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Sainte-Laguë method

The Webster method, also called the Sainte-Laguë method, is a highest averages apportionment method for allocating seats in a parliament among federal states, or among parties in a party-list proportional representation system. The Sainte-Laguë method shows a more equal seats-to-votes ratio for different sized parties among apportionment methods.

The method was first described in 1832 by American statesman and senator Daniel Webster. In 1842 the method was adopted for proportional allocation of seats in United States congressional apportionment (Act of 25 June 1842, ch 46, 5 Stat. 491). The same method was independently invented in 1910 by the French mathematician André Sainte-Laguë. It seems that French and European literature was unaware of Webster until after World War II. This is the reason for the double name.

Motivation
Proportional electoral systems attempt to distribute seats in proportion to the votes for each political party, i.e. a party with 30% of votes would receive 30% of seats. Exact proportionality is not possible because only whole seats can be distributed. Different apportionment methods, of which the Sainte-Laguë method is one, exist to distribute the seats according to the votes. Different apportionment methods show different levels of proportionality, apportionment paradoxes and political fragmentation. The Sainte-Laguë method minimizes the average seats-to-votes ratio deviation and empirically shows the best proportionality behavior and more equal seats-to-votes ratio for different sized parties among apportionment methods. Among other common methods, the D'Hondt method favours large parties and coalitions over small parties. While favoring large parties reduces political fragmentation, this can be achieved with electoral thresholds as well. The Sainte-Laguë method shows fewer apportionment paradoxes compared to largest remainder methods such as the Hare quota and other highest averages methods such as d'Hondt method.

Description
After all the votes have been tallied, successive quotients are calculated for each party. The formula for the quotient is

 * $$\text{quotient} = \frac V {2s+1}$$

where:

 * V is the total number of votes that party received, and
 * s is the number of seats that have been allocated so far to that party, initially 0 for all parties.

Whichever party has the highest quotient gets the next seat allocated, and their quotient is recalculated. The process is repeated until all seats have been allocated.

The Webster/Sainte-Laguë method does not ensure that a party receiving more than half the votes will win at least half the seats; nor does its modified form.

Often there is an electoral threshold; that is, in order to be allocated seats, a minimum percentage of votes must be gained.

Example
In this example, 230,000 voters decide the disposition of 8 seats among 4 parties. Since 8 seats are to be allocated, each party's total votes are divided by 1, then by 3, and 5 (and then, if necessary, by 7, 9, 11, 13, and so on by using the formula above) every time the number of votes is the biggest for the current round of calculation.

For comparison, the "True proportion" column shows the exact fractional numbers of seats due, calculated in proportion to the number of votes received. (For example, 100,000/230,000 × 8 = 3.48.)

The 8 highest entries (in the current round of calculation) are marked by asterisk: from 100,000 down to 16,000; for each, the corresponding party gets a seat.

The below chart is an easy way to perform the calculation:

In comparison, the D'Hondt method would allocate four seats to party A and no seats to party D, reflecting the D'Hondt method's overrepresentation of larger parties.

Properties
When apportioning seats in proportional representation, it is particularly important to avoid bias between large parties and small parties to avoid strategic voting. André Sainte-Laguë showed theoretically that the Sainte-Laguë method shows the lowest average bias in apportionment, confirmed by different theoretical and empirical ways. The European Parliament (Representation) Act 2003 stipulates each region must be allocated at least 3 seats and that the ratio of electors to seats is as nearly as possible the same for each, the Commission found the Sainte-Laguë method produced the smallest standard deviation when compared to the D'Hondt method and Hare quota.

Proportionality under Sainte-Laguë method
The seats-to-votes ratio $$a_i$$ for a political party $$i$$ is the ratio between the fraction of seats $$s_i$$ and the fraction of votes $$v_i$$ for that party:
 * $$a_i = \frac{s_i}{v_i}$$

The Sainte-Laguë method approximates proportionality by optimizing the seats-to-votes ratio among all parties $$i$$ with the least squares approach. First, the difference between the seats-to-votes ratio for a party and the ideal seats-to-votes ratio is calculated and squared to obtain the error for the party $$i$$. To achieve equal representation of each voter, the ideal ratio of seats share to votes share is $$1$$.

$$error_i = (a_i-a_{ideal})^2=\left(\frac{s_i}{v_i}-1\right)^2$$

Second, the error for each party is weighted according to the vote share of each party to represent each voter equally. In the last step, the errors for each party are summed up. This error is identical to the Sainte-Laguë Index.

$$error = \sum_i v_i*error_i=\sum_i {v_i*\left(\frac{s_i}{v_i}-1\right)^2}$$

It was shown that this error is minimized by the Sainte-Laguë method.

Modified Sainte-Laguë method
To reduce political fragmentation, some countries, e.g. Nepal, Norway and Sweden, change the quotient formula for parties with no seats (s = 0). These countries changed the quotient from V to V/1.4, though from the general 2018 elections onwards, Sweden has been using V/1.2. That is, the modified method changes the sequence of divisors used in this method from (1, 3, 5, 7, ...) to (1.4, 3, 5, 7, ...). This makes it more difficult for parties to earn only one seat, compared to the unmodified Sainte-Laguë's method. With the modified method, such small parties do not get any seats; these seats are instead given to a larger party.

Norway further amends this system by utilizing a two-tier proportionality. The number of members to be returned from each of Norway's 19 constituencies (former counties) depends on the population and area of the county: each inhabitant counts one point, while each km2 counts 1.8 points. Furthermore, one seat from each constituency is allocated according to the national distribution of votes.

History
Webster proposed the method in the United States Congress in 1832 for proportional allocation of seats in United States congressional apportionment. In 1842 the method was adopted (Act of June 25, 1842, ch 46, 5 Stat. 491). It was then replaced by Hamilton method and in 1911 the Webster method was reintroduced.

Webster and Sainte-Laguë methods should be treated as two methods with the same result, because the Webster method is used for allocating seats based on states' population, and the Sainte-Laguë based on parties' votes. Webster invented his method for legislative apportionment (allocating legislative seats to regions based on their share of the population) rather than elections (allocating legislative seats to parties based on their share of the votes) but this makes no difference to the calculations in the method.

Webster's method is defined in terms of a quota as in the largest remainder method; in this method, the quota is called a "divisor". For a given value of the divisor, the population count for each region is divided by this divisor and then rounded to give the number of legislators to allocate to that region. In order to make the total number of legislators come out equal to the target number, the divisor is adjusted to make the sum of allocated seats after being rounded give the required total.

One way to determine the correct value of the divisor would be to start with a very large divisor, so that no seats are allocated after rounding. Then the divisor may be successively decreased until one seat, two seats, three seats and finally the total number of seats are allocated. The number of allocated seats for a given region increases from s to s + 1 exactly when the divisor equals the population of the region divided by s + 1/2, so at each step the next region to get a seat will be the one with the largest value of this quotient. That means that this successive adjustment method for implementing Webster's method allocates seats in the same order to the same regions as the Sainte-Laguë method would allocate them.

In 1980 the German physicist Hans Schepers, at the time Head of the Data Processing Group of the German Bundestag, suggested that the distribution of seats according to d'Hondt be modified to avoid putting smaller parties at a disadvantage. German media started using the term Schepers Method and later German literature usually calls it Sainte-Laguë/Schepers.

Threshold for seats
An election threshold can be set to reduce political fragmentation, and any list party which does not receive at least a specified percentage of list votes will not be allocated any seats, even if it received enough votes to have otherwise receive a seat. Examples of countries using the Sainte-Laguë method with a threshold are Germany and New Zealand (5%), although the threshold does not apply if a party wins at least one electorate seat in New Zealand or three electorate seats in Germany. Sweden uses a modified Sainte-Laguë method with a 4% threshold, and a 12% threshold in individual constituencies (i.e. a political party can gain representation with a minuscule representation on the national stage, if its vote share in at least one constituency exceeded 12%). Norway has a threshold of 4% to qualify for leveling seats that are allocated according to the national distribution of votes. This means that even though a party is below the threshold of 4% nationally, they can still get seats from constituencies in which they are particularly popular.

Usage by country
The Webster/Sainte-Laguë method is currently used in Bosnia and Herzegovina, Ecuador, Indonesia, Iraq, Kosovo, Latvia, Nepal, New Zealand, Norway and Sweden. In Germany it is used on the federal level for the Bundestag, and on the state level for the legislatures of Baden-Württemberg, Bavaria, Bremen, Hamburg, North Rhine-Westphalia, Rhineland-Palatinate, and Schleswig-Holstein. In Denmark it is used for leveling seats in the Folketing, correcting the disproportionality of the D'Hondt method for the other seats.

Some cantons in Switzerland use the Sainte-Laguë method for biproportional apportionment between electoral districts and for votes to seats allocation.

The Webster/Sainte-Laguë method was used in Bolivia in 1993, in Poland in 2001, and the Palestinian Legislative Council in 2006. The United Kingdom Electoral Commission has used the method from 2003 to 2013 to distribute British seats in the European Parliament to constituent countries of the United Kingdom and the English regions.

The method has been proposed by the Green Party in Ireland as a reform for use in Dáil Éireann elections, and by the United Kingdom Conservative–Liberal Democrat coalition government in 2011 as the method for calculating the distribution of seats in elections to the House of Lords, the country's upper house of parliament.

Comparison to other methods
The method belongs to the class of highest-averages methods. It is similar to the Jefferson/D'Hondt method, but uses different divisors. The Jefferson/D'Hondt method favors larger parties while the Webster/Sainte-Laguë method doesn't. The Webster/Sainte-Laguë method is generally seen as more proportional, but risks an outcome where a party with more than half the votes can win fewer than half the seats.

When there are two parties, the Webster method is the unique divisor method which is identical to the Hamilton method.

title: Sainte-Laguë method


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Open-source software

Open-source software (OSS) is computer software that is released under a license in which the copyright holder grants users the rights to use, study, change, and distribute the software and its source code to anyone and for any purpose. Open-source software may be developed in a collaborative, public manner. Open-source software is a prominent example of open collaboration, meaning any capable user is able to participate online in development, making the number of possible contributors indefinite. The ability to examine the code facilitates public trust in the software.

Open-source software development can bring in diverse perspectives beyond those of a single company. A 2024 estimate of the value of open source software to firms is $8.8 trillion as firms would need to spend 3.5 times the amount they currently do without the use of open source software.

Open-source code can be used for studying and allows capable end users to adapt software to their personal needs in a similar way user scripts and custom style sheets allow for web sites, and eventually publish the modification as a fork for users with similar preferences, and directly submit possible improvements as pull requests.

Definitions
The Open Source Initiative's (OSI) definition is recognized by several governments internationally as the standard or de facto definition. OSI uses The Open Source Definition to determine whether it considers a software license open source. The definition was based on the Debian Free Software Guidelines, written and adapted primarily by Perens. Perens did not base his writing on the "four freedoms" from the Free Software Foundation (FSF), which were only widely available later.

Under Perens' definition, open source is a broad software license that makes source code available to the general public with relaxed or non-existent restrictions on the use and modification of the code. It is an explicit "feature" of open source that it puts very few restrictions on the use or distribution by any organization or user, in order to enable the rapid evolution of the software.

According to Feller et al. (2005), the terms "free software" and "open source software" should be applied to any "software products distributed under terms that allow users" to use, modify, and redistribute the software "in any manner they see fit, without requiring that they pay the author(s) of the software a royalty or fee for engaging in the listed activities."

Despite initially accepting it, Richard Stallman of the FSF now flatly opposes the term "Open Source" being applied to what they refer to as "free software". Although he agrees that the two terms describe "almost the same category of software", Stallman considers equating the terms incorrect and misleading. Stallman also opposes the professed pragmatism of the Open Source Initiative, as he fears that the free software ideals of freedom and community are threatened by compromising on the FSF's idealistic standards for software freedom. The FSF considers free software to be a subset of open-source software, and Richard Stallman explained that DRM software, for example, can be developed as open source, despite that it does not give its users freedom (it restricts them), and thus does not qualify as free software.

Development model
In his 1997 essay The Cathedral and the Bazaar, open-source influential contributor Eric S. Raymond suggests a model for developing OSS known as the bazaar model. Raymond likens the development of software by traditional methodologies to building a cathedral, with careful isolated work by individuals or small groups. He suggests that all software should be developed using the bazaar style, with differing agendas and approaches.

In the traditional model of development, which he called the cathedral model, development takes place in a centralized way. Roles are clearly defined. Roles include people dedicated to designing (the architects), people responsible for managing the project, and people responsible for implementation. Traditional software engineering follows the cathedral model.

The bazaar model, however, is different. In this model, roles are not clearly defined. Some proposed characteristics of software developed using the bazaar model should exhibit the following patterns:

Users should be treated as co-developers: The users are treated like co-developers and so they should have access to the source code of the software. Furthermore, users are encouraged to submit additions to the software, code fixes for the software, bug reports, documentation, etc. Having more co-developers increases the rate at which the software evolves. Linus's law states that given enough eyeballs all bugs are shallow. This means that if many users view the source code, they will eventually find all bugs and suggest how to fix them. Some users have advanced programming skills, and furthermore, each user's machine provides an additional testing environment. This new testing environment offers the ability to find and fix a new bug.

Early releases: The first version of the software should be released as early as possible so as to increase one's chances of finding co-developers early.

Frequent integration: Code changes should be integrated (merged into a shared code base) as often as possible so as to avoid the overhead of fixing a large number of bugs at the end of the project life cycle. Some open-source projects have nightly builds where integration is done automatically.

Several versions: There should be at least two versions of the software. There should be a buggier version with more features and a more stable version with fewer features. The buggy version (also called the development version) is for users who want the immediate use of the latest features and are willing to accept the risk of using code that is not yet thoroughly tested. The users can then act as co-developers, reporting bugs and providing bug fixes.

High modularization: The general structure of the software should be modular allowing for parallel development on independent components.

Dynamic decision-making structure: There is a need for a decision-making structure, whether formal or informal, that makes strategic decisions depending on changing user requirements and other factors. Compare with extreme programming.

The process of Open source development begins with a requirements elicitation where developers consider if they should add new features or if a bug needs to be fixed in their project. This is established by communicating with the OSS community through avenues such as bug reporting and tracking or mailing lists and project pages. Next, OSS developers select or are assigned to a task and identify a solution. Because there are often many different possible routes for solutions in OSS, the best solution must be chosen with careful consideration and sometimes even peer feedback. The developer then begins to develop and commit the code. The code is then tested and reviewed by peers. Developers can edit and evolve their code through feedback from continuous integration. Once the leadership and community are satisfied with the whole project, it can be partially released and user instruction can be documented. If the project is ready to be released, it is frozen, with only serious bug fixes or security repairs occurring. Finally, the project is fully released and only changed through minor bug fixes.

Advantages
Open source implementation of a standard can increase adoption of that standard. This creates developer loyalty as developers feel empowered and have a sense of ownership of the end product.

Moreover, lower costs of marketing and logistical services are needed for OSS. OSS can be a tool to promote a company's image, including its commercial products. The OSS development approach has helped produce reliable, high quality software quickly and inexpensively.

Open source development offers the potential to quicken innovation and create of social value. In France for instance, a policy that incentivized government to favor free open-source software increased to nearly 600,000 OSS contributions per year, generating social value by increasing the quantity and quality of open-source software. This policy also led to an estimated increase of up to 18% of tech startups and a 14% increase in the number of people employed in the IT sector.

OSS can be highly reliable when it has thousands of independent programmers testing and fixing bugs of the software. Open source is not dependent on the company or author that originally created it. Even if the company fails, the code continues to exist and be developed by its users.

OSS is flexible because modular systems allow programmers to build custom interfaces, or add new abilities to it and it is innovative since open-source programs are the product of collaboration among a large number of different programmers. The mix of divergent perspectives, corporate objectives, and personal goals speeds up innovation.

Moreover, free software can be developed in accordance with purely technical requirements. It does not require thinking about commercial pressure that often degrades the quality of the software. Commercial pressures make traditional software developers pay more attention to customers' requirements than to security requirements, since such features are somewhat invisible to the customer.

Development tools
In open source software development, tools are used to support the development of the product and the development process itself.

Version control systems such as Centralized Version control system (CVCS) and the distributed version control system (DVCS) are examples of tools, often open source, that help manage the source code files and the changes to those files for a software project in order to foster collaboration. CVCS are centralized with a central repository while DVCS are decentralized and have a local repository for every user. concurrent versions system (CVS) and later Subversion (SVN) and Git are examples of CVCS. The repositories are hosted and published on source-code-hosting facilities such as GitHub.

Open-source projects use utilities such as issue trackers to organize open-source software development. Commonly used bugtrackers include Bugzilla and Redmine.

Tools such as mailing lists and IRC provide means of coordination and discussion of bugs among developers. Project web pages, wiki pages, roadmap lists and newsgroups allow for the distribution of project information that focuses on end users.

Contributing
The basic roles OSS participants can fall into multiple categories, beginning with leadership at the center of the project who have control over its execution. Next are the core contributors with a great deal of experience and authority in the project who may guide the other contributors. Non-core contributors have less experience and authority, but regularly contribute and are vital to the project’s development. New contributors are the least experienced but with mentorship and guidance can become regular contributors.

Some possible ways of contributing to open source software include such roles as programming, user interface design and testing, web design, Bug triage, accessibility design and testing, UX design, code testing, and security review and testing. However, there are several ways of contributing to OSS projects even without coding skills. For example, some less technical ways of participating are documentation writing and editing, translation, project management, event organization and coordination, marketing, release management, community management, and public relations and outreach.

Funding is absolutely another terrific way that individuals and organizations choose to contribute to open source projects. Groups like Open Collective provide a means for individuals to contribute monthly to supporting their favorite projects. Organizations like the Sovereign Tech Fund is able to contribute to millions to supporting the tools the German Government uses. The National Science Foundation established a Pathways to Enable Open-Source Ecosystems (POSE) program to support open source innovation.

Industry participation
The adoption of open source software by industry is increasing over time. OSS is popular in several industries such as telecommunications, aerospace, healthcare, and media & entertainment due to the benefits it provides. Adoption of OSS is more likely in larger organizations and is dependent on the company’s IT usage, operating efficiencies, and the productivity of employees.

Industries are likely to use OSS due to back-office functionality, sales support, research and development, software features, quick deployment, portability across platforms and avoidance of commercial license management. Additionally, lower cost for hardware and ownership are also important benefits.

Prominent organizations
Organizations that contribute to the development and expansions of free and open source software movements exist all over the world. These organizations are dedicated to goals such as teaching and spreading technology. As listed by a former vice president of the open source initiative, some American organizations include the free software foundation, software freedom conservancy, the open source initiative and software in the public interest. Within Europe some notable organizations are Free Software Foundation Europe, open source projects EU (OSP) and openforum europe (OFE). One Australian organization is linux australia while Asia has open source asia and fossasia. free and open source software for africa (FOSSFA) and openafrica are African organizations and Central and South Asia has such organizations as FLISOL and GRUP de usuarios de software libre peru. Outside of these, many more organizations dedicated to the advancement of open source software exist.

Licensing
FOSS products are generally licensed under two types of licenses: permissive licensing and copyleft licensing. Both of these types of licenses are different than proprietary licensing in that they can allow more users access to the software and allow for the creation of derivative works as specified by the terms of the specific license, as each license has its own rules. Permissive licenses allow recipients of the software to implement the author's copyright rights without having to use the same license for distribution. Examples of this type of license include the BSD, MIT, and Apache licenses. Copyleft licenses are different in that they require recipients to use the same license for at least some parts of the distribution of their works. Strong copyleft licenses require all derivative works to use the same license while weak copyleft licenses require the use of the same license only under certain conditions. Examples of this type of license include the GNU family of licenses, and the MPL and EPL licenses. The similarities between these two categories of licensing include that they provide a broad grant of copyright rights, require that recipients preserve copyright notices, and that a copy of the license is provided to recipients with the code.

One important legal precedent for open source software was created in 2008, when the Jacobson v Katzer case enforced terms of the Artistic license, including attribution and identification of modifications. The ruling of this case cemented enforcement under copyright law when the conditions of the license were not followed. Because of the similarity of the Artistic license to other open source software licenses, the ruling created a precedent that applied widely.

Examples of free software license / open source licenses include Apache License, BSD license, GNU General Public License, GNU Lesser General Public License, MIT License, Eclipse Public License and Mozilla Public License.

Legal issues
Several gray areas exist within software regulation that have great impact on open source software, such as if software is a good or service, what can be considered a modification, governance through contract vs license, ownership and right of use. While there have been developments on these issues, they often lead to even more questions. The existence of these uncertainties in regulation has a negative impact on industries involved in technologies as a whole.

Within the legal history of software as a whole, there was much debate on whether to protect it as intellectual property under patent law, copyright law or establishing a unique regulation. Ultimately, copyright law became the standard with computer programs being considered a form of literary work, with some tweaks of unique regulation.

Software is generally considered source code and object code, with both being protectable, though there is legal variety in this definition. Some jurisdictions attempt to expand or reduce this conceptualization for their own purposes. For example, The European Court of Justice defines a computer program as not including the functionality of a program, the programing language, or the format of data files. By limiting protections of the different aspects of software, the law favors an open source approach to software use. The US especially has an open approach to software, with most open-source licenses originating there. However, this has increased the focus on patent rights within these licenses, which has seen backlash from the OSS community, who prefer other forms of IP protection.

Another issue includes technological protection measures (TPM) and digital rights management (DRM) techniques which were internationally legally recognized and protected in the 1996 World Intellectual Property Organization (WIPO) Treaty. Open source software proponents disliked these technologies as they constrained end-users potentially beyond copyright law. Europe responded to such complaints by putting TPM under legal controls, representing a victory for OSS supporters.

Economic/business implications
In open source communities, instead of owning the software produced, the producer owns the development of the evolving software. In this way, the future of the software is open, making ownership or intellectual property difficult within OSS. Licensing and branding can prevent others from stealing it, preserving its status as a public good. Open source software can be considered a public good as it is available to everyone and does not decrease in value for others when downloaded by one person. Open source software is unique in that it becomes more valuable as it is used and contributed to, instead of diminishing the resource. This is explained by concepts such as investment in reputation and network effects.

The economic model of open source software can be explained as developers contribute work to projects, creating public benefits. Developers choose projects based on the perceived benefits or costs, such as improved reputation or value of the project. The motivations of developers can come from many different places and reasons, but the important takeaway is that money is not the only or even most important incentivization.

Because economic theory mainly focuses on the consumption of scarce resources, the OSS dynamic can be hard to understand. In OSS, producers become consumers by reaping the rewards of contributing to a project. For example, a developer becomes well regarded by their peers for a successful contribution to an OSS project. The social benefits and interactions of OSS are difficult to account for in economic models as well. Furthermore, the innovation of technology creates constantly changing value discussions and outlooks, making economic model unable to predict social behavior.

Although OSS is theoretically challenging in economic models, it is explainable as a sustainable social activity that requires resources. These resources include time, money, technology and contributions. Many developers have used technology funded by organizations such as universities and governments, though these same organizations benefit from the work done by OSS. As OSS grows, hybrid systems containing OSS and proprietary systems are becoming more common.

Throughout the mid 2000s, more and more tech companies have begun to use OSS. For example, Dell's move of selling computers with GNU/Linux already installed. Microsoft itself has launched a Linux-based operating system despite previous animosity with the OSS movement. Despite these developments, these companies tend to only use OSS for certain purposes, leading to worries that OSS is being taken advantage of by corporations and not given anything in return.

Government uses
While many governments are interested in implementing and promoting open-source software due to the many benefits provided, a huge issue to be considered is cybersecurity. While accidental vulnerabilities are possible, so are attacks by outside agents. Because of these fears, governmental interest in contributing to the governance of software has become more prominent. However, these are the broad strokes of the issue, with each country having their own specific politicized interactions with open source software and their goals for its implementation. For example, the United States has focused on national security in regard to open source software implementation due to the perceived threat of the increase of open source software activity in countries like China and Russia, with the Department of Defense considering multiple criteria for using OSS. These criteria include: if it comes from and is maintained by trusted sources, whether it will continue to be maintained, if there are dependencies on sub-components in the software, component security and integrity, and foreign governmental influence.

Another issue for governments in regard to open source is their investments in technologies such as operating systems, semiconductors, cloud, and artificial intelligence. These technologies all have implications for global cooperation, again opening up security issues and political consequences. Many countries have to balance technological innovation with technological dependence in these partnerships. For example, after China’s open-source dependent company Huawei was prevented from using Google's android system in 2019, they began to create their own alternative operating system: Harmony OS.

Germany recently established a Sovereign Tech Fund, to help support the governance and maintenance of the software that they use.

History
In the early days of computing, such as the 1950s and into the 1960s, programmers and developers shared software to learn from each other and evolve the field of computing. For example, Unix included the operating system source code for users. Eventually, the commercialization of software in the years 1970–1980 began to prevent this practice. However, academics still often developed software collaboratively.

In response, the open source movement was born out of the work of skilled programmer enthusiasts, widely referred to as hackers or hacker culture. One of these enthusiasts, Richard Stallman, was a driving force behind the free software movement, which would later allow for the open source movement. In 1984, he resigned from MIT to create a free operating system, GNU, after the programmer culture in his lab was stifled by proprietary software preventing source code from being shared and improved upon. GNU was UNIX compatible, meaning that the programmer enthusiasts would still be familiar with how it worked. However, it quickly became apparent that there was some confusion with the label Stallman had chosen of free software, which he described as free as in free speech, not free beer, referring to the meaning of free as freedom rather than price. He later expanded this concept of freedom to the four essential freedoms. Through GNU, open source norms of incorporating others' source code, community bug fixes and suggestions of code for new features appeared. In 1985, Stallman founded the Free Software Foundation (FSF) to promote changes in software and to help write GNU. In order to prevent his work from being used in proprietary software, Stallman created the concept of copyleft, which allowed the use of his work by anyone, but under specific terms. To do this, he created the GNU General Public License (GNU GPL) in 1989, which was updated in 1991. In 1991, GNU was combined with the Linux kernel written by Linus Torvalds, as a kernel was missing in GNU. The operating system is now usually referred to as Linux. Throughout this whole period, there were many other free software projects and licenses around at the time, all with different ideas of what the concept of free software was and should be, as well as the morality of proprietary software, such as Berkeley Software Distribution, TeX, and the X Window System.

As free software developed, the Free Software Foundation began to look how to bring free software ideas and perceived benefits to the commercial software industry. It was concluded that FSF's social activism was not appealing to companies and they needed a way to rebrand the free software movement to emphasize the business potential of sharing and collaborating on software source code. The term open source was suggested by Christine Peterson in 1998 at a meeting of supporters of free software. Many in the group felt the name free software was confusing to newcomers and holding back industry interest and they readily accepted the new designation of open source, creating the Open Source Initiative (OSI) and the OSI definition of what open source software is. The Open Source Initiative's (OSI) definition is now recognized by several governments internationally as the standard or de facto definition. The definition was based on the Debian Free Software Guidelines, written and adapted primarily by Bruce Perens. The OSI definition differed from the free software definition in that it allows the inclusion of proprietary software and allows more liberties in its licensing. Some, such as Stallman, agree more with the original concept of free software as a result because it takes a strong moral stance against proprietary software, through there is much overlap between the two movements in terms of the operation of the software.

While the Open Source Initiative sought to encourage the use of the new term and evangelize the principles it adhered to, commercial software vendors found themselves increasingly threatened by the concept of freely distributed software and universal access to an application's source code, with an executive of Microsoft calling open source an intellectual property destroyer in 2001. However, while free and open-source software (FOSS) has historically played a role outside of mainstream private software development, companies as large as Microsoft have begun to develop official open source presences on the Internet. IBM, Oracle, and State Farm are just a few of the companies with a serious public stake in today's competitive open source market, marking a significant shift in the corporate philosophy concerning the development of FOSS.

Future
The future of the open source software community, and the free software community by extension, has become successful if not confused about what it stands for. For example, Android and Ubuntu are examples milestones of success in the open source software rise to prominence from the sidelines of technological innovation as it existed in the early 2000s. However, some in the community consider them failures in their representation of OSS due to issues such as the downplaying of the OSS center of Android by Google and its partners, the use of an Apache license that allowed forking and resulted in a loss of opportunities for collaboration within Android, the prioritization of convivence over freedom in Ubuntu, and features within Ubuntu that track users for marketing purposes.

The use of OSS has become more common in business with 78% of companies reporting that they run all or part of their operations on FOSS. The popularity of OSS has risen to the point that Microsoft, a once detractor of OSS, has included its use in their systems. However, this success has raised concerns that will determine the future of OSS as the community must answer questions such as what OSS is, what should it be, and what should be done to protect it, if it even needs protecting. All in all, while the free and open source revolution has slowed to a perceived equilibrium in the market place, that does not mean it is over as many theoretical discussions must take place to determine its future.

Closed source / proprietary software
Open source software differs from proprietary software in that it is publicly available, the license requires no fees, modifications and distributions are allowed under license specifications. All of this works to prevent a monopoly on any OSS product, which is a goal of proprietary software. Proprietary software limits their customers' choices to either committing to using that software, upgrading it or switching to other software, forcing customers to have their software preferences impacted by their monetary cost. The ideal case scenario for the proprietary software vendor would be a lock-in, where the customer does not or cannot switch software due to these costs and continues to buy products from that vendor.

Within proprietary software, bug fixes can only be provided by the vendor, moving platforms requires another purchase and the existence of the product relies on the vendor, who can discontinue it at any point. Additionally, proprietary software does not provide its source code and cannot be altered by users. For businesses, this can pose a security risk and source of frustration, as they cannot specialize the product to their needs, and there may be hidden threats or information leaks within the software that they cannot access or change.

Free software
Under OSI's definition, open source is a broad software license that makes source code available to the general public with relaxed or non-existent restrictions on the use and modification of the code. It is an explicit feature of open source that it puts very few restrictions on the use or distribution by any organization or user, in order to enable the rapid evolution of the software.

Richard Stallman, leader of the Free software movement and member of the free software foundation opposes the term open source being applied to what they refer to as free software. Although he agrees that the two terms describe almost the same category of software, Stallman considers equating the terms incorrect and misleading. He believes that the main difference is that by choosing one term over the other lets others know about what one's goals are: development (open source) or a social stance (free software). Nevertheless, there is significant overlap between open source software and free software. Stallman also opposes the professed pragmatism of the Open Source Initiative, as he fears that the free software ideals of freedom and community are threatened by compromising on the FSF's idealistic standards for software freedom. The FSF considers free software to be a subset of open-source software, and Richard Stallman explained that DRM software, for example, can be developed as open source, despite how it restricts its users, and thus does not qualify as free software.

The FSF said that the term open source fosters an ambiguity of a different kind such that it confuses the mere availability of the source with the freedom to use, modify, and redistribute it. On the other hand, the term free software was criticized for the ambiguity of the word free, which was seen as discouraging for business adoption, and for the historical ambiguous usage of the term.

Developers have used the alternative terms Free and Open Source Software (FOSS), or Free/Libre and Open Source Software (FLOSS), consequently, to describe open-source software that is also free software.

Source-available software
Software can be distributed with source code, which is a code that is readable. Software is source available when this source code is available to be seen. However to be source available or FOSS, the source code does not need to be accessible to all, just the users of that software. While all FOSS software is source available because this is a requirement made by the Open Source Definition, not all source available software is FOSS. For example, if the software doesn't meet other aspects of the Open Source Definition such as permitted modification or redistribution, even if the source code is available, the software is not FOSS.

Open-sourcing
A recent trend within software companies is open sourcing, or transitioning their previous proprietary software into open source software through releasing it under a open source license. Examples of companies who have done this are Google, Microsoft and Apple. Additionally, open sourcing can refer to programming open source software or installing open source software. Open sourcing can be beneficial in multiple ways, such as attracting more external contributors who bring new perspectives and problem solving capabilities. The downsides of open sourcing include the work that has to be done to maintaining the new community, such as making the base code easily understandable, setting up communication channels for new developers and creating documentation to allow new developers to easily join. However, a review of several open sourced projects found that although a newly open sourced project attracts many newcomers, a great amount are likely to soon leave the project and their forks are also likely to not be impactful.

Other
Other concepts that may share some similarities to open source are shareware, public domain software, freeware, and software viewers/readers that are freely available but do not provide source code. However, these differ from open source software in access to source code, licensing, copyright and fees.

Demographics
Despite being able to collaborate internationally, open source software contributors were found to mostly be located in large clusters such as Silicon Valley that largely collaborate within themselves. Possible reasons for this phenomenon may be that the OSS contributor demographic largely works in software, meaning that the OSS geographic location is closely related to that dispersion and collaborations could be encouraged through work and social networks. Code acceptance can be impacted by status within these social network clusters, creating unfair predispositions in code acceptance based on location. Barriers to international collaboration also include linguistic or cultural differences. Furthermore, each country has been shown to have a higher acceptance rate for code from contributors within their country except India, indicating a bias for culturally similar collaborators.

In 2021, the countries with the highest open source software contributions included the United States, China, Germany, India, and the UK, in that order. The counties with the highest OSS developers per capita from a study in 2021 include, in order, Iceland, Switzerland, Norway, Sweden, and Finland, while in 2008 the countries with top amount of estimated contributors in SourceForge were the United States, Germany, United Kingdom, Canada and France. Though there have been several studies done on the distribution and contributions of OSS developers, this is still an open field that can be measured in several different ways. For instance, Information and communication technology participation, population, wealth and proportion of access to the internet have been shown to be correlated with OSS contributions.

Although gender diversity has been found to enhance team productivity, women still face biases while contributing to open source software projects when their gender is identifiable. In 2002, only 1.5% of international open-source software developers were women, while women made up 28% of tech industry roles, demonstrating their low representation in the software field. Despite OSS contributions having no prerequisites, this gender bias may continue to exist due to the common belief of contributors that gender shouldn't matter, and the quality of code should be the only consideration for code acceptance, preventing the community from addressing the systemic disparities in female representation. However, a more recent figure of female OSS participation internationally calculated across 2005 to 2021 is 9.8%, with most being recent contributors, indicating that female participation may be growing.

Motivations
There are many motivations for contributing to the OSS community. For one, it is an opportunity to learn and practice multiple skills such as coding and other technology related abilities, but also fundamental skills such as communication and collaboration and practical skills needed to excel in technology related fields such as issue tracking or version control. Instead of learning through a classroom or a job, learning through contributing to OSS allows participants to learn at their own pace and follow what interests them. When contributing to OSS, you can learn the current industry best practices, technology and trends and even have the opportunity to contribute to the next big innovation as OSS grows increasingly popular within the tech field. Contributing to OSS without payment means there is no threat of being fired, though reputations can take a hit. On the other hand, a huge motivation to contribute to OSS is the reputation gained as you grow your public portfolio.

Disparities
Even though programming was originally seen as a female profession, there remains a large gap in computing. Social identity tends to be a large concern as women in the tech industry face insecurity about attracting unwanted male attention and harassment or being unfeminine in their technology knowledge, having a large impact on confidence. Some male tech participants make clear that they believe women fitting in within the culture is impossible, furthering the insecurity for women and their place in the tech industry. Additionally, even in a voluntary contribution environment like open source software, women tend to end up doing the less technical aspects of projects, such as manual testing or documentation despite women and men showing the same productivity in OSS contributions. Explicit biases include longer feedback time, more scrutinization of code and lower acceptance rate of code. Specifically in the open-source software community, women report that sexually offensive language is common and the women’s identity as female is given more attention that as an OSS contributor Bias is hard to address due to the belief that gender shouldn’t matter, with most contributors feeling that women getting special treatment is unfair and success should be dependent on skill, preventing any changes to be more inclusive.

Key projects
Open source software projects are built and maintained by a network of programmers, who may often be volunteers, and are widely used in free as well as commercial products.

Unix: Unix is an operating system created by AT&T that began as a precursor to open source software in that the free and open source software revolution began when developers began trying to create operating systems without Unix code. Unix was created in the 1960s, before the commercialization of software and before the concept of open source software was necessary, therefore it was not considered a true open source software project. It started as a research project before being commercialized in the mid 1980s. Before its commercialization, it represented many of the ideals held by the Free and Open source software revolution, including the decentralized collaboration of global users, rolling releases and a community culture of distaste towards proprietary software.

BSD: Berkely Software Distribution (BSD) is an operating system that began as a variant of Unix in 1978 that mixed Unix code with code from Berkely labs to increase functionality. As BSD was focused on increasing functionality, it would publicly share its greatest innovations with the main Unix operating system. This is an example of the free public code sharing that is a central characteristic of FOSS today. As Unix became commercialized in the 1980s, developers or members of the community who did not support proprietary software began to focus on BSD and turning it into an operating system that did not include any of Unix's code. The final version of BSD was released in 1995.

GNU: GNU is a free operating system created by Richard Stallman in 1984 with its name meaning Gnu's Not Unix. The idea was to create a Unix alternative operating system that was would be available for anyone to use and allow programmers to share code freely between them. However, the goal of GNU was not to only replace Unix, but to make a superior version that had more technological capabilities. It was released before the philosophical beliefs of the Free and Open source software revolution were truly defined. Because of its creation by prominent FOSS programmer Richard Stallman, GNU was heavily involved in FOSS activism, with one of the greatest achievements of GNU being the creation of the GNU General Public License or GPL, which allowed developers to release software that could be legally shared and modified.

Linux: Linux is an operating system kernel that was introduced in 1991 by Linus Torvalds. Linux was inspired by making a better version of the for profit operating service Minux. It was radically different than what other hackers were producing at the time due to it being totally free of cost and being decentralized. Later, Linux was put under the GPL license, allowing people to make money with Linux and bringing Linux into the FOSS community.

Apache: Apache began in 1995 as a collaboration between a group of developers releasing their own web server due to their frustration with NCSA HTTPd code base. The name Apache was used because of the several patches they applied to this code base. Within a year of its release, it became the worldwide leading web server. Soon, Apache came out with its own license, creating discord in the greater FOSS community, though ultimately proving successful. The Apache license allowed permitted members to directly access source code, a marked difference from GNU and Linux's approaches.

Extensions for non-software use
While the term open source applied originally only to the source code of software, it is now being applied to many other areas such as open source ecology, a movement to decentralize technologies so that any human can use them. However, it is often misapplied to other areas that have different and competing principles, which overlap only partially.

The same principles that underlie open-source software can be found in many other ventures, such as open source, open content, and open collaboration.

This "culture" or ideology takes the view that the principles apply more generally to facilitate concurrent input of different agendas, approaches, and priorities, in contrast with more centralized models of development such as those typically used in commercial companies.

Value
More than 90 percent of companies use open-source software as a component of their proprietary software. The decision to use open-source software, or even engage with open-source projects to improve existing open-source software, is typically a pragmatic business decision. When proprietary software is in direct competition with an open-source alternative, research has found conflicting results on the effect of the competition on the proprietary product's price and quality.

For decades, some companies have made servicing of an open-source software product for enterprise users their business model. These companies control an open-source software product, and instead of charging for licensing or use, charge for improvements, integration, and other servicing. Software as a service (SaaS) products based on open-source components are increasingly common.

Open-source software is preferred for scientific applications, because it increases transparency and aids in the validation and acceptance of scientific results.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Open-source software


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Aharonov–Bohm effect

The Aharonov–Bohm effect, sometimes called the Ehrenberg–Siday–Aharonov–Bohm effect, is a quantum-mechanical phenomenon in which an electrically charged particle is affected by an electromagnetic potential ($$\varphi$$, $$\mathbf{A}$$), despite being confined to a region in which both the magnetic field $$\mathbf{B}$$ and electric field $$\mathbf{E}$$ are zero. The underlying mechanism is the coupling of the electromagnetic potential with the complex phase of a charged particle's wave function, and the Aharonov–Bohm effect is accordingly illustrated by interference experiments.

The most commonly described case, sometimes called the Aharonov–Bohm solenoid effect, takes place when the wave function of a charged particle passing around a long solenoid experiences a phase shift as a result of the enclosed magnetic field, despite the magnetic field being negligible in the region through which the particle passes and the particle's wavefunction being negligible inside the solenoid. This phase shift has been observed experimentally. There are also magnetic Aharonov–Bohm effects on bound energies and scattering cross sections, but these cases have not been experimentally tested. An electric Aharonov–Bohm phenomenon was also predicted, in which a charged particle is affected by regions with different electrical potentials but zero electric field, but this has no experimental confirmation yet. A separate "molecular" Aharonov–Bohm effect was proposed for nuclear motion in multiply connected regions, but this has been argued to be a different kind of geometric phase as it is "neither nonlocal nor topological", depending only on local quantities along the nuclear path.

Werner Ehrenberg (1901–1975) and Raymond E. Siday first predicted the effect in 1949. Yakir Aharonov and David Bohm published their analysis in 1959. After publication of the 1959 paper, Bohm was informed of Ehrenberg and Siday's work, which was acknowledged and credited in Bohm and Aharonov's subsequent 1961 paper. The effect was confirmed experimentally, with a very large error, while Bohm was still alive. By the time the error was down to a respectable value, Bohm had died.

Significance
In the 18th and 19th centuries, physics was dominated by Newtonian dynamics, with its emphasis on forces. Electromagnetic phenomena were elucidated by a series of experiments involving the measurement of forces between charges, currents and magnets in various configurations. Eventually, a description arose according to which charges, currents and magnets acted as local sources of propagating force fields, which then acted on other charges and currents locally through the Lorentz force law. In this framework, because one of the observed properties of the electric field was that it was irrotational, and one of the observed properties of the magnetic field was that it was divergenceless, it was possible to express an electrostatic field as the gradient of a scalar potential (e.g. Coulomb's electrostatic potential, which is mathematically analogous to the classical gravitational potential) and a stationary magnetic field as the curl of a vector potential (then a new concept – the idea of a scalar potential was already well accepted by analogy with gravitational potential). The language of potentials generalised seamlessly to the fully dynamic case but, since all physical effects were describable in terms of the fields which were the derivatives of the potentials, potentials (unlike fields) were not uniquely determined by physical effects: potentials were only defined up to an arbitrary additive constant electrostatic potential and an irrotational stationary magnetic vector potential.

The Aharonov–Bohm effect is important conceptually because it bears on three issues apparent in the recasting of (Maxwell's) classical electromagnetic theory as a gauge theory, which before the advent of quantum mechanics could be argued to be a mathematical reformulation with no physical consequences. The Aharonov–Bohm thought experiments and their experimental realization imply that the issues were not just philosophical.

The three issues are:
 * 1) whether potentials are "physical" or just a convenient tool for calculating force fields;
 * 2) whether action principles are fundamental;
 * 3) the principle of locality.

Because of reasons like these, the Aharonov–Bohm effect was chosen by the New Scientist magazine as one of the "seven wonders of the quantum world".

Chen-Ning Yang considered the AB effect to be only direct experimental proof of the gauge principle. The philosophical import is that the magnetic four-potential $$(\phi, \mathbf A)$$ overdescribes the physics, in that all observable phenomena are the same after we perform a gauge transformation. The Maxwell fields underdescribe the physics, as it does not predict the AB effect. And just as the gauge principle predicts, the quantities that are invariant under gauge transforms are exactly the physically observable phenomena.

Potentials vs. fields
It is generally argued that the Aharonov–Bohm effect illustrates the physicality of electromagnetic potentials, Φ and A, in quantum mechanics. Classically it was possible to argue that only the electromagnetic fields are physical, while the electromagnetic potentials are purely mathematical constructs, that due to gauge freedom are not even unique for a given electromagnetic field.

However, Vaidman has challenged this interpretation by showing that the Aharonov–Bohm effect can be explained without the use of potentials so long as one gives a full quantum mechanical treatment to the source charges that produce the electromagnetic field. According to this view, the potential in quantum mechanics is just as physical (or non-physical) as it was classically. Aharonov, Cohen, and Rohrlich responded that the effect may be due to a local gauge potential or due to non-local gauge-invariant fields.

Two papers published in the journal Physical Review A in 2017 have demonstrated a quantum mechanical solution for the system. Their analysis shows that the phase shift can be viewed as generated by a solenoid's vector potential acting on the electron or the electron's vector potential acting on the solenoid or the electron and solenoid currents acting on the quantized vector potential.

Global action vs. local forces
Similarly, the Aharonov–Bohm effect illustrates that the Lagrangian approach to dynamics, based on energies, is not just a computational aid to the Newtonian approach, based on forces. Thus the Aharonov–Bohm effect validates the view that forces are an incomplete way to formulate physics, and potential energies must be used instead. In fact Richard Feynman complained that he had been taught electromagnetism from the perspective of electromagnetic fields, and he wished later in life he had been taught to think in terms of the electromagnetic potential instead, as this would be more fundamental. In Feynman's path-integral view of dynamics, the potential field directly changes the phase of an electron wave function, and it is these changes in phase that lead to measurable quantities.

Locality of electromagnetic effects
The Aharonov–Bohm effect shows that the local E and B fields do not contain full information about the electromagnetic field, and the electromagnetic four-potential, (Φ, A), must be used instead. By Stokes' theorem, the magnitude of the Aharonov–Bohm effect can be calculated using the electromagnetic fields alone, or using the four-potential alone. But when using just the electromagnetic fields, the effect depends on the field values in a region from which the test particle is excluded. In contrast, when using just the four-potential, the effect only depends on the potential in the region where the test particle is allowed. Therefore, one must either abandon the principle of locality, which most physicists are reluctant to do, or accept that the electromagnetic four-potential offers a more complete description of electromagnetism than the electric and magnetic fields can. On the other hand, the Aharonov–Bohm effect is crucially quantum mechanical; quantum mechanics is well known to feature non-local effects (albeit still disallowing superluminal communication), and Vaidman has argued that this is just a non-local quantum effect in a different form.

In classical electromagnetism the two descriptions were equivalent. With the addition of quantum theory, though, the electromagnetic potentials Φ and A are seen as being more fundamental. Despite this, all observable effects end up being expressible in terms of the electromagnetic fields, E and B. This is interesting because, while you can calculate the electromagnetic field from the four-potential, due to gauge freedom the reverse is not true.

Magnetic solenoid effect
The magnetic Aharonov–Bohm effect can be seen as a result of the requirement that quantum physics must be invariant with respect to the gauge choice for the electromagnetic potential, of which the magnetic vector potential $$\mathbf{A}$$ forms part.

Electromagnetic theory implies that a particle with electric charge $$q$$ traveling along some path $$P$$ in a region with zero magnetic field $$\mathbf{B}$$, but non-zero $$\mathbf{A}$$ (by $$\mathbf{B} = \mathbf{0} = \nabla \times \mathbf{A}$$), acquires a phase shift $$\varphi$$, given in SI units by

 * $$\varphi = \frac{q}{\hbar} \int_P \mathbf{A} \cdot d\mathbf{x},$$

Therefore, particles, with the same start and end points, but traveling along two different routes will acquire a phase difference $$\Delta \varphi$$ determined by the magnetic flux $$\Phi_B$$ through the area between the paths (via Stokes' theorem and $$\nabla \times \mathbf{A} = \mathbf{B}$$), and given by:

 * $$\Delta\varphi = \frac{q \, \Phi_B}{\hbar}.$$

In quantum mechanics the same particle can travel between two points by a variety of paths. Therefore, this phase difference can be observed by placing a solenoid between the slits of a double-slit experiment (or equivalent). An ideal solenoid (i.e. infinitely long and with a perfectly uniform current distribution) encloses a magnetic field $$\mathbf{B}$$, but does not produce any magnetic field outside of its cylinder, and thus the charged particle (e.g. an electron) passing outside experiences no magnetic field $$\mathbf{B}$$. (This idealization simplifies the analysis but it's important to realize that the Aharonov-Bohm effect does not rely on it, provided the magnetic flux returns outside the electron paths, for example if one path goes through a toroidal solenoid and the other around it, and the solenoid is shielded so that it produces no external magnetic field.) However, there is a (curl-free) vector potential $$\mathbf{A}$$ outside the solenoid with an enclosed flux, and so the relative phase of particles passing through one slit or the other is altered by whether the solenoid current is turned on or off. This corresponds to an observable shift of the interference fringes on the observation plane.

The same phase effect is responsible for the quantized-flux requirement in superconducting loops. This quantization occurs because the superconducting wave function must be single valued: its phase difference $$\Delta \varphi$$ around a closed loop must be an integer multiple of $$2\pi$$ (with the charge $$q=2e$$ for the electron Cooper pairs), and thus the flux must be a multiple of $$h/2e$$. The superconducting flux quantum was actually predicted prior to Aharonov and Bohm, by F. London in 1948 using a phenomenological model.

The first claimed experimental confirmation was by Robert G. Chambers in 1960, in an electron interferometer with a magnetic field produced by a thin iron whisker, and other early work is summarized in Olariu and Popèscu (1984). However, subsequent authors questioned the validity of several of these early results because the electrons may not have been completely shielded from the magnetic fields. An early experiment in which an unambiguous Aharonov–Bohm effect was observed by completely excluding the magnetic field from the electron path (with the help of a superconducting film) was performed by Tonomura et al. in 1986. The effect's scope and application continues to expand. Webb et al. (1985) demonstrated Aharonov–Bohm oscillations in ordinary, non-superconducting metallic rings; for a discussion, see Schwarzschild (1986) and Imry & Webb (1989). Bachtold et al. (1999) detected the effect in carbon nanotubes; for a discussion, see Kong et al. (2004).

Monopoles and Dirac strings
The magnetic Aharonov–Bohm effect is also closely related to Dirac's argument that the existence of a magnetic monopole can be accommodated by the existing magnetic source-free Maxwell's equations if both electric and magnetic charges are quantized.

A magnetic monopole implies a mathematical singularity in the vector potential, which can be expressed as a Dirac string of infinitesimal diameter that contains the equivalent of all of the 4πg flux from a monopole "charge" g. The Dirac string starts from, and terminates on, a magnetic monopole. Thus, assuming the absence of an infinite-range scattering effect by this arbitrary choice of singularity, the requirement of single-valued wave functions (as above) necessitates charge-quantization. That is, $$2\frac{q_\text{e}q_\text{m}}{\hbar c}$$ must be an integer (in cgs units) for any electric charge qe and magnetic charge qm.

Like the electromagnetic potential A the Dirac string is not gauge invariant (it moves around with fixed endpoints under a gauge transformation) and so is also not directly measurable.

Electric effect
Just as the phase of the wave function depends upon the magnetic vector potential, it also depends upon the scalar electric potential. By constructing a situation in which the electrostatic potential varies for two paths of a particle, through regions of zero electric field, an observable Aharonov–Bohm interference phenomenon from the phase shift has been predicted; again, the absence of an electric field means that, classically, there would be no effect.

From the Schrödinger equation, the phase of an eigenfunction with energy $$E$$ goes as $$e^{-iEt/\hbar}$$. The energy, however, will depend upon the electrostatic potential $$V$$ for a particle with charge $$q$$. In particular, for a region with constant potential $$V$$ (zero field), the electric potential energy $$qV$$ is simply added to $$E$$, resulting in a phase shift:

 * $$\Delta\varphi = -\frac{qVt}{\hbar} ,$$

where t is the time spent in the potential.

For example, we may have a pair of large flat conductors, connected to a battery of voltage $$\Delta V$$. Then, we can run a single electron double-slit experiment, with the two slits on the two sides of the pair of conductors. If the electron takes time $$t$$ to hit the screen, then we should observe a phase shift $$e\Delta V t/\hbar$$. By adjusting the battery voltage, we can horizontally shift the interference pattern on the screen.

The initial theoretical proposal for this effect suggested an experiment where charges pass through conducting cylinders along two paths, which shield the particles from external electric fields in the regions where they travel, but still allow a time dependent potential to be applied by charging the cylinders. This proved difficult to realize, however. Instead, a different experiment was proposed involving a ring geometry interrupted by tunnel barriers, with a constant bias voltage V relating the potentials of the two halves of the ring. This situation results in an Aharonov–Bohm phase shift as above, and was observed experimentally in 1998, albeit in a setup where the charges do traverse the electric field generated by the bias voltage. The original time dependent electric Aharonov–Bohm effect has not yet found experimental verification.

Gravitational effect
The Aharonov–Bohm phase shift due to the gravitational potential should also be possible to observe in theory, and in early 2022  an experiment was carried out to observe it based on an experimental design from 2012. In the experiment, ultra-cold rubidium atoms in superposition were launched vertically inside a vacuum tube and split with a laser so that one part would go higher than the other and then recombined back. Outside of the chamber at the top sits an axially symmetric mass that changes the gravitational potential. Thus, the part that goes higher should experience a greater change which manifests as an interference pattern when the wave packets recombine resulting in a measurable phase shift. Evidence of a match between the measurements and the predictions was found by the team. Multiple other tests have been proposed.

Non-abelian effect
In 1975 Tai-Tsun Wu and Chen-Ning Yang formulated the non-Abelian Aharonov–Bohm effect, and in 2019 this was experimentally reported in a system with light waves rather than the electron wave function. The effect was produced in two different ways. In one light went through a crystal in strong magnetic field and in another light was modulated using time-varying electrical signals. In both cases the phase shift was observed via an interference pattern which was also different depending if going forwards and backwards in time.

Aharonov–Bohm nano rings
Nano rings were created by accident while intending to make quantum dots. They have interesting optical properties associated with excitons and the Aharonov–Bohm effect. Application of these rings used as light capacitors or buffers includes photonic computing and communications technology. Analysis and measurement of geometric phases in mesoscopic rings is ongoing. It is even suggested they could be used to make a form of slow glass.

Several experiments, including some reported in 2012, show Aharonov–Bohm oscillations in charge density wave (CDW) current versus magnetic flux, of dominant period h/2e through CDW rings up to 85 μm in circumference above 77 K. This behavior is similar to that of the superconducting quantum interference devices (see SQUID).

Mathematical interpretation
The Aharonov–Bohm effect can be understood from the fact that one can only measure absolute values of the wave function. While this allows for measurement of phase differences through quantum interference experiments, there is no way to specify a wavefunction with constant absolute phase. In the absence of an electromagnetic field one can come close by declaring the eigenfunction of the momentum operator with zero momentum to be the function "1" (ignoring normalization problems) and specifying wave functions relative to this eigenfunction "1". In this representation the i-momentum operator is (up to a factor $$\hbar/i$$) the differential operator $$ \partial_i = \frac{\partial}{\partial x^i}$$. However, by gauge invariance, it is equally valid to declare the zero momentum eigenfunction to be $$ e^{-i\phi(x)} $$ at the cost of representing the i-momentum operator (up to a factor) as $$\nabla_i = \partial_i + i(\partial_i \phi)$$ i.e. with a pure gauge vector potential $$A = d\phi$$. There is no real asymmetry because representing the former in terms of the latter is just as messy as representing the latter in terms of the former. This means that it is physically more natural to describe wave "functions", in the language of differential geometry, as sections in a complex line bundle with a hermitian metric and a U(1)-connection $$\nabla$$. The curvature form of the connection, $$i F = \nabla\wedge\nabla $$, is, up to the factor i, the Faraday tensor of the electromagnetic field strength. The Aharonov–Bohm effect is then a manifestation of the fact that a connection with zero curvature (i.e. flat), need not be trivial since it can have monodromy along a topologically nontrivial path fully contained in the zero curvature (i.e. field-free) region. By definition this means that sections that are parallelly translated along a topologically non trivial path pick up a phase, so that covariant constant sections cannot be defined over the whole field-free region.

Given a trivialization of the line-bundle, a non-vanishing section, the U(1)-connection is given by the 1-form corresponding to the electromagnetic four-potential A as $$ \nabla = d + i A\,$$ where d means exterior derivation on the Minkowski space. The monodromy is the holonomy of the flat connection. The holonomy of a connection, flat or non flat, around a closed loop $$\gamma$$ is $$e^{i \int_\gamma A}$$ (one can show this does not depend on the trivialization but only on the connection). For a flat connection one can find a gauge transformation in any simply connected field free region(acting on wave functions and connections) that gauges away the vector potential. However, if the monodromy is nontrivial, there is no such gauge transformation for the whole outside region. In fact as a consequence of Stokes' theorem, the holonomy is determined by the magnetic flux through a surface $$\sigma$$ bounding the loop $$\gamma$$, but such a surface may exist only if $$\sigma$$ passes through a region of non trivial field:

 * $$ e^{i \int_{\partial \sigma} A} = e^{i \int_{\sigma} dA} = e^{i\int_\sigma F}$$

The monodromy of the flat connection only depends on the topological type of the loop in the field free region (in fact on the loops homology class). The holonomy description is general, however, and works inside as well as outside the superconductor. Outside of the conducting tube containing the magnetic field, the field strength $$F = 0$$. In other words, outside the tube the connection is flat, and the monodromy of the loop contained in the field-free region depends only on the winding number around the tube. The monodromy of the connection for a loop going round once (winding number 1) is the phase difference of a particle interfering by propagating left and right of the superconducting tube containing the magnetic field. If one wants to ignore the physics inside the superconductor and only describe the physics in the outside region, it becomes natural and mathematically convenient to describe the quantum electron by a section in a complex line bundle with an "external" flat connection $$\nabla$$ with monodromy

 * $$\alpha = $$ magnetic flux through the tube / $$(\hbar/e)$$

rather than an external EM field $$F$$. The Schrödinger equation readily generalizes to this situation by using the Laplacian of the connection for the (free) Hamiltonian

 * $$ H = \frac{1}{2m}\nabla^*\nabla $$.

Equivalently, one can work in two simply connected regions with cuts that pass from the tube towards or away from the detection screen. In each of these regions the ordinary free Schrödinger equations would have to be solved, but in passing from one region to the other, in only one of the two connected components of the intersection (effectively in only one of the slits) a monodromy factor $$ e^{i\alpha}$$ is picked up, which results in the shift in the interference pattern as one changes the flux.

Effects with similar mathematical interpretation can be found in other fields. For example, in classical statistical physics, quantization of a molecular motor motion in a stochastic environment can be interpreted as an Aharonov–Bohm effect induced by a gauge field acting in the space of control parameters.

title: Aharonov–Bohm effect


text:
Jack Frost

Jack Frost is a personification of frost, ice, snow, sleet, winter, and freezing cold. He is a variant of Old Man Winter who is held responsible for frosty weather, nipping the fingers and toes in such weather, coloring the foliage in autumn, and leaving fern-like patterns on cold windows in winter.

Starting in late 19th century literature, more developed characterizations of Jack Frost depict him as a sprite-like character, sometimes appearing as a sinister mischief-maker or as a hero.

Background
Jack Frost is traditionally said to leave the frosty, fern-like patterns on windows on cold winter mornings (window frost or fern frost) and nipping the extremities in cold weather. Over time, window frost has become far less prevalent in the modern world due to the advance of double-glazing, but Jack Frost remains a well-known figure in popular culture. He is sometimes described or depicted with paint brush and bucket coloring the autumnal foliage red, yellow, brown, and orange.

History
Tales of Jack Frost may originate from Anglo-Saxon and Norse winter customs. The Finnish equivalent Pakkasukko has an entire chapter named after him in Kalevala, the Finnish national epic compiled from their ancient oral tradition. In Swedish folklore, the equivalent is Kung Bore (King Bore); the name originating from Swedish 17'th century writer Olaus Rudbeck.

There are various other mythological beings who take on a similar role yet have a unique folklore to them. In Russia, he has taken on a different form as Grandfather Frost, and in Germany there is instead a different entity altogether known as Mrs. Holle. The Hindu Kush mountain range is named after stories of a resident giant who would kill (kesh) those who attempted to pass, and has been compared to England's Jack Frost.

The earliest reference to Jack Frost in literature is in the book 'Round About Our Coal Fire, or Christmas Entertainments' published in 1734.

Jack Frost is mentioned in many songs – such as the wintertime song "The Christmas Song" (aka "Chestnuts Roasting on an Open Fire") – and movies. He has been presented as a villain in some media and a hero in others.

Literature

 * Hannah Flagg Gould's (1789-1865) poem "The Frost" features a mischievous being responsible for the quieter phenomena of winter, beautiful ice paintings on windows but who also got upset at lack of gifts and caused the cold to break and ruin things.
 * In Margaret T. Canby's "Birdie and His Fairy Friends" (1874), there is a short story titled "The Frost Fairies." In this story, Jack Frost is the king of the Winter Spirits and is described as a kind fellow who wants to help children, whereas a king of a neighboring kingdom, King Winter, is cruel to them. The story tells the origins of how Jack Frost began to oversee the coloring of the leaves of the forest in fall.
 * In 1891 Helen Keller made her own reproduction of the story, titled The Frost King.
 * In Charles Sangster's "Little Jack Frost", published in The Aldine, (Vol.7, No.16, 1875) Jack Frost is a playful being who runs around playing pranks and 'nose-biting', coating places with snow before being chased off by Dame Nature for spring.
 * In L. Frank Baum's The Life and Adventures of Santa Claus (1902), Jack Frost is the son of the otherwise unnamed Frost King. He takes pleasure in nipping "scores of noses and ears and toes", but Santa Claus, who likes Jack (who he sees as a "jolly rogue") though he mistrusts him, asks him to spare the children. Jack says he will, if he can resist the temptation. The same Jack appears in "The Runaway Shadows", a short story by Baum. In this story, he has the power to freeze shadows, separating them from their owners, making them their own living entities.
 * In Laurell K. Hamilton's Meredith Gentry series, a character emerges as the original Jack Frost.
 * Jack Frost has appeared as a minor character in the Rupert Bear stories.
 * In the Rainbow Magic books by Daisy Meadows, Jack Frost is an antagonist who causes trouble in Fairyland. He is accompanied by pesky goblins who steal the fairies' important objects, and try to sabotage them.
 * Jack Frost also appears in "First Death in Nova Scotia", a poem by Elizabeth Bishop.
 * In the novel Hogfather by Terry Pratchett, Jack grows tired of "fern patterns" and paints more elaborate pictures on windows.
 * Jack Frost appears in The Veil trilogy of novels by Christopher Golden.
 * The Man Jack, an enigmatic assassin, calls himself Jack Frost in The Graveyard Book by Neil Gaiman.
 * The Stranger, a picture book by Chris Van Allsburg, stars Jack Frost as a lonely stranger with amnesia.
 * In Amy Wilson's The Lost Frost Girl, Jack Frost’s daughter Owl discovers powers she’s inherited from Jack and ventures into the world of the fae.

Comics

 * Jack Frost appears in Windsor McCay's comic strip Little Nemo in Slumberland, first being mentioned in a strip published in 1906 and then appearing in another strip published the subsequent year.
 * In comic books, Jack Frost appears as a superhero in works published by Timely Comics ( Marvel Comics) in the 1940s.
 * Marvel Comics had a second Jack Frost, the first alias of the original Blizzard.
 * Jack Frost is the alias of Dane McGowan one of the main characters from the 1990s Vertigo series The Invisibles.
 * In Jack of Fables (a Fables spinoff) the titular character became Jack Frost for a period of time. A second Jack Frost ("Jack too, or Jack two") appears as the son of Jack Horner and The Snow Queen.

Films

 * Jack Frost, a Russian film from 1964, has the title Morozko—the Russian equivalent of Jack Frost.
 * In the 1997 film Jack Frost, Scott MacDonald plays a serial killer turned into a snowman who continues his rampage. This movie spawned a sequel in 2000: Jack Frost 2: Revenge of the Mutant Killer Snowman.
 * In the 1998 film Jack Frost, Michael Keaton plays a human turned into a snowman by the name of Jack Frost.
 * Jack Frost appears as the primary antagonist in The Santa Clause 3: The Escape Clause played by Martin Short.
 * Jack Frost appears as the main protagonist of Rise of the Guardians, loosely based on the series Guardians of Childhood by William Joyce, voiced by Chris Pine.

Radio, animation, and television

 * Jack Frost appears in the children's radio serial The Cinnamon Bear.
 * Jack Frost appears as the title character in Ub Iwerks's 1934 animated short Jack Frost in the ComiColor Cartoons series.
 * Jack Frost is the title character (voiced by Robert Morse) of a Rankin/Bass TV special produced in 1979. The character (voiced by Paul Frees) also makes an appearance in the Christmas television special Frosty's Winter Wonderland and in Rudolph and Frosty's Christmas in July, voiced again by Paul Frees.

Video games
Jack Frost has appeared in many video games including:

 * Megami Tensei (game series including spin-offs such as Shin Megami Tensei, Jack Bros., and Persona; this iteration of Jack Frost is also the mascot for the games' developer, Atlus)
 * AdventureQuest
 * City of Villains
 * Granado Espada
 * Guild Wars
 * Killing Floor
 * Ragnarok Online
 * Rise of the Guardians: The Video Game (which is based on the film of the same name)
 * RuneScape
 * Scribblenauts Unlimited
 * The Sims FreePlay

Music

 * Jethro Tull has a song titled "Jack Frost and the Hooded Crow"
 * Saint Vitus depict Jack Frost as an evil spirit of winter on their album V
 * The radio station WRHS-FM 89.7 in Norridge, Illinois brands its Christmas music format "Jack Frost"
 * The name has been employed as a pseudonym by musicians Bob Dylan and Jack Dempsey
 * Jack Frost, an Austrian doom metal band
 * Jack Frost, an Australian duo
 * Jack Frost, a 1991 album by Jack Frost
 * The song "The Christmas Song" by Nat King Cole has the lyric "Jack Frost nipping at your nose"
 * The Burt Bacharach song "The Bell That Couldn't Jingle" features Jack Frost "freezing a teardrop"
 * The song "Intro" by Pusha T has the lyric "I’m Jack Frost of sellin’ that blast off"

title: Jack Frost


text:
Tomcats (2001 film)

Tomcats is a 2001 American sex comedy film written and directed by Gregory Poirier. It stars Jerry O'Connell, Shannon Elizabeth, and Jake Busey. This film also features Dakota Fanning in her film acting debut.

It is the first film released by Revolution Studios.

Plot
A group of men, Michael Delaney, Kyle Brenner, Steve and others, are talking before the wedding ceremony of one of their friends. They agree to a pact whereby they will all invest in a fund that only the last bachelor will inherit.

Michael goes to Las Vegas years later with Steve and his girlfriend. After going back to their room, he mistakenly thinks she wants him to say "I love you". He panics and kicks her out of the room and she dumps him. He goes on a bender and picks up a sultry redhead who encourages him to gamble at the roulette table. He goes on an amazing run of bad luck, losing everything and taking out credit from the casino. In the end, the casino intervenes and he is summoned to meet the pit boss Carlos. Carlos informs Michael how serious the trouble he is in and what will happen if he fails to repay the debt.

Michael realizes that he and Kyle are now the last men standing and stand to inherit the entire fund, which has swelled to an enormous amount. He approaches Kyle and learns that the latter is wealthy and a jerk. He decides that his only option is to set Kyle up so that Michael can inherit the fund. He learns on a drunken night that Kyle has only ever loved one woman, a lady named Natalie, whom he met at Steve's wife Tricia's sister's wedding.

Michael approaches Steve and Tricia and finds out where Natalie is. Unfortunately, they neglect to tell him that Natalie is a police detective. Natalie is working undercover as a streetwalker when Michael finds her and he inadvertently gets himself arrested for solicitation. During interrogation, he admits everything and is eventually released.

Natalie visits him and agrees to go in with his plan on the basis that she gets half the money. He finds out that Kyle's memory of Natalie and his night together left out some important details, notably that, when Natalie woke up, Kyle had bolted, leaving her with a bag of quarters to get a cab home.

Michael and Natalie decide to spend time together investigating Kyle in order to ensure their plan works. It all appears to be working until Michael realizes he is falling in love with Natalie. Natalie too is falling for Michael, but decides to go ahead with marrying Kyle. During the bachelor party, Kyle informs Michael that he has no intention of quitting womanizing and believes Natalie to be the perfect stay-at-home wife.

Before Michael can act and prevent the wedding, he is knocked unconscious by a bowling ball. He wakes up and discovers that his clothes are missing. He grabs some clothing from an unconscious stripper and tries to run to the civic hall to prevent the marriage. After a number of misadventures, Michael finally arrives, but learns that he is too late and Kyle has already married Natalie. He leaves, somewhat dejected.

Michael goes on to claim the fund and thereby settles his debt with Carlos. He later bumps into Kyle who is on a date with another girl and punches him as he believes Kyle to be cheating on Natalie. Kyle beats up Michael and then discloses that he has split with Natalie. On their wedding night, she knocked him unconscious and when he woke up, he found a bag of quarters. Michael runs off to find Natalie and finds her again working undercover. He proposes that they date and he gives up his tomcat ways. The pair ultimately marry. Kyle ends up with the librarian Michael was with earlier.

Box office
Tomcats debuted alongside Spy Kids, Someone Like You, and The Tailor of Panama. The film grossed $6.4 million in its opening weekend in the United States and Canada, debuting in fourth place. Overall, the film grossed $13.6 million in the United States and Canada, contributing to a $23.4 million worldwide total.

Critical response
Tomcats holds a 14% approval rating on review aggregator Rotten Tomatoes, based on 77 critic reviews with an average rating of 3.4/10. The website's critics consensus is: "Why even bother? You already know if you're going to see it or not." On Metacritic, the film has a weighted average score of 15 out of 100 based on 19 critics, indicating "overwhelming dislike". Audiences polled by CinemaScore gave the film an average grade of "C+" on an A+ to F scale. Peter Travers of Rolling Stone said "Tomcats is laced with such rampant misogyny that the laughs stick in your throat." The New York Times said, "The film is enthusiastically vulgar but not particularly funny, perhaps because it too often loses the distinction between gross-out humor and the merely gross."

title: Tomcats (2001 film)