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Clay Mathematics Institute

The Clay Mathematics Institute (CMI) is a private, non-profit foundation dedicated to increasing and disseminating mathematical knowledge. Formerly based in Peterborough, New Hampshire, the corporate address is now in Denver, Colorado. CMI's scientific activities are managed from the President's office in Oxford, United Kingdom. It gives out various awards and sponsorships to promising mathematicians. The institute was founded in 1998 through the sponsorship of Boston businessman Landon T. Clay. Harvard mathematician Arthur Jaffe was the first president of CMI.

While the institute is best known for its Millennium Prize Problems, it carries out a wide range of activities, including a postdoctoral program (ten Clay Research Fellows are supported currently ), conferences, workshops, and summer schools.

Governance
The institute is run according to a standard structure comprising a scientific advisory committee that decides on grant-awarding and research proposals, and a board of directors that oversees and approves the committee's decisions. , the board is made up of members of the Clay family, whereas the advisory committee is composed of Simon Donaldson, Michael Hopkins, Andrei Okounkov, Gigliola Staffilani, and Andrew Wiles. Martin R. Bridson is the current president of CMI.

2024 Clay Research Fellows
The Clay Mathematics Institute has announced that Ishan Levy and Mehtaab Sawhney have been awarded the 2024 Clay Research Fellowships. Both are completing their PhDs at the Massachusetts Institute of Technology and will start their five-year fellowships on July 1, 2024.

2024 Clay Research Conference and Workshops
The 2024 Clay Research Conference will be held on October 2, 2024, at the Mathematical Institute, University of Oxford. The conference will be accompanied by workshops from September 30 to October 4, 2024. Notable workshops include:
 * New Advances in the Langlands Program: Geometry and Arithmetic
 * New Frontiers in Probabilistic and Extremal Combinatorics
 * The P=W Conjecture in Non Abelian Hodge Theory

Awards and Recognitions
Daniel Graham from the University of Surrey won the Gold Medal for Mathematical Sciences at the 2024 STEM for Britain competition for his work on quantum authentication methods.

Millennium Prize Problems
The institute is best known for establishing the Millennium Prize Problems on May 24, 2000. These seven problems are considered by CMI to be "important classic questions that have resisted solution over the years." For each problem, the first person to solve it will be awarded US$1,000,000 by the CMI. In announcing the prize, CMI drew a parallel to Hilbert's problems, which were proposed in 1900, and had a substantial impact on 20th century mathematics. Of the initial 23 Hilbert problems, most of which have been solved, only the Riemann hypothesis (formulated in 1859) is included in the seven Millennium Prize Problems.

For each problem, the Institute had a professional mathematician write up an official statement of the problem, which will be the main standard against which a given solution will be measured. The seven problems are:

 * P versus NP
 * The Hodge conjecture
 * The Poincaré conjecture—solved, by Grigori Perelman
 * The Riemann hypothesis
 * Yang–Mills existence and mass gap
 * Navier–Stokes existence and smoothness
 * The Birch and Swinnerton-Dyer conjecture

Some of the mathematicians who were involved in the selection and presentation of the seven problems were Michael Atiyah, Enrico Bombieri, Alain Connes, Pierre Deligne, Charles Fefferman, John Milnor, David Mumford, Andrew Wiles, and Edward Witten.

The Clay Research Award
In recognition of major breakthroughs in mathematical research, the institute has an annual prize – the Clay Research Award. Its recipients to date are Ian Agol, Manindra Agrawal, Yves Benoist, Manjul Bhargava, Tristan Buckmaster, Danny Calegari, Alain Connes, Nils Dencker, Alex Eskin, David Gabai, Ben Green, Mark Gross, Larry Guth, Christopher Hacon, Richard S. Hamilton, Michael Harris, Philip Isett, Jeremy Kahn, Nets Katz, Laurent Lafforgue, Gérard Laumon, Aleksandr Logunov, Eugenia Malinnikova, Vladimir Markovic, James McKernan, Jason Miller, Maryam Mirzakhani, Ngô Bảo Châu, Rahul Pandharipande, Jonathan Pila, Jean-François Quint, Peter Scholze, Oded Schramm, Scott Sheffield, Bernd Siebert, Stanislav Smirnov, Terence Tao, Clifford Taubes, Richard Taylor, Maryna Viazovska, Vlad Vicol, Claire Voisin, Jean-Loup Waldspurger, Andrew Wiles, Geordie Williamson, Edward Witten and Wei Zhang.

Other activities
Besides the Millennium Prize Problems, the Clay Mathematics Institute supports mathematics via the awarding of research fellowships (which range from two to five years and are aimed at younger mathematicians), as well as shorter-term scholarships for programs, individual research, and book writing. The institute also has a yearly Clay Research Award, recognizing major breakthroughs in mathematical research. Finally, the institute organizes a number of summer schools, conferences, workshops, public lectures, and outreach activities aimed primarily at junior mathematicians (from the high school to the postdoctoral level). CMI publications are available in PDF form at most six months after they appear in print.

title: Clay Mathematics Institute


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Cerebral arteriovenous malformation

A cerebral arteriovenous malformation (cerebral AVM, CAVM, cAVM, brain AVM, or BAVM) is an abnormal connection between the arteries and veins in the brain—specifically, an arteriovenous malformation in the cerebrum.

Signs and symptoms
The most frequently observed problems related to a cerebral arteriovenous malformation (AVM) are headaches and seizures, cranial nerve afflictions including pinched nerve and palsy, backaches, neckaches, and nausea from coagulated blood that has made its way down to be dissolved in the cerebrospinal fluid. Perhaps 15% of the population at detection are asymptomatic. Other common symptoms are a pulsing noise in the head, progressive weakness, numbness and vision changes as well as debilitating, excruciating pain.

In serious cases, blood vessels rupture and cause bleeding within the brain (intracranial hemorrhage). In more than half of patients with AVM, this is the first symptom. Symptoms due to bleeding include loss of consciousness, sudden and severe headache, nausea, vomiting, incontinence, and blurred vision, amongst others. Impairments caused by local brain-tissue damage on the bleed site are also possible, including seizure, one-sided weakness (hemiparesis), a loss of touch sensation on one side of the body and deficits in language processing (aphasia). Ruptured AVMs are responsible for considerable mortality and morbidity.

AVMs in certain critical locations may stop the circulation of the cerebrospinal fluid, causing it to accumulate within the skull and giving rise to a clinical condition called hydrocephalus. A stiff neck can occur as the result of increased pressure within the skull and irritation of the meninges.

Pathophysiology
A cerebral AVM is an abnormal anastomosis (connection) between the arteries and veins in the human brain and are most commonly of prenatal origin. In a normal brain, oxygen-enriched blood from the heart travels in sequence through smaller blood vessels going from arteries, to arterioles and then capillaries. Oxygen is removed in the capillaries to be used by the brain. After the oxygen is removed, blood reaches venules and later veins which will take it back to the heart and lungs. A cerebral AVM causes blood to travel from arteries to veins through the abnormal connections, disrupting normal circulation.

Diagnosis

A cerebral AVM diagnosis is established by neuroimaging studies after a complete neurological and physical examination. Three main techniques are used to visualize the brain and search for an AVM: computed tomography (CT), magnetic resonance imaging (MRI), and cerebral angiography. A CT scan of the head is usually performed first when the subject is symptomatic. It can suggest the approximate site of the bleed. MRI is more sensitive than CT in the diagnosis, and provides better information about the exact location of the malformation. More detailed pictures of the tangle of blood vessels that compose an AVM can be obtained by using radioactive agents injected into the blood stream. If a CT is used in conjunction with an angiogram, this is called a computerized tomography angiogram; while, if MRI is used it is called magnetic resonance angiogram. The best images of a cerebral AVM are obtained through cerebral angiography. This procedure involves using a catheter, threaded through an artery up to the head, to deliver a contrast agent into the AVM. As the contrast agent flows through the AVM structure, a sequence of X-ray images are obtained.

Spetzler-Martin (SM) Grade
A common method of grading cerebral AVMs is the Spetzler-Martin (SM) grade. This system was designed to assess the patient's risk of neurological deficit after open surgical resection (surgical morbidity), based on characteristics of the AVM itself. Based on this system, AVMs may be classified as grades 1–5. This system was not intended to characterize risk of hemorrhage.

"Eloquent" is defined as areas within the brain that, if removed will result in loss of sensory processing or linguistic ability, minor paralysis, or paralysis. These include the basal ganglia, language cortices, sensorimotor regions, and white matter tracts. Importantly, eloquent areas are often defined differently across studies where deep cerebellar nuclei, cerebral peduncles, thalamus, hypothalamus, internal capsule, brainstem, and the visual cortex could be included.

The risk of post-surgical neurological deficit (difficulty with language, motor weakness, vision loss) increases with increasing Spetzler-Martin grade.

Supplemented Spetzler-Martin (SM-supp, Lawton-Young) Grade
A limitation of the Spetzler-Martin Grading system is that it does not include the following factors: Patient age, hemorrhage, diffuseness of nidus, and arterial supply. In 2010 a new supplemented Spetzler-Martin system (SM-supp, Lawton-Young) was devised adding these variables to the SM system. Under this new system AVMs are classified from grades 1–10. It has since been determined to have greater predictive accuracy than SM grades alone.

Treatment
Treatment depends on the location and size of the AVM and whether there is bleeding or not.

The treatment in the case of sudden bleeding is focused on restoration of vital function.

Medical
Anticonvulsant medications such as phenytoin are often used to control seizure; medications or procedures may be employed to relieve intracranial pressure. Eventually, curative treatment may be required to prevent recurrent hemorrhage. However, any type of intervention may also carry a risk of creating a neurological deficit.

Surgical
Surgical elimination of the blood vessels involved is the preferred curative treatment for many types of AVM. Surgery is performed by a neurosurgeon who temporarily removes part of the skull (craniotomy), separates the AVM from surrounding brain tissue, and resects the abnormal vessels. While surgery can result in an immediate, complete removal of the AVM, risks exist depending on the size and the location of the malformation. The AVM must be resected en bloc, for partial resection will likely cause severe hemorrhage. The preferred treatment of Spetzler-Martin grade 1 and 2 AVMs in young, healthy patients is surgical resection due to the relatively small risk of neurological damage compared to the high lifetime risk of hemorrhage. Grade 3 AVMs may or may not be amenable to surgery. Grade 4 and 5 AVMs are not usually surgically treated.

Radiosurgical
Radiosurgery has been widely used on small AVMs with considerable success. The Gamma Knife is an apparatus used to precisely apply a controlled radiation dosage to the volume of the brain occupied by the AVM. While this treatment does not require an incision and craniotomy (with their own inherent risks), three or more years may pass before the complete effects are known, during which time patients are at risk of bleeding. Complete obliteration of the AVM may or may not occur after several years, and repeat treatment may be needed. Radiosurgery is itself not without risk. In one large study, nine percent of patients had transient neurological symptoms, including headache, after radiosurgery for AVM. However, most symptoms resolved, and the long-term rate of neurological symptoms was 3.8%.

Neuroendovascular therapy
Embolization is performed by interventional neuroradiologists and the occlusion of blood vessels most commonly is obtained with ethylene vinyl alcohol copolymer (Onyx) or n-butyl cyanoacrylate. These substances are introduced by a radiographically guided catheter, and block vessels responsible for blood flow into the AVM. Embolization is frequently used as an adjunct to either surgery or radiation treatment. Embolization reduces the size of the AVM and during surgery it reduces the risk of bleeding. However, embolization alone may completely obliterate some AVMs. In high flow intranidal fistulas balloons can also be used to reduce the flow so that embolization can be done safely.

Risks
A first-of-its-kind controlled clinical trial by the National Institutes of Health and National Institute of Neurological Disorders and Stroke focuses on the risk of stroke or death in patients with an AVM who either did or did not undergo interventional eradication. Early results suggest that the invasive treatment of unruptured AVMs tends to yield worse results than the therapeutic (medical) management of symptoms. Because of the higher-than-expected experimental event rate (e.g. stroke or death), patient enrollment was halted by May 2013, while the study intended to follow participants (over a planned 5 to 10 years) to determine which approach seems to produce better long-term results.

Prognosis
The main risk is intracranial hemorrhage. This risk is difficult to quantify since many patients with asymptomatic AVMs will never come to medical attention. Small AVMs tend to bleed more often than do larger ones, the opposite of cerebral aneurysms. If a rupture or bleeding incident occurs, the blood may penetrate either into the brain tissue (cerebral hemorrhage) or into the subarachnoid space, which is located between the sheaths (meninges) surrounding the brain (subarachnoid hemorrhage). Bleeding may also extend into the ventricular system (intraventricular hemorrhage). Cerebral hemorrhage appears to be most common. One long-term study (mean follow up greater than 20 years) of over 150 symptomatic AVMs (either presenting with bleeding or seizures) found the risk of cerebral hemorrhage to be approximately 4% per year, slightly higher than the 2–4% seen in other studies. The earlier an AVM appears, the more likely it is to cause hemorrhage over one's lifetime; e.g. (assuming a 3% annual risk), an AVM appearing at 25 years of age indicates a 79% lifetime chance of hemorrhage, while one appearing at age 85 indicates only a 17% chance. Ruptured AVMs are a significant source of morbidity and mortality; following a rupture, as many as 29% of patients will die, with only 55% able to live independently.

Epidemiology
The annual new detection rate incidence of AVMs is approximately 1 per 100,000 a year. The point prevalence in adults is approximately 18 per 100,000. AVMs are more common in males than females, although in females pregnancy may start or worsen symptoms due to the increase in blood flow and volume it usually brings. There is a significant preponderance (15–20%) of AVM in patients with hereditary hemorrhagic telangiectasia (Osler–Weber–Rendu syndrome).

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

title: Cerebral arteriovenous malformation


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Comparative method

In linguistics, the comparative method is a technique for studying the development of languages by performing a feature-by-feature comparison of two or more languages with common descent from a shared ancestor and then extrapolating backwards to infer the properties of that ancestor. The comparative method may be contrasted with the method of internal reconstruction in which the internal development of a single language is inferred by the analysis of features within that language. Ordinarily, both methods are used together to reconstruct prehistoric phases of languages; to fill in gaps in the historical record of a language; to discover the development of phonological, morphological and other linguistic systems and to confirm or to refute hypothesised relationships between languages.

The comparative method emerged in the early 19th century with the birth of Indo-European studies, then took a definite scientific approach with the works of the Neogrammarians in the late 19th–early 20th century. Key contributions were made by the Danish scholars Rasmus Rask (1787–1832) and Karl Verner (1846–1896), and the German scholar Jacob Grimm (1785–1863). The first linguist to offer reconstructed forms from a proto-language was August Schleicher (1821–1868) in his Compendium der vergleichenden Grammatik der indogermanischen Sprachen, originally published in 1861. Here is Schleicher's explanation of why he offered reconstructed forms: "In the present work an attempt is made to set forth the inferred Indo-European original language side by side with its really existent derived languages. Besides the advantages offered by such a plan, in setting immediately before the eyes of the student the final results of the investigation in a more concrete form, and thereby rendering easier his insight into the nature of particular Indo-European languages, there is, I think, another of no less importance gained by it, namely that it shows the baselessness of the assumption that the non-Indian Indo-European languages were derived from Old-Indian (Sanskrit)."

Principles
The aim of the comparative method is to highlight and interpret systematic phonological and semantic correspondences between two or more attested languages. If those correspondences cannot be rationally explained as the result of linguistic universals or language contact (borrowings, areal influence, etc.), and if they are sufficiently numerous, regular, and systematic that they cannot be dismissed as chance similarities, then it must be assumed that they descend from a single parent language called the 'proto-language'.

A sequence of regular sound changes (along with their underlying sound laws) can then be postulated to explain the correspondences between the attested forms, which eventually allows for the reconstruction of a proto-language by the methodical comparison of "linguistic facts" within a generalized system of correspondences.

"Every linguistic fact is part of a whole in which everything is connected to everything else. One detail must not be linked to another detail, but one linguistic system to another."

Relation is considered to be "established beyond a reasonable doubt" if a reconstruction of the common ancestor is feasible.

"The ultimate proof of genetic relationship, and to many linguists' minds the only real proof, lies in a successful reconstruction of the ancestral forms from which the semantically corresponding cognates can be derived."

- Hans Henrich Hock In some cases, this reconstruction can only be partial, generally because the compared languages are too scarcely attested, the temporal distance between them and their proto-language is too deep, or their internal evolution render many of the sound laws obscure to researchers. In such case, a relation is considered plausible, but uncertain.

Terminology
Descent is defined as transmission across the generations: children learn a language from the parents' generation and, after being influenced by their peers, transmit it to the next generation, and so on. For example, a continuous chain of speakers across the centuries links Vulgar Latin to all of its modern descendants.

Two languages are genetically related if they descended from the same ancestor language. For example, Italian and French both come from Latin and therefore belong to the same family, the Romance languages. Having a large component of vocabulary from a certain origin is not sufficient to establish relatedness; for example, heavy borrowing from Arabic into Persian has caused more of the vocabulary of Modern Persian to be from Arabic than from the direct ancestor of Persian, Proto-Indo-Iranian, but Persian remains a member of the Indo-Iranian family and is not considered "related" to Arabic.

However, it is possible for languages to have different degrees of relatedness. English, for example, is related to both German and Russian but is more closely related to the former than to the latter. Although all three languages share a common ancestor, Proto-Indo-European, English and German also share a more recent common ancestor, Proto-Germanic, but Russian does not. Therefore, English and German are considered to belong to a subgroup of Indo-European that Russian does not belong to, the Germanic languages.

The division of related languages into subgroups is accomplished by finding shared linguistic innovations that differentiate them from the parent language. For instance, English and German both exhibit the effects of a collection of sound changes known as Grimm's Law, which Russian was not affected by. The fact that English and German share this innovation is seen as evidence of English and German's more recent common ancestor—since the innovation actually took place within that common ancestor, before English and German diverged into separate languages. On the other hand, shared retentions from the parent language are not sufficient evidence of a sub-group. For example, German and Russian both retain from Proto-Indo-European a contrast between the dative case and the accusative case, which English has lost. However, that similarity between German and Russian is not evidence that German is more closely related to Russian than to English but means only that the innovation in question, the loss of the accusative/dative distinction, happened more recently in English than the divergence of English from German.

Origin and development
In classical antiquity, Romans were aware of the similarities between Greek and Latin, but did not study them systematically. They sometimes explained them mythologically, as the result of Rome being a Greek colony speaking a debased dialect.

Even though grammarians of Antiquity had access to other languages around them (Oscan, Umbrian, Etruscan, Gaulish, Egyptian, Parthian...), they showed little interest in comparing, studying, or just documenting them. Comparison between languages really began after classical antiquity.

Early works
In the 9th or 10th century AD, Yehuda Ibn Quraysh compared the phonology and morphology of Hebrew, Aramaic and Arabic but attributed the resemblance to the Biblical story of Babel, with Abraham, Isaac and Joseph retaining Adam's language, with other languages at various removes becoming more altered from the original Hebrew.

In publications of 1647 and 1654, Marcus Zuerius van Boxhorn first described a rigorous methodology for historical linguistic comparisons and proposed the existence of an Indo-European proto-language, which he called "Scythian", unrelated to Hebrew but ancestral to Germanic, Greek, Romance, Persian, Sanskrit, Slavic, Celtic and Baltic languages. The Scythian theory was further developed by Andreas Jäger (1686) and William Wotton (1713), who made early forays to reconstruct the primitive common language. In 1710 and 1723, Lambert ten Kate first formulated the regularity of sound laws, introducing among others the term root vowel.

Another early systematic attempt to prove the relationship between two languages on the basis of similarity of grammar and lexicon was made by the Hungarian János Sajnovics in 1770, when he attempted to demonstrate the relationship between Sami and Hungarian. That work was later extended to all Finno-Ugric languages in 1799 by his countryman Samuel Gyarmathi. However, the origin of modern historical linguistics is often traced back to Sir William Jones, an English philologist living in India, who in 1786 made his famous observation:"The Sanscrit language, whatever be its antiquity, is of a wonderful structure; more perfect than the Greek, more copious than the Latin, and more exquisitely refined than either, yet bearing to both of them a stronger affinity, both in the roots of verbs and the forms of grammar, than could possibly have been produced by accident; so strong indeed, that no philologer could examine them all three, without believing them to have sprung from some common source, which, perhaps, no longer exists. There is a similar reason, though not quite so forcible, for supposing that both the Gothick and the Celtick, though blended with a very different idiom, had the same origin with the Sanscrit; and the old Persian might be added to the same family."

Comparative linguistics
The comparative method developed out of attempts to reconstruct the proto-language mentioned by Jones, which he did not name but subsequent linguists have labelled Proto-Indo-European (PIE). The first professional comparison between the Indo-European languages that were then known was made by the German linguist Franz Bopp in 1816. He did not attempt a reconstruction but demonstrated that Greek, Latin and Sanskrit shared a common structure and a common lexicon. In 1808, Friedrich Schlegel first stated the importance of using the eldest possible form of a language when trying to prove its relationships; in 1818, Rasmus Christian Rask developed the principle of regular sound-changes to explain his observations of similarities between individual words in the Germanic languages and their cognates in Greek and Latin. Jacob Grimm, better known for his Fairy Tales, used the comparative method in Deutsche Grammatik (published 1819–1837 in four volumes), which attempted to show the development of the Germanic languages from a common origin, which was the first systematic study of diachronic language change.

Both Rask and Grimm were unable to explain apparent exceptions to the sound laws that they had discovered. Although Hermann Grassmann explained one of the anomalies with the publication of Grassmann's law in 1862, Karl Verner made a methodological breakthrough in 1875, when he identified a pattern now known as Verner's law, the first sound-law based on comparative evidence showing that a phonological change in one phoneme could depend on other factors within the same word (such as neighbouring phonemes and the position of the accent ), which are now called conditioning environments.

Neo-grammarian approach
Similar discoveries made by the Junggrammatiker (usually translated as "Neogrammarians") at the University of Leipzig in the late 19th century led them to conclude that all sound changes were ultimately regular, resulting in the famous statement by Karl Brugmann and Hermann Osthoff in 1878 that "sound laws have no exceptions". That idea is fundamental to the modern comparative method since it necessarily assumes regular correspondences between sounds in related languages and thus regular sound changes from the proto-language. The Neogrammarian hypothesis led to the application of the comparative method to reconstruct Proto-Indo-European since Indo-European was then by far the most well-studied language family. Linguists working with other families soon followed suit, and the comparative method quickly became the established method for uncovering linguistic relationships.

Application
There is no fixed set of steps to be followed in the application of the comparative method, but some steps are suggested by Lyle Campbell and Terry Crowley, who are both authors of introductory texts in historical linguistics. This abbreviated summary is based on their concepts of how to proceed.

Step 1, assemble potential cognate lists
This step involves making lists of words that are likely cognates among the languages being compared. If there is a regularly-recurring match between the phonetic structure of basic words with similar meanings, a genetic kinship can probably then be established. For example, linguists looking at the Polynesian family might come up with a list similar to the following (their actual list would be much longer):

Borrowings or false cognates can skew or obscure the correct data. For example, English taboo is like the six Polynesian forms because of borrowing from Tongan into English, not because of a genetic similarity. That problem can usually be overcome by using basic vocabulary, such as kinship terms, numbers, body parts and pronouns. Nonetheless, even basic vocabulary can be sometimes borrowed. Finnish, for example, borrowed the word for "mother", äiti, from Proto-Germanic *aiþį̄ (compare to Gothic aiþei). English borrowed the pronouns "they", "them", and "their(s)" from Norse. Thai and various other East Asian languages borrowed their numbers from Chinese. An extreme case is represented by Pirahã, a Muran language of South America, which has been controversially claimed to have borrowed all of its pronouns from Nheengatu.

Step 2, establish correspondence sets
The next step involves determining the regular sound-correspondences exhibited by the lists of potential cognates. For example, in the Polynesian data above, it is apparent that words that contain t in most of the languages listed have cognates in Hawaiian with k in the same position. That is visible in multiple cognate sets: the words glossed as 'one', 'three', 'man' and 'taboo' all show the relationship. The situation is called a "regular correspondence" between k in Hawaiian and t in the other Polynesian languages. Similarly, a regular correspondence can be seen between Hawaiian and Rapanui h, Tongan and Samoan f, Maori ɸ, and Rarotongan ʔ.

Mere phonetic similarity, as between English day and Latin dies (both with the same meaning), has no probative value. English initial d- does not regularly match Latin d- since a large set of English and Latin non-borrowed cognates cannot be assembled such that English d repeatedly and consistently corresponds to Latin d at the beginning of a word, and whatever sporadic matches can be observed are due either to chance (as in the above example) or to borrowing (for example, Latin diabolus and English devil, both ultimately of Greek origin ). However, English and Latin exhibit a regular correspondence of t- : d- (in which "A : B" means "A corresponds to B"), as in the following examples:

If there are many regular correspondence sets of this kind (the more, the better), a common origin becomes a virtual certainty, particularly if some of the correspondences are non-trivial or unusual.

Step 3, discover which sets are in complementary distribution
During the late 18th to late 19th century, two major developments improved the method's effectiveness.

First, it was found that many sound changes are conditioned by a specific context. For example, in both Greek and Sanskrit, an aspirated stop evolved into an unaspirated one, but only if a second aspirate occurred later in the same word; this is Grassmann's law, first described for Sanskrit by Sanskrit grammarian Pāṇini and promulgated by Hermann Grassmann in 1863.

Second, it was found that sometimes sound changes occurred in contexts that were later lost. For instance, in Sanskrit velars (k-like sounds) were replaced by palatals (ch-like sounds) whenever the following vowel was *i or *e. Subsequent to this change, all instances of *e were replaced by a. The situation could be reconstructed only because the original distribution of e and a could be recovered from the evidence of other Indo-European languages. For instance, the Latin suffix que, "and", preserves the original *e vowel that caused the consonant shift in Sanskrit:

Verner's Law, discovered by Karl Verner c. 1875, provides a similar case: the voicing of consonants in Germanic languages underwent a change that was determined by the position of the old Indo-European accent. Following the change, the accent shifted to initial position. Verner solved the puzzle by comparing the Germanic voicing pattern with Greek and Sanskrit accent patterns.

This stage of the comparative method, therefore, involves examining the correspondence sets discovered in step 2 and seeing which of them apply only in certain contexts. If two (or more) sets apply in complementary distribution, they can be assumed to reflect a single original phoneme: "some sound changes, particularly conditioned sound changes, can result in a proto-sound being associated with more than one correspondence set".

For example, the following potential cognate list can be established for Romance languages, which descend from Latin:

They evidence two correspondence sets, k : k and ''k : :

Since French  occurs only before a where the other languages also have a, and French k occurs elsewhere, the difference is caused by different environments (being before a conditions the change), and the sets are complementary. They can, therefore, be assumed to reflect a single proto-phoneme (in this case *k, spelled ⟨c⟩ in Latin). The original Latin words are corpus, crudus, catena and captiare, all with an initial k. If more evidence along those lines were given, one might conclude that an alteration of the original k took place because of a different environment.

A more complex case involves consonant clusters in Proto-Algonquian. The Algonquianist Leonard Bloomfield used the reflexes of the clusters in four of the daughter languages to reconstruct the following correspondence sets:

Although all five correspondence sets overlap with one another in various places, they are not in complementary distribution and so Bloomfield recognised that a different cluster must be reconstructed for each set. His reconstructions were, respectively, *hk, *xk, *čk (=), *šk (=), and çk (in which x and ç are arbitrary symbols, rather than attempts to guess the phonetic value of the proto-phonemes).

Step 4, reconstruct proto-phonemes
Typology assists in deciding what reconstruction best fits the data. For example, the voicing of voiceless stops between vowels is common, but the devoicing of voiced stops in that environment is rare. If a correspondence -t- : -d- between vowels is found in two languages, the proto-phoneme is more likely to be *-t-, with a development to the voiced form in the second language. The opposite reconstruction would represent a rare type.

However, unusual sound changes occur. The Proto-Indo-European word for two, for example, is reconstructed as *dwō, which is reflected in Classical Armenian as erku. Several other cognates demonstrate a regular change *dw- → erk- in Armenian. Similarly, in Bearlake, a dialect of the Athabaskan language of Slavey, there has been a sound change of Proto-Athabaskan *ts → Bearlake . It is very unlikely that *dw- changed directly into erk- and *ts into , but they probably instead went through several intermediate steps before they arrived at the later forms. It is not phonetic similarity that matters for the comparative method but rather regular sound correspondences.

By the principle of economy, the reconstruction of a proto-phoneme should require as few sound changes as possible to arrive at the modern reflexes in the daughter languages. For example, Algonquian languages exhibit the following correspondence set:

The simplest reconstruction for this set would be either *m or *b. Both *m → b and *b → m are likely. Because m occurs in five of the languages and b in only one of them, if *b is reconstructed, it is necessary to assume five separate changes of *b → m, but if *m is reconstructed, it is necessary to assume only one change of *m → b and so *m would be most economical.

That argument assumes the languages other than Arapaho to be at least partly independent of one another. If they all formed a common subgroup, the development *b → m would have to be assumed to have occurred only once.

Step 5, examine the reconstructed system typologically
In the final step, the linguist checks to see how the proto-phonemes fit the known typological constraints. For example, a hypothetical system,

has only one voiced stop, *b, and although it has an alveolar and a velar nasal, *n and *ŋ, there is no corresponding labial nasal. However, languages generally maintain symmetry in their phonemic inventories. In this case, a linguist might attempt to investigate the possibilities that either what was earlier reconstructed as *b is in fact *m or that the *n and *ŋ are in fact *d and *g.

Even a symmetrical system can be typologically suspicious. For example, here is the traditional Proto-Indo-European stop inventory:

An earlier voiceless aspirated row was removed on grounds of insufficient evidence. Since the mid-20th century, a number of linguists have argued that this phonology is implausible and that it is extremely unlikely for a language to have a voiced aspirated (breathy voice) series without a corresponding voiceless aspirated series.

Thomas Gamkrelidze and Vyacheslav Ivanov provided a potential solution and argued that the series that are traditionally reconstructed as plain voiced should be reconstructed as glottalized: either implosive or ejective. The plain voiceless and voiced aspirated series would thus be replaced by just voiceless and voiced, with aspiration being a non-distinctive quality of both. That example of the application of linguistic typology to linguistic reconstruction has become known as the glottalic theory. It has a large number of proponents but is not generally accepted.

The reconstruction of proto-sounds logically precedes the reconstruction of grammatical morphemes (word-forming affixes and inflectional endings), patterns of declension and conjugation and so on. The full reconstruction of an unrecorded protolanguage is an open-ended task.

The history of historical linguistics
The limitations of the comparative method were recognized by the very linguists who developed it, but it is still seen as a valuable tool. In the case of Indo-European, the method seemed at least a partial validation of the centuries-old search for an Ursprache, the original language. The others were presumed to be ordered in a family tree, which was the tree model of the neogrammarians.

The archaeologists followed suit and attempted to find archaeological evidence of a culture or cultures that could be presumed to have spoken a proto-language, such as Vere Gordon Childe's The Aryans: a study of Indo-European origins, 1926. Childe was a philologist turned archaeologist. Those views culminated in the Siedlungsarchaologie, or "settlement-archaeology", of Gustaf Kossinna, becoming known as "Kossinna's Law". Kossinna asserted that cultures represent ethnic groups, including their languages, but his law was rejected after World War II. The fall of Kossinna's Law removed the temporal and spatial framework previously applied to many proto-languages. Fox concludes: "The Comparative Method as such is not, in fact, historical; it provides evidence of linguistic relationships to which we may give a historical interpretation.... [Our increased knowledge about the historical processes involved] has probably made historical linguists less prone to equate the idealizations required by the method with historical reality.... Provided we keep [the interpretation of the results and the method itself] apart, the Comparative Method can continue to be used in the reconstruction of earlier stages of languages."

Proto-languages can be verified in many historical instances, such as Latin. Although no longer a law, settlement-archaeology is known to be essentially valid for some cultures that straddle history and prehistory, such as the Celtic Iron Age (mainly Celtic) and Mycenaean civilization (mainly Greek). None of those models can be or have been completely rejected, but none is sufficient alone.

The Neogrammarian principle
The foundation of the comparative method, and of comparative linguistics in general, is the Neogrammarians' fundamental assumption that "sound laws have no exceptions". When it was initially proposed, critics of the Neogrammarians proposed an alternate position that summarised by the maxim "each word has its own history". Several types of change actually alter words in irregular ways. Unless identified, they may hide or distort laws and cause false perceptions of relationship.

Borrowing
All languages borrow words from other languages in various contexts. Loanwords imitate the form of the donor language, as in Finnic kuningas, from Proto-Germanic *kuningaz ('king'), with possible adaptations to the local phonology, as in Japanese sakkā, from English soccer. At first sight, borrowed words may mislead the investigator into seeing a genetic relationship, although they can more easily be identified with information on the historical stages of both the donor and receiver languages. Inherently, words that were borrowed from a common source (such as English coffee and Basque kafe, ultimately from Arabic qahwah) do share a genetic relationship, although limited to the history of this word.

Areal diffusion
Borrowing on a larger scale occurs in areal diffusion, when features are adopted by contiguous languages over a geographical area. The borrowing may be phonological, morphological or lexical. A false proto-language over the area may be reconstructed for them or may be taken to be a third language serving as a source of diffused features.

Several areal features and other influences may converge to form a Sprachbund, a wider region sharing features that appear to be related but are diffusional. For instance, the Mainland Southeast Asia linguistic area, before it was recognised, suggested several false classifications of such languages as Chinese, Thai and Vietnamese.

Random mutations
Sporadic changes, such as irregular inflections, compounding and abbreviation, do not follow any laws. For example, the Spanish words palabra ('word'), peligro ('danger') and milagro ('miracle') would have been parabla, periglo, miraglo by regular sound changes from the Latin parabŏla, perīcŭlum and mīrācŭlum, but the r and l changed places by sporadic metathesis.

Analogy
Analogy is the sporadic change of a feature to be like another feature in the same or a different language. It may affect a single word or be generalized to an entire class of features, such as a verb paradigm. An example is the Russian word for nine. The word, by regular sound changes from Proto-Slavic, should have been, but it is in fact. It is believed that the initial ' changed to ' under influence of the word for "ten" in Russian,.

Gradual application
Those who study contemporary language changes, such as William Labov, acknowledge that even a systematic sound change is applied at first inconsistently, with the percentage of its occurrence in a person's speech dependent on various social factors. The sound change seems to gradually spread in a process known as lexical diffusion. While it does not invalidate the Neogrammarians' axiom that "sound laws have no exceptions", the gradual application of the very sound laws shows that they do not always apply to all lexical items at the same time. Hock notes, "While it probably is true in the long run every word has its own history, it is not justified to conclude as some linguists have, that therefore the Neogrammarian position on the nature of linguistic change is falsified".

Non-inherited features
The comparative method cannot recover aspects of a language that were not inherited in its daughter idioms. For instance, the Latin declension pattern was lost in Romance languages, resulting in an impossibility to fully reconstruct such a feature via systematic comparison.

The tree model
The comparative method is used to construct a tree model (German Stammbaum) of language evolution, in which daughter languages are seen as branching from the proto-language, gradually growing more distant from it through accumulated phonological, morpho-syntactic, and lexical changes.

The presumption of a well-defined node

The tree model features nodes that are presumed to be distinct proto-languages existing independently in distinct regions during distinct historical times. The reconstruction of unattested proto-languages lends itself to that illusion since they cannot be verified, and the linguist is free to select whatever definite times and places seems best. Right from the outset of Indo-European studies, however, Thomas Young said: "It is not, however, very easy to say what the definition should be that should constitute a separate language, but it seems most natural to call those languages distinct, of which the one cannot be understood by common persons in the habit of speaking the other.... Still, however, it may remain doubtfull whether the Danes and the Swedes could not, in general, understand each other tolerably well... nor is it possible to say if the twenty ways of pronouncing the sounds, belonging to the Chinese characters, ought or ought not to be considered as so many languages or dialects.... But,... the languages so nearly allied must stand next to each other in a systematic order…"

The assumption of uniformity in a proto-language, implicit in the comparative method, is problematic. Even small language communities always have differences in dialect, whether they are based on area, gender, class or other factors. The Pirahã language of Brazil is spoken by only several hundred people but has at least two different dialects, one spoken by men and one by women. Campbell points out: "It is not so much that the comparative method 'assumes' no variation; rather, it is just that there is nothing built into the comparative method which would allow it to address variation directly.... This assumption of uniformity is a reasonable idealization; it does no more damage to the understanding of the language than, say, modern reference grammars do which concentrate on a language's general structure, typically leaving out consideration of regional or social variation."

Different dialects, as they evolve into separate languages, remain in contact with and influence one another. Even after they are considered distinct, languages near one another continue to influence one another and often share grammatical, phonological, and lexical innovations. A change in one language of a family may spread to neighboring languages, and multiple waves of change are communicated like waves across language and dialect boundaries, each with its own randomly delimited range. If a language is divided into an inventory of features, each with its own time and range (isoglosses), they do not all coincide. History and prehistory may not offer a time and place for a distinct coincidence, as may be the case for Proto-Italic, for which the proto-language is only a concept. However, Hock observes:

"The discovery in the late nineteenth century that isoglosses can cut across well-established linguistic boundaries at first created considerable attention and controversy. And it became fashionable to oppose a wave theory to a tree theory.... Today, however, it is quite evident that the phenomena referred to by these two terms are complementary aspects of linguistic change...."

Subjectivity of the reconstruction
The reconstruction of unknown proto-languages is inherently subjective. In the Proto-Algonquian example above, the choice of *m as the parent phoneme is only likely, not certain. It is conceivable that a Proto-Algonquian language with *b in those positions split into two branches, one that preserved *b and one that changed it to *m instead, and while the first branch developed only into Arapaho, the second spread out more widely and developed into all the other Algonquian tribes. It is also possible that the nearest common ancestor of the Algonquian languages used some other sound instead, such as *p, which eventually mutated to *b in one branch and to *m in the other.

Examples of strikingly complicated and even circular developments are indeed known to have occurred (such as Proto-Indo-European *t > Pre-Proto-Germanic *þ > Proto-Germanic *ð > Proto-West-Germanic *d > Old High German t in fater > Modern German Vater), but in the absence of any evidence or other reason to postulate a more complicated development, the preference of a simpler explanation is justified by the principle of parsimony, also known as Occam's razor. Since reconstruction involves many such choices, some linguists prefer to view the reconstructed features as abstract representations of sound correspondences, rather than as objects with a historical time and place.

The existence of proto-languages and the validity of the comparative method is verifiable if the reconstruction can be matched to a known language, which may be known only as a shadow in the loanwords of another language. For example, Finnic languages such as Finnish have borrowed many words from an early stage of Germanic, and the shape of the loans matches the forms that have been reconstructed for Proto-Germanic. Finnish kuningas 'king' and kaunis 'beautiful' match the Germanic reconstructions *kuningaz and *skauniz (> German König 'king', schön 'beautiful').

Additional models
The wave model was developed in the 1870s as an alternative to the tree model to represent the historical patterns of language diversification. Both the tree-based and the wave-based representations are compatible with the comparative method.

By contrast, some approaches are incompatible with the comparative method, including contentious glottochronology and even more controversial mass lexical comparison considered by most historical linguists to be flawed and unreliable.

title: Comparative method


text:
Council of Constance

The Council of Constance (Concilium Constantiense; Konzil von Konstanz) was an ecumenical council of the Catholic Church that was held from 1414 to 1418 in the Bishopric of Constance (Konstanz) in present-day Germany. The council ended the Western Schism by deposing or accepting the resignation of the remaining papal claimants and by electing Pope Martin V. It was the last papal election to take place outside of Italy.

The council also condemned Jan Hus as a heretic and facilitated his execution by the civil authority, and ruled on issues of national sovereignty, the rights of pagans and just war, in response to a conflict between the Grand Duchy of Lithuania, the Kingdom of Poland and the Order of the Teutonic Knights.

The council is also important for its role in the debates over ecclesial conciliarism and papal supremacy. Constance issued two particularly significant decrees regarding the constitution of the Catholic Church: Haec sancta (1415), which asserted the superiority of ecumenical councils over popes in at least certain situations, and Frequens (1417), which provided for councils to be held automatically every ten years. The status of these decrees proved controversial in the centuries after the council, and Frequens was never put into practice. Though Haec sancta, at least, continued to be accepted as binding by much of the church up to the 19th century, present-day Catholic theologians generally regard these decrees as either invalid or as practical responses to a particular situation without wider implications.

Origin and background
The council's main purpose was to end the Papal schism which had resulted from the confusion following the Avignon Papacy. Pope Gregory XI's return to Rome in 1377, followed by his death (in 1378) and the controversial election of his successor, Pope Urban VI, resulted in the defection of a number of cardinals and the election of a rival pope based at Avignon in 1378. After thirty years of schism, the rival courts convened the Council of Pisa seeking to resolve the situation by deposing the two claimant popes and electing a new one. The council claimed that in such a situation, a council of bishops had greater authority than just one bishop, even if he were the bishop of Rome. Though the elected Antipope Alexander V and his successor, Antipope John XXIII (not to be confused with the 20th-century Pope John XXIII), gained widespread support, especially at the cost of the Avignon antipope, the schism remained, now involving not two but three claimants: Gregory XII at Rome, Benedict XIII at Avignon, and John XXIII.

Therefore, many voices, including Sigismund, King of the Romans and of Hungary (and later Holy Roman Emperor), pressed for another council to resolve the issue. That council was called by John XXIII and was held from 16 November 1414 to 22 April 1418 in Constance, Germany. The council was attended by roughly 29 cardinals, 100 "learned doctors of law and divinity", 134 abbots, and 183 bishops and archbishops.

Participants
Sigismund arrived on Christmas Eve 1414 and exercised a profound and continuous influence on the course of the council in his capacity of imperial protector of the church. An innovation at the council was that instead of voting as individuals, the bishops voted in national blocs. The vote by nations was in great measure the initiative of the English, German, and French members. The legality of this measure, in imitation of the "nations" of the universities, was more than questionable, but during February 1415 it carried and thenceforth was accepted in practice, though never authorized by any formal decree of the council. The four "nations" consisted of England, France, Italy, and Germany, with Poles, Hungarians, Danes, and Scandinavians counted with the Germans. While the Italian representatives made up half of those in attendance, they were equal in influence to the English, who sent twenty deputies and three bishops. The Spanish deputies (from Portugal, Castile, Navarre and Aragon), initially absent, joined the council at the twenty-first session, constituting upon arrival the fifth nation.

Decrees and doctrinal status
Many members of the new assembly (comparatively few bishops, but many doctors of theology and of canon and civil law, procurators of bishops, deputies of universities, cathedral chapters, provosts, etc., agents and representatives of princes, etc.) strongly favored the voluntary abdication of all three popes, as did King Sigismund.

Although the Italian bishops who had accompanied John XXIII in large numbers supported his legitimacy, he grew increasingly more suspicious of the council. Partly in response to a fierce anonymous attack on his character from an Italian source, on 2 March 1415 he promised to resign. However, on 20 March he secretly fled the city and took refuge at Schaffhausen in territory of his friend Frederick, Duke of Austria-Tyrol.

The famous decree Haec sancta synodus, which gave primacy to the authority of the council and thus became a source for ecclesial conciliarism, was promulgated in the fifth session, 6 April 1415:

"Legitimately assembled in the holy Spirit, constituting a general council and representing the Catholic church militant, it has power immediately from Christ; and everyone of whatever state or dignity, even papal, is bound to obey it in those matters which pertain to the faith, the eradication of the said schism, and the general reform of the said church of God in head and members."

Haec sancta synodus marks the high-water mark of the Conciliar movement of reform.

The acts of the council were not made public until 1442, at the behest of the Council of Basel; they were printed in 1500. The creation of a book on how to die was ordered by the council, and thus written in 1415 under the title Ars moriendi.

Haec sancta is today generally considered invalid by the Catholic Church, on the basis that Gregory XII was the legitimate pope at the time and the decree was passed by the council in a session before his confirmation. On this reading, the first sessions of the Council of Constance represented an invalid and illicit assembly of bishops, gathered under the authority of an antipope. This historiography is of much later provenance than the council itself, however: the Pisan line represented by John XXIII had been considered legitimate not just by most of the Latin church at the time of the council, but also subsequently by Pope Martin V, who referred to John as "our predecessor" in contrast to the other two claimants, who were merely "popes so-called in their obediences". The specific argument distinguishing two parts in the council was seemingly first made by the 17th-century Sorbonne theologian André Duval, and remained a fringe view for some time before its vindication within the Catholic Church under the influence of 19th-century ultramontanism.

Ending the Western Schism
With the support of King Sigismund, enthroned before the high altar of the cathedral of Constance, the Council of Constance recommended that all three papal claimants abdicate, and that another be chosen. In part because of the constant presence of the King, other rulers demanded that they have a say in who would be pope.

Gregory XII then sent representatives to Constance, whom he granted full powers to summon, open, and preside over an Ecumenical Council; he also empowered them to present his resignation of the papacy. This would pave the way for the end of the Western Schism.

The legates were received by King Sigismund and by the assembled Bishops, and the King yielded the presidency of the proceedings to the papal legates, Cardinal Giovanni Dominici of Ragusa and Prince Carlo Malatesta. On 4 July 1415 the Bull of Gregory XII which appointed Dominici and Malatesta as his proxies at the council was formally read before the assembled Bishops. The cardinal then read a decree of Gregory XII which convoked the council and authorized its succeeding acts. Thereupon, the Bishops voted to accept the summons. Prince Malatesta immediately informed the council that he was empowered by a commission from Pope Gregory XII to resign the Papal Throne on the Pontiff's behalf. He asked the council whether they would prefer to receive the abdication at that point or at a later date. The Bishops voted to receive the Papal abdication immediately. Thereupon the commission by Gregory XII authorizing his proxy to resign the Papacy on his behalf was read and Malatesta, acting in the name of Gregory XII, pronounced the resignation of the papacy by Gregory XII and handed a written copy of the resignation to the assembly.

Former Pope Gregory XII was then created titular Cardinal Bishop of Porto and Santa Ruffina by the council, with rank immediately below the Pope (which made him the highest-ranking person in the church, since, due to his abdication, the See of Peter in Rome was vacant). Gregory XII's cardinals were accepted as true cardinals by the council, but the members of the council delayed electing a new pope for fear that a new pope would restrict further discussion of pressing issues in the church.

By the time the anti-popes were all deposed and the new Pope, Martin V, was elected, two years had passed since Gregory XII's abdication, and Gregory was already dead. The council took great care to protect the legitimacy of the succession, ratified all his acts, and a new pontiff was chosen. The new pope, Martin V, elected November 1417, soon asserted the absolute authority of the papal office.

Condemnation of Jan Hus
A second goal of the council was to continue the reforms begun at the Council of Pisa (1409). The reforms were largely directed against John Wycliffe, mentioned in the opening session and condemned in the eighth on 4 May 1415, and Jan Hus, along with their followers. Hus, summoned to Constance under a letter of safe conduct, was found guilty of heresy by the council and turned over to the secular court. "This holy synod of Constance, seeing that God's church has nothing more that it can do, relinquishes Jan Hus to the judgment of the secular authority and decrees that he is to be relinquished to the secular court." (Council of Constance Session 15 – 6 July 1415). The secular court sentenced him to be burned to death at the stake.

Jerome of Prague, a supporter of Hus, came to Constance to offer assistance but was similarly arrested, judged, found guilty of heresy and turned over to the same secular court, with the same outcome as Hus. Poggio Bracciolini attended the council and related the unfairness of the process against Jerome.

Paweł Włodkowic and the other Polish representatives to the Council of Constance publicly defended Hus.

Polish–Lithuanian–Teutonic conflict
In 1411, the First Peace of Thorn ended the Polish–Lithuanian–Teutonic War, in which the Teutonic Knights fought the Kingdom of Poland and Grand Duchy of Lithuania. However, the peace was not stable and further conflicts arose regarding demarcation of the Samogitian borders. The tensions erupted into the brief Hunger War in summer 1414. It was concluded that the disputes would be mediated by the Council of Constance.

The Polish-Lithuanian position was defended by Paulus Vladimiri, rector of the Jagiellonian University, who challenged legality of the Teutonic crusade against Lithuania. He argued that a forced conversion was incompatible with free will, which was an essential component of a genuine conversion. Therefore, the Knights could only wage a defensive war if pagans violated natural rights of the Christians. Vladimiri further stipulated that infidels had rights which had to be respected, and neither the Pope nor the Holy Roman Emperor had the authority to violate them. Lithuanians also brought a group of Samogitian representatives to testify to atrocities committed by the Knights.

The Dominican theologian John of Falkenberg proved to be the fiercest opponent of the Poles. In his Liber de doctrina, Falkenberg argued that"the Emperor has the right to slay even peaceful infidels simply because they are pagans. ... The Poles deserve death for defending infidels, and should be exterminated even more than the infidels; they should be deprived of their sovereignty and reduced to slavery."In Satira, he attacked Polish-Lithuanian King Jogaila, calling him a "mad dog" unworthy to be king. Falkenberg was condemned and imprisoned for such libel. Other opponents included Grand Master's proctor Peter Wormditt, Dominic of San Gimignano, John Urbach, Ardecino de Porta of Novara, and Bishop of Ciudad Rodrigo Andrew Escobar. They argued that the Knights were perfectly justified in their crusade as it was a sacred duty of Christians to spread the true faith. Cardinal Pierre d'Ailly published an independent opinion that attempted to somewhat balance both Polish and Teutonic positions.

The council established the Diocese of Samogitia, with its seat in Medininkai and subordinated to Lithuanian dioceses, and appointed Matthias of Trakai as the first bishop. Pope Martin V appointed the Lithuanians Jogaila and Vytautas, who were respectively King of Poland and Grand Duke of Lithuania, as vicars general in Pskov and Veliky Novgorod in recognition of their Catholicism. After another round of futile negotiations, the Gollub War broke out in 1422. It ended with the Treaty of Melno. Polish-Lithuanian-Teutonic wars continued for another hundred years.

Later status
Although Pope Martin V did not directly challenge the decrees of the council, his successor Eugenius IV repudiated an attempt by a faction at the Council of Basel to declare the provisions of Haec sancta and Frequens a matter of faith. His 1439 bull on the matter, Moyses vir Dei, was underwritten by the Council of Florence. In convening the Fifth Lateran Council (1512–17), Pope Julius II further pronounced that Frequens had lost its force; Lateran V is sometimes seen as having itself abrogated Haec sancta, though the reading is controversial. Either way, while Rome itself came to reject the provisions made by the council, significant parts of the Church, notably in France, continued to uphold the validity of its decisions long after the event: Haec sancta was reaffirmed in the Gallican Articles of 1682, and even during the First Vatican Council of 1869–70 the French-American bishop of St. Augustine, Florida, Augustin Vérot, attempted to read Haec sancta into the record of deliberations.

Despite the apparently definitive rejection of conciliarism at the First Vatican Council, the debate over the status of Constance was renewed in the 20th century. In the 1960s, in the context of the Second Vatican Council, the reformist Catholic theologian Hans Küng and the historian Paul de Vooght argued in defense of the dogmatic character of Haec sancta, suggesting that its terms could be reconciled with the definition of papal supremacy at Vatican I. Küng's argument received support from prelates such as Cardinal Franz König. Other Catholic historians adopted different views: Hubert Jedin considered Haec sancta to be an emergency measure with no binding validity beyond its immediate context, while Joseph Gill rejected the validity of the session that passed the decree altogether. The debate over Haec sancta subsided in the 1970s, however, without resolution.

title: Council of Constance


text:
Churches Uniting in Christ

Churches Uniting in Christ (CUIC) is an ecumenical organization that brings together mainline American denominations (including both predominantly white and predominantly black churches), and was inaugurated on January 20, 2002, in Memphis, Tennessee on the balcony of the Lorraine Motel. It is the successor organization to the Consultation on Church Union.

Origins
CUIC is the successor organization to the Consultation on Church Union (COCU), which had been founded in 1962. The original task of COCU was to negotiate a consensus between its nine (originally four) member communions (it also included three "advisory participant" churches). However, it never succeeded in this goal, despite making progress on several ecumenical fronts. At COCU's 18th plenary meeting in St. Louis, Missouri (January 1999), CUIC was proposed as a new relationship among the nine member communions. Each member communion voted to join CUIC over the next few years.

Inauguration
Heads of communion from each member of COCU (as well as the ELCA, a partner in mission and dialogue) inaugurated the group on the day before Martin Luther King Jr. Day in 2002 at the motel where he was killed. This particular location highlighted the group's focus on racism as a major dividing factor between and among churches.

Task forces
The Coordinating Council of CUIC created several task forces: Racial and Social Justice, Ministry, Young Adult and Local and Regional Ecumenism. Each task force represented an important part of early CUIC work. Local ecumenical liturgies were encouraged, and excitement initially built around "pilot programs" in Denver, Los Angeles, and Memphis. The Racial and Social Justice task force created gatherings and discussions on racial justice. The Ministry task force received much of the attention from church structures, however. The group had been given a mandate to complete work on reconciliation by 2007, and in 2003 began working on a document entitled "Mutual Recognition and Mutual Reconciliation of Ministries."

Mutual Recognition and Mutual Reconciliation of Ministries (MRMRM)
One of the most difficult issues concerning recognition and reconciliation of ministries was that of the historic episcopate. This was one of the issues that defeated proposals for union by COCU as well. The group approached this problem through dialogue, soliciting information from each member communion on the particularities of their theology and ecclesiology in order to come to a mutually acceptable conclusion.

CUIC released the seventh and final draft of the MRMRM document in June 2005. Much work was done in 2006 on this document, which focused on "Episkope," the oversight of ministry. The work culminated in a consultation on episkope in St. Louis in October 2006 involving the heads of communion of the members of CUIC. At this consultation, the MRMRM document was met with resistance, and concern was raised in particular that CUIC was focusing too narrowly on reconciliation of ministries and "not taking seriously our commitment to working on those issues of systemic racism that remain at the heart of our continuing and separated life as churches here in the United States."

Moravian Church (Northern Province)
The nine churches which inaugurated CUIC in 2002 were joined by the Moravian Church, Northern Province. The Moravians had been partners in mission and dialogue since 2002, but joined as a member communion after the October 2006 consultation on episcope.

Suspension of activities
In 2007, the African Methodist Episcopal Zion Church and the African Methodist Episcopal Church withdrew from CUIC. Neither body sent representatives to the CUIC plenary on January 11–14, 2008, though the AME Council of Bishops never voted to suspend membership officially. They felt the other churches were not doing enough to counter the history of racial injustice between black and white churches. In response to this, the remaining churches in CUIC decided in 2008 to suspend their work while they seek reconciliation with these churches. This work began with a group of representatives who revisited the 1999 document "Call to Christian Commitment and Action to Combat Racism," which is available on the current CUIC website. This also meant eliminating the position of Director as well as the suspension of the work of the CUIC task forces. As of 2012, CUIC no longer has physical offices, opting instead for a virtual office and storing the archives of both CUIC and COCU at Princeton Seminary's Henry Luce III Library.

Reconciliation efforts
The African Methodist Episcopal Church resumed its participation by the February 2010 plenary meeting, where CUIC moved to refocus on its eight marks of commitment and a shared concern for racial justice as a major dividing factor facing ecumenism. Although the African Methodist Episcopal Zion Church has not rejoined the group, efforts have continued to bring this communion back into membership. The Rev. Staccato Powell, an AMEZ pastor, preached at the 2011 CUIC plenary in Ft. Lauderdale, Florida as a part of these reconciliation efforts. Combating racism has again become a priority of CUIC. Concerns over the historic episcopate have been sidelined since 2008, though they may re-emerge. The group's focus on mutual reconciliation of ministries has been revisited in the light of racism and the impact that racism may have on exchanging ministers between denominations. Therefore, the coordinating council of CUIC created a consultation on race and ministry while also choosing to partner with the Samuel Dewitt Proctor Conference, a social justice organization involved in African American faith communities.

Purpose
The purpose of CUIC has always been unity (as reflected in their current slogan, "reconciling the baptized, seeking unity with justice"). This reflects one of the core scripture passages in the ecumenical movement, Jesus' prayer in John 17:21, "That they all may be one". CUIC has approached this goal of unity in various ways throughout its history.

Racism
Racism has been a primary focus of CUIC since 2002 (and, indeed, a primary focus of COCU alongside other forms of exclusion and prejudice, such as sexism and ableism). According to Dan Krutz, former president of CUIC, "Overcoming racism has been a focal point of CUIC since its beginning... Racism may be the biggest sin that divides churches." Even before the absence of the AME and AMEZ churches at the January 2011 plenary, some in CUIC had noticed the lack of commitment to racial reconciliation. Since 2008, however, racism has become an even more pressing concern. This has led CUIC to address issues of racism in the public sphere, including the killing of Trayvon Martin and the recovery from the 2010 Haiti earthquake.

Marks of Commitment
According to their website, one of the reasons for transitioning from COCU to CUIC is so that member churches "stop 'consulting' and start living their unity in Christ more fully." This means that each member communion in CUIC agrees to abide by the eight Marks of Commitment, which are summarized as follows:
 * Receive each other as Christ's church
 * Mutually recognize baptisms & members
 * Affirm apostolic creeds
 * Celebrate Eucharist together
 * Engage in mission & anti-racism
 * Promote wholeness & inclusion
 * Structure accountability, consultation & decision-making
 * Support ongoing theological dialogue

Full members

 * African Methodist Episcopal Church
 * African Methodist Episcopal Zion Church
 * Christian Church (Disciples of Christ)
 * Christian Methodist Episcopal Church
 * Episcopal Church (United States)
 * International Council of Community Churches
 * Moravian Church in North America
 * Presbyterian Church (USA)
 * United Church of Christ
 * United Methodist Church

Former partners in mission and dialogue

 * Evangelical Lutheran Church in America

Networking partners

 * National Council of Churches
 * World Council of Churches
 * Graymoor Ecumenical & Interreligious Institute
 * Samuel DeWitt Proctor Conference

title: Churches Uniting in Christ


text:
Canadian Unitarian Council

The Canadian Unitarian Council (Conseil unitarien du Canada) (CUC) is a liberal religious association of Unitarian, Universalist, and Unitarian Universalist congregations in Canada. It was formed on May 14, 1961, initially to be the national organization for Canadians belonging to the Unitarian Universalist Association (UUA) which formed a day later on May 15, 1961. Between 1961 and 2002, almost all member congregations of the CUC were also members of the UUA and most services to congregations in Canada were provided by the UUA. However, in 2002, the CUC formally became a separate entity from the UUA, although the UUA continues to provide ministerial settlement services and remains the primary source for education and theological resources. Some Canadian congregations have continued to be members of both the CUC and the UUA, while most congregations are only members of the CUC.

The Canadian Unitarian Council is the only national body for Unitarian, Universalist, and Unitarian Universalist congregations in Canada and was one of the seventeen members of the now defunct International Council of Unitarians and Universalists (1995–2021).

Organization

The CUC is made up of 43 member congregations and emerging groups, who are the legal owners of the organization, and who are, for governance and service delivery, divided into four regions: "BC" (British Columbia), "Western" (Alberta to Thunder Bay), "Central" (between Thunder Bay and Kingston), and "Eastern" (Kingston, Ottawa and everything east of that). However, for youth ministry, the "Central" and "Eastern" regions are combined to form a youth region known as "QuOM" (Quebec, Ontario and the Maritimes), giving the youth only three regions for their activities. The organization as a whole is governed by the CUC Board of Trustees (Board), whose mandate it is to govern in the best interests of the CUC's owners. The Board is made up of eight members who are elected by congregational delegates at the CUC's Annual General Meeting. This consists of two Trustees from each region, who are eligible to serve a maximum of two three-year terms. Board meetings also include Official Observers to the Board, who participate without a vote and represent UU Youth and Ministers.

Service delivery
As members of the CUC, congregations and emerging groups are served by volunteer Service Consultants, Congregational Networks, and a series of other committees. There are two directors of regional services, one for the Western two regions, and one for the Eastern two regions. Youth and young adults are served by a Youth and Young Adult Ministry Development staff of two.

Annual conference and meeting
Policies and business of the CUC are determined at the Annual Conference and Meeting (ACM), consisting of the Bi-Annual Conference, in which workshops are held, and the Annual General Meeting, in which business matters and plenary meetings are performed. The ACM features two addresses, a Keynote and a Confluence Lecture. The Confluence Lecture is comparable to the UUA's Ware Lecture in prestige. In early days this event simply consisted of the Annual General Meeting component as the Annual Conference component was not added to much later. And starting in 2017 the conference portion will only take place every second year. Past ACMs have been held in the following locations:
 * ^Not an ACM, but an "Annual General Meeting" and "Symposium", and unlike ACMs it was organized by the CUC and the Unitarian Universalist Ministers of Canada instead of a local congregation. #Not a keynote presenter or lecturer, rather a symposium "provocateur".  *Upcoming locations

Principles and sources
The CUC does not have a central creed in which members are required to believe, but they have found it useful to articulate their common values in what has become known as The Principles and Sources of our Religious Faith, which are currently based on the UUA's Principles and Sources with the addition of an 8th principle adopted by CUC members at a special meeting on November 27, 2021. The CUC had a task force whose mandate was to consider revising them.

The principles and sources as published in church literature and on the CUC website:

"The Principles and Sources of our Religious Faith

Principles

We, the member congregations of the Canadian Unitarian Council, covenant to affirm and promote:

 * The inherent worth and dignity of every person;
 * Justice, equity, and compassion in human relations;
 * Acceptance of one another and encouragement to spiritual growth in our congregations;
 * A free and responsible search for truth and meaning;
 * The right of conscience and the use of the democratic process within our congregations and in society at large;
 * The goal of world community with peace, liberty, and justice for all;
 * Respect for the interdependent web of all existence of which we are a part;
 * Individual and communal action that accountably dismantles racism and systemic barriers to full inclusion in ourselves and our institutions.

Sources

The living tradition which we share draws from many sources:

 * Direct experience of that transcending mystery and wonder, affirmed in all cultures, which moves us to a renewal of the spirit and an openness to the forces which create and uphold life;
 * Words and deeds of prophetic women and men which challenge us to confront powers and structures of evil with justice, compassion, and the transforming power of love;
 * Wisdom from the world's religions which inspires us in our ethical and spiritual life;
 * Jewish and Christian teachings which call us to respond to God's love by loving our neighbours as ourselves;
 * Humanist teachings which counsel us to heed the guidance of reason and the results of science, and warn us against idolatries of the mind and spirit;
 * Spiritual teachings of Earth-centred traditions which celebrate the sacred circle of life and instruct us to live in harmony with the rhythms of nature.

Grateful for the religious pluralism which enriches and ennobles our faith, we are inspired to deepen our understanding and expand our vision. As free congregations we enter into this covenant, promising to one another our mutual trust and support."

Formation and relationship to the Unitarian Universalist Association
The CUC formed on May 14, 1961, to be the national organization for Canadians within the about-to-form UUA (it formed a day later on May 15, 1961). And until 2002, almost all member congregations of the CUC were also members of the UUA and most services to CUC member congregations were provided by the UUA. However, after an agreement between the UUA and the CUC, since 2002 most services have been provided by the CUC to its own member congregations, with the UUA continuing to provide ministerial settlement services. And also since 2002, some Canadian congregations have continued to be members of both the UUA and CUC while others are members of only the CUC.

The Canadian Unitarian Universalist youth of the day disapproved of the 2002 change in relationship between the CUC and UUA. It is quite evident in the words of this statement, which was adopted by the attendees of the 2001 youth conference held at the Unitarian Church of Montreal:

"We the youth of Canada are deeply concerned about the direction the CUC seems to be taking. As stewards of our faith, adults have a responsibility to take into consideration the concerns of youth. We are opposed to making this massive jump in our evolutionary progress."

Canadian Unitarian Universalist Women's Association
The Canadian Unitarian Universalist Women's Association (CUUWA), established in May 2011, is a women's rights organization associated with the CUC. The CUUWA gained initial support from Prairie Women's Gathering and the Vancouver Island Women's retreat, and has since become a nationally-recognized organization.

Mission
Originally called the Canadian Unitarian Universalist Women's Federation, the organization aims to raise awareness for women's education, rights, and equality of income. The association also aims to change societal attitudes about women and inform society of the issues women have faced locally and internationally. As a part of their mission, the CUUWA circulates educational materials that highlight women's contributions to society. The organization hosts an annual general meeting during the Canadian Unitarian Council Annual Conference.

Name of CUC and playful abbreviation of Unitarian Universalist
While the name of the organization is the Canadian Unitarian Council, the CUC includes congregations with Unitarian, Universalist, Unitarian Universalist, and Universalist Unitarian in their names. Changing the name of the CUC has occasionally been debated, but there have been no successful motions. To recognize this diversity, some members of the CUC abbreviate Unitarian Universalist as U*U (and playfully read it as "You star, you"). Note, not all CUC members like this playful reading and so when these people write the abbreviation they leave out the star (*), just writing UU instead.

title: Canadian Unitarian Council