomial can be found using the quotient group
Gal
(
Q
(
2
,
3
)
/
Q
)
/
Gal
(
Q
(
3
)
/
Q
)
{\displaystyle {\text{Gal}}(\mathbb {Q} ({\sqrt {2}},{\sqrt {3}})/\mathbb {Q} )/{\text{Gal}}(\mathbb {Q} ({\sqrt {3}})/\mathbb {Q} )}
.
=== Roots of unity ===
The minimal polynomials in Q[x] of roots of unity are the cyclotomic polynomials. The roots of the minimal polynomial of 2cos(2π/n) are twice the real part of the primitive roots of unity.
=== Swinnerton-Dyer polynomials ===
The minimal polynomial in Q[x] of the sum of the square roots of the first n prime numbers is constructed analogously, and is called a Swinnerton-Dyer polynomial.
== See also ==
Ring of integers
Algebraic number field
Minimal polynomial (linear algebra)
== References ==
Weisstein, Eric W. "Algebraic Number Minimal Polynomial". MathWorld.
Minimal polynomial at PlanetMath.
Pinter, Charles C. A Book of Abstract Algebra. Dover Books on Mathematics Series. Dover Publications, 2010, p. 270–273. ISBN 978-0-486-47417-5 | Wikipedia/Minimal_polynomial_(field_theory) |
In mathematics, specifically algebraic geometry, a period or algebraic period is a complex number that can be expressed as an integral of an algebraic function over an algebraic domain. The periods are a class of numbers which includes, alongside the algebraic numbers, many well known mathematical constants such as the number π. Sums and products of periods remain periods, such that the periods
P
{\displaystyle {\mathcal {P}}}
form a ring.
Maxim Kontsevich and Don Zagier gave a survey of periods and introduced some conjectures about them.
Periods play an important role in the theory of differential equations and transcendental numbers as well as in open problems of modern arithmetical algebraic geometry. They also appear when computing the integrals that arise from Feynman diagrams, and there has been intensive work trying to understand the connections.
== Definition ==
A number
α
{\displaystyle \alpha }
is a period if it can be expressed as an integral of the form
α
=
∫
P
(
x
1
,
…
,
x
n
)
≥
0
Q
(
x
1
,
…
,
x
n
)
d
x
1
…
d
x
n
{\displaystyle \alpha =\int _{P(x_{1},\ldots ,x_{n})\geq 0}Q(x_{1},\ldots ,x_{n})\ \mathrm {d} x_{1}\ldots \mathrm {d} x_{n}}
where
P
{\displaystyle P}
is a polynomial and
Q
{\displaystyle Q}
a rational function on
R
n
{\displaystyle \mathbb {R} ^{n}}
with rational coefficients. A complex number is a period if its real and imaginary parts are periods.
An alternative definition allows
P
{\displaystyle P}
and
Q
{\displaystyle Q}
to be algebraic functions; this looks more general, but is equivalent. The coefficients of the rational functions and polynomials can also be generalised to algebraic numbers because irrational algebraic numbers are expressible in terms of areas of suitable domains.
In the other direction,
Q
{\displaystyle Q}
can be restricted to be the constant function
1
{\displaystyle 1}
or
−
1
{\displaystyle -1}
, by replacing the integrand with an integral of
±
1
{\displaystyle \pm 1}
over a region defined by a polynomial in additional variables.
In other words, a (nonnegative) period is the volume of a region in
R
n
{\displaystyle \mathbb {R} ^{n}}
defined by polynomial inequalities with rational coefficients.
== Properties and motivation ==
The periods are intended to bridge the gap between the well-behaved algebraic numbers, which form a class too narrow to include many common mathematical constants and the transcendental numbers, which are uncountable and apart from very few specific examples hard to describe. The latter are also not generally computable.
The ring of periods
P
{\displaystyle {\mathcal {P}}}
lies in between the fields of algebraic numbers
Q
¯
{\displaystyle \mathbb {\overline {Q}} }
and complex numbers
C
{\displaystyle \mathbb {C} }
and is countable. The periods themselves are all computable, and in particular definable. That is:
Q
¯
⊂
P
⊂
C
{\displaystyle \mathbb {\overline {Q}} \subset {\mathcal {P}}\subset \mathbb {C} }
.
Periods include some of those transcendental numbers, that can be described in an algorithmic way and only contain a finite amount of information.
== Numbers known to be periods ==
The following numbers are among the ones known to be periods:
== Open questions ==
Many of the constants known to be periods are also given by integrals of transcendental functions. Kontsevich and Zagier note that there "seems to be no universal rule explaining why certain infinite sums or integrals of transcendental functions are periods".
Kontsevich and Zagier conjectured that, if a period is given by two different integrals, then each integral can be transformed into the other using only the linearity of integrals (in both the integrand and the domain), changes of variables, and the Newton–Leibniz formula
∫
a
b
f
′
(
x
)
d
x
=
f
(
b
)
−
f
(
a
)
{\displaystyle \int _{a}^{b}f'(x)\,dx=f(b)-f(a)}
(or, more generally, the Stokes formula).
A useful property of algebraic numbers is that equality between two algebraic expressions can be determined algorithmically. The conjecture of Kontsevich and Zagier would imply that equality of periods is also decidable: inequality of computable reals is known recursively enumerable; and conversely if two integrals agree, then an algorithm could confirm so by trying all possible ways to transform one of them into the other one.
Further open questions consist of proving which known mathematical constants do not belong to the ring of periods. An example of a real number that is not a period is given by Chaitin's constant Ω. Any other non-computable number also gives an example of a real number that is not a period. It is also possible to construct artificial examples of computable numbers which are not periods. However there are no computable numbers proven not to be periods, which have not been artificially constructed for that purpose.
It is conjectured that 1/π, Euler's number e and the Euler–Mascheroni constant γ are not periods.
Kontsevich and Zagier suspect these problems to be very hard and remain open a long time.
== Extensions ==
The ring of periods can be widened to the ring of extended periods
P
^
{\displaystyle {\hat {\mathcal {P}}}}
by adjoining the element 1/π.
Permitting the integrand
Q
{\displaystyle Q}
to be the product of an algebraic function and the exponential of an algebraic function, results in another extension: the exponential periods
E
P
{\displaystyle {\mathcal {E}}{\mathcal {P}}}
. They also form a ring and are countable. It is
Q
¯
⊂
P
⊆
E
P
⊂
C
{\displaystyle {\overline {\mathbb {Q} }}\subset {\mathcal {P}}\subseteq {\mathcal {EP}}\subset \mathbb {C} }
.
The following numbers are among the ones known to be exponential periods:
== See also ==
Transcendental number theory
Mathematical constant
L-function
Jacobian variety
Gauss–Manin connection
Mixed motives (math)
Tannakian formalism
== References ==
== External links ==
PlanetMath: Period | Wikipedia/Period_(algebraic_geometry) |
In algebraic number theory, a fundamental unit is a generator (modulo the roots of unity) for the unit group of the ring of integers of a number field, when that group has rank 1 (i.e. when the unit group modulo its torsion subgroup is infinite cyclic). Dirichlet's unit theorem shows that the unit group has rank 1 exactly when the number field is a real quadratic field, a complex cubic field, or a totally imaginary quartic field. When the unit group has rank ≥ 1, a basis of it modulo its torsion is called a fundamental system of units. Some authors use the term fundamental unit to mean any element of a fundamental system of units, not restricting to the case of rank 1 (e.g. Neukirch 1999, p. 42).
== Real quadratic fields ==
For the real quadratic field
K
=
Q
(
d
)
{\displaystyle K=\mathbf {Q} ({\sqrt {d}})}
(with d square-free), the fundamental unit ε is commonly normalized so that ε > 1 (as a real number). Then it is uniquely characterized as the minimal unit among those that are greater than 1. If Δ denotes the discriminant of K, then the fundamental unit is
ε
=
a
+
b
Δ
2
{\displaystyle \varepsilon ={\frac {a+b{\sqrt {\Delta }}}{2}}}
where (a, b) is the smallest solution to
x
2
−
Δ
y
2
=
±
4
{\displaystyle x^{2}-\Delta y^{2}=\pm 4}
in positive integers. This equation is basically Pell's equation or the negative Pell equation and its solutions can be obtained similarly using the continued fraction expansion of
Δ
{\displaystyle {\sqrt {\Delta }}}
.
Whether or not x2 − Δy2 = −4 has a solution determines whether or not the class group of K is the same as its narrow class group, or equivalently, whether or not there is a unit of norm −1 in K. This equation is known to have a solution if, and only if, the period of the continued fraction expansion of
Δ
{\displaystyle {\sqrt {\Delta }}}
is odd. A simpler relation can be obtained using congruences: if Δ is divisible by a prime that is congruent to 3 modulo 4, then K does not have a unit of norm −1. However, the converse does not hold as shown by the example d = 34. In the early 1990s, Peter Stevenhagen proposed a probabilistic model that led him to a conjecture on how often the converse fails. Specifically, if D(X) is the number of real quadratic fields whose discriminant Δ < X is not divisible by a prime congruent to 3 modulo 4 and D−(X) is those who have a unit of norm −1, then
lim
X
→
∞
D
−
(
X
)
D
(
X
)
=
1
−
∏
j
≥
1
odd
(
1
−
2
−
j
)
.
{\displaystyle \lim _{X\rightarrow \infty }{\frac {D^{-}(X)}{D(X)}}=1-\prod _{j\geq 1{\text{ odd}}}\left(1-2^{-j}\right).}
In other words, the converse fails about 42% of the time. As of March 2012, a recent result towards this conjecture was provided by Étienne Fouvry and Jürgen Klüners who show that the converse fails between 33% and 59% of the time. In 2022, Peter Koymans and Carlo Pagano claimed a complete proof of Stevenhagen's conjecture.
== Cubic fields ==
If K is a complex cubic field then it has a unique real embedding and the fundamental unit ε can be picked uniquely such that |ε| > 1 in this embedding. If the discriminant Δ of K satisfies |Δ| ≥ 33, then
ϵ
3
>
|
Δ
|
−
27
4
.
{\displaystyle \epsilon ^{3}>{\frac {|\Delta |-27}{4}}.}
For example, the fundamental unit of
Q
(
2
3
)
{\displaystyle \mathbf {Q} ({\sqrt[{3}]{2}})}
is
ϵ
=
1
+
2
3
+
2
2
3
,
{\displaystyle \epsilon =1+{\sqrt[{3}]{2}}+{\sqrt[{3}]{2^{2}}},}
and
ϵ
3
≈
56.9
{\displaystyle \epsilon ^{3}\approx 56.9}
whereas the discriminant of this field is −108 thus
|
Δ
|
−
27
4
=
20.25
{\displaystyle {\frac {|\Delta |-27}{4}}=20.25}
so
ϵ
3
≈
56.9
>
20.25
{\displaystyle \epsilon ^{3}\approx 56.9>20.25}
.
== Notes ==
== References ==
Alaca, Şaban; Williams, Kenneth S. (2004), Introductory algebraic number theory, Cambridge University Press, ISBN 978-0-521-54011-7
Duncan Buell (1989), Binary quadratic forms: classical theory and modern computations, Springer-Verlag, pp. 92–93, ISBN 978-0-387-97037-0
Fouvry, Étienne; Klüners, Jürgen (2010), "On the negative Pell equation", Annals of Mathematics, 2 (3): 2035–2104, doi:10.4007/annals.2010.172.2035, MR 2726105
Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften, vol. 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, MR 1697859, Zbl 0956.11021
Stevenhagen, Peter (1993), "The number of real quadratic fields having units of negative norm", Experimental Mathematics, 2 (2): 121–136, CiteSeerX 10.1.1.27.3512, doi:10.1080/10586458.1993.10504272, MR 1259426
== External links ==
Weisstein, Eric W. "Fundamental Unit". MathWorld. | Wikipedia/Fundamental_unit_(number_theory) |
In mathematics, a basic semialgebraic set is a set defined by polynomial equalities and polynomial inequalities, and a semialgebraic set is a finite union of basic semialgebraic sets. A semialgebraic function is a function with a semialgebraic graph. Such sets and functions are mainly studied in real algebraic geometry which is the appropriate framework for algebraic geometry over the real numbers.
== Definition ==
Let
F
{\displaystyle \mathbb {F} }
be a real closed field (For example
F
{\displaystyle \mathbb {F} }
could be the field of real numbers
R
{\displaystyle \mathbb {R} }
).
A subset
S
{\displaystyle S}
of
F
n
{\displaystyle \mathbb {F} ^{n}}
is a semialgebraic set if it is a finite union of sets defined by polynomial equalities of the form
{
(
x
1
,
.
.
.
,
x
n
)
∈
F
n
∣
P
(
x
1
,
.
.
.
,
x
n
)
=
0
}
{\displaystyle \{(x_{1},...,x_{n})\in \mathbb {F} ^{n}\mid P(x_{1},...,x_{n})=0\}}
and of sets defined by polynomial inequalities of the form
{
(
x
1
,
.
.
.
,
x
n
)
∈
F
n
∣
P
(
x
1
,
.
.
.
,
x
n
)
>
0
}
.
{\displaystyle \{(x_{1},...,x_{n})\in \mathbb {F} ^{n}\mid P(x_{1},...,x_{n})>0\}.}
== Properties ==
Similarly to algebraic subvarieties, finite unions and intersections of semialgebraic sets are still semialgebraic sets. Furthermore, unlike subvarieties, the complement of a semialgebraic set is again semialgebraic. Finally, and most importantly, the Tarski–Seidenberg theorem says that they are also closed under the projection operation: in other words a semialgebraic set projected onto a linear subspace yields another semialgebraic set (as is the case for quantifier elimination). These properties together mean that semialgebraic sets form an o-minimal structure on R.
A semialgebraic set (or function) is said to be defined over a subring A of R if there is some description, as in the definition, where the polynomials can be chosen to have coefficients in A.
On a dense open subset of the semialgebraic set S, it is (locally) a submanifold. One can define the dimension of S to be the largest dimension at points at which it is a submanifold. It is not hard to see that a semialgebraic set lies inside an algebraic subvariety of the same dimension.
== See also ==
Łojasiewicz inequality
Existential theory of the reals
Subanalytic set
Piecewise algebraic space
== References ==
Bochnak, J.; Coste, M.; Roy, M.-F. (1998), Real algebraic geometry, Berlin: Springer-Verlag, ISBN 9783662037188.
Bierstone, Edward; Milman, Pierre D. (1988), "Semianalytic and subanalytic sets", Inst. Hautes Études Sci. Publ. Math., 67: 5–42, doi:10.1007/BF02699126, MR 0972342, S2CID 56006439.
van den Dries, L. (1998), Tame topology and o-minimal structures, Cambridge University Press, ISBN 9780521598385.
== External links ==
PlanetMath page | Wikipedia/Semialgebraic_set |
In topology, a branch of mathematics, the ends of a topological space are, roughly speaking, the connected components of the "ideal boundary" of the space. That is, each end represents a topologically distinct way to move to infinity within the space. Adding a point at each end yields a compactification of the original space, known as the end compactification.
The notion of an end of a topological space was introduced by Hans Freudenthal (1931).
== Definition ==
Let
X
{\displaystyle X}
be a topological space, and suppose that
is an ascending sequence of compact subsets of
X
{\displaystyle X}
whose interiors cover
X
{\displaystyle X}
. Then
X
{\displaystyle X}
has one end for every sequence
where each
U
n
{\displaystyle U_{n}}
is a connected component of
X
∖
K
n
{\displaystyle X\setminus K_{n}}
. The number of ends does not depend on the specific sequence
(
K
i
)
{\displaystyle (K_{i})}
of compact sets; there is a natural bijection between the sets of ends associated with any two such sequences.
Using this definition, a neighborhood of an end
(
U
i
)
{\displaystyle (U_{i})}
is an open set
V
{\displaystyle V}
such that
V
⊃
U
n
{\displaystyle V\supset U_{n}}
for some
n
{\displaystyle n}
. Such neighborhoods represent the neighborhoods of the corresponding point at infinity in the end compactification (this "compactification" is not always compact; the topological space X has to be connected and locally connected).
The definition of ends given above applies only to spaces
X
{\displaystyle X}
that possess an exhaustion by compact sets (that is,
X
{\displaystyle X}
must be hemicompact). However, it can be generalized as follows: let
X
{\displaystyle X}
be any topological space, and consider the direct system
{
K
}
{\displaystyle \{K\}}
of compact subsets of
X
{\displaystyle X}
and inclusion maps. There is a corresponding inverse system
{
π
0
(
X
∖
K
)
}
{\displaystyle \{\pi _{0}(X\setminus K)\}}
, where
π
0
(
Y
)
{\displaystyle \pi _{0}(Y)}
denotes the set of connected components of a space
Y
{\displaystyle Y}
, and each inclusion map
Y
→
Z
{\displaystyle Y\to Z}
induces a function
π
0
(
Y
)
→
π
0
(
Z
)
{\displaystyle \pi _{0}(Y)\to \pi _{0}(Z)}
. Then set of ends of
X
{\displaystyle X}
is defined to be the inverse limit of this inverse system.
Under this definition, the set of ends is a functor from the category of topological spaces, where morphisms are only proper continuous maps, to the category of sets. Explicitly, if
φ
:
X
→
Y
{\displaystyle \varphi :X\to Y}
is a proper map and
x
=
(
x
K
)
K
{\displaystyle x=(x_{K})_{K}}
is an end of
X
{\displaystyle X}
(i.e. each element
x
K
{\displaystyle x_{K}}
in the family is a connected component of
X
∖
K
{\displaystyle X\setminus K}
and they are compatible with maps induced by inclusions) then
φ
(
x
)
{\displaystyle \varphi (x)}
is the family
φ
∗
(
x
φ
−
1
(
K
′
)
)
{\displaystyle \varphi _{*}(x_{\varphi ^{-1}(K')})}
where
K
′
{\displaystyle K'}
ranges over compact subsets of Y and
φ
∗
{\displaystyle \varphi _{*}}
is the map induced by
φ
{\displaystyle \varphi }
from
π
0
(
X
∖
φ
−
1
(
K
′
)
)
{\displaystyle \pi _{0}(X\setminus \varphi ^{-1}(K'))}
to
π
0
(
Y
∖
K
′
)
{\displaystyle \pi _{0}(Y\setminus K')}
. Properness of
φ
{\displaystyle \varphi }
is used to ensure that each
φ
−
1
(
K
)
{\displaystyle \varphi ^{-1}(K)}
is compact in
X
{\displaystyle X}
.
The original definition above represents the special case where the direct system of compact subsets has a cofinal sequence.
== Examples ==
The set of ends of any compact space is the empty set.
The real line
R
{\displaystyle \mathbb {R} }
has two ends. For example, if we let Kn be the closed interval [−n, n], then the two ends are the sequences of open sets Un = (n, ∞) and Vn = (−∞, −n). These ends are usually referred to as "infinity" and "minus infinity", respectively.
If n > 1, then Euclidean space
R
n
{\displaystyle \mathbb {R} ^{n}}
has only one end. This is because
R
n
∖
K
{\displaystyle \mathbb {R} ^{n}\smallsetminus K}
has only one unbounded component for any compact set K.
More generally, if M is a compact manifold with boundary, then the number of ends of the interior of M is equal to the number of connected components of the boundary of M.
The union of n distinct rays emanating from the origin in
R
2
{\displaystyle \mathbb {R} ^{2}}
has n ends.
The infinite complete binary tree has uncountably many ends, corresponding to the uncountably many different descending paths starting at the root. (This can be seen by letting Kn be the complete binary tree of depth n.) These ends can be thought of as the "leaves" of the infinite tree. In the end compactification, the set of ends has the topology of a Cantor set.
== Ends of graphs and groups ==
In infinite graph theory, an end is defined slightly differently, as an equivalence class of semi-infinite paths in the graph, or as a haven, a function mapping finite sets of vertices to connected components of their complements. However, for locally finite graphs (graphs in which each vertex has finite degree), the ends defined in this way correspond one-for-one with the ends of topological spaces defined from the graph (Diestel & Kühn 2003).
The ends of a finitely generated group are defined to be the ends of the corresponding Cayley graph; this definition is insensitive to the choice of generating set. Every finitely-generated infinite group has either 1, 2, or infinitely many ends, and Stallings theorem about ends of groups provides a decomposition for groups with more than one end.
== Ends of a CW complex ==
For a path connected CW-complex, the ends can be characterized as homotopy classes of proper maps
R
+
→
X
{\displaystyle \mathbb {R} ^{+}\to X}
, called rays in X: more precisely, if between the restriction —to the subset
N
{\displaystyle \mathbb {N} }
— of any two of these maps exists a proper homotopy we say that they are equivalent and they define an equivalence class of proper rays. This set is called an end of X.
== References ==
Diestel, Reinhard; Kühn, Daniela (2003), "Graph-theoretical versus topological ends of graphs", Journal of Combinatorial Theory, Series B, 87 (1): 197–206, doi:10.1016/S0095-8956(02)00034-5, MR 1967888.
Freudenthal, Hans (1931), "Über die Enden topologischer Räume und Gruppen", Mathematische Zeitschrift, 33, Springer Berlin / Heidelberg: 692–713, doi:10.1007/BF01174375, ISSN 0025-5874, S2CID 120965216, Zbl 0002.05603
Ross Geoghegan, Topological methods in group theory, GTM-243 (2008), Springer ISBN 978-0-387-74611-1.
Scott, Peter; Wall, Terry; Wall, C. T. C. (1979). "Topological methods in group theory". Homological Group Theory. pp. 137–204. doi:10.1017/CBO9781107325449.007. ISBN 9781107325449. | Wikipedia/End_(topology) |
In topology, puncturing a manifold is removing a finite set of points from that manifold. The set of points can be small as a single point. In this case, the manifold is known as once-punctured. With the removal of a second point, it becomes twice-punctured, and so on.
Examples of punctured manifolds include the open disk (which is a sphere with a single puncture), the cylinder (which is a sphere with two punctures), and the Möbius strip (which is a projective plane with a single puncture).
== References ==
== Bibliography ==
Seifert, Herbert; Threlfall, William (1980). A Textbook of Topology. Pure and Applied Mathematics. Vol. 89. Translated by Goldman, Michael A. New York & London: Academic Press. p. 12. ISBN 0-12-634850-2. MR 0575168. | Wikipedia/Puncturing_(topology) |
In mathematics, the Teichmüller space
T
(
S
)
{\displaystyle T(S)}
of a (real) topological (or differential) surface
S
{\displaystyle S}
is a space that parametrizes complex structures on
S
{\displaystyle S}
up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmüller spaces are named after Oswald Teichmüller.
Each point in a Teichmüller space
T
(
S
)
{\displaystyle T(S)}
may be regarded as an isomorphism class of "marked" Riemann surfaces, where a "marking" is an isotopy class of homeomorphisms from
S
{\displaystyle S}
to itself. It can be viewed as a moduli space for marked hyperbolic structure on the surface, and this endows it with a natural topology for which it is homeomorphic to a ball of dimension
6
g
−
6
{\displaystyle 6g-6}
for a surface of genus
g
≥
2
{\displaystyle g\geq 2}
. In this way Teichmüller space can be viewed as the universal covering orbifold of the Riemann moduli space.
The Teichmüller space has a canonical complex manifold structure and a wealth of natural metrics. The study of geometric features of these various structures is an active body of research.
The sub-field of mathematics that studies the Teichmüller space is called Teichmüller theory.
== History ==
Moduli spaces for Riemann surfaces and related Fuchsian groups have been studied since the work of Bernhard Riemann (1826–1866), who knew that
6
g
−
6
{\displaystyle 6g-6}
parameters were needed to describe the variations of complex structures on a surface of genus
g
≥
2
{\displaystyle g\geq 2}
. The early study of Teichmüller space, in the late nineteenth–early twentieth century, was geometric and founded on the interpretation of Riemann surfaces as hyperbolic surfaces. Among the main contributors were Felix Klein, Henri Poincaré, Paul Koebe, Jakob Nielsen, Robert Fricke and Werner Fenchel.
The main contribution of Teichmüller to the study of moduli was the introduction of quasiconformal mappings to the subject. They allow us to give much more depth to the study of moduli spaces by endowing them with additional features that were not present in the previous, more elementary works. After World War II the subject was developed further in this analytic vein, in particular by Lars Ahlfors and Lipman Bers. The theory continues to be active, with numerous studies of the complex structure of Teichmüller space (introduced by Bers).
The geometric vein in the study of Teichmüller space was revived following the work of William Thurston in the late 1970s, who introduced a geometric compactification which he used in his study of the mapping class group of a surface. Other more combinatorial objects associated to this group (in particular the curve complex) have also been related to Teichmüller space, and this is a very active subject of research in geometric group theory.
== Definitions ==
=== Teichmüller space from complex structures ===
Let
S
{\displaystyle S}
be an orientable smooth surface (a differentiable manifold of dimension 2). Informally the Teichmüller space
T
(
S
)
{\displaystyle T(S)}
of
S
{\displaystyle S}
is the space of Riemann surface structures on
S
{\displaystyle S}
up to isotopy.
Formally it can be defined as follows. Two complex structures
X
,
Y
{\displaystyle X,Y}
on
S
{\displaystyle S}
are said to be equivalent if there is a diffeomorphism
f
∈
Diff
(
S
)
{\displaystyle f\in \operatorname {Diff} (S)}
such that:
It is holomorphic (the differential is complex linear at each point, for the structures
X
{\displaystyle X}
at the source and
Y
{\displaystyle Y}
at the target) ;
it is isotopic to the identity of
S
{\displaystyle S}
(there is a continuous map
γ
:
[
0
,
1
]
→
Diff
(
S
)
{\displaystyle \gamma :[0,1]\to \operatorname {Diff} (S)}
such that
γ
(
0
)
=
f
,
γ
(
1
)
=
I
d
{\displaystyle \gamma (0)=f,\gamma (1)=\mathrm {Id} }
).
Then
T
(
S
)
{\displaystyle T(S)}
is the space of equivalence classes of complex structures on
S
{\displaystyle S}
for this relation.
Another equivalent definition is as follows:
T
(
S
)
{\displaystyle T(S)}
is the space of pairs
(
X
,
g
)
{\displaystyle (X,g)}
where
X
{\displaystyle X}
is a Riemann surface and
g
:
S
→
X
{\displaystyle g:S\to X}
a diffeomorphism, and two pairs
(
X
,
g
)
,
(
Y
,
h
)
{\displaystyle (X,g),(Y,h)}
are regarded as equivalent if
h
∘
g
−
1
:
X
→
Y
{\displaystyle h\circ g^{-1}:X\to Y}
is isotopic to a holomorphic diffeomorphism. Such a pair is called a marked Riemann surface; the marking being the diffeomorphism; another definition of markings is by systems of curves.
There are two simple examples that are immediately computed from the Uniformization theorem: there is a unique complex structure on the sphere
S
2
{\displaystyle \mathbb {S} ^{2}}
(see Riemann sphere) and there are two on
R
2
{\displaystyle \mathbb {R} ^{2}}
(the complex plane and the unit disk) and in each case the group of positive diffeomorphisms is connected. Thus the Teichmüller space of
S
2
{\displaystyle \mathbb {S} ^{2}}
is a single point and that of
R
2
{\displaystyle \mathbb {R} ^{2}}
contains exactly two points.
A slightly more involved example is the open annulus, for which the Teichmüller space is the interval
[
0
,
1
)
{\displaystyle [0,1)}
(the complex structure associated to
λ
{\displaystyle \lambda }
is the Riemann surface
{
z
∈
C
:
λ
<
|
z
|
<
λ
−
1
}
{\displaystyle \{z\in \mathbb {C} :\lambda <|z|<\lambda ^{-1}\}}
).
=== The Teichmüller space of the torus and flat metrics ===
The next example is the torus
T
2
=
R
2
/
Z
2
.
{\displaystyle \mathbb {T} ^{2}=\mathbb {R} ^{2}/\mathbb {Z} ^{2}.}
In this case any complex structure can be realised by a Riemann surface of the form
C
/
(
Z
+
τ
Z
)
{\displaystyle \mathbb {C} /(\mathbb {Z} +\tau \mathbb {Z} )}
(a complex elliptic curve) for a complex number
τ
∈
H
{\displaystyle \tau \in \mathbb {H} }
where
H
=
{
z
∈
C
:
Im
(
z
)
>
0
}
,
{\displaystyle \mathbb {H} =\{z\in \mathbb {C} :\operatorname {Im} (z)>0\},}
is the complex upper half-plane. Then we have a bijection:
H
⟶
T
(
T
2
)
{\displaystyle \mathbb {H} \longrightarrow T(\mathbb {T} ^{2})}
τ
⟼
(
C
/
(
Z
+
τ
Z
)
,
(
x
,
y
)
↦
x
+
τ
y
)
{\displaystyle \tau \longmapsto (\mathbb {C} /(\mathbb {Z} +\tau \mathbb {Z} ),(x,y)\mapsto x+\tau y)}
and thus the Teichmüller space of
T
2
{\displaystyle \mathbb {T} ^{2}}
is
H
.
{\displaystyle \mathbb {H} .}
If we identify
C
{\displaystyle \mathbb {C} }
with the Euclidean plane then each point in Teichmüller space can also be viewed as a marked flat structure on
T
2
.
{\displaystyle \mathbb {T} ^{2}.}
Thus the Teichmüller space is in bijection with the set of pairs
(
M
,
f
)
{\displaystyle (M,f)}
where
M
{\displaystyle M}
is a flat surface and
f
:
T
2
→
M
{\displaystyle f:\mathbb {T} ^{2}\to M}
is a diffeomorphism up to isotopy on
f
{\displaystyle f}
.
=== Finite type surfaces ===
These are the surfaces for which Teichmüller space is most often studied, which include closed surfaces. A surface is of finite type if it is diffeomorphic to a compact surface minus a finite set. If
S
{\displaystyle S}
is a closed surface of genus
g
{\displaystyle g}
then the surface obtained by removing
k
{\displaystyle k}
points from
S
{\displaystyle S}
is usually denoted
S
g
,
k
{\displaystyle S_{g,k}}
and its Teichmüller space by
T
g
,
k
.
{\displaystyle T_{g,k}.}
=== Teichmüller spaces and hyperbolic metrics ===
Every finite type orientable surface other than the ones above admits complete Riemannian metrics of constant curvature
−
1
{\displaystyle -1}
. For a given surface of finite type there is a bijection between such metrics and complex structures as follows from the uniformisation theorem. Thus if
2
g
−
2
+
k
>
0
{\displaystyle 2g-2+k>0}
the Teichmüller space
T
g
,
k
{\displaystyle T_{g,k}}
can be realised as the set of marked hyperbolic surfaces of genus
g
{\displaystyle g}
with
k
{\displaystyle k}
cusps, that is the set of pairs
(
M
,
f
)
{\displaystyle (M,f)}
where
M
{\displaystyle M}
is a hyperbolic surface and
f
:
S
→
M
{\displaystyle f:S\to M}
is a diffeomorphism, modulo the equivalence relation where
(
M
,
f
)
{\displaystyle (M,f)}
and
(
N
,
g
)
{\displaystyle (N,g)}
are identified if
f
∘
g
−
1
{\displaystyle f\circ g^{-1}}
is isotopic to an isometry.
=== The topology on Teichmüller space ===
In all cases computed above there is an obvious topology on Teichmüller space. In the general case there are many natural ways to topologise
T
(
S
)
{\displaystyle T(S)}
, perhaps the simplest is via hyperbolic metrics and length functions.
If
α
{\displaystyle \alpha }
is a closed curve on
S
{\displaystyle S}
and
x
=
(
M
,
f
)
{\displaystyle x=(M,f)}
a marked hyperbolic surface then one
f
∗
α
{\displaystyle f_{*}\alpha }
is homotopic to a unique closed geodesic
α
x
{\displaystyle \alpha _{x}}
on
M
{\displaystyle M}
(up to parametrisation). The value at
x
{\displaystyle x}
of the length function associated to (the homotopy class of)
α
{\displaystyle \alpha }
is then:
ℓ
α
(
x
)
=
Length
(
α
x
)
.
{\displaystyle \ell _{\alpha }(x)=\operatorname {Length} (\alpha _{x}).}
Let
S
{\displaystyle {\mathcal {S}}}
be the set of simple closed curves on
S
{\displaystyle S}
. Then the map
T
(
S
)
→
R
S
{\displaystyle T(S)\to \mathbb {R} ^{\mathcal {S}}}
x
↦
(
ℓ
α
(
x
)
)
α
∈
S
{\displaystyle x\mapsto \left(\ell _{\alpha }(x)\right)_{\alpha \in {\mathcal {S}}}}
is an embedding. The space
R
S
{\displaystyle \mathbb {R} ^{\mathcal {S}}}
has the product topology and
T
(
S
)
{\displaystyle T(S)}
is endowed with the induced topology. With this topology
T
(
S
g
,
k
)
{\displaystyle T(S_{g,k})}
is homeomorphic to
R
6
g
−
6
+
2
k
.
{\displaystyle \mathbb {R} ^{6g-6+2k}.}
In fact one can obtain an embedding with
9
g
−
9
{\displaystyle 9g-9}
curves, and even
6
g
−
5
+
2
k
{\displaystyle 6g-5+2k}
. In both case one can use the embedding to give a geometric proof of the homeomorphism above.
=== More examples of small Teichmüller spaces ===
There is a unique complete hyperbolic metric of finite volume on the three-holed sphere and so the Teichmüller space of finite-volume complete metrics of constant curvature
T
(
S
0
,
3
)
{\displaystyle T(S_{0,3})}
is a point (this also follows from the dimension formula of the previous paragraph).
The Teichmüller spaces
T
(
S
0
,
4
)
{\displaystyle T(S_{0,4})}
and
T
(
S
1
,
1
)
{\displaystyle T(S_{1,1})}
are naturally realised as the upper half-plane, as can be seen using Fenchel–Nielsen coordinates.
=== Teichmüller space and conformal structures ===
Instead of complex structures or hyperbolic metrics one can define Teichmüller space using conformal structures. Indeed, conformal structures are the same as complex structures in two (real) dimensions. Moreover, the Uniformisation Theorem also implies that in each conformal class of Riemannian metrics on a surface there is a unique metric of constant curvature.
=== Teichmüller spaces as representation spaces ===
Yet another interpretation of Teichmüller space is as a representation space for surface groups. If
S
{\displaystyle S}
is hyperbolic, of finite type and
Γ
=
π
1
(
S
)
{\displaystyle \Gamma =\pi _{1}(S)}
is the fundamental group of
S
{\displaystyle S}
then Teichmüller space is in natural bijection with:
The set of injective representations
Γ
→
P
S
L
2
(
R
)
{\displaystyle \Gamma \to \mathrm {PSL} _{2}(\mathbb {R} )}
with discrete image, up to conjugation by an element of
P
S
L
2
(
R
)
{\displaystyle \mathrm {PSL} _{2}(\mathbb {R} )}
, if
S
{\displaystyle S}
is compact ;
In general, the set of such representations, with the added condition that those elements of
Γ
{\displaystyle \Gamma }
which are represented by curves freely homotopic to a puncture are sent to parabolic elements of
P
S
L
2
(
R
)
{\displaystyle \mathrm {PSL} _{2}(\mathbb {R} )}
, again up to conjugation by an element of
P
S
L
2
(
R
)
{\displaystyle \mathrm {PSL} _{2}(\mathbb {R} )}
.
The map sends a marked hyperbolic structure
(
M
,
f
)
{\displaystyle (M,f)}
to the composition
ρ
∘
f
∗
{\displaystyle \rho \circ f_{*}}
where
ρ
:
π
1
(
M
)
→
P
S
L
2
(
R
)
{\displaystyle \rho :\pi _{1}(M)\to \mathrm {PSL} _{2}(\mathbb {R} )}
is the monodromy of the hyperbolic structure and
f
∗
:
π
1
(
S
)
→
π
1
(
M
)
{\displaystyle f_{*}:\pi _{1}(S)\to \pi _{1}(M)}
is the isomorphism induced by
f
{\displaystyle f}
.
Note that this realises
T
(
S
)
{\displaystyle T(S)}
as a closed subset of
P
S
L
2
(
R
)
2
g
+
k
−
1
{\displaystyle \mathrm {PSL} _{2}(\mathbb {R} )^{2g+k-1}}
which endows it with a topology. This can be used to see the homeomorphism
T
(
S
)
≅
R
6
g
−
6
+
2
k
{\displaystyle T(S)\cong \mathbb {R} ^{6g-6+2k}}
directly.
This interpretation of Teichmüller space is generalised by higher Teichmüller theory, where the group
P
S
L
2
(
R
)
{\displaystyle \mathrm {PSL} _{2}(\mathbb {R} )}
is replaced by an arbitrary semisimple Lie group.
=== A remark on categories ===
All definitions above can be made in the topological category instead of the category of differentiable manifolds, and this does not change the objects.
=== Infinite-dimensional Teichmüller spaces ===
Surfaces which are not of finite type also admit hyperbolic structures, which can be parametrised by infinite-dimensional spaces (homeomorphic to
R
N
{\displaystyle \mathbb {R} ^{\mathbb {N} }}
). Another example of infinite-dimensional space related to Teichmüller theory is the Teichmüller space of a lamination by surfaces.
== Action of the mapping class group and relation to moduli space ==
=== The map to moduli space ===
There is a map from Teichmüller space to the moduli space of Riemann surfaces diffeomorphic to
S
{\displaystyle S}
, defined by
(
X
,
f
)
↦
X
{\displaystyle (X,f)\mapsto X}
. It is a covering map, and since
T
(
S
)
{\displaystyle T(S)}
is simply connected it is the orbifold universal cover for the moduli space.
=== Action of the mapping class group ===
The mapping class group of
S
{\displaystyle S}
is the coset group
M
C
G
(
S
)
{\displaystyle MCG(S)}
of the diffeomorphism group of
S
{\displaystyle S}
by the normal subgroup of those that are isotopic to the identity (the same definition can be made with homeomorphisms instead of diffeomorphisms and, for surfaces, this does not change the resulting group). The group of diffeomorphisms acts naturally on Teichmüller space by
g
⋅
(
X
,
f
)
↦
(
X
,
f
∘
g
−
1
)
.
{\displaystyle g\cdot (X,f)\mapsto (X,f\circ g^{-1}).}
If
γ
∈
M
C
G
(
S
)
{\displaystyle \gamma \in MCG(S)}
is a mapping class and
g
,
h
{\displaystyle g,h}
two diffeomorphisms representing it then they are isotopic. Thus the classes of
(
X
,
f
∘
g
−
1
)
{\displaystyle (X,f\circ g^{-1})}
and
(
X
,
f
∘
h
−
1
)
{\displaystyle (X,f\circ h^{-1})}
are the same in Teichmüller space, and the action above factorises through the mapping class group.
The action of the mapping class group
M
C
G
(
S
)
{\displaystyle MCG(S)}
on the Teichmüller space is properly discontinuous, and the quotient is the moduli space.
=== Fixed points ===
The Nielsen realisation problem asks whether any finite subgroup of the mapping class group has a global fixed point (a point fixed by all group elements) in Teichmüller space. In more classical terms the question is: can every finite subgroup of
M
C
G
(
S
)
{\displaystyle MCG(S)}
be realised as a group of isometries of some complete hyperbolic metric on
S
{\displaystyle S}
(or equivalently as a group of holomorphic diffeomorphisms of some complex structure). This was solved by Steven Kerckhoff.
== Coordinates ==
=== Fenchel–Nielsen coordinates ===
The Fenchel–Nielsen coordinates (so named after Werner Fenchel and Jakob Nielsen) on the Teichmüller space
T
(
S
)
{\displaystyle T(S)}
are associated to a pants decomposition of the surface
S
{\displaystyle S}
. This is a decomposition of
S
{\displaystyle S}
into pairs of pants, and to each curve in the decomposition is associated its length in the hyperbolic metric corresponding to the point in Teichmüller space, and another real parameter called the twist which is more involved to define.
In case of a closed surface of genus
g
{\displaystyle g}
there are
3
g
−
3
{\displaystyle 3g-3}
curves in a pants decomposition and we get
6
g
−
6
{\displaystyle 6g-6}
parameters, which is the dimension of
T
(
S
g
)
{\displaystyle T(S_{g})}
. The Fenchel–Nielsen coordinates in fact define a homeomorphism
T
(
S
g
)
→
]
0
,
+
∞
[
3
g
−
3
×
R
3
g
−
3
{\displaystyle T(S_{g})\to \left]0,+\infty \right[^{3g-3}\times \mathbb {R} ^{3g-3}}
.
In the case of a surface with punctures some pairs of pants are "degenerate" (they have a cusp) and give only two length and twist parameters. Again in this case the Fenchel–Nielsen coordinates define a homeomorphism
T
(
S
g
,
k
)
→
]
0
,
+
∞
[
3
g
−
3
+
k
×
R
3
g
−
3
+
k
{\displaystyle T(S_{g,k})\to \left]0,+\infty \right[^{3g-3+k}\times \mathbb {R} ^{3g-3+k}}
.
=== Shear coordinates ===
If
k
>
0
{\displaystyle k>0}
the surface
S
=
S
g
,
k
{\displaystyle S=S_{g,k}}
admits ideal triangulations (whose vertices are exactly the punctures). By the formula for the Euler characteristic such a triangulation has
4
g
−
4
+
2
k
{\displaystyle 4g-4+2k}
triangles. A hyperbolic structure
M
{\displaystyle M}
on
S
{\displaystyle S}
determines a (unique up to isotopy) diffeomorphism
S
→
M
{\displaystyle S\to M}
sending every triangle to a hyperbolic ideal triangle, thus a point in
T
(
S
)
{\displaystyle T(S)}
. The parameters for such a structure are the translation lengths for each pair of sides of the triangles glued in the triangulation. There are
6
g
−
6
+
3
k
{\displaystyle 6g-6+3k}
such parameters which can each take any value in
R
{\displaystyle \mathbb {R} }
, and the completeness of the structure corresponds to a linear equation and thus we get the right dimension
6
g
−
6
+
2
k
{\displaystyle 6g-6+2k}
. These coordinates are called shear coordinates.
For closed surfaces, a pair of pants can be decomposed as the union of two ideal triangles (it can be seen as an incomplete hyperbolic metric on the three-holed sphere). Thus we also get
3
g
−
3
{\displaystyle 3g-3}
shear coordinates on
T
(
S
g
)
{\displaystyle T(S_{g})}
.
=== Earthquakes ===
A simple earthquake path in Teichmüller space is a path determined by varying a single shear or length Fenchel–Nielsen coordinate (for a fixed ideal triangulation of a surface). The name comes from seeing the ideal triangles or the pants as tectonic plates and the shear as plate motion.
More generally one can do earthquakes along geodesic laminations. A theorem of Thurston then states that two points in Teichmüller space are joined by a unique earthquake path.
== Analytic theory ==
=== Quasiconformal mappings ===
A quasiconformal mapping between two Riemann surfaces is a homeomorphism which deforms the conformal structure in a bounded manner over the surface. More precisely it is differentiable almost everywhere and there is a constant
K
≥
1
{\displaystyle K\geq 1}
, called the dilatation, such that
|
f
z
|
+
|
f
z
¯
|
|
f
z
|
−
|
f
z
¯
|
≤
K
{\displaystyle {\frac {|f_{z}|+|f_{\bar {z}}|}{|f_{z}|-|f_{\bar {z}}|}}\leq K}
where
f
z
,
f
z
¯
{\displaystyle f_{z},f_{\bar {z}}}
are the derivatives in a conformal coordinate
z
{\displaystyle z}
and its conjugate
z
¯
{\displaystyle {\bar {z}}}
.
There are quasi-conformal mappings in every isotopy class and so an alternative definition for The Teichmüller space is as follows. Fix a Riemann surface
X
{\displaystyle X}
diffeomorphic to
S
{\displaystyle S}
, and Teichmüller space is in natural bijection with the marked surfaces
(
Y
,
g
)
{\displaystyle (Y,g)}
where
g
:
X
→
Y
{\displaystyle g:X\to Y}
is a quasiconformal mapping, up to the same equivalence relation as above.
=== Quadratic differentials and the Bers embedding ===
With the definition above, if
X
=
Γ
∖
H
2
{\displaystyle X=\Gamma \setminus \mathbb {H} ^{2}}
there is a natural map from Teichmüller space to the space of
Γ
{\displaystyle \Gamma }
-equivariant solutions to the Beltrami differential equation. These give rise, via the Schwarzian derivative, to quadratic differentials on
X
{\displaystyle X}
. The space of those is a complex space of complex dimension
3
g
−
3
{\displaystyle 3g-3}
, and the image of Teichmüller space is an open set. This map is called the Bers embedding.
A quadratic differential on
X
{\displaystyle X}
can be represented by a translation surface conformal to
X
{\displaystyle X}
.
=== Teichmüller mappings ===
Teichmüller's theorem states that between two marked Riemann surfaces
(
X
,
g
)
{\displaystyle (X,g)}
and
(
Y
,
h
)
{\displaystyle (Y,h)}
there is always a unique quasiconformal mapping
X
→
Y
{\displaystyle X\to Y}
in the isotopy class of
h
∘
g
−
1
{\displaystyle h\circ g^{-1}}
which has minimal dilatation. This map is called a Teichmüller mapping.
In the geometric picture this means that for every two diffeomorphic Riemann surfaces
X
,
Y
{\displaystyle X,Y}
and diffeomorphism
f
:
X
→
Y
{\displaystyle f:X\to Y}
there exists two polygons representing
X
,
Y
{\displaystyle X,Y}
and an affine map sending one to the other, which has smallest dilatation among all quasiconformal maps
X
→
Y
{\displaystyle X\to Y}
.
== Metrics ==
=== The Teichmüller metric ===
If
x
,
y
∈
T
(
S
)
{\displaystyle x,y\in T(S)}
and the Teichmüller mapping between them has dilatation
K
{\displaystyle K}
then the Teichmüller distance between them is by definition
1
2
log
K
{\displaystyle {\frac {1}{2}}\log K}
. This indeed defines a distance on
T
(
S
)
{\displaystyle T(S)}
which induces its topology, and for which it is complete. This is the metric most commonly used for the study of the metric geometry of Teichmüller space. In particular it is of interest to geometric group theorists.
There is a function similarly defined, using the Lipschitz constants of maps between hyperbolic surfaces instead of the quasiconformal dilatations, on
T
(
S
)
×
T
(
S
)
{\displaystyle T(S)\times T(S)}
, which is not symmetric.
=== The Weil–Petersson metric ===
Quadratic differentials on a Riemann surface
X
{\displaystyle X}
are identified with the cotangent space at
(
X
,
f
)
{\displaystyle (X,f)}
to Teichmüller space. The Weil–Petersson metric is the Riemannian metric defined by the
L
2
{\displaystyle L^{2}}
inner product on quadratic differentials.
== Compactifications ==
There are several inequivalent compactifications of Teichmüller spaces that have been studied. Several of the earlier compactifications depend on the choice of a point in Teichmüller space so are not invariant under the modular group, which can be inconvenient. William Thurston later found a compactification without this disadvantage, which has become the most widely used compactification.
=== Thurston compactification ===
By looking at the hyperbolic lengths of simple closed curves for each point in Teichmüller space and taking the closure in the (infinite-dimensional) projective space, Thurston (1988) introduced a compactification whose points at infinity correspond to projective measured laminations. The compactified space is homeomorphic to a closed ball. This Thurston compactification is acted on continuously by the modular group. In particular any element of the modular group has a fixed point in Thurston's compactification, which Thurston used in his classification of elements of the modular group.
=== Bers compactification ===
The Bers compactification is given by taking the closure of the image of the Bers embedding of Teichmüller space, studied by Bers (1970). The Bers embedding depends on the choice of a point in Teichmüller space so is not invariant under the modular group, and in fact the modular group does not act continuously on the Bers compactification.
=== Teichmüller compactification ===
The "points at infinity" in the Teichmüller compactification consist of geodesic rays (for the Teichmüller metric) starting at a fixed basepoint. This compactification depends on the choice of basepoint so is not acted on by the modular group, and in fact Kerckhoff showed that the action of the modular group on Teichmüller space does not extend to a continuous action on this compactification.
=== Gardiner–Masur compactification ===
Gardiner & Masur (1991) considered a compactification similar to the Thurston compactification, but using extremal length rather than hyperbolic length. The modular group acts continuously on this compactification, but they showed that their compactification has strictly more points at infinity.
== Large-scale geometry ==
There has been an extensive study of the geometric properties of Teichmüller space endowed with the Teichmüller metric. Known large-scale properties include:
Teichmüller space
T
(
S
g
,
k
)
{\displaystyle T(S_{g,k})}
contains flat subspaces of dimension
3
g
−
3
+
k
{\displaystyle 3g-3+k}
, and there are no higher-dimensional quasi-isometrically embedded flats.
In particular, if
g
>
1
{\displaystyle g>1}
or
g
=
1
,
k
>
1
{\displaystyle g=1,k>1}
or
g
=
0
,
k
>
4
{\displaystyle g=0,k>4}
then
T
(
S
g
,
k
)
{\displaystyle T(S_{g,k})}
is not hyperbolic.
On the other hand, Teichmüller space exhibits several properties characteristic of hyperbolic spaces, such as:
Some geodesics behave like they do in hyperbolic space.
Random walks on Teichmüller space converge almost surely to a point on the Thurston boundary.
Some of these features can be explained by the study of maps from Teichmüller space to the curve complex, which is known to be hyperbolic.
== Complex geometry ==
The Bers embedding gives
T
(
S
)
{\displaystyle T(S)}
a complex structure as an open subset of
C
3
g
−
3
.
{\displaystyle \mathbb {C} ^{3g-3}.}
=== Metrics coming from the complex structure ===
Since Teichmüller space is a complex manifold it carries a Carathéodory metric. Teichmüller space is Kobayashi hyperbolic and its Kobayashi metric coincides with the Teichmüller metric. This latter result is used in Royden's proof that the mapping class group is the full group of isometries for the Teichmüller metric.
The Bers embedding realises Teichmüller space as a domain of holomorphy and hence it also carries a Bergman metric.
=== Kähler metrics on Teichmüller space ===
The Weil–Petersson metric is Kähler but it is not complete.
Cheng and Yau showed that there is a unique complete Kähler–Einstein metric on Teichmüller space. It has constant negative scalar curvature.
Teichmüller space also carries a complete Kähler metric of bounded sectional curvature introduced by McMullen (2000) that is Kähler-hyperbolic.
=== Equivalence of metrics ===
With the exception of the incomplete Weil–Petersson metric, all metrics on Teichmüller space introduced here are quasi-isometric to each other.
== See also ==
Moduli of algebraic curves
p-adic Teichmüller theory
Inter-universal Teichmüller theory
Teichmüller modular form
== References ==
== Sources ==
Ahlfors, Lars V. (2006). Lectures on quasiconformal mappings. Second edition. With supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard. American Math. Soc. pp. viii+162. ISBN 978-0-8218-3644-6.
Bers, Lipman (1970), "On boundaries of Teichmüller spaces and on Kleinian groups. I", Annals of Mathematics, Second Series, 91 (3): 570–600, doi:10.2307/1970638, JSTOR 1970638, MR 0297992
Fathi, Albert; Laudenbach, François; Poenaru, Valentin (2012). Thurston's work on surfaces. Princeton University Press. pp. xvi+254. ISBN 978-0-691-14735-2. MR 3053012.
Gardiner, Frederic P.; Masur, Howard (1991), "Extremal length geometry of Teichmüller space", Complex Variables Theory Appl., 16 (2–3): 209–237, doi:10.1080/17476939108814480, MR 1099913
Imayoshi, Yôichi; Taniguchi, Masahiko (1992). An introduction to Teichmüller spaces. Springer. pp. xiv+279. ISBN 978-4-431-70088-3.
Kerckhoff, Steven P. (1983). "The Nielsen realization problem". Annals of Mathematics. Second Series. 117 (2): 235–265. CiteSeerX 10.1.1.353.3593. doi:10.2307/2007076. JSTOR 2007076. MR 0690845.
McMullen, Curtis T. (2000), "The moduli space of Riemann surfaces is Kähler hyperbolic", Annals of Mathematics, Second Series, 151 (1): 327–357, arXiv:math/0010022, doi:10.2307/121120, JSTOR 121120, MR 1745010, S2CID 8032847
Ratcliffe, John (2006). Foundations of hyperbolic manifolds, Second edition. Springer. pp. xii+779. ISBN 978-0387-33197-3.
Thurston, William P. (1988), "On the geometry and dynamics of diffeomorphisms of surfaces", Bulletin of the American Mathematical Society, New Series, 19 (2): 417–431, doi:10.1090/S0273-0979-1988-15685-6, MR 0956596
== Further reading ==
Bers, Lipman (1981), "Finite-dimensional Teichmüller spaces and generalizations", Bulletin of the American Mathematical Society, New Series, 5 (2): 131–172, doi:10.1090/S0273-0979-1981-14933-8, MR 0621883
Gardiner, Frederick P. (1987), Teichmüller theory and quadratic differentials, Pure and Applied Mathematics (New York), New York: John Wiley & Sons, ISBN 978-0-471-84539-3, MR 0903027
Hubbard, John Hamal (2006), Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1, Matrix Editions, Ithaca, NY, ISBN 978-0-9715766-2-9, MR 2245223
Papadopoulos, Athanase, ed. (2007–2016), Handbook of Teichmüller theory. Vols. I-V (PDF), IRMA Lectures in Mathematics and Theoretical Physics, vol. 11, 13, 17, 19, 26, European Mathematical Society (EMS), Zürich, doi:10.4171/029, ISBN 978-3-03719-029-6, MR 2284826, S2CID 9203341 The last volume contains translations of several of Teichmüller's papers.
Teichmüller, Oswald (1939), "Extremale quasikonforme Abbildungen und quadratische Differentiale", Abh. Preuss. Akad. Wiss. Math.-Nat. Kl., 1939 (22): 197, JFM 66.1252.01, MR 0003242
Teichmüller, Oswald (1982), Ahlfors, Lars V.; Gehring, Frederick W. (eds.), Gesammelte Abhandlungen, Berlin, New York: Springer-Verlag, ISBN 978-3-540-10899-3, MR 0649778
Voitsekhovskii, M.I. (2001) [1994], "Teichmüller space", Encyclopedia of Mathematics, EMS Press | Wikipedia/Teichmüller_theory |
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on
n
{\displaystyle n}
-dimensional Euclidean space
R
n
{\displaystyle \mathbb {R} ^{n}}
by the Euclidean metric.
== Definition ==
The Euclidean norm on
R
n
{\displaystyle \mathbb {R} ^{n}}
is the non-negative function
‖
⋅
‖
:
R
n
→
R
{\displaystyle \|\cdot \|:\mathbb {R} ^{n}\to \mathbb {R} }
defined by
‖
(
p
1
,
…
,
p
n
)
‖
:=
p
1
2
+
⋯
+
p
n
2
.
{\displaystyle \left\|\left(p_{1},\ldots ,p_{n}\right)\right\|~:=~{\sqrt {p_{1}^{2}+\cdots +p_{n}^{2}}}.}
Like all norms, it induces a canonical metric defined by
d
(
p
,
q
)
=
‖
p
−
q
‖
.
{\displaystyle d(p,q)=\|p-q\|.}
The metric
d
:
R
n
×
R
n
→
R
{\displaystyle d:\mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R} }
induced by the Euclidean norm is called the Euclidean metric or the Euclidean distance and the distance between points
p
=
(
p
1
,
…
,
p
n
)
{\displaystyle p=\left(p_{1},\ldots ,p_{n}\right)}
and
q
=
(
q
1
,
…
,
q
n
)
{\displaystyle q=\left(q_{1},\ldots ,q_{n}\right)}
is
d
(
p
,
q
)
=
‖
p
−
q
‖
=
(
p
1
−
q
1
)
2
+
(
p
2
−
q
2
)
2
+
⋯
+
(
p
i
−
q
i
)
2
+
⋯
+
(
p
n
−
q
n
)
2
.
{\displaystyle d(p,q)~=~\|p-q\|~=~{\sqrt {\left(p_{1}-q_{1}\right)^{2}+\left(p_{2}-q_{2}\right)^{2}+\cdots +\left(p_{i}-q_{i}\right)^{2}+\cdots +\left(p_{n}-q_{n}\right)^{2}}}.}
In any metric space, the open balls form a base for a topology on that space.
The Euclidean topology on
R
n
{\displaystyle \mathbb {R} ^{n}}
is the topology generated by these balls.
In other words, the open sets of the Euclidean topology on
R
n
{\displaystyle \mathbb {R} ^{n}}
are given by (arbitrary) unions of the open balls
B
r
(
p
)
{\displaystyle B_{r}(p)}
defined as
B
r
(
p
)
:=
{
x
∈
R
n
:
d
(
p
,
x
)
<
r
}
,
{\displaystyle B_{r}(p):=\left\{x\in \mathbb {R} ^{n}:d(p,x)<r\right\},}
for all real
r
>
0
{\displaystyle r>0}
and all
p
∈
R
n
,
{\displaystyle p\in \mathbb {R} ^{n},}
where
d
{\displaystyle d}
is the Euclidean metric.
== Properties ==
When endowed with this topology, the real line
R
{\displaystyle \mathbb {R} }
is a T5 space.
Given two subsets say
A
{\displaystyle A}
and
B
{\displaystyle B}
of
R
{\displaystyle \mathbb {R} }
with
A
¯
∩
B
=
A
∩
B
¯
=
∅
,
{\displaystyle {\overline {A}}\cap B=A\cap {\overline {B}}=\varnothing ,}
where
A
¯
{\displaystyle {\overline {A}}}
denotes the closure of
A
,
{\displaystyle A,}
there exist open sets
S
A
{\displaystyle S_{A}}
and
S
B
{\displaystyle S_{B}}
with
A
⊆
S
A
{\displaystyle A\subseteq S_{A}}
and
B
⊆
S
B
{\displaystyle B\subseteq S_{B}}
such that
S
A
∩
S
B
=
∅
.
{\displaystyle S_{A}\cap S_{B}=\varnothing .}
== See also ==
Hilbert space – Type of vector space in math
List of Banach spaces
List of topologies – List of concrete topologies and topological spaces
== References == | Wikipedia/Euclidean_topology |
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that distinguish topological spaces.
A subset of a topological space
X
{\displaystyle X}
is a connected set if it is a connected space when viewed as a subspace of
X
{\displaystyle X}
.
Some related but stronger conditions are path connected, simply connected, and
n
{\displaystyle n}
-connected. Another related notion is locally connected, which neither implies nor follows from connectedness.
== Formal definition ==
A topological space
X
{\displaystyle X}
is said to be disconnected if it is the union of two disjoint non-empty open sets. Otherwise,
X
{\displaystyle X}
is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article does not follow that practice.
For a topological space
X
{\displaystyle X}
the following conditions are equivalent:
X
{\displaystyle X}
is connected, that is, it cannot be divided into two disjoint non-empty open sets.
The only subsets of
X
{\displaystyle X}
which are both open and closed (clopen sets) are
X
{\displaystyle X}
and the empty set.
The only subsets of
X
{\displaystyle X}
with empty boundary are
X
{\displaystyle X}
and the empty set.
X
{\displaystyle X}
cannot be written as the union of two non-empty separated sets (sets for which each is disjoint from the other's closure).
All continuous functions from
X
{\displaystyle X}
to
{
0
,
1
}
{\displaystyle \{0,1\}}
are constant, where
{
0
,
1
}
{\displaystyle \{0,1\}}
is the two-point space endowed with the discrete topology.
Historically this modern formulation of the notion of connectedness (in terms of no partition of
X
{\displaystyle X}
into two separated sets) first appeared (independently) with N.J. Lennes, Frigyes Riesz, and Felix Hausdorff at the beginning of the 20th century. See (Wilder 1978) for details.
=== Connected components ===
Given some point
x
{\displaystyle x}
in a topological space
X
,
{\displaystyle X,}
the union of any collection of connected subsets such that each contains
x
{\displaystyle x}
will once again be a connected subset.
The connected component of a point
x
{\displaystyle x}
in
X
{\displaystyle X}
is the union of all connected subsets of
X
{\displaystyle X}
that contain
x
;
{\displaystyle x;}
it is the unique largest (with respect to
⊆
{\displaystyle \subseteq }
) connected subset of
X
{\displaystyle X}
that contains
x
.
{\displaystyle x.}
The maximal connected subsets (ordered by inclusion
⊆
{\displaystyle \subseteq }
) of a non-empty topological space are called the connected components of the space.
The components of any topological space
X
{\displaystyle X}
form a partition of
X
{\displaystyle X}
: they are disjoint, non-empty and their union is the whole space.
Every component is a closed subset of the original space. It follows that, in the case where their number is finite, each component is also an open subset. However, if their number is infinite, this might not be the case; for instance, the connected components of the set of the rational numbers are the one-point sets (singletons), which are not open. Proof: Any two distinct rational numbers
q
1
<
q
2
{\displaystyle q_{1}<q_{2}}
are in different components. Take an irrational number
q
1
<
r
<
q
2
,
{\displaystyle q_{1}<r<q_{2},}
and then set
A
=
{
q
∈
Q
:
q
<
r
}
{\displaystyle A=\{q\in \mathbb {Q} :q<r\}}
and
B
=
{
q
∈
Q
:
q
>
r
}
.
{\displaystyle B=\{q\in \mathbb {Q} :q>r\}.}
Then
(
A
,
B
)
{\displaystyle (A,B)}
is a separation of
Q
,
{\displaystyle \mathbb {Q} ,}
and
q
1
∈
A
,
q
2
∈
B
{\displaystyle q_{1}\in A,q_{2}\in B}
. Thus each component is a one-point set.
Let
Γ
x
{\displaystyle \Gamma _{x}}
be the connected component of
x
{\displaystyle x}
in a topological space
X
,
{\displaystyle X,}
and
Γ
x
′
{\displaystyle \Gamma _{x}'}
be the intersection of all clopen sets containing
x
{\displaystyle x}
(called quasi-component of
x
{\displaystyle x}
). Then
Γ
x
⊂
Γ
x
′
{\displaystyle \Gamma _{x}\subset \Gamma '_{x}}
where the equality holds if
X
{\displaystyle X}
is compact Hausdorff or locally connected.
=== Disconnected spaces ===
A space in which all components are one-point sets is called totally disconnected. Related to this property, a space
X
{\displaystyle X}
is called totally separated if, for any two distinct elements
x
{\displaystyle x}
and
y
{\displaystyle y}
of
X
{\displaystyle X}
, there exist disjoint open sets
U
{\displaystyle U}
containing
x
{\displaystyle x}
and
V
{\displaystyle V}
containing
y
{\displaystyle y}
such that
X
{\displaystyle X}
is the union of
U
{\displaystyle U}
and
V
{\displaystyle V}
. Clearly, any totally separated space is totally disconnected, but the converse does not hold. For example, take two copies of the rational numbers
Q
{\displaystyle \mathbb {Q} }
, and identify them at every point except zero. The resulting space, with the quotient topology, is totally disconnected. However, by considering the two copies of zero, one sees that the space is not totally separated. In fact, it is not even Hausdorff, and the condition of being totally separated is strictly stronger than the condition of being Hausdorff.
== Examples ==
The closed interval
[
0
,
2
)
{\displaystyle [0,2)}
in the standard subspace topology is connected; although it can, for example, be written as the union of
[
0
,
1
)
{\displaystyle [0,1)}
and
[
1
,
2
)
,
{\displaystyle [1,2),}
the second set is not open in the chosen topology of
[
0
,
2
)
.
{\displaystyle [0,2).}
The union of
[
0
,
1
)
{\displaystyle [0,1)}
and
(
1
,
2
]
{\displaystyle (1,2]}
is disconnected; both of these intervals are open in the standard topological space
[
0
,
1
)
∪
(
1
,
2
]
.
{\displaystyle [0,1)\cup (1,2].}
(
0
,
1
)
∪
{
3
}
{\displaystyle (0,1)\cup \{3\}}
is disconnected.
A convex subset of
R
n
{\displaystyle \mathbb {R} ^{n}}
is connected; it is actually simply connected.
A Euclidean plane excluding the origin,
(
0
,
0
)
,
{\displaystyle (0,0),}
is connected, but is not simply connected. The three-dimensional Euclidean space without the origin is connected, and even simply connected. In contrast, the one-dimensional Euclidean space without the origin is not connected.
A Euclidean plane with a straight line removed is not connected since it consists of two half-planes.
R
{\displaystyle \mathbb {R} }
, the space of real numbers with the usual topology, is connected.
The Sorgenfrey line is disconnected.
If even a single point is removed from
R
{\displaystyle \mathbb {R} }
, the remainder is disconnected. However, if even a countable infinity of points are removed from
R
n
{\displaystyle \mathbb {R} ^{n}}
, where
n
≥
2
,
{\displaystyle n\geq 2,}
the remainder is connected. If
n
≥
3
{\displaystyle n\geq 3}
, then
R
n
{\displaystyle \mathbb {R} ^{n}}
remains simply connected after removal of countably many points.
Any topological vector space, e.g. any Hilbert space or Banach space, over a connected field (such as
R
{\displaystyle \mathbb {R} }
or
C
{\displaystyle \mathbb {C} }
), is simply connected.
Every discrete topological space with at least two elements is disconnected, in fact such a space is totally disconnected. The simplest example is the discrete two-point space.
On the other hand, a finite set might be connected. For example, the spectrum of a discrete valuation ring consists of two points and is connected. It is an example of a Sierpiński space.
The Cantor set is totally disconnected; since the set contains uncountably many points, it has uncountably many components.
If a space
X
{\displaystyle X}
is homotopy equivalent to a connected space, then
X
{\displaystyle X}
is itself connected.
The topologist's sine curve is an example of a set that is connected but is neither path connected nor locally connected.
The general linear group
GL
(
n
,
R
)
{\displaystyle \operatorname {GL} (n,\mathbb {R} )}
(that is, the group of
n
{\displaystyle n}
-by-
n
{\displaystyle n}
real, invertible matrices) consists of two connected components: the one with matrices of positive determinant and the other of negative determinant. In particular, it is not connected. In contrast,
GL
(
n
,
C
)
{\displaystyle \operatorname {GL} (n,\mathbb {C} )}
is connected. More generally, the set of invertible bounded operators on a complex Hilbert space is connected.
The spectra of commutative local ring and integral domains are connected. More generally, the following are equivalent
The spectrum of a commutative ring
R
{\displaystyle R}
is connected
Every finitely generated projective module over
R
{\displaystyle R}
has constant rank.
R
{\displaystyle R}
has no idempotent
≠
0
,
1
{\displaystyle \neq 0,1}
(i.e.,
R
{\displaystyle R}
is not a product of two rings in a nontrivial way).
An example of a space that is not connected is a plane with an infinite line deleted from it. Other examples of disconnected spaces (that is, spaces which are not connected) include the plane with an annulus removed, as well as the union of two disjoint closed disks, where all examples of this paragraph bear the subspace topology induced by two-dimensional Euclidean space.
== Path connectedness ==
A path-connected space is a stronger notion of connectedness, requiring the structure of a path. A path from a point
x
{\displaystyle x}
to a point
y
{\displaystyle y}
in a topological space
X
{\displaystyle X}
is a continuous function
f
{\displaystyle f}
from the unit interval
[
0
,
1
]
{\displaystyle [0,1]}
to
X
{\displaystyle X}
with
f
(
0
)
=
x
{\displaystyle f(0)=x}
and
f
(
1
)
=
y
{\displaystyle f(1)=y}
. A path-component of
X
{\displaystyle X}
is an equivalence class of
X
{\displaystyle X}
under the equivalence relation which makes
x
{\displaystyle x}
equivalent to
y
{\displaystyle y}
if and only if there is a path from
x
{\displaystyle x}
to
y
{\displaystyle y}
. The space
X
{\displaystyle X}
is said to be path-connected (or pathwise connected or
0
{\displaystyle \mathbf {0} }
-connected) if there is exactly one path-component. For non-empty spaces, this is equivalent to the statement that there is a path joining any two points in
X
{\displaystyle X}
. Again, many authors exclude the empty space.
Every path-connected space is connected. The converse is not always true: examples of connected spaces that are not path-connected include the extended long line
L
∗
{\displaystyle L^{*}}
and the topologist's sine curve.
Subsets of the real line
R
{\displaystyle \mathbb {R} }
are connected if and only if they are path-connected; these subsets are the intervals and rays of
R
{\displaystyle \mathbb {R} }
.
Also, open subsets of
R
n
{\displaystyle \mathbb {R} ^{n}}
or
C
n
{\displaystyle \mathbb {C} ^{n}}
are connected if and only if they are path-connected.
Additionally, connectedness and path-connectedness are the same for finite topological spaces.
== Arc connectedness ==
A space
X
{\displaystyle X}
is said to be arc-connected or arcwise connected if any two topologically distinguishable points can be joined by an arc, which is an embedding
f
:
[
0
,
1
]
→
X
{\displaystyle f:[0,1]\to X}
. An arc-component of
X
{\displaystyle X}
is a maximal arc-connected subset of
X
{\displaystyle X}
; or equivalently an equivalence class of the equivalence relation of whether two points can be joined by an arc or by a path whose points are topologically indistinguishable.
Every Hausdorff space that is path-connected is also arc-connected; more generally this is true for a
Δ
{\displaystyle \Delta }
-Hausdorff space, which is a space where each image of a path is closed. An example of a space which is path-connected but not arc-connected is given by the line with two origins; its two copies of
0
{\displaystyle 0}
can be connected by a path but not by an arc.
Intuition for path-connected spaces does not readily transfer to arc-connected spaces. Let
X
{\displaystyle X}
be the line with two origins. The following are facts whose analogues hold for path-connected spaces, but do not hold for arc-connected spaces:
Continuous image of arc-connected space may not be arc-connected: for example, a quotient map from an arc-connected space to its quotient with countably many (at least 2) topologically distinguishable points cannot be arc-connected due to too small cardinality.
Arc-components may not be disjoint. For example,
X
{\displaystyle X}
has two overlapping arc-components.
Arc-connected product space may not be a product of arc-connected spaces. For example,
X
×
R
{\displaystyle X\times \mathbb {R} }
is arc-connected, but
X
{\displaystyle X}
is not.
Arc-components of a product space may not be products of arc-components of the marginal spaces. For example,
X
×
R
{\displaystyle X\times \mathbb {R} }
has a single arc-component, but
X
{\displaystyle X}
has two arc-components.
If arc-connected subsets have a non-empty intersection, then their union may not be arc-connected. For example, the arc-components of
X
{\displaystyle X}
intersect, but their union is not arc-connected.
== Local connectedness ==
A topological space is said to be locally connected at a point
x
{\displaystyle x}
if every neighbourhood of
x
{\displaystyle x}
contains a connected open neighbourhood. It is locally connected if it has a base of connected sets. It can be shown that a space
X
{\displaystyle X}
is locally connected if and only if every component of every open set of
X
{\displaystyle X}
is open.
Similarly, a topological space is said to be locally path-connected if it has a base of path-connected sets.
An open subset of a locally path-connected space is connected if and only if it is path-connected.
This generalizes the earlier statement about
R
n
{\displaystyle \mathbb {R} ^{n}}
and
C
n
{\displaystyle \mathbb {C} ^{n}}
, each of which is locally path-connected. More generally, any topological manifold is locally path-connected.
Locally connected does not imply connected, nor does locally path-connected imply path connected. A simple example of a locally connected (and locally path-connected) space that is not connected (or path-connected) is the union of two separated intervals in
R
{\displaystyle \mathbb {R} }
, such as
(
0
,
1
)
∪
(
2
,
3
)
{\displaystyle (0,1)\cup (2,3)}
.
A classic example of a connected space that is not locally connected is the so-called topologist's sine curve, defined as
T
=
{
(
0
,
0
)
}
∪
{
(
x
,
sin
(
1
x
)
)
:
x
∈
(
0
,
1
]
}
{\displaystyle T=\{(0,0)\}\cup \left\{\left(x,\sin \left({\tfrac {1}{x}}\right)\right):x\in (0,1]\right\}}
, with the Euclidean topology induced by inclusion in
R
2
{\displaystyle \mathbb {R} ^{2}}
.
== Set operations ==
The intersection of connected sets is not necessarily connected.
The union of connected sets is not necessarily connected, as can be seen by considering
X
=
(
0
,
1
)
∪
(
1
,
2
)
{\displaystyle X=(0,1)\cup (1,2)}
.
Each ellipse is a connected set, but the union is not connected, since it can be partitioned into two disjoint open sets
U
{\displaystyle U}
and
V
{\displaystyle V}
.
This means that, if the union
X
{\displaystyle X}
is disconnected, then the collection
{
X
i
}
{\displaystyle \{X_{i}\}}
can be partitioned into two sub-collections, such that the unions of the sub-collections are disjoint and open in
X
{\displaystyle X}
(see picture). This implies that in several cases, a union of connected sets is necessarily connected. In particular:
If the common intersection of all sets is not empty (
⋂
X
i
≠
∅
{\textstyle \bigcap X_{i}\neq \emptyset }
), then obviously they cannot be partitioned to collections with disjoint unions. Hence the union of connected sets with non-empty intersection is connected.
If the intersection of each pair of sets is not empty (
∀
i
,
j
:
X
i
∩
X
j
≠
∅
{\displaystyle \forall i,j:X_{i}\cap X_{j}\neq \emptyset }
) then again they cannot be partitioned to collections with disjoint unions, so their union must be connected.
If the sets can be ordered as a "linked chain", i.e. indexed by integer indices and
∀
i
:
X
i
∩
X
i
+
1
≠
∅
{\displaystyle \forall i:X_{i}\cap X_{i+1}\neq \emptyset }
, then again their union must be connected.
If the sets are pairwise-disjoint and the quotient space
X
/
{
X
i
}
{\displaystyle X/\{X_{i}\}}
is connected, then X must be connected. Otherwise, if
U
∪
V
{\displaystyle U\cup V}
is a separation of X then
q
(
U
)
∪
q
(
V
)
{\displaystyle q(U)\cup q(V)}
is a separation of the quotient space (since
q
(
U
)
,
q
(
V
)
{\displaystyle q(U),q(V)}
are disjoint and open in the quotient space).
The set difference of connected sets is not necessarily connected. However, if
X
⊇
Y
{\displaystyle X\supseteq Y}
and their difference
X
∖
Y
{\displaystyle X\setminus Y}
is disconnected (and thus can be written as a union of two open sets
X
1
{\displaystyle X_{1}}
and
X
2
{\displaystyle X_{2}}
), then the union of
Y
{\displaystyle Y}
with each such component is connected (i.e.
Y
∪
X
i
{\displaystyle Y\cup X_{i}}
is connected for all
i
{\displaystyle i}
).
== Theorems ==
Main theorem of connectedness: Let
X
{\displaystyle X}
and
Y
{\displaystyle Y}
be topological spaces and let
f
:
X
→
Y
{\displaystyle f:X\rightarrow Y}
be a continuous function. If
X
{\displaystyle X}
is (path-)connected then the image
f
(
X
)
{\displaystyle f(X)}
is (path-)connected. This result can be considered a generalization of the intermediate value theorem.
Every path-connected space is connected.
In a locally path-connected space, every open connected set is path-connected.
Every locally path-connected space is locally connected.
A locally path-connected space is path-connected if and only if it is connected.
The closure of a connected subset is connected. Furthermore, any subset between a connected subset and its closure is connected.
The connected components are always closed (but in general not open)
The connected components of a locally connected space are also open.
The connected components of a space are disjoint unions of the path-connected components (which in general are neither open nor closed).
Every quotient of a connected (resp. locally connected, path-connected, locally path-connected) space is connected (resp. locally connected, path-connected, locally path-connected).
Every product of a family of connected (resp. path-connected) spaces is connected (resp. path-connected).
Every open subset of a locally connected (resp. locally path-connected) space is locally connected (resp. locally path-connected).
Every manifold is locally path-connected.
Arc-wise connected space is path connected, but path-wise connected space may not be arc-wise connected
Continuous image of arc-wise connected set is arc-wise connected.
== Graphs ==
Graphs have path connected subsets, namely those subsets for which every pair of points has a path of edges joining them.
However, it is not always possible to find a topology on the set of points which induces the same connected sets. The 5-cycle graph (and any
n
{\displaystyle n}
-cycle with
n
>
3
{\displaystyle n>3}
odd) is one such example.
As a consequence, a notion of connectedness can be formulated independently of the topology on a space. To wit, there is a category of connective spaces consisting of sets with collections of connected subsets satisfying connectivity axioms; their morphisms are those functions which map connected sets to connected sets (Muscat & Buhagiar 2006). Topological spaces and graphs are special cases of connective spaces; indeed, the finite connective spaces are precisely the finite graphs.
However, every graph can be canonically made into a topological space, by treating vertices as points and edges as copies of the unit interval (see topological graph theory#Graphs as topological spaces). Then one can show that the graph is connected (in the graph theoretical sense) if and only if it is connected as a topological space.
== Stronger forms of connectedness ==
There are stronger forms of connectedness for topological spaces, for instance:
If there exist no two disjoint non-empty open sets in a topological space
X
{\displaystyle X}
,
X
{\displaystyle X}
must be connected, and thus hyperconnected spaces are also connected.
Since a simply connected space is, by definition, also required to be path connected, any simply connected space is also connected. If the "path connectedness" requirement is dropped from the definition of simple connectivity, a simply connected space does not need to be connected.
Yet stronger versions of connectivity include the notion of a contractible space. Every contractible space is path connected and thus also connected.
In general, any path connected space must be connected but there exist connected spaces that are not path connected. The deleted comb space furnishes such an example, as does the above-mentioned topologist's sine curve.
== See also ==
Connected component (graph theory) – Maximal subgraph whose vertices can reach each otherPages displaying short descriptions of redirect targets
Connectedness locus
Domain (mathematical analysis) – Connected open subset of a topological space
Extremally disconnected space – Topological space in which the closure of every open set is open
Locally connected space – Property of topological spaces
n-connected
Uniformly connected space – Type of uniform space
Pixel connectivity
== References ==
Wilder, R.L. (1978). "Evolution of the Topological Concept of "Connected"". American Mathematical Monthly. 85 (9): 720–726. doi:10.2307/2321676. JSTOR 2321676.
== Further reading == | Wikipedia/Connected_component_(topology) |
In topology, especially algebraic topology, the cone of a topological space
X
{\displaystyle X}
is intuitively obtained by stretching X into a cylinder and then collapsing one of its end faces to a point. The cone of X is denoted by
C
X
{\displaystyle CX}
or by
cone
(
X
)
{\displaystyle \operatorname {cone} (X)}
.
== Definitions ==
Formally, the cone of X is defined as:
C
X
=
(
X
×
[
0
,
1
]
)
∪
p
v
=
lim
→
(
(
X
×
[
0
,
1
]
)
↩
(
X
×
{
0
}
)
→
p
v
)
,
{\displaystyle CX=(X\times [0,1])\cup _{p}v\ =\ \varinjlim {\bigl (}(X\times [0,1])\hookleftarrow (X\times \{0\})\xrightarrow {p} v{\bigr )},}
where
v
{\displaystyle v}
is a point (called the vertex of the cone) and
p
{\displaystyle p}
is the projection to that point. In other words, it is the result of attaching the cylinder
X
×
[
0
,
1
]
{\displaystyle X\times [0,1]}
by its face
X
×
{
0
}
{\displaystyle X\times \{0\}}
to a point
v
{\displaystyle v}
along the projection
p
:
(
X
×
{
0
}
)
→
v
{\displaystyle p:{\bigl (}X\times \{0\}{\bigr )}\to v}
.
If
X
{\displaystyle X}
is a non-empty compact subspace of Euclidean space, the cone on
X
{\displaystyle X}
is homeomorphic to the union of segments from
X
{\displaystyle X}
to any fixed point
v
∉
X
{\displaystyle v\not \in X}
such that these segments intersect only in
v
{\displaystyle v}
itself. That is, the topological cone agrees with the geometric cone for compact spaces when the latter is defined. However, the topological cone construction is more general.
The cone is a special case of a join:
C
X
≃
X
⋆
{
v
}
=
{\displaystyle CX\simeq X\star \{v\}=}
the join of
X
{\displaystyle X}
with a single point
v
∉
X
{\displaystyle v\not \in X}
.: 76
== Examples ==
Here we often use a geometric cone (
C
X
{\displaystyle CX}
where
X
{\displaystyle X}
is a non-empty compact subspace of Euclidean space). The considered spaces are compact, so we get the same result up to homeomorphism.
The cone over a point p of the real line is a line-segment in
R
2
{\displaystyle \mathbb {R} ^{2}}
,
{
p
}
×
[
0
,
1
]
{\displaystyle \{p\}\times [0,1]}
.
The cone over two points {0, 1} is a "V" shape with endpoints at {0} and {1}.
The cone over a closed interval I of the real line is a filled-in triangle (with one of the edges being I), otherwise known as a 2-simplex (see the final example).
The cone over a polygon P is a pyramid with base P.
The cone over a disk is the solid cone of classical geometry (hence the concept's name).
The cone over a circle given by
{
(
x
,
y
,
z
)
∈
R
3
∣
x
2
+
y
2
=
1
and
z
=
0
}
{\displaystyle \{(x,y,z)\in \mathbb {R} ^{3}\mid x^{2}+y^{2}=1{\mbox{ and }}z=0\}}
is the curved surface of the solid cone:
{
(
x
,
y
,
z
)
∈
R
3
∣
x
2
+
y
2
=
(
z
−
1
)
2
and
0
≤
z
≤
1
}
.
{\displaystyle \{(x,y,z)\in \mathbb {R} ^{3}\mid x^{2}+y^{2}=(z-1)^{2}{\mbox{ and }}0\leq z\leq 1\}.}
This in turn is homeomorphic to the closed disc.
More general examples:: 77, Exercise.1
The cone over an n-sphere is homeomorphic to the closed (n + 1)-ball.
The cone over an n-ball is also homeomorphic to the closed (n + 1)-ball.
The cone over an n-simplex is an (n + 1)-simplex.
== Properties ==
All cones are path-connected since every point can be connected to the vertex point. Furthermore, every cone is contractible to the vertex point by the homotopy
h
t
(
x
,
s
)
=
(
x
,
(
1
−
t
)
s
)
{\displaystyle h_{t}(x,s)=(x,(1-t)s)}
.
The cone is used in algebraic topology precisely because it embeds a space as a subspace of a contractible space.
When X is compact and Hausdorff (essentially, when X can be embedded in Euclidean space), then the cone
C
X
{\displaystyle CX}
can be visualized as the collection of lines joining every point of X to a single point. However, this picture fails when X is not compact or not Hausdorff, as generally the quotient topology on
C
X
{\displaystyle CX}
will be finer than the set of lines joining X to a point.
== Cone functor ==
The map
X
↦
C
X
{\displaystyle X\mapsto CX}
induces a functor
C
:
T
o
p
→
T
o
p
{\displaystyle C\colon \mathbf {Top} \to \mathbf {Top} }
on the category of topological spaces Top. If
f
:
X
→
Y
{\displaystyle f\colon X\to Y}
is a continuous map, then
C
f
:
C
X
→
C
Y
{\displaystyle Cf\colon CX\to CY}
is defined by
(
C
f
)
(
[
x
,
t
]
)
=
[
f
(
x
)
,
t
]
{\displaystyle (Cf)([x,t])=[f(x),t]}
,
where square brackets denote equivalence classes.
== Reduced cone ==
If
(
X
,
x
0
)
{\displaystyle (X,x_{0})}
is a pointed space, there is a related construction, the reduced cone, given by
(
X
×
[
0
,
1
]
)
/
(
X
×
{
0
}
∪
{
x
0
}
×
[
0
,
1
]
)
{\displaystyle (X\times [0,1])/(X\times \left\{0\right\}\cup \left\{x_{0}\right\}\times [0,1])}
where we take the basepoint of the reduced cone to be the equivalence class of
(
x
0
,
0
)
{\displaystyle (x_{0},0)}
. With this definition, the natural inclusion
x
↦
(
x
,
1
)
{\displaystyle x\mapsto (x,1)}
becomes a based map. This construction also gives a functor, from the category of pointed spaces to itself.
== See also ==
Cone (disambiguation)
Suspension (topology)
Desuspension
Mapping cone (topology)
Join (topology)
== References ==
Allen Hatcher, Algebraic topology. Cambridge University Press, Cambridge, 2002. xii+544 pp. ISBN 0-521-79160-X and ISBN 0-521-79540-0
"Cone". PlanetMath. | Wikipedia/Cone_(topology) |
In mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with the space of all directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the point with the projectivized tangent space at that point. The metaphor is that of zooming in on a photograph to enlarge part of the picture, rather than referring to an explosion. The inverse operation is called blowing down.
Blowups are the most fundamental transformation in birational geometry, because every birational morphism between projective varieties is a blowup. The weak factorization theorem says that every birational map can be factored as a composition of particularly simple blowups. The Cremona group, the group of birational automorphisms of the plane, is generated by blowups.
Besides their importance in describing birational transformations, blowups are also an important way of constructing new spaces. For instance, most procedures for resolution of singularities proceed by blowing up singularities until they become smooth. A consequence of this is that blowups can be used to resolve the singularities of birational maps.
Classically, blowups were defined extrinsically, by first defining the blowup on spaces such as projective space using an explicit construction in coordinates and then defining blowups on other spaces in terms of an embedding. This is reflected in some of the terminology, such as the classical term monoidal transformation. Contemporary algebraic geometry treats blowing up as an intrinsic operation on an algebraic variety. From this perspective, a blowup is the universal (in the sense of category theory) way to turn a subvariety into a Cartier divisor.
A blowup can also be called monoidal transformation, locally quadratic transformation, dilatation, σ-process, or Hopf map.
== The blowup of a point in a plane ==
The simplest case of a blowup is the blowup of a point in a plane. Most of the general features of blowing up can be seen in this example.
The blowup has a synthetic description as an incidence correspondence. Recall that the Grassmannian
G
r
(
1
,
2
)
{\displaystyle \mathbf {Gr} (1,2)}
parametrizes the set of all lines through a point in the plane. The blowup of the projective plane
P
2
{\displaystyle \mathbf {P} ^{2}}
at the point
P
{\displaystyle P}
, which we will denote
X
{\displaystyle X}
, is
X
=
{
(
Q
,
ℓ
)
∣
P
,
Q
∈
ℓ
}
⊆
P
2
×
G
r
(
1
,
2
)
.
{\displaystyle X=\{(Q,\ell )\mid P,\,Q\in \ell \}\subseteq \mathbf {P} ^{2}\times \mathbf {Gr} (1,2).}
Here
Q
{\displaystyle Q}
denotes another point in
P
2
{\displaystyle \mathbf {P} ^{2}}
and
ℓ
{\displaystyle \ell }
is an element of the Grassmannian.
X
{\displaystyle X}
is a projective variety because it is a closed subvariety of a product of projective varieties. It comes with a natural morphism
π
{\displaystyle \pi }
to
P
2
{\displaystyle \mathbf {P} ^{2}}
that takes the pair
(
Q
,
ℓ
)
{\displaystyle (Q,\ell )}
to
Q
{\displaystyle Q}
. This morphism is an isomorphism on the open subset of all points
(
Q
,
ℓ
)
∈
X
{\displaystyle (Q,\ell )\in X}
with
Q
≠
P
{\displaystyle Q\neq P}
because the line
ℓ
{\displaystyle \ell }
is determined by those two points. When
Q
=
P
{\displaystyle Q=P}
, however, the line
ℓ
{\displaystyle \ell }
can be any line through
P
{\displaystyle P}
. These lines correspond to the space of directions through
P
{\displaystyle P}
, which is isomorphic to
P
1
{\displaystyle \mathbf {P} ^{1}}
. This
P
1
{\displaystyle \mathbf {P} ^{1}}
is called the exceptional divisor, and by definition it is the projectivized normal space at
P
{\displaystyle P}
. Because
P
{\displaystyle P}
is a point, the normal space is the same as the tangent space, so the exceptional divisor is isomorphic to the projectivized tangent space at
P
{\displaystyle P}
.
To give coordinates on the blowup, we can write down equations for the above incidence correspondence. Give
P
2
{\displaystyle \mathbf {P} ^{2}}
homogeneous coordinates
[
X
0
:
X
1
:
X
2
]
{\displaystyle [X_{0}:X_{1}:X_{2}]}
in which
P
{\displaystyle P}
is the point
[
P
0
:
P
1
:
P
2
]
{\displaystyle [P_{0}:P_{1}:P_{2}]}
. By projective duality,
G
r
(
1
,
2
)
{\displaystyle \mathbf {Gr} (1,2)}
is isomorphic to
P
2
{\displaystyle \mathbf {P} ^{2}}
, so we may give it homogeneous coordinates
[
L
0
:
L
1
:
L
2
]
{\displaystyle [L_{0}:L_{1}:L_{2}]}
. A line
ℓ
0
=
[
L
0
:
L
1
:
L
2
]
{\displaystyle \ell _{0}=[L_{0}:L_{1}:L_{2}]}
is the set of all
[
X
0
:
X
1
:
X
2
]
{\displaystyle [X_{0}:X_{1}:X_{2}]}
such that
X
0
L
0
+
X
1
L
1
+
X
2
L
2
=
0
{\displaystyle X_{0}L_{0}+X_{1}L_{1}+X_{2}L_{2}=0}
. Therefore, the blowup can be described as
X
=
{
(
[
X
0
:
X
1
:
X
2
]
,
[
L
0
:
L
1
:
L
2
]
)
∣
P
0
L
0
+
P
1
L
1
+
P
2
L
2
=
0
,
X
0
L
0
+
X
1
L
1
+
X
2
L
2
=
0
}
⊆
P
2
×
P
2
.
{\displaystyle X={\bigl \{}{\bigl (}[X_{0}:X_{1}:X_{2}],[L_{0}:L_{1}:L_{2}]{\bigr )}\mid P_{0}L_{0}+P_{1}L_{1}+P_{2}L_{2}=0,\,X_{0}L_{0}+X_{1}L_{1}+X_{2}L_{2}=0{\bigr \}}\subseteq \mathbf {P} ^{2}\times \mathbf {P} ^{2}.}
The blowup is an isomorphism away from
P
{\displaystyle P}
, and by working in the affine plane instead of the projective plane, we can give simpler equations for the blowup. After a projective transformation, we may assume that
P
=
[
0
:
0
:
1
]
{\displaystyle P=[0:0:1]}
. Write
x
{\displaystyle x}
and
y
{\displaystyle y}
for the coordinates on the affine plane
{
X
2
≠
0
}
{\displaystyle \{X_{2}\neq 0\}}
. The condition
P
∈
ℓ
{\displaystyle P\in \ell }
implies that
L
2
=
0
{\displaystyle L_{2}=0}
, so we may replace the Grassmannian with a
P
1
{\displaystyle \mathbf {P} ^{1}}
. Then the blowup is the variety
{
(
(
x
,
y
)
,
[
z
:
w
]
)
∣
x
z
+
y
w
=
0
}
⊆
A
2
×
P
1
.
{\displaystyle {\bigl \{}{\bigl (}(x,y),[z:w]{\bigr )}\mid xz+yw=0{\bigr \}}\subseteq \mathbf {A} ^{2}\times \mathbf {P} ^{1}.}
It is more common to change coordinates so as to reverse one of the signs. Then the blowup can be written as
{
(
(
x
,
y
)
,
[
z
:
w
]
)
∣
det
[
x
y
w
z
]
=
0
}
.
{\displaystyle \left\{{\bigl (}(x,y),[z:w]{\bigr )}\mid \det {\begin{bmatrix}x&y\\w&z\end{bmatrix}}=0\right\}.}
This equation is easier to generalize than the previous one.
The blowup can be easily visualized if we remove the infinity point of the Grassmannian, e.g. by setting
w
=
1
{\displaystyle w=1}
, and obtain the standard saddle surface
y
=
x
z
{\displaystyle y=xz}
in 3D space.
The blowup can also be described by directly using coordinates on the normal space to the point. Again we work on the affine plane
A
2
{\displaystyle \mathbf {A} ^{2}}
. The normal space to the origin is the vector space
m
/
m
2
{\displaystyle {\mathfrak {m}}/{\mathfrak {m}}^{2}}
, where
m
=
(
x
,
y
)
{\displaystyle {\mathfrak {m}}=(x,y)}
is the maximal ideal of the origin. Algebraically, the projectivization of this vector space is Proj of its symmetric algebra, that is,
X
=
Proj
⨁
r
=
0
∞
Sym
k
[
x
,
y
]
r
m
/
m
2
.
{\displaystyle X=\operatorname {Proj} \bigoplus _{r=0}^{\infty }\operatorname {Sym} _{k[x,y]}^{r}{\mathfrak {m}}/{\mathfrak {m}}^{2}.}
In this example, this has a concrete description as
X
=
Proj
k
[
x
,
y
]
[
z
,
w
]
/
(
x
z
−
y
w
)
,
{\displaystyle X=\operatorname {Proj} k[x,y][z,w]/(xz-yw),}
where
x
{\displaystyle x}
and
y
{\displaystyle y}
have degree 0 and
z
{\displaystyle z}
and
w
{\displaystyle w}
have degree 1.
Over the real or complex numbers, the blowup has a topological description as the connected sum
P
2
#
P
2
{\displaystyle \mathbf {P} ^{2}\#\mathbf {P} ^{2}}
. Assume that
P
{\displaystyle P}
is the origin in
A
2
⊆
P
2
{\displaystyle \mathbf {A} ^{2}\subseteq \mathbf {P} ^{2}}
, and write
L
{\displaystyle L}
for the line at infinity.
A
2
∖
{
0
}
{\displaystyle \mathbf {A} ^{2}\backslash \{0\}}
has an inversion map
t
{\displaystyle {t}}
which sends
(
x
,
y
)
{\displaystyle (x,y)}
to
(
x
|
x
|
2
+
|
y
|
2
,
y
|
x
|
2
+
|
y
|
2
)
{\displaystyle \left({\frac {x}{\vert x\vert ^{2}+\vert y\vert ^{2}}},{\frac {y}{\vert x\vert ^{2}+\vert y\vert ^{2}}}\right)}
.
t
{\displaystyle t}
is the circle inversion with respect to the unit sphere
S
{\displaystyle S}
: It fixes
S
{\displaystyle S}
, preserves each line through the origin, and exchanges the inside of the sphere with the outside.
t
{\displaystyle t}
extends to a continuous map
P
2
∖
{
0
}
→
A
2
{\displaystyle \mathbf {P} ^{2}\backslash \{0\}\to \mathbf {A} ^{2}}
by sending the line at infinity to the origin. This extension, which we also denote
t
{\displaystyle t}
, can be used to construct the blowup. Let
C
{\displaystyle C}
denote the complement of the unit ball. The blowup
X
{\displaystyle X}
is the manifold obtained by attaching two copies of
C
{\displaystyle C}
along
S
{\displaystyle S}
.
X
{\displaystyle X}
comes with a map
π
{\displaystyle \pi }
to
P
2
{\displaystyle \mathbf {P} ^{2}}
which is the identity on the first copy of
C
{\displaystyle C}
and
t
{\displaystyle t}
on the second copy of
C
{\displaystyle C}
. This map is an isomorphism away from
P
{\displaystyle P}
, and the fiber over
P
{\displaystyle P}
is the line at infinity in the second copy of
C
{\displaystyle C}
. Each point in this line corresponds to a unique line through the origin, so the fiber over
π
{\displaystyle \pi }
corresponds to the possible normal directions through the origin.
For
C
P
2
{\displaystyle \mathbf {CP} ^{2}}
this process ought to produce an oriented manifold. In order to make this happen, the two copies of
C
{\displaystyle C}
should be given opposite orientations. In symbols,
X
{\displaystyle X}
is
C
P
2
#
C
P
2
¯
{\displaystyle \mathbf {CP} ^{2}\#{\overline {\mathbf {CP} ^{2}}}}
, where
C
P
2
¯
{\displaystyle {\overline {\mathbf {CP} ^{2}}}}
is
C
P
2
{\displaystyle \mathbf {CP} ^{2}}
with the opposite of the standard orientation.
== Blowing up points in complex space ==
Let Z be the origin in n-dimensional complex space, Cn. That is, Z is the point where the n coordinate functions
x
1
,
…
,
x
n
{\displaystyle x_{1},\ldots ,x_{n}}
simultaneously vanish. Let Pn - 1 be (n - 1)-dimensional complex projective space with homogeneous coordinates
y
1
,
…
,
y
n
{\displaystyle y_{1},\ldots ,y_{n}}
. Let
C
n
~
{\displaystyle {\tilde {\mathbf {C} ^{n}}}}
be the subset of Cn × Pn - 1 that satisfies simultaneously the equations
x
i
y
j
=
x
j
y
i
{\displaystyle x_{i}y_{j}=x_{j}y_{i}}
for i, j = 1, ..., n. The projection
π
:
C
n
×
P
n
−
1
→
C
n
{\displaystyle \pi :\mathbf {C} ^{n}\times \mathbf {P} ^{n-1}\to \mathbf {C} ^{n}}
naturally induces a holomorphic map
π
:
C
n
~
→
C
n
.
{\displaystyle \pi :{\tilde {\mathbf {C} ^{n}}}\to \mathbf {C} ^{n}.}
This map π (or, often, the space
C
n
~
{\displaystyle {\tilde {\mathbf {C} ^{n}}}}
) is called the blow-up (variously spelled blow up or blowup) of Cn.
The exceptional divisor E is defined as the inverse image of the blow-up locus Z under π. It is easy to see that
E
=
Z
×
P
n
−
1
⊆
C
n
×
P
n
−
1
{\displaystyle E=Z\times \mathbf {P} ^{n-1}\subseteq \mathbf {C} ^{n}\times \mathbf {P} ^{n-1}}
is a copy of projective space. It is an effective divisor. Away from E, π is an isomorphism between
C
n
~
∖
E
{\displaystyle {\tilde {\mathbf {C} ^{n}}}\setminus E}
and Cn \ Z; it is a birational map between
C
n
~
{\displaystyle {\tilde {\mathbf {C} ^{n}}}}
and Cn.
If instead we consider the holomorphic projection
q
:
C
n
~
→
P
n
−
1
{\displaystyle q\colon {\tilde {\mathbf {C} ^{n}}}\to \mathbf {P} ^{n-1}}
we obtain the tautological line bundle of
P
n
−
1
{\displaystyle \mathbf {P} ^{n-1}}
and we can identify the exceptional divisor
{
Z
}
×
P
n
−
1
{\displaystyle \lbrace Z\rbrace \times \mathbf {P} ^{n-1}}
with its zero section, namely
0
:
P
n
−
1
→
O
P
n
−
1
{\displaystyle \mathbf {0} \colon \mathbf {P} ^{n-1}\to {\mathcal {O}}_{\mathbf {P} ^{n-1}}}
which assigns to each point
p
{\displaystyle p}
the zero element
0
p
{\displaystyle \mathbf {0} _{p}}
in the fiber over
p
{\displaystyle p}
.
== Blowing up submanifolds in complex manifolds ==
More generally, one can blow up any codimension-
k
{\displaystyle k}
complex submanifold
Z
{\displaystyle Z}
of
C
n
{\displaystyle \mathbf {C} ^{n}}
. Suppose that
Z
{\displaystyle Z}
is the locus of the equations
x
1
=
⋯
=
x
k
=
0
{\displaystyle x_{1}=\cdots =x_{k}=0}
, and let
y
1
,
…
,
y
k
{\displaystyle y_{1},\ldots ,y_{k}}
be homogeneous coordinates on
P
k
−
1
{\displaystyle \mathbf {P} ^{k-1}}
. Then the blow-up
C
~
n
{\displaystyle {\tilde {\mathbf {C} }}^{n}}
is the locus of the equations
x
i
y
j
=
x
j
y
i
{\displaystyle x_{i}y_{j}=x_{j}y_{i}}
for all
1
≤
i
,
j
≤
k
{\displaystyle 1\leq i,j\leq k}
, in the space
C
n
×
P
k
−
1
{\displaystyle \mathbf {C} ^{n}\times \mathbf {P} ^{k-1}}
.
More generally still, one can blow up any submanifold of any complex manifold
X
{\displaystyle X}
by applying this construction locally. The effect is, as before, to replace the blow-up locus
Z
{\displaystyle Z}
with the exceptional divisor
E
{\displaystyle E}
. In other words, the blow-up map
π
:
X
~
→
X
{\displaystyle \pi :{\tilde {X}}\to X}
is a birational mapping which, away from
E
{\displaystyle E}
, induces an isomorphism, and, on
E
{\displaystyle E}
, a locally trivial fibration with fiber
P
k
−
1
{\displaystyle \mathbf {P} ^{k-1}}
. Indeed, the restriction
π
|
E
:
E
→
Z
{\displaystyle \pi |_{E}:E\to Z}
is naturally seen as the projectivization of the normal bundle of
Z
{\displaystyle Z}
in
X
{\displaystyle X}
.
Since
E
{\displaystyle E}
is a smooth divisor (which has co-dim 1), its normal bundle is a line bundle. It is not difficult to show that
E
{\displaystyle E}
intersects itself negatively. This means that its normal bundle possesses no holomorphic sections;
E
{\displaystyle E}
is the only smooth complex representative of its homology class in
X
~
{\displaystyle {\tilde {X}}}
. (Suppose
E
{\displaystyle E}
could be perturbed off itself to another complex submanifold in the same class. Then the two submanifolds would intersect positively — as complex submanifolds always do — contradicting the negative self-intersection of
E
{\displaystyle E}
.) This is why the divisor is called exceptional.
Let
V
{\displaystyle V}
be some submanifold of
X
{\displaystyle X}
other than
Z
{\displaystyle Z}
. If
V
{\displaystyle V}
is disjoint from
Z
{\displaystyle Z}
, then it is essentially unaffected by blowing up along
Z
{\displaystyle Z}
. However, if it intersects
Z
{\displaystyle Z}
, then there are two distinct analogues of
V
{\displaystyle V}
in the blow-up
X
~
{\displaystyle {\tilde {X}}}
. One is the proper (or strict) transform, which is the closure of
π
−
1
(
V
∖
Z
)
{\displaystyle \pi ^{-1}(V\setminus Z)}
; its normal bundle in
X
~
{\displaystyle {\tilde {X}}}
is typically different from that of
V
{\displaystyle V}
in
X
{\displaystyle X}
. The other is the total transform, which incorporates some or all of
E
{\displaystyle E}
; it is essentially the pullback of
V
{\displaystyle V}
in cohomology.
== Blowing up schemes ==
To pursue blow-up in its greatest generality, let X be a scheme, and let
I
{\displaystyle {\mathcal {I}}}
be a coherent sheaf of ideals on X. The blow-up of X with respect to
I
{\displaystyle {\mathcal {I}}}
is a scheme
X
~
{\displaystyle {\tilde {X}}}
along with a morphism
π
:
X
~
→
X
{\displaystyle \pi \colon {\tilde {X}}\rightarrow X}
such that the pullback
π
−
1
I
⋅
O
X
~
{\displaystyle \pi ^{-1}{\mathcal {I}}\cdot {\mathcal {O}}_{\tilde {X}}}
is an invertible sheaf, characterized by this universal property: for any morphism f: Y → X such that
f
−
1
I
⋅
O
Y
{\displaystyle f^{-1}{\mathcal {I}}\cdot {\mathcal {O}}_{Y}}
is an invertible sheaf, f factors uniquely through π.
Notice that
X
~
=
P
r
o
j
⨁
n
=
0
∞
I
n
{\displaystyle {\tilde {X}}=\mathbf {Proj} \bigoplus _{n=0}^{\infty }{\mathcal {I}}^{n}}
has this property; this is how the blow-up is constructed (see also Rees algebra). Here Proj is the Proj construction on graded sheaves of commutative rings.
=== Exceptional divisors ===
The exceptional divisor of a blowup
π
:
Bl
I
X
→
X
{\displaystyle \pi :\operatorname {Bl} _{\mathcal {I}}X\to X}
is the subscheme defined by the inverse image of the ideal sheaf
I
{\displaystyle {\mathcal {I}}}
, which is sometimes denoted
π
−
1
I
⋅
O
Bl
I
X
{\displaystyle \pi ^{-1}{\mathcal {I}}\cdot {\mathcal {O}}_{\operatorname {Bl} _{\mathcal {I}}X}}
. It follows from the definition of the blow up in terms of Proj that this subscheme E is defined by the ideal sheaf
⨁
n
=
0
∞
I
n
+
1
{\displaystyle \textstyle \bigoplus _{n=0}^{\infty }{\mathcal {I}}^{n+1}}
. This ideal sheaf is also the relative
O
(
1
)
{\displaystyle {\mathcal {O}}(1)}
for π.
π is an isomorphism away from the exceptional divisor, but the exceptional divisor need not be in the exceptional locus of π. That is, π may be an isomorphism on E. This happens, for example, in the trivial situation where
I
{\displaystyle {\mathcal {I}}}
is already an invertible sheaf. In particular, in such cases the morphism π does not determine the exceptional divisor. Another situation where the exceptional locus can be strictly smaller than the exceptional divisor is when X has singularities. For instance, let X be the affine cone over P1 × P1. X can be given as the vanishing locus of xw − yz in A4. The ideals (x, y) and (x, z) define two planes, each of which passes through the vertex of X. Away from the vertex, these planes are hypersurfaces in X, so the blowup is an isomorphism there. The exceptional locus of the blowup of either of these planes is therefore centered over the vertex of the cone, and consequently it is strictly smaller than the exceptional divisor.
== Further examples ==
=== Blowups of linear subspaces ===
Let
P
n
{\displaystyle \mathbf {P} ^{n}}
be n-dimensional projective space. Fix a linear subspace L of codimension d. There are several explicit ways to describe the blowup of
P
n
{\displaystyle \mathbf {P} ^{n}}
along L. Suppose that
P
n
{\displaystyle \mathbf {P} ^{n}}
has coordinates
X
0
,
…
,
X
n
{\displaystyle X_{0},\dots ,X_{n}}
. After changing coordinates, we may assume that
L
=
{
X
n
−
d
+
1
=
⋯
=
X
n
=
0
}
{\displaystyle L=\{X_{n-d+1}=\dots =X_{n}=0\}}
. The blowup may be embedded in
P
n
×
P
n
−
d
{\displaystyle \mathbf {P} ^{n}\times \mathbf {P} ^{n-d}}
. Let
Y
0
,
…
,
Y
n
−
d
{\displaystyle Y_{0},\dots ,Y_{n-d}}
be coordinates on the second factor. Because L is defined by a regular sequence, the blowup is determined by the vanishing of the two-by-two minors of the matrix
(
X
0
⋯
X
n
−
d
Y
0
⋯
Y
n
−
d
)
.
{\displaystyle {\begin{pmatrix}X_{0}&\cdots &X_{n-d}\\Y_{0}&\cdots &Y_{n-d}\end{pmatrix}}.}
This system of equations is equivalent to asserting that the two rows are linearly dependent. A point
P
∈
P
n
{\displaystyle P\in \mathbf {P} ^{n}}
is in L if and only if, when its coordinates are substituted in the first row of the matrix above, that row is zero. In this case, there are no conditions on Q. If, however, that row is non-zero, then linear dependence implies that the second row is a scalar multiple of the first and therefore that there is a unique point
Q
∈
P
n
−
d
{\displaystyle Q\in \mathbf {P} ^{n-d}}
such that
(
P
,
Q
)
{\displaystyle (P,Q)}
is in the blowup.
This blowup can also be given a synthetic description as the incidence correspondence
{
(
P
,
M
)
:
P
∈
M
,
L
⊆
M
}
⊆
P
n
×
Gr
(
n
,
n
−
d
+
1
)
,
{\displaystyle \{(P,M)\colon P\in M,\,L\subseteq M\}\subseteq \mathbf {P} ^{n}\times \operatorname {Gr} (n,n-d+1),}
where
Gr
{\displaystyle \operatorname {Gr} }
denotes the Grassmannian of
(
n
−
d
+
1
)
{\displaystyle (n-d+1)}
-dimensional subspaces in
P
n
{\displaystyle \mathbf {P} ^{n}}
. To see the relation with the previous coordinatization, observe that the set of all
M
∈
Gr
(
n
,
n
−
d
+
1
)
{\displaystyle M\in \operatorname {Gr} (n,n-d+1)}
that contain L is isomorphic to a projective space
P
n
−
d
{\displaystyle \mathbf {P} ^{n-d}}
. This is because each subspace M is the linear join of L and a point Q not in L, and two points Q and Q' determine the same M if and only if they have the same image under the projection of
P
n
{\displaystyle \mathbf {P} ^{n}}
away from L. Therefore, the Grassmannian may be replaced by a copy of
P
n
−
d
{\displaystyle \mathbf {P} ^{n-d}}
. When
P
∉
L
{\displaystyle P\not \in L}
, there is only one subspace M containing P, the linear join of P and L. In the coordinates above, this is the case where
(
X
0
,
…
,
X
n
−
d
)
{\displaystyle (X_{0},\dots ,X_{n-d})}
is not the zero vector. The case
P
∈
L
{\displaystyle P\in L}
corresponds to
(
X
0
,
…
,
X
n
−
d
)
{\displaystyle (X_{0},\dots ,X_{n-d})}
being the zero vector, and in this case, any Q is allowed, that is, any M containing L is possible.
=== Blowing up intersections of curves scheme-theoretically ===
Let
f
,
g
∈
C
[
x
,
y
,
z
]
{\displaystyle f,g\in \mathbb {C} [x,y,z]}
be generic homogeneous polynomials of degree
d
{\displaystyle d}
(meaning their associated projective varieties intersects at
d
2
{\displaystyle d^{2}}
points by Bézout's theorem). The following projective morphism of schemes gives a model of blowing up
P
2
{\displaystyle \mathbb {P} ^{2}}
at
d
2
{\displaystyle d^{2}}
points:
Proj
(
C
[
s
,
t
]
[
x
,
y
,
z
]
(
s
f
(
x
,
y
,
z
)
+
t
g
(
x
,
y
,
z
)
)
)
↓
Proj
(
C
[
x
,
y
,
z
]
)
{\displaystyle {\begin{matrix}{\textbf {Proj}}\left({\dfrac {\mathbb {C} [s,t][x,y,z]}{(sf(x,y,z)+tg(x,y,z))}}\right)\\\downarrow \\{\textbf {Proj}}(\mathbb {C} [x,y,z])\end{matrix}}}
Looking at the fibers explains why this is true: if we take a point
p
=
[
x
0
:
x
1
:
x
2
]
{\displaystyle p=[x_{0}:x_{1}:x_{2}]}
then the pullback diagram
Proj
(
C
[
s
,
t
]
s
f
(
p
)
+
t
g
(
p
)
)
→
Proj
(
C
[
s
,
t
]
[
x
,
y
,
z
]
(
s
f
(
x
,
y
,
z
)
+
t
g
(
x
,
y
,
z
)
)
)
↓
↓
Spec
(
C
)
→
[
x
0
:
x
1
:
x
2
]
Proj
(
C
[
x
,
y
,
z
]
)
{\displaystyle {\begin{matrix}{\textbf {Proj}}\left({\dfrac {\mathbb {C} [s,t]}{sf(p)+tg(p)}}\right)&\rightarrow &{\textbf {Proj}}\left({\dfrac {\mathbb {C} [s,t][x,y,z]}{(sf(x,y,z)+tg(x,y,z))}}\right)\\\downarrow &&\downarrow \\{\textbf {Spec}}(\mathbb {C} )&{\xrightarrow {[x_{0}:x_{1}:x_{2}]}}&{\textbf {Proj}}(\mathbb {C} [x,y,z])\end{matrix}}}
tells us the fiber is a point whenever
f
(
p
)
≠
0
{\displaystyle f(p)\neq 0}
or
g
(
p
)
≠
0
{\displaystyle g(p)\neq 0}
and the fiber is
P
1
{\displaystyle \mathbb {P} ^{1}}
if
f
(
p
)
=
g
(
p
)
=
0
{\displaystyle f(p)=g(p)=0}
.
== Related constructions ==
In the blow-up of Cn described above, there was nothing essential about the use of complex numbers; blow-ups can be performed over any field. For example, the real blow-up of R2 at the origin results in the Möbius strip; correspondingly, the blow-up of the two-sphere S2 results in the real projective plane.
Deformation to the normal cone is a blow-up technique used to prove many results in algebraic geometry. Given a scheme X and a closed subscheme V, one blows up
V
×
{
0
}
in
Y
=
X
×
C
or
X
×
P
1
{\displaystyle V\times \{0\}\ {\text{in}}\ Y=X\times \mathbf {C} \ {\text{or}}\ X\times \mathbf {P} ^{1}}
Then
Y
~
→
C
{\displaystyle {\tilde {Y}}\to \mathbf {C} }
is a fibration. The general fiber is naturally isomorphic to X, while the central fiber is a union of two schemes: one is the blow-up of X along V, and the other is the normal cone of V with its fibers completed to projective spaces.
Blow-ups can also be performed in the symplectic category, by endowing the symplectic manifold with a compatible almost complex structure and proceeding with a complex blow-up. This makes sense on a purely topological level; however, endowing the blow-up with a symplectic form requires some care, because one cannot arbitrarily extend the symplectic form across the exceptional divisor E. One must alter the symplectic form in a neighborhood of E, or perform the blow-up by cutting out a neighborhood of Z and collapsing the boundary in a well-defined way. This is best understood using the formalism of symplectic cutting, of which symplectic blow-up is a special case. Symplectic cutting, together with the inverse operation of symplectic summation, is the symplectic analogue of deformation to the normal cone along a smooth divisor.
== See also ==
Infinitely near point
Resolution of singularities
== References ==
Fulton, William (1998). Intersection Theory. Springer-Verlag. ISBN 0-387-98549-2.
Griffiths, Phillip; Harris, Joseph (1978). Principles of Algebraic Geometry. John Wiley & Sons. ISBN 0-471-32792-1.
Hartshorne, Robin (1977). Algebraic Geometry. Springer-Verlag. ISBN 0-387-90244-9.
McDuff, Dusa; Salamon, Dietmar (1998). Introduction to Symplectic Topology. Oxford University Press. ISBN 0-19-850451-9. | Wikipedia/Monoidal_transformation |
In mathematics, the Enriques–Kodaira classification groups compact complex surfaces into ten classes, each parametrized by a moduli space. For most of the classes the moduli spaces are well understood, but for the class of surfaces of general type the moduli spaces seem too complicated to describe explicitly, though some components are known.
Max Noether began the systematic study of algebraic surfaces, and Guido Castelnuovo proved important parts of the classification. Federigo Enriques (1914, 1949) described the classification of complex projective surfaces. Kunihiko Kodaira (1964, 1966, 1968a, 1968b) later extended the classification to include non-algebraic compact surfaces. The analogous classification of surfaces in positive characteristic was begun by David Mumford (1969) and completed by Enrico Bombieri and David Mumford (1976, 1977); it is similar to the characteristic 0 projective case, except that one also gets singular and supersingular Enriques surfaces in characteristic 2, and quasi-hyperelliptic surfaces in characteristics 2 and 3.
== Statement of the classification ==
The Enriques–Kodaira classification of compact complex surfaces states that every nonsingular minimal compact complex surface is of exactly one of the 10 types listed on this page; in other words, it is one of the rational, ruled (genus > 0), type VII, K3, Enriques, Kodaira, toric, hyperelliptic, properly quasi-elliptic, or general type surfaces.
For the 9 classes of surfaces other than general type, there is a fairly complete description of what all the surfaces look like (which for class VII depends on the global spherical shell conjecture, still unproved in 2024). For surfaces of general type not much is known about their explicit classification, though many examples have been found.
The classification of algebraic surfaces in positive characteristic (Mumford 1969, Mumford & Bombieri 1976, 1977) is similar to that of algebraic surfaces in characteristic 0, except that there are no Kodaira surfaces or surfaces of type VII, and there are some extra families of Enriques surfaces in characteristic 2, and hyperelliptic surfaces in characteristics 2 and 3, and in Kodaira dimension 1 in characteristics 2 and 3 one also allows quasielliptic fibrations. These extra families can be understood as follows: In characteristic 0 these surfaces are the quotients of surfaces by finite groups, but in finite characteristics it is also possible to take quotients by finite group schemes that are not étale.
Oscar Zariski constructed some surfaces in positive characteristic that are unirational but not rational, derived from inseparable extensions (Zariski surfaces). In positive characteristic Serre showed that
h
0
(
Ω
)
{\displaystyle h^{0}(\Omega )}
may differ from
h
1
(
O
)
{\displaystyle h^{1}({\mathcal {O}})}
, and Igusa showed that even when they are equal they may be greater than the irregularity (the dimension of the Picard variety).
== Invariants of surfaces ==
=== Hodge numbers and Kodaira dimension ===
The most important invariants of a compact complex surfaces used in the classification can be given in terms of the dimensions of various coherent sheaf cohomology groups. The basic ones are the plurigenera and the Hodge numbers defined as follows:
K is the canonical line bundle whose sections are the holomorphic 2-forms.
P
n
=
dim
H
0
(
K
n
)
,
n
⩾
1
{\displaystyle P_{n}=\dim H^{0}(K^{n}),n\geqslant 1}
are called the plurigenera. They are birational invariants, i.e., invariant under blowing up. Using Seiberg–Witten theory, Robert Friedman and John Morgan showed that for complex manifolds they only depend on the underlying oriented smooth 4-manifold. For non-Kähler surfaces the plurigenera are determined by the fundamental group, but for Kähler surfaces there are examples of surfaces that are homeomorphic but have different plurigenera and Kodaira dimensions. The individual plurigenera are not often used; the most important thing about them is their growth rate, measured by the Kodaira dimension.
κ
{\displaystyle \kappa }
is the Kodaira dimension: it is
−
∞
{\displaystyle -\infty }
(sometimes written −1) if the plurigenera are all 0, and is otherwise the smallest number (0, 1, or 2 for surfaces) such that
P
n
/
n
κ
{\displaystyle P_{n}/n^{\kappa }}
is bounded. Enriques did not use this definition: instead he used the values of
P
12
{\displaystyle P_{12}}
and
K
⋅
K
=
c
1
2
{\displaystyle K\cdot K=c_{1}^{2}}
. These determine the Kodaira dimension given the following correspondence:
κ
=
−
∞
⟷
P
12
=
0
κ
=
0
⟷
P
12
=
1
κ
=
1
⟷
P
12
>
1
and
K
⋅
K
=
0
κ
=
2
⟷
P
12
>
1
and
K
⋅
K
>
0
{\displaystyle {\begin{aligned}\kappa =-\infty &\longleftrightarrow P_{12}=0\\\kappa =0&\longleftrightarrow P_{12}=1\\\kappa =1&\longleftrightarrow P_{12}>1{\text{ and }}K\cdot K=0\\\kappa =2&\longleftrightarrow P_{12}>1{\text{ and }}K\cdot K>0\\\end{aligned}}}
h
i
,
j
=
dim
H
j
(
X
,
Ω
i
)
,
{\displaystyle h^{i,j}=\dim H^{j}(X,\Omega ^{i}),}
where
Ω
i
{\displaystyle \Omega ^{i}}
is the sheaf of holomorphic i-forms, are the Hodge numbers, often arranged in the Hodge diamond:
h
0
,
0
h
1
,
0
h
0
,
1
h
2
,
0
h
1
,
1
h
0
,
2
h
2
,
1
h
1
,
2
h
2
,
2
{\displaystyle {\begin{matrix}&&h^{0,0}&&\\&h^{1,0}&&h^{0,1}&\\h^{2,0}&&h^{1,1}&&h^{0,2}\\&h^{2,1}&&h^{1,2}&\\&&h^{2,2}&&\\\end{matrix}}}
By Serre duality
h
i
,
j
=
h
2
−
i
,
2
−
j
{\displaystyle h^{i,j}=h^{2-i,2-j}}
and
h
0
,
0
=
h
2
,
2
=
1.
{\displaystyle h^{0,0}=h^{2,2}=1.}
The Hodge numbers of a complex surface depend only on the oriented real cohomology ring of the surface, and are invariant under birational transformations except for
h
1
,
1
{\displaystyle h^{1,1}}
which increases by 1 under blowing up a single point.
If the surface is Kähler then
h
i
,
j
=
h
j
,
i
{\displaystyle h^{i,j}=h^{j,i}}
and there are only three independent Hodge numbers.
If the surface is compact then
h
1
,
0
{\displaystyle h^{1,0}}
equals
h
0
,
1
{\displaystyle h^{0,1}}
or
h
0
,
1
−
1.
{\displaystyle h^{0,1}-1.}
=== Invariants related to Hodge numbers ===
There are many invariants that (at least for complex surfaces) can be written as linear combinations of the Hodge numbers, as follows:
Betti numbers: defined by
b
i
=
dim
H
i
(
S
)
,
0
⩽
i
⩽
4.
{\displaystyle b_{i}=\dim H^{i}(S),0\leqslant i\leqslant 4.}
{
b
0
=
b
4
=
1
b
1
=
b
3
=
h
1
,
0
+
h
0
,
1
=
h
2
,
1
+
h
1
,
2
b
2
=
h
2
,
0
+
h
1
,
1
+
h
0
,
2
{\displaystyle {\begin{cases}b_{0}=b_{4}=1\\b_{1}=b_{3}=h^{1,0}+h^{0,1}=h^{2,1}+h^{1,2}\\b_{2}=h^{2,0}+h^{1,1}+h^{0,2}\end{cases}}}
In characteristic p > 0 the Betti numbers are defined using l-adic cohomology and need not satisfy these relations.
Euler characteristic or Euler number:
e
=
b
0
−
b
1
+
b
2
−
b
3
+
b
4
.
{\displaystyle e=b_{0}-b_{1}+b_{2}-b_{3}+b_{4}.}
The irregularity is defined as the dimension of the Picard variety and the Albanese variety and denoted by q. For complex surfaces (but not always for surfaces of prime characteristic)
q
=
h
0
,
1
.
{\displaystyle q=h^{0,1}.}
The geometric genus:
p
g
=
h
0
,
2
=
h
2
,
0
=
P
1
.
{\displaystyle p_{g}=h^{0,2}=h^{2,0}=P_{1}.}
The arithmetic genus:
p
a
=
p
g
−
q
=
h
0
,
2
−
h
0
,
1
.
{\displaystyle p_{a}=p_{g}-q=h^{0,2}-h^{0,1}.}
The holomorphic Euler characteristic of the trivial bundle (usually differs from the Euler number e defined above):
χ
=
p
g
−
q
+
1
=
h
0
,
2
−
h
0
,
1
+
1.
{\displaystyle \chi =p_{g}-q+1=h^{0,2}-h^{0,1}+1.}
By Noether's formula it is also equal to the Todd genus
1
12
(
c
1
2
+
c
2
)
.
{\displaystyle {\tfrac {1}{12}}(c_{1}^{2}+c_{2}).}
The signature of the second cohomology group for complex surfaces is denoted by
τ
{\displaystyle \tau }
:
τ
=
4
χ
−
e
=
∑
i
,
j
(
−
1
)
j
h
i
,
j
.
{\displaystyle \tau =4\chi -e=\sum \nolimits _{i,j}(-1)^{j}h^{i,j}.}
b
±
{\displaystyle b^{\pm }}
are the dimensions of the maximal positive and negative definite subspaces of
H
2
,
{\displaystyle H^{2},}
so:
{
b
+
+
b
−
=
b
2
b
+
−
b
−
=
τ
{\displaystyle {\begin{cases}b^{+}+b^{-}=b_{2}\\b^{+}-b^{-}=\tau \end{cases}}}
c2 = e and
c
1
2
=
K
2
=
12
χ
−
e
{\displaystyle c_{1}^{2}=K^{2}=12\chi -e}
are the Chern numbers, defined as the integrals of various polynomials in the Chern classes over the manifold.
=== Other invariants ===
There are further invariants of compact complex surfaces that are not used so much in the classification. These include algebraic invariants such as the Picard group Pic(X) of divisors modulo linear equivalence, its quotient the Néron–Severi group NS(X) with rank the Picard number ρ, topological invariants such as the fundamental group π1 and the integral homology and cohomology groups, and invariants of the underlying smooth 4-manifold such as the Seiberg–Witten invariants and Donaldson invariants.
== Minimal models and blowing up ==
Any surface is birational to a non-singular surface, so for most purposes it is enough to classify the non-singular surfaces.
Given any point on a surface, we can form a new surface by blowing up this point, which means roughly that we replace it by a copy of the projective line. For the purpose of this article, a non-singular surface X is called minimal if it cannot be obtained from another non-singular surface by blowing up a point. By Castelnuovo's contraction theorem, this is equivalent to saying that X has no (−1)-curves (smooth rational curves with self-intersection number −1). (In the more modern terminology of the minimal model program, a smooth projective surface X would be called minimal if its canonical line bundle KX is nef. A smooth projective surface has a minimal model in that stronger sense if and only if its Kodaira dimension is nonnegative.)
Every surface X is birational to a minimal non-singular surface, and this minimal non-singular surface is unique if X has Kodaira dimension at least 0 or is not algebraic. Algebraic surfaces of Kodaira dimension
−
∞
{\displaystyle -\infty }
may be birational to more than one minimal non-singular surface, but it is easy to describe the relation between these minimal surfaces. For example, P1 × P1 blown up at a point is isomorphic to P2 blown up twice. So to classify all compact complex surfaces up to birational isomorphism it is (more or less) enough to classify the minimal non-singular ones.
== Surfaces of Kodaira dimension −∞ ==
Algebraic surfaces of Kodaira dimension
−
∞
{\displaystyle -\infty }
can be classified as follows. If q > 0 then the map to the Albanese variety has fibers that are projective lines (if the surface is minimal) so the surface is a ruled surface. If q = 0 this argument does not work as the Albanese variety is a point, but in this case Castelnuovo's theorem implies that the surface is rational.
For non-algebraic surfaces Kodaira found an extra class of surfaces, called type VII, which are still not well understood.
=== Rational surfaces ===
Rational surface means surface birational to the complex projective plane P2. These are all algebraic. The minimal rational surfaces are P2 itself and the Hirzebruch surfaces Σn for n = 0 or n ≥ 2. (The Hirzebruch surface Σn is the P1 bundle over P1 associated to the sheaf O(0) + O(n). The surface Σ0 is isomorphic to P1 × P1, and Σ1 is isomorphic to P2 blown up at a point so is not minimal.)
Invariants: The plurigenera are all 0 and the fundamental group is trivial.
Hodge diamond:
Examples: P2, P1 × P1 = Σ0, Hirzebruch surfaces Σn, quadrics, cubic surfaces, del Pezzo surfaces, Veronese surface. Many of these examples are non-minimal.
=== Ruled surfaces of genus > 0 ===
Ruled surfaces of genus g have a smooth morphism to a curve of genus g whose fibers are lines P1. They are all algebraic.
(The ones of genus 0 are the Hirzebruch surfaces and are rational.) Any ruled surface is birationally equivalent to P1 × C for a unique curve C, so the classification of ruled surfaces up to birational equivalence is essentially the same as the classification of curves. A ruled surface not isomorphic to P1 × P1 has a unique ruling (P1 × P1 has two).
Invariants: The plurigenera are all 0.
Hodge diamond:
Examples: The product of any curve of genus > 0 with P1.
=== Surfaces of class VII ===
These surfaces are never algebraic or Kähler. The minimal ones with b2 = 0 have been classified by Bogomolov, and are either Hopf surfaces or Inoue surfaces. Examples with positive second Betti number include Inoue-Hirzebruch surfaces, Enoki surfaces, and more generally Kato surfaces. The global spherical shell conjecture implies that all minimal class VII surfaces with positive second Betti number are Kato surfaces, which would more or less complete the classification of the type VII surfaces.
Invariants: q = 1, h1,0 = 0. All plurigenera are 0.
Hodge diamond:
== Surfaces of Kodaira dimension 0 ==
These surfaces are classified by starting with Noether's formula
12
χ
=
c
2
+
c
1
2
.
{\displaystyle 12\chi =c_{2}+c_{1}^{2}.}
For Kodaira dimension 0, K has zero intersection number with itself, so
c
1
2
=
0.
{\displaystyle c_{1}^{2}=0.}
Using
χ
=
h
0
,
0
−
h
0
,
1
+
h
0
,
2
c
2
=
2
−
2
b
1
+
b
2
{\displaystyle {\begin{aligned}\chi &=h^{0,0}-h^{0,1}+h^{0,2}\\c_{2}&=2-2b_{1}+b_{2}\end{aligned}}}
we arrive at:
10
+
12
h
0
,
2
=
8
h
0
,
1
+
2
(
2
h
0
,
1
−
b
1
)
+
b
2
{\displaystyle 10+12h^{0,2}=8h^{0,1}+2\left(2h^{0,1}-b_{1}\right)+b_{2}}
Moreover since κ = 0 we have:
h
0
,
2
=
{
1
K
=
0
0
otherwise
{\displaystyle h^{0,2}={\begin{cases}1&K=0\\0&{\text{otherwise}}\end{cases}}}
combining this with the previous equation gives:
8
h
0
,
1
+
2
(
2
h
0
,
1
−
b
1
)
+
b
2
=
{
22
K
=
0
10
otherwise
{\displaystyle 8h^{0,1}+2\left(2h^{0,1}-b_{1}\right)+b_{2}={\begin{cases}22&K=0\\10&{\text{otherwise}}\end{cases}}}
In general 2h0,1 ≥ b1, so three terms on the left are non-negative integers and there are only a few solutions to this equation.
For algebraic surfaces 2h0,1 − b1 is an even integer between 0 and 2pg.
For compact complex surfaces 2h0,1 − b1 = 0 or 1.
For Kähler surfaces 2h0,1 − b1 = 0 and h1,0 = h0,1.
Most solutions to these conditions correspond to classes of surfaces, as in the following table:
=== K3 surfaces ===
These are the minimal compact complex surfaces of Kodaira dimension 0 with q = 0 and trivial canonical line bundle. They are all Kähler manifolds. All K3 surfaces are diffeomorphic, and their diffeomorphism class is an important example of a smooth spin simply connected 4-manifold.
Invariants: The second cohomology group H2(X, Z) is isomorphic to the unique even unimodular lattice II3,19 of dimension 22 and signature −16.
Hodge diamond:
Examples:
Degree 4 hypersurfaces in P3(C)
Kummer surfaces. These are obtained by quotienting out an abelian surface by the automorphism a → −a, then blowing up the 16 singular points.
A marked K3 surface is a K3 surface together with an isomorphism from II3,19 to H2(X, Z). The moduli space of marked K3 surfaces is connected non-Hausdorff smooth analytic space of dimension 20. The algebraic K3 surfaces form a countable collection of 19-dimensional subvarieties of it.
=== Abelian surfaces and 2-dimensional complex tori ===
The two-dimensional complex tori include the abelian surfaces. One-dimensional complex tori are just elliptic curves and are all algebraic, but Riemann discovered that most complex tori of dimension 2 are not algebraic. The algebraic ones are exactly the 2-dimensional abelian varieties. Most of their theory is a special case of the theory of higher-dimensional tori or abelian varieties. Criteria to be a product of two elliptic curves (up to isogeny) were a popular study in the nineteenth century.
Invariants: The plurigenera are all 1. The surface is diffeomorphic to S1 × S1 × S1 × S1 so the fundamental group is Z4.
Hodge diamond:
Examples: A product of two elliptic curves. The Jacobian of a genus 2 curve. Any quotient of C2 by a lattice.
=== Kodaira surfaces ===
These are never algebraic, though they have non-constant meromorphic functions. They are usually divided into two subtypes: primary Kodaira surfaces with trivial canonical bundle, and secondary Kodaira surfaces which are quotients of these by finite groups of orders 2, 3, 4, or 6, and which have non-trivial canonical bundles. The secondary Kodaira surfaces have the same relation to primary ones that Enriques surfaces have to K3 surfaces, or bielliptic surfaces have to abelian surfaces.
Invariants: If the surface is the quotient of a primary Kodaira surface by a group of order k = 1, 2, 3, 4, 6, then the plurigenera Pn are 1 if n is divisible by k and 0 otherwise.
Hodge diamond:
Examples: Take a non-trivial line bundle over an elliptic curve, remove the zero section, then quotient out the fibers by Z acting as multiplication by powers of some complex number z. This gives a primary Kodaira surface.
=== Enriques surfaces ===
These are the complex surfaces such that q = 0 and the canonical line bundle is non-trivial, but has trivial square. Enriques surfaces are all algebraic (and therefore Kähler). They are quotients of K3 surfaces by a group of order 2 and their theory is similar to that of algebraic K3 surfaces.
Invariants: The plurigenera Pn are 1 if n is even and 0 if n is odd. The fundamental group has order 2. The second cohomology group H2(X, Z) is isomorphic to the sum of the unique even unimodular lattice II1,9 of dimension 10 and signature −8 and a group of order 2.
Hodge diamond:
Marked Enriques surfaces form a connected 10-dimensional family, which has been described explicitly.
In characteristic 2 there are some extra families of Enriques surfaces called singular and supersingular Enriques surfaces; see the article on Enriques surfaces for details.
=== Hyperelliptic (or bielliptic) surfaces ===
Over the complex numbers these are quotients of a product of two elliptic curves by a finite group of automorphisms. The finite group can be Z/2Z, Z/2Z + Z/2Z, Z/3Z, Z/3Z + Z/3Z, Z/4Z, Z/4Z + Z/2Z, or Z/6Z, giving seven families of such surfaces.
Hodge diamond:
Over fields of characteristics 2 or 3 there are some extra families given by taking quotients by a non-etale group scheme; see the article on hyperelliptic surfaces for details.
== Surfaces of Kodaira dimension 1 ==
An elliptic surface is a surface equipped with an elliptic fibration (a surjective holomorphic map to a curve B such that all but finitely many fibers are smooth irreducible curves of genus 1). The generic fiber in such a fibration is a genus 1 curve over the function field of B. Conversely, given a genus 1 curve over the function field of a curve, its relative minimal model is an elliptic surface. Kodaira and others have given a fairly complete description of all elliptic surfaces. In particular, Kodaira gave a complete list of the possible singular fibers. The theory of elliptic surfaces is analogous to the theory of proper regular models of elliptic curves over discrete valuation rings (e.g., the ring of p-adic integers) and Dedekind domains (e.g., the ring of integers of a number field).
In finite characteristic 2 and 3 one can also get quasi-elliptic surfaces, whose fibers may almost all be rational curves with a single node, which are "degenerate elliptic curves".
Every surface of Kodaira dimension 1 is an elliptic surface (or a quasielliptic surface in characteristics 2 or 3), but the converse is not true: an elliptic surface can have Kodaira dimension
−
∞
{\displaystyle -\infty }
, 0, or 1. All Enriques surfaces, all hyperelliptic surfaces, all Kodaira surfaces, some K3 surfaces, some abelian surfaces, and some rational surfaces are elliptic surfaces, and these examples have Kodaira dimension less than 1. An elliptic surface whose base curve B is of genus at least 2 always has Kodaira dimension 1, but the Kodaira dimension can be 1 also for some elliptic surfaces with B of genus 0 or 1.
Invariants:
c
1
2
=
0
,
c
2
⩾
0.
{\displaystyle c_{1}^{2}=0,c_{2}\geqslant 0.}
Example: If E is an elliptic curve and B is a curve of genus at least 2, then E×B is an elliptic surface of Kodaira dimension 1.
== Surfaces of Kodaira dimension 2 (surfaces of general type) ==
These are all algebraic, and in some sense most surfaces are in this class. Gieseker showed that there is a coarse moduli scheme for surfaces of general type; this means that for any fixed values of the Chern numbers c21 and c2, there is a quasi-projective scheme classifying the surfaces of general type with those Chern numbers. However it is a very difficult problem to describe these schemes explicitly, and there are very few pairs of Chern numbers for which this has been done (except when the scheme is empty!)
Invariants: There are several conditions that the Chern numbers of a minimal complex surface of general type must satisfy:
c
1
2
,
c
2
>
0
{\displaystyle c_{1}^{2},c_{2}>0}
c
1
2
⩽
3
c
2
{\displaystyle c_{1}^{2}\leqslant 3c_{2}}
(the Bogomolov–Miyaoka–Yau inequality)
5
c
1
2
−
c
2
+
36
⩾
0
{\displaystyle 5c_{1}^{2}-c_{2}+36\geqslant 0}
(the Noether inequality)
c
1
2
+
c
2
≡
0
mod
1
2.
{\displaystyle c_{1}^{2}+c_{2}\equiv 0{\bmod {1}}2.}
Most pairs of integers satisfying these conditions are the Chern numbers for some complex surface of general type.
Examples: The simplest examples are the product of two curves of genus at least 2, and a hypersurface of degree at least 5 in P3. There are a large number of other constructions known. However, there is no known construction that can produce "typical" surfaces of general type for large Chern numbers; in fact it is not even known if there is any reasonable concept of a "typical" surface of general type. There are many other examples that have been found, including most Hilbert modular surfaces, fake projective planes, Barlow surfaces, and so on.
== See also ==
List of algebraic surfaces
== References ==
Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004), Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 4, Springer-Verlag, Berlin, doi:10.1007/978-3-642-57739-0, ISBN 978-3-540-00832-3, MR 2030225 – the standard reference book for compact complex surfaces
Beauville, Arnaud (1996), Complex algebraic surfaces, London Mathematical Society Student Texts, vol. 34 (2nd ed.), Cambridge University Press, doi:10.1017/CBO9780511623936, ISBN 978-0-521-49510-3, MR 1406314; (ISBN 978-0-521-49842-5 softcover) – including a more elementary introduction to the classification
Bombieri, Enrico; Mumford, David (1977), "Enriques' classification of surfaces in char. p. II", Complex analysis and algebraic geometry, Tokyo: Iwanami Shoten, pp. 23–42, MR 0491719
Bombieri, E.; Mumford, D. (1977). "Enriques' Classification of Surfaces in Char. P, II". Complex Analysis and Algebraic Geometry. pp. 23–42. doi:10.1017/CBO9780511569197.004. ISBN 9780521217774.
Bombieri, Enrico; Mumford, David (1976), "Enriques' classification of surfaces in char. p. III." (PDF), Inventiones Mathematicae, 35: 197–232, Bibcode:1976InMat..35..197B, doi:10.1007/BF01390138, MR 0491720, S2CID 122816845
Enriques, Federigo (1914), "Sulla classificazione delle superficie algebriche e particolarmente sulle superficie di genere p1=1", Atti. Acc. Lincei V Ser., 23
Enriques, Federigo (1949), Le Superficie Algebriche, Nicola Zanichelli, Bologna, MR 0031770
Kodaira, Kunihiko (1964), "On the structure of compact complex analytic surfaces. I", American Journal of Mathematics, 86 (4): 751–798, doi:10.2307/2373157, JSTOR 2373157, MR 0187255
Kodaira, Kunihiko (1966), "On the structure of compact complex analytic surfaces. II", American Journal of Mathematics, 88 (3): 682–721, doi:10.2307/2373150, JSTOR 2373150, MR 0205280, PMC 300219, PMID 16578569
Kodaira, Kunihiko (1968a), "On the structure of compact complex analytic surfaces. III", American Journal of Mathematics, 90 (1): 55–83, doi:10.2307/2373426, JSTOR 2373426, MR 0228019
Kodaira, Kunihiko (1968b), "On the structure of complex analytic surfaces. IV", American Journal of Mathematics, 90 (4): 1048–1066, doi:10.2307/2373289, JSTOR 2373289, MR 0239114
Mumford, David (1969), "Enriques' classification of surfaces in char p I", Global Analysis (Papers in Honor of K. Kodaira), Tokyo: Univ. Tokyo Press, pp. 325–339, doi:10.1515/9781400871230-019, ISBN 978-1-4008-7123-0, JSTOR j.ctt13x10qw.21, MR 0254053
Reid, Miles (1997), "Chapters on algebraic surfaces", Complex algebraic geometry (Park City, UT, 1993), IAS/Park City Math. Ser., vol. 3, Providence, R.I.: American Mathematical Society, pp. 3–159, arXiv:alg-geom/9602006, Bibcode:1996alg.geom..2006R, doi:10.1090/pcms/003/02, ISBN 9780821811450, MR 1442522, S2CID 116933286
Shafarevich, Igor R.; Averbuh, Boris G.; Vaĭnberg, Ju. R.; Zhizhchenko, A. B.; Manin, Yuri I.; Moishezon, Boris G.; Tjurina, Galina N.; Tjurin, Andrei N. (1967) [1965], "Algebraic surfaces", Proceedings of the Steklov Institute of Mathematics, 75, Providence, R.I.: American Mathematical Society: 1–215, ISBN 978-0-8218-1875-6, MR 0190143
Van de Ven, Antonius (1978), "On the Enriques classification of algebraic surfaces", Séminaire Bourbaki, 29e année (1976/77), Lecture Notes in Math., vol. 677, Berlin, New York: Springer-Verlag, pp. 237–251, MR 0521772
Lang, William E. "Quasi-elliptic surfaces in characteristic three", Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 12 (1979) no. 4, pp. 473-500. doi : 10.24033/asens.1373. Theorem 4.3 of this article classifies the Hodge numbers of a quasi-hyperelliptic surface in characteristic three.
== External links ==
le superficie algebriche is an interactive visualisation of the Enriques--Kodaira classification, by Pieter Belmans and Johan Commelin | Wikipedia/Classification_of_algebraic_surfaces |
In mathematics, the Weil conjectures were highly influential proposals by André Weil (1949). They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory.
The conjectures concern the generating functions (known as local zeta functions) derived from counting points on algebraic varieties over finite fields. A variety V over a finite field with q elements has a finite number of rational points (with coordinates in the original field), as well as points with coordinates in any finite extension of the original field. The generating function has coefficients derived from the numbers Nk of points over the extension field with qk elements.
Weil conjectured that such zeta functions for smooth varieties are rational functions, satisfy a certain functional equation, and have their zeros in restricted places. The last two parts were consciously modelled on the Riemann zeta function, a kind of generating function for prime integers, which obeys a functional equation and (conjecturally) has its zeros restricted by the Riemann hypothesis. The rationality was proved by Bernard Dwork (1960), the functional equation by Alexander Grothendieck (1965), and the analogue of the Riemann hypothesis by Pierre Deligne (1974).
== Background and history ==
The earliest antecedent of the Weil conjectures is by Carl Friedrich Gauss and appears in section VII of his Disquisitiones Arithmeticae (Mazur 1974), concerned with roots of unity and Gaussian periods. In article 358, he moves on from the periods that build up towers of quadratic extensions, for the construction of regular polygons; and assumes that p is a prime number congruent to 1 modulo 3. Then there is a cyclic cubic field inside the cyclotomic field of pth roots of unity, and a normal integral basis of periods for the integers of this field (an instance of the Hilbert–Speiser theorem). Gauss constructs the order-3 periods, corresponding to the cyclic group (Z/pZ)× of non-zero residues modulo p under multiplication and its unique subgroup of index three. Gauss lets
R
{\displaystyle {\mathfrak {R}}}
,
R
′
{\displaystyle {\mathfrak {R}}'}
, and
R
″
{\displaystyle {\mathfrak {R}}''}
be its cosets. Taking the periods (sums of roots of unity) corresponding to these cosets applied to exp(2πi/p), he notes that these periods have a multiplication table that is accessible to calculation. Products are linear combinations of the periods, and he determines the coefficients. He sets, for example,
(
R
R
)
{\displaystyle ({\mathfrak {R}}{\mathfrak {R}})}
equal to the number of elements of Z/pZ which are in
R
{\displaystyle {\mathfrak {R}}}
and which, after being increased by one, are also in
R
{\displaystyle {\mathfrak {R}}}
. He proves that this number and related ones are the coefficients of the products of the periods. To see the relation of these sets to the Weil conjectures, notice that if α and α + 1 are both in
R
{\displaystyle {\mathfrak {R}}}
, then there exist x and y in Z/pZ such that x3 = α and y3 = α + 1; consequently, x3 + 1 = y3. Therefore
(
R
R
)
{\displaystyle ({\mathfrak {R}}{\mathfrak {R}})}
is related to the number of solutions to x3 + 1 = y3 in the finite field Z/pZ. The other coefficients have similar interpretations. Gauss's determination of the coefficients of the products of the periods therefore counts the number of points on these elliptic curves, and as a byproduct he proves the analog of the Riemann hypothesis.
The Weil conjectures in the special case of algebraic curves were conjectured by Emil Artin (1924). The case of curves over finite fields was proved by Weil, finishing the project started by Hasse's theorem on elliptic curves over finite fields. Their interest was obvious enough from within number theory: they implied upper bounds for exponential sums, a basic concern in analytic number theory (Moreno 2001).
What was really eye-catching, from the point of view of other mathematical areas, was the proposed connection with algebraic topology. Given that finite fields are discrete in nature, and topology speaks only about the continuous, the detailed formulation of Weil (based on working out some examples) was striking and novel. It suggested that geometry over finite fields should fit into well-known patterns relating to Betti numbers, the Lefschetz fixed-point theorem and so on.
The analogy with topology suggested that a new homological theory be set up applying within algebraic geometry. This took two decades (it was a central aim of the work and school of Alexander Grothendieck) building up on initial suggestions from Serre. The rationality part of the conjectures was proved first by Bernard Dwork (1960), using p-adic methods. Grothendieck (1965) and his collaborators established the rationality conjecture, the functional equation and the link to Betti numbers by using the properties of étale cohomology, a new cohomology theory developed by Grothendieck and Michael Artin for attacking the Weil conjectures, as outlined in Grothendieck (1960).
Of the four conjectures, the analogue of the Riemann hypothesis was the hardest to prove. Motivated by the proof of Serre (1960) of an analogue of the Weil conjectures for Kähler manifolds, Grothendieck envisioned a proof based on his standard conjectures on algebraic cycles (Kleiman 1968). However, Grothendieck's standard conjectures remain open (except for the hard Lefschetz theorem, which was proved by Deligne by extending his work on the Weil conjectures), and the analogue of the Riemann hypothesis was proved by Deligne (1974), using the étale cohomology theory but circumventing the use of standard conjectures by an ingenious argument.
Deligne (1980) found and proved a generalization of the Weil conjectures, bounding the weights of the pushforward of a sheaf.
== Statement of the Weil conjectures ==
Suppose that X is a non-singular n-dimensional projective algebraic variety over the field Fq with q elements. The zeta function ζ(X, s) of X is by definition
ζ
(
X
,
s
)
=
exp
(
∑
m
=
1
∞
N
m
m
q
−
m
s
)
{\displaystyle \zeta (X,s)=\exp \left(\sum _{m=1}^{\infty }{\frac {N_{m}}{m}}q^{-ms}\right)}
where Nm is the number of points of X defined over the degree m extension Fqm of Fq.
The Weil conjectures state:
1. (Rationality) ζ(X, s) is a rational function of T = q−s. More precisely, ζ(X, s) can be written as a finite alternating product
∏
i
=
0
2
n
P
i
(
q
−
s
)
(
−
1
)
i
+
1
=
P
1
(
T
)
⋯
P
2
n
−
1
(
T
)
P
0
(
T
)
⋯
P
2
n
(
T
)
,
{\displaystyle \prod _{i=0}^{2n}P_{i}(q^{-s})^{(-1)^{i+1}}={\frac {P_{1}(T)\dotsb P_{2n-1}(T)}{P_{0}(T)\dotsb P_{2n}(T)}},}
where each Pi(T) is an integral polynomial. Furthermore, P0(T) = 1 − T, P2n(T) = 1 − qnT, and for 1 ≤ i ≤ 2n − 1, Pi(T) factors over C as
∏
j
(
1
−
α
i
j
T
)
{\displaystyle \textstyle \prod _{j}(1-\alpha _{ij}T)}
for some numbers αij.
2. (Functional equation and Poincaré duality) The zeta function satisfies
ζ
(
X
,
n
−
s
)
=
±
q
n
E
/
2
−
E
s
ζ
(
X
,
s
)
{\displaystyle \zeta (X,n-s)=\pm q^{nE/2-Es}\zeta (X,s)}
or equivalently
ζ
(
X
,
q
−
n
T
−
1
)
=
±
q
n
E
/
2
T
E
ζ
(
X
,
T
)
{\displaystyle \zeta (X,q^{-n}T^{-1})=\pm q^{nE/2}T^{E}\zeta (X,T)}
where E is the Euler characteristic of X. In particular, for each i, the numbers α2n−i,1, α2n−i,2, ... equal the numbers qn/αi,1, qn/αi,2, ... in some order.
3. (Riemann hypothesis) |αi,j| = qi/2 for all 1 ≤ i ≤ 2n − 1 and all j. This implies that all zeros of Pk(T) lie on the "critical line" of complex numbers s with real part k/2.
4. (Betti numbers) If X is a (good) "reduction mod p" of a non-singular projective variety Y defined over a number field embedded in the field of complex numbers, then the degree of Pi is the ith Betti number of the space of complex points of Y.
== Examples ==
=== The projective line ===
The simplest example (other than a point) is to take X to be the projective line. The number of points of X over a field with qm elements is just Nm = qm + 1 (where the "+ 1" comes from the "point at infinity"). The zeta function is just
1
(
1
−
q
−
s
)
(
1
−
q
1
−
s
)
.
{\displaystyle {\frac {1}{(1-q^{-s})(1-q^{1-s})}}.}
It is easy to check all parts of the Weil conjectures directly. For example, the corresponding complex variety is the Riemann sphere and its initial Betti numbers are 1, 0, 1.
=== Projective space ===
It is not much harder to do n-dimensional projective space. The number of points of X over a field with qm elements is just Nm = 1 + qm + q2m + ⋯ + qnm. The zeta function is just
1
(
1
−
q
−
s
)
(
1
−
q
1
−
s
)
…
(
1
−
q
n
−
s
)
.
{\displaystyle {\frac {1}{(1-q^{-s})(1-q^{1-s})\dots (1-q^{n-s})}}.}
It is again easy to check all parts of the Weil conjectures directly. (Complex projective space gives the relevant Betti numbers, which nearly determine the answer.)
The number of points on the projective line and projective space are so easy to calculate because they can be written as disjoint unions of a finite number of copies of affine spaces. It is also easy to prove the Weil conjectures for other spaces, such as Grassmannians and flag varieties, which have the same "paving" property.
=== Elliptic curves ===
These give the first non-trivial cases of the Weil conjectures (proved by Hasse). If E is an elliptic curve over a finite field with q elements, then the number of points of E defined over the field with qm elements is 1 − αm − βm + qm, where α and β are complex conjugates with absolute value √q.
The zeta function is
(
1
−
α
q
−
s
)
(
1
−
β
q
−
s
)
(
1
−
q
−
s
)
(
1
−
q
1
−
s
)
.
{\displaystyle {\frac {(1-\alpha q^{-s})(1-\beta q^{-s})}{(1-q^{-s})(1-q^{1-s})}}.}
The Betti numbers are given by the torus, 1,2,1, and the numerator is a quadratic.
=== Hyperelliptic curves ===
As an example, consider the hyperelliptic curve
C
:
y
2
+
y
=
x
5
,
{\displaystyle C:y^{2}+y=x^{5},}
which is of genus
g
=
2
{\displaystyle g=2}
and dimension
n
=
1
{\displaystyle n=1}
. At first viewed as a curve
C
/
Q
{\displaystyle C/\mathbb {Q} }
defined over the rational numbers
Q
{\displaystyle \mathbb {Q} }
, this curve has good reduction at all primes
5
≠
q
∈
P
{\displaystyle 5\neq q\in \mathbb {P} }
. So, after reduction modulo
q
≠
5
{\displaystyle q\neq 5}
, one obtains a hyperelliptic curve
C
/
F
q
:
y
2
+
h
(
x
)
y
=
f
(
x
)
{\displaystyle C/{\bf {F}}_{q}:y^{2}+h(x)y=f(x)}
of genus 2, with
h
(
x
)
=
1
,
f
(
x
)
=
x
5
∈
F
q
[
x
]
{\displaystyle h(x)=1,f(x)=x^{5}\in {\bf {F}}_{q}[x]}
. Taking
q
=
41
{\displaystyle q=41}
as an example, the Weil polynomials
P
i
(
T
)
{\displaystyle P_{i}(T)}
,
i
=
0
,
1
,
2
,
{\displaystyle i=0,1,2,}
and the zeta function of
C
/
F
41
{\displaystyle C/{\bf {F}}_{41}}
assume the form
ζ
(
C
/
F
41
,
s
)
=
P
1
(
T
)
P
0
(
T
)
⋅
P
2
(
T
)
=
1
−
9
⋅
T
+
71
⋅
T
2
−
9
⋅
41
⋅
T
3
+
41
2
⋅
T
4
(
1
−
T
)
(
1
−
41
⋅
T
)
.
{\displaystyle \zeta (C/{\bf {F}}_{41},s)={\frac {P_{1}(T)}{P_{0}(T)\cdot P_{2}(T)}}={\frac {1-9\cdot T+71\cdot T^{2}-9\cdot 41\cdot T^{3}+41^{2}\cdot T^{4}}{(1-T)(1-41\cdot T)}}.}
The values
c
1
=
−
9
{\displaystyle c_{1}=-9}
and
c
2
=
71
{\displaystyle c_{2}=71}
can be determined by counting the numbers of solutions
(
x
,
y
)
{\displaystyle (x,y)}
of
y
2
+
y
=
x
5
{\displaystyle y^{2}+y=x^{5}}
over
F
41
{\displaystyle {\bf {F}}_{41}}
and
F
41
2
{\displaystyle {\bf {F}}_{41^{2}}}
, respectively, and adding 1 to each of these two numbers to allow for the point at infinity
∞
{\displaystyle \infty }
. This counting yields
N
1
=
33
{\displaystyle N_{1}=33}
and
N
2
=
1743
{\displaystyle N_{2}=1743}
. It follows:
c
1
=
N
1
−
1
−
q
=
33
−
1
−
41
=
−
9
{\displaystyle c_{1}=N_{1}-1-q=33-1-41=-9}
and
c
2
=
(
N
2
−
1
−
q
2
+
c
1
2
)
/
2
=
(
1743
−
1
−
41
2
+
(
−
9
)
2
)
/
2
=
71.
{\displaystyle c_{2}=(N_{2}-1-q^{2}+c_{1}^{2})/2=(1743-1-41^{2}+(-9)^{2})/2=71.}
The zeros of
P
1
(
T
)
{\displaystyle P_{1}(T)}
are
z
1
:=
0.12305
+
−
1
⋅
0.09617
{\displaystyle z_{1}:=0.12305+{\sqrt {-1}}\cdot 0.09617}
and
z
2
:=
−
0.01329
+
−
1
⋅
0.15560
{\displaystyle z_{2}:=-0.01329+{\sqrt {-1}}\cdot 0.15560}
(the decimal expansions of these real and imaginary parts are cut off after the fifth decimal place) together with their complex conjugates
z
3
:=
z
¯
1
{\displaystyle z_{3}:={\bar {z}}_{1}}
and
z
4
:=
z
¯
2
{\displaystyle z_{4}:={\bar {z}}_{2}}
. So, in the factorisation
P
1
(
T
)
=
∏
j
=
1
4
(
1
−
α
1
,
j
T
)
{\displaystyle P_{1}(T)=\prod _{j=1}^{4}(1-\alpha _{1,j}T)}
, we have
α
1
,
j
=
1
/
z
j
{\displaystyle \alpha _{1,j}=1/z_{j}}
. As stated in the third part (Riemann hypothesis) of the Weil conjectures,
|
α
1
,
j
|
=
41
{\displaystyle |\alpha _{1,j}|={\sqrt {41}}}
for
j
=
1
,
2
,
3
,
4
{\displaystyle j=1,2,3,4}
.
The non-singular, projective, complex manifold that belongs to
C
/
Q
{\displaystyle C/\mathbb {Q} }
has the Betti numbers
B
0
=
1
,
B
1
=
2
g
=
4
,
B
2
=
1
{\displaystyle B_{0}=1,B_{1}=2g=4,B_{2}=1}
. As described in part four of the Weil conjectures, the (topologically defined!) Betti numbers coincide with the degrees of the Weil polynomials
P
i
(
T
)
{\displaystyle P_{i}(T)}
, for all primes
q
≠
5
{\displaystyle q\neq 5}
:
d
e
g
(
P
i
)
=
B
i
,
i
=
0
,
1
,
2
{\displaystyle {\rm {deg}}(P_{i})=B_{i},\,i=0,1,2}
.
=== Abelian surfaces ===
An Abelian surface is a two-dimensional Abelian variety. This is, they are projective varieties that also have the structure of a group, in a way that is compatible with the group composition and taking inverses. Elliptic curves represent one-dimensional Abelian varieties. As an example of an Abelian surface defined over a finite field, consider the Jacobian variety
X
:=
Jac
(
C
/
F
41
)
{\displaystyle X:={\text{Jac}}(C/{\bf {F}}_{41})}
of the genus 2 curve
C
/
F
41
:
y
2
+
y
=
x
5
,
{\displaystyle C/{\bf {F}}_{41}:y^{2}+y=x^{5},}
which was introduced in the section on hyperelliptic curves. The dimension of
X
{\displaystyle X}
equals the genus of
C
{\displaystyle C}
, so
n
=
2
{\displaystyle n=2}
. There are algebraic integers
α
1
,
…
,
α
4
{\displaystyle \alpha _{1},\ldots ,\alpha _{4}}
such that
the polynomial
P
(
x
)
=
∏
j
=
1
4
(
x
−
α
j
)
{\displaystyle P(x)=\prod _{j=1}^{4}(x-\alpha _{j})}
has coefficients in
Z
{\displaystyle \mathbb {Z} }
;
M
m
:=
|
Jac
(
C
/
F
41
m
)
|
=
∏
j
=
1
4
(
1
−
α
j
m
)
{\displaystyle M_{m}:=|{\text{Jac}}(C/{\bf {F}}_{41^{m}})|=\prod _{j=1}^{4}(1-\alpha _{j}^{m})}
for all
m
∈
N
{\displaystyle m\in \mathbb {N} }
; and
|
α
j
|
=
41
{\displaystyle |\alpha _{j}|={\sqrt {41}}}
for
j
=
1
,
…
,
4
{\displaystyle j=1,\ldots ,4}
.
The zeta-function of
X
{\displaystyle X}
is given by
ζ
(
X
,
s
)
=
∏
i
=
0
4
P
i
(
q
−
s
)
(
−
1
)
i
+
1
=
P
1
(
T
)
⋅
P
3
(
T
)
P
0
(
T
)
⋅
P
2
(
T
)
⋅
P
4
(
T
)
,
{\displaystyle \zeta (X,s)=\prod _{i=0}^{4}P_{i}(q^{-s})^{(-1)^{i+1}}={\frac {P_{1}(T)\cdot P_{3}(T)}{P_{0}(T)\cdot P_{2}(T)\cdot P_{4}(T)}},}
where
q
=
41
{\displaystyle q=41}
,
T
=
q
−
s
=
d
e
f
exp
(
−
s
⋅
log
(
41
)
)
{\displaystyle T=q^{-s}\,{\stackrel {\rm {def}}{=}}\,{\text{exp}}(-s\cdot {\text{log}}(41))}
, and
s
{\displaystyle s}
represents the complex variable of the zeta-function. The Weil polynomials
P
i
(
T
)
{\displaystyle P_{i}(T)}
have the following specific form (Kahn 2020):
P
i
(
T
)
=
∏
1
≤
j
1
<
j
2
<
…
<
j
i
−
1
<
j
i
≤
4
(
1
−
α
j
1
⋅
…
⋅
α
j
i
T
)
{\displaystyle P_{i}(T)=\prod _{1\leq j_{1}<j_{2}<\ldots <j_{i-1}<j_{i}\leq 4}(1-\alpha _{j_{1}}\cdot \ldots \cdot \alpha _{j_{i}}T)}
for
i
=
0
,
1
,
…
,
4
{\displaystyle i=0,1,\ldots ,4}
, and
P
1
(
T
)
=
∏
j
=
1
4
(
1
−
α
j
T
)
=
1
−
9
⋅
T
+
71
⋅
T
2
−
9
⋅
41
⋅
T
3
+
41
2
⋅
T
4
{\displaystyle P_{1}(T)=\prod _{j=1}^{4}(1-\alpha _{j}T)=1-9\cdot T+71\cdot T^{2}-9\cdot 41\cdot T^{3}+41^{2}\cdot T^{4}}
is the same for the curve
C
{\displaystyle C}
(see section above) and its Jacobian variety
X
{\displaystyle X}
. This is, the inverse roots of
P
i
(
T
)
{\displaystyle P_{i}(T)}
are the products
α
j
1
⋅
…
⋅
α
j
i
{\displaystyle \alpha _{j_{1}}\cdot \ldots \cdot \alpha _{j_{i}}}
that consist of
i
{\displaystyle i}
many, different inverse roots of
P
1
(
T
)
{\displaystyle P_{1}(T)}
. Hence, all coefficients of the polynomials
P
i
(
T
)
{\displaystyle P_{i}(T)}
can be expressed as polynomial functions of the parameters
c
1
=
−
9
{\displaystyle c_{1}=-9}
,
c
2
=
71
{\displaystyle c_{2}=71}
and
q
=
41
{\displaystyle q=41}
appearing in
P
1
(
T
)
=
1
+
c
1
T
+
c
2
T
2
+
q
c
1
T
3
+
q
2
T
4
.
{\displaystyle P_{1}(T)=1+c_{1}T+c_{2}T^{2}+qc_{1}T^{3}+q^{2}T^{4}.}
Calculating these polynomial functions for the coefficients of the
P
i
(
T
)
{\displaystyle P_{i}(T)}
shows that
P
0
(
T
)
=
1
−
T
P
1
(
T
)
=
1
−
3
2
⋅
T
+
71
⋅
T
2
−
3
2
⋅
41
⋅
T
3
+
41
2
⋅
T
4
P
2
(
T
)
=
(
1
−
41
⋅
T
)
2
⋅
(
1
+
11
⋅
T
+
3
⋅
7
⋅
41
⋅
T
2
+
11
⋅
41
2
⋅
T
3
+
41
4
⋅
T
4
)
P
3
(
T
)
=
1
−
3
2
⋅
41
⋅
T
+
71
⋅
41
2
⋅
T
2
−
3
2
⋅
41
4
⋅
T
3
+
41
6
⋅
T
4
P
4
(
T
)
=
1
−
41
2
⋅
T
{\displaystyle {\begin{alignedat}{2}P_{0}(T)&=1-T\\P_{1}(T)&=1-3^{2}\cdot T+71\cdot T^{2}-3^{2}\cdot 41\cdot T^{3}+41^{2}\cdot T^{4}\\P_{2}(T)&=(1-41\cdot T)^{2}\cdot (1+11\cdot T+3\cdot 7\cdot 41\cdot T^{2}+11\cdot 41^{2}\cdot T^{3}+41^{4}\cdot T^{4})\\P_{3}(T)&=1-3^{2}\cdot 41\cdot T+71\cdot 41^{2}\cdot T^{2}-3^{2}\cdot 41^{4}\cdot T^{3}+41^{6}\cdot T^{4}\\P_{4}(T)&=1-41^{2}\cdot T\end{alignedat}}}
Polynomial
P
1
{\displaystyle P_{1}}
allows for calculating the numbers of elements of the Jacobian variety
Jac
(
C
)
{\displaystyle {\text{Jac}}(C)}
over the finite field
F
41
{\displaystyle {\bf {F}}_{41}}
and its field extension
F
41
2
{\displaystyle {\bf {F}}_{41^{2}}}
:
M
1
=
d
e
f
|
Jac
(
C
/
F
41
)
|
=
P
1
(
1
)
=
∏
j
=
1
4
[
1
−
α
j
T
]
T
=
1
=
[
1
−
9
⋅
T
+
71
⋅
T
2
−
9
⋅
41
⋅
T
3
+
41
2
⋅
T
4
]
T
=
1
=
1
−
9
+
71
−
9
⋅
41
+
41
2
=
1375
=
5
3
⋅
11
, and
M
2
=
d
e
f
|
Jac
(
C
/
F
41
2
)
|
=
∏
j
=
1
4
[
1
−
α
j
2
T
]
T
=
1
=
[
1
+
61
⋅
T
+
3
⋅
587
⋅
T
2
+
61
⋅
41
2
⋅
T
3
+
41
4
⋅
T
4
]
T
=
1
=
2930125
=
5
3
⋅
11
⋅
2131.
{\displaystyle {\begin{alignedat}{2}M_{1}&\;{\overset {\underset {\mathrm {def} }{}}{=}}\;|{\text{Jac}}(C/{\bf {F}}_{41})|=P_{1}(1)=\prod _{j=1}^{4}[1-\alpha _{j}T]_{T=1}\\&=[1-9\cdot T+71\cdot T^{2}-9\cdot 41\cdot T^{3}+41^{2}\cdot T^{4}]_{T=1}=1-9+71-9\cdot 41+41^{2}=1375=5^{3}\cdot 11{\text{, and}}\\M_{2}&\;{\overset {\underset {\mathrm {def} }{}}{=}}\;|{\text{Jac}}(C/{\bf {F}}_{41^{2}})|=\prod _{j=1}^{4}[1-\alpha _{j}^{2}T]_{T=1}\\&=[1+61\cdot T+3\cdot 587\cdot T^{2}+61\cdot 41^{2}\cdot T^{3}+41^{4}\cdot T^{4}]_{T=1}=2930125=5^{3}\cdot 11\cdot 2131.\end{alignedat}}}
The inverses
α
i
,
j
{\displaystyle \alpha _{i,j}}
of the zeros of
P
i
(
T
)
{\displaystyle P_{i}(T)}
do have the expected absolute value of
41
i
/
2
{\displaystyle 41^{i/2}}
(Riemann hypothesis). Moreover, the maps
α
i
,
j
⟼
41
2
/
α
i
,
j
,
{\displaystyle \alpha _{i,j}\longmapsto 41^{2}/\alpha _{i,j},}
j
=
1
,
…
,
deg
P
i
,
{\displaystyle j=1,\ldots ,\deg P_{i},}
correlate the inverses of the zeros of
P
i
(
T
)
{\displaystyle P_{i}(T)}
and the inverses of the zeros of
P
4
−
i
(
T
)
{\displaystyle P_{4-i}(T)}
. A non-singular, complex, projective, algebraic variety
Y
{\displaystyle Y}
with good reduction at the prime 41 to
X
=
Jac
(
C
/
F
41
)
{\displaystyle X={\text{Jac}}(C/{\bf {F}}_{41})}
must have Betti numbers
B
0
=
B
4
=
1
,
B
1
=
B
3
=
4
,
B
2
=
6
{\displaystyle B_{0}=B_{4}=1,B_{1}=B_{3}=4,B_{2}=6}
, since these are the degrees of the polynomials
P
i
(
T
)
.
{\displaystyle P_{i}(T).}
The Euler characteristic
E
{\displaystyle E}
of
X
{\displaystyle X}
is given by the alternating sum of these degrees/Betti numbers:
E
=
1
−
4
+
6
−
4
+
1
=
0
{\displaystyle E=1-4+6-4+1=0}
.
By taking the logarithm of
ζ
(
Jac
(
C
/
F
41
)
,
s
)
=
exp
(
∑
m
=
1
∞
M
m
m
(
41
−
s
)
m
)
=
∏
i
=
0
4
P
i
(
41
−
s
)
(
−
1
)
i
+
1
=
P
1
(
T
)
⋅
P
3
(
T
)
P
0
(
T
)
⋅
P
2
(
T
)
⋅
P
4
(
T
)
,
{\displaystyle \zeta ({\text{Jac}}(C/{\bf {F}}_{41}),s)\,=\,\exp \left(\sum _{m=1}^{\infty }{\frac {M_{m}}{m}}(41^{-s})^{m}\right)\,=\,\prod _{i=0}^{4}\,P_{i}(41^{-s})^{(-1)^{i+1}}={\frac {P_{1}(T)\cdot P_{3}(T)}{P_{0}(T)\cdot P_{2}(T)\cdot P_{4}(T)}},}
it follows that
∑
m
=
1
∞
M
m
m
(
41
−
s
)
m
=
log
(
P
1
(
T
)
⋅
P
3
(
T
)
P
0
(
T
)
⋅
P
2
(
T
)
⋅
P
4
(
T
)
)
=
1375
⋅
T
+
2930125
/
2
⋅
T
2
+
4755796375
/
3
⋅
T
3
+
7984359145125
/
4
⋅
T
4
+
13426146538750000
/
5
⋅
T
5
+
O
(
T
6
)
.
{\displaystyle {\begin{alignedat}{2}\sum _{m=1}^{\infty }&{\frac {M_{m}}{m}}(41^{-s})^{m}\,=\,\log \left({\frac {P_{1}(T)\cdot P_{3}(T)}{P_{0}(T)\cdot P_{2}(T)\cdot P_{4}(T)}}\right)\\&=1375\cdot T+2930125/2\cdot T^{2}+4755796375/3\cdot T^{3}+7984359145125/4\cdot T^{4}+13426146538750000/5\cdot T^{5}+O(T^{6}).\end{alignedat}}}
Aside from the values
M
1
{\displaystyle M_{1}}
and
M
2
{\displaystyle M_{2}}
already known, you can read off from this Taylor series all other numbers
M
m
{\displaystyle M_{m}}
,
m
∈
N
{\displaystyle m\in \mathbb {N} }
, of
F
41
m
{\displaystyle {\bf {F}}_{41^{m}}}
-rational elements of the Jacobian variety, defined over
F
41
{\displaystyle {\bf {F}}_{41}}
, of the curve
C
/
F
41
{\displaystyle C/{\bf {F}}_{41}}
: for instance,
M
3
=
4755796375
=
5
3
⋅
11
⋅
61
⋅
56701
{\displaystyle M_{3}=4755796375=5^{3}\cdot 11\cdot 61\cdot 56701}
and
M
4
=
7984359145125
=
3
4
⋅
5
3
⋅
11
⋅
2131
⋅
33641
{\displaystyle M_{4}=7984359145125=3^{4}\cdot 5^{3}\cdot 11\cdot 2131\cdot 33641}
. In doing so,
m
1
|
m
2
{\displaystyle m_{1}|m_{2}}
always implies
M
m
1
|
M
m
2
{\displaystyle M_{m_{1}}|M_{m_{2}}}
since then,
Jac
(
C
/
F
41
m
1
)
{\displaystyle {\text{Jac}}(C/{\bf {F}}_{41^{m_{1}}})}
is a subgroup of
Jac
(
C
/
F
41
m
2
)
{\displaystyle {\text{Jac}}(C/{\bf {F}}_{41^{m_{2}}})}
.
== Weil cohomology ==
Weil suggested that the conjectures would follow from the existence of a suitable "Weil cohomology theory" for varieties over finite fields, similar to the usual cohomology with rational coefficients for complex varieties. His idea was that if F is the Frobenius automorphism over the finite field, then the number of points of the variety X over the field of order qm is the number of fixed points of Fm (acting on all points of the variety X defined over the algebraic closure). In algebraic topology the number of fixed points of an automorphism can be worked out using the Lefschetz fixed-point theorem, given as an alternating sum of traces on the cohomology groups. So if there were similar cohomology groups for varieties over finite fields, then the zeta function could be expressed in terms of them.
The first problem with this is that the coefficient field for a Weil cohomology theory cannot be the rational numbers. To see this consider the case of a supersingular elliptic curve over a finite field of characteristic p. The endomorphism ring of this is an order in a quaternion algebra over the rationals, and should act on the first cohomology group, which should be a 2-dimensional vector space over the coefficient field by analogy with the case of a complex elliptic curve. However a quaternion algebra over the rationals cannot act on a 2-dimensional vector space over the rationals. The same argument eliminates the possibility of the coefficient field being the reals or the p-adic numbers, because the quaternion algebra is still a division algebra over these fields. However it does not eliminate the possibility that the coefficient field is the field of ℓ-adic numbers for some prime ℓ ≠ p, because over these fields the division algebra splits and becomes a matrix algebra, which can act on a 2-dimensional vector space. Grothendieck and Michael Artin managed to construct suitable cohomology theories over the field of ℓ-adic numbers for each prime ℓ ≠ p, called ℓ-adic cohomology.
== Grothendieck's proofs of three of the four conjectures ==
By the end of 1964 Grothendieck together with Artin and Jean-Louis Verdier (and the earlier 1960 work by Dwork) proved the Weil conjectures apart from the most difficult third conjecture above (the "Riemann hypothesis" conjecture) (Grothendieck 1965). The general theorems about étale cohomology allowed Grothendieck to prove an analogue of the Lefschetz fixed-point formula for the ℓ-adic cohomology theory, and by applying it to the Frobenius automorphism F he was able to prove the conjectured formula for the zeta function:
ζ
(
s
)
=
P
1
(
T
)
⋯
P
2
n
−
1
(
T
)
P
0
(
T
)
P
2
(
T
)
⋯
P
2
n
(
T
)
{\displaystyle \zeta (s)={\frac {P_{1}(T)\cdots P_{2n-1}(T)}{P_{0}(T)P_{2}(T)\cdots P_{2n}(T)}}}
where each polynomial Pi is the determinant of I − TF on the ℓ-adic cohomology group Hi.
The rationality of the zeta function follows immediately. The functional equation for the zeta function follows from Poincaré duality for ℓ-adic cohomology, and the relation with complex Betti numbers of a lift follows from a comparison theorem between ℓ-adic and ordinary cohomology for complex varieties.
More generally, Grothendieck proved a similar formula for the zeta function (or "generalized L-function") of a sheaf F0:
Z
(
X
0
,
F
0
,
t
)
=
∏
x
∈
|
X
0
|
det
(
1
−
F
x
∗
t
deg
(
x
)
∣
F
0
)
−
1
{\displaystyle Z(X_{0},F_{0},t)=\prod _{x\in |X_{0}|}\det(1-F_{x}^{*}t^{\deg(x)}\mid F_{0})^{-1}}
as a product over cohomology groups:
Z
(
X
0
,
F
0
,
t
)
=
∏
i
det
(
1
−
F
∗
t
∣
H
c
i
(
F
)
)
(
−
1
)
i
+
1
{\displaystyle Z(X_{0},F_{0},t)=\prod _{i}\det(1-F^{*}t\mid H_{c}^{i}(F))^{(-1)^{i+1}}}
The special case of the constant sheaf gives the usual zeta function.
== Deligne's first proof of the Riemann hypothesis conjecture ==
Verdier (1974), Serre (1975), Katz (1976) and Freitag & Kiehl (1988) gave expository accounts of the first proof of Deligne (1974). Much of the background in ℓ-adic cohomology is described in (Deligne 1977).
Deligne's first proof of the remaining third Weil conjecture (the "Riemann hypothesis conjecture") used the following steps:
=== Use of Lefschetz pencils ===
Grothendieck expressed the zeta function in terms of the trace of Frobenius on ℓ-adic cohomology groups, so the Weil conjectures for a d-dimensional variety V over a finite field with q elements depend on showing that the eigenvalues α of Frobenius acting on the ith ℓ-adic cohomology group Hi(V) of V have absolute values |α| = qi/2 (for an embedding of the algebraic elements of Qℓ into the complex numbers).
After blowing up V and extending the base field, one may assume that the variety V has a morphism onto the projective line P1, with a finite number of singular fibers with very mild (quadratic) singularities. The theory of monodromy of Lefschetz pencils, introduced for complex varieties (and ordinary cohomology) by Lefschetz (1924), and extended by Grothendieck (1972) and Deligne & Katz (1973) to ℓ-adic cohomology, relates the cohomology of V to that of its fibers. The relation depends on the space Ex of vanishing cycles, the subspace of the cohomology Hd−1(Vx) of a non-singular fiber Vx, spanned by classes that vanish on singular fibers.
The Leray spectral sequence relates the middle cohomology group of V to the cohomology of the fiber and base. The hard part to deal with is more or less a group H1(P1, j*E) = H1c(U,E), where U is the points the projective line with non-singular fibers, and j is the inclusion of U into the projective line, and E is the sheaf with fibers the spaces Ex of vanishing cycles.
=== The key estimate ===
The heart of Deligne's proof is to show that the sheaf E over U is pure, in other words to find the absolute values of the eigenvalues of Frobenius on its stalks. This is done by studying the zeta functions of the even powers Ek of E and applying Grothendieck's formula for the zeta functions as alternating products over cohomology groups. The crucial idea of considering even k powers of E was inspired by the paper Rankin (1939), who used a similar idea with k = 2 for bounding the Ramanujan tau function. Langlands (1970, section 8) pointed out that a generalization of Rankin's result for higher even values of k would imply the Ramanujan conjecture, and Deligne realized that in the case of zeta functions of varieties, Grothendieck's theory of zeta functions of sheaves provided an analogue of this generalization.
The poles of the zeta function of Ek are found using Grothendieck's formula
Z
(
U
,
E
k
,
T
)
=
det
(
1
−
F
∗
T
∣
H
c
1
(
E
k
)
)
det
(
1
−
F
∗
T
∣
H
c
0
(
E
k
)
)
det
(
1
−
F
∗
T
∣
H
c
2
(
E
k
)
)
{\displaystyle Z(U,E^{k},T)={\frac {\det(1-F^{*}T\mid H_{c}^{1}(E^{k}))}{\det(1-F^{*}T\mid H_{c}^{0}(E^{k}))\det(1-F^{*}T\mid H_{c}^{2}(E^{k}))}}}
and calculating the cohomology groups in the denominator explicitly. The H0c term is usually just 1 as U is usually not compact, and the H2c can be calculated explicitly as follows. Poincaré duality relates H2c(Ek) to H0(Ek), which is in turn the space of covariants of the monodromy group, which is the geometric fundamental group of U acting on the fiber of Ek at a point. The fiber of E has a bilinear form induced by cup product, which is antisymmetric if d is even, and makes E into a symplectic space. (This is a little inaccurate: Deligne did later show that E∩E⊥ = 0 by using the hard Lefschetz theorem, this requires the Weil conjectures, and the proof of the Weil conjectures really has to use a slightly more complicated argument with E/E∩E⊥ rather than E.) An argument of Kazhdan and Margulis shows that the image of the monodromy group acting on E, given by the Picard–Lefschetz formula, is Zariski dense in a symplectic group and therefore has the same invariants, which are well known from classical invariant theory. Keeping track of the action of Frobenius in this calculation shows that its eigenvalues are all qk(d−1)/2+1, so the zeta function of Z(Ek,T) has poles only at T = 1/qk(d−1)/2+1.
The Euler product for the zeta function of Ek is
Z
(
E
k
,
T
)
=
∏
x
1
Z
(
E
x
k
,
T
)
{\displaystyle Z(E^{k},T)=\prod _{x}{\frac {1}{Z(E_{x}^{k},T)}}}
If k is even then all the coefficients of the factors on the right (considered as power series in T) are non-negative; this follows by writing
1
det
(
1
−
T
deg
(
x
)
F
x
∣
E
k
)
=
exp
(
∑
n
>
0
T
n
n
Trace
(
F
x
n
∣
E
)
k
)
{\displaystyle {\frac {1}{\det(1-T^{\deg(x)}F_{x}\mid E^{k})}}=\exp \left(\sum _{n>0}{\frac {T^{n}}{n}}\operatorname {Trace} (F_{x}^{n}\mid E)^{k}\right)}
and using the fact that the traces of powers of F are rational, so their k powers are non-negative as k is even. Deligne proves the rationality of the traces by relating them to numbers of points of varieties, which are always (rational) integers.
The powers series for Z(Ek, T) converges for T less than the absolute value 1/qk(d−1)/2+1 of its only possible pole. When k is even the coefficients of all its Euler factors are non-negative, so that each of the Euler factors has coefficients bounded by a constant times the coefficients of Z(Ek, T) and therefore converges on the same region and has no poles in this region. So for k even the polynomials Z(Ekx, T) have no zeros in this region, or in other words the eigenvalues of Frobenius on the stalks of Ek have absolute value at most qk(d−1)/2+1.
This estimate can be used to find the absolute value of any eigenvalue α of Frobenius on a fiber of E as follows. For any integer k, αk is an eigenvalue of Frobenius on a stalk of Ek, which for k even is bounded by q1+k(d−1)/2. So
|
α
k
|
≤
q
k
(
d
−
1
)
/
2
+
1
{\displaystyle |\alpha ^{k}|\leq q^{k(d-1)/2+1}}
As this is true for arbitrarily large even k, this implies that
|
α
|
≤
q
(
d
−
1
)
/
2
.
{\displaystyle |\alpha |\leq q^{(d-1)/2}.}
Poincaré duality then implies that
|
α
|
=
q
(
d
−
1
)
/
2
.
{\displaystyle |\alpha |=q^{(d-1)/2}.}
=== Completion of the proof ===
The deduction of the Riemann hypothesis from this estimate is mostly a fairly straightforward use of standard techniques and is done as follows.
The eigenvalues of Frobenius on H1c(U,E) can now be estimated as they are the zeros of the zeta function of the sheaf E. This zeta function can be written as an Euler product of zeta functions of the stalks of E, and using the estimate for the eigenvalues on these stalks shows that this product converges for |T| < q−d/2−1/2, so that there are no zeros of the zeta function in this region. This implies that the eigenvalues of Frobenius on E are at most qd/2+1/2 in absolute value (in fact it will soon be seen that they have absolute value exactly qd/2). This step of the argument is very similar to the usual proof that the Riemann zeta function has no zeros with real part greater than 1, by writing it as an Euler product.
The conclusion of this is that the eigenvalues α of the Frobenius of a variety of even dimension d on the middle cohomology group satisfy
|
α
|
≤
q
d
/
2
+
1
/
2
{\displaystyle |\alpha |\leq q^{d/2+1/2}}
To obtain the Riemann hypothesis one needs to eliminate the 1/2 from the exponent. This can be done as follows. Applying this estimate to any even power Vk of V and using the Künneth formula shows that the eigenvalues of Frobenius on the middle cohomology of a variety V of any dimension d satisfy
|
α
k
|
≤
q
k
d
/
2
+
1
/
2
{\displaystyle |\alpha ^{k}|\leq q^{kd/2+1/2}}
As this is true for arbitrarily large even k, this implies that
|
α
|
≤
q
d
/
2
{\displaystyle |\alpha |\leq q^{d/2}}
Poincaré duality then implies that
|
α
|
=
q
d
/
2
.
{\displaystyle |\alpha |=q^{d/2}.}
This proves the Weil conjectures for the middle cohomology of a variety. The Weil conjectures for the cohomology below the middle dimension follow from this by applying the weak Lefschetz theorem, and the conjectures for cohomology above the middle dimension then follow from Poincaré duality.
== Deligne's second proof ==
Deligne (1980) found and proved a generalization of the Weil conjectures, bounding the weights of the pushforward of a sheaf. In practice it is this generalization rather than the original Weil conjectures that is mostly used in applications, such as the hard Lefschetz theorem. Much of the second proof is a rearrangement of the ideas of his first proof. The main extra idea needed is an argument closely related to the theorem of Jacques Hadamard and Charles Jean de la Vallée Poussin, used by Deligne to show that various L-series do not have zeros with real part 1.
A constructible sheaf on a variety over a finite field is called pure of weight β if for all points x the eigenvalues of the Frobenius at x all have absolute value N(x)β/2, and is called mixed of weight ≤ β if it can be written as repeated extensions by pure sheaves with weights ≤ β.
Deligne's theorem states that if f is a morphism of schemes of finite type over a finite field, then Rif! takes mixed sheaves of weight ≤ β to mixed sheaves of weight ≤ β + i.
The original Weil conjectures follow by taking f to be a morphism from a smooth projective variety to a point and considering the constant sheaf Qℓ on the variety. This gives an upper bound on the absolute values of the eigenvalues of Frobenius, and Poincaré duality then shows that this is also a lower bound.
In general Rif! does not take pure sheaves to pure sheaves. However it does when a suitable form of Poincaré duality holds, for example if f is smooth and proper, or if one works with perverse sheaves rather than sheaves as in Beilinson, Bernstein & Deligne (1982).
Inspired by the work of Witten (1982) on Morse theory, Laumon (1987) found another proof, using Deligne's ℓ-adic Fourier transform, which allowed him to simplify Deligne's proof by avoiding the use of the method of Hadamard and de la Vallée Poussin. His proof generalizes the classical calculation of the absolute value of Gauss sums using the fact that the norm of a Fourier transform has a simple relation to the norm of the original function. Kiehl & Weissauer (2001) used Laumon's proof as the basis for their exposition of Deligne's theorem. Katz (2001) gave a further simplification of Laumon's proof, using monodromy in the spirit of Deligne's first proof. Kedlaya (2006) gave another proof using the Fourier transform, replacing etale cohomology with rigid cohomology.
== Applications ==
Deligne (1980) was able to prove the hard Lefschetz theorem over finite fields using his second proof of the Weil conjectures.
Deligne (1971) had previously shown that the Ramanujan–Petersson conjecture follows from the Weil conjectures.
Deligne (1974, section 8) used the Weil conjectures to prove estimates for exponential sums.
Nick Katz and William Messing (1974) were able to prove the Künneth type standard conjecture over finite fields using Deligne's proof of the Weil conjectures.
== References ==
Artin, Emil (1924), "Quadratische Körper im Gebiete der höheren Kongruenzen. II. Analytischer Teil", Mathematische Zeitschrift, 19 (1): 207–246, doi:10.1007/BF01181075, ISSN 0025-5874, S2CID 117936362
Beilinson, Alexander A.; Bernstein, Joseph; Deligne, Pierre (1982), "Faisceaux pervers", Analysis and topology on singular spaces, I (Luminy, 1981), Astérisque, vol. 100, Paris: Société Mathématique de France, pp. 5–171, MR 0751966
Deligne, Pierre (1971), "Formes modulaires et représentations l-adiques", Séminaire Bourbaki vol. 1968/69 Exposés 347-363, Lecture Notes in Mathematics, vol. 179, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0058801, ISBN 978-3-540-05356-9
Deligne, Pierre (1974), "La conjecture de Weil. I", Publications Mathématiques de l'IHÉS, 43 (43): 273–307, doi:10.1007/BF02684373, ISSN 1618-1913, MR 0340258, S2CID 123139343
Deligne, Pierre, ed. (1977), Séminaire de Géométrie Algébrique du Bois Marie — Cohomologie étale (SGA 4.5), Lecture Notes in Mathematics (in French), vol. 569, Berlin: Springer-Verlag, doi:10.1007/BFb0091516, ISBN 978-0-387-08066-6, archived from the original on 2009-05-15, retrieved 2010-02-03
Deligne, Pierre (1980), "La conjecture de Weil. II", Publications Mathématiques de l'IHÉS, 52 (52): 137–252, doi:10.1007/BF02684780, ISSN 1618-1913, MR 0601520, S2CID 189769469
Deligne, Pierre; Katz, Nicholas (1973), Groupes de monodromie en géométrie algébrique. II, Lecture Notes in Mathematics, Vol. 340, vol. 340, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0060505, ISBN 978-3-540-06433-6, MR 0354657
Dwork, Bernard (1960), "On the rationality of the zeta function of an algebraic variety", American Journal of Mathematics, 82 (3), American Journal of Mathematics, Vol. 82, No. 3: 631–648, doi:10.2307/2372974, ISSN 0002-9327, JSTOR 2372974, MR 0140494
Freitag, Eberhard; Kiehl, Reinhardt (1988), Étale cohomology and the Weil conjecture, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 13, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-662-02541-3, ISBN 978-3-540-12175-6, MR 0926276
Grothendieck, Alexander (1960), "The cohomology theory of abstract algebraic varieties", Proc. Internat. Congress Math. (Edinburgh, 1958), Cambridge University Press, pp. 103–118, MR 0130879
Grothendieck, Alexander (1995) [1965], "Formule de Lefschetz et rationalité des fonctions L", Séminaire Bourbaki, vol. 9, Paris: Société Mathématique de France, pp. 41–55, MR 1608788
Grothendieck, Alexander (1972), Groupes de monodromie en géométrie algébrique. I, Lecture Notes in Mathematics, Vol. 288, vol. 288, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0068688, ISBN 978-3-540-05987-5, MR 0354656
Kahn, Bruno (2020), "The zeta function of an abelian variety", Zeta and L-Functions of Varieties and Motives, London Mathematical Society Lecture Note Series, Cambridge: Cambridge University Press, pp. 52–53, doi:10.1017/9781108691536, ISBN 978-1-108-70339-0
Katz, Nicholas M. (1976), "An overview of Deligne's proof of the Riemann hypothesis for varieties over finite fields", Mathematical developments arising from Hilbert problems, Proc. Sympos. Pure Math., vol. XXVIII, Providence, R. I.: American Mathematical Society, pp. 275–305, MR 0424822
Katz, Nicholas (2001), "L-functions and monodromy: four lectures on Weil II", Advances in Mathematics, 160 (1): 81–132, doi:10.1006/aima.2000.1979, MR 1831948
Katz, Nicholas M.; Messing, William (1974), "Some consequences of the Riemann hypothesis for varieties over finite fields", Inventiones Mathematicae, 23: 73–77, Bibcode:1974InMat..23...73K, doi:10.1007/BF01405203, ISSN 0020-9910, MR 0332791, S2CID 121989640
Kedlaya, Kiran S. (2006), "Fourier transforms and p-adic 'Weil II'", Compositio Mathematica, 142 (6): 1426–1450, arXiv:math/0210149, doi:10.1112/S0010437X06002338 (inactive 5 January 2025), ISSN 0010-437X, MR 2278753, S2CID 5233570{{citation}}: CS1 maint: DOI inactive as of January 2025 (link)
Kiehl, Reinhardt; Weissauer, Rainer (2001), Weil conjectures, perverse sheaves and l'adic Fourier transform, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 42, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-662-04576-3, ISBN 978-3-540-41457-5, MR 1855066
Kleiman, Steven L. (1968), "Algebraic cycles and the Weil conjectures", Dix esposés sur la cohomologie des schémas, Amsterdam: North-Holland, pp. 359–386, MR 0292838
Langlands, Robert P. (1970), "Problems in the theory of automorphic forms", Lectures in modern analysis and applications, III, Lecture Notes in Math, vol. 170, Berlin, New York: Springer-Verlag, pp. 18–61, doi:10.1007/BFb0079065, ISBN 978-3-540-05284-5, MR 0302614
Laumon, Gérard (1987), "Transformation de Fourier, constantes d'équations fonctionnelles et conjecture de Weil", Publications Mathématiques de l'IHÉS, 65 (65): 131–210, doi:10.1007/BF02698937, ISSN 1618-1913, MR 0908218, S2CID 119951352
Lefschetz, Solomon (1924), L'Analysis situs et la géométrie algébrique, Collection de Monographies publiée sous la Direction de M. Émile Borel (in French), Paris: Gauthier-Villars Reprinted in Lefschetz, Solomon (1971), Selected papers, New York: Chelsea Publishing Co., ISBN 978-0-8284-0234-7, MR 0299447
Mazur, Barry (1974), "Eigenvalues of Frobenius acting on algebraic varieties over finite fields", in Hartshorne, Robin (ed.), Algebraic Geometry, Arcata 1974, Proceedings of symposia in pure mathematics, vol. 29, ISBN 0-8218-1429-X
Moreno, O. (2001) [1994], "Bombieri–Weil bound", Encyclopedia of Mathematics, EMS Press
Rankin, Robert A.; Hardy, G. H. (1939), "Contributions to the theory of Ramanujan's function τ and similar arithmetical functions. II. The order of the Fourier coefficients of integral modular forms", Proceedings of the Cambridge Philosophical Society, 35 (3): 357–372, Bibcode:1939PCPS...35..357R, doi:10.1017/S0305004100021101, MR 0000411, S2CID 251097961
Serre, Jean-Pierre (1960), "Analogues kählériens de certaines conjectures de Weil", Annals of Mathematics, Second Series, 71 (2), The Annals of Mathematics, Vol. 71, No. 2: 392–394, doi:10.2307/1970088, ISSN 0003-486X, JSTOR 1970088, MR 0112163
Serre, Jean-Pierre (1975), "Valeurs propers des endomorphismes de Frobenius [d'après P. Deligne]", Séminaire Bourbaki vol. 1973/74 Exposés 436–452, Lecture Notes in Mathematics, vol. 431, pp. 190–204, doi:10.1007/BFb0066371, ISBN 978-3-540-07023-8
Verdier, Jean-Louis (1974), "Indépendance par rapport a ℓ des polynômes caractéristiques des endomorphismes de frobenius de la cohomologie ℓ-adique", Séminaire Bourbaki vol. 1972/73 Exposés 418–435, Lecture Notes in Mathematics, vol. 383, Springer Berlin / Heidelberg, pp. 98–115, doi:10.1007/BFb0057304, ISBN 978-3-540-06796-2
Weil, André (1949), "Numbers of solutions of equations in finite fields", Bulletin of the American Mathematical Society, 55 (5): 497–508, doi:10.1090/S0002-9904-1949-09219-4, ISSN 0002-9904, MR 0029393 Reprinted in Oeuvres Scientifiques/Collected Papers by André Weil ISBN 0-387-90330-5
Witten, Edward (1982), "Supersymmetry and Morse theory", Journal of Differential Geometry, 17 (4): 661–692, doi:10.4310/jdg/1214437492, ISSN 0022-040X, MR 0683171
== External links ==
== References == | Wikipedia/Weil_conjectures |
In algebraic geometry, a branch of mathematics, an adequate equivalence relation is an equivalence relation on algebraic cycles of smooth projective varieties used to obtain a well-working theory of such cycles, and in particular, well-defined intersection products. Pierre Samuel formalized the concept of an adequate equivalence relation in 1958. Since then it has become central to theory of motives. For every adequate equivalence relation, one may define the category of pure motives with respect to that relation.
Possible (and useful) adequate equivalence relations include rational, algebraic, homological and numerical equivalence. They are called "adequate" because dividing out by the equivalence relation is functorial, i.e. push-forward (with change of codimension) and pull-back of cycles is well-defined. Codimension 1 cycles modulo rational equivalence form the classical group of divisors modulo linear equivalence. All cycles modulo rational equivalence form the Chow ring.
== Definition ==
Let Z*(X) := Z[X] be the free abelian group on the algebraic cycles of X. Then an adequate equivalence relation is a family of equivalence relations, ~X on Z*(X), one for each smooth projective variety X, satisfying the following three conditions:
(Linearity) The equivalence relation is compatible with addition of cycles.
(Moving lemma) If
α
,
β
∈
Z
∗
(
X
)
{\displaystyle \alpha ,\beta \in Z^{*}(X)}
are cycles on X, then there exists a cycle
α
′
∈
Z
∗
(
X
)
{\displaystyle \alpha '\in Z^{*}(X)}
such that
α
{\displaystyle \alpha }
~X
α
′
{\displaystyle \alpha '}
and
α
′
{\displaystyle \alpha '}
intersects
β
{\displaystyle \beta }
properly.
(Push-forwards) Let
α
∈
Z
∗
(
X
)
{\displaystyle \alpha \in Z^{*}(X)}
and
β
∈
Z
∗
(
X
×
Y
)
{\displaystyle \beta \in Z^{*}(X\times Y)}
be cycles such that
β
{\displaystyle \beta }
intersects
α
×
Y
{\displaystyle \alpha \times Y}
properly. If
α
{\displaystyle \alpha }
~X 0, then
(
π
Y
)
∗
(
β
⋅
(
α
×
Y
)
)
{\displaystyle (\pi _{Y})_{*}(\beta \cdot (\alpha \times Y))}
~Y 0, where
π
Y
:
X
×
Y
→
Y
{\displaystyle \pi _{Y}:X\times Y\to Y}
is the projection.
The push-forward cycle in the last axiom is often denoted
β
(
α
)
:=
(
π
Y
)
∗
(
β
⋅
(
α
×
Y
)
)
{\displaystyle \beta (\alpha ):=(\pi _{Y})_{*}(\beta \cdot (\alpha \times Y))}
If
β
{\displaystyle \beta }
is the graph of a function, then this reduces to the push-forward of the function. The generalizations of functions from X to Y to cycles on X × Y are known as correspondences. The last axiom allows us to push forward cycles by a correspondence.
== Examples of equivalence relations ==
The most common equivalence relations, listed from strongest to weakest, are gathered in the following table.
== Notes ==
== References ==
Kleiman, Steven L. (1972), "Motives", in Oort, F. (ed.), Algebraic geometry, Oslo 1970 (Proc. Fifth Nordic Summer-School in Math., Oslo, 1970), Groningen: Wolters-Noordhoff, pp. 53–82, MR 0382267
Jannsen, U. (2000), "Equivalence relations on algebraic cycles", The Arithmetic and Geometry of Algebraic Cycles, NATO, 200, Kluwer Ac. Publ. Co.: 225–260 | Wikipedia/Algebraic_equivalence |
In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variables.
For example,
{
3
x
+
2
y
−
z
=
1
2
x
−
2
y
+
4
z
=
−
2
−
x
+
1
2
y
−
z
=
0
{\displaystyle {\begin{cases}3x+2y-z=1\\2x-2y+4z=-2\\-x+{\frac {1}{2}}y-z=0\end{cases}}}
is a system of three equations in the three variables x, y, z. A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. In the example above, a solution is given by the ordered triple
(
x
,
y
,
z
)
=
(
1
,
−
2
,
−
2
)
,
{\displaystyle (x,y,z)=(1,-2,-2),}
since it makes all three equations valid.
Linear systems are a fundamental part of linear algebra, a subject used in most modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics. A system of non-linear equations can often be approximated by a linear system (see linearization), a helpful technique when making a mathematical model or computer simulation of a relatively complex system.
Very often, and in this article, the coefficients and solutions of the equations are constrained to be real or complex numbers, but the theory and algorithms apply to coefficients and solutions in any field. For other algebraic structures, other theories have been developed. For coefficients and solutions in an integral domain, such as the ring of integers, see Linear equation over a ring. For coefficients and solutions that are polynomials, see Gröbner basis. For finding the "best" integer solutions among many, see Integer linear programming. For an example of a more exotic structure to which linear algebra can be applied, see Tropical geometry.
== Elementary examples ==
=== Trivial example ===
The system of one equation in one unknown
2
x
=
4
{\displaystyle 2x=4}
has the solution
x
=
2.
{\displaystyle x=2.}
However, most interesting linear systems have at least two equations.
=== Simple nontrivial example ===
The simplest kind of nontrivial linear system involves two equations and two variables:
2
x
+
3
y
=
6
4
x
+
9
y
=
15
.
{\displaystyle {\begin{alignedat}{5}2x&&\;+\;&&3y&&\;=\;&&6&\\4x&&\;+\;&&9y&&\;=\;&&15&.\end{alignedat}}}
One method for solving such a system is as follows. First, solve the top equation for
x
{\displaystyle x}
in terms of
y
{\displaystyle y}
:
x
=
3
−
3
2
y
.
{\displaystyle x=3-{\frac {3}{2}}y.}
Now substitute this expression for x into the bottom equation:
4
(
3
−
3
2
y
)
+
9
y
=
15.
{\displaystyle 4\left(3-{\frac {3}{2}}y\right)+9y=15.}
This results in a single equation involving only the variable
y
{\displaystyle y}
. Solving gives
y
=
1
{\displaystyle y=1}
, and substituting this back into the equation for
x
{\displaystyle x}
yields
x
=
3
2
{\displaystyle x={\frac {3}{2}}}
. This method generalizes to systems with additional variables (see "elimination of variables" below, or the article on elementary algebra.)
== General form ==
A general system of m linear equations with n unknowns and coefficients can be written as
{
a
11
x
1
+
a
12
x
2
+
⋯
+
a
1
n
x
n
=
b
1
a
21
x
1
+
a
22
x
2
+
⋯
+
a
2
n
x
n
=
b
2
⋮
a
m
1
x
1
+
a
m
2
x
2
+
⋯
+
a
m
n
x
n
=
b
m
,
{\displaystyle {\begin{cases}a_{11}x_{1}+a_{12}x_{2}+\dots +a_{1n}x_{n}=b_{1}\\a_{21}x_{1}+a_{22}x_{2}+\dots +a_{2n}x_{n}=b_{2}\\\vdots \\a_{m1}x_{1}+a_{m2}x_{2}+\dots +a_{mn}x_{n}=b_{m},\end{cases}}}
where
x
1
,
x
2
,
…
,
x
n
{\displaystyle x_{1},x_{2},\dots ,x_{n}}
are the unknowns,
a
11
,
a
12
,
…
,
a
m
n
{\displaystyle a_{11},a_{12},\dots ,a_{mn}}
are the coefficients of the system, and
b
1
,
b
2
,
…
,
b
m
{\displaystyle b_{1},b_{2},\dots ,b_{m}}
are the constant terms.
Often the coefficients and unknowns are real or complex numbers, but integers and rational numbers are also seen, as are polynomials and elements of an abstract algebraic structure.
=== Vector equation ===
One extremely helpful view is that each unknown is a weight for a column vector in a linear combination.
x
1
[
a
11
a
21
⋮
a
m
1
]
+
x
2
[
a
12
a
22
⋮
a
m
2
]
+
⋯
+
x
n
[
a
1
n
a
2
n
⋮
a
m
n
]
=
[
b
1
b
2
⋮
b
m
]
{\displaystyle x_{1}{\begin{bmatrix}a_{11}\\a_{21}\\\vdots \\a_{m1}\end{bmatrix}}+x_{2}{\begin{bmatrix}a_{12}\\a_{22}\\\vdots \\a_{m2}\end{bmatrix}}+\dots +x_{n}{\begin{bmatrix}a_{1n}\\a_{2n}\\\vdots \\a_{mn}\end{bmatrix}}={\begin{bmatrix}b_{1}\\b_{2}\\\vdots \\b_{m}\end{bmatrix}}}
This allows all the language and theory of vector spaces (or more generally, modules) to be brought to bear. For example, the collection of all possible linear combinations of the vectors on the left-hand side (LHS) is called their span, and the equations have a solution just when the right-hand vector is within that span. If every vector within that span has exactly one expression as a linear combination of the given left-hand vectors, then any solution is unique. In any event, the span has a basis of linearly independent vectors that do guarantee exactly one expression; and the number of vectors in that basis (its dimension) cannot be larger than m or n, but it can be smaller. This is important because if we have m independent vectors a solution is guaranteed regardless of the right-hand side (RHS), and otherwise not guaranteed.
=== Matrix equation ===
The vector equation is equivalent to a matrix equation of the form
A
x
=
b
{\displaystyle A\mathbf {x} =\mathbf {b} }
where A is an m×n matrix, x is a column vector with n entries, and b is a column vector with m entries.
A
=
[
a
11
a
12
⋯
a
1
n
a
21
a
22
⋯
a
2
n
⋮
⋮
⋱
⋮
a
m
1
a
m
2
⋯
a
m
n
]
,
x
=
[
x
1
x
2
⋮
x
n
]
,
b
=
[
b
1
b
2
⋮
b
m
]
.
{\displaystyle A={\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{bmatrix}},\quad \mathbf {x} ={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}},\quad \mathbf {b} ={\begin{bmatrix}b_{1}\\b_{2}\\\vdots \\b_{m}\end{bmatrix}}.}
The number of vectors in a basis for the span is now expressed as the rank of the matrix.
== Solution set ==
A solution of a linear system is an assignment of values to the variables
x
1
,
x
2
,
…
,
x
n
{\displaystyle x_{1},x_{2},\dots ,x_{n}}
such that each of the equations is satisfied. The set of all possible solutions is called the solution set.
A linear system may behave in any one of three possible ways:
The system has infinitely many solutions.
The system has a unique solution.
The system has no solution.
=== Geometric interpretation ===
For a system involving two variables (x and y), each linear equation determines a line on the xy-plane. Because a solution to a linear system must satisfy all of the equations, the solution set is the intersection of these lines, and is hence either a line, a single point, or the empty set.
For three variables, each linear equation determines a plane in three-dimensional space, and the solution set is the intersection of these planes. Thus the solution set may be a plane, a line, a single point, or the empty set. For example, as three parallel planes do not have a common point, the solution set of their equations is empty; the solution set of the equations of three planes intersecting at a point is single point; if three planes pass through two points, their equations have at least two common solutions; in fact the solution set is infinite and consists in all the line passing through these points.
For n variables, each linear equation determines a hyperplane in n-dimensional space. The solution set is the intersection of these hyperplanes, and is a flat, which may have any dimension lower than n.
=== General behavior ===
In general, the behavior of a linear system is determined by the relationship between the number of equations and the number of unknowns. Here, "in general" means that a different behavior may occur for specific values of the coefficients of the equations.
In general, a system with fewer equations than unknowns has infinitely many solutions, but it may have no solution. Such a system is known as an underdetermined system.
In general, a system with the same number of equations and unknowns has a single unique solution.
In general, a system with more equations than unknowns has no solution. Such a system is also known as an overdetermined system.
In the first case, the dimension of the solution set is, in general, equal to n − m, where n is the number of variables and m is the number of equations.
The following pictures illustrate this trichotomy in the case of two variables:
The first system has infinitely many solutions, namely all of the points on the blue line. The second system has a single unique solution, namely the intersection of the two lines. The third system has no solutions, since the three lines share no common point.
It must be kept in mind that the pictures above show only the most common case (the general case). It is possible for a system of two equations and two unknowns to have no solution (if the two lines are parallel), or for a system of three equations and two unknowns to be solvable (if the three lines intersect at a single point).
A system of linear equations behave differently from the general case if the equations are linearly dependent, or if it is inconsistent and has no more equations than unknowns.
== Properties ==
=== Independence ===
The equations of a linear system are independent if none of the equations can be derived algebraically from the others. When the equations are independent, each equation contains new information about the variables, and removing any of the equations increases the size of the solution set. For linear equations, logical independence is the same as linear independence.
For example, the equations
3
x
+
2
y
=
6
and
6
x
+
4
y
=
12
{\displaystyle 3x+2y=6\;\;\;\;{\text{and}}\;\;\;\;6x+4y=12}
are not independent — they are the same equation when scaled by a factor of two, and they would produce identical graphs. This is an example of equivalence in a system of linear equations.
For a more complicated example, the equations
x
−
2
y
=
−
1
3
x
+
5
y
=
8
4
x
+
3
y
=
7
{\displaystyle {\begin{alignedat}{5}x&&\;-\;&&2y&&\;=\;&&-1&\\3x&&\;+\;&&5y&&\;=\;&&8&\\4x&&\;+\;&&3y&&\;=\;&&7&\end{alignedat}}}
are not independent, because the third equation is the sum of the other two. Indeed, any one of these equations can be derived from the other two, and any one of the equations can be removed without affecting the solution set. The graphs of these equations are three lines that intersect at a single point.
=== Consistency ===
A linear system is inconsistent if it has no solution, and otherwise, it is said to be consistent. When the system is inconsistent, it is possible to derive a contradiction from the equations, that may always be rewritten as the statement 0 = 1.
For example, the equations
3
x
+
2
y
=
6
and
3
x
+
2
y
=
12
{\displaystyle 3x+2y=6\;\;\;\;{\text{and}}\;\;\;\;3x+2y=12}
are inconsistent. In fact, by subtracting the first equation from the second one and multiplying both sides of the result by 1/6, we get 0 = 1. The graphs of these equations on the xy-plane are a pair of parallel lines.
It is possible for three linear equations to be inconsistent, even though any two of them are consistent together. For example, the equations
x
+
y
=
1
2
x
+
y
=
1
3
x
+
2
y
=
3
{\displaystyle {\begin{alignedat}{7}x&&\;+\;&&y&&\;=\;&&1&\\2x&&\;+\;&&y&&\;=\;&&1&\\3x&&\;+\;&&2y&&\;=\;&&3&\end{alignedat}}}
are inconsistent. Adding the first two equations together gives 3x + 2y = 2, which can be subtracted from the third equation to yield 0 = 1. Any two of these equations have a common solution. The same phenomenon can occur for any number of equations.
In general, inconsistencies occur if the left-hand sides of the equations in a system are linearly dependent, and the constant terms do not satisfy the dependence relation. A system of equations whose left-hand sides are linearly independent is always consistent.
Putting it another way, according to the Rouché–Capelli theorem, any system of equations (overdetermined or otherwise) is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix. If, on the other hand, the ranks of these two matrices are equal, the system must have at least one solution. The solution is unique if and only if the rank equals the number of variables. Otherwise the general solution has k free parameters where k is the difference between the number of variables and the rank; hence in such a case there is an infinitude of solutions. The rank of a system of equations (that is, the rank of the augmented matrix) can never be higher than [the number of variables] + 1, which means that a system with any number of equations can always be reduced to a system that has a number of independent equations that is at most equal to [the number of variables] + 1.
=== Equivalence ===
Two linear systems using the same set of variables are equivalent if each of the equations in the second system can be derived algebraically from the equations in the first system, and vice versa. Two systems are equivalent if either both are inconsistent or each equation of each of them is a linear combination of the equations of the other one. It follows that two linear systems are equivalent if and only if they have the same solution set.
== Solving a linear system ==
There are several algorithms for solving a system of linear equations.
=== Describing the solution ===
When the solution set is finite, it is reduced to a single element. In this case, the unique solution is described by a sequence of equations whose left-hand sides are the names of the unknowns and right-hand sides are the corresponding values, for example
(
x
=
3
,
y
=
−
2
,
z
=
6
)
{\displaystyle (x=3,\;y=-2,\;z=6)}
. When an order on the unknowns has been fixed, for example the alphabetical order the solution may be described as a vector of values, like
(
3
,
−
2
,
6
)
{\displaystyle (3,\,-2,\,6)}
for the previous example.
To describe a set with an infinite number of solutions, typically some of the variables are designated as free (or independent, or as parameters), meaning that they are allowed to take any value, while the remaining variables are dependent on the values of the free variables.
For example, consider the following system:
x
+
3
y
−
2
z
=
5
3
x
+
5
y
+
6
z
=
7
{\displaystyle {\begin{alignedat}{7}x&&\;+\;&&3y&&\;-\;&&2z&&\;=\;&&5&\\3x&&\;+\;&&5y&&\;+\;&&6z&&\;=\;&&7&\end{alignedat}}}
The solution set to this system can be described by the following equations:
x
=
−
7
z
−
1
and
y
=
3
z
+
2
.
{\displaystyle x=-7z-1\;\;\;\;{\text{and}}\;\;\;\;y=3z+2{\text{.}}}
Here z is the free variable, while x and y are dependent on z. Any point in the solution set can be obtained by first choosing a value for z, and then computing the corresponding values for x and y.
Each free variable gives the solution space one degree of freedom, the number of which is equal to the dimension of the solution set. For example, the solution set for the above equation is a line, since a point in the solution set can be chosen by specifying the value of the parameter z. An infinite solution of higher order may describe a plane, or higher-dimensional set.
Different choices for the free variables may lead to different descriptions of the same solution set. For example, the solution to the above equations can alternatively be described as follows:
y
=
−
3
7
x
+
11
7
and
z
=
−
1
7
x
−
1
7
.
{\displaystyle y=-{\frac {3}{7}}x+{\frac {11}{7}}\;\;\;\;{\text{and}}\;\;\;\;z=-{\frac {1}{7}}x-{\frac {1}{7}}{\text{.}}}
Here x is the free variable, and y and z are dependent.
=== Elimination of variables ===
The simplest method for solving a system of linear equations is to repeatedly eliminate variables. This method can be described as follows:
In the first equation, solve for one of the variables in terms of the others.
Substitute this expression into the remaining equations. This yields a system of equations with one fewer equation and unknown.
Repeat steps 1 and 2 until the system is reduced to a single linear equation.
Solve this equation, and then back-substitute until the entire solution is found.
For example, consider the following system:
{
x
+
3
y
−
2
z
=
5
3
x
+
5
y
+
6
z
=
7
2
x
+
4
y
+
3
z
=
8
{\displaystyle {\begin{cases}x+3y-2z=5\\3x+5y+6z=7\\2x+4y+3z=8\end{cases}}}
Solving the first equation for x gives
x
=
5
+
2
z
−
3
y
{\displaystyle x=5+2z-3y}
, and plugging this into the second and third equation yields
{
y
=
3
z
+
2
y
=
7
2
z
+
1
{\displaystyle {\begin{cases}y=3z+2\\y={\tfrac {7}{2}}z+1\end{cases}}}
Since the LHS of both of these equations equal y, equating the RHS of the equations. We now have:
3
z
+
2
=
7
2
z
+
1
⇒
z
=
2
{\displaystyle {\begin{aligned}3z+2={\tfrac {7}{2}}z+1\\\Rightarrow z=2\end{aligned}}}
Substituting z = 2 into the second or third equation gives y = 8, and the values of y and z into the first equation yields x = −15. Therefore, the solution set is the ordered triple
(
x
,
y
,
z
)
=
(
−
15
,
8
,
2
)
{\displaystyle (x,y,z)=(-15,8,2)}
.
=== Row reduction ===
In row reduction (also known as Gaussian elimination), the linear system is represented as an augmented matrix
[
1
3
−
2
5
3
5
6
7
2
4
3
8
]
.
{\displaystyle \left[{\begin{array}{rrr|r}1&3&-2&5\\3&5&6&7\\2&4&3&8\end{array}}\right]{\text{.}}}
This matrix is then modified using elementary row operations until it reaches reduced row echelon form. There are three types of elementary row operations:
Type 1: Swap the positions of two rows.
Type 2: Multiply a row by a nonzero scalar.
Type 3: Add to one row a scalar multiple of another.
Because these operations are reversible, the augmented matrix produced always represents a linear system that is equivalent to the original.
There are several specific algorithms to row-reduce an augmented matrix, the simplest of which are Gaussian elimination and Gauss–Jordan elimination. The following computation shows Gauss–Jordan elimination applied to the matrix above:
[
1
3
−
2
5
3
5
6
7
2
4
3
8
]
∼
[
1
3
−
2
5
0
−
4
12
−
8
2
4
3
8
]
∼
[
1
3
−
2
5
0
−
4
12
−
8
0
−
2
7
−
2
]
∼
[
1
3
−
2
5
0
1
−
3
2
0
−
2
7
−
2
]
∼
[
1
3
−
2
5
0
1
−
3
2
0
0
1
2
]
∼
[
1
3
−
2
5
0
1
0
8
0
0
1
2
]
∼
[
1
3
0
9
0
1
0
8
0
0
1
2
]
∼
[
1
0
0
−
15
0
1
0
8
0
0
1
2
]
.
{\displaystyle {\begin{aligned}\left[{\begin{array}{rrr|r}1&3&-2&5\\3&5&6&7\\2&4&3&8\end{array}}\right]&\sim \left[{\begin{array}{rrr|r}1&3&-2&5\\0&-4&12&-8\\2&4&3&8\end{array}}\right]\sim \left[{\begin{array}{rrr|r}1&3&-2&5\\0&-4&12&-8\\0&-2&7&-2\end{array}}\right]\sim \left[{\begin{array}{rrr|r}1&3&-2&5\\0&1&-3&2\\0&-2&7&-2\end{array}}\right]\\&\sim \left[{\begin{array}{rrr|r}1&3&-2&5\\0&1&-3&2\\0&0&1&2\end{array}}\right]\sim \left[{\begin{array}{rrr|r}1&3&-2&5\\0&1&0&8\\0&0&1&2\end{array}}\right]\sim \left[{\begin{array}{rrr|r}1&3&0&9\\0&1&0&8\\0&0&1&2\end{array}}\right]\sim \left[{\begin{array}{rrr|r}1&0&0&-15\\0&1&0&8\\0&0&1&2\end{array}}\right].\end{aligned}}}
The last matrix is in reduced row echelon form, and represents the system x = −15, y = 8, z = 2. A comparison with the example in the previous section on the algebraic elimination of variables shows that these two methods are in fact the same; the difference lies in how the computations are written down.
=== Cramer's rule ===
Cramer's rule is an explicit formula for the solution of a system of linear equations, with each variable given by a quotient of two determinants. For example, the solution to the system
x
+
3
y
−
2
z
=
5
3
x
+
5
y
+
6
z
=
7
2
x
+
4
y
+
3
z
=
8
{\displaystyle {\begin{alignedat}{7}x&\;+&\;3y&\;-&\;2z&\;=&\;5\\3x&\;+&\;5y&\;+&\;6z&\;=&\;7\\2x&\;+&\;4y&\;+&\;3z&\;=&\;8\end{alignedat}}}
is given by
x
=
|
5
3
−
2
7
5
6
8
4
3
|
|
1
3
−
2
3
5
6
2
4
3
|
,
y
=
|
1
5
−
2
3
7
6
2
8
3
|
|
1
3
−
2
3
5
6
2
4
3
|
,
z
=
|
1
3
5
3
5
7
2
4
8
|
|
1
3
−
2
3
5
6
2
4
3
|
.
{\displaystyle x={\frac {\,{\begin{vmatrix}5&3&-2\\7&5&6\\8&4&3\end{vmatrix}}\,}{\,{\begin{vmatrix}1&3&-2\\3&5&6\\2&4&3\end{vmatrix}}\,}},\;\;\;\;y={\frac {\,{\begin{vmatrix}1&5&-2\\3&7&6\\2&8&3\end{vmatrix}}\,}{\,{\begin{vmatrix}1&3&-2\\3&5&6\\2&4&3\end{vmatrix}}\,}},\;\;\;\;z={\frac {\,{\begin{vmatrix}1&3&5\\3&5&7\\2&4&8\end{vmatrix}}\,}{\,{\begin{vmatrix}1&3&-2\\3&5&6\\2&4&3\end{vmatrix}}\,}}.}
For each variable, the denominator is the determinant of the matrix of coefficients, while the numerator is the determinant of a matrix in which one column has been replaced by the vector of constant terms.
Though Cramer's rule is important theoretically, it has little practical value for large matrices, since the computation of large determinants is somewhat cumbersome. (Indeed, large determinants are most easily computed using row reduction.)
Further, Cramer's rule has very poor numerical properties, making it unsuitable for solving even small systems reliably, unless the operations are performed in rational arithmetic with unbounded precision.
=== Matrix solution ===
If the equation system is expressed in the matrix form
A
x
=
b
{\displaystyle A\mathbf {x} =\mathbf {b} }
, the entire solution set can also be expressed in matrix form. If the matrix A is square (has m rows and n=m columns) and has full rank (all m rows are independent), then the system has a unique solution given by
x
=
A
−
1
b
{\displaystyle \mathbf {x} =A^{-1}\mathbf {b} }
where
A
−
1
{\displaystyle A^{-1}}
is the inverse of A. More generally, regardless of whether m=n or not and regardless of the rank of A, all solutions (if any exist) are given using the Moore–Penrose inverse of A, denoted
A
+
{\displaystyle A^{+}}
, as follows:
x
=
A
+
b
+
(
I
−
A
+
A
)
w
{\displaystyle \mathbf {x} =A^{+}\mathbf {b} +\left(I-A^{+}A\right)\mathbf {w} }
where
w
{\displaystyle \mathbf {w} }
is a vector of free parameters that ranges over all possible n×1 vectors. A necessary and sufficient condition for any solution(s) to exist is that the potential solution obtained using
w
=
0
{\displaystyle \mathbf {w} =\mathbf {0} }
satisfy
A
x
=
b
{\displaystyle A\mathbf {x} =\mathbf {b} }
— that is, that
A
A
+
b
=
b
.
{\displaystyle AA^{+}\mathbf {b} =\mathbf {b} .}
If this condition does not hold, the equation system is inconsistent and has no solution. If the condition holds, the system is consistent and at least one solution exists. For example, in the above-mentioned case in which A is square and of full rank,
A
+
{\displaystyle A^{+}}
simply equals
A
−
1
{\displaystyle A^{-1}}
and the general solution equation simplifies to
x
=
A
−
1
b
+
(
I
−
A
−
1
A
)
w
=
A
−
1
b
+
(
I
−
I
)
w
=
A
−
1
b
{\displaystyle \mathbf {x} =A^{-1}\mathbf {b} +\left(I-A^{-1}A\right)\mathbf {w} =A^{-1}\mathbf {b} +\left(I-I\right)\mathbf {w} =A^{-1}\mathbf {b} }
as previously stated, where
w
{\displaystyle \mathbf {w} }
has completely dropped out of the solution, leaving only a single solution. In other cases, though,
w
{\displaystyle \mathbf {w} }
remains and hence an infinitude of potential values of the free parameter vector
w
{\displaystyle \mathbf {w} }
give an infinitude of solutions of the equation.
=== Other methods ===
While systems of three or four equations can be readily solved by hand (see Cracovian), computers are often used for larger systems. The standard algorithm for solving a system of linear equations is based on Gaussian elimination with some modifications. Firstly, it is essential to avoid division by small numbers, which may lead to inaccurate results. This can be done by reordering the equations if necessary, a process known as pivoting. Secondly, the algorithm does not exactly do Gaussian elimination, but it computes the LU decomposition of the matrix A. This is mostly an organizational tool, but it is much quicker if one has to solve several systems with the same matrix A but different vectors b.
If the matrix A has some special structure, this can be exploited to obtain faster or more accurate algorithms. For instance, systems with a symmetric positive definite matrix can be solved twice as fast with the Cholesky decomposition. Levinson recursion is a fast method for Toeplitz matrices. Special methods exist also for matrices with many zero elements (so-called sparse matrices), which appear often in applications.
A completely different approach is often taken for very large systems, which would otherwise take too much time or memory. The idea is to start with an initial approximation to the solution (which does not have to be accurate at all), and to change this approximation in several steps to bring it closer to the true solution. Once the approximation is sufficiently accurate, this is taken to be the solution to the system. This leads to the class of iterative methods. For some sparse matrices, the introduction of randomness improves the speed of the iterative methods. One example of an iterative method is the Jacobi method, where the matrix
A
{\displaystyle A}
is split into its diagonal component
D
{\displaystyle D}
and its non-diagonal component
L
+
U
{\displaystyle L+U}
. An initial guess
x
(
0
)
{\displaystyle {\mathbf {x}}^{(0)}}
is used at the start of the algorithm. Each subsequent guess is computed using the iterative equation:
x
(
k
+
1
)
=
D
−
1
(
b
−
(
L
+
U
)
x
(
k
)
)
{\displaystyle {\mathbf {x}}^{(k+1)}=D^{-1}({\mathbf {b}}-(L+U){\mathbf {x}}^{(k)})}
When the difference between guesses
x
(
k
)
{\displaystyle {\mathbf {x}}^{(k)}}
and
x
(
k
+
1
)
{\displaystyle {\mathbf {x}}^{(k+1)}}
is sufficiently small, the algorithm is said to have converged on the solution.
There is also a quantum algorithm for linear systems of equations.
== Homogeneous systems ==
A system of linear equations is homogeneous if all of the constant terms are zero:
a
11
x
1
+
a
12
x
2
+
⋯
+
a
1
n
x
n
=
0
a
21
x
1
+
a
22
x
2
+
⋯
+
a
2
n
x
n
=
0
⋮
a
m
1
x
1
+
a
m
2
x
2
+
⋯
+
a
m
n
x
n
=
0.
{\displaystyle {\begin{alignedat}{7}a_{11}x_{1}&&\;+\;&&a_{12}x_{2}&&\;+\cdots +\;&&a_{1n}x_{n}&&\;=\;&&&0\\a_{21}x_{1}&&\;+\;&&a_{22}x_{2}&&\;+\cdots +\;&&a_{2n}x_{n}&&\;=\;&&&0\\&&&&&&&&&&\vdots \;\ &&&\\a_{m1}x_{1}&&\;+\;&&a_{m2}x_{2}&&\;+\cdots +\;&&a_{mn}x_{n}&&\;=\;&&&0.\\\end{alignedat}}}
A homogeneous system is equivalent to a matrix equation of the form
A
x
=
0
{\displaystyle A\mathbf {x} =\mathbf {0} }
where A is an m × n matrix, x is a column vector with n entries, and 0 is the zero vector with m entries.
=== Homogeneous solution set ===
Every homogeneous system has at least one solution, known as the zero (or trivial) solution, which is obtained by assigning the value of zero to each of the variables. If the system has a non-singular matrix (det(A) ≠ 0) then it is also the only solution. If the system has a singular matrix then there is a solution set with an infinite number of solutions. This solution set has the following additional properties:
If u and v are two vectors representing solutions to a homogeneous system, then the vector sum u + v is also a solution to the system.
If u is a vector representing a solution to a homogeneous system, and r is any scalar, then ru is also a solution to the system.
These are exactly the properties required for the solution set to be a linear subspace of Rn. In particular, the solution set to a homogeneous system is the same as the null space of the corresponding matrix A.
=== Relation to nonhomogeneous systems ===
There is a close relationship between the solutions to a linear system and the solutions to the corresponding homogeneous system:
A
x
=
b
and
A
x
=
0
.
{\displaystyle A\mathbf {x} =\mathbf {b} \qquad {\text{and}}\qquad A\mathbf {x} =\mathbf {0} .}
Specifically, if p is any specific solution to the linear system Ax = b, then the entire solution set can be described as
{
p
+
v
:
v
is any solution to
A
x
=
0
}
.
{\displaystyle \left\{\mathbf {p} +\mathbf {v} :\mathbf {v} {\text{ is any solution to }}A\mathbf {x} =\mathbf {0} \right\}.}
Geometrically, this says that the solution set for Ax = b is a translation of the solution set for Ax = 0. Specifically, the flat for the first system can be obtained by translating the linear subspace for the homogeneous system by the vector p.
This reasoning only applies if the system Ax = b has at least one solution. This occurs if and only if the vector b lies in the image of the linear transformation A.
== See also ==
Arrangement of hyperplanes
Iterative refinement – Method to improve accuracy of numerical solutions to systems of linear equations
Coates graph – A mathematical graph for solution of linear equations
LAPACK – Software library for numerical linear algebra
Linear equation over a ring
Linear least squares – Least squares approximation of linear functions to dataPages displaying short descriptions of redirect targets
Matrix decomposition – Representation of a matrix as a product
Matrix splitting – Representation of a matrix as a sum
NAG Numerical Library – Software library of numerical-analysis algorithms
Rybicki Press algorithm – An algorithm for inverting a matrix
Simultaneous equations – Set of equations to be solved togetherPages displaying short descriptions of redirect targets
== References ==
== Bibliography ==
Anton, Howard (1987), Elementary Linear Algebra (5th ed.), New York: Wiley, ISBN 0-471-84819-0
Beauregard, Raymond A.; Fraleigh, John B. (1973), A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields, Boston: Houghton Mifflin Company, ISBN 0-395-14017-X
Burden, Richard L.; Faires, J. Douglas (1993), Numerical Analysis (5th ed.), Boston: Prindle, Weber and Schmidt, ISBN 0-534-93219-3
Cullen, Charles G. (1990), Matrices and Linear Transformations, MA: Dover, ISBN 978-0-486-66328-9
Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3rd ed.), Baltimore: Johns Hopkins University Press, ISBN 0-8018-5414-8
Harper, Charlie (1976), Introduction to Mathematical Physics, New Jersey: Prentice-Hall, ISBN 0-13-487538-9
Harrow, Aram W.; Hassidim, Avinatan; Lloyd, Seth (2009), "Quantum Algorithm for Linear Systems of Equations", Physical Review Letters, 103 (15): 150502, arXiv:0811.3171, Bibcode:2009PhRvL.103o0502H, doi:10.1103/PhysRevLett.103.150502, PMID 19905613, S2CID 5187993
Sterling, Mary J. (2009), Linear Algebra for Dummies, Indianapolis, Indiana: Wiley, ISBN 978-0-470-43090-3
Whitelaw, T. A. (1991), Introduction to Linear Algebra (2nd ed.), CRC Press, ISBN 0-7514-0159-5
== Further reading ==
Axler, Sheldon Jay (1997). Linear Algebra Done Right (2nd ed.). Springer-Verlag. ISBN 0-387-98259-0.
Lay, David C. (August 22, 2005). Linear Algebra and Its Applications (3rd ed.). Addison Wesley. ISBN 978-0-321-28713-7.
Meyer, Carl D. (February 15, 2001). Matrix Analysis and Applied Linear Algebra. Society for Industrial and Applied Mathematics (SIAM). ISBN 978-0-89871-454-8. Archived from the original on March 1, 2001.
Poole, David (2006). Linear Algebra: A Modern Introduction (2nd ed.). Brooks/Cole. ISBN 0-534-99845-3.
Anton, Howard (2005). Elementary Linear Algebra (Applications Version) (9th ed.). Wiley International.
Leon, Steven J. (2006). Linear Algebra With Applications (7th ed.). Pearson Prentice Hall.
Strang, Gilbert (2005). Linear Algebra and Its Applications.
Peng, Richard; Vempala, Santosh S. (2024). "Solving Sparse Linear Systems Faster than Matrix Multiplication". Comm. ACM. 67 (7): 79–86. arXiv:2007.10254. doi:10.1145/3615679.
== External links ==
Media related to System of linear equations at Wikimedia Commons | Wikipedia/Linear_simultaneous_equations |
A successive-approximation ADC (or SAR ADC) is a type of analog-to-digital converter (ADC) that digitizes each sample from a continuous analog waveform using a binary search through all possible quantization levels.
== History ==
The SAR ADC was first used for experimental pulse-code modulation (PCM) by Bell Labs in the 1940s. In 1954, Bernard Gordon introduced the first commercial vacuum tube SAR ADC, converting 50,000 11-bit samples per second.
== Algorithm ==
The successive-approximation analog-to-digital converter circuit typically contains four chief subcircuits:
A sample-and-hold circuit that acquires the input voltage Vin.
An analog voltage comparator that compares Vin to the output of a digital-to-analog converter (DAC).
A successive-approximation register that is updated by results of the comparator to provide the DAC with a digital code whose accuracy increases each successive iteration.
A DAC that supplies the comparator with an analog voltage relative to the reference voltage Vref (which corresponds to the full-scale range of the ADC) and proportional to the digital code of the SAR.
The successive-approximation register is initialized with 1 in the most significant bit (MSB) and zeroes in the lower bits. The register's code is fed into the DAC, which provides an analog equivalent of its digital code (initially 1/2Vref) to the comparator for comparison with the sampled input voltage. If this analog voltage exceeds Vin, then the comparator causes the SAR to reset this bit; otherwise, the bit is left as 1. Then the next bit is set to 1 and the same test is done, continuing this binary search until every bit in the SAR has been tested. The resulting code is the digital approximated output of the sampled input voltage.
The algorithm's objective for the nth iteration is to approximately digitize the input voltage to an accuracy of 1⁄2n relative to the reference voltage. To show this mathematically, the normalized input voltage is represented as x in [−1, 1] by letting Vin = xVref. The algorithm starts with an initial approximation of x0 = 0 and during each iteration i produces the following approximation:ith approximation: xi = xi−1 − sgn(xi−1 − x)/2iwhere the binary signum function sgn mathematically represents the comparison of the previous iteration's approximation xi-1 with the normalized input voltage x:
s
g
n
(
x
i
−
1
−
x
)
=
{
+
1
if
x
i
−
1
≥
x
,
−
1
if
x
i
−
1
<
x
.
{\displaystyle sgn(x_{i-1}-x)={\begin{cases}+1&{\text{if }}x_{i-1}\geq x,\\-1&{\text{if }}x_{i-1}<x.\end{cases}}}
It follows using mathematical induction that the approximation of the nth iteration theoretically has a bounded accuracy of: |xn − x| ≤ 1/2n.
=== Inaccuracies in non-ideal analog circuits ===
When implemented as a real analog circuit, circuit inaccuracies and noise may cause the binary search algorithm to incorrectly remove values it believes Vin cannot be, so a successive-approximation ADC might not output the closest value. It is very important for the DAC to accurately produce all 2n analog values for comparison against the unknown Vin in order to produce a best match estimate. The maximal error can easily exceed several LSBs, especially as the error between the actual and ideal 2n becomes large. Manufacturers may characterize the accuracy using an effective number of bits (ENOB) smaller than the actual number of output bits.
As of 2001, the component-matching limitations of the DAC generally limited the linearity to about 12 bits in practical designs and mandated some form of trimming or calibration to achieve the necessary linearity for more than 12 bits. And since kT/C noise is inversely proportional to capacitance, low noise demands a large input capacitance (which costs chip area and requires a more powerful drive buffer), which has motivated proposals around noise cancellation. For comparison, for a Vref of 5 V, the least significant bit of a 16-bit converter corresponds to 76 µV, which is around the 64 µVrms noise of a 1 pF (large for on-chip) capacitor at room temperature. As of 2012, SAR ADCs are limited to 18 bits, while delta-sigma ADCs (which can be 24 bits) are better suited if more than 16 bits are needed. SAR ADCs are commonly found on microcontrollers because they are easy to integrate into a mixed-signal process, but suffer from inaccuracies from the internal reference voltage resistor ladder and clock and signal noise from the rest of the microcontroller, so external ADC chips may provide better accuracy.
=== Examples ===
Example 1: The steps to converting an analog input to 9-bit digital, using successive-approximation, are shown here for all voltages from 5 V to 0 V in 0.1 V iterations. Since the reference voltage is 5 V, when the input voltage is also 5 V, all bits are set. As the voltage is decreased to 4.9 V, only some of the least significant bits are cleared. The MSB will remain set until the input is one half the reference voltage, 2.5 V.
The binary weights assigned to each bit, starting with the MSB, are 2.5, 1.25, 0.625, 0.3125, 0.15625, 0.078125, 0.0390625, 0.01953125, 0.009765625. All of these add up to 4.990234375, meaning binary 111111111, or one LSB less than 5.
When the analog input is being compared to the internal DAC output, it effectively is being compared to each of these binary weights, starting with the 2.5 V and either keeping it or clearing it as a result. Then by adding the next weight to the previous result, comparing again, and repeating until all the bits and their weights have been compared to the input, the result, a binary number representing the analog input, is found.
Example 2: The working of a 4-bit successive-approximation ADC is illustrated below. The MSB is initially set to 1 whereas the remaining digits are set to zero. If the input voltage is lower than the value stored in the register, on the next clock cycle, the register changes its value to that illustrated in the figure by following the green line. If the input voltage is higher, then on the next clock cycle, the register changes its value to that illustrated in the figure by following the red line. The simplified structure of this type of ADC that acts on 2n volts range can be expressed as an algorithm:
Initialize register with MSB set to 1 and all other values set to zero.
In the nth clock cycle, if voltage is higher than digital equivalent voltage of the number in register, the (n+1)th digit from the left is set to 1. If the voltage were lower than digital equivalent voltage, then nth digit from left is set to zero and the next digit is set to 1. To perform a conversion, an N-bit ADC requires N such clock cycles excluding the initial state.
The successive-approximation ADC can be alternatively explained by first uniformly assigning each digital output to corresponding ranges as shown. It can be seen that the algorithm essentially divides the voltage range into two regions and checks which of the two regions the input voltage belongs to. Successive steps involve taking the identified region from before and further dividing the region into two and continuing identification. This occurs until all possible choices of digital representations are exhausted, leaving behind an identified region that corresponds to only one of the digital representations.
=== Variants ===
Counter type ADC: The D to A converter can be easily turned around to provide the inverse function A to D conversion. The principle is to adjust the DAC's input code until the DAC's output comes within ±1⁄2 LSB to the analog input which is to be converted to binary digital form.
Servo tracking ADC: It is an improved version of a counting ADC. The circuit consists of an up-down counter with the comparator controlling the direction of the count. The analog output of the DAC is compared with the analog input. If the input is greater than the DAC output signal, the output of the comparator goes high and the counter is caused to count up. The tracking ADC has the advantage of being simple. The disadvantage, however, is the time needed to stabilize as a new conversion value is directly proportional to the rate at which the analog signal changes.
== Charge-redistribution successive-approximation ADC ==
One of the most common SAR ADC implementations uses a charge-scaling DAC consisting of an array of individually-switched capacitors sized in powers of two and an additional duplicate of the smallest capacitor, for a total of N+1 capacitors for N bits. Thus if the largest capacitance is C, then the array's total capacitance is 2C. The switched capacitor array acts as both the sample-and-hold element and the DAC. Redistributing their charge will adjust their net voltage, which is fed into the negative input of a comparator (whose positive input is always grounded) to perform the binary search using the following steps:
Discharge: The capacitors are discharged. (Note, discharging to comparator's offset voltage will automatically provide offset cancellation.)
Sampling: The capacitors are switched to the input signal Vin. After a brief sampling period, the capacitors will hold a charge equal to their respective capacitance times Vin (and minus the offset voltage upon each of them), so the array holds a total charge of 2C·Vin.
Hold: The capacitors are then switched to ground. This provides the comparator's negative input with a voltage of −Vin.
Conversion: the actual conversion process proceeds with the following steps in each iteration, starting with the largest capacitor as the test capacitor for the MSB, and then testing each next smaller capacitor in order for each bit of lower significance:
Redistribution: The current test capacitor is switched to Vref. The test capacitor forms a charge divider with the remainder of the array whose ratio depends on the capacitor's relative size. In the first iteration, the ratio is 1:1, so the comparator's negative input becomes −Vin + Vref⁄2. On the ith iteration, the ratio will be 1:2i−1, so the ith iteration of this redistribution step effectively adds Vref⁄2i to the voltage.
Comparison: The comparator's output determines the bit's value for to the current test capacitor. In the first iteration, if Vin is greater than Vref⁄2, then the comparator will output a digital 1 and otherwise output a digital 0.
Update Switch: A digital 0 result will leave the current test capacitor connected to Vref for subsequent iterations, while a digital 1 result will switch the capacitor back to ground. Thus, each ith iteration may or may not add Vref⁄2i to the comparator's negative input voltage. For instance, the voltage at the end of the first iteration will be −Vin + MSB·Vref⁄2.
End Of Conversion: After all capacitors are tested in the same manner, the comparator's negative input voltage will have converged as close as possible (given the resolution of the DAC) to the comparator's offset voltage.
== See also ==
Quantization noise
Digital-to-analog converter
== References ==
== Further reading ==
CMOS Circuit Design, Layout, and Simulation, 3rd Edition; R. J. Baker; Wiley-IEEE; 1208 pages; 2010; ISBN 978-0-470-88132-3
Data Conversion Handbook; Analog Devices; Newnes; 976 pages; 2004; ISBN 978-0750678414
== External links ==
Understanding SAR ADCs: Their Architecture and Comparison with Other ADCs - Maxim
Choose the right A/D converter for your application - TI | Wikipedia/Successive-approximation_ADC |
For small angles, the trigonometric functions sine, cosine, and tangent can be calculated with reasonable accuracy by the following simple approximations:
sin
θ
≈
tan
θ
≈
θ
,
cos
θ
≈
1
−
1
2
θ
2
≈
1
,
{\displaystyle {\begin{aligned}\sin \theta &\approx \tan \theta \approx \theta ,\\[5mu]\cos \theta &\approx 1-{\tfrac {1}{2}}\theta ^{2}\approx 1,\end{aligned}}}
provided the angle is measured in radians. Angles measured in degrees must first be converted to radians by multiplying them by
π
/
180
{\displaystyle \pi /180}
.
These approximations have a wide range of uses in branches of physics and engineering, including mechanics, electromagnetism, optics, cartography, astronomy, and computer science. One reason for this is that they can greatly simplify differential equations that do not need to be answered with absolute precision.
There are a number of ways to demonstrate the validity of the small-angle approximations. The most direct method is to truncate the Maclaurin series for each of the trigonometric functions. Depending on the order of the approximation,
cos
θ
{\displaystyle \textstyle \cos \theta }
is approximated as either
1
{\displaystyle 1}
or as
1
−
1
2
θ
2
{\textstyle 1-{\frac {1}{2}}\theta ^{2}}
.
== Justifications ==
=== Geometric ===
For a small angle, H and A are almost the same length, and therefore cos θ is nearly 1. The segment d (in red to the right) is the difference between the lengths of the hypotenuse, H, and the adjacent side, A, and has length
H
−
H
2
−
O
2
{\displaystyle \textstyle H-{\sqrt {H^{2}-O^{2}}}}
, which for small angles is approximately equal to
O
2
/
2
H
≈
1
2
θ
2
H
{\displaystyle \textstyle O^{2}\!/2H\approx {\tfrac {1}{2}}\theta ^{2}H}
. As a second-order approximation,
cos
θ
≈
1
−
θ
2
2
.
{\displaystyle \cos {\theta }\approx 1-{\frac {\theta ^{2}}{2}}.}
The opposite leg, O, is approximately equal to the length of the blue arc, s. The arc s has length θA, and by definition sin θ = O/H and tan θ = O/A, and for a small angle, O ≈ s and H ≈ A, which leads to:
sin
θ
=
O
H
≈
O
A
=
tan
θ
=
O
A
≈
s
A
=
A
θ
A
=
θ
.
{\displaystyle \sin \theta ={\frac {O}{H}}\approx {\frac {O}{A}}=\tan \theta ={\frac {O}{A}}\approx {\frac {s}{A}}={\frac {A\theta }{A}}=\theta .}
Or, more concisely,
sin
θ
≈
tan
θ
≈
θ
.
{\displaystyle \sin \theta \approx \tan \theta \approx \theta .}
=== Calculus ===
Using the squeeze theorem, we can prove that
lim
θ
→
0
sin
(
θ
)
θ
=
1
,
{\displaystyle \lim _{\theta \to 0}{\frac {\sin(\theta )}{\theta }}=1,}
which is a formal restatement of the approximation
sin
(
θ
)
≈
θ
{\displaystyle \sin(\theta )\approx \theta }
for small values of θ.
A more careful application of the squeeze theorem proves that
lim
θ
→
0
tan
(
θ
)
θ
=
1
,
{\displaystyle \lim _{\theta \to 0}{\frac {\tan(\theta )}{\theta }}=1,}
from which we conclude that
tan
(
θ
)
≈
θ
{\displaystyle \tan(\theta )\approx \theta }
for small values of θ.
Finally, L'Hôpital's rule tells us that
lim
θ
→
0
cos
(
θ
)
−
1
θ
2
=
lim
θ
→
0
−
sin
(
θ
)
2
θ
=
−
1
2
,
{\displaystyle \lim _{\theta \to 0}{\frac {\cos(\theta )-1}{\theta ^{2}}}=\lim _{\theta \to 0}{\frac {-\sin(\theta )}{2\theta }}=-{\frac {1}{2}},}
which rearranges to
cos
(
θ
)
≈
1
−
θ
2
2
{\textstyle \cos(\theta )\approx 1-{\frac {\theta ^{2}}{2}}}
for small values of θ. Alternatively, we can use the double angle formula
cos
2
A
≡
1
−
2
sin
2
A
{\displaystyle \cos 2A\equiv 1-2\sin ^{2}A}
. By letting
θ
=
2
A
{\displaystyle \theta =2A}
, we get that
cos
θ
=
1
−
2
sin
2
θ
2
≈
1
−
θ
2
2
{\textstyle \cos \theta =1-2\sin ^{2}{\frac {\theta }{2}}\approx 1-{\frac {\theta ^{2}}{2}}}
.
=== Algebraic ===
The Taylor series expansions of trigonometric functions sine, cosine, and tangent near zero are:
sin
θ
=
θ
−
1
6
θ
3
+
1
120
θ
5
−
⋯
,
cos
θ
=
1
−
1
2
θ
2
+
1
24
θ
4
−
⋯
,
tan
θ
=
θ
+
1
3
θ
3
+
2
15
θ
5
+
⋯
.
{\displaystyle {\begin{aligned}\sin \theta &=\theta -{\frac {1}{6}}\theta ^{3}+{\frac {1}{120}}\theta ^{5}-\cdots ,\\[6mu]\cos \theta &=1-{\frac {1}{2}}{\theta ^{2}}+{\frac {1}{24}}\theta ^{4}-\cdots ,\\[6mu]\tan \theta &=\theta +{\frac {1}{3}}\theta ^{3}+{\frac {2}{15}}\theta ^{5}+\cdots .\end{aligned}}}
where
θ
{\displaystyle \theta }
is the angle in radians. For very small angles, higher powers of
θ
{\displaystyle \theta }
become extremely small, for instance if
θ
=
0.01
{\displaystyle \theta =0.01}
, then
θ
3
=
0.000
001
{\displaystyle \theta ^{3}=0.000\,001}
, just one ten-thousandth of
θ
{\displaystyle \theta }
. Thus for many purposes it suffices to drop the cubic and higher terms and approximate the sine and tangent of a small angle using the radian measure of the angle,
sin
θ
≈
tan
θ
≈
θ
{\displaystyle \sin \theta \approx \tan \theta \approx \theta }
, and drop the quadratic term and approximate the cosine as
cos
θ
≈
1
{\displaystyle \cos \theta \approx 1}
.
If additional precision is needed the quadratic and cubic terms can also be included,
sin
θ
≈
θ
−
1
6
θ
3
{\displaystyle \sin \theta \approx \theta -{\tfrac {1}{6}}\theta ^{3}}
,
cos
θ
≈
1
−
1
2
θ
2
{\displaystyle \cos \theta \approx 1-{\tfrac {1}{2}}\theta ^{2}}
, and
tan
θ
≈
θ
+
1
3
θ
3
{\displaystyle \tan \theta \approx \theta +{\tfrac {1}{3}}\theta ^{3}}
.
==== Dual numbers ====
One may also use dual numbers, defined as numbers in the form
a
+
b
ε
{\displaystyle a+b\varepsilon }
, with
a
,
b
∈
R
{\displaystyle a,b\in \mathbb {R} }
and
ε
{\displaystyle \varepsilon }
satisfying by definition
ε
2
=
0
{\displaystyle \varepsilon ^{2}=0}
and
ε
≠
0
{\displaystyle \varepsilon \neq 0}
. By using the MacLaurin series of cosine and sine, one can show that
cos
(
θ
ε
)
=
1
{\displaystyle \cos(\theta \varepsilon )=1}
and
sin
(
θ
ε
)
=
θ
ε
{\displaystyle \sin(\theta \varepsilon )=\theta \varepsilon }
. Furthermore, it is not hard to prove that the Pythagorean identity holds:
sin
2
(
θ
ε
)
+
cos
2
(
θ
ε
)
=
(
θ
ε
)
2
+
1
2
=
θ
2
ε
2
+
1
=
θ
2
⋅
0
+
1
=
1
{\displaystyle \sin ^{2}(\theta \varepsilon )+\cos ^{2}(\theta \varepsilon )=(\theta \varepsilon )^{2}+1^{2}=\theta ^{2}\varepsilon ^{2}+1=\theta ^{2}\cdot 0+1=1}
== Error of the approximations ==
Near zero, the relative error of the approximations
cos
θ
≈
1
{\displaystyle \cos \theta \approx 1}
,
sin
θ
≈
θ
{\displaystyle \sin \theta \approx \theta }
, and
tan
θ
≈
θ
{\displaystyle \tan \theta \approx \theta }
is quadratic in
θ
{\displaystyle \theta }
: for each order of magnitude smaller the angle is, the relative error of these approximations shrinks by two orders of magnitude. The approximation
cos
θ
≈
1
−
1
2
θ
2
{\displaystyle \textstyle \cos \theta \approx 1-{\tfrac {1}{2}}\theta ^{2}}
has relative error which is quartic in
θ
{\displaystyle \theta }
: for each order of magnitude smaller the angle is, the relative error shrinks by four orders of magnitude.
Figure 3 shows the relative errors of the small angle approximations. The angles at which the relative error exceeds 1% are as follows:
cos
θ
≈
1
{\displaystyle \cos \theta \approx 1}
at about 0.14 radians (8.1°)
tan
θ
≈
θ
{\displaystyle \tan \theta \approx \theta }
at about 0.17 radians (9.9°)
sin
θ
≈
θ
{\displaystyle \sin \theta \approx \theta }
at about 0.24 radians (14.0°)
cos
θ
≈
1
−
1
2
θ
2
{\displaystyle \textstyle \cos \theta \approx 1-{\tfrac {1}{2}}\theta ^{2}}
at about 0.66 radians (37.9°)
== Slide-rule approximations ==
Many slide rules – especially "trig" and higher models – include an "ST" (sines and tangents) or "SRT" (sines, radians, and tangents) scale on the front or back of the slide, for computing with sines and tangents of angles smaller than about 0.1 radian.
The right-hand end of the ST or SRT scale cannot be accurate to three decimal places for both arcsine(0.1) = 5.74 degrees and arctangent(0.1) = 5.71 degrees, so sines and tangents of angles near 5 degrees are given with somewhat worse than the usual expected "slide-rule accuracy". Some slide rules, such as the K&E Deci-Lon in the photo, calibrate to be accurate for radian conversion, at 5.73 degrees (off by nearly 0.4% for the tangent and 0.2% for the sine for angles around 5 degrees). Others are calibrated to 5.725 degrees, to balance the sine and tangent errors at below 0.3%.
== Angle sum and difference ==
The angle addition and subtraction theorems reduce to the following when one of the angles is small (β ≈ 0):
== Specific uses ==
=== Astronomy ===
In astronomy, the angular size or angle subtended by the image of a distant object is often only a few arcseconds (denoted by the symbol ″), so it is well suited to the small angle approximation. The linear size (D) is related to the angular size (X) and the distance from the observer (d) by the simple formula:
D
=
X
d
206
265
″
{\displaystyle D=X{\frac {d}{206\,265{''}}}}
where X is measured in arcseconds.
The quantity 206265″ is approximately equal to the number of arcseconds in a circle (1296000″), divided by 2π, or, the number of arcseconds in 1 radian.
The exact formula is
D
=
d
tan
(
X
2
π
1
296
000
″
)
{\displaystyle D=d\tan \left(X{\frac {2\pi }{1\,296\,000{''}}}\right)}
and the above approximation follows when tan X is replaced by X.
For example, the parsec is defined by the value of d when D=1 AU, X=1 arcsecond, but the definition used is the small-angle approximation (the first equation above).
=== Motion of a pendulum ===
The second-order cosine approximation is especially useful in calculating the potential energy of a pendulum, which can then be applied with a Lagrangian to find the indirect (energy) equation of motion. When calculating the period of a simple pendulum, the small-angle approximation for sine is used to allow the resulting differential equation to be solved easily by comparison with the differential equation describing simple harmonic motion.
=== Optics ===
In optics, the small-angle approximations form the basis of the paraxial approximation.
=== Wave interference ===
The sine and tangent small-angle approximations are used in relation to the double-slit experiment or a diffraction grating to develop simplified equations like the following, where y is the distance of a fringe from the center of maximum light intensity, m is the order of the fringe, D is the distance between the slits and projection screen, and d is the distance between the slits:
y
≈
m
λ
D
d
{\displaystyle y\approx {\frac {m\lambda D}{d}}}
=== Structural mechanics ===
The small-angle approximation also appears in structural mechanics, especially in stability and bifurcation analyses (mainly of axially-loaded columns ready to undergo buckling). This leads to significant simplifications, though at a cost in accuracy and insight into the true behavior.
=== Piloting ===
The 1 in 60 rule used in air navigation has its basis in the small-angle approximation, plus the fact that one radian is approximately 60 degrees.
=== Interpolation ===
The formulas for addition and subtraction involving a small angle may be used for interpolating between trigonometric table values:
Example: sin(0.755)
sin
(
0.755
)
=
sin
(
0.75
+
0.005
)
≈
sin
(
0.75
)
+
(
0.005
)
cos
(
0.75
)
≈
(
0.6816
)
+
(
0.005
)
(
0.7317
)
≈
0.6853.
{\displaystyle {\begin{aligned}\sin(0.755)&=\sin(0.75+0.005)\\&\approx \sin(0.75)+(0.005)\cos(0.75)\\&\approx (0.6816)+(0.005)(0.7317)\\&\approx 0.6853.\end{aligned}}}
where the values for sin(0.75) and cos(0.75) are obtained from trigonometric table. The result is accurate to the four digits given.
== See also ==
Skinny triangle
Versine
Exsecant
== References == | Wikipedia/Small-angle_approximation |
In philosophy of science, idealization is the process by which scientific models assume facts about the phenomenon being modeled that are strictly false but make models easier to understand or solve. That is, it is determined whether the phenomenon approximates an "ideal case," then the model is applied to make a prediction based on that ideal case.
If an approximation is accurate, the model will have high predictive power; for example, it is not usually necessary to account for air resistance when determining the acceleration of a falling bowling ball, and doing so would be more complicated. In this case, air resistance is idealized to be zero. Although this is not strictly true, it is a good approximation because its effect is negligible compared to that of gravity.
Idealizations may allow predictions to be made when none otherwise could be. For example, the approximation of air resistance as zero was the only option before the formulation of Stokes' law allowed the calculation of drag forces. Many debates surrounding the usefulness of a particular model are about the appropriateness of different idealizations.
== Early use ==
Galileo used the concept of idealization in order to formulate the law of free fall. Galileo, in his study of bodies in motion, set up experiments that assumed frictionless surfaces and spheres of perfect roundness. The crudity of ordinary objects has the potential to obscure their mathematical essence, and idealization is used to combat this tendency.
The most well-known example of idealization in Galileo's experiments is in his analysis of motion. Galileo predicted that if a perfectly round and smooth ball were rolled along a perfectly smooth horizontal plane, there would be nothing to stop the ball (in fact, it would slide instead of roll, because rolling requires friction). This hypothesis is predicated on the assumption that there is no air resistance.
== Other examples ==
=== Mathematics ===
Geometry involves the process of idealization because it studies ideal entities, forms and figures. Perfect circles, spheres, straight lines and angles are abstractions that help us think about and investigate the world.
=== Science ===
An example of the use of idealization in physics is in Boyle's Gas Law:
Given any x and any y, if all the molecules in y are perfectly elastic and spherical, possess equal masses and volumes, have negligible size, and exert no forces on one another except during collisions, then if x is a gas and y is a given mass of x which is trapped in a vessel of variable size and the temperature of y is kept constant, then any decrease of the volume of y increases the pressure of y proportionally, and vice versa.
In physics, people will often solve for Newtonian systems without friction. While we know that friction is present in actual systems, solving the model without friction can provide insights to the behavior of actual systems where the force of friction is negligible.
=== Social science ===
It has been argued by the "Poznań School" (in Poland) that Karl Marx used idealization in the social sciences (see the works written by Leszek Nowak). Similarly, in economic models individuals are assumed to make maximally rational choices. This assumption, although known to be violated by actual humans, can often lead to insights about the behavior of human populations.
In psychology, idealization refers to a defence mechanism in which a person perceives another to be better (or have more desirable attributes) than would actually be supported by the evidence. This sometimes occurs in child custody conflicts. The child of a single parent frequently may imagine ("idealize") the (ideal) absent parent to have those characteristics of a perfect parent. However, the child may find imagination is favorable to reality. Upon meeting that parent, the child may be happy for a while, but disappointed later when learning that the parent does not actually nurture, support and protect as the former caretaker parent had.
A notable proponent of idealization in both the natural sciences and the social sciences was the economist Milton Friedman. In his view, the standard by which we should assess an empirical theory is the accuracy of the predictions that that theory makes. This amounts to an instrumentalist conception of science, including social science. He also argues against the criticism that we should reject an empirical theory if we find that the assumptions of that theory are not realistic, in the sense of being imperfect descriptions of reality. This criticism is wrongheaded, Friedman claims, because the assumptions of any empirical theory are necessarily unrealistic, since such a theory must abstract from the particular details of each instance of the phenomenon that the theory seeks to explain. This leads him to the conclusion that “[t]ruly important and significant hypotheses will be found to have ‘assumptions’ that are wildly inaccurate descriptive representations of reality, and, in general, the more significant the theory, the more unrealistic the assumptions (in this sense).” Consistently with this, he makes the case for seeing the assumptions of neoclassical positive economics as not importantly different from the idealizations that are employed in natural science, drawing a comparison between treating a falling body as if it were falling in a vacuum and viewing firms as if they were rational actors seeking to maximize expected returns.
Against this instrumentalist conception, which judges empirical theories on the basis of their predictive success, the social theorist Jon Elster has argued that an explanation in the social sciences is more convincing when it ‘opens the black box’ — that is to say, when the explanation specifies a chain of events leading from the independent variable to the dependent variable. The more detailed this chain, argues Elster, the less likely it is that the explanation specifying that chain is neglecting a hidden variable that could account for both the independent variable and the dependent variable. Relatedly, he also contends that social-scientific explanations should be formulated in terms of causal mechanisms, which he defines as “frequently occurring and easily recognizable causal patterns that are triggered under generally unknown conditions or with indeterminate consequences.” All this informs Elster's disagreement with rational-choice theory in general and Friedman in particular. On Elster's analysis, Friedman is right to argue that criticizing the assumptions of an empirical theory as unrealistic is misguided, but he is mistaken to defend on this basis the value of rational-choice theory in social science (especially economics). Elster presents two reasons for why this is the case: first, because rational-choice theory does not illuminate “a mechanism that brings about non-intentionally the same outcome that a superrational agent could have calculated intentionally”, a mechanism “that would simulate rationality”; and second, because rational-choice explanations do not provide precise, pinpoint predictions, comparable to those of quantum mechanics. When a theory can predict outcomes that precisely, then, Elster contends, we have reason to believe that theory is true. Accordingly, Elster wonders whether the as-if assumptions of rational-choice theory help explain any social or political phenomenon.
=== Science education ===
In science education, idealized science can be thought of as engaging students in the practices of science and doing so authentically, which means allowing for the messiness of scientific work without needing to be immersed in the complexity of professional science and its esoteric content. This helps the student develop the mindset of a scientist as well as their habits and dispositions. Idealized science is especially important for learning science because of the deeply cognitively and materially distributed nature of modern science, where most science is done by larger groups of scientists. One example is a 2016 gravitational waves paper listing over a thousand authors and more than a hundred science institutions. By simplifying the content, students can engage in all aspects of scientific work and not just add one small piece of the whole project. Idealized Science also helps to dispel the notion that science simply follows a single set scientific method. Instead, idealized science provides a framework for the iterative nature of scientific work, the reliance on critique, and the social aspects that help continually guide the work.
== Limits on use ==
While idealization is used extensively by certain scientific disciplines, it has been rejected by others. For instance, Edmund Husserl recognized the importance of idealization but opposed its application to the study of the mind, holding that mental phenomena do not lend themselves to idealization.
Although idealization is considered one of the essential elements of modern science, it is nonetheless the source of continued controversy in the literature of the philosophy of science. For example, Nancy Cartwright suggested that Galilean idealization presupposes tendencies or capacities in nature and that this allows for generalization beyond what is the ideal case.
There is continued philosophical concern over how Galileo's idealization method assists in the description of the behavior of individuals or objects in the real world. Since the laws created through idealization (such as the ideal gas law) describe only the behavior of ideal bodies, these laws can only be used to predict the behavior of real bodies when a considerable number of factors have been physically eliminated (e.g. through shielding conditions) or ignored. Laws that account for these factors are usually more complicated and in some cases have not yet been developed.
== See also ==
Spherical cow
== References ==
== Further reading ==
William F, Barr, A Pragmatic Analysis of Idealization in Physics, Philosophy of Science, Vol. 41, No. 1, pg 48, Mar. 1974.
Krzysztof Brzechczyn, (ed.), Idealization XIII: Modeling in History, Amsterdam-New York: Rodopi, 2009.
Nancy Cartwright, How the Laws of Physics Lie, Clarendon Press:Oxford 1983
Francesco Coniglione, Between Abstraction and Idealization: Scientific Practice and Philosophical Awareness, in F. Coniglione, R. Poli and R. Rollinger (Eds.), Idealization XI: Historical Studies on Abstraction, Atlanta-Amsterdam:Rodopi 2004, pp. 59–110.
Craig Dilworth, The Metaphysics of Science: An Account of Modern Science in Terms of Principles, Laws and Theories, Springer:Dordrecht 2007 (2a ed.)
Andrzej Klawiter, Why Did Husserl Not Become the Galileo of the Science of Consciousness?, in F. Coniglione, R. Poli and R. Rollinger, (Eds.), Idealization XI: Historical Studies on Abstraction, Poznań Studies in the Philosophy of the Sciences and the Humanities, Vol. 82, Rodopi:Atlanta-Amsterdam 2004, pp. 253–271.
Mansoor Niaz, The Role of Idealization in Science and Its Implications for Science Education, Journal of Science Education and Technology, Vol. 8, No. 2, 1999, pp. 145–150.
Leszek Nowak, The Structure of Idealization. Towards a Systematic Interpretation of the Marxian Idea of Science, Dordrecht:Reidel 1980
Leszek Nowak and Izabella Nowakowa, Idealization X: The Richness of Idealization, Amsterdam / Atlanta: Rodopi 2000. | Wikipedia/Idealization_(philosophy_of_science) |
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.
The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent functions. Their reciprocals are respectively the cosecant, the secant, and the cotangent functions, which are less used. Each of these six trigonometric functions has a corresponding inverse function, and an analog among the hyperbolic functions.
The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extend the sine and cosine functions to functions whose domain is the whole real line, geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used; then the domain of the other functions is the real line with some isolated points removed. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations. This allows extending the domain of sine and cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane with some isolated points removed.
== Notation ==
Conventionally, an abbreviation of each trigonometric function's name is used as its symbol in formulas. Today, the most common versions of these abbreviations are "sin" for sine, "cos" for cosine, "tan" or "tg" for tangent, "sec" for secant, "csc" or "cosec" for cosecant, and "cot" or "ctg" for cotangent. Historically, these abbreviations were first used in prose sentences to indicate particular line segments or their lengths related to an arc of an arbitrary circle, and later to indicate ratios of lengths, but as the function concept developed in the 17th–18th century, they began to be considered as functions of real-number-valued angle measures, and written with functional notation, for example sin(x). Parentheses are still often omitted to reduce clutter, but are sometimes necessary; for example the expression
sin
x
+
y
{\displaystyle \sin x+y}
would typically be interpreted to mean
(
sin
x
)
+
y
,
{\displaystyle (\sin x)+y,}
so parentheses are required to express
sin
(
x
+
y
)
.
{\displaystyle \sin(x+y).}
A positive integer appearing as a superscript after the symbol of the function denotes exponentiation, not function composition. For example
sin
2
x
{\displaystyle \sin ^{2}x}
and
sin
2
(
x
)
{\displaystyle \sin ^{2}(x)}
denote
(
sin
x
)
2
,
{\displaystyle (\sin x)^{2},}
not
sin
(
sin
x
)
.
{\displaystyle \sin(\sin x).}
This differs from the (historically later) general functional notation in which
f
2
(
x
)
=
(
f
∘
f
)
(
x
)
=
f
(
f
(
x
)
)
.
{\displaystyle f^{2}(x)=(f\circ f)(x)=f(f(x)).}
In contrast, the superscript
−
1
{\displaystyle -1}
is commonly used to denote the inverse function, not the reciprocal. For example
sin
−
1
x
{\displaystyle \sin ^{-1}x}
and
sin
−
1
(
x
)
{\displaystyle \sin ^{-1}(x)}
denote the inverse trigonometric function alternatively written
arcsin
x
.
{\displaystyle \arcsin x\,.}
The equation
θ
=
sin
−
1
x
{\displaystyle \theta =\sin ^{-1}x}
implies
sin
θ
=
x
,
{\displaystyle \sin \theta =x,}
not
θ
⋅
sin
x
=
1.
{\displaystyle \theta \cdot \sin x=1.}
In this case, the superscript could be considered as denoting a composed or iterated function, but negative superscripts other than
−
1
{\displaystyle {-1}}
are not in common use.
== Right-angled triangle definitions ==
If the acute angle θ is given, then any right triangles that have an angle of θ are similar to each other. This means that the ratio of any two side lengths depends only on θ. Thus these six ratios define six functions of θ, which are the trigonometric functions. In the following definitions, the hypotenuse is the length of the side opposite the right angle, opposite represents the side opposite the given angle θ, and adjacent represents the side between the angle θ and the right angle.
Various mnemonics can be used to remember these definitions.
In a right-angled triangle, the sum of the two acute angles is a right angle, that is, 90° or π/2 radians. Therefore
sin
(
θ
)
{\displaystyle \sin(\theta )}
and
cos
(
90
∘
−
θ
)
{\displaystyle \cos(90^{\circ }-\theta )}
represent the same ratio, and thus are equal. This identity and analogous relationships between the other trigonometric functions are summarized in the following table.
== Radians versus degrees ==
In geometric applications, the argument of a trigonometric function is generally the measure of an angle. For this purpose, any angular unit is convenient. One common unit is degrees, in which a right angle is 90° and a complete turn is 360° (particularly in elementary mathematics).
However, in calculus and mathematical analysis, the trigonometric functions are generally regarded more abstractly as functions of real or complex numbers, rather than angles. In fact, the functions sin and cos can be defined for all complex numbers in terms of the exponential function, via power series, or as solutions to differential equations given particular initial values (see below), without reference to any geometric notions. The other four trigonometric functions (tan, cot, sec, csc) can be defined as quotients and reciprocals of sin and cos, except where zero occurs in the denominator. It can be proved, for real arguments, that these definitions coincide with elementary geometric definitions if the argument is regarded as an angle in radians. Moreover, these definitions result in simple expressions for the derivatives and indefinite integrals for the trigonometric functions. Thus, in settings beyond elementary geometry, radians are regarded as the mathematically natural unit for describing angle measures.
When radians (rad) are employed, the angle is given as the length of the arc of the unit circle subtended by it: the angle that subtends an arc of length 1 on the unit circle is 1 rad (≈ 57.3°), and a complete turn (360°) is an angle of 2π (≈ 6.28) rad. For real number x, the notation sin x, cos x, etc. refers to the value of the trigonometric functions evaluated at an angle of x rad. If units of degrees are intended, the degree sign must be explicitly shown (sin x°, cos x°, etc.). Using this standard notation, the argument x for the trigonometric functions satisfies the relationship x = (180x/π)°, so that, for example, sin π = sin 180° when we take x = π. In this way, the degree symbol can be regarded as a mathematical constant such that 1° = π/180 ≈ 0.0175.
== Unit-circle definitions ==
The six trigonometric functions can be defined as coordinate values of points on the Euclidean plane that are related to the unit circle, which is the circle of radius one centered at the origin O of this coordinate system. While right-angled triangle definitions allow for the definition of the trigonometric functions for angles between 0 and
π
2
{\textstyle {\frac {\pi }{2}}}
radians (90°), the unit circle definitions allow the domain of trigonometric functions to be extended to all positive and negative real numbers.
Let
L
{\displaystyle {\mathcal {L}}}
be the ray obtained by rotating by an angle θ the positive half of the x-axis (counterclockwise rotation for
θ
>
0
,
{\displaystyle \theta >0,}
and clockwise rotation for
θ
<
0
{\displaystyle \theta <0}
). This ray intersects the unit circle at the point
A
=
(
x
A
,
y
A
)
.
{\displaystyle \mathrm {A} =(x_{\mathrm {A} },y_{\mathrm {A} }).}
The ray
L
,
{\displaystyle {\mathcal {L}},}
extended to a line if necessary, intersects the line of equation
x
=
1
{\displaystyle x=1}
at point
B
=
(
1
,
y
B
)
,
{\displaystyle \mathrm {B} =(1,y_{\mathrm {B} }),}
and the line of equation
y
=
1
{\displaystyle y=1}
at point
C
=
(
x
C
,
1
)
.
{\displaystyle \mathrm {C} =(x_{\mathrm {C} },1).}
The tangent line to the unit circle at the point A, is perpendicular to
L
,
{\displaystyle {\mathcal {L}},}
and intersects the y- and x-axes at points
D
=
(
0
,
y
D
)
{\displaystyle \mathrm {D} =(0,y_{\mathrm {D} })}
and
E
=
(
x
E
,
0
)
.
{\displaystyle \mathrm {E} =(x_{\mathrm {E} },0).}
The coordinates of these points give the values of all trigonometric functions for any arbitrary real value of θ in the following manner.
The trigonometric functions cos and sin are defined, respectively, as the x- and y-coordinate values of point A. That is,
cos
θ
=
x
A
{\displaystyle \cos \theta =x_{\mathrm {A} }\quad }
and
sin
θ
=
y
A
.
{\displaystyle \quad \sin \theta =y_{\mathrm {A} }.}
In the range
0
≤
θ
≤
π
/
2
{\displaystyle 0\leq \theta \leq \pi /2}
, this definition coincides with the right-angled triangle definition, by taking the right-angled triangle to have the unit radius OA as hypotenuse. And since the equation
x
2
+
y
2
=
1
{\displaystyle x^{2}+y^{2}=1}
holds for all points
P
=
(
x
,
y
)
{\displaystyle \mathrm {P} =(x,y)}
on the unit circle, this definition of cosine and sine also satisfies the Pythagorean identity.
cos
2
θ
+
sin
2
θ
=
1.
{\displaystyle \cos ^{2}\theta +\sin ^{2}\theta =1.}
The other trigonometric functions can be found along the unit circle as
tan
θ
=
y
B
{\displaystyle \tan \theta =y_{\mathrm {B} }\quad }
and
cot
θ
=
x
C
,
{\displaystyle \quad \cot \theta =x_{\mathrm {C} },}
csc
θ
=
y
D
{\displaystyle \csc \theta \ =y_{\mathrm {D} }\quad }
and
sec
θ
=
x
E
.
{\displaystyle \quad \sec \theta =x_{\mathrm {E} }.}
By applying the Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is
tan
θ
=
sin
θ
cos
θ
,
cot
θ
=
cos
θ
sin
θ
,
sec
θ
=
1
cos
θ
,
csc
θ
=
1
sin
θ
.
{\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }},\quad \cot \theta ={\frac {\cos \theta }{\sin \theta }},\quad \sec \theta ={\frac {1}{\cos \theta }},\quad \csc \theta ={\frac {1}{\sin \theta }}.}
Since a rotation of an angle of
±
2
π
{\displaystyle \pm 2\pi }
does not change the position or size of a shape, the points A, B, C, D, and E are the same for two angles whose difference is an integer multiple of
2
π
{\displaystyle 2\pi }
. Thus trigonometric functions are periodic functions with period
2
π
{\displaystyle 2\pi }
. That is, the equalities
sin
θ
=
sin
(
θ
+
2
k
π
)
{\displaystyle \sin \theta =\sin \left(\theta +2k\pi \right)\quad }
and
cos
θ
=
cos
(
θ
+
2
k
π
)
{\displaystyle \quad \cos \theta =\cos \left(\theta +2k\pi \right)}
hold for any angle θ and any integer k. The same is true for the four other trigonometric functions. By observing the sign and the monotonicity of the functions sine, cosine, cosecant, and secant in the four quadrants, one can show that
2
π
{\displaystyle 2\pi }
is the smallest value for which they are periodic (i.e.,
2
π
{\displaystyle 2\pi }
is the fundamental period of these functions). However, after a rotation by an angle
π
{\displaystyle \pi }
, the points B and C already return to their original position, so that the tangent function and the cotangent function have a fundamental period of
π
{\displaystyle \pi }
. That is, the equalities
tan
θ
=
tan
(
θ
+
k
π
)
{\displaystyle \tan \theta =\tan(\theta +k\pi )\quad }
and
cot
θ
=
cot
(
θ
+
k
π
)
{\displaystyle \quad \cot \theta =\cot(\theta +k\pi )}
hold for any angle θ and any integer k.
== Algebraic values ==
The algebraic expressions for the most important angles are as follows:
sin
0
=
sin
0
∘
=
0
2
=
0
{\displaystyle \sin 0=\sin 0^{\circ }\quad ={\frac {\sqrt {0}}{2}}=0}
(zero angle)
sin
π
6
=
sin
30
∘
=
1
2
=
1
2
{\displaystyle \sin {\frac {\pi }{6}}=\sin 30^{\circ }={\frac {\sqrt {1}}{2}}={\frac {1}{2}}}
sin
π
4
=
sin
45
∘
=
2
2
=
1
2
{\displaystyle \sin {\frac {\pi }{4}}=\sin 45^{\circ }={\frac {\sqrt {2}}{2}}={\frac {1}{\sqrt {2}}}}
sin
π
3
=
sin
60
∘
=
3
2
{\displaystyle \sin {\frac {\pi }{3}}=\sin 60^{\circ }={\frac {\sqrt {3}}{2}}}
sin
π
2
=
sin
90
∘
=
4
2
=
1
{\displaystyle \sin {\frac {\pi }{2}}=\sin 90^{\circ }={\frac {\sqrt {4}}{2}}=1}
(right angle)
Writing the numerators as square roots of consecutive non-negative integers, with a denominator of 2, provides an easy way to remember the values.
Such simple expressions generally do not exist for other angles which are rational multiples of a right angle.
For an angle which, measured in degrees, is a multiple of three, the exact trigonometric values of the sine and the cosine may be expressed in terms of square roots. These values of the sine and the cosine may thus be constructed by ruler and compass.
For an angle of an integer number of degrees, the sine and the cosine may be expressed in terms of square roots and the cube root of a non-real complex number. Galois theory allows a proof that, if the angle is not a multiple of 3°, non-real cube roots are unavoidable.
For an angle which, expressed in degrees, is a rational number, the sine and the cosine are algebraic numbers, which may be expressed in terms of nth roots. This results from the fact that the Galois groups of the cyclotomic polynomials are cyclic.
For an angle which, expressed in degrees, is not a rational number, then either the angle or both the sine and the cosine are transcendental numbers. This is a corollary of Baker's theorem, proved in 1966.
If the sine of an angle is a rational number then the cosine is not necessarily a rational number, and vice-versa. However if the tangent of an angle is rational then both the sine and cosine of the double angle will be rational.
=== Simple algebraic values ===
The following table lists the sines, cosines, and tangents of multiples of 15 degrees from 0 to 90 degrees.
== Definitions in analysis ==
G. H. Hardy noted in his 1908 work A Course of Pure Mathematics that the definition of the trigonometric functions in terms of the unit circle is not satisfactory, because it depends implicitly on a notion of angle that can be measured by a real number. Thus in modern analysis, trigonometric functions are usually constructed without reference to geometry.
Various ways exist in the literature for defining the trigonometric functions in a manner suitable for analysis; they include:
Using the "geometry" of the unit circle, which requires formulating the arc length of a circle (or area of a sector) analytically.
By a power series, which is particularly well-suited to complex variables.
By using an infinite product expansion.
By inverting the inverse trigonometric functions, which can be defined as integrals of algebraic or rational functions.
As solutions of a differential equation.
=== Definition by differential equations ===
Sine and cosine can be defined as the unique solution to the initial value problem:
d
d
x
sin
x
=
cos
x
,
d
d
x
cos
x
=
−
sin
x
,
sin
(
0
)
=
0
,
cos
(
0
)
=
1.
{\displaystyle {\frac {d}{dx}}\sin x=\cos x,\ {\frac {d}{dx}}\cos x=-\sin x,\ \sin(0)=0,\ \cos(0)=1.}
Differentiating again,
d
2
d
x
2
sin
x
=
d
d
x
cos
x
=
−
sin
x
{\textstyle {\frac {d^{2}}{dx^{2}}}\sin x={\frac {d}{dx}}\cos x=-\sin x}
and
d
2
d
x
2
cos
x
=
−
d
d
x
sin
x
=
−
cos
x
{\textstyle {\frac {d^{2}}{dx^{2}}}\cos x=-{\frac {d}{dx}}\sin x=-\cos x}
, so both sine and cosine are solutions of the same ordinary differential equation
y
″
+
y
=
0
.
{\displaystyle y''+y=0\,.}
Sine is the unique solution with y(0) = 0 and y′(0) = 1; cosine is the unique solution with y(0) = 1 and y′(0) = 0.
One can then prove, as a theorem, that solutions
cos
,
sin
{\displaystyle \cos ,\sin }
are periodic, having the same period. Writing this period as
2
π
{\displaystyle 2\pi }
is then a definition of the real number
π
{\displaystyle \pi }
which is independent of geometry.
Applying the quotient rule to the tangent
tan
x
=
sin
x
/
cos
x
{\displaystyle \tan x=\sin x/\cos x}
,
d
d
x
tan
x
=
cos
2
x
+
sin
2
x
cos
2
x
=
1
+
tan
2
x
,
{\displaystyle {\frac {d}{dx}}\tan x={\frac {\cos ^{2}x+\sin ^{2}x}{\cos ^{2}x}}=1+\tan ^{2}x\,,}
so the tangent function satisfies the ordinary differential equation
y
′
=
1
+
y
2
.
{\displaystyle y'=1+y^{2}\,.}
It is the unique solution with y(0) = 0.
=== Power series expansion ===
The basic trigonometric functions can be defined by the following power series expansions. These series are also known as the Taylor series or Maclaurin series of these trigonometric functions:
sin
x
=
x
−
x
3
3
!
+
x
5
5
!
−
x
7
7
!
+
⋯
=
∑
n
=
0
∞
(
−
1
)
n
(
2
n
+
1
)
!
x
2
n
+
1
cos
x
=
1
−
x
2
2
!
+
x
4
4
!
−
x
6
6
!
+
⋯
=
∑
n
=
0
∞
(
−
1
)
n
(
2
n
)
!
x
2
n
.
{\displaystyle {\begin{aligned}\sin x&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \\[6mu]&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\\[8pt]\cos x&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \\[6mu]&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}.\end{aligned}}}
The radius of convergence of these series is infinite. Therefore, the sine and the cosine can be extended to entire functions (also called "sine" and "cosine"), which are (by definition) complex-valued functions that are defined and holomorphic on the whole complex plane.
Term-by-term differentiation shows that the sine and cosine defined by the series obey the differential equation discussed previously, and conversely one can obtain these series from elementary recursion relations derived from the differential equation.
Being defined as fractions of entire functions, the other trigonometric functions may be extended to meromorphic functions, that is functions that are holomorphic in the whole complex plane, except some isolated points called poles. Here, the poles are the numbers of the form
(
2
k
+
1
)
π
2
{\textstyle (2k+1){\frac {\pi }{2}}}
for the tangent and the secant, or
k
π
{\displaystyle k\pi }
for the cotangent and the cosecant, where k is an arbitrary integer.
Recurrences relations may also be computed for the coefficients of the Taylor series of the other trigonometric functions. These series have a finite radius of convergence. Their coefficients have a combinatorial interpretation: they enumerate alternating permutations of finite sets.
More precisely, defining
Un, the nth up/down number,
Bn, the nth Bernoulli number, and
En, is the nth Euler number,
one has the following series expansions:
tan
x
=
∑
n
=
0
∞
U
2
n
+
1
(
2
n
+
1
)
!
x
2
n
+
1
=
∑
n
=
1
∞
(
−
1
)
n
−
1
2
2
n
(
2
2
n
−
1
)
B
2
n
(
2
n
)
!
x
2
n
−
1
=
x
+
1
3
x
3
+
2
15
x
5
+
17
315
x
7
+
⋯
,
for
|
x
|
<
π
2
.
{\displaystyle {\begin{aligned}\tan x&{}=\sum _{n=0}^{\infty }{\frac {U_{2n+1}}{(2n+1)!}}x^{2n+1}\\[8mu]&{}=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}2^{2n}\left(2^{2n}-1\right)B_{2n}}{(2n)!}}x^{2n-1}\\[5mu]&{}=x+{\frac {1}{3}}x^{3}+{\frac {2}{15}}x^{5}+{\frac {17}{315}}x^{7}+\cdots ,\qquad {\text{for }}|x|<{\frac {\pi }{2}}.\end{aligned}}}
csc
x
=
∑
n
=
0
∞
(
−
1
)
n
+
1
2
(
2
2
n
−
1
−
1
)
B
2
n
(
2
n
)
!
x
2
n
−
1
=
x
−
1
+
1
6
x
+
7
360
x
3
+
31
15120
x
5
+
⋯
,
for
0
<
|
x
|
<
π
.
{\displaystyle {\begin{aligned}\csc x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n+1}2\left(2^{2n-1}-1\right)B_{2n}}{(2n)!}}x^{2n-1}\\[5mu]&=x^{-1}+{\frac {1}{6}}x+{\frac {7}{360}}x^{3}+{\frac {31}{15120}}x^{5}+\cdots ,\qquad {\text{for }}0<|x|<\pi .\end{aligned}}}
sec
x
=
∑
n
=
0
∞
U
2
n
(
2
n
)
!
x
2
n
=
∑
n
=
0
∞
(
−
1
)
n
E
2
n
(
2
n
)
!
x
2
n
=
1
+
1
2
x
2
+
5
24
x
4
+
61
720
x
6
+
⋯
,
for
|
x
|
<
π
2
.
{\displaystyle {\begin{aligned}\sec x&=\sum _{n=0}^{\infty }{\frac {U_{2n}}{(2n)!}}x^{2n}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}E_{2n}}{(2n)!}}x^{2n}\\[5mu]&=1+{\frac {1}{2}}x^{2}+{\frac {5}{24}}x^{4}+{\frac {61}{720}}x^{6}+\cdots ,\qquad {\text{for }}|x|<{\frac {\pi }{2}}.\end{aligned}}}
cot
x
=
∑
n
=
0
∞
(
−
1
)
n
2
2
n
B
2
n
(
2
n
)
!
x
2
n
−
1
=
x
−
1
−
1
3
x
−
1
45
x
3
−
2
945
x
5
−
⋯
,
for
0
<
|
x
|
<
π
.
{\displaystyle {\begin{aligned}\cot x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}2^{2n}B_{2n}}{(2n)!}}x^{2n-1}\\[5mu]&=x^{-1}-{\frac {1}{3}}x-{\frac {1}{45}}x^{3}-{\frac {2}{945}}x^{5}-\cdots ,\qquad {\text{for }}0<|x|<\pi .\end{aligned}}}
=== Continued fraction expansion ===
The following continued fractions are valid in the whole complex plane:
sin
x
=
x
1
+
x
2
2
⋅
3
−
x
2
+
2
⋅
3
x
2
4
⋅
5
−
x
2
+
4
⋅
5
x
2
6
⋅
7
−
x
2
+
⋱
{\displaystyle \sin x={\cfrac {x}{1+{\cfrac {x^{2}}{2\cdot 3-x^{2}+{\cfrac {2\cdot 3x^{2}}{4\cdot 5-x^{2}+{\cfrac {4\cdot 5x^{2}}{6\cdot 7-x^{2}+\ddots }}}}}}}}}
cos
x
=
1
1
+
x
2
1
⋅
2
−
x
2
+
1
⋅
2
x
2
3
⋅
4
−
x
2
+
3
⋅
4
x
2
5
⋅
6
−
x
2
+
⋱
{\displaystyle \cos x={\cfrac {1}{1+{\cfrac {x^{2}}{1\cdot 2-x^{2}+{\cfrac {1\cdot 2x^{2}}{3\cdot 4-x^{2}+{\cfrac {3\cdot 4x^{2}}{5\cdot 6-x^{2}+\ddots }}}}}}}}}
tan
x
=
x
1
−
x
2
3
−
x
2
5
−
x
2
7
−
⋱
=
1
1
x
−
1
3
x
−
1
5
x
−
1
7
x
−
⋱
{\displaystyle \tan x={\cfrac {x}{1-{\cfrac {x^{2}}{3-{\cfrac {x^{2}}{5-{\cfrac {x^{2}}{7-\ddots }}}}}}}}={\cfrac {1}{{\cfrac {1}{x}}-{\cfrac {1}{{\cfrac {3}{x}}-{\cfrac {1}{{\cfrac {5}{x}}-{\cfrac {1}{{\cfrac {7}{x}}-\ddots }}}}}}}}}
The last one was used in the historically first proof that π is irrational.
=== Partial fraction expansion ===
There is a series representation as partial fraction expansion where just translated reciprocal functions are summed up, such that the poles of the cotangent function and the reciprocal functions match:
π
cot
π
x
=
lim
N
→
∞
∑
n
=
−
N
N
1
x
+
n
.
{\displaystyle \pi \cot \pi x=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {1}{x+n}}.}
This identity can be proved with the Herglotz trick.
Combining the (–n)th with the nth term lead to absolutely convergent series:
π
cot
π
x
=
1
x
+
2
x
∑
n
=
1
∞
1
x
2
−
n
2
.
{\displaystyle \pi \cot \pi x={\frac {1}{x}}+2x\sum _{n=1}^{\infty }{\frac {1}{x^{2}-n^{2}}}.}
Similarly, one can find a partial fraction expansion for the secant, cosecant and tangent functions:
π
csc
π
x
=
∑
n
=
−
∞
∞
(
−
1
)
n
x
+
n
=
1
x
+
2
x
∑
n
=
1
∞
(
−
1
)
n
x
2
−
n
2
,
{\displaystyle \pi \csc \pi x=\sum _{n=-\infty }^{\infty }{\frac {(-1)^{n}}{x+n}}={\frac {1}{x}}+2x\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{x^{2}-n^{2}}},}
π
2
csc
2
π
x
=
∑
n
=
−
∞
∞
1
(
x
+
n
)
2
,
{\displaystyle \pi ^{2}\csc ^{2}\pi x=\sum _{n=-\infty }^{\infty }{\frac {1}{(x+n)^{2}}},}
π
sec
π
x
=
∑
n
=
0
∞
(
−
1
)
n
(
2
n
+
1
)
(
n
+
1
2
)
2
−
x
2
,
{\displaystyle \pi \sec \pi x=\sum _{n=0}^{\infty }(-1)^{n}{\frac {(2n+1)}{(n+{\tfrac {1}{2}})^{2}-x^{2}}},}
π
tan
π
x
=
2
x
∑
n
=
0
∞
1
(
n
+
1
2
)
2
−
x
2
.
{\displaystyle \pi \tan \pi x=2x\sum _{n=0}^{\infty }{\frac {1}{(n+{\tfrac {1}{2}})^{2}-x^{2}}}.}
=== Infinite product expansion ===
The following infinite product for the sine is due to Leonhard Euler, and is of great importance in complex analysis:
sin
z
=
z
∏
n
=
1
∞
(
1
−
z
2
n
2
π
2
)
,
z
∈
C
.
{\displaystyle \sin z=z\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{n^{2}\pi ^{2}}}\right),\quad z\in \mathbb {C} .}
This may be obtained from the partial fraction decomposition of
cot
z
{\displaystyle \cot z}
given above, which is the logarithmic derivative of
sin
z
{\displaystyle \sin z}
. From this, it can be deduced also that
cos
z
=
∏
n
=
1
∞
(
1
−
z
2
(
n
−
1
/
2
)
2
π
2
)
,
z
∈
C
.
{\displaystyle \cos z=\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{(n-1/2)^{2}\pi ^{2}}}\right),\quad z\in \mathbb {C} .}
=== Euler's formula and the exponential function ===
Euler's formula relates sine and cosine to the exponential function:
e
i
x
=
cos
x
+
i
sin
x
.
{\displaystyle e^{ix}=\cos x+i\sin x.}
This formula is commonly considered for real values of x, but it remains true for all complex values.
Proof: Let
f
1
(
x
)
=
cos
x
+
i
sin
x
,
{\displaystyle f_{1}(x)=\cos x+i\sin x,}
and
f
2
(
x
)
=
e
i
x
.
{\displaystyle f_{2}(x)=e^{ix}.}
One has
d
f
j
(
x
)
/
d
x
=
i
f
j
(
x
)
{\displaystyle df_{j}(x)/dx=if_{j}(x)}
for j = 1, 2. The quotient rule implies thus that
d
/
d
x
(
f
1
(
x
)
/
f
2
(
x
)
)
=
0
{\displaystyle d/dx\,(f_{1}(x)/f_{2}(x))=0}
. Therefore,
f
1
(
x
)
/
f
2
(
x
)
{\displaystyle f_{1}(x)/f_{2}(x)}
is a constant function, which equals 1, as
f
1
(
0
)
=
f
2
(
0
)
=
1.
{\displaystyle f_{1}(0)=f_{2}(0)=1.}
This proves the formula.
One has
e
i
x
=
cos
x
+
i
sin
x
e
−
i
x
=
cos
x
−
i
sin
x
.
{\displaystyle {\begin{aligned}e^{ix}&=\cos x+i\sin x\\[5pt]e^{-ix}&=\cos x-i\sin x.\end{aligned}}}
Solving this linear system in sine and cosine, one can express them in terms of the exponential function:
sin
x
=
e
i
x
−
e
−
i
x
2
i
cos
x
=
e
i
x
+
e
−
i
x
2
.
{\displaystyle {\begin{aligned}\sin x&={\frac {e^{ix}-e^{-ix}}{2i}}\\[5pt]\cos x&={\frac {e^{ix}+e^{-ix}}{2}}.\end{aligned}}}
When x is real, this may be rewritten as
cos
x
=
Re
(
e
i
x
)
,
sin
x
=
Im
(
e
i
x
)
.
{\displaystyle \cos x=\operatorname {Re} \left(e^{ix}\right),\qquad \sin x=\operatorname {Im} \left(e^{ix}\right).}
Most trigonometric identities can be proved by expressing trigonometric functions in terms of the complex exponential function by using above formulas, and then using the identity
e
a
+
b
=
e
a
e
b
{\displaystyle e^{a+b}=e^{a}e^{b}}
for simplifying the result.
Euler's formula can also be used to define the basic trigonometric function directly, as follows, using the language of topological groups. The set
U
{\displaystyle U}
of complex numbers of unit modulus is a compact and connected topological group, which has a neighborhood of the identity that is homeomorphic to the real line. Therefore, it is isomorphic as a topological group to the one-dimensional torus group
R
/
Z
{\displaystyle \mathbb {R} /\mathbb {Z} }
, via an isomorphism
e
:
R
/
Z
→
U
.
{\displaystyle e:\mathbb {R} /\mathbb {Z} \to U.}
In pedestrian terms
e
(
t
)
=
exp
(
2
π
i
t
)
{\displaystyle e(t)=\exp(2\pi it)}
, and this isomorphism is unique up to taking complex conjugates.
For a nonzero real number
a
{\displaystyle a}
(the base), the function
t
↦
e
(
t
/
a
)
{\displaystyle t\mapsto e(t/a)}
defines an isomorphism of the group
R
/
a
Z
→
U
{\displaystyle \mathbb {R} /a\mathbb {Z} \to U}
. The real and imaginary parts of
e
(
t
/
a
)
{\displaystyle e(t/a)}
are the cosine and sine, where
a
{\displaystyle a}
is used as the base for measuring angles. For example, when
a
=
2
π
{\displaystyle a=2\pi }
, we get the measure in radians, and the usual trigonometric functions. When
a
=
360
{\displaystyle a=360}
, we get the sine and cosine of angles measured in degrees.
Note that
a
=
2
π
{\displaystyle a=2\pi }
is the unique value at which the derivative
d
d
t
e
(
t
/
a
)
{\displaystyle {\frac {d}{dt}}e(t/a)}
becomes a unit vector with positive imaginary part at
t
=
0
{\displaystyle t=0}
. This fact can, in turn, be used to define the constant
2
π
{\displaystyle 2\pi }
.
=== Definition via integration ===
Another way to define the trigonometric functions in analysis is using integration. For a real number
t
{\displaystyle t}
, put
θ
(
t
)
=
∫
0
t
d
τ
1
+
τ
2
=
arctan
t
{\displaystyle \theta (t)=\int _{0}^{t}{\frac {d\tau }{1+\tau ^{2}}}=\arctan t}
where this defines this inverse tangent function. Also,
π
{\displaystyle \pi }
is defined by
1
2
π
=
∫
0
∞
d
τ
1
+
τ
2
{\displaystyle {\frac {1}{2}}\pi =\int _{0}^{\infty }{\frac {d\tau }{1+\tau ^{2}}}}
a definition that goes back to Karl Weierstrass.
On the interval
−
π
/
2
<
θ
<
π
/
2
{\displaystyle -\pi /2<\theta <\pi /2}
, the trigonometric functions are defined by inverting the relation
θ
=
arctan
t
{\displaystyle \theta =\arctan t}
. Thus we define the trigonometric functions by
tan
θ
=
t
,
cos
θ
=
(
1
+
t
2
)
−
1
/
2
,
sin
θ
=
t
(
1
+
t
2
)
−
1
/
2
{\displaystyle \tan \theta =t,\quad \cos \theta =(1+t^{2})^{-1/2},\quad \sin \theta =t(1+t^{2})^{-1/2}}
where the point
(
t
,
θ
)
{\displaystyle (t,\theta )}
is on the graph of
θ
=
arctan
t
{\displaystyle \theta =\arctan t}
and the positive square root is taken.
This defines the trigonometric functions on
(
−
π
/
2
,
π
/
2
)
{\displaystyle (-\pi /2,\pi /2)}
. The definition can be extended to all real numbers by first observing that, as
θ
→
π
/
2
{\displaystyle \theta \to \pi /2}
,
t
→
∞
{\displaystyle t\to \infty }
, and so
cos
θ
=
(
1
+
t
2
)
−
1
/
2
→
0
{\displaystyle \cos \theta =(1+t^{2})^{-1/2}\to 0}
and
sin
θ
=
t
(
1
+
t
2
)
−
1
/
2
→
1
{\displaystyle \sin \theta =t(1+t^{2})^{-1/2}\to 1}
. Thus
cos
θ
{\displaystyle \cos \theta }
and
sin
θ
{\displaystyle \sin \theta }
are extended continuously so that
cos
(
π
/
2
)
=
0
,
sin
(
π
/
2
)
=
1
{\displaystyle \cos(\pi /2)=0,\sin(\pi /2)=1}
. Now the conditions
cos
(
θ
+
π
)
=
−
cos
(
θ
)
{\displaystyle \cos(\theta +\pi )=-\cos(\theta )}
and
sin
(
θ
+
π
)
=
−
sin
(
θ
)
{\displaystyle \sin(\theta +\pi )=-\sin(\theta )}
define the sine and cosine as periodic functions with period
2
π
{\displaystyle 2\pi }
, for all real numbers.
Proving the basic properties of sine and cosine, including the fact that sine and cosine are analytic, one may first establish the addition formulae. First,
arctan
s
+
arctan
t
=
arctan
s
+
t
1
−
s
t
{\displaystyle \arctan s+\arctan t=\arctan {\frac {s+t}{1-st}}}
holds, provided
arctan
s
+
arctan
t
∈
(
−
π
/
2
,
π
/
2
)
{\displaystyle \arctan s+\arctan t\in (-\pi /2,\pi /2)}
, since
arctan
s
+
arctan
t
=
∫
−
s
t
d
τ
1
+
τ
2
=
∫
0
s
+
t
1
−
s
t
d
τ
1
+
τ
2
{\displaystyle \arctan s+\arctan t=\int _{-s}^{t}{\frac {d\tau }{1+\tau ^{2}}}=\int _{0}^{\frac {s+t}{1-st}}{\frac {d\tau }{1+\tau ^{2}}}}
after the substitution
τ
→
s
+
τ
1
−
s
τ
{\displaystyle \tau \to {\frac {s+\tau }{1-s\tau }}}
. In particular, the limiting case as
s
→
∞
{\displaystyle s\to \infty }
gives
arctan
t
+
π
2
=
arctan
(
−
1
/
t
)
,
t
∈
(
−
∞
,
0
)
.
{\displaystyle \arctan t+{\frac {\pi }{2}}=\arctan(-1/t),\quad t\in (-\infty ,0).}
Thus we have
sin
(
θ
+
π
2
)
=
−
1
t
1
+
(
−
1
/
t
)
2
=
−
1
1
+
t
2
=
−
cos
(
θ
)
{\displaystyle \sin \left(\theta +{\frac {\pi }{2}}\right)={\frac {-1}{t{\sqrt {1+(-1/t)^{2}}}}}={\frac {-1}{\sqrt {1+t^{2}}}}=-\cos(\theta )}
and
cos
(
θ
+
π
2
)
=
1
1
+
(
−
1
/
t
)
2
=
t
1
+
t
2
=
sin
(
θ
)
.
{\displaystyle \cos \left(\theta +{\frac {\pi }{2}}\right)={\frac {1}{\sqrt {1+(-1/t)^{2}}}}={\frac {t}{\sqrt {1+t^{2}}}}=\sin(\theta ).}
So the sine and cosine functions are related by translation over a quarter period
π
/
2
{\displaystyle \pi /2}
.
=== Definitions using functional equations ===
One can also define the trigonometric functions using various functional equations.
For example, the sine and the cosine form the unique pair of continuous functions that satisfy the difference formula
cos
(
x
−
y
)
=
cos
x
cos
y
+
sin
x
sin
y
{\displaystyle \cos(x-y)=\cos x\cos y+\sin x\sin y\,}
and the added condition
0
<
x
cos
x
<
sin
x
<
x
for
0
<
x
<
1.
{\displaystyle 0<x\cos x<\sin x<x\quad {\text{ for }}\quad 0<x<1.}
=== In the complex plane ===
The sine and cosine of a complex number
z
=
x
+
i
y
{\displaystyle z=x+iy}
can be expressed in terms of real sines, cosines, and hyperbolic functions as follows:
sin
z
=
sin
x
cosh
y
+
i
cos
x
sinh
y
cos
z
=
cos
x
cosh
y
−
i
sin
x
sinh
y
{\displaystyle {\begin{aligned}\sin z&=\sin x\cosh y+i\cos x\sinh y\\[5pt]\cos z&=\cos x\cosh y-i\sin x\sinh y\end{aligned}}}
By taking advantage of domain coloring, it is possible to graph the trigonometric functions as complex-valued functions. Various features unique to the complex functions can be seen from the graph; for example, the sine and cosine functions can be seen to be unbounded as the imaginary part of
z
{\displaystyle z}
becomes larger (since the color white represents infinity), and the fact that the functions contain simple zeros or poles is apparent from the fact that the hue cycles around each zero or pole exactly once. Comparing these graphs with those of the corresponding Hyperbolic functions highlights the relationships between the two.
== Periodicity and asymptotes ==
The sine and cosine functions are periodic, with period
2
π
{\displaystyle 2\pi }
, which is the smallest positive period:
sin
(
z
+
2
π
)
=
sin
(
z
)
,
cos
(
z
+
2
π
)
=
cos
(
z
)
.
{\displaystyle \sin(z+2\pi )=\sin(z),\quad \cos(z+2\pi )=\cos(z).}
Consequently, the cosecant and secant also have
2
π
{\displaystyle 2\pi }
as their period.
The functions sine and cosine also have semiperiods
π
{\displaystyle \pi }
, and
sin
(
z
+
π
)
=
−
sin
(
z
)
,
cos
(
z
+
π
)
=
−
cos
(
z
)
{\displaystyle \sin(z+\pi )=-\sin(z),\quad \cos(z+\pi )=-\cos(z)}
and consequently
tan
(
z
+
π
)
=
tan
(
z
)
,
cot
(
z
+
π
)
=
cot
(
z
)
.
{\displaystyle \tan(z+\pi )=\tan(z),\quad \cot(z+\pi )=\cot(z).}
Also,
sin
(
x
+
π
/
2
)
=
cos
(
x
)
,
cos
(
x
+
π
/
2
)
=
−
sin
(
x
)
{\displaystyle \sin(x+\pi /2)=\cos(x),\quad \cos(x+\pi /2)=-\sin(x)}
(see Complementary angles).
The function
sin
(
z
)
{\displaystyle \sin(z)}
has a unique zero (at
z
=
0
{\displaystyle z=0}
) in the strip
−
π
<
ℜ
(
z
)
<
π
{\displaystyle -\pi <\Re (z)<\pi }
. The function
cos
(
z
)
{\displaystyle \cos(z)}
has the pair of zeros
z
=
±
π
/
2
{\displaystyle z=\pm \pi /2}
in the same strip. Because of the periodicity, the zeros of sine are
π
Z
=
{
…
,
−
2
π
,
−
π
,
0
,
π
,
2
π
,
…
}
⊂
C
.
{\displaystyle \pi \mathbb {Z} =\left\{\dots ,-2\pi ,-\pi ,0,\pi ,2\pi ,\dots \right\}\subset \mathbb {C} .}
There zeros of cosine are
π
2
+
π
Z
=
{
…
,
−
3
π
2
,
−
π
2
,
π
2
,
3
π
2
,
…
}
⊂
C
.
{\displaystyle {\frac {\pi }{2}}+\pi \mathbb {Z} =\left\{\dots ,-{\frac {3\pi }{2}},-{\frac {\pi }{2}},{\frac {\pi }{2}},{\frac {3\pi }{2}},\dots \right\}\subset \mathbb {C} .}
All of the zeros are simple zeros, and both functions have derivative
±
1
{\displaystyle \pm 1}
at each of the zeros.
The tangent function
tan
(
z
)
=
sin
(
z
)
/
cos
(
z
)
{\displaystyle \tan(z)=\sin(z)/\cos(z)}
has a simple zero at
z
=
0
{\displaystyle z=0}
and vertical asymptotes at
z
=
±
π
/
2
{\displaystyle z=\pm \pi /2}
, where it has a simple pole of residue
−
1
{\displaystyle -1}
. Again, owing to the periodicity, the zeros are all the integer multiples of
π
{\displaystyle \pi }
and the poles are odd multiples of
π
/
2
{\displaystyle \pi /2}
, all having the same residue. The poles correspond to vertical asymptotes
lim
x
→
π
−
tan
(
x
)
=
+
∞
,
lim
x
→
π
+
tan
(
x
)
=
−
∞
.
{\displaystyle \lim _{x\to \pi ^{-}}\tan(x)=+\infty ,\quad \lim _{x\to \pi ^{+}}\tan(x)=-\infty .}
The cotangent function
cot
(
z
)
=
cos
(
z
)
/
sin
(
z
)
{\displaystyle \cot(z)=\cos(z)/\sin(z)}
has a simple pole of residue 1 at the integer multiples of
π
{\displaystyle \pi }
and simple zeros at odd multiples of
π
/
2
{\displaystyle \pi /2}
. The poles correspond to vertical asymptotes
lim
x
→
0
−
cot
(
x
)
=
−
∞
,
lim
x
→
0
+
cot
(
x
)
=
+
∞
.
{\displaystyle \lim _{x\to 0^{-}}\cot(x)=-\infty ,\quad \lim _{x\to 0^{+}}\cot(x)=+\infty .}
== Basic identities ==
Many identities interrelate the trigonometric functions. This section contains the most basic ones; for more identities, see List of trigonometric identities. These identities may be proved geometrically from the unit-circle definitions or the right-angled-triangle definitions (although, for the latter definitions, care must be taken for angles that are not in the interval [0, π/2], see Proofs of trigonometric identities). For non-geometrical proofs using only tools of calculus, one may use directly the differential equations, in a way that is similar to that of the above proof of Euler's identity. One can also use Euler's identity for expressing all trigonometric functions in terms of complex exponentials and using properties of the exponential function.
=== Parity ===
The cosine and the secant are even functions; the other trigonometric functions are odd functions. That is:
sin
(
−
x
)
=
−
sin
x
cos
(
−
x
)
=
cos
x
tan
(
−
x
)
=
−
tan
x
cot
(
−
x
)
=
−
cot
x
csc
(
−
x
)
=
−
csc
x
sec
(
−
x
)
=
sec
x
.
{\displaystyle {\begin{aligned}\sin(-x)&=-\sin x\\\cos(-x)&=\cos x\\\tan(-x)&=-\tan x\\\cot(-x)&=-\cot x\\\csc(-x)&=-\csc x\\\sec(-x)&=\sec x.\end{aligned}}}
=== Periods ===
All trigonometric functions are periodic functions of period 2π. This is the smallest period, except for the tangent and the cotangent, which have π as smallest period. This means that, for every integer k, one has
sin
(
x
+
2
k
π
)
=
sin
x
cos
(
x
+
2
k
π
)
=
cos
x
tan
(
x
+
k
π
)
=
tan
x
cot
(
x
+
k
π
)
=
cot
x
csc
(
x
+
2
k
π
)
=
csc
x
sec
(
x
+
2
k
π
)
=
sec
x
.
{\displaystyle {\begin{array}{lrl}\sin(x+&2k\pi )&=\sin x\\\cos(x+&2k\pi )&=\cos x\\\tan(x+&k\pi )&=\tan x\\\cot(x+&k\pi )&=\cot x\\\csc(x+&2k\pi )&=\csc x\\\sec(x+&2k\pi )&=\sec x.\end{array}}}
See Periodicity and asymptotes.
=== Pythagorean identity ===
The Pythagorean identity, is the expression of the Pythagorean theorem in terms of trigonometric functions. It is
sin
2
x
+
cos
2
x
=
1
{\displaystyle \sin ^{2}x+\cos ^{2}x=1}
.
Dividing through by either
cos
2
x
{\displaystyle \cos ^{2}x}
or
sin
2
x
{\displaystyle \sin ^{2}x}
gives
tan
2
x
+
1
=
sec
2
x
{\displaystyle \tan ^{2}x+1=\sec ^{2}x}
1
+
cot
2
x
=
csc
2
x
{\displaystyle 1+\cot ^{2}x=\csc ^{2}x}
and
sec
2
x
+
csc
2
x
=
sec
2
x
csc
2
x
{\displaystyle \sec ^{2}x+\csc ^{2}x=\sec ^{2}x\csc ^{2}x}
.
=== Sum and difference formulas ===
The sum and difference formulas allow expanding the sine, the cosine, and the tangent of a sum or a difference of two angles in terms of sines and cosines and tangents of the angles themselves. These can be derived geometrically, using arguments that date to Ptolemy (see Angle sum and difference identities). One can also produce them algebraically using Euler's formula.
Sum
sin
(
x
+
y
)
=
sin
x
cos
y
+
cos
x
sin
y
,
cos
(
x
+
y
)
=
cos
x
cos
y
−
sin
x
sin
y
,
tan
(
x
+
y
)
=
tan
x
+
tan
y
1
−
tan
x
tan
y
.
{\displaystyle {\begin{aligned}\sin \left(x+y\right)&=\sin x\cos y+\cos x\sin y,\\[5mu]\cos \left(x+y\right)&=\cos x\cos y-\sin x\sin y,\\[5mu]\tan(x+y)&={\frac {\tan x+\tan y}{1-\tan x\tan y}}.\end{aligned}}}
Difference
sin
(
x
−
y
)
=
sin
x
cos
y
−
cos
x
sin
y
,
cos
(
x
−
y
)
=
cos
x
cos
y
+
sin
x
sin
y
,
tan
(
x
−
y
)
=
tan
x
−
tan
y
1
+
tan
x
tan
y
.
{\displaystyle {\begin{aligned}\sin \left(x-y\right)&=\sin x\cos y-\cos x\sin y,\\[5mu]\cos \left(x-y\right)&=\cos x\cos y+\sin x\sin y,\\[5mu]\tan(x-y)&={\frac {\tan x-\tan y}{1+\tan x\tan y}}.\end{aligned}}}
When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulae.
sin
2
x
=
2
sin
x
cos
x
=
2
tan
x
1
+
tan
2
x
,
cos
2
x
=
cos
2
x
−
sin
2
x
=
2
cos
2
x
−
1
=
1
−
2
sin
2
x
=
1
−
tan
2
x
1
+
tan
2
x
,
tan
2
x
=
2
tan
x
1
−
tan
2
x
.
{\displaystyle {\begin{aligned}\sin 2x&=2\sin x\cos x={\frac {2\tan x}{1+\tan ^{2}x}},\\[5mu]\cos 2x&=\cos ^{2}x-\sin ^{2}x=2\cos ^{2}x-1=1-2\sin ^{2}x={\frac {1-\tan ^{2}x}{1+\tan ^{2}x}},\\[5mu]\tan 2x&={\frac {2\tan x}{1-\tan ^{2}x}}.\end{aligned}}}
These identities can be used to derive the product-to-sum identities.
By setting
t
=
tan
1
2
θ
,
{\displaystyle t=\tan {\tfrac {1}{2}}\theta ,}
all trigonometric functions of
θ
{\displaystyle \theta }
can be expressed as rational fractions of
t
{\displaystyle t}
:
sin
θ
=
2
t
1
+
t
2
,
cos
θ
=
1
−
t
2
1
+
t
2
,
tan
θ
=
2
t
1
−
t
2
.
{\displaystyle {\begin{aligned}\sin \theta &={\frac {2t}{1+t^{2}}},\\[5mu]\cos \theta &={\frac {1-t^{2}}{1+t^{2}}},\\[5mu]\tan \theta &={\frac {2t}{1-t^{2}}}.\end{aligned}}}
Together with
d
θ
=
2
1
+
t
2
d
t
,
{\displaystyle d\theta ={\frac {2}{1+t^{2}}}\,dt,}
this is the tangent half-angle substitution, which reduces the computation of integrals and antiderivatives of trigonometric functions to that of rational fractions.
=== Derivatives and antiderivatives ===
The derivatives of trigonometric functions result from those of sine and cosine by applying the quotient rule. The values given for the antiderivatives in the following table can be verified by differentiating them. The number C is a constant of integration.
Note: For
0
<
x
<
π
{\displaystyle 0<x<\pi }
the integral of
csc
x
{\displaystyle \csc x}
can also be written as
−
arsinh
(
cot
x
)
,
{\displaystyle -\operatorname {arsinh} (\cot x),}
and for the integral of
sec
x
{\displaystyle \sec x}
for
−
π
/
2
<
x
<
π
/
2
{\displaystyle -\pi /2<x<\pi /2}
as
arsinh
(
tan
x
)
,
{\displaystyle \operatorname {arsinh} (\tan x),}
where
arsinh
{\displaystyle \operatorname {arsinh} }
is the inverse hyperbolic sine.
Alternatively, the derivatives of the 'co-functions' can be obtained using trigonometric identities and the chain rule:
d
cos
x
d
x
=
d
d
x
sin
(
π
/
2
−
x
)
=
−
cos
(
π
/
2
−
x
)
=
−
sin
x
,
d
csc
x
d
x
=
d
d
x
sec
(
π
/
2
−
x
)
=
−
sec
(
π
/
2
−
x
)
tan
(
π
/
2
−
x
)
=
−
csc
x
cot
x
,
d
cot
x
d
x
=
d
d
x
tan
(
π
/
2
−
x
)
=
−
sec
2
(
π
/
2
−
x
)
=
−
csc
2
x
.
{\displaystyle {\begin{aligned}{\frac {d\cos x}{dx}}&={\frac {d}{dx}}\sin(\pi /2-x)=-\cos(\pi /2-x)=-\sin x\,,\\{\frac {d\csc x}{dx}}&={\frac {d}{dx}}\sec(\pi /2-x)=-\sec(\pi /2-x)\tan(\pi /2-x)=-\csc x\cot x\,,\\{\frac {d\cot x}{dx}}&={\frac {d}{dx}}\tan(\pi /2-x)=-\sec ^{2}(\pi /2-x)=-\csc ^{2}x\,.\end{aligned}}}
== Inverse functions ==
The trigonometric functions are periodic, and hence not injective, so strictly speaking, they do not have an inverse function. However, on each interval on which a trigonometric function is monotonic, one can define an inverse function, and this defines inverse trigonometric functions as multivalued functions. To define a true inverse function, one must restrict the domain to an interval where the function is monotonic, and is thus bijective from this interval to its image by the function. The common choice for this interval, called the set of principal values, is given in the following table. As usual, the inverse trigonometric functions are denoted with the prefix "arc" before the name or its abbreviation of the function.
The notations sin−1, cos−1, etc. are often used for arcsin and arccos, etc. When this notation is used, inverse functions could be confused with multiplicative inverses. The notation with the "arc" prefix avoids such a confusion, though "arcsec" for arcsecant can be confused with "arcsecond".
Just like the sine and cosine, the inverse trigonometric functions can also be expressed in terms of infinite series. They can also be expressed in terms of complex logarithms.
== Applications ==
=== Angles and sides of a triangle ===
In this section A, B, C denote the three (interior) angles of a triangle, and a, b, c denote the lengths of the respective opposite edges. They are related by various formulas, which are named by the trigonometric functions they involve.
==== Law of sines ====
The law of sines states that for an arbitrary triangle with sides a, b, and c and angles opposite those sides A, B and C:
sin
A
a
=
sin
B
b
=
sin
C
c
=
2
Δ
a
b
c
,
{\displaystyle {\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}={\frac {2\Delta }{abc}},}
where Δ is the area of the triangle,
or, equivalently,
a
sin
A
=
b
sin
B
=
c
sin
C
=
2
R
,
{\displaystyle {\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R,}
where R is the triangle's circumradius.
It can be proved by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in triangulation, a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.
==== Law of cosines ====
The law of cosines (also known as the cosine formula or cosine rule) is an extension of the Pythagorean theorem:
c
2
=
a
2
+
b
2
−
2
a
b
cos
C
,
{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos C,}
or equivalently,
cos
C
=
a
2
+
b
2
−
c
2
2
a
b
.
{\displaystyle \cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}.}
In this formula the angle at C is opposite to the side c. This theorem can be proved by dividing the triangle into two right ones and using the Pythagorean theorem.
The law of cosines can be used to determine a side of a triangle if two sides and the angle between them are known. It can also be used to find the cosines of an angle (and consequently the angles themselves) if the lengths of all the sides are known.
==== Law of tangents ====
The law of tangents says that:
tan
A
−
B
2
tan
A
+
B
2
=
a
−
b
a
+
b
{\displaystyle {\frac {\tan {\frac {A-B}{2}}}{\tan {\frac {A+B}{2}}}}={\frac {a-b}{a+b}}}
.
==== Law of cotangents ====
If s is the triangle's semiperimeter, (a + b + c)/2, and r is the radius of the triangle's incircle, then rs is the triangle's area. Therefore Heron's formula implies that:
r
=
1
s
(
s
−
a
)
(
s
−
b
)
(
s
−
c
)
{\displaystyle r={\sqrt {{\frac {1}{s}}(s-a)(s-b)(s-c)}}}
.
The law of cotangents says that:
cot
A
2
=
s
−
a
r
{\displaystyle \cot {\frac {A}{2}}={\frac {s-a}{r}}}
It follows that
cot
A
2
s
−
a
=
cot
B
2
s
−
b
=
cot
C
2
s
−
c
=
1
r
.
{\displaystyle {\frac {\cot {\dfrac {A}{2}}}{s-a}}={\frac {\cot {\dfrac {B}{2}}}{s-b}}={\frac {\cot {\dfrac {C}{2}}}{s-c}}={\frac {1}{r}}.}
=== Periodic functions ===
The trigonometric functions are also important in physics. The sine and the cosine functions, for example, are used to describe simple harmonic motion, which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string. The sine and cosine functions are one-dimensional projections of uniform circular motion.
Trigonometric functions also prove to be useful in the study of general periodic functions. The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light waves.
Under rather general conditions, a periodic function f (x) can be expressed as a sum of sine waves or cosine waves in a Fourier series. Denoting the sine or cosine basis functions by φk, the expansion of the periodic function f (t) takes the form:
f
(
t
)
=
∑
k
=
1
∞
c
k
φ
k
(
t
)
.
{\displaystyle f(t)=\sum _{k=1}^{\infty }c_{k}\varphi _{k}(t).}
For example, the square wave can be written as the Fourier series
f
square
(
t
)
=
4
π
∑
k
=
1
∞
sin
(
(
2
k
−
1
)
t
)
2
k
−
1
.
{\displaystyle f_{\text{square}}(t)={\frac {4}{\pi }}\sum _{k=1}^{\infty }{\sin {\big (}(2k-1)t{\big )} \over 2k-1}.}
In the animation of a square wave at top right it can be seen that just a few terms already produce a fairly good approximation. The superposition of several terms in the expansion of a sawtooth wave are shown underneath.
== History ==
While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period. The chord function was defined by Hipparchus of Nicaea (180–125 BCE) and Ptolemy of Roman Egypt (90–165 CE). The functions of sine and versine (1 – cosine) are closely related to the jyā and koti-jyā functions used in Gupta period Indian astronomy (Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic and then from Arabic to Latin. (See Aryabhata's sine table.)
All six trigonometric functions in current use were known in Islamic mathematics by the 9th century, as was the law of sines, used in solving triangles. Al-Khwārizmī (c. 780–850) produced tables of sines and cosines. Circa 860, Habash al-Hasib al-Marwazi defined the tangent and the cotangent, and produced their tables. Muhammad ibn Jābir al-Harrānī al-Battānī (853–929) defined the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°. The trigonometric functions were later studied by mathematicians including Omar Khayyám, Bhāskara II, Nasir al-Din al-Tusi, Jamshīd al-Kāshī (14th century), Ulugh Beg (14th century), Regiomontanus (1464), Rheticus, and Rheticus' student Valentinus Otho.
Madhava of Sangamagrama (c. 1400) made early strides in the analysis of trigonometric functions in terms of infinite series. (See Madhava series and Madhava's sine table.)
The tangent function was brought to Europe by Giovanni Bianchini in 1467 in trigonometry tables he created to support the calculation of stellar coordinates.
The terms tangent and secant were first introduced by the Danish mathematician Thomas Fincke in his book Geometria rotundi (1583).
The 17th century French mathematician Albert Girard made the first published use of the abbreviations sin, cos, and tan in his book Trigonométrie.
In a paper published in 1682, Gottfried Leibniz proved that sin x is not an algebraic function of x. Though defined as ratios of sides of a right triangle, and thus appearing to be rational functions, Leibnitz result established that they are actually transcendental functions of their argument. The task of assimilating circular functions into algebraic expressions was accomplished by Euler in his Introduction to the Analysis of the Infinite (1748). His method was to show that the sine and cosine functions are alternating series formed from the even and odd terms respectively of the exponential series. He presented "Euler's formula", as well as near-modern abbreviations (sin., cos., tang., cot., sec., and cosec.).
A few functions were common historically, but are now seldom used, such as the chord, versine (which appeared in the earliest tables), haversine, coversine, half-tangent (tangent of half an angle), and exsecant. List of trigonometric identities shows more relations between these functions.
crd
θ
=
2
sin
1
2
θ
,
vers
θ
=
1
−
cos
θ
=
2
sin
2
1
2
θ
,
hav
θ
=
1
2
vers
θ
=
sin
2
1
2
θ
,
covers
θ
=
1
−
sin
θ
=
vers
(
1
2
π
−
θ
)
,
exsec
θ
=
sec
θ
−
1.
{\displaystyle {\begin{aligned}\operatorname {crd} \theta &=2\sin {\tfrac {1}{2}}\theta ,\\[5mu]\operatorname {vers} \theta &=1-\cos \theta =2\sin ^{2}{\tfrac {1}{2}}\theta ,\\[5mu]\operatorname {hav} \theta &={\tfrac {1}{2}}\operatorname {vers} \theta =\sin ^{2}{\tfrac {1}{2}}\theta ,\\[5mu]\operatorname {covers} \theta &=1-\sin \theta =\operatorname {vers} {\bigl (}{\tfrac {1}{2}}\pi -\theta {\bigr )},\\[5mu]\operatorname {exsec} \theta &=\sec \theta -1.\end{aligned}}}
Historically, trigonometric functions were often combined with logarithms in compound functions like the logarithmic sine, logarithmic cosine, logarithmic secant, logarithmic cosecant, logarithmic tangent and logarithmic cotangent.
== Etymology ==
The word sine derives from Latin sinus, meaning "bend; bay", and more specifically "the hanging fold of the upper part of a toga", "the bosom of a garment", which was chosen as the translation of what was interpreted as the Arabic word jaib, meaning "pocket" or "fold" in the twelfth-century translations of works by Al-Battani and al-Khwārizmī into Medieval Latin.
The choice was based on a misreading of the Arabic written form j-y-b (جيب), which itself originated as a transliteration from Sanskrit jīvā, which along with its synonym jyā (the standard Sanskrit term for the sine) translates to "bowstring", being in turn adopted from Ancient Greek χορδή "string".
The word tangent comes from Latin tangens meaning "touching", since the line touches the circle of unit radius, whereas secant stems from Latin secans—"cutting"—since the line cuts the circle.
The prefix "co-" (in "cosine", "cotangent", "cosecant") is found in Edmund Gunter's Canon triangulorum (1620), which defines the cosinus as an abbreviation of the sinus complementi (sine of the complementary angle) and proceeds to define the cotangens similarly.
== See also ==
== Notes ==
== References ==
== External links ==
"Trigonometric functions", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Visionlearning Module on Wave Mathematics
GonioLab Visualization of the unit circle, trigonometric and hyperbolic functions
q-Sine Article about the q-analog of sin at MathWorld
q-Cosine Article about the q-analog of cos at MathWorld | Wikipedia/Circular_function |
A hypertranscendental function or transcendentally transcendental function is a transcendental analytic function which is not the solution of an algebraic differential equation with coefficients in
Z
{\displaystyle \mathbb {Z} }
(the integers) and with algebraic initial conditions.
== History ==
The term 'transcendentally transcendental' was introduced by E. H. Moore in 1896; the term 'hypertranscendental' was introduced by D. D. Morduhai-Boltovskoi in 1914.
== Definition ==
One standard definition (there are slight variants) defines solutions of differential equations of the form
F
(
x
,
y
,
y
′
,
⋯
,
y
(
n
)
)
=
0
{\displaystyle F\left(x,y,y',\cdots ,y^{(n)}\right)=0}
,
where
F
{\displaystyle F}
is a polynomial with constant coefficients, as algebraically transcendental or differentially algebraic. Transcendental functions which are not algebraically transcendental are transcendentally transcendental. Hölder's theorem shows that the gamma function is in this category.
Hypertranscendental functions usually arise as the solutions to functional equations, for example the gamma function.
== Examples ==
=== Hypertranscendental functions ===
The zeta functions of algebraic number fields, in particular, the Riemann zeta function
The gamma function (cf. Hölder's theorem)
=== Transcendental but not hypertranscendental functions ===
The exponential function, logarithm, and the trigonometric and hyperbolic functions.
The generalized hypergeometric functions, including special cases such as Bessel functions (except some special cases which are algebraic).
=== Non-transcendental (algebraic) functions ===
All algebraic functions, in particular polynomials.
== See also ==
Hypertranscendental number
== Notes ==
== References ==
Loxton, J.H., Poorten, A.J. van der, "A class of hypertranscendental functions", Aequationes Mathematicae, Periodical volume 16
Mahler, K., "Arithmetische Eigenschaften einer Klasse transzendental-transzendenter Funktionen", Math. Z. 32 (1930) 545-585.
Morduhaĭ-Boltovskoĭ, D. (1949), "On hypertranscendental functions and hypertranscendental numbers", Doklady Akademii Nauk SSSR, New Series (in Russian), 64: 21–24, MR 0028347 | Wikipedia/Hypertranscendental_function |
In mathematics, an algebraic differential equation is a differential equation that can be expressed by means of differential algebra. There are several such notions, according to the concept of differential algebra used.
The intention is to include equations formed by means of differential operators, in which the coefficients are rational functions of the variables (e.g. the hypergeometric equation). Algebraic differential equations are widely used in computer algebra and number theory.
A simple concept is that of a polynomial vector field, in other words a vector field expressed with respect to a standard co-ordinate basis as the first partial derivatives with polynomial coefficients. This is a type of first-order algebraic differential operator.
== Formulations ==
Derivations D can be used as algebraic analogues of the formal part of differential calculus, so that algebraic differential equations make sense in commutative rings.
The theory of differential fields was set up to express differential Galois theory in algebraic terms.
The Weyl algebra W of differential operators with polynomial coefficients can be considered; certain modules M can be used to express differential equations, according to the presentation of M.
The concept of Koszul connection is something that transcribes easily into algebraic geometry, giving an algebraic analogue of the way systems of differential equations are geometrically represented by vector bundles with connections.
The concept of jet can be described in purely algebraic terms, as was done in part of Grothendieck's EGA project.
The theory of D-modules is a global theory of linear differential equations, and has been developed to include substantive results in the algebraic theory (including a Riemann-Hilbert correspondence for higher dimensions).
== Algebraic solutions ==
It is usually not the case that the general solution of an algebraic differential equation is an algebraic function: solving equations typically produces novel transcendental functions. The case of algebraic solutions is however of considerable interest; the classical Schwarz list deals with the case of the hypergeometric equation. In differential Galois theory the case of algebraic solutions is that in which the differential Galois group G is finite (equivalently, of dimension 0, or of a finite monodromy group for the case of Riemann surfaces and linear equations). This case stands in relation with the whole theory roughly as invariant theory does to group representation theory. The group G is in general difficult to compute, the understanding of algebraic solutions is an indication of upper bounds for G.
== External links ==
Mikhalev, A.V.; Pankrat'ev, E.V. (2001) [1994], "Differential algebra", Encyclopedia of Mathematics, EMS Press
Mikhalev, A.V.; Pankrat'ev, E.V. (2001) [1994], "Extension of a differential field", Encyclopedia of Mathematics, EMS Press | Wikipedia/Algebraic_differential_equation |
In mathematics, specifically transcendental number theory, Schanuel's conjecture is a conjecture about the transcendence degree of certain field extensions of the rational numbers
Q
{\displaystyle \mathbb {Q} }
, which would establish the transcendence of a large class of numbers, for which this is currently unknown. It is due to Stephen Schanuel and was published by Serge Lang in 1966.
== Statement ==
Schanuel's conjecture can be given as follows:
== Consequences ==
Schanuel's conjecture, if proven, would generalize most known results in transcendental number theory and establish a large class of numbers transcendental. Special cases of Schanuel's conjecture include:
=== Lindemann-Weierstrass theorem ===
Considering Schanuel's conjecture for only
n
=
1
{\displaystyle n=1}
gives that for nonzero complex numbers
z
{\displaystyle z}
, at least one of the numbers
z
{\displaystyle z}
and
e
z
{\displaystyle e^{z}}
must be transcendental. This was proved by Ferdinand von Lindemann in 1882.
If the numbers
z
1
,
.
.
.
,
z
n
{\displaystyle z_{1},...,z_{n}}
are taken to be all algebraic and linearly independent over
Q
{\displaystyle \mathbb {Q} }
then the
e
z
1
,
.
.
.
,
e
z
n
{\displaystyle e^{z_{1}},...,e^{z_{n}}}
result to be transcendental and algebraically independent over
Q
{\displaystyle \mathbb {Q} }
. The first proof for this more general result was given by Carl Weierstrass in 1885.
This so-called Lindemann–Weierstrass theorem implies the transcendence of the numbers e and π. It also follows that for algebraic numbers
α
{\displaystyle \alpha }
not equal to 0 or 1, both
e
α
{\displaystyle e^{\alpha }}
and
ln
(
α
)
{\displaystyle \ln(\alpha )}
are transcendental. It further gives the transcendence of the trigonometric functions at nonzero algebraic values.
=== Baker's theorem ===
Another special case was proved by Alan Baker in 1966: If complex numbers
λ
1
,
.
.
.
,
λ
n
{\displaystyle \lambda _{1},...,\lambda _{n}}
are chosen to be linearly independent over the rational numbers
Q
{\displaystyle \mathbb {Q} }
such that
e
λ
1
,
.
.
.
,
e
λ
n
{\displaystyle e^{\lambda _{1}},...,e^{\lambda _{n}}}
are algebraic, then
λ
1
,
.
.
.
,
λ
n
{\displaystyle \lambda _{1},...,\lambda _{n}}
are also linearly independent over the algebraic numbers
Q
¯
{\displaystyle \mathbb {\overline {Q}} }
.
Schanuel's conjecture would strengthen this result, implying that
λ
1
,
.
.
.
,
λ
n
{\displaystyle \lambda _{1},...,\lambda _{n}}
would also be algebraically independent over
Q
{\displaystyle \mathbb {Q} }
(and equivalently over
Q
¯
{\displaystyle \mathbb {\overline {Q}} }
).
=== Gelfond-Schneider theorem ===
In 1934 it was proved by Aleksander Gelfond and Theodor Schneider that if
α
{\displaystyle \alpha }
and
β
{\displaystyle \beta }
are two algebraic complex numbers with
α
∉
{
0
,
1
}
{\displaystyle \alpha \notin \{0,1\}}
and
β
∉
Q
{\displaystyle \beta \notin \mathbb {Q} }
, then
α
β
{\displaystyle \alpha ^{\beta }}
is transcendental.
This establishes the transcendence of numbers like Hilbert's constant
2
2
{\displaystyle 2^{\sqrt {2}}}
and Gelfond's constant
e
π
{\displaystyle e^{\pi }}
.
The Gelfond–Schneider theorem follows from Schanuel's conjecture by setting
n
=
3
{\displaystyle n=3}
and
z
1
=
β
,
z
2
=
ln
α
,
z
3
=
β
ln
α
{\displaystyle z_{1}=\beta ,z_{2}=\ln \alpha ,z_{3}=\beta \ln \alpha }
. It also would follow from the strengthened version of Baker's theorem above.
=== Four exponentials conjecture ===
The currently unproven four exponentials conjecture would also follow from Schanuel's conjecture: If
z
1
,
z
2
{\displaystyle z_{1},z_{2}}
and
w
1
,
w
2
{\displaystyle w_{1},w_{2}}
are two pairs of complex numbers, with each pair being linearly independent over the rational numbers, then at least one of the following four numbers is transcendental:
e
z
1
w
1
,
e
z
1
w
2
,
e
z
2
w
1
,
e
z
2
w
2
.
{\displaystyle e^{z_{1}w_{1}},e^{z_{1}w_{2}},e^{z_{2}w_{1}},e^{z_{2}w_{2}}.}
The four exponential conjecture would imply that for any irrational number
t
{\displaystyle t}
, at least one of the numbers
2
t
{\displaystyle 2^{t}}
and
3
t
{\displaystyle 3^{t}}
is transcendental. It also implies that if
t
{\displaystyle t}
is a positive real number such that both
2
t
{\displaystyle 2^{t}}
and
3
t
{\displaystyle 3^{t}}
are integers, then
t
{\displaystyle t}
itself must be an integer. The related six exponentials theorem has been proven.
=== Other consequences ===
Schanuel's conjecture, if proved, would also establish many nontrivial combinations of e, π, algebraic numbers and elementary functions to be transcendental:
e
+
π
,
e
π
,
e
π
2
,
e
e
,
π
e
,
π
2
,
π
π
,
π
π
π
,
log
π
,
log
log
2
,
sin
e
,
.
.
.
{\displaystyle e+\pi ,e\pi ,e^{\pi ^{2}},e^{e},\pi ^{e},\pi ^{\sqrt {2}},\pi ^{\pi },\pi ^{\pi ^{\pi }},\,\log \pi ,\,\log \log 2,\,\sin e,...}
In particular it would follow that e and π are algebraically independent simply by setting
z
1
=
1
{\displaystyle z_{1}=1}
and
z
2
=
i
π
{\displaystyle z_{2}=i\pi }
.
Euler's identity states that
e
i
π
+
1
=
0
{\displaystyle e^{i\pi }+1=0}
. If Schanuel's conjecture is true then this is, in some precise sense involving exponential rings, the only non-trivial relation between e, π, and i over the complex numbers.
== Related conjectures and results ==
The converse Schanuel conjecture is the following statement:
A version of Schanuel's conjecture for formal power series, also by Schanuel, was proven by James Ax in 1971. It states:
Although ostensibly a problem in number theory, Schanuel's conjecture has implications in model theory as well. Angus Macintyre and Alex Wilkie, for example, proved that the theory of the real field with exponentiation,
R
{\displaystyle \mathbb {R} }
exp, is decidable provided Schanuel's conjecture is true. In fact, to prove this result, they only needed the real version of the conjecture, which is as follows:
This would be a positive solution to Tarski's exponential function problem.
A related conjecture called the uniform real Schanuel's conjecture essentially says the same but puts a bound on the integers mi. The uniform real version of the conjecture is equivalent to the standard real version. Macintyre and Wilkie showed that a consequence of Schanuel's conjecture, which they dubbed the Weak Schanuel's conjecture, was equivalent to the decidability of
R
{\displaystyle \mathbb {R} }
exp. This conjecture states that there is a computable upper bound on the norm of non-singular solutions to systems of exponential polynomials; this is, non-obviously, a consequence of Schanuel's conjecture for the reals.
It is also known that Schanuel's conjecture would be a consequence of conjectural results in the theory of motives. In this setting Grothendieck's period conjecture for an abelian variety A states that the transcendence degree of its period matrix is the same as the dimension of the associated Mumford–Tate group, and what is known by work of Pierre Deligne is that the dimension is an upper bound for the transcendence degree. Bertolin has shown how a generalised period conjecture includes Schanuel's conjecture.
== Zilber's pseudo-exponentiation ==
While a proof of Schanuel's conjecture seems a long way off, connections with model theory have prompted a surge of research on the conjecture.
In 2004, Boris Zilber systematically constructed exponential fields Kexp that are algebraically closed and of characteristic zero, and such that one of these fields exists for each uncountable cardinality. He axiomatised these fields and, using Hrushovski's construction and techniques inspired by work of Shelah on categoricity in infinitary logics, proved that this theory of "pseudo-exponentiation" has a unique model in each uncountable cardinal. Schanuel's conjecture is part of this axiomatisation, and so the natural conjecture that the unique model of cardinality continuum is actually isomorphic to the complex exponential field implies Schanuel's conjecture. In fact, Zilber showed that this conjecture holds if and only if both Schanuel's conjecture and the Exponential-Algebraic Closedness conjecture hold. As this construction can also give models with counterexamples of Schanuel's conjecture, this method cannot prove Schanuel's conjecture.
== See also ==
Four exponentials conjecture
Algebraic independence
List of unsolved problems in mathematics
Existential Closedness conjecture
Zilber-Pink conjecture
Pregeometry
== References ==
== Sources ==
Weierstrass, K. (1885), "Zu Lindemann's Abhandlung. "Über die Ludolph'sche Zahl".", Sitzungsberichte der Königlich Preussischen Akademie der Wissen-schaften zu Berlin, 5: 1067–1085
== External links ==
Weisstein, Eric W. "Schanuel's Conjecture". MathWorld. | Wikipedia/Schanuel's_conjecture |
Numerical methods for linear least squares entails the numerical analysis of linear least squares problems.
== Introduction ==
A general approach to the least squares problem
m
i
n
‖
y
−
X
β
‖
2
{\displaystyle \operatorname {\,min} \,{\big \|}\mathbf {y} -X{\boldsymbol {\beta }}{\big \|}^{2}}
can be described as follows. Suppose that we can find an n by m matrix S
such that XS is an
orthogonal projection onto the image of X. Then a solution to our minimization problem is given by
β
=
S
y
{\displaystyle {\boldsymbol {\beta }}=S\mathbf {y} }
simply because
X
β
=
X
(
S
y
)
=
(
X
S
)
y
{\displaystyle X{\boldsymbol {\beta }}=X(S\mathbf {y} )=(XS)\mathbf {y} }
is exactly a sought for orthogonal projection of
y
{\displaystyle \mathbf {y} }
onto an image of X
(see the picture below and note that as explained in the
next section the image of X is just a subspace generated by column vectors of X).
A few popular ways to find such a matrix S are described below.
== Inverting the matrix of the normal equations ==
The equation
(
X
T
X
)
β
=
X
T
y
{\displaystyle (\mathbf {X} ^{\rm {T}}\mathbf {X} )\beta =\mathbf {X} ^{\rm {T}}y}
is known as the normal equation.
The algebraic solution of the normal equations with a full-rank matrix XTX can be written as
β
^
=
(
X
T
X
)
−
1
X
T
y
=
X
+
y
{\displaystyle {\hat {\boldsymbol {\beta }}}=(\mathbf {X} ^{\rm {T}}\mathbf {X} )^{-1}\mathbf {X} ^{\rm {T}}\mathbf {y} =\mathbf {X} ^{+}\mathbf {y} }
where X+ is the Moore–Penrose pseudoinverse of X. Although this equation is correct and can work in many applications, it is not computationally efficient to invert the normal-equations matrix (the Gramian matrix). An exception occurs in numerical smoothing and differentiation where an analytical expression is required.
If the matrix XTX is well-conditioned and positive definite, implying that it has full rank, the normal equations can be solved directly by using the Cholesky decomposition RTR, where R is an upper triangular matrix, giving:
R
T
R
β
^
=
X
T
y
.
{\displaystyle R^{\rm {T}}R{\hat {\boldsymbol {\beta }}}=X^{\rm {T}}\mathbf {y} .}
The solution is obtained in two stages, a forward substitution step, solving for z:
R
T
z
=
X
T
y
,
{\displaystyle R^{\rm {T}}\mathbf {z} =X^{\rm {T}}\mathbf {y} ,}
followed by a backward substitution, solving for
β
^
{\displaystyle {\hat {\boldsymbol {\beta }}}}
:
R
β
^
=
z
.
{\displaystyle R{\hat {\boldsymbol {\beta }}}=\mathbf {z} .}
Both substitutions are facilitated by the triangular nature of R.
== Orthogonal decomposition methods ==
Orthogonal decomposition methods of solving the least squares problem are slower than the normal equations method but are more numerically stable because they avoid forming the product XTX.
The residuals are written in matrix notation as
r
=
y
−
X
β
^
.
{\displaystyle \mathbf {r} =\mathbf {y} -X{\hat {\boldsymbol {\beta }}}.}
The matrix X is subjected to an orthogonal decomposition, e.g., the QR decomposition as follows.
X
=
Q
(
R
0
)
{\displaystyle X=Q{\begin{pmatrix}R\\0\end{pmatrix}}\ }
,
where Q is an m×m orthogonal matrix (QTQ=I) and R is an n×n upper triangular matrix with
r
i
i
>
0
{\displaystyle r_{ii}>0}
.
The residual vector is left-multiplied by QT.
Q
T
r
=
Q
T
y
−
(
Q
T
Q
)
(
R
0
)
β
^
=
[
(
Q
T
y
)
n
−
R
β
^
(
Q
T
y
)
m
−
n
]
=
[
u
v
]
{\displaystyle Q^{\rm {T}}\mathbf {r} =Q^{\rm {T}}\mathbf {y} -\left(Q^{\rm {T}}Q\right){\begin{pmatrix}R\\0\end{pmatrix}}{\hat {\boldsymbol {\beta }}}={\begin{bmatrix}\left(Q^{\rm {T}}\mathbf {y} \right)_{n}-R{\hat {\boldsymbol {\beta }}}\\\left(Q^{\rm {T}}\mathbf {y} \right)_{m-n}\end{bmatrix}}={\begin{bmatrix}\mathbf {u} \\\mathbf {v} \end{bmatrix}}}
Because Q is orthogonal, the sum of squares of the residuals, s, may be written as:
s
=
‖
r
‖
2
=
r
T
r
=
r
T
Q
Q
T
r
=
u
T
u
+
v
T
v
{\displaystyle s=\|\mathbf {r} \|^{2}=\mathbf {r} ^{\rm {T}}\mathbf {r} =\mathbf {r} ^{\rm {T}}QQ^{\rm {T}}\mathbf {r} =\mathbf {u} ^{\rm {T}}\mathbf {u} +\mathbf {v} ^{\rm {T}}\mathbf {v} }
Since v doesn't depend on β, the minimum value of s is attained when the upper block, u, is zero. Therefore, the parameters are found by solving:
R
β
^
=
(
Q
T
y
)
n
.
{\displaystyle R{\hat {\boldsymbol {\beta }}}=\left(Q^{\rm {T}}\mathbf {y} \right)_{n}.}
These equations are easily solved as R is upper triangular.
An alternative decomposition of X is the singular value decomposition (SVD)
X
=
U
Σ
V
T
{\displaystyle X=U\Sigma V^{\rm {T}}\ }
,
where U is m by m orthogonal matrix, V is n by n orthogonal matrix and
Σ
{\displaystyle \Sigma }
is an m by n matrix with all its elements outside of the main diagonal equal to 0. The pseudoinverse of
Σ
{\displaystyle \Sigma }
is easily obtained by inverting its non-zero diagonal elements and transposing. Hence,
X
X
+
=
U
Σ
V
T
V
Σ
+
U
T
=
U
P
U
T
,
{\displaystyle \mathbf {X} \mathbf {X} ^{+}=U\Sigma V^{\rm {T}}V\Sigma ^{+}U^{\rm {T}}=UPU^{\rm {T}},}
where P is obtained from
Σ
{\displaystyle \Sigma }
by replacing its non-zero diagonal elements with ones. Since
(
X
X
+
)
∗
=
X
X
+
{\displaystyle (\mathbf {X} \mathbf {X} ^{+})^{*}=\mathbf {X} \mathbf {X} ^{+}}
(the property of pseudoinverse), the matrix
U
P
U
T
{\displaystyle UPU^{\rm {T}}}
is an orthogonal projection onto the image (column-space) of X. In accordance with a general approach described in the introduction above (find XS which is an orthogonal projection),
S
=
X
+
{\displaystyle S=\mathbf {X} ^{+}}
,
and thus,
β
=
V
Σ
+
U
T
y
{\displaystyle \beta =V\Sigma ^{+}U^{\rm {T}}\mathbf {y} }
is a solution of a least squares problem. This method is the most computationally intensive, but is particularly useful if the normal equations matrix, XTX, is very ill-conditioned (i.e. if its condition number multiplied by the machine's relative round-off error is appreciably large). In that case, including the smallest singular values in the inversion merely adds numerical noise to the solution. This can be cured with the truncated SVD approach, giving a more stable and exact answer, by explicitly setting to zero all singular values below a certain threshold and so ignoring them, a process closely related to factor analysis.
== Discussion ==
The numerical methods for linear least squares are important because linear regression models are among the most important types of model, both as formal statistical models and for exploration of data-sets. The majority of statistical computer packages contain facilities for regression analysis that make use of linear least squares computations. Hence it is appropriate that considerable effort has been devoted to the task of ensuring that these computations are undertaken efficiently and with due regard to round-off error.
Individual statistical analyses are seldom undertaken in isolation, but rather are part of a sequence of investigatory steps. Some of the topics involved in considering numerical methods for linear least squares relate to this point. Thus important topics can be
Computations where a number of similar, and often nested, models are considered for the same data-set. That is, where models with the same dependent variable but different sets of independent variables are to be considered, for essentially the same set of data-points.
Computations for analyses that occur in a sequence, as the number of data-points increases.
Special considerations for very extensive data-sets.
Fitting of linear models by least squares often, but not always, arise in the context of statistical analysis. It can therefore be important that considerations of computation efficiency for such problems extend to all of the auxiliary quantities required for such analyses, and are not restricted to the formal solution of the linear least squares problem.
Matrix calculations, like any other, are affected by rounding errors. An early summary of these effects, regarding the choice of computation methods for matrix inversion, was provided by Wilkinson.
== See also ==
Numerical linear algebra
Numerical methods for non-linear least squares
== References ==
== Further reading ==
Ake Björck(1996), Numerical Methods for Least Squares Problems, SIAM.
R. W. Farebrother, Linear Least Squares Computations, CRC Press, 1988.
Barlow, Jesse L. (1993), "Chapter 9: Numerical aspects of Solving Linear Least Squares Problems", in Rao, C. R. (ed.), Computational Statistics, Handbook of Statistics, vol. 9, North-Holland, ISBN 0-444-88096-8
Björck, Åke (1996). Numerical methods for least squares problems. Philadelphia: SIAM. ISBN 0-89871-360-9.
Goodall, Colin R. (1993), "Chapter 13: Computation using the QR decomposition", in Rao, C. R. (ed.), Computational Statistics, Handbook of Statistics, vol. 9, North-Holland, ISBN 0-444-88096-8
National Physical Laboratory (1961), "Chapter 1: Linear Equations and Matrices: Direct Methods", Modern Computing Methods, Notes on Applied Science, vol. 16 (2nd ed.), Her Majesty's Stationery Office
National Physical Laboratory (1961), "Chapter 2: Linear Equations and Matrices: Direct Methods on Automatic Computers", Modern Computing Methods, Notes on Applied Science, vol. 16 (2nd ed.), Her Majesty's Stationery Office | Wikipedia/Numerical_methods_for_linear_least_squares |
In computing, a cache-oblivious algorithm (or cache-transcendent algorithm) is an algorithm designed to take advantage of a processor cache without having the size of the cache (or the length of the cache lines, etc.) as an explicit parameter. An optimal cache-oblivious algorithm is a cache-oblivious algorithm that uses the cache optimally (in an asymptotic sense, ignoring constant factors). Thus, a cache-oblivious algorithm is designed to perform well, without modification, on multiple machines with different cache sizes, or for a memory hierarchy with different levels of cache having different sizes. Cache-oblivious algorithms are contrasted with explicit loop tiling, which explicitly breaks a problem into blocks that are optimally sized for a given cache.
Optimal cache-oblivious algorithms are known for matrix multiplication, matrix transposition, sorting, and several other problems. Some more general algorithms, such as Cooley–Tukey FFT, are optimally cache-oblivious under certain choices of parameters. As these algorithms are only optimal in an asymptotic sense (ignoring constant factors), further machine-specific tuning may be required to obtain nearly optimal performance in an absolute sense. The goal of cache-oblivious algorithms is to reduce the amount of such tuning that is required.
Typically, a cache-oblivious algorithm works by a recursive divide-and-conquer algorithm, where the problem is divided into smaller and smaller subproblems. Eventually, one reaches a subproblem size that fits into the cache, regardless of the cache size. For example, an optimal cache-oblivious matrix multiplication is obtained by recursively dividing each matrix into four sub-matrices to be multiplied, multiplying the submatrices in a depth-first fashion. In tuning for a specific machine, one may use a hybrid algorithm which uses loop tiling tuned for the specific cache sizes at the bottom level but otherwise uses the cache-oblivious algorithm.
== History ==
The idea (and name) for cache-oblivious algorithms was conceived by Charles E. Leiserson as early as 1996 and first published by Harald Prokop in his master's thesis at the Massachusetts Institute of Technology in 1999. There were many predecessors, typically analyzing specific problems; these are discussed in detail in Frigo et al. 1999. Early examples cited include Singleton 1969 for a recursive Fast Fourier Transform, similar ideas in Aggarwal et al. 1987, Frigo 1996 for matrix multiplication and LU decomposition, and Todd Veldhuizen 1996 for matrix algorithms in the Blitz++ library.
== Idealized cache model ==
In general, a program can be made more cache-conscious:
Temporal locality, where the algorithm fetches the same pieces of memory multiple times;
Spatial locality, where the subsequent memory accesses are adjacent or nearby memory addresses.
Cache-oblivious algorithms are typically analyzed using an idealized model of the cache, sometimes called the cache-oblivious model. This model is much easier to analyze than a real cache's characteristics (which have complex associativity, replacement policies, etc.), but in many cases is provably within a constant factor of a more realistic cache's performance. It is different than the external memory model because cache-oblivious algorithms do not know the block size or the cache size.
In particular, the cache-oblivious model is an abstract machine (i.e., a theoretical model of computation). It is similar to the RAM machine model which replaces the Turing machine's infinite tape with an infinite array. Each location within the array can be accessed in
O
(
1
)
{\displaystyle O(1)}
time, similar to the random-access memory on a real computer. Unlike the RAM machine model, it also introduces a cache: the second level of storage between the RAM and the CPU. The other differences between the two models are listed below. In the cache-oblivious model:
Memory is broken into blocks of
B
{\displaystyle B}
objects each.
A load or a store between main memory and a CPU register may now be serviced from the cache.
If a load or a store cannot be serviced from the cache, it is called a cache miss.
A cache miss results in one block being loaded from the main memory into the cache. Namely, if the CPU tries to access word
w
{\displaystyle w}
and
x
{\displaystyle x}
is the line containing
w
{\displaystyle w}
, then
x
{\displaystyle x}
is loaded into the cache. If the cache was previously full, then a line will be evicted as well (see replacement policy below).
The cache holds
M
{\displaystyle M}
objects, where
M
=
Ω
(
B
2
)
{\displaystyle M=\Omega (B^{2})}
. This is also known as the tall cache assumption.
The cache is fully associative: each line can be loaded into any location in the cache.
The replacement policy is optimal. In other words, the cache is assumed to be given the entire sequence of memory accesses during algorithm execution. If it needs to evict a line at time
t
{\displaystyle t}
, it will look into its sequence of future requests and evict the line whose first access is furthest in the future. This can be emulated in practice with the Least Recently Used policy, which is shown to be within a small constant factor of the offline optimal replacement strategy
To measure the complexity of an algorithm that executes within the cache-oblivious model, we measure the number of cache misses that the algorithm experiences. Because the model captures the fact that accessing elements in the cache is much faster than accessing things in main memory, the running time of the algorithm is defined only by the number of memory transfers between the cache and main memory. This is similar to the external memory model, which all of the features above, but cache-oblivious algorithms are independent of cache parameters (
B
{\displaystyle B}
and
M
{\displaystyle M}
). The benefit of such an algorithm is that what is efficient on a cache-oblivious machine is likely to be efficient across many real machines without fine-tuning for particular real machine parameters. For many problems, an optimal cache-oblivious algorithm will also be optimal for a machine with more than two memory hierarchy levels.
== Examples ==
The simplest cache-oblivious algorithm presented in Frigo et al. is an out-of-place matrix transpose operation (in-place algorithms have also been devised for transposition, but are much more complex for non-square matrices). Given m×n array A and n×m array B, we would like to store the transpose of A in B. The naive solution traverses one array in row-major order and another in column-major. The result is that when the matrices are large, we get a cache miss on every step of the column-wise traversal. The total number of cache misses is
Θ
(
m
n
)
{\displaystyle \Theta (mn)}
.
The cache-oblivious algorithm has optimal work complexity
O
(
m
n
)
{\displaystyle O(mn)}
and optimal cache complexity
O
(
1
+
m
n
/
B
)
{\displaystyle O(1+mn/B)}
. The basic idea is to reduce the transpose of two large matrices into the transpose of small (sub)matrices. We do this by dividing the matrices in half along their larger dimension until we just have to perform the transpose of a matrix that will fit into the cache. Because the cache size is not known to the algorithm, the matrices will continue to be divided recursively even after this point, but these further subdivisions will be in cache. Once the dimensions m and n are small enough so an input array of size
m
×
n
{\displaystyle m\times n}
and an output array of size
n
×
m
{\displaystyle n\times m}
fit into the cache, both row-major and column-major traversals result in
O
(
m
n
)
{\displaystyle O(mn)}
work and
O
(
m
n
/
B
)
{\displaystyle O(mn/B)}
cache misses. By using this divide and conquer approach we can achieve the same level of complexity for the overall matrix.
(In principle, one could continue dividing the matrices until a base case of size 1×1 is reached, but in practice one uses a larger base case (e.g. 16×16) in order to amortize the overhead of the recursive subroutine calls.)
Most cache-oblivious algorithms rely on a divide-and-conquer approach. They reduce the problem, so that it eventually fits in cache no matter how small the cache is, and end the recursion at some small size determined by the function-call overhead and similar cache-unrelated optimizations, and then use some cache-efficient access pattern to merge the results of these small, solved problems.
Like external sorting in the external memory model, cache-oblivious sorting is possible in two variants: funnelsort, which resembles mergesort; and cache-oblivious distribution sort, which resembles quicksort. Like their external memory counterparts, both achieve a running time of
O
(
N
B
log
M
B
N
B
)
{\displaystyle O\left({\tfrac {N}{B}}\log _{\tfrac {M}{B}}{\tfrac {N}{B}}\right)}
, which matches a lower bound and is thus asymptotically optimal.
== Practicality ==
An empirical comparison of 2 RAM-based, 1 cache-aware, and 2 cache-oblivious algorithms implementing priority queues found that:
Cache-oblivious algorithms performed worse than RAM-based and cache-aware algorithms when data fits into main memory.
The cache-aware algorithm did not seem significantly more complex to implement than the cache-oblivious algorithms, and offered the best performance in all cases tested in the study.
Cache oblivious algorithms outperformed RAM-based algorithms when data size exceeded the size of main memory.
Another study compared hash tables (as RAM-based or cache-unaware), B-trees (as cache-aware), and a cache-oblivious data structure referred to as a "Bender set". For both execution time and memory usage, the hash table was best, followed by the B-tree, with the Bender set the worst in all cases. The memory usage for all tests did not exceed main memory. The hash tables were described as easy to implement, while the Bender set "required a greater amount of effort to implement correctly".
== See also ==
Cache-oblivious distribution sort
External memory algorithm
Funnelsort
== References == | Wikipedia/Cache-oblivious_algorithm |
In mathematics, the generalized minimal residual method (GMRES) is an iterative method for the numerical solution of an indefinite nonsymmetric system of linear equations. The method approximates the solution by the vector in a Krylov subspace with minimal residual. The Arnoldi iteration is used to find this vector.
The GMRES method was developed by Yousef Saad and Martin H. Schultz in 1986. It is a generalization and improvement of the MINRES method due to Paige and Saunders in 1975. The MINRES method requires that the matrix is symmetric, but has the advantage that it only requires handling of three vectors. GMRES is a special case of the DIIS method developed by Peter Pulay in 1980. DIIS is applicable to non-linear systems.
== The method ==
Denote the Euclidean norm of any vector v by
‖
v
‖
{\displaystyle \|v\|}
. Denote the (square) system of linear equations to be solved by
A
x
=
b
.
{\displaystyle Ax=b.}
The matrix A is assumed to be invertible of size m-by-m. Furthermore, it is assumed that b is normalized, i.e., that
‖
b
‖
=
1
{\displaystyle \|b\|=1}
.
The n-th Krylov subspace for this problem is
K
n
=
K
n
(
A
,
r
0
)
=
span
{
r
0
,
A
r
0
,
A
2
r
0
,
…
,
A
n
−
1
r
0
}
.
{\displaystyle K_{n}=K_{n}(A,r_{0})=\operatorname {span} \,\{r_{0},Ar_{0},A^{2}r_{0},\ldots ,A^{n-1}r_{0}\}.\,}
where
r
0
=
b
−
A
x
0
{\displaystyle r_{0}=b-Ax_{0}}
is the initial residual given an initial guess
x
0
≠
0
{\displaystyle x_{0}\neq 0}
. Clearly
r
0
=
b
{\displaystyle r_{0}=b}
if
x
0
=
0
{\displaystyle x_{0}=0}
.
GMRES approximates the exact solution of
A
x
=
b
{\displaystyle Ax=b}
by the vector
x
n
∈
x
0
+
K
n
{\displaystyle x_{n}\in x_{0}+K_{n}}
that minimizes the Euclidean norm of the residual
r
n
=
b
−
A
x
n
{\displaystyle r_{n}=b-Ax_{n}}
.
The vectors
r
0
,
A
r
0
,
…
A
n
−
1
r
0
{\displaystyle r_{0},Ar_{0},\ldots A^{n-1}r_{0}}
might be close to linearly dependent, so instead of this basis, the Arnoldi iteration is used to find orthonormal vectors
q
1
,
q
2
,
…
,
q
n
{\displaystyle q_{1},q_{2},\ldots ,q_{n}\,}
which form a basis for
K
n
{\displaystyle K_{n}}
. In particular,
q
1
=
‖
r
0
‖
2
−
1
r
0
{\displaystyle q_{1}=\|r_{0}\|_{2}^{-1}r_{0}}
.
Therefore, the vector
x
n
∈
x
0
+
K
n
{\displaystyle x_{n}\in x_{0}+K_{n}}
can be written as
x
n
=
x
0
+
Q
n
y
n
{\displaystyle x_{n}=x_{0}+Q_{n}y_{n}}
with
y
n
∈
R
n
{\displaystyle y_{n}\in \mathbb {R} ^{n}}
, where
Q
n
{\displaystyle Q_{n}}
is the m-by-n matrix formed by
q
1
,
…
,
q
n
{\displaystyle q_{1},\ldots ,q_{n}}
. In other words, finding the n-th approximation of the solution (i.e.,
x
n
{\displaystyle x_{n}}
) is reduced to finding the vector
y
n
{\displaystyle y_{n}}
, which is determined via minimizing the residue as described below.
The Arnoldi process also constructs
H
~
n
{\displaystyle {\tilde {H}}_{n}}
, an (
n
+
1
{\displaystyle n+1}
)-by-
n
{\displaystyle n}
upper Hessenberg matrix which satisfies
A
Q
n
=
Q
n
+
1
H
~
n
{\displaystyle AQ_{n}=Q_{n+1}{\tilde {H}}_{n}\,}
an equality which is used to simplify the calculation of
y
n
{\displaystyle y_{n}}
(see § Solving the least squares problem). Note that, for symmetric matrices, a symmetric tri-diagonal matrix is actually achieved, resulting in the MINRES method.
Because columns of
Q
n
{\displaystyle Q_{n}}
are orthonormal, we have
‖
r
n
‖
=
‖
b
−
A
x
n
‖
=
‖
b
−
A
(
x
0
+
Q
n
y
n
)
‖
=
‖
r
0
−
A
Q
n
y
n
‖
=
‖
β
q
1
−
A
Q
n
y
n
‖
=
‖
β
q
1
−
Q
n
+
1
H
~
n
y
n
‖
=
‖
Q
n
+
1
(
β
e
1
−
H
~
n
y
n
)
‖
=
‖
β
e
1
−
H
~
n
y
n
‖
{\displaystyle {\begin{aligned}\left\|r_{n}\right\|&=\left\|b-Ax_{n}\right\|\\&=\left\|b-A(x_{0}+Q_{n}y_{n})\right\|\\&=\left\|r_{0}-AQ_{n}y_{n}\right\|\\&=\left\|\beta q_{1}-AQ_{n}y_{n}\right\|\\&=\left\|\beta q_{1}-Q_{n+1}{\tilde {H}}_{n}y_{n}\right\|\\&=\left\|Q_{n+1}(\beta e_{1}-{\tilde {H}}_{n}y_{n})\right\|\\&=\left\|\beta e_{1}-{\tilde {H}}_{n}y_{n}\right\|\end{aligned}}}
where
e
1
=
(
1
,
0
,
0
,
…
,
0
)
T
{\displaystyle e_{1}=(1,0,0,\ldots ,0)^{T}\,}
is the first vector in the standard basis of
R
n
+
1
{\displaystyle \mathbb {R} ^{n+1}}
, and
β
=
‖
r
0
‖
,
{\displaystyle \beta =\|r_{0}\|\,,}
r
0
{\displaystyle r_{0}}
being the first trial residual vector (usually
b
{\displaystyle b}
). Hence,
x
n
{\displaystyle x_{n}}
can be found by minimizing the Euclidean norm of the residual
r
n
=
H
~
n
y
n
−
β
e
1
.
{\displaystyle r_{n}={\tilde {H}}_{n}y_{n}-\beta e_{1}.}
This is a linear least squares problem of size n.
This yields the GMRES method. On the
n
{\displaystyle n}
-th iteration:
calculate
q
n
{\displaystyle q_{n}}
with the Arnoldi method;
find the
y
n
{\displaystyle y_{n}}
which minimizes
‖
r
n
‖
{\displaystyle \|r_{n}\|}
;
compute
x
n
=
x
0
+
Q
n
y
n
{\displaystyle x_{n}=x_{0}+Q_{n}y_{n}}
;
repeat if the residual is not yet small enough.
At every iteration, a matrix-vector product
A
q
n
{\displaystyle Aq_{n}}
must be computed. This costs about
2
m
2
{\displaystyle 2m^{2}}
floating-point operations for general dense matrices of size
m
{\displaystyle m}
, but the cost can decrease to
O
(
m
)
{\displaystyle O(m)}
for sparse matrices. In addition to the matrix-vector product,
O
(
n
m
)
{\displaystyle O(nm)}
floating-point operations must be computed at the n -th iteration.
== Convergence ==
The nth iterate minimizes the residual in the Krylov subspace
K
n
{\displaystyle K_{n}}
. Since every subspace is contained in the next subspace, the residual does not increase. After m iterations, where m is the size of the matrix A, the Krylov space Km is the whole of Rm and hence the GMRES method arrives at the exact solution. However, the idea is that after a small number of iterations (relative to m), the vector xn is already a good approximation to the
exact solution.
This does not happen in general. Indeed, a theorem of Greenbaum, Pták and Strakoš states that for every nonincreasing sequence a1, ..., am−1, am = 0, one can find a matrix A such that the ‖rn‖ = an for all n, where rn is the residual defined above. In particular, it is possible to find a matrix for which the residual stays constant for m − 1 iterations, and only drops to zero at the last iteration.
In practice, though, GMRES often performs well. This can be proven in specific situations. If the symmetric part of A, that is
(
A
T
+
A
)
/
2
{\displaystyle (A^{T}+A)/2}
, is positive definite, then
‖
r
n
‖
≤
(
1
−
λ
min
2
(
1
/
2
(
A
T
+
A
)
)
λ
max
(
A
T
A
)
)
n
/
2
‖
r
0
‖
,
{\displaystyle \|r_{n}\|\leq \left(1-{\frac {\lambda _{\min }^{2}(1/2(A^{T}+A))}{\lambda _{\max }(A^{T}A)}}\right)^{n/2}\|r_{0}\|,}
where
λ
m
i
n
(
M
)
{\displaystyle \lambda _{\mathrm {min} }(M)}
and
λ
m
a
x
(
M
)
{\displaystyle \lambda _{\mathrm {max} }(M)}
denote the smallest and largest eigenvalue of the matrix
M
{\displaystyle M}
, respectively.
If A is symmetric and positive definite, then we even have
‖
r
n
‖
≤
(
κ
2
(
A
)
2
−
1
κ
2
(
A
)
2
)
n
/
2
‖
r
0
‖
.
{\displaystyle \|r_{n}\|\leq \left({\frac {\kappa _{2}(A)^{2}-1}{\kappa _{2}(A)^{2}}}\right)^{n/2}\|r_{0}\|.}
where
κ
2
(
A
)
{\displaystyle \kappa _{2}(A)}
denotes the condition number of A in the Euclidean norm.
In the general case, where A is not positive definite, we have
‖
r
n
‖
‖
b
‖
≤
inf
p
∈
P
n
‖
p
(
A
)
‖
≤
κ
2
(
V
)
inf
p
∈
P
n
max
λ
∈
σ
(
A
)
|
p
(
λ
)
|
,
{\displaystyle {\frac {\|r_{n}\|}{\|b\|}}\leq \inf _{p\in P_{n}}\|p(A)\|\leq \kappa _{2}(V)\inf _{p\in P_{n}}\max _{\lambda \in \sigma (A)}|p(\lambda )|,\,}
where Pn denotes the set of polynomials of degree at most n with p(0) = 1, V is the matrix appearing in the spectral decomposition of A, and σ(A) is the spectrum of A. Roughly speaking, this says that fast convergence occurs when the eigenvalues of A are clustered away from the origin and A is not too far from normality.
All these inequalities bound only the residuals instead of the actual error, that is, the distance between the current iterate xn and the exact solution.
== Extensions of the method ==
Like other iterative methods, GMRES is usually combined with a preconditioning method in order to speed up convergence.
The cost of the iterations grow as O(n2), where n is the iteration number. Therefore, the method is sometimes restarted after a number, say k, of iterations, with xk as initial guess. The resulting method is called GMRES(k) or Restarted GMRES. For non-positive definite matrices, this method may suffer from stagnation in convergence as the restarted subspace is often close to the earlier subspace.
The shortcomings of GMRES and restarted GMRES are addressed by the recycling of Krylov subspace in the GCRO type methods such as GCROT and GCRODR.
Recycling of Krylov subspaces in GMRES can also speed up convergence when sequences of linear systems need to be solved.
== Comparison with other solvers ==
The Arnoldi iteration reduces to the Lanczos iteration for symmetric matrices. The corresponding Krylov subspace method is the minimal residual method (MinRes) of Paige and Saunders. Unlike the unsymmetric case, the MinRes method is given by a three-term recurrence relation. It can be shown that there is no Krylov subspace method for general matrices, which is given by a short recurrence relation and yet minimizes the norms of the residuals, as GMRES does.
Another class of methods builds on the unsymmetric Lanczos iteration, in particular the BiCG method. These use a three-term recurrence relation, but they do not attain the minimum residual, and hence the residual does not decrease monotonically for these methods. Convergence is not even guaranteed.
The third class is formed by methods like CGS and BiCGSTAB. These also work with a three-term recurrence relation (hence, without optimality) and they can even terminate prematurely without achieving convergence. The idea behind these methods is to choose the generating polynomials of the iteration sequence suitably.
None of these three classes is the best for all matrices; there are always examples in which one class outperforms the other. Therefore, multiple solvers are tried in practice to see which one is the best for a given problem.
== Solving the least squares problem ==
One part of the GMRES method is to find the vector
y
n
{\displaystyle y_{n}}
which minimizes
‖
H
~
n
y
n
−
β
e
1
‖
.
{\displaystyle \left\|{\tilde {H}}_{n}y_{n}-\beta e_{1}\right\|.}
Note that
H
~
n
{\displaystyle {\tilde {H}}_{n}}
is an (n + 1)-by-n matrix, hence it gives an over-constrained linear system of n+1 equations for n unknowns.
The minimum can be computed using a QR decomposition: find an (n + 1)-by-(n + 1) orthogonal matrix Ωn and an (n + 1)-by-n upper triangular matrix
R
~
n
{\displaystyle {\tilde {R}}_{n}}
such that
Ω
n
H
~
n
=
R
~
n
.
{\displaystyle \Omega _{n}{\tilde {H}}_{n}={\tilde {R}}_{n}.}
The triangular matrix has one more row than it has columns, so its bottom row consists of zero. Hence, it can be decomposed as
R
~
n
=
[
R
n
0
]
,
{\displaystyle {\tilde {R}}_{n}={\begin{bmatrix}R_{n}\\0\end{bmatrix}},}
where
R
n
{\displaystyle R_{n}}
is an n-by-n (thus square) triangular matrix.
The QR decomposition can be updated cheaply from one iteration to the next, because the Hessenberg matrices differ only by a row of zeros and a column:
H
~
n
+
1
=
[
H
~
n
h
n
+
1
0
h
n
+
2
,
n
+
1
]
,
{\displaystyle {\tilde {H}}_{n+1}={\begin{bmatrix}{\tilde {H}}_{n}&h_{n+1}\\0&h_{n+2,n+1}\end{bmatrix}},}
where hn+1 = (h1,n+1, ..., hn+1,n+1)T. This implies that premultiplying the Hessenberg matrix with Ωn, augmented with zeroes and a row with multiplicative identity, yields almost a triangular matrix:
[
Ω
n
0
0
1
]
H
~
n
+
1
=
[
R
n
r
n
+
1
0
ρ
0
σ
]
{\displaystyle {\begin{bmatrix}\Omega _{n}&0\\0&1\end{bmatrix}}{\tilde {H}}_{n+1}={\begin{bmatrix}R_{n}&r_{n+1}\\0&\rho \\0&\sigma \end{bmatrix}}}
This would be triangular if σ is zero. To remedy this, one needs the Givens rotation
G
n
=
[
I
n
0
0
0
c
n
s
n
0
−
s
n
c
n
]
{\displaystyle G_{n}={\begin{bmatrix}I_{n}&0&0\\0&c_{n}&s_{n}\\0&-s_{n}&c_{n}\end{bmatrix}}}
where
c
n
=
ρ
ρ
2
+
σ
2
and
s
n
=
σ
ρ
2
+
σ
2
.
{\displaystyle c_{n}={\frac {\rho }{\sqrt {\rho ^{2}+\sigma ^{2}}}}\quad {\text{and}}\quad s_{n}={\frac {\sigma }{\sqrt {\rho ^{2}+\sigma ^{2}}}}.}
With this Givens rotation, we form
Ω
n
+
1
=
G
n
[
Ω
n
0
0
1
]
.
{\displaystyle \Omega _{n+1}=G_{n}{\begin{bmatrix}\Omega _{n}&0\\0&1\end{bmatrix}}.}
Indeed,
Ω
n
+
1
H
~
n
+
1
=
[
R
n
r
n
+
1
0
r
n
+
1
,
n
+
1
0
0
]
{\displaystyle \Omega _{n+1}{\tilde {H}}_{n+1}={\begin{bmatrix}R_{n}&r_{n+1}\\0&r_{n+1,n+1}\\0&0\end{bmatrix}}}
is a triangular matrix with
r
n
+
1
,
n
+
1
=
ρ
2
+
σ
2
{\textstyle r_{n+1,n+1}={\sqrt {\rho ^{2}+\sigma ^{2}}}}
.
Given the QR decomposition, the minimization problem is easily solved by noting that
‖
H
~
n
y
n
−
β
e
1
‖
=
‖
Ω
n
(
H
~
n
y
n
−
β
e
1
)
‖
=
‖
R
~
n
y
n
−
β
Ω
n
e
1
‖
.
{\displaystyle {\begin{aligned}\left\|{\tilde {H}}_{n}y_{n}-\beta e_{1}\right\|&=\left\|\Omega _{n}({\tilde {H}}_{n}y_{n}-\beta e_{1})\right\|\\&=\left\|{\tilde {R}}_{n}y_{n}-\beta \Omega _{n}e_{1}\right\|.\end{aligned}}}
Denoting the vector
β
Ω
n
e
1
{\displaystyle \beta \Omega _{n}e_{1}}
by
g
~
n
=
[
g
n
γ
n
]
{\displaystyle {\tilde {g}}_{n}={\begin{bmatrix}g_{n}\\\gamma _{n}\end{bmatrix}}}
with gn ∈ Rn and γn ∈ R, this is
‖
H
~
n
y
n
−
β
e
1
‖
=
‖
R
~
n
y
n
−
β
Ω
n
e
1
‖
=
‖
[
R
n
0
]
y
n
−
[
g
n
γ
n
]
‖
.
{\displaystyle {\begin{aligned}\left\|{\tilde {H}}_{n}y_{n}-\beta e_{1}\right\|&=\left\|{\tilde {R}}_{n}y_{n}-\beta \Omega _{n}e_{1}\right\|\\&=\left\|{\begin{bmatrix}R_{n}\\0\end{bmatrix}}y_{n}-{\begin{bmatrix}g_{n}\\\gamma _{n}\end{bmatrix}}\right\|.\end{aligned}}}
The vector y that minimizes this expression is given by
y
n
=
R
n
−
1
g
n
.
{\displaystyle y_{n}=R_{n}^{-1}g_{n}.}
Again, the vectors
g
n
{\displaystyle g_{n}}
are easy to update.
== Example code ==
=== Regular GMRES (MATLAB / GNU Octave) ===
== See also ==
Biconjugate gradient method
== References == | Wikipedia/Generalized_minimal_residual_method |
Because matrix multiplication is such a central operation in many numerical algorithms, much work has been invested in making matrix multiplication algorithms efficient. Applications of matrix multiplication in computational problems are found in many fields including scientific computing and pattern recognition and in seemingly unrelated problems such as counting the paths through a graph. Many different algorithms have been designed for multiplying matrices on different types of hardware, including parallel and distributed systems, where the computational work is spread over multiple processors (perhaps over a network).
Directly applying the mathematical definition of matrix multiplication gives an algorithm that takes time on the order of n3 field operations to multiply two n × n matrices over that field (Θ(n3) in big O notation). Better asymptotic bounds on the time required to multiply matrices have been known since the Strassen's algorithm in the 1960s, but the optimal time (that is, the computational complexity of matrix multiplication) remains unknown. As of April 2024, the best announced bound on the asymptotic complexity of a matrix multiplication algorithm is O(n2.371552) time, given by Williams, Xu, Xu, and Zhou. This improves on the bound of O(n2.3728596) time, given by Alman and Williams. However, this algorithm is a galactic algorithm because of the large constants and cannot be realized practically.
== Iterative algorithm ==
The definition of matrix multiplication is that if C = AB for an n × m matrix A and an m × p matrix B, then C is an n × p matrix with entries
c
i
j
=
∑
k
=
1
m
a
i
k
b
k
j
.
{\displaystyle c_{ij}=\sum _{k=1}^{m}a_{ik}b_{kj}.}
From this, a simple algorithm can be constructed which loops over the indices i from 1 through n and j from 1 through p, computing the above using a nested loop:
This algorithm takes time Θ(nmp) (in asymptotic notation). A common simplification for the purpose of algorithm analysis is to assume that the inputs are all square matrices of size n × n, in which case the running time is Θ(n3), i.e., cubic in the size of the dimension.
=== Cache behavior ===
The three loops in iterative matrix multiplication can be arbitrarily swapped with each other without an effect on correctness or asymptotic running time. However, the order can have a considerable impact on practical performance due to the memory access patterns and cache use of the algorithm;
which order is best also depends on whether the matrices are stored in row-major order, column-major order, or a mix of both.
In particular, in the idealized case of a fully associative cache consisting of M bytes and b bytes per cache line (i.e. M/b cache lines), the above algorithm is sub-optimal for A and B stored in row-major order. When n > M/b, every iteration of the inner loop (a simultaneous sweep through a row of A and a column of B) incurs a cache miss when accessing an element of B. This means that the algorithm incurs Θ(n3) cache misses in the worst case. As of 2010, the speed of memories compared to that of processors is such that the cache misses, rather than the actual calculations, dominate the running time for sizable matrices.
The optimal variant of the iterative algorithm for A and B in row-major layout is a tiled version, where the matrix is implicitly divided into square tiles of size √M by √M:
In the idealized cache model, this algorithm incurs only Θ(n3/b √M) cache misses; the divisor b √M amounts to several orders of magnitude on modern machines, so that the actual calculations dominate the running time, rather than the cache misses.
== Divide-and-conquer algorithm ==
An alternative to the iterative algorithm is the divide-and-conquer algorithm for matrix multiplication. This relies on the block partitioning
C
=
(
C
11
C
12
C
21
C
22
)
,
A
=
(
A
11
A
12
A
21
A
22
)
,
B
=
(
B
11
B
12
B
21
B
22
)
,
{\displaystyle C={\begin{pmatrix}C_{11}&C_{12}\\C_{21}&C_{22}\\\end{pmatrix}},\,A={\begin{pmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\\\end{pmatrix}},\,B={\begin{pmatrix}B_{11}&B_{12}\\B_{21}&B_{22}\\\end{pmatrix}},}
which works for all square matrices whose dimensions are powers of two, i.e., the shapes are 2n × 2n for some n. The matrix product is now
(
C
11
C
12
C
21
C
22
)
=
(
A
11
A
12
A
21
A
22
)
(
B
11
B
12
B
21
B
22
)
=
(
A
11
B
11
+
A
12
B
21
A
11
B
12
+
A
12
B
22
A
21
B
11
+
A
22
B
21
A
21
B
12
+
A
22
B
22
)
{\displaystyle {\begin{pmatrix}C_{11}&C_{12}\\C_{21}&C_{22}\\\end{pmatrix}}={\begin{pmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\\\end{pmatrix}}{\begin{pmatrix}B_{11}&B_{12}\\B_{21}&B_{22}\\\end{pmatrix}}={\begin{pmatrix}A_{11}B_{11}+A_{12}B_{21}&A_{11}B_{12}+A_{12}B_{22}\\A_{21}B_{11}+A_{22}B_{21}&A_{21}B_{12}+A_{22}B_{22}\\\end{pmatrix}}}
which consists of eight multiplications of pairs of submatrices, followed by an addition step. The divide-and-conquer algorithm computes the smaller multiplications recursively, using the scalar multiplication c11 = a11b11 as its base case.
The complexity of this algorithm as a function of n is given by the recurrence
T
(
1
)
=
Θ
(
1
)
;
{\displaystyle T(1)=\Theta (1);}
T
(
n
)
=
8
T
(
n
/
2
)
+
Θ
(
n
2
)
,
{\displaystyle T(n)=8T(n/2)+\Theta (n^{2}),}
accounting for the eight recursive calls on matrices of size n/2 and Θ(n2) to sum the four pairs of resulting matrices element-wise. Application of the master theorem for divide-and-conquer recurrences shows this recursion to have the solution Θ(n3), the same as the iterative algorithm.
=== Non-square matrices ===
A variant of this algorithm that works for matrices of arbitrary shapes and is faster in practice splits matrices in two instead of four submatrices, as follows.
Splitting a matrix now means dividing it into two parts of equal size, or as close to equal sizes as possible in the case of odd dimensions.
=== Cache behavior ===
The cache miss rate of recursive matrix multiplication is the same as that of a tiled iterative version, but unlike that algorithm, the recursive algorithm is cache-oblivious: there is no tuning parameter required to get optimal cache performance, and it behaves well in a multiprogramming environment where cache sizes are effectively dynamic due to other processes taking up cache space.
(The simple iterative algorithm is cache-oblivious as well, but much slower in practice if the matrix layout is not adapted to the algorithm.)
The number of cache misses incurred by this algorithm, on a machine with M lines of ideal cache, each of size b bytes, is bounded by: 13
Θ
(
m
+
n
+
p
+
m
n
+
n
p
+
m
p
b
+
m
n
p
b
M
)
{\displaystyle \Theta \left(m+n+p+{\frac {mn+np+mp}{b}}+{\frac {mnp}{b{\sqrt {M}}}}\right)}
== Sub-cubic algorithms ==
Algorithms exist that provide better running times than the straightforward ones. The first to be discovered was Strassen's algorithm, devised by Volker Strassen in 1969 and often referred to as "fast matrix multiplication". It is based on a way of multiplying two 2×2 matrices which requires only 7 multiplications (instead of the usual 8), at the expense of several additional addition and subtraction operations. Applying this recursively gives an algorithm with a multiplicative cost of
O
(
n
log
2
7
)
≈
O
(
n
2.807
)
{\displaystyle O(n^{\log _{2}7})\approx O(n^{2.807})}
. Strassen's algorithm is more complex, and the numerical stability is reduced compared to the naïve algorithm, but it is faster in cases where n > 100 or so and appears in several libraries, such as BLAS. It is very useful for large matrices over exact domains such as finite fields, where numerical stability is not an issue.
Since Strassen's algorithm is actually used in practical numerical software and computer algebra systems, improving on the constants hidden in the big-O notation has its merits. A table that compares key aspects of the improved version based on recursive multiplication of 2×2-block matrices via 7 block matrix multiplications follows. As usual,
n
{\displaystyle n}
gives the dimensions of the matrix and
M
{\displaystyle M}
designates the memory size.
It is known that a Strassen-like algorithm with a 2×2-block matrix step requires at least 7 block matrix multiplications. In 1976 Probert showed that such an algorithm requires at least 15 additions (including subtractions); however, a hidden assumption was that the blocks and the 2×2-block matrix are represented in the same basis. Karstadt and Schwartz computed in different bases and traded 3 additions for less expensive basis transformations. They also proved that one cannot go below 12 additions per step using different bases. In subsequent work Beniamini et el. applied this base-change trick to more general decompositions than 2×2-block matrices and improved the leading constant for their run times.
It is an open question in theoretical computer science how well Strassen's algorithm can be improved in terms of asymptotic complexity. The matrix multiplication exponent, usually denoted
ω
{\displaystyle \omega }
, is the smallest real number for which any
n
×
n
{\displaystyle n\times n}
matrix over a field can be multiplied together using
n
ω
+
o
(
1
)
{\displaystyle n^{\omega +o(1)}}
field operations. The current best bound on
ω
{\displaystyle \omega }
is
ω
<
2.371552
{\displaystyle \omega <2.371552}
, by Williams, Xu, Xu, and Zhou. This algorithm, like all other recent algorithms in this line of research, is a generalization of the Coppersmith–Winograd algorithm, which was given by Don Coppersmith and Shmuel Winograd in 1990. The conceptual idea of these algorithms is similar to Strassen's algorithm: a way is devised for multiplying two k × k-matrices with fewer than k3 multiplications, and this technique is applied recursively. However, the constant coefficient hidden by the big-O notation is so large that these algorithms are only worthwhile for matrices that are too large to handle on present-day computers. Victor Pan proposed so-called feasible sub-cubic matrix multiplication algorithms with an exponent slightly above 2.77, but in return with a much smaller hidden constant coefficient.
Freivalds' algorithm is a simple Monte Carlo algorithm that, given matrices A, B and C, verifies in Θ(n2) time if AB = C.
=== AlphaTensor ===
In 2022, DeepMind introduced AlphaTensor, a neural network that used a single-player game analogy to invent thousands of matrix multiplication algorithms, including some previously discovered by humans and some that were not. Operations were restricted to the non-commutative ground field(normal arithmetic) and finite field
Z
/
2
Z
{\displaystyle \mathbb {Z} /2\mathbb {Z} }
(mod 2 arithmetic). The best "practical" (explicit low-rank decomposition of a matrix multiplication tensor) algorithm found ran in O(n2.778). Finding low-rank decompositions of such tensors (and beyond) is NP-hard; optimal multiplication even for 3×3 matrices remains unknown, even in commutative field. On 4×4 matrices, AlphaTensor unexpectedly discovered a solution with 47 multiplication steps, an improvement over the 49 required with Strassen’s algorithm of 1969, albeit restricted to mod 2 arithmetic. Similarly, AlphaTensor solved 5×5 matrices with 96 rather than Strassen's 98 steps. Based on the surprising discovery that such improvements exist, other researchers were quickly able to find a similar independent 4×4 algorithm, and separately tweaked Deepmind's 96-step 5×5 algorithm down to 95 steps in mod 2 arithmetic and to 97 in normal arithmetic. Some algorithms were completely new: for example, (4, 5, 5) was improved to 76 steps from a baseline of 80 in both normal and mod 2 arithmetic.
== Parallel and distributed algorithms ==
=== Shared-memory parallelism ===
The divide-and-conquer algorithm sketched earlier can be parallelized in two ways for shared-memory multiprocessors. These are based on the fact that the eight recursive matrix multiplications in
(
A
11
B
11
+
A
12
B
21
A
11
B
12
+
A
12
B
22
A
21
B
11
+
A
22
B
21
A
21
B
12
+
A
22
B
22
)
{\displaystyle {\begin{pmatrix}A_{11}B_{11}+A_{12}B_{21}&A_{11}B_{12}+A_{12}B_{22}\\A_{21}B_{11}+A_{22}B_{21}&A_{21}B_{12}+A_{22}B_{22}\\\end{pmatrix}}}
can be performed independently of each other, as can the four summations (although the algorithm needs to "join" the multiplications before doing the summations). Exploiting the full parallelism of the problem, one obtains an algorithm that can be expressed in fork–join style pseudocode:
Here, fork is a keyword that signal a computation may be run in parallel with the rest of the function call, while join waits for all previously "forked" computations to complete. partition achieves its goal by pointer manipulation only.
This algorithm has a critical path length of Θ(log2 n) steps, meaning it takes that much time on an ideal machine with an infinite number of processors; therefore, it has a maximum possible speedup of Θ(n3/log2 n) on any real computer. The algorithm isn't practical due to the communication cost inherent in moving data to and from the temporary matrix T, but a more practical variant achieves Θ(n2) speedup, without using a temporary matrix.
=== Communication-avoiding and distributed algorithms ===
On modern architectures with hierarchical memory, the cost of loading and storing input matrix elements tends to dominate the cost of arithmetic. On a single machine this is the amount of data transferred between RAM and cache, while on a distributed memory multi-node machine it is the amount transferred between nodes; in either case it is called the communication bandwidth. The naïve algorithm using three nested loops uses Ω(n3) communication bandwidth.
Cannon's algorithm, also known as the 2D algorithm, is a communication-avoiding algorithm that partitions each input matrix into a block matrix whose elements are submatrices of size √M/3 by √M/3, where M is the size of fast memory. The naïve algorithm is then used over the block matrices, computing products of submatrices entirely in fast memory. This reduces communication bandwidth to O(n3/√M), which is asymptotically optimal (for algorithms performing Ω(n3) computation).
In a distributed setting with p processors arranged in a √p by √p 2D mesh, one submatrix of the result can be assigned to each processor, and the product can be computed with each processor transmitting O(n2/√p) words, which is asymptotically optimal assuming that each node stores the minimum O(n2/p) elements. This can be improved by the 3D algorithm, which arranges the processors in a 3D cube mesh, assigning every product of two input submatrices to a single processor. The result submatrices are then generated by performing a reduction over each row. This algorithm transmits O(n2/p2/3) words per processor, which is asymptotically optimal. However, this requires replicating each input matrix element p1/3 times, and so requires a factor of p1/3 more memory than is needed to store the inputs. This algorithm can be combined with Strassen to further reduce runtime. "2.5D" algorithms provide a continuous tradeoff between memory usage and communication bandwidth. On modern distributed computing environments such as MapReduce, specialized multiplication algorithms have been developed.
=== Algorithms for meshes ===
There are a variety of algorithms for multiplication on meshes. For multiplication of two n×n on a standard two-dimensional mesh using the 2D Cannon's algorithm, one can complete the multiplication in 3n-2 steps although this is reduced to half this number for repeated computations. The standard array is inefficient because the data from the two matrices does not arrive simultaneously and it must be padded with zeroes.
The result is even faster on a two-layered cross-wired mesh, where only 2n-1 steps are needed. The performance improves further for repeated computations leading to 100% efficiency. The cross-wired mesh array may be seen as a special case of a non-planar (i.e. multilayered) processing structure.
In a 3D mesh with n3 processing elements, two matrices can be multiplied in
O
(
log
n
)
{\displaystyle {\mathcal {O}}(\log n)}
using the DNS algorithm.
== See also ==
Computational complexity of mathematical operations
Computational complexity of matrix multiplication
CYK algorithm § Valiant's algorithm
Matrix chain multiplication
Method of Four Russians
Multiplication algorithm
Sparse matrix–vector multiplication
== References ==
== Further reading == | Wikipedia/Matrix_multiplication_algorithm |
Automatically Tuned Linear Algebra Software (ATLAS) is a software library for linear algebra. It provides a mature open source implementation of BLAS APIs for C and FORTRAN 77.
ATLAS is often recommended as a way to automatically generate an optimized BLAS library. While its performance often trails that of specialized libraries written for one specific hardware platform, it is often the first or even only optimized BLAS implementation available on new systems and is a large improvement over the generic BLAS available at Netlib. For this reason, ATLAS is sometimes used as a performance baseline for comparison with other products.
ATLAS runs on most Unix-like operating systems and on Microsoft Windows (using Cygwin). It is released under a BSD-style license without advertising clause, and many well-known mathematics applications including MATLAB, Mathematica, Scilab, SageMath, and some builds of GNU Octave may use it.
== Functionality ==
ATLAS provides a full implementation of the BLAS APIs as well as some additional functions from LAPACK, a higher-level library built on top of BLAS. In BLAS, functionality is divided into three groups called levels 1, 2 and 3.
Level 1 contains vector operations of the form
y
←
α
x
+
y
{\displaystyle \mathbf {y} \leftarrow \alpha \mathbf {x} +\mathbf {y} \!}
as well as scalar dot products and vector norms, among other things.
Level 2 contains matrix-vector operations of the form
y
←
α
A
x
+
β
y
{\displaystyle \mathbf {y} \leftarrow \alpha A\mathbf {x} +\beta \mathbf {y} \!}
as well as solving
T
x
=
y
{\displaystyle T\mathbf {x} =\mathbf {y} }
for
x
{\displaystyle \mathbf {x} }
with
T
{\displaystyle T}
being triangular, among other things.
Level 3 contains matrix-matrix operations such as the widely used General Matrix Multiply (GEMM) operation
C
←
α
A
B
+
β
C
{\displaystyle C\leftarrow \alpha AB+\beta C\!}
as well as solving
B
←
α
T
−
1
B
{\displaystyle B\leftarrow \alpha T^{-1}B}
for triangular matrices
T
{\displaystyle T}
, among other things.
== Optimization approach ==
The optimization approach is called Automated Empirical Optimization of Software (AEOS), which identifies four fundamental approaches to computer assisted optimization of which ATLAS employs three:
Parameterization—searching over the parameter space of a function, used for blocking factor, cache edge, etc.
Multiple implementation—searching through various approaches to implementing the same function, e.g., for SSE support before intrinsics made them available in C code
Code generation—programs that write programs incorporating what knowledge they can about what will produce the best performance for the system
Optimization of the level 1 BLAS uses parameterization and multiple implementation
Every ATLAS level 1 BLAS function has its own kernel. Since it would be difficult to maintain thousands of cases in ATLAS there is little architecture specific optimization for Level 1 BLAS. Instead multiple implementation is relied upon to allow for compiler optimization to produce high performance implementation for the system.
Optimization of the level 2 BLAS uses parameterization and multiple implementation
With
N
2
{\displaystyle N^{2}}
data and
N
2
{\displaystyle N^{2}}
operations to perform the function is usually limited by bandwidth to memory, and thus there is not much opportunity for optimization
All routines in the ATLAS level 2 BLAS are built from two Level 2 BLAS kernels:
GEMV—matrix by vector multiply update:
y
←
α
A
x
+
β
y
{\displaystyle \mathbf {y} \leftarrow \alpha A\mathbf {x} +\beta \mathbf {y} \!}
GER—general rank 1 update from an outer product:
A
←
α
x
y
T
+
A
{\displaystyle A\leftarrow \alpha \mathbf {x} \mathbf {y} ^{T}+A\!}
Optimization of the level 3 BLAS uses code generation and the other two techniques
Since we have
N
3
{\displaystyle N^{3}}
ops with only
N
2
{\displaystyle N^{2}}
data, there are many opportunities for optimization
== Level 3 BLAS ==
Most of the Level 3 BLAS is derived from GEMM, so that is the primary focus of the optimization.
O
(
n
3
)
{\displaystyle O(n^{3})}
operations vs.
O
(
n
2
)
{\displaystyle O(n^{2})}
data
The intuition that the
n
3
{\displaystyle n^{3}}
operations will dominate over the
n
2
{\displaystyle n^{2}}
data accesses only works for roughly square matrices.
The real measure should be some kind of surface area to volume.
The difference becomes important for very non-square matrices.
=== Can it afford to copy? ===
Copying the inputs allows the data to be arranged in a way that provides optimal access for the kernel functions,
but this comes at the cost of allocating temporary space, and an extra read and write of the inputs.
So the first question GEMM faces is, can it afford to copy the inputs?
If so,
Put into block major format with good alignment
Take advantage of user contributed kernels and cleanup
Handle the transpose cases with the copy: make everything into TN (transpose - no-transpose)
Deal with α in the copy
If not,
Use the nocopy version
Make no assumptions on the stride of matrix A and B in memory
Handle all transpose cases explicitly
No guarantee about alignment of data
Support α specific code
Run the risk of TLB issues, bad strides, etc.
The actual decision is made through a simple heuristic which checks for "skinny cases".
=== Cache edge ===
For 2nd Level Cache blocking a single cache edge parameter is used.
The high level choose an order to traverse the blocks: ijk, jik, ikj, jki, kij, kji.
These need not be the same order as the product is done within a block.
Typically chosen orders are ijk or jik.
For jik the ideal situation would be to copy A and the NB wide panel of B.
For ijk swap the role of A and B.
Choosing the bigger of M or N for the outer loop reduces the footprint of the copy.
But for large K ATLAS does not even allocate such a large amount of memory.
Instead it defines a parameter, Kp, to give best use of the L2 cache.
Panels are limited to Kp in length.
It first tries to allocate (in the jik case)
M
⋅
p
+
N
B
⋅
K
p
+
N
B
⋅
N
B
{\displaystyle M\cdot p+NB\cdot Kp+NB\cdot NB}
.
If that fails it tries
2
⋅
K
p
⋅
N
B
+
N
B
⋅
N
B
{\displaystyle 2\cdot Kp\cdot NB+NB\cdot NB}
.
(If that fails it uses the no-copy version of GEMM, but this case is unlikely for reasonable choices of cache edge.)
Kp is a function of cache edge and NB.
== LAPACK ==
When integrating the ATLAS BLAS with LAPACK an important consideration is the choice of blocking factor for LAPACK. If the ATLAS blocking factor is small enough the blocking factor of LAPACK could be set to match that of ATLAS.
To take advantage of recursive factorization, ATLAS provides replacement routines for some LAPACK routines. These simply overwrite the corresponding LAPACK routines from Netlib.
== Need for installation ==
Installing ATLAS on a particular platform is a challenging process which is typically done by a system vendor or a local expert and made available to a wider audience.
For many systems, architectural default parameters are available; these are essentially saved searches plus the results of hand tuning.
If the arch defaults work they will likely get 10-15% better performance than the install search. On such systems the installation process is greatly simplified.
== References ==
== External links ==
Automatically Tuned Linear Algebra Software on SourceForge
User contribution to ATLAS
A Collaborative guide to ATLAS Development
The FAQ has links to the Quick reference guide to BLAS and Quick reference to ATLAS LAPACK API reference
Microsoft Visual C++ Howto Archived 2007-09-28 at the Wayback Machine for ATLAS | Wikipedia/Automatically_Tuned_Linear_Algebra_Software |
In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. The main types of inequality are less than and greater than (denoted by < and >, respectively the less-than and greater-than signs).
== Notation ==
There are several different notations used to represent different kinds of inequalities:
The notation a < b means that a is less than b.
The notation a > b means that a is greater than b.
In either case, a is not equal to b. These relations are known as strict inequalities, meaning that a is strictly less than or strictly greater than b. Equality is excluded.
In contrast to strict inequalities, there are two types of inequality relations that are not strict:
The notation a ≤ b or a ⩽ b or a ≦ b means that a is less than or equal to b (or, equivalently, at most b, or not greater than b).
The notation a ≥ b or a ⩾ b or a ≧ b means that a is greater than or equal to b (or, equivalently, at least b, or not less than b).
In the 17th and 18th centuries, personal notations or typewriting signs were used to signal inequalities. For example, In 1670, John Wallis used a single horizontal bar above rather than below the < and >.
Later in 1734, ≦ and ≧, known as "less than (greater-than) over equal to" or "less than (greater than) or equal to with double horizontal bars", first appeared in Pierre Bouguer's work . After that, mathematicians simplified Bouguer's symbol to "less than (greater than) or equal to with one horizontal bar" (≤), or "less than (greater than) or slanted equal to" (⩽).
The relation not greater than can also be represented by
a
≯
b
,
{\displaystyle a\ngtr b,}
the symbol for "greater than" bisected by a slash, "not". The same is true for not less than,
a
≮
b
.
{\displaystyle a\nless b.}
The notation a ≠ b means that a is not equal to b; this inequation sometimes is considered a form of strict inequality. It does not say that one is greater than the other; it does not even require a and b to be member of an ordered set.
In engineering sciences, less formal use of the notation is to state that one quantity is "much greater" than another, normally by several orders of magnitude.
The notation a ≪ b means that a is much less than b.
The notation a ≫ b means that a is much greater than b.
This implies that the lesser value can be neglected with little effect on the accuracy of an approximation (such as the case of ultrarelativistic limit in physics).
In all of the cases above, any two symbols mirroring each other are symmetrical; a < b and b > a are equivalent, etc.
== Properties on the number line ==
Inequalities are governed by the following properties. All of these properties also hold if all of the non-strict inequalities (≤ and ≥) are replaced by their corresponding strict inequalities (< and >) and — in the case of applying a function — monotonic functions are limited to strictly monotonic functions.
=== Converse ===
The relations ≤ and ≥ are each other's converse, meaning that for any real numbers a and b:
=== Transitivity ===
The transitive property of inequality states that for any real numbers a, b, c:
If either of the premises is a strict inequality, then the conclusion is a strict inequality:
=== Addition and subtraction ===
A common constant c may be added to or subtracted from both sides of an inequality. So, for any real numbers a, b, c:
In other words, the inequality relation is preserved under addition (or subtraction) and the real numbers are an ordered group under addition.
=== Multiplication and division ===
The properties that deal with multiplication and division state that for any real numbers, a, b and non-zero c:
In other words, the inequality relation is preserved under multiplication and division with positive constant, but is reversed when a negative constant is involved. More generally, this applies for an ordered field. For more information, see § Ordered fields.
=== Additive inverse ===
The property for the additive inverse states that for any real numbers a and b:
=== Multiplicative inverse ===
If both numbers are positive, then the inequality relation between the multiplicative inverses is opposite of that between the original numbers. More specifically, for any non-zero real numbers a and b that are both positive (or both negative):
All of the cases for the signs of a and b can also be written in chained notation, as follows:
=== Applying a function to both sides ===
Any monotonically increasing function, by its definition, may be applied to both sides of an inequality without breaking the inequality relation (provided that both expressions are in the domain of that function). However, applying a monotonically decreasing function to both sides of an inequality means the inequality relation would be reversed. The rules for the additive inverse, and the multiplicative inverse for positive numbers, are both examples of applying a monotonically decreasing function.
If the inequality is strict (a < b, a > b) and the function is strictly monotonic, then the inequality remains strict. If only one of these conditions is strict, then the resultant inequality is non-strict. In fact, the rules for additive and multiplicative inverses are both examples of applying a strictly monotonically decreasing function.
A few examples of this rule are:
Raising both sides of an inequality to a power n > 0 (equiv., −n < 0), when a and b are positive real numbers:
Taking the natural logarithm on both sides of an inequality, when a and b are positive real numbers: (this is true because the natural logarithm is a strictly increasing function.)
== Formal definitions and generalizations ==
A (non-strict) partial order is a binary relation ≤ over a set P which is reflexive, antisymmetric, and transitive. That is, for all a, b, and c in P, it must satisfy the three following clauses:
a ≤ a (reflexivity)
if a ≤ b and b ≤ a, then a = b (antisymmetry)
if a ≤ b and b ≤ c, then a ≤ c (transitivity)
A set with a partial order is called a partially ordered set. Those are the very basic axioms that every kind of order has to satisfy.
A strict partial order is a relation < that satisfies
a ≮ a (irreflexivity),
if a < b, then b ≮ a (asymmetry),
if a < b and b < c, then a < c (transitivity),
where ≮ means that < does not hold.
Some types of partial orders are specified by adding further axioms, such as:
Total order: For every a and b in P, a ≤ b or b ≤ a .
Dense order: For all a and b in P for which a < b, there is a c in P such that a < c < b.
Least-upper-bound property: Every non-empty subset of P with an upper bound has a least upper bound (supremum) in P.
=== Ordered fields ===
If (F, +, ×) is a field and ≤ is a total order on F, then (F, +, ×, ≤) is called an ordered field if and only if:
a ≤ b implies a + c ≤ b + c;
0 ≤ a and 0 ≤ b implies 0 ≤ a × b.
Both
(
Q
,
+
,
×
,
≤
)
{\displaystyle (\mathbb {Q} ,+,\times ,\leq )}
and
(
R
,
+
,
×
,
≤
)
{\displaystyle (\mathbb {R} ,+,\times ,\leq )}
are ordered fields, but ≤ cannot be defined in order to make
(
C
,
+
,
×
,
≤
)
{\displaystyle (\mathbb {C} ,+,\times ,\leq )}
an ordered field, because −1 is the square of i and would therefore be positive.
Besides being an ordered field, R also has the Least-upper-bound property. In fact, R can be defined as the only ordered field with that quality.
== Chained notation ==
The notation a < b < c stands for "a < b and b < c", from which, by the transitivity property above, it also follows that a < c. By the above laws, one can add or subtract the same number to all three terms, or multiply or divide all three terms by same nonzero number and reverse all inequalities if that number is negative. Hence, for example, a < b + e < c is equivalent to a − e < b < c − e.
This notation can be generalized to any number of terms: for instance, a1 ≤ a2 ≤ ... ≤ an means that ai ≤ ai+1 for i = 1, 2, ..., n − 1. By transitivity, this condition is equivalent to ai ≤ aj for any 1 ≤ i ≤ j ≤ n.
When solving inequalities using chained notation, it is possible and sometimes necessary to evaluate the terms independently. For instance, to solve the inequality 4x < 2x + 1 ≤ 3x + 2, it is not possible to isolate x in any one part of the inequality through addition or subtraction. Instead, the inequalities must be solved independently, yielding x < 1/2 and x ≥ −1 respectively, which can be combined into the final solution −1 ≤ x < 1/2.
Occasionally, chained notation is used with inequalities in different directions, in which case the meaning is the logical conjunction of the inequalities between adjacent terms. For example, the defining condition of a zigzag poset is written as a1 < a2 > a3 < a4 > a5 < a6 > ... . Mixed chained notation is used more often with compatible relations, like <, =, ≤. For instance, a < b = c ≤ d means that a < b, b = c, and c ≤ d. This notation exists in a few programming languages such as Python. In contrast, in programming languages that provide an ordering on the type of comparison results, such as C, even homogeneous chains may have a completely different meaning.
== Sharp inequalities ==
An inequality is said to be sharp if it cannot be relaxed and still be valid in general. Formally, a universally quantified inequality φ is called sharp if, for every valid universally quantified inequality ψ, if ψ ⇒ φ holds, then ψ ⇔ φ also holds. For instance, the inequality ∀a ∈ R. a2 ≥ 0 is sharp, whereas the inequality ∀a ∈ R. a2 ≥ −1 is not sharp.
== Inequalities between means ==
There are many inequalities between means. For example, for any positive numbers a1, a2, ..., an we have
H
≤
G
≤
A
≤
Q
,
{\displaystyle H\leq G\leq A\leq Q,}
where they represent the following means of the sequence:
Harmonic mean :
H
=
n
1
a
1
+
1
a
2
+
⋯
+
1
a
n
{\displaystyle H={\frac {n}{{\frac {1}{a_{1}}}+{\frac {1}{a_{2}}}+\cdots +{\frac {1}{a_{n}}}}}}
Geometric mean :
G
=
a
1
⋅
a
2
⋯
a
n
n
{\displaystyle G={\sqrt[{n}]{a_{1}\cdot a_{2}\cdots a_{n}}}}
Arithmetic mean :
A
=
a
1
+
a
2
+
⋯
+
a
n
n
{\displaystyle A={\frac {a_{1}+a_{2}+\cdots +a_{n}}{n}}}
Quadratic mean :
Q
=
a
1
2
+
a
2
2
+
⋯
+
a
n
2
n
{\displaystyle Q={\sqrt {\frac {a_{1}^{2}+a_{2}^{2}+\cdots +a_{n}^{2}}{n}}}}
== Cauchy–Schwarz inequality ==
The Cauchy–Schwarz inequality states that for all vectors u and v of an inner product space it is true that
|
⟨
u
,
v
⟩
|
2
≤
⟨
u
,
u
⟩
⋅
⟨
v
,
v
⟩
,
{\displaystyle |\langle \mathbf {u} ,\mathbf {v} \rangle |^{2}\leq \langle \mathbf {u} ,\mathbf {u} \rangle \cdot \langle \mathbf {v} ,\mathbf {v} \rangle ,}
where
⟨
⋅
,
⋅
⟩
{\displaystyle \langle \cdot ,\cdot \rangle }
is the inner product. Examples of inner products include the real and complex dot product; In Euclidean space Rn with the standard inner product, the Cauchy–Schwarz inequality is
(
∑
i
=
1
n
u
i
v
i
)
2
≤
(
∑
i
=
1
n
u
i
2
)
(
∑
i
=
1
n
v
i
2
)
.
{\displaystyle {\biggl (}\sum _{i=1}^{n}u_{i}v_{i}{\biggr )}^{2}\leq {\biggl (}\sum _{i=1}^{n}u_{i}^{2}{\biggr )}{\biggl (}\sum _{i=1}^{n}v_{i}^{2}{\biggr )}.}
== Power inequalities ==
A power inequality is an inequality containing terms of the form ab, where a and b are real positive numbers or variable expressions. They often appear in mathematical olympiads exercises.
Examples:
For any real x,
e
x
≥
1
+
x
.
{\displaystyle e^{x}\geq 1+x.}
If x > 0 and p > 0, then
x
p
−
1
p
≥
ln
(
x
)
≥
1
−
1
x
p
p
.
{\displaystyle {\frac {x^{p}-1}{p}}\geq \ln(x)\geq {\frac {1-{\frac {1}{x^{p}}}}{p}}.}
In the limit of p → 0, the upper and lower bounds converge to ln(x).
If x > 0, then
x
x
≥
(
1
e
)
1
e
.
{\displaystyle x^{x}\geq \left({\frac {1}{e}}\right)^{\frac {1}{e}}.}
If x > 0, then
x
x
x
≥
x
.
{\displaystyle x^{x^{x}}\geq x.}
If x, y, z > 0, then
(
x
+
y
)
z
+
(
x
+
z
)
y
+
(
y
+
z
)
x
>
2.
{\displaystyle \left(x+y\right)^{z}+\left(x+z\right)^{y}+\left(y+z\right)^{x}>2.}
For any real distinct numbers a and b,
e
b
−
e
a
b
−
a
>
e
(
a
+
b
)
/
2
.
{\displaystyle {\frac {e^{b}-e^{a}}{b-a}}>e^{(a+b)/2}.}
If x, y > 0 and 0 < p < 1, then
x
p
+
y
p
>
(
x
+
y
)
p
.
{\displaystyle x^{p}+y^{p}>\left(x+y\right)^{p}.}
If x, y, z > 0, then
x
x
y
y
z
z
≥
(
x
y
z
)
(
x
+
y
+
z
)
/
3
.
{\displaystyle x^{x}y^{y}z^{z}\geq \left(xyz\right)^{(x+y+z)/3}.}
If a, b > 0, then
a
a
+
b
b
≥
a
b
+
b
a
.
{\displaystyle a^{a}+b^{b}\geq a^{b}+b^{a}.}
If a, b > 0, then
a
e
a
+
b
e
b
≥
a
e
b
+
b
e
a
.
{\displaystyle a^{ea}+b^{eb}\geq a^{eb}+b^{ea}.}
If a, b, c > 0, then
a
2
a
+
b
2
b
+
c
2
c
≥
a
2
b
+
b
2
c
+
c
2
a
.
{\displaystyle a^{2a}+b^{2b}+c^{2c}\geq a^{2b}+b^{2c}+c^{2a}.}
If a, b > 0, then
a
b
+
b
a
>
1.
{\displaystyle a^{b}+b^{a}>1.}
== Well-known inequalities ==
Mathematicians often use inequalities to bound quantities for which exact formulas cannot be computed easily. Some inequalities are used so often that they have names:
== Complex numbers and inequalities ==
The set of complex numbers
C
{\displaystyle \mathbb {C} }
with its operations of addition and multiplication is a field, but it is impossible to define any relation ≤ so that
(
C
,
+
,
×
,
≤
)
{\displaystyle (\mathbb {C} ,+,\times ,\leq )}
becomes an ordered field. To make
(
C
,
+
,
×
,
≤
)
{\displaystyle (\mathbb {C} ,+,\times ,\leq )}
an ordered field, it would have to satisfy the following two properties:
if a ≤ b, then a + c ≤ b + c;
if 0 ≤ a and 0 ≤ b, then 0 ≤ ab.
Because ≤ is a total order, for any number a, either 0 ≤ a or a ≤ 0 (in which case the first property above implies that 0 ≤ −a). In either case 0 ≤ a2; this means that i2 > 0 and 12 > 0; so −1 > 0 and 1 > 0, which means (−1 + 1) > 0; contradiction.
However, an operation ≤ can be defined so as to satisfy only the first property (namely, "if a ≤ b, then a + c ≤ b + c"). Sometimes the lexicographical order definition is used:
a ≤ b, if
Re(a) < Re(b), or
Re(a) = Re(b) and Im(a) ≤ Im(b)
It can easily be proven that for this definition a ≤ b implies a + c ≤ b + c.
== Systems of inequalities ==
Systems of linear inequalities can be simplified by Fourier–Motzkin elimination.
The cylindrical algebraic decomposition is an algorithm that allows testing whether a system of polynomial equations and inequalities has solutions, and, if solutions exist, describing them. The complexity of this algorithm is doubly exponential in the number of variables. It is an active research domain to design algorithms that are more efficient in specific cases.
== See also ==
Binary relation
Bracket (mathematics), for the use of similar ‹ and › signs as brackets
Inclusion (set theory)
Inequation
Interval (mathematics)
List of inequalities
List of triangle inequalities
Partially ordered set
Relational operators, used in programming languages to denote inequality
== References ==
== Sources ==
Hardy, G., Littlewood J. E., Pólya, G. (1999). Inequalities. Cambridge Mathematical Library, Cambridge University Press. ISBN 0-521-05206-8.{{cite book}}: CS1 maint: multiple names: authors list (link)
Beckenbach, E. F., Bellman, R. (1975). An Introduction to Inequalities. Random House Inc. ISBN 0-394-01559-2.{{cite book}}: CS1 maint: multiple names: authors list (link)
Drachman, Byron C., Cloud, Michael J. (1998). Inequalities: With Applications to Engineering. Springer-Verlag. ISBN 0-387-98404-6.{{cite book}}: CS1 maint: multiple names: authors list (link)
Grinshpan, A. Z. (2005), "General inequalities, consequences, and applications", Advances in Applied Mathematics, 34 (1): 71–100, doi:10.1016/j.aam.2004.05.001
Murray S. Klamkin. "'Quickie' inequalities" (PDF). Math Strategies. Archived (PDF) from the original on 2022-10-09.
Arthur Lohwater (1982). "Introduction to Inequalities". Online e-book in PDF format.
Harold Shapiro (2005). "Mathematical Problem Solving". The Old Problem Seminar. Kungliga Tekniska högskolan.
"3rd USAMO". Archived from the original on 2008-02-03.
Pachpatte, B. G. (2005). Mathematical Inequalities. North-Holland Mathematical Library. Vol. 67 (first ed.). Amsterdam, the Netherlands: Elsevier. ISBN 0-444-51795-2. ISSN 0924-6509. MR 2147066. Zbl 1091.26008.
Ehrgott, Matthias (2005). Multicriteria Optimization. Springer-Berlin. ISBN 3-540-21398-8.
Steele, J. Michael (2004). The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities. Cambridge University Press. ISBN 978-0-521-54677-5.
== External links ==
"Inequality", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Graph of Inequalities by Ed Pegg, Jr.
AoPS Wiki entry about Inequalities | Wikipedia/Systems_of_polynomial_inequalities |
Wenjun Wu's method is an algorithm for solving multivariate polynomial equations introduced in the late 1970s by the Chinese mathematician Wen-Tsun Wu. This method is based on the mathematical concept of characteristic set introduced in the late 1940s by J.F. Ritt. It is fully independent of the Gröbner basis method, introduced by Bruno Buchberger (1965), even if Gröbner bases may be used to compute characteristic sets.
Wu's method is powerful for mechanical theorem proving in elementary geometry, and provides a complete decision process for certain classes of problem. It has been used in research in his laboratory (KLMM, Key Laboratory of Mathematics Mechanization in Chinese Academy of Science) and around the world. The main trends of research on Wu's method concern systems of polynomial equations of positive dimension and differential algebra where Ritt's results have been made effective. Wu's method has been applied in various scientific fields, like biology, computer vision, robot kinematics and especially automatic proofs in geometry.
== Informal description ==
Wu's method uses polynomial division to solve problems of the form:
∀
x
,
y
,
z
,
…
I
(
x
,
y
,
z
,
…
)
⟹
f
(
x
,
y
,
z
,
…
)
{\displaystyle \forall x,y,z,\dots I(x,y,z,\dots )\implies f(x,y,z,\dots )\,}
where f is a polynomial equation and I is a conjunction of polynomial equations. The algorithm is complete for such problems over the complex domain.
The core idea of the algorithm is that you can divide one polynomial by another to give a remainder. Repeated division results in either the remainder vanishing (in which case the I implies f statement is true), or an irreducible remainder is left behind (in which case the statement is false).
More specifically, for an ideal I in the ring k[x1, ..., xn] over a field k, a (Ritt) characteristic set C of I is composed of a set of polynomials in I, which is in triangular shape: polynomials in C have distinct main variables (see the formal definition below). Given a characteristic set C of I, one can decide if a polynomial f is zero modulo I. That is, the membership test is checkable for I, provided a characteristic set of I.
== Ritt characteristic set ==
A Ritt characteristic set is a finite set of polynomials in triangular form of an ideal. This triangular set satisfies
certain minimal condition with respect to the Ritt ordering, and it preserves many interesting geometrical properties
of the ideal. However it may not be its system of generators.
=== Notation ===
Let R be the multivariate polynomial ring k[x1, ..., xn] over a field k.
The variables are ordered linearly according to their subscript: x1 < ... < xn.
For a non-constant polynomial p in R, the greatest variable effectively presenting in p, called main variable or class, plays a particular role:
p can be naturally regarded as a univariate polynomial in its main variable xk with coefficients in k[x1, ..., xk−1].
The degree of p as a univariate polynomial in its main variable is also called its main degree.
=== Triangular set ===
A set T of non-constant polynomials is called a triangular set if all polynomials in T have distinct main variables. This generalizes triangular systems of linear equations in a natural way.
=== Ritt ordering ===
For two non-constant polynomials p and q, we say p is smaller than q with respect to Ritt ordering and written as p <r q, if one of the following assertions holds:
(1) the main variable of p is smaller than the main variable of q, that is, mvar(p) < mvar(q),
(2) p and q have the same main variable, and the main degree of p is less than the main degree of q, that is, mvar(p) = mvar(q) and mdeg(p) < mdeg(q).
In this way, (k[x1, ..., xn],<r) forms a well partial order. However, the Ritt ordering is not a total order:
there exist polynomials p and q such that neither p <r q nor p >r q. In this case, we say that p and q are not comparable.
The Ritt ordering is comparing the rank of p and q. The rank, denoted by rank(p), of a non-constant polynomial p is defined to be a power of
its main variable: mvar(p)mdeg(p) and ranks are compared by comparing first the variables and then, in case of equality of the variables, the degrees.
=== Ritt ordering on triangular sets ===
A crucial generalization on Ritt ordering is to compare triangular sets.
Let T = { t1, ..., tu} and S = { s1, ..., sv} be two triangular sets
such that polynomials in T and S are sorted increasingly according to their main variables.
We say T is smaller than S w.r.t. Ritt ordering if one of the following assertions holds
there exists k ≤ min(u, v) such that rank(ti) = rank(si) for 1 ≤ i < k and tk <r sk,
u > v and rank(ti) = rank(si) for 1 ≤ i ≤ v.
Also, there exists incomparable triangular sets w.r.t Ritt ordering.
=== Ritt characteristic set ===
Let I be a non-zero ideal of k[x1, ..., xn]. A subset T of I is a Ritt characteristic set of I if one of the following conditions holds:
T consists of a single nonzero constant of k,
T is a triangular set and T is minimal w.r.t Ritt ordering in the set of all triangular sets contained in I.
A polynomial ideal may possess (infinitely) many characteristic sets, since Ritt ordering is a partial order.
== Wu characteristic set ==
The Ritt–Wu process, first devised by Ritt, subsequently modified by Wu, computes not a Ritt characteristic but an extended one, called Wu characteristic set or ascending chain.
A non-empty subset T of the ideal ⟨F⟩ generated by F is a Wu characteristic set of F if one of the following condition holds
T = {a} with a being a nonzero constant,
T is a triangular set and there exists a subset G of ⟨F⟩ such that ⟨F⟩ = ⟨G⟩ and every polynomial in G is pseudo-reduced to zero with respect to T.
Wu characteristic set is defined to the set F of polynomials, rather to the ideal ⟨F⟩ generated by F. Also it can be shown that a Ritt characteristic set T of ⟨F⟩ is a Wu characteristic set of F. Wu characteristic sets can be computed by Wu's algorithm CHRST-REM, which only requires pseudo-remainder computations and no factorizations are needed.
Wu's characteristic set method has exponential complexity; improvements in computing efficiency by weak chains, regular chains, saturated chain were introduced
== Decomposing algebraic varieties ==
An application is an algorithm for solving systems of algebraic equations by means of characteristic sets. More precisely, given a finite subset F of polynomials, there is an algorithm to compute characteristic sets T1, ..., Te such that:
V
(
F
)
=
W
(
T
1
)
∪
⋯
∪
W
(
T
e
)
,
{\displaystyle V(F)=W(T_{1})\cup \cdots \cup W(T_{e}),}
where W(Ti) is the difference of V(Ti) and V(hi), here hi is the product of initials of the polynomials in Ti.
== See also ==
Regular chain
Mathematics-Mechanization Platform
== References ==
P. Aubry, M. Moreno Maza (1999) Triangular Sets for Solving Polynomial Systems: a Comparative Implementation of Four Methods. J. Symb. Comput. 28(1–2): 125–154
David A. Cox, John B. Little, Donal O'Shea. Ideals, Varieties, and Algorithms. 2007.
Hua-Shan, Liu (24 August 2005). "WuRittSolva: Implementation of Wu-Ritt Characteristic Set Method". Wolfram Library Archive. Wolfram. Retrieved 17 November 2012.
Heck, André (2003). Introduction to Maple (3. ed.). New York: Springer. pp. 105, 508. ISBN 9780387002309.
Ritt, J. (1966). Differential Algebra. New York, Dover Publications.
Dongming Wang (1998). Elimination Methods. Springer-Verlag, Wien, Springer-Verlag
Dongming Wang (2004). Elimination Practice, Imperial College Press, London ISBN 1-86094-438-8
Wu, W. T. (1984). Basic principles of mechanical theorem proving in elementary geometries. J. Syst. Sci. Math. Sci., 4, 207–35
Wu, W. T. (1987). A zero structure theorem for polynomial equations solving. MM Research Preprints, 1, 2–12
Xiaoshan, Gao; Chunming, Yuan; Guilin, Zhang (2009). "Ritt-Wu's characteristic set method for ordinary difference polynomial systems with arbitrary ordering". Acta Mathematica Scientia. 29 (4): 1063–1080. CiteSeerX 10.1.1.556.9549. doi:10.1016/S0252-9602(09)60086-2.
== External links ==
wsolve Maple package
The Characteristic Set Method | Wikipedia/Wu's_method_of_characteristic_set |
In computer algebra, a regular semi-algebraic system is a particular kind of triangular system of multivariate polynomials over a real closed field.
== Introduction ==
Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. The notion of a regular semi-algebraic system is an adaptation of the concept of a regular chain focusing on solutions of the real analogue: semi-algebraic systems.
Any semi-algebraic system
S
{\displaystyle S}
can be decomposed into finitely many regular semi-algebraic systems
S
1
,
…
,
S
e
{\displaystyle S_{1},\ldots ,S_{e}}
such that a point (with real coordinates) is a solution of
S
{\displaystyle S}
if and only if it is a solution of one of the systems
S
1
,
…
,
S
e
{\displaystyle S_{1},\ldots ,S_{e}}
.
== Formal definition ==
Let
T
{\displaystyle T}
be a regular chain of
k
[
x
1
,
…
,
x
n
]
{\displaystyle \mathbf {k} [x_{1},\ldots ,x_{n}]}
for some ordering of the variables
x
=
x
1
,
…
,
x
n
{\displaystyle \mathbf {x} =x_{1},\ldots ,x_{n}}
and a real closed field
k
{\displaystyle \mathbf {k} }
. Let
u
=
u
1
,
…
,
u
d
{\displaystyle \mathbf {u} =u_{1},\ldots ,u_{d}}
and
y
=
y
1
,
…
,
y
n
−
d
{\displaystyle \mathbf {y} =y_{1},\ldots ,y_{n-d}}
designate respectively the variables of
x
{\displaystyle \mathbf {x} }
that are free and algebraic with respect to
T
{\displaystyle T}
. Let
P
⊂
k
[
x
]
{\displaystyle P\subset \mathbf {k} [\mathbf {x} ]}
be finite such that each polynomial in
P
{\displaystyle P}
is regular with respect to the saturated ideal of
T
{\displaystyle T}
. Define
P
>
:=
{
p
>
0
∣
p
∈
P
}
{\displaystyle P_{>}:=\{p>0\mid p\in P\}}
. Let
Q
{\displaystyle {\mathcal {Q}}}
be a quantifier-free formula of
k
[
x
]
{\displaystyle \mathbf {k} [\mathbf {x} ]}
involving only the variables of
u
{\displaystyle \mathbf {u} }
. We say that
R
:=
[
Q
,
T
,
P
>
]
{\displaystyle R:=[{\mathcal {Q}},T,P_{>}]}
is a regular semi-algebraic system if the following three conditions hold.
Q
{\displaystyle {\mathcal {Q}}}
defines a non-empty open semi-algebraic set
S
{\displaystyle S}
of
k
d
{\displaystyle \mathbf {k} ^{d}}
,
the regular system
[
T
,
P
]
{\displaystyle [T,P]}
specializes well at every point
u
{\displaystyle u}
of
S
{\displaystyle S}
,
at each point
u
{\displaystyle u}
of
S
{\displaystyle S}
, the specialized system
[
T
(
u
)
,
P
(
u
)
>
]
{\displaystyle [T(u),P(u)_{>}]}
has at least one real zero.
The zero set of
R
{\displaystyle R}
, denoted by
Z
k
(
R
)
{\displaystyle Z_{\mathbf {k} }(R)}
, is defined as the set of points
(
u
,
y
)
∈
k
d
×
k
n
−
d
{\displaystyle (u,y)\in \mathbf {k} ^{d}\times \mathbf {k} ^{n-d}}
such that
Q
(
u
)
{\displaystyle {\mathcal {Q}}(u)}
is true and
t
(
u
,
y
)
=
0
,
p
(
u
,
y
)
>
0
{\displaystyle t(u,y)=0,p(u,y)>0}
, for all
t
∈
T
{\displaystyle t\in T}
and all
p
∈
P
{\displaystyle p\in P}
. Observe that
Z
k
(
R
)
{\displaystyle Z_{\mathbf {k} }(R)}
has dimension
d
{\displaystyle d}
in the affine space
k
n
{\displaystyle \mathbf {k} ^{n}}
.
== See also ==
Real algebraic geometry
== References == | Wikipedia/Regular_semi-algebraic_system |
In mathematics and particularly in algebra, a system of equations (either linear or nonlinear) is called consistent if there is at least one set of values for the unknowns that satisfies each equation in the system—that is, when substituted into each of the equations, they make each equation hold true as an identity. In contrast, a linear or non linear equation system is called inconsistent if there is no set of values for the unknowns that satisfies all of the equations.
If a system of equations is inconsistent, then the equations cannot be true together leading to contradictory information, such as the false statements 2 = 1, or
x
3
+
y
3
=
5
{\displaystyle x^{3}+y^{3}=5}
and
x
3
+
y
3
=
6
{\displaystyle x^{3}+y^{3}=6}
(which implies 5 = 6).
Both types of equation system, inconsistent and consistent, can be any of overdetermined (having more equations than unknowns), underdetermined (having fewer equations than unknowns), or exactly determined.
== Simple examples ==
=== Underdetermined and consistent ===
The system
x
+
y
+
z
=
3
,
x
+
y
+
2
z
=
4
{\displaystyle {\begin{aligned}x+y+z&=3,\\x+y+2z&=4\end{aligned}}}
has an infinite number of solutions, all of them having z = 1 (as can be seen by subtracting the first equation from the second), and all of them therefore having x + y = 2 for any values of x and y.
The nonlinear system
x
2
+
y
2
+
z
2
=
10
,
x
2
+
y
2
=
5
{\displaystyle {\begin{aligned}x^{2}+y^{2}+z^{2}&=10,\\x^{2}+y^{2}&=5\end{aligned}}}
has an infinitude of solutions, all involving
z
=
±
5
.
{\displaystyle z=\pm {\sqrt {5}}.}
Since each of these systems has more than one solution, it is an indeterminate system .
=== Underdetermined and inconsistent ===
The system
x
+
y
+
z
=
3
,
x
+
y
+
z
=
4
{\displaystyle {\begin{aligned}x+y+z&=3,\\x+y+z&=4\end{aligned}}}
has no solutions, as can be seen by subtracting the first equation from the second to obtain the impossible 0 = 1.
The non-linear system
x
2
+
y
2
+
z
2
=
17
,
x
2
+
y
2
+
z
2
=
14
{\displaystyle {\begin{aligned}x^{2}+y^{2}+z^{2}&=17,\\x^{2}+y^{2}+z^{2}&=14\end{aligned}}}
has no solutions, because if one equation is subtracted from the other we obtain the impossible 0 = 3.
=== Exactly determined and consistent ===
The system
x
+
y
=
3
,
x
+
2
y
=
5
{\displaystyle {\begin{aligned}x+y&=3,\\x+2y&=5\end{aligned}}}
has exactly one solution: x = 1, y = 2
The nonlinear system
x
+
y
=
1
,
x
2
+
y
2
=
1
{\displaystyle {\begin{aligned}x+y&=1,\\x^{2}+y^{2}&=1\end{aligned}}}
has the two solutions (x, y) = (1, 0) and (x, y) = (0, 1), while
x
3
+
y
3
+
z
3
=
10
,
x
3
+
2
y
3
+
z
3
=
12
,
3
x
3
+
5
y
3
+
3
z
3
=
34
{\displaystyle {\begin{aligned}x^{3}+y^{3}+z^{3}&=10,\\x^{3}+2y^{3}+z^{3}&=12,\\3x^{3}+5y^{3}+3z^{3}&=34\end{aligned}}}
has an infinite number of solutions because the third equation is the first equation plus twice the second one and hence contains no independent information; thus any value of z can be chosen and values of x and y can be found to satisfy the first two (and hence the third) equations.
=== Exactly determined and inconsistent ===
The system
x
+
y
=
3
,
4
x
+
4
y
=
10
{\displaystyle {\begin{aligned}x+y&=3,\\4x+4y&=10\end{aligned}}}
has no solutions; the inconsistency can be seen by multiplying the first equation by 4 and subtracting the second equation to obtain the impossible 0 = 2.
Likewise,
x
3
+
y
3
+
z
3
=
10
,
x
3
+
2
y
3
+
z
3
=
12
,
3
x
3
+
5
y
3
+
3
z
3
=
32
{\displaystyle {\begin{aligned}x^{3}+y^{3}+z^{3}&=10,\\x^{3}+2y^{3}+z^{3}&=12,\\3x^{3}+5y^{3}+3z^{3}&=32\end{aligned}}}
is an inconsistent system because the first equation plus twice the second minus the third contains the contradiction 0 = 2.
=== Overdetermined and consistent ===
The system
x
+
y
=
3
,
x
+
2
y
=
7
,
4
x
+
6
y
=
20
{\displaystyle {\begin{aligned}x+y&=3,\\x+2y&=7,\\4x+6y&=20\end{aligned}}}
has a solution, x = –1, y = 4, because the first two equations do not contradict each other and the third equation is redundant (since it contains the same information as can be obtained from the first two equations by multiplying each through by 2 and summing them).
The system
x
+
2
y
=
7
,
3
x
+
6
y
=
21
,
7
x
+
14
y
=
49
{\displaystyle {\begin{aligned}x+2y&=7,\\3x+6y&=21,\\7x+14y&=49\end{aligned}}}
has an infinitude of solutions since all three equations give the same information as each other (as can be seen by multiplying through the first equation by either 3 or 7). Any value of y is part of a solution, with the corresponding value of x being 7 – 2y.
The nonlinear system
x
2
−
1
=
0
,
y
2
−
1
=
0
,
(
x
−
1
)
(
y
−
1
)
=
0
{\displaystyle {\begin{aligned}x^{2}-1&=0,\\y^{2}-1&=0,\\(x-1)(y-1)&=0\end{aligned}}}
has the three solutions (x, y) = (1, –1), (–1, 1), (1, 1).
=== Overdetermined and inconsistent ===
The system
x
+
y
=
3
,
x
+
2
y
=
7
,
4
x
+
6
y
=
21
{\displaystyle {\begin{aligned}x+y&=3,\\x+2y&=7,\\4x+6y&=21\end{aligned}}}
is inconsistent because the last equation contradicts the information embedded in the first two, as seen by multiplying each of the first two through by 2 and summing them.
The system
x
2
+
y
2
=
1
,
x
2
+
2
y
2
=
2
,
2
x
2
+
3
y
2
=
4
{\displaystyle {\begin{aligned}x^{2}+y^{2}&=1,\\x^{2}+2y^{2}&=2,\\2x^{2}+3y^{2}&=4\end{aligned}}}
is inconsistent because the sum of the first two equations contradicts the third one.
== Criteria for consistency ==
As can be seen from the above examples, consistency versus inconsistency is a different issue from comparing the numbers of equations and unknowns.
=== Linear systems ===
A linear system is consistent if and only if its coefficient matrix has the same rank as does its augmented matrix (the coefficient matrix with an extra column added, that column being the column vector of constants).
=== Nonlinear systems ===
== References == | Wikipedia/Inconsistent_equations |
In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning, such as preserving distances, angles, or ratios (scale). More specifically, it is a function whose domain and range are sets of points – most often a real coordinate space,
R
2
{\displaystyle \mathbb {R} ^{2}}
or
R
3
{\displaystyle \mathbb {R} ^{3}}
– such that the function is bijective so that its inverse exists. The study of geometry may be approached by the study of these transformations, such as in transformation geometry.
== Classifications ==
Geometric transformations can be classified by the dimension of their operand sets (thus distinguishing between, say, planar transformations and spatial transformations). They can also be classified according to the properties they preserve:
Displacements preserve distances and oriented angles (e.g., translations);
Isometries preserve angles and distances (e.g., Euclidean transformations);
Similarities preserve angles and ratios between distances (e.g., resizing);
Affine transformations preserve parallelism (e.g., scaling, shear);
Projective transformations preserve collinearity;
Each of these classes contains the previous one.
Möbius transformations using complex coordinates on the plane (as well as circle inversion) preserve the set of all lines and circles, but may interchange lines and circles.
Conformal transformations preserve angles, and are, in the first order, similarities.
Equiareal transformations, preserve areas in the planar case or volumes in the three dimensional case. and are, in the first order, affine transformations of determinant 1.
Homeomorphisms (bicontinuous transformations) preserve the neighborhoods of points.
Diffeomorphisms (bidifferentiable transformations) are the transformations that are affine in the first order; they contain the preceding ones as special cases, and can be further refined.
Transformations of the same type form groups that may be sub-groups of other transformation groups.
== Opposite group actions ==
Many geometric transformations are expressed with linear algebra. The bijective linear transformations are elements of a general linear group. The linear transformation A is non-singular. For a row vector v, the matrix product vA gives another row vector w = vA.
The transpose of a row vector v is a column vector vT, and the transpose of the above equality is
w
T
=
(
v
A
)
T
=
A
T
v
T
.
{\displaystyle w^{T}=(vA)^{T}=A^{T}v^{T}.}
Here AT provides a left action on column vectors.
In transformation geometry there are compositions AB. Starting with a row vector v, the right action of the composed transformation is w = vAB. After transposition,
w
T
=
(
v
A
B
)
T
=
(
A
B
)
T
v
T
=
B
T
A
T
v
T
.
{\displaystyle w^{T}=(vAB)^{T}=(AB)^{T}v^{T}=B^{T}A^{T}v^{T}.}
Thus for AB the associated left group action is
B
T
A
T
.
{\displaystyle B^{T}A^{T}.}
In the study of opposite groups, the distinction is made between opposite group actions because commutative groups are the only groups for which these opposites are equal.
== Active and passive transformations ==
== See also ==
Coordinate transformation
Erlangen program
Symmetry (geometry)
Motion
Reflection
Rigid transformation
Rotation
Topology
Transformation matrix
== References ==
== Further reading ==
Adler, Irving (2012) [1966], A New Look at Geometry, Dover, ISBN 978-0-486-49851-5
Dienes, Z. P.; Golding, E. W. (1967) . Geometry Through Transformations (3 vols.): Geometry of Distortion, Geometry of Congruence, and Groups and Coordinates. New York: Herder and Herder.
David Gans – Transformations and geometries.
Hilbert, David; Cohn-Vossen, Stephan (1952). Geometry and the Imagination (2nd ed.). Chelsea. ISBN 0-8284-1087-9. {{cite book}}: ISBN / Date incompatibility (help)
John McCleary (2013) Geometry from a Differentiable Viewpoint, Cambridge University Press ISBN 978-0-521-11607-7
Modenov, P. S.; Parkhomenko, A. S. (1965) . Geometric Transformations (2 vols.): Euclidean and Affine Transformations, and Projective Transformations. New York: Academic Press.
A. N. Pressley – Elementary Differential Geometry.
Yaglom, I. M. (1962, 1968, 1973, 2009) . Geometric Transformations (4 vols.). Random House (I, II & III), MAA (I, II, III & IV). | Wikipedia/Transformation_(geometry) |
A telecommunications network is a group of nodes interconnected by telecommunications links that are used to exchange messages between the nodes. The links may use a variety of technologies based on the methodologies of circuit switching, message switching, or packet switching, to pass messages and signals.
Multiple nodes may cooperate to pass the message from an originating node to the destination node, via multiple network hops. For this routing function, each node in the network is assigned a network address for identification and locating it on the network. The collection of addresses in the network is called the address space of the network.
Examples of telecommunications networks include computer networks, the Internet, the public switched telephone network (PSTN), the global Telex network, the aeronautical ACARS network, and the wireless radio networks of cell phone telecommunication providers.
== Network structure ==
this is the structure of network general, every telecommunications network conceptually consists of three parts, or planes (so-called because they can be thought of as being and often are, separate overlay networks):
The data plane (also user plane, bearer plane, or forwarding plane) carries the network's users' traffic, the actual payload.
The control plane carries control information (also known as signaling).
The management plane carries the operations, administration and management traffic required for network management. The management plane is sometimes considered a part of the control plane.
== Data networks ==
Data networks are used extensively throughout the world for communication between individuals and organizations. Data networks can be connected to allow users seamless access to resources that are hosted outside of the particular provider they are connected to. The Internet is the best example of the internetworking of many data networks from different organizations.
Terminals attached to IP networks like the Internet are addressed using IP addresses. Protocols of the Internet protocol suite (TCP/IP) provide the control and routing of messages across the and IP data network. There are many different network structures that IP can be used across to efficiently route messages, for example:
Wide area networks (WAN)
Metropolitan area networks (MAN)
Local area networks (LAN)
There are three features that differentiate MANs from LANs or WANs:
The area of the network size is between LANs and WANs. The MAN will have a physical area between 5 and 50 km in diameter.
MANs do not generally belong to a single organization. The equipment that interconnects the network, the links, and the MAN itself are often owned by an association or a network provider that provides or leases the service to others.
A MAN is a means for sharing resources at high speeds within the network. It often provides connections to WAN networks for access to resources outside the scope of the MAN.
Data center networks also rely highly on TCP/IP for communication across machines. They connect thousands of servers, are designed to be highly robust, provide low latency and high bandwidth. Data center network topology plays a significant role in determining the level of failure resiliency, ease of incremental expansion, communication bandwidth and latency.
== Capacity and speed ==
In analogy to the improvements in the speed and capacity of digital computers, provided by advances in semiconductor technology and expressed in the bi-yearly doubling of transistor density, which is described empirically by Moore's law, the capacity and speed of telecommunications networks have followed similar advances, for similar reasons. In telecommunication, this is expressed in Edholm's law, proposed by and named after Phil Edholm in 2004. This empirical law holds that the bandwidth of telecommunication networks doubles every 18 months, which has proven to be true since the 1970s. The trend is evident in the Internet, cellular (mobile), wireless and wired local area networks (LANs), and personal area networks. This development is the consequence of rapid advances in the development of metal-oxide-semiconductor technology.
== See also ==
Transcoder free operation
== References == | Wikipedia/Telecommunications_network |
A pantograph (or "pan" or "panto") is an apparatus mounted on the roof of an electric train, tram or trolley buses to collect power through contact with an overhead line. The term stems from the resemblance of some styles to the mechanical pantographs used for copying handwriting and drawings.
The pantograph is a common type of current collector; typically, a single or double wire is used, with the return current running through the rails. Other types of current collectors include the bow collector and the trolley pole.
== Invention ==
The pantograph, with a low-friction, replaceable graphite contact strip or "shoe" to minimise lateral stress on the contact wire, first appeared in the late 19th century. Early versions include the bow collector, invented in 1889 by Walter Reichel, chief engineer at Siemens & Halske in Germany, and a flat slide-pantograph first used in 1895 by the Baltimore and Ohio Railroad.
The familiar diamond-shaped roller pantograph was devised and patented by John Q. Brown of the Key System shops for their commuter trains which ran between San Francisco and the East Bay section of the San Francisco Bay Area in California. They appear in photographs of the first day of service, 26 October 1903. For many decades thereafter, the same diamond shape was used by electric-rail systems around the world and remains in use by some today.
The pantograph was an improvement on the simple trolley pole, which prevailed up to that time, primarily because the pantograph allows an electric-rail vehicle to travel at much higher speeds without losing contact with the overhead lines, e.g. due to dewirement of the trolley pole.
Notwithstanding this, trolley pole current collection was used successfully at up to 140 km/h (90 mph) on the Electroliner vehicles of the Chicago North Shore and Milwaukee Railroad, also known as the North Shore Line.
== Modern use ==
The most common type of pantograph today is the so-called half-pantograph (sometimes Z-shaped), which evolved to provide a more compact and responsive single-arm design at high speeds as trains got faster. Louis Faiveley invented this type of pantograph in 1955. The half-pantograph can be seen in use on everything from very fast trains (such as the TGV) to low-speed urban tram systems. The design operates with equal efficiency in either direction of motion, as demonstrated by the Swiss and Austrian railways whose newest high-performance locomotives, the Re 460 and Taurus, operate with them set in the opposite direction. In Europe the geometry and shape of the pantographs are specified by CENELEC, the European Committee for Electrotechnical Standardization.
While a pantograph is mainly used to power a railway traction unit, there are certain cases where it has a function other than traction:
Mechanical measurements and tests of new catenary (with or without voltage), on a catenary and contact line inspection car;
General power supply of a measuring car;
Power supply of an air-conditioned train (for example: pantograph mounted on a RENFE van for the 3,000 V DC power supply of the rest of the train via the UIC train line, or heating line), in the absence of a locomotive;
Power supply of a restaurant car when parked on a siding in the absence of a locomotive, under a catenary electrified with 15 kV AC 16+2⁄3 Hz; this system is used on restaurant cars of the Swiss and German railways;
Grounding of the catenary during work carried out from certain work vehicles.
== Technical details ==
The electric transmission system for modern electric rail systems consists of an upper, weight-carrying wire (known as a catenary) from which is suspended a contact wire. The pantograph is spring-loaded and pushes a contact shoe up against the underside of the contact wire to draw the current needed to run the train. The steel rails of the tracks act as the electrical return. As the train moves, the contact shoe slides along the wire and can set up standing waves in the wires which break the contact and degrade current collection. This means that on some systems adjacent pantographs are not permitted.
Pantographs are the successor technology to trolley poles, which were widely used on early streetcar systems. Trolley poles are still used by trolleybuses, whose freedom of movement and need for a two-wire circuit makes pantographs impractical, and some streetcar networks, such as the Toronto streetcar system, which have frequent turns sharp enough to require additional freedom of movement in their current collection to ensure unbroken contact. However, many of these networks, including Toronto's, are undergoing upgrades to accommodate pantograph operation.
Pantographs with overhead wires are now the dominant form of current collection for modern electric trains in city street, main and high speed lines because, although more fragile than a third rail system, they are safer for public (protection by distance), they may also allow higher voltages (especially AC ones) and higher speed.
Pantographs are typically operated by compressed air from the vehicle's braking system, either to raise the unit and hold it against the conductor or, when springs are used to effect the extension, to lower it. As a precaution against loss of pressure in the second case, the arm is held in the down position by a catch. For high-voltage systems, the same air supply is used to "blow out" the electric arc when roof-mounted circuit breakers are used.
== Single and double pantographs ==
Pantographs may have either a single or a double arm. Double-arm pantographs are usually heavier, requiring more power to raise and lower, but may also be more fault-tolerant.
On railways of the former USSR, the most widely used pantographs are those with a double arm ("made of two rhombs"), but, since the late 1990s, there have been some single-arm pantographs on Russian railways. Some streetcars use double-arm pantographs, among them the Russian KTM-5, KTM-8, LVS-86 and many other Russian-made trams, as well as some Euro-PCC trams in Belgium. American streetcars use either trolley poles or single-arm pantographs.
== Metro systems and overhead lines ==
Most rapid transit systems are powered by a third rail, but some use pantographs, particularly ones that involve extensive above-ground running. Most hybrid metro-tram or 'pre-metro' lines whose routes include tracks on city streets or in other publicly accessible areas, such as (formerly) line 51 of the Amsterdam Metro, the MBTA Green Line, RTA Rapid Transit in Cleveland, Frankfurt am Main U-Bahn, and San Francisco's Muni Metro, use overhead wire, as a standard third rail would obstruct street traffic and present too great a risk of electrocution.
Among the various exceptions are several tram systems, such as the ones in Bordeaux, Angers, Reims and Dubai that use a proprietary underground system developed by Alstom, called APS, which only applies power to segments of track that are completely covered by the tram. This system was originally designed to be used in the historic centre of Bordeaux because an overhead wire system would cause a visual intrusion. Similar systems that avoid overhead lines have been developed by Bombardier, AnsaldoBreda, CAF, and others. These may consist of physical ground-level infrastructure, or use energy stored in battery packs to travel over short distances without overhead wiring.
Overhead pantographs are sometimes used as alternatives to third rails because third rails can ice over in certain winter weather conditions. The MBTA Blue Line uses pantograph power for the entire section of its route that runs on the surface, while switching to third rail power before entering the underground portion of its route. The entire metro systems of Sydney, Madrid, Barcelona, Porto, Shanghai, Hong Kong, Seoul, Kobe, Fukuoka, Sendai, Jaipur, Chennai, Mumbai and Delhi use overhead wiring and pantographs (as well as certain lines of the metro systems in Beijing, Chongqing, Noida, Hyderabad, Jakarta, Tokyo, Osaka, Nagoya, Singapore, Sapporo, Budapest, and Mexico City). Pantographs were also used on the Nord-Sud Company rapid transit lines in Paris until the other operating company of the time, Compagnie du chemin de fer métropolitain de Paris, bought out the company and replaced all overhead wiring with the standard third rail system used on other lines.
Numerous railway lines use both third rail and overhead power collection along different portions of their routes, generally for historical reasons. They include the North London line and West London lines of London Overground, the Northern City Line of Great Northern, three of the five lines in the Rotterdam Metro network, Metro-North Railroad's New Haven Line, and the Chicago Transit Authority's Yellow Line. In this last case, the overhead portion was a remnant of the Chicago North Shore and Milwaukee Railroad's high-speed Skokie Valley Route, and was the only line on the entire Chicago subway system to utilize pantograph collection for any length. As such, the line required railcars that featured pantographs as well as third rail shoes, and since the overhead was a very small portion of the system, only a few cars would be so equipped. The changeover occurred at the grade crossing at East Prairie, the former site of the Crawford-East Prairie station. Here, trains bound for Dempster-Skokie would raise their pantographs, while those bound for Howard would lower theirs, doing so at speed in both instances. In 2005, due to the cost and unique maintenance needs for what only represented a very small portion of the system, the overhead system was removed and replaced with the same third rail power that was used throughout the rest of the system, which allowed all of Chicago's railcars to operate on the line. All the pantographs were removed from the Skokie equipped cars.
Until 2010, the Oslo Metro line 1 changed from third rail to overhead line power at Frøen station. Due to the many level crossings, it was deemed difficult to install a third rail on the rest of the older line's single track. After 2010 third rails were used in spite of level crossings. The third rails have gaps, but there are two contact shoes.
== Three-phase supply ==
On some systems using three phase power supply, locomotives and power cars have two pantographs with the third-phase circuit provided by the running rails. In 1901 an experimental high-speed installation, another design from Walter Reichel at Siemens & Halske, used three vertically mounted overhead wires with the collectors mounted on horizontally extending pantographs.
== Inclined pantographs ==
On lines where open wagons are loaded from above, the overhead line may be offset to allow this; the pantographs are then mounted at an angle to the vertical.
== Weaknesses ==
Contact between a pantograph and an overhead line is usually assured through a block of graphite. This material conducts electricity while working as a lubricant. As graphite is brittle, pieces can break off during operation. Poorly-built pantographs can seize the overhead wire and tear it down, and poor-condition wires can damage the pantograph. To prevent this, a pantograph monitoring station can be used. At sustained high speeds, above 300 km/h (190 mph), friction can cause the contact strip to become red hot, which in turn can cause excessive arcing and eventual failure.
In the UK, the pantographs (Brecknell Willis and Stone Faiveley) of vehicles are raised by air pressure and the graphite contact "carbons" create an air gallery in the pantograph head which release the air if a graphite strip is lost, activating the automatic drop device and lowering the pantograph to prevent damage. Newer electric traction units may use more sophisticated methods which detect the disturbances caused by arcing at the point of contact when the graphite strips are damaged. There are not always two pantographs on an electric multiple unit but, in cases where there are, the other one can be used if one is damaged; an example of this situation would be a Class 390 Pendolino. The rear pantograph in relation to the direction of travel is often used as to avoid damaging both pantographs in case of entanglements: if the front pantograph was used, debris from an entanglement could cause damage to the rear pantograph, rendering both pantographs and the vehicle inoperable.
== Automatic dropping device ==
Automatic dropping device (ADD) is a device that automatically lowers the pantograph on electric trains to prevent additional damages in case of obstructions or emergencies. It is also known as pantograph dropping device.
In Europe, the automatic dropping device is mandatory for trains with operational speeds of 160 km/h or 120 km/h and higher if one or several pantographs are used respectively. Otherwise, the train operators are free to install these device. The damage that causes the pantograph to fall can include the strip head, the pantograph head and other parts.
The ADD mostly uses a pneumatic system to detect a damage. For example, a broken contact strip will cause a pressure drop in the air tube inside. On non-pneumatic pantographs (mostly tramway or metro), solutions as a pin on the pantograph head can pull a cable which lower pantograph. Nevertheless, infrastructure are free to impose it and performances (distance, force, time...) are not defined
Time reaction and reaction to quit overhead contact line are defined for main line use.
This function can be different to wearing detection made of a tube or insert place higher than ADD tube.
== Other ==
Several pantograph head profiles exist to meet electrical and mechanical infrastructure gauges of each country. In Europe, two interoperable profiles were defined with 1600 and 1950 mm length.
An infrastructure can accept several head profiles, especially with conductive or insulated horns. The Variopanto is a head with two possible lengths.
Flat pantographs are emerging to fit on narrow roof gauges, allowing flat roofs and reduced aerodynamic impact. They are made of an insulated lower arm contrary to a "classic" pantograph fixed on insulators. Example: ETR1000 pantograph
== See also ==
== References == | Wikipedia/Pantograph_(rail) |
In number theory, an additive function is an arithmetic function f(n) of the positive integer variable n such that whenever a and b are coprime, the function applied to the product ab is the sum of the values of the function applied to a and b:
f
(
a
b
)
=
f
(
a
)
+
f
(
b
)
.
{\displaystyle f(ab)=f(a)+f(b).}
== Completely additive ==
An additive function f(n) is said to be completely additive if
f
(
a
b
)
=
f
(
a
)
+
f
(
b
)
{\displaystyle f(ab)=f(a)+f(b)}
holds for all positive integers a and b, even when they are not coprime. Totally additive is also used in this sense by analogy with totally multiplicative functions. If f is a completely additive function then f(1) = 0.
Every completely additive function is additive, but not vice versa.
== Examples ==
Examples of arithmetic functions which are completely additive are:
The restriction of the logarithmic function to
N
.
{\displaystyle \mathbb {N} .}
The multiplicity of a prime factor p in n, that is the largest exponent m for which pm divides n.
a0(n) – the sum of primes dividing n counting multiplicity, sometimes called sopfr(n), the potency of n or the integer logarithm of n (sequence A001414 in the OEIS). For example:
a0(4) = 2 + 2 = 4
a0(20) = a0(22 · 5) = 2 + 2 + 5 = 9
a0(27) = 3 + 3 + 3 = 9
a0(144) = a0(24 · 32) = a0(24) + a0(32) = 8 + 6 = 14
a0(2000) = a0(24 · 53) = a0(24) + a0(53) = 8 + 15 = 23
a0(2003) = 2003
a0(54,032,858,972,279) = 1240658
a0(54,032,858,972,302) = 1780417
a0(20,802,650,704,327,415) = 1240681
The function Ω(n), defined as the total number of prime factors of n, counting multiple factors multiple times, sometimes called the "Big Omega function" (sequence A001222 in the OEIS). For example;
Ω(1) = 0, since 1 has no prime factors
Ω(4) = 2
Ω(16) = Ω(2·2·2·2) = 4
Ω(20) = Ω(2·2·5) = 3
Ω(27) = Ω(3·3·3) = 3
Ω(144) = Ω(24 · 32) = Ω(24) + Ω(32) = 4 + 2 = 6
Ω(2000) = Ω(24 · 53) = Ω(24) + Ω(53) = 4 + 3 = 7
Ω(2001) = 3
Ω(2002) = 4
Ω(2003) = 1
Ω(54,032,858,972,279) = Ω(11 ⋅ 19932 ⋅ 1236661) = 4
Ω(54,032,858,972,302) = Ω(2 ⋅ 72 ⋅ 149 ⋅ 2081 ⋅ 1778171) = 6
Ω(20,802,650,704,327,415) = Ω(5 ⋅ 7 ⋅ 112 ⋅ 19932 ⋅ 1236661) = 7.
Examples of arithmetic functions which are additive but not completely additive are:
ω(n), defined as the total number of distinct prime factors of n (sequence A001221 in the OEIS). For example:
ω(4) = 1
ω(16) = ω(24) = 1
ω(20) = ω(22 · 5) = 2
ω(27) = ω(33) = 1
ω(144) = ω(24 · 32) = ω(24) + ω(32) = 1 + 1 = 2
ω(2000) = ω(24 · 53) = ω(24) + ω(53) = 1 + 1 = 2
ω(2001) = 3
ω(2002) = 4
ω(2003) = 1
ω(54,032,858,972,279) = 3
ω(54,032,858,972,302) = 5
ω(20,802,650,704,327,415) = 5
a1(n) – the sum of the distinct primes dividing n, sometimes called sopf(n) (sequence A008472 in the OEIS). For example:
a1(1) = 0
a1(4) = 2
a1(20) = 2 + 5 = 7
a1(27) = 3
a1(144) = a1(24 · 32) = a1(24) + a1(32) = 2 + 3 = 5
a1(2000) = a1(24 · 53) = a1(24) + a1(53) = 2 + 5 = 7
a1(2001) = 55
a1(2002) = 33
a1(2003) = 2003
a1(54,032,858,972,279) = 1238665
a1(54,032,858,972,302) = 1780410
a1(20,802,650,704,327,415) = 1238677
== Multiplicative functions ==
From any additive function
f
(
n
)
{\displaystyle f(n)}
it is possible to create a related multiplicative function
g
(
n
)
,
{\displaystyle g(n),}
which is a function with the property that whenever
a
{\displaystyle a}
and
b
{\displaystyle b}
are coprime then:
g
(
a
b
)
=
g
(
a
)
×
g
(
b
)
.
{\displaystyle g(ab)=g(a)\times g(b).}
One such example is
g
(
n
)
=
2
f
(
n
)
.
{\displaystyle g(n)=2^{f(n)}.}
Likewise if
f
(
n
)
{\displaystyle f(n)}
is completely additive, then
g
(
n
)
=
2
f
(
n
)
{\displaystyle g(n)=2^{f(n)}}
is completely multiplicative. More generally, we could consider the function
g
(
n
)
=
c
f
(
n
)
{\displaystyle g(n)=c^{f(n)}}
, where
c
{\displaystyle c}
is a nonzero real constant.
== Summatory functions ==
Given an additive function
f
{\displaystyle f}
, let its summatory function be defined by
M
f
(
x
)
:=
∑
n
≤
x
f
(
n
)
{\textstyle {\mathcal {M}}_{f}(x):=\sum _{n\leq x}f(n)}
. The average of
f
{\displaystyle f}
is given exactly as
M
f
(
x
)
=
∑
p
α
≤
x
f
(
p
α
)
(
⌊
x
p
α
⌋
−
⌊
x
p
α
+
1
⌋
)
.
{\displaystyle {\mathcal {M}}_{f}(x)=\sum _{p^{\alpha }\leq x}f(p^{\alpha })\left(\left\lfloor {\frac {x}{p^{\alpha }}}\right\rfloor -\left\lfloor {\frac {x}{p^{\alpha +1}}}\right\rfloor \right).}
The summatory functions over
f
{\displaystyle f}
can be expanded as
M
f
(
x
)
=
x
E
(
x
)
+
O
(
x
⋅
D
(
x
)
)
{\displaystyle {\mathcal {M}}_{f}(x)=xE(x)+O({\sqrt {x}}\cdot D(x))}
where
E
(
x
)
=
∑
p
α
≤
x
f
(
p
α
)
p
−
α
(
1
−
p
−
1
)
D
2
(
x
)
=
∑
p
α
≤
x
|
f
(
p
α
)
|
2
p
−
α
.
{\displaystyle {\begin{aligned}E(x)&=\sum _{p^{\alpha }\leq x}f(p^{\alpha })p^{-\alpha }(1-p^{-1})\\D^{2}(x)&=\sum _{p^{\alpha }\leq x}|f(p^{\alpha })|^{2}p^{-\alpha }.\end{aligned}}}
The average of the function
f
2
{\displaystyle f^{2}}
is also expressed by these functions as
M
f
2
(
x
)
=
x
E
2
(
x
)
+
O
(
x
D
2
(
x
)
)
.
{\displaystyle {\mathcal {M}}_{f^{2}}(x)=xE^{2}(x)+O(xD^{2}(x)).}
There is always an absolute constant
C
f
>
0
{\displaystyle C_{f}>0}
such that for all natural numbers
x
≥
1
{\displaystyle x\geq 1}
,
∑
n
≤
x
|
f
(
n
)
−
E
(
x
)
|
2
≤
C
f
⋅
x
D
2
(
x
)
.
{\displaystyle \sum _{n\leq x}|f(n)-E(x)|^{2}\leq C_{f}\cdot xD^{2}(x).}
Let
ν
(
x
;
z
)
:=
1
x
#
{
n
≤
x
:
f
(
n
)
−
A
(
x
)
B
(
x
)
≤
z
}
.
{\displaystyle \nu (x;z):={\frac {1}{x}}\#\!\left\{n\leq x:{\frac {f(n)-A(x)}{B(x)}}\leq z\right\}\!.}
Suppose that
f
{\displaystyle f}
is an additive function with
−
1
≤
f
(
p
α
)
=
f
(
p
)
≤
1
{\displaystyle -1\leq f(p^{\alpha })=f(p)\leq 1}
such that as
x
→
∞
{\displaystyle x\rightarrow \infty }
,
B
(
x
)
=
∑
p
≤
x
f
2
(
p
)
/
p
→
∞
.
{\displaystyle B(x)=\sum _{p\leq x}f^{2}(p)/p\rightarrow \infty .}
Then
ν
(
x
;
z
)
∼
G
(
z
)
{\displaystyle \nu (x;z)\sim G(z)}
where
G
(
z
)
{\displaystyle G(z)}
is the Gaussian distribution function
G
(
z
)
=
1
2
π
∫
−
∞
z
e
−
t
2
/
2
d
t
.
{\displaystyle G(z)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{z}e^{-t^{2}/2}dt.}
Examples of this result related to the prime omega function and the numbers of prime divisors of shifted primes include the following for fixed
z
∈
R
{\displaystyle z\in \mathbb {R} }
where the relations hold for
x
≫
1
{\displaystyle x\gg 1}
:
#
{
n
≤
x
:
ω
(
n
)
−
log
log
x
≤
z
(
log
log
x
)
1
/
2
}
∼
x
G
(
z
)
,
{\displaystyle \#\{n\leq x:\omega (n)-\log \log x\leq z(\log \log x)^{1/2}\}\sim xG(z),}
#
{
p
≤
x
:
ω
(
p
+
1
)
−
log
log
x
≤
z
(
log
log
x
)
1
/
2
}
∼
π
(
x
)
G
(
z
)
.
{\displaystyle \#\{p\leq x:\omega (p+1)-\log \log x\leq z(\log \log x)^{1/2}\}\sim \pi (x)G(z).}
== See also ==
Sigma additivity
Prime omega function
Multiplicative function
Arithmetic function
== References ==
== Further reading == | Wikipedia/Additive_function |
In number theory, a multiplicative function is an arithmetic function
f
{\displaystyle f}
of a positive integer
n
{\displaystyle n}
with the property that
f
(
1
)
=
1
{\displaystyle f(1)=1}
and
f
(
a
b
)
=
f
(
a
)
f
(
b
)
{\displaystyle f(ab)=f(a)f(b)}
whenever
a
{\displaystyle a}
and
b
{\displaystyle b}
are coprime.
An arithmetic function is said to be completely multiplicative (or totally multiplicative) if
f
(
1
)
=
1
{\displaystyle f(1)=1}
and
f
(
a
b
)
=
f
(
a
)
f
(
b
)
{\displaystyle f(ab)=f(a)f(b)}
holds for all positive integers
a
{\displaystyle a}
and
b
{\displaystyle b}
, even when they are not coprime.
== Examples ==
Some multiplicative functions are defined to make formulas easier to write:
1
(
n
)
{\displaystyle 1(n)}
: the constant function defined by
1
(
n
)
=
1
{\displaystyle 1(n)=1}
Id
(
n
)
{\displaystyle \operatorname {Id} (n)}
: the identity function, defined by
Id
(
n
)
=
n
{\displaystyle \operatorname {Id} (n)=n}
Id
k
(
n
)
{\displaystyle \operatorname {Id} _{k}(n)}
: the power functions, defined by
Id
k
(
n
)
=
n
k
{\displaystyle \operatorname {Id} _{k}(n)=n^{k}}
for any complex number
k
{\displaystyle k}
. As special cases we have
Id
0
(
n
)
=
1
(
n
)
{\displaystyle \operatorname {Id} _{0}(n)=1(n)}
, and
Id
1
(
n
)
=
Id
(
n
)
{\displaystyle \operatorname {Id} _{1}(n)=\operatorname {Id} (n)}
.
ε
(
n
)
{\displaystyle \varepsilon (n)}
: the function defined by
ε
(
n
)
=
1
{\displaystyle \varepsilon (n)=1}
if
n
=
1
{\displaystyle n=1}
and
0
{\displaystyle 0}
otherwise; this is the unit function, so called because it is the multiplicative identity for Dirichlet convolution. Sometimes written as
u
(
n
)
{\displaystyle u(n)}
; not to be confused with
μ
(
n
)
{\displaystyle \mu (n)}
.
λ
(
n
)
{\displaystyle \lambda (n)}
: the Liouville function,
λ
(
n
)
=
(
−
1
)
Ω
(
n
)
{\displaystyle \lambda (n)=(-1)^{\Omega (n)}}
, where
Ω
(
n
)
{\displaystyle \Omega (n)}
is the total number of primes (counted with multiplicity) dividing
n
{\displaystyle n}
The above functions are all completely multiplicative.
1
C
(
n
)
{\displaystyle 1_{C}(n)}
: the indicator function of the set
C
⊆
Z
{\displaystyle C\subseteq \mathbb {Z} }
. This function is multiplicative precisely when
C
{\displaystyle C}
is closed under multiplication of coprime elements. There are also other sets (not closed under multiplication) that give rise to such functions, such as the set of square-free numbers.
Other examples of multiplicative functions include many functions of importance in number theory, such as:
gcd
(
n
,
k
)
{\displaystyle \gcd(n,k)}
: the greatest common divisor of
n
{\displaystyle n}
and
k
{\displaystyle k}
, as a function of
n
{\displaystyle n}
, where
k
{\displaystyle k}
is a fixed integer
φ
(
n
)
{\displaystyle \varphi (n)}
: Euler's totient function, which counts the positive integers coprime to (but not bigger than)
n
{\displaystyle n}
μ
(
n
)
{\displaystyle \mu (n)}
: the Möbius function, the parity (
−
1
{\displaystyle -1}
for odd,
+
1
{\displaystyle +1}
for even) of the number of prime factors of square-free numbers;
0
{\displaystyle 0}
if
n
{\displaystyle n}
is not square-free
σ
k
(
n
)
{\displaystyle \sigma _{k}(n)}
: the divisor function, which is the sum of the
k
{\displaystyle k}
-th powers of all the positive divisors of
n
{\displaystyle n}
(where
k
{\displaystyle k}
may be any complex number). As special cases we have
σ
0
(
n
)
=
d
(
n
)
{\displaystyle \sigma _{0}(n)=d(n)}
, the number of positive divisors of
n
{\displaystyle n}
,
σ
1
(
n
)
=
σ
(
n
)
{\displaystyle \sigma _{1}(n)=\sigma (n)}
, the sum of all the positive divisors of
n
{\displaystyle n}
.
σ
k
∗
(
n
)
{\displaystyle \sigma _{k}^{*}(n)}
: the sum of the
k
{\displaystyle k}
-th powers of all unitary divisors of
n
{\displaystyle n}
σ
k
∗
(
n
)
=
∑
d
∣
n
gcd
(
d
,
n
/
d
)
=
1
d
k
{\displaystyle \sigma _{k}^{*}(n)\,=\!\!\sum _{d\,\mid \,n \atop \gcd(d,\,n/d)=1}\!\!\!d^{k}}
a
(
n
)
{\displaystyle a(n)}
: the number of non-isomorphic abelian groups of order
n
{\displaystyle n}
γ
(
n
)
{\displaystyle \gamma (n)}
, defined by
γ
(
n
)
=
(
−
1
)
ω
(
n
)
{\displaystyle \gamma (n)=(-1)^{\omega (n)}}
, where the additive function
ω
(
n
)
{\displaystyle \omega (n)}
is the number of distinct primes dividing
n
{\displaystyle n}
τ
(
n
)
{\displaystyle \tau (n)}
: the Ramanujan tau function
All Dirichlet characters are completely multiplicative functions, for example
(
n
/
p
)
{\displaystyle (n/p)}
, the Legendre symbol, considered as a function of
n
{\displaystyle n}
where
p
{\displaystyle p}
is a fixed prime number
An example of a non-multiplicative function is the arithmetic function
r
2
(
n
)
{\displaystyle r_{2}(n)}
, the number of representations of
n
{\displaystyle n}
as a sum of squares of two integers, positive, negative, or zero, where in counting the number of ways, reversal of order is allowed. For example:
and therefore
r
2
(
1
)
=
4
≠
1
{\displaystyle r_{2}(1)=4\neq 1}
. This shows that the function is not multiplicative. However,
r
2
(
n
)
/
4
{\displaystyle r_{2}(n)/4}
is multiplicative.
In the On-Line Encyclopedia of Integer Sequences, sequences of values of a multiplicative function have the keyword "mult".
See arithmetic function for some other examples of non-multiplicative functions.
== Properties ==
A multiplicative function is completely determined by its values at the powers of prime numbers, a consequence of the fundamental theorem of arithmetic. Thus, if n is a product of powers of distinct primes, say n = pa qb ..., then
f(n) = f(pa) f(qb) ...
This property of multiplicative functions significantly reduces the need for computation, as in the following examples for n = 144 = 24 · 32:
d
(
144
)
=
σ
0
(
144
)
=
σ
0
(
2
4
)
σ
0
(
3
2
)
=
(
1
0
+
2
0
+
4
0
+
8
0
+
16
0
)
(
1
0
+
3
0
+
9
0
)
=
5
⋅
3
=
15
{\displaystyle d(144)=\sigma _{0}(144)=\sigma _{0}(2^{4})\,\sigma _{0}(3^{2})=(1^{0}+2^{0}+4^{0}+8^{0}+16^{0})(1^{0}+3^{0}+9^{0})=5\cdot 3=15}
σ
(
144
)
=
σ
1
(
144
)
=
σ
1
(
2
4
)
σ
1
(
3
2
)
=
(
1
1
+
2
1
+
4
1
+
8
1
+
16
1
)
(
1
1
+
3
1
+
9
1
)
=
31
⋅
13
=
403
{\displaystyle \sigma (144)=\sigma _{1}(144)=\sigma _{1}(2^{4})\,\sigma _{1}(3^{2})=(1^{1}+2^{1}+4^{1}+8^{1}+16^{1})(1^{1}+3^{1}+9^{1})=31\cdot 13=403}
σ
∗
(
144
)
=
σ
∗
(
2
4
)
σ
∗
(
3
2
)
=
(
1
1
+
16
1
)
(
1
1
+
9
1
)
=
17
⋅
10
=
170
{\displaystyle \sigma ^{*}(144)=\sigma ^{*}(2^{4})\,\sigma ^{*}(3^{2})=(1^{1}+16^{1})(1^{1}+9^{1})=17\cdot 10=170}
Similarly, we have:
φ
(
144
)
=
φ
(
2
4
)
φ
(
3
2
)
=
8
⋅
6
=
48
{\displaystyle \varphi (144)=\varphi (2^{4})\,\varphi (3^{2})=8\cdot 6=48}
In general, if f(n) is a multiplicative function and a, b are any two positive integers, then
Every completely multiplicative function is a homomorphism of monoids and is completely determined by its restriction to the prime numbers.
== Convolution ==
If f and g are two multiplicative functions, one defines a new multiplicative function
f
∗
g
{\displaystyle f*g}
, the Dirichlet convolution of f and g, by
(
f
∗
g
)
(
n
)
=
∑
d
|
n
f
(
d
)
g
(
n
d
)
{\displaystyle (f\,*\,g)(n)=\sum _{d|n}f(d)\,g\left({\frac {n}{d}}\right)}
where the sum extends over all positive divisors d of n.
With this operation, the set of all multiplicative functions turns into an abelian group; the identity element is ε. Convolution is commutative, associative, and distributive over addition.
Relations among the multiplicative functions discussed above include:
μ
∗
1
=
ε
{\displaystyle \mu *1=\varepsilon }
(the Möbius inversion formula)
(
μ
Id
k
)
∗
Id
k
=
ε
{\displaystyle (\mu \operatorname {Id} _{k})*\operatorname {Id} _{k}=\varepsilon }
(generalized Möbius inversion)
φ
∗
1
=
Id
{\displaystyle \varphi *1=\operatorname {Id} }
d
=
1
∗
1
{\displaystyle d=1*1}
σ
=
Id
∗
1
=
φ
∗
d
{\displaystyle \sigma =\operatorname {Id} *1=\varphi *d}
σ
k
=
Id
k
∗
1
{\displaystyle \sigma _{k}=\operatorname {Id} _{k}*1}
Id
=
φ
∗
1
=
σ
∗
μ
{\displaystyle \operatorname {Id} =\varphi *1=\sigma *\mu }
Id
k
=
σ
k
∗
μ
{\displaystyle \operatorname {Id} _{k}=\sigma _{k}*\mu }
The Dirichlet convolution can be defined for general arithmetic functions, and yields a ring structure, the Dirichlet ring.
The Dirichlet convolution of two multiplicative functions is again multiplicative. A proof of this fact is given by the following expansion for relatively prime
a
,
b
∈
Z
+
{\displaystyle a,b\in \mathbb {Z} ^{+}}
:
(
f
∗
g
)
(
a
b
)
=
∑
d
|
a
b
f
(
d
)
g
(
a
b
d
)
=
∑
d
1
|
a
∑
d
2
|
b
f
(
d
1
d
2
)
g
(
a
b
d
1
d
2
)
=
∑
d
1
|
a
f
(
d
1
)
g
(
a
d
1
)
×
∑
d
2
|
b
f
(
d
2
)
g
(
b
d
2
)
=
(
f
∗
g
)
(
a
)
⋅
(
f
∗
g
)
(
b
)
.
{\displaystyle {\begin{aligned}(f\ast g)(ab)&=\sum _{d|ab}f(d)g\left({\frac {ab}{d}}\right)\\&=\sum _{d_{1}|a}\sum _{d_{2}|b}f(d_{1}d_{2})g\left({\frac {ab}{d_{1}d_{2}}}\right)\\&=\sum _{d_{1}|a}f(d_{1})g\left({\frac {a}{d_{1}}}\right)\times \sum _{d_{2}|b}f(d_{2})g\left({\frac {b}{d_{2}}}\right)\\&=(f\ast g)(a)\cdot (f\ast g)(b).\end{aligned}}}
=== Dirichlet series for some multiplicative functions ===
∑
n
≥
1
μ
(
n
)
n
s
=
1
ζ
(
s
)
{\displaystyle \sum _{n\geq 1}{\frac {\mu (n)}{n^{s}}}={\frac {1}{\zeta (s)}}}
∑
n
≥
1
φ
(
n
)
n
s
=
ζ
(
s
−
1
)
ζ
(
s
)
{\displaystyle \sum _{n\geq 1}{\frac {\varphi (n)}{n^{s}}}={\frac {\zeta (s-1)}{\zeta (s)}}}
∑
n
≥
1
d
(
n
)
2
n
s
=
ζ
(
s
)
4
ζ
(
2
s
)
{\displaystyle \sum _{n\geq 1}{\frac {d(n)^{2}}{n^{s}}}={\frac {\zeta (s)^{4}}{\zeta (2s)}}}
∑
n
≥
1
2
ω
(
n
)
n
s
=
ζ
(
s
)
2
ζ
(
2
s
)
{\displaystyle \sum _{n\geq 1}{\frac {2^{\omega (n)}}{n^{s}}}={\frac {\zeta (s)^{2}}{\zeta (2s)}}}
More examples are shown in the article on Dirichlet series.
== Rational arithmetical functions ==
An arithmetical function f is said to be a rational arithmetical function of order
(
r
,
s
)
{\displaystyle (r,s)}
if there exists completely multiplicative functions g1,...,gr,
h1,...,hs such that
f
=
g
1
∗
⋯
∗
g
r
∗
h
1
−
1
∗
⋯
∗
h
s
−
1
,
{\displaystyle f=g_{1}\ast \cdots \ast g_{r}\ast h_{1}^{-1}\ast \cdots \ast h_{s}^{-1},}
where the inverses are with respect to the Dirichlet convolution. Rational arithmetical functions of order
(
1
,
1
)
{\displaystyle (1,1)}
are known as totient functions, and rational arithmetical functions of order
(
2
,
0
)
{\displaystyle (2,0)}
are known as quadratic functions or specially multiplicative functions. Euler's function
φ
(
n
)
{\displaystyle \varphi (n)}
is a totient function, and the divisor function
σ
k
(
n
)
{\displaystyle \sigma _{k}(n)}
is a quadratic function.
Completely multiplicative functions are rational arithmetical functions of order
(
1
,
0
)
{\displaystyle (1,0)}
. Liouville's function
λ
(
n
)
{\displaystyle \lambda (n)}
is completely multiplicative. The Möbius function
μ
(
n
)
{\displaystyle \mu (n)}
is a rational arithmetical function of order
(
0
,
1
)
{\displaystyle (0,1)}
.
By convention, the identity element
ε
{\displaystyle \varepsilon }
under the Dirichlet convolution is a rational arithmetical function of order
(
0
,
0
)
{\displaystyle (0,0)}
.
All rational arithmetical functions are multiplicative. A multiplicative function f is a rational arithmetical function of order
(
r
,
s
)
{\displaystyle (r,s)}
if and only if its Bell series is of the form
f
p
(
x
)
=
∑
n
=
0
∞
f
(
p
n
)
x
n
=
(
1
−
h
1
(
p
)
x
)
(
1
−
h
2
(
p
)
x
)
⋯
(
1
−
h
s
(
p
)
x
)
(
1
−
g
1
(
p
)
x
)
(
1
−
g
2
(
p
)
x
)
⋯
(
1
−
g
r
(
p
)
x
)
{\displaystyle {\displaystyle f_{p}(x)=\sum _{n=0}^{\infty }f(p^{n})x^{n}={\frac {(1-h_{1}(p)x)(1-h_{2}(p)x)\cdots (1-h_{s}(p)x)}{(1-g_{1}(p)x)(1-g_{2}(p)x)\cdots (1-g_{r}(p)x)}}}}
for all prime numbers
p
{\displaystyle p}
.
The concept of a rational arithmetical function originates from R. Vaidyanathaswamy (1931).
== Busche-Ramanujan identities ==
A multiplicative function
f
{\displaystyle f}
is said to be specially multiplicative
if there is a completely multiplicative function
f
A
{\displaystyle f_{A}}
such that
f
(
m
)
f
(
n
)
=
∑
d
∣
(
m
,
n
)
f
(
m
n
/
d
2
)
f
A
(
d
)
{\displaystyle f(m)f(n)=\sum _{d\mid (m,n)}f(mn/d^{2})f_{A}(d)}
for all positive integers
m
{\displaystyle m}
and
n
{\displaystyle n}
, or equivalently
f
(
m
n
)
=
∑
d
∣
(
m
,
n
)
f
(
m
/
d
)
f
(
n
/
d
)
μ
(
d
)
f
A
(
d
)
{\displaystyle f(mn)=\sum _{d\mid (m,n)}f(m/d)f(n/d)\mu (d)f_{A}(d)}
for all positive integers
m
{\displaystyle m}
and
n
{\displaystyle n}
, where
μ
{\displaystyle \mu }
is the Möbius function.
These are known as Busche-Ramanujan identities.
In 1906, E. Busche stated the identity
σ
k
(
m
)
σ
k
(
n
)
=
∑
d
∣
(
m
,
n
)
σ
k
(
m
n
/
d
2
)
d
k
,
{\displaystyle \sigma _{k}(m)\sigma _{k}(n)=\sum _{d\mid (m,n)}\sigma _{k}(mn/d^{2})d^{k},}
and, in 1915, S. Ramanujan gave the inverse form
σ
k
(
m
n
)
=
∑
d
∣
(
m
,
n
)
σ
k
(
m
/
d
)
σ
k
(
n
/
d
)
μ
(
d
)
d
k
{\displaystyle \sigma _{k}(mn)=\sum _{d\mid (m,n)}\sigma _{k}(m/d)\sigma _{k}(n/d)\mu (d)d^{k}}
for
k
=
0
{\displaystyle k=0}
. S. Chowla gave the inverse form for general
k
{\displaystyle k}
in 1929, see P. J. McCarthy (1986). The study of Busche-Ramanujan identities begun from an attempt to better understand the special cases given by Busche and Ramanujan.
It is known that quadratic functions
f
=
g
1
∗
g
2
{\displaystyle f=g_{1}\ast g_{2}}
satisfy the Busche-Ramanujan identities with
f
A
=
g
1
g
2
{\displaystyle f_{A}=g_{1}g_{2}}
. Quadratic functions are exactly the same as specially multiplicative functions. Totients satisfy a restricted Busche-Ramanujan identity. For further details, see R. Vaidyanathaswamy (1931).
== Multiplicative function over Fq[X] ==
Let A = Fq[X], the polynomial ring over the finite field with q elements. A is a principal ideal domain and therefore A is a unique factorization domain.
A complex-valued function
λ
{\displaystyle \lambda }
on A is called multiplicative if
λ
(
f
g
)
=
λ
(
f
)
λ
(
g
)
{\displaystyle \lambda (fg)=\lambda (f)\lambda (g)}
whenever f and g are relatively prime.
=== Zeta function and Dirichlet series in Fq[X] ===
Let h be a polynomial arithmetic function (i.e. a function on set of monic polynomials over A). Its corresponding Dirichlet series is defined to be
D
h
(
s
)
=
∑
f
monic
h
(
f
)
|
f
|
−
s
,
{\displaystyle D_{h}(s)=\sum _{f{\text{ monic}}}h(f)|f|^{-s},}
where for
g
∈
A
,
{\displaystyle g\in A,}
set
|
g
|
=
q
deg
(
g
)
{\displaystyle |g|=q^{\deg(g)}}
if
g
≠
0
,
{\displaystyle g\neq 0,}
and
|
g
|
=
0
{\displaystyle |g|=0}
otherwise.
The polynomial zeta function is then
ζ
A
(
s
)
=
∑
f
monic
|
f
|
−
s
.
{\displaystyle \zeta _{A}(s)=\sum _{f{\text{ monic}}}|f|^{-s}.}
Similar to the situation in N, every Dirichlet series of a multiplicative function h has a product representation (Euler product):
D
h
(
s
)
=
∏
P
(
∑
n
=
0
∞
h
(
P
n
)
|
P
|
−
s
n
)
,
{\displaystyle D_{h}(s)=\prod _{P}\left(\sum _{n\mathop {=} 0}^{\infty }h(P^{n})|P|^{-sn}\right),}
where the product runs over all monic irreducible polynomials P. For example, the product representation of the zeta function is as for the integers:
ζ
A
(
s
)
=
∏
P
(
1
−
|
P
|
−
s
)
−
1
.
{\displaystyle \zeta _{A}(s)=\prod _{P}(1-|P|^{-s})^{-1}.}
Unlike the classical zeta function,
ζ
A
(
s
)
{\displaystyle \zeta _{A}(s)}
is a simple rational function:
ζ
A
(
s
)
=
∑
f
|
f
|
−
s
=
∑
n
∑
deg
(
f
)
=
n
q
−
s
n
=
∑
n
(
q
n
−
s
n
)
=
(
1
−
q
1
−
s
)
−
1
.
{\displaystyle \zeta _{A}(s)=\sum _{f}|f|^{-s}=\sum _{n}\sum _{\deg(f)=n}q^{-sn}=\sum _{n}(q^{n-sn})=(1-q^{1-s})^{-1}.}
In a similar way, If f and g are two polynomial arithmetic functions, one defines f * g, the Dirichlet convolution of f and g, by
(
f
∗
g
)
(
m
)
=
∑
d
∣
m
f
(
d
)
g
(
m
d
)
=
∑
a
b
=
m
f
(
a
)
g
(
b
)
,
{\displaystyle {\begin{aligned}(f*g)(m)&=\sum _{d\mid m}f(d)g\left({\frac {m}{d}}\right)\\&=\sum _{ab=m}f(a)g(b),\end{aligned}}}
where the sum is over all monic divisors d of m, or equivalently over all pairs (a, b) of monic polynomials whose product is m. The identity
D
h
D
g
=
D
h
∗
g
{\displaystyle D_{h}D_{g}=D_{h*g}}
still holds.
== Multivariate ==
Multivariate functions can be constructed using multiplicative model estimators. Where a matrix function of A is defined as
D
N
=
N
2
×
N
(
N
+
1
)
/
2
{\displaystyle D_{N}=N^{2}\times N(N+1)/2}
a sum can be distributed across the product
y
t
=
∑
(
t
/
T
)
1
/
2
u
t
=
∑
(
t
/
T
)
1
/
2
G
t
1
/
2
ϵ
t
{\displaystyle y_{t}=\sum (t/T)^{1/2}u_{t}=\sum (t/T)^{1/2}G_{t}^{1/2}\epsilon _{t}}
For the efficient estimation of Σ(.), the following two nonparametric regressions can be considered:
y
~
t
2
=
y
t
2
g
t
=
σ
2
(
t
/
T
)
+
σ
2
(
t
/
T
)
(
ϵ
t
2
−
1
)
,
{\displaystyle {\tilde {y}}_{t}^{2}={\frac {y_{t}^{2}}{g_{t}}}=\sigma ^{2}(t/T)+\sigma ^{2}(t/T)(\epsilon _{t}^{2}-1),}
and
y
t
2
=
σ
2
(
t
/
T
)
+
σ
2
(
t
/
T
)
(
g
t
ϵ
t
2
−
1
)
.
{\displaystyle y_{t}^{2}=\sigma ^{2}(t/T)+\sigma ^{2}(t/T)(g_{t}\epsilon _{t}^{2}-1).}
Thus it gives an estimate value of
L
t
(
τ
;
u
)
=
∑
t
=
1
T
K
h
(
u
−
t
/
T
)
[
l
n
τ
+
y
t
2
g
t
τ
]
{\displaystyle L_{t}(\tau ;u)=\sum _{t=1}^{T}K_{h}(u-t/T){\begin{bmatrix}ln\tau +{\frac {y_{t}^{2}}{g_{t}\tau }}\end{bmatrix}}}
with a local likelihood function for
y
t
2
{\displaystyle y_{t}^{2}}
with known
g
t
{\displaystyle g_{t}}
and unknown
σ
2
(
t
/
T
)
{\displaystyle \sigma ^{2}(t/T)}
.
== Generalizations ==
An arithmetical function
f
{\displaystyle f}
is
quasimultiplicative if there exists a nonzero constant
c
{\displaystyle c}
such that
c
f
(
m
n
)
=
f
(
m
)
f
(
n
)
{\displaystyle c\,f(mn)=f(m)f(n)}
for all positive integers
m
,
n
{\displaystyle m,n}
with
(
m
,
n
)
=
1
{\displaystyle (m,n)=1}
. This concept originates by Lahiri (1972).
An arithmetical function
f
{\displaystyle f}
is semimultiplicative
if there exists a nonzero constant
c
{\displaystyle c}
, a positive integer
a
{\displaystyle a}
and
a multiplicative function
f
m
{\displaystyle f_{m}}
such that
f
(
n
)
=
c
f
m
(
n
/
a
)
{\displaystyle f(n)=cf_{m}(n/a)}
for all positive integers
n
{\displaystyle n}
(under the convention that
f
m
(
x
)
=
0
{\displaystyle f_{m}(x)=0}
if
x
{\displaystyle x}
is not a positive integer.) This concept is due to David Rearick (1966).
An arithmetical function
f
{\displaystyle f}
is Selberg multiplicative if
for each prime
p
{\displaystyle p}
there exists a function
f
p
{\displaystyle f_{p}}
on nonnegative integers with
f
p
(
0
)
=
1
{\displaystyle f_{p}(0)=1}
for
all but finitely many primes
p
{\displaystyle p}
such that
f
(
n
)
=
∏
p
f
p
(
ν
p
(
n
)
)
{\displaystyle f(n)=\prod _{p}f_{p}(\nu _{p}(n))}
for all positive integers
n
{\displaystyle n}
, where
ν
p
(
n
)
{\displaystyle \nu _{p}(n)}
is the exponent of
p
{\displaystyle p}
in the canonical factorization of
n
{\displaystyle n}
.
See Selberg (1977).
It is known that the classes of semimultiplicative and Selberg multiplicative functions coincide. They both satisfy the arithmetical identity
f
(
m
)
f
(
n
)
=
f
(
(
m
,
n
)
)
f
(
[
m
,
n
]
)
{\displaystyle f(m)f(n)=f((m,n))f([m,n])}
for all positive integers
m
,
n
{\displaystyle m,n}
. See Haukkanen (2012).
It is well known and easy to see that multiplicative functions are quasimultiplicative functions with
c
=
1
{\displaystyle c=1}
and quasimultiplicative functions are semimultiplicative functions with
a
=
1
{\displaystyle a=1}
.
== See also ==
Euler product
Bell series
Lambert series
== References ==
See chapter 2 of Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001
P. J. McCarthy, Introduction to Arithmetical Functions, Universitext. New York: Springer-Verlag, 1986.
Hafner, Christian M.; Linton, Oliver (2010). "Efficient estimation of a multivariate multiplicative volatility model" (PDF). Journal of Econometrics. 159 (1): 55–73. doi:10.1016/j.jeconom.2010.04.007. S2CID 54812323.
P. Haukkanen (2003). "Some characterizations of specially multiplicative functions". Int. J. Math. Math. Sci. 2003 (37): 2335–2344. doi:10.1155/S0161171203301139.
P. Haukkanen (2012). "Extensions of the class of multiplicative functions". East–West Journal of Mathematics. 14 (2): 101–113.
DB Lahiri (1972). "Hypo-multiplicative number-theoretic functions". Aequationes Mathematicae. 8 (3): 316–317. doi:10.1007/BF01844515.
D. Rearick (1966). "Semi-multiplicative functions". Duke Math. J. 33: 49–53.
L. Tóth (2013). "Two generalizations of the Busche-Ramanujan identities". International Journal of Number Theory. 9 (5): 1301–1311. arXiv:1301.3331. doi:10.1142/S1793042113500280.
R. Vaidyanathaswamy (1931). "The theory of multiplicative arithmetic functions". Transactions of the American Mathematical Society. 33 (2): 579–662. doi:10.1090/S0002-9947-1931-1501607-1.
S. Ramanujan, Some formulae in the analytic theory of numbers. Messenger 45 (1915), 81--84.
E. Busche, Lösung einer Aufgabe über Teileranzahlen. Mitt. Math. Ges. Hamb. 4, 229--237 (1906)
A. Selberg: Remarks on multiplicative functions. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976), pp. 232–241, Springer, 1977.
== External links ==
Multiplicative function at PlanetMath.
== References == | Wikipedia/Multiplicative_function |
In mathematics, the L-functions of number theory are expected to have several characteristic properties, one of which is that they satisfy certain functional equations. There is an elaborate theory of what these equations should be, much of which is still conjectural.
== Introduction ==
A prototypical example, the Riemann zeta function has a functional equation relating its value at the complex number s with its value at 1 − s. In every case this relates to some value ζ(s) that is only defined by analytic continuation from the infinite series definition. That is, writing – as is conventional – σ for the real part of s, the functional equation relates the cases
σ > 1 and σ < 0,
and also changes a case with
0 < σ < 1
in the critical strip to another such case, reflected in the line σ = ½. Therefore, use of the functional equation is basic, in order to study the zeta-function in the whole complex plane.
The functional equation in question for the Riemann zeta function takes the simple form
Z
(
s
)
=
Z
(
1
−
s
)
{\displaystyle Z(s)=Z(1-s)\,}
where Z(s) is ζ(s) multiplied by a gamma-factor, involving the gamma function. This is now read as an 'extra' factor in the Euler product for the zeta-function, corresponding to the infinite prime. Just the same shape of functional equation holds for the Dedekind zeta function of a number field K, with an appropriate gamma-factor that depends only on the embeddings of K (in algebraic terms, on the tensor product of K with the real field).
There is a similar equation for the Dirichlet L-functions, but this time relating them in pairs:
Λ
(
s
,
χ
)
=
ε
Λ
(
1
−
s
,
χ
∗
)
{\displaystyle \Lambda (s,\chi )=\varepsilon \Lambda (1-s,\chi ^{*})}
with χ a primitive Dirichlet character, χ* its complex conjugate, Λ the L-function multiplied by a gamma-factor, and ε a complex number of absolute value 1, of shape
G
(
χ
)
|
G
(
χ
)
|
{\displaystyle G(\chi ) \over {\left|G(\chi )\right\vert }}
where G(χ) is a Gauss sum formed from χ. This equation has the same function on both sides if and only if χ is a real character, taking values in {0,1,−1}. Then ε must be 1 or −1, and the case of the value −1 would imply a zero of Λ(s) at s = ½. According to the theory (of Gauss, in effect) of Gauss sums, the value is always 1, so no such simple zero can exist (the function is even about the point).
== Theory of functional equations ==
A unified theory of such functional equations was given by Erich Hecke, and the theory was taken up again in Tate's thesis by John Tate. Hecke found generalised characters of number fields, now called Hecke characters, for which his proof (based on theta functions) also worked. These characters and their associated L-functions are now understood to be strictly related to complex multiplication, as the Dirichlet characters are to cyclotomic fields.
There are also functional equations for the local zeta-functions, arising at a fundamental level for the (analogue of) Poincaré duality in étale cohomology. The Euler products of the Hasse–Weil zeta-function for an algebraic variety V over a number field K, formed by reducing modulo prime ideals to get local zeta-functions, are conjectured to have a global functional equation; but this is currently considered out of reach except in special cases. The definition can be read directly out of étale cohomology theory, again; but in general some assumption coming from automorphic representation theory seems required to get the functional equation. The Taniyama–Shimura conjecture was a particular case of this as general theory. By relating the gamma-factor aspect to Hodge theory, and detailed studies of the expected ε factor, the theory as empirical has been brought to quite a refined state, even if proofs are missing.
== See also ==
Explicit formula (L-function)
Riemann–Siegel formula (particular approximate functional equation)
== References ==
== External links ==
Weisstein, Eric W. "Functional Equation". MathWorld. | Wikipedia/Functional_equation_(L-function) |
In systems engineering, software engineering, and computer science, a function model or functional model is a structured representation of the functions (activities, actions, processes, operations) within the modeled system or subject area.
A function model, similar with the activity model or process model, is a graphical representation of an enterprise's function within a defined scope. The purposes of the function model are to describe the functions and processes, assist with discovery of information needs, help identify opportunities, and establish a basis for determining product and service costs.
== History ==
The function model in the field of systems engineering and software engineering originates in the 1950s and 1960s, but the origin of functional modelling of organizational activity goes back to the late 19th century.
In the late 19th century the first diagrams appeared that pictured business activities, actions, processes, or operations, and in the first half of the 20th century the first structured methods for documenting business process activities emerged. One of those methods was the flow process chart, introduced by Frank Gilbreth to members of American Society of Mechanical Engineers (ASME) in 1921 with the presentation, entitled “Process Charts—First Steps in Finding the One Best Way”. Gilbreth's tools quickly found their way into industrial engineering curricula.
The emergence of the field of systems engineering can be traced back to Bell Telephone Laboratories in the 1940s. The need to identify and manipulate the properties of a system as a whole, which in complex engineering projects may greatly differ from the sum of the parts' properties, motivated various industries to apply the discipline. One of the first to define the function model in this field was the British engineer William Gosling. In his book The design of engineering systems (1962, p. 25) he stated:
A functional model must thus achieve two aims in order to be of use. It must furnish a throughput description mechanics capable of completely defining the first and last throughput states, and perhaps some of the intervening states. It must also offer some means by which any input, correctly described in terms of this mechanics, can be used to generate an output which is an equally correct description of the output which the actual system would have given for the input concerned. It may also be noted that there are two other things which a functional model may do, but which are not necessary to all functional models. Thus such a system may, but need not, describe the system throughputs other than at the input and output, and it may also contain a description of the operation which each element carries out on the throughput, but once again this is not.
One of the first well defined function models, was the functional flow block diagram (FFBD) developed by the defense-related TRW Incorporated in the 1950s. In the 1960s it was exploited by the NASA to visualize the time sequence of events in a space systems and flight missions. It is further widely used in classical systems engineering to show the order of execution of system functions.
== Functional modeling topics ==
=== Functional perspective ===
In systems engineering and software engineering a function model is created with a functional modeling perspective. The functional perspective is one of the perspectives possible in business process modelling, other perspectives are for example behavioural, organisational or informational.
A functional modeling perspective concentrates on describing the dynamic process. The main concept in this modeling perspective is the process, this could be a function, transformation, activity, action, task etc. A well-known example of a modeling language employing this perspective is data flow diagrams.
The perspective uses four symbols to describe a process, these being:
Process: Illustrates transformation from input to output.
Store: Data-collection or some sort of material.
Flow: Movement of data or material in the process.
External Entity: External to the modeled system, but interacts with it.
Now, with these symbols, a process can be represented as a network of these symbols. This decomposed process is a DFD, data flow diagram.
In Dynamic Enterprise Modeling a division is made in the Control model, Function Model, Process model and Organizational model.
=== Functional decomposition ===
Functional decomposition refers broadly to the process of resolving a functional relationship into its constituent parts in such a way that the original function can be reconstructed from those parts by function composition. In general, this process of decomposition is undertaken either for the purpose of gaining insight into the identity of the constituent components, or for the purpose of obtaining a compressed representation of the global function, a task which is feasible only when the constituent processes possess a certain level of modularity.
Functional decomposition has a prominent role in computer programming, where a major goal is to modularize processes to the greatest extent possible. For example, a library management system may be broken up into an inventory module, a patron information module, and a fee assessment module. In the early decades of computer programming, this was manifested as the "art of subroutining," as it was called by some prominent practitioners.
Functional decomposition of engineering systems is a method for analyzing engineered systems. The basic idea is to try to divide a system in such a way that each block of the block diagram can be described without an "and" or "or" in the description.
This exercise forces each part of the system to have a pure function. When a system is composed of pure functions, they can be reused, or replaced. A usual side effect is that the interfaces between blocks become simple and generic. Since the interfaces usually become simple, it is easier to replace a pure function with a related, similar function.
== Functional modeling methods ==
The functional approach is extended in multiple diagrammic techniques and modeling notations. This section gives an overview of the important techniques in chronological order.
=== Function block diagram ===
A functional block diagram is a block diagram, that describes the functions and interrelationships of a system. The functional block diagram can picture:
Functions of a system pictured by blocks
Input of a block pictured with lines, and
Relationships between 9 functions
Functional sequences and paths for matter and or signals
The block diagram can use additional schematic symbols to show particular properties.
Specific function block diagram are the classic functional flow block diagram, and the Function Block Diagram (FBD) used in the design of programmable logic controllers.
=== Functional flow block diagram ===
The functional flow block diagram (FFBD) is a multi-tier, time-sequenced, step-by-step flow diagram of the system's functional flow.
The diagram is developed in the 1950s and widely used in classical systems engineering. The functional flow block diagram is also referred to as Functional Flow Diagram, functional block diagram, and functional flow.
Functional flow block diagrams (FFBD) usually define the detailed, step-by-step operational and support sequences for systems, but they are also used effectively to define processes in developing and producing systems. The software development processes also use FFBDs extensively. In the system context, the functional flow steps may include combinations of hardware, software, personnel, facilities, and/or procedures.
In the FFBD method, the functions are organized and depicted by their logical order of execution. Each function is shown with respect to its logical relationship to the execution and completion of other functions. A node labeled with the function name depicts each function. Arrows from left to right show the order of execution of the functions. Logic symbols represent sequential or parallel execution of functions.
=== HIPO and oPO ===
HIPO for hierarchical input process output is a popular 1970s systems analysis design aid and documentation technique for representing the modules of a system as a hierarchy and for documenting each module.
It was used to develop requirements, construct the design, and support implementation of an expert system to demonstrate automated rendezvous. Verification was then conducted systematically because of the method of design and implementation.
The overall design of the system is documented using HIPO charts or structure charts. The structure chart is similar in appearance to an organizational chart, but has been modified to show additional detail. Structure charts can be used to display several types of information, but are used most commonly to diagram either data structures or code structures.
=== N2 Chart ===
The N2 Chart is a diagram in the shape of a matrix, representing functional or physical interfaces between system elements. It is used to systematically identify, define, tabulate, design, and analyze functional and physical interfaces. It applies to system interfaces and hardware and/or software interfaces.
The N2 diagram has been used extensively to develop data interfaces, primarily in the software areas. However, it can also be used to develop hardware interfaces. The basic N2 chart is shown in Figure 2. The system functions are placed on the diagonal; the remainder of the squares in the N × N matrix represent the interface inputs and outputs.
=== Structured Analysis and Design Technique ===
Structured Analysis and Design Technique (SADT) is a software engineering methodology for describing systems as a hierarchy of functions, a diagrammatic notation for constructing a sketch for a software application. It offers building blocks to represent entities and activities, and a variety of arrows to relate boxes. These boxes and arrows have an associated informal semantics. SADT can be used as a functional analysis tool of a given process, using successive levels of details. The SADT method allows to define user needs for IT developments, which is used in industrial Information Systems, but also to explain and to present an activity's manufacturing processes, procedures.
The SADT supplies a specific functional view of any enterprise by describing the functions and their relationships in a company. These functions fulfill the objectives of a company, such as sales, order planning, product design, part manufacturing, and human resource management. The SADT can depict simple functional relationships and can reflect data and control flow relationships between different functions. The IDEF0 formalism is based on SADT, developed by Douglas T. Ross in 1985.
=== IDEF0 ===
IDEF0 is a function modeling methodology for describing manufacturing functions, which offers a functional modeling language for the analysis, development, re-engineering, and integration of information systems; business processes; or software engineering analysis. It is part of the IDEF family of modeling languages in the field of software engineering, and is built on the functional modeling language building SADT.
The IDEF0 Functional Modeling method is designed to model the decisions, actions, and activities of an organization or system. It was derived from the established graphic modeling language structured analysis and design technique (SADT) developed by Douglas T. Ross and SofTech, Inc. In its original form, IDEF0 includes both a definition of a graphical modeling language (syntax and semantics) and a description of a comprehensive methodology for developing models. The US Air Force commissioned the SADT developers to develop a function model method for analyzing and communicating the functional perspective of a system. IDEF0 should assist in organizing system analysis and promote effective communication between the analyst and the customer through simplified graphical devices.
=== Axiomatic design ===
Axiomatic design is a top down hierarchical functional decomposition process used as a solution synthesis framework for the analysis, development, re-engineering, and integration of products, information systems, business processes or software engineering solutions. Its structure is suited mathematically to analyze coupling between functions in order to optimize the architectural robustness of potential functional solution models.
== Related types of models ==
In the field of systems and software engineering numerous specific function and functional models and close related models have been defined. Here only a few general types will be explained.
=== Business function model ===
A Business Function Model (BFM) is a general description or category of operations performed routinely to carry out an organization's mission. They "provide a conceptual structure for the identification of general business functions". It can show the critical business processes in the context of the business area functions. The processes in the business function model must be consistent with the processes in the value chain models. Processes are a group of related business activities performed to produce an end product or to provide a service. Unlike business functions that are performed on a continual basis, processes are characterized by the fact that they have a specific beginning and an end point marked by the delivery of a desired output. The figure on the right depicts the relationship between the business processes, business functions, and the business area's business reference model.
=== Business Process Model and Notation ===
Business Process Model and Notation (BPMN) is a graphical representation for specifying business processes in a workflow. BPMN was developed by Business Process Management Initiative (BPMI), and is currently maintained by the Object Management Group since the two organizations merged in 2005. The current version of BPMN is 2.0.
The Business Process Model and Notation (BPMN) specification provides a graphical notation for specifying business processes in a Business Process Diagram (BPD). The objective of BPMN is to support business process management for both technical users and business users by providing a notation that is intuitive to business users yet able to represent complex process semantics. The BPMN specification also provides a mapping between the graphics of the notation to the underlying constructs of execution languages, particularly BPEL4WS.
=== Business reference model ===
A Business reference model is a reference model, concentrating on the functional and organizational aspects of the core business of an enterprise, service organization or government agency. In enterprise engineering a business reference model is part of an Enterprise Architecture Framework or Architecture Framework, which defines how to organize the structure and views associated with an Enterprise Architecture.
A reference model in general is a model of something that embodies the basic goal or idea of something and can then be looked at as a reference for various purposes. A business reference model is a means to describe the business operations of an organization, independent of the organizational structure that perform them. Other types of business reference model can also depict the relationship between the business processes, business functions, and the business area's business reference model. These reference model can be constructed in layers, and offer a foundation for the analysis of service components, technology, data, and performance.
=== Operator function model ===
The Operator Function Model (OFM) is proposed as an alternative to traditional task analysis techniques used by human factors engineers. An operator function model attempts to represent in mathematical form how an operator might decompose a complex system into simpler parts and coordinate control actions and system configurations so that acceptable overall system performance is achieved. The model represents basic issues of knowledge representation, information flow, and decision making in complex systems. Miller (1985) suggests that the network structure can be thought of as a possible representation of an operator's internal model of the system plus a control structure which specifies how the model is used to solve the decision problems that comprise operator control functions.
== See also ==
Bus Functional Model
Business process modeling
Data and information visualization
Data model
Enterprise modeling
Functional Software Architecture
Multilevel Flow Modeling
Polynomial function model
Rational function model
Scientific modeling
Unified Modeling Language
View model
== References ==
This article incorporates public domain material from the National Institute of Standards and Technology
This article incorporates public domain material from Operator Function Model (OFM). Federal Aviation Administration. | Wikipedia/Functional_model |
Böttcher's equation, named after Lucjan Böttcher, is the functional equation
F
(
h
(
z
)
)
=
(
F
(
z
)
)
n
{\displaystyle F(h(z))=(F(z))^{n}}
where
h is a given analytic function with a superattracting fixed point of order n at a, (that is,
h
(
z
)
=
a
+
c
(
z
−
a
)
n
+
O
(
(
z
−
a
)
n
+
1
)
,
{\displaystyle h(z)=a+c(z-a)^{n}+O((z-a)^{n+1})~,}
in a neighbourhood of a), with n ≥ 2
F is a sought function.
The logarithm of this functional equation amounts to Schröder's equation.
== Solution ==
Solution of functional equation is a function in implicit form.
Lucian Emil Böttcher sketched a proof in 1904 on the existence of solution: an analytic function F in a neighborhood of the fixed point a, such that:
F
(
a
)
=
0
{\displaystyle F(a)=0}
This solution is sometimes called:
the Böttcher coordinate
the Böttcher function
the Boettcher map.
The complete proof was published by Joseph Ritt in 1920, who was unaware of the original formulation.
Böttcher's coordinate (the logarithm of the Schröder function) conjugates h(z) in a neighbourhood of the fixed point to the function zn. An especially important case is when h(z) is a polynomial of degree n, and a = ∞ .
=== Explicit ===
One can explicitly compute Böttcher coordinates for:
power maps
z
→
z
d
{\displaystyle z\to z^{d}}
Chebyshev polynomials
==== Examples ====
For the function h and n=2
h
(
x
)
=
x
2
1
−
2
x
2
{\displaystyle h(x)={\frac {x^{2}}{1-2x^{2}}}}
the Böttcher function F is:
F
(
x
)
=
x
1
+
x
2
{\displaystyle F(x)={\frac {x}{1+x^{2}}}}
== Applications ==
Böttcher's equation plays a fundamental role in the part of holomorphic dynamics which studies iteration of polynomials of one complex variable.
Global properties of the Böttcher coordinate were studied by Fatou
and Douady and Hubbard.
== See also ==
Schröder's equation
External ray
== References == | Wikipedia/Böttcher's_equation |
In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the 3rd power: 1000 = 103 = 10 × 10 × 10. More generally, if x = by, then y is the logarithm of x to base b, written logb x, so log10 1000 = 3. As a single-variable function, the logarithm to base b is the inverse of exponentiation with base b.
The logarithm base 10 is called the decimal or common logarithm and is commonly used in science and engineering. The natural logarithm has the number e ≈ 2.718 as its base; its use is widespread in mathematics and physics because of its very simple derivative. The binary logarithm uses base 2 and is widely used in computer science, information theory, music theory, and photography. When the base is unambiguous from the context or irrelevant it is often omitted, and the logarithm is written log x.
Logarithms were introduced by John Napier in 1614 as a means of simplifying calculations. They were rapidly adopted by navigators, scientists, engineers, surveyors, and others to perform high-accuracy computations more easily. Using logarithm tables, tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition. This is possible because the logarithm of a product is the sum of the logarithms of the factors:
log
b
(
x
y
)
=
log
b
x
+
log
b
y
,
{\displaystyle \log _{b}(xy)=\log _{b}x+\log _{b}y,}
provided that b, x and y are all positive and b ≠ 1. The slide rule, also based on logarithms, allows quick calculations without tables, but at lower precision. The present-day notion of logarithms comes from Leonhard Euler, who connected them to the exponential function in the 18th century, and who also introduced the letter e as the base of natural logarithms.
Logarithmic scales reduce wide-ranging quantities to smaller scopes. For example, the decibel (dB) is a unit used to express ratio as logarithms, mostly for signal power and amplitude (of which sound pressure is a common example). In chemistry, pH is a logarithmic measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and of geometric objects called fractals. They help to describe frequency ratios of musical intervals, appear in formulas counting prime numbers or approximating factorials, inform some models in psychophysics, and can aid in forensic accounting.
The concept of logarithm as the inverse of exponentiation extends to other mathematical structures as well. However, in general settings, the logarithm tends to be a multi-valued function. For example, the complex logarithm is the multi-valued inverse of the complex exponential function. Similarly, the discrete logarithm is the multi-valued inverse of the exponential function in finite groups; it has uses in public-key cryptography.
== Motivation ==
Addition, multiplication, and exponentiation are three of the most fundamental arithmetic operations. The inverse of addition is subtraction, and the inverse of multiplication is division. Similarly, a logarithm is the inverse operation of exponentiation. Exponentiation is when a number b, the base, is raised to a certain power y, the exponent, to give a value x; this is denoted
b
y
=
x
.
{\displaystyle b^{y}=x.}
For example, raising 2 to the power of 3 gives 8:
2
3
=
8.
{\displaystyle 2^{3}=8.}
The logarithm of base b is the inverse operation, that provides the output y from the input x. That is,
y
=
log
b
x
{\displaystyle y=\log _{b}x}
is equivalent to
x
=
b
y
{\displaystyle x=b^{y}}
if b is a positive real number. (If b is not a positive real number, both exponentiation and logarithm can be defined but may take several values, which makes definitions much more complicated.)
One of the main historical motivations of introducing logarithms is the formula
log
b
(
x
y
)
=
log
b
x
+
log
b
y
,
{\displaystyle \log _{b}(xy)=\log _{b}x+\log _{b}y,}
by which tables of logarithms allow multiplication and division to be reduced to addition and subtraction, a great aid to calculations before the invention of computers.
== Definition ==
Given a positive real number b such that b ≠ 1, the logarithm of a positive real number x with respect to base b is the exponent by which b must be raised to yield x. In other words, the logarithm of x to base b is the unique real number y such that
b
y
=
x
{\displaystyle b^{y}=x}
.
The logarithm is denoted "logb x" (pronounced as "the logarithm of x to base b", "the base-b logarithm of x", or most commonly "the log, base b, of x").
An equivalent and more succinct definition is that the function logb is the inverse function to the function
x
↦
b
x
{\displaystyle x\mapsto b^{x}}
.
=== Examples ===
log2 16 = 4, since 24 = 2 × 2 × 2 × 2 = 16.
Logarithms can also be negative:
log
2
1
2
=
−
1
{\textstyle \log _{2}\!{\frac {1}{2}}=-1}
since
2
−
1
=
1
2
1
=
1
2
.
{\textstyle 2^{-1}={\frac {1}{2^{1}}}={\frac {1}{2}}.}
log10 150 is approximately 2.176, which lies between 2 and 3, just as 150 lies between 102 = 100 and 103 = 1000.
For any base b, logb b = 1 and logb 1 = 0, since b1 = b and b0 = 1, respectively.
== Logarithmic identities ==
Several important formulas, sometimes called logarithmic identities or logarithmic laws, relate logarithms to one another.
=== Product, quotient, power, and root ===
The logarithm of a product is the sum of the logarithms of the numbers being multiplied; the logarithm of the ratio of two numbers is the difference of the logarithms. The logarithm of the p-th power of a number is p times the logarithm of the number itself; the logarithm of a p-th root is the logarithm of the number divided by p. The following table lists these identities with examples. Each of the identities can be derived after substitution of the logarithm definitions
x
=
b
log
b
x
{\displaystyle x=b^{\,\log _{b}x}}
or
y
=
b
log
b
y
{\displaystyle y=b^{\,\log _{b}y}}
in the left hand sides. In the following formulas,
x
{\displaystyle x}
and
y
{\displaystyle y}
are positive real numbers and
p
{\displaystyle p}
is an integer greater than 1.
=== Change of base ===
The logarithm logb x can be computed from the logarithms of x and b with respect to an arbitrary base k using the following formula:
log
b
x
=
log
k
x
log
k
b
.
{\displaystyle \log _{b}x={\frac {\log _{k}x}{\log _{k}b}}.}
Typical scientific calculators calculate the logarithms to bases 10 and e. Logarithms with respect to any base b can be determined using either of these two logarithms by the previous formula:
log
b
x
=
log
10
x
log
10
b
=
log
e
x
log
e
b
.
{\displaystyle \log _{b}x={\frac {\log _{10}x}{\log _{10}b}}={\frac {\log _{e}x}{\log _{e}b}}.}
Given a number x and its logarithm y = logb x to an unknown base b, the base is given by:
b
=
x
1
y
,
{\displaystyle b=x^{\frac {1}{y}},}
which can be seen from taking the defining equation
x
=
b
log
b
x
=
b
y
{\displaystyle x=b^{\,\log _{b}x}=b^{y}}
to the power of
1
y
.
{\displaystyle {\tfrac {1}{y}}.}
== Particular bases ==
Among all choices for the base, three are particularly common. These are b = 10, b = e (the irrational mathematical constant e ≈ 2.71828183 ), and b = 2 (the binary logarithm). In mathematical analysis, the logarithm base e is widespread because of analytical properties explained below. On the other hand, base 10 logarithms (the common logarithm) are easy to use for manual calculations in the decimal number system:
log
10
(
10
x
)
=
log
10
10
+
log
10
x
=
1
+
log
10
x
.
{\displaystyle \log _{10}\,(\,10\,x\,)\ =\;\log _{10}10\ +\;\log _{10}x\ =\ 1\,+\,\log _{10}x\,.}
Thus, log10 (x) is related to the number of decimal digits of a positive integer x: The number of digits is the smallest integer strictly bigger than log10 (x) .
For example, log10(5986) is approximately 3.78 . The next integer above it is 4, which is the number of digits of 5986. Both the natural logarithm and the binary logarithm are used in information theory, corresponding to the use of nats or bits as the fundamental units of information, respectively.
Binary logarithms are also used in computer science, where the binary system is ubiquitous; in music theory, where a pitch ratio of two (the octave) is ubiquitous and the number of cents between any two pitches is a scaled version of the binary logarithm, or log 2 times 1200, of the pitch ratio (that is, 100 cents per semitone in conventional equal temperament), or equivalently the log base 21/1200 ; and in photography rescaled base 2 logarithms are used to measure exposure values, light levels, exposure times, lens apertures, and film speeds in "stops".
The abbreviation log x is often used when the intended base can be inferred based on the context or discipline, or when the base is indeterminate or immaterial. Common logarithms (base 10), historically used in logarithm tables and slide rules, are a basic tool for measurement and computation in many areas of science and engineering; in these contexts log x still often means the base ten logarithm. In mathematics log x usually refers to the natural logarithm (base e).
In computer science and information theory, log often refers to binary logarithms (base 2). The following table lists common notations for logarithms to these bases. The "ISO notation" column lists designations suggested by the International Organization for Standardization.
== History ==
The history of logarithms in seventeenth-century Europe saw the discovery of a new function that extended the realm of analysis beyond the scope of algebraic methods. The method of logarithms was publicly propounded by John Napier in 1614, in a book titled Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Canon of Logarithms). Prior to Napier's invention, there had been other techniques of similar scopes, such as the prosthaphaeresis or the use of tables of progressions, extensively developed by Jost Bürgi around 1600. Napier coined the term for logarithm in Middle Latin, logarithmus, literally meaning 'ratio-number', derived from the Greek logos 'proportion, ratio, word' + arithmos 'number'.
The common logarithm of a number is the index of that power of ten which equals the number. Speaking of a number as requiring so many figures is a rough allusion to common logarithm, and was referred to by Archimedes as the "order of a number". The first real logarithms were heuristic methods to turn multiplication into addition, thus facilitating rapid computation. Some of these methods used tables derived from trigonometric identities. Such methods are called prosthaphaeresis.
Invention of the function now known as the natural logarithm began as an attempt to perform a quadrature of a rectangular hyperbola by Grégoire de Saint-Vincent, a Belgian Jesuit residing in Prague. Archimedes had written The Quadrature of the Parabola in the third century BC, but a quadrature for the hyperbola eluded all efforts until Saint-Vincent published his results in 1647. The relation that the logarithm provides between a geometric progression in its argument and an arithmetic progression of values, prompted A. A. de Sarasa to make the connection of Saint-Vincent's quadrature and the tradition of logarithms in prosthaphaeresis, leading to the term "hyperbolic logarithm", a synonym for natural logarithm. Soon the new function was appreciated by Christiaan Huygens, and James Gregory. The notation Log y was adopted by Gottfried Wilhelm Leibniz in 1675, and the next year he connected it to the integral
∫
d
y
y
.
{\textstyle \int {\frac {dy}{y}}.}
Before Euler developed his modern conception of complex natural logarithms, Roger Cotes had a nearly equivalent result when he showed in 1714 that
log
(
cos
θ
+
i
sin
θ
)
=
i
θ
.
{\displaystyle \log(\cos \theta +i\sin \theta )=i\theta .}
== Logarithm tables, slide rules, and historical applications ==
By simplifying difficult calculations before calculators and computers became available, logarithms contributed to the advance of science, especially astronomy. They were critical to advances in surveying, celestial navigation, and other domains. Pierre-Simon Laplace called logarithms
"...[a]n admirable artifice which, by reducing to a few days the labour of many months, doubles the life of the astronomer, and spares him the errors and disgust inseparable from long calculations."
As the function f(x) = bx is the inverse function of logb x, it has been called an antilogarithm. Nowadays, this function is more commonly called an exponential function.
=== Log tables ===
A key tool that enabled the practical use of logarithms was the table of logarithms. The first such table was compiled by Henry Briggs in 1617, immediately after Napier's invention but with the innovation of using 10 as the base. Briggs' first table contained the common logarithms of all integers in the range from 1 to 1000, with a precision of 14 digits. Subsequently, tables with increasing scope were written. These tables listed the values of log10 x for any number x in a certain range, at a certain precision. Base-10 logarithms were universally used for computation, hence the name common logarithm, since numbers that differ by factors of 10 have logarithms that differ by integers. The common logarithm of x can be separated into an integer part and a fractional part, known as the characteristic and mantissa. Tables of logarithms need only include the mantissa, as the characteristic can be easily determined by counting digits from the decimal point. The characteristic of 10 · x is one plus the characteristic of x, and their mantissas are the same. Thus using a three-digit log table, the logarithm of 3542 is approximated by
log
10
3542
=
log
10
(
1000
⋅
3.542
)
=
3
+
log
10
3.542
≈
3
+
log
10
3.54
{\displaystyle {\begin{aligned}\log _{10}3542&=\log _{10}(1000\cdot 3.542)\\&=3+\log _{10}3.542\\&\approx 3+\log _{10}3.54\end{aligned}}}
Greater accuracy can be obtained by interpolation:
log
10
3542
≈
3
+
log
10
3.54
+
0.2
(
log
10
3.55
−
log
10
3.54
)
{\displaystyle \log _{10}3542\approx {}3+\log _{10}3.54+0.2(\log _{10}3.55-\log _{10}3.54)}
The value of 10x can be determined by reverse look up in the same table, since the logarithm is a monotonic function.
=== Computations ===
The product and quotient of two positive numbers c and d were routinely calculated as the sum and difference of their logarithms. The product cd or quotient c/d came from looking up the antilogarithm of the sum or difference, via the same table:
c
d
=
10
log
10
c
10
log
10
d
=
10
log
10
c
+
log
10
d
{\displaystyle cd=10^{\,\log _{10}c}\,10^{\,\log _{10}d}=10^{\,\log _{10}c\,+\,\log _{10}d}}
and
c
d
=
c
d
−
1
=
10
log
10
c
−
log
10
d
.
{\displaystyle {\frac {c}{d}}=cd^{-1}=10^{\,\log _{10}c\,-\,\log _{10}d}.}
For manual calculations that demand any appreciable precision, performing the lookups of the two logarithms, calculating their sum or difference, and looking up the antilogarithm is much faster than performing the multiplication by earlier methods such as prosthaphaeresis, which relies on trigonometric identities.
Calculations of powers and roots are reduced to multiplications or divisions and lookups by
c
d
=
(
10
log
10
c
)
d
=
10
d
log
10
c
{\displaystyle c^{d}=\left(10^{\,\log _{10}c}\right)^{d}=10^{\,d\log _{10}c}}
and
c
d
=
c
1
d
=
10
1
d
log
10
c
.
{\displaystyle {\sqrt[{d}]{c}}=c^{\frac {1}{d}}=10^{{\frac {1}{d}}\log _{10}c}.}
Trigonometric calculations were facilitated by tables that contained the common logarithms of trigonometric functions.
=== Slide rules ===
Another critical application was the slide rule, a pair of logarithmically divided scales used for calculation. The non-sliding logarithmic scale, Gunter's rule, was invented shortly after Napier's invention. William Oughtred enhanced it to create the slide rule—a pair of logarithmic scales movable with respect to each other. Numbers are placed on sliding scales at distances proportional to the differences between their logarithms. Sliding the upper scale appropriately amounts to mechanically adding logarithms, as illustrated here:
For example, adding the distance from 1 to 2 on the lower scale to the distance from 1 to 3 on the upper scale yields a product of 6, which is read off at the lower part. The slide rule was an essential calculating tool for engineers and scientists until the 1970s, because it allows, at the expense of precision, much faster computation than techniques based on tables.
== Analytic properties ==
A deeper study of logarithms requires the concept of a function. A function is a rule that, given one number, produces another number. An example is the function producing the x-th power of b from any real number x, where the base b is a fixed number. This function is written as f(x) = b x. When b is positive and unequal to 1, we show below that f is invertible when considered as a function from the reals to the positive reals.
=== Existence ===
Let b be a positive real number not equal to 1 and let f(x) = b x.
It is a standard result in real analysis that any continuous strictly monotonic function is bijective between its domain and range. This fact follows from the intermediate value theorem. Now, f is strictly increasing (for b > 1), or strictly decreasing (for 0 < b < 1), is continuous, has domain
R
{\displaystyle \mathbb {R} }
, and has range
R
>
0
{\displaystyle \mathbb {R} _{>0}}
. Therefore, f is a bijection from
R
{\displaystyle \mathbb {R} }
to
R
>
0
{\displaystyle \mathbb {R} _{>0}}
. In other words, for each positive real number y, there is exactly one real number x such that
b
x
=
y
{\displaystyle b^{x}=y}
.
We let
log
b
:
R
>
0
→
R
{\displaystyle \log _{b}\colon \mathbb {R} _{>0}\to \mathbb {R} }
denote the inverse of f. That is, logb y is the unique real number x such that
b
x
=
y
{\displaystyle b^{x}=y}
. This function is called the base-b logarithm function or logarithmic function (or just logarithm).
=== Characterization by the product formula ===
The function logb x can also be essentially characterized by the product formula
log
b
(
x
y
)
=
log
b
x
+
log
b
y
.
{\displaystyle \log _{b}(xy)=\log _{b}x+\log _{b}y.}
More precisely, the logarithm to any base b > 1 is the only increasing function f from the positive reals to the reals satisfying f(b) = 1 and
f
(
x
y
)
=
f
(
x
)
+
f
(
y
)
.
{\displaystyle f(xy)=f(x)+f(y).}
=== Graph of the logarithm function ===
As discussed above, the function logb is the inverse to the exponential function
x
↦
b
x
{\displaystyle x\mapsto b^{x}}
. Therefore, their graphs correspond to each other upon exchanging the x- and the y-coordinates (or upon reflection at the diagonal line x = y), as shown at the right: a point (t, u = bt) on the graph of f yields a point (u, t = logb u) on the graph of the logarithm and vice versa. As a consequence, logb (x) diverges to infinity (gets bigger than any given number) if x grows to infinity, provided that b is greater than one. In that case, logb(x) is an increasing function. For b < 1, logb (x) tends to minus infinity instead. When x approaches zero, logb x goes to minus infinity for b > 1 (plus infinity for b < 1, respectively).
=== Derivative and antiderivative ===
Analytic properties of functions pass to their inverses. Thus, as f(x) = bx is a continuous and differentiable function, so is logb y. Roughly, a continuous function is differentiable if its graph has no sharp "corners". Moreover, as the derivative of f(x) evaluates to ln(b) bx by the properties of the exponential function, the chain rule implies that the derivative of logb x is given by
d
d
x
log
b
x
=
1
x
ln
b
.
{\displaystyle {\frac {d}{dx}}\log _{b}x={\frac {1}{x\ln b}}.}
That is, the slope of the tangent touching the graph of the base-b logarithm at the point (x, logb (x)) equals 1/(x ln(b)).
The derivative of ln(x) is 1/x; this implies that ln(x) is the unique antiderivative of 1/x that has the value 0 for x = 1. It is this very simple formula that motivated to qualify as "natural" the natural logarithm; this is also one of the main reasons of the importance of the constant e.
The derivative with a generalized functional argument f(x) is
d
d
x
ln
f
(
x
)
=
f
′
(
x
)
f
(
x
)
.
{\displaystyle {\frac {d}{dx}}\ln f(x)={\frac {f'(x)}{f(x)}}.}
The quotient at the right hand side is called the logarithmic derivative of f. Computing f'(x) by means of the derivative of ln(f(x)) is known as logarithmic differentiation. The antiderivative of the natural logarithm ln(x) is:
∫
ln
(
x
)
d
x
=
x
ln
(
x
)
−
x
+
C
.
{\displaystyle \int \ln(x)\,dx=x\ln(x)-x+C.}
Related formulas, such as antiderivatives of logarithms to other bases can be derived from this equation using the change of bases.
=== Integral representation of the natural logarithm ===
The natural logarithm of t can be defined as the definite integral:
ln
t
=
∫
1
t
1
x
d
x
.
{\displaystyle \ln t=\int _{1}^{t}{\frac {1}{x}}\,dx.}
This definition has the advantage that it does not rely on the exponential function or any trigonometric functions; the definition is in terms of an integral of a simple reciprocal. As an integral, ln(t) equals the area between the x-axis and the graph of the function 1/x, ranging from x = 1 to x = t. This is a consequence of the fundamental theorem of calculus and the fact that the derivative of ln(x) is 1/x. Product and power logarithm formulas can be derived from this definition. For example, the product formula ln(tu) = ln(t) + ln(u) is deduced as:
ln
(
t
u
)
=
∫
1
t
u
1
x
d
x
=
(
1
)
∫
1
t
1
x
d
x
+
∫
t
t
u
1
x
d
x
=
(
2
)
ln
(
t
)
+
∫
1
u
1
w
d
w
=
ln
(
t
)
+
ln
(
u
)
.
{\displaystyle {\begin{aligned}\ln(tu)&=\int _{1}^{tu}{\frac {1}{x}}\,dx\\&{\stackrel {(1)}{=}}\int _{1}^{t}{\frac {1}{x}}\,dx+\int _{t}^{tu}{\frac {1}{x}}\,dx\\&{\stackrel {(2)}{=}}\ln(t)+\int _{1}^{u}{\frac {1}{w}}\,dw\\&=\ln(t)+\ln(u).\end{aligned}}}
The equality (1) splits the integral into two parts, while the equality (2) is a change of variable (w = x/t). In the illustration below, the splitting corresponds to dividing the area into the yellow and blue parts. Rescaling the left hand blue area vertically by the factor t and shrinking it by the same factor horizontally does not change its size. Moving it appropriately, the area fits the graph of the function f(x) = 1/x again. Therefore, the left hand blue area, which is the integral of f(x) from t to tu is the same as the integral from 1 to u. This justifies the equality (2) with a more geometric proof.
The power formula ln(tr) = r ln(t) may be derived in a similar way:
ln
(
t
r
)
=
∫
1
t
r
1
x
d
x
=
∫
1
t
1
w
r
(
r
w
r
−
1
d
w
)
=
r
∫
1
t
1
w
d
w
=
r
ln
(
t
)
.
{\displaystyle {\begin{aligned}\ln(t^{r})&=\int _{1}^{t^{r}}{\frac {1}{x}}dx\\&=\int _{1}^{t}{\frac {1}{w^{r}}}\left(rw^{r-1}\,dw\right)\\&=r\int _{1}^{t}{\frac {1}{w}}\,dw\\&=r\ln(t).\end{aligned}}}
The second equality uses a change of variables (integration by substitution), w = x1/r.
The sum over the reciprocals of natural numbers,
1
+
1
2
+
1
3
+
⋯
+
1
n
=
∑
k
=
1
n
1
k
,
{\displaystyle 1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}=\sum _{k=1}^{n}{\frac {1}{k}},}
is called the harmonic series. It is closely tied to the natural logarithm: as n tends to infinity, the difference,
∑
k
=
1
n
1
k
−
ln
(
n
)
,
{\displaystyle \sum _{k=1}^{n}{\frac {1}{k}}-\ln(n),}
converges (i.e. gets arbitrarily close) to a number known as the Euler–Mascheroni constant γ = 0.5772.... This relation aids in analyzing the performance of algorithms such as quicksort.
=== Transcendence of the logarithm ===
Real numbers that are not algebraic are called transcendental; for example, π and e are such numbers, but
2
−
3
{\displaystyle {\sqrt {2-{\sqrt {3}}}}}
is not. Almost all real numbers are transcendental. The logarithm is an example of a transcendental function. The Gelfond–Schneider theorem asserts that logarithms usually take transcendental, i.e. "difficult" values.
== Calculation ==
Logarithms are easy to compute in some cases, such as log10 (1000) = 3. In general, logarithms can be calculated using power series or the arithmetic–geometric mean, or be retrieved from a precalculated logarithm table that provides a fixed precision. Newton's method, an iterative method to solve equations approximately, can also be used to calculate the logarithm, because its inverse function, the exponential function, can be computed efficiently. Using look-up tables, CORDIC-like methods can be used to compute logarithms by using only the operations of addition and bit shifts. Moreover, the binary logarithm algorithm calculates lb(x) recursively, based on repeated squarings of x, taking advantage of the relation
log
2
(
x
2
)
=
2
log
2
|
x
|
.
{\displaystyle \log _{2}\left(x^{2}\right)=2\log _{2}|x|.}
=== Power series ===
==== Taylor series ====
For any real number z that satisfies 0 < z ≤ 2, the following formula holds:
ln
(
z
)
=
(
z
−
1
)
1
1
−
(
z
−
1
)
2
2
+
(
z
−
1
)
3
3
−
(
z
−
1
)
4
4
+
⋯
=
∑
k
=
1
∞
(
−
1
)
k
+
1
(
z
−
1
)
k
k
.
{\displaystyle {\begin{aligned}\ln(z)&={\frac {(z-1)^{1}}{1}}-{\frac {(z-1)^{2}}{2}}+{\frac {(z-1)^{3}}{3}}-{\frac {(z-1)^{4}}{4}}+\cdots \\&=\sum _{k=1}^{\infty }(-1)^{k+1}{\frac {(z-1)^{k}}{k}}.\end{aligned}}}
Equating the function ln(z) to this infinite sum (series) is shorthand for saying that the function can be approximated to a more and more accurate value by the following expressions (known as partial sums):
(
z
−
1
)
,
(
z
−
1
)
−
(
z
−
1
)
2
2
,
(
z
−
1
)
−
(
z
−
1
)
2
2
+
(
z
−
1
)
3
3
,
…
{\displaystyle (z-1),\ \ (z-1)-{\frac {(z-1)^{2}}{2}},\ \ (z-1)-{\frac {(z-1)^{2}}{2}}+{\frac {(z-1)^{3}}{3}},\ \ldots }
For example, with z = 1.5 the third approximation yields 0.4167, which is about 0.011 greater than ln(1.5) = 0.405465, and the ninth approximation yields 0.40553, which is only about 0.0001 greater. The nth partial sum can approximate ln(z) with arbitrary precision, provided the number of summands n is large enough.
In elementary calculus, the series is said to converge to the function ln(z), and the function is the limit of the series. It is the Taylor series of the natural logarithm at z = 1. The Taylor series of ln(z) provides a particularly useful approximation to ln(1 + z) when z is small, |z| < 1, since then
ln
(
1
+
z
)
=
z
−
z
2
2
+
z
3
3
−
⋯
≈
z
.
{\displaystyle \ln(1+z)=z-{\frac {z^{2}}{2}}+{\frac {z^{3}}{3}}-\cdots \approx z.}
For example, with z = 0.1 the first-order approximation gives ln(1.1) ≈ 0.1, which is less than 5% off the correct value 0.0953.
==== Inverse hyperbolic tangent ====
Another series is based on the inverse hyperbolic tangent function:
ln
(
z
)
=
2
⋅
artanh
z
−
1
z
+
1
=
2
(
z
−
1
z
+
1
+
1
3
(
z
−
1
z
+
1
)
3
+
1
5
(
z
−
1
z
+
1
)
5
+
⋯
)
,
{\displaystyle \ln(z)=2\cdot \operatorname {artanh} \,{\frac {z-1}{z+1}}=2\left({\frac {z-1}{z+1}}+{\frac {1}{3}}{\left({\frac {z-1}{z+1}}\right)}^{3}+{\frac {1}{5}}{\left({\frac {z-1}{z+1}}\right)}^{5}+\cdots \right),}
for any real number z > 0. Using sigma notation, this is also written as
ln
(
z
)
=
2
∑
k
=
0
∞
1
2
k
+
1
(
z
−
1
z
+
1
)
2
k
+
1
.
{\displaystyle \ln(z)=2\sum _{k=0}^{\infty }{\frac {1}{2k+1}}\left({\frac {z-1}{z+1}}\right)^{2k+1}.}
This series can be derived from the above Taylor series. It converges quicker than the Taylor series, especially if z is close to 1. For example, for z = 1.5, the first three terms of the second series approximate ln(1.5) with an error of about 3×10−6. The quick convergence for z close to 1 can be taken advantage of in the following way: given a low-accuracy approximation y ≈ ln(z) and putting
A
=
z
exp
(
y
)
,
{\displaystyle A={\frac {z}{\exp(y)}},}
the logarithm of z is:
ln
(
z
)
=
y
+
ln
(
A
)
.
{\displaystyle \ln(z)=y+\ln(A).}
The better the initial approximation y is, the closer A is to 1, so its logarithm can be calculated efficiently. A can be calculated using the exponential series, which converges quickly provided y is not too large. Calculating the logarithm of larger z can be reduced to smaller values of z by writing z = a · 10b, so that ln(z) = ln(a) + b · ln(10).
A closely related method can be used to compute the logarithm of integers. Putting
z
=
n
+
1
n
{\displaystyle \textstyle z={\frac {n+1}{n}}}
in the above series, it follows that:
ln
(
n
+
1
)
=
ln
(
n
)
+
2
∑
k
=
0
∞
1
2
k
+
1
(
1
2
n
+
1
)
2
k
+
1
.
{\displaystyle \ln(n+1)=\ln(n)+2\sum _{k=0}^{\infty }{\frac {1}{2k+1}}\left({\frac {1}{2n+1}}\right)^{2k+1}.}
If the logarithm of a large integer n is known, then this series yields a fast converging series for log(n+1), with a rate of convergence of
(
1
2
n
+
1
)
2
{\textstyle \left({\frac {1}{2n+1}}\right)^{2}}
.
=== Arithmetic–geometric mean approximation ===
The arithmetic–geometric mean yields high-precision approximations of the natural logarithm. Sasaki and Kanada showed in 1982 that it was particularly fast for precisions between 400 and 1000 decimal places, while Taylor series methods were typically faster when less precision was needed. In their work ln(x) is approximated to a precision of 2−p (or p precise bits) by the following formula (due to Carl Friedrich Gauss):
ln
(
x
)
≈
π
2
M
(
1
,
2
2
−
m
/
x
)
−
m
ln
(
2
)
.
{\displaystyle \ln(x)\approx {\frac {\pi }{2\,\mathrm {M} \!\left(1,2^{2-m}/x\right)}}-m\ln(2).}
Here M(x, y) denotes the arithmetic–geometric mean of x and y. It is obtained by repeatedly calculating the average (x + y)/2 (arithmetic mean) and
x
y
{\textstyle {\sqrt {xy}}}
(geometric mean) of x and y then let those two numbers become the next x and y. The two numbers quickly converge to a common limit which is the value of M(x, y). m is chosen such that
x
2
m
>
2
p
/
2
.
{\displaystyle x\,2^{m}>2^{p/2}.\,}
to ensure the required precision. A larger m makes the M(x, y) calculation take more steps (the initial x and y are farther apart so it takes more steps to converge) but gives more precision. The constants π and ln(2) can be calculated with quickly converging series.
=== Feynman's algorithm ===
While at Los Alamos National Laboratory working on the Manhattan Project, Richard Feynman developed a bit-processing algorithm to compute the logarithm that is similar to long division and was later used in the Connection Machine. The algorithm relies on the fact that every real number x where 1 < x < 2 can be represented as a product of distinct factors of the form 1 + 2−k. The algorithm sequentially builds that product P, starting with P = 1 and k = 1: if P · (1 + 2−k) < x, then it changes P to P · (1 + 2−k). It then increases
k
{\displaystyle k}
by one regardless. The algorithm stops when k is large enough to give the desired accuracy. Because log(x) is the sum of the terms of the form log(1 + 2−k) corresponding to those k for which the factor 1 + 2−k was included in the product P, log(x) may be computed by simple addition, using a table of log(1 + 2−k) for all k. Any base may be used for the logarithm table.
== Applications ==
Logarithms have many applications inside and outside mathematics. Some of these occurrences are related to the notion of scale invariance. For example, each chamber of the shell of a nautilus is an approximate copy of the next one, scaled by a constant factor. This gives rise to a logarithmic spiral. Benford's law on the distribution of leading digits can also be explained by scale invariance. Logarithms are also linked to self-similarity. For example, logarithms appear in the analysis of algorithms that solve a problem by dividing it into two similar smaller problems and patching their solutions. The dimensions of self-similar geometric shapes, that is, shapes whose parts resemble the overall picture are also based on logarithms. Logarithmic scales are useful for quantifying the relative change of a value as opposed to its absolute difference. Moreover, because the logarithmic function log(x) grows very slowly for large x, logarithmic scales are used to compress large-scale scientific data. Logarithms also occur in numerous scientific formulas, such as the Tsiolkovsky rocket equation, the Fenske equation, or the Nernst equation.
=== Logarithmic scale ===
Scientific quantities are often expressed as logarithms of other quantities, using a logarithmic scale. For example, the decibel is a unit of measurement associated with logarithmic-scale quantities. It is based on the common logarithm of ratios—10 times the common logarithm of a power ratio or 20 times the common logarithm of a voltage ratio. It is used to quantify the attenuation or amplification of electrical signals, to describe power levels of sounds in acoustics, and the absorbance of light in the fields of spectrometry and optics. The signal-to-noise ratio describing the amount of unwanted noise in relation to a (meaningful) signal is also measured in decibels. In a similar vein, the peak signal-to-noise ratio is commonly used to assess the quality of sound and image compression methods using the logarithm.
The strength of an earthquake is measured by taking the common logarithm of the energy emitted at the quake. This is used in the moment magnitude scale or the Richter magnitude scale. For example, a 5.0 earthquake releases 32 times (101.5) and a 6.0 releases 1000 times (103) the energy of a 4.0. Apparent magnitude measures the brightness of stars logarithmically. In chemistry the negative of the decimal logarithm, the decimal cologarithm, is indicated by the letter p. For instance, pH is the decimal cologarithm of the activity of hydronium ions (the form hydrogen ions H+ take in water). The activity of hydronium ions in neutral water is 10−7 mol·L−1, hence a pH of 7. Vinegar typically has a pH of about 3. The difference of 4 corresponds to a ratio of 104 of the activity, that is, vinegar's hydronium ion activity is about 10−3 mol·L−1.
Semilog (log–linear) graphs use the logarithmic scale concept for visualization: one axis, typically the vertical one, is scaled logarithmically. For example, the chart at the right compresses the steep increase from 1 million to 1 trillion to the same space (on the vertical axis) as the increase from 1 to 1 million. In such graphs, exponential functions of the form f(x) = a · bx appear as straight lines with slope equal to the logarithm of b. Log-log graphs scale both axes logarithmically, which causes functions of the form f(x) = a · xk to be depicted as straight lines with slope equal to the exponent k. This is applied in visualizing and analyzing power laws.
=== Psychology ===
Logarithms occur in several laws describing human perception: Hick's law proposes a logarithmic relation between the time individuals take to choose an alternative and the number of choices they have. Fitts's law predicts that the time required to rapidly move to a target area is a logarithmic function of the ratio between the distance to a target and the size of the target. In psychophysics, the Weber–Fechner law proposes a logarithmic relationship between stimulus and sensation such as the actual vs. the perceived weight of an item a person is carrying. (This "law", however, is less realistic than more recent models, such as Stevens's power law.)
Psychological studies found that individuals with little mathematics education tend to estimate quantities logarithmically, that is, they position a number on an unmarked line according to its logarithm, so that 10 is positioned as close to 100 as 100 is to 1000. Increasing education shifts this to a linear estimate (positioning 1000 10 times as far away) in some circumstances, while logarithms are used when the numbers to be plotted are difficult to plot linearly.
=== Probability theory and statistics ===
Logarithms arise in probability theory: the law of large numbers dictates that, for a fair coin, as the number of coin-tosses increases to infinity, the observed proportion of heads approaches one-half. The fluctuations of this proportion about one-half are described by the law of the iterated logarithm.
Logarithms also occur in log-normal distributions. When the logarithm of a random variable has a normal distribution, the variable is said to have a log-normal distribution. Log-normal distributions are encountered in many fields, wherever a variable is formed as the product of many independent positive random variables, for example in the study of turbulence.
Logarithms are used for maximum-likelihood estimation of parametric statistical models. For such a model, the likelihood function depends on at least one parameter that must be estimated. A maximum of the likelihood function occurs at the same parameter-value as a maximum of the logarithm of the likelihood (the "log likelihood"), because the logarithm is an increasing function. The log-likelihood is easier to maximize, especially for the multiplied likelihoods for independent random variables.
Benford's law describes the occurrence of digits in many data sets, such as heights of buildings. According to Benford's law, the probability that the first decimal-digit of an item in the data sample is d (from 1 to 9) equals log10 (d + 1) − log10 (d), regardless of the unit of measurement. Thus, about 30% of the data can be expected to have 1 as first digit, 18% start with 2, etc. Auditors examine deviations from Benford's law to detect fraudulent accounting.
The logarithm transformation is a type of data transformation used to bring the empirical distribution closer to the assumed one.
=== Computational complexity ===
Analysis of algorithms is a branch of computer science that studies the performance of algorithms (computer programs solving a certain problem). Logarithms are valuable for describing algorithms that divide a problem into smaller ones, and join the solutions of the subproblems.
For example, to find a number in a sorted list, the binary search algorithm checks the middle entry and proceeds with the half before or after the middle entry if the number is still not found. This algorithm requires, on average, log2 (N) comparisons, where N is the list's length. Similarly, the merge sort algorithm sorts an unsorted list by dividing the list into halves and sorting these first before merging the results. Merge sort algorithms typically require a time approximately proportional to N · log(N). The base of the logarithm is not specified here, because the result only changes by a constant factor when another base is used. A constant factor is usually disregarded in the analysis of algorithms under the standard uniform cost model.
A function f(x) is said to grow logarithmically if f(x) is (exactly or approximately) proportional to the logarithm of x. (Biological descriptions of organism growth, however, use this term for an exponential function.) For example, any natural number N can be represented in binary form in no more than log2 N + 1 bits. In other words, the amount of memory needed to store N grows logarithmically with N.
=== Entropy and chaos ===
Entropy is broadly a measure of the disorder of some system. In statistical thermodynamics, the entropy S of some physical system is defined as
S
=
−
k
∑
i
p
i
ln
(
p
i
)
.
{\displaystyle S=-k\sum _{i}p_{i}\ln(p_{i}).\,}
The sum is over all possible states i of the system in question, such as the positions of gas particles in a container. Moreover, pi is the probability that the state i is attained and k is the Boltzmann constant. Similarly, entropy in information theory measures the quantity of information. If a message recipient may expect any one of N possible messages with equal likelihood, then the amount of information conveyed by any one such message is quantified as log2 N bits.
Lyapunov exponents use logarithms to gauge the degree of chaoticity of a dynamical system. For example, for a particle moving on an oval billiard table, even small changes of the initial conditions result in very different paths of the particle. Such systems are chaotic in a deterministic way, because small measurement errors of the initial state predictably lead to largely different final states. At least one Lyapunov exponent of a deterministically chaotic system is positive.
=== Fractals ===
Logarithms occur in definitions of the dimension of fractals. Fractals are geometric objects that are self-similar in the sense that small parts reproduce, at least roughly, the entire global structure. The Sierpinski triangle (pictured) can be covered by three copies of itself, each having sides half the original length. This makes the Hausdorff dimension of this structure ln(3)/ln(2) ≈ 1.58. Another logarithm-based notion of dimension is obtained by counting the number of boxes needed to cover the fractal in question.
=== Music ===
Logarithms are related to musical tones and intervals. In equal temperament tunings, the frequency ratio depends only on the interval between two tones, not on the specific frequency, or pitch, of the individual tones. In the 12-tone equal temperament tuning common in modern Western music, each octave (doubling of frequency) is broken into twelve equally spaced intervals called semitones. For example, if the note A has a frequency of 440 Hz then the note B-flat has a frequency of 466 Hz. The interval between A and B-flat is a semitone, as is the one between B-flat and B (frequency 493 Hz). Accordingly, the frequency ratios agree:
466
440
≈
493
466
≈
1.059
≈
2
12
.
{\displaystyle {\frac {466}{440}}\approx {\frac {493}{466}}\approx 1.059\approx {\sqrt[{12}]{2}}.}
Intervals between arbitrary pitches can be measured in octaves by taking the base-2 logarithm of the frequency ratio, can be measured in equally tempered semitones by taking the base-21/12 logarithm (12 times the base-2 logarithm), or can be measured in cents, hundredths of a semitone, by taking the base-21/1200 logarithm (1200 times the base-2 logarithm). The latter is used for finer encoding, as it is needed for finer measurements or non-equal temperaments.
=== Number theory ===
Natural logarithms are closely linked to counting prime numbers (2, 3, 5, 7, 11, ...), an important topic in number theory. For any integer x, the quantity of prime numbers less than or equal to x is denoted π(x). The prime number theorem asserts that π(x) is approximately given by
x
ln
(
x
)
,
{\displaystyle {\frac {x}{\ln(x)}},}
in the sense that the ratio of π(x) and that fraction approaches 1 when x tends to infinity. As a consequence, the probability that a randomly chosen number between 1 and x is prime is inversely proportional to the number of decimal digits of x. A far better estimate of π(x) is given by the offset logarithmic integral function Li(x), defined by
L
i
(
x
)
=
∫
2
x
1
ln
(
t
)
d
t
.
{\displaystyle \mathrm {Li} (x)=\int _{2}^{x}{\frac {1}{\ln(t)}}\,dt.}
The Riemann hypothesis, one of the oldest open mathematical conjectures, can be stated in terms of comparing π(x) and Li(x). The Erdős–Kac theorem describing the number of distinct prime factors also involves the natural logarithm.
The logarithm of n factorial, n! = 1 · 2 · ... · n, is given by
ln
(
n
!
)
=
ln
(
1
)
+
ln
(
2
)
+
⋯
+
ln
(
n
)
.
{\displaystyle \ln(n!)=\ln(1)+\ln(2)+\cdots +\ln(n).}
This can be used to obtain Stirling's formula, an approximation of n! for large n.
== Generalizations ==
=== Complex logarithm ===
All the complex numbers a that solve the equation
e
a
=
z
{\displaystyle e^{a}=z}
are called complex logarithms of z, when z is (considered as) a complex number. A complex number is commonly represented as z = x + iy, where x and y are real numbers and i is an imaginary unit, the square of which is −1. Such a number can be visualized by a point in the complex plane, as shown at the right. The polar form encodes a non-zero complex number z by its absolute value, that is, the (positive, real) distance r to the origin, and an angle between the real (x) axis Re and the line passing through both the origin and z. This angle is called the argument of z.
The absolute value r of z is given by
r
=
x
2
+
y
2
.
{\displaystyle \textstyle r={\sqrt {x^{2}+y^{2}}}.}
Using the geometrical interpretation of sine and cosine and their periodicity in 2π, any complex number z may be denoted as
z
=
x
+
i
y
=
r
(
cos
φ
+
i
sin
φ
)
=
r
(
cos
(
φ
+
2
k
π
)
+
i
sin
(
φ
+
2
k
π
)
)
,
{\displaystyle {\begin{aligned}z&=x+iy\\&=r(\cos \varphi +i\sin \varphi )\\&=r(\cos(\varphi +2k\pi )+i\sin(\varphi +2k\pi )),\end{aligned}}}
for any integer number k. Evidently the argument of z is not uniquely specified: both φ and φ' = φ + 2kπ are valid arguments of z for all integers k, because adding 2kπ radians or k⋅360° to φ corresponds to "winding" around the origin counter-clock-wise by k turns. The resulting complex number is always z, as illustrated at the right for k = 1. One may select exactly one of the possible arguments of z as the so-called principal argument, denoted Arg(z), with a capital A, by requiring φ to belong to one, conveniently selected turn, e.g. −π < φ ≤ π or 0 ≤ φ < 2π. These regions, where the argument of z is uniquely determined are called branches of the argument function.
Euler's formula connects the trigonometric functions sine and cosine to the complex exponential:
e
i
φ
=
cos
φ
+
i
sin
φ
.
{\displaystyle e^{i\varphi }=\cos \varphi +i\sin \varphi .}
Using this formula, and again the periodicity, the following identities hold:
z
=
r
(
cos
φ
+
i
sin
φ
)
=
r
(
cos
(
φ
+
2
k
π
)
+
i
sin
(
φ
+
2
k
π
)
)
=
r
e
i
(
φ
+
2
k
π
)
=
e
ln
(
r
)
e
i
(
φ
+
2
k
π
)
=
e
ln
(
r
)
+
i
(
φ
+
2
k
π
)
=
e
a
k
,
{\displaystyle {\begin{aligned}z&=r\left(\cos \varphi +i\sin \varphi \right)\\&=r\left(\cos(\varphi +2k\pi )+i\sin(\varphi +2k\pi )\right)\\&=re^{i(\varphi +2k\pi )}\\&=e^{\ln(r)}e^{i(\varphi +2k\pi )}\\&=e^{\ln(r)+i(\varphi +2k\pi )}=e^{a_{k}},\end{aligned}}}
where ln(r) is the unique real natural logarithm, ak denote the complex logarithms of z, and k is an arbitrary integer. Therefore, the complex logarithms of z, which are all those complex values ak for which the ak-th power of e equals z, are the infinitely many values
a
k
=
ln
(
r
)
+
i
(
φ
+
2
k
π
)
,
{\displaystyle a_{k}=\ln(r)+i(\varphi +2k\pi ),}
for arbitrary integers k.
Taking k such that φ + 2kπ is within the defined interval for the principal arguments, then ak is called the principal value of the logarithm, denoted Log(z), again with a capital L. The principal argument of any positive real number x is 0; hence Log(x) is a real number and equals the real (natural) logarithm. However, the above formulas for logarithms of products and powers do not generalize to the principal value of the complex logarithm.
The illustration at the right depicts Log(z), confining the arguments of z to the interval (−π, π]. This way the corresponding branch of the complex logarithm has discontinuities all along the negative real x axis, which can be seen in the jump in the hue there. This discontinuity arises from jumping to the other boundary in the same branch, when crossing a boundary, i.e. not changing to the corresponding k-value of the continuously neighboring branch. Such a locus is called a branch cut. Dropping the range restrictions on the argument makes the relations "argument of z", and consequently the "logarithm of z", multi-valued functions.
=== Inverses of other exponential functions ===
Exponentiation occurs in many areas of mathematics and its inverse function is often referred to as the logarithm. For example, the logarithm of a matrix is the (multi-valued) inverse function of the matrix exponential. Another example is the p-adic logarithm, the inverse function of the p-adic exponential. Both are defined via Taylor series analogous to the real case. In the context of differential geometry, the exponential map maps the tangent space at a point of a manifold to a neighborhood of that point. Its inverse is also called the logarithmic (or log) map.
In the context of finite groups exponentiation is given by repeatedly multiplying one group element b with itself. The discrete logarithm is the integer n solving the equation
b
n
=
x
,
{\displaystyle b^{n}=x,}
where x is an element of the group. Carrying out the exponentiation can be done efficiently, but the discrete logarithm is believed to be very hard to calculate in some groups. This asymmetry has important applications in public key cryptography, such as for example in the Diffie–Hellman key exchange, a routine that allows secure exchanges of cryptographic keys over unsecured information channels. Zech's logarithm is related to the discrete logarithm in the multiplicative group of non-zero elements of a finite field.
Further logarithm-like inverse functions include the double logarithm ln(ln(x)), the super- or hyper-4-logarithm (a slight variation of which is called iterated logarithm in computer science), the Lambert W function, and the logit. They are the inverse functions of the double exponential function, tetration, of f(w) = wew, and of the logistic function, respectively.
=== Related concepts ===
From the perspective of group theory, the identity log(cd) = log(c) + log(d) expresses a group isomorphism between positive reals under multiplication and reals under addition. Logarithmic functions are the only continuous isomorphisms between these groups. By means of that isomorphism, the Haar measure (Lebesgue measure) dx on the reals corresponds to the Haar measure dx/x on the positive reals. The non-negative reals not only have a multiplication, but also have addition, and form a semiring, called the probability semiring; this is in fact a semifield. The logarithm then takes multiplication to addition (log multiplication), and takes addition to log addition (LogSumExp), giving an isomorphism of semirings between the probability semiring and the log semiring.
Logarithmic one-forms df/f appear in complex analysis and algebraic geometry as differential forms with logarithmic poles.
The polylogarithm is the function defined by
Li
s
(
z
)
=
∑
k
=
1
∞
z
k
k
s
.
{\displaystyle \operatorname {Li} _{s}(z)=\sum _{k=1}^{\infty }{z^{k} \over k^{s}}.}
It is related to the natural logarithm by Li1 (z) = −ln(1 − z). Moreover, Lis (1) equals the Riemann zeta function ζ(s).
== See also ==
Decimal exponent (dex)
Exponential function
Index of logarithm articles
== Notes ==
== References ==
== External links ==
Media related to Logarithm at Wikimedia Commons
The dictionary definition of logarithm at Wiktionary
Quotations related to History of logarithms at Wikiquote
A lesson on logarithms can be found on Wikiversity
Weisstein, Eric W., "Logarithm", MathWorld
Khan Academy: Logarithms, free online micro lectures
"Logarithmic function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Colin Byfleet, Educational video on logarithms, retrieved 12 October 2010
Edward Wright, Translation of Napier's work on logarithms, archived from the original on 3 December 2002, retrieved 12 October 2010
Glaisher, James Whitbread Lee (1911), "Logarithm" , in Chisholm, Hugh (ed.), Encyclopædia Britannica, vol. 16 (11th ed.), Cambridge University Press, pp. 868–77 | Wikipedia/Logarithm_function |
The Abel equation, named after Niels Henrik Abel, is a type of functional equation of the form
f
(
h
(
x
)
)
=
h
(
x
+
1
)
{\displaystyle f(h(x))=h(x+1)}
or
α
(
f
(
x
)
)
=
α
(
x
)
+
1
{\displaystyle \alpha (f(x))=\alpha (x)+1}
.
The forms are equivalent when α is invertible. h or α control the iteration of f.
== Equivalence ==
The second equation can be written
α
−
1
(
α
(
f
(
x
)
)
)
=
α
−
1
(
α
(
x
)
+
1
)
.
{\displaystyle \alpha ^{-1}(\alpha (f(x)))=\alpha ^{-1}(\alpha (x)+1)\,.}
Taking x = α−1(y), the equation can be written
f
(
α
−
1
(
y
)
)
=
α
−
1
(
y
+
1
)
.
{\displaystyle f(\alpha ^{-1}(y))=\alpha ^{-1}(y+1)\,.}
For a known function f(x) , a problem is to solve the functional equation for the function α−1 ≡ h, possibly satisfying additional requirements, such as α−1(0) = 1.
The change of variables sα(x) = Ψ(x), for a real parameter s, brings Abel's equation into the celebrated Schröder's equation, Ψ(f(x)) = s Ψ(x) .
The further change F(x) = exp(sα(x)) into Böttcher's equation, F(f(x)) = F(x)s.
The Abel equation is a special case of (and easily generalizes to) the translation equation,
ω
(
ω
(
x
,
u
)
,
v
)
=
ω
(
x
,
u
+
v
)
,
{\displaystyle \omega (\omega (x,u),v)=\omega (x,u+v)~,}
e.g., for
ω
(
x
,
1
)
=
f
(
x
)
{\displaystyle \omega (x,1)=f(x)}
,
ω
(
x
,
u
)
=
α
−
1
(
α
(
x
)
+
u
)
{\displaystyle \omega (x,u)=\alpha ^{-1}(\alpha (x)+u)}
. (Observe ω(x,0) = x.)
The Abel function α(x) further provides the canonical coordinate for Lie advective flows (one parameter Lie groups).
== History ==
Initially, the equation in the more general form
was reported. Even in the case of a single variable, the equation is non-trivial, and admits special analysis.
In the case of a linear transfer function, the solution is expressible compactly.
== Special cases ==
The equation of tetration is a special case of Abel's equation, with f = exp.
In the case of an integer argument, the equation encodes a recurrent procedure, e.g.,
α
(
f
(
f
(
x
)
)
)
=
α
(
x
)
+
2
,
{\displaystyle \alpha (f(f(x)))=\alpha (x)+2~,}
and so on,
α
(
f
n
(
x
)
)
=
α
(
x
)
+
n
.
{\displaystyle \alpha (f_{n}(x))=\alpha (x)+n~.}
== Solutions ==
The Abel equation has at least one solution on
E
{\displaystyle E}
if and only if for all
x
∈
E
{\displaystyle x\in E}
and all
n
∈
N
{\displaystyle n\in \mathbb {N} }
,
f
n
(
x
)
≠
x
{\displaystyle f^{n}(x)\neq x}
, where
f
n
=
f
∘
f
∘
.
.
.
∘
f
{\displaystyle f^{n}=f\circ f\circ ...\circ f}
, is the function f iterated n times.
We have the following existence and uniqueness theorem: Theorem B
Let
h
:
R
→
R
{\displaystyle h:\mathbb {R} \to \mathbb {R} }
be analytic, meaning it has a Taylor expansion. To find: real analytic solutions
α
:
R
→
C
{\displaystyle \alpha :\mathbb {R} \to \mathbb {C} }
of the Abel equation
α
∘
h
=
α
+
1
{\textstyle \alpha \circ h=\alpha +1}
.
=== Existence ===
A real analytic solution
α
{\displaystyle \alpha }
exists if and only if both of the following conditions hold:
h
{\displaystyle h}
has no fixed points, meaning there is no
y
∈
R
{\displaystyle y\in \mathbb {R} }
such that
h
(
y
)
=
y
{\displaystyle h(y)=y}
.
The set of critical points of
h
{\displaystyle h}
, where
h
′
(
y
)
=
0
{\displaystyle h'(y)=0}
, is bounded above if
h
(
y
)
>
y
{\displaystyle h(y)>y}
for all
y
{\displaystyle y}
, or bounded below if
h
(
y
)
<
y
{\displaystyle h(y)<y}
for all
y
{\displaystyle y}
.
=== Uniqueness ===
The solution is essentially unique in the sense that there exists a canonical solution
α
0
{\displaystyle \alpha _{0}}
with the following properties:
The set of critical points of
α
0
{\displaystyle \alpha _{0}}
is bounded above if
h
(
y
)
>
y
{\displaystyle h(y)>y}
for all
y
{\displaystyle y}
, or bounded below if
h
(
y
)
<
y
{\displaystyle h(y)<y}
for all
y
{\displaystyle y}
.
This canonical solution generates all other solutions. Specifically, the set of all real analytic solutions is given by
{
α
0
+
β
∘
α
0
|
β
:
R
→
R
is analytic, with period 1
}
.
{\displaystyle \{\alpha _{0}+\beta \circ \alpha _{0}|\beta :\mathbb {R} \to \mathbb {R} {\text{ is analytic, with period 1}}\}.}
=== Approximate solution ===
Analytic solutions (Fatou coordinates) can be approximated by asymptotic expansion of a function defined by power series in the sectors around a parabolic fixed point. The analytic solution is unique up to a constant.
== See also ==
Functional equation
Schröder's equation
Böttcher's equation
Infinite compositions of analytic functions
Iterated function
Shift operator
Superfunction
== References ==
M. Kuczma, Functional Equations in a Single Variable, Polish Scientific Publishers, Warsaw (1968).
M. Kuczma, Iterative Functional Equations. Vol. 1017. Cambridge University Press, 1990. | Wikipedia/Abel_equation |
Cauchy's functional equation is the functional equation:
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
.
{\displaystyle f(x+y)=f(x)+f(y).\ }
A function
f
{\displaystyle f}
that solves this equation is called an additive function. Over the rational numbers, it can be shown using elementary algebra that there is a single family of solutions, namely
f
:
x
↦
c
x
{\displaystyle f\colon x\mapsto cx}
for any rational constant
c
.
{\displaystyle c.}
Over the real numbers, the family of linear maps
f
:
x
↦
c
x
,
{\displaystyle f:x\mapsto cx,}
now with
c
{\displaystyle c}
an arbitrary real constant, is likewise a family of solutions; however there can exist other solutions not of this form that are extremely complicated. However, any of a number of regularity conditions, some of them quite weak, will preclude the existence of these pathological solutions. For example, an additive function
f
:
R
→
R
{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }
is linear if:
f
{\displaystyle f}
is continuous (Cauchy, 1821). In fact, it suffices for
f
{\displaystyle f}
to be continuous at one point (Darboux, 1875).
f
(
x
)
≥
0
{\displaystyle f(x)\geq 0}
or
f
(
x
)
≤
0
{\displaystyle f(x)\leq 0}
for all
x
≥
0
{\displaystyle x\geq 0}
.
f
{\displaystyle f}
is monotonic on any interval.
f
{\displaystyle f}
is bounded above or below on any interval.
f
{\displaystyle f}
is Lebesgue measurable.
f
(
x
n
+
1
)
=
x
n
f
(
x
)
{\displaystyle f\left(x^{n+1}\right)=x^{n}f(x)}
for all real
x
{\displaystyle x}
and some positive integer
n
{\displaystyle n}
.
The graph of
f
{\displaystyle f}
is not dense in
R
2
{\displaystyle \mathbb {R} ^{2}}
.
On the other hand, if no further conditions are imposed on
f
,
{\displaystyle f,}
then (assuming the axiom of choice) there are infinitely many other functions that satisfy the equation. This was proved in 1905 by Georg Hamel using Hamel bases. Such functions are sometimes called Hamel functions.
The fifth problem on Hilbert's list is a generalisation of this equation. Functions where there exists a real number
c
{\displaystyle c}
such that
f
(
c
x
)
≠
c
f
(
x
)
{\displaystyle f(cx)\neq cf(x)}
are known as Cauchy-Hamel functions and are used in Dehn-Hadwiger invariants which are used in the extension of Hilbert's third problem from 3D to higher dimensions.
This equation is sometimes referred to as Cauchy's additive functional equation to distinguish it from the other functional equations introduced by Cauchy in 1821, the exponential functional equation
f
(
x
+
y
)
=
f
(
x
)
f
(
y
)
,
{\displaystyle f(x+y)=f(x)f(y),}
the logarithmic functional equation
f
(
x
y
)
=
f
(
x
)
+
f
(
y
)
,
{\displaystyle f(xy)=f(x)+f(y),}
and the multiplicative functional equation
f
(
x
y
)
=
f
(
x
)
f
(
y
)
.
{\displaystyle f(xy)=f(x)f(y).}
== Solutions over the rational numbers ==
A simple argument, involving only elementary algebra, demonstrates that the set of additive maps
f
:
V
→
W
{\displaystyle f\colon V\to W}
, where
V
,
W
{\displaystyle V,W}
are vector spaces over an extension field of
Q
{\displaystyle \mathbb {Q} }
, is identical to the set of
Q
{\displaystyle \mathbb {Q} }
-linear maps from
V
{\displaystyle V}
to
W
{\displaystyle W}
.
Theorem: Let
f
:
V
→
W
{\displaystyle f\colon V\to W}
be an additive function. Then
f
{\displaystyle f}
is
Q
{\displaystyle \mathbb {Q} }
-linear.
Proof: We want to prove that any solution
f
:
V
→
W
{\displaystyle f\colon V\to W}
to Cauchy’s functional equation,
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
{\displaystyle f(x+y)=f(x)+f(y)}
, satisfies
f
(
q
v
)
=
q
f
(
v
)
{\displaystyle f(qv)=qf(v)}
for any
q
∈
Q
{\displaystyle q\in \mathbb {Q} }
and
v
∈
V
{\displaystyle v\in V}
. Let
v
∈
V
{\displaystyle v\in V}
.
First note
f
(
0
)
=
f
(
0
+
0
)
=
f
(
0
)
+
f
(
0
)
{\displaystyle f(0)=f(0+0)=f(0)+f(0)}
, hence
f
(
0
)
=
0
{\displaystyle f(0)=0}
, and therewith
0
=
f
(
0
)
=
f
(
v
+
(
−
v
)
)
=
f
(
v
)
+
f
(
−
v
)
{\displaystyle 0=f(0)=f(v+(-v))=f(v)+f(-v)}
from which follows
f
(
−
v
)
=
−
f
(
v
)
{\displaystyle f(-v)=-f(v)}
.
Via induction,
f
(
m
v
)
=
m
f
(
v
)
{\displaystyle f(mv)=mf(v)}
is proved for any
m
∈
N
∪
{
0
}
{\displaystyle m\in \mathbb {N} \cup \{0\}}
.
For any negative integer
m
∈
Z
{\displaystyle m\in \mathbb {Z} }
we know
−
m
∈
N
{\displaystyle -m\in \mathbb {N} }
, therefore
f
(
m
v
)
=
f
(
(
−
m
)
(
−
v
)
)
=
(
−
m
)
f
(
−
v
)
=
(
−
m
)
(
−
f
(
v
)
)
=
m
f
(
v
)
{\displaystyle f(mv)=f((-m)(-v))=(-m)f(-v)=(-m)(-f(v))=mf(v)}
. Thus far we have proved
f
(
m
v
)
=
m
f
(
v
)
{\displaystyle f(mv)=mf(v)}
for any
m
∈
Z
{\displaystyle m\in \mathbb {Z} }
.
Let
n
∈
N
{\displaystyle n\in \mathbb {N} }
, then
f
(
v
)
=
f
(
n
n
−
1
v
)
=
n
f
(
n
−
1
v
)
{\displaystyle f(v)=f(nn^{-1}v)=nf(n^{-1}v)}
and hence
f
(
n
−
1
v
)
=
n
−
1
f
(
v
)
.
{\displaystyle f(n^{-1}v)=n^{-1}f(v).}
Finally, any
q
∈
Q
{\displaystyle q\in \mathbb {Q} }
has a representation
q
=
m
n
{\displaystyle q={\frac {m}{n}}}
with
m
∈
Z
{\displaystyle m\in \mathbb {Z} }
and
n
∈
N
{\displaystyle n\in \mathbb {N} }
, so, putting things together,
f
(
q
v
)
=
f
(
m
n
v
)
=
f
(
1
n
(
m
v
)
)
=
1
n
f
(
m
v
)
=
1
n
m
f
(
v
)
=
q
f
(
v
)
{\displaystyle f(qv)=f\left({\frac {m}{n}}\,v\right)=f\left({\frac {1}{n}}\,(mv)\right)={\frac {1}{n}}\,f(mv)={\frac {1}{n}}\,m\,f(v)=qf(v)}
, q.e.d.
== Properties of nonlinear solutions over the real numbers ==
We prove below that any other solutions must be highly pathological functions.
In particular, it is shown that any other solution must have the property that its graph
{
(
x
,
f
(
x
)
)
|
x
∈
R
}
{\displaystyle \{(x,f(x))\vert x\in \mathbb {R} \}}
is dense in
R
2
,
{\displaystyle \mathbb {R} ^{2},}
that is, that any disk in the plane (however small) contains a point from the graph.
From this it is easy to prove the various conditions given in the introductory paragraph.
== Existence of nonlinear solutions over the real numbers ==
The linearity proof given above also applies to
f
:
α
Q
→
R
,
{\displaystyle f\colon \alpha \mathbb {Q} \to \mathbb {R} ,}
where
α
Q
{\displaystyle \alpha \mathbb {Q} }
is a scaled copy of the rationals. This shows that only linear solutions are permitted when the domain of
f
{\displaystyle f}
is restricted to such sets. Thus, in general, we have
f
(
α
q
)
=
f
(
α
)
q
{\displaystyle f(\alpha q)=f(\alpha )q}
for all
α
∈
R
{\displaystyle \alpha \in \mathbb {R} }
and
q
∈
Q
.
{\displaystyle q\in \mathbb {Q} .}
However, as we will demonstrate below, highly pathological solutions can be found for functions
f
:
R
→
R
{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }
based on these linear solutions, by viewing the reals as a vector space over the field of rational numbers. Note, however, that this method is nonconstructive, relying as it does on the existence of a (Hamel) basis for any vector space, a statement proved using Zorn's lemma. (In fact, the existence of a basis for every vector space is logically equivalent to the axiom of choice.) There exist models such as the Solovay model where all sets of reals are measurable which are consistent with ZF + DC, and therein all solutions are linear.
To show that solutions other than the ones defined by
f
(
x
)
=
f
(
1
)
x
{\displaystyle f(x)=f(1)x}
exist, we first note that because every vector space has a basis, there is a basis for
R
{\displaystyle \mathbb {R} }
over the field
Q
,
{\displaystyle \mathbb {Q} ,}
i.e. a set
B
⊂
R
{\displaystyle {\mathcal {B}}\subset \mathbb {R} }
with the property that any
x
∈
R
{\displaystyle x\in \mathbb {R} }
can be expressed uniquely as
x
=
∑
i
∈
I
λ
i
x
i
,
{\textstyle x=\sum _{i\in I}{\lambda _{i}x_{i}},}
where
{
x
i
}
i
∈
I
{\displaystyle \{x_{i}\}_{i\in I}}
is a finite subset of
B
,
{\displaystyle {\mathcal {B}},}
and each
λ
i
{\displaystyle \lambda _{i}}
is in
Q
.
{\displaystyle \mathbb {Q} .}
We note that because no explicit basis for
R
{\displaystyle \mathbb {R} }
over
Q
{\displaystyle \mathbb {Q} }
can be written down, the pathological solutions defined below likewise cannot be expressed explicitly.
As argued above, the restriction of
f
{\displaystyle f}
to
x
i
Q
{\displaystyle x_{i}\mathbb {Q} }
must be a linear map for each
x
i
∈
B
.
{\displaystyle x_{i}\in {\mathcal {B}}.}
Moreover, because
x
i
q
↦
f
(
x
i
)
q
{\displaystyle x_{i}q\mapsto f(x_{i})q}
for
q
∈
Q
,
{\displaystyle q\in \mathbb {Q} ,}
it is clear that
f
(
x
i
)
x
i
{\displaystyle f(x_{i}) \over x_{i}}
is the constant of proportionality. In other words,
f
:
x
i
Q
→
R
{\displaystyle f\colon x_{i}\mathbb {Q} \to \mathbb {R} }
is the map
ξ
↦
[
f
(
x
i
)
/
x
i
]
ξ
.
{\displaystyle \xi \mapsto [f(x_{i})/x_{i}]\xi .}
Since any
x
∈
R
{\displaystyle x\in \mathbb {R} }
can be expressed as a unique (finite) linear combination of the
x
i
{\displaystyle x_{i}}
s, and
f
:
R
→
R
{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }
is additive,
f
(
x
)
{\displaystyle f(x)}
is well-defined for all
x
∈
R
{\displaystyle x\in \mathbb {R} }
and is given by:
f
(
x
)
=
f
(
∑
i
∈
I
λ
i
x
i
)
=
∑
i
∈
I
f
(
x
i
λ
i
)
=
∑
i
∈
I
f
(
x
i
)
λ
i
.
{\displaystyle f(x)=f{\Big (}\sum _{i\in I}\lambda _{i}x_{i}{\Big )}=\sum _{i\in I}f(x_{i}\lambda _{i})=\sum _{i\in I}f(x_{i})\lambda _{i}.}
It is easy to check that
f
{\displaystyle f}
is a solution to Cauchy's functional equation given a definition of
f
{\displaystyle f}
on the basis elements,
f
:
B
→
R
.
{\displaystyle f\colon {\mathcal {B}}\to \mathbb {R} .}
Moreover, it is clear that every solution is of this form. In particular, the solutions of the functional equation are linear if and only if
f
(
x
i
)
x
i
{\displaystyle f(x_{i}) \over x_{i}}
is constant over all
x
i
∈
B
.
{\displaystyle x_{i}\in {\mathcal {B}}.}
Thus, in a sense, despite the inability to exhibit a nonlinear solution, "most" (in the sense of cardinality) solutions to the Cauchy functional equation are actually nonlinear and pathological.
== See also ==
Antilinear map – Conjugate homogeneous additive map
Homogeneous function – Function with a multiplicative scaling behaviour
Minkowski functional – Function made from a set
Semilinear map – homomorphism between modules, paired with the associated homomorphism between the respective base ringsPages displaying wikidata descriptions as a fallback
== References ==
Kuczma, Marek (2009). An introduction to the theory of functional equations and inequalities. Cauchy's equation and Jensen's inequality. Basel: Birkhäuser. ISBN 9783764387495.
Hamel, Georg (1905). "Eine Basis aller Zahlen und die unstetigen Lösungen der Funktionalgleichung: f(x+y) = f(x) + f(y)". Mathematische Annalen.
== External links ==
Solution to the Cauchy Equation Rutgers University
The Hunt for Addi(c)tive Monster
Martin Sleziak; et al. (2013). "Overview of basic facts about Cauchy functional equation". StackExchange. Retrieved 20 December 2015. | Wikipedia/Cauchy's_functional_equation |
In mathematics, an even function is a real function such that
f
(
−
x
)
=
f
(
x
)
{\displaystyle f(-x)=f(x)}
for every
x
{\displaystyle x}
in its domain. Similarly, an odd function is a function such that
f
(
−
x
)
=
−
f
(
x
)
{\displaystyle f(-x)=-f(x)}
for every
x
{\displaystyle x}
in its domain.
They are named for the parity of the powers of the power functions which satisfy each condition: the function
f
(
x
)
=
x
n
{\displaystyle f(x)=x^{n}}
is even if n is an even integer, and it is odd if n is an odd integer.
Even functions are those real functions whose graph is self-symmetric with respect to the y-axis, and odd functions are those whose graph is self-symmetric with respect to the origin.
If the domain of a real function is self-symmetric with respect to the origin, then the function can be uniquely decomposed as the sum of an even function and an odd function.
== Early history ==
The concept of even and odd functions appears to date back to the early 18th century, with Leonard Euler playing a significant role in their formalization. Euler introduced the concepts of even and odd functions (using Latin terms pares and impares) in his work Traiectoriarum Reciprocarum Solutio from 1727. Before Euler, however, Isaac Newton had already developed geometric means of deriving coefficients of power series when writing the Principia (1687), and included algebraic techniques in an early draft of his Quadrature of Curves, though he removed it before publication in 1706. It is also noteworthy that Newton didn't explicitly name or focus on the even-odd decomposition, his work with power series would have involved understanding properties related to even and odd powers.
== Definition and examples ==
Evenness and oddness are generally considered for real functions, that is real-valued functions of a real variable. However, the concepts may be more generally defined for functions whose domain and codomain both have a notion of additive inverse. This includes abelian groups, all rings, all fields, and all vector spaces. Thus, for example, a real function could be odd or even (or neither), as could a complex-valued function of a vector variable, and so on.
The given examples are real functions, to illustrate the symmetry of their graphs.
=== Even functions ===
A real function f is even if, for every x in its domain, −x is also in its domain and: p. 11
f
(
−
x
)
=
f
(
x
)
{\displaystyle f(-x)=f(x)}
or equivalently
f
(
x
)
−
f
(
−
x
)
=
0.
{\displaystyle f(x)-f(-x)=0.}
Geometrically, the graph of an even function is symmetric with respect to the y-axis, meaning that its graph remains unchanged after reflection about the y-axis.
Examples of even functions are:
The absolute value
x
↦
|
x
|
,
{\displaystyle x\mapsto |x|,}
x
↦
x
2
,
{\displaystyle x\mapsto x^{2},}
x
↦
x
n
{\displaystyle x\mapsto x^{n}}
for any even integer
n
,
{\displaystyle n,}
cosine
cos
,
{\displaystyle \cos ,}
hyperbolic cosine
cosh
,
{\displaystyle \cosh ,}
Gaussian function
x
↦
exp
(
−
x
2
)
.
{\displaystyle x\mapsto \exp(-x^{2}).}
=== Odd functions ===
A real function f is odd if, for every x in its domain, −x is also in its domain and: p. 72
f
(
−
x
)
=
−
f
(
x
)
{\displaystyle f(-x)=-f(x)}
or equivalently
f
(
x
)
+
f
(
−
x
)
=
0.
{\displaystyle f(x)+f(-x)=0.}
Geometrically, the graph of an odd function has rotational symmetry with respect to the origin, meaning that its graph remains unchanged after rotation of 180 degrees about the origin.
If
x
=
0
{\displaystyle x=0}
is in the domain of an odd function
f
(
x
)
{\displaystyle f(x)}
, then
f
(
0
)
=
0
{\displaystyle f(0)=0}
.
Examples of odd functions are:
The sign function
x
↦
sgn
(
x
)
,
{\displaystyle x\mapsto \operatorname {sgn}(x),}
The identity function
x
↦
x
,
{\displaystyle x\mapsto x,}
x
↦
x
n
{\displaystyle x\mapsto x^{n}}
for any odd integer
n
,
{\displaystyle n,}
x
↦
x
n
{\displaystyle x\mapsto {\sqrt[{n}]{x}}}
for any odd positive integer
n
,
{\displaystyle n,}
sine
sin
,
{\displaystyle \sin ,}
hyperbolic sine
sinh
,
{\displaystyle \sinh ,}
The error function
erf
.
{\displaystyle \operatorname {erf} .}
== Basic properties ==
=== Uniqueness ===
If a function is both even and odd, it is equal to 0 everywhere it is defined.
If a function is odd, the absolute value of that function is an even function.
=== Addition and subtraction ===
The sum of two even functions is even.
The sum of two odd functions is odd.
The difference between two odd functions is odd.
The difference between two even functions is even.
The sum of an even and odd function is not even or odd, unless one of the functions is equal to zero over the given domain.
=== Multiplication and division ===
The product and quotient of two even functions is an even function.
This implies that the product of any number of even functions is also even.
This implies that the reciprocal of an even function is also even.
The product and quotient of two odd functions is an even function.
The product and both quotients of an even function and an odd function is an odd function.
This implies that the reciprocal of an odd function is odd.
=== Composition ===
The composition of two even functions is even.
The composition of two odd functions is odd.
The composition of an even function and an odd function is even.
The composition of any function with an even function is even (but not vice versa).
=== Inverse function ===
If an odd function is invertible, then its inverse is also odd.
== Even–odd decomposition ==
If a real function has a domain that is self-symmetric with respect to the origin, it may be uniquely decomposed as the sum of an even and an odd function, which are called respectively the even part (or the even component) and the odd part (or the odd component) of the function, and are defined by
f
even
(
x
)
=
f
(
x
)
+
f
(
−
x
)
2
,
{\displaystyle f_{\text{even}}(x)={\frac {f(x)+f(-x)}{2}},}
and
f
odd
(
x
)
=
f
(
x
)
−
f
(
−
x
)
2
.
{\displaystyle f_{\text{odd}}(x)={\frac {f(x)-f(-x)}{2}}.}
It is straightforward to verify that
f
even
{\displaystyle f_{\text{even}}}
is even,
f
odd
{\displaystyle f_{\text{odd}}}
is odd, and
f
=
f
even
+
f
odd
.
{\displaystyle f=f_{\text{even}}+f_{\text{odd}}.}
This decomposition is unique since, if
f
(
x
)
=
g
(
x
)
+
h
(
x
)
,
{\displaystyle f(x)=g(x)+h(x),}
where g is even and h is odd, then
g
=
f
even
{\displaystyle g=f_{\text{even}}}
and
h
=
f
odd
,
{\displaystyle h=f_{\text{odd}},}
since
2
f
e
(
x
)
=
f
(
x
)
+
f
(
−
x
)
=
g
(
x
)
+
g
(
−
x
)
+
h
(
x
)
+
h
(
−
x
)
=
2
g
(
x
)
,
2
f
o
(
x
)
=
f
(
x
)
−
f
(
−
x
)
=
g
(
x
)
−
g
(
−
x
)
+
h
(
x
)
−
h
(
−
x
)
=
2
h
(
x
)
.
{\displaystyle {\begin{aligned}2f_{\text{e}}(x)&=f(x)+f(-x)=g(x)+g(-x)+h(x)+h(-x)=2g(x),\\2f_{\text{o}}(x)&=f(x)-f(-x)=g(x)-g(-x)+h(x)-h(-x)=2h(x).\end{aligned}}}
For example, the hyperbolic cosine and the hyperbolic sine may be regarded as the even and odd parts of the exponential function, as the first one is an even function, the second one is odd, and
e
x
=
cosh
(
x
)
⏟
f
even
(
x
)
+
sinh
(
x
)
⏟
f
odd
(
x
)
{\displaystyle e^{x}=\underbrace {\cosh(x)} _{f_{\text{even}}(x)}+\underbrace {\sinh(x)} _{f_{\text{odd}}(x)}}
.
Fourier's sine and cosine transforms also perform even–odd decomposition by representing a function's odd part with sine waves (an odd function) and the function's even part with cosine waves (an even function).
== Further algebraic properties ==
Any linear combination of even functions is even, and the even functions form a vector space over the reals. Similarly, any linear combination of odd functions is odd, and the odd functions also form a vector space over the reals. In fact, the vector space of all real functions is the direct sum of the subspaces of even and odd functions. This is a more abstract way of expressing the property in the preceding section.
The space of functions can be considered a graded algebra over the real numbers by this property, as well as some of those above.
The even functions form a commutative algebra over the reals. However, the odd functions do not form an algebra over the reals, as they are not closed under multiplication.
== Analytic properties ==
A function's being odd or even does not imply differentiability, or even continuity. For example, the Dirichlet function is even, but is nowhere continuous.
In the following, properties involving derivatives, Fourier series, Taylor series are considered, and these concepts are thus supposed to be defined for the considered functions.
=== Basic analytic properties ===
The derivative of an even function is odd.
The derivative of an odd function is even.
If an odd function is integrable over a bounded symmetric interval
[
−
A
,
A
]
{\displaystyle [-A,A]}
, the integral over that interval is zero; that is
∫
−
A
A
f
(
x
)
d
x
=
0
{\displaystyle \int _{-A}^{A}f(x)\,dx=0}
.
This implies that the Cauchy principal value of an odd function over the entire real line is zero.
If an even function is integrable over a bounded symmetric interval
[
−
A
,
A
]
{\displaystyle [-A,A]}
, the integral over that interval is twice the integral from 0 to A; that is
∫
−
A
A
f
(
x
)
d
x
=
2
∫
0
A
f
(
x
)
d
x
{\displaystyle \int _{-A}^{A}f(x)\,dx=2\int _{0}^{A}f(x)\,dx}
.
This property is also true for the improper integral when
A
=
∞
{\displaystyle A=\infty }
, provided the integral from 0 to
∞
{\displaystyle \infty }
converges.
=== Series ===
The Maclaurin series of an even function includes only even powers.
The Maclaurin series of an odd function includes only odd powers.
The Fourier series of a periodic even function includes only cosine terms.
The Fourier series of a periodic odd function includes only sine terms.
The Fourier transform of a purely real-valued even function is real and even. (see Fourier analysis § Symmetry properties)
The Fourier transform of a purely real-valued odd function is imaginary and odd. (see Fourier analysis § Symmetry properties)
== Harmonics ==
In signal processing, harmonic distortion occurs when a sine wave signal is sent through a memory-less nonlinear system, that is, a system whose output at time t only depends on the input at time t and does not depend on the input at any previous times. Such a system is described by a response function
V
out
(
t
)
=
f
(
V
in
(
t
)
)
{\displaystyle V_{\text{out}}(t)=f(V_{\text{in}}(t))}
. The type of harmonics produced depend on the response function f:
When the response function is even, the resulting signal will consist of only even harmonics of the input sine wave;
0
f
,
2
f
,
4
f
,
6
f
,
…
{\displaystyle 0f,2f,4f,6f,\dots }
The fundamental is also an odd harmonic, so will not be present.
A simple example is a full-wave rectifier.
The
0
f
{\displaystyle 0f}
component represents the DC offset, due to the one-sided nature of even-symmetric transfer functions.
When it is odd, the resulting signal will consist of only odd harmonics of the input sine wave;
1
f
,
3
f
,
5
f
,
…
{\displaystyle 1f,3f,5f,\dots }
The output signal will be half-wave symmetric.
A simple example is clipping in a symmetric push-pull amplifier.
When it is asymmetric, the resulting signal may contain either even or odd harmonics;
1
f
,
2
f
,
3
f
,
…
{\displaystyle 1f,2f,3f,\dots }
Simple examples are a half-wave rectifier, and clipping in an asymmetrical class-A amplifier.
This does not hold true for more complex waveforms. A sawtooth wave contains both even and odd harmonics, for instance. After even-symmetric full-wave rectification, it becomes a triangle wave, which, other than the DC offset, contains only odd harmonics.
== Generalizations ==
=== Multivariate functions ===
Even symmetry:
A function
f
:
R
n
→
R
{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }
is called even symmetric if:
f
(
x
1
,
x
2
,
…
,
x
n
)
=
f
(
−
x
1
,
−
x
2
,
…
,
−
x
n
)
for all
x
1
,
…
,
x
n
∈
R
{\displaystyle f(x_{1},x_{2},\ldots ,x_{n})=f(-x_{1},-x_{2},\ldots ,-x_{n})\quad {\text{for all }}x_{1},\ldots ,x_{n}\in \mathbb {R} }
Odd symmetry:
A function
f
:
R
n
→
R
{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }
is called odd symmetric if:
f
(
x
1
,
x
2
,
…
,
x
n
)
=
−
f
(
−
x
1
,
−
x
2
,
…
,
−
x
n
)
for all
x
1
,
…
,
x
n
∈
R
{\displaystyle f(x_{1},x_{2},\ldots ,x_{n})=-f(-x_{1},-x_{2},\ldots ,-x_{n})\quad {\text{for all }}x_{1},\ldots ,x_{n}\in \mathbb {R} }
=== Complex-valued functions ===
The definitions for even and odd symmetry for complex-valued functions of a real argument are similar to the real case. In signal processing, a similar symmetry is sometimes considered, which involves complex conjugation.
Conjugate symmetry:
A complex-valued function of a real argument
f
:
R
→
C
{\displaystyle f:\mathbb {R} \to \mathbb {C} }
is called conjugate symmetric if
f
(
x
)
=
f
(
−
x
)
¯
for all
x
∈
R
{\displaystyle f(x)={\overline {f(-x)}}\quad {\text{for all }}x\in \mathbb {R} }
A complex valued function is conjugate symmetric if and only if its real part is an even function and its imaginary part is an odd function.
A typical example of a conjugate symmetric function is the cis function
x
→
e
i
x
=
cos
x
+
i
sin
x
{\displaystyle x\to e^{ix}=\cos x+i\sin x}
Conjugate antisymmetry:
A complex-valued function of a real argument
f
:
R
→
C
{\displaystyle f:\mathbb {R} \to \mathbb {C} }
is called conjugate antisymmetric if:
f
(
x
)
=
−
f
(
−
x
)
¯
for all
x
∈
R
{\displaystyle f(x)=-{\overline {f(-x)}}\quad {\text{for all }}x\in \mathbb {R} }
A complex valued function is conjugate antisymmetric if and only if its real part is an odd function and its imaginary part is an even function.
=== Finite length sequences ===
The definitions of odd and even symmetry are extended to N-point sequences (i.e. functions of the form
f
:
{
0
,
1
,
…
,
N
−
1
}
→
R
{\displaystyle f:\left\{0,1,\ldots ,N-1\right\}\to \mathbb {R} }
) as follows:: p. 411
Even symmetry:
A N-point sequence is called conjugate symmetric if
f
(
n
)
=
f
(
N
−
n
)
for all
n
∈
{
1
,
…
,
N
−
1
}
.
{\displaystyle f(n)=f(N-n)\quad {\text{for all }}n\in \left\{1,\ldots ,N-1\right\}.}
Such a sequence is often called a palindromic sequence; see also Palindromic polynomial.
Odd symmetry:
A N-point sequence is called conjugate antisymmetric if
f
(
n
)
=
−
f
(
N
−
n
)
for all
n
∈
{
1
,
…
,
N
−
1
}
.
{\displaystyle f(n)=-f(N-n)\quad {\text{for all }}n\in \left\{1,\ldots ,N-1\right\}.}
Such a sequence is sometimes called an anti-palindromic sequence; see also Antipalindromic polynomial.
== See also ==
Hermitian function for a generalization in complex numbers
Taylor series
Fourier series
Holstein–Herring method
Parity (physics)
== Notes ==
== References ==
Gelfand, I. M.; Glagoleva, E. G.; Shnol, E. E. (2002) [1969], Functions and Graphs, Mineola, N.Y: Dover Publications | Wikipedia/Even_function |
In physics, Maxwell's equations in curved spacetime govern the dynamics of the electromagnetic field in curved spacetime (where the metric may not be the Minkowski metric) or where one uses an arbitrary (not necessarily Cartesian) coordinate system. These equations can be viewed as a generalization of the vacuum Maxwell's equations which are normally formulated in the local coordinates of flat spacetime. But because general relativity dictates that the presence of electromagnetic fields (or energy/matter in general) induce curvature in spacetime, Maxwell's equations in flat spacetime should be viewed as a convenient approximation.
When working in the presence of bulk matter, distinguishing between free and bound electric charges may facilitate analysis. When the distinction is made, they are called the macroscopic Maxwell's equations. Without this distinction, they are sometimes called the "microscopic" Maxwell's equations for contrast.
The electromagnetic field admits a coordinate-independent geometric description, and Maxwell's equations expressed in terms of these geometric objects are the same in any spacetime, curved or not. Also, the same modifications are made to the equations of flat Minkowski space when using local coordinates that are not rectilinear. For example, the equations in this article can be used to write Maxwell's equations in spherical coordinates. For these reasons, it may be useful to think of Maxwell's equations in Minkowski space as a special case of the general formulation.
== Summary ==
In general relativity, the metric tensor
g
α
β
{\displaystyle g_{\alpha \beta }}
is no longer a constant (like
η
α
β
{\displaystyle \eta _{\alpha \beta }}
as in Examples of metric tensor) but can vary in space and time, and the equations of electromagnetism in vacuum become
F
α
β
=
∂
α
A
β
−
∂
β
A
α
,
D
μ
ν
=
1
μ
0
g
μ
α
F
α
β
g
β
ν
−
g
c
,
J
μ
=
∂
ν
D
μ
ν
,
f
μ
=
F
μ
ν
J
ν
,
{\displaystyle {\begin{aligned}F_{\alpha \beta }&=\partial _{\alpha }A_{\beta }-\partial _{\beta }A_{\alpha },\\{\mathcal {D}}^{\mu \nu }&={\frac {1}{\mu _{0}}}\,g^{\mu \alpha }\,F_{\alpha \beta }\,g^{\beta \nu }\,{\frac {\sqrt {-g}}{c}},\\J^{\mu }&=\partial _{\nu }{\mathcal {D}}^{\mu \nu },\\f_{\mu }&=F_{\mu \nu }\,J^{\nu },\end{aligned}}}
where
f
μ
{\displaystyle f_{\mu }}
is the density of the Lorentz force,
g
α
β
{\displaystyle g^{\alpha \beta }}
is the inverse of the metric tensor
g
α
β
{\displaystyle g_{\alpha \beta }}
, and
g
{\displaystyle g}
is the determinant of the metric tensor. Notice that
A
α
{\displaystyle A_{\alpha }}
and
F
α
β
{\displaystyle F_{\alpha \beta }}
are (ordinary) tensors, while
D
μ
ν
{\displaystyle {\mathcal {D}}^{\mu \nu }}
,
J
ν
{\displaystyle J^{\nu }}
, and
f
μ
{\displaystyle f_{\mu }}
are tensor densities of weight +1. Despite the use of partial derivatives, these equations are invariant under arbitrary curvilinear coordinate transformations. Thus, if one replaced the partial derivatives with covariant derivatives, the extra terms thereby introduced would cancel out (see Manifest covariance § Example).
== Electromagnetic potential ==
The electromagnetic potential is a covariant vector Aα, which is the undefined primitive of electromagnetism. Being a covariant vector, its components transform from one coordinate system to another according to
A
¯
β
(
x
¯
)
=
∂
x
γ
∂
x
¯
β
A
γ
(
x
)
.
{\displaystyle {\bar {A}}_{\beta }({\bar {x}})={\frac {\partial x^{\gamma }}{\partial {\bar {x}}^{\beta }}}A_{\gamma }(x).}
== Electromagnetic field ==
The electromagnetic field is a covariant antisymmetric tensor of degree 2, which can be defined in terms of the electromagnetic potential by
F
α
β
=
∂
α
A
β
−
∂
β
A
α
.
{\displaystyle F_{\alpha \beta }=\partial _{\alpha }A_{\beta }-\partial _{\beta }A_{\alpha }.}
To see that this equation is invariant, we transform the coordinates as described in the classical treatment of tensors:
F
¯
α
β
=
∂
A
¯
β
∂
x
¯
α
−
∂
A
¯
α
∂
x
¯
β
=
∂
∂
x
¯
α
(
∂
x
γ
∂
x
¯
β
A
γ
)
−
∂
∂
x
¯
β
(
∂
x
δ
∂
x
¯
α
A
δ
)
=
∂
2
x
γ
∂
x
¯
α
∂
x
¯
β
A
γ
+
∂
x
γ
∂
x
¯
β
∂
A
γ
∂
x
¯
α
−
∂
2
x
δ
∂
x
¯
β
∂
x
¯
α
A
δ
−
∂
x
δ
∂
x
¯
α
∂
A
δ
∂
x
¯
β
=
∂
x
γ
∂
x
¯
β
∂
x
δ
∂
x
¯
α
∂
A
γ
∂
x
δ
−
∂
x
δ
∂
x
¯
α
∂
x
γ
∂
x
¯
β
∂
A
δ
∂
x
γ
=
∂
x
δ
∂
x
¯
α
∂
x
γ
∂
x
¯
β
(
∂
A
γ
∂
x
δ
−
∂
A
δ
∂
x
γ
)
=
∂
x
δ
∂
x
¯
α
∂
x
γ
∂
x
¯
β
F
δ
γ
.
{\displaystyle {\begin{aligned}{\bar {F}}_{\alpha \beta }&={\frac {\partial {\bar {A}}_{\beta }}{\partial {\bar {x}}^{\alpha }}}-{\frac {\partial {\bar {A}}_{\alpha }}{\partial {\bar {x}}^{\beta }}}\\&={\frac {\partial }{\partial {\bar {x}}^{\alpha }}}\left({\frac {\partial x^{\gamma }}{\partial {\bar {x}}^{\beta }}}A_{\gamma }\right)-{\frac {\partial }{\partial {\bar {x}}^{\beta }}}\left({\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}A_{\delta }\right)\\&={\frac {\partial ^{2}x^{\gamma }}{\partial {\bar {x}}^{\alpha }\partial {\bar {x}}^{\beta }}}A_{\gamma }+{\frac {\partial x^{\gamma }}{\partial {\bar {x}}^{\beta }}}{\frac {\partial A_{\gamma }}{\partial {\bar {x}}^{\alpha }}}-{\frac {\partial ^{2}x^{\delta }}{\partial {\bar {x}}^{\beta }\partial {\bar {x}}^{\alpha }}}A_{\delta }-{\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}{\frac {\partial A_{\delta }}{\partial {\bar {x}}^{\beta }}}\\&={\frac {\partial x^{\gamma }}{\partial {\bar {x}}^{\beta }}}{\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}{\frac {\partial A_{\gamma }}{\partial x^{\delta }}}-{\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}{\frac {\partial x^{\gamma }}{\partial {\bar {x}}^{\beta }}}{\frac {\partial A_{\delta }}{\partial x^{\gamma }}}\\&={\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}{\frac {\partial x^{\gamma }}{\partial {\bar {x}}^{\beta }}}\left({\frac {\partial A_{\gamma }}{\partial x^{\delta }}}-{\frac {\partial A_{\delta }}{\partial x^{\gamma }}}\right)\\&={\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}{\frac {\partial x^{\gamma }}{\partial {\bar {x}}^{\beta }}}F_{\delta \gamma }.\end{aligned}}}
This definition implies that the electromagnetic field satisfies
∂
λ
F
μ
ν
+
∂
μ
F
ν
λ
+
∂
ν
F
λ
μ
=
0
,
{\displaystyle \partial _{\lambda }F_{\mu \nu }+\partial _{\mu }F_{\nu \lambda }+\partial _{\nu }F_{\lambda \mu }=0,}
which incorporates Faraday's law of induction and Gauss's law for magnetism. This is seen from
∂
λ
F
μ
ν
+
∂
μ
F
ν
λ
+
∂
ν
F
λ
μ
=
∂
λ
∂
μ
A
ν
−
∂
λ
∂
ν
A
μ
+
∂
μ
∂
ν
A
λ
−
∂
μ
∂
λ
A
ν
+
∂
ν
∂
λ
A
μ
−
∂
ν
∂
μ
A
λ
=
0.
{\displaystyle \partial _{\lambda }F_{\mu \nu }+\partial _{\mu }F_{\nu \lambda }+\partial _{\nu }F_{\lambda \mu }=\partial _{\lambda }\partial _{\mu }A_{\nu }-\partial _{\lambda }\partial _{\nu }A_{\mu }+\partial _{\mu }\partial _{\nu }A_{\lambda }-\partial _{\mu }\partial _{\lambda }A_{\nu }+\partial _{\nu }\partial _{\lambda }A_{\mu }-\partial _{\nu }\partial _{\mu }A_{\lambda }=0.}
Thus, the right-hand side of that Maxwell law is zero identically, meaning that the classic EM field theory leaves no room for magnetic monopoles or currents of such to act as sources of the field.
Although there appear to be 64 equations in Faraday–Gauss, it actually reduces to just four independent equations. Using the antisymmetry of the electromagnetic field, one can either reduce to an identity (0 = 0) or render redundant all the equations except for those with {λ, μ, ν} being either {1, 2, 3}, {2, 3, 0}, {3, 0, 1}, or {0, 1, 2}.
The Faraday–Gauss equation is sometimes written
F
[
μ
ν
;
λ
]
=
F
[
μ
ν
,
λ
]
=
1
6
(
∂
λ
F
μ
ν
+
∂
μ
F
ν
λ
+
∂
ν
F
λ
μ
−
∂
λ
F
ν
μ
−
∂
μ
F
λ
ν
−
∂
ν
F
μ
λ
)
=
1
3
(
∂
λ
F
μ
ν
+
∂
μ
F
ν
λ
+
∂
ν
F
λ
μ
)
=
0
,
{\displaystyle F_{[\mu \nu ;\lambda ]}=F_{[\mu \nu ,\lambda ]}={\frac {1}{6}}(\partial _{\lambda }F_{\mu \nu }+\partial _{\mu }F_{\nu \lambda }+\partial _{\nu }F_{\lambda \mu }-\partial _{\lambda }F_{\nu \mu }-\partial _{\mu }F_{\lambda \nu }-\partial _{\nu }F_{\mu \lambda })={\frac {1}{3}}(\partial _{\lambda }F_{\mu \nu }+\partial _{\mu }F_{\nu \lambda }+\partial _{\nu }F_{\lambda \mu })=0,}
where a semicolon indicates a covariant derivative, a comma indicates a partial derivative, and square brackets indicate anti-symmetrization (see Ricci calculus for the notation). The covariant derivative of the electromagnetic field is
F
α
β
;
γ
=
F
α
β
,
γ
−
Γ
μ
α
γ
F
μ
β
−
Γ
μ
β
γ
F
α
μ
,
{\displaystyle F_{\alpha \beta ;\gamma }=F_{\alpha \beta ,\gamma }-{\Gamma ^{\mu }}_{\alpha \gamma }F_{\mu \beta }-{\Gamma ^{\mu }}_{\beta \gamma }F_{\alpha \mu },}
where Γαβγ is the Christoffel symbol, which is symmetric in its lower indices.
== Electromagnetic displacement ==
The electric displacement field D and the auxiliary magnetic field H form an antisymmetric contravariant rank-2 tensor density of weight +1. In vacuum, this is given by
D
μ
ν
=
1
μ
0
g
μ
α
F
α
β
g
β
ν
−
g
c
.
{\displaystyle {\mathcal {D}}^{\mu \nu }={\frac {1}{\mu _{0}}}\,g^{\mu \alpha }\,F_{\alpha \beta }\,g^{\beta \nu }\,{\frac {\sqrt {-g}}{c}}.}
This equation is the only place where the metric (and thus gravity) enters into the theory of electromagnetism. Furthermore, the equation is invariant under a change of scale, that is, multiplying the metric by a constant has no effect on this equation. Consequently, gravity can only affect electromagnetism by changing the speed of light relative to the global coordinate system being used. Light is only deflected by gravity because it is slower near massive bodies. So it is as if gravity increased the index of refraction of space near massive bodies.
More generally, in materials where the magnetization–polarization tensor is non-zero, we have
D
μ
ν
=
1
μ
0
g
μ
α
F
α
β
g
β
ν
−
g
c
−
M
μ
ν
.
{\displaystyle {\mathcal {D}}^{\mu \nu }={\frac {1}{\mu _{0}}}\,g^{\mu \alpha }\,F_{\alpha \beta }\,g^{\beta \nu }\,{\frac {\sqrt {-g}}{c}}-{\mathcal {M}}^{\mu \nu }.}
The transformation law for electromagnetic displacement is
D
¯
μ
ν
=
∂
x
¯
μ
∂
x
α
∂
x
¯
ν
∂
x
β
D
α
β
det
[
∂
x
σ
∂
x
¯
ρ
]
,
{\displaystyle {\bar {\mathcal {D}}}^{\mu \nu }={\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}\,{\frac {\partial {\bar {x}}^{\nu }}{\partial x^{\beta }}}\,{\mathcal {D}}^{\alpha \beta }\,\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right],}
where the Jacobian determinant is used. If the magnetization-polarization tensor is used, it has the same transformation law as the electromagnetic displacement.
== Electric current ==
The electric current is the divergence of the electromagnetic displacement. In vacuum,
J
μ
=
∂
ν
D
μ
ν
.
{\displaystyle J^{\mu }=\partial _{\nu }{\mathcal {D}}^{\mu \nu }.}
If magnetization–polarization is used, then this just gives the free portion of the current
J
free
μ
=
∂
ν
D
μ
ν
.
{\displaystyle J_{\text{free}}^{\mu }=\partial _{\nu }{\mathcal {D}}^{\mu \nu }.}
This incorporates Ampere's law and Gauss's law.
In either case, the fact that the electromagnetic displacement is antisymmetric implies that the electric current is automatically conserved:
∂
μ
J
μ
=
∂
μ
∂
ν
D
μ
ν
=
0
,
{\displaystyle \partial _{\mu }J^{\mu }=\partial _{\mu }\partial _{\nu }{\mathcal {D}}^{\mu \nu }=0,}
because the partial derivatives commute.
The Ampere–Gauss definition of the electric current is not sufficient to determine its value because the electromagnetic potential (from which it was ultimately derived) has not been given a value. Instead, the usual procedure is to equate the electric current to some expression in terms of other fields, mainly the electron and proton, and then solve for the electromagnetic displacement, electromagnetic field, and electromagnetic potential.
The electric current is a contravariant vector density, and as such it transforms as follows:
J
¯
μ
=
∂
x
¯
μ
∂
x
α
J
α
det
[
∂
x
σ
∂
x
¯
ρ
]
.
{\displaystyle {\bar {J}}^{\mu }={\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}J^{\alpha }\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right].}
Verification of this transformation law:
J
¯
μ
=
∂
∂
x
¯
ν
(
D
¯
μ
ν
)
=
∂
∂
x
¯
ν
(
∂
x
¯
μ
∂
x
α
∂
x
¯
ν
∂
x
β
D
α
β
det
[
∂
x
σ
∂
x
¯
ρ
]
)
=
∂
2
x
¯
μ
∂
x
¯
ν
∂
x
α
∂
x
¯
ν
∂
x
β
D
α
β
det
[
∂
x
σ
∂
x
¯
ρ
]
+
∂
x
¯
μ
∂
x
α
∂
2
x
¯
ν
∂
x
¯
ν
∂
x
β
D
α
β
det
[
∂
x
σ
∂
x
¯
ρ
]
+
∂
x
¯
μ
∂
x
α
∂
x
¯
ν
∂
x
β
∂
D
α
β
∂
x
¯
ν
det
[
∂
x
σ
∂
x
¯
ρ
]
+
∂
x
¯
μ
∂
x
α
∂
x
¯
ν
∂
x
β
D
α
β
∂
∂
x
¯
ν
det
[
∂
x
σ
∂
x
¯
ρ
]
=
∂
2
x
¯
μ
∂
x
β
∂
x
α
D
α
β
det
[
∂
x
σ
∂
x
¯
ρ
]
+
∂
x
¯
μ
∂
x
α
∂
2
x
¯
ν
∂
x
¯
ν
∂
x
β
D
α
β
det
[
∂
x
σ
∂
x
¯
ρ
]
+
∂
x
¯
μ
∂
x
α
∂
D
α
β
∂
x
β
det
[
∂
x
σ
∂
x
¯
ρ
]
+
∂
x
¯
μ
∂
x
α
∂
x
¯
ν
∂
x
β
D
α
β
det
[
∂
x
σ
∂
x
¯
ρ
]
∂
x
¯
ρ
∂
x
σ
∂
2
x
σ
∂
x
¯
ν
∂
x
¯
ρ
=
0
+
∂
x
¯
μ
∂
x
α
∂
2
x
¯
ν
∂
x
¯
ν
∂
x
β
D
α
β
det
[
∂
x
σ
∂
x
¯
ρ
]
+
∂
x
¯
μ
∂
x
α
J
α
det
[
∂
x
σ
∂
x
¯
ρ
]
+
∂
x
¯
μ
∂
x
α
D
α
β
det
[
∂
x
σ
∂
x
¯
ρ
]
∂
x
¯
ρ
∂
x
σ
∂
2
x
σ
∂
x
β
∂
x
¯
ρ
=
∂
x
¯
μ
∂
x
α
J
α
det
[
∂
x
σ
∂
x
¯
ρ
]
+
∂
x
¯
μ
∂
x
α
D
α
β
det
[
∂
x
σ
∂
x
¯
ρ
]
(
∂
2
x
¯
ν
∂
x
¯
ν
∂
x
β
+
∂
x
¯
ρ
∂
x
σ
∂
2
x
σ
∂
x
β
∂
x
¯
ρ
)
.
{\displaystyle {\begin{aligned}{\bar {J}}^{\mu }&={\frac {\partial }{\partial {\bar {x}}^{\nu }}}\left({\bar {\mathcal {D}}}^{\mu \nu }\right)\\[6pt]&={\frac {\partial }{\partial {\bar {x}}^{\nu }}}\left({\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}{\frac {\partial {\bar {x}}^{\nu }}{\partial x^{\beta }}}{\mathcal {D}}^{\alpha \beta }\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right]\right)\\[6pt]&={\frac {\partial ^{2}{\bar {x}}^{\mu }}{\partial {\bar {x}}^{\nu }\partial x^{\alpha }}}{\frac {\partial {\bar {x}}^{\nu }}{\partial x^{\beta }}}{\mathcal {D}}^{\alpha \beta }\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right]+{\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}{\frac {\partial ^{2}{\bar {x}}^{\nu }}{\partial {\bar {x}}^{\nu }\partial x^{\beta }}}{\mathcal {D}}^{\alpha \beta }\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right]+{\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}{\frac {\partial {\bar {x}}^{\nu }}{\partial x^{\beta }}}{\frac {\partial {\mathcal {D}}^{\alpha \beta }}{\partial {\bar {x}}^{\nu }}}\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right]+{\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}{\frac {\partial {\bar {x}}^{\nu }}{\partial x^{\beta }}}{\mathcal {D}}^{\alpha \beta }{\frac {\partial }{\partial {\bar {x}}^{\nu }}}\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right]\\[6pt]&={\frac {\partial ^{2}{\bar {x}}^{\mu }}{\partial x^{\beta }\partial x^{\alpha }}}{\mathcal {D}}^{\alpha \beta }\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right]+{\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}{\frac {\partial ^{2}{\bar {x}}^{\nu }}{\partial {\bar {x}}^{\nu }\partial x^{\beta }}}{\mathcal {D}}^{\alpha \beta }\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right]+{\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}{\frac {\partial {\mathcal {D}}^{\alpha \beta }}{\partial x^{\beta }}}\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right]+{\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}{\frac {\partial {\bar {x}}^{\nu }}{\partial x^{\beta }}}{\mathcal {D}}^{\alpha \beta }\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right]{\frac {\partial {\bar {x}}^{\rho }}{\partial x^{\sigma }}}{\frac {\partial ^{2}x^{\sigma }}{\partial {\bar {x}}^{\nu }\partial {\bar {x}}^{\rho }}}\\[6pt]&=0+{\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}{\frac {\partial ^{2}{\bar {x}}^{\nu }}{\partial {\bar {x}}^{\nu }\partial x^{\beta }}}{\mathcal {D}}^{\alpha \beta }\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right]+{\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}J^{\alpha }\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right]+{\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}{\mathcal {D}}^{\alpha \beta }\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right]{\frac {\partial {\bar {x}}^{\rho }}{\partial x^{\sigma }}}{\frac {\partial ^{2}x^{\sigma }}{\partial x^{\beta }\partial {\bar {x}}^{\rho }}}\\[6pt]&={\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}J^{\alpha }\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right]+{\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}{\mathcal {D}}^{\alpha \beta }\det \left[{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\rho }}}\right]\left({\frac {\partial ^{2}{\bar {x}}^{\nu }}{\partial {\bar {x}}^{\nu }\partial x^{\beta }}}+{\frac {\partial {\bar {x}}^{\rho }}{\partial x^{\sigma }}}{\frac {\partial ^{2}x^{\sigma }}{\partial x^{\beta }\partial {\bar {x}}^{\rho }}}\right).\end{aligned}}}
So all that remains is to show that
∂
2
x
¯
ν
∂
x
¯
ν
∂
x
β
+
∂
x
¯
ρ
∂
x
σ
∂
2
x
σ
∂
x
β
∂
x
¯
ρ
=
0
,
{\displaystyle {\frac {\partial ^{2}{\bar {x}}^{\nu }}{\partial {\bar {x}}^{\nu }\partial x^{\beta }}}+{\frac {\partial {\bar {x}}^{\rho }}{\partial x^{\sigma }}}{\frac {\partial ^{2}x^{\sigma }}{\partial x^{\beta }\partial {\bar {x}}^{\rho }}}=0,}
which is a version of a known theorem (see Inverse functions and differentiation § Higher derivatives).
∂
2
x
¯
ν
∂
x
¯
ν
∂
x
β
+
∂
x
¯
ρ
∂
x
σ
∂
2
x
σ
∂
x
β
∂
x
¯
ρ
=
∂
x
σ
∂
x
¯
ν
∂
2
x
¯
ν
∂
x
σ
∂
x
β
+
∂
x
¯
ν
∂
x
σ
∂
2
x
σ
∂
x
β
∂
x
¯
ν
=
∂
x
σ
∂
x
¯
ν
∂
2
x
¯
ν
∂
x
β
∂
x
σ
+
∂
2
x
σ
∂
x
β
∂
x
¯
ν
∂
x
¯
ν
∂
x
σ
=
∂
∂
x
β
(
∂
x
σ
∂
x
¯
ν
∂
x
¯
ν
∂
x
σ
)
=
∂
∂
x
β
(
∂
x
¯
ν
∂
x
¯
ν
)
=
∂
∂
x
β
(
4
)
=
0.
{\displaystyle {\begin{aligned}&{\frac {\partial ^{2}{\bar {x}}^{\nu }}{\partial {\bar {x}}^{\nu }\partial x^{\beta }}}+{\frac {\partial {\bar {x}}^{\rho }}{\partial x^{\sigma }}}{\frac {\partial ^{2}x^{\sigma }}{\partial x^{\beta }\partial {\bar {x}}^{\rho }}}\\[6pt]{}={}&{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\nu }}}{\frac {\partial ^{2}{\bar {x}}^{\nu }}{\partial x^{\sigma }\partial x^{\beta }}}+{\frac {\partial {\bar {x}}^{\nu }}{\partial x^{\sigma }}}{\frac {\partial ^{2}x^{\sigma }}{\partial x^{\beta }\partial {\bar {x}}^{\nu }}}\\[6pt]{}={}&{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\nu }}}{\frac {\partial ^{2}{\bar {x}}^{\nu }}{\partial x^{\beta }\partial x^{\sigma }}}+{\frac {\partial ^{2}x^{\sigma }}{\partial x^{\beta }\partial {\bar {x}}^{\nu }}}{\frac {\partial {\bar {x}}^{\nu }}{\partial x^{\sigma }}}\\[6pt]{}={}&{\frac {\partial }{\partial x^{\beta }}}\left({\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\nu }}}{\frac {\partial {\bar {x}}^{\nu }}{\partial x^{\sigma }}}\right)\\[6pt]{}={}&{\frac {\partial }{\partial x^{\beta }}}\left({\frac {\partial {\bar {x}}^{\nu }}{\partial {\bar {x}}^{\nu }}}\right)\\[6pt]{}={}&{\frac {\partial }{\partial x^{\beta }}}\left(\mathbf {4} \right)\\[6pt]{}={}&0.\end{aligned}}}
== Lorentz force density ==
The density of the Lorentz force is a covariant vector density given by
f
μ
=
F
μ
ν
J
ν
.
{\displaystyle f_{\mu }=F_{\mu \nu }J^{\nu }.}
The force on a test particle subject only to gravity and electromagnetism is
d
p
α
d
t
=
Γ
α
γ
β
p
β
d
x
γ
d
t
+
q
F
α
γ
d
x
γ
d
t
,
{\displaystyle {\frac {dp_{\alpha }}{dt}}=\Gamma _{\alpha \gamma }^{\beta }p_{\beta }{\frac {dx^{\gamma }}{dt}}+qF_{\alpha \gamma }{\frac {dx^{\gamma }}{dt}},}
where pα is the linear 4-momentum of the particle, t is any time coordinate parameterizing the world line of the particle, Γβαγ is the Christoffel symbol (gravitational force field), and q is the electric charge of the particle.
This equation is invariant under a change in the time coordinate; just multiply by
d
t
/
d
t
¯
{\displaystyle dt/d{\bar {t}}}
and use the chain rule. It is also invariant under a change in the x coordinate system.
Using the transformation law for the Christoffel symbol,
Γ
¯
α
γ
β
=
∂
x
¯
β
∂
x
ϵ
∂
x
δ
∂
x
¯
α
∂
x
ζ
∂
x
¯
γ
Γ
δ
ζ
ϵ
+
∂
x
¯
β
∂
x
η
∂
2
x
η
∂
x
¯
α
∂
x
¯
γ
,
{\displaystyle {\bar {\Gamma }}_{\alpha \gamma }^{\beta }={\frac {\partial {\bar {x}}^{\beta }}{\partial x^{\epsilon }}}{\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}{\frac {\partial x^{\zeta }}{\partial {\bar {x}}^{\gamma }}}\Gamma _{\delta \zeta }^{\epsilon }+{\frac {\partial {\bar {x}}^{\beta }}{\partial x^{\eta }}}{\frac {\partial ^{2}x^{\eta }}{\partial {\bar {x}}^{\alpha }\partial {\bar {x}}^{\gamma }}},}
we get
d
p
¯
α
d
t
−
Γ
¯
α
γ
β
p
¯
β
d
x
¯
γ
d
t
−
q
F
¯
α
γ
d
x
¯
γ
d
t
=
d
d
t
(
∂
x
δ
∂
x
¯
α
p
δ
)
−
(
∂
x
¯
β
∂
x
θ
∂
x
δ
∂
x
¯
α
∂
x
ι
∂
x
¯
γ
Γ
δ
ι
θ
+
∂
x
¯
β
∂
x
η
∂
2
x
η
∂
x
¯
α
∂
x
¯
γ
)
∂
x
ϵ
∂
x
¯
β
p
ϵ
∂
x
¯
γ
∂
x
ζ
d
x
ζ
d
t
−
q
∂
x
δ
∂
x
¯
α
F
δ
ζ
d
x
ζ
d
t
=
∂
x
δ
∂
x
¯
α
(
d
p
δ
d
t
−
Γ
δ
ζ
ϵ
p
ϵ
d
x
ζ
d
t
−
q
F
δ
ζ
d
x
ζ
d
t
)
+
d
d
t
(
∂
x
δ
∂
x
¯
α
)
p
δ
−
(
∂
x
¯
β
∂
x
η
∂
2
x
η
∂
x
¯
α
∂
x
¯
γ
)
∂
x
ϵ
∂
x
¯
β
p
ϵ
∂
x
¯
γ
∂
x
ζ
d
x
ζ
d
t
=
0
+
d
d
t
(
∂
x
δ
∂
x
¯
α
)
p
δ
−
∂
2
x
ϵ
∂
x
¯
α
∂
x
¯
γ
p
ϵ
d
x
¯
γ
d
t
=
0.
{\displaystyle {\begin{aligned}&{\frac {d{\bar {p}}_{\alpha }}{dt}}-{\bar {\Gamma }}_{\alpha \gamma }^{\beta }{\bar {p}}_{\beta }{\frac {d{\bar {x}}^{\gamma }}{dt}}-q{\bar {F}}_{\alpha \gamma }{\frac {d{\bar {x}}^{\gamma }}{dt}}\\[6pt]{}={}&{\frac {d}{dt}}\left({\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}p_{\delta }\right)-\left({\frac {\partial {\bar {x}}^{\beta }}{\partial x^{\theta }}}{\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}{\frac {\partial x^{\iota }}{\partial {\bar {x}}^{\gamma }}}\Gamma _{\delta \iota }^{\theta }+{\frac {\partial {\bar {x}}^{\beta }}{\partial x^{\eta }}}{\frac {\partial ^{2}x^{\eta }}{\partial {\bar {x}}^{\alpha }\partial {\bar {x}}^{\gamma }}}\right){\frac {\partial x^{\epsilon }}{\partial {\bar {x}}^{\beta }}}p_{\epsilon }{\frac {\partial {\bar {x}}^{\gamma }}{\partial x^{\zeta }}}{\frac {dx^{\zeta }}{dt}}-q{\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}F_{\delta \zeta }{\frac {dx^{\zeta }}{dt}}\\[6pt]{}={}&{\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}\left({\frac {dp_{\delta }}{dt}}-\Gamma _{\delta \zeta }^{\epsilon }p_{\epsilon }{\frac {dx^{\zeta }}{dt}}-qF_{\delta \zeta }{\frac {dx^{\zeta }}{dt}}\right)+{\frac {d}{dt}}\left({\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}\right)p_{\delta }-\left({\frac {\partial {\bar {x}}^{\beta }}{\partial x^{\eta }}}{\frac {\partial ^{2}x^{\eta }}{\partial {\bar {x}}^{\alpha }\partial {\bar {x}}^{\gamma }}}\right){\frac {\partial x^{\epsilon }}{\partial {\bar {x}}^{\beta }}}p_{\epsilon }{\frac {\partial {\bar {x}}^{\gamma }}{\partial x^{\zeta }}}{\frac {dx^{\zeta }}{dt}}\\[6pt]{}={}&0+{\frac {d}{dt}}\left({\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\alpha }}}\right)p_{\delta }-{\frac {\partial ^{2}x^{\epsilon }}{\partial {\bar {x}}^{\alpha }\partial {\bar {x}}^{\gamma }}}p_{\epsilon }{\frac {d{\bar {x}}^{\gamma }}{dt}}\\[6pt]{}={}&0.\end{aligned}}}
== Lagrangian ==
In vacuum, the Lagrangian density for classical electrodynamics (in joules per cubic meter) is a scalar density
L
=
−
1
4
μ
0
F
α
β
F
α
β
−
g
c
+
A
α
J
α
,
{\displaystyle {\mathcal {L}}=-{\frac {1}{4\mu _{0}}}\,F_{\alpha \beta }\,F^{\alpha \beta }\,{\frac {\sqrt {-g}}{c}}+A_{\alpha }\,J^{\alpha },}
where
F
α
β
=
g
α
γ
F
γ
δ
g
δ
β
.
{\displaystyle F^{\alpha \beta }=g^{\alpha \gamma }F_{\gamma \delta }g^{\delta \beta }.}
The 4-current should be understood as an abbreviation of many terms expressing the electric currents of other charged fields in terms of their variables.
If we separate free currents from bound currents, the Lagrangian becomes
L
=
−
1
4
μ
0
F
α
β
F
α
β
−
g
c
+
A
α
J
free
α
+
1
2
F
α
β
M
α
β
.
{\displaystyle {\mathcal {L}}=-{\frac {1}{4\mu _{0}}}\,F_{\alpha \beta }\,F^{\alpha \beta }\,{\frac {\sqrt {-g}}{c}}+A_{\alpha }\,J_{\text{free}}^{\alpha }+{\frac {1}{2}}\,F_{\alpha \beta }\,{\mathcal {M}}^{\alpha \beta }.}
== Electromagnetic stress–energy tensor ==
As part of the source term in the Einstein field equations, the electromagnetic stress–energy tensor is a covariant symmetric tensor
T
μ
ν
=
−
1
μ
0
(
F
μ
α
g
α
β
F
β
ν
−
1
4
g
μ
ν
F
σ
α
g
α
β
F
β
ρ
g
ρ
σ
)
,
{\displaystyle T_{\mu \nu }=-{\frac {1}{\mu _{0}}}\left(F_{\mu \alpha }g^{\alpha \beta }F_{\beta \nu }-{\frac {1}{4}}g_{\mu \nu }F_{\sigma \alpha }g^{\alpha \beta }F_{\beta \rho }g^{\rho \sigma }\right),}
using a metric of signature (−, +, +, +). If using the metric with signature (+, −, −, −), the expression for
T
μ
ν
{\displaystyle T_{\mu \nu }}
will have opposite sign. The stress–energy tensor is trace-free:
T
μ
ν
g
μ
ν
=
0
{\displaystyle T_{\mu \nu }g^{\mu \nu }=0}
because electromagnetism propagates at the local invariant speed, and is conformal-invariant.
In the expression for the conservation of energy and linear momentum, the electromagnetic stress–energy tensor is best represented as a mixed tensor density
T
μ
ν
=
T
μ
γ
g
γ
ν
−
g
c
.
{\displaystyle {\mathfrak {T}}_{\mu }^{\nu }=T_{\mu \gamma }g^{\gamma \nu }{\frac {\sqrt {-g}}{c}}.}
From the equations above, one can show that
T
μ
ν
;
ν
+
f
μ
=
0
,
{\displaystyle {{\mathfrak {T}}_{\mu }^{\nu }}_{;\nu }+f_{\mu }=0,}
where the semicolon indicates a covariant derivative.
This can be rewritten as
−
T
μ
ν
,
ν
=
−
Γ
μ
ν
σ
T
σ
ν
+
f
μ
,
{\displaystyle -{{\mathfrak {T}}_{\mu }^{\nu }}_{,\nu }=-\Gamma _{\mu \nu }^{\sigma }{\mathfrak {T}}_{\sigma }^{\nu }+f_{\mu },}
which says that the decrease in the electromagnetic energy is the same as the work done by the electromagnetic field on the gravitational field plus the work done on matter (via the Lorentz force), and similarly the rate of decrease in the electromagnetic linear momentum is the electromagnetic force exerted on the gravitational field plus the Lorentz force exerted on matter.
Derivation of conservation law:
T
μ
ν
;
ν
+
f
μ
=
−
1
μ
0
(
F
μ
α
;
ν
g
α
β
F
β
γ
g
γ
ν
+
F
μ
α
g
α
β
F
β
γ
;
ν
g
γ
ν
−
1
2
δ
μ
ν
F
σ
α
;
ν
g
α
β
F
β
ρ
g
ρ
σ
)
−
g
c
+
1
μ
0
F
μ
α
g
α
β
F
β
γ
;
ν
g
γ
ν
−
g
c
=
−
1
μ
0
(
F
μ
α
;
ν
F
α
ν
−
1
2
F
σ
α
;
μ
F
α
σ
)
−
g
c
=
−
1
μ
0
(
(
−
F
ν
μ
;
α
−
F
α
ν
;
μ
)
F
α
ν
−
1
2
F
σ
α
;
μ
F
α
σ
)
−
g
c
=
−
1
μ
0
(
F
μ
ν
;
α
F
α
ν
−
F
α
ν
;
μ
F
α
ν
+
1
2
F
σ
α
;
μ
F
σ
α
)
−
g
c
=
−
1
μ
0
(
F
μ
α
;
ν
F
ν
α
−
1
2
F
α
ν
;
μ
F
α
ν
)
−
g
c
=
−
1
μ
0
(
−
F
μ
α
;
ν
F
α
ν
+
1
2
F
σ
α
;
μ
F
α
σ
)
−
g
c
,
{\displaystyle {\begin{aligned}{{\mathfrak {T}}_{\mu }^{\nu }}_{;\nu }+f_{\mu }&=-{\frac {1}{\mu _{0}}}\left(F_{\mu \alpha ;\nu }g^{\alpha \beta }F_{\beta \gamma }g^{\gamma \nu }+F_{\mu \alpha }g^{\alpha \beta }F_{\beta \gamma ;\nu }g^{\gamma \nu }-{\frac {1}{2}}\delta _{\mu }^{\nu }F_{\sigma \alpha ;\nu }g^{\alpha \beta }F_{\beta \rho }g^{\rho \sigma }\right){\frac {\sqrt {-g}}{c}}+{\frac {1}{\mu _{0}}}F_{\mu \alpha }g^{\alpha \beta }F_{\beta \gamma ;\nu }g^{\gamma \nu }{\frac {\sqrt {-g}}{c}}\\&=-{\frac {1}{\mu _{0}}}\left(F_{\mu \alpha ;\nu }F^{\alpha \nu }-{\frac {1}{2}}F_{\sigma \alpha ;\mu }F^{\alpha \sigma }\right){\frac {\sqrt {-g}}{c}}\\&=-{\frac {1}{\mu _{0}}}\left(\left(-F_{\nu \mu ;\alpha }-F_{\alpha \nu ;\mu }\right)F^{\alpha \nu }-{\frac {1}{2}}F_{\sigma \alpha ;\mu }F^{\alpha \sigma }\right){\frac {\sqrt {-g}}{c}}\\&=-{\frac {1}{\mu _{0}}}\left(F_{\mu \nu ;\alpha }F^{\alpha \nu }-F_{\alpha \nu ;\mu }F^{\alpha \nu }+{\frac {1}{2}}F_{\sigma \alpha ;\mu }F^{\sigma \alpha }\right){\frac {\sqrt {-g}}{c}}\\&=-{\frac {1}{\mu _{0}}}\left(F_{\mu \alpha ;\nu }F^{\nu \alpha }-{\frac {1}{2}}F_{\alpha \nu ;\mu }F^{\alpha \nu }\right){\frac {\sqrt {-g}}{c}}\\&=-{\frac {1}{\mu _{0}}}\left(-F_{\mu \alpha ;\nu }F^{\alpha \nu }+{\frac {1}{2}}F_{\sigma \alpha ;\mu }F^{\alpha \sigma }\right){\frac {\sqrt {-g}}{c}},\end{aligned}}}
which is zero because it is the negative of itself (see four lines above).
== Electromagnetic wave equation ==
The nonhomogeneous electromagnetic wave equation in terms of the field tensor is modified from the special-relativity form to
◻
F
a
b
=
def
F
a
b
;
d
d
=
−
2
R
a
c
b
d
F
c
d
+
R
a
e
F
e
b
−
R
b
e
F
e
a
+
J
a
;
b
−
J
b
;
a
,
{\displaystyle \Box F_{ab}\ {\stackrel {\text{def}}{=}}\ F_{ab;}{}^{d}{}_{d}=-2R_{acbd}F^{cd}+R_{ae}F^{e}{}_{b}-R_{be}F^{e}{}_{a}+J_{a;b}-J_{b;a},}
where Racbd is the covariant form of the Riemann tensor, and
◻
{\displaystyle \Box }
is a generalization of the d'Alembertian operator for covariant derivatives. Using
◻
A
a
=
A
a
;
b
b
.
{\displaystyle \Box A^{a}={{A^{a;}}^{b}}_{b}.}
Maxwell's source equations can be written in terms of the 4-potential [ref. 2, p. 569] as
◻
A
a
−
A
b
;
a
b
=
−
μ
0
J
a
{\displaystyle \Box A^{a}-{A^{b;a}}_{b}=-\mu _{0}J^{a}}
or, assuming the generalization of the Lorenz gauge in curved spacetime,
A
a
;
a
=
0
,
◻
A
a
=
−
μ
0
J
a
+
R
a
b
A
b
,
{\displaystyle {\begin{aligned}{A^{a}}_{;a}&=0,\\\Box A^{a}&=-\mu _{0}J^{a}+{R^{a}}_{b}A^{b},\end{aligned}}}
where
R
a
b
=
def
R
s
a
s
b
{\displaystyle R_{ab}\ {\stackrel {\text{def}}{=}}\ {R^{s}}_{asb}}
is the Ricci curvature tensor.
This is the same form of the wave equation as in flat spacetime, except that the derivatives are replaced by covariant derivatives and there is an additional term proportional to the curvature. The wave equation in this form also bears some resemblance to the Lorentz force in curved spacetime, where Aa plays the role of the 4-position.
For the case of a metric signature in the form (+, −, −, −), the derivation of the wave equation in curved spacetime is carried out in the article.
== Nonlinearity of Maxwell's equations in a dynamic spacetime ==
When Maxwell's equations are treated in a background-independent manner, that is, when the spacetime metric is taken to be a dynamical variable dependent on the electromagnetic field, then the electromagnetic wave equation and Maxwell's equations are nonlinear. This can be seen by noting that the curvature tensor depends on the stress–energy tensor through the Einstein field equation
G
a
b
=
8
π
G
c
4
T
a
b
,
{\displaystyle G_{ab}={\frac {8\pi G}{c^{4}}}T_{ab},}
where
G
a
b
=
def
R
a
b
−
1
2
R
g
a
b
{\displaystyle G_{ab}\ {\stackrel {\text{def}}{=}}\ R_{ab}-{\frac {1}{2}}Rg_{ab}}
is the Einstein tensor, G is the Newtonian constant of gravitation, gab is the metric tensor, and R (scalar curvature) is the trace of the Ricci curvature tensor. The stress–energy tensor is composed of the stress–energy from particles, but also stress–energy from the electromagnetic field. This generates the nonlinearity.
== Geometric formulation ==
In the differential geometric formulation of the electromagnetic field, the antisymmetric Faraday tensor can be considered as the Faraday 2-form
F
{\displaystyle \mathbf {F} }
. In this view, one of Maxwell's two equations is
d
F
=
0
,
{\displaystyle \mathrm {d} \mathbf {F} =0,}
where
d
{\displaystyle \mathrm {d} }
is the exterior derivative operator. This equation is completely coordinate- and metric-independent and says that the electromagnetic flux through a closed two-dimensional surface in space–time is topological, more precisely, depends only on its homology class (a generalization of the integral form of Gauss law and Maxwell–Faraday equation, as the homology class in Minkowski space is automatically 0). By the Poincaré lemma, this equation implies (at least locally) that there exists a 1-form
A
{\displaystyle \mathbf {A} }
satisfying
F
=
d
A
.
{\displaystyle \mathbf {F} =\mathrm {d} \mathbf {A} .}
The other equation is
d
⋆
F
=
J
.
{\displaystyle \mathrm {d} {\star }\mathbf {F} =\mathbf {J} .}
In this context,
J
{\displaystyle \mathbf {J} }
is the current 3-form (or even more precise, twisted 3-form), and the star
⋆
{\displaystyle \star }
denotes the Hodge star operator. The dependence of Maxwell's equation on the metric of spacetime lies in the Hodge star operator
⋆
{\displaystyle \star }
on 2-forms, which is conformally invariant. Written this way, Maxwell's equation is the same in any space–time, manifestly coordinate-invariant, and convenient to use (even in Minkowski space or Euclidean space and time, especially with curvilinear coordinates).
An alternative geometric interpretation is that the Faraday 2-form
F
{\displaystyle \mathbf {F} }
is (up to a factor
i
{\displaystyle i}
) the curvature 2-form
F
(
∇
)
{\displaystyle F(\nabla )}
of a U(1)-connection
∇
{\displaystyle \nabla }
on a principal U(1)-bundle whose sections represent charged fields. The connection is much like the vector potential, since every connection can be written as
∇
=
∇
0
+
i
A
{\displaystyle \nabla =\nabla _{0}+iA}
for a "base" connection
∇
0
{\displaystyle \nabla _{0}}
, and
F
=
F
0
+
d
A
.
{\displaystyle \mathbf {F} =\mathbf {F} _{0}+\mathrm {d} \mathbf {A} .}
In this view, the Maxwell "equation"
d
F
=
0
{\displaystyle \mathrm {d} \mathbf {F} =0}
is a mathematical identity known as the Bianchi identity. The equation
d
⋆
F
=
J
{\displaystyle \mathrm {d} {\star }\mathbf {F} =\mathbf {J} }
is the only equation with any physical content in this formulation. This point of view is particularly natural when considering charged fields or quantum mechanics. It can be interpreted as saying that, much like gravity can be understood as being the result of the necessity of a connection to parallel transport vectors at different points, electromagnetic phenomena, or more subtle quantum effects like the Aharonov–Bohm effect, can be understood as a result from the necessity of a connection to parallel transport charged fields or wave sections at different points. In fact, just as the Riemann tensor is the holonomy of the Levi-Civita connection along an infinitesimal closed curve, the curvature of the connection is the holonomy of the U(1)-connection.
== See also ==
Electromagnetic wave equation
Inhomogeneous electromagnetic wave equation
Mathematical descriptions of the electromagnetic field
Covariant formulation of classical electromagnetism
Theoretical motivation for general relativity
Introduction to the mathematics of general relativity
Electrovacuum solution
Paradox of radiation of charged particles in a gravitational field
== Notes ==
== References ==
Einstein, A. (1961). Relativity: The Special and General Theory. New York: Crown. ISBN 0-517-02961-8. {{cite book}}: ISBN / Date incompatibility (help)
Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ISBN 0-7167-0344-0.
Landau, L. D.; Lifshitz, E. M. (1975). Classical Theory of Fields (Fourth Revised English ed.). Oxford: Pergamon. ISBN 0-08-018176-7.
Feynman, R. P.; Moringo, F. B.; Wagner, W. G. (1995). Feynman Lectures on Gravitation. Addison-Wesley. ISBN 0-201-62734-5.
== External links ==
Electromagnetic fields in curved spacetimes | Wikipedia/Maxwell's_equations_in_curved_spacetime |
In electrical engineering, a transformer is a passive component that transfers electrical energy from one electrical circuit to another circuit, or multiple circuits. A varying current in any coil of the transformer produces a varying magnetic flux in the transformer's core, which induces a varying electromotive force (EMF) across any other coils wound around the same core. Electrical energy can be transferred between separate coils without a metallic (conductive) connection between the two circuits. Faraday's law of induction, discovered in 1831, describes the induced voltage effect in any coil due to a changing magnetic flux encircled by the coil.
Transformers are used to change AC voltage levels, such transformers being termed step-up or step-down type to increase or decrease voltage level, respectively. Transformers can also be used to provide galvanic isolation between circuits as well as to couple stages of signal-processing circuits. Since the invention of the first constant-potential transformer in 1885, transformers have become essential for the transmission, distribution, and utilization of alternating current electric power. A wide range of transformer designs is encountered in electronic and electric power applications. Transformers range in size from RF transformers less than a cubic centimeter in volume, to units weighing hundreds of tons used to interconnect the power grid.
== Principles ==
Ideal transformer equations
By Faraday's law of induction:
where
V
{\displaystyle V}
is the instantaneous voltage,
N
{\displaystyle N}
is the number of turns in a winding, dΦ/dt is the derivative of the magnetic flux Φ through one turn of the winding over time (t), and subscripts P and S denotes primary and secondary.
Combining the ratio of eq. 1 & eq. 2:
where for a step-up transformer a < 1 and for a step-down transformer a > 1.
By the law of conservation of energy, apparent, real and reactive power are each conserved in the input and output:
where
S
{\displaystyle S}
is apparent power and
I
{\displaystyle I}
is current.
Combining Eq. 3 & Eq. 4 with this endnote gives the ideal transformer identity:
where
L
P
{\displaystyle L_{\text{P}}}
is the primary winding self-inductance and
L
S
{\displaystyle L_{\text{S}}}
is the secondary winding self-inductance.
By Ohm's law and ideal transformer identity:
where
Z
L
{\displaystyle Z_{\text{L}}}
is the load impedance of the secondary circuit &
Z
L
′
{\displaystyle Z'_{\text{L}}}
is the apparent load or driving point impedance of the primary circuit, the superscript
′
{\displaystyle '}
denoting referred to the primary.
=== Ideal transformer ===
An ideal transformer is linear, lossless and perfectly coupled. Perfect coupling implies infinitely high core magnetic permeability and winding inductance and zero net magnetomotive force (i.e. ipnp − isns = 0).
A varying current in the transformer's primary winding creates a varying magnetic flux in the transformer core, which is also encircled by the secondary winding. This varying flux at the secondary winding induces a varying electromotive force or voltage in the secondary winding. This electromagnetic induction phenomenon is the basis of transformer action and, in accordance with Lenz's law, the secondary current so produced creates a flux equal and opposite to that produced by the primary winding.
The windings are wound around a core of infinitely high magnetic permeability so that all of the magnetic flux passes through both the primary and secondary windings. With a voltage source connected to the primary winding and a load connected to the secondary winding, the transformer currents flow in the indicated directions and the core magnetomotive force cancels to zero.
According to Faraday's law, since the same magnetic flux passes through both the primary and secondary windings in an ideal transformer, a voltage is induced in each winding proportional to its number of turns. The transformer winding voltage ratio is equal to the winding turns ratio.
An ideal transformer is a reasonable approximation for a typical commercial transformer, with voltage ratio and winding turns ratio both being inversely proportional to the corresponding current ratio.
The load impedance referred to the primary circuit is equal to the turns ratio squared times the secondary circuit load impedance.
=== Real transformer ===
==== Deviations from ideal transformer ====
The ideal transformer model neglects many basic linear aspects of real transformers, including unavoidable losses and inefficiencies.
(a) Core losses, collectively called magnetizing current losses, consisting of
Hysteresis losses due to nonlinear magnetic effects in the transformer core, and
Eddy current losses due to joule heating in the core that are proportional to the square of the transformer's applied voltage.
(b) Unlike the ideal model, the windings in a real transformer have non-zero resistances and inductances associated with:
Joule losses due to resistance in the primary and secondary windings
Leakage flux that escapes from the core and passes through one winding only resulting in primary and secondary reactive impedance.
(c) similar to an inductor, parasitic capacitance and self-resonance phenomenon due to the electric field distribution. Three kinds of parasitic capacitance are usually considered and the closed-loop equations are provided
Capacitance between adjacent turns in any one layer;
Capacitance between adjacent layers;
Capacitance between the core and the layer(s) adjacent to the core;
Inclusion of capacitance into the transformer model is complicated, and is rarely attempted; the 'real' transformer model's equivalent circuit shown below does not include parasitic capacitance. However, the capacitance effect can be measured by comparing open-circuit inductance, i.e. the inductance of a primary winding when the secondary circuit is open, to a short-circuit inductance when the secondary winding is shorted.
==== Leakage flux ====
The ideal transformer model assumes that all flux generated by the primary winding links all the turns of every winding, including itself. In practice, some flux traverses paths that take it outside the windings. Such flux is termed leakage flux, and results in leakage inductance in series with the mutually coupled transformer windings. Leakage flux results in energy being alternately stored in and discharged from the magnetic fields with each cycle of the power supply. It is not directly a power loss, but results in inferior voltage regulation, causing the secondary voltage not to be directly proportional to the primary voltage, particularly under heavy load. Transformers are therefore normally designed to have very low leakage inductance.
In some applications increased leakage is desired, and long magnetic paths, air gaps, or magnetic bypass shunts may deliberately be introduced in a transformer design to limit the short-circuit current it will supply. Leaky transformers may be used to supply loads that exhibit negative resistance, such as electric arcs, mercury- and sodium- vapor lamps and neon signs or for safely handling loads that become periodically short-circuited such as electric arc welders.: 485
Air gaps are also used to keep a transformer from saturating, especially audio-frequency transformers in circuits that have a DC component flowing in the windings. A saturable reactor exploits saturation of the core to control alternating current.
Knowledge of leakage inductance is also useful when transformers are operated in parallel. It can be shown that if the percent impedance and associated winding leakage reactance-to-resistance (X/R) ratio of two transformers were
the same, the transformers would share the load power in proportion to their respective ratings. However, the impedance tolerances of commercial transformers are significant. Also, the impedance and X/R ratio of different capacity transformers tends to vary.
==== Equivalent circuit ====
Referring to the diagram, a practical transformer's physical behavior may be represented by an equivalent circuit model, which can incorporate an ideal transformer.
Winding joule losses and leakage reactance are represented by the following series loop impedances of the model:
Primary winding: RP, XP
Secondary winding: RS, XS.
In normal course of circuit equivalence transformation, RS and XS are in practice usually referred to the primary side by multiplying these impedances by the turns ratio squared, (NP/NS) 2 = a2.
Core loss and reactance is represented by the following shunt leg impedances of the model:
Core or iron losses: RC
Magnetizing reactance: XM.
RC and XM are collectively termed the magnetizing branch of the model.
Core losses are caused mostly by hysteresis and eddy current effects in the core and are proportional to the square of the core flux for operation at a given frequency.: 142–143 The finite permeability core requires a magnetizing current IM to maintain mutual flux in the core. Magnetizing current is in phase with the flux, the relationship between the two being non-linear due to saturation effects. However, all impedances of the equivalent circuit shown are by definition linear and such non-linearity effects are not typically reflected in transformer equivalent circuits.: 142 With sinusoidal supply, core flux lags the induced EMF by 90°. With open-circuited secondary winding, magnetizing branch current I0 equals transformer no-load current.
The resulting model, though sometimes termed 'exact' equivalent circuit based on linearity assumptions, retains a number of approximations. Analysis may be simplified by assuming that magnetizing branch impedance is relatively high and relocating the branch to the left of the primary impedances. This introduces error but allows combination of primary and referred secondary resistances and reactance by simple summation as two series impedances.
Transformer equivalent circuit impedance and transformer ratio parameters can be derived from the following tests: open-circuit test, short-circuit test, winding resistance test, and transformer ratio test.
=== Transformer EMF equation ===
If the flux in the core is purely sinusoidal, the relationship for either winding between its rms voltage Erms of the winding, and the supply frequency f, number of turns N, core cross-sectional area A in m2 and peak magnetic flux density Bpeak in Wb/m2 or T (tesla) is given by the universal EMF equation:
E
rms
=
2
π
f
N
A
B
peak
2
≈
4.44
f
N
A
B
peak
{\displaystyle E_{\text{rms}}={\frac {2\pi fNAB_{\text{peak}}}{\sqrt {2}}}\approx 4.44fNAB_{\text{peak}}}
=== Polarity ===
A dot convention is often used in transformer circuit diagrams, nameplates or terminal markings to define the relative polarity of transformer windings. Positively increasing instantaneous current entering the primary winding's 'dot' end induces positive polarity voltage exiting the secondary winding's 'dot' end. Three-phase transformers used in electric power systems will have a nameplate that indicate the phase relationships between their terminals. This may be in the form of a phasor diagram, or using an alpha-numeric code to show the type of internal connection (wye or delta) for each winding.
=== Effect of frequency ===
The EMF of a transformer at a given flux increases with frequency. By operating at higher frequencies, transformers can be physically more compact because a given core is able to transfer more power without reaching saturation and fewer turns are needed to achieve the same impedance. However, properties such as core loss and conductor skin effect also increase with frequency. Aircraft and military equipment employ 400 Hz power supplies which reduce core and winding weight. Conversely, frequencies used for some railway electrification systems were much lower (e.g. 16.7 Hz and 25 Hz) than normal utility frequencies (50–60 Hz) for historical reasons concerned mainly with the limitations of early electric traction motors. Consequently, the transformers used to step-down the high overhead line voltages were much larger and heavier for the same power rating than those required for the higher frequencies.
Operation of a transformer at its designed voltage but at a higher frequency than intended will lead to reduced magnetizing current. At a lower frequency, the magnetizing current will increase. Operation of a large transformer at other than its design frequency may require assessment of voltages, losses, and cooling to establish if safe operation is practical. Transformers may require protective relays to protect the transformer from overvoltage at higher than rated frequency.
One example is in traction transformers used for electric multiple unit and high-speed train service operating across regions with different electrical standards. The converter equipment and traction transformers have to accommodate different input frequencies and voltage (ranging from as high as 50 Hz down to 16.7 Hz and rated up to 25 kV).
At much higher frequencies the transformer core size required drops dramatically: a physically small transformer can handle power levels that would require a massive iron core at mains frequency. The development of switching power semiconductor devices made switch-mode power supplies viable, to generate a high frequency, then change the voltage level with a small transformer.
Transformers for higher frequency applications such as SMPS typically use core materials with much lower hysteresis and eddy-current losses than those for 50/60 Hz. Primary examples are iron-powder and ferrite cores. The lower frequency-dependant losses of these cores often is at the expense of flux density at saturation. For instance, ferrite saturation occurs at a substantially lower flux density than laminated iron.
Large power transformers are vulnerable to insulation failure due to transient voltages with high-frequency components, such as caused in switching or by lightning.
=== Energy losses ===
Transformer energy losses are dominated by winding and core losses. Transformers' efficiency tends to improve with increasing transformer capacity. The efficiency of typical distribution transformers is between about 98 and 99 percent.
As transformer losses vary with load, it is often useful to tabulate no-load loss, full-load loss, half-load loss, and so on. Hysteresis and eddy current losses are constant at all load levels and dominate at no load, while winding loss increases as load increases. The no-load loss can be significant, so that even an idle transformer constitutes a drain on the electrical supply. Designing energy efficient transformers for lower loss requires a larger core, good-quality silicon steel, or even amorphous steel for the core and thicker wire, increasing initial cost. The choice of construction represents a trade-off between initial cost and operating cost.
Transformer losses arise from:
Winding joule losses
Current flowing through a winding's conductor causes joule heating due to the resistance of the wire. As frequency increases, skin effect and proximity effect causes the winding's resistance and, hence, losses to increase.
Core losses
Hysteresis losses
Each time the magnetic field is reversed, a small amount of energy is lost due to hysteresis within the core, caused by motion of the magnetic domains within the steel. According to Steinmetz's formula, the heat energy due to hysteresis is given by
W
h
≈
η
β
max
1.6
{\displaystyle W_{\text{h}}\approx \eta \beta _{\text{max}}^{1.6}}
and,
hysteresis loss is thus given by
P
h
≈
W
h
f
≈
η
f
β
max
1.6
{\displaystyle P_{\text{h}}\approx {W}_{\text{h}}f\approx \eta {f}\beta _{\text{max}}^{1.6}}
where, f is the frequency, η is the hysteresis coefficient and βmax is the maximum flux density, the empirical exponent of which varies from about 1.4 to 1.8 but is often given as 1.6 for iron. For more detailed analysis, see Magnetic core and Steinmetz's equation.
Eddy current losses
Eddy currents are induced in the conductive metal transformer core by the changing magnetic field, and this current flowing through the resistance of the iron dissipates energy as heat in the core. The eddy current loss is a complex function of the square of supply frequency and inverse square of the material thickness. Eddy current losses can be reduced by making the core of a stack of laminations (thin plates) electrically insulated from each other, rather than a solid block; all transformers operating at low frequencies use laminated or similar cores.
Magnetostriction related transformer hum
Magnetic flux in a ferromagnetic material, such as the core, causes it to physically expand and contract slightly with each cycle of the magnetic field, an effect known as magnetostriction, the frictional energy of which produces an audible noise known as mains hum or "transformer hum". This transformer hum is especially objectionable in transformers supplied at power frequencies and in high-frequency flyback transformers associated with television CRTs.
Stray losses
Leakage inductance is by itself largely lossless, since energy supplied to its magnetic fields is returned to the supply with the next half-cycle. However, any leakage flux that intercepts nearby conductive materials such as the transformer's support structure will give rise to eddy currents and be converted to heat.
Radiative
There are also radiative losses due to the oscillating magnetic field but these are usually small.
Mechanical vibration and audible noise transmission
In addition to magnetostriction, the alternating magnetic field causes fluctuating forces between the primary and secondary windings. This energy incites vibration transmission in interconnected metalwork, thus amplifying audible transformer hum.
== Construction ==
=== Cores ===
Closed-core transformers are constructed in 'core form' or 'shell form'. When windings surround the core, the transformer is core form; when windings are surrounded by the core, the transformer is shell form. Shell form design may be more prevalent than core form design for distribution transformer applications due to the relative ease in stacking the core around winding coils. Core form design tends to, as a general rule, be more economical, and therefore more prevalent, than shell form design for high voltage power transformer applications at the lower end of their voltage and power rating ranges (less than or equal to, nominally, 230 kV or 75 MVA). At higher voltage and power ratings, shell form transformers tend to be more prevalent. Shell form design tends to be preferred for extra-high voltage and higher MVA applications because, though more labor-intensive to manufacture, shell form transformers are characterized as having inherently better kVA-to-weight ratio, better short-circuit strength characteristics and higher immunity to transit damage.
==== Laminated steel cores ====
Transformers for use at power or audio frequencies typically have cores made of high permeability silicon steel. The steel has a permeability many times that of free space and the core thus serves to greatly reduce the magnetizing current and confine the flux to a path which closely couples the windings. Early transformer developers soon realized that cores constructed from solid iron resulted in prohibitive eddy current losses, and their designs mitigated this effect with cores consisting of bundles of insulated iron wires. Later designs constructed the core by stacking layers of thin steel laminations, a principle that has remained in use. Each lamination is insulated from its neighbors by a thin non-conducting layer of insulation. The transformer universal EMF equation can be used to calculate the core cross-sectional area for a preferred level of magnetic flux.
The effect of laminations is to confine eddy currents to highly elliptical paths that enclose little flux, and so reduce their magnitude. Thinner laminations reduce losses, but are more laborious and expensive to construct. Thin laminations are generally used on high-frequency transformers, with some of very thin steel laminations able to operate up to 10 kHz.
One common design of laminated core is made from interleaved stacks of E-shaped steel sheets capped with I-shaped pieces, leading to its name of E-I transformer. Such a design tends to exhibit more losses, but is very economical to manufacture. The cut-core or C-core type is made by winding a steel strip around a rectangular form and then bonding the layers together. It is then cut in two, forming two C shapes, and the core assembled by binding the two C halves together with a steel strap. They have the advantage that the flux is always oriented parallel to the metal grains, reducing reluctance.
A steel core's remanence means that it retains a static magnetic field when power is removed. When power is then reapplied, the residual field will cause a high inrush current until the effect of the remaining magnetism is reduced, usually after a few cycles of the applied AC waveform. Overcurrent protection devices such as fuses must be selected to allow this harmless inrush to pass.
On transformers connected to long, overhead power transmission lines, induced currents due to geomagnetic disturbances during solar storms can cause saturation of the core and operation of transformer protection devices.
Distribution transformers can achieve low no-load losses by using cores made with low-loss high-permeability silicon steel or amorphous (non-crystalline) metal alloy. The higher initial cost of the core material is offset over the life of the transformer by its lower losses at light load.
==== Solid cores ====
Powdered iron cores are used in circuits such as switch-mode power supplies that operate above mains frequencies and up to a few tens of kilohertz. These materials combine high magnetic permeability with high bulk electrical resistivity. For frequencies extending beyond the VHF band, cores made from non-conductive magnetic ceramic materials called ferrites are common. Some radio-frequency transformers also have movable cores (sometimes called 'slugs') which allow adjustment of the coupling coefficient (and bandwidth) of tuned radio-frequency circuits.
==== Toroidal cores ====
Toroidal transformers are built around a ring-shaped core, which, depending on operating frequency, is made from a long strip of silicon steel or permalloy wound into a coil, powdered iron, or ferrite. A strip construction ensures that the grain boundaries are optimally aligned, improving the transformer's efficiency by reducing the core's reluctance. The closed ring shape eliminates air gaps inherent in the construction of an E-I core. : 485 The cross-section of the ring is usually square or rectangular, but more expensive cores with circular cross-sections are also available. The primary and secondary coils are often wound concentrically to cover the entire surface of the core. This minimizes the length of wire needed and provides screening to minimize the core's magnetic field from generating electromagnetic interference.
Toroidal transformers are more efficient than the cheaper laminated E-I types for a similar power level. Other advantages compared to E-I types, include smaller size (about half), lower weight (about half), less mechanical hum (making them superior in audio amplifiers), lower exterior magnetic field (about one tenth), low off-load losses (making them more efficient in standby circuits), single-bolt mounting, and greater choice of shapes. The main disadvantages are higher cost and limited power capacity (see Classification parameters below). Because of the lack of a residual gap in the magnetic path, toroidal transformers also tend to exhibit higher inrush current, compared to laminated E-I types.
Ferrite toroidal cores are used at higher frequencies, typically between a few tens of kilohertz to hundreds of megahertz, to reduce losses, physical size, and weight of inductive components. A drawback of toroidal transformer construction is the higher labor cost of winding. This is because it is necessary to pass the entire length of a coil winding through the core aperture each time a single turn is added to the coil. As a consequence, toroidal transformers rated more than a few kVA are uncommon. Relatively few toroids are offered with power ratings above 10 kVA, and practically none above 25 kVA. Small distribution transformers may achieve some of the benefits of a toroidal core by splitting it and forcing it open, then inserting a bobbin containing primary and secondary windings.
==== Air cores ====
A transformer can be produced by placing the windings near each other, an arrangement termed an "air-core" transformer. An air-core transformer eliminates loss due to hysteresis in the core material. The magnetizing inductance is drastically reduced by the lack of a magnetic core, resulting in large magnetizing currents and losses if used at low frequencies. Air-core transformers are unsuitable for use in power distribution, but are frequently employed in radio-frequency applications. Air cores are also used for resonant transformers such as tesla coils, where they can achieve reasonably low loss despite the low magnetizing inductance.
=== Windings ===
The electrical conductor used for the windings depends upon the application, but in all cases the individual turns must be electrically insulated from each other to ensure that the current travels throughout every turn. For small transformers, in which currents are low and the potential difference between adjacent turns is small, the coils are often wound from enameled magnet wire. Larger power transformers may be wound with copper rectangular strip conductors insulated by oil-impregnated paper and blocks of pressboard.
High-frequency transformers operating in the tens to hundreds of kilohertz often have windings made of braided Litz wire to minimize the skin-effect and proximity effect losses. Large power transformers use multiple-stranded conductors as well, since even at low power frequencies non-uniform distribution of current would otherwise exist in high-current windings. Each strand is individually insulated, and the strands are arranged so that at certain points in the winding, or throughout the whole winding, each portion occupies different relative positions in the complete conductor. The transposition equalizes the current flowing in each strand of the conductor, and reduces eddy current losses in the winding itself. The stranded conductor is also more flexible than a solid conductor of similar size, aiding manufacture.
The windings of signal transformers minimize leakage inductance and stray capacitance to improve high-frequency response. Coils are split into sections, and those sections interleaved between the sections of the other winding.
Power-frequency transformers may have taps at intermediate points on the winding, usually on the higher voltage winding side, for voltage adjustment. Taps may be manually reconnected, or a manual or automatic switch may be provided for changing taps. Automatic on-load tap changers are used in electric power transmission or distribution, on equipment such as arc furnace transformers, or for automatic voltage regulators for sensitive loads. Audio-frequency transformers, used for the distribution of audio to public address loudspeakers, have taps to allow adjustment of impedance to each speaker. A center-tapped transformer is often used in the output stage of an audio power amplifier in a push-pull circuit. Modulation transformers in AM transmitters are very similar.
=== Cooling ===
It is a rule of thumb that the life expectancy of electrical insulation is halved for about every 7 °C to 10 °C increase in operating temperature (an instance of the application of the Arrhenius equation).
Small dry-type and liquid-immersed transformers are often self-cooled by natural convection and radiation heat dissipation. As power ratings increase, transformers are often cooled by forced-air cooling, forced-oil cooling, water-cooling, or combinations of these. Large transformers are filled with transformer oil that both cools and insulates the windings. Transformer oil is often a highly refined mineral oil that cools the windings and insulation by circulating within the transformer tank. The mineral oil and paper insulation system has been extensively studied and used for more than 100 years. It is estimated that 50% of power transformers will survive 50 years of use, that the average age of failure of power transformers is about 10 to 15 years, and that about 30% of power transformer failures are due to insulation and overloading failures. Prolonged operation at elevated temperature degrades insulating properties of winding insulation and dielectric coolant, which not only shortens transformer life but can ultimately lead to catastrophic transformer failure. With a great body of empirical study as a guide, transformer oil testing including dissolved gas analysis provides valuable maintenance information.
Building regulations in many jurisdictions require indoor liquid-filled transformers to either use dielectric fluids that are less flammable than oil, or be installed in fire-resistant rooms. Air-cooled dry transformers can be more economical where they eliminate the cost of a fire-resistant transformer room.
The tank of liquid-filled transformers often has radiators through which the liquid coolant circulates by natural convection or fins. Some large transformers employ electric fans for forced-air cooling, pumps for forced-liquid cooling, or have heat exchangers for water-cooling. An oil-immersed transformer may be equipped with a Buchholz relay, which, depending on severity of gas accumulation due to internal arcing, is used to either trigger an alarm or de-energize the transformer. Oil-immersed transformer installations usually include fire protection measures such as walls, oil containment, and fire-suppression sprinkler systems.
Polychlorinated biphenyls (PCBs) have properties that once favored their use as a dielectric coolant, though concerns over their environmental persistence led to a widespread ban on their use.
Today, non-toxic, stable silicone-based oils, or fluorinated hydrocarbons may be used where the expense of a fire-resistant liquid offsets additional building cost for a transformer vault. However, the long life span of transformers can mean that the potential for exposure can be high long after banning.
Some transformers are gas-insulated. Their windings are enclosed in sealed, pressurized tanks and often cooled by nitrogen or sulfur hexafluoride gas.
Experimental power transformers in the 500–1,000 kVA range have been built with liquid nitrogen or helium cooled superconducting windings, which eliminates winding losses without affecting core losses.
=== Insulation ===
Insulation must be provided between the individual turns of the windings, between the windings, between windings and core, and at the terminals of the winding.
Inter-turn insulation of small transformers may be a layer of insulating varnish on the wire. Layer of paper or polymer films may be inserted between layers of windings, and between primary and secondary windings. A transformer may be coated or dipped in a polymer resin to improve the strength of windings and protect them from moisture or corrosion. The resin may be impregnated into the winding insulation using combinations of vacuum and pressure during the coating process, eliminating all air voids in the winding. In the limit, the entire coil may be placed in a mold, and resin cast around it as a solid block, encapsulating the windings.
Large oil-filled power transformers use windings wrapped with insulating paper, which is impregnated with oil during assembly of the transformer. Oil-filled transformers use highly refined mineral oil to insulate and cool the windings and core.
Construction of oil-filled transformers requires that the insulation covering the windings be thoroughly dried of residual moisture before the oil is introduced. Drying may be done by circulating hot air around the core, by circulating externally heated transformer oil, or by vapor-phase drying (VPD) where an evaporated solvent transfers heat by condensation on the coil and core. For small transformers, resistance heating by injection of current into the windings is used.
=== Bushings ===
Larger transformers are provided with high-voltage insulated bushings made of polymers or porcelain. A large bushing can be a complex structure since it must provide careful control of the electric field gradient without letting the transformer leak oil.
== Classification parameters ==
Transformers can be classified in many ways, such as the following:
Power rating: From a fraction of a volt-ampere (VA) to over a thousand MVA.
Duty of a transformer: Continuous, short-time, intermittent, periodic, varying.
Frequency range: Power-frequency, audio-frequency, or radio-frequency.
Voltage class: From a few volts to hundreds of kilovolts.
Cooling type: Dry or liquid-immersed; self-cooled, forced air-cooled;forced oil-cooled, water-cooled.
Application: power supply, impedance matching, output voltage and current stabilizer, pulse, circuit isolation, power distribution, rectifier, arc furnace, amplifier output, etc..
Basic magnetic form: Core form, shell form, concentric, sandwich.
Constant-potential transformer descriptor: Step-up, step-down, isolation.
General winding configuration: By IEC vector group, two-winding combinations of the phase designations delta, wye or star, and zigzag; autotransformer, Scott-T
Rectifier phase-shift winding configuration: 2-winding, 6-pulse; 3-winding, 12-pulse; . . ., n-winding, [n − 1]·6-pulse; polygon; etc.
K-factor: A measure of how well the transformer can withstand harmonic loads.
== Applications ==
Various specific electrical application designs require a variety of transformer types. Although they all share the basic characteristic transformer principles, they are customized in construction or electrical properties for certain installation requirements or circuit conditions.
In electric power transmission, transformers allow transmission of electric power at high voltages, which reduces the loss due to heating of the wires. This allows generating plants to be located economically at a distance from electrical consumers. All but a tiny fraction of the world's electrical power has passed through a series of transformers by the time it reaches the consumer.
In many electronic devices, a transformer is used to convert voltage from the distribution wiring to convenient values for the circuit requirements, either directly at the power line frequency or through a switch mode power supply.
Signal and audio transformers are used to couple stages of amplifiers and to match devices such as microphones and record players to the input of amplifiers. Audio transformers allowed telephone circuits to carry on a two-way conversation over a single pair of wires. A balun transformer converts a signal that is referenced to ground to a signal that has balanced voltages to ground, such as between external cables and internal circuits. Isolation transformers prevent leakage of current into the secondary circuit and are used in medical equipment and at construction sites. Resonant transformers are used for coupling between stages of radio receivers, or in high-voltage Tesla coils.
== History ==
=== Discovery of induction ===
Electromagnetic induction, the principle of the operation of the transformer, was discovered independently by Michael Faraday in 1831 and Joseph Henry in 1832. Only Faraday furthered his experiments to the point of working out the equation describing the relationship between EMF and magnetic flux now known as Faraday's law of induction:
|
E
|
=
|
d
Φ
B
d
t
|
,
{\displaystyle |{\mathcal {E}}|=\left|{{\mathrm {d} \Phi _{\text{B}}} \over \mathrm {d} t}\right|,}
where
|
E
|
{\displaystyle |{\mathcal {E}}|}
is the magnitude of the EMF in volts and ΦB is the magnetic flux through the circuit in webers.
Faraday performed early experiments on induction between coils of wire, including winding a pair of coils around an iron ring, thus creating the first toroidal closed-core transformer. However he only applied individual pulses of current to his transformer, and never discovered the relation between the turns ratio and EMF in the windings.
=== Induction coils ===
The first type of transformer to see wide use was the induction coil, invented by Irish-Catholic Rev. Nicholas Callan of Maynooth College, Ireland in 1836. He was one of the first researchers to realize the more turns the secondary winding has in relation to the primary winding, the larger the induced secondary EMF will be. Induction coils evolved from scientists' and inventors' efforts to get higher voltages from batteries. Since batteries produce direct current (DC) rather than AC, induction coils relied upon vibrating electrical contacts that regularly interrupted the current in the primary to create the flux changes necessary for induction. Between the 1830s and the 1870s, efforts to build better induction coils, mostly by trial and error, slowly revealed the basic principles of transformers.
=== First alternating current transformers ===
By the 1870s, efficient generators producing alternating current (AC) were available, and it was found AC could power an induction coil directly, without an interrupter.
In 1876, Russian engineer Pavel Yablochkov invented a lighting system based on a set of induction coils where the primary windings were connected to a source of AC. The secondary windings could be connected to several 'electric candles' (arc lamps) of his own design. The coils Yablochkov employed functioned essentially as transformers.
In 1878, the Ganz factory, Budapest, Hungary, began producing equipment for electric lighting and, by 1883, had installed over fifty systems in Austria-Hungary. Their AC systems used arc and incandescent lamps, generators, and other equipment.
In 1882, Lucien Gaulard and John Dixon Gibbs first exhibited a device with an initially widely criticized laminated plate open iron core called a 'secondary generator' in London, then sold the idea to the Westinghouse company in the United States in 1886. They also exhibited the invention in Turin, Italy in 1884, where it was highly successful and adopted for an electric lighting system. Their open-core device used a fixed 1:1 ratio to supply a series circuit for the utilization load (lamps). However, the voltage of their system was controlled by moving the iron core in or out.
==== Early series circuit transformer distribution ====
Induction coils with open magnetic circuits are inefficient at transferring power to loads. Until about 1880, the paradigm for AC power transmission from a high voltage supply to a low voltage load was a series circuit. Open-core transformers with a ratio near 1:1 were connected with their primaries in series to allow use of a high voltage for transmission while presenting a low voltage to the lamps. The inherent flaw in this method was that turning off a single lamp (or other electric device) affected the voltage supplied to all others on the same circuit. Many adjustable transformer designs were introduced to compensate for this problematic characteristic of the series circuit, including those employing methods of adjusting the core or bypassing the magnetic flux around part of a coil.
Efficient, practical transformer designs did not appear until the 1880s, but within a decade, the transformer would be instrumental in the war of the currents, and in seeing AC distribution systems triumph over their DC counterparts, a position in which they have remained dominant ever since.
=== Closed-core transformers and parallel power distribution ===
In the autumn of 1884, Károly Zipernowsky, Ottó Bláthy and Miksa Déri (ZBD), three Hungarian engineers associated with the Ganz Works, had determined that open-core devices were impracticable, as they were incapable of reliably regulating voltage. The Ganz factory had also in the autumn of 1884 made delivery of the world's first five high-efficiency AC transformers, the first of these units having been shipped on September 16, 1884. This first unit had been manufactured to the following specifications: 1,400 W, 40 Hz, 120:72 V, 11.6:19.4 A, ratio 1.67:1, one-phase, shell form. In their joint 1885 patent applications for novel transformers (later called ZBD transformers), they described two designs with closed magnetic circuits where copper windings were either wound around an iron wire ring core or surrounded by an iron wire core. The two designs were the first application of the two basic transformer constructions in common use to this day, termed "core form" or "shell form" .
In both designs, the magnetic flux linking the primary and secondary windings traveled almost entirely within the confines of the iron core, with no intentional path through air (see Toroidal cores below). The new transformers were 3.4 times more efficient than the open-core bipolar devices of Gaulard and Gibbs. The ZBD patents included two other major interrelated innovations: one concerning the use of parallel connected, instead of series connected, utilization loads, the other concerning the ability to have high turns ratio transformers such that the supply network voltage could be much higher (initially 1,400 to 2,000 V) than the voltage of utilization loads (100 V initially preferred). When employed in parallel connected electric distribution systems, closed-core transformers finally made it technically and economically feasible to provide electric power for lighting in homes, businesses and public spaces. Bláthy had suggested the use of closed cores, Zipernowsky had suggested the use of parallel shunt connections, and Déri had performed the experiments; In early 1885, the three engineers also eliminated the problem of eddy current losses with the invention of the lamination of electromagnetic cores.
Transformers today are designed on the principles discovered by the three engineers. They also popularized the word 'transformer' to describe a device for altering the EMF of an electric current although the term had already been in use by 1882. In 1886, the ZBD engineers designed, and the Ganz factory supplied electrical equipment for, the world's first power station that used AC generators to power a parallel connected common electrical network, the steam-powered Rome-Cerchi power plant.
=== Westinghouse improvements ===
Building on the advancement of AC technology in Europe, George Westinghouse founded the Westinghouse Electric in Pittsburgh, Pennsylvania, on January 8, 1886. The new firm became active in developing alternating current (AC) electric infrastructure throughout the United States.
The Edison Electric Light Company held an option on the US rights for the ZBD transformers, requiring Westinghouse to pursue alternative designs on the same principles. George Westinghouse had bought Gaulard and Gibbs' patents for $50,000 in February 1886. He assigned to William Stanley the task of redesign the Gaulard and Gibbs transformer for commercial use in United States. Stanley's first patented design was for induction coils with single cores of soft iron and adjustable gaps to regulate the EMF present in the secondary winding (see image). This design was first used commercially in the US in 1886 but Westinghouse was intent on improving the Stanley design to make it (unlike the ZBD type) easy and cheap to produce.
Westinghouse, Stanley and associates soon developed a core that was easier to manufacture, consisting of a stack of thin 'E‑shaped' iron plates insulated by thin sheets of paper or other insulating material. Pre-wound copper coils could then be slid into place, and straight iron plates laid in to create a closed magnetic circuit. Westinghouse obtained a patent for the new low-cost design in 1887.
=== Other early transformer designs ===
In 1889, Russian-born engineer Mikhail Dolivo-Dobrovolsky developed the first three-phase transformer at the Allgemeine Elektricitäts-Gesellschaft ('General Electricity Company') in Germany.
In 1891, Nikola Tesla invented the Tesla coil, an air-cored, dual-tuned resonant transformer for producing very high voltages at high frequency.
Audio frequency transformers ("repeating coils") were used by early experimenters in the development of the telephone.
== See also ==
== Notes ==
== References ==
== Bibliography ==
Beeman, Donald, ed. (1955). Industrial Power Systems Handbook. McGraw-Hill.
Calvert, James (2001). "Inside Transformers". University of Denver. Archived from the original on May 9, 2007. Retrieved May 19, 2007.
Coltman, J. W. (Jan 1988). "The Transformer". Scientific American. 258 (1): 86–95. Bibcode:1988SciAm.258a..86C. doi:10.1038/scientificamerican0188-86. OSTI 6851152.
Coltman, J. W. (Jan–Feb 2002). "History: The Transformer". IEEE Industry Applications Magazine. 8 (1): 8–15. doi:10.1109/2943.974352. S2CID 18160717.
Brenner, Egon; Javid, Mansour (1959). "Chapter 18–Circuits with Magnetic Coupling". Analysis of Electric Circuits. McGraw-Hill. pp. 586–622.
CEGB, (Central Electricity Generating Board) (1982). Modern Power Station Practice. Pergamon. ISBN 978-0-08-016436-6.
Crosby, D. (1958). "The Ideal Transformer". IRE Transactions on Circuit Theory. 5 (2): 145. doi:10.1109/TCT.1958.1086447.
Daniels, A. R. (1985). Introduction to Electrical Machines. Macmillan. ISBN 978-0-333-19627-4.
De Keulenaer, Hans; Chapman, David; Fassbinder, Stefan; McDermott, Mike (2001). The Scope for Energy Saving in the EU through the Use of Energy-Efficient Electricity Distribution Transformers (PDF). 16th International Conference and Exhibition on Electricity Distribution (CIRED 2001). Institution of Engineering and Technology. doi:10.1049/cp:20010853. Archived from the original (PDF) on 4 March 2016. Retrieved 10 July 2014.
Del Vecchio, Robert M.; Poulin, Bertrand; Feghali, Pierre T.M.; Shah, Dilipkumar; Ahuja, Rajendra (2002). Transformer Design Principles: With Applications to Core-Form Power Transformers. Boca Raton: CRC Press. ISBN 978-90-5699-703-8.
Fink, Donald G.; Beatty, H. Wayne, eds. (1978). Standard Handbook for Electrical Engineers (11th ed.). McGraw Hill. ISBN 978-0-07-020974-9.
Gottlieb, Irving (1998). Practical Transformer Handbook: for Electronics, Radio and Communications Engineers. Elsevier. ISBN 978-0-7506-3992-7.
Guarnieri, M. (2013). "Who Invented the Transformer?". IEEE Industrial Electronics Magazine. 7 (4): 56–59. doi:10.1109/MIE.2013.2283834. S2CID 27936000.
Halacsy, A. A.; Von Fuchs, G. H. (April 1961). "Transformer Invented 75 Years Ago". Transactions of the American Institute of Electrical Engineers. Part III: Power Apparatus and Systems. 80 (3): 121–125. doi:10.1109/AIEEPAS.1961.4500994. S2CID 51632693.
Hameyer, Kay (2004). Electrical Machines I: Basics, Design, Function, Operation (PDF). RWTH Aachen University Institute of Electrical Machines. Archived from the original (PDF) on 2013-02-10.
Hammond, John Winthrop (1941). Men and Volts: The Story of General Electric. J. B. Lippincott Company. pp. see esp. 106–107, 178, 238.
Harlow, James (2004). Electric Power Transformer Engineering (PDF). CRC Press. ISBN 0-8493-1704-5.
Hughes, Thomas P. (1993). Networks of Power: Electrification in Western Society, 1880-1930. Baltimore: The Johns Hopkins University Press. p. 96. ISBN 978-0-8018-2873-7. Retrieved Sep 9, 2009.
Heathcote, Martin (1998). J & P Transformer Book (12th ed.). Newnes. ISBN 978-0-7506-1158-9.
Hindmarsh, John (1977). Electrical Machines and Their Applications (4th ed.). Exeter: Pergamon. ISBN 978-0-08-030573-8.
Kothari, D.P.; Nagrath, I.J. (2010). Electric Machines (4th ed.). Tata McGraw-Hill. ISBN 978-0-07-069967-0.
Kulkarni, S. V.; Khaparde, S. A. (2004). Transformer Engineering: Design and Practice. CRC Press. ISBN 978-0-8247-5653-6.
McLaren, Peter (1984). Elementary Electric Power and Machines. Ellis Horwood. ISBN 978-0-470-20057-5.
McLyman, Colonel William (2004). "Chapter 3". Transformer and Inductor Design Handbook. CRC. ISBN 0-8247-5393-3.
Pansini, Anthony (1999). Electrical Transformers and Power Equipment. CRC Press. ISBN 978-0-88173-311-2.
Parker, M. R; Ula, S.; Webb, W. E. (2005). "§2.5.5 'Transformers' & §10.1.3 'The Ideal Transformer'". In Whitaker, Jerry C. (ed.). The Electronics Handbook (2nd ed.). Taylor & Francis. pp. 172, 1017. ISBN 0-8493-1889-0.
Ryan, H. M. (2004). High Voltage Engineering and Testing. CRC Press. ISBN 978-0-85296-775-1.
== External links ==
General links: | Wikipedia/Transformer |
The gyrator–capacitor model - sometimes also the capacitor-permeance model - is a lumped-element model for magnetic circuits, that can be used in place of the more common resistance–reluctance model. The model makes permeance elements analogous to electrical capacitance (see magnetic capacitance section) rather than electrical resistance (see magnetic reluctance). Windings are represented as gyrators, interfacing between the electrical circuit and the magnetic model.
The primary advantage of the gyrator–capacitor model compared to the magnetic reluctance model is that the model preserves the correct values of energy flow, storage and dissipation. The gyrator–capacitor model is an example of a group of analogies that preserve energy flow across energy domains by making power conjugate pairs of variables in the various domains analogous. It fills the same role as the impedance analogy for the mechanical domain.
== Nomenclature ==
Magnetic circuit may refer to either the physical magnetic circuit or the model magnetic circuit. Elements and dynamical variables that are part of the model magnetic circuit have names that start with the adjective magnetic, although this convention is not strictly followed. Elements or dynamical variables in the model magnetic circuit may not have a one to one correspondence with components in the physical magnetic circuit. Symbols for elements and variables that are part of the model magnetic circuit may be written with a subscript of M. For example,
C
M
{\displaystyle C_{M}}
would be a magnetic capacitor in the model circuit.
Electrical elements in an associated electrical circuit may be brought into the magnetic model for ease of analysis. Model elements in the magnetic circuit that represent electrical elements are typically the electrical dual of the electrical elements. This is because transducers between the electrical and magnetic domains in this model are usually represented by gyrators. A gyrator will transform an element into its dual. For example, a magnetic inductance may represent an electrical capacitance.
== Summary of analogy between magnetic circuits and electrical circuits ==
The following table summarizes the mathematical analogy between electrical circuit theory and magnetic circuit theory.
== Gyrator ==
A gyrator is a two-port element used in network analysis. The gyrator is the complement of the transformer; whereas in a transformer, a voltage on one port will transform to a proportional voltage on the other port, in a gyrator, a voltage on one port will transform to a current on the other port, and vice versa.
The role gyrators play in the gyrator–capacitor model is as transducers between the electrical energy domain and the magnetic energy domain. An emf in the electrical domain is analogous to an mmf in the magnetic domain, and a transducer doing such a conversion would be represented as a transformer. However, real electro-magnetic transducers usually behave as gyrators. A transducer from the magnetic domain to the electrical domain will obey Faraday's law of induction, that is, a rate of change of magnetic flux (a magnetic current in this analogy) produces a proportional emf in the electrical domain. Similarly, a transducer from the electrical domain to the magnetic domain will obey Ampère's circuital law, that is, an electric current will produce a mmf.
A winding of N turns is modeled by a gyrator with a gyration resistance of N ohms.: 100
Transducers that are not based on magnetic induction may not be represented by a gyrator. For instance, a Hall effect sensor is modelled by a transformer.
== Magnetic voltage ==
Magnetic voltage,
v
m
{\displaystyle v_{m}}
, is an alternate name for magnetomotive force (mmf),
F
{\displaystyle {\mathcal {F}}}
(SI unit: A or amp-turn), which is analogous to electrical voltage in an electric circuit.: 42 : 5 Not all authors use the term magnetic voltage. The magnetomotive force applied to an element between point A and point B is equal to the line integral through the component of the magnetic field strength,
H
{\displaystyle \mathbf {H} }
.
v
m
=
F
=
−
∫
A
B
H
⋅
d
l
{\displaystyle v_{m}={\mathcal {F}}=-\int _{A}^{B}\mathbf {H} \cdot d\mathbf {l} }
The resistance–reluctance model uses the same equivalence between magnetic voltage and magnetomotive force.
== Magnetic current ==
Magnetic current,
i
m
{\displaystyle i_{m}}
, is an alternate name for the time rate of change of flux,
Φ
˙
{\displaystyle {\dot {\Phi }}}
(SI unit: Wb/sec or volts), which is analogous to electrical current in an electric circuit.: 2429 : 37 In the physical circuit,
Φ
˙
{\displaystyle {\dot {\Phi }}}
, is magnetic displacement current.: 37 The magnetic current flowing through an element of cross section,
S
{\displaystyle S}
, is the area integral of the magnetic flux density
B
{\displaystyle \mathbf {B} }
.
i
m
=
Φ
˙
=
d
d
t
∫
S
B
⋅
d
S
{\displaystyle i_{m}={\dot {\Phi }}={\frac {d}{dt}}\int _{S}\mathbf {B} \cdot d\mathbf {S} }
The resistance–reluctance model uses a different equivalence, taking magnetic current to be an alternate name for flux,
Φ
{\displaystyle \Phi }
. This difference in the definition of magnetic current is the fundamental difference between the gyrator-capacitor model and the resistance–reluctance model. The definition of magnetic current and magnetic voltage imply the definitions of the other magnetic elements.: 35
== Magnetic capacitance ==
Magnetic capacitance is an alternate name for permeance, (SI unit: H). It is represented by a capacitance in the model magnetic circuit. Some authors use
C
M
{\displaystyle C_{\mathrm {M} }}
to denote magnetic capacitance while others use
P
{\displaystyle P}
and refer to the capacitance as a permeance. Permeance of an element is an extensive property defined as the magnetic flux,
Φ
{\displaystyle \Phi }
, through the cross sectional surface of the element divided by the magnetomotive force,
F
{\displaystyle {\mathcal {F}}}
, across the element': 6
C
M
=
P
=
∫
B
⋅
d
S
∫
H
⋅
d
l
=
Φ
F
{\displaystyle C_{\mathrm {M} }=P={\frac {\int \mathbf {B} \cdot d\mathbf {S} }{\int \mathbf {H} \cdot d\mathbf {l} }}={\frac {\Phi }{\mathcal {F}}}}
For a bar of uniform cross-section, magnetic capacitance is given by,
C
M
=
P
=
μ
r
μ
0
S
l
{\displaystyle C_{\mathrm {M} }=P=\mu _{\mathrm {r} }\mu _{0}{\frac {S}{l}}}
where:
μ
r
μ
0
=
μ
{\displaystyle \mu _{\mathrm {r} }\mu _{0}=\mu }
is the magnetic permeability,
S
{\displaystyle S}
is the element cross-section, and
l
{\displaystyle l}
is the element length.
For phasor analysis, the magnetic permeability and the permeance are complex values.
Permeance is the reciprocal of reluctance.
== Magnetic inductance ==
In the context of the gyrator-capacitor model of a magnetic circuit, magnetic inductance
L
M
{\displaystyle L_{\mathrm {M} }}
(SI unit: F) is the analogy to inductance in an electrical circuit.
For phasor analysis the magnetic inductive reactance is:
x
L
=
ω
L
M
{\displaystyle x_{\mathrm {L} }=\omega L_{\mathrm {M} }}
where:
L
M
{\displaystyle L_{\mathrm {M} }}
is the magnetic inductance
ω
{\displaystyle \omega }
is the angular frequency of the magnetic circuit
In the complex form it is a positive imaginary number:
j
x
L
=
j
ω
L
M
{\displaystyle jx_{\mathrm {L} }=j\omega L_{\mathrm {M} }}
The magnetic potential energy sustained by magnetic inductance varies with the frequency of oscillations in electric fields. The average power in a given period is equal to zero. Due to its dependence on frequency, magnetic inductance is mainly observable in magnetic circuits which operate at VHF and/or UHF frequencies.
The notion of magnetic inductance is employed in analysis and computation of circuit behavior in the gyrator–capacitor model in a way analogous to inductance in electrical circuits.
A magnetic inductor can represent an electrical capacitor.: 43 A shunt capacitance in the electrical circuit, such as intra-winding capacitance can be represented as a series inductance in the magnetic circuit.
== Examples ==
=== Three phase transformer ===
This example shows a three-phase transformer modeled by the gyrator-capacitor approach. The transformer in this example has three primary windings and three secondary windings. The magnetic circuit is split into seven reluctance or permeance elements. Each winding is modeled by a gyrator. The gyration resistance of each gyrator is equal to the number of turns on the associated winding. Each permeance element is modeled by a capacitor. The value of each capacitor in farads is the same as the inductance of the associated permeance in henrys.
N1, N2, and N3 are the number of turns in the three primary windings. N4, N5, and N6 are the number of turns in the three secondary windings. Φ1, Φ2, and Φ3 are the fluxes in the three vertical elements. Magnetic flux in each permeance element in webers is numerically equal to the charge in the associated capacitance in coulombs. The energy in each permeance element is the same as the energy in the associated capacitor.
The schematic shows a three phase generator and a three phase load in addition to the schematic of the transformer model.
=== Transformer with gap and leakage flux ===
The gyrator-capacitor approach can accommodate leakage inductance and air gaps in the magnetic circuit. Gaps and leakage flux have a permeance which can be added to the equivalent circuit as capacitors. The permeance of the gap is computed in the same way as the substantive elements, except a relative permeability of unity is used. The permeance of the leakage flux may be difficult to compute due to complex geometry. It may be computed from other considerations such as measurements or specifications.
CPL and CSL represent the primary and secondary leakage inductance respectively. CGAP represents the air gap permeance.
== Magnetic impedance ==
=== Magnetic complex impedance ===
Magnetic complex impedance, also called full magnetic resistance, is the quotient of a complex sinusoidal magnetic tension (magnetomotive force,
F
{\displaystyle {\mathcal {F}}}
) on a passive magnetic circuit and the resulting complex sinusoidal magnetic current (
Φ
˙
{\displaystyle {\dot {\Phi }}}
) in the circuit. Magnetic impedance is analogous to electrical impedance.
Magnetic complex impedance (SI unit: S) is determined by:
Z
M
=
F
Φ
˙
=
z
M
e
j
ϕ
{\displaystyle Z_{M}={\frac {\mathcal {F}}{\dot {\Phi }}}=z_{M}e^{j\phi }}
where
z
M
{\displaystyle z_{M}}
is the modulus of
Z
M
{\displaystyle Z_{M}}
and
ϕ
{\displaystyle \phi }
is its phase. The argument of a complex magnetic impedance is equal to the difference of the phases of the magnetic tension and magnetic current.
Complex magnetic impedance can be presented in following form:
Z
M
=
z
M
e
j
ϕ
=
z
M
cos
ϕ
+
j
z
M
sin
ϕ
=
r
M
+
j
x
M
{\displaystyle Z_{M}=z_{M}e^{j\phi }=z_{M}\cos \phi +jz_{M}\sin \phi =r_{M}+jx_{M}}
where
r
M
=
z
M
cos
ϕ
{\displaystyle r_{M}=z_{M}\cos \phi }
is the real part of the complex magnetic impedance, called the effective magnetic resistance, and
x
M
=
z
M
sin
ϕ
{\displaystyle x_{M}=z_{M}\sin \phi }
is the imaginary part of the complex magnetic impedance, called the reactive magnetic resistance.
The magnetic impedance is equal to
z
M
=
r
M
2
+
x
M
2
,
{\displaystyle z_{M}={\sqrt {r_{M}^{2}+x_{M}^{2}}},}
ϕ
=
arctan
x
M
r
M
{\displaystyle \phi =\arctan {\frac {x_{M}}{r_{M}}}}
==== Magnetic effective resistance ====
Magnetic effective resistance is the real component of complex magnetic impedance. This causes a magnetic circuit to lose magnetic potential energy. Active power in a magnetic circuit equals the product of magnetic effective resistance
r
M
{\displaystyle r_{\mathrm {M} }}
and magnetic current squared
I
M
2
{\displaystyle I_{\mathrm {M} }^{2}}
.
P
=
r
M
I
M
2
{\displaystyle P=r_{\mathrm {M} }I_{\mathrm {M} }^{2}}
The magnetic effective resistance on a complex plane appears as the side of the resistance triangle for magnetic circuit of an alternating current. The effective magnetic resistance is bounding with the effective magnetic conductance
g
M
{\displaystyle g_{\mathrm {M} }}
by the expression
g
M
=
r
M
z
M
2
{\displaystyle g_{\mathrm {M} }={\frac {r_{\mathrm {M} }}{z_{\mathrm {M} }^{2}}}}
where
z
M
{\displaystyle z_{\mathrm {M} }}
is the full magnetic impedance of a magnetic circuit.
==== Magnetic reactance ====
Magnetic reactance is the parameter of a passive magnetic circuit, or an element of the circuit, which is equal to the square root of the difference of squares of the magnetic complex impedance and magnetic effective resistance to a magnetic current, taken with the sign plus, if the magnetic current lags behind the magnetic tension in phase, and with the sign minus, if the magnetic current leads the magnetic tension in phase.
Magnetic reactance is the component of magnetic complex impedance of the alternating current circuit, which produces the phase shift between a magnetic current and magnetic tension in the circuit. It is measured in units of
1
Ω
{\displaystyle {\tfrac {1}{\Omega }}}
and is denoted by
x
{\displaystyle x}
(or
X
{\displaystyle X}
). It may be inductive
x
L
=
ω
L
M
{\displaystyle x_{L}=\omega L_{M}}
or capacitive
x
C
=
1
ω
C
M
{\displaystyle x_{C}={\tfrac {1}{\omega C_{M}}}}
, where
ω
{\displaystyle \omega }
is the angular frequency of a magnetic current,
L
M
{\displaystyle L_{M}}
is the magnetic inductiance of a circuit,
C
M
{\displaystyle C_{M}}
is the magnetic capacitance of a circuit. The magnetic reactance of an undeveloped circuit with the inductance and the capacitance which are connected in series, is equal:
x
=
x
L
−
x
C
=
ω
L
M
−
1
ω
C
M
{\textstyle x=x_{L}-x_{C}=\omega L_{M}-{\frac {1}{\omega C_{M}}}}
. If
x
L
=
x
C
{\displaystyle x_{L}=x_{C}}
, then the net reactance
x
=
0
{\displaystyle x=0}
and resonance takes place in the circuit. In the general case
x
=
z
2
−
r
2
{\textstyle x={\sqrt {z^{2}-r^{2}}}}
. When an energy loss is absent (
r
=
0
{\displaystyle r=0}
),
x
=
z
{\displaystyle x=z}
. The angle of the phase shift in a magnetic circuit
ϕ
=
arctan
x
r
{\textstyle \phi =\arctan {\frac {x}{r}}}
. On a complex plane, the magnetic reactance appears as the side of the resistance triangle for circuit of an alternating current.
== Limitations of the analogy ==
The limitations of this analogy between magnetic circuits and electric circuits include the following;
The current in typical electric circuits is confined to the circuit, with very little "leakage". In typical magnetic circuits not all of the magnetic field is confined to the magnetic circuit because magnetic permeability also exists outside materials (see vacuum permeability). Thus, there may be significant "leakage flux" in the space outside the magnetic cores. If the leakage flux is small compared to the main circuit, it can often be represented as additional elements. In extreme cases, a lumped-element model may not be appropriate at all, and field theory is used instead.
Magnetic circuits are nonlinear; the permeance in a magnetic circuit is not constant, unlike capacitance in an electrical circuit, but varies depending on the magnetic field. At high magnetic fluxes the ferromagnetic materials used for the cores of magnetic circuits saturate, limiting further increase of the magnetic flux, so above this level the permeance decreases rapidly. In addition, the flux in ferromagnetic materials is subject to hysteresis; it depends not just on the instantaneous MMF but also on the history of MMF. After the source of the magnetic flux is turned off, remanent magnetism is left in ferromagnetic materials, creating flux with no MMF.
== References == | Wikipedia/Gyrator–capacitor_model |
Electric potential energy is a potential energy (measured in joules) that results from conservative Coulomb forces and is associated with the configuration of a particular set of point charges within a defined system. An object may be said to have electric potential energy by virtue of either its own electric charge or its relative position to other electrically charged objects.
The term "electric potential energy" is used to describe the potential energy in systems with time-variant electric fields, while the term "electrostatic potential energy" is used to describe the potential energy in systems with time-invariant electric fields.
== Definition ==
The electric potential energy of a system of point charges is defined as the work required to assemble this system of charges by bringing them close together, as in the system from an infinite distance. Alternatively, the electric potential energy of any given charge or system of charges is termed as the total work done by an external agent in bringing the charge or the system of charges from infinity to the present configuration without undergoing any acceleration.
The electrostatic potential energy can also be defined from the electric potential as follows:
== Units ==
The SI unit of electric potential energy is joule (named after the English physicist James Prescott Joule). In the CGS system the erg is the unit of energy, being equal to 10−7 Joules. Also electronvolts may be used, 1 eV = 1.602×10−19 Joules.
== Electrostatic potential energy of one point charge ==
=== One point charge q in the presence of another point charge Q ===
The electrostatic potential energy, UE, of one point charge q at position r in the presence of a point charge Q, taking an infinite separation between the charges as the reference position, is:
U
E
(
r
)
=
1
4
π
ε
0
q
Q
r
{\displaystyle U_{E}(\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}{\frac {qQ}{r}}}
where r is the distance between the point charges q and Q, and q and Q are the charges (not the absolute values of the charges—i.e., an electron would have a negative value of charge when placed in the formula). The following outline of proof states the derivation from the definition of electric potential energy and Coulomb's law to this formula.
=== One point charge q in the presence of n point charges Qi ===
The electrostatic potential energy, UE, of one point charge q in the presence of n point charges Qi, taking an infinite separation between the charges as the reference position, is:
U
E
(
r
)
=
q
4
π
ε
0
∑
i
=
1
n
Q
i
r
i
,
{\displaystyle U_{E}(r)={\frac {q}{4\pi \varepsilon _{0}}}\sum _{i=1}^{n}{\frac {Q_{i}}{r_{i}}},}
where ri is the distance between the point charges q and Qi, and q and Qi are the assigned values of the charges.
== Electrostatic potential energy stored in a system of point charges ==
The electrostatic potential energy UE stored in a system of N charges q1, q2, …, qN at positions r1, r2, …, rN respectively, is:
where, for each i value, V(ri) is the electrostatic potential due to all point charges except the one at ri, and is equal to:
V
(
r
i
)
=
k
e
∑
j
≠
i
j
=
1
N
q
j
r
i
j
,
{\displaystyle V(\mathbf {r} _{i})=k_{e}\sum _{\stackrel {j=1}{j\neq i}}^{N}{\frac {q_{j}}{r_{ij}}},}
where rij is the distance between qi and qj.
=== Energy stored in a system of one point charge ===
The electrostatic potential energy of a system containing only one point charge is zero, as there are no other sources of electrostatic force against which an external agent must do work in moving the point charge from infinity to its final location.
A common question arises concerning the interaction of a point charge with its own electrostatic potential. Since this interaction doesn't act to move the point charge itself, it doesn't contribute to the stored energy of the system.
=== Energy stored in a system of two point charges ===
Consider bringing a point charge, q, into its final position near a point charge, Q1. The electric potential V(r) due to Q1 is
V
(
r
)
=
k
e
Q
1
r
{\displaystyle V(\mathbf {r} )=k_{e}{\frac {Q_{1}}{r}}}
Hence we obtain, the electrostatic potential energy of q in the potential of Q1 as
U
E
=
1
4
π
ε
0
q
Q
1
r
1
{\displaystyle U_{E}={\frac {1}{4\pi \varepsilon _{0}}}{\frac {qQ_{1}}{r_{1}}}}
where r1 is the separation between the two point charges.
=== Energy stored in a system of three point charges ===
The electrostatic potential energy of a system of three charges should not be confused with the electrostatic potential energy of Q1 due to two charges Q2 and Q3, because the latter doesn't include the electrostatic potential energy of the system of the two charges Q2 and Q3.
The electrostatic potential energy stored in the system of three charges is:
U
E
=
1
4
π
ε
0
[
Q
1
Q
2
r
12
+
Q
1
Q
3
r
13
+
Q
2
Q
3
r
23
]
{\displaystyle U_{\mathrm {E} }={\frac {1}{4\pi \varepsilon _{0}}}\left[{\frac {Q_{1}Q_{2}}{r_{12}}}+{\frac {Q_{1}Q_{3}}{r_{13}}}+{\frac {Q_{2}Q_{3}}{r_{23}}}\right]}
== Energy stored in an electrostatic field distribution in vacuum ==
The energy density, or energy per unit volume,
d
U
d
V
{\textstyle {\frac {dU}{dV}}}
, of the electrostatic field of a continuous charge distribution is:
u
e
=
d
U
d
V
=
1
2
ε
0
|
E
|
2
.
{\displaystyle u_{e}={\frac {dU}{dV}}={\frac {1}{2}}\varepsilon _{0}\left|{\mathbf {E} }\right|^{2}.}
== Energy stored in electronic elements ==
Some elements in a circuit can convert energy from one form to another. For example, a resistor converts electrical energy to heat. This is known as the Joule effect. A capacitor stores it in its electric field. The total electrostatic potential energy stored in a capacitor is given by
U
E
=
1
2
Q
V
=
1
2
C
V
2
=
Q
2
2
C
{\displaystyle U_{E}={\frac {1}{2}}QV={\frac {1}{2}}CV^{2}={\frac {Q^{2}}{2C}}}
where C is the capacitance, V is the electric potential difference, and Q the charge stored in the capacitor.
The total electrostatic potential energy may also be expressed in terms of the electric field in the form
U
E
=
1
2
∫
V
E
⋅
D
d
V
{\displaystyle U_{E}={\frac {1}{2}}\int _{V}\mathrm {E} \cdot \mathrm {D} \,dV}
where
D
{\displaystyle \mathrm {D} }
is the electric displacement field within a dielectric material and integration is over the entire volume of the dielectric.
The total electrostatic potential energy stored within a charged dielectric may also be expressed in terms of a continuous volume charge,
ρ
{\displaystyle \rho }
,
U
E
=
1
2
∫
V
ρ
Φ
d
V
{\displaystyle U_{E}={\frac {1}{2}}\int _{V}\rho \Phi \,dV}
where integration is over the entire volume of the dielectric.
These latter two expressions are valid only for cases when the smallest increment of charge is zero (
d
q
→
0
{\displaystyle dq\to 0}
) such as dielectrics in the presence of metallic electrodes or dielectrics containing many charges.
Note that a virtual experiment based on the energy transfer between capacitor plates reveals that an additional term should be taken into account when dealing with semiconductors for instance. While this extra energy cancels when dealing with insulators, the derivation predicts that it cannot be ignored as it may exceed the polarization energy.
== Notes ==
== References ==
== External links ==
Media related to Electric potential energy at Wikimedia Commons | Wikipedia/Electric_Potential_Energy |
In physics, the magnetomotive force (abbreviated mmf or MMF, symbol
F
{\displaystyle {\mathcal {F}}}
) is a quantity appearing in the equation for the magnetic flux in a magnetic circuit, Hopkinson's law. It is the property of certain substances or phenomena that give rise to magnetic fields:
F
=
Φ
R
,
{\displaystyle {\mathcal {F}}=\Phi {\mathcal {R}},}
where Φ is the magnetic flux and
R
{\displaystyle {\mathcal {R}}}
is the reluctance of the circuit. It can be seen that the magnetomotive force plays a role in this equation analogous to the voltage V in Ohm's law, V = IR, since it is the cause of magnetic flux in a magnetic circuit:
F
=
N
I
{\displaystyle {\mathcal {F}}=NI}
where N is the number of turns in a coil and I is the electric current through the coil.
F
=
Φ
R
{\displaystyle {\mathcal {F}}=\Phi {\mathcal {R}}}
where Φ is the magnetic flux and
R
{\displaystyle {\mathcal {R}}}
is the magnetic reluctance
F
=
H
L
{\displaystyle {\mathcal {F}}=HL}
where H is the magnetizing force (the strength of the magnetizing field) and L is the mean length of a solenoid or the circumference of a toroid.
== Units ==
The SI unit of mmf is the ampere, the same as the unit of current (analogously the units of emf and voltage are both the volt). Informally, and frequently, this unit is stated as the ampere-turn to avoid confusion with current. This was the unit name in the MKS system. Occasionally, the cgs system unit of the gilbert may also be encountered.
== History ==
The term magnetomotive force was coined by Henry Augustus Rowland in 1880. Rowland intended this to indicate a direct analogy with electromotive force. The idea of a magnetic analogy to electromotive force can be found much earlier in the work of Michael Faraday (1791–1867) and it is hinted at by James Clerk Maxwell (1831–1879). However, Rowland coined the term and was the first to make explicit an Ohm's law for magnetic circuits in 1873.
Ohm's law for magnetic circuits is sometimes referred to as Hopkinson's law rather than Rowland's law as some authors attribute the law to John Hopkinson instead of Rowland. According to a review of magnetic circuit analysis methods this is an incorrect attribution originating from an 1885 paper by Hopkinson. Furthermore, Hopkinson actually cites Rowland's 1873 paper in this work.
== References ==
== Bibliography ==
=== Cited sources ===
Hon, Giora; Goldstein, Bernard R, "Symmetry and asymmetry in electrodynamics from Rowland to Einstein", Studies in History and Philosophy of Modern Physics, vol. 37, iss. 4, pp. 635–660, Elsevier December 2006.
Hopkinson, John, "Magnetisation of iron", Philosophical Transactions of the Royal Society, vol. 176, pp. 455–469, 1885.
Lambert, Mathieu; Mahseredjian, Jean; Martínez-Duró, Manuel; Sirois, Frédéric, "Magnetic circuits within electric circuits: critical review of existing methods and new mutator implementations", IEEE Transactions on Power Delivery, vol. 30, iss. 6, pp. 2427–2434, December 2015.
Newell, David B.; Tiesinga, Eite, eds. (2019). NIST Special Publication 330: The International System of Units (SI) (Standards publication) (2019 ed.). National Institute of Standards and Technology. doi:10.6028/NIST.SP.330-2019.
Rowland, Henry A, "On magnetic permeability and the maximum magnetism of iron, steel, and nickel", Philosophical Magazine, series 4, vol. 46, no. 304, pp. 140–159, August 1873.
Rowland, Henry A, "On the general equations of electro-magnetic action, with application to a new theory of magnetic attractions, and to the theory of the magnetic rotation of the plane of polarization of light" (part 2), American Journal of Mathematics, vol. 3, nos. 1–2, pp. 89–113, March 1880.
Schmidt, Robert Munnig; Schitter, Georg, "Electromechanical actuators", ch. 5 in Schmidt, Robert Munnig; Schitter, Georg; Rankers, Adrian; van Eijk, Jan, The Design of High Performance Mechatronics, IOS Press, 2014 ISBN 1614993688.
Thompson, Silvanus Phillips, The Electromagnet and Electromagnetic Mechanism, Cambridge University Press, 2011 (first published 1891) ISBN 1108029213.
Smith, R.J. (1966), Circuits, Devices and Systems, Chapter 15, Wiley International Edition, New York. Library of Congress Catalog Card No. 66-17612
Waygood, Adrian, An Introduction to Electrical Science, Routledge, 2013 ISBN 1135071136.
=== General references ===
The Penguin Dictionary of Physics, 1977, ISBN 0-14-051071-0
A Textbook of Electrical Technology, 2008, ISBN 81-219-2440-5 | Wikipedia/Magnetomotive_force |
In relativistic physics, the electromagnetic stress–energy tensor is the contribution to the stress–energy tensor due to the electromagnetic field. The stress–energy tensor describes the flow of energy and momentum in spacetime. The electromagnetic stress–energy tensor contains the negative of the classical Maxwell stress tensor that governs the electromagnetic interactions.
== Definition ==
=== ISQ convention ===
The electromagnetic stress–energy tensor in the International System of Quantities (ISQ), which underlies the SI, is
T
μ
ν
=
1
μ
0
[
F
μ
α
F
ν
α
−
1
4
η
μ
ν
F
α
β
F
α
β
]
,
{\displaystyle T^{\mu \nu }={\frac {1}{\mu _{0}}}\left[F^{\mu \alpha }F^{\nu }{}_{\alpha }-{\frac {1}{4}}\eta ^{\mu \nu }F_{\alpha \beta }F^{\alpha \beta }\right]\,,}
where
F
μ
ν
{\displaystyle F^{\mu \nu }}
is the electromagnetic tensor and where
η
μ
ν
{\displaystyle \eta _{\mu \nu }}
is the Minkowski metric tensor of metric signature (− + + +) and the Einstein summation convention over repeated indices is used.
Explicitly in matrix form:
T
μ
ν
=
[
u
1
c
S
x
1
c
S
y
1
c
S
z
1
c
S
x
−
σ
xx
−
σ
xy
−
σ
xz
1
c
S
y
−
σ
yx
−
σ
yy
−
σ
yz
1
c
S
z
−
σ
zx
−
σ
zy
−
σ
zz
]
,
{\displaystyle T^{\mu \nu }={\begin{bmatrix}u&{\frac {1}{c}}S_{\text{x}}&{\frac {1}{c}}S_{\text{y}}&{\frac {1}{c}}S_{\text{z}}\\{\frac {1}{c}}S_{\text{x}}&-\sigma _{\text{xx}}&-\sigma _{\text{xy}}&-\sigma _{\text{xz}}\\{\frac {1}{c}}S_{\text{y}}&-\sigma _{\text{yx}}&-\sigma _{\text{yy}}&-\sigma _{\text{yz}}\\{\frac {1}{c}}S_{\text{z}}&-\sigma _{\text{zx}}&-\sigma _{\text{zy}}&-\sigma _{\text{zz}}\end{bmatrix}},}
where
u
=
1
2
(
ϵ
0
E
2
+
1
μ
0
B
2
)
{\displaystyle u={\frac {1}{2}}\left(\epsilon _{0}\mathbf {E} ^{2}+{\frac {1}{\mu _{0}}}\mathbf {B} ^{2}\right)}
is the volumetric energy density,
S
=
1
μ
0
E
×
B
{\displaystyle \mathbf {S} ={\frac {1}{\mu _{0}}}\mathbf {E} \times \mathbf {B} }
is the Poynting vector,
σ
i
j
=
ϵ
0
E
i
E
j
+
1
μ
0
B
i
B
j
−
1
2
(
ϵ
0
E
2
+
1
μ
0
B
2
)
δ
i
j
{\displaystyle \sigma _{ij}=\epsilon _{0}E_{i}E_{j}+{\frac {1}{\mu _{0}}}B_{i}B_{j}-{\frac {1}{2}}\left(\epsilon _{0}\mathbf {E} ^{2}+{\frac {1}{\mu _{0}}}\mathbf {B} ^{2}\right)\delta _{ij}}
is the Maxwell stress tensor, and
c
{\displaystyle c}
is the speed of light. Thus, each component of
T
μ
ν
{\displaystyle T^{\mu \nu }}
is dimensionally equivalent to pressure (with SI unit pascal).
=== Gaussian CGS conventions ===
The in the Gaussian system (shown here with a prime) that correspond to the permittivity of free space and permeability of free space are
ϵ
0
′
=
1
4
π
,
μ
0
′
=
4
π
{\displaystyle \epsilon _{0}'={\frac {1}{4\pi }},\quad \mu _{0}'=4\pi }
then:
T
μ
ν
=
1
4
π
[
F
′
μ
α
F
′
ν
α
−
1
4
η
μ
ν
F
α
β
′
F
′
α
β
]
{\displaystyle T^{\mu \nu }={\frac {1}{4\pi }}\left[F'^{\mu \alpha }F'^{\nu }{}_{\alpha }-{\frac {1}{4}}\eta ^{\mu \nu }F'_{\alpha \beta }F'^{\alpha \beta }\right]}
and in explicit matrix form:
T
μ
ν
=
[
u
1
c
S
x
1
c
S
y
1
c
S
z
1
c
S
x
−
σ
xx
−
σ
xy
−
σ
xz
1
c
S
y
−
σ
yx
−
σ
yy
−
σ
yz
1
c
S
z
−
σ
zx
−
σ
zy
−
σ
zz
]
{\displaystyle T^{\mu \nu }={\begin{bmatrix}u&{\frac {1}{c}}S_{\text{x}}&{\frac {1}{c}}S_{\text{y}}&{\frac {1}{c}}S_{\text{z}}\\{\frac {1}{c}}S_{\text{x}}&-\sigma _{\text{xx}}&-\sigma _{\text{xy}}&-\sigma _{\text{xz}}\\{\frac {1}{c}}S_{\text{y}}&-\sigma _{\text{yx}}&-\sigma _{\text{yy}}&-\sigma _{\text{yz}}\\{\frac {1}{c}}S_{\text{z}}&-\sigma _{\text{zx}}&-\sigma _{\text{zy}}&-\sigma _{\text{zz}}\end{bmatrix}}}
where the energy density becomes
u
=
1
8
π
(
E
′
2
+
B
′
2
)
{\displaystyle u={\frac {1}{8\pi }}\left(\mathbf {E} '^{2}+\mathbf {B} '^{2}\right)}
and the Poynting vector becomes
S
=
c
4
π
E
′
×
B
′
.
{\displaystyle \mathbf {S} ={\frac {c}{4\pi }}\mathbf {E} '\times \mathbf {B} '.}
The stress–energy tensor for an electromagnetic field in a dielectric medium is less well understood and is the subject of the Abraham–Minkowski controversy.
The element
T
μ
ν
{\displaystyle T^{\mu \nu }}
of the stress–energy tensor represents the flux of the component with index
μ
{\displaystyle \mu }
of the four-momentum of the electromagnetic field,
P
μ
{\displaystyle P^{\mu }}
, going through a hyperplane. It represents the contribution of electromagnetism to the source of the gravitational field (curvature of spacetime) in general relativity.
== Algebraic properties ==
The electromagnetic stress–energy tensor has several algebraic properties:
The symmetry of the tensor is as for a general stress–energy tensor in general relativity. The trace of the energy–momentum tensor is a Lorentz scalar; the electromagnetic field (and in particular electromagnetic waves) has no Lorentz-invariant energy scale, so its energy–momentum tensor must have a vanishing trace. This tracelessness eventually relates to the masslessness of the photon.
== Conservation laws ==
The electromagnetic stress–energy tensor allows a compact way of writing the conservation laws of linear momentum and energy in electromagnetism. The divergence of the stress–energy tensor is:
∂
ν
T
μ
ν
+
η
μ
ρ
f
ρ
=
0
{\displaystyle \partial _{\nu }T^{\mu \nu }+\eta ^{\mu \rho }\,f_{\rho }=0\,}
where
f
ρ
{\displaystyle f_{\rho }}
is the (4D) Lorentz force per unit volume on matter.
This equation is equivalent to the following 3D conservation laws
∂
u
e
m
∂
t
+
∇
⋅
S
+
J
⋅
E
=
0
∂
p
e
m
∂
t
−
∇
⋅
σ
+
ρ
E
+
J
×
B
=
0
⇔
ϵ
0
μ
0
∂
S
∂
t
−
∇
⋅
σ
+
f
=
0
{\displaystyle {\begin{aligned}{\frac {\partial u_{\mathrm {em} }}{\partial t}}+\mathbf {\nabla } \cdot \mathbf {S} +\mathbf {J} \cdot \mathbf {E} &=0\\{\frac {\partial \mathbf {p} _{\mathrm {em} }}{\partial t}}-\mathbf {\nabla } \cdot \sigma +\rho \mathbf {E} +\mathbf {J} \times \mathbf {B} &=0\ \Leftrightarrow \ \epsilon _{0}\mu _{0}{\frac {\partial \mathbf {S} }{\partial t}}-\nabla \cdot \mathbf {\sigma } +\mathbf {f} =0\end{aligned}}}
respectively describing the electromagnetic energy density
u
e
m
=
1
2
(
ϵ
0
E
2
+
1
μ
0
B
2
)
{\displaystyle u_{\mathrm {em} }={\frac {1}{2}}\left(\epsilon _{0}\mathbf {E} ^{2}+{\frac {1}{\mu _{0}}}\mathbf {B} ^{2}\right)}
and electromagnetic momentum density
p
e
m
=
S
c
2
,
{\displaystyle \mathbf {p} _{\mathrm {em} }={\mathbf {S} \over {c^{2}}},}
where
J
{\displaystyle \mathbf {J} }
is the electric current density,
ρ
{\displaystyle \rho }
the electric charge density, and
f
{\displaystyle \mathbf {f} }
is the Lorentz force density.
== See also ==
Ricci calculus
Covariant formulation of classical electromagnetism
Mathematical descriptions of the electromagnetic field
Maxwell's equations
Maxwell's equations in curved spacetime
General relativity
Einstein field equations
Magnetohydrodynamics
Vector calculus
== References == | Wikipedia/Electromagnetic_stress–energy_tensor |
An electrical network is an interconnection of electrical components (e.g., batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e.g., voltage sources, current sources, resistances, inductances, capacitances). An electrical circuit is a network consisting of a closed loop, giving a return path for the current. Thus all circuits are networks, but not all networks are circuits (although networks without a closed loop are often imprecisely referred to as "circuits").
A resistive network is a network containing only resistors and ideal current and voltage sources. Analysis of resistive networks is less complicated than analysis of networks containing capacitors and inductors. If the sources are constant (DC) sources, the result is a DC network. The effective resistance and current distribution properties of arbitrary resistor networks can be modeled in terms of their graph measures and geometrical properties.
A network that contains active electronic components is known as an electronic circuit. Such networks are generally nonlinear and require more complex design and analysis tools.
== Classification ==
=== By passivity ===
An active network contains at least one voltage source or current source that can supply energy to the network indefinitely. A passive network does not contain an active source.
An active network contains one or more sources of electromotive force. Practical examples of such sources include a battery or a generator. Active elements can inject power to the circuit, provide power gain, and control the current flow within the circuit.
Passive networks do not contain any sources of electromotive force. They consist of passive elements like resistors and capacitors.
=== By linearity ===
Linear electrical networks, a special type consisting only of sources (voltage or current), linear lumped elements (resistors, capacitors, inductors), and linear distributed elements (transmission lines), have the property that signals are linearly superimposable. They are thus more easily analyzed, using powerful frequency domain methods such as Laplace transforms, to determine DC response, AC response, and transient response.
Passive networks are generally taken to be linear, but there are exceptions. For instance, an inductor with an iron core can be driven into saturation if driven with a large enough current. In this region, the behaviour of the inductor is very non-linear.
=== By lumpiness ===
Discrete passive components (resistors, capacitors and inductors) are called lumped elements because all of their, respectively, resistance, capacitance and inductance is assumed to be located ("lumped") at one place. This design philosophy is called the lumped-element model and networks so designed are called lumped-element circuits. This is the conventional approach to circuit design. At high enough frequencies, or for long enough circuits (such as power transmission lines), the lumped assumption no longer holds because there is a significant fraction of a wavelength across the component dimensions. A new design model is needed for such cases called the distributed-element model. Networks designed to this model are called distributed-element circuits.
A distributed-element circuit that includes some lumped components is called a semi-lumped design. An example of a semi-lumped circuit is the combline filter.
== Classification of sources ==
Sources can be classified as independent sources and dependent sources.
=== Independent ===
An ideal independent source maintains the same voltage or current regardless of the other elements present in the circuit. Its value is either constant (DC) or sinusoidal (AC). The strength of voltage or current is not changed by any variation in the connected network.
=== Dependent ===
Dependent sources depend upon a particular element of the circuit for delivering the power or voltage or current depending upon the type of source it is.
== Applying electrical laws ==
A number of electrical laws apply to all linear resistive networks. These include:
Kirchhoff's current law: The sum of all currents entering a node is equal to the sum of all currents leaving the node.
Kirchhoff's voltage law: The directed sum of the electrical potential differences around a loop must be zero.
Ohm's law: The voltage across a resistor is equal to the product of the resistance and the current flowing through it.
Norton's theorem: Any network of voltage or current sources and resistors is electrically equivalent to an ideal current source in parallel with a single resistor.
Thévenin's theorem: Any network of voltage or current sources and resistors is electrically equivalent to a single voltage source in series with a single resistor.
Superposition theorem: In a linear network with several independent sources, the response in a particular branch when all the sources are acting simultaneously is equal to the linear sum of individual responses calculated by taking one independent source at a time.
Applying these laws results in a set of simultaneous equations that can be solved either algebraically or numerically. The laws can generally be extended to networks containing reactances. They cannot be used in networks that contain nonlinear or time-varying components.
== Design methods ==
To design any electrical circuit, either analog or digital, electrical engineers need to be able to predict the voltages and currents at all places within the circuit. Simple linear circuits can be analyzed by hand using complex number theory. In more complex cases the circuit may be analyzed with specialized computer programs or estimation techniques such as the piecewise-linear model.
Circuit simulation software, such as HSPICE (an analog circuit simulator), and languages such as VHDL-AMS and verilog-AMS allow engineers to design circuits without the time, cost and risk of error involved in building circuit prototypes.
== Network simulation software ==
More complex circuits can be analyzed numerically with software such as SPICE or GNUCAP, or symbolically using software such as SapWin.
=== Linearization around operating point ===
When faced with a new circuit, the software first tries to find a steady state solution, that is, one where all nodes conform to Kirchhoff's current law and the voltages across and through each element of the circuit conform to the voltage/current equations governing that element.
Once the steady state solution is found, the operating points of each element in the circuit are known. For a small signal analysis, every non-linear element can be linearized around its operation point to obtain the small-signal estimate of the voltages and currents. This is an application of Ohm's Law. The resulting linear circuit matrix can be solved with Gaussian elimination.
=== Piecewise-linear approximation ===
Software such as the PLECS interface to Simulink uses piecewise-linear approximation of the equations governing the elements of a circuit. The circuit is treated as a completely linear network of ideal diodes. Every time a diode switches from on to off or vice versa, the configuration of the linear network changes. Adding more detail to the approximation of equations increases the accuracy of the simulation, but also increases its running time.
== See also ==
Digital circuit
Ground (electricity)
Impedance
Load
Memristor
Open-circuit voltage
Short circuit
Voltage drop
=== Representation ===
Circuit diagram
Schematic
Netlist
=== Design and analysis methodologies ===
Network analysis (electrical circuits)
Mathematical methods in electronics
Superposition theorem
Topology (electronics)
Mesh analysis
Prototype filter
=== Measurement ===
Network analyzer (electrical)
Network analyzer (AC power)
Continuity test
=== Analogies ===
Hydraulic analogy
Mechanical–electrical analogies
Impedance analogy (Maxwell analogy)
Mobility analogy (Firestone analogy)
Through and across analogy (Trent analogy)
=== Specific topologies ===
Bridge circuit
LC circuit
RC circuit
RL circuit
RLC circuit
Potential divider
Series and parallel circuits
== References == | Wikipedia/Electrical_network |
Physics of Fluids is a monthly peer-reviewed scientific journal covering fluid dynamics, established by the American Institute of Physics in 1958, and is published by AIP Publishing. The journal focus is the dynamics of gases, liquids, and complex or multiphase fluids—and the journal contains original research resulting from theoretical, computational, and experimental studies.
== History ==
From 1958 through 1988, the journal included plasma physics. From 1989 until 1993, the journal split into Physics of Fluids A covering fluid dynamics, and Physics of Fluids B, on plasma physics. In 1994, the latter was renamed Physics of Plasmas, and the former continued under its original name, Physics of Fluids.
The journal was originally published by the American Institute of Physics in cooperation with the American Physical Society's Division of Fluid Dynamics. In 2016, the American Institute of Physics became the sole publisher. From 1985 to 2015, Physics of Fluids published the Gallery of Fluid Motion, containing award-winning photographs, images, and visual streaming media of fluid flow.
With funding from the American Institute of Physics the annual "François Naftali Frenkiel Award" was established by the American Physical Society in 1984 to reward a young scientist who published a paper containing significant contributions to fluid dynamics during the previous year. The award-winning paper was chosen from Physics of Fluids until 2016, but is presently chosen from Physical Review Fluids. Similarly, the invited papers from plenary talks at the annual American Physical Society Division of Fluid Dynamics were formerly published in Physics of Fluids but, since 2016, are now published in either this journal or Physical Review Fluids.
== Reception ==
Physics of Fluids A, Physics of Fluids B, and Physics of Fluids were ranked 3, 4, and 6, respectively based on their citation impact from 1981 to 2004 within the category of journals on the physics of fluids and plasmas. According to the Journal Citation Reports, the journal has a 2023 impact factor of 4.1.
== Editors-in-chief ==
The following persons are or have been editors-in-chief:
1958–1981: François Naftali Frenkiel
1982–1997: Andreas Acrivos
1998–2015: John Kim, L. Gary Leal
2016–present: Alan Jeffrey Giacomin
== See also ==
List of fluid mechanics journals
== References ==
== Further reading ==
Scott, John T. (2008), "Fifty years of Physics of Fluids", Physics of Fluids, 20 (1): 011301–011301–4, Bibcode:2008PhFl...20a1301S, doi:10.1063/1.2832774, ISSN 1070-6631
Kim, John; Leal, L. Gary (2008), "Editorial: Fifty years of Physics of Fluids", Physics of Fluids, 20 (1): 010401–010401–5, Bibcode:2008PhFl...20a0401K, doi:10.1063/1.2832366, ISSN 1070-6631
== External links ==
Official website
Physics of Fluids' Gallery of fluid motion | Wikipedia/Physics_of_Fluids |
In electromagnetism, Jefimenko's equations (named after Oleg D. Jefimenko) give the electric field and magnetic field due to a distribution of electric charges and electric current in space, that takes into account the propagation delay (retarded time) of the fields due to the finite speed of light and relativistic effects. Therefore, they can be used for moving charges and currents. They are the particular solutions to Maxwell's equations for any arbitrary distribution of charges and currents.
== Equations ==
=== Electric and magnetic fields ===
Jefimenko's equations give the electric field E and magnetic field B produced by an arbitrary charge or current distribution, of charge density ρ and current density J:
E
(
r
,
t
)
=
1
4
π
ε
0
∫
[
r
−
r
′
|
r
−
r
′
|
3
ρ
(
r
′
,
t
r
)
+
r
−
r
′
|
r
−
r
′
|
2
1
c
∂
ρ
(
r
′
,
t
r
)
∂
t
−
1
|
r
−
r
′
|
1
c
2
∂
J
(
r
′
,
t
r
)
∂
t
]
d
V
′
,
{\displaystyle \mathbf {E} (\mathbf {r} ,t)={\frac {1}{4\pi \varepsilon _{0}}}\int \left[{\frac {\mathbf {r} -\mathbf {r} '}{|\mathbf {r} -\mathbf {r} '|^{3}}}\rho (\mathbf {r} ',t_{r})+{\frac {\mathbf {r} -\mathbf {r} '}{|\mathbf {r} -\mathbf {r} '|^{2}}}{\frac {1}{c}}{\frac {\partial \rho (\mathbf {r} ',t_{r})}{\partial t}}-{\frac {1}{|\mathbf {r} -\mathbf {r} '|}}{\frac {1}{c^{2}}}{\frac {\partial \mathbf {J} (\mathbf {r} ',t_{r})}{\partial t}}\right]dV',}
B
(
r
,
t
)
=
−
μ
0
4
π
∫
[
r
−
r
′
|
r
−
r
′
|
3
×
J
(
r
′
,
t
r
)
+
r
−
r
′
|
r
−
r
′
|
2
×
1
c
∂
J
(
r
′
,
t
r
)
∂
t
]
d
V
′
,
{\displaystyle \mathbf {B} (\mathbf {r} ,t)=-{\frac {\mu _{0}}{4\pi }}\int \left[{\frac {\mathbf {r} -\mathbf {r} '}{|\mathbf {r} -\mathbf {r} '|^{3}}}\times \mathbf {J} (\mathbf {r} ',t_{r})+{\frac {\mathbf {r} -\mathbf {r} '}{|\mathbf {r} -\mathbf {r} '|^{2}}}\times {\frac {1}{c}}{\frac {\partial \mathbf {J} (\mathbf {r} ',t_{r})}{\partial t}}\right]dV',}
where r′ is a point in the charge distribution, r is a point in space, and
t
r
=
t
−
|
r
−
r
′
|
c
{\displaystyle t_{r}=t-{\frac {|\mathbf {r} -\mathbf {r} '|}{c}}}
is the retarded time. There are similar expressions for D and H.
These equations are the time-dependent generalization of Coulomb's law and the Biot–Savart law to electrodynamics, which were originally true only for electrostatic and magnetostatic fields, and steady currents.
=== Origin from retarded potentials ===
Jefimenko's equations can be found from the retarded potentials φ and A:
φ
(
r
,
t
)
=
1
4
π
ε
0
∫
ρ
(
r
′
,
t
r
)
|
r
−
r
′
|
d
V
′
,
A
(
r
,
t
)
=
μ
0
4
π
∫
J
(
r
′
,
t
r
)
|
r
−
r
′
|
d
V
′
,
{\displaystyle {\begin{aligned}&\varphi (\mathbf {r} ,t)={\dfrac {1}{4\pi \varepsilon _{0}}}\int {\dfrac {\rho (\mathbf {r} ',t_{r})}{|\mathbf {r} -\mathbf {r} '|}}dV',\\&\mathbf {A} (\mathbf {r} ,t)={\dfrac {\mu _{0}}{4\pi }}\int {\dfrac {\mathbf {J} (\mathbf {r} ',t_{r})}{|\mathbf {r} -\mathbf {r} '|}}dV',\end{aligned}}}
which are the solutions to Maxwell's equations in the potential formulation, then substituting in the definitions of the electromagnetic potentials themselves:
E
=
−
∇
φ
−
∂
A
∂
t
,
B
=
∇
×
A
{\displaystyle \mathbf {E} =-\nabla \varphi -{\dfrac {\partial \mathbf {A} }{\partial t}}\,,\quad \mathbf {B} =\nabla \times \mathbf {A} }
and using the relation
c
2
=
1
ε
0
μ
0
{\displaystyle c^{2}={\frac {1}{\varepsilon _{0}\mu _{0}}}}
replaces the potentials φ and A by the fields E and B.
== Heaviside–Feynman formula ==
The Heaviside–Feynman formula, also known as the Jefimenko–Feynman formula, can be seen as the point-like electric charge version of Jefimenko's equations. Actually, it can be (non trivially) deduced from them using Dirac functions, or using the Liénard-Wiechert potentials. It is mostly known from The Feynman Lectures on Physics, where it was used to introduce and describe the origin of electromagnetic radiation. The formula provides a natural generalization of the Coulomb's law for cases where the source charge is moving:
E
=
−
q
4
π
ε
0
[
e
r
′
r
′
2
+
r
′
c
d
d
t
(
e
r
′
r
′
2
)
+
1
c
2
d
2
d
t
2
e
r
′
]
{\displaystyle \mathbf {E} ={\frac {-q}{4\pi \varepsilon _{0}}}\left[{\frac {\mathbf {e} _{r'}}{r'^{2}}}+{\frac {r'}{c}}{\frac {d}{dt}}\left({\frac {\mathbf {e} _{r'}}{r'^{2}}}\right)+{\frac {1}{c^{2}}}{\frac {d^{2}}{dt^{2}}}\mathbf {e} _{r'}\right]}
B
=
−
e
r
′
×
E
c
{\displaystyle \mathbf {B} =-\mathbf {e} _{r'}\times {\frac {\mathbf {E} }{c}}}
Here,
E
{\displaystyle \mathbf {E} }
and
B
{\displaystyle \mathbf {B} }
are the electric and magnetic fields respectively,
q
{\displaystyle q}
is the electric charge,
ε
0
{\displaystyle \varepsilon _{0}}
is the vacuum permittivity (electric field constant) and
c
{\displaystyle c}
is the speed of light. The vector
e
r
′
{\displaystyle \mathbf {e} _{r'}}
is a unit vector pointing from the observer to the charge and
r
′
{\displaystyle r'}
is the distance between observer and charge. Since the electromagnetic field propagates at the speed of light, both these quantities are evaluated at the retarded time
t
−
r
′
/
c
{\displaystyle t-r'/c}
.
The first term in the formula for
E
{\displaystyle \mathbf {E} }
represents the Coulomb's law for the static electric field. The second term is the time derivative of the first Coulombic term multiplied by
r
′
c
{\displaystyle {\frac {r'}{c}}}
which is the propagation time of the electric field. Heuristically, this can be regarded as nature "attempting" to forecast what the present field would be by linear extrapolation to the present time. The last term, proportional to the second derivative of the unit direction vector
e
r
′
{\displaystyle e_{r'}}
, is sensitive to charge motion perpendicular to the line of sight. It can be shown that the electric field generated by this term is proportional to
a
t
/
r
′
{\displaystyle a_{t}/r'}
, where
a
t
{\displaystyle a_{t}}
is the transverse acceleration in the retarded time. As it decreases only as
1
/
r
′
{\displaystyle 1/r'}
with distance compared to the standard
1
/
r
′
2
{\displaystyle 1/r'^{2}}
Coulombic behavior, this term is responsible for the long-range electromagnetic radiation caused by the accelerating charge.
The Heaviside–Feynman formula can be derived from Maxwell's equations using the technique of the retarded potential. It allows, for example, the derivation of the Larmor formula for overall radiation power of the accelerating charge.
== Discussion ==
There is a widespread interpretation of Maxwell's equations indicating that spatially varying electric and magnetic fields can cause each other to change in time, thus giving rise to a propagating electromagnetic wave (electromagnetism). However, Jefimenko's equations show an alternative point of view. Jefimenko says, "...neither Maxwell's equations nor their solutions indicate an existence of causal links between electric and magnetic fields. Therefore, we must conclude that an electromagnetic field is a dual entity always having an electric and a magnetic component simultaneously created by their common sources: time-variable electric charges and currents."
As pointed out by McDonald, Jefimenko's equations seem to appear first in 1962 in the second edition of Panofsky and Phillips's classic textbook. David Griffiths, however, clarifies that "the earliest explicit statement of which I am aware was by Oleg Jefimenko, in 1966" and characterizes equations in Panofsky and Phillips's textbook as only "closely related expressions". According to Andrew Zangwill, the equations analogous to Jefimenko's but in the Fourier frequency domain were first derived by George Adolphus Schott in his treatise Electromagnetic Radiation (University Press, Cambridge, 1912).
Essential features of these equations are easily observed which is that the right hand sides involve "retarded" time which reflects the "causality" of the expressions. In other words, the left side of each equation is actually "caused" by the right side, unlike the normal differential expressions for Maxwell's equations where both sides take place simultaneously. In the typical expressions for Maxwell's equations there is no doubt that both sides are equal to each other, but as Jefimenko notes, "... since each of these equations connects quantities simultaneous in time, none of these equations can represent a causal relation."
== See also ==
Liénard–Wiechert potential
== Notes == | Wikipedia/Jefimenko's_equations |
The London equations, developed by brothers Fritz and Heinz London in 1935, are constitutive relations for a superconductor relating its superconducting current to electromagnetic fields in and around it. Whereas Ohm's law is the simplest constitutive relation for an ordinary conductor, the London equations are the simplest meaningful description of superconducting phenomena, and form the genesis of almost any modern introductory text on the subject. A major triumph of the equations is their ability to explain the Meissner effect, wherein a material exponentially expels all internal magnetic fields as it crosses the superconducting threshold.
== Description ==
There are two London equations when expressed in terms of measurable fields:
∂
j
s
∂
t
=
n
s
e
2
m
E
,
∇
×
j
s
=
−
n
s
e
2
m
B
.
{\displaystyle {\frac {\partial \mathbf {j} _{\rm {s}}}{\partial t}}={\frac {n_{\rm {s}}e^{2}}{m}}\mathbf {E} ,\qquad \mathbf {\nabla } \times \mathbf {j} _{\rm {s}}=-{\frac {n_{\rm {s}}e^{2}}{m}}\mathbf {B} .}
Here
j
s
{\displaystyle {\mathbf {j} }_{\rm {s}}}
is the (superconducting) current density, E and B are respectively the electric and magnetic fields within the superconductor,
e
{\displaystyle e\,}
is the charge of an electron or proton,
m
{\displaystyle m\,}
is electron mass, and
n
s
{\displaystyle n_{\rm {s}}\,}
is a phenomenological constant loosely associated with a number density of superconducting carriers.
The two equations can be combined into a single "London Equation"
in terms of a specific vector potential
A
s
{\displaystyle \mathbf {A} _{\rm {s}}}
which has been gauge fixed to the "London gauge", giving:
j
s
=
−
n
s
e
2
m
A
s
.
{\displaystyle \mathbf {j} _{s}=-{\frac {n_{\rm {s}}e^{2}}{m}}\mathbf {A} _{\rm {s}}.}
In the London gauge, the vector potential obeys the following requirements, ensuring that it can be interpreted as a current density:
∇
⋅
A
s
=
0
,
{\displaystyle \nabla \cdot \mathbf {A} _{\rm {s}}=0,}
A
s
=
0
{\displaystyle \mathbf {A} _{\rm {s}}=0}
in the superconductor bulk,
A
s
⋅
n
^
=
0
,
{\displaystyle \mathbf {A} _{\rm {s}}\cdot {\hat {\mathbf {n} }}=0,}
where
n
^
{\displaystyle {\hat {\mathbf {n} }}}
is the normal vector at the surface of the superconductor.
The first requirement, also known as Coulomb gauge condition, leads to the constant superconducting electron density
ρ
˙
s
=
0
{\displaystyle {\dot {\rho }}_{\rm {s}}=0}
as expected from the continuity equation. The second requirement is consistent with the fact that supercurrent flows near the surface. The third requirement ensures no accumulation of superconducting electrons on the surface. These requirements do away with all gauge freedom and uniquely determine the vector potential. One can also write the London equation in terms of an arbitrary gauge
A
{\displaystyle \mathbf {A} }
by simply defining
A
s
=
(
A
+
∇
ϕ
)
{\displaystyle \mathbf {A} _{\rm {s}}=(\mathbf {A} +\nabla \phi )}
, where
ϕ
{\displaystyle \phi }
is a scalar function and
∇
ϕ
{\displaystyle \nabla \phi }
is the change in gauge which shifts the arbitrary gauge to the London gauge.
The vector potential expression holds for magnetic fields that vary slowly in space.
== London penetration depth ==
If the second of London's equations is manipulated by applying Ampere's law,
∇
×
B
=
μ
0
j
{\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {j} }
,
then it can be turned into the Helmholtz equation for magnetic field:
∇
2
B
=
1
λ
s
2
B
{\displaystyle \nabla ^{2}\mathbf {B} ={\frac {1}{\lambda _{\rm {s}}^{2}}}\mathbf {B} }
where the inverse of the laplacian eigenvalue:
λ
s
≡
m
μ
0
n
s
e
2
{\displaystyle \lambda _{\rm {s}}\equiv {\sqrt {\frac {m}{\mu _{0}n_{\rm {s}}e^{2}}}}}
is the characteristic length scale,
λ
s
{\displaystyle \lambda _{\rm {s}}}
, over which external magnetic fields are exponentially suppressed: it is called the London penetration depth: typical values are from 50 to 500 nm.
For example, consider a superconductor within free space where the magnetic field outside the superconductor is a constant value pointed parallel to the superconducting boundary plane in the z direction. If x leads perpendicular to the boundary then the solution inside the superconductor may be shown to be
B
z
(
x
)
=
B
0
e
−
x
/
λ
s
.
{\displaystyle B_{z}(x)=B_{0}e^{-x/\lambda _{\rm {s}}}.\,}
From here the physical meaning of the London penetration depth can perhaps most easily be discerned.
== Rationale ==
=== Original arguments ===
While it is important to note that the above equations cannot be formally derived,
the Londons did follow a certain intuitive logic in the formulation of their theory. Substances across a stunningly wide range of composition behave roughly according to Ohm's law, which states that current is proportional to electric field. However, such a linear relationship is impossible in a superconductor for, almost by definition, the electrons in a superconductor flow with no resistance whatsoever. To this end, the London brothers imagined electrons as if they were free electrons under the influence of a uniform external electric field. According to the Lorentz force law
F
=
m
v
˙
=
−
e
E
−
e
v
×
B
{\displaystyle \mathbf {F} =m{\dot {\mathbf {v} }}=-e\mathbf {E} -e\mathbf {v} \times \mathbf {B} }
these electrons should encounter a uniform force, and thus they should in fact accelerate uniformly. Assume that the electrons in the superconductor are now driven by an electric field, then according to the definition of current density
j
s
=
−
n
s
e
v
s
{\displaystyle \mathbf {j} _{\rm {s}}=-n_{\rm {s}}e\mathbf {v} _{\rm {s}}}
we should have
∂
j
s
∂
t
=
−
n
s
e
∂
v
∂
t
=
n
s
e
2
m
E
{\displaystyle {\frac {\partial \mathbf {j} _{s}}{\partial t}}=-n_{\rm {s}}e{\frac {\partial \mathbf {v} }{\partial t}}={\frac {n_{\rm {s}}e^{2}}{m}}\mathbf {E} }
This is the first London equation. To obtain the second equation, take the curl of the first London equation and apply Faraday's law,
∇
×
E
=
−
∂
B
∂
t
{\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}
,
to obtain
∂
∂
t
(
∇
×
j
s
+
n
s
e
2
m
B
)
=
0.
{\displaystyle {\frac {\partial }{\partial t}}\left(\nabla \times \mathbf {j} _{\rm {s}}+{\frac {n_{\rm {s}}e^{2}}{m}}\mathbf {B} \right)=0.}
As it currently stands, this equation permits both constant and exponentially decaying solutions. The Londons recognized from the Meissner effect that constant nonzero solutions were nonphysical, and thus postulated that not only was the time derivative of the above expression equal to zero, but also that the expression in the parentheses must be identically zero:
∇
×
j
s
+
n
s
e
2
m
B
=
0
{\displaystyle \nabla \times \mathbf {j} _{\rm {s}}+{\frac {n_{\rm {s}}e^{2}}{m}}\mathbf {B} =0}
This results in the second London equation and
j
s
=
−
n
s
e
2
m
A
s
{\displaystyle \mathbf {j} _{s}=-{\frac {n_{\rm {s}}e^{2}}{m}}\mathbf {A} _{\rm {s}}}
(up to a gauge transformation which is fixed by choosing "London gauge") since the magnetic field is defined through
B
=
∇
×
A
s
.
{\displaystyle B=\nabla \times A_{\rm {s}}.}
Additionally, according to Ampere's law
∇
×
B
=
μ
0
j
s
{\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {j} _{\rm {s}}}
, one may derive that:
∇
×
(
∇
×
B
)
=
∇
×
μ
0
j
s
=
−
μ
0
n
s
e
2
m
B
.
{\displaystyle \nabla \times (\nabla \times \mathbf {B} )=\nabla \times \mu _{0}\mathbf {j} _{\rm {s}}=-{\frac {\mu _{0}n_{\rm {s}}e^{2}}{m}}\mathbf {B} .}
On the other hand, since
∇
⋅
B
=
0
{\displaystyle \nabla \cdot \mathbf {B} =0}
, we have
∇
×
(
∇
×
B
)
=
−
∇
2
B
{\displaystyle \nabla \times (\nabla \times \mathbf {B} )=-\nabla ^{2}\mathbf {B} }
, which leads to the spatial distribution of magnetic field obeys :
∇
2
B
=
1
λ
s
2
B
{\displaystyle \nabla ^{2}\mathbf {B} ={\frac {1}{\lambda _{\rm {s}}^{2}}}\mathbf {B} }
with penetration depth
λ
s
=
m
μ
0
n
s
e
2
{\displaystyle \lambda _{\rm {s}}={\sqrt {\frac {m}{\mu _{0}n_{\rm {s}}e^{2}}}}}
. In one dimension, such Helmholtz equation has the solution form
B
z
(
x
)
=
B
0
e
−
x
/
λ
s
.
{\displaystyle B_{z}(x)=B_{0}e^{-x/\lambda _{\rm {s}}}.\,}
Inside the superconductor
(
x
>
0
)
{\displaystyle (x>0)}
, the magnetic field exponetially decay, which well explains the Meissner effect. With the magnetic field distribution, we can use Ampere's law
∇
×
B
=
μ
0
j
s
{\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {j} _{\rm {s}}}
again to see that the supercurrent
j
s
{\displaystyle \mathbf {j} _{\rm {s}}}
also flows near the surface of superconductor, as expected from the requirement for interpreting
j
s
{\displaystyle \mathbf {j} _{\rm {s}}}
as physical current.
While the above rationale holds for superconductor, one may also argue in the same way for a perfect conductor. However, one important fact that distinguishes the superconductor from perfect conductor is that perfect conductor does not exhibit Meissner effect for
T
<
T
c
{\displaystyle T<T_{c}}
. In fact, the postulation
∇
×
j
s
+
n
s
e
2
m
B
=
0
{\displaystyle \nabla \times \mathbf {j} _{\rm {s}}+{\frac {n_{\rm {s}}e^{2}}{m}}\mathbf {B} =0}
does not hold for a perfect conductor. Instead, the time derivative must be kept and cannot be simply removed. This results in the fact that the time derivative of
B
{\displaystyle \mathbf {B} }
field (instead of
B
{\displaystyle \mathbf {B} }
field) obeys:
∇
2
∂
B
∂
t
=
1
λ
s
2
∂
B
∂
t
.
{\displaystyle \nabla ^{2}{\frac {\partial \mathbf {B} }{\partial t}}={\frac {1}{\lambda _{\rm {s}}^{2}}}{\frac {\partial \mathbf {B} }{\partial t}}.}
For
T
<
T
c
{\displaystyle T<T_{c}}
, deep inside a perfect conductor we have
B
˙
=
0
{\displaystyle {\dot {\mathbf {B} }}=0}
rather than
B
=
0
{\displaystyle \mathbf {B} =0}
as the superconductor. Consequently, whether the magnetic flux inside a perfect conductor will vanish depends on the initial condition (whether it's zero-field cooled or not).
=== Canonical momentum arguments ===
It is also possible to justify the London equations by other means.
Current density is defined according to the equation
j
s
=
−
n
s
e
v
s
.
{\displaystyle \mathbf {j} _{\rm {s}}=-n_{\rm {s}}e\mathbf {v} _{\rm {s}}.}
Taking this expression from a classical description to a quantum mechanical one, we must replace values
j
s
{\displaystyle \mathbf {j} _{\rm {s}}}
and
v
s
{\displaystyle \mathbf {v} _{\rm {s}}}
by the expectation values of their operators. The velocity operator
v
s
=
1
m
(
p
+
e
A
s
)
{\displaystyle \mathbf {v} _{\rm {s}}={\frac {1}{m}}\left(\mathbf {p} +e\mathbf {A} _{\rm {s}}\right)}
is defined by dividing the gauge-invariant, kinematic momentum operator by the particle mass m. Note we are using
−
e
{\displaystyle -e}
as the electron charge.
We may then make this replacement in the equation above. However, an important assumption from the microscopic theory of superconductivity is that the superconducting state of a system is the ground state, and according to a theorem of Bloch's,
in such a state the canonical momentum p is zero. This leaves
j
=
−
n
s
e
2
m
A
s
,
{\displaystyle \mathbf {j} =-{\frac {n_{\rm {s}}e^{2}}{m}}\mathbf {A} _{\rm {s}},}
which is the London equation according to the second formulation above.
== References == | Wikipedia/London_equations |
Electric potential energy is a potential energy (measured in joules) that results from conservative Coulomb forces and is associated with the configuration of a particular set of point charges within a defined system. An object may be said to have electric potential energy by virtue of either its own electric charge or its relative position to other electrically charged objects.
The term "electric potential energy" is used to describe the potential energy in systems with time-variant electric fields, while the term "electrostatic potential energy" is used to describe the potential energy in systems with time-invariant electric fields.
== Definition ==
The electric potential energy of a system of point charges is defined as the work required to assemble this system of charges by bringing them close together, as in the system from an infinite distance. Alternatively, the electric potential energy of any given charge or system of charges is termed as the total work done by an external agent in bringing the charge or the system of charges from infinity to the present configuration without undergoing any acceleration.
The electrostatic potential energy can also be defined from the electric potential as follows:
== Units ==
The SI unit of electric potential energy is joule (named after the English physicist James Prescott Joule). In the CGS system the erg is the unit of energy, being equal to 10−7 Joules. Also electronvolts may be used, 1 eV = 1.602×10−19 Joules.
== Electrostatic potential energy of one point charge ==
=== One point charge q in the presence of another point charge Q ===
The electrostatic potential energy, UE, of one point charge q at position r in the presence of a point charge Q, taking an infinite separation between the charges as the reference position, is:
U
E
(
r
)
=
1
4
π
ε
0
q
Q
r
{\displaystyle U_{E}(\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}{\frac {qQ}{r}}}
where r is the distance between the point charges q and Q, and q and Q are the charges (not the absolute values of the charges—i.e., an electron would have a negative value of charge when placed in the formula). The following outline of proof states the derivation from the definition of electric potential energy and Coulomb's law to this formula.
=== One point charge q in the presence of n point charges Qi ===
The electrostatic potential energy, UE, of one point charge q in the presence of n point charges Qi, taking an infinite separation between the charges as the reference position, is:
U
E
(
r
)
=
q
4
π
ε
0
∑
i
=
1
n
Q
i
r
i
,
{\displaystyle U_{E}(r)={\frac {q}{4\pi \varepsilon _{0}}}\sum _{i=1}^{n}{\frac {Q_{i}}{r_{i}}},}
where ri is the distance between the point charges q and Qi, and q and Qi are the assigned values of the charges.
== Electrostatic potential energy stored in a system of point charges ==
The electrostatic potential energy UE stored in a system of N charges q1, q2, …, qN at positions r1, r2, …, rN respectively, is:
where, for each i value, V(ri) is the electrostatic potential due to all point charges except the one at ri, and is equal to:
V
(
r
i
)
=
k
e
∑
j
≠
i
j
=
1
N
q
j
r
i
j
,
{\displaystyle V(\mathbf {r} _{i})=k_{e}\sum _{\stackrel {j=1}{j\neq i}}^{N}{\frac {q_{j}}{r_{ij}}},}
where rij is the distance between qi and qj.
=== Energy stored in a system of one point charge ===
The electrostatic potential energy of a system containing only one point charge is zero, as there are no other sources of electrostatic force against which an external agent must do work in moving the point charge from infinity to its final location.
A common question arises concerning the interaction of a point charge with its own electrostatic potential. Since this interaction doesn't act to move the point charge itself, it doesn't contribute to the stored energy of the system.
=== Energy stored in a system of two point charges ===
Consider bringing a point charge, q, into its final position near a point charge, Q1. The electric potential V(r) due to Q1 is
V
(
r
)
=
k
e
Q
1
r
{\displaystyle V(\mathbf {r} )=k_{e}{\frac {Q_{1}}{r}}}
Hence we obtain, the electrostatic potential energy of q in the potential of Q1 as
U
E
=
1
4
π
ε
0
q
Q
1
r
1
{\displaystyle U_{E}={\frac {1}{4\pi \varepsilon _{0}}}{\frac {qQ_{1}}{r_{1}}}}
where r1 is the separation between the two point charges.
=== Energy stored in a system of three point charges ===
The electrostatic potential energy of a system of three charges should not be confused with the electrostatic potential energy of Q1 due to two charges Q2 and Q3, because the latter doesn't include the electrostatic potential energy of the system of the two charges Q2 and Q3.
The electrostatic potential energy stored in the system of three charges is:
U
E
=
1
4
π
ε
0
[
Q
1
Q
2
r
12
+
Q
1
Q
3
r
13
+
Q
2
Q
3
r
23
]
{\displaystyle U_{\mathrm {E} }={\frac {1}{4\pi \varepsilon _{0}}}\left[{\frac {Q_{1}Q_{2}}{r_{12}}}+{\frac {Q_{1}Q_{3}}{r_{13}}}+{\frac {Q_{2}Q_{3}}{r_{23}}}\right]}
== Energy stored in an electrostatic field distribution in vacuum ==
The energy density, or energy per unit volume,
d
U
d
V
{\textstyle {\frac {dU}{dV}}}
, of the electrostatic field of a continuous charge distribution is:
u
e
=
d
U
d
V
=
1
2
ε
0
|
E
|
2
.
{\displaystyle u_{e}={\frac {dU}{dV}}={\frac {1}{2}}\varepsilon _{0}\left|{\mathbf {E} }\right|^{2}.}
== Energy stored in electronic elements ==
Some elements in a circuit can convert energy from one form to another. For example, a resistor converts electrical energy to heat. This is known as the Joule effect. A capacitor stores it in its electric field. The total electrostatic potential energy stored in a capacitor is given by
U
E
=
1
2
Q
V
=
1
2
C
V
2
=
Q
2
2
C
{\displaystyle U_{E}={\frac {1}{2}}QV={\frac {1}{2}}CV^{2}={\frac {Q^{2}}{2C}}}
where C is the capacitance, V is the electric potential difference, and Q the charge stored in the capacitor.
The total electrostatic potential energy may also be expressed in terms of the electric field in the form
U
E
=
1
2
∫
V
E
⋅
D
d
V
{\displaystyle U_{E}={\frac {1}{2}}\int _{V}\mathrm {E} \cdot \mathrm {D} \,dV}
where
D
{\displaystyle \mathrm {D} }
is the electric displacement field within a dielectric material and integration is over the entire volume of the dielectric.
The total electrostatic potential energy stored within a charged dielectric may also be expressed in terms of a continuous volume charge,
ρ
{\displaystyle \rho }
,
U
E
=
1
2
∫
V
ρ
Φ
d
V
{\displaystyle U_{E}={\frac {1}{2}}\int _{V}\rho \Phi \,dV}
where integration is over the entire volume of the dielectric.
These latter two expressions are valid only for cases when the smallest increment of charge is zero (
d
q
→
0
{\displaystyle dq\to 0}
) such as dielectrics in the presence of metallic electrodes or dielectrics containing many charges.
Note that a virtual experiment based on the energy transfer between capacitor plates reveals that an additional term should be taken into account when dealing with semiconductors for instance. While this extra energy cancels when dealing with insulators, the derivation predicts that it cannot be ignored as it may exceed the polarization energy.
== Notes ==
== References ==
== External links ==
Media related to Electric potential energy at Wikimedia Commons | Wikipedia/Electrostatic_energy |
Topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.
A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of topological spaces, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. The following are basic examples of topological properties: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connectedness, which allows distinguishing a circle from two non-intersecting circles.
The ideas underlying topology go back to Gottfried Wilhelm Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. Leonhard Euler's Seven Bridges of Königsberg problem and polyhedron formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, although, it was not until the first decades of the 20th century that the idea of a topological space was developed.
== Motivation ==
The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. For example, the square and the circle have many properties in common: they are both one-dimensional objects (from a topological point of view) and both separate the plane into two parts, the part inside and the part outside.
In one of the first papers in topology, Leonhard Euler demonstrated that it was impossible to find a route through the town of Königsberg (now Kaliningrad) that would cross each of its seven bridges exactly once. This result did not depend on the lengths of the bridges or on their distance from one another, but only on connectivity properties: which bridges connect to which islands or riverbanks. This Seven Bridges of Königsberg problem led to the branch of mathematics known as graph theory.
Similarly, the hairy ball theorem of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating a cowlick." This fact is immediately convincing to most people, even though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere. As with the Bridges of Königsberg, the result does not depend on the shape of the sphere; it applies to any kind of smooth blob, as long as it has no holes.
To deal with these problems that do not rely on the exact shape of the objects, one must be clear about just what properties these problems do rely on. From this need arises the notion of homeomorphism. The impossibility of crossing each bridge just once applies to any arrangement of bridges homeomorphic to those in Königsberg, and the hairy ball theorem applies to any space homeomorphic to a sphere.
Intuitively, two spaces are homeomorphic if one can be deformed into the other without cutting or gluing. A famous example, known as the "Topologist's Breakfast", is that a topologist cannot distinguish a coffee mug from a doughnut; a sufficiently pliable doughnut could be reshaped to a coffee cup by creating a dimple and progressively enlarging it while shrinking the hole into a handle.
Homeomorphism can be considered the most basic topological equivalence. Another is homotopy equivalence. This is harder to describe without getting technical, but the essential notion is that two objects are homotopy equivalent if they both result from "squishing" some larger object.
== History ==
Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries. Among these are certain questions in geometry investigated by Leonhard Euler. His 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750, Euler wrote to a friend that he had realized the importance of the edges of a polyhedron. This led to his polyhedron formula, V − E + F = 2 (where V, E, and F respectively indicate the number of vertices, edges, and faces of the polyhedron). Some authorities regard this analysis as the first theorem, signaling the birth of topology.
Further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti. Listing introduced the term "Topologie" in Vorstudien zur Topologie, written in his native German, in 1847, having used the word for ten years in correspondence before its first appearance in print. The English form "topology" was used in 1883 in Listing's obituary in the journal Nature to distinguish "qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated".
Their work was corrected, consolidated and greatly extended by Henri Poincaré. In 1895, he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology.
The development of topology in the 20th century was marked by significant advances in both foundational theory and its application to other fields of mathematics. Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a special case of a general topological space, with any given topological space potentially giving rise to many distinct metric spaces. In 1914, Felix Hausdorff coined the term "topological space" and defined what is now called a Hausdorff space. Currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski.
Modern topology depends strongly on the ideas of set theory, developed by Georg Cantor in the later part of the 19th century. In addition to establishing the basic ideas of set theory, Cantor considered point sets in Euclidean space as part of his study of Fourier series. For further developments, see point-set topology and algebraic topology.
The 2022 Abel Prize was awarded to Dennis Sullivan "for his groundbreaking contributions to topology in its broadest sense, and in particular its algebraic, geometric and dynamical aspects".
== Concepts ==
=== Topologies on sets ===
The term topology also refers to a specific mathematical idea central to the area of mathematics called topology. Informally, a topology describes how elements of a set relate spatially to each other. The same set can have different topologies. For instance, the real line, the complex plane, and the Cantor set can be thought of as the same set with different topologies.
Formally, let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X if:
Both the empty set and X are elements of τ.
Any union of elements of τ is an element of τ.
Any intersection of finitely many elements of τ is an element of τ.
If τ is a topology on X, then the pair (X, τ) is called a topological space. The notation Xτ may be used to denote a set X endowed with the particular topology τ. By definition, every topology is a π-system.
The members of τ are called open sets in X. A subset of X is said to be closed if its complement is in τ (that is, its complement is open). A subset of X may be open, closed, both (a clopen set), or neither. The empty set and X itself are always both closed and open. An open subset of X which contains a point x is called an open neighborhood of x.
=== Continuous functions and homeomorphisms ===
A function or map from one topological space to another is called continuous if the inverse image of any open set is open. If the function maps the real numbers to the real numbers (both spaces with the standard topology), then this definition of continuous is equivalent to the definition of continuous in calculus. If a continuous function is one-to-one and onto, and if the inverse of the function is also continuous, then the function is called a homeomorphism and the domain of the function is said to be homeomorphic to the range. Another way of saying this is that the function has a natural extension to the topology. If two spaces are homeomorphic, they have identical topological properties and are considered topologically the same. The cube and the sphere are homeomorphic, as are the coffee cup and the doughnut. However, the sphere is not homeomorphic to the doughnut.
=== Manifolds ===
While topological spaces can be extremely varied and exotic, many areas of topology focus on the more familiar class of spaces known as manifolds. A manifold is a topological space that resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighborhood that is homeomorphic to the Euclidean space of dimension n. Lines and circles, but not figure eights, are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces, although not all surfaces are manifolds. Examples include the plane, the sphere, and the torus, which can all be realized without self-intersection in three dimensions, and the Klein bottle and real projective plane, which cannot (that is, all their realizations are surfaces that are not manifolds).
== Topics ==
=== General topology ===
General topology is the branch of topology dealing with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.
The basic object of study is topological spaces, which are sets equipped with a topology, that is, a family of subsets, called open sets, which is closed under finite intersections and (finite or infinite) unions. The fundamental concepts of topology, such as continuity, compactness, and connectedness, can be defined in terms of open sets. Intuitively, continuous functions take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words nearby, arbitrarily small, and far apart can all be made precise by using open sets. Several topologies can be defined on a given space. Changing a topology consists of changing the collection of open sets. This changes which functions are continuous and which subsets are compact or connected.
Metric spaces are an important class of topological spaces where the distance between any two points is defined by a function called a metric. In a metric space, an open set is a union of open disks, where an open disk of radius r centered at x is the set of all points whose distance to x is less than r. Many common spaces are topological spaces whose topology can be defined by a metric. This is the case of the real line, the complex plane, real and complex vector spaces and Euclidean spaces. Having a metric simplifies many proofs.
=== Algebraic topology ===
Algebraic topology is a branch of mathematics that uses tools from algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.
The most important of these invariants are homotopy groups, homology, and cohomology.
Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group.
=== Differential topology ===
Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.
More specifically, differential topology considers the properties and structures that require only a smooth structure on a manifold to be defined. Smooth manifolds are "softer" than manifolds with extra geometric structures, which can act as obstructions to certain types of equivalences and deformations that exist in differential topology. For instance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on the same smooth manifold – that is, one can smoothly "flatten out" certain manifolds, but it might require distorting the space and affecting the curvature or volume.
=== Geometric topology ===
Geometric topology is a branch of topology that primarily focuses on low-dimensional manifolds (that is, spaces of dimensions 2, 3, and 4) and their interaction with geometry, but it also includes some higher-dimensional topology. Some examples of topics in geometric topology are orientability, handle decompositions, local flatness, crumpling and the planar and higher-dimensional Schönflies theorem.
In high-dimensional topology, characteristic classes are a basic invariant, and surgery theory is a key theory.
Low-dimensional topology is strongly geometric, as reflected in the uniformization theorem in 2 dimensions – every surface admits a constant curvature metric; geometrically, it has one of 3 possible geometries: positive curvature/spherical, zero curvature/flat, and negative curvature/hyperbolic – and the geometrization conjecture (now theorem) in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of eight possible geometries.
2-dimensional topology can be studied as complex geometry in one variable (Riemann surfaces are complex curves) – by the uniformization theorem every conformal class of metrics is equivalent to a unique complex one, and 4-dimensional topology can be studied from the point of view of complex geometry in two variables (complex surfaces), though not every 4-manifold admits a complex structure.
=== Generalizations ===
Occasionally, one needs to use the tools of topology but a "set of points" is not available. In pointless topology one considers instead the lattice of open sets as the basic notion of the theory, while Grothendieck topologies are structures defined on arbitrary categories that allow the definition of sheaves on those categories and with that the definition of general cohomology theories.
== Applications ==
=== Biology ===
Topology has been used to study various biological systems including molecules and nanostructure (e.g., membraneous objects). In particular, circuit topology and knot theory have been extensively applied to classify and compare the topology of folded proteins and nucleic acids. Circuit topology classifies folded molecular chains based on the pairwise arrangement of their intra-chain contacts and chain crossings. Knot theory, a branch of topology, is used in biology to study the effects of certain enzymes on DNA. These enzymes cut, twist, and reconnect the DNA, causing knotting with observable effects such as slower electrophoresis.
=== Computer science ===
Topological data analysis uses techniques from algebraic topology to determine the large-scale structure of a set (for instance, determining if a cloud of points is spherical or toroidal). The main method used by topological data analysis is to:
Replace a set of data points with a family of simplicial complexes, indexed by a proximity parameter.
Analyse these topological complexes via algebraic topology – specifically, via the theory of persistent homology.
Encode the persistent homology of a data set in the form of a parameterized version of a Betti number, which is called a barcode.
Several branches of programming language semantics, such as domain theory, are formalized using topology. In this context, Steve Vickers, building on work by Samson Abramsky and Michael B. Smyth, characterizes topological spaces as Boolean or Heyting algebras over open sets, which are characterized as semidecidable (equivalently, finitely observable) properties.
=== Physics ===
Topology is relevant to physics in areas such as condensed matter physics, quantum field theory, quantum computing and physical cosmology.
The topological dependence of mechanical properties in solids is of interest in the disciplines of mechanical engineering and materials science. Electrical and mechanical properties depend on the arrangement and network structures of molecules and elementary units in materials. The compressive strength of crumpled topologies is studied in attempts to understand the high strength to weight of such structures that are mostly empty space. Topology is of further significance in Contact mechanics where the dependence of stiffness and friction on the dimensionality of surface structures is the subject of interest with applications in multi-body physics.
A topological quantum field theory (or topological field theory or TQFT) is a quantum field theory that computes topological invariants. Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory, the theory of four-manifolds in algebraic topology, and the theory of moduli spaces in algebraic geometry. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for work related to topological field theory.
The topological classification of Calabi–Yau manifolds has important implications in string theory, as different manifolds can sustain different kinds of strings.
In topological quantum computers, the qubits are stored in topological properties, that are by definition invariant with respect to homotopies.
In cosmology, topology can be used to describe the overall shape of the universe. This area of research is commonly known as spacetime topology.
In condensed matter, a relevant application to topological physics comes from the possibility of obtaining a one-way current, which is a current protected from backscattering. It was first discovered in electronics with the famous quantum Hall effect, and then generalized in other areas of physics, for instance in photonics by F.D.M Haldane.
=== Robotics ===
The possible positions of a robot can be described by a manifold called configuration space. In the area of motion planning, one finds paths between two points in configuration space. These paths represent a motion of the robot's joints and other parts into the desired pose.
=== Games and puzzles ===
Disentanglement puzzles are based on topological aspects of the puzzle's shapes and components.
=== Fiber art ===
In order to create a continuous join of pieces in a modular construction, it is necessary to create an unbroken path in an order that surrounds each piece and traverses each edge only once. This process is an application of the Eulerian path.
== Resources and research ==
=== Major journals ===
Geometry & Topology- a mathematic research journal focused on geometry and topology, and their applications, published by Mathematical Sciences Publishers.
Journal of Topology- a scientific journal which publishes papers of high quality and significance in topology, geometry, and adjacent areas of mathematics.
=== Major books ===
Munkres, James R. (2000). Topology (2nd ed.). Upper Saddle River, NJ: Prentice Hall. ISBN 978-0-13-181629-9
Willard, Stephen (2016). General topology. Dover books on mathematics. Mineola, N.Y: Dover publications. ISBN 978-0-486-43479-7
Armstrong, M. A. (1983). Basic topology. Undergraduate texts in mathematics. New York: Springer-Verlag. ISBN 978-0-387-90839-7
John Kelley "General Topology" Springer, 1979.
== See also ==
== References ==
=== Citations ===
=== Bibliography ===
Aleksandrov, P.S. (1969) [1956]. "Chapter XVIII Topology". In Aleksandrov, A.D.; Kolmogorov, A.N.; Lavrent'ev, M.A. (eds.). Mathematics / Its Content, Methods and Meaning (2nd ed.). The M.I.T. Press.
Croom, Fred H. (1989). Principles of Topology. Saunders College Publishing. ISBN 978-0-03-029804-2.
Richeson, D. (2008). Euler's Gem: The Polyhedron Formula and the Birth of Topology. Princeton University Press.
== Further reading ==
Ryszard Engelking, General Topology, Heldermann Verlag, Sigma Series in Pure Mathematics, December 1989, ISBN 3-88538-006-4.
Bourbaki; Elements of Mathematics: General Topology, Addison–Wesley (1966).
Breitenberger, E. (2006). "Johann Benedict Listing". In James, I.M. (ed.). History of Topology. North Holland. ISBN 978-0-444-82375-5.
Kelley, John L. (1975) [1955]. General Topology. Graduate Texts in Mathematics. Vol. 27 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-90125-1. OCLC 1365153.
Brown, Ronald (2006). Topology and Groupoids. Booksurge. ISBN 978-1-4196-2722-4. (Provides a well-motivated, geometric account of general topology, and shows the use of groupoids in discussing van Kampen's theorem, covering spaces, and orbit spaces.)
Wacław Sierpiński, General Topology, Dover Publications, 2000, ISBN 0-486-41148-6
Pickover, Clifford A. (2006). The Möbius Strip: Dr. August Möbius's Marvelous Band in Mathematics, Games, Literature, Art, Technology, and Cosmology. Thunder's Mouth Press. ISBN 978-1-56025-826-1. (Provides a popular introduction to topology and geometry)
Gemignani, Michael C. (1990) [1967]. Elementary Topology (2nd ed.). Dover Publications Inc. ISBN 978-0-486-66522-1.
== External links ==
"Topology, general". Encyclopedia of Mathematics. EMS Press. 2001 [1994].
Elementary Topology: A First Course Viro, Ivanov, Netsvetaev, Kharlamov.
The Topological Zoo at The Geometry Center.
Topology Atlas
Topology Course Lecture Notes Aisling McCluskey and Brian McMaster, Topology Atlas.
Topology Glossary
Moscow 1935: Topology moving towards America, a historical essay by Hassler Whitney. | Wikipedia/topology |
In abstract algebra, a cover is one instance of some mathematical structure mapping onto another instance, such as a group (trivially) covering a subgroup. This should not be confused with the concept of a cover in topology.
When some object X is said to cover another object Y, the cover is given by some surjective and structure-preserving map f : X → Y. The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which X and Y are instances. In order to be interesting, the cover is usually endowed with additional properties, which are highly dependent on the context.
== Examples ==
A classic result in semigroup theory due to D. B. McAlister states that every inverse semigroup has an E-unitary cover; besides being surjective, the homomorphism in this case is also idempotent separating, meaning that in its kernel an idempotent and non-idempotent never belong to the same equivalence class.; something slightly stronger has actually be shown for inverse semigroups: every inverse semigroup admits an F-inverse cover. McAlister's covering theorem generalizes to orthodox semigroups: every orthodox semigroup has a unitary cover.
Examples from other areas of algebra include the Frattini cover of a profinite group and the universal cover of a Lie group.
== Modules ==
If F is some family of modules over some ring R, then an F-cover of a module M is a homomorphism X→M with the following properties:
X is in the family F
X→M is surjective
Any surjective map from a module in the family F to M factors through X
Any endomorphism of X commuting with the map to M is an automorphism.
In general an F-cover of M need not exist, but if it does exist then it is unique up to (non-unique) isomorphism.
Examples include:
Projective covers (always exist over perfect rings)
flat covers (always exist)
torsion-free covers (always exist over integral domains)
injective covers
== See also ==
Embedding
== Notes ==
== References ==
Howie, John M. (1995). Fundamentals of Semigroup Theory. Clarendon Press. ISBN 0-19-851194-9. | Wikipedia/Cover_(algebra) |
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivalently, a bijection is a relation between two sets such that each element of either set is paired with exactly one element of the other set.
A function is bijective if it is invertible; that is, a function
f
:
X
→
Y
{\displaystyle f:X\to Y}
is bijective if and only if there is a function
g
:
Y
→
X
,
{\displaystyle g:Y\to X,}
the inverse of f, such that each of the two ways for composing the two functions produces an identity function:
g
(
f
(
x
)
)
=
x
{\displaystyle g(f(x))=x}
for each
x
{\displaystyle x}
in
X
{\displaystyle X}
and
f
(
g
(
y
)
)
=
y
{\displaystyle f(g(y))=y}
for each
y
{\displaystyle y}
in
Y
.
{\displaystyle Y.}
For example, the multiplication by two defines a bijection from the integers to the even numbers, which has the division by two as its inverse function.
A function is bijective if and only if it is both injective (or one-to-one)—meaning that each element in the codomain is mapped from at most one element of the domain—and surjective (or onto)—meaning that each element of the codomain is mapped from at least one element of the domain. The term one-to-one correspondence must not be confused with one-to-one function, which means injective but not necessarily surjective.
The elementary operation of counting establishes a bijection from some finite set to the first natural numbers (1, 2, 3, ...), up to the number of elements in the counted set. It results that two finite sets have the same number of elements if and only if there exists a bijection between them. More generally, two sets are said to have the same cardinal number if there exists a bijection between them.
A bijective function from a set to itself is also called a permutation, and the set of all permutations of a set forms its symmetric group.
Some bijections with further properties have received specific names, which include automorphisms, isomorphisms, homeomorphisms, diffeomorphisms, permutation groups, and most geometric transformations. Galois correspondences are bijections between sets of mathematical objects of apparently very different nature.
== Definition ==
For a binary relation pairing elements of set X with elements of set Y to be a bijection, four properties must hold:
each element of X must be paired with at least one element of Y,
no element of X may be paired with more than one element of Y,
each element of Y must be paired with at least one element of X, and
no element of Y may be paired with more than one element of X.
Satisfying properties (1) and (2) means that a pairing is a function with domain X. It is more common to see properties (1) and (2) written as a single statement: Every element of X is paired with exactly one element of Y. Functions which satisfy property (3) are said to be "onto Y " and are called surjections (or surjective functions). Functions which satisfy property (4) are said to be "one-to-one functions" and are called injections (or injective functions). With this terminology, a bijection is a function which is both a surjection and an injection, or using other words, a bijection is a function which is both "one-to-one" and "onto".
== Examples ==
=== Batting line-up of a baseball or cricket team ===
Consider the batting line-up of a baseball or cricket team (or any list of all the players of any sports team where every player holds a specific spot in a line-up). The set X will be the players on the team (of size nine in the case of baseball) and the set Y will be the positions in the batting order (1st, 2nd, 3rd, etc.) The "pairing" is given by which player is in what position in this order. Property (1) is satisfied since each player is somewhere in the list. Property (2) is satisfied since no player bats in two (or more) positions in the order. Property (3) says that for each position in the order, there is some player batting in that position and property (4) states that two or more players are never batting in the same position in the list.
=== Seats and students of a classroom ===
In a classroom there are a certain number of seats. A group of students enter the room and the instructor asks them to be seated. After a quick look around the room, the instructor declares that there is a bijection between the set of students and the set of seats, where each student is paired with the seat they are sitting in. What the instructor observed in order to reach this conclusion was that:
Every student was in a seat (there was no one standing),
No student was in more than one seat,
Every seat had someone sitting there (there were no empty seats), and
No seat had more than one student in it.
The instructor was able to conclude that there were just as many seats as there were students, without having to count either set.
== More mathematical examples ==
For any set X, the identity function 1X: X → X, 1X(x) = x is bijective.
The function f: R → R, f(x) = 2x + 1 is bijective, since for each y there is a unique x = (y − 1)/2 such that f(x) = y. More generally, any linear function over the reals, f: R → R, f(x) = ax + b (where a is non-zero) is a bijection. Each real number y is obtained from (or paired with) the real number x = (y − b)/a.
The function f: R → (−π/2, π/2), given by f(x) = arctan(x) is bijective, since each real number x is paired with exactly one angle y in the interval (−π/2, π/2) so that tan(y) = x (that is, y = arctan(x)). If the codomain (−π/2, π/2) was made larger to include an integer multiple of π/2, then this function would no longer be onto (surjective), since there is no real number which could be paired with the multiple of π/2 by this arctan function.
The exponential function, g: R → R, g(x) = ex, is not bijective: for instance, there is no x in R such that g(x) = −1, showing that g is not onto (surjective). However, if the codomain is restricted to the positive real numbers
R
+
≡
(
0
,
∞
)
{\displaystyle \mathbb {R} ^{+}\equiv \left(0,\infty \right)}
, then g would be bijective; its inverse (see below) is the natural logarithm function ln.
The function h: R → R+, h(x) = x2 is not bijective: for instance, h(−1) = h(1) = 1, showing that h is not one-to-one (injective). However, if the domain is restricted to
R
0
+
≡
[
0
,
∞
)
{\displaystyle \mathbb {R} _{0}^{+}\equiv \left[0,\infty \right)}
, then h would be bijective; its inverse is the positive square root function.
By Schröder–Bernstein theorem, given any two sets X and Y, and two injective functions f: X → Y and g: Y → X, there exists a bijective function h: X → Y.
== Inverses ==
A bijection f with domain X (indicated by f: X → Y in functional notation) also defines a converse relation starting in Y and going to X (by turning the arrows around). The process of "turning the arrows around" for an arbitrary function does not, in general, yield a function, but properties (3) and (4) of a bijection say that this inverse relation is a function with domain Y. Moreover, properties (1) and (2) then say that this inverse function is a surjection and an injection, that is, the inverse function exists and is also a bijection. Functions that have inverse functions are said to be invertible. A function is invertible if and only if it is a bijection.
Stated in concise mathematical notation, a function f: X → Y is bijective if and only if it satisfies the condition
for every y in Y there is a unique x in X with y = f(x).
Continuing with the baseball batting line-up example, the function that is being defined takes as input the name of one of the players and outputs the position of that player in the batting order. Since this function is a bijection, it has an inverse function which takes as input a position in the batting order and outputs the player who will be batting in that position.
== Composition ==
The composition
g
∘
f
{\displaystyle g\,\circ \,f}
of two bijections f: X → Y and g: Y → Z is a bijection, whose inverse is given by
g
∘
f
{\displaystyle g\,\circ \,f}
is
(
g
∘
f
)
−
1
=
(
f
−
1
)
∘
(
g
−
1
)
{\displaystyle (g\,\circ \,f)^{-1}\;=\;(f^{-1})\,\circ \,(g^{-1})}
.
Conversely, if the composition
g
∘
f
{\displaystyle g\,\circ \,f}
of two functions is bijective, it only follows that f is injective and g is surjective.
== Cardinality ==
If X and Y are finite sets, then there exists a bijection between the two sets X and Y if and only if X and Y have the same number of elements. Indeed, in axiomatic set theory, this is taken as the definition of "same number of elements" (equinumerosity), and generalising this definition to infinite sets leads to the concept of cardinal number, a way to distinguish the various sizes of infinite sets.
== Properties ==
A function f: R → R is bijective if and only if its graph meets every horizontal and vertical line exactly once.
If X is a set, then the bijective functions from X to itself, together with the operation of functional composition (
∘
{\displaystyle \circ }
), form a group, the symmetric group of X, which is denoted variously by S(X), SX, or X! (X factorial).
Bijections preserve cardinalities of sets: for a subset A of the domain with cardinality |A| and subset B of the codomain with cardinality |B|, one has the following equalities:
|f(A)| = |A| and |f−1(B)| = |B|.
If X and Y are finite sets with the same cardinality, and f: X → Y, then the following are equivalent:
f is a bijection.
f is a surjection.
f is an injection.
For a finite set S, there is a bijection between the set of possible total orderings of the elements and the set of bijections from S to S. That is to say, the number of permutations of elements of S is the same as the number of total orderings of that set—namely, n!.
== Category theory ==
Bijections are precisely the isomorphisms in the category Set of sets and set functions. However, the bijections are not always the isomorphisms for more complex categories. For example, in the category Grp of groups, the morphisms must be homomorphisms since they must preserve the group structure, so the isomorphisms are group isomorphisms which are bijective homomorphisms.
== Generalization to partial functions ==
The notion of one-to-one correspondence generalizes to partial functions, where they are called partial bijections, although partial bijections are only required to be injective. The reason for this relaxation is that a (proper) partial function is already undefined for a portion of its domain; thus there is no compelling reason to constrain its inverse to be a total function, i.e. defined everywhere on its domain. The set of all partial bijections on a given base set is called the symmetric inverse semigroup.
Another way of defining the same notion is to say that a partial bijection from A to B is any relation
R (which turns out to be a partial function) with the property that R is the graph of a bijection f:A′→B′, where A′ is a subset of A and B′ is a subset of B.
When the partial bijection is on the same set, it is sometimes called a one-to-one partial transformation. An example is the Möbius transformation simply defined on the complex plane, rather than its completion to the extended complex plane.
== Gallery ==
== See also ==
Ax–Grothendieck theorem
Bijection, injection and surjection
Bijective numeration
Bijective proof
Category theory
Multivalued function
== Notes ==
== References ==
This topic is a basic concept in set theory and can be found in any text which includes an introduction to set theory. Almost all texts that deal with an introduction to writing proofs will include a section on set theory, so the topic may be found in any of these:
Hall, Marshall Jr. (1959). The Theory of Groups. MacMillan.
Wolf (1998). Proof, Logic and Conjecture: A Mathematician's Toolbox. Freeman.
Sundstrom (2003). Mathematical Reasoning: Writing and Proof. Prentice-Hall.
Smith; Eggen; St.Andre (2006). A Transition to Advanced Mathematics (6th Ed.). Thomson (Brooks/Cole).
Schumacher (1996). Chapter Zero: Fundamental Notions of Abstract Mathematics. Addison-Wesley.
O'Leary (2003). The Structure of Proof: With Logic and Set Theory. Prentice-Hall.
Morash. Bridge to Abstract Mathematics. Random House.
Maddox (2002). Mathematical Thinking and Writing. Harcourt/ Academic Press.
Lay (2001). Analysis with an introduction to proof. Prentice Hall.
Gilbert; Vanstone (2005). An Introduction to Mathematical Thinking. Pearson Prentice-Hall.
Fletcher; Patty. Foundations of Higher Mathematics. PWS-Kent.
Iglewicz; Stoyle. An Introduction to Mathematical Reasoning. MacMillan.
Devlin, Keith (2004). Sets, Functions, and Logic: An Introduction to Abstract Mathematics. Chapman & Hall/ CRC Press.
D'Angelo; West (2000). Mathematical Thinking: Problem Solving and Proofs. Prentice Hall.
Cupillari (1989). The Nuts and Bolts of Proofs. Wadsworth. ISBN 9780534103200.
Bond. Introduction to Abstract Mathematics. Brooks/Cole.
Barnier; Feldman (2000). Introduction to Advanced Mathematics. Prentice Hall.
Ash. A Primer of Abstract Mathematics. MAA.
== External links ==
"Bijection", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Weisstein, Eric W. "Bijection". MathWorld.
Earliest Uses of Some of the Words of Mathematics: entry on Injection, Surjection and Bijection has the history of Injection and related terms. | Wikipedia/Bijective_function |
In mathematics, the graph of a function
f
{\displaystyle f}
is the set of ordered pairs
(
x
,
y
)
{\displaystyle (x,y)}
, where
f
(
x
)
=
y
.
{\displaystyle f(x)=y.}
In the common case where
x
{\displaystyle x}
and
f
(
x
)
{\displaystyle f(x)}
are real numbers, these pairs are Cartesian coordinates of points in a plane and often form a curve.
The graphical representation of the graph of a function is also known as a plot.
In the case of functions of two variables – that is, functions whose domain consists of pairs
(
x
,
y
)
{\displaystyle (x,y)}
–, the graph usually refers to the set of ordered triples
(
x
,
y
,
z
)
{\displaystyle (x,y,z)}
where
f
(
x
,
y
)
=
z
{\displaystyle f(x,y)=z}
. This is a subset of three-dimensional space; for a continuous real-valued function of two real variables, its graph forms a surface, which can be visualized as a surface plot.
In science, engineering, technology, finance, and other areas, graphs are tools used for many purposes. In the simplest case one variable is plotted as a function of another, typically using rectangular axes; see Plot (graphics) for details.
A graph of a function is a special case of a relation.
In the modern foundations of mathematics, and, typically, in set theory, a function is actually equal to its graph. However, it is often useful to see functions as mappings, which consist not only of the relation between input and output, but also which set is the domain, and which set is the codomain. For example, to say that a function is onto (surjective) or not the codomain should be taken into account. The graph of a function on its own does not determine the codomain. It is common to use both terms function and graph of a function since even if considered the same object, they indicate viewing it from a different perspective.
== Definition ==
Given a function
f
:
X
→
Y
{\displaystyle f:X\to Y}
from a set X (the domain) to a set Y (the codomain), the graph of the function is the set
G
(
f
)
=
{
(
x
,
f
(
x
)
)
:
x
∈
X
}
,
{\displaystyle G(f)=\{(x,f(x)):x\in X\},}
which is a subset of the Cartesian product
X
×
Y
{\displaystyle X\times Y}
. In the definition of a function in terms of set theory, it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its domain, its codomain and its graph.
== Examples ==
=== Functions of one variable ===
The graph of the function
f
:
{
1
,
2
,
3
}
→
{
a
,
b
,
c
,
d
}
{\displaystyle f:\{1,2,3\}\to \{a,b,c,d\}}
defined by
f
(
x
)
=
{
a
,
if
x
=
1
,
d
,
if
x
=
2
,
c
,
if
x
=
3
,
{\displaystyle f(x)={\begin{cases}a,&{\text{if }}x=1,\\d,&{\text{if }}x=2,\\c,&{\text{if }}x=3,\end{cases}}}
is the subset of the set
{
1
,
2
,
3
}
×
{
a
,
b
,
c
,
d
}
{\displaystyle \{1,2,3\}\times \{a,b,c,d\}}
G
(
f
)
=
{
(
1
,
a
)
,
(
2
,
d
)
,
(
3
,
c
)
}
.
{\displaystyle G(f)=\{(1,a),(2,d),(3,c)\}.}
From the graph, the domain
{
1
,
2
,
3
}
{\displaystyle \{1,2,3\}}
is recovered as the set of first component of each pair in the graph
{
1
,
2
,
3
}
=
{
x
:
∃
y
,
such that
(
x
,
y
)
∈
G
(
f
)
}
{\displaystyle \{1,2,3\}=\{x:\ \exists y,{\text{ such that }}(x,y)\in G(f)\}}
.
Similarly, the range can be recovered as
{
a
,
c
,
d
}
=
{
y
:
∃
x
,
such that
(
x
,
y
)
∈
G
(
f
)
}
{\displaystyle \{a,c,d\}=\{y:\exists x,{\text{ such that }}(x,y)\in G(f)\}}
.
The codomain
{
a
,
b
,
c
,
d
}
{\displaystyle \{a,b,c,d\}}
, however, cannot be determined from the graph alone.
The graph of the cubic polynomial on the real line
f
(
x
)
=
x
3
−
9
x
{\displaystyle f(x)=x^{3}-9x}
is
{
(
x
,
x
3
−
9
x
)
:
x
is a real number
}
.
{\displaystyle \{(x,x^{3}-9x):x{\text{ is a real number}}\}.}
If this set is plotted on a Cartesian plane, the result is a curve (see figure).
=== Functions of two variables ===
The graph of the trigonometric function
f
(
x
,
y
)
=
sin
(
x
2
)
cos
(
y
2
)
{\displaystyle f(x,y)=\sin(x^{2})\cos(y^{2})}
is
{
(
x
,
y
,
sin
(
x
2
)
cos
(
y
2
)
)
:
x
and
y
are real numbers
}
.
{\displaystyle \{(x,y,\sin(x^{2})\cos(y^{2})):x{\text{ and }}y{\text{ are real numbers}}\}.}
If this set is plotted on a three dimensional Cartesian coordinate system, the result is a surface (see figure).
Oftentimes it is helpful to show with the graph, the gradient of the function and several level curves. The level curves can be mapped on the function surface or can be projected on the bottom plane. The second figure shows such a drawing of the graph of the function:
f
(
x
,
y
)
=
−
(
cos
(
x
2
)
+
cos
(
y
2
)
)
2
.
{\displaystyle f(x,y)=-(\cos(x^{2})+\cos(y^{2}))^{2}.}
== See also ==
== References ==
== Further reading ==
== External links ==
Weisstein, Eric W. "Function Graph." From MathWorld—A Wolfram Web Resource. | Wikipedia/Function_graph |
In mathematics, the simplicial approximation theorem is a foundational result for algebraic topology, guaranteeing that continuous mappings can be (by a slight deformation) approximated by ones that are piecewise of the simplest kind. It applies to mappings between spaces that are built up from simplices—that is, finite simplicial complexes. The general continuous mapping between such spaces can be represented approximately by the type of mapping that is (affine-) linear on each simplex into another simplex, at the cost (i) of sufficient barycentric subdivision of the simplices of the domain, and (ii) replacement of the actual mapping by a homotopic one.
This theorem was first proved by L.E.J. Brouwer, by use of the Lebesgue covering theorem (a result based on compactness). It served to put the homology theory of the time—the first decade of the twentieth century—on a rigorous basis, since it showed that the topological effect (on homology groups) of continuous mappings could in a given case be expressed in a finitary way. This must be seen against the background of a realisation at the time that continuity was in general compatible with the pathological, in some other areas. This initiated, one could say, the era of combinatorial topology.
There is a further simplicial approximation theorem for homotopies, stating that a homotopy between continuous mappings can likewise be approximated by a combinatorial version.
== Formal statement of the theorem ==
Let
K
{\displaystyle K}
and
L
{\displaystyle L}
be two simplicial complexes. A simplicial mapping
f
:
K
→
L
{\displaystyle f:K\to L}
is called a simplicial approximation of a continuous function
F
:
|
K
|
→
|
L
|
{\displaystyle F:|K|\to |L|}
if for every point
x
∈
|
K
|
{\displaystyle x\in |K|}
,
|
f
|
(
x
)
{\displaystyle |f|(x)}
belongs to the minimal closed simplex of
L
{\displaystyle L}
containing the point
F
(
x
)
{\displaystyle F(x)}
. If
f
{\displaystyle f}
is a simplicial approximation to a continuous map
F
{\displaystyle F}
, then the geometric realization of
f
{\displaystyle f}
,
|
f
|
{\displaystyle |f|}
is necessarily homotopic to
F
{\displaystyle F}
.
The simplicial approximation theorem states that given any continuous map
F
:
|
K
|
→
|
L
|
{\displaystyle F:|K|\to |L|}
there exists a natural number
n
0
{\displaystyle n_{0}}
such that for all
n
≥
n
0
{\displaystyle n\geq n_{0}}
there exists a simplicial approximation
f
:
B
d
n
K
→
L
{\displaystyle f:\mathrm {Bd} ^{n}K\to L}
to
F
{\displaystyle F}
(where
B
d
K
{\displaystyle \mathrm {Bd} \;K}
denotes the barycentric subdivision of
K
{\displaystyle K}
, and
B
d
n
K
{\displaystyle \mathrm {Bd} ^{n}K}
denotes the result of applying barycentric subdivision
n
{\displaystyle n}
times.), in other words, if
K
{\displaystyle K}
and
L
{\displaystyle L}
are simplicial complexes and
f
:
|
K
|
→
|
L
|
{\displaystyle f:|K|\to |L|}
is a continuous function, then there is a subdivision
K
′
{\displaystyle K'}
of
K
{\displaystyle K}
and a simplicial map
g
:
K
′
→
L
{\displaystyle g:K'\to L}
which is homotopic to
f
{\displaystyle f}
. Moreover, if
ϵ
:
|
L
|
→
R
{\displaystyle \epsilon :|L|\to {\mathbb {R}}}
is a positive continuous map, then there are subdivisions
K
′
,
L
′
{\displaystyle K',L'}
of
K
,
L
{\displaystyle K,L}
and a simplicial map
g
:
K
′
→
L
′
{\displaystyle g:K'\to L'}
such that
g
{\displaystyle g}
is
ϵ
{\displaystyle \epsilon }
-homotopic to
f
{\displaystyle f}
; that is, there is a homotopy
H
:
|
K
|
×
[
0
,
1
]
→
|
L
|
{\displaystyle H:|K|\times [0,1]\to |L|}
from
f
{\displaystyle f}
to
g
{\displaystyle g}
such that
d
i
a
m
(
H
(
x
×
[
0
,
1
]
)
)
<
ϵ
(
f
(
x
)
)
{\displaystyle \mathrm {diam} (H(x\times [0,1]))<\epsilon (f(x))}
for all
x
∈
|
K
|
{\displaystyle x\in |K|}
. So, we may consider the simplicial approximation theorem as a piecewise linear analog of Whitney approximation theorem.
== References ==
"Simplicial complex", Encyclopedia of Mathematics, EMS Press, 2001 [1994] | Wikipedia/Simplicial_approximation_theorem |
The grid cell topology is studied in digital topology as part of the theoretical basis for (low-level) algorithms in computer image analysis or computer graphics.
The elements of the n-dimensional grid cell topology (n ≥ 1) are all n-dimensional grid cubes and their k-dimensional faces ( for 0 ≤ k ≤ n−1); between these a partial order A ≤ B is defined if A is a subset of B (and thus also dim(A) ≤ dim(B)). The grid cell topology is the Alexandrov topology (open sets are up-sets) with respect to this partial order. (See also poset topology.)
Alexandrov and Hopf first introduced the grid cell topology, for the two-dimensional case, within an exercise in their text Topologie I (1935).
A recursive method to obtain n-dimensional grid cells and an intuitive definition for
grid cell manifolds can be found in Chen, 2004. It is related to digital manifolds.
== See also ==
Pixel connectivity
== References ==
Digital Geometry: Geometric Methods for Digital Image Analysis, by Reinhard Klette and Azriel Rosenfeld, Morgan Kaufmann Pub, May 2004, (The Morgan Kaufmann Series in Computer Graphics) ISBN 1-55860-861-3
Topologie I, by Paul Alexandroff and Heinz Hopf, Springer, Berlin, 1935, xiii + 636 pp.
Chen, L. (2004). Discrete Surfaces and Manifolds: A Theory of Digital-Discrete Geometry and Topology. SP Computing. ISBN 0-9755122-1-8. | Wikipedia/Grid_cell_topology |
A COVID‑19 vaccine is a vaccine intended to provide acquired immunity against severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), the virus that causes coronavirus disease 2019 (COVID‑19).
Knowledge about the structure and function of previous coronaviruses causing diseases like severe acute respiratory syndrome (SARS) and Middle East respiratory syndrome (MERS) accelerated the development of various vaccine platforms in early 2020. In 2020, the first COVID‑19 vaccines were developed and made available to the public through emergency authorizations and conditional approvals. However, immunity from the vaccines wanes over time, requiring people to get booster doses of the vaccine to maintain protection against COVID‑19.
The COVID‑19 vaccines are widely credited for their role in reducing the spread of COVID‑19 and reducing the severity and death caused by COVID‑19. Many countries implemented phased distribution plans that prioritized those at highest risk of complications, such as the elderly, and those at high risk of exposure and transmission, such as healthcare workers.
Common side effects of COVID‑19 vaccines include soreness, redness, rash, inflammation at the injection site, fatigue, headache, myalgia (muscle pain), and arthralgia (joint pain), which resolve without medical treatment within a few days. COVID‑19 vaccination is safe for people who are pregnant or are breastfeeding.
As of August 2024, 13.72 billion doses of COVID‑19 vaccines have been administered worldwide, based on official reports from national public health agencies. By December 2020, more than 10 billion vaccine doses had been preordered by countries, with about half of the doses purchased by high-income countries comprising 14% of the world's population.
Despite the extremely rapid development of effective mRNA and viral vector vaccines, worldwide vaccine equity has not been achieved. The development and use of whole inactivated virus (WIV) and protein-based vaccines have also been recommended, especially for use in developing countries.
The 2023 Nobel Prize in Physiology or Medicine was awarded to Katalin Karikó and Drew Weissman for the development of effective mRNA vaccines against COVID‑19.
== Background ==
Prior to COVID‑19, a vaccine for an infectious disease had never been produced in less than several years – and no vaccine existed for preventing a coronavirus infection in humans. However, vaccines have been produced against several animal diseases caused by coronaviruses, including (as of 2003) infectious bronchitis virus in birds, canine coronavirus, and feline coronavirus. Previous projects to develop vaccines for viruses in the family Coronaviridae that affect humans have been aimed at severe acute respiratory syndrome (SARS) and Middle East respiratory syndrome (MERS). Vaccines against SARS and MERS have been tested in non-human animals.
According to studies published in 2005 and 2006, the identification and development of novel vaccines and medicines to treat SARS was a priority for governments and public health agencies around the world at that time. There is no cure or protective vaccine proven to be safe and effective against SARS in humans. There is also no proven vaccine against MERS. When MERS became prevalent, it was believed that existing SARS research might provide a useful template for developing vaccines and therapeutics against a MERS-CoV infection. As of March 2020, there was one (DNA-based) MERS vaccine that completed Phase I clinical trials in humans, and three others in progress, all being viral-vectored vaccines: two adenoviral-vectored (ChAdOx1-MERS, BVRS-GamVac) and one MVA-vectored (MVA-MERS-S).
Vaccines that use an inactive or weakened virus that has been grown in eggs typically take more than a decade to develop. In contrast, mRNA is a molecule that can be made quickly, and research on mRNA to fight diseases was begun decades before the COVID‑19 pandemic by scientists such as Drew Weissman and Katalin Karikó, who tested on mice. Moderna began human testing of an mRNA vaccine in 2015. Viral vector vaccines were also developed for the COVID‑19 pandemic after the technology was previously cleared for Ebola.
As multiple COVID‑19 vaccines have been authorized or licensed for use, real-world vaccine effectiveness (RWE) is being assessed using case control and observational studies. A study is investigating the long-lasting protection against SARS-CoV-2 provided by the mRNA vaccines.
== Vaccine technologies ==
The initial focus of SARS-CoV-2 vaccines was on preventing symptomatic, often severe, illness. Most of the first COVID‑19 vaccines were two-dose vaccines, with the exception single-dose vaccines Convidecia and the Janssen COVID‑19 vaccine, and vaccines with three-dose schedules, Razi Cov Pars and Soberana.
As of July 2021, at least nine different technology platforms were under research and development to create an effective vaccine against COVID‑19. Most of the platforms of vaccine candidates in clinical trials are focused on the coronavirus spike protein (S protein) and its variants as the primary antigen of COVID‑19 infection, since the S protein triggers strong B-cell and T-cell immune responses. However, other coronavirus proteins are also being investigated for vaccine development, like the nucleocapsid, because they also induce a robust T-cell response and their genes are more conserved and recombine less frequently. Future generations of COVID‑19 vaccines targeting more conserved genomic regions could be used to treat future variations of SARS-CoV-2, or any similar coronavirus epidemic/pandemic.
Platforms developed in 2020 involved nucleic acid technologies (nucleoside-modified messenger RNA and DNA), non-replicating viral vectors, peptides, recombinant proteins, live attenuated viruses, and inactivated viruses.
Many vaccine technologies being developed for COVID‑19 use "next-generation" strategies for precise targeting of COVID‑19 infection mechanisms. Several of the synthetic vaccines use a 2P mutation to lock the spike protein into its prefusion configuration, stimulating an adaptive immune response to the virus before it attaches to a human cell. Vaccine platforms in development may improve flexibility for antigen manipulation and effectiveness for targeting mechanisms of COVID‑19 infection in susceptible population subgroups, such as healthcare workers, the elderly, children, pregnant women, and people with weakened immune systems.
=== mRNA vaccines ===
Several COVID‑19 vaccines, such as the Pfizer–BioNTech and Moderna vaccines, use RNA to stimulate an immune response. When introduced into human tissue, the vaccine contains either self-replicating RNA or messenger RNA (mRNA), which both cause cells to express the SARS-CoV-2 spike protein. This teaches the immune system how to identify and destroy the corresponding pathogen. RNA vaccines often use nucleoside-modified messenger RNA. The delivery of mRNA is achieved by a coformulation of the molecule into lipid nanoparticles, which protect the RNA strands and help their absorption into the cells.
RNA vaccines are the first COVID‑19 vaccines to be authorized in the United Kingdom, the United States, and the European Union. Authorized vaccines of this type include the Pfizer–BioNTech and Moderna vaccines. The CVnCoV RNA vaccine from CureVac failed in clinical trials.
Severe allergic reactions are rare. In December 2020, 1,893,360 first doses of Pfizer–BioNTech COVID‑19 vaccine administration resulted in 175 cases of severe allergic reactions, of which 21 were anaphylaxis. For 4,041,396 Moderna COVID‑19 vaccine dose administrations in December 2020 and January 2021, only ten cases of anaphylaxis were reported. Lipid nanoparticles (LNPs) were most likely responsible for the allergic reactions.
=== Adenovirus vector vaccines ===
These vaccines are examples of non-replicating viral vector vaccines using an adenovirus shell containing DNA that encodes a SARS‑CoV‑2 protein. The viral vector-based vaccines against COVID‑19 are non-replicating, meaning that they do not make new virus particles but rather produce only the antigen that elicits a systemic immune response.
Authorized vaccines of this type include the Oxford–AstraZeneca COVID‑19 vaccine, the Sputnik V COVID‑19 vaccine, Convidecia, and the Janssen COVID‑19 vaccine.
Convidecia and Janssen are both single dose vaccines that can be stored under ordinary refrigeration for several months.
Sputnik V uses Ad26 for its first dose, which is the same as Janssen's only dose, and Ad5 for the second dose, which is the same as Convidecia's only dose.
In August 2021, the developers of Sputnik V proposed, in view of the Delta case surge, that Pfizer test the Ad26 component (termed its 'Light' version) as a booster shot.
=== Inactivated virus vaccines ===
Inactivated vaccines consist of virus particles that are grown in culture and then killed using a method such as heat or formaldehyde to lose disease-producing capacity while still stimulating an immune response.
Inactivated virus vaccines authorized in China include the Chinese CoronaVac and the Sinopharm BIBP and WIBP vaccines; there is also the Indian Covaxin, the Russian CoviVac, the Kazakh vaccine QazVac, and the Iranian COVIran Barekat. Vaccines in clinical trials include the Valneva COVID‑19 vaccine.
=== Subunit vaccines ===
Subunit vaccines present one or more antigens without introducing whole pathogen particles. The antigens involved are often protein subunits, but they can be any molecule fragment of the pathogen.
The authorized vaccines of this type include the peptide vaccine EpiVacCorona, ZF2001, MVC-COV1901, Corbevax, the Sanofi–GSK vaccine, and Soberana 02 (a conjugate vaccine). Bimervax (selvacovatein) was approved for use as a booster vaccine in the European Union in March 2023.
The V451 vaccine was in clinical trials that were terminated after it was found that the vaccine may potentially cause incorrect results for subsequent HIV testing.
=== Virus-like particle vaccines ===
The authorized vaccines of this type include the Novavax COVID‑19 vaccine.
=== Other types ===
Additional types of vaccines that are in clinical trials include multiple DNA plasmid vaccines, at least two lentivirus vector vaccines, a conjugate vaccine, and a vesicular stomatitis virus displaying the SARS‑CoV‑2 spike protein.
Scientists investigated whether existing vaccines for unrelated conditions could prime the immune system and lessen the severity of COVID‑19 infections. There is experimental evidence that the BCG vaccine for tuberculosis has non-specific effects on the immune system, but there is no evidence that this vaccine is effective against COVID‑19.
== List of authorized vaccines ==
== Delivery methods ==
Most coronavirus vaccines are administered by intramuscular injection, with further vaccine delivery methods being studied for future coronavirus vaccines.
=== Intranasal ===
Intranasal vaccines target mucosal immunity in the nasal mucosa, which is a portal for viral entry into the body. These vaccines are designed to stimulate nasal immune factors, such as IgA. In addition to inhibiting the virus, nasal vaccines provide ease of administration because no needles (or needle phobia) are involved.
A variety of intranasal COVID‑19 vaccines are undergoing clinical trials. The first authorised intranasal vaccine was Razi Cov Pars in Iran at the end of October 2021. The first viral component of Sputnik V vaccine was authorised in Russia as Sputnik Nasal in April 2022. In September 2022, India and China approved two nasal COVID‑19 vaccines (iNCOVACC and Convidecia), which may (as boosters) also reduce transmission (potentially via sterilizing immunity). In December 2022, China approved a second intranasal vaccine as a booster, trade name Pneucolin.
=== Autologous ===
Aivita Biomedical is developing an experimental autologous dendritic cell COVID‑19 vaccine kit where the vaccine is prepared and incubated at the point-of-care using cells from the intended recipient. The vaccine is undergoing small phase I and phase II clinical studies.
== Universal vaccine ==
A universal coronavirus vaccine would be effective against all coronaviruses and possibly other viruses. The concept was publicly endorsed by NIAID director Anthony Fauci, virologist Jeffery K. Taubenberger, and David M. Morens. In March 2022, the White House released the "National COVID‑19 Preparedness Plan", which recommended accelerating the development of a universal coronavirus vaccine.
One attempt at such a vaccine is being developed at the Walter Reed Army Institute of Research. It uses a spike ferritin-based nanoparticle (SpFN). This vaccine began a Phase I clinical trial in April 2022. Results of this trial were published in May 2024. Other universal vaccines that have entered clinical trial include OVX033 (France), PanCov (France), pEVAC-PS (UK), and VBI-2902 (Canada).
Another strategy is to attach vaccine fragments from multiple strains to a nanoparticle scaffold. One theory is that a broader range of strains can be vaccinated against by targeting the receptor-binding domain, rather than the whole spike protein.
== Formulation ==
As of September 2020, eleven of the vaccine candidates in clinical development use adjuvants to enhance immunogenicity. An immunological adjuvant is a substance formulated with a vaccine to elevate the immune response to an antigen, such as the COVID‑19 virus or influenza virus. Specifically, an adjuvant may be used in formulating a COVID‑19 vaccine candidate to boost its immunogenicity and efficacy to reduce or prevent COVID‑19 infection in vaccinated individuals. Adjuvants used in COVID‑19 vaccine formulation may be particularly effective for technologies using the inactivated COVID‑19 virus and recombinant protein-based or vector-based vaccines. Aluminum salts, known as "alum", were the first adjuvant used for licensed vaccines and are the adjuvant of choice in some 80% of adjuvanted vaccines. The alum adjuvant initiates diverse molecular and cellular mechanisms to enhance immunogenicity, including the release of proinflammatory cytokines.
In June 2024, the US Food and Drug Administration advised the manufacturers of the licensed and authorized COVID-19 vaccines that the COVID-19 vaccines (2024-2025 Formula) for use in the United States beginning in fall 2024 should be monovalent JN.1 vaccines.
As of May 2025, the World Health Organization recommends that monovalent JN.1 or KP.2 vaccines remain appropriate vaccine antigens and that monovalent LP.8.1 is a suitable alternative vaccine antigen. The European Medicines Agency recommends updating COVID-19 vaccines to target LP.8.1 for the 2025/2026 vaccination campaign.
== Planning and development ==
Since January 2020, vaccine development has been expedited via unprecedented collaboration in the multinational pharmaceutical industry and between governments.
Multiple steps along the entire development path are evaluated, including:
the level of acceptable toxicity of the vaccine (its safety),
targeting vulnerable populations,
the need for vaccine efficacy breakthroughs,
the duration of vaccination protection,
special delivery systems (such as oral or nasal, rather than by injection),
dose regimen,
stability and storage characteristics,
emergency use authorization before formal licensing,
optimal manufacturing for scaling to billions of doses, and
dissemination of the licensed vaccine.
=== Challenges ===
There have been several unique challenges with COVID‑19 vaccine development.
Timelines for conducting clinical research – normally a sequential process requiring years – are being compressed into safety, efficacy, and dosing trials running simultaneously over months, potentially compromising safety assurance. For example, Chinese vaccine developers and the Chinese Center for Disease Control and Prevention began their efforts in January 2020, and by March they were pursuing numerous candidates on short timelines.
The rapid development and urgency of producing a vaccine for the COVID‑19 pandemic were expected to increase the risks and failure rate of delivering a safe, effective vaccine. Additionally, research at universities is obstructed by physical distancing and the closing of laboratories.
Vaccines must progress through several phases of clinical trials to test for safety, immunogenicity, effectiveness, dose levels, and adverse effects of the candidate vaccine. Vaccine developers have to invest resources internationally to find enough participants for Phase II–III clinical trials when the virus has proved to be a "moving target" of changing transmission rates across and within countries, forcing companies to compete for trial participants.
Clinical trial organizers may also encounter people unwilling to be vaccinated due to vaccine hesitancy or disbelief in the science of the vaccine technology and its ability to prevent infection. As new vaccines are developed during the COVID‑19 pandemic, licensure of COVID‑19 vaccine candidates requires submission of a full dossier of information on development and manufacturing quality.
=== Organizations ===
Internationally, the Access to COVID‑19 Tools Accelerator is a G20 and World Health Organization (WHO) initiative announced in April 2020. It is a cross-discipline support structure to enable partners to share resources and knowledge. It comprises four pillars, each managed by two to three collaborating partners: Vaccines (also called "COVAX"), Diagnostics, Therapeutics, and Health Systems Connector. The WHO's April 2020 "R&D Blueprint (for the) novel Coronavirus" documented a "large, international, multi-site, individually randomized controlled clinical trial" to allow "the concurrent evaluation of the benefits and risks of each promising candidate vaccine within 3–6 months of it being made available for the trial." The WHO vaccine coalition will prioritize which vaccines should go into Phase II and III clinical trials and determine harmonized Phase III protocols for all vaccines achieving the pivotal trial stage.
National governments have also been involved in vaccine development. Canada announced funding for 96 projects for the development and production of vaccines at Canadian companies and universities, with plans to establish a "vaccine bank" that could be used if another coronavirus outbreak occurs, support clinical trials, and develop manufacturing and supply chains for vaccines.
China provided low-rate loans to one vaccine developer through its central bank and "quickly made land available for the company" to build production plants. Three Chinese vaccine companies and research institutes are supported by the government for financing research, conducting clinical trials, and manufacturing.
The United Kingdom government formed a COVID‑19 vaccine task force in April 2020 to stimulate local efforts for accelerated development of a vaccine through collaborations between industries, universities, and government agencies. The UK's Vaccine Taskforce contributed to every phase of development, from research to manufacturing.
In the United States, the Biomedical Advanced Research and Development Authority (BARDA), a federal agency funding disease-fighting technology, announced investments to support American COVID‑19 vaccine development and the manufacturing of the most promising candidates. In May 2020, the government announced funding for a fast-track program called Operation Warp Speed. By March 2021, BARDA had funded an estimated $19.3 billion in COVID‑19 vaccine development.
Large pharmaceutical companies with experience in making vaccines at scale, including Johnson & Johnson, AstraZeneca, and GlaxoSmithKline (GSK), formed alliances with biotechnology companies, governments, and universities to accelerate progress toward effective vaccines.
== Clinical research ==
=== Post-vaccination complications ===
== History ==
In November 2021, the full nucleotide sequences of the AstraZeneca and Pfizer/BioNTech vaccines were released by the UK Medicines and Healthcare products Regulatory Agency in response to a freedom of information request.
== Effectiveness ==
An analysis involving more than 20 million adults found that vaccinated people had a lower risk of long COVID compared to those who had not had a COVID-19 vaccine.
=== Duration of immunity ===
As of 2021, available evidence shows that fully vaccinated individuals and those previously infected with SARS-CoV-2 have a low risk of subsequent infection for at least six months. There is insufficient data to determine an antibody titer threshold that indicates when an individual is protected from infection. Multiple studies show that antibody titers are associated with protection at the population level, but individual protection titers remain unknown. For some populations, such as the elderly and the immunocompromised, protection levels may be reduced after both vaccination and infection. Available evidence indicates that the level of protection may not be the same for all variants of the virus.
As of December 2021, there are no FDA-authorized or approved tests that providers or the public can use to determine if a person is protected from infection reliably.
As of March 2022, elderly residents' protection against severe illness, hospitalization, and death in English care homes was high immediately after vaccination, but protection declined significantly in the months following vaccination. Protection among care home staff, who were younger, declined much more slowly. Regular boosters are recommended for older people, and boosters for care home residents every six months appear reasonable.
The US Centers for Disease Control and Prevention (CDC) recommends a fourth dose of the Pfizer mRNA vaccine as of March 2022 for "certain immunocompromised individuals and people over the age of 50".
=== Immune evasion by variants ===
In contrast to other investigated prior variants, the SARS-CoV-2 Omicron variant and its BA.4/5 subvariants have evaded immunity induced by vaccines, which may lead to breakthrough infections despite recent vaccination. Nevertheless, vaccines are thought to provide protection against severe illness, hospitalizations, and deaths due to Omicron.
==== Vaccine adjustments ====
=== Effectiveness against transmission ===
As of 2022, fully vaccinated individuals with breakthrough infections with the SARS-CoV-2 delta (B.1.617.2) variant were found to have a peak viral load similar to unvaccinated cases and could transmit infection in household settings.
=== Mix and match ===
According to studies, the combination of two different COVID‑19 vaccines, also called heterologous vaccination, cross-vaccination, or the mix-and-match method, provides protection equivalent to that of mRNA vaccines, including protection against the Delta variant. Individuals who receive the combination of two different vaccines produce strong immune responses, with side effects no worse than those caused by standard regimens.
=== Drug interactions ===
Methotrexate reduces the immune response to COVID-19 vaccines, making them less effective. Pausing methotrexate for two weeks following COVID-19 vaccination may result in improved immunity. Not taking the medicine for two weeks might result in a minor increase of inflammatory disease flares in some people.
== Adverse events ==
For most people, the side effects, also called adverse effects, from COVID‑19 vaccines are mild and can be managed at home. The adverse effects of the COVID‑19 vaccination are similar to those of other vaccines, and severe adverse effects are rare. Adverse effects from the vaccine are higher than placebo, but placebo arms of vaccine trials still reported adverse effects that can be attributed to the nocebo effect.
All vaccines that are administered via intramuscular injection, including COVID‑19 vaccines, have side effects related to the mild trauma associated with the procedure and the introduction of a foreign substance into the body. These include soreness, redness, rash, and inflammation at the injection site. Other common side effects include fatigue, headache, myalgia (muscle pain), and arthralgia (joint pain), all of which generally resolve without medical treatment within a few days. Like any other vaccine, some people are allergic to one or more ingredients in COVID‑19 vaccines. Typical side effects are stronger and more common in younger people and in subsequent doses, and up to 20% of people report a disruptive level of side effects after the second dose of an mRNA vaccine. These side effects are less common or weaker in inactivated vaccines. COVID‑19 vaccination-related enlargement of lymph nodes happens in 11.6% of those who received one dose of the vaccine and in 16% of those who received two doses.
Experiments in mice show that intramuscular injections of lipid excipient nanoparticles (an inactive substance that serves as the vehicle or medium) cause particles to enter the blood plasma and many organs, with higher concentrations found in the liver and lower concentrations in the spleen, adrenal glands, and ovaries. The highest concentration of nanoparticles was found at the injection site itself.
COVID‑19 vaccination is safe for breastfeeding people. Temporary changes to the menstrual cycle in young women have been reported. However, these changes are "small compared with natural variation and quickly reverse." In one study, women who received both doses of a two-dose vaccine during the same menstrual cycle (an atypical situation) may see their next period begin a couple of days late. They have about twice the usual risk of a clinically significant delay (about 10% of these women, compared to about 4% of unvaccinated women). Cycle lengths return to normal after two menstrual cycles post-vaccination. Women who received doses in separate cycles had approximately the same natural variation in cycle lengths as unvaccinated women. Other temporary menstrual effects have been reported, such as heavier than normal menstrual bleeding after vaccination.
Serious adverse events associated COVID‑19 vaccines are generally rare but of high interest to the public. The official databases of reported adverse events include
the World Health Organization's VigiBase;
the United States Vaccine Adverse Events Reporting System (VAERS);
the United Kingdom's Yellow Card Scheme;
the European Medicines Agency's EudraVigilance system, which operates a regular transfer of data on suspected adverse drug reactions occurring in the EU to WHO's Uppsala Monitoring Centre.
Increased public awareness of these reporting systems and the extra reporting requirements under US FDA Emergency Use Authorization rules have increased reported adverse events. Serious side effects are an ongoing area of study, and resources have been allocated to try to better understand them. Research currently indicates that the rate and type of side effects are lower-risk than infection. For example, although vaccination may trigger some side effects, the effects experienced from an infection could be worse. Neurological side effects from getting COVID‑19 are hundreds of times more likely than from vaccination.
Documented rare serious effects include:
anaphylaxis, a severe type of allergic reaction. Anaphylaxis affects one person per 250,000 to 400,000 doses administered. According to a 2022 systematic review, the mortality rate of people with anaphylaxis following COVID‐19 vaccination was 0.5%.
blood clots (thrombosis). These vaccine-induced immune thrombocytopenia and thrombosis are associated with vaccines using an adenovirus system (Janssen and Oxford-AstraZeneca). These affect about one person per 100,000.
myocarditis and pericarditis, or inflammation of the heart. There is a rare risk of myocarditis (inflammation of the heart muscle) or pericarditis (inflammation of the membrane covering the heart) after the mRNA COVID‑19 vaccines (Moderna or Pfizer-BioNTech). The risk of myocarditis after COVID‑19 vaccination is estimated to be 0.3 to 5 cases per 100,000 persons, with the highest risk in young males. In an Israeli nation-wide population-based study (in which the Pfizer-BioNTech vaccine was exclusively given), the incidence rate of myocarditis was 54 cases out of 2.5 million vaccine recipients, with an overall incidence rate of 2 cases per 100,000 persons, with the highest incidence seen in young males (aged 16 to 29) at 10 cases per 100,000 vaccine recipients. Of the cases of myocarditis seen, 76% were mild in severity, with one case of cardiogenic shock (heart failure) and one death (in a person with a preexisting heart condition) reported within the 83-day follow-up period. COVID‑19 vaccines may protect against myocarditis due to subsequent COVID‑19 infection. The risk of myocarditis and pericarditis is significantly higher (up to 11 times higher with respect to myocarditis) after COVID‑19 infection as compared to COVID‑19 vaccination, with the possible exception of younger men (less than 40 years old) who may have a higher risk of myocarditis after the second Moderna mRNA vaccine (an additional 97 cases of myocarditis per 1 million persons vaccinated). The mortality rate from myocarditis post-vaccination is extremely low. According to a 2022 study, of patients diagnosed with myocarditis (in both vaccination and COVID-19 cohort) 1.07% were hospitalized and 0.015% died.
thrombotic thrombocytopenia and other autoimmune diseases, which have been reported as adverse events after the COVID‑19 vaccine.
There are rare reports of subjective hearing changes, including tinnitus, after vaccination.
== Society and culture ==
=== Distribution ===
Note about the table in this section: number and percentage of people who have received at least one dose of a COVID‑19 vaccine (unless noted otherwise). May include vaccination of non-citizens, which can push totals beyond 100% of the local population. The table is updated daily by a bot.
=== Access ===
Countries have extremely unequal access to the COVID‑19 vaccine. Vaccine equity has not been achieved or even approximated. The inequity has harmed both countries with poor access and countries with good access.
Nations pledged to buy doses of the COVID‑19 vaccines before the doses were available. Though high-income nations represent only 14% of the global population, as of 15 November 2020, they had contracted to buy 51% of all pre-sold doses. Some high-income nations bought more doses than would be necessary to vaccinate their entire populations.
In January 2021, WHO Director-General Tedros Adhanom Ghebreyesus warned of problems with equitable distribution: "More than 39 million doses of vaccine have now been administered in at least 49 higher-income countries. Just 25 doses have been given in one lowest-income country. Not 25 million; not 25 thousand; just 25."
In March 2021, it was revealed that the US attempted to convince Brazil not to purchase the Sputnik V COVID‑19 vaccine, fearing "Russian influence" in Latin America. Some nations involved in long-standing territorial disputes have reportedly had their access to vaccines blocked by competing nations; Palestine has accused Israel of blocking vaccine delivery to Gaza, while Taiwan has suggested that China has hampered its efforts to procure vaccine doses.
A single dose of the COVID‑19 vaccines by AstraZeneca would cost 47 Egyptian pounds (EGP), and the authorities are selling them for between 100 and 200 EGP. A report by the Carnegie Endowment for International Peace cited the poverty rate in Egypt as around 29.7 percent, which constitutes approximately 30.5 million people, and claimed that about 15 million Egyptians would be unable to gain access to the luxury of vaccination. A human rights lawyer, Khaled Ali, launched a lawsuit against the government, forcing them to provide vaccinations free of charge to all members of the public.
According to immunologist Anthony Fauci, mutant strains of the virus and limited vaccine distribution pose continuing risks, and he said, "we have to get the entire world vaccinated, not just our own country." Edward Bergmark and Arick Wierson are calling for a global vaccination effort and wrote that the wealthier nations' "me-first" mentality could ultimately backfire because the spread of the virus in poorer countries would lead to more variants, against which the vaccines could be less effective.
In March 2021, the United States, Britain, European Union member states, and some other members of the World Trade Organization (WTO) blocked a push by more than eighty developing countries to waive COVID‑19 vaccine patent rights in an effort to boost production of vaccines for poor nations. On 5 May 2021, the US government under President Joe Biden announced that it supports waiving intellectual property protections for COVID‑19 vaccines. The Members of the European Parliament have backed a motion demanding the temporary lifting of intellectual property rights for COVID‑19 vaccines.
In a meeting in April 2021, the World Health Organization's emergency committee addressed concerns of persistent inequity in global vaccine distribution. Although 9 percent of the world's population lives in the 29 poorest countries, these countries had received only 0.3% of all vaccines administered as of May 2021. In March 2021, Brazilian journalism agency Agência Pública reported that the country vaccinated about twice as many people who declare themselves white than black and noted that mortality from COVID‑19 is higher in the black population.
In May 2021, UNICEF made an urgent appeal to industrialized nations to pool their excess COVID‑19 vaccine capacity to make up for a 125-million-dose gap in the COVAX program. The program mostly relied on the Oxford–AstraZeneca COVID‑19 vaccine produced by the Serum Institute of India, which faced serious supply problems due to increased domestic vaccine needs in India from March to June 2021. Only a limited amount of vaccines can be distributed efficiently, and the shortfall of vaccines in South America and parts of Asia is due to a lack of expedient donations by richer nations. International aid organizations have pointed at Nepal, Sri Lanka, and the Maldives, as well as Argentina, Brazil, and some parts of the Caribbean, as problem areas where vaccines are in short supply. In mid-May 2021, UNICEF was also critical of the fact that most proposed donations of Moderna and Pfizer vaccines were not slated for delivery until the second half of 2021 or early in 2022.
In July 2021, the heads of the World Bank Group, the International Monetary Fund, the World Health Organization, and the World Trade Organization said in a joint statement: "As many countries are struggling with new variants and a third wave of COVID‑19 infections, accelerating access to vaccines becomes even more critical to ending the pandemic everywhere and achieving broad-based growth. We are deeply concerned about the limited vaccines, therapeutics, diagnostics, and support for deliveries available to developing countries." In July 2021, The BMJ reported that countries had thrown out over 250,000 vaccine doses as supply exceeded demand and strict laws prevented the sharing of vaccines. A survey by The New York Times found that over a million doses of vaccine had been thrown away in ten U.S. states because federal regulations prohibit recalling them, preventing their redistribution abroad. Furthermore, doses donated close to expiration often cannot be administered quickly enough by recipient countries and end up having to be discarded. To help overcome this problem, the Prime Minister of India, Narendra Modi, announced that they would make their digital vaccination management platform, CoWIN, open to the global community. He also announced that India would also release the source code for the contact tracing app Aarogya Setu for developers around the world. Around 142 countries, including Afghanistan, Bangladesh, Bhutan, the Maldives, Guyana, Antigua and Barbuda, St. Kitts and Nevis, and Zambia, expressed their interest in the application for COVID management.
Amnesty International and Oxfam International have criticized the support of vaccine monopolies by the governments of producing countries, noting that this is dramatically increasing the dose price by five times and often much more, creating an economic barrier to access for poor countries. Médecins Sans Frontières (Doctors without Borders) has also criticized vaccine monopolies and repeatedly called for their suspension, supporting the TRIPS waiver. The waiver was first proposed in October 2020 and has support from most countries, but was delayed by opposition from the EU (especially Germany; major EU countries such as France, Italy, and Spain support the exemption), the UK, Norway, and Switzerland, among others. MSF called for a Day of Action in September 2021 to put pressure on the WTO Minister's meeting in November, which was expected to discuss the TRIPS IP waiver.
In August 2021, to reduce unequal distribution between rich and poor countries, the WHO called for a moratorium on booster doses at least until the end of September. However, in August, the United States government announced plans to offer booster doses eight months after the initial course to the general population, starting with priority groups. Before the announcement, the WHO harshly criticized this type of decision, citing the lack of evidence for the need for boosters, except for patients with specific conditions. At this time, vaccine coverage of at least one dose was 58% in high-income countries and only 1.3% in low-income countries, and 1.14 million Americans had already received an unauthorized booster dose. US officials argued that waning efficacy against mild and moderate disease might indicate reduced protection against severe disease in the coming months. Israel, France, Germany, and the United Kingdom have also started planning boosters for specific groups. In September 2021, more than 140 former world leaders and Nobel laureates, including former President of France François Hollande, former Prime Minister of the United Kingdom Gordon Brown, former Prime Minister of New Zealand Helen Clark, and Professor Joseph Stiglitz, called on the candidates to be the next German chancellor to declare themselves in favor of waiving intellectual property rules for COVID‑19 vaccines and transferring vaccine technologies. In November 2021, nursing unions in 28 countries filed a formal appeal with the United Nations over the refusal of the UK, EU, Norway, Switzerland, and Singapore to temporarily waive patents for COVID‑19 vaccines.
During his first international trip, the President of Peru, Pedro Castillo, spoke at the seventy-sixth session of the United Nations General Assembly on 21 September 2021, proposing the creation of an international treaty signed by world leaders and pharmaceutical companies to guarantee universal vaccine access, arguing that "The battle against the pandemic has shown us the failure of the international community to cooperate under the principle of solidarity."
Optimizing the societal benefit of vaccination may benefit from a strategy that is tailored to the state of the pandemic, the demographics of a country, the age of the recipients, the availability of vaccines, and the individual risk for severe disease. In the UK, the interval between prime and booster doses was extended to vaccinate as many people as early as possible. Many countries are starting to give an additional booster shot to the immunosuppressed and the elderly, and research predicts an additional benefit of personalizing vaccine doses in the setting of limited vaccine availability when a wave of virus Variants of Concern hits a country.
Despite the extremely rapid development of effective mRNA and viral vector vaccines, vaccine equity has not been achieved. The World Health Organization called for 70 percent of the global population to be vaccinated by mid-2022, but as of March 2022, it was estimated that only one percent of the 10 billion doses given worldwide had been administered in low-income countries. An additional 6 billion vaccinations may be needed to fill vaccine access gaps, particularly in developing countries. Given the projected availability of newer vaccines, the development and use of whole inactivated virus (WIV) and protein-based vaccines are also recommended. Organizations such as the Developing Countries Vaccine Manufacturers Network could help to support the production of such vaccines in developing countries, with lower production costs and greater ease of deployment.
While vaccines substantially reduce the probability and severity of infection, it is still possible for fully vaccinated people to contract and spread COVID‑19. Public health agencies have recommended that vaccinated people continue using preventive measures (wear face masks, social distance, wash hands) to avoid infecting others, especially vulnerable people, particularly in areas with high community spread. Governments have indicated that such recommendations will be reduced as vaccination rates increase and community spread declines.
==== Economics ====
Vaccine inequity damages the global economy, disrupting the global supply chain. Most vaccines were reserved for wealthy countries; as of September 2021, some countries have more vaccines than are needed to fully vaccinate their populations. When people are under-vaccinated, needlessly die, experience disability, and live under lockdown restrictions, they cannot supply the same goods and services. This harms the economies of under-vaccinated and over-vaccinated countries alike. Since rich countries have larger economies, rich countries may lose more money to vaccine inequity than poor ones, though the poor ones will lose a higher percentage of GDP and experience longer-term effects. High-income countries would profit an estimated US$4.80 for every $1 spent on giving vaccines to lower-income countries.
The International Monetary Fund sees the vaccine divide between rich and poor nations as a serious obstacle to a global economic recovery. Vaccine inequity disproportionately affects refuge-providing states, as they tend to be poorer, and refugees and displaced people are economically more vulnerable even within those low-income states, so they have suffered more economically from vaccine inequity.
=== Liability ===
Several governments agreed to shield pharmaceutical companies like Pfizer and Moderna from negligence claims related to COVID‑19 vaccines (and treatments), as in previous pandemics, when governments also took on liability for such claims.
In the US, these liability shields took effect on 4 February 2020, when the US Secretary of Health and Human Services, Alex Azar, published a notice of declaration under the Public Readiness and Emergency Preparedness Act (PREP Act) for medical countermeasures against COVID‑19, covering "any vaccine, used to treat, diagnose, cure, prevent, or mitigate COVID‑19, or the transmission of SARS-CoV-2 or a virus mutating therefrom". The declaration precludes "liability claims alleging negligence by a manufacturer in creating a vaccine, or negligence by a health care provider in prescribing the wrong dose, absent willful misconduct." In other words, absent "willful misconduct", these companies cannot be sued for money damages for any injuries that occur between 2020 and 2024 from the administration of vaccines and treatments related to COVID‑19. The declaration is effective in the United States through 1 October 2024.
In December 2020, the UK government granted Pfizer legal indemnity for its COVID‑19 vaccine.
In the European Union, the COVID‑19 vaccines were granted a conditional marketing authorization, which does not exempt manufacturers from civil and administrative liability claims. The EU conditional marketing authorizations were changed to standard authorizations in September 2022. While the purchasing contracts with vaccine manufacturers remain secret, they do not contain liability exemptions, even for side effects not known at the time of licensure.
The Bureau of Investigative Journalism, a nonprofit news organization, reported in an investigation that unnamed officials in some countries, such as Argentina and Brazil, said that Pfizer demanded guarantees against costs of legal cases due to adverse effects in the form of liability waivers and sovereign assets such as federal bank reserves, embassy buildings, or military bases, going beyond what was expected from other countries, such as the US. During the pandemic parliamentary inquiry in Brazil, Pfizer's representative said that its terms for Brazil are the same as for all other countries with which it has signed deals.
On 13 December 2022, the governor of Florida, Ron DeSantis, said that he would petition the state supreme court to convene a grand jury to investigate possible violations in respect to COVID‑19 vaccines, and declared that his government would be able to get "the data whether they [the companies] want to give it or not".
In November 2023, the US state of Texas sued Pfizer under section 17.47 of the Texas Deceptive Trade Practices Act, alleging that the company misled the public about its COVID-19 vaccine by hiding risks while making false claims about its effectiveness. In June 2024, the US state of Kansas similarly sued Pfizer under the Kansas Consumer Protection Act, making similar allegations.
=== Controversy ===
In June 2021, a report revealed that the UB-612 vaccine, developed by the US-based Covaxx, was a for-profit venture initiated by Blackwater founder Erik Prince. In a series of text messages to Paul Behrends, the close associate recruited for the Covaxx project, Prince described the profit-making possibilities of selling the COVID‑19 vaccines. Covaxx provided no data from the clinical trials on safety or efficacy it conducted in Taiwan. The responsibility of creating distribution networks was assigned to an Abu Dhabi-based entity, which was mentioned as "Windward Capital" on the Covaxx letterhead but was actually Windward Holdings. The firm's sole shareholder, who handled "professional, scientific and technical activities", was Erik Prince. In March 2021, Covaxx raised $1.35 billion in a private placement.
=== Misinformation and hesitancy ===
The United States Department of Defense (DoD) undertook a disinformation campaign in the Philippines, later expanded to Central Asia and the Middle East, which sought to discredit China, in particular its Sinovac vaccine, disseminating hashtags of #ChinaIsTheVirus and posts claiming that the Sinovac vaccine contained gelatin from pork and therefore was haram or forbidden for purposes of Islamic law.
== See also ==
COVID‑19 drug development
COVID‑19 drug repurposing research
== Notes ==
== References ==
== Further reading ==
=== Vaccine protocols ===
"Protocol mRNA-1273-P301" (PDF). Moderna. Archived from the original (PDF) on 28 September 2020. Retrieved 21 September 2020.
"Protocol C4591001 PF-07302048 (BNT162 RNA-Based COVID-19 Vaccines)" (PDF). Pfizer. Archived from the original (PDF) on 6 December 2021. Retrieved 6 August 2022.
"Protocol AZD1222 – D8110C00001" (PDF). AstraZeneca.
"Protocol VAC31518COV3001; Phase 3 (Ensemble)" (PDF). Janssen Vaccines & Prevention.
"Protocol VAC31518COV3009; Phase 3 (Ensemble 2)" (PDF). Janssen Vaccines & Prevention.
"Protocol VAT00008 – Study of Monovalent and Bivalent Recombinant Protein Vaccines against COVID-19 in Adults 18 Years of Age and Older Protocol" (PDF). Sanofi Pasteur. Archived from the original (PDF) on 23 July 2021. Retrieved 30 July 2021.
"Protocol 2019nCoV-301". Novavax.
== External links ==
"COVID-19 vaccine tracker and landscape". World Health Organization (WHO).
M.I.T. Lecture 10: Kizzmekia Corbett, Vaccines on YouTube
M.I.T. Lecture 12: Dan Barouch, Covid-19 Vaccine Development on YouTube
"The 5 Stages of COVID-19 Vaccine Development: What You Need to Know About How a Clinical Trial Works". Johnson & Johnson. September 2020.
"Coronavirus vaccine – weekly summary of Yellow Card reporting". UK Medicines and Healthcare products Regulatory Agency (MHRA).
"COVID-19 vaccine safety report - 05-10-2023". Therapeutic Goods Administration (TGA). 4 October 2023.
Kolata G, Mueller B (15 January 2022). "Halting Progress and Happy Accidents: How mRNA Vaccines Were Made". The New York Times. Archived from the original on 15 January 2022.
"COVID-19 Vaccine Information Statement". U.S. Centers for Disease Control and Prevention (CDC). 16 October 2024. | Wikipedia/COVID-19_vaccine |
In mathematics, the Bing–Borsuk conjecture states that every
n
{\displaystyle n}
-dimensional homogeneous absolute neighborhood retract space is a topological manifold. The conjecture has been proved for dimensions 1 and 2, and it is known that the 3-dimensional version of the conjecture implies the Poincaré conjecture.
== Definitions ==
A topological space is homogeneous if, for any two points
m
1
,
m
2
∈
M
{\displaystyle m_{1},m_{2}\in M}
, there is a homeomorphism of
M
{\displaystyle M}
which takes
m
1
{\displaystyle m_{1}}
to
m
2
{\displaystyle m_{2}}
.
A metric space
M
{\displaystyle M}
is an absolute neighborhood retract (ANR) if, for every closed embedding
f
:
M
→
N
{\displaystyle f:M\rightarrow N}
(where
N
{\displaystyle N}
is a metric space), there exists an open neighbourhood
U
{\displaystyle U}
of the image
f
(
M
)
{\displaystyle f(M)}
which retracts to
f
(
M
)
{\displaystyle f(M)}
.
There is an alternate statement of the Bing–Borsuk conjecture: suppose
M
{\displaystyle M}
is embedded in
R
m
+
n
{\displaystyle \mathbb {R} ^{m+n}}
for some
m
≥
3
{\displaystyle m\geq 3}
and this embedding can be extended to an embedding of
M
×
(
−
ε
,
ε
)
{\displaystyle M\times (-\varepsilon ,\varepsilon )}
. If
M
{\displaystyle M}
has a mapping cylinder neighbourhood
N
=
C
φ
{\displaystyle N=C_{\varphi }}
of some map
φ
:
∂
N
→
M
{\displaystyle \varphi :\partial N\rightarrow M}
with mapping cylinder projection
π
:
N
→
M
{\displaystyle \pi :N\rightarrow M}
, then
π
{\displaystyle \pi }
is an approximate fibration.
== History ==
The conjecture was first made in a paper by R. H. Bing and Karol Borsuk in 1965, who proved it for
n
=
1
{\displaystyle n=1}
and 2.
Włodzimierz Jakobsche showed in 1978 that, if the Bing–Borsuk conjecture is true in dimension 3, then the Poincaré conjecture must also be true.
The Busemann conjecture states that every Busemann
G
{\displaystyle G}
-space is a topological manifold. It is a special case of the Bing–Borsuk conjecture. The Busemann conjecture is known to be true for dimensions 1 to 4.
== References == | Wikipedia/Bing–Borsuk_conjecture |
In topology, a graph manifold (in German: Graphenmannigfaltigkeit) is a 3-manifold which is obtained by gluing some circle bundles. They were discovered and classified by the German topologist Friedhelm Waldhausen in 1967. This definition allows a very convenient combinatorial description as a graph whose vertices are the fundamental parts and (decorated) edges stand for the description of the gluing, hence the name.
Two very important classes of examples are given by the Seifert bundles and the Solv manifolds. This leads to a more modern definition: a graph manifold is either a Solv manifold, a manifold having only Seifert pieces in its JSJ decomposition, or connect sums of the previous two categories. From this perspective, Waldhausen's article can be seen as the first breakthrough towards the discovery of JSJ decomposition.
One of the numerous consequences of the Thurston-Perelman geometrization theorem is that graph manifolds are precisely the 3-manifolds whose Gromov norm vanishes.
== References ==
Waldhausen, Friedhelm (1967), "Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. I", Inventiones Mathematicae, 3 (4): 308–333, doi:10.1007/BF01402956, ISSN 0020-9910, MR 0235576
Waldhausen, Friedhelm (1967), "Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. II", Inventiones Mathematicae, 4 (2): 87–117, doi:10.1007/BF01425244, ISSN 0020-9910, MR 0235576 | Wikipedia/Graph_manifold |
In mathematics, Thurston's geometrization conjecture (now a theorem) states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries (Euclidean, spherical, or hyperbolic).
In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by William Thurston (1982) as part of his 24 questions, and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture.
Thurston's hyperbolization theorem implies that Haken manifolds satisfy the geometrization conjecture. Thurston announced a proof in the 1980s, and since then, several complete proofs have appeared in print.
Grigori Perelman announced a proof of the full geometrization conjecture in 2003 using Ricci flow with surgery in two papers posted at the arxiv.org preprint server. Perelman's papers were studied by several independent groups that produced books and online manuscripts filling in the complete details of his arguments. Verification was essentially complete in time for Perelman to be awarded the 2006 Fields Medal for his work, and in 2010 the Clay Mathematics Institute awarded him its 1 million USD prize for solving the Poincaré conjecture, though Perelman declined both awards.
The Poincaré conjecture and the spherical space form conjecture are corollaries of the geometrization conjecture, although there are shorter proofs of the former that do not lead to the geometrization conjecture.
== The conjecture ==
A 3-manifold is called closed if it is compact – without "punctures" or "missing endpoints" – and has no boundary ("edge").
Every closed 3-manifold has a prime decomposition: this means it is the connected sum ("a gluing together") of prime 3-manifolds. This reduces much of the study of 3-manifolds to the case of prime 3-manifolds: those that cannot be written as a non-trivial connected sum.
Here is a statement of Thurston's conjecture:
Every oriented prime closed 3-manifold can be cut along tori, so that the interior of each of the resulting manifolds has a geometric structure with finite volume.
There are 8 possible geometric structures in 3 dimensions. There is a unique minimal way of cutting an irreducible oriented 3-manifold along tori into pieces that are Seifert manifolds or atoroidal called the JSJ decomposition, which is not quite the same as the decomposition in the geometrization conjecture, because some of the pieces in the JSJ decomposition might not have finite volume geometric structures. (For example, the mapping torus of an Anosov map of a torus has a finite volume solv structure, but its JSJ decomposition cuts it open along one torus to produce a product of a torus and a unit interval, and the interior of this has no finite volume geometric structure.)
For non-oriented manifolds the easiest way to state a geometrization conjecture is to first take the oriented double cover. It is also possible to work directly with non-orientable manifolds, but this gives some extra complications: it may be necessary to cut along projective planes and Klein bottles as well as spheres and tori, and manifolds with a projective plane boundary component usually have no geometric structure.
In 2 dimensions, every closed surface has a geometric structure consisting of a metric with constant curvature; it is not necessary to cut the manifold up first. Specifically, every closed surface is diffeomorphic to a quotient of S2, E2, or H2.
== The eight Thurston geometries ==
A model geometry is a simply connected smooth manifold X together with a transitive action of a Lie group G on X with compact stabilizers.
A model geometry is called maximal if G is maximal among groups acting smoothly and transitively on X with compact stabilizers. Sometimes this condition is included in the definition of a model geometry.
A geometric structure on a manifold M is a diffeomorphism from M to X/Γ for some model geometry X, where Γ is a discrete subgroup of G acting freely on X ; this is a special case of a complete (G,X)-structure. If a given manifold admits a geometric structure, then it admits one whose model is maximal.
A 3-dimensional model geometry X is relevant to the geometrization conjecture if it is maximal and if there is at least one compact manifold with a geometric structure modelled on X. Thurston classified the 8 model geometries satisfying these conditions; they are listed below and are sometimes called Thurston geometries. (There are also uncountably many model geometries without compact quotients.)
There is some connection with the Bianchi groups: the 3-dimensional Lie groups. Most Thurston geometries can be realized as a left invariant metric on a Bianchi group. However S2 × R cannot be, Euclidean space corresponds to two different Bianchi groups, and there are an uncountable number of solvable non-unimodular Bianchi groups, most of which give model geometries with no compact representatives.
=== Spherical geometry S3 ===
The point stabilizer is O(3, R), and the group G is the 6-dimensional Lie group O(4, R), with 2 components. The corresponding manifolds are exactly the closed 3-manifolds with finite fundamental group. Examples include the 3-sphere, the Poincaré homology sphere, Lens spaces. This geometry can be modeled as a left invariant metric on the Bianchi group of type IX. Manifolds with this geometry are all compact, orientable, and have the structure of a Seifert fiber space (often in several ways). The complete list of such manifolds is given in the article on spherical 3-manifolds. Under Ricci flow, manifolds with this geometry collapse to a point in finite time.
=== Euclidean geometry E3 ===
The point stabilizer is O(3, R), and the group G is the 6-dimensional Lie group R3 × O(3, R), with 2 components. Examples are the 3-torus, and more generally the mapping torus of a finite-order automorphism of the 2-torus; see torus bundle. There are exactly 10 finite closed 3-manifolds with this geometry, 6 orientable and 4 non-orientable. This geometry can be modeled as a left invariant metric on the Bianchi groups of type I or VII0. Finite volume manifolds with this geometry are all compact, and have the structure of a Seifert fiber space (sometimes in two ways). The complete list of such manifolds is given in the article on Seifert fiber spaces. Under Ricci flow, manifolds with Euclidean geometry remain invariant.
=== Hyperbolic geometry H3 ===
The point stabilizer is O(3, R), and the group G is the 6-dimensional Lie group O+(1, 3, R), with 2 components. There are enormous numbers of examples of these, and their classification is not completely understood. The example with smallest volume is the Weeks manifold. Other examples are given by the Seifert–Weber space, or "sufficiently complicated" Dehn surgeries on links, or most Haken manifolds. The geometrization conjecture implies that a closed 3-manifold is hyperbolic if and only if it is irreducible, atoroidal, and has infinite fundamental group. This geometry can be modeled as a left invariant metric on the Bianchi group of type V or VIIh≠0. Under Ricci flow, manifolds with hyperbolic geometry expand.
=== The geometry of S2 × R ===
The point stabilizer is O(2, R) × Z/2Z, and the group G is O(3, R) × R × Z/2Z, with 4 components. The four finite volume manifolds with this geometry are: S2 × S1, the mapping torus of the antipode map of S2, the connected sum of two copies of 3-dimensional projective space, and the product of S1 with two-dimensional projective space. The first two are mapping tori of the identity map and antipode map of the 2-sphere, and are the only examples of 3-manifolds that are prime but not irreducible. The third is the only example of a non-trivial connected sum with a geometric structure. This is the only model geometry that cannot be realized as a left invariant metric on a 3-dimensional Lie group. Finite volume manifolds with this geometry are all compact and have the structure of a Seifert fiber space (often in several ways). Under normalized Ricci flow manifolds with this geometry converge to a 1-dimensional manifold.
=== The geometry of H2 × R ===
The point stabilizer is O(2, R) × Z/2Z, and the group G is O+(1, 2, R) × R × Z/2Z, with 4 components. Examples include the product of a hyperbolic surface with a circle, or more generally the mapping torus of an isometry of a hyperbolic surface. Finite volume manifolds with this geometry have the structure of a Seifert fiber space if they are orientable. (If they are not orientable the natural fibration by circles is not necessarily a Seifert fibration: the problem is that some fibers may "reverse orientation"; in other words their neighborhoods look like fibered solid Klein bottles rather than solid tori.) The classification of such (oriented) manifolds is given in the article on Seifert fiber spaces. This geometry can be modeled as a left invariant metric on the Bianchi group of type III. Under normalized Ricci flow manifolds with this geometry converge to a 2-dimensional manifold.
=== The geometry of the universal cover of SL(2, R) ===
The universal cover of SL(2, R) is denoted
S
L
~
(
2
,
R
)
{\displaystyle {\widetilde {\rm {SL}}}(2,\mathbf {R} )}
. It fibers over H2, and the space is sometimes called "Twisted H2 × R". The group G has 2 components. Its identity component has the structure
(
R
×
S
L
~
2
(
R
)
)
/
Z
{\displaystyle (\mathbf {R} \times {\widetilde {\rm {SL}}}_{2}(\mathbf {R} ))/\mathbf {Z} }
. The point stabilizer is O(2,R).
Examples of these manifolds include: the manifold of unit vectors of the tangent bundle of a hyperbolic surface, and more generally the Brieskorn homology spheres (excepting the 3-sphere and the Poincaré dodecahedral space). This geometry can be modeled as a left invariant metric on the Bianchi group of type VIII or III. Finite volume manifolds with this geometry are orientable and have the structure of a Seifert fiber space. The classification of such manifolds is given in the article on Seifert fiber spaces. Under normalized Ricci flow manifolds with this geometry converge to a 2-dimensional manifold.
=== Nil geometry ===
This fibers over E2, and so is sometimes known as "Twisted E2 × R". It is the geometry of the Heisenberg group. The point stabilizer is O(2, R). The group G has 2 components, and is a semidirect product of the 3-dimensional Heisenberg group by the group O(2, R) of isometries of a circle. Compact manifolds with this geometry include the mapping torus of a Dehn twist of a 2-torus, or the quotient of the Heisenberg group by the "integral Heisenberg group". This geometry can be modeled as a left invariant metric on the Bianchi group of type II. Finite volume manifolds with this geometry are compact and orientable and have the structure of a Seifert fiber space. The classification of such manifolds is given in the article on Seifert fiber spaces. Under normalized Ricci flow, compact manifolds with this geometry converge to R2 with the flat metric.
=== Sol geometry ===
This geometry (also called Solv geometry) fibers over the line with fiber the plane, and is the geometry of the identity component of the group G. The point stabilizer is the dihedral group of order 8. The group G has 8 components, and is the group of maps from 2-dimensional Minkowski space to itself that are either isometries or multiply the metric by −1. The identity component has a normal subgroup R2 with quotient R, where R acts on R2 with 2 (real) eigenspaces, with distinct real eigenvalues of product 1. This is the Bianchi group of type VI0 and the geometry can be modeled as a left invariant metric on this group. All finite volume manifolds with solv geometry are compact. The compact manifolds with solv geometry are either the mapping torus of an Anosov map of the 2-torus (such a map is an automorphism of the 2-torus given by an invertible 2 by 2 matrix whose eigenvalues are real and distinct, such as
(
2
1
1
1
)
{\displaystyle \left({\begin{array}{*{20}c}2&1\\1&1\\\end{array}}\right)}
), or quotients of these by groups of order at most 8. The eigenvalues of the automorphism of the torus generate an order of a real quadratic field, and the solv manifolds can be classified in terms of the units and ideal classes of this order.
Under normalized Ricci flow compact manifolds with this geometry converge (rather slowly) to R1.
== Uniqueness ==
A closed 3-manifold has a geometric structure of at most one of the 8 types above, but finite volume non-compact 3-manifolds can occasionally have more than one type of geometric structure. (Nevertheless, a manifold can have many different geometric structures of the same type; for example, a surface of genus at least 2 has a continuum of different hyperbolic metrics.) More precisely, if M is a manifold with a finite volume geometric structure, then the type of geometric structure is almost determined as follows, in terms of the fundamental group π1(M):
If π1(M) is finite then the geometric structure on M is spherical, and M is compact.
If π1(M) is virtually cyclic but not finite then the geometric structure on M is S2×R, and M is compact.
If π1(M) is virtually abelian but not virtually cyclic then the geometric structure on M is Euclidean, and M is compact.
If π1(M) is virtually nilpotent but not virtually abelian then the geometric structure on M is nil geometry, and M is compact.
If π1(M) is virtually solvable but not virtually nilpotent then the geometric structure on M is solv geometry, and M is compact.
If π1(M) has an infinite normal cyclic subgroup but is not virtually solvable then the geometric structure on M is either H2×R or the universal cover of SL(2, R). The manifold M may be either compact or non-compact. If it is compact, then the 2 geometries can be distinguished by whether or not π1(M) has a finite index subgroup that splits as a semidirect product of the normal cyclic subgroup and something else. If the manifold is non-compact, then the fundamental group cannot distinguish the two geometries, and there are examples (such as the complement of a trefoil knot) where a manifold may have a finite volume geometric structure of either type.
If π1(M) has no infinite normal cyclic subgroup and is not virtually solvable then the geometric structure on M is hyperbolic, and M may be either compact or non-compact.
Infinite volume manifolds can have many different types of geometric structure: for example, R3 can have 6 of the different geometric structures listed above, as 6 of the 8 model geometries are homeomorphic to it. Moreover if the volume does not have to be finite there are an infinite number of new geometric structures with no compact models; for example, the geometry of almost any non-unimodular 3-dimensional Lie group.
There can be more than one way to decompose a closed 3-manifold into pieces with geometric structures. For example:
Taking connected sums with several copies of S3 does not change a manifold.
The connected sum of two projective 3-spaces has a S2×R geometry, and is also the connected sum of two pieces with S3 geometry.
The product of a surface of negative curvature and a circle has a geometric structure, but can also be cut along tori to produce smaller pieces that also have geometric structures. There are many similar examples for Seifert fiber spaces.
It is possible to choose a "canonical" decomposition into pieces with geometric structure, for example by first cutting the manifold into prime pieces in a minimal way, then cutting these up using the smallest possible number of tori. However this minimal decomposition is not necessarily the one produced by Ricci flow; in fact, the Ricci flow can cut up a manifold into geometric pieces in many inequivalent ways, depending on the choice of initial metric.
== History ==
The Fields Medal was awarded to Thurston in 1982 partially for his proof of the geometrization conjecture for Haken manifolds.
In 1982, Richard S. Hamilton showed that given a closed 3-manifold with a metric of positive Ricci curvature, the Ricci flow would collapse the manifold to a point in finite time, which proves the geometrization conjecture for this case as the metric becomes "almost round" just before the collapse. He later developed a program to prove the geometrization conjecture by Ricci flow with surgery. The idea is that the Ricci flow will in general produce singularities, but one may be able to continue the Ricci flow past the singularity by using surgery to change the topology of the manifold. Roughly speaking, the Ricci flow contracts positive curvature regions and expands negative curvature regions, so it should kill off the pieces of the manifold with the "positive curvature" geometries S3 and S2 × R, while what is left at large times should have a thick–thin decomposition into a "thick" piece with hyperbolic geometry and a "thin" graph manifold.
In 2003, Grigori Perelman announced a proof of the geometrization conjecture by showing that the Ricci flow can indeed be continued past the singularities, and has the behavior described above.
One component of Perelman's proof was a novel collapsing theorem in Riemannian geometry. Perelman did not release any details on the proof of this result (Theorem 7.4 in the preprint 'Ricci flow with surgery on three-manifolds'). Beginning with Shioya and Yamaguchi, there are now several different proofs of Perelman's collapsing theorem, or variants thereof. Shioya and Yamaguchi's formulation was used in the first fully detailed formulations of Perelman's work.
A second route to the last part of Perelman's proof of geometrization is the method of Laurent Bessières and co-authors, which uses Thurston's hyperbolization theorem for Haken manifolds and Gromov's norm for 3-manifolds. A book by the same authors with complete details of their version of the proof has been published by the European Mathematical Society.
== Higher dimensions ==
In four dimensions, only a rather restricted class of closed 4-manifolds admit a geometric decomposition. However, lists of maximal model geometries can still be given.
The four-dimensional maximal model geometries were classified by Richard Filipkiewicz in 1983. They number eighteen, plus one countably infinite family: their usual names are E4, Nil4, Nil3 × E1, Sol4m,n (a countably infinite family), Sol40, Sol41, H3 × E1,
S
L
~
{\displaystyle {\widetilde {\rm {SL}}}}
× E1, H2 × E2, H2 × H2, H4, H2(C) (a complex hyperbolic space), F4 (the tangent bundle of the hyperbolic plane), S2 × E2, S2 × H2, S3 × E1, S4, CP2 (the complex projective plane), and S2 × S2. No closed manifold admits the geometry F4, but there are manifolds with proper decomposition including an F4 piece.
The five-dimensional maximal model geometries were classified by Andrew Geng in 2016. There are 53 individual geometries and six infinite families. Some new phenomena not observed in lower dimensions occur, including two uncountable families of geometries and geometries with no compact quotients.
== Footnotes ==
== Notes ==
== References ==
L. Bessieres, G. Besson, M. Boileau, S. Maillot, J. Porti, 'Geometrisation of 3-manifolds', EMS Tracts in Mathematics, volume 13. European Mathematical Society, Zurich, 2010. [1]
M. Boileau Geometrization of 3-manifolds with symmetries
F. Bonahon Geometric structures on 3-manifolds Handbook of Geometric Topology (2002) Elsevier.
Cao, Huai-Dong; Zhu, Xi-Ping (2006). "A complete proof of the Poincaré and geometrization conjectures—application of the Hamilton–Perelman theory of the Ricci flow". Asian Journal of Mathematics. 10 (2): 165–492. doi:10.4310/ajm.2006.v10.n2.a2. MR 2233789. Zbl 1200.53057.– – (2006). "Erratum". Asian Journal of Mathematics. 10 (4): 663–664. doi:10.4310/AJM.2006.v10.n4.e2. MR 2282358.– – (2006). "Hamilton–Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture". arXiv:math/0612069.
Allen Hatcher: Notes on Basic 3-Manifold Topology 2000
J. Isenberg, M. Jackson, Ricci flow of locally homogeneous geometries on a Riemannian manifold, J. Diff. Geom. 35 (1992) no. 3 723–741.
Kleiner, Bruce; Lott, John (2008). "Notes on Perelman's papers". Geometry & Topology. 12 (5). Updated for corrections in 2011 & 2013: 2587–2855. arXiv:math/0605667. doi:10.2140/gt.2008.12.2587. MR 2460872. Zbl 1204.53033.
John W. Morgan. Recent progress on the Poincaré conjecture and the classification of 3-manifolds. Bulletin Amer. Math. Soc. 42 (2005) no. 1, 57–78 (expository article explains the eight geometries and geometrization conjecture briefly, and gives an outline of Perelman's proof of the Poincaré conjecture)
Morgan, John W.; Fong, Frederick Tsz-Ho (2010). Ricci Flow and Geometrization of 3-Manifolds. University Lecture Series. ISBN 978-0-8218-4963-7. Retrieved 2010-09-26.
Morgan, John; Tian, Gang (2014). The geometrization conjecture. Clay Mathematics Monographs. Vol. 5. Cambridge, MA: Clay Mathematics Institute. ISBN 978-0-8218-5201-9. MR 3186136.
Perelman, Grisha (2002). "The entropy formula for the Ricci flow and its geometric applications". arXiv:math/0211159.
Perelman, Grisha (2003). "Ricci flow with surgery on three-manifolds". arXiv:math/0303109.
Perelman, Grisha (2003). "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds". arXiv:math/0307245.
Scott, Peter The geometries of 3-manifolds. (errata) Bull. London Math. Soc. 15 (1983), no. 5, 401–487.
Thurston, William P. (1982). "Three-dimensional manifolds, Kleinian groups and hyperbolic geometry". Bulletin of the American Mathematical Society. New Series. 6 (3): 357–381. doi:10.1090/S0273-0979-1982-15003-0. ISSN 0002-9904. MR 0648524. This gives the original statement of the conjecture.
William Thurston. Three-dimensional geometry and topology. Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. x+311 pp. ISBN 0-691-08304-5 (in depth explanation of the eight geometries and the proof that there are only eight)
William Thurston. The Geometry and Topology of Three-Manifolds, 1980 Princeton lecture notes on geometric structures on 3-manifolds.
== External links ==
"The Geometry of 3-Manifolds (video)". Archived from the original on January 27, 2010. Retrieved January 20, 2010. A public lecture on the Poincaré and geometrization conjectures, given by C. McMullen at Harvard in 2006. | Wikipedia/Thurston's_geometrization_conjecture |
Cancer immunotherapy (immuno-oncotherapy) is the stimulation of the immune system to treat cancer, improving the immune system's natural ability to fight the disease. It is an application of the fundamental research of cancer immunology (immuno-oncology) and a growing subspecialty of oncology.
Cancer immunotherapy exploits the fact that cancer cells often have tumor antigens, molecules on their surface that can bind to antibody proteins or T-cell receptors, triggering an immune system response. The tumor antigens are often proteins or other macromolecules (e.g., carbohydrates). Normal antibodies bind to external pathogens, but the modified immunotherapy antibodies bind to the tumor antigens marking and identifying the cancer cells for the immune system to inhibit or kill. The clinical success of cancer immunotherapy is highly variable between different forms of cancer; for instance, certain subtypes of gastric cancer react well to the approach whereas immunotherapy is not effective for other subtypes.
In 2018, American immunologist James P. Allison and Japanese immunologist Tasuku Honjo received the Nobel Prize in Physiology or Medicine for their discovery of cancer therapy by inhibition of negative immune regulation.
== History ==
"During the 17th and 18th centuries, various forms of immunotherapy in cancer became widespread... In the 18th and 19th centuries, septic dressings enclosing ulcerative tumours were used for the treatment of cancer. Surgical wounds were left open to facilitate the development of infection, and purulent sores were created deliberately... One of the most well-known effects of microorganisms on ... cancer was reported in 1891, when an American surgeon, William Coley, inoculated patients having inoperable tumours with [ Streptococcus pyogenes ]." "Coley [had] thoroughly reviewed the literature available at that time and found 38 reports of cancer patients with accidental or iatrogenic feverish erysipelas. In 12 patients, the sarcoma or carcinoma had completely disappeared; the others had substantially improved. Coley decided to attempt the therapeutic use of iatrogenic erysipelas..." "Coley developed a toxin that contained heat-killed bacteria [ Streptococcus pyogenes and Serratia marcescens ]. Until 1963, this treatment was used for the treatment of sarcoma." "Coley injected more than 1000 cancer patients with bacteria or bacterial products." 51.9% of [Coley's] patients with inoperable soft-tissue sarcomas showed complete tumour regression and survived for more than 5 years, and 21.2% of the patients had no clinical evidence of tumour at least 20 years after this treatment..." Research continued in the 20th century under Maria O'Connor Hornung at Tulane Medical School.
In the 1980's, researchers at the National Cancer Institute's Center for Cancer Research (CCR) began exploring the then-heretical idea that a patient’s immune system could be harnessed to fight cancer. These researchers included Michael Potter, Ira Pastan, and Steven Rosenberg who developed approaches including monoclonal antibody-based immunotoxins, checkpoint blockade drugs, cytokine-based therapies, and adoptive cell therapy studies.
== Types and categories ==
There are several types of immunotherapy used to treat cancer:
Immune checkpoint inhibitors: drugs that block immune system checkpoints to allow immune cells to respond more strongly to the cancer.
T-cell transfer therapy: a treatment that takes T-cells from the tumor and selects or changes them in the lab to better attack cancer cells, then reintroduces them into the patient.
Monoclonal antibodies: designed to bind to specific targets on cancer cells, marking cancer cells so that they will be better seen and destroyed by the immune system.
Treatment vaccines: also known as therapeutic cancer vaccines, help the immune system learn to recognize and react to mutant proteins specific to the tumor and destroy cancer cells containing them.
Immune system modulators: agents that enhance the body’s immune response against cancer.
Immunotherapies can be categorized as active or passive based on their ability to engage the host immune system against cancer. Active immunotherapy specifically targets tumor cells via the immune system. Examples include therapeutic cancer vaccines (also known as treatment vaccines, which are designed to boost the body's immune system to fight cancer), CAR-T cells, and targeted antibody therapies. In contrast, passive immunotherapy does not directly target tumor cells, but enhances the ability of the immune system to attack cancer cells. Examples include checkpoint inhibitors and cytokines.
Active cellular therapies aim to destroy cancer cells by recognition of distinct markers known as antigens. In cancer vaccines, the goal is to generate an immune response to these antigens through a vaccine. Currently, only one vaccine (sipuleucel-T for prostate cancer) has been approved. In cell-mediated therapies like CAR-T cell therapy, immune cells are extracted from the patient, genetically engineered to recognize tumor-specific antigens, and returned to the patient. Cell types that can be used in this way are natural killer (NK) cells, lymphokine-activated killer cells, cytotoxic T cells, and dendritic cells. Finally, specific antibodies can be developed that recognize cancer cells and target them for destruction by the immune system. Examples of such antibodies include rituximab (targeting CD-20), trastuzumab (targeting HER-2), and cetuximab (targeting EGFR).
Passive antibody therapies aim to increase the activity of the immune system without specifically targeting cancer cells. For example, cytokines directly stimulate the immune system and increase immune activity. Checkpoint inhibitors target proteins (immune checkpoints) that normally dampen the immune response. This enhances the ability of the immune system to attack cancer cells. Current research is identifying new potential targets to enhance immune function. Approved checkpoint inhibitors include antibodies such as ipilimumab, nivolumab, and pembrolizumab.
== Cellular immunotherapy ==
=== Dendritic cell therapy ===
Dendritic cell therapy provokes anti-tumor responses by causing dendritic cells to present tumor antigens to lymphocytes, which activates them, priming them to kill other cells that present the antigen. Dendritic cells are antigen-presenting cells (APCs) in the mammalian immune system. In cancer treatment, they aid cancer antigen targeting. The only approved cellular cancer therapy based on dendritic cells is sipuleucel-T.
One method of inducing dendritic cells to present tumor antigens is by vaccination with autologous tumor lysates or short peptides (small parts of the protein that correspond to the protein antigens on cancer cells). These peptides are often given in combination with adjuvants (highly immunogenic substances) to increase the immune and anti-tumor responses. Other adjuvants include proteins or other chemicals that attract and/or activate dendritic cells, such as granulocyte-macrophage colony-stimulating factor (GM-CSF). The most common sources of antigens used for dendritic cell vaccine in glioblastoma (GBM) as an aggressive brain tumor were whole tumor lysate, CMV antigen RNA and tumor-associated peptides like EGFRvIII.
Dendritic cells can also be activated in vivo by making tumor cells express GM-CSF. This can be achieved by either genetically engineering tumor cells to produce GM-CSF or by infecting tumor cells with an oncolytic virus that expresses GM-CSF.
Another strategy is to remove dendritic cells from the blood of a patient and activate them outside the body. The dendritic cells are activated in the presence of tumor antigens, which may be a single tumor-specific peptide/protein or a tumor cell lysate (a solution of broken-down tumor cells). These cells (with optional adjuvants) are infused and provoke an immune response.
Dendritic cell therapies include the use of antibodies that bind to receptors on the surface of dendritic cells. Antigens can be added to the antibody and can induce the dendritic cells to mature and provide immunity to the tumor. Dendritic cell receptors such as TLR3, TLR7, TLR8 or CD40 have been used as antibody targets. Dendritic cell-NK cell interface also has an important role in immunotherapy. The design of new dendritic cell-based vaccination strategies should also encompass NK cell-stimulating potency. It is critical to systematically incorporate NK cells monitoring as an outcome in antitumor DC-based clinical trials.
==== Drugs ====
Sipuleucel-T (Provenge) was approved for treatment of asymptomatic or minimally symptomatic metastatic castration-resistant prostate cancer in 2010. The treatment consists of removal of antigen-presenting cells from blood by leukapheresis and growing them with the fusion protein PA2024 made from GM-CSF and prostate-specific prostatic acid phosphatase (PAP) and reinfused. This process is repeated three times.
=== Adoptive T-cell therapy ===
Adoptive T cell therapy is a form of passive immunization by the transfusion of T-cells. They are found in blood and tissue and typically activate when they find foreign pathogens. Activation occurs when the T-cell's surface receptors encounter cells that display parts of foreign proteins (either on their surface or intracellularly). These can be either infected cells or other antigen-presenting cells (APCs). The latter are found in normal tissue and in tumor tissue, where they are known as tumor-infiltrating lymphocytes (TILs). They are activated by the presence of APCs such as dendritic cells that present tumor antigens. Although these cells can attack tumors, the tumor microenvironment is highly immunosuppressive, interfering with immune-mediated tumour death.
Multiple ways of producing tumour-destroying T-cells have been developed. Most commonly, T-cells specific to a tumor antigen can be removed from a tumor sample (TILs) or filtered from blood. The T-cells can optionally be modified in various ways, cultured and infused into patients. T cells can be modified via genetic engineering, producing CAR-T cell or TCR T cells or by exposing the T cells to tumor antigens in a non-immunosuppressive environment, that they recognize as foreign and learn to attack.
Another approach is transfer of haploidentical γδ T cells or natural killer cells from a healthy donor. The major advantage of this approach is that these cells do not cause graft-versus-host disease. The disadvantage is that transferred cells frequently have impaired function.
==== Tumor-derived T cell therapy ====
The simplest example involves removing TILs from a tumor, culturing but not modifying them, and infusing the result back into the tumour. The first therapy of this type, Lifileucel, achieved US Food and Drug Administration (FDA) approval in February 2024.
==== CAR-T cell therapy ====
The premise of CAR-T immunotherapy is to modify T cells to recognize cancer cells in order to target and destroy them. Scientists harvest T cells from people, genetically alter them to add a chimeric antigen receptor (CAR) that specifically recognizes cancer cells, then infuse the resulting CAR-T cells into patients to attack their tumors.
Tisagenlecleucel (Kymriah), a chimeric antigen receptor (CAR-T) therapy, was approved by the FDA in 2017 to treat acute lymphoblastic leukemia (ALL). This treatment removes CD19 positive cells (B-cells) from the body (including the diseased cells, but also normal antibody-producing cells).
Axicabtagene ciloleucel (Yescarta) is another CAR-T therapeutic, approved in 2017 for treatment of diffuse large B-cell lymphoma (DLBCL).
==== Multifunctional alginate scaffolds ====
Multifunctional alginate scaffolds for T cell engineering and release (MASTER) is a technique for in situ engineering, replication and release of genetically engineered T cells. It is an evolution of CAR T cell therapy. T cells are extracted from the patient and mixed with a genetically engineered virus that contains a cancer-targeting gene (as with CAR T). The mixture is then added to a MASTER (scaffold), which absorbs them. The MASTER contains antibodies that activate the T cells and interleukins that trigger cell proliferation. The MASTER is then implanted into the patient. The activated T cells interact with the viruses to become CAR T cells. The interleukins stimulate these CAR T cells to proliferate, and the CAR T cells exit the MASTER to attack the cancer. The technique takes hours instead of weeks. And because the cells are younger, they last longer in the body, show stronger potency against cancer, and display fewer markers of exhaustion. These features were demonstrated in mouse models. The treatment was more effective and longer-lasting against lymphoma.
==== T cell receptor T cell therapy ====
== Antibody therapy ==
=== Antibody types ===
==== Conjugation ====
Two types are used in cancer treatments:
Naked monoclonal antibodies are antibodies without added elements. Most antibody therapies use this antibody type.
Conjugated monoclonal antibodies are joined to another molecule, which is either cytotoxic or radioactive. The toxic chemicals are those typically used as chemotherapy drugs, but other toxins can be used. The antibody binds to specific antigens on cancer cell surfaces, directing the therapy to the tumor. Radioactive compound-linked antibodies are referred to as radiolabelled. Chemolabelled or immunotoxins antibodies are tagged with chemotherapeutic molecules or toxins, respectively. Research has also demonstrated conjugation of a TLR agonist to an anti-tumor monoclonal antibody.
==== Fc regions ====
Fc's ability to bind Fc receptors is important because it allows antibodies to activate the immune system. Fc regions are varied: they exist in numerous subtypes and can be further modified, for example with the addition of sugars in a process called glycosylation. Changes in the Fc region can alter an antibody's ability to engage Fc receptors and, by extension, will determine the type of immune response that the antibody triggers. For example, immune checkpoint blockers targeting PD-1 are antibodies designed to bind PD-1 expressed by T cells and reactivate these cells to eliminate tumors. Anti-PD-1 drugs contain not only a Fab region that binds PD-1 but also an Fc region. Experimental work indicates that the Fc portion of cancer immunotherapy drugs can affect the outcome of treatment. For example, anti-PD-1 drugs with Fc regions that bind inhibitory Fc receptors can have decreased therapeutic efficacy. Imaging studies have further shown that the Fc region of anti-PD-1 drugs can bind Fc receptors expressed by tumor-associated macrophages. This process removes the drugs from their intended targets (i.e. PD-1 molecules expressed on the surface of T cells) and limits therapeutic efficacy. Furthermore, antibodies targeting the co-stimulatory protein CD40 require engagement with selective Fc receptors for optimal therapeutic efficacy. Together, these studies underscore the importance of Fc status in antibody-based immune checkpoint targeting strategies.
==== Human/non-human antibodies ====
Antibodies can come from a variety of sources, including human cells, mice, and a combination of the two (chimeric antibodies). Different sources of antibodies can provoke different kinds of immune responses. For example, the human immune system can recognize mouse antibodies (also known as murine antibodies) and trigger an immune response against them. This could reduce the effectiveness of the antibodies as a treatment and cause an immune reaction. Chimeric antibodies attempt to reduce murine antibodies' immunogenicity by replacing part of the antibody with the corresponding human counterpart. Humanized antibodies are almost completely human; only the complementarity determining regions of the variable regions are derived from murine sources. Human antibodies have been produced using unmodified human DNA.
=== Mechanism of action ===
==== Antibody-dependent cell-mediated cytotoxicity (ADCC) ====
Antibody-dependent cell-mediated cytotoxicity (ADCC) requires antibodies to bind to target cell surfaces. Antibodies are formed of a binding region (Fab) and the Fc region that can be detected by immune system cells via their Fc surface receptors. Fc receptors are found on many immune system cells, including NK cells. When NK cells encounter antibody-coated cells, the latter's Fc regions interact with their Fc receptors, releasing perforin and granzyme B to kill the tumor cell. Examples include rituximab, ofatumumab, elotuzumab, and alemtuzumab. Antibodies under development have altered Fc regions that have higher affinity for a specific type of Fc receptor, FcγRIIIA, which can dramatically increase effectiveness.
=== Anti-CD47 therapy ===
Many tumor cells overexpress CD47 to escape immunosurveilance of host immune system. CD47 binds to its receptor signal-regulatory protein alpha (SIRPα) and downregulate phagocytosis of tumor cell. Therefore, anti-CD47 therapy aims to restore clearance of tumor cells. Additionally, growing evidence supports the employment of tumor antigen-specific T cell response in response to anti-CD47 therapy. A number of therapeutics are being developed, including anti-CD47 antibodies, engineered decoy receptors, anti-SIRPα antibodies and bispecific agents. As of 2017, wide range of solid and hematologic malignancies were being clinically tested.
=== Anti-GD2 antibodies ===
Carbohydrate antigens on the surface of cells can be used as targets for immunotherapy. GD2 is a ganglioside found on the surface of many types of cancer cell including neuroblastoma, retinoblastoma, melanoma, small cell lung cancer, brain tumors, osteosarcoma, rhabdomyosarcoma, Ewing's sarcoma, liposarcoma, fibrosarcoma, leiomyosarcoma and other soft tissue sarcomas. It is not usually expressed on the surface of normal tissues, making it a good target for immunotherapy. As of 2014, clinical trials were underway.
==== Complement Activation ====
The complement system includes blood proteins that can cause cell death after an antibody binds to the cell surface (the classical complement pathway, among the ways of complement activation). Generally, the system deals with foreign pathogens but can be activated with therapeutic antibodies in cancer. The system can be triggered if the antibody is chimeric, humanized, or human; as long as it contains the IgG1 Fc region. Complement can lead to cell death by activation of the membrane attack complex, known as complement-dependent cytotoxicity; enhancement of antibody-dependent cell-mediated cytotoxicity; and CR3-dependent cellular cytotoxicity. Complement-dependent cytotoxicity occurs when antibodies bind to the cancer cell surface, the C1 complex binds to these antibodies and subsequently, protein pores are formed in cancer cell membrane.
Blocking
Antibody therapies can also function by binding to proteins and physically blocking them from interacting with other proteins. Checkpoint inhibitors (CTLA-4, PD-1, and PD-L1) operate by this mechanism. Briefly, checkpoint inhibitors are proteins that normally help to slow immune responses and prevent the immune system from attacking normal cells. Checkpoint inhibitors bind these proteins and prevent them from functioning normally, which increases the activity of the immune system. Examples include durvalumab, ipilimumab, nivolumab, and pembrolizumab.
=== FDA-approved antibodies ===
==== Alemtuzumab ====
Alemtuzumab (Campath-1H) is an anti-CD52 humanized IgG1 monoclonal antibody indicated for the treatment of fludarabine-refractory chronic lymphocytic leukemia (CLL), cutaneous T-cell lymphoma, peripheral T-cell lymphoma and T-cell prolymphocytic leukemia. CD52 is found on >95% of peripheral blood lymphocytes (both T-cells and B-cells) and monocytes, but its function in lymphocytes is unknown. It binds to CD52 and initiates its cytotoxic effect by complement fixation and ADCC mechanisms. Due to the antibody target (cells of the immune system), common complications of alemtuzumab therapy are infection, toxicity and myelosuppression.
==== Atezolizumab ====
==== Atezolizumab/hyaluronidase ====
==== Avelumab ====
==== Durvalumab ====
Durvalumab (Imfinzi) is a human immunoglobulin G1 kappa (IgG1κ) monoclonal antibody that blocks the interaction of programmed cell death ligand 1 (PD-L1) with the PD-1 and CD80 (B7.1) molecules. Durvalumab is approved for the treatment of patients with locally advanced or metastatic urothelial carcinoma who:
have disease progression during or following platinum-containing chemotherapy.
have disease progression within 12 months of neoadjuvant or adjuvant treatment with platinum-containing chemotherapy.
On 16 February 2018, the Food and Drug Administration approved durvalumab for patients with unresectable stage III non-small cell lung cancer (NSCLC) whose disease has not progressed following concurrent platinum-based chemotherapy and radiation therapy.
==== Elotuzumab ====
==== Ipilimumab ====
Ipilimumab (Yervoy) is a human IgG1 antibody that binds the surface protein CTLA4. In normal physiology T-cells are activated by two signals: the T-cell receptor binding to an antigen-MHC complex and T-cell surface receptor CD28 binding to CD80 or CD86 proteins. CTLA4 binds to CD80 or CD86, preventing the binding of CD28 to these surface proteins and therefore negatively regulates the activation of T-cells.
Active cytotoxic T-cells are required for the immune system to attack melanoma cells. Normally inhibited active melanoma-specific cytotoxic T-cells can produce an effective anti-tumor response. Ipilimumab can cause a shift in the ratio of regulatory T-cells to cytotoxic T-cells to increase the anti-tumor response. Regulatory T-cells inhibit other T-cells, which may benefit the tumor.
==== Nivolumab ====
Nivolumab is a human IgG4 antibody that prevents T-cell inactivation by blocking the binding of programmed cell death 1 ligand 1 or programmed cell death 1 ligand 2 (PD-L1 or PD-L2), a protein expressed by cancer cells, with PD-1, a protein found on the surface of activated T-cells. Nivolumab is used in advanced melanoma, metastatic renal cell carcinoma, advanced lung cancer, advanced head and neck cancer, and Hodgkin's lymphoma.
==== Ofatumumab ====
Ofatumumab is a second generation human IgG1 antibody that binds to CD20. It is used in the treatment of chronic lymphocytic leukemia (CLL) because the cancerous cells of CLL are usually CD20-expressing B-cells. Unlike rituximab, which binds to a large loop of the CD20 protein, ofatumumab binds to a separate, small loop. This may explain their different characteristics. Compared to rituximab, ofatumumab induces complement-dependent cytotoxicity at a lower dose with less immunogenicity.
==== Pembrolizumab ====
As of 2019, pembrolizumab, which blocks PD-1, programmed cell death protein 1, has been used via intravenous infusion to treat inoperable or metastatic melanoma, metastatic non-small cell lung cancer (NSCLC) in certain situations, as a second-line treatment for head and neck squamous cell carcinoma (HNSCC), after platinum-based chemotherapy, and for the treatment of adult and pediatric patients with refractory classic Hodgkin's lymphoma (cHL). It is also indicated for certain patients with urothelial carcinoma, stomach cancer and cervical cancer.
==== Rituximab ====
Rituximab is a chimeric monoclonal IgG1 antibody specific for CD20, developed from its parent antibody Ibritumomab. As with ibritumomab, rituximab targets CD20, making it effective in treating certain B-cell malignancies. These include aggressive and indolent lymphomas such as diffuse large B-cell lymphoma and follicular lymphoma and leukemias such as B-cell chronic lymphocytic leukemia. Although the function of CD20 is relatively unknown, CD20 may be a calcium channel involved in B-cell activation. The antibody's mode of action is primarily through the induction of ADCC and complement-mediated cytotoxicity. Other mechanisms include apoptosis and cellular growth arrest. Rituximab also increases the sensitivity of cancerous B-cells to chemotherapy.
==== Trastuzumab ====
== Immune checkpoint antibody therapy or immune checkpoint blockade ==
Immune checkpoints affect the immune system function. Immune checkpoints can be stimulatory or inhibitory. Tumors can use these checkpoints to protect themselves from immune system attacks. Checkpoint therapies approved as of 2012 block inhibitory checkpoint receptors. Blockade of negative feedback signaling to immune cells thus results in an enhanced immune response against tumors. As of 2020, immune checkpoint blockade therapies have varied effectiveness. In Hodgkin lymphoma and natural killer T-cell lymphoma, response rates are high, at 50–60%. Response rates are quite low for breast and prostate cancers, however. A major challenge are the large variations in responses to immunocheckpoint inhibitors, some patients showing spectacular clinical responses while no positive effects are seen in others. A plethora of possible reasons for the absence of efficacy in many patients have been proposed, but the biomedical community has still to begin to find consensus in this respect. For instance, a recent paper documented that infection with Helicobacter pylori would negatively influence the effects of immunocheckpoint inhibitors in gastric cancer., but this notion was quickly challenged by others.
One ligand-receptor interaction under investigation is the interaction between the transmembrane programmed cell death 1 protein (PDCD1, PD-1; also known as CD279) and its ligand, PD-1 ligand 1 (PD-L1, CD274). PD-L1 on the cell surface binds to PD1 on an immune cell surface, which inhibits immune cell activity. Among PD-L1 functions is a key regulatory role on T cell activities. It appears that (cancer-mediated) upregulation of PD-L1 on the cell surface may inhibit T cells that might otherwise attack. PD-L1 on cancer cells also inhibits FAS- and interferon-dependent apoptosis, protecting cells from cytotoxic molecules produced by T cells. Antibodies that bind to either PD-1 or PD-L1 and therefore block the interaction may allow the T-cells to attack the tumor.
=== CTLA-4 blockade ===
The first checkpoint antibody approved by the FDA was ipilimumab, approved in 2011 to treat melanoma. It blocks the immune checkpoint molecule CTLA-4. As of 2012, clinical trials have also shown some benefits of anti-CTLA-4 therapy on lung cancer or pancreatic cancer, specifically in combination with other drugs. In on-going trials the combination of CTLA-4 blockade with PD-1 or PD-L1 inhibitors is tested on different types of cancer.
However, as of 2015 it is known that patients treated with checkpoint blockade (specifically CTLA-4 blocking antibodies), or a combination of check-point blocking antibodies, are at high risk of having immune-related adverse events such as dermatologic, gastrointestinal, endocrine, or hepatic autoimmune reactions. These are most likely due to the breadth of the induced T-cell activation when anti-CTLA-4 antibodies are administered by injection in the bloodstream.
A 2024 cohort study of ICI use during pregnancy showed no overreporting of specific adverse effects on pregnancy, fetal, and/or newborn outcomes, interestingly.
Using a mouse model of bladder cancer, researchers have found that a local injection of a low dose anti-CTLA-4 in the tumour area had the same tumour inhibiting capacity as when the antibody was delivered in the blood. At the same time the levels of circulating antibodies were lower, suggesting that local administration of the anti-CTLA-4 therapy might result in fewer adverse events.
=== PD-1 inhibitors ===
Initial clinical trial results with IgG4 PD1 antibody nivolumab were published in 2010. It was approved in 2014. Nivolumab is approved to treat melanoma, lung cancer, kidney cancer, bladder cancer, head and neck cancer, and Hodgkin's lymphoma. A 2016 clinical trial for non-small cell lung cancer failed to meet its primary endpoint for treatment in the first-line setting, but is FDA-approved in subsequent lines of therapy.
Pembrolizumab (Keytruda) is another PD1 inhibitor that was approved by the FDA in 2014. Pembrolizumab is approved to treat melanoma and lung cancer.
Antibody BGB-A317 is a PD-1 inhibitor (designed to not bind Fc gamma receptor I) in early clinical trials.
=== PD-L1 inhibitors ===
In May 2016, PD-L1 inhibitor atezolizumab was approved for treating bladder cancer.
Anti-PD-L1 antibodies currently in development include avelumab and durvalumab, in addition to an inhibitory affimer.
=== CISH ===
=== Combinations ===
Many cancer patients do not respond to immune checkpoint blockade. Response rate may be improved by combining that with additional therapies, including those that stimulate T cell infiltration. For example, targeted therapies such as radiotherapy, vasculature targeting agents, and immunogenic chemotherapy can improve immune checkpoint blockade response in animal models.
Combining immunotherapies such as PD1 and CTLA4 inhibitors can create to durable responses.
Combinatorial ablation and immunotherapy enhances the immunostimulating response and has synergistic effects for metastatic cancer treatment.
Combining checkpoint immunotherapies with pharmaceutical agents has the potential to improve response, and as of 2018 were a target of clinical investigation. Immunostimulatory drugs such as CSF-1R inhibitors and TLR agonists have been effective.
Two independent 2024 clinical trials reported that combinations of JAK inhibitors with anti–PD-1 immunotherapy could improve efficacy. A phase 2 trial investigated the combination as a first-line therapy for metastatic non-small-cell lung cancer. Administration of itacitinib after treatment with pembrolizumab improved therapeutic response. A separate phase 1/2 trial with patients with relapsed/refractory Hodgkin’s lymphoma combined ruxolitinib and nivolumab, yielding improved clinical efficacy in patients who had previously failed checkpoint blockade immunotherapy.
== Cytokine therapy ==
Cytokines are proteins produced by many types of cells present within a tumor. They can modulate immune responses. The tumor often employs them to allow it to grow and reduce the immune response. These immune-modulating effects allow them to be used as drugs to provoke an immune response. Two commonly used cytokines are interferons and interleukins.
Interleukin-2 and interferon-α are cytokines, proteins that regulate and coordinate the behavior of the immune system. They have the ability to enhance anti-tumor activity and thus can be used as passive cancer treatments. Interferon-α is used in the treatment of hairy-cell leukaemia, AIDS-related Kaposi's sarcoma, follicular lymphoma, chronic myeloid leukaemia and malignant melanoma. Interleukin-2 is used in the treatment of malignant melanoma and renal cell carcinoma.
=== Interferon ===
Interferons are produced by the immune system. They are usually involved in anti-viral response, but also have use for cancer. They fall in three groups: type I (IFNα and IFNβ), type II (IFNγ) and type III (IFNλ). IFNα has been approved for use in hairy-cell leukaemia, AIDS-related Kaposi's sarcoma, follicular lymphoma, chronic myeloid leukaemia and melanoma. Type I and II IFNs have been researched extensively and although both types promote anti-tumor immune system effects, only type I IFNs have been shown to be clinically effective. IFNλ shows promise for its anti-tumor effects in animal models.
Unlike type I IFNs, Interferon gamma is not approved yet for the treatment of any cancer. However, improved survival was observed when Interferon gamma was administered to patients with bladder carcinoma and melanoma cancers. The most promising result was achieved in patients with stage 2 and 3 of ovarian carcinoma. The in vitro study of IFN-gamma in cancer cells is more extensive and results indicate anti-proliferative activity of IFN-gamma leading to the growth inhibition or cell death, generally induced by apoptosis but sometimes by autophagy.
=== Interleukin ===
Interleukins have an array of immune system effects. Interleukin-2 is used in the treatment of malignant melanoma and renal cell carcinoma. In normal physiology it promotes both effector T cells and T-regulatory cells, but its exact mechanism of action is unknown.
== Genetic pre-treatment testing for therapeutic significance ==
Because of the high cost of many of immunotherapy medications and the reluctance of medical insurance companies to prepay for their prescriptions various test methods have been proposed, to attempt to forecast the effectiveness of these medications. In some cases the FDA has approved genetic tests for medication specific to certain genetic markers. For example, the FDA approved BRAF-associated medication for metastatic melanoma, to be administered to patients after testing for the BRAF genetic mutation.
As of 2018, the detection of PD-L1 protein seemed to be an indication of cancer susceptible to several immunotherapy medications, but research found that both the lack of this protein or its inclusion in the cancerous tissue was inconclusive, due to the little-understood varying quantities of the protein during different times and locations within the infected cells and tissue.
In 2018, some genetic indications such as Tumor Mutational Burden (TMB, the number of mutations within a targeted genetic region in the cancerous cell's DNA), and microsatellite instability (MSI, the quantity of impaired DNA mismatch leading to probable mutations), have been approved by the FDA as good indicators for the probability of effective treatment of immunotherapy medication for certain cancers, but research is still in progress. As of 2020, the patient prioritization for immunotherapy based on TMB was still highly controversial.
Tests of this sort are being widely advertised for general cancer treatment and are expensive. In the past, some genetic testing for cancer treatment has been involved in scams such as the Duke University Cancer Fraud scandal, or claimed to be hoaxes.
== Research ==
=== Oncolytic virus ===
An oncolytic virus is a virus that preferentially infects and kills cancer cells. As the infected cancer cells are destroyed by oncolysis, they release new infectious virus particles or virions to help destroy the remaining tumour. Oncolytic viruses are thought not only to cause direct destruction of the tumour cells, but also to stimulate host anti-tumour immune responses for long-term immunotherapy.
The potential of viruses as anti-cancer agents was first realized in the early twentieth century, although coordinated research efforts did not begin until the 1960s. A number of viruses including adenovirus, reovirus, measles, herpes simplex, Newcastle disease virus and vaccinia have now been clinically tested as oncolytic agents. T-Vec is the first FDA-approved oncolytic virus for the treatment of melanoma. A number of other oncolytic viruses are in Phase II-III development.
=== Polysaccharides ===
Certain compounds found in mushrooms, primarily polysaccharides, can up-regulate the immune system and may have anti-cancer properties. For example, beta-glucans such as lentinan have been shown in laboratory studies to stimulate macrophage, NK cells, T cells and immune system cytokines and have been investigated in clinical trials as immunologic adjuvants.
=== Neoantigens ===
Many tumors express mutations. These mutations potentially create new targetable antigens (neoantigens) for use in T-cell immunotherapy. The presence of CD8+ T cells in cancer lesions, as identified using RNA sequencing data, is higher in tumors with a high mutational burden. The level of transcripts associated with the cytolytic activity of natural killer cells and T cells positively correlates with mutational load in many human tumors. In non–small cell lung cancer patients treated with lambrolizumab, mutational load shows a strong correlation with clinical response. In melanoma patients treated with ipilimumab, the long-term benefit is also associated with a higher mutational load, although less significantly. The predicted MHC binding neoantigens in patients with a long-term clinical benefit were enriched for a series of tetrapeptide motifs that were not found in tumors of patients with no or minimal clinical benefit. However, human neoantigens identified in other studies do not show the bias toward tetrapeptide signatures.
=== Polysaccharide-K ===
In the 1980s, Japan's Ministry of Health, Labour and Welfare approved polysaccharide-K extracted from the mushroom, Coriolus versicolor, to stimulate the immune systems of patients undergoing chemotherapy. It is a dietary supplement in the US and other jurisdictions.
== See also ==
Cancer vaccine
Antigen 5T4
Coley's toxins
Combinatorial ablation and immunotherapy
Cryoimmunotherapy
Photoimmunotherapy
Radioimmunotherapy
== References ==
== External links ==
A primer on "Immunotherapy to Treat Cancer", NIH
Immunotherapy – Using the Immune System to Treat Cancer Archived 4 April 2017 at the Wayback Machine
Cancer Research Institute – What is Cancer Immunotherapy
Association for Immunotherapy of Cancer
Society for Immunotherapy of Cancer
"And Then There Were Five". Economist.
"Discover the Science of Immuno-Oncology". Bristol-Myers Squibb. Archived from the original on 10 October 2014. Retrieved 13 March 2014.
Eggermont A, Finn O (September 2012). "Advances in immuno-oncology. Foreword". Annals of Oncology. 23 (Suppl 8): viii5. doi:10.1093/annonc/mds255. PMID 22918929.
"Cancer Immunotherapy in Gujarat" | Wikipedia/Cancer_immunotherapy |
In mathematics, the Zeeman conjecture or Zeeman's collapsibility conjecture asks whether given a finite contractible 2-dimensional CW complex
K
{\displaystyle K}
, the space
K
×
[
0
,
1
]
{\displaystyle K\times [0,1]}
is collapsible. It can nowadays be restated as the claim that for any 2-complex G which is homotopic to a point, there is an interval I such that some barycentric subdivision of G × I is contractible.
The conjecture, due to Christopher Zeeman, implies the Poincaré conjecture and the Andrews–Curtis conjecture.
== References ==
Matveev, Sergei (2007), "1.3.4 Zeeman's Collapsing Conjecture", Algorithmic Topology and Classification of 3-Manifolds, Algorithms and Computation in Mathematics, vol. 9, Springer, pp. 46–58, ISBN 9783540458999 | Wikipedia/Zeeman_conjecture |
Geometry & Topology is a peer-refereed, international mathematics research journal devoted to geometry and topology, and their applications. It is currently based at the University of Warwick, United Kingdom, and published by Mathematical Sciences Publishers, a nonprofit academic publishing organisation.
It was founded in 1997 by a group of topologists who were dissatisfied with recent substantial rises in subscription prices of journals published by major publishing corporations. The aim was to set up a high-quality journal, capable of competing with existing journals, but with substantially lower subscription fees. The journal was open-access for its first ten years of existence and was available free to individual users, although institutions were required to pay modest subscription fees for both online access and for printed volumes. At present, an online subscription is required to view full-text PDF copies of articles in the most recent three volumes; articles older than that are open-access, at which point copies of the published articles are uploaded to the arXiv. A traditional printed version is also published, at present on an annual basis.
The journal has grown to be well respected in its field, and has in recent years published a number of important papers, in particular proofs of the Property P conjecture and the Birman conjecture.
== References ==
Walter Neumann on the Success of Geometry & Topology, May 2010, Sciencewatch.com, Thomson Reuters
== External links ==
Geometry & Topology
MSP Open Access Policy | Wikipedia/Geometry_and_Topology |
In mathematics, the surgery structure set
S
(
X
)
{\displaystyle {\mathcal {S}}(X)}
is the basic object in the study of manifolds which are homotopy equivalent to a closed manifold X. It is a concept which helps to answer the question whether two homotopy equivalent manifolds are diffeomorphic (or PL-homeomorphic or homeomorphic). There are different versions of the structure set depending on the category (DIFF, PL or TOP) and whether Whitehead torsion is taken into account or not.
== Definition ==
Let X be a closed smooth (or PL- or topological) manifold of dimension n. We call two homotopy equivalences
f
i
:
M
i
→
X
{\displaystyle f_{i}:M_{i}\to X}
from closed manifolds
M
i
{\displaystyle M_{i}}
of dimension
n
{\displaystyle n}
to
X
{\displaystyle X}
(
i
=
0
,
1
{\displaystyle i=0,1}
) equivalent if there exists a cobordism
(
W
;
M
0
,
M
1
)
{\displaystyle {\mathcal {}}(W;M_{0},M_{1})}
together with a map
(
F
;
f
0
,
f
1
)
:
(
W
;
M
0
,
M
1
)
→
(
X
×
[
0
,
1
]
;
X
×
{
0
}
,
X
×
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1
}
)
{\displaystyle (F;f_{0},f_{1}):(W;M_{0},M_{1})\to (X\times [0,1];X\times \{0\},X\times \{1\})}
such that
F
{\displaystyle F}
,
f
0
{\displaystyle f_{0}}
and
f
1
{\displaystyle f_{1}}
are homotopy equivalences.
The structure set
S
h
(
X
)
{\displaystyle {\mathcal {S}}^{h}(X)}
is the set of equivalence classes of homotopy equivalences
f
:
M
→
X
{\displaystyle f:M\to X}
from closed manifolds of dimension n to X.
This set has a preferred base point:
i
d
:
X
→
X
{\displaystyle id:X\to X}
.
There is also a version which takes Whitehead torsion into account. If we require in the definition above the homotopy equivalences F,
f
0
{\displaystyle f_{0}}
and
f
1
{\displaystyle f_{1}}
to be simple homotopy equivalences then we obtain the simple structure set
S
s
(
X
)
{\displaystyle {\mathcal {S}}^{s}(X)}
.
== Remarks ==
Notice that
(
W
;
M
0
,
M
1
)
{\displaystyle (W;M_{0},M_{1})}
in the definition of
S
h
(
X
)
{\displaystyle {\mathcal {S}}^{h}(X)}
resp.
S
s
(
X
)
{\displaystyle {\mathcal {S}}^{s}(X)}
is an h-cobordism resp. an s-cobordism. Using the s-cobordism theorem we obtain another description for the simple structure set
S
s
(
X
)
{\displaystyle {\mathcal {S}}^{s}(X)}
, provided that n>4: The simple structure set
S
s
(
X
)
{\displaystyle {\mathcal {S}}^{s}(X)}
is the set of equivalence classes of homotopy equivalences
f
:
M
→
X
{\displaystyle f:M\to X}
from closed manifolds
M
{\displaystyle M}
of dimension n to X with respect to the following equivalence relation. Two homotopy equivalences
f
i
:
M
i
→
X
{\displaystyle f_{i}:M_{i}\to X}
(i=0,1) are equivalent if there exists a
diffeomorphism (or PL-homeomorphism or homeomorphism)
g
:
M
0
→
M
1
{\displaystyle g:M_{0}\to M_{1}}
such that
f
1
∘
g
{\displaystyle f_{1}\circ g}
is homotopic to
f
0
{\displaystyle f_{0}}
.
As long as we are dealing with differential manifolds, there is in general no canonical group structure on
S
s
(
X
)
{\displaystyle {\mathcal {S}}^{s}(X)}
. If we deal with topological manifolds, it is possible to endow
S
s
(
X
)
{\displaystyle {\mathcal {S}}^{s}(X)}
with a preferred structure of an abelian group (see chapter 18 in the book of Ranicki).
Notice that a manifold M is diffeomorphic (or PL-homeomorphic or homeomorphic) to a closed manifold X if and only if there exists a simple homotopy equivalence
ϕ
:
M
→
X
{\displaystyle \phi :M\to X}
whose equivalence class is the base point in
S
s
(
X
)
{\displaystyle {\mathcal {S}}^{s}(X)}
. Some care is necessary because it may be possible that a given simple homotopy equivalence
ϕ
:
M
→
X
{\displaystyle \phi :M\to X}
is not homotopic to a diffeomorphism (or PL-homeomorphism or homeomorphism) although M and X are diffeomorphic (or PL-homeomorphic or homeomorphic). Therefore, it is also necessary to study the operation of the group of homotopy classes of simple self-equivalences of X on
S
s
(
X
)
{\displaystyle {\mathcal {S}}^{s}(X)}
.
The basic tool to compute the simple structure set is the surgery exact sequence.
== Examples ==
Topological Spheres: The generalized Poincaré conjecture in the topological category says that
S
s
(
S
n
)
{\displaystyle {\mathcal {S}}^{s}(S^{n})}
only consists of the base point. This conjecture was proved by Smale (n > 4), Freedman (n = 4) and Perelman (n = 3).
Exotic Spheres: The classification of exotic spheres by Kervaire and Milnor gives
S
s
(
S
n
)
=
θ
n
=
π
n
(
P
L
/
O
)
{\displaystyle {\mathcal {S}}^{s}(S^{n})=\theta _{n}=\pi _{n}(PL/O)}
for n > 4 (smooth category).
== References ==
Browder, William (1972), Surgery on simply-connected manifolds, Berlin, New York: Springer-Verlag, MR 0358813
Ranicki, Andrew (2002), Algebraic and Geometric Surgery, Oxford Mathematical Monographs, Clarendon Press, ISBN 978-0-19-850924-0, MR 2061749
Wall, C. T. C. (1999), Surgery on compact manifolds, Mathematical Surveys and Monographs, vol. 69 (2nd ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0942-6, MR 1687388
Ranicki, Andrew (1992), Algebraic L-theory and topological manifolds (PDF), Cambridge Tracts in Mathematics 102, CUP, ISBN 0-521-42024-5, MR 1211640
== External links ==
Andrew Ranicki's homepage
Shmuel Weinberger's homepage | Wikipedia/Surgery_structure_set |
In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. Every such cohomology theory is representable, as follows from Brown's representability theorem. This means that, given a cohomology theory
E
∗
:
CW
o
p
→
Ab
{\displaystyle {\mathcal {E}}^{*}:{\text{CW}}^{op}\to {\text{Ab}}}
,there exist spaces
E
k
{\displaystyle E^{k}}
such that evaluating the cohomology theory in degree
k
{\displaystyle k}
on a space
X
{\displaystyle X}
is equivalent to computing the homotopy classes of maps to the space
E
k
{\displaystyle E^{k}}
, that is
E
k
(
X
)
≅
[
X
,
E
k
]
{\displaystyle {\mathcal {E}}^{k}(X)\cong \left[X,E^{k}\right]}
.Note there are several different categories of spectra leading to many technical difficulties, but they all determine the same homotopy category, known as the stable homotopy category. This is one of the key points for introducing spectra because they form a natural home for stable homotopy theory.
== The definition of a spectrum ==
There are many variations of the definition: in general, a spectrum is any sequence
X
n
{\displaystyle X_{n}}
of pointed topological spaces or pointed simplicial sets together with the structure maps
S
1
∧
X
n
→
X
n
+
1
{\displaystyle S^{1}\wedge X_{n}\to X_{n+1}}
, where
∧
{\displaystyle \wedge }
is the smash product. The smash product of a pointed space
X
{\displaystyle X}
with a circle is homeomorphic to the reduced suspension of
X
{\displaystyle X}
, denoted
Σ
X
{\displaystyle \Sigma X}
.
The following is due to Frank Adams (1974): a spectrum (or CW-spectrum) is a sequence
E
:=
{
E
n
}
n
∈
N
{\displaystyle E:=\{E_{n}\}_{n\in \mathbb {N} }}
of CW complexes together with inclusions
Σ
E
n
→
E
n
+
1
{\displaystyle \Sigma E_{n}\to E_{n+1}}
of the suspension
Σ
E
n
{\displaystyle \Sigma E_{n}}
as a subcomplex of
E
n
+
1
{\displaystyle E_{n+1}}
.
For other definitions, see symmetric spectrum and simplicial spectrum.
=== Homotopy groups of a spectrum ===
Some of the most important invariants of a spectrum are its homotopy groups. These groups mirror the definition of the stable homotopy groups of spaces since the structure of the suspension maps is integral in its definition. Given a spectrum
E
{\displaystyle E}
define the homotopy group
π
n
(
E
)
{\displaystyle \pi _{n}(E)}
as the colimit
π
n
(
E
)
=
lim
→
k
π
n
+
k
(
E
k
)
=
lim
→
(
⋯
→
π
n
+
k
(
E
k
)
→
π
n
+
k
+
1
(
E
k
+
1
)
→
⋯
)
{\displaystyle {\begin{aligned}\pi _{n}(E)&=\lim _{\to k}\pi _{n+k}(E_{k})\\&=\lim _{\to }\left(\cdots \to \pi _{n+k}(E_{k})\to \pi _{n+k+1}(E_{k+1})\to \cdots \right)\end{aligned}}}
where the maps are induced from the composition of the map
Σ
:
π
n
+
k
(
E
n
)
→
π
n
+
k
+
1
(
Σ
E
n
)
{\displaystyle \Sigma :\pi _{n+k}(E_{n})\to \pi _{n+k+1}(\Sigma E_{n})}
(that is,
[
S
n
+
k
,
E
n
]
→
[
S
n
+
k
+
1
,
Σ
E
n
]
{\displaystyle [S^{n+k},E_{n}]\to [S^{n+k+1},\Sigma E_{n}]}
given by functoriality of
Σ
{\displaystyle \Sigma }
) and the structure map
Σ
E
n
→
E
n
+
1
{\displaystyle \Sigma E_{n}\to E_{n+1}}
. A spectrum is said to be connective if its
π
k
{\displaystyle \pi _{k}}
are zero for negative k.
== Examples ==
=== Eilenberg–Maclane spectrum ===
Consider singular cohomology
H
n
(
X
;
A
)
{\displaystyle H^{n}(X;A)}
with coefficients in an abelian group
A
{\displaystyle A}
. For a CW complex
X
{\displaystyle X}
, the group
H
n
(
X
;
A
)
{\displaystyle H^{n}(X;A)}
can be identified with the set of homotopy classes of maps from
X
{\displaystyle X}
to
K
(
A
,
n
)
{\displaystyle K(A,n)}
, the Eilenberg–MacLane space with homotopy concentrated in degree
n
{\displaystyle n}
. We write this as
[
X
,
K
(
A
,
n
)
]
=
H
n
(
X
;
A
)
{\displaystyle [X,K(A,n)]=H^{n}(X;A)}
Then the corresponding spectrum
H
A
{\displaystyle HA}
has
n
{\displaystyle n}
-th space
K
(
A
,
n
)
{\displaystyle K(A,n)}
; it is called the Eilenberg–MacLane spectrum of
A
{\displaystyle A}
. Note this construction can be used to embed any ring
R
{\displaystyle R}
into the category of spectra. This embedding forms the basis of spectral geometry, a model for derived algebraic geometry. One of the important properties of this embedding are the isomorphisms
π
i
(
H
(
R
/
I
)
∧
R
H
(
R
/
J
)
)
≅
H
i
(
R
/
I
⊗
L
R
/
J
)
≅
Tor
i
R
(
R
/
I
,
R
/
J
)
{\displaystyle {\begin{aligned}\pi _{i}(H(R/I)\wedge _{R}H(R/J))&\cong H_{i}\left(R/I\otimes ^{\mathbf {L} }R/J\right)\\&\cong \operatorname {Tor} _{i}^{R}(R/I,R/J)\end{aligned}}}
showing the category of spectra keeps track of the derived information of commutative rings, where the smash product acts as the derived tensor product. Moreover, Eilenberg–Maclane spectra can be used to define theories such as topological Hochschild homology for commutative rings, a more refined theory than classical Hochschild homology.
=== Topological complex K-theory ===
As a second important example, consider topological K-theory. At least for X compact,
K
0
(
X
)
{\displaystyle K^{0}(X)}
is defined to be the Grothendieck group of the monoid of complex vector bundles on X. Also,
K
1
(
X
)
{\displaystyle K^{1}(X)}
is the group corresponding to vector bundles on the suspension of X. Topological K-theory is a generalized cohomology theory, so it gives a spectrum. The zeroth space is
Z
×
B
U
{\displaystyle \mathbb {Z} \times BU}
while the first space is
U
{\displaystyle U}
. Here
U
{\displaystyle U}
is the infinite unitary group and
B
U
{\displaystyle BU}
is its classifying space. By Bott periodicity we get
K
2
n
(
X
)
≅
K
0
(
X
)
{\displaystyle K^{2n}(X)\cong K^{0}(X)}
and
K
2
n
+
1
(
X
)
≅
K
1
(
X
)
{\displaystyle K^{2n+1}(X)\cong K^{1}(X)}
for all n, so all the spaces in the topological K-theory spectrum are given by either
Z
×
B
U
{\displaystyle \mathbb {Z} \times BU}
or
U
{\displaystyle U}
. There is a corresponding construction using real vector bundles instead of complex vector bundles, which gives an 8-periodic spectrum.
=== Sphere spectrum ===
One of the quintessential examples of a spectrum is the sphere spectrum
S
{\displaystyle \mathbb {S} }
. This is a spectrum whose homotopy groups are given by the stable homotopy groups of spheres, so
π
n
(
S
)
=
π
n
S
{\displaystyle \pi _{n}(\mathbb {S} )=\pi _{n}^{\mathbb {S} }}
We can write down this spectrum explicitly as
S
i
=
S
i
{\displaystyle \mathbb {S} _{i}=S^{i}}
where
S
0
=
{
0
,
1
}
{\displaystyle \mathbb {S} _{0}=\{0,1\}}
. Note the smash product gives a product structure on this spectrum
S
n
∧
S
m
≃
S
n
+
m
{\displaystyle S^{n}\wedge S^{m}\simeq S^{n+m}}
induces a ring structure on
S
{\displaystyle \mathbb {S} }
. Moreover, if considering the category of symmetric spectra, this forms the initial object, analogous to
Z
{\displaystyle \mathbb {Z} }
in the category of commutative rings.
=== Thom spectra ===
Another canonical example of spectra come from the Thom spectra representing various cobordism theories. This includes real cobordism
M
O
{\displaystyle MO}
, complex cobordism
M
U
{\displaystyle MU}
, framed cobordism, spin cobordism
M
S
p
i
n
{\displaystyle MSpin}
, string cobordism
M
S
t
r
i
n
g
{\displaystyle MString}
, and so on. In fact, for any topological group
G
{\displaystyle G}
there is a Thom spectrum
M
G
{\displaystyle MG}
.
=== Suspension spectrum ===
A spectrum may be constructed out of a space. The suspension spectrum of a space
X
{\displaystyle X}
, denoted
Σ
∞
X
{\displaystyle \Sigma ^{\infty }X}
is a spectrum
X
n
=
S
n
∧
X
{\displaystyle X_{n}=S^{n}\wedge X}
(the structure maps are the identity.) For example, the suspension spectrum of the 0-sphere is the sphere spectrum discussed above. The homotopy groups of this spectrum are then the stable homotopy groups of
X
{\displaystyle X}
, so
π
n
(
Σ
∞
X
)
=
π
n
S
(
X
)
{\displaystyle \pi _{n}(\Sigma ^{\infty }X)=\pi _{n}^{\mathbb {S} }(X)}
The construction of the suspension spectrum implies every space can be considered as a cohomology theory. In fact, it defines a functor
Σ
∞
:
h
CW
→
h
Spectra
{\displaystyle \Sigma ^{\infty }:h{\text{CW}}\to h{\text{Spectra}}}
from the homotopy category of CW complexes to the homotopy category of spectra. The morphisms are given by
[
Σ
∞
X
,
Σ
∞
Y
]
=
colim
→
n
[
Σ
n
X
,
Σ
n
Y
]
{\displaystyle [\Sigma ^{\infty }X,\Sigma ^{\infty }Y]={\underset {\to n}{\operatorname {colim} {}}}[\Sigma ^{n}X,\Sigma ^{n}Y]}
which by the Freudenthal suspension theorem eventually stabilizes. By this we mean
[
Σ
N
X
,
Σ
N
Y
]
≃
[
Σ
N
+
1
X
,
Σ
N
+
1
Y
]
≃
⋯
{\displaystyle \left[\Sigma ^{N}X,\Sigma ^{N}Y\right]\simeq \left[\Sigma ^{N+1}X,\Sigma ^{N+1}Y\right]\simeq \cdots }
and
[
Σ
∞
X
,
Σ
∞
Y
]
≃
[
Σ
N
X
,
Σ
N
Y
]
{\displaystyle \left[\Sigma ^{\infty }X,\Sigma ^{\infty }Y\right]\simeq \left[\Sigma ^{N}X,\Sigma ^{N}Y\right]}
for some finite integer
N
{\displaystyle N}
. For a CW complex
X
{\displaystyle X}
there is an inverse construction
Ω
∞
{\displaystyle \Omega ^{\infty }}
which takes a spectrum
E
{\displaystyle E}
and forms a space
Ω
∞
E
=
colim
→
n
Ω
n
E
n
{\displaystyle \Omega ^{\infty }E={\underset {\to n}{\operatorname {colim} {}}}\Omega ^{n}E_{n}}
called the infinite loop space of the spectrum. For a CW complex
X
{\displaystyle X}
Ω
∞
Σ
∞
X
=
colim
→
Ω
n
Σ
n
X
{\displaystyle \Omega ^{\infty }\Sigma ^{\infty }X={\underset {\to }{\operatorname {colim} {}}}\Omega ^{n}\Sigma ^{n}X}
and this construction comes with an inclusion
X
→
Ω
n
Σ
n
X
{\displaystyle X\to \Omega ^{n}\Sigma ^{n}X}
for every
n
{\displaystyle n}
, hence gives a map
X
→
Ω
∞
Σ
∞
X
{\displaystyle X\to \Omega ^{\infty }\Sigma ^{\infty }X}
which is injective. Unfortunately, these two structures, with the addition of the smash product, lead to significant complexity in the theory of spectra because there cannot exist a single category of spectra which satisfies a list of five axioms relating these structures. The above adjunction is valid only in the homotopy categories of spaces and spectra, but not always with a specific category of spectra (not the homotopy category).
=== Ω-spectrum ===
An Ω-spectrum is a spectrum such that the adjoint of the structure map (i.e., the map
X
n
→
Ω
X
n
+
1
{\displaystyle X_{n}\to \Omega X_{n+1}}
) is a weak equivalence. The K-theory spectrum of a ring is an example of an Ω-spectrum.
=== Ring spectrum ===
A ring spectrum is a spectrum X such that the diagrams that describe ring axioms in terms of smash products commute "up to homotopy" (
S
0
→
X
{\displaystyle S^{0}\to X}
corresponds to the identity.) For example, the spectrum of topological K-theory is a ring spectrum. A module spectrum may be defined analogously.
For many more examples, see the list of cohomology theories.
== Functions, maps, and homotopies of spectra ==
There are three natural categories whose objects are spectra, whose morphisms are the functions, or maps, or homotopy classes defined below.
A function between two spectra E and F is a sequence of maps from En to Fn that commute with the
maps ΣEn → En+1 and ΣFn → Fn+1.
Given a spectrum
E
n
{\displaystyle E_{n}}
, a subspectrum
F
n
{\displaystyle F_{n}}
is a sequence of subcomplexes that is also a spectrum. As each i-cell in
E
j
{\displaystyle E_{j}}
suspends to an (i + 1)-cell in
E
j
+
1
{\displaystyle E_{j+1}}
, a cofinal subspectrum is a subspectrum for which each cell of the parent spectrum is eventually contained in the subspectrum after a finite number of suspensions. Spectra can then be turned into a category by defining a map of spectra
f
:
E
→
F
{\displaystyle f:E\to F}
to be a function from a cofinal subspectrum
G
{\displaystyle G}
of
E
{\displaystyle E}
to
F
{\displaystyle F}
, where two such functions represent the same map if they coincide on some cofinal subspectrum. Intuitively such a map of spectra does not need to be everywhere defined, just eventually become defined, and two maps that coincide on a cofinal subspectrum are said to be equivalent.
This gives the category of spectra (and maps), which is a major tool. There is a natural embedding of the category of pointed CW complexes into this category: it takes
Y
{\displaystyle Y}
to the suspension spectrum in which the nth complex is
Σ
n
Y
{\displaystyle \Sigma ^{n}Y}
.
The smash product of a spectrum
E
{\displaystyle E}
and a pointed complex
X
{\displaystyle X}
is a spectrum given by
(
E
∧
X
)
n
=
E
n
∧
X
{\displaystyle (E\wedge X)_{n}=E_{n}\wedge X}
(associativity of the smash product yields immediately that this is indeed a spectrum). A homotopy of maps between spectra corresponds to a map
(
E
∧
I
+
)
→
F
{\displaystyle (E\wedge I^{+})\to F}
, where
I
+
{\displaystyle I^{+}}
is the disjoint union
[
0
,
1
]
⊔
{
∗
}
{\displaystyle [0,1]\sqcup \{*\}}
with
∗
{\displaystyle *}
taken to be the basepoint.
The stable homotopy category, or homotopy category of (CW) spectra is defined to be the category whose objects are spectra and whose morphisms are homotopy classes of maps between spectra. Many other definitions of spectrum, some appearing very different, lead to equivalent stable homotopy categories.
Finally, we can define the suspension of a spectrum by
(
Σ
E
)
n
=
E
n
+
1
{\displaystyle (\Sigma E)_{n}=E_{n+1}}
. This translation suspension is invertible, as we can desuspend too, by setting
(
Σ
−
1
E
)
n
=
E
n
−
1
{\displaystyle (\Sigma ^{-1}E)_{n}=E_{n-1}}
.
== The triangulated homotopy category of spectra ==
The stable homotopy category is additive: maps can be added by using a variant of the track addition used to define homotopy groups. Thus homotopy classes from one spectrum to another form an abelian group. Furthermore the stable homotopy category is triangulated (Vogt (1970)), the shift being given by suspension and the distinguished triangles by the mapping cone sequences of spectra
X
→
Y
→
Y
∪
C
X
→
(
Y
∪
C
X
)
∪
C
Y
≅
Σ
X
{\displaystyle X\rightarrow Y\rightarrow Y\cup CX\rightarrow (Y\cup CX)\cup CY\cong \Sigma X}
.
== Smash products of spectra ==
The smash product of spectra extends the smash product of CW complexes. It makes the stable homotopy category into a monoidal category; in other words it behaves like the (derived) tensor product of abelian groups. A major problem with the smash product is that obvious ways of defining it make it associative and commutative only up to homotopy. Some more recent definitions of spectra, such as symmetric spectra, eliminate this problem, and give a symmetric monoidal structure at the level of maps, before passing to homotopy classes.
The smash product is compatible with the triangulated category structure. In particular the smash product of a distinguished triangle with a spectrum is a distinguished triangle.
== Generalized homology and cohomology of spectra ==
We can define the (stable) homotopy groups of a spectrum to be those given by
π
n
E
=
[
Σ
n
S
,
E
]
{\displaystyle \displaystyle \pi _{n}E=[\Sigma ^{n}\mathbb {S} ,E]}
,
where
S
{\displaystyle \mathbb {S} }
is the sphere spectrum and
[
X
,
Y
]
{\displaystyle [X,Y]}
is the set of homotopy classes of maps from
X
{\displaystyle X}
to
Y
{\displaystyle Y}
.
We define the generalized homology theory of a spectrum E by
E
n
X
=
π
n
(
E
∧
X
)
=
[
Σ
n
S
,
E
∧
X
]
{\displaystyle E_{n}X=\pi _{n}(E\wedge X)=[\Sigma ^{n}\mathbb {S} ,E\wedge X]}
and define its generalized cohomology theory by
E
n
X
=
[
Σ
−
n
X
,
E
]
.
{\displaystyle \displaystyle E^{n}X=[\Sigma ^{-n}X,E].}
Here
X
{\displaystyle X}
can be a spectrum or (by using its suspension spectrum) a space.
== Technical complexities with spectra ==
One of the canonical complexities while working with spectra and defining a category of spectra comes from the fact each of these categories cannot satisfy five seemingly obvious axioms concerning the infinite loop space of a spectrum
Q
{\displaystyle Q}
Q
:
Top
∗
→
Top
∗
{\displaystyle Q:{\text{Top}}_{*}\to {\text{Top}}_{*}}
sending
Q
X
=
colim
→
n
Ω
n
Σ
n
X
{\displaystyle QX=\mathop {\text{colim}} _{\to n}\Omega ^{n}\Sigma ^{n}X}
, a pair of adjoint functors
Σ
∞
:
Top
∗
⇆
Spectra
∗
:
Ω
∞
{\displaystyle \Sigma ^{\infty }:{\text{Top}}_{*}\leftrightarrows {\text{Spectra}}_{*}:\Omega ^{\infty }}
, and the smash product
∧
{\displaystyle \wedge }
in both the category of spaces and the category of spectra. If we let
Top
∗
{\displaystyle {\text{Top}}_{*}}
denote the category of based, compactly generated, weak Hausdorff spaces, and
Spectra
∗
{\displaystyle {\text{Spectra}}_{*}}
denote a category of spectra, the following five axioms can never be satisfied by the specific model of spectra:
Spectra
∗
{\displaystyle {\text{Spectra}}_{*}}
is a symmetric monoidal category with respect to the smash product
∧
{\displaystyle \wedge }
The functor
Σ
∞
{\displaystyle \Sigma ^{\infty }}
is left-adjoint to
Ω
∞
{\displaystyle \Omega ^{\infty }}
The unit for the smash product
∧
{\displaystyle \wedge }
is the sphere spectrum
Σ
∞
S
0
=
S
{\displaystyle \Sigma ^{\infty }S^{0}=\mathbb {S} }
Either there is a natural transformation
ϕ
:
(
Ω
∞
E
)
∧
(
Ω
∞
E
′
)
→
Ω
∞
(
E
∧
E
′
)
{\displaystyle \phi :\left(\Omega ^{\infty }E\right)\wedge \left(\Omega ^{\infty }E'\right)\to \Omega ^{\infty }\left(E\wedge E'\right)}
or a natural transformation
γ
:
(
Σ
∞
E
)
∧
(
Σ
∞
E
′
)
→
Σ
∞
(
E
∧
E
′
)
{\displaystyle \gamma :\left(\Sigma ^{\infty }E\right)\wedge \left(\Sigma ^{\infty }E'\right)\to \Sigma ^{\infty }\left(E\wedge E'\right)}
which commutes with the unit object in both categories, and the commutative and associative isomorphisms in both categories.
There is a natural weak equivalence
θ
:
Ω
∞
Σ
∞
X
→
Q
X
{\displaystyle \theta :\Omega ^{\infty }\Sigma ^{\infty }X\to QX}
for
X
∈
Ob
(
Top
∗
)
{\displaystyle X\in \operatorname {Ob} ({\text{Top}}_{*})}
which means that there is a commuting diagram:
X
→
η
Ω
∞
Σ
∞
X
=
↓
↓
θ
X
→
i
Q
X
{\displaystyle {\begin{matrix}X&\xrightarrow {\eta } &\Omega ^{\infty }\Sigma ^{\infty }X\\{\mathord {=}}\downarrow &&\downarrow \theta \\X&\xrightarrow {i} &QX\end{matrix}}}
where
η
{\displaystyle \eta }
is the unit map in the adjunction.
Because of this, the study of spectra is fractured based upon the model being used. For an overview, check out the article cited above.
== History ==
A version of the concept of a spectrum was introduced in the 1958 doctoral dissertation of Elon Lages Lima. His advisor Edwin Spanier wrote further on the subject in 1959. Spectra were adopted by Michael Atiyah and George W. Whitehead in their work on generalized homology theories in the early 1960s. The 1964 doctoral thesis of J. Michael Boardman gave a workable definition of a category of spectra and of maps (not just homotopy classes) between them, as useful in stable homotopy theory as the category of CW complexes is in the unstable case. (This is essentially the category described above, and it is still used for many purposes: for other accounts, see Adams (1974) or Rainer Vogt (1970).) Important further theoretical advances have however been made since 1990, improving vastly the formal properties of spectra. Consequently, much recent literature uses modified definitions of spectrum: see Michael Mandell et al. (2001) for a unified treatment of these new approaches.
== See also ==
Ring spectrum
Symmetric spectrum
G-spectrum
Mapping spectrum
Suspension (topology)
Adams spectral sequence
== References ==
=== Introductory ===
Adams, J. Frank (1974). Stable homotopy and generalised homology. University of Chicago Press. ISBN 9780226005249.
Elmendorf, Anthony D.; Kříž, Igor; Mandell, Michael A.; May, J. Peter (1995), "Modern foundations for stable homotopy theory" (PDF), in James., Ioan M. (ed.), Handbook of algebraic topology, Amsterdam: North-Holland, pp. 213–253, CiteSeerX 10.1.1.55.8006, doi:10.1016/B978-044481779-2/50007-9, ISBN 978-0-444-81779-2, MR 1361891
=== Modern articles developing the theory ===
Mandell, Michael A.; May, J. Peter; Schwede, Stefan; Shipley, Brooke (2001), "Model categories of diagram spectra", Proceedings of the London Mathematical Society, Series 3, 82 (2): 441–512, CiteSeerX 10.1.1.22.3815, doi:10.1112/S0024611501012692, MR 1806878, S2CID 551246
=== Historically relevant articles ===
Atiyah, Michael F. (1961). "Bordism and cobordism". Proceedings of the Cambridge Philosophical Society. 57 (2): 200–8. doi:10.1017/s0305004100035064. S2CID 122937421.
Lima, Elon Lages (1959), "The Spanier–Whitehead duality in new homotopy categories", Summa Brasil. Math., 4: 91–148, MR 0116332
Lima, Elon Lages (1960), "Stable Postnikov invariants and their duals", Summa Brasil. Math., 4: 193–251
Vogt, Rainer (1970), Boardman's stable homotopy category, Lecture Notes Series, No. 21, Matematisk Institut, Aarhus Universitet, Aarhus, MR 0275431
Whitehead, George W. (1962), "Generalized homology theories", Transactions of the American Mathematical Society, 102 (2): 227–283, doi:10.1090/S0002-9947-1962-0137117-6
== External links ==
Spectral Sequences - Allen Hatcher - contains excellent introduction to spectra and applications for constructing Adams spectral sequence
An untitled book project about symmetric spectra
"Are spectra really the same as cohomology theories?". | Wikipedia/Spectrum_(homotopy_theory) |
In mathematics, obstruction theory is a name given to two different mathematical theories, both of which yield cohomological invariants.
In the original work of Stiefel and Whitney, characteristic classes were defined as obstructions to the existence of certain fields of linear independent vectors. Obstruction theory turns out to be an application of cohomology theory to the problem of constructing a cross-section of a bundle.
== In homotopy theory ==
The older meaning for obstruction theory in homotopy theory relates to the procedure, inductive with respect to dimension, for extending a continuous mapping defined on a simplicial complex, or CW complex. It is traditionally called Eilenberg obstruction theory, after Samuel Eilenberg. It involves cohomology groups with coefficients in homotopy groups to define obstructions to extensions. For example, with a mapping from a simplicial complex X to another, Y, defined initially on the 0-skeleton of X (the vertices of X), an extension to the 1-skeleton will be possible whenever the image of the 0-skeleton will belong to the same path-connected component of Y. Extending from the 1-skeleton to the 2-skeleton means defining the mapping on each solid triangle from X, given the mapping already defined on its boundary edges. Likewise, then extending the mapping to the 3-skeleton involves extending the mapping to each solid 3-simplex of X, given the mapping already defined on its boundary.
At some point, say extending the mapping from the (n-1)-skeleton of X to the n-skeleton of X, this procedure might be impossible. In that case, one can assign to each n-simplex the homotopy class
π
n
−
1
(
Y
)
{\displaystyle \pi _{n-1}(Y)}
of the mapping already defined on its boundary, (at least one of which will be non-zero). These assignments define an n-cochain with coefficients in
π
n
−
1
(
Y
)
{\displaystyle \pi _{n-1}(Y)}
. Amazingly, this cochain turns out to be a cocycle and so defines a cohomology class in the nth cohomology group of X with coefficients in
π
n
−
1
(
Y
)
{\displaystyle \pi _{n-1}(Y)}
. When this cohomology class is equal to 0, it turns out that the mapping may be modified within its homotopy class on the (n-1)-skeleton of X so that the mapping may be extended to the n-skeleton of X. If the class is not equal to zero, it is called the obstruction to extending the mapping over the n-skeleton, given its homotopy class on the (n-1)-skeleton.
=== Obstruction to extending a section of a principal bundle ===
==== Construction ====
Suppose that B is a simply connected simplicial complex and that p : E → B is a fibration with fiber F. Furthermore, assume that we have a partially defined section σn : Bn → E on the n-skeleton of B.
For every (n + 1)-simplex Δ in B, σn can be restricted to the boundary ∂Δ (which is a topological n-sphere). Because p sends each σn(∂Δ) back to ∂Δ, σn defines a map from the n-sphere to p−1(Δ). Because fibrations satisfy the homotopy lifting property, and Δ is contractible; p−1(Δ) is homotopy equivalent to F. So this partially defined section assigns an element of πn(F) to every (n + 1)-simplex. This is precisely the data of a πn(F)-valued simplicial cochain of degree n + 1 on B, i.e. an element of Cn + 1(B; πn(F)). This cochain is called the obstruction cochain because it being the zero means that all of these elements of πn(F) are trivial, which means that our partially defined section can be extended to the (n + 1)-skeleton by using the homotopy between (the partially defined section on the boundary of each Δ) and the constant map.
The fact that this cochain came from a partially defined section (as opposed to an arbitrary collection of maps from all the boundaries of all the (n + 1)-simplices) can be used to prove that this cochain is a cocycle. If one started with a different partially defined section σn that agreed with the original on the (n − 1)-skeleton, then one can also prove that the resulting cocycle would differ from the first by a coboundary. Therefore we have a well-defined element of the cohomology group Hn + 1(B; πn(F)) such that if a partially defined section on the (n + 1)-skeleton exists that agrees with the given choice on the (n − 1)-skeleton, then this cohomology class must be trivial.
The converse is also true if one allows such things as homotopy sections, i.e. a map σ : B → E such that p ∘ σ is homotopic (as opposed to equal) to the identity map on B. Thus it provides a complete invariant of the existence of sections up to homotopy on the (n + 1)-skeleton.
==== Applications ====
By inducting over n, one can construct a first obstruction to a section as the first of the above cohomology classes that is non-zero.
This can be used to find obstructions to trivializations of principal bundles.
Because any map can be turned into a fibration, this construction can be used to see if there are obstructions to the existence of a lift (up to homotopy) of a map into B to a map into E even if p : E → B is not a fibration.
It is crucial to the construction of Postnikov systems.
== In geometric topology ==
In geometric topology, obstruction theory is concerned with when a topological manifold has a piecewise linear structure, and when a piecewise linear manifold has a differential structure.
In dimension at most 2 (Rado), and 3 (Moise), the notions of topological manifolds and piecewise linear manifolds coincide. In dimension 4 they are not the same.
In dimensions at most 6 the notions of piecewise linear manifolds and differentiable manifolds coincide.
== In surgery theory ==
The two basic questions of surgery theory are whether a topological space with n-dimensional Poincaré duality is homotopy equivalent to an n-dimensional manifold, and also whether a homotopy equivalence of n-dimensional manifolds is homotopic to a diffeomorphism. In both cases there are two obstructions for n>9, a primary topological K-theory obstruction to the existence of a vector bundle: if this vanishes there exists a normal map, allowing the definition of the secondary surgery obstruction in algebraic L-theory to performing surgery on the normal map to obtain a homotopy equivalence.
== See also ==
Kirby–Siebenmann class
Wall's finiteness obstruction
== References ==
Husemöller, Dale (1994), Fibre Bundles, Springer Verlag, ISBN 0-387-94087-1
Steenrod, Norman (1951), The Topology of Fibre Bundles, Princeton University Press, ISBN 0-691-08055-0 {{citation}}: ISBN / Date incompatibility (help)
Scorpan, Alexandru (2005). The wild world of 4-manifolds. American Mathematical Society. ISBN 0-8218-3749-4. | Wikipedia/Obstruction_theory |
In the field of topology, the signature is an integer invariant which is defined for an oriented manifold M of dimension divisible by four.
This invariant of a manifold has been studied in detail, starting with Rokhlin's theorem for 4-manifolds, and Hirzebruch signature theorem.
== Definition ==
Given a connected and oriented manifold M of dimension 4k, the cup product gives rise to a quadratic form Q on the 'middle' real cohomology group
H
2
k
(
M
,
R
)
{\displaystyle H^{2k}(M,\mathbf {R} )}
.
The basic identity for the cup product
α
p
⌣
β
q
=
(
−
1
)
p
q
(
β
q
⌣
α
p
)
{\displaystyle \alpha ^{p}\smile \beta ^{q}=(-1)^{pq}(\beta ^{q}\smile \alpha ^{p})}
shows that with p = q = 2k the product is symmetric. It takes values in
H
4
k
(
M
,
R
)
{\displaystyle H^{4k}(M,\mathbf {R} )}
.
If we assume also that M is compact, Poincaré duality identifies this with
H
0
(
M
,
R
)
{\displaystyle H_{0}(M,\mathbf {R} )}
which can be identified with
R
{\displaystyle \mathbf {R} }
. Therefore the cup product, under these hypotheses, does give rise to a symmetric bilinear form on H2k(M,R); and therefore to a quadratic form Q. The form Q is non-degenerate due to Poincaré duality, as it pairs non-degenerately with itself. More generally, the signature can be defined in this way for any general compact polyhedron with 4n-dimensional Poincaré duality.
The signature
σ
(
M
)
{\displaystyle \sigma (M)}
of M is by definition the signature of Q, that is,
σ
(
M
)
=
n
+
−
n
−
{\displaystyle \sigma (M)=n_{+}-n_{-}}
where any diagonal matrix defining Q has
n
+
{\displaystyle n_{+}}
positive entries and
n
−
{\displaystyle n_{-}}
negative entries. If M is not connected, its signature is defined to be the sum of the signatures of its connected components.
== Other dimensions ==
If M has dimension not divisible by 4, its signature is usually defined to be 0. There are alternative generalization in L-theory: the signature can be interpreted as the 4k-dimensional (simply connected) symmetric L-group
L
4
k
,
{\displaystyle L^{4k},}
or as the 4k-dimensional quadratic L-group
L
4
k
,
{\displaystyle L_{4k},}
and these invariants do not always vanish for other dimensions. The Kervaire invariant is a mod 2 (i.e., an element of
Z
/
2
{\displaystyle \mathbf {Z} /2}
) for framed manifolds of dimension 4k+2 (the quadratic L-group
L
4
k
+
2
{\displaystyle L_{4k+2}}
), while the de Rham invariant is a mod 2 invariant of manifolds of dimension 4k+1 (the symmetric L-group
L
4
k
+
1
{\displaystyle L^{4k+1}}
); the other dimensional L-groups vanish.
=== Kervaire invariant ===
When
d
=
4
k
+
2
=
2
(
2
k
+
1
)
{\displaystyle d=4k+2=2(2k+1)}
is twice an odd integer (singly even), the same construction gives rise to an antisymmetric bilinear form. Such forms do not have a signature invariant; if they are non-degenerate, any two such forms are equivalent. However, if one takes a quadratic refinement of the form, which occurs if one has a framed manifold, then the resulting ε-quadratic forms need not be equivalent, being distinguished by the Arf invariant. The resulting invariant of a manifold is called the Kervaire invariant.
== Properties ==
Compact oriented manifolds M and N satisfy
σ
(
M
⊔
N
)
=
σ
(
M
)
+
σ
(
N
)
{\displaystyle \sigma (M\sqcup N)=\sigma (M)+\sigma (N)}
by definition, and satisfy
σ
(
M
×
N
)
=
σ
(
M
)
σ
(
N
)
{\displaystyle \sigma (M\times N)=\sigma (M)\sigma (N)}
by a Künneth formula.
If M is an oriented boundary, then
σ
(
M
)
=
0
{\displaystyle \sigma (M)=0}
.
René Thom (1954) showed that the signature of a manifold is a cobordism invariant, and in particular is given by some linear combination of its Pontryagin numbers. For example, in four dimensions, it is given by
p
1
3
{\displaystyle {\frac {p_{1}}{3}}}
. Friedrich Hirzebruch (1954) found an explicit expression for this linear combination as the L genus of the manifold.
William Browder (1962) proved that a simply connected compact polyhedron with 4n-dimensional Poincaré duality is homotopy equivalent to a manifold if and only if its signature satisfies the expression of the Hirzebruch signature theorem.
Rokhlin's theorem says that the signature of a 4-dimensional simply connected manifold with a spin structure is divisible by 16.
== See also ==
Hirzebruch signature theorem
Genus of a multiplicative sequence
Rokhlin's theorem
== References == | Wikipedia/Signature_(topology) |
In the mathematical surgery theory the surgery exact sequence is the main technical tool to calculate the surgery structure set of a compact manifold in dimension
>
4
{\displaystyle >4}
. The surgery structure set
S
(
X
)
{\displaystyle {\mathcal {S}}(X)}
of a compact
n
{\displaystyle n}
-dimensional manifold
X
{\displaystyle X}
is a pointed set which classifies
n
{\displaystyle n}
-dimensional manifolds within the homotopy type of
X
{\displaystyle X}
.
The basic idea is that in order to calculate
S
(
X
)
{\displaystyle {\mathcal {S}}(X)}
it is enough to understand the other terms in the sequence, which are usually easier to determine. These are on one hand the normal invariants which form generalized cohomology groups, and hence one can use standard tools of algebraic topology to calculate them at least in principle. On the other hand, there are the L-groups which are defined algebraically in terms of quadratic forms or in terms of chain complexes with quadratic structure. A great deal is known about these groups. Another part of the sequence are the surgery obstruction maps from normal invariants to the L-groups. For these maps there are certain characteristic classes formulas, which enable to calculate them in some cases. Knowledge of these three components, that means: the normal maps, the L-groups and the surgery obstruction maps is enough to determine the structure set (at least up to extension problems).
In practice one has to proceed case by case, for each manifold
X
{\displaystyle {\mathcal {}}X}
it is a unique task to determine the surgery exact sequence, see some examples below. Also note that there are versions of the surgery exact sequence depending on the category of manifolds we work with: smooth (DIFF), PL, or topological manifolds and whether we take Whitehead torsion into account or not (decorations
s
{\displaystyle s}
or
h
{\displaystyle h}
).
The original 1962 work of Browder and Novikov on the existence and uniqueness of manifolds within a simply-connected homotopy type was reformulated by Sullivan in 1966 as a surgery exact sequence.
In 1970 Wall developed non-simply-connected surgery theory and the surgery exact sequence for manifolds with arbitrary fundamental group.
== Definition ==
The surgery exact sequence is defined as
⋯
→
N
∂
(
X
×
I
)
→
L
n
+
1
(
π
1
(
X
)
)
→
S
(
X
)
→
N
(
X
)
→
L
n
(
π
1
(
X
)
)
{\displaystyle \cdots \to {\mathcal {N}}_{\partial }(X\times I)\to L_{n+1}(\pi _{1}(X))\to {\mathcal {S}}(X)\to {\mathcal {N}}(X)\to L_{n}(\pi _{1}(X))}
where:
the entries
N
∂
(
X
×
I
)
{\displaystyle {\mathcal {N}}_{\partial }(X\times I)}
and
N
(
X
)
{\displaystyle {\mathcal {N}}(X)}
are the abelian groups of normal invariants,
the entries
L
n
+
1
(
π
1
(
X
)
)
{\displaystyle {\mathcal {}}L_{n+1}(\pi _{1}(X))}
and
L
n
(
π
1
(
X
)
)
{\displaystyle {\mathcal {}}L_{n}(\pi _{1}(X))}
are the L-groups associated to the group ring
Z
[
π
1
(
X
)
]
{\displaystyle \mathbb {Z} [\pi _{1}(X)]}
,
the maps
θ
:
N
∂
(
X
×
I
)
→
L
n
+
1
(
π
1
(
X
)
)
{\displaystyle \theta \colon {\mathcal {N}}_{\partial }(X\times I)\to L_{n+1}(\pi _{1}(X))}
and
θ
:
N
(
X
)
→
L
n
(
π
1
(
X
)
)
{\displaystyle \theta \colon {\mathcal {N}}(X)\to L_{n}(\pi _{1}(X))}
are the surgery obstruction maps,
the arrows
∂
:
L
n
+
1
(
π
1
(
X
)
)
→
S
(
X
)
{\displaystyle \partial \colon L_{n+1}(\pi _{1}(X))\to {\mathcal {S}}(X)}
and
η
:
S
(
X
)
→
N
(
X
)
{\displaystyle \eta \colon {\mathcal {S}}(X)\to {\mathcal {N}}(X)}
will be explained below.
== Versions ==
There are various versions of the surgery exact sequence. One can work in either of the three categories of manifolds: differentiable (smooth), PL, topological. Another possibility is to work with the decorations
s
{\displaystyle s}
or
h
{\displaystyle h}
.
== The entries ==
=== Normal invariants ===
A degree one normal map
(
f
,
b
)
:
M
→
X
{\displaystyle (f,b)\colon M\to X}
consists of the following data: an
n
{\displaystyle n}
-dimensional oriented closed manifold
M
{\displaystyle M}
, a map
f
{\displaystyle f}
which is of degree one (that means
f
∗
(
[
M
]
)
=
[
X
]
{\displaystyle f_{*}([M])=[X]}
), and a bundle map
b
:
T
M
⊕
ε
k
→
ξ
{\displaystyle b\colon TM\oplus \varepsilon ^{k}\to \xi }
from the stable tangent bundle of
M
{\displaystyle M}
to some bundle
ξ
{\displaystyle \xi }
over
X
{\displaystyle X}
. Two such maps are equivalent if there exists a normal bordism between them (that means a bordism of the sources covered by suitable bundle data). The equivalence classes of degree one normal maps are called normal invariants.
When defined like this the normal invariants
N
(
X
)
{\displaystyle {\mathcal {N}}(X)}
are just a pointed set, with the base point given by
(
i
d
,
i
d
)
{\displaystyle (id,id)}
. However the Pontrjagin-Thom construction gives
N
(
X
)
{\displaystyle {\mathcal {N}}(X)}
a structure of an abelian group. In fact we have a non-natural bijection
N
(
X
)
≅
[
X
,
G
/
O
]
{\displaystyle {\mathcal {N}}(X)\cong [X,G/O]}
where
G
/
O
{\displaystyle G/O}
denotes the homotopy fiber of the map
J
:
B
O
→
B
G
{\displaystyle J\colon BO\to BG}
, which is an infinite loop space and hence maps into it define a generalized cohomology theory. There are corresponding identifications of the normal invariants with
[
X
,
G
/
P
L
]
{\displaystyle [X,G/PL]}
when working with PL-manifolds and with
[
X
,
G
/
T
O
P
]
{\displaystyle [X,G/TOP]}
when working with topological manifolds.
=== L-groups ===
The
L
{\displaystyle L}
-groups are defined algebraically in terms of quadratic forms or in terms of chain complexes with quadratic structure. See the main article for more details. Here only the properties of the L-groups described below will be important.
=== Surgery obstruction maps ===
The map
θ
:
N
(
X
)
→
L
n
(
π
1
(
X
)
)
{\displaystyle \theta \colon {\mathcal {N}}(X)\to L_{n}(\pi _{1}(X))}
is in the first instance a set-theoretic map (that means not necessarily a homomorphism) with the following property (when
n
≥
5
{\displaystyle n\geq 5}
:
A degree one normal map
(
f
,
b
)
:
M
→
X
{\displaystyle (f,b)\colon M\to X}
is normally cobordant to a homotopy equivalence if and only if the image
θ
(
f
,
b
)
=
0
{\displaystyle \theta (f,b)=0}
in
L
n
(
Z
[
π
1
(
X
)
]
)
{\displaystyle L_{n}(\mathbb {Z} [\pi _{1}(X)])}
.
=== The normal invariants arrow ===
η
:
S
(
X
)
→
N
(
X
)
{\displaystyle \eta \colon {\mathcal {S}}(X)\to {\mathcal {N}}(X)}
Any homotopy equivalence
f
:
M
→
X
{\displaystyle f\colon M\to X}
defines a degree one normal map.
=== The surgery obstruction arrow ===
∂
:
L
n
+
1
(
π
1
(
X
)
)
→
S
(
X
)
{\displaystyle \partial \colon L_{n+1}(\pi _{1}(X))\to {\mathcal {S}}(X)}
This arrow describes in fact an action of the group
L
n
+
1
(
π
1
(
X
)
)
{\displaystyle L_{n+1}(\pi _{1}(X))}
on the set
S
(
X
)
{\displaystyle {\mathcal {S}}(X)}
rather than just a map. The definition is based on the realization theorem for the elements of the
L
{\displaystyle L}
-groups which reads as follows:
Let
M
{\displaystyle M}
be an
n
{\displaystyle n}
-dimensional manifold with
π
1
(
M
)
≅
π
1
(
X
)
{\displaystyle \pi _{1}(M)\cong \pi _{1}(X)}
and let
x
∈
L
n
+
1
(
π
1
(
X
)
)
{\displaystyle x\in L_{n+1}(\pi _{1}(X))}
. Then there exists a degree one normal map of manifolds with boundary
(
F
,
B
)
:
(
W
,
M
,
M
′
)
→
(
M
×
I
,
M
×
0
,
M
×
1
)
{\displaystyle (F,B)\colon (W,M,M')\to (M\times I,M\times 0,M\times 1)}
with the following properties:
1.
θ
(
F
,
B
)
=
x
∈
L
n
+
1
(
π
1
(
X
)
)
{\displaystyle \theta (F,B)=x\in L_{n+1}(\pi _{1}(X))}
2.
F
0
:
M
→
M
×
0
{\displaystyle F_{0}\colon M\to M\times 0}
is a diffeomorphism
3.
F
1
:
M
′
→
M
×
1
{\displaystyle F_{1}\colon M'\to M\times 1}
is a homotopy equivalence of closed manifolds
Let
f
:
M
→
X
{\displaystyle f\colon M\to X}
represent an element in
S
(
X
)
{\displaystyle {\mathcal {S}}(X)}
and let
x
∈
L
n
+
1
(
π
1
(
X
)
)
{\displaystyle x\in L_{n+1}(\pi _{1}(X))}
. Then
∂
(
f
,
x
)
{\displaystyle \partial (f,x)}
is defined as
f
∘
F
1
:
M
′
→
X
{\displaystyle f\circ F_{1}\colon M'\to X}
.
== The exactness ==
Recall that the surgery structure set is only a pointed set and that the surgery obstruction map
θ
{\displaystyle \theta }
might not be a homomorphism. Hence it is necessary to explain what is meant when talking about the "exact sequence". So the surgery exact sequence is an exact sequence in the following sense:
For a normal invariant
z
∈
N
(
X
)
{\displaystyle z\in {\mathcal {N}}(X)}
we have
z
∈
I
m
(
η
)
{\displaystyle z\in \mathrm {Im} (\eta )}
if and only if
θ
(
z
)
=
0
{\displaystyle \theta (z)=0}
. For two manifold structures
x
1
,
x
2
∈
S
(
X
)
{\displaystyle x_{1},x_{2}\in {\mathcal {S}}(X)}
we have
η
(
x
1
)
=
η
(
x
2
)
{\displaystyle \eta (x_{1})=\eta (x_{2})}
if and only if there exists
u
∈
L
n
+
1
(
π
1
(
X
)
)
{\displaystyle u\in L_{n+1}(\pi _{1}(X))}
such that
∂
(
u
,
x
1
)
=
x
2
{\displaystyle \partial (u,x_{1})=x_{2}}
. For an element
u
∈
L
n
+
1
(
π
1
(
X
)
)
{\displaystyle u\in L_{n+1}(\pi _{1}(X))}
we have
∂
(
u
,
i
d
)
=
i
d
{\displaystyle \partial (u,\mathrm {id} )=\mathrm {id} }
if and only if
u
∈
I
m
(
θ
)
{\displaystyle u\in \mathrm {Im} (\theta )}
.
== Versions revisited ==
In the topological category the surgery obstruction map can be made into a homomorphism. This is achieved by putting an alternative abelian group structure on the normal invariants as described here. Moreover, the surgery exact sequence can be identified with the algebraic surgery exact sequence of Ranicki which is an exact sequence of abelian groups by definition. This gives the structure set
S
(
X
)
{\displaystyle {\mathcal {S}}(X)}
the structure of an abelian group. Note, however, that there is to this date no satisfactory geometric description of this abelian group structure.
== Classification of manifolds ==
The answer to the organizing questions of the surgery theory can be formulated in terms of the surgery exact sequence. In both cases the answer is given in the form of a two-stage obstruction theory.
The existence question. Let
X
{\displaystyle X}
be a finite Poincaré complex. It is homotopy equivalent to a manifold if and only if the following two conditions are satisfied. Firstly,
X
{\displaystyle X}
must have a vector bundle reduction of its Spivak normal fibration. This condition can be also formulated as saying that the set of normal invariants
N
(
X
)
{\displaystyle {\mathcal {N}}(X)}
is non-empty. Secondly, there must be a normal invariant
x
∈
N
(
X
)
{\displaystyle x\in {\mathcal {N}}(X)}
such that
θ
(
x
)
=
0
{\displaystyle \theta (x)=0}
. Equivalently, the surgery obstruction map
θ
:
N
(
X
)
→
L
n
(
π
1
(
X
)
)
{\displaystyle \theta \colon {\mathcal {N}}(X)\rightarrow L_{n}(\pi _{1}(X))}
hits
0
∈
L
n
(
π
1
(
X
)
)
{\displaystyle 0\in L_{n}(\pi _{1}(X))}
.
The uniqueness question. Let
f
:
M
→
X
{\displaystyle f\colon M\to X}
and
f
′
:
M
′
→
X
{\displaystyle f'\colon M'\to X}
represent two elements in the surgery structure set
S
(
X
)
{\displaystyle {\mathcal {S}}(X)}
. The question whether they represent the same element can be answered in two stages as follows. First there must be a normal cobordism between the degree one normal maps induced by
f
{\displaystyle {\mathcal {}}f}
and
f
′
{\displaystyle {\mathcal {}}f'}
, this means
η
(
f
)
=
η
(
f
′
)
{\displaystyle {\mathcal {}}\eta (f)=\eta (f')}
in
N
(
X
)
{\displaystyle {\mathcal {N}}(X)}
. Denote the normal cobordism
(
F
,
B
)
:
(
W
,
M
,
M
′
)
→
(
X
×
I
,
X
×
0
,
X
×
1
)
{\displaystyle (F,B)\colon (W,M,M')\to (X\times I,X\times 0,X\times 1)}
. If the surgery obstruction
θ
(
F
,
B
)
{\displaystyle {\mathcal {}}\theta (F,B)}
in
L
n
+
1
(
π
1
(
X
)
)
{\displaystyle {\mathcal {}}L_{n+1}(\pi _{1}(X))}
to make this normal cobordism to an h-cobordism (or s-cobordism) relative to the boundary vanishes then
f
{\displaystyle {\mathcal {}}f}
and
f
′
{\displaystyle {\mathcal {}}f'}
in fact represent the same element in the surgery structure set.
== Quinn's surgery fibration ==
In his thesis written under the guidance of Browder, Frank Quinn introduced a fiber sequence so that the surgery long exact sequence is the induced sequence on homotopy groups.
== Examples ==
=== 1. Homotopy spheres ===
This is an example in the smooth category,
n
≥
5
{\displaystyle n\geq 5}
.
The idea of the surgery exact sequence is implicitly present already in the original article of Kervaire and Milnor on the groups of homotopy spheres. In the present terminology we have
S
D
I
F
F
(
S
n
)
=
Θ
n
{\displaystyle {\mathcal {S}}^{DIFF}(S^{n})=\Theta ^{n}}
N
D
I
F
F
(
S
n
)
=
Ω
n
a
l
m
{\displaystyle {\mathcal {N}}^{DIFF}(S^{n})=\Omega _{n}^{alm}}
the cobordism group of almost framed
n
{\displaystyle n}
manifolds,
N
∂
D
I
F
F
(
S
n
×
I
)
=
Ω
n
+
1
a
l
m
{\displaystyle {\mathcal {N}}_{\partial }^{DIFF}(S^{n}\times I)=\Omega _{n+1}^{alm}}
L
n
(
1
)
=
Z
,
0
,
Z
2
,
0
{\displaystyle L_{n}(1)=\mathbb {Z} ,0,\mathbb {Z} _{2},0}
where
n
≡
0
,
1
,
2
,
3
{\displaystyle n\equiv 0,1,2,3}
mod
4
{\displaystyle 4}
(recall the
4
{\displaystyle 4}
-periodicity of the L-groups)
The surgery exact sequence in this case is an exact sequence of abelian groups. In addition to the above identifications we have
b
P
n
+
1
=
k
e
r
(
η
:
S
D
I
F
F
(
S
n
)
→
N
D
I
F
F
(
S
n
)
)
=
c
o
k
e
r
(
θ
:
N
∂
D
I
F
F
(
S
n
×
I
)
→
L
n
+
1
(
1
)
)
{\displaystyle bP^{n+1}=\mathrm {ker} (\eta \colon {\mathcal {S}}^{DIFF}(S^{n})\to {\mathcal {N}}^{DIFF}(S^{n}))=\mathrm {coker} (\theta \colon {\mathcal {N}}_{\partial }^{DIFF}(S^{n}\times I)\to L_{n+1}(1))}
Because the odd-dimensional L-groups are trivial one obtains these exact sequences:
0
→
Θ
4
i
→
Ω
4
i
a
l
m
→
Z
→
b
P
4
i
→
0
{\displaystyle 0\to \Theta ^{4i}\to \Omega _{4i}^{alm}\to \mathbb {Z} \to bP^{4i}\to 0}
0
→
Θ
4
i
−
2
→
Ω
4
i
−
2
a
l
m
→
Z
/
2
→
b
P
4
i
−
2
→
0
{\displaystyle 0\to \Theta ^{4i-2}\to \Omega _{4i-2}^{alm}\to \mathbb {Z} /2\to bP^{4i-2}\to 0}
0
→
b
P
2
j
→
Θ
2
j
−
1
→
Ω
2
j
−
1
a
l
m
→
0
{\displaystyle 0\to bP^{2j}\to \Theta ^{2j-1}\to \Omega _{2j-1}^{alm}\to 0}
The results of Kervaire and Milnor are obtained by studying the middle map in the first two sequences and by relating the groups
Ω
i
a
l
m
{\displaystyle \Omega _{i}^{alm}}
to stable homotopy theory.
=== 2. Topological spheres ===
The generalized Poincaré conjecture in dimension
n
{\displaystyle n}
can be phrased as saying that
S
T
O
P
(
S
n
)
=
0
{\displaystyle {\mathcal {S}}^{TOP}(S^{n})=0}
. It has been proved for any
n
{\displaystyle n}
by the work of Smale, Freedman and Perelman. From the surgery exact sequence for
S
n
{\displaystyle S^{n}}
for
n
≥
5
{\displaystyle n\geq 5}
in the topological category we see that
θ
:
N
T
O
P
(
S
n
)
→
L
n
(
1
)
{\displaystyle \theta \colon {\mathcal {N}}^{TOP}(S^{n})\to L_{n}(1)}
is an isomorphism. (In fact this can be extended to
n
≥
1
{\displaystyle n\geq 1}
by some ad-hoc methods.)
=== 3. Complex projective spaces in the topological category ===
The complex projective space
C
P
n
{\displaystyle \mathbb {C} P^{n}}
is a
(
2
n
)
{\displaystyle (2n)}
-dimensional topological manifold with
π
1
(
C
P
n
)
=
1
{\displaystyle \pi _{1}(\mathbb {C} P^{n})=1}
. In addition it is known that in the case
π
1
(
X
)
=
1
{\displaystyle \pi _{1}(X)=1}
in the topological category the surgery obstruction map
θ
{\displaystyle \theta }
is always surjective. Hence we have
0
→
S
T
O
P
(
C
P
n
)
→
N
T
O
P
(
C
P
n
)
→
L
2
n
(
1
)
→
0
{\displaystyle 0\to {\mathcal {S}}^{TOP}(\mathbb {C} P^{n})\to {\mathcal {N}}^{TOP}(\mathbb {C} P^{n})\to L_{2n}(1)\to 0}
From the work of Sullivan one can calculate
N
(
C
P
n
)
≅
⊕
i
=
1
⌊
n
/
2
⌋
Z
⊕
⊕
i
=
1
⌊
(
n
+
1
)
/
2
⌋
Z
2
{\displaystyle {\mathcal {N}}(\mathbb {C} P^{n})\cong \oplus _{i=1}^{\lfloor n/2\rfloor }\mathbb {Z} \oplus \oplus _{i=1}^{\lfloor (n+1)/2\rfloor }\mathbb {Z} _{2}}
and hence
S
(
C
P
n
)
≅
⊕
i
=
1
⌊
(
n
−
1
)
/
2
⌋
Z
⊕
⊕
i
=
1
⌊
n
/
2
⌋
Z
2
{\displaystyle {\mathcal {S}}(\mathbb {C} P^{n})\cong \oplus _{i=1}^{\lfloor (n-1)/2\rfloor }\mathbb {Z} \oplus \oplus _{i=1}^{\lfloor n/2\rfloor }\mathbb {Z} _{2}}
=== 4. Aspherical manifolds in the topological category ===
An aspherical
n
{\displaystyle n}
-dimensional manifold
X
{\displaystyle X}
is an
n
{\displaystyle n}
-manifold such that
π
i
(
X
)
=
0
{\displaystyle \pi _{i}(X)=0}
for
i
≥
2
{\displaystyle i\geq 2}
. Hence the only non-trivial homotopy group is
π
1
(
X
)
{\displaystyle \pi _{1}(X)}
One way to state the Borel conjecture is to say that for such
X
{\displaystyle X}
we have that the Whitehead group
W
h
(
π
1
(
X
)
)
{\displaystyle Wh(\pi _{1}(X))}
is trivial and that
S
(
X
)
=
0
{\displaystyle {\mathcal {S}}(X)=0}
This conjecture was proven in many special cases - for example when
π
1
(
X
)
{\displaystyle \pi _{1}(X)}
is
Z
n
{\displaystyle \mathbb {Z} ^{n}}
, when it is the fundamental group of a negatively curved manifold or when it is a word-hyperbolic group or a CAT(0)-group.
The statement is equivalent to showing that the surgery obstruction map to the right of the surgery structure set is injective and the surgery obstruction map to the left of the surgery structure set is surjective. Most of the proofs of the above-mentioned results are done by studying these maps or by studying the assembly maps with which they can be identified. See more details in Borel conjecture, Farrell-Jones Conjecture.
== References ==
Browder, William (1972), Surgery on simply-connected manifolds, Berlin, New York: Springer-Verlag, MR 0358813
Lück, Wolfgang (2002), A basic introduction to surgery theory (PDF), ICTP Lecture Notes Series 9, Band 1, of the school "High-dimensional manifold theory" in Trieste, May/June 2001, Abdus Salam International Centre for Theoretical Physics, Trieste 1-224
Ranicki, Andrew (1992), Algebraic L-theory and topological manifolds (PDF), Cambridge Tracts in Mathematics, vol. 102, Cambridge University Press
Ranicki, Andrew (2002), Algebraic and Geometric Surgery (PDF), Oxford Mathematical Monographs, Clarendon Press, ISBN 978-0-19-850924-0, MR 2061749
Wall, C. T. C. (1999), Surgery on compact manifolds, Mathematical Surveys and Monographs, vol. 69 (2nd ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0942-6, MR 1687388 | Wikipedia/Surgery_exact_sequence |
In geometric topology, the Borel conjecture (named for Armand Borel) asserts that an aspherical closed manifold is determined by its fundamental group, up to homeomorphism. It is a rigidity conjecture, asserting that a weak, algebraic notion of equivalence (namely, homotopy equivalence) should imply a stronger, topological notion (namely, homeomorphism).
== Precise formulation of the conjecture ==
Let
M
{\displaystyle M}
and
N
{\displaystyle N}
be closed and aspherical topological manifolds, and let
f
:
M
→
N
{\displaystyle f\colon M\to N}
be a homotopy equivalence. The Borel conjecture states that the map
f
{\displaystyle f}
is homotopic to a homeomorphism. Since aspherical manifolds with isomorphic fundamental groups are homotopy equivalent, the Borel conjecture implies that aspherical closed manifolds are determined, up to homeomorphism, by their fundamental groups.
This conjecture is false if topological manifolds and homeomorphisms are replaced by smooth manifolds and diffeomorphisms; counterexamples can be constructed by taking a connected sum with an exotic sphere.
== The origin of the conjecture ==
In a May 1953 letter to Jean-Pierre Serre, Armand Borel raised the question whether two aspherical manifolds with isomorphic fundamental groups are homeomorphic. A positive answer to the question "Is every homotopy equivalence between closed aspherical manifolds homotopic to a homeomorphism?" is referred to as the "so-called Borel Conjecture" in a 1986 paper of Jonathan Rosenberg.
== Motivation for the conjecture ==
A basic question is the following: if two closed manifolds are homotopy equivalent, are they homeomorphic? This is not true in general: there are homotopy equivalent lens spaces which are not homeomorphic.
Nevertheless, there are classes of manifolds for which homotopy equivalences between them can be homotoped to homeomorphisms. For instance, the Mostow rigidity theorem states that a homotopy equivalence between closed hyperbolic manifolds is homotopic to an isometry—in particular, to a homeomorphism. The Borel conjecture is a topological reformulation of Mostow rigidity, weakening the hypothesis from hyperbolic manifolds to aspherical manifolds, and similarly weakening the conclusion from an isometry to a homeomorphism.
== Relationship to other conjectures ==
The Borel conjecture implies the Novikov conjecture for the special case in which the reference map
f
:
M
→
B
G
{\displaystyle f\colon M\to BG}
is a homotopy equivalence.
The Poincaré conjecture asserts that a closed manifold homotopy equivalent to
S
3
{\displaystyle S^{3}}
, the 3-sphere, is homeomorphic to
S
3
{\displaystyle S^{3}}
. This is not a special case of the Borel conjecture, because
S
3
{\displaystyle S^{3}}
is not aspherical. Nevertheless, the Borel conjecture for the 3-torus
T
3
=
S
1
×
S
1
×
S
1
{\displaystyle T^{3}=S^{1}\times S^{1}\times S^{1}}
implies the Poincaré conjecture for
S
3
{\displaystyle S^{3}}
.
== References ==
== Further reading ==
Matthias Kreck, and Wolfgang Lück, The Novikov conjecture. Geometry and algebra. Oberwolfach Seminars, 33. Birkhäuser Verlag, Basel, 2005. | Wikipedia/Borel_conjecture |
In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link. It is often conceptualized as two steps: drilling then filling.
== Definitions ==
Given a 3-manifold
M
{\displaystyle M}
and a link
L
⊂
M
{\displaystyle L\subset M}
, the manifold
M
{\displaystyle M}
drilled along
L
{\displaystyle L}
is obtained by removing an open tubular neighborhood of
L
{\displaystyle L}
from
M
{\displaystyle M}
. If
L
=
L
1
∪
⋯
∪
L
k
{\displaystyle L=L_{1}\cup \dots \cup L_{k}}
, the drilled manifold has
k
{\displaystyle k}
torus boundary components
T
1
∪
⋯
∪
T
k
{\displaystyle T_{1}\cup \dots \cup T_{k}}
. The manifold
M
{\displaystyle M}
drilled along
L
{\displaystyle L}
is also known as the link complement, since if one removed the corresponding closed tubular neighborhood from
M
{\displaystyle M}
, one obtains a manifold diffeomorphic to
M
∖
L
{\displaystyle M\setminus L}
.
Given a 3-manifold whose boundary is made of 2-tori
T
1
∪
⋯
∪
T
k
{\displaystyle T_{1}\cup \dots \cup T_{k}}
, we may glue in one solid torus by a homeomorphism (resp. diffeomorphism) of its boundary to each of the torus boundary components
T
i
{\displaystyle T_{i}}
of the original 3-manifold. There are many inequivalent ways of doing this, in general. This process is called Dehn filling.
Dehn surgery on a 3-manifold containing a link consists of drilling out a tubular neighbourhood of the link together with Dehn filling on all the components of the boundary corresponding to the link.
In order to describe a Dehn surgery, one picks two oriented simple closed curves
m
i
{\displaystyle m_{i}}
and
ℓ
i
{\displaystyle \ell _{i}}
on the corresponding boundary torus
T
i
{\displaystyle T_{i}}
of the drilled 3-manifold, where
m
i
{\displaystyle m_{i}}
is a meridian of
L
i
{\displaystyle L_{i}}
(a curve staying in a small ball in
M
{\displaystyle M}
and having linking number +1 with
L
i
{\displaystyle L_{i}}
or, equivalently, a curve that bounds a disc that intersects once the component
L
i
{\displaystyle L_{i}}
) and
ℓ
i
{\displaystyle \ell _{i}}
is a longitude of
T
i
{\displaystyle T_{i}}
(a curve travelling once along
L
i
{\displaystyle L_{i}}
or, equivalently, a curve on
T
i
{\displaystyle T_{i}}
such that the algebraic intersection
⟨
ℓ
i
,
m
i
⟩
{\displaystyle \langle \ell _{i},m_{i}\rangle }
is equal to +1).
The curves
m
i
{\displaystyle m_{i}}
and
ℓ
i
{\displaystyle \ell _{i}}
generate the fundamental group of the torus
T
i
{\displaystyle T_{i}}
, and they form a basis of its first homology group. This gives any simple closed curve
γ
i
{\displaystyle \gamma _{i}}
on the torus
T
i
{\displaystyle T_{i}}
two coordinates
a
i
{\displaystyle a_{i}}
and
b
i
{\displaystyle b_{i}}
, so that
[
γ
i
]
=
[
a
i
ℓ
i
+
b
i
m
i
]
{\displaystyle [\gamma _{i}]=[a_{i}\ell _{i}+b_{i}m_{i}]}
. These coordinates only depend on the homotopy class of
γ
i
{\displaystyle \gamma _{i}}
.
We can specify a homeomorphism of the boundary of a solid torus to
T
i
{\displaystyle T_{i}}
by having the meridian curve of the solid torus map to a curve homotopic to
γ
i
{\displaystyle \gamma _{i}}
. As long as the meridian maps to the surgery slope
[
γ
i
]
{\displaystyle [\gamma _{i}]}
, the resulting Dehn surgery will yield a 3-manifold that will not depend on the specific gluing (up to homeomorphism). The ratio
b
i
/
a
i
∈
Q
∪
{
∞
}
{\displaystyle b_{i}/a_{i}\in \mathbb {Q} \cup \{\infty \}}
is called the surgery coefficient of
L
i
{\displaystyle L_{i}}
.
In the case of links in the 3-sphere or more generally an oriented integral homology sphere, there is a canonical choice of the longitudes
ℓ
i
{\displaystyle \ell _{i}}
: every longitude is chosen so that it is null-homologous in the knot complement—equivalently, if it is the boundary of a Seifert surface.
When the ratios
b
i
/
a
i
{\displaystyle b_{i}/a_{i}}
are all integers (note that this condition does not depend on the choice of the longitudes, since it corresponds to the new meridians intersecting exactly once the ancient meridians), the surgery is called an integral surgery.
Such surgeries are closely related to handlebodies, cobordism and Morse functions.
== Examples ==
If all surgery coefficients are infinite, then each new meridian
γ
i
{\displaystyle \gamma _{i}}
is homotopic to the ancient meridian
m
i
{\displaystyle m_{i}}
. Therefore the homeomorphism-type of the manifold is unchanged by the surgery.
If
M
{\displaystyle M}
is the 3-sphere,
L
{\displaystyle L}
is the unknot, and the surgery coefficient is
0
{\displaystyle 0}
, then the surgered 3-manifold is
S
2
×
S
1
{\displaystyle \mathbb {S} ^{2}\times \mathbb {S} ^{1}}
.
If
M
{\displaystyle M}
is the 3-sphere,
L
{\displaystyle L}
is the unknot, and the surgery coefficient is
b
/
a
{\displaystyle b/a}
, then the surgered 3-manifold is the lens space
L
(
b
,
a
)
{\displaystyle L(b,a)}
. In particular if the surgery coefficient is of the form
±
1
/
r
{\displaystyle \pm 1/r}
, then the surgered 3-manifold is still the 3-sphere.
If
M
{\displaystyle M}
is the 3-sphere,
L
{\displaystyle L}
is the right-handed trefoil knot, and the surgery coefficient is
+
1
{\displaystyle +1}
, then the surgered 3-manifold is the Poincaré dodecahedral space.
== Results ==
Every closed, orientable, connected 3-manifold is obtained by performing Dehn surgery on a link in the 3-sphere. This result, the Lickorish–Wallace theorem, was first proven by Andrew H. Wallace in 1960 and independently by W. B. R. Lickorish in a stronger form in 1962. Via the now well-known relation between genuine surgery and cobordism, this result is equivalent to the theorem that the oriented cobordism group of 3-manifolds is trivial, a theorem originally proved by Vladimir Abramovich Rokhlin in 1951.
Since orientable 3-manifolds can all be generated by suitably decorated links, one might ask how distinct surgery presentations of a given 3-manifold might be related. The answer is called the Kirby calculus.
== See also ==
Hyperbolic Dehn surgery
Tubular neighborhood
Surgery on manifolds, in the general sense, also called spherical modification.
== Footnotes ==
== References ==
Dehn, Max (1938), "Die Gruppe der Abbildungsklassen", Acta Mathematica, 69 (1): 135–206, doi:10.1007/BF02547712.
Thom, René (1954), "Quelques propriétés globales des variétés différentiables", Commentarii Mathematici Helvetici, 28: 17–86, doi:10.1007/BF02566923, MR 0061823, S2CID 120243638
Rolfsen, Dale (1976), Knots and links (PDF), Mathematics lecture series, vol. 346, Berkeley, California: Publish or Perish, ISBN 9780914098164
Kirby, Rob (1978), "A calculus for framed links in S3", Inventiones Mathematicae, 45 (1): 35–56, Bibcode:1978InMat..45...35K, doi:10.1007/BF01406222, MR 0467753, S2CID 120770295.
Fenn, Roger; Rourke, Colin (1979), "On Kirby's calculus of links", Topology, 18 (1): 1–15, doi:10.1016/0040-9383(79)90010-7, MR 0528232.
Gompf, Robert; Stipsicz, András (1999), 4-Manifolds and Kirby Calculus, Graduate Studies in Mathematics, vol. 20, Providence, RI: American Mathematical Society, doi:10.1090/gsm/020, ISBN 0-8218-0994-6, MR 1707327. | Wikipedia/Dehn_surgery |
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiable function on a manifold will reflect the topology quite directly. Morse theory allows one to find CW structures and handle decompositions on manifolds and to obtain substantial information about their homology.
Before Morse, Arthur Cayley and James Clerk Maxwell had developed some of the ideas of Morse theory in the context of topography. Morse originally applied his theory to geodesics (critical points of the energy functional on the space of paths). These techniques were used in Raoul Bott's proof of his periodicity theorem.
The analogue of Morse theory for complex manifolds is Picard–Lefschetz theory.
== Basic concepts ==
To illustrate, consider a mountainous landscape surface
M
{\displaystyle M}
(more generally, a manifold). If
f
{\displaystyle f}
is the function
M
→
R
{\displaystyle M\to \mathbb {R} }
giving the elevation of each point, then the inverse image of a point in
R
{\displaystyle \mathbb {R} }
is a contour line (more generally, a level set). Each connected component of a contour line is either a point, a simple closed curve, or a closed curve with double point(s). Contour lines may also have points of higher order (triple points, etc.), but these are unstable and may be removed by a slight deformation of the landscape. Double points in contour lines occur at saddle points, or passes, where the surrounding landscape curves up in one direction and down in the other.
Imagine flooding this landscape with water. When the water reaches elevation
a
{\displaystyle a}
, the underwater surface is
M
a
=
def
f
−
1
(
−
∞
,
a
]
{\displaystyle M^{a}\,{\stackrel {\text{def}}{=}}\,f^{-1}(-\infty ,a]}
, the points with elevation
a
{\displaystyle a}
or below. Consider how the topology of this surface changes as the water rises. It appears unchanged except when
a
{\displaystyle a}
passes the height of a critical point, where the gradient of
f
{\displaystyle f}
is
0
{\displaystyle 0}
(more generally, the Jacobian matrix acting as a linear map between tangent spaces does not have maximal rank). In other words, the topology of
M
a
{\displaystyle M^{a}}
does not change except when the water either (1) starts filling a basin, (2) covers a saddle (a mountain pass), or (3) submerges a peak.
To these three types of critical points—basins, passes, and peaks (i.e. minima, saddles, and maxima)—one associates a number called the index, the number of independent directions in which
f
{\displaystyle f}
decreases from the point. More precisely, the index of a non-degenerate critical point
p
{\displaystyle p}
of
f
{\displaystyle f}
is the dimension of the largest subspace of the tangent space to
M
{\displaystyle M}
at
p
{\displaystyle p}
on which the Hessian of
f
{\displaystyle f}
is negative definite. The indices of basins, passes, and peaks are
0
,
1
,
{\displaystyle 0,1,}
and
2
,
{\displaystyle 2,}
respectively.
Considering a more general surface, let
M
{\displaystyle M}
be a torus oriented as in the picture, with
f
{\displaystyle f}
again taking a point to its height above the plane. One can again analyze how the topology of the underwater surface
M
a
{\displaystyle M^{a}}
changes as the water level
a
{\displaystyle a}
rises.
Starting from the bottom of the torus, let
p
,
q
,
r
,
{\displaystyle p,q,r,}
and
s
{\displaystyle s}
be the four critical points of index
0
,
1
,
1
,
{\displaystyle 0,1,1,}
and
2
{\displaystyle 2}
corresponding to the basin, two saddles, and peak, respectively. When
a
{\displaystyle a}
is less than
f
(
p
)
=
0
,
{\displaystyle f(p)=0,}
then
M
a
{\displaystyle M^{a}}
is the empty set. After
a
{\displaystyle a}
passes the level of
p
,
{\displaystyle p,}
when
0
<
a
<
f
(
q
)
,
{\displaystyle 0<a<f(q),}
then
M
a
{\displaystyle M^{a}}
is a disk, which is homotopy equivalent to a point (a 0-cell) which has been "attached" to the empty set. Next, when
a
{\displaystyle a}
exceeds the level of
q
,
{\displaystyle q,}
and
f
(
q
)
<
a
<
f
(
r
)
,
{\displaystyle f(q)<a<f(r),}
then
M
a
{\displaystyle M^{a}}
is a cylinder, and is homotopy equivalent to a disk with a 1-cell attached (image at left). Once
a
{\displaystyle a}
passes the level of
r
,
{\displaystyle r,}
and
f
(
r
)
<
a
<
f
(
s
)
,
{\displaystyle f(r)<a<f(s),}
then
M
a
{\displaystyle M^{a}}
is a torus with a disk removed, which is homotopy equivalent to a cylinder with a 1-cell attached (image at right). Finally, when
a
{\displaystyle a}
is greater than the critical level of
s
,
{\displaystyle s,}
M
a
{\displaystyle M^{a}}
is a torus, i.e. a torus with a disk (a 2-cell) removed and re-attached.
This illustrates the following rule: the topology of
M
a
{\displaystyle M^{a}}
does not change except when
a
{\displaystyle a}
passes the height of a critical point; at this point, a
γ
{\displaystyle \gamma }
-cell is attached to
M
a
{\displaystyle M^{a}}
, where
γ
{\displaystyle \gamma }
is the index of the point. This does not address what happens when two critical points are at the same height, which can be resolved by a slight perturbation of
f
.
{\displaystyle f.}
In the case of a landscape or a manifold embedded in Euclidean space, this perturbation might simply be tilting slightly, rotating the coordinate system.
One must take care to make the critical points non-degenerate. To see what can pose a problem, let
M
=
R
{\displaystyle M=\mathbb {R} }
and let
f
(
x
)
=
x
3
.
{\displaystyle f(x)=x^{3}.}
Then
0
{\displaystyle 0}
is a critical point of
f
,
{\displaystyle f,}
but the topology of
M
a
{\displaystyle M^{a}}
does not change when
a
{\displaystyle a}
passes
0.
{\displaystyle 0.}
The problem is that the second derivative is
f
″
(
0
)
=
0
{\displaystyle f''(0)=0}
—that is, the Hessian of
f
{\displaystyle f}
vanishes and the critical point is degenerate. This situation is unstable, since by slightly deforming
f
{\displaystyle f}
to
f
(
x
)
=
x
3
+
ϵ
x
{\displaystyle f(x)=x^{3}+\epsilon x}
, the degenerate critical point is either removed (
ϵ
>
0
{\displaystyle \epsilon >0}
) or breaks up into two non-degenerate critical points (
ϵ
<
0
{\displaystyle \epsilon <0}
).
== Formal development ==
For a real-valued smooth function
f
:
M
→
R
{\displaystyle f:M\to \mathbb {R} }
on a differentiable manifold
M
,
{\displaystyle M,}
the points where the differential of
f
{\displaystyle f}
vanishes are called critical points of
f
{\displaystyle f}
and their images under
f
{\displaystyle f}
are called critical values. If at a critical point
p
{\displaystyle p}
the matrix of second partial derivatives (the Hessian matrix) is non-singular, then
p
{\displaystyle p}
is called a non-degenerate critical point; if the Hessian is singular then
p
{\displaystyle p}
is a degenerate critical point.
For the functions
f
(
x
)
=
a
+
b
x
+
c
x
2
+
d
x
3
+
⋯
{\displaystyle f(x)=a+bx+cx^{2}+dx^{3}+\cdots }
from
R
{\displaystyle \mathbb {R} }
to
R
,
{\displaystyle \mathbb {R} ,}
f
{\displaystyle f}
has a critical point at the origin if
b
=
0
,
{\displaystyle b=0,}
which is non-degenerate if
c
≠
0
{\displaystyle c\neq 0}
(that is,
f
{\displaystyle f}
is of the form
a
+
c
x
2
+
⋯
{\displaystyle a+cx^{2}+\cdots }
) and degenerate if
c
=
0
{\displaystyle c=0}
(that is,
f
{\displaystyle f}
is of the form
a
+
d
x
3
+
⋯
{\displaystyle a+dx^{3}+\cdots }
). A less trivial example of a degenerate critical point is the origin of the monkey saddle.
The index of a non-degenerate critical point
p
{\displaystyle p}
of
f
{\displaystyle f}
is the dimension of the largest subspace of the tangent space to
M
{\displaystyle M}
at
p
{\displaystyle p}
on which the Hessian is negative definite. This corresponds to the intuitive notion that the index is the number of directions in which
f
{\displaystyle f}
decreases. The degeneracy and index of a critical point are independent of the choice of the local coordinate system used, as shown by Sylvester's Law.
=== Morse lemma ===
Let
p
{\displaystyle p}
be a non-degenerate critical point of
f
:
M
→
R
.
{\displaystyle f\colon M\to \mathbb {R} .}
Then there exists a chart
(
x
1
,
x
2
,
…
,
x
n
)
{\displaystyle \left(x_{1},x_{2},\ldots ,x_{n}\right)}
in a neighborhood
U
{\displaystyle U}
of
p
{\displaystyle p}
such that
x
i
(
p
)
=
0
{\displaystyle x_{i}(p)=0}
for all
i
{\displaystyle i}
and
f
(
x
)
=
f
(
p
)
−
x
1
2
−
⋯
−
x
γ
2
+
x
γ
+
1
2
+
⋯
+
x
n
2
{\displaystyle f(x)=f(p)-x_{1}^{2}-\cdots -x_{\gamma }^{2}+x_{\gamma +1}^{2}+\cdots +x_{n}^{2}}
throughout
U
.
{\displaystyle U.}
Here
γ
{\displaystyle \gamma }
is equal to the index of
f
{\displaystyle f}
at
p
{\displaystyle p}
. As a corollary of the Morse lemma, one sees that non-degenerate critical points are isolated. (Regarding an extension to the complex domain see Complex Morse Lemma. For a generalization, see Morse–Palais lemma).
=== Fundamental theorems ===
A smooth real-valued function on a manifold
M
{\displaystyle M}
is a Morse function if it has no degenerate critical points. A basic result of Morse theory says that almost all functions are Morse functions. Technically, the Morse functions form an open, dense subset of all smooth functions
M
→
R
{\displaystyle M\to \mathbb {R} }
in the
C
2
{\displaystyle C^{2}}
topology. This is sometimes expressed as "a typical function is Morse" or "a generic function is Morse".
As indicated before, we are interested in the question of when the topology of
M
a
=
f
−
1
(
−
∞
,
a
]
{\displaystyle M^{a}=f^{-1}(-\infty ,a]}
changes as
a
{\displaystyle a}
varies. Half of the answer to this question is given by the following theorem.
Theorem. Suppose
f
{\displaystyle f}
is a smooth real-valued function on
M
,
{\displaystyle M,}
a
<
b
,
{\displaystyle a<b,}
f
−
1
[
a
,
b
]
{\displaystyle f^{-1}[a,b]}
is compact, and there are no critical values between
a
{\displaystyle a}
and
b
.
{\displaystyle b.}
Then
M
a
{\displaystyle M^{a}}
is diffeomorphic to
M
b
,
{\displaystyle M^{b},}
and
M
b
{\displaystyle M^{b}}
deformation retracts onto
M
a
.
{\displaystyle M^{a}.}
It is also of interest to know how the topology of
M
a
{\displaystyle M^{a}}
changes when
a
{\displaystyle a}
passes a critical point. The following theorem answers that question.
Theorem. Suppose
f
{\displaystyle f}
is a smooth real-valued function on
M
{\displaystyle M}
and
p
{\displaystyle p}
is a non-degenerate critical point of
f
{\displaystyle f}
of index
γ
,
{\displaystyle \gamma ,}
and that
f
(
p
)
=
q
.
{\displaystyle f(p)=q.}
Suppose
f
−
1
[
q
−
ε
,
q
+
ε
]
{\displaystyle f^{-1}[q-\varepsilon ,q+\varepsilon ]}
is compact and contains no critical points besides
p
.
{\displaystyle p.}
Then
M
q
+
ε
{\displaystyle M^{q+\varepsilon }}
is homotopy equivalent to
M
q
−
ε
{\displaystyle M^{q-\varepsilon }}
with a
γ
{\displaystyle \gamma }
-cell attached.
These results generalize and formalize the 'rule' stated in the previous section.
Using the two previous results and the fact that there exists a Morse function on any differentiable manifold, one can prove that any differentiable manifold is a CW complex with an
n
{\displaystyle n}
-cell for each critical point of index
n
.
{\displaystyle n.}
To do this, one needs the technical fact that one can arrange to have a single critical point on each critical level, which is usually proven by using gradient-like vector fields to rearrange the critical points.
=== Morse inequalities ===
Morse theory can be used to prove some strong results on the homology of manifolds. The number of critical points of index
γ
{\displaystyle \gamma }
of
f
:
M
→
R
{\displaystyle f:M\to \mathbb {R} }
is equal to the number of
γ
{\displaystyle \gamma }
cells in the CW structure on
M
{\displaystyle M}
obtained from "climbing"
f
.
{\displaystyle f.}
Using the fact that the alternating sum of the ranks of the homology groups of a topological space is equal to the alternating sum of the ranks of the chain groups from which the homology is computed, then by using the cellular chain groups (see cellular homology) it is clear that the Euler characteristic
χ
(
M
)
{\displaystyle \chi (M)}
is equal to the sum
∑
(
−
1
)
γ
C
γ
=
χ
(
M
)
{\displaystyle \sum (-1)^{\gamma }C^{\gamma }\,=\chi (M)}
where
C
γ
{\displaystyle C^{\gamma }}
is the number of critical points of index
γ
.
{\displaystyle \gamma .}
Also by cellular homology, the rank of the
n
{\displaystyle n}
th homology group of a CW complex
M
{\displaystyle M}
is less than or equal to the number of
n
{\displaystyle n}
-cells in
M
.
{\displaystyle M.}
Therefore, the rank of the
γ
{\displaystyle \gamma }
th homology group, that is, the Betti number
b
γ
(
M
)
{\displaystyle b_{\gamma }(M)}
, is less than or equal to the number of critical points of index
γ
{\displaystyle \gamma }
of a Morse function on
M
.
{\displaystyle M.}
These facts can be strengthened to obtain the Morse inequalities:
C
γ
−
C
γ
−
1
±
⋯
+
(
−
1
)
γ
C
0
≥
b
γ
(
M
)
−
b
γ
−
1
(
M
)
±
⋯
+
(
−
1
)
γ
b
0
(
M
)
.
{\displaystyle C^{\gamma }-C^{\gamma -1}\pm \cdots +(-1)^{\gamma }C^{0}\geq b_{\gamma }(M)-b_{\gamma -1}(M)\pm \cdots +(-1)^{\gamma }b_{0}(M).}
In particular, for any
γ
∈
{
0
,
…
,
n
=
dim
M
}
,
{\displaystyle \gamma \in \{0,\ldots ,n=\dim M\},}
one has
C
γ
≥
b
γ
(
M
)
.
{\displaystyle C^{\gamma }\geq b_{\gamma }(M).}
This gives a powerful tool to study manifold topology. Suppose on a closed manifold there exists a Morse function
f
:
M
→
R
{\displaystyle f:M\to \mathbb {R} }
with precisely k critical points. In what way does the existence of the function
f
{\displaystyle f}
restrict
M
{\displaystyle M}
? The case
k
=
2
{\displaystyle k=2}
was studied by Georges Reeb in 1952; the Reeb sphere theorem states that
M
{\displaystyle M}
is homeomorphic to a sphere
S
n
.
{\displaystyle S^{n}.}
The case
k
=
3
{\displaystyle k=3}
is possible only in a small number of low dimensions, and M is homeomorphic to an Eells–Kuiper manifold.
In 1982 Edward Witten developed an analytic approach to the Morse inequalities by considering the de Rham complex for the perturbed operator
d
t
=
e
−
t
f
d
e
t
f
.
{\displaystyle d_{t}=e^{-tf}de^{tf}.}
=== Application to classification of closed 2-manifolds ===
Morse theory has been used to classify closed 2-manifolds up to diffeomorphism. If
M
{\displaystyle M}
is oriented, then
M
{\displaystyle M}
is classified by its genus
g
{\displaystyle g}
and is diffeomorphic to a sphere with
g
{\displaystyle g}
handles: thus if
g
=
0
,
{\displaystyle g=0,}
M
{\displaystyle M}
is diffeomorphic to the 2-sphere; and if
g
>
0
,
{\displaystyle g>0,}
M
{\displaystyle M}
is diffeomorphic to the connected sum of
g
{\displaystyle g}
2-tori. If
N
{\displaystyle N}
is unorientable, it is classified by a number
g
>
0
{\displaystyle g>0}
and is diffeomorphic to the connected sum of
g
{\displaystyle g}
real projective spaces
R
P
2
.
{\displaystyle \mathbf {RP} ^{2}.}
In particular two closed 2-manifolds are homeomorphic if and only if they are diffeomorphic.
=== Morse homology ===
Morse homology is a particularly easy way to understand the homology of smooth manifolds. It is defined using a generic choice of Morse function and Riemannian metric. The basic theorem is that the resulting homology is an invariant of the manifold (that is, independent of the function and metric) and isomorphic to the singular homology of the manifold; this implies that the Morse and singular Betti numbers agree and gives an immediate proof of the Morse inequalities. An infinite dimensional analog of Morse homology in symplectic geometry is known as Floer homology.
== Morse–Bott theory ==
The notion of a Morse function can be generalized to consider functions that have nondegenerate manifolds of critical points. A Morse–Bott function is a smooth function on a manifold whose critical set is a closed submanifold and whose Hessian is non-degenerate in the normal direction. (Equivalently, the kernel of the Hessian at a critical point equals the tangent space to the critical submanifold.) A Morse function is the special case where the critical manifolds are zero-dimensional (so the Hessian at critical points is non-degenerate in every direction, that is, has no kernel).
The index is most naturally thought of as a pair
(
i
−
,
i
+
)
,
{\displaystyle \left(i_{-},i_{+}\right),}
where
i
−
{\displaystyle i_{-}}
is the dimension of the unstable manifold at a given point of the critical manifold, and
i
+
{\displaystyle i_{+}}
is equal to
i
−
{\displaystyle i_{-}}
plus the dimension of the critical manifold. If the Morse–Bott function is perturbed by a small function on the critical locus, the index of all critical points of the perturbed function on a critical manifold of the unperturbed function will lie between
i
−
{\displaystyle i_{-}}
and
i
+
.
{\displaystyle i_{+}.}
Morse–Bott functions are useful because generic Morse functions are difficult to work with; the functions one can visualize, and with which one can easily calculate, typically have symmetries. They often lead to positive-dimensional critical manifolds. Raoul Bott used Morse–Bott theory in his original proof of the Bott periodicity theorem.
Round functions are examples of Morse–Bott functions, where the critical sets are (disjoint unions of) circles.
Morse homology can also be formulated for Morse–Bott functions; the differential in Morse–Bott homology is computed by a spectral sequence. Frederic Bourgeois sketched an approach in the course of his work on a Morse–Bott version of symplectic field theory, but this work was never published due to substantial analytic difficulties.
== See also ==
== References ==
== Further reading ==
Bott, Raoul (1988). "Morse Theory Indomitable". Publications Mathématiques de l'IHÉS. 68: 99–114. doi:10.1007/bf02698544. S2CID 54005577.
Bott, Raoul (1982). "Lectures on Morse theory, old and new". Bulletin of the American Mathematical Society. (N.S.). 7 (2): 331–358. doi:10.1090/s0273-0979-1982-15038-8.
Cayley, Arthur (1859). "On Contour and Slope Lines" (PDF). The Philosophical Magazine. 18 (120): 264–268.
Guest, Martin (2001). "Morse Theory in the 1990s". arXiv:math/0104155.
Hirsch, M. (1994). Differential Topology (2nd ed.). Springer.
Kosinski, Antoni A. (19 October 2007). Differential Manifolds. Dover Book on Mathematics (Reprint of 1993 ed.). Mineola, New York: Dover Publications. ISBN 978-0-486-46244-8. OCLC 853621933.
Lang, Serge (1999). Fundamentals of Differential Geometry. Graduate Texts in Mathematics. Vol. 191. New York: Springer-Verlag. ISBN 978-0-387-98593-0. OCLC 39379395.
Matsumoto, Yukio (2002). An Introduction to Morse Theory.
Maxwell, James Clerk (1870). "On Hills and Dales" (PDF). The Philosophical Magazine. 40 (269): 421–427.
Milnor, John (1963). Morse Theory. Princeton University Press. ISBN 0-691-08008-9. {{cite book}}: ISBN / Date incompatibility (help) A classic advanced reference in mathematics and mathematical physics.
Milnor, John (1965). Lectures on the h-cobordism theorem (PDF).
Morse, Marston (1934). The Calculus of Variations in the Large. American Mathematical Society Colloquium Publication. Vol. 18. New York.{{cite book}}: CS1 maint: location missing publisher (link)
Schwarz, Matthias (1993). Morse Homology. Birkhäuser. ISBN 9780817629045. | Wikipedia/Morse_function |
In mathematics, specifically in surgery theory, the surgery obstructions define a map
θ
:
N
(
X
)
→
L
n
(
π
1
(
X
)
)
{\displaystyle \theta \colon {\mathcal {N}}(X)\to L_{n}(\pi _{1}(X))}
from the normal invariants to the L-groups which is in the first instance a set-theoretic map (that means not necessarily a homomorphism) with the following property when
n
≥
5
{\displaystyle n\geq 5}
:
A degree-one normal map
(
f
,
b
)
:
M
→
X
{\displaystyle (f,b)\colon M\to X}
is normally cobordant to a homotopy equivalence if and only if the image
θ
(
f
,
b
)
=
0
{\displaystyle \theta (f,b)=0}
in
L
n
(
Z
[
π
1
(
X
)
]
)
{\displaystyle L_{n}(\mathbb {Z} [\pi _{1}(X)])}
.
== Sketch of the definition ==
The surgery obstruction of a degree-one normal map has a relatively complicated definition.
Consider a degree-one normal map
(
f
,
b
)
:
M
→
X
{\displaystyle (f,b)\colon M\to X}
. The idea in deciding the question whether it is normally cobordant to a homotopy equivalence is to try to systematically improve
(
f
,
b
)
{\displaystyle (f,b)}
so that the map
f
{\displaystyle f}
becomes
m
{\displaystyle m}
-connected (that means the homotopy groups
π
∗
(
f
)
=
0
{\displaystyle \pi _{*}(f)=0}
for
∗
≤
m
{\displaystyle *\leq m}
) for high
m
{\displaystyle m}
. It is a consequence of Poincaré duality that if we can achieve this for
m
>
⌊
n
/
2
⌋
{\displaystyle m>\lfloor n/2\rfloor }
then the map
f
{\displaystyle f}
already is a homotopy equivalence. The word systematically above refers to the fact that one tries to do surgeries on
M
{\displaystyle M}
to kill elements of
π
i
(
f
)
{\displaystyle \pi _{i}(f)}
. In fact it is more convenient to use homology of the universal covers to observe how connected the map
f
{\displaystyle f}
is. More precisely, one works with the surgery kernels
K
i
(
M
~
)
:=
k
e
r
{
f
∗
:
H
i
(
M
~
)
→
H
i
(
X
~
)
}
{\displaystyle K_{i}({\tilde {M}}):=\mathrm {ker} \{f_{*}\colon H_{i}({\tilde {M}})\rightarrow H_{i}({\tilde {X}})\}}
which one views as
Z
[
π
1
(
X
)
]
{\displaystyle \mathbb {Z} [\pi _{1}(X)]}
-modules. If all these vanish, then the map
f
{\displaystyle f}
is a homotopy equivalence. As a consequence of Poincaré duality on
M
{\displaystyle M}
and
X
{\displaystyle X}
there is a
Z
[
π
1
(
X
)
]
{\displaystyle \mathbb {Z} [\pi _{1}(X)]}
-modules Poincaré duality
K
n
−
i
(
M
~
)
≅
K
i
(
M
~
)
{\displaystyle K^{n-i}({\tilde {M}})\cong K_{i}({\tilde {M}})}
, so one only has to watch half of them, that means those for which
i
≤
⌊
n
/
2
⌋
{\displaystyle i\leq \lfloor n/2\rfloor }
.
Any degree-one normal map can be made
⌊
n
/
2
⌋
{\displaystyle \lfloor n/2\rfloor }
-connected by the process called surgery below the middle dimension. This is the process of killing elements of
K
i
(
M
~
)
{\displaystyle K_{i}({\tilde {M}})}
for
i
<
⌊
n
/
2
⌋
{\displaystyle i<\lfloor n/2\rfloor }
described here when we have
p
+
q
=
n
{\displaystyle p+q=n}
such that
i
=
p
<
⌊
n
/
2
⌋
{\displaystyle i=p<\lfloor n/2\rfloor }
. After this is done there are two cases.
1. If
n
=
2
k
{\displaystyle n=2k}
then the only nontrivial homology group is the kernel
K
k
(
M
~
)
:=
k
e
r
{
f
∗
:
H
k
(
M
~
)
→
H
k
(
X
~
)
}
{\displaystyle K_{k}({\tilde {M}}):=\mathrm {ker} \{f_{*}\colon H_{k}({\tilde {M}})\rightarrow H_{k}({\tilde {X}})\}}
. It turns out that the cup-product pairings on
M
{\displaystyle M}
and
X
{\displaystyle X}
induce a cup-product pairing on
K
k
(
M
~
)
{\displaystyle K_{k}({\tilde {M}})}
. This defines a symmetric bilinear form in case
k
=
2
l
{\displaystyle k=2l}
and a skew-symmetric bilinear form in case
k
=
2
l
+
1
{\displaystyle k=2l+1}
. It turns out that these forms can be refined to
ε
{\displaystyle \varepsilon }
-quadratic forms, where
ε
=
(
−
1
)
k
{\displaystyle \varepsilon =(-1)^{k}}
. These
ε
{\displaystyle \varepsilon }
-quadratic forms define elements in the L-groups
L
n
(
π
1
(
X
)
)
{\displaystyle L_{n}(\pi _{1}(X))}
.
2. If
n
=
2
k
+
1
{\displaystyle n=2k+1}
the definition is more complicated. Instead of a quadratic form one obtains from the geometry a quadratic formation, which is a kind of automorphism of quadratic forms. Such a thing defines an element in the odd-dimensional L-group
L
n
(
π
1
(
X
)
)
{\displaystyle L_{n}(\pi _{1}(X))}
.
If the element
θ
(
f
,
b
)
{\displaystyle \theta (f,b)}
is zero in the L-group surgery can be done on
M
{\displaystyle M}
to modify
f
{\displaystyle f}
to a homotopy equivalence.
Geometrically the reason why this is not always possible is that performing surgery in the middle dimension to kill an element in
K
k
(
M
~
)
{\displaystyle K_{k}({\tilde {M}})}
possibly creates an element in
K
k
−
1
(
M
~
)
{\displaystyle K_{k-1}({\tilde {M}})}
when
n
=
2
k
{\displaystyle n=2k}
or in
K
k
(
M
~
)
{\displaystyle K_{k}({\tilde {M}})}
when
n
=
2
k
+
1
{\displaystyle n=2k+1}
. So this possibly destroys what has already been achieved. However, if
θ
(
f
,
b
)
{\displaystyle \theta (f,b)}
is zero, surgeries can be arranged in such a way that this does not happen.
== Example ==
In the simply connected case the following happens.
If
n
=
2
k
+
1
{\displaystyle n=2k+1}
there is no obstruction.
If
n
=
4
l
{\displaystyle n=4l}
then the surgery obstruction can be calculated as the difference of the signatures of M and X.
If
n
=
4
l
+
2
{\displaystyle n=4l+2}
then the surgery obstruction is the Arf-invariant of the associated kernel quadratic form over
Z
2
{\displaystyle \mathbb {Z} _{2}}
.
== References ==
Browder, William (1972), Surgery on simply-connected manifolds, Berlin, New York: Springer-Verlag, MR 0358813
Lück, Wolfgang (2002), A basic introduction to surgery theory (PDF), ICTP Lecture Notes Series 9, Band 1, of the school "High-dimensional manifold theory" in Trieste, May/June 2001, Abdus Salam International Centre for Theoretical Physics, Trieste 1-224
Ranicki, Andrew (2002), Algebraic and Geometric Surgery, Oxford Mathematical Monographs, Clarendon Press, ISBN 978-0-19-850924-0, MR 2061749
Wall, C. T. C. (1999), Surgery on compact manifolds, Mathematical Surveys and Monographs, vol. 69 (2nd ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0942-6, MR 1687388 | Wikipedia/Surgery_obstruction |
In single-variable differential calculus, the fundamental increment lemma is an immediate consequence of the definition of the derivative
f
′
(
a
)
{\textstyle f'(a)}
of a function
f
{\textstyle f}
at a point
a
{\textstyle a}
:
f
′
(
a
)
=
lim
h
→
0
f
(
a
+
h
)
−
f
(
a
)
h
.
{\displaystyle f'(a)=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}.}
The lemma asserts that the existence of this derivative implies the existence of a function
φ
{\displaystyle \varphi }
such that
lim
h
→
0
φ
(
h
)
=
0
and
f
(
a
+
h
)
=
f
(
a
)
+
f
′
(
a
)
h
+
φ
(
h
)
h
{\displaystyle \lim _{h\to 0}\varphi (h)=0\qquad {\text{and}}\qquad f(a+h)=f(a)+f'(a)h+\varphi (h)h}
for sufficiently small but non-zero
h
{\textstyle h}
. For a proof, it suffices to define
φ
(
h
)
=
f
(
a
+
h
)
−
f
(
a
)
h
−
f
′
(
a
)
{\displaystyle \varphi (h)={\frac {f(a+h)-f(a)}{h}}-f'(a)}
and verify this
φ
{\displaystyle \varphi }
meets the requirements.
The lemma says, at least when
h
{\displaystyle h}
is sufficiently close to zero, that the difference quotient
f
(
a
+
h
)
−
f
(
a
)
h
{\displaystyle {\frac {f(a+h)-f(a)}{h}}}
can be written as the derivative f' plus an error term
φ
(
h
)
{\displaystyle \varphi (h)}
that vanishes at
h
=
0
{\displaystyle h=0}
.
That is, one has
f
(
a
+
h
)
−
f
(
a
)
h
=
f
′
(
a
)
+
φ
(
h
)
.
{\displaystyle {\frac {f(a+h)-f(a)}{h}}=f'(a)+\varphi (h).}
== Differentiability in higher dimensions ==
In that the existence of
φ
{\displaystyle \varphi }
uniquely characterises the number
f
′
(
a
)
{\displaystyle f'(a)}
, the fundamental increment lemma can be said to characterise the differentiability of single-variable functions. For this reason, a generalisation of the lemma can be used in the definition of differentiability in multivariable calculus. In particular, suppose f maps some subset of
R
n
{\displaystyle \mathbb {R} ^{n}}
to
R
{\displaystyle \mathbb {R} }
. Then f is said to be differentiable at a if there is a linear function
M
:
R
n
→
R
{\displaystyle M:\mathbb {R} ^{n}\to \mathbb {R} }
and a function
Φ
:
D
→
R
,
D
⊆
R
n
∖
{
0
}
,
{\displaystyle \Phi :D\to \mathbb {R} ,\qquad D\subseteq \mathbb {R} ^{n}\smallsetminus \{\mathbf {0} \},}
such that
lim
h
→
0
Φ
(
h
)
=
0
and
f
(
a
+
h
)
−
f
(
a
)
=
M
(
h
)
+
Φ
(
h
)
⋅
‖
h
‖
{\displaystyle \lim _{\mathbf {h} \to 0}\Phi (\mathbf {h} )=0\qquad {\text{and}}\qquad f(\mathbf {a} +\mathbf {h} )-f(\mathbf {a} )=M(\mathbf {h} )+\Phi (\mathbf {h} )\cdot \Vert \mathbf {h} \Vert }
for non-zero h sufficiently close to 0. In this case, M is the unique derivative (or total derivative, to distinguish from the directional and partial derivatives) of f at a. Notably, M is given by the Jacobian matrix of f evaluated at a.
We can write the above equation in terms of the partial derivatives
∂
f
∂
x
i
{\displaystyle {\frac {\partial f}{\partial x_{i}}}}
as
f
(
a
+
h
)
−
f
(
a
)
=
∑
i
=
1
n
∂
f
(
a
)
∂
x
i
+
Φ
(
h
)
⋅
‖
h
‖
{\displaystyle f(\mathbf {a} +\mathbf {h} )-f(\mathbf {a} )=\displaystyle \sum _{i=1}^{n}{\frac {\partial f(a)}{\partial x_{i}}}+\Phi (\mathbf {h} )\cdot \Vert \mathbf {h} \Vert }
== See also ==
Generalizations of the derivative
== References ==
Talman, Louis (2007-09-12). "Differentiability for Multivariable Functions" (PDF). Archived from the original (PDF) on 2010-06-20. Retrieved 2012-06-28.
Stewart, James (2008). Calculus (7th ed.). Cengage Learning. p. 942. ISBN 978-0538498845.
Folland, Gerald. "Derivatives and Linear Approximation" (PDF). | Wikipedia/Fundamental_increment_lemma |
A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, and research. Theories can be scientific, falling within the realm of empirical and testable knowledge, or they may belong to non-scientific disciplines, such as philosophy, art, or sociology. In some cases, theories may exist independently of any formal discipline.
In modern science, the term "theory" refers to scientific theories, a well-confirmed type of explanation of nature, made in a way consistent with the scientific method, and fulfilling the criteria required by modern science. Such theories are described in such a way that scientific tests should be able to provide empirical support for it, or empirical contradiction ("falsify") of it. Scientific theories are the most reliable, rigorous, and comprehensive form of scientific knowledge, in contrast to more common uses of the word "theory" that imply that something is unproven or speculative (which in formal terms is better characterized by the word hypothesis). Scientific theories are distinguished from hypotheses, which are individual empirically testable conjectures, and from scientific laws, which are descriptive accounts of the way nature behaves under certain conditions.
Theories guide the enterprise of finding facts rather than of reaching goals, and are neutral concerning alternatives among values.: 131 A theory can be a body of knowledge, which may or may not be associated with particular explanatory models. To theorize is to develop this body of knowledge.: 46
The word theory or "in theory" is sometimes used outside of science to refer to something which the speaker did not experience or test before. In science, this same concept is referred to as a hypothesis, and the word "hypothetically" is used both inside and outside of science. In its usage outside of science, the word "theory" is very often contrasted to "practice" (from Greek praxis, πρᾶξις) a Greek term for doing, which is opposed to theory. A "classical example" of the distinction between "theoretical" and "practical" uses the discipline of medicine: medical theory involves trying to understand the causes and nature of health and sickness, while the practical side of medicine is trying to make people healthy. These two things are related but can be independent, because it is possible to research health and sickness without curing specific patients, and it is possible to cure a patient without knowing how the cure worked.
== Ancient usage ==
The English word theory derives from a technical term in philosophy in Ancient Greek. As an everyday word, theoria, θεωρία, meant "looking at, viewing, beholding", but in more technical contexts it came to refer to contemplative or speculative understandings of natural things, such as those of natural philosophers, as opposed to more practical ways of knowing things, like that of skilled orators or artisans. English-speakers have used the word theory since at least the late 16th century. Modern uses of the word theory derive from the original definition, but have taken on new shades of meaning, still based on the idea of a theory as a thoughtful and rational explanation of the general nature of things.
Although it has more mundane meanings in Greek, the word θεωρία apparently developed special uses early in the recorded history of the Greek language. In the book From Religion to Philosophy, Francis Cornford suggests that the Orphics used the word theoria to mean "passionate sympathetic contemplation". Pythagoras changed the word to mean "the passionless contemplation of rational, unchanging truth" of mathematical knowledge, because he considered this intellectual pursuit the way to reach the highest plane of existence. Pythagoras emphasized subduing emotions and bodily desires to help the intellect function at the higher plane of theory. Thus, it was Pythagoras who gave the word theory the specific meaning that led to the classical and modern concept of a distinction between theory (as uninvolved, neutral thinking) and practice.
Aristotle's terminology, as already mentioned, contrasts theory with praxis or practice, and this contrast exists till today. For Aristotle, both practice and theory involve thinking, but the aims are different. Theoretical contemplation considers things humans do not move or change, such as nature, so it has no human aim apart from itself and the knowledge it helps create. On the other hand, praxis involves thinking, but always with an aim to desired actions, whereby humans cause change or movement themselves for their own ends. Any human movement that involves no conscious choice and thinking could not be an example of praxis or doing.
== Formality ==
Theories are analytical tools for understanding, explaining, and making predictions about a given subject matter. There are theories in many and varied fields of study, including the arts and sciences. A formal theory is syntactic in nature and is only meaningful when given a semantic component by applying it to some content (e.g., facts and relationships of the actual historical world as it is unfolding). Theories in various fields of study are often expressed in natural language, but can be constructed in such a way that their general form is identical to a theory as it is expressed in the formal language of mathematical logic. Theories may be expressed mathematically, symbolically, or in common language, but are generally expected to follow principles of rational thought or logic.
Theory is constructed of a set of sentences that are thought to be true statements about the subject under consideration. However, the truth of any one of these statements is always relative to the whole theory. Therefore, the same statement may be true with respect to one theory, and not true with respect to another. This is, in ordinary language, where statements such as "He is a terrible person" cannot be judged as true or false without reference to some interpretation of who "He" is and for that matter what a "terrible person" is under the theory.
Sometimes two theories have exactly the same explanatory power because they make the same predictions. A pair of such theories is called indistinguishable or observationally equivalent, and the choice between them reduces to convenience or philosophical preference.
The form of theories is studied formally in mathematical logic, especially in model theory. When theories are studied in mathematics, they are usually expressed in some formal language and their statements are closed under application of certain procedures called rules of inference. A special case of this, an axiomatic theory, consists of axioms (or axiom schemata) and rules of inference. A theorem is a statement that can be derived from those axioms by application of these rules of inference. Theories used in applications are abstractions of observed phenomena and the resulting theorems provide solutions to real-world problems. Obvious examples include arithmetic (abstracting concepts of number), geometry (concepts of space), and probability (concepts of randomness and likelihood).
Gödel's incompleteness theorem shows that no consistent, recursively enumerable theory (that is, one whose theorems form a recursively enumerable set) in which the concept of natural numbers can be expressed, can include all true statements about them. As a result, some domains of knowledge cannot be formalized, accurately and completely, as mathematical theories. (Here, formalizing accurately and completely means that all true propositions—and only true propositions—are derivable within the mathematical system.) This limitation, however, in no way precludes the construction of mathematical theories that formalize large bodies of scientific knowledge.
=== Underdetermination ===
A theory is underdetermined (also called indeterminacy of data to theory) if a rival, inconsistent theory is at least as consistent with the evidence. Underdetermination is an epistemological issue about the relation of evidence to conclusions.
A theory that lacks supporting evidence is generally, more properly, referred to as a hypothesis.
=== Intertheoretic reduction and elimination ===
If a new theory better explains and predicts a phenomenon than an old theory (i.e., it has more explanatory power), we are justified in believing that the newer theory describes reality more correctly. This is called an intertheoretic reduction because the terms of the old theory can be reduced to the terms of the new one. For instance, our historical understanding about sound, light and heat have been reduced to wave compressions and rarefactions, electromagnetic waves, and molecular kinetic energy, respectively. These terms, which are identified with each other, are called intertheoretic identities. When an old and new theory are parallel in this way, we can conclude that the new one describes the same reality, only more completely.
When a new theory uses new terms that do not reduce to terms of an older theory, but rather replace them because they misrepresent reality, it is called an intertheoretic elimination. For instance, the obsolete scientific theory that put forward an understanding of heat transfer in terms of the movement of caloric fluid was eliminated when a theory of heat as energy replaced it. Also, the theory that phlogiston is a substance released from burning and rusting material was eliminated with the new understanding of the reactivity of oxygen.
=== Versus theorems ===
Theories are distinct from theorems. A theorem is derived deductively from axioms (basic assumptions) according to a formal system of rules, sometimes as an end in itself and sometimes as a first step toward being tested or applied in a concrete situation; theorems are said to be true in the sense that the conclusions of a theorem are logical consequences of the axioms. Theories are abstract and conceptual, and are supported or challenged by observations in the world. They are 'rigorously tentative', meaning that they are proposed as true and expected to satisfy careful examination to account for the possibility of faulty inference or incorrect observation. Sometimes theories are incorrect, meaning that an explicit set of observations contradicts some fundamental objection or application of the theory, but more often theories are corrected to conform to new observations, by restricting the class of phenomena the theory applies to or changing the assertions made. An example of the former is the restriction of classical mechanics to phenomena involving macroscopic length scales and particle speeds much lower than the speed of light.
== Theory–practice relationship ==
Theory is often distinguished from practice or praxis. The question of whether theoretical models of work are relevant to work itself is of interest to scholars of professions such as medicine, engineering, law, and management.: 802
The gap between theory and practice has been framed as a knowledge transfer where there is a task of translating research knowledge to be application in practice, and ensuring that practitioners are made aware of it. Academics have been criticized for not attempting to transfer the knowledge they produce to practitioners.: 804 Another framing supposes that theory and knowledge seek to understand different problems and model the world in different words (using different ontologies and epistemologies). Another framing says that research does not produce theory that is relevant to practice.: 803
In the context of management, Van de Van and Johnson propose a form of engaged scholarship where scholars examine problems that occur in practice, in an interdisciplinary fashion, producing results that create both new practical results as well as new theoretical models, but targeting theoretical results shared in an academic fashion.: 815 They use a metaphor of "arbitrage" of ideas between disciplines, distinguishing it from collaboration.: 803
== Scientific ==
In science, the term "theory" refers to "a well-substantiated explanation of some aspect of the natural world, based on a body of facts that have been repeatedly confirmed through observation and experiment." Theories must also meet further requirements, such as the ability to make falsifiable predictions with consistent accuracy across a broad area of scientific inquiry, and production of strong evidence in favor of the theory from multiple independent sources (consilience).
The strength of a scientific theory is related to the diversity of phenomena it can explain, which is measured by its ability to make falsifiable predictions with respect to those phenomena. Theories are improved (or replaced by better theories) as more evidence is gathered, so that accuracy in prediction improves over time; this increased accuracy corresponds to an increase in scientific knowledge. Scientists use theories as a foundation to gain further scientific knowledge, as well as to accomplish goals such as inventing technology or curing diseases.
=== Definitions from scientific organizations ===
The United States National Academy of Sciences defines scientific theories as follows:The formal scientific definition of "theory" is quite different from the everyday meaning of the word. It refers to a comprehensive explanation of some aspect of nature that is supported by a vast body of evidence. Many scientific theories are so well established that no new evidence is likely to alter them substantially. For example, no new evidence will demonstrate that the Earth does not orbit around the sun (heliocentric theory), or that living things are not made of cells (cell theory), that matter is not composed of atoms, or that the surface of the Earth is not divided into solid plates that have moved over geological timescales (the theory of plate tectonics) ... One of the most useful properties of scientific theories is that they can be used to make predictions about natural events or phenomena that have not yet been observed.
From the American Association for the Advancement of Science:
A scientific theory is a well-substantiated explanation of some aspect of the natural world, based on a body of facts that have been repeatedly confirmed through observation and experiment. Such fact-supported theories are not "guesses" but reliable accounts of the real world. The theory of biological evolution is more than "just a theory." It is as factual an explanation of the universe as the atomic theory of matter or the germ theory of disease. Our understanding of gravity is still a work in progress. But the phenomenon of gravity, like evolution, is an accepted fact.
The term theory is not appropriate for describing scientific models or untested, but intricate hypotheses.
=== Philosophical views ===
The logical positivists thought of scientific theories as deductive theories—that a theory's content is based on some formal system of logic and on basic axioms. In a deductive theory, any sentence which is a logical consequence of one or more of the axioms is also a sentence of that theory. This is called the received view of theories.
In the semantic view of theories, which has largely replaced the received view, theories are viewed as scientific models. A model is an abstract and informative representation of reality (a "model of reality"), similar to the way that a map is a graphical model that represents the territory of a city or country. In this approach, theories are a specific category of models that fulfill the necessary criteria. (See Theories as models for further discussion.)
=== In physics ===
In physics the term theory is generally used for a mathematical framework—derived from a small set of basic postulates (usually symmetries, like equality of locations in space or in time, or identity of electrons, etc.)—which is capable of producing experimental predictions for a given category of physical systems. One good example is classical electromagnetism, which encompasses results derived from gauge symmetry (sometimes called gauge invariance) in a form of a few equations called Maxwell's equations. The specific mathematical aspects of classical electromagnetic theory are termed "laws of electromagnetism", reflecting the level of consistent and reproducible evidence that supports them. Within electromagnetic theory generally, there are numerous hypotheses about how electromagnetism applies to specific situations. Many of these hypotheses are already considered adequately tested, with new ones always in the making and perhaps untested.
=== Regarding the term "theoretical" ===
Certain tests may be infeasible or technically difficult. As a result, theories may make predictions that have not been confirmed or proven incorrect. These predictions may be described informally as "theoretical". They can be tested later, and if they are incorrect, this may lead to revision, invalidation, or rejection of the theory.
== Mathematical ==
In mathematics, the term theory is used differently than its use in science ─ necessarily so, since mathematics contains no explanations of natural phenomena per se, even though it may help provide insight into natural systems or be inspired by them. In the general sense, a mathematical theory is a branch of mathematics devoted to some specific topics or methods, such as set theory, number theory, group theory, probability theory, game theory, control theory, perturbation theory, etc., such as might be appropriate for a single textbook.
In mathematical logic, a theory has a related but different sense: it is the collection of the theorems that can be deduced from a given set of axioms, given a given set of inference rules.
== Philosophical ==
A theory can be either descriptive as in science, or prescriptive (normative) as in philosophy. The latter are those whose subject matter consists not of empirical data, but rather of ideas. At least some of the elementary theorems of a philosophical theory are statements whose truth cannot necessarily be scientifically tested through empirical observation.
A field of study is sometimes named a "theory" because its basis is some initial set of assumptions describing the field's approach to the subject. These assumptions are the elementary theorems of the particular theory, and can be thought of as the axioms of that field. Some commonly known examples include set theory and number theory; however literary theory, critical theory, and music theory are also of the same form.
=== Metatheory ===
One form of philosophical theory is a metatheory or meta-theory. A metatheory is a theory whose subject matter is some other theory or set of theories. In other words, it is a theory about theories. Statements made in the metatheory about the theory are called metatheorems.
== Political ==
A political theory is an ethical theory about the law and government. Often the term "political theory" refers to a general view, or specific ethic, political belief or attitude, thought about politics.
== Jurisprudential ==
In social science, jurisprudence is the philosophical theory of law. Contemporary philosophy of law addresses problems internal to law and legal systems, and problems of law as a particular social institution.
== Examples ==
Most of the following are scientific theories. Some are not, but rather encompass a body of knowledge or art, such as Music theory and Visual Arts Theories.
Anthropology:
Carneiro's circumscription theory
Astronomy:
Alpher–Bethe–Gamow theory —
B2FH Theory —
Copernican theory —
Newton's theory of gravitation —
Hubble's law —
Kepler's laws of planetary motion Ptolemaic theory
Biology:
Cell theory —
Chemiosmotic theory —
Evolution —
Germ theory —
Symbiogenesis
Chemistry:
Molecular theory —
Kinetic theory of gases —
Molecular orbital theory —
Valence bond theory —
Transition state theory —
RRKM theory —
Chemical graph theory —
Flory–Huggins solution theory —
Marcus theory —
Lewis theory (successor to Brønsted–Lowry acid–base theory) —
HSAB theory —
Debye–Hückel theory —
Thermodynamic theory of polymer elasticity —
Reptation theory —
Polymer field theory —
Møller–Plesset perturbation theory —
density functional theory —
Frontier molecular orbital theory —
Polyhedral skeletal electron pair theory —
Baeyer strain theory —
Quantum theory of atoms in molecules —
Collision theory —
Ligand field theory (successor to Crystal field theory) —
Variational transition-state theory —
Benson group increment theory —
Specific ion interaction theory
Climatology:
Climate change theory (general study of climate changes)
anthropogenic climate change (ACC)/
anthropogenic global warming (AGW) theories (due to human activity)
Computer Science:
Automata theory —
Queueing theory
Cosmology:
Big Bang Theory —
Cosmic inflation —
Loop quantum gravity —
Superstring theory —
Supergravity —
Supersymmetric theory —
Multiverse theory —
Holographic principle —
Quantum gravity —
M-theory
Economics:
Macroeconomic theory —
Microeconomic theory —
Law of Supply and demand
Education:
Constructivist theory —
Critical pedagogy theory —
Education theory —
Multiple intelligence theory —
Progressive education theory
Engineering:
Circuit theory —
Control theory —
Signal theory —
Systems theory —
Information theory
Film:
Film theory
Geology:
Plate tectonics
Humanities:
Critical theory
Jurisprudence or 'Legal theory':
Natural law —
Legal positivism —
Legal realism —
Critical legal studies
Law: see Jurisprudence; also Case theory
Linguistics:
X-bar theory —
Government and Binding —
Principles and parameters —
Universal grammar
Literature:
Literary theory
Mathematics:
Approximation theory —
Arakelov theory —
Asymptotic theory —
Bifurcation theory —
Catastrophe theory —
Category theory —
Chaos theory —
Choquet theory —
Coding theory —
Combinatorial game theory —
Computability theory —
Computational complexity theory —
Deformation theory —
Dimension theory —
Ergodic theory —
Field theory —
Galois theory —
Game theory —
Gauge theory —
Graph theory —
Group theory —
Hodge theory —
Homology theory —
Homotopy theory —
Ideal theory —
Intersection theory —
Invariant theory —
Iwasawa theory —
K-theory —
KK-theory —
Knot theory —
L-theory —
Lie theory —
Littlewood–Paley theory —
Matrix theory —
Measure theory —
Model theory —
Module theory —
Morse theory —
Nevanlinna theory —
Number theory —
Obstruction theory —
Operator theory —
Order theory —
PCF theory —
Perturbation theory —
Potential theory —
Probability theory —
Ramsey theory —
Rational choice theory —
Representation theory —
Ring theory —
Set theory —
Shape theory —
Small cancellation theory —
Spectral theory —
Stability theory —
Stable theory —
Sturm–Liouville theory —
Surgery theory —
Twistor theory —
Yang–Mills theory
Music:
Music theory
Philosophy:
Proof theory —
Speculative reason —
Theory of truth —
Type theory —
Value theory —
Virtue theory
Physics:
Acoustic theory —
Antenna theory —
Atomic theory —
BCS theory —
Conformal field theory —
Dirac hole theory —
Dynamo theory —
Landau theory —
M-theory —
Perturbation theory —
Theory of relativity (successor to classical mechanics) —
Gauge theory —
Quantum field theory —
Scattering theory —
String theory —
Quantum information theory
Psychology:
Cognitive dissonance theory —
Attachment theory —
Object permanence —
Poverty of stimulus —
Attribution theory —
Self-fulfilling prophecy —
Stockholm syndrome
Public Budgeting:
Incrementalism —
Zero-based budgeting
Public Administration:
Organizational theory
Semiotics:
Intertheoricity –
Transferogenesis
Sociology:
Critical theory —
Engaged theory —
Social theory —
Sociological theory –
Social capital theory
Statistics:
Extreme value theory
Theatre:
Performance theory
Visual Arts:
Aesthetics —
Art educational theory —
Architecture —
Composition —
Anatomy —
Color theory —
Perspective —
Visual perception —
Geometry —
Manifolds
Other:
Obsolete scientific theories
== See also ==
== Notes ==
== References ==
=== Citations ===
=== Sources ===
== Further reading ==
Eisenhardt, K. M., & Graebner, M. E. (2007). Theory building from cases: Opportunities and challenges. Academy of management journal, 50(1), 25-32.
== External links ==
"How science works: Even theories change", Understanding Science by the University of California Museum of Paleontology.
What is a Theory? | Wikipedia/Mathematical_theory |
William Thurston's elliptization conjecture states that a closed 3-manifold with finite fundamental group is spherical, i.e. has a Riemannian metric of constant positive sectional curvature.
== Relation to other conjectures ==
A 3-manifold with a Riemannian metric of constant positive sectional curvature is covered by the 3-sphere, moreover the group of covering transformations are isometries of the 3-sphere.
If the original 3-manifold had in fact a trivial fundamental group, then it is homeomorphic to the 3-sphere (via the covering map). Thus, proving the elliptization conjecture would prove the Poincaré conjecture as a corollary. In fact, the elliptization conjecture is logically equivalent to two simpler conjectures: the Poincaré conjecture and the spherical space form conjecture.
The elliptization conjecture is a special case of Thurston's geometrization conjecture, which was proved in 2003 by G. Perelman.
== References ==
For the proof of the conjectures, see the references in the articles on geometrization conjecture or Poincaré conjecture.
William Thurston. Three-dimensional geometry and topology. Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. x+311 pp. ISBN 0-691-08304-5.
William Thurston. The Geometry and Topology of Three-Manifolds, 1980 Princeton lecture notes on geometric structures on 3-manifolds, that states his elliptization conjecture near the beginning of section 3. | Wikipedia/Elliptization_conjecture |
In mathematics, a Kleinian model is a model of a three-dimensional hyperbolic manifold N by the quotient space
H
3
/
Γ
{\displaystyle \mathbb {H} ^{3}/\Gamma }
where
Γ
{\displaystyle \Gamma }
is a discrete subgroup of PSL(2,C). Here, the subgroup
Γ
{\displaystyle \Gamma }
, a Kleinian group, is defined so that it is isomorphic to the fundamental group
π
1
(
N
)
{\displaystyle \pi _{1}(N)}
of the surface N. Many authors use the terms Kleinian group and Kleinian model interchangeably, letting one stand for the other. The concept is named after Felix Klein.
In less technical terms, a Kleinian model it is a way of assigning coordinates to a hyperbolic manifold, or a three-dimensional space in which every point locally resembles hyperbolic space. A Kleinian model is created by taking three-dimensional hyperbolic space and treating two points as equivalent if and only if they can be reached from each other by applying a member of a group action of a Kleinian group on the space. A Kleinian group is any discrete subgroup, consisting only of isolated points, of orientation-preserving isometries of hyperbolic 3-space. The group action of a group is a set of functions on a set which, roughly speaking, have the same structure as a group.
Many properties of Kleinian models are in direct analogy to those of Fuchsian models; however, overall, the theory is less well developed. A number of unsolved conjectures on Kleinian models are the analogs to theorems on Fuchsian models.
== See also ==
Hyperbolic 3-manifold
== References ==
=== Sources ===
Matsuzaki, Katsuhiro; Taniguchi, Masahiko (1998). Hyperbolic Manifolds and Kleinian Groups. Clarendon Press. ISBN 0-19-850062-9.
Elstrodt, Jürgen; Grunewald, Fritz; Mennicke, Jens (1997). Groups Acting on Hyperbolic Space. Springer. ISBN 3-540-62745-6. | Wikipedia/Kleinian_model |
In mathematics, the braid group on n strands (denoted
B
n
{\displaystyle B_{n}}
), also known as the Artin braid group, is the group whose elements are equivalence classes of n-braids (e.g. under ambient isotopy), and whose group operation is composition of braids (see § Introduction). Example applications of braid groups include knot theory, where any knot may be represented as the closure of certain braids (a result known as Alexander's theorem); in mathematical physics where Artin's canonical presentation of the braid group corresponds to the Yang–Baxter equation (see § Basic properties); and in monodromy invariants of algebraic geometry.
== Introduction ==
In this introduction let n = 4; the generalization to other values of n will be straightforward. Consider two sets of four items lying on a table, with the items in each set being arranged in a vertical line, and such that one set sits next to the other. (In the illustrations below, these are the black dots.) Using four strands, each item of the first set is connected with an item of the second set so that a one-to-one correspondence results. Such a connection is called a braid. Often some strands will have to pass over or under others, and this is crucial: the following two connections are different braids:
On the other hand, two such connections which can be made to look the same by "pulling the strands" are considered the same braid:
All strands are required to move from left to right; knots like the following are not considered braids:
Any two braids can be composed by drawing the first next to the second, identifying the four items in the middle, and connecting corresponding strands:
Another example:
The composition of the braids σ and τ is written as στ.
The set of all braids on four strands is denoted by
B
4
{\displaystyle B_{4}}
. The above composition of braids is indeed a group operation. The identity element is the braid consisting of four parallel horizontal strands, and the inverse of a braid consists of that braid which "undoes" whatever the first braid did, which is obtained by flipping a diagram such as the ones above across a vertical line going through its centre. (The first two example braids above are inverses of each other.)
== Applications ==
Braid theory has recently been applied to fluid mechanics, specifically to the field of chaotic mixing in fluid flows. The braiding of (2 + 1)-dimensional space-time trajectories formed by motion of physical rods, periodic orbits or "ghost rods", and almost-invariant sets has been used to estimate the topological entropy of several engineered and naturally occurring fluid systems, via the use of Nielsen–Thurston classification.
Another field of intense investigation involving braid groups and related topological concepts in the context of quantum physics is in the theory and (conjectured) experimental implementation of the proposed particles anyons. These may well end up forming the basis for error-corrected quantum computing and so their abstract study is currently of fundamental importance in quantum information.
== Formal treatment ==
To put the above informal discussion of braid groups on firm ground, one needs to use the homotopy concept of algebraic topology, defining braid groups as fundamental groups of a configuration space. Alternatively, one can define the braid group purely algebraically via the braid relations, keeping the pictures in mind only to guide the intuition.
To explain how to reduce a braid group in the sense of Artin to a fundamental group, we consider a connected manifold
X
{\displaystyle X}
of dimension at least 2. The symmetric product of
n
{\displaystyle n}
copies of
X
{\displaystyle X}
means the quotient of
X
n
{\displaystyle X^{n}}
, the
n
{\displaystyle n}
-fold Cartesian product of
X
{\displaystyle X}
by the permutation action of the symmetric group on
n
{\displaystyle n}
strands operating on the indices of coordinates. That is, an ordered
n
{\displaystyle n}
-tuple is in the same orbit as any other that is a re-ordered version of it.
A path in the
n
{\displaystyle n}
-fold symmetric product is the abstract way of discussing
n
{\displaystyle n}
points of
X
{\displaystyle X}
, considered as an unordered
n
{\displaystyle n}
-tuple, independently tracing out
n
{\displaystyle n}
strings. Since we must require that the strings never pass through each other, it is necessary that we pass to the subspace
Y
{\displaystyle Y}
of the symmetric product, of orbits of
n
{\displaystyle n}
-tuples of distinct points. That is, we remove all the subspaces of
X
n
{\displaystyle X^{n}}
defined by conditions
x
i
=
x
j
{\displaystyle x_{i}=x_{j}}
for all
1
≤
i
<
j
≤
n
{\displaystyle 1\leq i<j\leq n}
. This is invariant under the symmetric group, and
Y
{\displaystyle Y}
is the quotient by the symmetric group of the non-excluded
n
{\displaystyle n}
-tuples. Under the dimension condition
Y
{\displaystyle Y}
will be connected.
With this definition, then, we can call the braid group of
X
{\displaystyle X}
with
n
{\displaystyle n}
strings the fundamental group of
Y
{\displaystyle Y}
(for any choice of base point – this is well-defined up to isomorphism). The case where
X
{\displaystyle X}
is the Euclidean plane is the original one of Artin. In some cases it can be shown that the higher homotopy groups of
Y
{\displaystyle Y}
are trivial.
=== Closed braids ===
When X is the plane, the braid can be closed, i.e., corresponding ends can be connected in pairs, to form a link, i.e., a possibly intertwined union of possibly knotted loops in three dimensions. The number of components of the link can be anything from 1 to n, depending on the permutation of strands determined by the link. A theorem of J. W. Alexander demonstrates that every link can be obtained in this way as the "closure" of a braid. Compare with string links.
Different braids can give rise to the same link, just as different crossing diagrams can give rise to the same knot. In 1935, Andrey Markov Jr. described two moves on braid diagrams that yield equivalence in the corresponding closed braids. A single-move version of Markov's theorem, was published by in 1997.
Vaughan Jones originally defined his polynomial as a braid invariant and then showed that it depended only on the class of the closed braid.
The Markov theorem gives necessary and sufficient conditions under which the closures of two braids are equivalent links.
=== Braid index ===
The "braid index" is the least number of strings needed to make a closed braid representation of a link. It is equal to the least number of Seifert circles in any projection of a knot.
== History ==
Braid groups were introduced explicitly by Emil Artin in 1925, although (as Wilhelm Magnus pointed out in 1974) they were already implicit in Adolf Hurwitz's work on monodromy from 1891.
Braid groups may be described by explicit presentations, as was shown by Emil Artin in 1947. Braid groups are also understood by a deeper mathematical interpretation: as the fundamental group of certain configuration spaces.
As Magnus says, Hurwitz gave the interpretation of a braid group as the fundamental group of a configuration space (cf. braid theory), an interpretation that was lost from view until it was rediscovered by Ralph Fox and Lee Neuwirth in 1962.
== Basic properties ==
=== Generators and relations ===
Consider the following three braids:
Every braid in
B
4
{\displaystyle B_{4}}
can be written as a composition of a number of these braids and their inverses. In other words, these three braids generate the group
B
4
{\displaystyle B_{4}}
. To see this, an arbitrary braid is scanned from left to right for crossings. Numbering the strands beginning at the top, whenever a crossing of strands
i
{\displaystyle i}
and
i
+
1
{\displaystyle i+1}
is encountered,
σ
i
{\displaystyle \sigma _{i}}
or
σ
i
−
1
{\displaystyle \sigma _{i}^{-1}}
is written down, depending on whether strand
i
{\displaystyle i}
moves over or under strand
i
+
1
{\displaystyle i+1}
. Upon reaching the right end, the braid has been written as a product of the
σ
i
{\displaystyle \sigma _{i}}
and their inverses.
It is clear that
(i)
σ
1
σ
3
=
σ
3
σ
1
{\displaystyle \sigma _{1}\sigma _{3}=\sigma _{3}\sigma _{1}}
,
while the following two relations are not quite as obvious:
(iia)
σ
1
σ
2
σ
1
=
σ
2
σ
1
σ
2
{\displaystyle \sigma _{1}\sigma _{2}\sigma _{1}=\sigma _{2}\sigma _{1}\sigma _{2}}
,
(iib)
σ
2
σ
3
σ
2
=
σ
3
σ
2
σ
3
{\displaystyle \sigma _{2}\sigma _{3}\sigma _{2}=\sigma _{3}\sigma _{2}\sigma _{3}}
(these relations can be appreciated best by drawing the braid on a piece of paper). It can be shown that all other relations among the braids
σ
1
{\displaystyle \sigma _{1}}
,
σ
2
{\displaystyle \sigma _{2}}
and
σ
3
{\displaystyle \sigma _{3}}
already follow from these relations and the group axioms.
Generalising this example to
n
{\displaystyle n}
strands, the group
B
n
{\displaystyle B_{n}}
can be abstractly defined via the following presentation:
B
n
=
⟨
σ
1
,
…
,
σ
n
−
1
∣
σ
i
σ
i
+
1
σ
i
=
σ
i
+
1
σ
i
σ
i
+
1
,
σ
i
σ
j
=
σ
j
σ
i
⟩
,
{\displaystyle B_{n}=\left\langle \sigma _{1},\ldots ,\sigma _{n-1}\mid \sigma _{i}\sigma _{i+1}\sigma _{i}=\sigma _{i+1}\sigma _{i}\sigma _{i+1},\sigma _{i}\sigma _{j}=\sigma _{j}\sigma _{i}\right\rangle ,}
where in the first group of relations
1
≤
i
≤
n
−
2
{\displaystyle 1\leq i\leq n-2}
and in the second group of relations
|
i
−
j
|
≥
2
{\displaystyle |i-j|\geq 2}
. This presentation leads to generalisations of braid groups called Artin groups. The cubic relations, known as the braid relations, play an important role in the theory of Yang–Baxter equations.
=== Further properties ===
The braid group
B
1
{\displaystyle B_{1}}
is trivial,
B
2
{\displaystyle B_{2}}
is the infinite cyclic group
Z
{\displaystyle \mathbb {Z} }
, and
B
3
{\displaystyle B_{3}}
is isomorphic to the knot group of the trefoil knot – in particular, it is an infinite non-abelian group.
The n-strand braid group
B
n
{\displaystyle B_{n}}
embeds as a subgroup into the
(
n
+
1
)
{\displaystyle (n+1)}
-strand braid group
B
n
+
1
{\displaystyle B_{n+1}}
by adding an extra strand that does not cross any of the first n strands. The increasing union of the braid groups with all
n
≥
1
{\displaystyle n\geq 1}
is the infinite braid group
B
∞
{\displaystyle B_{\infty }}
.
All non-identity elements of
B
n
{\displaystyle B_{n}}
have infinite order; i.e.,
B
n
{\displaystyle B_{n}}
is torsion-free.
There is a left-invariant linear order on
B
n
{\displaystyle B_{n}}
called the Dehornoy order.
For
n
≥
3
{\displaystyle n\geq 3}
,
B
n
{\displaystyle B_{n}}
contains a subgroup isomorphic to the free group on two generators.
There is a homomorphism
B
n
→
Z
{\displaystyle B_{n}\to \mathbb {Z} }
defined by σi ↦ 1. So for instance, the braid σ2σ3σ1−1σ2σ3 is mapped to 1 + 1 − 1 + 1 + 1 = 3. This map corresponds to the abelianization of the braid group. Since σik ↦ k, then σik is the identity if and only if
k
=
0
{\displaystyle k=0}
. This proves that the generators have infinite order.
== Interactions ==
=== Relation with symmetric group and the pure braid group ===
By forgetting how the strands twist and cross, every braid on n strands determines a permutation on n elements. This assignment is onto and compatible with composition, and therefore becomes a surjective group homomorphism Bn → Sn from the braid group onto the symmetric group. The image of the braid σi ∈ Bn is the transposition si = (i, i+1) ∈ Sn. These transpositions generate the symmetric group, satisfy the braid group relations, and have order 2. This transforms the Artin presentation of the braid group into the Coxeter presentation of the symmetric group:
S
n
=
⟨
s
1
,
…
,
s
n
−
1
|
s
i
s
i
+
1
s
i
=
s
i
+
1
s
i
s
i
+
1
,
s
i
s
j
=
s
j
s
i
for
|
i
−
j
|
≥
2
,
s
i
2
=
1
⟩
.
{\displaystyle S_{n}=\left\langle s_{1},\ldots ,s_{n-1}|s_{i}s_{i+1}s_{i}=s_{i+1}s_{i}s_{i+1},s_{i}s_{j}=s_{j}s_{i}{\text{ for }}|i-j|\geq 2,s_{i}^{2}=1\right\rangle .}
The kernel of the homomorphism Bn → Sn is the subgroup of Bn called the pure braid group on n strands and denoted Pn. This can be seen as the fundamental group of the space of n-tuples of distinct points of the Euclidean plane. In a pure braid, the beginning and the end of each strand are in the same position. Pure braid groups fit into a short exact sequence
1
→
F
n
−
1
→
P
n
→
P
n
−
1
→
1.
{\displaystyle 1\to F_{n-1}\to P_{n}\to P_{n-1}\to 1.}
This sequence splits and therefore pure braid groups are realized as iterated semi-direct products of free groups.
=== Relation between B3 and the modular group ===
The braid group
B
3
{\displaystyle B_{3}}
is the universal central extension of the modular group
P
S
L
(
2
,
Z
)
{\displaystyle \mathrm {PSL} (2,\mathbb {Z} )}
, with these sitting as lattices inside the (topological) universal covering group
S
L
(
2
,
R
)
¯
→
P
S
L
(
2
,
R
)
{\displaystyle {\overline {\mathrm {SL} (2,\mathbb {R} )}}\to \mathrm {PSL} (2,\mathbb {R} )}
.
Furthermore, the modular group has trivial center, and thus the modular group is isomorphic to the quotient group of
B
3
{\displaystyle B_{3}}
modulo its center,
Z
(
B
3
)
,
{\displaystyle Z(B_{3}),}
and equivalently, to the group of inner automorphisms of
B
3
{\displaystyle B_{3}}
.
Here is a construction of this isomorphism. Define
a
=
σ
1
σ
2
σ
1
,
b
=
σ
1
σ
2
{\displaystyle a=\sigma _{1}\sigma _{2}\sigma _{1},\quad b=\sigma _{1}\sigma _{2}}
.
From the braid relations it follows that
a
2
=
b
3
{\displaystyle a^{2}=b^{3}}
. Denoting this latter product as
c
{\displaystyle c}
, one may verify from the braid relations that
σ
1
c
σ
1
−
1
=
σ
2
c
σ
2
−
1
=
c
{\displaystyle \sigma _{1}c\sigma _{1}^{-1}=\sigma _{2}c\sigma _{2}^{-1}=c}
implying that
c
{\displaystyle c}
is in the center of
B
3
{\displaystyle B_{3}}
. Let
C
{\displaystyle C}
denote the subgroup of
B
3
{\displaystyle B_{3}}
generated by c, since C ⊂ Z(B3), it is a normal subgroup and one may take the quotient group B3/C. We claim B3/C ≅ PSL(2, Z); this isomorphism can be given an explicit form. The cosets σ1C and σ2C map to
σ
1
C
↦
R
=
[
1
1
0
1
]
σ
2
C
↦
L
−
1
=
[
1
0
−
1
1
]
{\displaystyle \sigma _{1}C\mapsto R={\begin{bmatrix}1&1\\0&1\end{bmatrix}}\qquad \sigma _{2}C\mapsto L^{-1}={\begin{bmatrix}1&0\\-1&1\end{bmatrix}}}
where L and R are the standard left and right moves on the Stern–Brocot tree; it is well known that these moves generate the modular group.
Alternately, one common presentation for the modular group is
⟨
v
,
p
|
v
2
=
p
3
=
1
⟩
{\displaystyle \langle v,p\,|\,v^{2}=p^{3}=1\rangle }
where
v
=
[
0
1
−
1
0
]
,
p
=
[
0
1
−
1
1
]
.
{\displaystyle v={\begin{bmatrix}0&1\\-1&0\end{bmatrix}},\qquad p={\begin{bmatrix}0&1\\-1&1\end{bmatrix}}.}
Mapping a to v and b to p yields a surjective group homomorphism B3 → PSL(2, Z).
The center of B3 is equal to C, a consequence of the facts that c is in the center, the modular group has trivial center, and the above surjective homomorphism has kernel C.
=== Relationship to the mapping class group and classification of braids ===
The braid group Bn can be shown to be isomorphic to the mapping class group of a punctured disk with n punctures. This is most easily visualized by imagining each puncture as being connected by a string to the boundary of the disk; each mapping homomorphism that permutes two of the punctures can then be seen to be a homotopy of the strings, that is, a braiding of these strings.
Via this mapping class group interpretation of braids, each braid may be classified as periodic, reducible or pseudo-Anosov.
=== Connection to knot theory ===
If a braid is given and one connects the first left-hand item to the first right-hand item using a new string, the second left-hand item to the second right-hand item etc. (without creating any braids in the new strings), one obtains a link, and sometimes a knot. Alexander's theorem in braid theory states that the converse is true as well: every knot and every link arises in this fashion from at least one braid; such a braid can be obtained by cutting the link. Since braids can be concretely given as words in the generators σi, this is often the preferred method of entering knots into computer programs.
=== Computational aspects ===
The word problem for the braid relations is efficiently solvable and there exists a normal form for elements of Bn in terms of the generators σ1, ..., σn−1. (In essence, computing the normal form of a braid is the algebraic analogue of "pulling the strands" as illustrated in our second set of images above.) The free GAP computer algebra system can carry out computations in Bn if the elements are given in terms of these generators. There is also a package called CHEVIE for GAP3 with special support for braid groups. The word problem is also efficiently solved via the Lawrence–Krammer representation.
In addition to the word problem, there are several known hard computational problems that could implement braid groups, applications in cryptography have been suggested.
== Actions ==
In analogy with the action of the symmetric group by permutations, in various mathematical settings there exists a natural action of the braid group on n-tuples of objects or on the n-folded tensor product that involves some "twists". Consider an arbitrary group G and let X be the set of all n-tuples of elements of G whose product is the identity element of G. Then Bn acts on X in the following fashion:
σ
i
(
x
1
,
…
,
x
i
−
1
,
x
i
,
x
i
+
1
,
…
,
x
n
)
=
(
x
1
,
…
,
x
i
−
1
,
x
i
+
1
,
x
i
+
1
−
1
x
i
x
i
+
1
,
x
i
+
2
,
…
,
x
n
)
.
{\displaystyle \sigma _{i}\left(x_{1},\ldots ,x_{i-1},x_{i},x_{i+1},\ldots ,x_{n}\right)=\left(x_{1},\ldots ,x_{i-1},x_{i+1},x_{i+1}^{-1}x_{i}x_{i+1},x_{i+2},\ldots ,x_{n}\right).}
Thus the elements xi and xi+1 exchange places and, in addition, xi is twisted by the inner automorphism corresponding to xi+1 – this ensures that the product of the components of x remains the identity element. It may be checked that the braid group relations are satisfied and this formula indeed defines a group action of Bn on X. As another example, a braided monoidal category is a monoidal category with a braid group action. Such structures play an important role in modern mathematical physics and lead to quantum knot invariants.
=== Representations ===
Elements of the braid group Bn can be represented more concretely by matrices. One classical such representation is Burau representation, where the matrix entries are single variable Laurent polynomials. It had been a long-standing question whether Burau representation was faithful, but the answer turned out to be negative for n ≥ 5. More generally, it was a major open problem whether braid groups were linear. In 1990, Ruth Lawrence described a family of more general "Lawrence representations" depending on several parameters. In 1996, Chetan Nayak and Frank Wilczek posited that in analogy to projective representations of SO(3), the projective representations of the braid group have a physical meaning for certain quasiparticles in the fractional quantum hall effect. Around 2001 Stephen Bigelow and Daan Krammer independently proved that all braid groups are linear. Their work used the Lawrence–Krammer representation of dimension
n
(
n
−
1
)
/
2
{\displaystyle n(n-1)/2}
depending on the variables q and t. By suitably specializing these variables, the braid group
B
n
{\displaystyle B_{n}}
may be realized as a subgroup of the general linear group over the complex numbers.
== Infinitely generated braid groups ==
There are many ways to generalize this notion to an infinite number of strands. The simplest way is to take the direct limit of braid groups, where the attaching maps
f
:
B
n
→
B
n
+
1
{\displaystyle f\colon B_{n}\to B_{n+1}}
send the
n
−
1
{\displaystyle n-1}
generators of
B
n
{\displaystyle B_{n}}
to the first
n
−
1
{\displaystyle n-1}
generators of
B
n
+
1
{\displaystyle B_{n+1}}
(i.e., by attaching a trivial strand). This group, however, admits no metrizable topology while remaining continuous.
Paul Fabel has shown that there are two topologies that can be imposed on the resulting group each of whose completion yields a different group. The first is a very tame group and is isomorphic to the mapping class group of the infinitely punctured disk—a discrete set of punctures limiting to the boundary of the disk.
The second group can be thought of the same as with finite braid groups. Place a strand at each of the points
(
0
,
1
/
n
)
{\displaystyle (0,1/n)}
and the set of all braids—where a braid is defined to be a collection of paths from the points
(
0
,
1
/
n
,
0
)
{\displaystyle (0,1/n,0)}
to the points
(
0
,
1
/
n
,
1
)
{\displaystyle (0,1/n,1)}
so that the function yields a permutation on endpoints—is isomorphic to this wilder group. An interesting fact is that the pure braid group in this group is isomorphic to both the inverse limit of finite pure braid groups
P
n
{\displaystyle P_{n}}
and to the fundamental group of the Hilbert cube minus the set
{
(
x
i
)
i
∈
N
∣
x
i
=
x
j
for some
i
≠
j
}
.
{\displaystyle \{(x_{i})_{i\in \mathbb {N} }\mid x_{i}=x_{j}{\text{ for some }}i\neq j\}.}
== Cohomology ==
The cohomology of a group
G
{\displaystyle G}
is defined as the cohomology of the corresponding Eilenberg–MacLane classifying space,
K
(
G
,
1
)
{\displaystyle K(G,1)}
, which is a CW complex uniquely determined by
G
{\displaystyle G}
up to homotopy. A classifying space for the braid group
B
n
{\displaystyle B_{n}}
is the nth unordered configuration space of
R
2
{\displaystyle \mathbb {R} ^{2}}
, that is, the space of all sets of
n
{\displaystyle n}
distinct unordered points in the plane:
UConf
n
(
R
2
)
=
{
{
u
1
,
.
.
.
,
u
n
}
:
u
i
∈
R
2
,
u
i
≠
u
j
for
i
≠
j
}
{\displaystyle \operatorname {UConf} _{n}(\mathbb {R} ^{2})=\{\{u_{1},...,u_{n}\}:u_{i}\in \mathbb {R} ^{2},u_{i}\neq u_{j}{\text{ for }}i\neq j\}}
.
So by definition
The calculations for coefficients in
Z
/
2
Z
{\displaystyle \mathbb {Z} /2\mathbb {Z} }
can be found in Fuks (1970).
Similarly, a classifying space for the pure braid group
P
n
{\displaystyle P_{n}}
is
Conf
n
(
R
2
)
{\displaystyle \operatorname {Conf} _{n}(\mathbb {R} ^{2})}
, the nth ordered configuration space of
R
2
{\displaystyle \mathbb {R} ^{2}}
. In 1968 Vladimir Arnold showed that the integral cohomology of the pure braid group
P
n
{\displaystyle P_{n}}
is the quotient of the exterior algebra generated by the collection of degree-one classes
ω
i
j
1
≤
i
<
j
≤
n
{\displaystyle \omega _{ij}\;\;1\leq i<j\leq n}
, subject to the relations
ω
k
,
ℓ
ω
ℓ
,
m
+
ω
ℓ
,
m
ω
m
,
k
+
ω
m
,
k
ω
k
,
ℓ
=
0.
{\displaystyle \omega _{k,\ell }\omega _{\ell ,m}+\omega _{\ell ,m}\omega _{m,k}+\omega _{m,k}\omega _{k,\ell }=0.}
== See also ==
Artin–Tits group
Braided monoidal category
Braided vector space
Braided Hopf algebra
Knot theory
Non-commutative cryptography
Spherical braid group
== References ==
== Further reading ==
== External links ==
"Braid group". PlanetMath.
CRAG: CRyptography and Groups computation library from the Stevens University's Algebraic Cryptography Center
Macauley, M. Lecture 1.3: Groups in science, art, and mathematics. Visual Group Theory. Clemson University.
Bigelow, Stephen. "Exploration of B5 Java applet". Archived from the original on 4 June 2013. Retrieved 1 November 2007.
Lipmaa, Helger, Cryptography and Braid Groups page, archived from the original on 3 August 2009
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Scherich, Nancy. Representations of the Braid Groups. Dance Your PhD. expanded further in Behind the Math of "Dance Your PhD," Part 1: The Braid Groups. | Wikipedia/Braid_theory |
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