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# Journal of African Languages and Linguistics
Ed. by Ameka, Felix K. / Amha, Azeb
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Volume 39, Issue 1
# The background marker ná in Barayin
Joseph Lovestrand
Published Online: 2018-04-26 | DOI: https://doi.org/10.1515/jall-2018-0001
## Abstract
This article gives a first account of the background marker in Barayin, an East Chadic language spoken in the Guera region of Chad. The article describes the marker’s syntactic distribution and the semantic and pragmatic contexts it occurs in. It commonly occurs following a sentence-initial noun phrase or adverbial, and it also commonly follows a sentence-initial dependent clause such as a conditional clause. The material preceding is background information which provides a context for the interpretation of the following proposition, which is the main point of the communication.
Keywords: information structure; background; Chadic; topic; focus
## 1 Introduction
Barayin [bva] is a Chadic language spoken by an estimated 5,000 people in the Guera region near the center of the Republic of Chad. The first grammatical analysis of the language (Jalkiya dialect) is by Lovestrand (2012). This article expands that work by giving a first account of the syntactic distribution, function and meaning of the marker –-the single most common word in Barayin texts.
The marker in Barayin divides a sentence into two parts. In this sense, the marker is similar to what Levinsohn (2012: 74) calls a “spacer”, and it creates what Güldemann (2010) calls a “bisected” structure. There is often (about half of the time) a noticeable pause following the marker . In other words, the marker appears to create a unit with the preceding material that can be phonologically separated from the following material.
The material preceding can be a “term” or a “proposition”. A term can be a noun phrase, prepositional phrase, or adverb. The term is normally either an argument selected by the verb (e.g., subject or object) or an adjunct (e.g., locative or adverbial). A proposition is a larger constituent which can be either a full clause or a clause that appears to have a gap filled by the term on the other side of the marker . The marker can divide a term and a proposition in either order, or occur between two propositions or two terms. In rare cases, the marker appears to occur in a sentence-initial position. In all cases, the material preceding can be described as background information which gives the addressee the appropriate context for understanding what follows .
Examples (1) and (2) show two of the most common places where the marker (glossed na) occurs. In example (1) it follows a sentence-initial argument of the verb, and is followed by the remainder of the finite clause. In example (2), it follows one finite clause, and is followed by another. Example (3) shows the third type, where separates two terms. The fourth type of construction using is shown in example (4). In this example the noun following appears to be the object/patient of the proposition preceding . The fifth type, sentence-initial , will be discussed in Section 2.5.1
(1)
mijjo ná sule makid-a-ti penden-ji person na prog arrange-ipfv-obj.3sg.f bow-poss.3sg.m
‘The man was arranging his bow.’ (Carnivores 31)
(2)
ni kol-eyi ŋ ammi ná dop-a je ammi sbj.3pl go-ipfv obl water na find-pfv part water
‘They went for water, and they found it.’ (Mosso 9)
(3)
tande ná suk de genne yesterday na market rel.sg.f ours
‘Yesterday was our market.’ (Yesterday 1)
(4)
ti wonn-eyi ná non-geti sbj.3sg.f know-ipfv na child-poss.3sg.f
‘It’s her child that she knows. (Not other children.)’
This introductory section contains a brief overview of the morphosyntax (Section 1.1) and information structure (Section 1.2) of Barayin, an introduction to background markers in other languages (Section 1.3), and an overview of the data used for this study (Section 1.4). The marker , like similar markers in other languages, is noteworthy for its distribution in what appears to be several distinct, but related contexts. Not only does the marker occur in a variety of syntactic positions, it also occurs in what appears to be a wide variety of semantic and pragmatic contexts. Section 2 contains a detailed description of the syntactic distribution of the marker . Many of the diverse semantic and pragmatic contexts where the marker occurs in natural speech are illustrated in Section 3. The variety of contexts where occurs are all analyzable as following background information. Section 4 is a brief conclusion.
## 1.1 Barayin morphosyntax overview
Barayin is SVO in its unmarked word order, as are most Chadic languages (Frajzyngier 1996: 15; Newman 2006: 199; Schuh 2003: 58). Indirect objects and adjuncts follow the object, and the final positions in a sentence are occupied by a negative marker and a question marker. There is also a pre-subject clause-internal position which will be discussed in Section 1.2. The following simplified template gives an overview of the basic clause structure:
Figure 1:
Simplified word order template for Barayin.
Subjects normally occur immediately before the predicate, whether the predicate is verbal or non-verbal, and whether the subject is nominal or pronominal. An overt subject is not obligatory and can be omitted in any case where the speaker deems the context clear enough for the hearer to discern the unstated subject. Indirect objects, locative arguments, adjuncts and adverbs typically occur following the direct object (if present) in a SVOX pattern. Interrogative mood (yes/no question) can be expressed by intonation, or is marked by a clause-final marker. Negation is expressed through a marker do in the pentultimate position. It can only be followed by an interrogative marker. Reported speech clauses (whether direct or indirect quotation) are typically preceded by a quotative which indexes the person, number and gender features of the speaker.
There are several morphologically distinct pronominal paradigms including: independent pronouns, subject proclitics, direct object suffixes and indirect object suffixes. These forms are given in Table 1. Each paradigm has ten pronouns. There is one dual (inclusive) form in the first person, and an inclusive/exclusive distinction in the first person plural. Note that the first person plural inclusive forms are bimorphemic. They are made up of the combination of a first person dual inclusive form with the enclitic (glossed pl) with a low tone–-not to be confused with the background marker with a high tone. Second and third person singular forms (and any agreement-sensitive words like adjectives and demonstratives) are distinguished for gender (masculine and feminine), but plurals are not.
Table 1:
Pronominal forms in Barayin.
Independent pronouns have the same distribution as a noun phrase. They can function as a subject or direct object, but do so more rarely than other forms. They are used in prepositional phrases, and are also the vocative form. Subject pronouns are not prefixes, but they are phonologically dependent on the following or preceding word, and have a limited syntactic distribution. They typically occur immediately before the verb (or non-verbal predicate). Only a limited number of adverbial words can intervene between a subject pronoun and the predicate. Third person subject pronouns can combine with one of three adnominal demonstratives (gi sg.m, di sg.f, ni pl) to create a demonstrative pronoun with the same distribution as a noun phrase or Independent pronoun: ka gi, ti di, and ni ni.
Direct and indirect object suffixes can have a pronominal function when they occur without a co-referential noun phrase in the same clause. They can also function as agreement markers when they occur with a co-referential noun phrase under certain discourse conditions. In a pattern similar to differential object marking, direct object suffixes normally co-occur with a (co-referential) definite nominal direct object, and do not co-occur if the nominal direct object is indefinite. However, there are some exceptions to this pattern. For more discussion, see Lovestrand (2012: 135).
Tense, aspect, and mood (TAM) are primarily encoded in verbal suffixes. TAM suffixes precede pronominal suffixes in the verbal morphology, and are often subject to suppletion or deletion when a pronominal suffix is present. TAM suffixes cannot combine to create complex TAM forms. The seven TAM suffixes are shown in Table 2 with the label describing their primary function. Future tense is expressed by a construction in which an oblique preposition ŋ is followed by a verb in the infinitive form.
Table 2:
Verbal TAM suffixes.
Barayin, like all Chadic languages, is a tonal language. Tone is essentially lexical in function. Questions can be formed by raising the tone and elongating the final vowel of a declarative clause. A similar intonational pattern occurs on the final vowel of a relative clause if followed by a demonstrative (Lovestrand 2012: 67). There are not any clear correlations between tone and any grammatical or information-structural functions. In the orthographic representation of the language used here, tonal marking is normally omitted.
## 1.2 Barayin information structure overview
Topic is used here in its more restricted sense as “the thing which the proposition expressed by the sentence is about” (Lambrecht 1994: 118). It is important to keep this definition of topic in mind since several studies on markers similar to in Central Chadic languages have used a much broader definition of topic (Section 1.3). Focus is defined as “the semantic component of a pragmatically structured proposition whereby the assertion differs from the presupposition” (Lambrecht 1994: 213).
The study of information structure in Barayin is still very limited, but some preliminary assumptions can be made that will be helpful in the discussions throughout this article. Barayin does not have any particles dedicated to marking topic and focus of a particular noun phrase. It is generally the case that the topic will be a pronominal subject proclitic, or an assumed but unspoken subject.
It is likely the case that post-nominal demonstratives have an information structure function, but this issue has not yet been explored. There are two other information structure features of the language that are not in the scope of this study, but will be mentioned here briefly: the pre-subject position and a preverbal marker joo or doo with contrastive meaning. The pre-subject position (Figure 1) clearly has some type of pragmatic meaning, but it is not well-understood. Interrogative pronouns, which are inherently focus words (Lambrecht 1994: 283), often occur in situ without any additional marking, however, they can also be preposed in a position before the subject, as in example (5).
(5)
ma Mariam mi$\underset{˜}{n}$-ga who Mariam slap-obj.3sg.m
‘Who did Mariam hit?’
Based on example (5), it would be plausible to suggest that the pre-subject position is a type of focus position. That analysis is less plausible for the rare cases of pre-subject object placement, as in jeedo ‘mountain’ in example (6). In the context, the mountain has already been mentioned in the preceding sentence, and the clause containing the object in a pre-subject position is a type of tail-head linkage followed by the marker (Section 3.8).
(6)
jeedo ti di iŋ daw-o-geti=nà ná mountain sbj.3sg.f dem.sg.f sbj.1du.incl occupy?-inf-poss.3sg.f=pl na ŋ daw-o-geti=nà ná talaŋ obl occupy-inf-poss=pl na how
‘[After several years they said: Our mountain here na, we should inhabit it. Then they said:] That mountain, we should inhabit it na, but how?’ (History 16)
Adverbs and adjuncts can also appear in a position before the subject, as in example (7). Adverbs that occur in this context are apparently not focus elements either.
(7)
tande mejere kol-eyi ŋ app-o ŋ ammi ... yesterday people go-ipfv obl dig-inf obl water
‘[I will tell you the story about Mosso... the story about the water.] The other day, some people went to drill for water...’ (Mosso 3)
Another possibility is that there is more than one pre-subject position. Further testing would be required to know if more than one unmarked pre-subject constituent can occur, and in what orders they can occur in. Since the information structure status of the pre-subject position is not the point of this article, the only relevant point to be made is that the pre-subject position(s) occur(s) structurally after any element marked by . In other words, elements marked by occur in a clause-external position before all of the elements of the word order template given in Figure 1. Example (8) shows followed by a pre-subject adverb. The example starts with a clause-external locative expression marked by . The first element of the clause following is a pre-subject adverbial direkt.
(8)
[ ] [Pre-subject] [Subject] [ Predicate ] ŋ suk Alay ná direkt ki ŋ jaŋg-o obl market(Ar.) Alay na directly(Fr.) sbj.2sg.m obl descend-inf
‘At the Alay market, you will keep going straight down.’ (Directions 10)
The second information structure element to mention is the pre-verbal particle joo or doo, glossed foc for ‘contrastive focus’. Its meaning is still poorly understood, but it appears to have scope over the entire predication, not over a single noun phrase or other constituent. It can optionally occur in negative sentences, primarily those with a subjunctive verb, such as the imperative in example (9) or the prohibitive in example (10). However, joo/doo can also occur without negation when there is a contrastive meaning, as in example (11).
(9)
joo/doo kol-u do foc go-sbjv neg
‘Don’t go!’ (Lovestrand 2012: 186)
(10)
nandaŋga doo $\underset{˜}{n}$oom-u iŋ aka do children foc play-sbjv asoc fire neg
‘Children should not play with fire.’ (Lovestrand 2012: 186)
(11)
ŋ nandiy-aŋ-ju ŋ gar-eyi ŋ jappa wo obl children.pl-nmlz-poss.1sg sbj.1sg study(Ar.)-ipfv obl church but sonde joo ŋ doy-eyi malumi-ya-ŋ now foc sbj.1sg study-ipfv islam(Ar.)-pl-nmlz
‘During my childhood, I went to church, but now I follow Islam.’ (Lovestrand 2012: 95)
## 1.3 Background markers in other languages
The term background is to be understood in the sense of Ameka (1991). Ameka describes a so-called “terminal” particle in Ewe which marks constituents that “typically carry information that a speaker wants an addressee to assume in order for him/her to process the rest of the discourse more easily” and which function as the “the domain within which the rest of the predication should be interpreted” (Ameka 1991: 152, 154). He concludes that the “invariant function of the terminal particles is to mark background information” (Ameka 1991: 152). In Ameka’s approach (and linguists writing on similar particles seem to agree), it is assumed that background is not incompatible with topic (“what the sentence is about”). His critique of those who have analyzed as a topic marker is not that it is contradictory, but that it is incomplete. Background can be understood as a larger category that includes the notion of topic (“what the sentence is about”), but is broader in that it also includes the function of sentence-initial dependent clauses and other types of information. In contrast, Ameka is clear that background and focus are incompatible (Ameka 1991: 152-153).
Many authors have described a particle with a similar function in Central Chadic languages. A relatively early study of the Central Chadic language Zulgo labels the marker a “Topic Marker” (Haller and Watters, 1984). Their use of “topic” comes from Chafe (1976: 51) who writes that “‘Real’ topics (in topic-prominent languages) are not so much ‘what the sentence is about’ as ‘the frame within which the sentence holds’.” Although they take the label “topic” from Chafe, Haller and Watters (1984) propose a definition based on what Dik (1981: 19) calls “theme”: “The Theme specifies the universe of discourse with respect to which the subsequent predication is presented as relevant.” Haller and Watters (1984) argue that this concept gives a unified account of all of the uses of the marker in Zulgo. Haiman (1978) proposes a similar understanding of “topic” in his analysis of Hua, a Papuan language, claiming that the identical morphosyntactic marking for conditional protases and “topic” noun phrases not only shows that conditional clauses are “topics” in Hua, but that conditional clauses should be universally considered “topics”.2 This use of the label “topic” is not followed in this article primarily to avoid potential confusion with topic as “what the sentence is about”. For example, Givn (1990: 846) argues against the use of “topic” for adverbial clauses by Haiman (1978) and Thompson and Longacre (1985) because of a clash with his definition of topic as “what the sentence is about”. Chafe saw this potential confusion over terminology as a significant issue, and later changed his terminology from “topic” to “starting point” (Chafe 1987: 22). Nonetheless, there is an obvious similarity between the concept of background used in this article, and the sense of “topic” as used by Chafe (1976), Haiman (1978) and other authors.
Several other linguists have followed Haller and Watters (1984) in using the label “Topic Marker” for a similar marker in other Central Chadic languages (Buwal: Viljoen (2013;2015: 612); Gemzek: Scherrer (2012); Mofu-Gudur: Hollingsworth and Peck (1992); Hollingsworth (1985); Moloko and Muyang: Smith (2003); Ouldemè and Vamè: Kinnaird (1999); Wandala: Pohlig and Pohlig (1994)). However, most of these studies offer a slightly different descriptions of the meaning of “Topic Marker”. For example, Hollingsworth and Peck (1992), Scherrer (2012) and Smith (2003) prefer a description in terms of “point of departure”. Point of departure has a two-part definition: (1) “It establishes a starting point for the communication;” and (2) It “cohesively anchors the subsequent clause(s) to something which is already in the context (i.e., to something accessible in the hearer’s mental representation)” (Levinsohn 2012: 40). Again, this definition is very similar to Ameka’s “background”, Chafe’s “topic” and Dik’s “theme”, but it is more restricted by an emphasis on a link between the “point of departure” and the context (see Section 3.8).
Other works on Central Chadic languages describe a marker with a similar distribution and function, but avoid giving any particular label to the marker (Gidar: Frajzyngier (2008: 379, 385–387, 437, 441); Hdi: Frajzyngier (2002: 391); Lamang: Wolff (1983: 244–247, 2015: 296-298); Mbuko: Gravina (2003); Wandala: Fluckiger and Whaley (1983)). At least two other labels have been proposed for these markers in Central Chadic languages: “spacer” (Dooley and Levinsohn 2001: 128) and “comment-clause marker” (Frajzyngier 2012).
In East Chadic languages, the branch Barayin belongs to, there has been very little study done on information structure and discourse particles. However, Lele has a marker na which, in addition to functioning as a demonstrative, also has a similar distribution to in Barayin (Frajzyngier 2001: 333–335, 420-423, 464–467). Schuh (2005: 89–95) identifies similar (unnamed) particles in four West Chadic languages: Bole, Ngamo, Karekare, Ngizim. These markers are also discussed by Güldemann (2016: 558–565) who cites Gimba (2005) as a source on the same marker in Bole, as does Zimmermann (2011: 1174–1176) in his discussion of the Bole data.
Outside of Chadic languages, a strikingly similar type of marker, also , is found in Bagirmi. Jacob (2010: 126) describes in Bagirmi as a “background marker” in addition to its functions as a determiner and a marker at the end of a relative clause. Jacob’s analysis is also discussed by Güldemann (2016: 565–567). Bagirmi is a Nilo-Saharan language that borders Barayin. Boujol and Clupot (1941: 34) report that the Barayin were previously in a vassal-suzerain relationship with the Bagirmi, and some older Barayin speakers still say they know how to speak Bagirmi. This seems to indicate that the similarity between Barayin and Bagirmi could potentially be at least partially explained by language contact.
Farther away, but still in Africa, Güldemann (2016: 567–572) draws attention to several publications about a particle la in Dagbani and other Oti-Volta (Niger-Congo) languages of Ghana. The marker is sometimes described as a focus marker, which happens to be isomorphic with a definite marker. Güldemann (2016: 570) concludes that the marker “is best analyzed as a background marker” on par with those in Chadic languages and Bagirmi. In his discussion of the background marker in Ewe (also a Niger-Congo language of Ghana), Ameka (1991: 168) lists a few other West African languages that have a similar marker, and suggests that background markers also occur in Polish, Thai and Japanese.
Besides their functional similarities, these markers are also similar to each other in that they can follow both sentence-initial noun phrases, and sentence-initial finite clauses. For example, “in Zulgo the particle ka, which is clearly used to mark a topicalized phrasal element, can also be used to mark clausal elements which at first glance appear to be cases of subordination” (Haller and Watters 1984: 27). The particle in Ewe “occurs at the end of preposed adverbial and nominal phrases” and “also occurs at the end of various kinds of initial dependent clauses, for example conditionals” (Ameka 1991: 145–146). It is very common for background markers to be isomorphic with a demonstrative or definite marker, as well as a marker following relative clauses. However, this is not the case in Barayin. The particle is distinct from the demonstratives gi, di and ni. In Barayin, these demonstratives (not the background marker ) optionally occur in the position at the end of a relative clause.
Güldemann (2016: 571) warns that “la-like particles are numerous across Oti-Volta but they do not all share the same functional and syntactic profile. So whatever the analysis of la in Dagbani and Gurene, it must not be transferred rashly to related languages with similar particles...” Similarly, Ameka (1991: 168) points out the need for further investigation of these similarities, but cautions that a “prerequisite for such a research is the systematic documentation and analysis of the data in individual languages.” This study of in Barayin contributes to the need for documentation of discourse particles in individual languages, leaving aside the intriguing comparative research for future work.
## 1.4 Data sources
The data used for the analysis of the particle in Barayin consists primarily of transcribed monologues, most of which are appended to Lovestrand (2012). Two additional more-recently transcribed stories were added to the corpus since they contain some uses of not seen in the other texts. These two stories are not yet published. Throughout this article, data extracted from these texts are referenced by a one word abbreviation of the title and the line number. The abbreviations are in Table 3. These nine transcribed texts will be referred to as the “corpus”.
Table 3:
Abbreviations for corpus references.
The particle occurs rarely in elicited language data of isolated phrases, but it is very common in the corpus of transcribed recordings of natural speech. In 599 (arbitrarily) divided lines of text (3963 words), it occurs 348 times. On average, more than half of the lines have the particle. In a slightly larger collection of fifteen texts, the marker is the most common word (690 occurrences), followed by the preposition ŋ (612 occurrences), and then the third person subject pronouns (294 sg.m, 232 pl, 168 sg.f).
## 2 Syntax of the marker ná
The five syntactic environments where can occur (the first four of which are shown in examples (1) through (4) above) are represented more abstractly by the following abbreviations (cf., Haller and Watters 1984: 43)
• term--proposition
• proposition--proposition
• term--term
• proposition--term
• -proposition
The first two syntactic environments (term--proposition and proposition--proposition) are very common in Barayin and other languages. These are described in detail in Sections 2.1 and 2.2. The third and fourth types have a smaller constituent, often a noun phrase or an interrogative word, following . These are discussed in Sections 2.3 and 2.4. Neither of these two structures is very common in the corpus of monologues. In Section 2.7 it is proposed that a single term following should be analyzed as functioning as a predicate in its own right. Therefore the distribution of can be described as always preceding a predication. Examples of in a sentence-initial position are discussed in Section 2.5. This is the least common construction, only occurring twice in the corpus. Section 2.6 contains examples of more than one being used in the same sentence. A proposed formal analysis of the syntactic structure of is presented in Section 2.7.
## 2.1 Term-ná-proposition
This section gives examples of the term--proposition construction. In many of the instances of this construction, the term occurring before the marker can be identified as a subject, object or indirect object of the proposition following the . In some cases, a pronominal element co-referential with the -marked term occurs in the following proposition. In example (12), the word mijjo ‘man/person’ before the marker can be identified as the subject of the following clause. The independent pronoun kalla in the subject position of the following proposition is co-referential with mijjo.
(12)
mijjo ná kalla bala inda-ji person na 3sg.m without(Ar.) have(Ar.)-poss.3sg.m
‘The man, he has nothing.’ (Carnivores 25)
It is not the case that a marked sentence-initial term always has to have its grammatical function in the following proposition identified by an overt pronoun. In the seven texts appended to Lovestrand (2012), there are eighteen examples where the -marked term in the sentence-initial position can be identified as the subject of the following proposition. In only two of those cases does the following proposition have a co-referential subject pronoun. In most cases, there is no overt subject in the proposition following , as in example (13). In this case, the subject role of the -marked term is identified by its semantic role and the gap in the subject position of the following clause.
(13)
inu ná ŋ kol-o ŋ duw-o-ji 1sg na obl go-inf obl see-inf-poss.3sg.m
‘I will go to see it.’ (Carnivores 48)
It is much less common for a term marked by to be identified as the direct or indirect object of the following proposition. In the seven texts appended to Lovestrand (2012), there are only three examples of a direct object occurring in this construction. In one case, example (14), the following clause has a pronominal object suffix that is co-referential with the sentence-initial term.
(14)
mijjo ná joo jel-ga=nà iŋ nílla do person na foc put-obj.3sg.m=pl asoc 2pl neg
‘The man... You should not put him with you.’ (Carnivores 57-58)
In the other two cases, one of which is shown in example (15), the following clause does not have an object suffix or pronoun co-indexing the sentence-initial term. The word ragga ‘mat’ is understood to have the semantic role of patient, which is the semantic role normally assigned to a direct object by this verb in this context.
(15)
ragga ná iŋ t-eyi t-ii do mat na sbj.1du.incl eat-ipfv eat-inf neg
‘We didn’t eat the mat.’ (Girl 56)
In the same seven texts, there is only one example of a -marked sentence-initial constituent that can be identified as the indirect object of the following proposition. In example (16), the verb of the proposition has an indirect object suffix (glossed dat) that is co-referential with the sentence-initial constituent.
(16)
aya kaw ná ni joo duw-aya je ragga $\underset{˜}{n}$erwa 1du.incl also na sbj.3pl foc put-dat.1du.incl part mat skin gi bas dem.sg.m only(Ar.)
‘Us too, they put down this leather mat for us.’ (Girl 35–36)
Adjunctival prepositional phrases or adverbs also commonly occur in the sentence-initial position before the marker , as in example (17).
(17)
iŋ bodo ná ni ni naa marbo ti di ná asoc night na sbj.3pl dem.pl quot.3pl girl sbj.3sg.f dem.sg.f na sent-eti aj-o refusal-poss.3sg.f come-inf
‘That night, they said: That girl refuses to come.’ (Girl 22–23)
It is also common for locatives indicating source or setting to appear in a sentence-initial position followed by the marker , as in examples (18) and (19).
(18)
Baro ná ni s-eyi Barlo Baro na sbj.3pl come-ipfv Barlo
‘From Baro, they came to Barlo.’ (History 2)
(19)
ŋ suk Alay ná direkt ki ŋ jaŋg-o obl market(Ar.) Alay na directly(Fr.) sbj.2sg.m obl descend-inf
‘At the Alay market, you keep going straight down.’ (Directions 10)
In example (20), a sentence-initial ideophone is followed by . Ideophones in Barayin generally have the same syntactic distribution as adverbs.
(20)
ratatatatata ná iss-a-jo luwa raga ideo na pour-ipfv-dtrv above mat
‘Plop plop plop. It fell onto the mat.’ (Girl 29)
## 2.2 Proposition-ná-proposition
In addition to marking a single sentence-initial term, the marker can also appear between two finite clauses, as seen in examples (21) and (22).
(21)
ka t-aa je ná ka kol-u sbj.3sg.m eat-pfv part na sbj.3sg.m go-sbjv
‘When he has eaten, he should leave.’ (Lovestrand 2012: 110)
(22)
to ki s-etta ná i$\underset{˜}{n}$o ŋ n-ii cond sbj.2sg.m come-prf na boule obl cook-inf
‘When you arrive, the boule will be ready.’ (Lovestrand 2012: 208)
The marker never occurs at the end of a sentence. An adverbial clause marked by a clause-initial to ‘if/when’ (like that in example (22)) can occur either before or after the main clause. The subordinating conjunction to is called a conditional marker but, as will be seen in Section 3.4, it can have either conditional (‘if’) or temporal (‘when’) meaning. When the conditional clause follows the main clause, it is not possible for the marker to occur at the end of the adverbial clause, as is shown in example (23). The marker cannot be described syntactically as a clause-final marker.
(23)
i$\underset{˜}{n}$o ŋ n-ii to ki s-etta (*ná) boule obl cook-inf cond sbj.2sg.m come-prf (na)
‘The boule will be ready when you arrive.’
## 2.3 Term-ná-term
In all of the examples of term--term structures examined, the first term is a noun phrase. The second term can also be a noun phrase, as in examples (24) and (25). Example (24) is an identificational construction where the pronominal preceding is co-referential with the noun phrase following (Section 3.6).
(24)
ti di ná non-ju di sbj.3sg.f dem.sg.f na child-poss.1sg dem.sg.f
‘This is my daughter.’ (Lovestrand 2012: 208)
In example (25), the noun phrase preceding is the unspoken subject of the nominal predicate following . The noun a$\underset{˜}{n}$a (often with a possessive suffix indexing the subject) is the standard way to express existence or presence in Barayin (Lovestrand 2012: 205–207). In this case, even though only a single word follows , that single word must be analyzed as a non-verbal predicate, just as it would if it were a single verb. This analysis can also plausibly be applied to the noun phrase following in example (24), as will be proposed in Section 2.7.
(25)
ragga-jiŋ ná a$\underset{˜}{n}$a-geti mat-poss.2pl na presence-poss.3sg.f
‘Your mat was still there.’ (Girl 51)
In example (26) the second term is an adjective. However, in this particular context the adjective does not modify the sentence-initial noun. In the story, a husband is searching for his wife who was taken back by her family. The wife does not know whether or not to respond to his calls. In this example, the grandmother speaks to the daughter empathizing with her since, in this situation involving her husband, it is difficult to know what to do. The adjective ‘difficult’ is functioning as a predicate describing the situation. It is not modifying the husband.
(26)
meeri ná tega-gu husband na difficult-sg.m
‘With your husband, it’s hard [to refuse].’ (Loori 144)
## 2.4 Proposition-ná-term
There are fewer examples of the proposition--term construction than the types described above. In examples (27) and (28), the word following is an interrogative word. Content questions are often formed using an interrogative word in situ or in a sentence-initial position. However, interrogatives can also occur in a clause-final position when preceded by . In this construction, the speaker first gives all of the presupposed elements of the question, followed by the marker , and then the appropriate question word.
(27)
mapana ki d-ii-ga ná talaŋ thing sbj.2sg.m kill-pfv-obj.3sg.m na how
‘How did you kill this thing?’ (lit., thing you killed it na how) (Carnivores 68)
(28)
de ŋ aj-o ŋ t-ii=nà ná mo gi saŋ rel.sg.m obl come-inf obl eat-inf=pl na what dem.sg.m q
‘...which we will come to eat, what [is it]?’ (Carnivores 29)
There are also a few examples where the term following is not an interrogative word. In example (29), the final term is an ideophone.
(29)
l-ega ná kiŋkil send-ipfv na ideo
‘They put down a lot!’ (Loori 134)
Example (30) is an elicited example modeled after similar sentences documented in other Chadic languages (e.g., Buwal: Viljoen (2013: 607); Mbuko: Gravina (2003: 7)). The marker separates a clause-final direct object from the rest of the clause.
(30)
ka t-eyi ná suu sbj.3sg.m eat-ipfv na meat
‘It’s meat that he eats.’
## 2.5 Sentence-initial ná (ná-proposition)
In just two cases, the particle appears to occur in a sentence-initial position. These are shown in examples (31) and (32).
(31)
a.
kaa d-aa d-ii de Njamena teyi da quot.3sg.m walk-pfv walk-inf rel.sg.f N’Djamena like.that ???
‘He said: You walk the walk of N’Djamena like that?’
b.
taa njamena njamena njamena njamena quot.3sg.f N’Djamena N’Djamena N’Djamena N’Djamena
‘She said: N’Djamena. N’Djamena. N’Djamena. N’Djamena.’
c.
ná taa gi hay killa duw-ga nopuno ge na quot.3sg.f dem.sg.m hey 2sg.m see-obj.3sg.m goat rel.sg.m dogo alli until there
‘Then she said: Hey, you see that goat over there?’ (Bulmi 58-60)
(32)
a.
kalas ná sonde hiya ka duwa joo wal-lo that’s.all(Ar.) na now so(Ar.) sbj.3sg.m lion foc spend.a.year-obl je maŋa part bush
‘So now the lion returned to the jungle.’
b.
aya=nà mijjo att-u ge siidi 1du.incl=pl person remain-sbjv rel.sg.m home
‘We humans stayed at home.’
c.
ná sidiki jeedo sidiki ti di na story again story sbj.3sg.f dem.sg.f
‘So, this story is yet another story.’ (Loori 252-254)
The analysis of the clauses in examples (31c) and (32c) as a distinct sentence from the preceding clause is based both on the meaning of the clauses and the prosody. In both examples, there is a noticeable pause before . A pause preceding is rare, if it at all occurs elsewhere. In terms of meaning, there is no obvious logical connection between the preceding clause and the clause followed by .
## 2.6 Multiple instances of ná in one sentence
It is not uncommon for more than one to appear in the same sentence. Güldemann (2016: 567) points out that this also happens in Bagirmi, but not in the four West Chadic languages he studied. In Barayin, this partitioning of the pre-clausal space can take several forms. For example, it can be a sequence of sentence-initial terms, each marked with as in example (33) where the two terms are co-referential.
(33)
mejere ná abbo-ya-tiya alli ná ganda t-ii-ga people na neighbor-pl-poss.1du.incl there na inside eat-ipfv-obj.3sg.m
‘...Those people, our neighbors there, are eating it.’ (Girl 33)
It is also possible to string together more than one clause marked by as in example (34).3
(34)
a.
att-e mijjo ná remain-prf person na
‘When only the man was left,’
b.
ni sul-eyi ŋ doo de ni sul-lo je ná sbj.3pl sit-ipfv obl place rel.sg.f sbj.3pl sit-obl part na
‘they sat where they sat before,’
c.
ni gas-eyi ... sbj.3pl say-ipfv
‘and they said...’ (Carnivores 23–25)
The use of the marker can create even more complex sentences. The marker occurs four times in example (35). It first is used with three consecutive terms (a location, the subject of the following clause, and a modifier of that subject), and then it occurs again between that clause and the next clause.
(35)
min Gili ná mejera-tiga ná sina ná juk-eyi ná naa from(Ar.) Gili na people-poss.3pl na other na stand-ipfv na quot.3pl ane kol-u duw-ga jeedo 1pl.excl go-sbjv see-obj.3sg.m mountain
‘From Gili, their people, some of them, got up, and they said: We should go see the mountain...’ (History 6–7)
## 2.7 Formal syntactic structure of ná
The syntactic distribution of can be generalized by stating that must always be followed by a clause or predicate. This approach would allow to be treated as a type of complementizer. In an X-bar theoretic analysis, it could be postulated that is the head of a CP projection, of which the following material is the complement, and the preceding material is the specifier. This analysis aligns with the strong pattern of left-headedness in the language. The specifier position must then allow a variety of lexical categories such as NP, PP, AdvP and S (or IP). The sister of C, its complement, would be either S (or IP) or CP when more than one appears in the same sentence. Figure 2 is a model of this proposed analysis applied to example (35) with multiple instances of .
Figure 2:
Structure of example (35).
In examples where is followed by a term, that term is analyzed as a non-verbal predicate with an unspoken subject. An existential clause consisting of a single noun phrase or adjective seems plausible since Barayin has both non-verbal predicates (Lovestrand 2012: 204-210) and clauses with no overt subject. In example (4), ‘She knows na child’, and similar examples, the single word following (e.g., ‘child’) would then have to be interpreted as having an existential predicative function (i.e., ‘It is her child.’). This structure is shows in Figure 3.
Figure 3:
Structure of example (4).
It is interesting to note that in this proposed analysis there is a mismatch between the syntactic structure and the prosodic and semantic structures. The marker forms a tighter syntactic constituent with the following material than the preceding, even though it forms a prosodic constituent with the preceding material, and its meaning might also be said to scope leftward (cf. Cysouw 2005).
## 3 Semantic and pragmatic contexts where ná occurs in Barayin
Section 2 above presents the varying types of syntactic constituents that can occur before and after the marker . This section gives an overview of several types of semantic and pragmatic material that can occur before the marker . Each subsection gives an example of a semantic or pragmatic context where is used. The contexts given are not necessarily exhaustive. The examples are meant to give an overall impression of the variety of contexts that the marker can occur in.
The types of information that can be marked by include: marked topics, vocatives, ordinal and temporal adverbials, conditional clauses, other finite background clauses, and the presupposition of a maximal backgrounding construction. In most cases, the use of is obligatory in the sense that the same information structure or pragmatic effect is not achieved without the marker. However, in the case of conditional clauses and ordinal and temporal phrases, the use of appears to be somewhat redundant, such that its removal does not obviously change the meaning of the clause in any way.
Section 3.7 briefly discusses the possible use of a sentence-initial to background the preceding discourse beyond one term or proposition. Section 3.8 explores a slightly different issue: how sentences with are used in creating discourse coherence between sentences by restating information from a previous sentence in a pattern that can be described as tail-head linkage or point of departure.
## 3.1 Marked topics
Recall that topic in this article is defined as “what the sentence is about”, and that it is generally the case in Barayin that the subject of a clause is also the topic. However, in certain cases, the topic is marked by . Topic has always been considered a part of the concept of background (Section 1.3). Presenting a topic as background gives the addressee the context in which the following information is relevant, making it easier for the proposition to be interpreted. Backgrounding serves as a way to signal that a non-subject is the topic, or that a subject is a new or contrastive topic.
In example (36), the hyena runs back to the group of animals to report on what he has just learned about the human. The hyena is giving information about the human to the other animals. The direct quote in (36b) begins with the noun phrase “person” followed by and co-referenced by a direct object suffix. Since the human is both the topic and the direct object, it is marked by . Otherwise the subject would be the topic by default.
(36)
a.
ka gor-eyi s-eyi sbj.3sg.m run-ipfv come-ipfv
‘He (Hyena) ran back.’
b.
kaa mijjo ná joo jel-ga=nà iŋ nílla do quot.3sg.m person na foc put-obj.3sg.m=pl asoc 2pl neg
‘He said: The man, you should not put him with you.’
c.
‘The man is evil.’ (Carnivores 56-59)
In some works on similar particles in other Chadic languages, constructions containing a marked sentence-initial term, like example (36b), are given a free translation beginning with “As for$\cdots$”. In general, the free translations I give are meant to be the English equivalent of the French translation given by the Barayin speaker who translated the text. My intuition is that, in many instances, the “As for...” translation would be a misinterpretation of the information structure. Chafe (1976: 49) analyzes the “As for...” construction in English is an example of what he calls “focus of contrast” (i.e., contrastive focus), and not the equivalent of what he calls “topic” (here called background). A reviewer suggests that “As for...” constructions could also be interpreted as a case of contrastive topic, depending on the context. In either case, the “As for...” English sentences are an appropriate free translation in some, but not all uses of the marker following a noun phrase in Barayin.
Example (37) is from another story where two pairs of animals are at odds because one pair ate a mat they should not have eaten, and they fear what consequences may come. After referring to the mat, they ask the other pair what they will say once the group arrives at their destination. In response, the innocent pair make a comment about their mat (which they did not eat). The pair of transgressors react to this by starting a fight. In order to make the mat the topic of their response, it is introduced as a sentence-initial constituent followed by in example (37c). The mat is understood to be the patient/object of the following clause, but (unlike example (36)) there is no co-referential pronominal element in the following clause. The role of the mat in the following clause is inferred by the semantics and the gap in the object position.
(37)
a.
bulmi kaa gi hyena quot.3sg.m dem.sg.m
‘Hyena said this:’
b.
ane ... to ane kol-e ŋ gas-o ná gi 1pl.excl cond 1pl.excl go-prf obl say-inf na dem.sg.m
‘We... when we arrive, what we will say is this:’
c.
ragga ná iŋ t-eyi t-ii do mat na sbj.1du.incl eat-ipfv eat-inf neg
‘We didn’t eat the mat.’ (lit., The mat, we didn’t eat.) (Girl 54-56)
Since subjects are by default the topic, they do not need to be morphosyntactically marked for this function. However, if a subject is new in the discourse (or not the topic of the previous sentence), it may be followed by . Cross-linguistically it is not surprising to find that new topics are morphosyntactically marked (Givón 2001: 254). In example (38), from the Carnivores narrative, the scene switches from a conversation between the animals about the man as they wonder how he will be able to hunt (example (38a)), to the activity of the man (example (38b)). At this point in the narrative, the subject is marked with . Following , there is no overt subject. Note that in the next sentence with the same subject and the same topic, the subject is omitted altogether (example (38c)).
(38)
a.
jekk-a=nà duw-ga=nà duw-o atti leave-pfv=pl see-obj.3sg.m=pl see-inf so
‘Let’s just watch him (the man).’
b.
mijjo ná sule makid-a-ti pendeŋ-ji person na prog arrange-ipfv-obj.3sg.f bow-poss.3sg.m
‘The man sat arranging his bow.’
c.
gow-a-ŋ kese-ji gather-pfv-obj.3pl arrow-poss.3sg.m
‘[He] gathered his arrows.’ (Carnivores 30–32)
A similar construction occurs in example (39) from the same narrative. There is a shift in the discourse from the hyena’s thoughts (as he watches the man hunt) to the actions of the hyena (when he decides to return to the other animals to report what he has seen).
(39)
bulmi ná juk-eyi maalaŋ hyena na stand-ipfv slowness
‘The hyena slowly got up.’ (Carnivores 46–47)
Another context where topical subjects are marked by is in an identificational construction of the form NP NP, where the first NP is the subject and the second is the predicate. This construction is discussed in Section 3.6.
An adjunct can also be marked as a topic in this way. One monologue is a response to the speaker being asked about what she did yesterday. The first sentence, example (40), begins with a -marked adverb ‘yesterday’ which refers to both the wider discourse topic, as well as the topic of the sentence.
(40)
tande ná suk de genne yesterday na market rel.sg.f ours
‘Yesterday was our market.’ (Yesterday 1)
A topical noun phrase marked by does not necessarily have any grammatical function in the following proposition. As described in Section2.3, the noun phrase in example (41) (repeated from example (26)) is not modified by the following adjective, i.e., it is not the subject of the adjectival predicate. Rather, this noun phrase refers more generally to the type of situation under discussion. The term following (a non-verbal predicate with an omitted subject) comments that this type of situation is difficult.
(41)
meeri ná tega-gu husband na difficult-sg.m
‘With your husband, it’s hard [to refuse].’ (Loori 144)
## 3.2 Vocative
In most cases in the corpus, a pronoun, noun or proper noun in reported speech that is co-referential with the addressee is not marked by any particle. However, in a few cases a vocative element referring to the addressee occurs in a sentence-initial position marked by , as in example (42b).
(42)
a.
ni kol-e ŋ doo ta way-o ná sbj.3pl go-prf obl place purp pass.time-inf na
‘They went to the place where they would spend the afternoon.’
b.
naa nílla ná ragga-jiŋ ná a$\underset{˜}{n}$a-geti quot.3pl 2pl na mat-poss.2pl na presence-poss.3sg.f
‘They said: You! Your mat was still there.’ (Girl 50–51)
When vocatives are marked by in the examples found in the corpus, the addressee also has some role in the following clause (e.g., possessor of the subject ‘mat’ in example (42b)). However, it is possible to elicit a sentence like example (43) in which a vocative noun phrase marked by has no role in the following clause.
(43)
Musa ná inu ŋ kol-o Moussa na 1sg obl go-inf
‘Moussa, I am going to leave.’ (Said to someone named Moussa.)
The use of following vocatives contrasts with the background marker in Ewe. Ameka (1991: 155) notes that in Ewe cannot follow vocatives. Ameka explains this fact in Ewe by claiming that vocatives are inherently not background information: “Vocatives cannot be said to constitute a setting for the rest of the utterance... This confirms the view that the terminal particles mark background information in a clause.” There are several possible approaches to understanding this discrepancy. In a linguistic relativity approach, one might postulate that the concept of background is conceptually different for speakers of different languages and cultures. A second approach would be to analyze the vocative use of in Barayin as an additional sense or function. In other words, is polysemous in Barayin. In either of those analyses it would be expected that would occur with essentially all vocatives. However, the marker is only used with a small percentage of vocatives. A third, more plausible approach would be to assume that there are some rare cases in which vocatives can also be topics, and that it is precisely when the topic is co-referential with the addressee that a vocative can be marked by a background marker. In this sense, it would be assumed that in example (43), Moussa is not only the person being addressed, but is also somehow the topic. For example, it is because of something Moussa did that the speaker is leaving.
## 3.3 Ordinal and temporal discourse information
Sentence-initial adverbial phrases marked by normally have a function of marking temporal or ordinal progression in a narrative. This use of is similar to what Ameka (1991) describes as marking “connectives” in discourse. Ordinal and temporal discourse information is background information in that it serves the purpose of helping the addressee process the following preposition by placing into its temporal context in the narrative.
For example, part of the Carnivores narrative involves each of the characters taking turns to hunt food for the group. Throughout this part of the text, most of the phrases used for ordinal numbering, such as those in examples (44), (45) and (46), are in a sentence-initial position marked by .
(44)
de ta siidi ná maarum kol-eyi rel.sg.f purp two na panther go-ipfv
‘Second, the panther went.’ (Carnivores 13)
(45)
de ta subu ná ni gisir-a-gi bulmi rel.sg.f purp three na sbj.3pl send.out-ipfv-obj.3sg.m hyena
‘Third, they sent out the hyena.’ (Carnivores 15)
(46)
de ta pudu ná balaw kol-eyi d-eyi rel.sg.f purp four na wolf go-ipfv kill-ipfv
‘Fourth, the wolf went and killed [something].’ (Carnivores 18)
It is not the case that all such ordinal expressions must necessarily be followed by . As mentioned in Section 1.2, an adjunct can also be in a clause-internal pre-subject position without being marked by . In the same story as examples (44), (45) and (46), the first and the fifth in the series of animals hunting begin with an ordinal expression without the marker , as seen in example (47). The semantic content of ordinal expression in this context already suggests that it is background information. It can be (redundantly) marked as background for clarity, but it does not necessarily need to be marked.
(47)
ta dawsu baŋa kol-eyi d-eyi purp five dog go-ipfv kill-ipfv
‘Fifth, the dog went and killed [something].’ (Carnivores 20)
In a similar construction in example (48b), a temporal adverbial (prepositional phrase) in a sentence-initial position indicating the temporal progression of the storyline is marked with .
(48)
a.
ti kol-a duw-e ŋ ger-geti siidi sbj.3sg.f go-pfv go.to.bed-prf obl home-poss.3sg.f own
‘She went to bed in her own hut.’
b.
iŋ bodo ná ni ni naa marbo ti di asoc night na sbj.3pl dem.sg.f quot.3pl girl sbj.3sg.f dem.sg.f ná sent-eti aj-o na refusal-poss.3sg.f come-inf
‘That night, they said: That girl refuses to come.’ (Girl 21–23)
Again, adjuncts can also occur in a pre-subject position without the marker . This can be seen in example (49) where an identical adverbial phrase appears in a sentence-initial position without the marker .
(49)
iŋ bodo ni dow-eyi ná nopuno juk-eyi asoc night sbj.3pl sleep-ipfv na goat stand-ipfv
‘That night, while they slept, the goat got up.’ (Girl 27)
## 3.4 Conditional (if/when) clause
Part of the motivation for the concept of background is to describe grammatical marking that is used for both topics and conditional clauses. Sentence-initial conditional clauses are background information in that they serve the purpose of giving a context for interpreting the following proposition. When the marker occurs between two finite clauses, the preceding clause is often marked by the clause-initial subordinator to. The conjunction to marks subordinate clauses of either hypothetical ‘if’ or sequential ‘when’ meaning. It is common in Chadic languages for the same word to allow both conditional and temporal interpretations (Frajzyngier 1996: 313, 327). When the conditional/sequential clause (protasis) precedes the main clause (apodosis), it may be followed by , as in example (50) and (51).
(50)
a.
to ki gus-e kaye mala ŋ bu-ji ŋ cond sbj.2sg.m exit-prf here ??? obl mouth-poss.3sg.m obl golmo-jiŋ ná house-poss.2pl na
‘When you’ve gone out of the entrance to your house,’
b.
ki ŋ pid-o-geti ŋ ara 2.sg.m obl take-inf-poss.3sg.m obl path
‘you will take the main road.’ (Directions 3–4)
(51)
to ki wonni-ga ger-ne do ná ki ŋ cond sbj.2sg.m know-obj.3sg.m home-poss.1pl.excl neg na sbj.2sg.m obl
‘If you don’t know our house, you will ask them.’ (Directions 28)
It is not the case that sentence-initial conditional/sequential clauses are always followed by . In example (52) and (53), the conditional clause is not followed by . Since conditional clauses are already marked by the clause-initial to, it is clear that they are background information even when is absent. Like the use of with temporal and ordinal markers (Section 3.3), the use of with conditional clauses is optional because conditional clauses are inherently background information.
(52)
wo sonde [to ní kol-e siidi] ní ta ŋ gas-o but now cond sbj.2pl go-prf home sbj.2pl cert obl say-inf ni-ya mo sbj.2pl-quot what
‘And now, when you go home, what will you say?’ (Girl 53)
(53)
[to ane kol-e] ŋ gas-o ... cond 1pl.excl go-prf obl say-inf
‘When we arrive, we will say...’ (Girl 55)
## 3.5 Other finite background clauses
The use of the marker between clauses is not restricted to clauses marked by to. The marker can also follow an otherwise unmarked finite clause. The temporal relationship between a finite -marked clause and the following clause can generally be predicted from the TAM marking on the verb in the -marked clause, removing the need for temporal conjunctions like ‘while’ and ‘then’. If the verb in the -marked clause has perfect marking, it will have an anterior (‘having done X then Y’) reading. If the verb in the -marked clause has imperfective marking, it will normally have a simultaneous reading (‘while X, Y’). The perfect-marked ‘then’ clauses typically contain repeated information (see tail-head linkage in Section 3.8) which does not advance the progression of the text, but repeats the temporal or narrative context for the proposition following . Simultaneous ‘while’ clauses can carry new information, but this new information serves to set the stage for the more salient storyline event in the following proposition.
In example (54b), the clause preceding has a sequential reading similar to the to-marked clause in example (50). The action described by the verb ‘reach/arrive’ with a perfect suffix is understood to be completed before the action described by the following clause begins.
(54)
a.
ŋ pid-o-ti ŋ chari Atiya ti di obl take-inf-poss.3sg.f obl path Atiya sbj.3sg.f dem.3sg.f
‘[You] will take the Atiya path.’
b.
ki an-e ná ki ŋ jaŋg-o ŋ kol-o ŋ sbj.2sg.m reach-prf na sbj.2sg.m obl descend-inf obl go-inf obl jamiye de paa-tu mosque(Ar.) rel.sg.f big-sg.f
‘When you’ve arrived, you will go down to the big mosque.’ (Directions 6–8)
In contrast, when the verb of a -marked clause takes the imperfective TAM suffix, the most likely interpretation will be that the event predicated in the clause preceding is ongoing during the action or state expressed by the second clause in the proposition--proposition construction. This use of imperfective aspect to express simultaneity is a common feature of background information in a narrative (Givón 2001: 339). In example (55a), the dialogue between the four main characters ends, and example (55b) follows with the progression of the narration. The first clause in example (55b) (preceding ) has an imperfective verb, and the action described by the verb of this clause (‘sleeping’) is ongoing while the more salient action described by the second verb (‘getting up’) occurs.
(55)
a.
iŋ dow-u=nà sokka da iŋ ta sbj.1du.incl go.to.bed-sbjv=pl again then sbj.1du.incl cert s-aa=nà come-sbjv=pl
‘We should go to bed and come again later.’
b.
iŋ bodo ni dow-eyi ná nopuno juk-eyi asoc night sbj.3pl sleep-ipfv na goat stand-ipfv
‘That night, while they slept, the goat got up.’ (Girl 26–27)
A similar construction occurs in example (56c) from a different story. The verb of the clause preceded by has an imperfective TAM suffix, and describes a state that is ongoing throughout the action described by the following clause (example (56d)).
(56)
a.
illa att-e mijjo except remain-prf man
‘Only the man was left’
b.
att-e mijjo ná remain-prf man na
‘When only the man was left,’
c.
ni sul-eyi ŋ doo de ni sul-lo je ná sbj.3pl sit-ipfv obl place rel.sg.f sbj.3pl sit-obl part na
‘As they sat (or were sitting) where they sat before,’
d.
ni gas-eyi naa mijjo ná kalla wala inda-ji sbj.3pl say-ipfv quot.3pl man na 3sg.m neg have-poss.3sg.m
‘they said: The man, he has nothing.’ (Carnivores 23–25)
In this way, the interaction of with tense-aspect marking has a function similar to that of temporal conjunctions like ‘while’ and ‘then’. Such conjunctions are generally not used in Barayin, with the exception of loanwords from Chadian Arabic.
## 3.6 Maximal backgrounding (Presupposition-Focus)
Barayin does not have a dedicated focus marker with scope over a noun phrase or similar constituent. One way of expressing focus is through a structure in which a single term follows the background marker (proposition--term or term--term). The use of this structure in a focus context is what Jacob (2010) calls “indirect focus marking”, and what Güldemann (2016) calls “maximal backgrounding” and “indirect focalization”. “The crucial requirement for backgrounding to assume the central role for focalization is that it is MAXIMAL in the sense that it removes all but one potential focus host from the assertion domain” (Güldemann 2016: 577, emphasis in original).
Interrogative sentences with an interrogative pronoun inherently involve focus on the question word, and the rest of the utterance is pragmatically presupposed (Lambrecht 1994: 283). Two examples of this were given in Section 2.4, one of which is repeated in example (57). All of the information before is background or presupposed, and the single question word after is in focus.
(57)
mapana ki d-ii-ga ná talaŋ thing sbj.2sg.m kill-pfv-obj.3sg.m na how
‘How did you kill this thing?’ (lit., thing you killed, how?) (Carnivores 68)
One other place where maximal backgrounding occurs in the corpus is in reported speech. In one type of reported speech, the verb of speech or quotative marker is followed by a demonstrative gi which is in a cataphoric relationship of apposition with the following clause of reported speech. This can be seen in example (58b). A similar construction is seen in example (58c), however, in this example, the marker separates the cataphoric demonstrative from the rest of the clause in a proposition--term structure. The explanation for this is that example (58c) is a response to the question in example (58a): ‘What will you say?’. Thus the complement of the verb ‘say’ in example (58c) is in focus in the response to the question.
(58)
a.
wo sonde to ní kol-e siidi ní ta ŋ gas-o but now cond sbj.2pl go-prf home sbj.2pl cert obl say-inf ni-ya mo sbj.2pl-quot what
‘And now, when you go home, what will you say?’
b.
bulmi kaa gi hyena quot.3sg.m dem.sg.m
‘Hyena said this:’
c.
ane ... to ane kol-e ŋ gas-o ná gi 1pl.excl cond 1pl.excl go-prf obl say-inf na dem.sg.m
‘When we arrive, we will say this:’
d.
ragga ná ŋ t-eyi t-ii rdo mat na sbj.1du.incl eat-ipfv eat-inf neg
‘We didn’t eat the mat.’ (Girl 53, 55–56)
Although it rarely occurs in the corpus, it is possible to construct other types of declarative sentences of the same syntactic structure and information structure. Similar examples have been documented in several Central Chadic languages (e.g., Buwal (Viljoen 2015: 29), Muyang (Smith 2003: 4), Mbuko (Gravina 2003: 8), Zulgo (Haller and Watters 1984: 38), and Mofu-Gudur (Hollingsworth 1985: 5)). In example (59) (repeated from Section 2.4), the speaker understood there to be contrastive focus on the utterance-final noun phrase following the marker . The French equivalent given was Il ne mange que la viande. In other words, suu ‘meat’ is selected as the object of teyi ‘eat’, in contrast to anything else in the set of possible things to eat in that context.
(59)
ka t-eyi ná suu sbj.3sg.m eat-ipfv na meat
‘It’s meat that he eats. (He’s not eating anything else.)’
In example (60), the marker separates an infinitival (nominalized) verbal complement phrase from the rest of the proposition. The interpretation is similar. There is contrastive focus on the complement of the verb ‘know’, namely the infinitival verb phrase ‘making boule’. She knows how to prepare boule, but she does not know how to do other things.
(60)
ti wonn-eyi ná gan-o ŋ i$\underset{˜}{n}$o sbj.3sg.f know-ipfv na do-inf obl boule
‘What she knows is how to make boule.’
A final example of maximal backgrounding is found in identificational sentences. Identificational sentences equate a pronominal demonstrative with a noun phrase, such as “That is Joe Smith” (Higgins, 1973; Heller and Wolter, 2008; Moltmann, 2013). This construction is used for the purpose of giving the identity of a particular (known) referent. It could be the response to a question such as “Who is that?” or “What is this?” In the identificational sentence in Barayin, the demonstrative pronoun consists of a subject proclitic followed by an adnominal demonstrative. In examples (61) and (62), the marker follows the subject-demonstrative pronominal. This is followed by a noun phrase giving the identity of the subject. The explanation for the use of in this structure is that the pronominal element is presupposed or background information, and the constituent following is in focus, perhaps in response to a question about the identity of the subject.
(61)
ka gi ná sek ge ŋ gera gi sbj.3sg.m dem.sg.m na chief(Ar.) rel.sg.m obl village dem.sg.m
‘This is the chief of the village.’ (Lovestrand 2012: 208)
(62)
ti di ná non-ju di sbj.3sg.f dem.sg.f na child-poss.1sg dem.sg.f
‘This is my daughter.’ (Lovestrand 2012: 208)
The pair of examples (63) and (64) show that, while focus information can occur following , it cannot occur in the constituent preceding . The identificational construction (maximal backgrounding) in example (63) would be a natural way to identify someone as the chief when presenting a group of people one at a time. That is, in a context where the referents are already given, and the identity is the new information. In contrast, in example (64), the context is that someone has asked about the identity of the chief. In this context, the subject pronominal is the answer to a question (it is in focus), and it cannot be followed by . Likewise interrogative words cannot be placed on their own before . In Barayin, it is not possible to place focused items before the marker . The same is true of the background marker in Ewe (Ameka 1991: 152–153) and in Zulgo (Haller and Watters 1984: 29–30).
(63)
ka gi ná mon sbj.3sg.m dem.sg.m na chief
‘(Presenting a group of people one-by-one) This man is the chief.’
(64)
ka gi (*ná) mon sbj.3sg.m dem.sg.m na chief
‘(Who is the chief?) The chief is this man.’
In identificational constructions, the pronominal preceding is not only presupposed, but is also the topic. Identificational constructions in Barayin are both topic-comment and presupposition-focus. In other words, the information structure of an identificational construction is topic--focus.
## 3.7 General backgrounding of previous discourse
The use of in a sentence-initial position is quite rare, only occurring twice in the corpus. These two cases are shown in examples (31) and (32) in Section 2.5. The limited data makes it difficult to precisely describe the function of the sentence-initial use of . Viljoen (2015: 46) states that when a similar discourse particle occurs in a sentence-initial position in the Central Chadic language Buwal, it does so “in order to give prominence to a theme-line event. It also indicates that the previous information is backgrounded with respect to what follows.” The sentence-initial in example (31) could plausibly be giving “prominence” to the following clause, but the context does not give any satisfactory reasons why this clause should be particularly prominent in the narrative. In example (32), the clause following the sentence-initial is the final clause of the narrative–-a postscript signaling that the narrator has finished her story. There is no reason to think of this sentence as “prominent”. At this point, the most that can be said about the rare use of sentence-intial is that it does not indicate any close pragmatic or logical connection with the preceding clause or term. It could plausibly be thought of as indicating that the reason for the following sentence is explained by the general context of the preceding discourse. This would clearly explain its use in the final sentence of the narrative in example (32), and could plausibly be applied to example (31) as well.
When the marker follows an element repeated from the end of the preceding sentence this creates tail-head linkage. Tail-head linkage is a structure where “the tail of one sentence$\cdots$ is recapitulated as the head or the beginning of the following sentence” (Longacre and Hwang 2012: 7). This pattern can be used as a discourse strategy for showing the continuity or progression of a sequence of sentences in a text. The repeated information serves as background for the following proposition. This discourse function is seen not only in a term--proposition construction, but also in a proposition--proposition construction.
Example (65) is part of an oral history of the settlement of Barayin territory. The narrator repeatedly states that the travelers arrived in a village, and then repeats the name of that village with the marker at the beginning of the next sentence describing their move to the next new village. Note that all of the examples in (65) are sequential in the original text, one following immediately after the other with no intervening material.
(65)
a.
min Duŋgur ná ni s-eyi jel-eyi Alaw from(Ar.) Dungur na sbj.3pl come-ipfv put-ipfv Alaw
‘From Dungur, they put [some people] at Alaw.’
b.
Alaw ná ni kol-eyi jel-eyi Wore Alaw na sbj.3pl go-ipfv put-ipfv Wore
‘From Alaw, they put [some people] at Wore.’
c.
Wore ná ni kol-eyi jel-eyi Bose Wore na sbj.3pl go-ipfv put-ipfv Bose
‘From Wore, they put [some people] at Bose.’
d.
Bose ná ni kol-eyi jel-eyi ŋ Bela Bose na sbj.3pl go-ipfv put-ipfv obl Bela
‘From Bose, they put [some people] at Bela.’
e.
ŋ Bela ná ni s-eyi jel-a-ti Mebra obl Bela na sbj.3pl come-ipfv put-ipfv-obj.3sg.f Mebra
‘From Bela, they put someone at Mebra.’
f.
min Mebra ná ni kol-eyi jel-eyi mejere Dakro from(Ar.) Mebra na sbj.3pl go-ipfv put-ipfv people Dakro
‘After Mebra, they put some people at Dakro.’ (History 33-38)
Example (66) is part of the one procedural text in the corpus. Step-by-step directions are given to walk from one part of Mongo to another. The marker often follows constituents referring to landmarks that were given in the previous sentence (in prepositional phrases and noun phrases). The main clause then explains the action to take once having arrived at the landmark, as is done with the Alay market in example (66).
(66)
a.
ki ŋ tirs-o ŋ bu-geti ŋ suk Alay sbj.2sg.m obl arrive-inf obl mouth-poss.3sg.f obl market Alay
‘You will arrive at the entrance to the Alay market.’
b.
ŋ suk Alay ná direkt ki ŋ jaŋg-o obl market Alay na directly sbj.2sg.m obl descend-inf
‘At the Alay market, you will keep going straight down.’ (Direction 9–10)
In the same text the marker also follows finite clauses (proposition--proposition) that repeat the verb from the previous sentence. The verb ‘ask’ which appears in a future tense construction in example (67a) is repeated in example (67b) with a perfect TAM suffix.
(67)
a.
ki ŋ kett-o-jiga nilla sbj.2sg.m obl ask-inf-poss.3pl 3pl
b.
minde kett-e nilla ná nilla ná ŋ gas-o-geti ŋ after ask-prf 3pl na 3pl na obl say-inf-poss.3sg.f obl bu-ji ŋ golmo mouth-poss.3sg.m obl house
‘After having asked them, they will show you the entrance to the house.’ (Direction 28–29)
In examples (68) and (69), the clause marked by does not repeat verbatim the verb of the previous clause, but the clause does provide an obvious semantic link or anchor. For this reason, they are not cases of tail-head linkage, but more general “points of departure” which have a similar discourse function (e.g., Lambrecht 1994: 44, 51; Levinsohn 2012: 40; Weil 1844: 25).
(68)
a.
ŋ pid-e-ti ŋ chari Atiya ti di obl take-prf-obj.3sg.f obl path Atiya sbj.3sg.f dem.sg.f
‘You will take the Atiya road.’
b.
ki an-e ná ki ŋ jaŋg-o ŋ kol-o sbj.2sg.m reach-prf na sbj.2sg.m obl descend-inf obl mosque(Ar.) ŋ jamiye de paa-tu rel.3sg.f big-sg.f
‘When you’ve arrived, you will go down to the big mosque.’ (Directions 6–7)
(69)
a.
ki ŋ kol-o ŋ jamiye de lokud-o sbj.2sg.m obl go-inf obl mosque(Ar.) rel.sg.f descend-inf
‘You will go to the lower mosque.’
b.
ki an-e ŋ jamiye de paa-tu tilla ná sbj.2sg.m reach-prf obl mosque(Ar.) rel.sg.f big-sg.f 3sg.f na ki ŋ kol-o... sbj.2sg.m obl go-inf
‘Once you’ve reached the big mosque, you will go...’ (Direction 14–15)
## 4 Conclusion
The particle in Barayin separates a sentence into two parts, marking the preceding element as what Ameka (1991) refers to as background information (which has also been called “topic” (Chafe 1976)). Background information serves to help the addressee understand a proposition by giving the context in which that information should be interpreted. This can include clarifying the topic (“what the sentence is about”), giving temporal or ordinal information that clarifies where the proposition fits into the discourse structure, giving situational information about what is occurring at the time of the proposition, or repeating previously given information in a tail-head linkage structure. Backgrounding also serves to indirectly mark focus items by placing all the presupposed information before the background marker. The single term occurring after the background marker is understood to be in focus.
Similar markers occur in many other Chadic languages, West African languages and possibly in languages spoken in other parts of the world. However, these markers do not all necessarily “share the same functional and syntactic profile” (Güldemann 2016: 571). This presents a challenge for descriptive and comparative studies. What does the claim that is a background marker mean if background markers do not all function the same way cross-linguistically? How should we understand the fact that the background marker in Ewe cannot follow vocatives, but the background marker in Barayin can (Section 3.2)? Should the use of to mark the end of relative clauses in Ewe (and other languages) be understood as another type of background, or a separate function of the same marker (Ameka 1991: 152)? Do such differences between these markers mean that these are polyfunctional morphemes with language-specific lexicalized functions within the general semantic range of background information? Or do speakers of these language define background in different ways? Are there other factors that indirectly influence the way background markers can be used? Why do some languages allow more than one background marker in the same sentence, while others do not (Güldemann 2016: 567)? One potential way forward in gaining a deeper understanding of these markers could be to apply the “semantic mapping” method (Haspelmath, 2003). This approach would allow detailed comparisons of the similarities and differences in the functions of background markers in different languages.
This article helps clarify these typological questions and contributes towards finding satisfactory answers by giving a detailed analysis of the syntactic distribution and common semantic and pragmatic contexts of in Barayin. This article is also a first step towards understanding the information-structural features of Barayin grammar. More generally, this study of Barayin makes another contribution to the documentation of a minority language in the underdocumented group of East Chadic languages.
## Abbreviations
asoc ‘associative (with/and)’, cert ‘certainty’, cond ‘conditional (if/when)’, dat ‘dative/indirect object’, dem ‘(adnominal) demonstrative’, dtrv ‘detransitivizer’, du ‘dual’, excl ‘exclusive’, f ‘feminine’, foc ‘(contrastive) focus’, ideo ‘ideophone’, incl ‘inclusive’, inf ‘infinitive’, ipfv ‘imperfective’, m ‘masculine’, neg ‘negation’, obj ‘direct object’, obl ‘oblique’, part ‘(unidentified) particle’, pfv ‘perfective’ pl ‘plural’, prog ‘progressive’, poss ‘possessive’, prf ‘perfect’, purp ‘purposive’, q ‘question/interrogative mood’, quot ‘quotative’, rel ‘relative clause marker’, sbj ‘subject’, sbjv ‘subjunctive’, sg ‘singular’, 1 ‘first person’, 2 ‘second person’, 3 ‘third person’, (Ar.) ‘Chadian Arabic loan word’, (Fr.) ‘French loan word’
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## Footnotes
• 1
Examples are given in a simplified orthographic representation. More detailed phonetic transcriptions of most examples can be found in Lovestrand (2012). The set of recordings referred to as the “corpus” in Section 1.4 can be accessed via the website of the Endangered Languages Archive at SOAS: https://elar.soas.ac.uk/Collection/MPI1035101.
• 2
This approach was adopted by Seiler (1983), Thompson and Longacre (1985) and de Vries (1995). Haller and Watters (1984) also mention that Schuh (1972) made similar observations about a particle in Ngizim (West Chadic).
• 3
Note that in example (34), the apparent subject of the verb att-o ‘remain’ follows the verb. This is also seen with this verb in example (56). This marked word order occurs often with this particular verb, but not with other verbs. This is similar to languages in the Senegambian region where it is common for the verb ‘remain’ to be the only verb to allow what is sometimes called a “presentational” or “thetic” structure in which the sole argument of the verb is presented postverbally, losing the properties of a typical subject (Creissels et al. 2015: 69–71).
Published Online: 2018-04-26
Published in Print: 2018-05-25
Citation Information: Journal of African Languages and Linguistics, Volume 39, Issue 1, Pages 1–39, ISSN (Online) 1613-3811, ISSN (Print) 0167-6164,
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© 2018 Walter de Gruyter GmbH, Berlin/Boston.
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## A minimally configured Windows laptop
November 22, 2020 Windows No comments , , , , , ,
I’ve now installed enough that my new Windows machine is minimally functional (LaTex, Linux, and Mathematica), with enough installed that I can compile any of my latex based books, or standalone content for blog posts. My list of installed extras includes:
• Brother HL-2170W (printer driver)
• Windows Terminal
• GPL Ghostscript (for MaTeX, latex labels in Mathematica figures.)
• Wolfram Mathematica
• Firefox
• Chrome
• Visual Studio
• Python
• Julia
• Discord
• OBS Studio
• MikTeX
• SumatraPDF
• GVim
• Git
• PowerShell (7)
• Ubuntu
• Dropbox
Some notes:
• On Windows, for my LaTeX work, I used to use MikTex + cygwin. The cygwin dependency was for my makefile dependencies (gnu-make+perl). With this new machine, I tried WSL2. I’m running my bash shells within the new Windows Terminal, which is far superior to the old cmd.
• Putty is no longer required. Windows Terminal does the job very nicely. It does terminal emulation well enough that I can even ssh into a Linux machine and use screen within my Linux session, and my .screenrc just works. Very nice.
• SumatraPDF is for latex reverse tex lookup. i.e. I can double click on pdf content, and up pops the editor with the latex file. Last time I used Sumatra, I had to configure it to use GVim (notepad used to be the default I think.) Now it seems to be the default (to my suprise.)
• I will probably uninstall Git, as it seems superfluous given all the repos I want to access are cloned within my bash file system.
• I used to use GVim extensively on Windows, but most of my editing has been in vim in the bash shell. I expect I’ll now only use it for reverse tex (–synctex) lookup editing.
WSL2 has very impressive integration. A really nice demo of that was access of synctex lookup. Here’s a screenshot that shows it in action:
I invoked the windows pdf viewer within a bash shell in the Ubuntu VM, using the following:
pjoot@DESKTOP-6J7L1NS:~/project/blogit$alias pdfview alias pdfview='/mnt/c/Users/peete/AppData/Local/SumatraPDF/SumatraPDF.exe' pjoot@DESKTOP-6J7L1NS:~/project/blogit$ pdfview fibonacci.pdf
The Ubuntu filesystem directory has the fibonacci.synctex.gz reverse lookup index that Summatra is able to read. Note that this file, after unzipping, has only Linux paths (/home/pjoot/…), but Summatra is able to use those without any trouble, and pops up the (Windows executable) editor on the files after I double click on the file. This sequence is pretty convoluted:
• Linux bash ->
• invoke Windows pdf viewer ->
• that program reading Linux files ->
• it invokes a windows editor (presumably using the Linux path), and that editor magically knows the path to the Linux file that it has to edit.
### C64 Nostalgia.
Incidentally, does anybody else still have their 6402 assembly programming references? I’ve kept mine all these years, moving them around house to house, and taking a peek in them every few years, but I really ought to toss them! I’m sure I couldn’t even give them away.
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It is becoming increasingly common to see large collections of network data objects — that is, data sets in which a network is viewed as a fundamental unit of observation. As a result, there is a pressing need to develop network-based analogues of even many of the most basic tools already standard for scalar and vector data. In this paper, our focus is on averages of unlabeled, undirected networks with edge weights. Specifically, we (i) characterize a certain notion of the space of all such networks, (ii) describe key topological and geometric properties of this space relevant to doing probability and statistics thereupon, and (iii) use these properties to establish the asymptotic behavior of a generalized notion of an empirical mean under sampling from a distribution supported on this space. Our results rely on a combination of tools from geometry, probability theory, and statistical shape analysis. In particular, the lack of vertex labeling necessitates working with a quotient space modding out permutations of labels. This results in a nontrivial geometry for the space of unlabeled networks, which in turn is found to have important implications on the types of probabilistic and statistical results that may be obtained and the techniques needed to obtain them.
This monograph aims at providing an introduction to key concepts, algorithms, and theoretical frameworks in machine learning, including supervised and unsupervised learning, statistical learning theory, probabilistic graphical models and approximate inference. The intended readership consists of electrical engineers with a background in probability and linear algebra. The treatment builds on first principles, and organizes the main ideas according to clearly defined categories, such as discriminative and generative models, frequentist and Bayesian approaches, exact and approximate inference, directed and undirected models, and convex and non-convex optimization. The mathematical framework uses information-theoretic measures as a unifying tool. The text offers simple and reproducible numerical examples providing insights into key motivations and conclusions. Rather than providing exhaustive details on the existing myriad solutions in each specific category, for which the reader is referred to textbooks and papers, this monograph is meant as an entry point for an engineer into the literature on machine learning.
There is a great need for technologies that can predict the mortality of patients in intensive care units with both high accuracy and accountability. We present joint end-to-end neural network architectures that combine long short-term memory (LSTM) and a latent topic model to simultaneously train a classifier for mortality prediction and learn latent topics indicative of mortality from textual clinical notes. For topic interpretability, the topic modeling layer has been carefully designed as a single-layer network with constraints inspired by LDA. Experiments on the MIMIC-III dataset show that our models significantly outperform prior models that are based on LDA topics in mortality prediction. However, we achieve limited success with our method for interpreting topics from the trained models by looking at the neural network weights.
We introduce TensorFlow Agents, an efficient infrastructure paradigm for building parallel reinforcement learning algorithms in TensorFlow. We simulate multiple environments in parallel, and group them to perform the neural network computation on a batch rather than individual observations. This allows the TensorFlow execution engine to parallelize computation, without the need for manual synchronization. Environments are stepped in separate Python processes to progress them in parallel without interference of the global interpreter lock. As part of this project, we introduce BatchPPO, an efficient implementation of the proximal policy optimization algorithm. By open sourcing TensorFlow Agents, we hope to provide a flexible starting point for future projects that accelerates future research in the field.
Convolutional sparse representations are a form of sparse representation with a dictionary that has a structure that is equivalent to convolution with a set of linear filters. While effective algorithms have recently been developed for the convolutional sparse coding problem, the corresponding dictionary learning problem is substantially more challenging. Furthermore, although a number of different approaches have been proposed, the absence of thorough comparisons between them makes it difficult to determine which of them represents the current state of the art. The present work both addresses this deficiency and proposes some new approaches that outperform existing ones in certain contexts. A thorough set of performance comparisons indicates a very wide range of performance differences among the existing and proposed methods, and clearly identifies those that are the most effective.
The number of component classifiers chosen for an ensemble has a great impact on its prediction ability. In this paper, we use a geometric framework for a priori determining the ensemble size, applicable to most of the existing batch and online ensemble classifiers. There are only a limited number of studies on the ensemble size considering Majority Voting (MV) and Weighted Majority Voting (WMV). Almost all of them are designed for batch-mode, barely addressing online environments. The big data dimensions and resource limitations in terms of time and memory make the determination of the ensemble size crucial, especially for online environments. Our framework proves, for the MV aggregation rule, that the more strong components we can add to the ensemble the more accurate predictions we can achieve. On the other hand, for the WMV aggregation rule, we prove the existence of an ideal number of components equal to the number of class labels, with the premise that components are completely independent of each other and strong enough. While giving the exact definition for a strong and independent classifier in the context of an ensemble is a challenging task, our proposed geometric framework provides a theoretical explanation of diversity and its impact on the accuracy of predictions. We conduct an experimental evaluation with two different scenarios to show the practical value of our theorems.
Approximate computing aims for efficient execution of workflows where an approximate output is sufficient instead of the exact output. The idea behind approximate computing is to compute over a representative sample instead of the entire input dataset. Thus, approximate computing – based on the chosen sample size – can make a systematic trade-off between the output accuracy and computation efficiency. Unfortunately, the state-of-the-art systems for approximate computing primarily target batch analytics, where the input data remains unchanged during the course of sampling. Thus, they are not well-suited for stream analytics. This motivated the design of StreamApprox – a stream analytics system for approximate computing. To realize this idea, we designed an online stratified reservoir sampling algorithm to produce approximate output with rigorous error bounds. Importantly, our proposed algorithm is generic and can be applied to two prominent types of stream processing systems: (1) batched stream processing such as Apache Spark Streaming, and (2) pipelined stream processing such as Apache Flink. We evaluated StreamApprox using a set of microbenchmarks and real-world case studies. Our results show that Spark- and Flink-based StreamApprox systems achieve a speedup of $1.15\times$$3\times$ compared to the respective native Spark Streaming and Flink executions, with varying sampling fraction of $80\%$ to $10\%$. Furthermore, we have also implemented an improved baseline in addition to the native execution baseline – a Spark-based approximate computing system leveraging the existing sampling modules in Apache Spark. Compared to the improved baseline, our results show that StreamApprox achieves a speedup $1.1\times$$2.4\times$ while maintaining the same accuracy level.
Residual Network (ResNet) is the state-of-the-art architecture that realizes successful training of really deep neural network. It is also known that good weight initialization of neural network avoids problem of vanishing/exploding gradients. In this paper, simplified models of ResNets are analyzed. We argue that goodness of ResNet is correlated with the fact that ResNets are relatively insensitive to choice of initial weights. We also demonstrate how batch normalization improves backpropagation of deep ResNets without tuning initial values of weights.
Recent advances in deep learning have led various applications to unprecedented achievements, which could potentially bring higher intelligence to a broad spectrum of mobile and ubiquitous applications. Although existing studies have demonstrated the effectiveness and feasibility of running deep neural network inference operations on mobile and embedded devices, they overlooked the reliability of mobile computing models. Reliability measurements such as predictive uncertainty estimations are key factors for improving the decision accuracy and user experience. In this work, we propose RDeepSense, the first deep learning model that provides well-calibrated uncertainty estimations for resource-constrained mobile and embedded devices. RDeepSense enables the predictive uncertainty by adopting a tunable proper scoring rule as the training criterion and dropout as the implicit Bayesian approximation, which theoretically proves its correctness.To reduce the computational complexity, RDeepSense employs efficient dropout and predictive distribution estimation instead of model ensemble or sampling-based method for inference operations. We evaluate RDeepSense with four mobile sensing applications using Intel Edison devices. Results show that RDeepSense can reduce around 90% of the energy consumption while producing superior uncertainty estimations and preserving at least the same model accuracy compared with other state-of-the-art methods.
The role of sentiment analysis is increasingly emerging to study software developers’ emotions by mining crowd-generated content within social software engineering tools. However, off-the-shelf sentiment analysis tools have been trained on non-technical domains and general-purpose social media, thus resulting in misclassifications of technical jargon and problem reports. Here, we present Senti4SD, a classifier specifically trained to support sentiment analysis in developers’ communication channels. Senti4SD is trained and validated using a gold standard of Stack Overflow questions, answers, and comments manually annotated for sentiment polarity. It exploits a suite of both lexicon- and keyword-based features, as well as semantic features based on word embedding. With respect to a mainstream off-the-shelf tool, which we use as a baseline, Senti4SD reduces the misclassifications of neutral and positive posts as emotionally negative. To encourage replications, we release a lab package including the classifier, the word embedding space, and the gold standard with annotation guidelines.
Training a Deep Neural Network (DNN) from scratch requires a large amount of labeled data. For a classification task where only small amount of training data is available, a common solution is to perform fine-tuning on a DNN which is pre-trained with related source data. This consecutive training process is time consuming and does not consider explicitly the relatedness between different source and target tasks. In this paper, we propose a novel method to jointly fine-tune a Deep Neural Network with source data and target data. By adding an Optimal Transport loss (OT loss) between source and target classifier predictions as a constraint on the source classifier, the proposed Joint Transfer Learning Network (JTLN) can effectively learn useful knowledge for target classification from source data. Furthermore, by using different kind of metric as cost matrix for the OT loss, JTLN can incorporate different prior knowledge about the relatedness between target categories and source categories. We carried out experiments with JTLN based on Alexnet on image classification datasets and the results verify the effectiveness of the proposed JTLN in comparison with standard consecutive fine-tuning. This Joint Transfer Learning with OT loss is general and can also be applied to other kind of Neural Networks.
Unordered feature sets are a nonstandard data structure that traditional neural networks are incapable of addressing in a principled manner. Providing a concatenation of features in an arbitrary order may lead to the learning of spurious patterns or biases that do not actually exist. Another complication is introduced if the number of features varies between each set. We propose convolutional deep averaging networks (CDANs) for classifying and learning representations of datasets whose instances comprise variable-size, unordered feature sets. CDANs are efficient, permutation-invariant, and capable of accepting sets of arbitrary size. We emphasize the importance of nonlinear feature embeddings for obtaining effective CDAN classifiers and illustrate their advantages in experiments versus linear embeddings and alternative permutation-invariant and -equivariant architectures.
Sparse coding (SC) is attracting more and more attention due to its comprehensive theoretical studies and its excellent performance in many signal processing applications. However, most existing sparse coding algorithms are nonconvex and are thus prone to becoming stuck into bad local minima, especially when there are outliers and noisy data. To enhance the learning robustness, in this paper, we propose a unified framework named Self-Paced Sparse Coding (SPSC), which gradually include matrix elements into SC learning from easy to complex. We also generalize the self-paced learning schema into different levels of dynamic selection on samples, features and elements respectively. Experimental results on real-world data demonstrate the efficacy of the proposed algorithms.
We propose an interdependent random geometric graph (RGG) model for interdependent networks. Based on this model, we study the robustness of two interdependent spatially embedded networks where interdependence exists between geographically nearby nodes in the two networks. We study the emergence of the giant mutual component in two interdependent RGGs as node densities increase, and define the percolation threshold as a pair of node densities above which the giant mutual component first appears. In contrast to the case for a single RGG, where the percolation threshold is a unique scalar for a given connection distance, for two interdependent RGGs, multiple pairs of percolation thresholds may exist, given that a smaller node density in one RGG may increase the minimum node density in the other RGG in order for a giant mutual component to exist. We derive analytical upper bounds on the percolation thresholds of two interdependent RGGs by discretization, and obtain $99\%$ confidence intervals for the percolation thresholds by simulation. Based on these results, we derive conditions for the interdependent RGGs to be robust under random failures and geographical attacks.
We consider a model of two interdependent networks, where every node in one network depends on one or more supply nodes in the other network and a node fails if it loses all of its supply nodes. We develop algorithms to compute the failure probability of a path, and obtain the most reliable path between a pair of nodes in a network, under the condition that each supply node fails independently with a given probability. Our work generalizes the classical shared risk group model, by considering multiple risks associated with a node and letting a node fail if all the risks occur. Moreover, we study the diverse routing problem by considering two paths between a pair of nodes. We define two paths to be $d$-failure resilient if at least one path survives after removing $d$ or fewer supply nodes, which generalizes the concept of disjoint paths in a single network, and risk-disjoint paths in a classical shared risk group model. We compute the probability that both paths fail, and develop algorithms to compute the most reliable pair of paths.
Gated Recurrent Unit (GRU) is a recently published variant of the Long Short-Term Memory (LSTM) network, designed to solve the vanishing gradient and exploding gradient problems. However, its main objective is to solve the long-term dependency problem in Recurrent Neural Networks (RNNs), which prevents the network to connect an information from previous iteration with the current iteration. This study proposes a modification on the GRU model, having Support Vector Machine (SVM) as its classifier instead of the Softmax function. The classifier is responsible for the output of a network in a classification problem. SVM was chosen over Softmax for its computational efficiency. To evaluate the proposed model, it will be used for intrusion detection, with the dataset from Kyoto University’s honeypot system in 2013 which will serve as both its training and testing data.
We discuss that how the majority of traditional modeling approaches are following the idealism point of view in scientific modeling, which follow the set theoretical notions of models based on abstract universals. We show that while successful in many classical modeling domains, there are fundamental limits to the application of set theoretical models in dealing with complex systems with many potential aspects or properties depending on the perspectives. As an alternative to abstract universals, we propose a conceptual modeling framework based on concrete universals that can be interpreted as a category theoretical approach to modeling. We call this modeling framework pre-specific modeling. We further, discuss how a certain group of mathematical and computational methods, along with ever-growing data streams are able to operationalize the concept of pre-specific modeling.
If we cannot store all edges in a graph stream, which edges should we store to estimate the triangle count accurately? Counting triangles (i.e., cycles of length three) is a fundamental graph problem with many applications in social network analysis, web mining, anomaly detection, etc. Recently, much effort has been made to accurately estimate global and local triangle counts in streaming settings with limited space. Although existing methods use sampling techniques without considering temporal dependencies in edges, we observe temporal locality in real dynamic graphs. That is, future edges are more likely to form triangles with recent edges than with older edges. In this work, we propose a single-pass streaming algorithm called Waiting-Room Sampling (WRS) for global and local triangle counting. WRS exploits the temporal locality by always storing the most recent edges, which future edges are more likely to from triangles with, in the waiting room, while it uses reservoir sampling for the remaining edges. Our theoretical and empirical analyses show that WRS is: (a) Fast and ‘any time’: runs in linear time, always maintaining and updating estimates while new edges arrive, (b) Effective: yields up to 47% smaller estimation error than its best competitors, and (c) Theoretically sound: gives unbiased estimates with small variances under the temporal locality.
Reinforcement Learning is divided in two main paradigms: model-free and model-based. Each of these two paradigms has strengths and limitations, and has been successfully applied to real world domains that are appropriate to its corresponding strengths. In this paper, we present a new approach aimed at bridging the gap between these two paradigms. We aim to take the best of the two paradigms and combine them in an approach that is at the same time data-efficient and cost-savvy. We do so by learning a probabilistic dynamics model and leveraging it as a prior for the intertwined model-free optimization. As a result, our approach can exploit the generality and structure of the dynamics model, but is also capable of ignoring its inevitable inaccuracies, by directly incorporating the evidence provided by the direct observation of the cost. As a proof-of-concept, we demonstrate on simulated tasks that our approach outperforms purely model-based and model-free approaches, as well as the approach of simply switching from a model-based to a model-free setting.
Multivariate time-series modeling and forecasting is an important problem with numerous applications. Traditional approaches such as VAR (vector auto-regressive) models and more recent approaches such as RNNs (recurrent neural networks) are indispensable tools in modeling time-series data. In many multivariate time series modeling problems, there is usually a significant linear dependency component, for which VARs are suitable, and a nonlinear component, for which RNNs are suitable. Modeling such times series with only VAR or only RNNs can lead to poor predictive performance or complex models with large training times. In this work, we propose a hybrid model called R2N2 (Residual RNN), which first models the time series with a simple linear model (like VAR) and then models its residual errors using RNNs. R2N2s can be trained using existing algorithms for VARs and RNNs. Through an extensive empirical evaluation on two real world datasets (aviation and climate domains), we show that R2N2 is competitive, usually better than VAR or RNN, used alone. We also show that R2N2 is faster to train as compared to an RNN, while requiring less number of hidden units.
Chatbots are a rapidly expanding application of dialogue systems with companies switching to bot services for customer support, and new applications for users interested in casual conversation. One style of casual conversation is argument, many people love nothing more than a good argument. Moreover, there are a number of existing corpora of argumentative dialogues, annotated for agreement and disagreement, stance, sarcasm and argument quality. This paper introduces Debbie, a novel arguing bot, that selects arguments from conversational corpora, and aims to use them appropriately in context. We present an initial working prototype of Debbie, with some preliminary evaluation and describe future work.
Graph processing is becoming increasingly prevalent across many application domains. In spite of this prevalence, there is little research about how graphs are actually used in practice. We conducted an online survey aimed at understanding: (i) the types of graphs users have; (ii) the graph computations users run; (iii) the types of graph software users use; and (iv) the major challenges users face when processing their graphs. We describe the responses of the participants to our questions, highlighting common patterns and challenges. The participants’ responses revealed surprising facts about graph processing in practice, which we hope can guide future research.
A central question in the era of ‘big data’ is what to do with the enormous amount of information. One possibility is to characterize it through statistics, e.g., averages, or classify it using machine learning, in order to understand the general structure of the overall data. The perspective in this paper is the opposite, namely that most of the value in the information in some applications is in the parts that deviate from the average, that are unusual, atypical. We define what we mean by ‘atypical’ in an axiomatic way as data that can be encoded with fewer bits in itself rather than using the code for the typical data. We show that this definition has good theoretical properties. We then develop an implementation based on universal source coding, and apply this to a number of real world data sets.
Semi-supervised active clustering (SSAC) utilizes the knowledge of a domain expert to cluster data points by interactively making pairwise ‘same-cluster’ queries. However, it is impractical to ask human oracles to answer every pairwise query. In this paper, we study the influence of allowing ‘not-sure’ answers from a weak oracle and propose algorithms to efficiently handle uncertainties. Different types of model assumptions are analyzed to cover realistic scenarios of oracle abstraction. In the first model, random-weak oracle, an oracle randomly abstains with a certain probability. We also proposed two distance-weak oracle models which simulate the case of getting confused based on the distance between two points in a pairwise query. For each weak oracle model, we show that a small query complexity is adequate for the effective $k$ means clustering with high probability. Sufficient conditions for the guarantee include a $\gamma$-margin property of the data, and an existence of a point close to each cluster center. Furthermore, we provide a sample complexity with a reduced effect of the cluster’s margin and only a logarithmic dependency on the data dimension. Our results allow significantly less number of same-cluster queries if the margin of the clusters is tight, i.e. $\gamma \approx 1$. Experimental results on synthetic data show the effective performance of our approach in overcoming uncertainties.
Convolutional highways are deep networks based on multiple stacked convolutional layers for feature preprocessing. We introduce an evolutionary algorithm (EA) for optimization of the structure and hyperparameters of convolutional highways and demonstrate the potential of this optimization setting on the well-known MNIST data set. The (1+1)-EA employs Rechenberg’s mutation rate control and a niching mechanism to overcome local optima adapts the optimization approach. An experimental study shows that the EA is capable of improving the state-of-the-art network contribution and of evolving highway networks from scratch.
Distributed Stream Processing Systems (DSPS) like Apache Storm and Spark Streaming enable composition of continuous dataflows that execute persistently over data streams. They are used by Internet of Things (IoT) applications to analyze sensor data from Smart City cyber-infrastructure, and make active utility management decisions. As the ecosystem of such IoT applications that leverage shared urban sensor streams continue to grow, applications will perform duplicate pre-processing and analytics tasks. This offers the opportunity to collaboratively reuse the outputs of overlapping dataflows, thereby improving the resource efficiency. In this paper, we propose \emph{dataflow reuse algorithms} that given a submitted dataflow, identifies the intersection of reusable tasks and streams from a collection of running dataflows to form a \emph{merged dataflow}. Similar algorithms to unmerge dataflows when they are removed are also proposed. We implement these algorithms for the popular Apache Storm DSPS, and validate their performance and resource savings for 35 synthetic dataflows based on public OPMW workflows with diverse arrival and departure distributions, and on 21 real IoT dataflows from RIoTBench.
Mixture models are among the most popular tools for model based clustering. However, when the dimension and the number of clusters is large, the estimation as well as the interpretation of the clusters become challenging. We propose a reduced-dimension mixture model, where the $K$ components parameters are combinations of words from a small dictionary – say $H$ words with $H \ll K$. Including a Nonnegative Matrix Factorization (NMF) in the EM algorithm allows to simultaneously estimate the dictionary and the parameters of the mixture. We propose the acronym NMF-EM for this algorithm. This original approach is motivated by passengers clustering from ticketing data: we apply NMF-EM to ticketing data from two Transdev public transport networks. In this case, the words are easily interpreted as typical slots in a timetable.
In a recent decade, ImageNet has become the most notable and powerful benchmark database in computer vision and machine learning community. As ImageNet has emerged as a representative benchmark for evaluating the performance of novel deep learning models, its evaluation tends to include only quantitative measures such as error rate, rather than qualitative analysis. Thus, there are few studies that analyze the failure cases of deep learning models in ImageNet, though there are numerous works analyzing the networks themselves and visualizing them. In this abstract, we qualitatively analyze the failure cases of ImageNet classification results from recent deep learning model, and categorize these cases according to the certain image patterns. Through this failure analysis, we believe that it can be discovered what the final challenges are in ImageNet database, which the current deep learning model is still vulnerable to.
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# Protein Design
We previously learned about the problem of predicting the folding of a protein, that is, given a chain of amino acids, find its final 3D structure. This time we’re interested in the reverse problem, that is, given a 3D structure, find some chain of amino-acid that would lead to that structure once fully folded. This problem is called Protein Design.
In this post we’ll focus on mathematical models for this problem, studying its computational complexity and discuss possible solutions.
### Mathematical Modeling
The 3D structure of a protein can be divided into two parts: the backbone and the side chains. In the model proposed by Piece and Winfree [3], we assume the backbone is rigid and that we’ll try to find the amino-acids for the side chains such that it minimizes some energy function.
This means we’re not really trying to predict the whole chain of amino acids, but a subset of those amino that will end up on the side chains.
The amino acids on the side-chain can have a specific orientation [2, 3], known as rotamer, which in turn can be represented by a single value, its dihedral angle. We can define some arbitrary order for these amino acids and label them with an index, which we call position.
At each position there are multiple rotamers possible and we need to select them to minimize the overall energy function. More formally, for each position i, let $R_i$ be the set of rotamers available and $r_i \in R_i$ the chosen rotamer.
The model assumes the cost function of the structure is pairwise decomposable, meaning that we can account for the interaction of each pair independently when calculating the total energy:
Where $E(r_i, r_j)$ is the energy cost of the interaction between positions i and j, assuming rotamers $r_i$ and $r_j$ respectivelly. The definition of E can be based on molecular dynamics such as AMBER.
In [3], the authors call this optimization problem PRODES (PROtein DESign).
### PRODES is NP-Hard
Pierce and Winfree [3] prove that PRODES is NP-Hard. We’ll provide an informal idea of the proof here.
First we need to prove that the decision version of PRODES is NP-Hard. The decision version of PRODES is, given a value K, determine whether there is a set of rotamers such that the energy cost is less or equal K. We’ll call it PRODESd. We can then prove that PRODESd is NP-complete by showing that it belongs to the NP complexity class and by reducing, in polynomial time, some known NP-complete problem to it.
We claim that this problem is in NP because given an instance for the problem we can verify in polynomial time whether it is a solution (i.e. returns true), since we just need to evaluate the cost function and verify whether it’s less than K.
We can reduce the 3-SAT problem, known to be NP-complete, to PRODESd. The idea is that we can map every instance of the 3-SAT to an instance of PRODESd and show that the result is “true” for that instance of 3-SAT if and only if the result is “true” for the mapped instance of PRODESd.
Let’s start by formalizing PRODESd as a graph problem, with an example shown in the picture below:
a) has 3 positions with their sets of rotamers. b) each rotamer becomes a vertex grouped into sets. We pick exactly one vertex per set and try to minimize the total cost of the edges associated with the selected vertices. Image source: [3]
Now, given a 3-SAT instance, we create a vertex set for each clause $C_i$ (containing a vertex for each literal), and a vertex set for each variable $x_i$ (containing vertices T and F). For each literal $x_i$ we add an edge of weight 1 to the vertex F of the set corresponding to variable $x_i$. Conversely, for each negated literal, $\bar x_i$, we add an edge of weight 1 to the vertex T. All other edges have weight 0.
For example, the instance $(x_1 + \bar x_2 + x_3) \cdot ( \bar x_1 + x_2 + x_3) \cdot (\bar x_3)$ yields the following graph where only edges of non-zero weight are shown:
Source: [3]
We claim that this 3-SAT instance is satisfiable if and only if the PRODESd is true for K=0. The idea is the following: for each vertex set corresponding to a variable we pick either T or F, which corresponds to assigning true or false to the variable. For each vertex set corresponding to a clause we pick a literal that will evaluate to true (and hence make the clause true). It’s possible to show that if 3-SAT is satisfiable, there’s a solution for PRODESd that avoids any edge with non-zero weight.
### Integer Linear Programming
Now that we know that PRODES is unlikely to have an efficient algorithm to solve it, we can attempt to obtain exact solutions using integer linear programming model. Let’s start with some definitions:
We can define our variables as:
The object function of our model becomes:
Finally, the constraints are:
Equation (1) says we should pick exactly one rotamer for each position. Constraints (2) and (3) enforce that $x_{i, j} = 1$ if and only if $r_i = r_j = 1$.
Note: the LaTeX used to generate the images above are available here.
### Conclusion
The study of protein prediction led me to protein design, which is a much simpler problem, even though from the computational complexity perspective it’s still an intractable problem.
The model we studied is very simple and makes a lot of assumptions, but it’s interesting as a theoretical computer science problem. Still I’m not sure how useful it is in practice.
### References
[1] Wikipedia – Protein design
[2] Wikipedia – Rotamer
[3] Protein Design is NP-hard – Niles A. Pierce, Erik Winfree
## One thought on “Protein Design”
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Math Help - homomorphism
1. homomorphism
a. Find all subgroups of S_3, and determine which are normal
b. Find all subgroups of the quaternion group, and determine which are normal
2. Originally Posted by jin_nzzang
a. Find all subgroups of S_3, and determine which are normal
b. Find all subgroups of the quaternion group, and determine which are normal
$S_3$ has order 6 and so it can only have subgroup of order 1, 2, 3 or 6 (can you see why?).
Now, $<(123)> = <(132)> = A_3$ is a subgroup of order 3. Just try every element of the group too see if this is normal or not - for example, $(12)(123)(12)^{-1} = (12)(123)(12) = (132) \in A_3$. Further, this is the only subgroup of order 3, as every element in $S_3$ that is not in $A_3$ has order 2, and so cannot be in a group of order 3 (the order of an element in a group MUST divide the order of the group.
I will leave you too find the subgroups of order 2, and I will also leave you too see if they are normal or not as it really isn't too hard. However, as a short cut you should notice that once you have proved whether one of these 2-groups is normal or not you need not do it for the other ones because they are essentially the same, just with permuted symbols.
For the quaternion group notice that $$ is a subgroup of order 4, and so the element $i$ cannot be contained in any other proper subgroup of $Q_8$ (as the order of the subgroup must divide 8 and 4 is the largest number other than 8 which does that). The same goes for $j$ and $k$. That gives us 3 proper subgroups. There is one more, which I shall let you find.
I shall also leave you to find out whether these subgroups are normal or not - all you need to do to find out if the subgroup $H$ is normal is to take every element $h \in H$ and see if $ghg^{-1} \in H$ holds for all $g \in G$. If it does $H$ is normal, else it is not.
Having types all that, can I ask you one thing: do you know of the result regarding normality and subgroups of index 2?
Also, what does the title have to do with the content of the post - you don't do anything with homomorphisms here!
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# University of web
In response to popular non-existent demand, we are publishing the most rigorous class that we require all first-year PhD students to briefly skim: Introduction to web3.
Of course, remember that I founded this institution with the goals of:
1. fomenting resentment against web3
2. perpetuating Big Tech, who rewards me handsomely with tungsten cubes to criticize any perceived threat to their continued dominance
3. doing well for a college course, which is what this is all actually for
TL;DR: web3 is overhyped, inefficient, and dangerous.
# Why is it called web3?
## web1.0
Archaeologists specializing in the prehistoric times of the 1990s agree that the World Wide Web began as universities and nerds hosting their own websites and mailservers. Despite the simplicity of this early version of the internet, many of the infrastructural components of the modern web were developed and iterated in a familiar alphabet soup: DNS, HTML, CSS, HTTP, etc.
## web2.0
Then we saw the emergence of tech companies that offered to ease the burden of having to run your own services. These companies would make it easy to connect with others and generate content for the world to see. You wouldn’t have to host your own website or mailserver anymore. You could just worry about producing content, and not the underlying infrastructure that was becoming increasingly complex as the internet was expanding.
In our current web2.0 era, we saw the rise of Facebook, Google, Instagram, Twitter, Reddit, Discord, WeChat, and so on. These platforms helped to faciliate an explosion in the internet’s global adoption. The user experience of communicating with others and finding information online was a lot easier and faster. The internet was no longer a niche thing.
Though these services have made our lives convenient, a lot of internet activity is now taking place under a few tech conglomerates. This has fueled concerns over privacy, censorship, and user autonomy.
## web3
web3 is a proposed set of technologies — largely centered around cryptocurrencies and the applications of crypto — that will define the next generation of the internet. In response to centralization of the internet that occurred under web2.0, proponents of web3 argue that a new internet built on immutable blockchain technology will improve privacy, avoid censorship, and create new economic opportunities.
web3 isn’t magic. It is merely a bundle of blockchain technologies working together so you can buy cryptocurrency. Nestled between two industries notoriously vulnerable to hype, crypto’s positioning at the center of finance and computer science can easily confound and disorient even the most technically-adept. So to fully explore the risks of web3, we need to start from the beginning.
# Is it effective?
## Bitcoin
Bitcoin was published in 2009 by the pseudonymous Satoshi Nakamoto, whose real identity remains the subject of speculation.
CAUTION – The following content distills a very technical topic
The basis of cryptocurrency is that the individual’s digital wallet is a kind of one-man show. You don’t need a banking apparatus to transact your crypto because each digital wallet has a unique address associated with it. You may have heard that crypto is like a digitized form of cash, which is a fairly appropriate way to describe it.
But unlike cash, crypto’s decentralized nature makes it resistant to inflation and intentional devaluation because it has no intrinsic governing body.
Decentralization is a buzzword that we’ll return to several times. In this context, decentralization refers to the lack of control any singular entity is supposed to have over the cryptocurrency as a whole. Whereas you’re probably familiar with the notion of a central bank dictating monetary policy – i.e., the US dollar and the Federal Reserve – crypto has no de jure authority.
### Proof-of-work
Though promising as described above, Bitcoin has fallen way short of its ambitions. Bitcoin relies upon a mechanism called “proof-of-work” to validate new nodes of transaction data added to the blockchain.
A blockchain is a digital ledger that cannot be edited once it is recorded. The Wikipedia page on blockchains is a good place to learn more.
This is a very computationally intensive process, which is to say that it requires an enormous amount of computing power. Bitcoin was designed to become more computationally intensive depending on the total computing power of the network. This was supposed to prevent a single actor from amassing a huge number of resources to dominate the blockchain, which is called a 51% attack.
a 51% attack is defined as a group of blockchain miners controlling the majority of the computing power available to the blockchain, and is therefore able to manipulate the blockchain. Imagine if a massive bunch of rogue voters decided to write-in their own candidate during an election, bypassing the typical party system.
So as the number of eager Bitcoin miners swelled in the past decade, the efficiency of the network plummeted. But as it turns out, these diminished returns are still marginally profitable, which makes mining exclusively favorable to miners who are willing to spend money on power-hungry mining rigs. Bitcoin alone consumes roughly the same amount of electricity as Argentina1, a country of over 45 million people.
The decentralized nature of Bitcoin paralyzes any attempt to improve the protocol. Imagine trying to change the way email works or switching the United States to the metric system; these endeavors are defeated by the inflexibility of the problem. Countless developers have attempted to “fork” the source code for Bitcoin and develop their own cryptocurrency to solve Bitcoin’s perceived flaws. Few of these offshoots have become successful, and most are generally indistinguishable from each other since creating a cryptocurrency is relatively trivial given the open-source nature of the field.
In fact, the issues plaguing Bitcoin are so endemic that within the crypto community, it is often regarded as inferior to other supposedly better currencies. Despite its poor reputation, Bitcoin remains one of the most popular crypto currencies because its skyrocketing price and notoriety in recent years makes it an attractive investment to many newcomers, who are inundated with advertisements to “invest in crypto today!”
## Ethereum
Ethereum is tapped to become the next big cryptocurrency superseding Bitcoin. Its blockchain is more versatile than that of Bitcoin’s, with the ability to handle and store arbitrary data beyond Ethereum transactions alone.
To this effect, people envision Ethereum being capable of being far more than a mere currency. In theory, it could store legal records and medical histories, although glaring complications make this far from trivial. Ethereum also advertises a consensus mechanism that would not require the insane power demands of Bitcoin: proof-of-stake.
### Proof-of-stake
Proof-of-stake is an alternative to proof-of-work whereby agents are issued transactions to validate depending on how much cryptocurrency they hold. Ethereum requires prospective validators to hold a measly 32 ETH to be eligible for this scheme. If you wanted to purchase 32 ETH sometime in the last year with fiat currency, it would have costed you a figure somewhere between $55,000 and$155,000 depending on your (un)lucky timing.
Despite proof-of-stake being designed to supposedly thwart the risk of a malicious actor by requiring that potential validators post some amount of collateral, this creates a system whereby only the most early adopters or wealthy individuals have the ability to participate in proof-of-stake. Given the typical American household has approximately $5,300 in savings,2 it is rare that a normal person just getting interested in crypto today can participate in proof-of-stake by idly investing money they likely don’t even have in a volatile, alien currency and running servers capable of handling Ethereum transactions. The odds are likely even less if we discard the assumption that our Ethereum dilettante isn’t from an developed economy. And while proof-of-stake is an order of magnitude more power efficient than proof-of-work, proof-of-stake still has an enormous energy demand. It also remains incredibly slow compared to established transaction systems, and inches Ethereum towards a level of centralization that cryptocurrencies ostensibly try to avoid. ## Transaction fees Cryptocurrency transactions require immense computing power to validate transaction data. To solve this, Bitcoin and Ethereum operate an auction system whereby users pay miners fees to process their transactions as quickly as possible. Let’s look in particular at Ethereum, which has exploded in popularity in the past few years. These fees, commonly known as “gas fees” within the Ethereum user base, have ranged sharply over the past few months: The more people trying to transact, the higher the fees become; a mechanism analogous to surge pricing on apps such as Uber or Lyft. These transaction fees make it implausible that mainstream crypto will be used to buy and sell goods and services, as they were originally intended. But unlike Uber and Lyft, ‘surge pricing’ is not a local event, and extends universally to the entire protocol. Imagine paying a higher ‘cryptocurrency processing fee’ at a local restaurant because people in a neighboring country are rushing to preorder movie tickets online. ## Paradox of centralization One of the most popularly cited advantages of web3 is decentralization. Decentralization in this case is the idea that the internet of the future will no longer be controlled by a handful of technology conglomerates that regulate and/or censor content If the internet was ‘decentralized,’ we would not have to worry about censorship. If we were to embed blockchain technology into the core of the internet, centralized power structures should eventually decay. But why exactly would this happen? It’s grandiose mental gymnastics. Centralization emerges in free markets because of economies of scale. And especially when Bitcoin and Ethereum have such high barriers to entry, the catalysts of centralization are even more apparent. Despite the hype surrounding decentralization, the cogs of web3 are ironically deeply centralized.3 We have: • Coinbase for trading and exchanging cryptocurrencies (89 million registered users) Un-fun fact: Mt. Gox used to handle upwards of 70% of all Bitcoin transactions in 2014. Then it suddenly shut down after people discovered that customers’ Bitcoins in Mt. Gox’s wallet had been systematically stolen and mismanaged for years. • OpenSea for trading NFTs • Metamask for Ethereum-based authentication • and company X to achieve basic function Y, etc. And most tragically, the blockchain is often so inefficient at scale that these organizations rely upon ordinary, boring technologies like SQL databases and C++ to handle transactions and user data. And where do these databases and code exist? On centralized computing resources like Amazon Web Services or Microsoft Azure. There are some genuinely effective implementations of a decentralized architecture. Folding@home, which uses volunteer computing power to simulate protein folding for medical research, is one of them. So is the Tor relay network, which enables millions of people to navigate the internet with anonymity and to evade heavy-handed electronic surveillance. Torrenting is an especially powerful example, because people attaining the contents of the torrent are simultaneously enabling that content to spread more easily. ## Renewable energy whataboutism Mining cryptocurrencies and validating blockchains consume an enormous amount of energy because their decentralized architecture demands a lot of redundant computational work. Crypto advocates defend cryptocurrency with two general stratagems: 1. Crypto is only as dirty as the energy it relies upon. This argument ignores the immense electronics waste that crypto generates — mining rigs are often several thousand dollars and consist of several high-end graphic cards originally designed for gaming. But more importantly, this argument shifts the blame of crypto’s wasted energy usage onto the world’s failure to adopt renewable energy. But cryptocurrencies do not accelerate the world’s shift to renewables, they instead delay it.The energy that goes into powering the Ethereum or Bitcoin network could have been used to power electric buses, tea kettles, and LED lightbulbs.4 In this sense, cryptocurrency is nullifying the effect of renewable energy, because it uses electrical capacity that could have gone to a far more productive cause. Granted, transaction speed and energy usage are loosely related, but the mere existence of a blockchain fundamentally demands energy. If the cost of electricity drops due to cheaper solar, wind, or nuclear power becoming readily available, then miners would simply scale up their activity in response. This is a classical demonstration of induced demand — the phenomenon where increased supply prompts increased demand. A city widens their highway (which takes great cost and time) to fight congestion; traffic instead increases to match the capacity of the road and congestion remains an issue. Eli Whitney thought his cotton gin would eliminate the need for slaves; instead, it made enslaved cotton production so efficient that plantations bought considerably more slaves to accomodate the throughput of the cotton gin. 2. The global banking industry consumes way more energy than crypto. People have different ideas of what the global banking industry is because the term itself is ambiguous. Is the global banking industry simply a collection of the largest banks, their subsidiaries, and payment processors such as Visa and Mastercard? Does it include the greater institutions that support the global banking industry, such as various national governments and the companies responsible for building millions of cash registers and payment keypads? Would it include the safeguards of international financial stability, such as the US military? The goalposts of this debate constantly shift and blur any attempt to answer the question of whether crypto or the global banking system is more efficient. If we were to adopt this whataboutism in the opposite direction, we can argue that crypto itself is complicit in the emissions of semiconductor manufacturers and rare-earth metal mining companies that provide the components and raw materials ultimately used to build mining rigs. What we do observe with certainty is that the gritty global banking system that the world relies on today provides financial services for nearly eight billion people transacting trillions of dollars per day. If crypto were to support the same global audience at its present stage of development, it’s doubtful that this system would be much more sustainable at scale. ## Crypto to the rescue of crypto? As we have seen, there are a bunch of fundamental mechanical problems with cryptocurrencies, whether they use proof-of-work or proof-of-stake. They spawn stratified power structures that are deeply antithetical to the spirit of the movement that promotes them. Because cryptocurrencies are open source, we have seen countless instances of developers forking a cryptocurrency and building their own implementation. This is especially easy when a nicely typeset $$\LaTeX$$ whitepaper is all you need to appear legitimate to gullible early adopters. The barriers to entry for a new cryptocurrency are so low that these derivative coins are often called shitcoins. But beyond those who are purely motivated by amusement or the hope of ‘getting rich quick,’ there are some developers who are working on hard-fought efforts to address the fundamental issues of cryptocurrencies and decentralized consensus protocols. You might hear of these technologies and movements – radix, sharding, DAOs, DeFi, layer 2. Here at the University of web3, we have decided to take a simple approach to these proposed technologies and ask ourselves: what problem are they truly solving? Is it too easy to confuse imitation for innovation, because many of these newer technologies ultimately only help catch crypto up to the performance and centralization of our established financial system? For instance, Ethereum proudly advertises that it will reduce its energy consumption by 99.95% once it fully adopts proof-of-stake. But as we have seen, proof-of-stake is a trade-off that further increases the centralization and barriers-to-entry of the Ethereum network. # Is it safe? web3 is built on technologies that are saturated with scams and fraud. ## Case study: non-fungible tokens (NFTs) In 2021, non-fungible tokens emerged as the keystone in a series of unfortunate events. A mechanism intended to somehow introduce scarcity to digital objects, NFTs bewilder any rational attempt to decipher their purpose. Non-fungible essentially means “non-substitutable” or “non-replicable”, and in this case means stamping unique serial numbers onto tokens hosted on the Ethereum blockchain. But this sense of uniqueness is completely lost upon the implementation of NFTs, which are little more than URLs pointing to an image file. If the URLs eventually no longer point to the artwork, then the NFT becomes an empty carcass since the blockchain can’t be retroactively edited. This implication of perpetual storage is a daunting responsibility to maintain, but we have already seen centralized services like NFT.STORAGE gladly advertise perpetual storage with no warning asterisks. The NFTs themselves are stored on “Filecoin storage providers,” which are basically just organizations with lots of hard drive server space who are hosting NFTs so long as they receive an appropriate return on investment. Becoming a Filecoin storage provider isn’t a trivial matter of installing some ordinary open-source software, it requires getting in touch with a sales team and building a power-hungry server, which again is antithetical to the whole “democratization of art” part of the NFT verbiage. From the Filecoin provider documentation, you’ll need: • 8+ core CPU • >128 gigabytes of RAM • one of these GPUs • a terabyte of NVMe-based SSD storage for caching • …and some other hard drives to serve as your actual server storage The CPU, RAM, GPU, and drives each cost more than a PlayStation 5.5 And this doesn’t begin to consider the cost of powering a +400 watt server6 that probably demands a faster internet connection than the one in a typical home. Those who can realistically afford and operate this kind of hardware belong to a technical minority, and their only incentive to hold your NFTs safely is their estimated return on investment. So now that we’ve addressed the worrying infrastructure of NFTs, let’s get back to the user side of things. Though the interface and user experience of NFT auction houses like OpenSea resemble that of an art auction, they are even more unscrupulous. Since cryptocurrency wallets are largely anonymous and at least pseudonymous,7 let’s see how we can synthesize some demand for a masterpiece in seven steps. 1. Alice, attracted to the NFT hype and possessing no artistic talent, mints an NFT for her ugly or stolen creation. 2. Alice publishes a page on OpenSea. 3. Alice observes no demand for her artwork. 4. Alice buys more Ethereum and deposits it into newly spawned wallets. 5. Alice uses her fresh wallets to bid on her worthless artwork and create the illusion of urgency and demand.8 6. Bob sees rapid interest and gets duped into buying Alice’s artwork. 7. Alice collects her profits. If Bob was swayed into buying the NFT, then Alice’s crypto would have gone back into her original wallet anyway. The big problem here is that both Alice and Bob need to spend their legitimate money to conduct this whole seven-act tragedy. The cost of minting the NFT is a volatile figure, but there are artists who have incurred net losses of several hundred dollars in various fees9 from minting their NFTs (which sometimes are merely stolen or lightly edited artwork). Meanwhile, they have provided upward pressure on the ETH:USD exchange rate and liquidity to cryptocurrency traders by having to buy Ethereum. ## Case study: rubber-hose cryptanalysis A bank offers many protections for its customers. If somebody steals your credit card, you can contact your bank’s fraud department to scrub the charges and you won’t have to pay anything. If your bank fails and you live in the United States, the federal government guarantees your savings are secure. “But wait! Consumer protections that guard against bank failure are an irrelevant counterargument against cryptocurrency’s safety, because cryptocurrencies operate without any need for banks! Why would you need to be protected from a bank robbery, if bank robbery can’t happen with crypto?” But in the cryptocurrency world, you are your own bank. There is nothing protecting you from poor memory, scammers, hackers, or people who are willing to beat you with a rubber hose and coerce you into revealing your wallet’s key. In fact, it is this complete lack of protection that forms the basis of why people created banks in the first place — so that their money could be safely kept away from them in a more secure location. If you put your account’s seed phrase on a flash drive and then forgot where that flash drive was, you’re SOL. There are horror stories of people who bought Bitcoin ten years ago, became millionaires unexpectedly, but are locked out of their newfound wealth because they can’t remember the decryption key to their hard drive, or they lost the storage device containing their wallet. Cryptocurrency transactions are not reversible. There isn’t any mechanism to revert a charge that you were misled into approving. This is one of the largest reasons why so many scams are conducted with cryptocurrency, because there is no law enforcement agency that can effectively halt a transaction or reverse it. # Is it wanted? At best, web3 developers have noble, if grandiose, motivations to improve financial equity or the freedom of the internet. At worst, it seems that web3 is nothing more than trying to lure normal people to buy cryptocurrency. This is the catch: by entering the web3 space, you provide liquidity to the Bitcoin and Ethereum market. As a striking figure, consider that users spent about$100 million in Ethereum gas fees to buy $200m in Bored Ape NFTs.10 The incentives in the web3 space are far removed from actually improving the quality of life issues we’ve seen with web2.0. As the price of Bitcoin and Ethereum has shot up by orders of magnitude in the last five years, we have seen countless news stories of random people who have become billionaires when considering their crypto net worth. The issue is that there are not enough buyers for these individuals to actually liquidate their holdings into a usable state of, say, US dollars.11 And this newfound wealth is tied to a highly speculative price. A price that is entirely contrived, and is tied to a currency that you cannot really use in a purposeful day-to-day manner. As such, the “whales” need somebody to buy into the crypto-hype, so they can wind their position down. a whale is a large holder of cryptocurrency who can manipulate a coin’s valuation through their sheer size This has been historically termed the “greater fool theory.” To stimulate demand for cryptocurrency, these bagholders need to create means for ordinary people to get involved. If you are particularly cynical, you can interpret the goals of NFTs in such a way. The greater fool theory posits that an already inflated asset can be resold at an even higher price, and generate profits for the original holder of said asset Remember the January 2021 GameStop fiasco? If you peer into the r/wallstreetbets threads, you’ll find countless users exhorting their fellow witless GME shareholders to “hold the line” and not sell their shares. All the while, those same users were quietly exiting their position and selling their shares to the next fool on the chopping block, such as the founder of University of web3. # So what’s next? The issue with web3, NFTs, or cryptocurrencies is that they fall way short of achieving their stated objectives. Cryptocurrencies are unable to solve the problems of the banking industry, web3 is unable to solve the problems of the tech industry, NFTs are unable to solve the problems of the art industry. This is because each of these industries suffers from dark patterns of human behavior, and web3 is not immune to such behavior. web3 does not prevent bad actors from exploiting and misleading others. The technologies surrounding web3 are instruments of human naivete and opportunism. Because of wild speculation and FOMO on behalf of venture capitalists eager to take a measured bet on such audacious claims, we have seen web3 startups explode in valuation despite delivering little value. Browse some of the technologies listed on this page and their landing pages, and try to convince yourself that you — as the potential user — are the intended audience rather than an investor. But there are still conceivable ways to dig us out of this mess. For instance, we can pin down the terrible externalities of crypto by instituting carbon taxes that will, among other things, disincentivize mining activity since its profitability is inextricably tied to the cost of electricity. But won’t this just cause miners to relocate to some country that doesn’t have a carbon tax? Sure, but the cost of shifting mining infrastructure is not cheap, and the margins of crypto mining are not particularly lucrative nowadays. This won’t outright halt emissions from cryptocurrency, but it may cause its inefficiencies to become so apparent that its users reconsider its utility, or at the very least hasten its transition to less polluting mechanisms. There is a direction that the internet is heading in, which unlike web3, does not need venture backing to sustainably grow or a minority of technocrat zealots to buy into. For better or worse, the actual web3 might be a network of tightening walled gardens and federations. Much of the internet has congealed into pools of user-produced content that are becoming increasingly inaccessible and myopic. Think of Reddit, Facebook, or Discord communities of thousands of people, whose activities are impossible to index on a search engine. Think of how the web became less navigable with artificial moats like login pages and paywalls, and is now being further entrenched with ideological barriers like echo chambers and content algorithms. An analogy we can use to express this trend is how our known universe is composed of galaxies that are drifting ever farther from each other, while each galaxy is crunching inward into a more compact form — the internet has created ever-tightening ingroups while separating people at large from each other. I myself know of two big ‘galaxies’ in the form of the Western internet and the Chinese internet, and we are seeing the schism of the alt-right internet taking hideous shape today. But I digress. At no point of this tectonic shift will a web3 of cryptocurrencies and NFTs have a naturally relevant seat. It will remain tomorrow as it lives on today: a messy dream. May 2022 Presented by Eric Cheng for 80-445 at Carnegie Mellon University. Opinions expressed are my own and do not necessarily reflect views of any affiliated people or organizations. No matter if you found reading this enjoyable, or feel as though I got something seriously wrong, I would be glad to talk further. Thank you to Simon Cullen for reading drafts of this. # Footnotes 1 According to researchers from the Cambridge Centre for Alternative Finance 2 According to the Federal Reserve’s Survey of Consumer Finances, American households had a median savings of$5,300 in 2019.
3
4
An interesting visualizer on Ethereum’s energy consumption
5
\$499 in the United States as of May 2022
6
Using the official supported GPU list and their average TDP, we can extrapolate the power requirements of the system at large.
7
See this Stack Exchange thread on the difference between anonymity and pseudonymity.
10
Data from this tweet by Will Papper
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However, the option to create them isn’t directly visible on the interface of these applications. The intended use when these characters were added to Unicode was to allow chemical and algebra formulas and phonetics to be written without markup, but produce true superscripts and subscripts. Computer languages have restrictions for which characters can appear in an identifier. perl -MCPAN -e shell install Unicode::Subscript. cpanm Unicode::Subscript. Apr 12, 2016 · R: subscript text as a variable. Page 212, subscript for O in chemical formula was unreadable. Resolution: Revise your code to ensure that you do not specify more than one subscript for a one-dimensional table item. , _text_ or *text*. We demonstrate the effectiveness of our. Hi! If a subscript has two or more characters, it is displayed wrongly in GeoGebra Android Apps. I am trying to enter subscript numbers into a database of mineralogical formulae. Other superscript and subscript characters. The “proper” LaTeX command for underscore is \textunderscore , but the LaTeX 2. There was a significant positive correlation between W[subscript max] and O[subscript 2] pulse R = 0. hi Alexander and thanks for your comment. 12p) pointsize. click on the box and type the text that has to go in the subscript. An array subscript is not part of the variable name. It is also possible to add several superscripts and subscripts to a text element using the expression function and the symbols ^ and [] as shown in the previous examples. How to make text superscript or subscript. Answer to Find R and theta θ given the components Upper R Subscript x Baseline equals 16. Design Flaws in R #3 — Zero Subscripts. R also provides a special subscripting method (double brackets) to extract the actual data (in this case a vector) from the data frame: > temps ['max'] max 1 59. The weird looking thing against each name is the markdown syntax to write hat, subscript, sum, limits and beta in the equation of linear model. R has several ways to subscript (that is, extract specific elements from a vector). Therefore, the resulting G r would have units J (or kJ) / mol. Enclose the character with the " sub " tag. Subscript text can be used for chemical formulas, like H2O to be written as H 2 O. For example, if we have a sequence of numbers and want to use a variable to represent them, we often use a subscript to indicate the index in the sequence. MarinStatsLectures-R Programming & Statistics 175,418 views 15:16 Microsoft excel shortcut: how to do superscript and subscript in graph or chart - Duration: 1:40. If you don't see the text transform right away in the campaign editor, select to Save it and then Preview the campaign. Subscript using VBA Code. remove background (remove backgroud colour and border lines, but does not remove grid lines). Next message: Warnes, Gregory R: "[R] [R-pkgs] New package: mcgibbsit, an MCMC run length diagnostic" Previous message: Peter Dalgaard: "Re: [R] Unexpected behaviour of identical" Maybe in reply to: Rajarshi Guha: "[R] using subscripts in a plot title with 2 lines" Next in thread: Paul Murrell: "Re: [R] using subscripts in a plot title with 2. Use scalar subscript values to select individual elements or subarrays from an array. ACM 22 (1975) 380-381. I've tried several codes… Hello everyone I need help to finish some hist3D plots using plot3D package. 9 at 30C in freshly prepared crude extracts. Description. You can create a quick. Code (Text):. When an array element isreferenced, the subscript expression designates the desired element byits position in the data. Modify makefile so that you can use make view etc to compile and view essential. Using superscript and subscript as part of the output is essential when creating formulas or presenting certain other kinds of information. substring is compatible with S, with first and last instead of start and stop. This page has the following sections: Vector Matrix and Data Frame List Replacement Matrix and Array Resources. Then feed that into the procedure you were hoping to invoke. Insert a superscript symbol. Subscript AB denotes diffusivity of species A diffusing into species B. Why use subscripts instead of different variables? Well, the variables in science often represent something specific. To correct this error. In R, a negative is an instruction to remove an element from a vector. If you find that you often need to add characters above or below the normal line of type, you may want to consider adding the Subscript and Superscript buttons to the visual editor for your own convenience. For example, if R denotes such a collection of names then the ith name in the collection may be referenced by R i (i. Among the properties of these objects are subscript and superscript. Can anyone help?. Array Subscripts. No need to use extra packages. For example, MD[5][3][9] identifies an element in a three-dimensional array called MD. O, o as third subscript: the terminal not mentioned is open-circuit R, r as Þrst subscript: reverse. You can create a quick. Fit Smooth Curve to Plot of Data in R (Example) In this tutorial you’ll learn how to draw a smooth line to a scatterplot in the R programming language. But with R[subscript 0] being the mean of a random variable, much more information is contained in the entire probability distribution. Mulliken Symbols for Irreducible Representations (R. 2 Greek letters 1. Tags: Alt Code Keyboard. However, the option to create them isn’t directly visible on the interface of these applications. The most common way is directly, using the square bracket operator: > x[4] [1] 4 In this example, the user has said "give me the fourth element," and R has said, "you get a vector whose first (and only) element is 4. For subscript, press Ctrl and the Equal sign. It seems natural that we should use two subscripts for a matrix. In the monotonic standard orthography, subscript iota is not used. For example, if R denotes such a collection of names then the ith name in the collection may be referenced by R i (i. Hope this helps!. 5 to be a subscript to PM. hi Alexander and thanks for your comment. Using superscript and subscript as part of the output is essential when creating formulas or presenting certain other kinds of information. Resolution: Revise your code to ensure that you do not specify more than one subscript for a one-dimensional table item. Active 7 years, 4 months ago. As an example; need to indicate the units of measurement for some data and therefore need to be able to pull data from the table such as the following: kg/m2, m/s2 etc. Ms Word shortcut for Subscript and Superscript. Check the meaning of subscripts. The array subscript is created by using the matrix index operator, or by using the [key. Insert Subscript or Superscript. With a vector, we subscript with a single index. The underscore character _ is ordinarily used in TeX to indicate a subscript in maths mode; if you type _, on its own, in the course of ordinary text, TeX will complain. For example, to subscript 2 in a mathematical equation like this (X 2), you'll need to:. Subscript and Superscript are important while formatting text in Word, Excel, and PowerPoint. Stata version is the same. O[subscript 2] pulse was found to be 13. R Subscript Xy Superscript 2 Baseline Equals 0. for example. Highlight the text you want to make superscript or subscript. A small letter or number placed slightly lower than the normal text. Fernandez, An order of primeness, F(p) N. See full list on programmingr. Comparison of J[subscript I][subscript c] and J-R curves for short crack and tensilely loaded specimen geometries of a high strength structural steel Author: J A Joyce ; E M Hackett ; C Roe ; U. 0 The ya-tag and ra-tag, or y and r subscript , and the s after vowels and consonants, were still in force. 01369v1 [cs. They are organized into seven classes based on their role in a mathematical expression. Hydrocomp Incorporated Mountain View, California 940UO Grant No. So R simply removes all zeros from a vector used as a subscript, producing a shorter vector. Fish Rensselaer Polytechnic Institute Troy, NY 12180 Abstract A new method for propagating arbitrary failure modes is presented. With a vector, we subscript with a single index. API documentation R package. Suppose we have the text in the Sheet1 and in the D2 cell then apply the subscript by using the below VBA code. , Cu^2+^ renders Cu 2+). Please share how this acce. Subscript AB denotes diffusivity of species A diffusing into species B. Subscripts are used in various ways, not as specifically as exponents. Suppose that we want r 2 to be the label on the y axis and x i to be the label on the x axis. Keyboard Shortcuts This information is available directly in the RStudio IDE under the Tools menu: Tools → Keyboard Shortcuts Help. Thanks, Humberto. The tag defines the subscript text. We present a conditional generative model to learn variation in cell and nuclear morphology and the location of subcellular structures from microscopy images. 1 Latin letters and Arabic numerals 1. I have avr program for the project and when compiling the following warning is displayed and the problem is that theere is no data arriving in the receiver and the program gets stuck around there as the led i put before this donot blink. This requires only typing both commands after 'Mt'. Getting subscripts from Excel into R. Wellington Pereira Guedes on 3 Oct 2017. With editing enabled, the subscript and superscript options will also be enabled. Subscript out of range (Visual Basic) 07/20/2015; 2 minutes to read +3; In this article. 2 $\begingroup$ My question is concerning a. Subscript and Superscript I figured out to type subscripts and superscripts if you're using Microsoft Word on your Mac Subscript: + (=/+) then type what you want Superscript: shift + + (=/+) then type what you want it keeps you in either subscript or supercript mode when you use these shortcuts. Subscript and superscript are integral part of any scientific manuscript. Parker, Primes with a prime subscript, J. Description. 1 Class 0 (Ord) symbols: Simple / ordinary ("noun") 1. 0 for Rcpp v0. 3,March26,2018. You enclose the characters that you want to superscript or subscript in curly brackets {}. Ram: you had a k-5 subscript with k starting at 1. in Figure 5-D. We have added a subscript "p" to the specific heat capacity to remind us that this value only applies to a constant pressure process. Enable that in your field and then read about the richValue property and how to set it using an array of Span objects. Getting subscripts from Excel into R. Indicating: $Economical value of the respective characteristic of the cohort. The expression function turns R code into text like this:. The Subscript shortcut is Ctrl + = on a PC and Ctrl + Cmd + + on a Mac. The usage is pretty easy, you can basically type the name of the letter and put a backslash in front of it. You can do this through the Font dialog box, but there is a much faster way. To install Unicode::Subscript, simply copy and paste either of the commands in to your terminal. Post a new example: Submit your example. In the range p[subscript T] = 5 – 10 GeV/c, the charged particle yield in the most central PbPb collisions is suppressed by up to a factor of 7. Usually, the subscript is placed in brackets following the array name. You can easily switch between superscript, subscript, and normal text in Microsoft Word. LaTeX handles superscripted superscripts and all of that stuff in the natural way. A Unity ID allows you to buy and/or subscribe to Unity products and services, shop in the Asset Store and participate in the Unity community. The subscripts can be integers or matrices ( 2d arrays ). Ms Word shortcut for Subscript and Superscript. Subscript - hindi meaning of नीचे की लिखावट. Many of the in equations in science, mathematics and engineering contains subscripts and superscript. Insert Subscript or Superscript. Using subscript indices is flexible, compact, universally technically supported, and intuitive. An occasional fifth letter in a Nasdaq-traded company's ticker symbol that identifies the stock as a rights offering. It's still clunky for that if I remember right, but at least it's possible there. Sex Discrimination in Employment: Zarda, Bostock, and R. Hydrocomp Incorporated Mountain View, California 940UO Grant No. Cheerz, Lars ----- Create and Deploy Rich Internet Apps outside the browser with Adobe(R)AIR(TM) software. Letter Subscript: Reserve SPE Subscript: Subscript Definition: Greek and Numerical: formation 100% saturated with water (used in R 0 only) 1: p,pri: primary 1,2,3. Table of contents:. Not from inside Spotfire. Bold text is produced using a pair of double asterisks (**text**). writethe subscript the candidate numerals are written in subscript in the cells. In Python, there is a method called maketrans. Indicating:$ Economical value of the respective characteristic of the cohort. Use multiple languages including R, Python, and SQL. if it is possible to store text in a table that includes subscript / superscript characters. A two-dimensional array with n columns and m rows, is addressed with a subscript of the form [i, j], where 0 £ i < n and 0 £ j < m. In the image below, subscript text is used to write down the formula of Carbon dioxide. Subscript and Superscript are important while formatting text in Word, Excel, and PowerPoint. 0024 ohm turn Full winding: R secondary R turn 240. For my graduate work, it was recommended for people to use Microsoft Visio to recreate their circuits with nice formatting and the capability to do subscripts for labels. To get an expression exp to appear as a subscript, you just type _{exp}. The easy way is to use the multiplot function, defined at the bottom of this page. Sex Discrimination in Employment: Zarda, Bostock, and R. I don't have any problem doing that if PM2. Brought to you by: http://www. 2008-09-21 at 2:34 pm 17 comments. What is the meaning of a decimal in a subscript in a chemical formulas? I came across the fractional subscripts in a publication, [1] for example: We introduce oxygen into a ferromagnetic metallic glass to form a $\ce{Co_{28. Use multiple languages including R, Python, and SQL. It is designed for inter-converting force and mass units in a number of equations. Accelerated rise in [CO. Apr 12, 2016 · R: subscript text as a variable. Inline text will be italic if surrounded by underscores or asterisks, e. Here, we have subscript operator, ‘[]’, which aside from having higher precedence than indirection operator (*) is same as indirection operator. Harris Funeral Homes (Decision June 15, 2020) Jun 17, 2020 Does the Civil Rights Act of 1964 protect against employment discrimination on the basis of sexual orientation or gender identity?. In latest Word versions: Word 2007 and Word 2010, you will find the superscript and subscript options right on the ribbon. Letter Subscript: Reserve SPE Subscript: Subscript Definition: Greek and Numerical: formation 100% saturated with water (used in R 0 only) 1: p,pri: primary 1,2,3. Resolution: Revise your code to ensure that you do not specify more than one subscript for a one-dimensional table item. In MTEXT: For subscript (text below the line) enter ^ before the text you wish to place below the line, then highlight the ^ and the text, right click and then Stack. Subscripting is also used to assign values to data object elements. With this view, we show - again by simple arguments - that R[subscript 0] can be greater than 1. Shortcut Image Reference. It even does the right thing when something has both a subscript and a superscript. It is usually smaller than the rest of the text. A special note on margins You noticed the code par(mar =c(5, 6, 5, 5)) I called above before I plotted my figure. This is italic. Sex Discrimination in Employment: Zarda, Bostock, and R. Use one of the following shortcuts: Superscript: Ctrl + Shift + Plus; Subscript: Ctrl + Plus. That should fix the problem. 1-4 , License: GPL-3 Community examples. For superscript, simply press Ctrl + Shift + + (press and hold Ctrl and Shift, then press +). Description. Error: invalid subscript type ‘list’. SubScript is my capstone project. H_0, Omega_b, Omega_dm, Omega_\\Lambda Omega_r in the form of latex My code is something like this import PySimpleGUI as sg sg. Then feed that into the procedure you were hoping to invoke. As you can see in Figure 2, our plot title contains a subscript. 144 time-saving Hotkeys for ALT Digits Symbol Codes. Likewise subscripts in R involve square brackets [] so x i is written x[i]. R Subscript Xy Superscript 2 Baseline Equals 0. Dear R help list members, I am experiencing difficulty in trying to generate a subscript '2' in an axis label. Superscript in a sentence - Use "superscript" in a sentence 1. No need to use extra packages. You want to put multiple graphs on one page. You should not expect it to work in a data. I'm using mathcad prime 3. Wellington Pereira Guedes on 3 Oct 2017. 7 s -1 in purified enzyme from Atriplex glabriuscula at 25 C (Badger and Collatz 1977), and 3. In the Code View popup, find the character you wish to make subscript. How to Enter a Subscript in Excel. The lowest subscript value for a dimension is always 0, and the highest subscript value is returned by the GetUpperBound method for that dimension. Design Flaws in R #3 — Zero Subscripts. #1 gstein1231, Dec 21, 2009. For example, using the R code below: the line plot (lp) will live in the first row and spans over two columns the box plot (bxp) and the dot plot (dp) will be first arranged and will live in the second row with two different columns. Subscript In Access Nov 18, 2004. I hope I helped. A composition is capable of curing via condensation reaction. Subscript text appears half a character below the normal line and is sometimes rendered in a smaller font. The array subscript is created by using the [key. GitHub Gist: instantly share code, notes, and snippets. Select the Superscript or Subscript button in the Font group. In that, I would like 2. While R's use of negative subscripts is unconventional, it makes sense in context. Description. plotting axis labels with both Greek symbols from vector and subscripts. Of course, you could simply give the commands a new, shorter name if you wish to write less. Data Types: double. 6 An introductory session. 0 and still, contrary to popular belief, the probability of an exponentially growing pandemic may be. In some situations, you want to display R code but not evaluate it. An occasional fifth letter in a Nasdaq-traded company's ticker symbol that identifies the stock as a rights offering. stackexchange. Modify makefile so that you can use make view etc to compile and view essential. The gas constant used by aerodynamicists is derived from the universal gas constant, but has a unique value. I am trying to enter subscript numbers into a database of mineralogical formulae. Tables :: Subscript / Superscript Characters In A Table Nov 1, 2014. Active 7 years, 4 months ago. Comparison of J[subscript I][subscript c] and J-R curves for short crack and tensilely loaded specimen geometries of a high strength structural steel Author: J A Joyce ; E M Hackett ; C Roe ; U. Subscripts & Superscripts. These are located below the font type selection box. R \int U \biguplus L \bigoplus W Q \bigvee \prod H \oint T \bigcap N \bigotimes V ‘ \bigwedge \coprod RR \iint S \bigcup J \bigodot F \bigsqcup 5 Standard Function. This example demonstrates how do I Subscript and SuperScript a string in android. Arrays with multiple dimensions are addressed by specifying a subscript expression for each dimension. Page 212, subscript for O in chemical formula was unreadable. To the best of my knowledge subscripts on vector data types is not compliant OpenCL. Hello everybody. 09 90 428 520 600 700 780 800 927 t 240 200 180 160 120 100 80 60 40 20 v t N2 N2 t 1 excitation 2 Im t N2 t 1 N2 excitation 1 turn 1 R turn 0. Not from inside Spotfire. 12p) pointsize. Precisely speaking, it is Pandoc’s Markdown. Example 3: Adding Multiple Superscripts & Subscripts to Plot. #1 gstein1231, Dec 21, 2009. Below is the summary of shortcuts in image format. Mulliken Symbols for Irreducible Representations (R. Or, select the existing text that you want to format as a superscript or subscript. the first address and the index is what is most important. So be in either of those apps and start typing as usual, then when you hit the point where you want to insert superscript or subscript text just do the following: Pull down the “Format” menu and go to “Font” Select the “Baseline” submenu and choose either “Superscript” or “Subscript”. On output, Subscript, Superscript and Subsuperscript format respectively as subscripts, superscripts and subsuperscripts: Notes To enter both a subscript and superscript, start with one or the other and then switch to the other position with :. For example if I wanted to display out H2O and subscript the 2. I am trying to enter subscript numbers into a database of mineralogical formulae. To Superscript: Repeat all above steps with a carat '^' instead of an underscore. Use scalar subscript values to select individual elements or subarrays from an array. Hello everyone I need help to finish some hist3D plots using plot3D package. Plot[f, {x, -1, 1}, AxesLabel -> {x, HoldForm[Subscript[f, i][x]]}] Of course, to enter the Subscript expression in your label, you can still use the keyboard shortcuts that are mentioned by @Szabolcs. Resolution: Revise your code to ensure that you do not specify more than one subscript for a one-dimensional table item. in Figure 5-D. The new legal = > advisor at our parent company is sub-scripting the "circle-R" after the = > brand-name; this is also how it's done on our parent company's product = > literature. Same fix either way - use the unlist() function to unpack the contents of an R list into an R vector. Superscript and subscript allow you to type characters that appear above or below the normal text line. If anyone can help me I ap. For example: $(a^n) ^{r + s} = a^{nr + ns}$. U+2095 ₕ Latin Subscript Small Letter H. 6 Comments. perl -MCPAN -e shell install Unicode::Subscript. In that, I would like 2. refers to an expression, either scalar or vector, for selecting one or more columns from the operand. Array Subscripts. GitHub Gist: instantly share code, notes, and snippets. The carat (^) character is used to raise something, and the underscore (_) is for lowering. More R Subscript. The most common way is directly, using the square bracket operator: > x[4] [1] 4 In this example, the user has said "give me the fourth element," and R has said, "you get a vector whose first (and only) element is 4. Block Quotes. Subscripts and Indices. Improving accuracy OpenCV HOG people detector. It even does the right thing when something has both a subscript and a superscript. Answer to Find R and theta θ given the components Upper R Subscript x Baseline equals 16. Superscript and subscript allow you to type characters that appear above or below the normal text line. News GCC 10. 6 An introductory session. These characters appear smaller than standard text, and are traditionally used for footnotes, endnotes, and mathematical notation. a bunch of CG's brought up by Mexican's who sold crack on the streets to support the members of subscript's CS:S addiction. This is especially the case in science subjects such as chemistry or physics. org/posting-guide. For example, to subscript 2 in a mathematical equation like this (X 2), you’ll need to:. It shouldn't change the resulting value because adding the subscript "r" to n (n r) just makes "n" a pure number instead of being "n moles". On output, Subscript, Superscript and Subsuperscript format respectively as subscripts, superscripts and subsuperscripts: Notes To enter both a subscript and superscript, start with one or the other and then switch to the other position with :. 75184 (P < 0. An array subscript allows Mathcad to display the value of a particular element in an array. Subscript and Superscript are important while formatting text in Word, Excel, and PowerPoint. What is the partial derivative, how do you compute it, and what does it mean. Separate them with a comma. Use one of the following shortcuts: Superscript: Ctrl + Shift + Plus; Subscript: Ctrl + Plus. U+2095 ₕ Latin Subscript Small Letter H. 26 days ago by. In some ways vectors in R are more like sets than arrays. I Depends on R version ( 2. For example, if R denotes such a collection of names then the ith name in the collection may be referenced by R i (i. As an example; need to indicate the units of measurement for some data and therefore need to be able to pull data from the table such as the following: kg/m2, m/s2 etc. A composition is capable of curing via condensation reaction. As you can see in Figure 2, our plot title contains a subscript. Brought to you by: http://www. if it is possible to store text in a table that includes subscript / superscript characters. To use either of them, select the character(s) you wish to convert to subscript or superscript, click on either icon on the menu bar, and voila. It’s always better to give than to receive. You can easily switch between superscript, subscript, and normal text in Microsoft Word. R also provides a special subscripting method (double brackets) to extract the actual data (in this case a vector) from the data frame: > temps ['max'] max 1 59. How to simply create superscript and subscript in all versions of Word. 0 (June, 1993) Encodings; HTML Entity (decimal) ₂ HTML Entity (hex) ₂ How to type in. To subscript a character in equation editor: 1. Tables :: Subscript / Superscript Characters In A Table Nov 1, 2014. Instead, SAS creates variable names by concatenating the array name and the numbers 1, 2, 3, … n. R Markdown supports a reproducible workflow for dozens of static and dynamic output formats including HTML, PDF, MS Word, Beamer, HTML5 slides, Tufte-style handouts, books, dashboards, shiny applications, scientific articles, websites, and more. R: subscript text as a variable. The carat (^) character is used to raise something, and the underscore (_) is for lowering. The WordPress codex section on enabling hidden MCE buttons, which demonstrates how to filter the button list. for example. This page has the following sections: Vector Matrix and Data Frame List Replacement Matrix and Array Resources. I am using the following code: p <- plot_ly(x = x_vec, y = y_v…. They are organized into seven classes based on their role in a mathematical expression. GitHub Gist: instantly share code, notes, and snippets. 1/3 was sent on food. Mulliken Symbols for Irreducible Representations (R. When nesting subscripts/superscripts, however, remember that each command must refer to a single element; this can be a single letter or number, as in the examples above, or a more complex mathematical expression collected in braces or brackets. MATLAB uses the caret (^) to denote superscript and the underscore (_) to denote subscript. Table of contents:. LaTeX handles superscripted superscripts and all of that stuff in the natural way. Please, see the attached image. Alan Wood’s Unicode Resources Test for Unicode support in Web browsers Superscripts and Subscripts U+2070 – U+209F (8304–8351). Remember there is a big difference between a constant index and a dynamic index. Place your cursor where you want to insert the superscript or subscript. LaTeX symbols have either names (denoted by backslash) or special characters. 6 Comments. Subscript and Superscript I figured out to type subscripts and superscripts if you're using Microsoft Word on your Mac Subscript: + (=/+) then type what you want Superscript: shift + + (=/+) then type what you want it keeps you in either subscript or supercript mode when you use these shortcuts. Mulliken, J. These characters appear smaller than standard text, and are traditionally used for footnotes, endnotes, and mathematical notation. To undo superscript or subscript formatting, select your text and press Ctrl+Spacebar. To use R under Windows the procedure to follow is basically the same. sub ] 1 operator [] shall be a non-static member function with exactly one parameter an arbitrary number of parameters. Table of contents:. To Superscript: Repeat all above steps with a carat '^' instead of an underscore. Attributes. Usually, the subscript is placed in brackets following the array name. Then feed that into the procedure you were hoping to invoke. subscript A means of referring to particular elements in an ordered collection of elements. The problem is that I don't know how to include superscripts and subscripts in the axis names of this kind of plots. The “proper” LaTeX command for underscore is \textunderscore , but the LaTeX 2. If you want to keep the shortcuts in offline, right click and save the image to your computer. , _text_ or *text*. For example, AR[5] identifies element number 5 in an array called AR. > > > xNA <- c(1:2, NA, 3:4) > yNA <- c(3:5, NA, 6) >. If you use superscript or subscript a lot, you might want to know the keyboard shortcut to save you rooting around in sub-menus. Remember we changed ORIGIN from 0 to 1. We use superscripts and subscripts to distinguish between different SNPs and different alleles within SNPs, respectively. A subscript is a number placed below normal text when you're writing or typing, such as in the chemical formula H 2 O for water. Hi everyone, Is there a way to do subscripts or superscripts in text blocks in notion? For all of Notion's incredible features, I feel like a lot of perhaps simpler and more commonplace ones are overlooked. Insert Subscript or Superscript. This is referred to as the subscript operator. We have to know in what cases to use those subscripts and superscripts in text mode, because it seems to me sometimes it can do much harm if to use it not in a correct way! Recently, I was writing a dissertation about usage of this scripts with a help of custom writers and this topic was really interesting to explore. However, the option to create them isn’t directly visible on the interface of these applications. 2008-09-21 at 2:34 pm 17 comments. The WordPress codex section on enabling hidden MCE buttons, which demonstrates how to filter the button list. For example, to subscript 2 in a mathematical equation like this (X 2), you'll need to:. I have avr program for the project and when compiling the following warning is displayed and the problem is that theere is no data arriving in the receiver and the program gets stuck around there as the led i put before this donot blink. The sign always denotes the character of the charge on the first component written in the subscript to. 1-4, License: GPL-3 Community examples. Explanation: In an array the base address i. 5 happens to be at the end of the string:. Just in case, make extra room in the margins!. it is found in chromatography. No need to use extra packages. Example 3: Adding Multiple Superscripts & Subscripts to Plot. This is y 2. September 11, 2017 at 6:15 AM. Can anyone help?. Use one of the following shortcuts: Superscript: Ctrl + Shift + Plus; Subscript: Ctrl + Plus. , are commonly used for inverse hyperbolic trigonometric functions (area hyperbolic functions), even though they are misnomers, since the prefix arc is the abbreviation for arcus, while the prefix ar stands for area. Dear R help list members, I am experiencing difficulty in trying to generate a subscript '2' in an axis label. September 11, 2017 at 6:15 AM. That should fix the problem. Step 1 − Create a new project in Android Studio, go to File ⇒ New Project and fill all required details to create a new project. 09 command \_ is an established alias. Toshowintegrationbetween x and 1 forexample, wetype$\int^\infty_x$. Complete Parts (a) Through (e). If you want to keep the shortcuts in offline, right click and save the image to your computer. Insert Subscript or Superscript. hello there, ladies and gentlemen. 0 The ya-tag and ra-tag, or y and r subscript , and the s after vowels and consonants, were still in force. To learn more about how to use subscripts, check out our documentation. – (‘g sub c’ or ‘g subscript c’) is an often misunderstood factor in engineering. Page 212, subscript for O in chemical formula was unreadable. Step 2 − Add the following code to res/layout/activity_main. Sign in Register R, Plot axis label text tips; by Trent Biggs; Last updated about 4 years ago; Hide Comments (–) Share Hide Toolbars. 4 Ways To Type Superscript and Subscript On a Mac If you need to raise characters above or lower them below the baseline, you can do it one of four ways. I have the following code: mtext(text="X[1]") How can I change "[1]" to be subscript? (see picture below) subscript. As the entertainment software industry moves away from single boxed purchases to a subscription model, service providers must continually engage their current user base to maintain their subscription volume (and revenue). a bunch of CG's brought up by Mexican's who sold crack on the streets to support the members of subscript's CS:S addiction. In MTEXT: For subscript (text below the line) enter ^ before the text you wish to place below the line, then highlight the ^ and the text, right click and then Stack. Subscript using VBA Code. Can anyone help me out here?. R has several ways to subscript (that is, extract specific elements from a vector). Next message: Warnes, Gregory R: "[R] [R-pkgs] New package: mcgibbsit, an MCMC run length diagnostic" Previous message: Peter Dalgaard: "Re: [R] Unexpected behaviour of identical" Maybe in reply to: Rajarshi Guha: "[R] using subscripts in a plot title with 2 lines" Next in thread: Paul Murrell: "Re: [R] using subscripts in a plot title with 2. H 2 O), but can also be used for something as simple as a date or other ordinal number (e. Table 6Example, leading and trailing superscript and subscript. As third subscript: with a speciÞed resistance between the terminal not mentioned and the reference terminal (RMS), (rms) root-mean-square value S, s as first or second subscript: source terminal (FETs only). Is there a way to write subscripts and superscripts? Similar Threads. Example 3: Adding Multiple Superscripts & Subscripts to Plot. 09 command \_ is an established alias. stackexchange. 7 s -1 in purified enzyme from Atriplex glabriuscula at 25 C (Badger and Collatz 1977), and 3. 75184 (P < 0. The sign always denotes the character of the charge on the first component written in the subscript to. To undo superscript or subscript formatting, select your text and press Ctrl+Spacebar. Although subscripts that reference particular elements are positive, negative subscripts are legal. In R, the operator for superscripts is ‘hat’ (or, more correctly, ‘caret’) so ‘r squared’ is written r^2. Hi! If a subscript has two or more characters, it is displayed wrongly in GeoGebra Android Apps. It shouldn't change the resulting value because adding the subscript "r" to n (n r) just makes "n" a pure number instead of being "n moles". Pressing the respective shortcut again will get you back to normal text. While we typically aim to apply functions to vectors as a whole, there are circumstances where we want to select only some of the elements of a vector. Thanks, Humberto. 7 > temps [ ['max']] [1] 59. 3 The stress indices in these tables provide only the maximum stresses at certain general locations due to internal pressure. 144 time-saving Hotkeys for ALT Digits Symbol Codes. In the evaluation of stresses in or adjacent to vessel openings and connections, it is often necessary. Johanson John C. Subscript and superscript are important when you are dealing with different types of formulas. 2008-09-21 at 2:34 pm 17 comments. API documentation R package. Hyperbolic functions The abbreviations arcsinh, arccosh, etc. These characters appear smaller than standard text, and are traditionally used for footnotes, endnotes, and mathematical notation. 5 Markdown syntax. In the range p[subscript T] = 5 – 10 GeV/c, the charged particle yield in the most central PbPb collisions is suppressed by up to a factor of 7. (Both r's, {eq}\theta{/eq}, and z are subscripts) (a) Is this flow incompressible? (b) Is this flow irrotational? (c) Find an expression for the velocity potential function. It is also possible to add several superscripts and subscripts to a text element using the expression function and the symbols ^ and [] as shown in the previous examples. You use the same characters as are used for subscripts and superscripts when specifying boundaries. Chemical reaction also requires subscript and superscripts. R Markdown is an easy-to-write plain text format for creating dynamic documents and reports. change_look_and_feel('Topanga') layout = [. R Pubs by RStudio. If you use superscript or subscript a lot, you might want to know the keyboard shortcut to save you rooting around in sub-menus. 50 Nbsp Nequals 55 SSR Equals (Type An Integer Or A Decimal. This is y 2. Question: Compute SSR, SSE, S Subscript E Superscript 2 , And The Coefficient Of Determination, Given The Following Statistics Computed From A Random Sample Of Pairs Of X And Y Observations. You can add text that appears smaller and slightly below (subscript) or above (superscript) your main text quite easily in Word. The i subscript is throwing me off. That means that you don't have to use the insert equation feature. This is italic. Automate all the things! Web Scraping with R (Examples) Monte Carlo Simulation in R Connecting R to Databases Animation & Graphics Manipulating Data Frames Matrix Algebra Operations Sampling Statistics Common Errors. Error: invalid subscript type 'list'. Is there a way to write subscripts and superscripts? Similar Threads. I have the following code: mtext(text="X[1]") How can I change "[1]" to be subscript? (see picture below) subscript. Hello everyone I need help to finish some hist3D plots using plot3D package. Corresponds to the positions in the data object to be excluded: R 1. You want to put multiple graphs on one page. Arrays with multiple dimensions are addressed by specifying a subscript expression for each dimension. But with R[subscript 0] being the mean of a random variable, much more information is contained in the entire probability distribution. 5 you would type: Matrix_1[1,1= and get a result of Matrix_1 1,1 =1. Method 2: Superscript and subscript keyboard shortcuts. T or P) indent paragraph indent (e. I have the following code: mtext(text="X[1]") How can I change "[1]" to be subscript? (see picture below) subscript. Error: invalid subscript type 'list'. 002) we can conclude that the best. the vector x with the element x[2] removed. Hello everyone I need help to finish some hist3D plots using plot3D package. On the other hand, superscripts in a chemical equation are the notations for a positive or negative ionic charge. Subscript text appears half a character below the normal line, and is sometimes rendered in a smaller font. Then feed that into the procedure you were hoping to invoke. Table 6Example, leading and trailing superscript and subscript. Subscript out of range. 12p) pointsize. 5 Subscripting [ over. It does seem to call the "int operator[](int n) const" when a[2] is called from within a function that specifies that a is a const A&, but calls the "int& operator[](int n)" in any other case, irrespective of whether a[2] is an rvalue or lvalue. Table 6Example, leading and trailing superscript and subscript. Keyboard Shortcuts This information is available directly in the RStudio IDE under the Tools menu: Tools → Keyboard Shortcuts Help. Subscript out of range. Type an underscore '_'. September 11, 2017 at 6:15 AM. Same fix either way - use the unlist() function to unpack the contents of an R list into an R vector. Resolution: Revise your code to ensure that you do not specify more than one subscript for a one-dimensional table item. X 2) and other technical fields (e. Question: subscript out of bounds in seurat expression matrix [SOLVED] 0. It is also possible to add several superscripts and subscripts to a text element using the expression function and the symbols ^ and [] as shown in the previous examples. 2] Atmospheric levels of carbon dioxide, a major greenhouse gas, climbed much faster in the last four years than during previous years, according to findings from some 30 stations around the globe. While (currently) unusual, subscripting might be a useful trick for clearer writing, compared to omitting such information or using standard cumbersome circumlocutions. Examples: Domain: R +:= {x ∈ R | x > 0}, the set of positive real numbers. COBCH0309 Malformed subscript. a bunch of CG's brought up by Mexican's who sold crack on the streets to support the members of subscript's CS:S addiction. Sign in Register Superscripts in R & Rmarkdown; by Nathan Brouwer; Last updated over 3 years ago; Hide Comments (-) Share Hide Toolbars. Subscript and Superscript are important while formatting text in Word, Excel, and PowerPoint. When applied to a pointer, the subscript expression is always an lvalue. This is referred to as the subscript operator. You can easily switch between superscript, subscript, and normal text in Microsoft Word. The most common way is directly, using the square bracket operator: > x[4] [1] 4 In this example, the user has said "give me the fourth element," and R has said, "you get a vector whose first (and only) element is 4. As far as I know this is the correct syntax for markdown subscript. Identify the symbol of the cation (first part of the name) and the anion. It even does the right thing when something has both a subscript and a superscript. This shortcut works in Microsoft Word and PowerPoint to quickly create (or remove) subscripts. This page has the following sections: Vector Matrix and Data Frame List Replacement Matrix and Array Resources. The weird looking thing against each name is the markdown syntax to write hat, subscript, sum, limits and beta in the equation of linear model. Sign in Register Superscripts in R & Rmarkdown; by Nathan Brouwer; Last updated over 3 years ago; Hide Comments (–) Share Hide Toolbars. The array subscript is created by using the matrix index operator, or by using the [key. LaTeX handles superscripted superscripts and all of that stuff in the natural way. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. To correct this error. There are many flavors of Markdown invented by different people, and Pandoc’s flavor is the most comprehensive one to our knowledge. You can create a quick. > "circle-R" symbol in superscript after the brand-name. Numerical subscripts in R start at 1, and continue up to the length of the vector. Letter Subscript: Reserve SPE Subscript: Subscript Definition: Greek and Numerical: formation 100% saturated with water (used in R 0 only) 1: p,pri: primary 1,2,3. That should fix the problem. 1/3 was sent on food. This printed form is the origin of the term but it is also used when the “subscript” is written on the same line. I've tried several codes… Hello everyone I need help to finish some hist3D plots using plot3D package. Use multiple languages including R, Python, and SQL. Description. I can't find a reference to which is correct (or if both are = > acceptable). Method 2: Superscript and subscript keyboard shortcuts. The easy way is to use the multiplot function, defined at the bottom of this page. Re: LaTeX representation of hollow "R" symbol for real space Post by localghost » Mon May 24, 2010 9:47 am I don't think that your math institute sets the standard [1,2]. 1 Inline formatting. More R Subscript. With editing enabled, the subscript and superscript options will also be enabled. The sign always denotes the character of the charge on the first component written in the subscript to. Is there a way to write subscripts and superscripts? Similar Threads. CPAN shell. The most common way is directly, using the square bracket operator: > x[4] [1] 4 In this example, the user has said "give me the fourth element," and R has said, "you get a vector whose first (and only) element is 4. You can do this through the Font dialog box, but there is a much faster way. Chemical reaction also requires subscript and superscripts. Subscripts and Indices. Then feed that into the procedure you were hoping to invoke. To get exp to appear as a superscript, you type ^{exp}. Displaying Blocks of Code Without Evaluating. Looking for online definition of subscript or what subscript stands for? subscript is listed in the World's largest and most authoritative dictionary database of abbreviations and acronyms Subscript - What does subscript stand for?. It's still clunky for that if I remember right, but at least it's possible there. 0 for Rcpp v0. Use one of the following shortcuts: Superscript: Ctrl + Shift + Plus; Subscript: Ctrl + Plus. As far as I know this is the correct syntax for markdown subscript. So if positive. It is purely representational therefore when it is time to publish/print a nice table. Mulliken, J. In Python, there is a method called maketrans. The problem is that I don't know how to include superscripts and subscripts in the axis names of this kind of plots. Use one of the following shortcuts: Superscript: Ctrl + Shift + Plus; Subscript: Ctrl + Plus. Subscript is a small character or string that sits below the line of text. The intended use when these characters were added to Unicode was to allow chemical and algebra formulas and phonetics to be written without markup, but produce true superscripts and subscripts. Adaptive control method of clutch torque during clutch slip engagement Jinrap Park 1and Seibum Choi Abstract—To precisely control clutch torque during clutch slip engagement, a clutch friction model is required. 5 happens to be at the end of the string:. R Markdown supports a reproducible workflow for dozens of static and dynamic output formats including HTML, PDF, MS Word, Beamer, HTML5 slides, Tufte-style handouts, books, dashboards, shiny applications, scientific articles, websites, and more. ACM 22 (1975) 380-381. 50 Nbsp Nequals 55 SSR Equals (Type An Integer Or A Decimal. Then launch R by double clicking on the icon. Data Types: double. The LOG function is a built-in function in Excel that is categorized as a Math/Trig Function. Here is the description of all the attributes of this tag − subscriptshift − To specify the minimum space to shift the subscript below the baseline of the expression. H 2 O), but can also be used for something as simple as a date or other ordinal number (e. For example, in early versions of the C and C++ languages, identifiers were restricted to a sequence of one or more ASCII letters, digits, which may not appear as the first character, and underscores. It is usually smaller than the rest of the text. Then feed that into the procedure you were hoping to invoke. The subscript R denotes the respective values of the reference. What is the meaning of a decimal in a subscript in a chemical formulas? I came across the fractional subscripts in a publication, [1] for example: We introduce oxygen into a ferromagnetic metallic glass to form a$\ce{Co_{28. Attributes. Using subscript indices is flexible, compact, universally technically supported, and intuitive. R Pubs by RStudio. , _text_ or *text*. If your locale is set to some other languages (in particular Korean, Chinese, or Japanese, and possibly others) and your MATLAB is new enough, then you can include characters up to char(65535) in the source code directly, inside character vectors or string objects or comments. However, the option to create them isn’t directly visible on the interface of these applications. Accelerated rise in [CO. I am trying to enter subscript numbers into a database of mineralogical formulae. Superscripts and Subscripts — Unicode Character Table 2070-209f. Adaptive control method of clutch torque during clutch slip engagement Jinrap Park 1and Seibum Choi Abstract—To precisely control clutch torque during clutch slip engagement, a clutch friction model is required. In R, the operator for superscripts is ‘hat’ (or, more correctly, ‘caret’) so ‘r squared’ is written r^2. Plot[f, {x, -1, 1}, AxesLabel -> {x, HoldForm[Subscript[f, i][x]]}] Of course, to enter the Subscript expression in your label, you can still use the keyboard shortcuts that are mentioned by @Szabolcs. Sub Subscripts (). Many of the in equations in science, mathematics and engineering contains subscripts and superscript. September 11, 2017 at 6:15 AM. n] is as follows Electromagnetic response of a circular DB cylinder in the presence of chiral and chiral nihility metamaterials. 12p) pointsize. 31 ml/beat at W[subscript max]. TheNotSoShort IntroductiontoLATEX2" OrLATEX2" in139minutes byTobiasOetiker HubertPartl,IreneHynaandElisabethSchlegl Version6. Sign in Register Superscripts in R & Rmarkdown; by Nathan Brouwer; Last updated over 3 years ago; Hide Comments (-) Share Hide Toolbars. ; The Latin Extended-C block contains one additional superscript, ⱽ, and one additional subscript ⱼ. 002) we can conclude that the best. To subscript a character in equation editor: 1. place before the character and after the character Example : Super Product ™ would be displayed as Super Product ™. The difference between superscript / subscript and numerator / denominator glyphs. I have the following code: mtext(text="X[1]") How can I change "[1]" to be subscript? (see picture below) subscript. Subscript Law is a nonprofit with a mission to enhance the legal and political decision-making of Americans. 09 90 428 520 600 700 780 800 927 t 240 200 180 160 120 100 80 60 40 20 v t N2 N2 t 1 excitation 2 Im t N2 t 1 N2 excitation 1 turn 1 R turn 0. An earlier poster said that the row names may be different, but in my case, I don’t have row names for either dataset. For example, using the R code below: the line plot (lp) will live in the first row and spans over two columns the box plot (bxp) and the dot plot (dp) will be first arranged and will live in the second row with two different columns. Description. Active 7 years, 4 months ago. You need select the text and click on the icon to make it super or subscript. Subscript[r, i] Subscript[6, i] Is there any way for me to get the value I assigned to Subscript[r,i]? Is this the wrong way to use Mathematica?. Same fix either way – use the unlist() function to unpack the contents of an R list into an R vector.
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## Probability of Transit
Transiting Planets. Credit: NASA
Transiting planets are valuable items to explore the properties of planetary atmospheres. Planet searches like Kepler that focous on fields of sky tend to reap rewards amongst dimmer stars simply because there are many more dim stars in a given patch of the sky than bright ones. Transiting planets around bright stars are of particular value, though, as the increased brightness makes the system easier to study.
Radial velocity surveys tend to monitor brighter stars since spectroscopy is even more severely limited by stellar brightness than photometry, but it is not limited to observing patches of sky – telescopes performing Doppler spectroscopy tend to observe a single object at a time due to technical and physical limitations. Radial velocity surveys are also much less sensitive to the inclination angle of a planet orbit with respect to the plane of the sky. The planet doesn’t have to transit to be spectroscopically detectable. As such, radial velocity surveys tend to generate discoveries of planet candidates with unknown inclinations and true masses, but around much brighter stars than those planets discovered by the transit method.
As such, planet candidates discovered by radial velocity, especially planet candidates in short orbital periods are excellent targets for follow-up observations to attempt to detect transits. Transiting planets that have been discovered first through radial velocity have been of great scientific interest due to their host stellar brightness and thus ease of study. If more such systems are found, it would be of great benefit to understanding extrasolar planet atmosphere. While only a hand-full of transiting planets have been discovered first through radial velocity, they all orbit bright stars and are some of the best-characterised planets outside our solar system.
The probability that a planet will transit is, as has been discussed previously, given by
$\displaystyle P_{tr} = \frac{R_*}{a}$
where a is the semi-major axis of the planet orbit. This is the distance between the centre of the star and the centre of the planet. However, due to the inclination degeneracy – the reoccurring evil villain constantly plaguing radial velocity science – the star-planet separation is unknown. Remember that the period of the RV curve gives only the orbital period of the planet. If the orbital period is held constant, increasing the mass of the planet increases the star-planet separation. An increase in the total system mass requires greater separation between the two bodies to preserve the same orbital period.
For example, if radial velocity observations of a star reveal the presence of a mp sin i = 1 ME planet candidate, but the inclination is actually extremely low such that the true mass of the companion is in the stellar regime, then because the mutual gravitational attraction between the two stars will be much greater than the mutual gravitational attraction between the star and an Earth-mass planet at the same period, the two stars must have a wider separation, otherwise their orbital period would be smaller.
Mathematically, the true semi-major axis is given by
$\displaystyle a = \left(\frac{G[M_*+M_{\text{pl}}(i)]}{4\pi^2}\right)^{1/3}T^{2/3}$
Where G is the gravitational constant, and Mpl(i) is the mass of the planet at a given inclination i, and T is the period of the system. It is worth noting that the true semi-major axis is not significantly different from the minimum semi-major axis as long as the mass of the star is much greater than the mass of the planet – which is typically the case.
The fact that the true semi-major axis is a function of the unknown inclination makes for an interesting clarification: The probability that a planet of unknown inclination will transit is not simply given by Rstar/a, but is only approximated by it. If we assume that the distribution of planet masses is uniform (and extending through into the brown dwarf mass regime), then you would expect a planet with a minimum mass equal to Earth to have a much greater chance of being a bona-fide planet than a planet with a minimum-mass of 10 MJ, simply because there is a greater range of inclinations the former planet can be while still remaining in the planetary mass regime. Taking this a step further, even if both the Earth-mass planet candidate and the 10 Jupiter-mass planet candidate have the same orbital period, the probability that the latter planet transits ends up being less than the Earth-mass planet simply because of its high mass. Since its inclination is unknown, the probability that its mass is so high that the true semi-major axis is noticeably larger than the minimum semi-major axis is much higher, resulting in a likely lower transit probability.
Except it turns out that the mass distribution of planets and brown dwarfs isn’t constant. Earth-sized planets are significantly more common than Jupiter-sized planets, and super-Jupiters appear rare. It isn’t clear yet what the mass distribution planets actually is, with significant uncertainty in the sub-Neptune regime, but it is clear that for a highly accurate estimate of the transit probability, the inclination distribution cannot be thought of as completely random as it is fundamentally tied to the planet mass distribution.
Planet Mass Distribution given by Ida & Lin (Left) and Mordasini (Right)
Consider the case of a super-Jovian planet candidate, perhaps with a minimum mass of 7 or 8 Jupiter-masses. Because a significant fraction of physically allowable inclinations would place the true mass planet into a mass regime that is in reality sparsely populated, it is less likely that the planet candidate’s orbit is in those inclinations. It is thus more likely that the planet candidate’s orbit is edge-on than would be expected from the probability function of randomly oriented orbits. As such, the transit probability of a super-Jovian planet is actually boosted by ~20 – 50% over what you would expect from Ptr = Rstar/a. If this is the case, then we would expect to find an excess in the fraction of transiting planets in this mass regime then would be expected purely from the standard transit probability function. Indeed this is what we see.
Candidate planets with masses in the terrestrial planet regime are similarly affected, with broadened transit probabilies owing to the fact that terrestrial planets are more common than higher mass planets, arguing in favour of a higher inclination than the random inclination distribution function.
On the other hand, planet or brown dwarf candidates of minimum masses in the most sparsely populated region of the mass distribution are unlikely to truly have that mass. They are quite likely in orbits with low inclinations and with much higher true masses. The transit probability for companion candidates with minimum masses in this mass regime are actually reduced from the standard transit probability function.
Geometric and a posteriori transit probabilities
In the table above, taken from this preprint, we see that the geometric transit probability, Ptr,0, can be much less than the a posteriori transit probability, Ptr. The transit probability for 55 Cnc e, for example, jumps up from 28% to 36%. With these higher a posteriori transit probabilities, these short-period low-mass planets should be followed-up for transits. If transits are found, it would be of significant benefit to the extrasolar planet field.
In summary, there are various additional effects that can cause the a posteriori transit probability to be significantly different from the geometric transit probability. Planets with only minimum masses known can be more accurately assigned a transit probability when taking into account the uneven planetary mass distribution. Low-mass planets and super-Jupiters are more likely to transit than their geometric transit probability because a significant range of the inclination space is consumed by planets of masses that are simply rare. These planet candidates are more promising targets for transit follow-up than, for example, Jupiter-mass planets or intermediate-mass brown dwarfs.
## The Real Ones
On the last post, we looked at recovering a periodic signal from a radial velocity plot and interpreting it as a planet. Now let’s look at a few of the complications involved in this.
A powerful statistical tool used to get an idea of what kind of periodic signals are in your radial velocity data set is a Lomb–Scargle periodogram (the mathematical details for the interested reader may be found here, but it is sufficiently complex to warrant skipping over in the interests of maintaining reader attention and reasonable post length). In the interests of brevity, further references to the Lomb-Scargle periodogram will be shortened to simply “periodogram.”
The purpose of this periodogram is to give an indication of how likely an arbitrary periodicity is in a data set whose data points need not be equally spaced (as is frequently the case in astronomy for a variety of reasons). Periodicities that are strongly represented in the data are assigned a higher “power,” where periodicities that are not present or only weakly present are given a lower power.
Let’s look at an example using a radial velocity data set for BD-08 2823 (source) If we calculate a periodogram for the data set, we come up with this
BD-08 2823 RV Data Periodogram
The dashed line represents a 0.1% false alarm probability (FAP). A clear, obvious peak is seen at 1 day, 230 days and ~700 days, implying that periodicities of 1 day, 230 days, and ~700 days are present in the data. Creating a one-planet model with a Saturn-mass planet at 238 days produces a nice fit. After subtracting this signal from the data, we’re left with the residuals. Now we may run up a periodogram of the residuals and see what’s left in the data.
BD-08 2823 Periodogram of Residuals
We see three noteworthy things. First and foremost is the emergence of a new peak in the periodogram that was not strongly present before at 5.6 days. We also see that the peak at 1 day remains. Lastly we see that the peak toward 700 days has weakened and moved further out. It would seem to suggest the 700-day signal is perhaps not real, or was an artifact of the 238-day signal.
Why was the 5.6-day signal not present in the first periodogram? The answer may lie in it’s mass: the planet has a mere 14 Earth-masses. It’s RV signal is completely dominated by the Saturn-mass planet. The giant planet forces the shape of the RV diagram and the signal of the second planet is just dragged along, superimposed on the larger signal.
On the radial velocity data plot, the two-planet fit we have come to looks like this:
BD-08 2823 Two-Planet Fit
It is important to realise that the obvious sine curve is not necessarily a bold line, but there is a second periodicity in there going up and down frantically, once every 5.6 days, compared to the Saturn-mass planet, at 237 days.
The fit has a reduced chi-squared of χ2 = 3.2, and a scatter of σO-C = 4.3 m s-1. There’s no obvious structure to the residuals and the scatter is not terribly bad, so any new signals will likely indicate planets of low mass. Let’s check in on the periodogram of the residuals to the two-planet fit and see what may be left in the data.
Periodogram of Residuals to 2-Planet Fit
That signal out toward a thousand days is stubbornly refusing to go away, despite a low χ2. It may either not be real, or it may be indicative of a low-amplitude signal with a rather long period.
Also noteworthy is that the periodicity at one day continues to exist, rather strongly. This periodicity is what’s known as an alias. Because the telescope observes only at night, the observations are roughly evenly spaced – there are (on average) twelve hour gaps between each data point. Therefore a sine curve with a period of 24 hours can be made to fit the data. To illustrate this, consider this (completely made up) data set:
Fake Signal
There’s no doubt that the data is well-fitted by the sine curve, but there is no real evidence that the periodicity proposed by it arises from a real, physical origin. What’s more, a sine curve with half this period could also equally well fit the data. So could a sine curve with a third of this period, and so on. There are mathematically an infinite number of aliases at ever-shortening periods that can be fit to this data.
Generally, if you observe a system with a frequency of $f_o$, and there exists a true signal with a frequency of $f_t$, then aliases will exist at frequencies $f_{t+i} * f_o$, where i is an integer.
Therefore we see that these aliases are caused by the sampling rate. If we could get data between the data points already available, if we could double our observation frequency, we could break this degeneracy. But the problem for telescopes on Earth is that the star is not actually up in the sky more than half the day, and a given portion of the time it is up could be during daylight hours. Therefore the radial velocity data sets of most stars can be plagued with short-period aliases since there is typically a small window of a few hours to observe any given star. It must be noted that as the seasons change and the stars are in different places in the sky at night, that window of availability will shift around a bit, allowing one some leverage in breaking these degeneracies. Ultimately, telescopes in multiple locations around the world (or one in space) would sufficiently break these degeneracies.
A real example of aliases exist in this example from an Alpha Arietis data set. In this case, the alias is not nearly so straightforward. Two signals of periods 0.445 days and 0.571 days can be modelled to fit the data.
Alpha Arietis RV Alias
So which of these two signals correspond to an actual planet? It turns out neither of them do: these radial velocity variations are caused by pulsations on the star – contracting and expansion of the star produces Keplerian-like signals in radial velocity data, too. That’s yet another thing to watch out for. This can be detected with simultaneous photometry of the star. If there is a photometric periodicity that is equivalent to your radial velocity periodicity, avoid claiming a planet at this period as if your academic credibility depends on it.
Additional observations could easily break this degeneracy, provided they are planned at times where the two signals do not overlap.
We see therefore that it is important to keep in mind that a low FAP speaks only to whether or not the signal is real, and not where or what it actually came from. The one-day periodicity is surely present in the data, but it is not of physical origin. It can also be extremely hard to tell whether or not a signal at a given period is actually an alias of another, more real period. There are times when the peak of an alias in the periodogram can be higher than the actual, real period. For reasons that include these, radial velocity fits must be considered fairly preliminary. New data may provide drastic revisions to the orbital periods of proposed planets if signals are exposed to be aliases.
Confusion over aliases have occurred before in literature. HD 156668 b and 55 Cnc e have both had their orbital periods considerably revised after it was realised that their published periods were, in fact, aliases. In the case of 55 Cnc e, the new, de-aliased orbital period ended up being vindicated after transits were detected). The GJ 581 data set, for example, is severely limited by sampling aliases that have spawned controversies over the possible existence of additional planets in that system.
In summary, periodograms are a useful tool to provide the user with a starting point when fitting Keplerian signals to radial velocity data, but they cannot distinguish real signals from aliases. Many observations with a diverse sampling rate are necessary to disentangle aliases from true planetary signals. Ultimately, a cautious approach to fitting signals to radial velocity data works best.
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# Calculus/Integration techniques/Partial Fraction Decomposition
← Integration techniques/Trigonometric Integrals Calculus Integration techniques/Tangent Half Angle → Integration techniques/Partial Fraction Decomposition
Suppose we want to find ${\displaystyle \int {\frac {3x+1}{x^{2}+x}}dx}$ . One way to do this is to simplify the integrand by finding constants ${\displaystyle A}$ and ${\displaystyle B}$ so that
${\displaystyle {\frac {3x+1}{x^{2}+x}}={\frac {3x+1}{x(x+1)}}={\frac {A}{x}}+{\frac {B}{x+1}}}$ .
This can be done by cross multiplying the fraction which gives
${\displaystyle {\frac {3x+1}{x(x+1)}}={\frac {A(x+1)+Bx}{x(x+1)}}}$
As both sides have the same denominator we must have
${\displaystyle 3x+1=A(x+1)+Bx}$
This is an equation for ${\displaystyle x}$ so it must hold whatever value ${\displaystyle x}$ is. If we put in ${\displaystyle x=0}$ we get ${\displaystyle A=1}$ and putting ${\displaystyle x=-1}$ gives ${\displaystyle =-B=-2}$ so ${\displaystyle B=2}$ . So we see that
${\displaystyle {\frac {3x+1}{x^{2}+x}}={\frac {1}{x}}+{\frac {2}{x+1}}}$
Returning to the original integral
${\displaystyle \int {\frac {3x+1}{x^{2}+x}}dx}$ ${\displaystyle =\int {\frac {dx}{x}}+\int {\frac {2}{x+1}}dx}$ ${\displaystyle =\int {\frac {dx}{x}}+2\int {\frac {dx}{x+1}}}$ ${\displaystyle =\ln |x|+2\ln {\Big |}x+1{\Big |}+C}$
Rewriting the integrand as a sum of simpler fractions has allowed us to reduce the initial integral to a sum of simpler integrals. In fact this method works to integrate any rational function.
## Method of Partial FractionsEdit
To decompose the rational function ${\displaystyle {\frac {P(x)}{Q(x)}}}$ :
• Step 1 Use long division (if necessary) to ensure that the degree of ${\displaystyle P(x)}$ is less than the degree of ${\displaystyle Q(x)}$ (see Breaking up a rational function in section
1.1).
• Step 2 Factor Q(x) as far as possible.
• Step 3 Write down the correct form for the partial fraction decomposition (see below) and solve for the constants.
To factor Q(x) we have to write it as a product of linear factors (of the form ${\displaystyle ax+b}$ ) and irreducible quadratic factors (of the form ${\displaystyle ax^{2}+bx+c}$ with ${\displaystyle b^{2}-4ac<0}$ ).
Some of the factors could be repeated. For instance if ${\displaystyle Q(x)=x^{3}-6x^{2}+9x}$ we factor ${\displaystyle Q(x)}$ as
${\displaystyle Q(x)=x(x^{2}-6x+9)=x(x-3)(x-3)=x(x-3)^{2}}$
It is important that in each quadratic factor we have ${\displaystyle b^{2}-4ac<0}$ , otherwise it is possible to factor that quadratic piece further. For example if ${\displaystyle Q(x)=x^{3}-3x^{2}+2x}$ then we can write
${\displaystyle Q(x)=x(x^{2}-3x+2)=x(x-1)(x-2)}$
We will now show how to write ${\displaystyle {\frac {P(x)}{Q(x)}}}$ as a sum of terms of the form
${\displaystyle {\frac {A}{(ax+b)^{k}}}}$ and ${\displaystyle {\frac {Ax+B}{(ax^{2}+bx+c)^{k}}}}$
Exactly how to do this depends on the factorization of ${\displaystyle Q(x)}$ and we now give four cases that can occur.
### Q(x) is a product of linear factors with no repeatsEdit
This means that ${\displaystyle Q(x)=(a_{1}x+b_{1})(a_{2}x+b_{2})\cdots (a_{n}x+b_{n})}$ where no factor is repeated and no factor is a multiple of another.
For each linear term we write down something of the form ${\displaystyle {\frac {A}{(ax+b)}}}$ , so in total we write
${\displaystyle {\frac {P(x)}{Q(x)}}={\frac {A_{1}}{a_{1}x+b_{1}}}+{\frac {A_{2}}{a_{2}x+b_{2}}}+\cdots +{\frac {A_{n}}{a_{n}x+b_{n}}}}$
Example 1 Find ${\displaystyle \int {\frac {1+x^{2}}{(x+3)(x+5)(x+7)}}dx}$ Here we have ${\displaystyle P(x)=1+x^{2}\ ,\ Q(x)=(x+3)(x+5)(x+7)}$ and Q(x) is a product of linear factors. So we write ${\displaystyle {\frac {1+x^{2}}{(x+3)(x+5)(x+7)}}={\frac {A}{x+3}}+{\frac {B}{x+5}}+{\frac {C}{x+7}}}$ Multiply both sides by the denominator ${\displaystyle 1+x^{2}=A(x+5)(x+7)+B(x+3)(x+7)+C(x+3)(x+5)}$ Substitute in three values of x to get three equations for the unknown constants, ${\displaystyle {\begin{matrix}x=-3&1+3^{2}=2\cdot 4A\\x=-5&1+5^{2}=-2\cdot 2B\\x=-7&1+7^{2}=(-4)\cdot (-2)C\end{matrix}}}$ so ${\displaystyle A={\tfrac {5}{4}}\ ,\ B=-{\tfrac {13}{2}}\ ,\ C={\tfrac {25}{4}}}$ , and ${\displaystyle {\frac {1+x^{2}}{(x+3)(x+5)(x+7)}}={\frac {5}{4x+12}}-{\frac {13}{2x+10}}+{\frac {25}{4x+28}}}$ We can now integrate the left hand side. ${\displaystyle \int {\frac {1+x^{2}}{(x+3)(x+5)(x+7)}}dx={\tfrac {5}{4}}\ln {\Big |}x+3{\Big |}-{\tfrac {13}{2}}\ln {\Big |}x+5{\Big |}+{\tfrac {25}{4}}\ln {\Big |}x+7{\Big |}+C}$
#### ExercisesEdit
Evaluate the following by the method partial fraction decomposition.
1. ${\displaystyle \int {\frac {2x+11}{(x+6)(x+5)}}dx}$
${\displaystyle \ln {\Big |}x+6{\Big |}+\ln {\Big |}x+5{\Big |}+C}$
2. ${\displaystyle \int {\frac {7x^{2}-5x+6}{(x-1)(x-3)(x-7)}}dx}$
${\displaystyle {\tfrac {2}{3}}\ln {\Big |}x-1{\Big |}-{\tfrac {27}{4}}\ln {\Big |}x-3{\Big |}+{\tfrac {157}{12}}\ln {\Big |}x-7{\Big |}+C}$
Solutions
### Q(x) is a product of linear factors some of which are repeatedEdit
If ${\displaystyle (ax+b)}$ appears in the factorisation of ${\displaystyle Q(x)}$ k-times then instead of writing the piece ${\displaystyle {\frac {A}{ax+b}}}$ we use the more complicated expression
${\displaystyle {\frac {A_{1}}{ax+b}}+{\frac {A_{2}}{(ax+b)^{2}}}+{\frac {A_{3}}{(ax+b)^{3}}}+\cdots +{\frac {A_{k}}{(ax+b)^{k}}}}$
Example 2 Find ${\displaystyle \int {\frac {dx}{(x+1)(x+2)^{2}}}}$ Here ${\displaystyle P(x)=1}$ and ${\displaystyle Q(x)=(x+1)(x+2)^{2}}$ We write ${\displaystyle {\frac {1}{(x+1)(x+2)^{2}}}={\frac {A}{x+1}}+{\frac {B}{x+2}}+{\frac {C}{(x+2)^{2}}}}$ Multiply both sides by the denominator ${\displaystyle 1=A(x+2)^{2}+B(x+1)(x+2)+C(x+1)}$ Substitute in three values of ${\displaystyle x}$ to get 3 equations for the unknown constants, ${\displaystyle {\begin{matrix}x=0&1=2^{2}A+2B+C\\x=-1&1=A\\x=-2&1=-C\end{matrix}}}$ so ${\displaystyle A=1\ ,\ B=-1\ ,\ C=-1}$ and ${\displaystyle {\frac {1}{(x+1)(x+2)^{2}}}={\frac {1}{x+1}}-{\frac {1}{x+2}}-{\frac {1}{(x+2)^{2}}}}$ We can now integrate the left hand side. ${\displaystyle \int {\frac {dx}{(x+1)(x+2)^{2}}}=\ln \left|{\frac {1}{x+1}}\right|-\ln \left|{\frac {1}{x+2}}\right|+{\frac {1}{x+2}}+C}$ We now simplify the fuction with the property of Logarithms. ${\displaystyle \ln \left|{\frac {1}{x+1}}\right|-\ln \left|{\frac {1}{x+2}}\right|+{\frac {1}{x+2}}+C=\ln \left|{\frac {x+2}{x+1}}\right|+{\frac {1}{x+2}}+C}$
#### ExerciseEdit
3. Evaluate ${\displaystyle \int {\frac {x^{2}-x+2}{x(x+2)^{2}}}dx}$ using the method of partial fractions.
${\displaystyle {\frac {\ln {\Big |}x(x+2){\Big |}}{2}}+{\frac {4}{x+2}}+C}$
Solution
### Q(x) contains some quadratic pieces which are not repeatedEdit
If ${\displaystyle ax^{2}+bx+c}$ appears we use ${\displaystyle {\frac {Ax+B}{ax^{2}+bx+c}}}$ .
#### ExercisesEdit
Evaluate the following using the method of partial fractions.
4. ${\displaystyle \int {\frac {2}{(x+2)(x^{2}+3)}}dx}$
${\displaystyle \ln \left({\sqrt[{7}]{\frac {(x+2)^{2}}{x^{2}+3}}}\right)+{\frac {4\arctan {\Big (}{\tfrac {x}{\sqrt {3}}}{\Big )}}{7{\sqrt {3}}}}+C}$
5. ${\displaystyle \int {\frac {dx}{(x+2)(x^{2}+2)}}}$
${\displaystyle \ln \left({\sqrt[{12}]{\frac {(x+2)^{2}}{x^{2}+2}}}\right)+{\frac {{\sqrt {2}}\arctan {\Big (}{\tfrac {x}{\sqrt {2}}}{\Big )}}{6}}+C}$
Solutions
### Q(x) contains some repeated quadratic factorsEdit
If ${\displaystyle ax^{2}+bx+c}$ appears k-times then use
${\displaystyle {\frac {A_{1}x+B_{1}}{ax^{2}+bx+c}}+{\frac {A_{2}x+B_{2}}{(ax^{2}+bx+c)^{2}}}+{\frac {A_{3}x+B_{3}}{(ax^{2}+bx+c)^{3}}}+\cdots +{\frac {A_{k}x+B_{k}}{(ax^{2}+bx+c)^{k}}}}$
#### ExerciseEdit
Evaluate the following using the method of partial fractions.
6. ${\displaystyle \int {\frac {dx}{(x-1)(x^{2}+1)^{2}}}}$
${\displaystyle {\frac {1-x}{4(x^{2}+1)}}+{\tfrac {1}{8}}\ln \left({\frac {(x-1)^{2}}{x^{2}+1}}\right)-{\frac {\arctan(x)}{2}}+C}$
Solution
← Integration techniques/Trigonometric Integrals Calculus Integration techniques/Tangent Half Angle → Integration techniques/Partial Fraction Decomposition
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17The Laws of Induction
17–1The physics of induction
In the last chapter we described many phenomena which show that the effects of induction are quite complicated and interesting. Now we want to discuss the fundamental principles which govern these effects. We have already defined the emf in a conducting circuit as the total accumulated force on the charges throughout the length of the loop. More specifically, it is the tangential component of the force per unit charge, integrated along the wire once around the circuit. This quantity is equal, therefore, to the total work done on a single charge that travels once around the circuit.
We have also given the “flux rule,” which says that the emf is equal to the rate at which the magnetic flux through such a conducting circuit is changing. Let’s see if we can understand why that might be. First, we’ll consider a case in which the flux changes because a circuit is moved in a steady field.
In Fig. 17–1 we show a simple loop of wire whose dimensions can be changed. The loop has two parts, a fixed U-shaped part (a) and a movable crossbar (b) that can slide along the two legs of the U. There is always a complete circuit, but its area is variable. Suppose we now place the loop in a uniform magnetic field with the plane of the U perpendicular to the field. According to the rule, when the crossbar is moved there should be in the loop an emf that is proportional to the rate of change of the flux through the loop. This emf will cause a current in the loop. We will assume that there is enough resistance in the wire that the currents are small. Then we can neglect any magnetic field from this current.
The flux through the loop is $wLB$, so the “flux rule” would give for the emf—which we write as $\emf$— \begin{equation*} \emf=wB\,\ddt{L}{t}=wBv, \end{equation*} where $v$ is the speed of translation of the crossbar.
Now we should be able to understand this result from the magnetic $\FLPv\times\FLPB$ forces on the charges in the moving crossbar. These charges will feel a force, tangential to the wire, equal to $vB$ per unit charge. It is constant along the length $w$ of the crossbar and zero elsewhere, so the integral is \begin{equation*} \emf=wvB, \end{equation*} which is the same result we got from the rate of change of the flux.
The argument just given can be extended to any case where there is a fixed magnetic field and the wires are moved. One can prove, in general, that for any circuit whose parts move in a fixed magnetic field the emf is the time derivative of the flux, regardless of the shape of the circuit.
On the other hand, what happens if the loop is stationary and the magnetic field is changed? We cannot deduce the answer to this question from the same argument. It was Faraday’s discovery—from experiment—that the “flux rule” is still correct no matter why the flux changes. The force on electric charges is given in complete generality by $\FLPF=q(\FLPE+\FLPv\times\FLPB)$; there are no new special “forces due to changing magnetic fields.” Any forces on charges at rest in a stationary wire come from the $\FLPE$ term. Faraday’s observations led to the discovery that electric and magnetic fields are related by a new law: in a region where the magnetic field is changing with time, electric fields are generated. It is this electric field which drives the electrons around the wire—and so is responsible for the emf in a stationary circuit when there is a changing magnetic flux.
The general law for the electric field associated with a changing magnetic field is $$\label{Eq:II:17:1} \FLPcurl{\FLPE}=-\ddp{\FLPB}{t}.$$ We will call this Faraday’s law. It was discovered by Faraday but was first written in differential form by Maxwell, as one of his equations. Let’s see how this equation gives the “flux rule” for circuits.
Using Stokes’ theorem, this law can be written in integral form as $$\label{Eq:II:17:2} \oint_\Gamma\FLPE\cdot d\FLPs= \int_S(\FLPcurl{\FLPE})\cdot\FLPn\,da= -\int_S\ddp{\FLPB}{t}\cdot\FLPn\,da,$$ \begin{aligned} \oint_\Gamma\FLPE\cdot d\FLPs &= \int_S(\FLPcurl{\FLPE})\cdot\FLPn\,da\\[1ex] &= -\int_S\ddp{\FLPB}{t}\cdot\FLPn\,da, \end{aligned} \label{Eq:II:17:2} where, as usual, $\Gamma$ is any closed curve and $S$ is any surface bounded by it. Here, remember, $\Gamma$ is a mathematical curve fixed in space, and $S$ is a fixed surface. Then the time derivative can be taken outside the integral and we have \begin{aligned} \oint_\Gamma\FLPE\cdot d\FLPs& =-\ddt{}{t}\int_S\FLPB\cdot\FLPn\,da \\[1ex] &=-\ddt{}{t}(\text{flux through S}). \end{aligned} \label{Eq:II:17:3} Applying this relation to a curve $\Gamma$ that follows a fixed circuit of conductor, we get the “flux rule” once again. The integral on the left is the emf, and that on the right is the negative rate of change of the flux linked by the circuit. So Eq. (17.1) applied to a fixed circuit is equivalent to the “flux rule.”
So the “flux rule”—that the emf in a circuit is equal to the rate of change of the magnetic flux through the circuit—applies whether the flux changes because the field changes or because the circuit moves (or both). The two possibilities—“circuit moves” or “field changes”—are not distinguished in the statement of the rule. Yet in our explanation of the rule we have used two completely distinct laws for the two cases—$\FLPv\times\FLPB$ for “circuit moves” and $\FLPcurl{\FLPE}=-\ddpl{\FLPB}{t}$ for “field changes.”
We know of no other place in physics where such a simple and accurate general principle requires for its real understanding an analysis in terms of two different phenomena. Usually such a beautiful generalization is found to stem from a single deep underlying principle. Nevertheless, in this case there does not appear to be any such profound implication. We have to understand the “rule” as the combined effects of two quite separate phenomena.
We must look at the “flux rule” in the following way. In general, the force per unit charge is $\FLPF/q=\FLPE+\FLPv\times\FLPB$. In moving wires there is the force from the second term. Also, there is an $\FLPE$-field if there is somewhere a changing magnetic field. They are independent effects, but the emf around the loop of wire is always equal to the rate of change of magnetic flux through it.
17–2Exceptions to the “flux rule”
We will now give some examples, due in part to Faraday, which show the importance of keeping clearly in mind the distinction between the two effects responsible for induced emf’s. Our examples involve situations to which the “flux rule” cannot be applied—either because there is no wire at all or because the path taken by induced currents moves about within an extended volume of a conductor.
We begin by making an important point: The part of the emf that comes from the $\FLPE$-field does not depend on the existence of a physical wire (as does the $\FLPv\times\FLPB$ part). The $\FLPE$-field can exist in free space, and its line integral around any imaginary line fixed in space is the rate of change of the flux of $\FLPB$ through that line. (Note that this is quite unlike the $\FLPE$-field produced by static charges, for in that case the line integral of $\FLPE$ around a closed loop is always zero.)
Now we will describe a situation in which the flux through a circuit does not change, but there is nevertheless an emf. Figure 17–2 shows a conducting disc which can be rotated on a fixed axis in the presence of a magnetic field. One contact is made to the shaft and another rubs on the outer periphery of the disc. A circuit is completed through a galvanometer. As the disc rotates, the “circuit,” in the sense of the place in space where the currents are, is always the same. But the part of the “circuit” in the disc is in material which is moving. Although the flux through the “circuit” is constant, there is still an emf, as can be observed by the deflection of the galvanometer. Clearly, here is a case where the $\FLPv\times\FLPB$ force in the moving disc gives rise to an emf which cannot be equated to a change of flux.
Now we consider, as an opposite example, a somewhat unusual situation in which the flux through a “circuit” (again in the sense of the place where the current is) changes but where there is no emf. Imagine two metal plates with slightly curved edges, as shown in Fig. 17–3, placed in a uniform magnetic field perpendicular to their surfaces. Each plate is connected to one of the terminals of a galvanometer, as shown. The plates make contact at one point $P$, so there is a complete circuit. If the plates are now rocked through a small angle, the point of contact will move to $P'$. If we imagine the “circuit” to be completed through the plates on the dotted line shown in the figure, the magnetic flux through this circuit changes by a large amount as the plates are rocked back and forth. Yet the rocking can be done with small motions, so that $\FLPv\times\FLPB$ is very small and there is practically no emf. The “flux rule” does not work in this case. It must be applied to circuits in which the material of the circuit remains the same. When the material of the circuit is changing, we must return to the basic laws. The correct physics is always given by the two basic laws \begin{align*} &\FLPF=q(\FLPE+\FLPv\times\FLPB),\\[1ex] &\FLPcurl{\FLPE}=-\ddp{\FLPB}{t}. \end{align*}
17–3Particle acceleration by an induced electric field; the betatron
We have said that the electromotive force generated by a changing magnetic field can exist even without conductors; that is, there can be magnetic induction without wires. We may still imagine an electromotive force around an arbitrary mathematical curve in space. It is defined as the tangential component of $\FLPE$ integrated around the curve. Faraday’s law says that this line integral is equal to minus the rate of change of the magnetic flux through the closed curve, Eq. (17.3).
Fig. 17–4.An electron accelerating in an axially symmetric, increasing magnetic field.
As an example of the effect of such an induced electric field, we want now to consider the motion of an electron in a changing magnetic field. We imagine a magnetic field which, everywhere on a plane, points in a vertical direction, as shown in Fig. 17–4. The magnetic field is produced by an electromagnet, but we will not worry about the details. For our example we will imagine that the magnetic field is symmetric about some axis, i.e., that the strength of the magnetic field will depend only on the distance from the axis. The magnetic field is also varying with time. We now imagine an electron that is moving in this field on a path that is a circle of constant radius with its center at the axis of the field. (We will see later how this motion can be arranged.) Because of the changing magnetic field, there will be an electric field $\FLPE$ tangential to the electron’s orbit which will drive it around the circle. Because of the symmetry, this electric field will have the same value everywhere on the circle. If the electron’s orbit has the radius $r$, the line integral of $\FLPE$ around the orbit is equal to minus the rate of change of the magnetic flux through the circle. The line integral of $\FLPE$ is just its magnitude times the circumference of the circle, $2\pi r$. The magnetic flux must, in general, be obtained from an integral. For the moment, we let $B_{\text{av}}$ represent the average magnetic field in the interior of the circle; then the flux is this average magnetic field times the area of the circle. We will have \begin{equation*} 2\pi rE=\ddt{}{t}(B_{\text{av}}\cdot\pi r^2). \end{equation*}
Since we are assuming $r$ is constant, $E$ is proportional to the time derivative of the average field: $$\label{Eq:II:17:4} E=\frac{r}{2}\,\ddt{B_{\text{av}}}{t}.$$ The electron will feel the electric force $q\FLPE$ and will be accelerated by it. Remembering that the relativistically correct equation of motion is that the rate of change of the momentum is proportional to the force, we have $$\label{Eq:II:17:5} qE=\ddt{p}{t}.$$
For the circular orbit we have assumed, the electric force on the electron is always in the direction of its motion, so its total momentum will be increasing at the rate given by Eq. (17.5). Combining Eqs. (17.5) and (17.4), we may relate the rate of change of momentum to the change of the average magnetic field: $$\label{Eq:II:17:6} \ddt{p}{t}=\frac{qr}{2}\,\ddt{B_{\text{av}}}{t}.$$ Integrating with respect to $t$, we find for the electron’s momentum $$\label{Eq:II:17:7} p=p_0+\frac{qr}{2}\,\Delta B_{\text{av}},$$ where $p_0$ is the momentum with which the electrons start out, and $\Delta B_{\text{av}}$, is the subsequent change in $B_{\text{av}}$. The operation of a betatron—a machine for accelerating electrons to high energies—is based on this idea.
To see how the betatron operates in detail, we must now examine how the electron can be constrained to move on a circle. We have discussed in Chapter 11 of Vol. I the principle involved. If we arrange that there is a magnetic field $\FLPB$ at the orbit of the electron, there will be a transverse force $q\FLPv\times\FLPB$ which, for a suitably chosen $\FLPB$, can cause the electron to keep moving on its assumed orbit. In the betatron this transverse force causes the electron to move in a circular orbit of constant radius. We can find out what the magnetic field at the orbit must be by using again the relativistic equation of motion, but this time, for the transverse component of the force. In the betatron (see Fig. 17–4), $\FLPB$ is at right angles to $\FLPv$, so the transverse force is $qvB$. Thus the force is equal to the rate of change of the transverse component $p_t$ of the momentum: $$\label{Eq:II:17:8} qvB=\ddt{p_t}{t}.$$ When a particle is moving in a circle, the rate of change of its transverse momentum is equal to the magnitude of the total momentum times $\omega$, the angular velocity of rotation (following the arguments of Chapter 11, Vol. I): $$\label{Eq:II:17:9} \ddt{p_t}{t}=\omega p,$$ where, since the motion is circular, $$\label{Eq:II:17:10} \omega=\frac{v}{r}.$$ Setting the magnetic force equal to the transverse acceleration, we have $$\label{Eq:II:17:11} qvB_{\text{orbit}}=p\,\frac{v}{r},$$ where $B_{\text{orbit}}$ is the field at the radius $r$.
As the betatron operates, the momentum of the electron grows in proportion to $B_{\text{av}}$, according to Eq. (17.7), and if the electron is to continue to move in its proper circle, Eq. (17.11) must continue to hold as the momentum of the electron increases. The value of $B_{\text{orbit}}$ must increase in proportion to the momentum $p$. Comparing Eq. (17.11) with Eq. (17.7), which determines $p$, we see that the following relation must hold between $B_{\text{av}}$, the average magnetic field inside the orbit at the radius $r$, and the magnetic field $B_{\text{orbit}}$ at the orbit: $$\label{Eq:II:17:12} \Delta B_{\text{av}}=2\,\Delta B_{\text{orbit}}.$$ The correct operation of a betatron requires that the average magnetic field inside the orbit increases at twice the rate of the magnetic field at the orbit itself. In these circumstances, as the energy of the particle is increased by the induced electric field the magnetic field at the orbit increases at just the rate required to keep the particle moving in a circle.
The betatron is used to accelerate electrons to energies of tens of millions of volts, or even to hundreds of millions of volts. However, it becomes impractical for the acceleration of electrons to energies much higher than a few hundred million volts for several reasons. One of them is the practical difficulty of attaining the required high average value for the magnetic field inside the orbit. Another is that Eq. (17.6) is no longer correct at very high energies because it does not include the loss of energy from the particle due to its radiation of electromagnetic energy (the so-called synchrotron radiation discussed in Chapter 34, Vol. I). For these reasons, the acceleration of electrons to the highest energies—to many billions of electron volts—is accomplished by means of a different kind of machine, called a synchrotron.
We would now like to describe for you an apparent paradox. A paradox is a situation which gives one answer when analyzed one way, and a different answer when analyzed another way, so that we are left in somewhat of a quandary as to actually what should happen. Of course, in physics there are never any real paradoxes because there is only one correct answer; at least we believe that nature will act in only one way (and that is the right way, naturally). So in physics a paradox is only a confusion in our own understanding. Here is our paradox.
Imagine that we construct a device like that shown in Fig. 17–5. There is a thin, circular plastic disc supported on a concentric shaft with excellent bearings, so that it is quite free to rotate. On the disc is a coil of wire in the form of a short solenoid concentric with the axis of rotation. This solenoid carries a steady current $I$ provided by a small battery, also mounted on the disc. Near the edge of the disc and spaced uniformly around its circumference are a number of small metal spheres insulated from each other and from the solenoid by the plastic material of the disc. Each of these small conducting spheres is charged with the same electrostatic charge $Q$. Everything is quite stationary, and the disc is at rest. Suppose now that by some accident—or by prearrangement—the current in the solenoid is interrupted, without, however, any intervention from the outside. So long as the current continued, there was a magnetic flux through the solenoid more or less parallel to the axis of the disc. When the current is interrupted, this flux must go to zero. There will, therefore, be an electric field induced which will circulate around in circles centered at the axis. The charged spheres on the perimeter of the disc will all experience an electric field tangential to the perimeter of the disc. This electric force is in the same sense for all the charges and so will result in a net torque on the disc. From these arguments we would expect that as the current in the solenoid disappears, the disc would begin to rotate. If we knew the moment of inertia of the disc, the current in the solenoid, and the charges on the small spheres, we could compute the resulting angular velocity.
But we could also make a different argument. Using the principle of the conservation of angular momentum, we could say that the angular momentum of the disc with all its equipment is initially zero, and so the angular momentum of the assembly should remain zero. There should be no rotation when the current is stopped. Which argument is correct? Will the disc rotate or will it not? We will leave this question for you to think about.
We should warn you that the correct answer does not depend on any nonessential feature, such as the asymmetric position of a battery, for example. In fact, you can imagine an ideal situation such as the following: The solenoid is made of superconducting wire through which there is a current. After the disc has been carefully placed at rest, the temperature of the solenoid is allowed to rise slowly. When the temperature of the wire reaches the transition temperature between superconductivity and normal conductivity, the current in the solenoid will be brought to zero by the resistance of the wire. The flux will, as before, fall to zero, and there will be an electric field around the axis. We should also warn you that the solution is not easy, nor is it a trick. When you figure it out, you will have discovered an important principle of electromagnetism.
17–5Alternating-current generator
In the remainder of this chapter we apply the principles of Section 17–1 to analyze a number of the phenomena discussed in Chapter 16. We first look in more detail at the alternating-current generator. Such a generator consists basically of a coil of wire rotating in a uniform magnetic field. The same result can also be achieved by a fixed coil in a magnetic field whose direction rotates in the manner described in the last chapter. We will consider only the former case. Suppose we have a circular coil of wire which can be turned on an axis along one of its diameters. Let this coil be located in a uniform magnetic field perpendicular to the axis of rotation, as in Fig. 17–6. We also imagine that the two ends of the coil are brought to external connections through some kind of sliding contacts.
Due to the rotation of the coil, the magnetic flux through it will be changing. The circuit of the coil will therefore have an emf in it. Let $S$ be the area of the coil and $\theta$ the angle between the magnetic field and the normal to the plane of the coil.1 The flux through the coil is then $$\label{Eq:II:17:13} BS\cos\theta.$$ If the coil is rotating at the uniform angular velocity $\omega$, $\theta$ varies with time as $\theta=\omega t$.
Each turn of the coil will have an emf equal to the rate of change of this flux. If the coil has $N$ turns of wire the total emf will be $N$ times larger, so $$\label{Eq:II:17:14} \emf=-N\,\ddt{}{t}(BS\cos\omega t)=NBS\omega\sin\omega t.$$ \begin{align} \emf&=-N\,\ddt{}{t}(BS\cos\omega t)\notag\\[1ex] \label{Eq:II:17:14} &=NBS\omega\sin\omega t. \end{align}
If we bring the wires from the generator to a point some distance from the rotating coil, where the magnetic field is zero, or at least is not varying with time, the curl of $\FLPE$ in this region will be zero and we can define an electric potential. In fact, if there is no current being drawn from the generator, the potential difference $V$ between the two wires will be equal to the emf in the rotating coil. That is, \begin{equation*} V=NBS\omega\sin\omega t=V_0\sin\omega t. \end{equation*} The potential difference between the wires varies as $\sin\omega t$. Such a varying potential difference is called an alternating voltage.
Since there is an electric field between the wires, they must be electrically charged. It is clear that the emf of the generator has pushed some excess charges out to the wire until the electric field from them is strong enough to exactly counterbalance the induction force. Seen from outside the generator, the two wires appear as though they had been electrostatically charged to the potential difference $V$, and as though the charge was being changed with time to give an alternating potential difference. There is also another difference from an electrostatic situation. If we connect the generator to an external circuit that permits passage of a current, we find that the emf does not permit the wires to be discharged but continues to provide charge to the wires as current is drawn from them, attempting to keep the wires always at the same potential difference. If, in fact, the generator is connected in a circuit whose total resistance is $R$, the current through the circuit will be proportional to the emf of the generator and inversely proportional to $R$. Since the emf has a sinusoidal time variation, so also does the current. There is an alternating current \begin{equation*} I=\frac{\emf}{R}=\frac{V_0}{R}\sin\omega t. \end{equation*} The schematic diagram of such a circuit is shown in Fig. 17–7.
We can also see that the emf determines how much energy is supplied by the generator. Each charge in the wire is receiving energy at the rate $\FLPF\cdot\FLPv$, where $\FLPF$ is the force on the charge and $\FLPv$ is its velocity. Now let the number of moving charges per unit length of the wire be $n$; then the power being delivered into any element $ds$ of the wire is \begin{equation*} \FLPF\cdot\FLPv n\,ds. \end{equation*} For a wire, $\FLPv$ is always along $d\FLPs$, so we can rewrite the power as \begin{equation*} nv\FLPF\cdot d\FLPs. \end{equation*} The total power being delivered to the complete circuit is the integral of this expression around the complete loop: $$\label{Eq:II:17:15} \text{Power}=\oint nv\FLPF\cdot d\FLPs.$$ Now remember that $qnv$ is the current $I$, and that the emf is defined as the integral of $F/q$ around the circuit. We get the result $$\label{Eq:II:17:16} \text{Power from a generator}=\emf I.$$
When there is a current in the coil of the generator, there will also be mechanical forces on it. In fact, we know that the torque on the coil is proportional to its magnetic moment, to the magnetic field strength $B$, and to the sine of the angle between. The magnetic moment is the current in the coil times its area. Therefore the torque is $$\label{Eq:II:17:17} \tau=NISB\sin\theta.$$ The rate at which mechanical work must be done to keep the coil rotating is the angular velocity $\omega$ times the torque: $$\label{Eq:II:17:18} \ddt{W}{t}=\omega\tau=\omega NISB\sin\theta.$$ Comparing this equation with Eq. (17.14), we see that the rate of mechanical work required to rotate the coil against the magnetic forces is just equal to $\emf I$, the rate at which electrical energy is delivered by the emf of the generator. All of the mechanical energy used up in the generator appears as electrical energy in the circuit.
As another example of the currents and forces due to an induced emf, let’s analyze what happens in the setup described in Section 17–1, and shown in Fig. 17–1. There are two parallel wires and a sliding crossbar located in a uniform magnetic field perpendicular to the plane of the parallel wires. Now let’s assume that the “bottom” of the U (the left side in the figure) is made of wires of high resistance, while the two side wires are made of a good conductor like copper—then we don’t need to worry about the change of the circuit resistance as the crossbar is moved. As before, the emf in the circuit is $$\label{Eq:II:17:19} \emf=vBw.$$ The current in the circuit is proportional to this emf and inversely proportional to the resistance of the circuit: $$\label{Eq:II:17:20} I=\frac{\emf}{R}=\frac{vBw}{R}.$$
Because of this current there will be a magnetic force on the crossbar that is proportional to its length, to the current in it, and to the magnetic field, such that $$\label{Eq:II:17:21} F=BIw.$$ Taking $I$ from Eq. (17.20), we have for the force $$\label{Eq:II:17:22} F=\frac{B^2w^2}{R}\,v.$$ We see that the force is proportional to the velocity of the crossbar. The direction of the force, as you can easily see, is opposite to its velocity. Such a “velocity-proportional” force, which is like the force of viscosity, is found whenever induced currents are produced by moving conductors in a magnetic field. The examples of eddy currents we gave in the last chapter also produced forces on the conductors proportional to the velocity of the conductor, even though such situations, in general, give a complicated distribution of currents which is difficult to analyze.
It is often convenient in the design of mechanical systems to have damping forces which are proportional to the velocity. Eddy-current forces provide one of the most convenient ways of getting such a velocity-dependent force. An example of the application of such a force is found in the conventional domestic wattmeter. In the wattmeter there is a thin aluminum disc that rotates between the poles of a permanent magnet. This disc is driven by a small electric motor whose torque is proportional to the power being consumed in the electrical circuit of the house. Because of the eddy-current forces in the disc, there is a resistive force proportional to the velocity. In equilibrium, the velocity is therefore proportional to the rate of consumption of electrical energy. By means of a counter attached to the rotating disc, a record is kept of the number of revolutions it makes. This count is an indication of the total energy consumption, i.e., the number of watthours used.
We may also point out that Eq. (17.22) shows that the force from induced currents—that is, any eddy-current force—is inversely proportional to the resistance. The force will be larger, the better the conductivity of the material. The reason, of course, is that an emf produces more current if the resistance is low, and the stronger currents represent greater mechanical forces.
We can also see from our formulas how mechanical energy is converted into electrical energy. As before, the electrical energy supplied to the resistance of the circuit is the product $\emf I$. The rate at which work is done in moving the conducting crossbar is the force on the bar times its velocity. Using Eq. (17.22) for the force, the rate of doing work is \begin{equation*} \ddt{W}{t}=\frac{v^2B^2w^2}{R}. \end{equation*} We see that this is indeed equal to the product $\emf I$ we would get from Eqs. (17.19) and (17.20). Again the mechanical work appears as electrical energy.
17–6Mutual inductance
We now want to consider a situation in which there are fixed coils of wire but changing magnetic fields. When we described the production of magnetic fields by currents, we considered only the case of steady currents. But so long as the currents are changed slowly, the magnetic field will at each instant be nearly the same as the magnetic field of a steady current. We will assume in the discussion of this section that the currents are always varying sufficiently slowly that this is true.
In Fig. 17–8 is shown an arrangement of two coils which demonstrates the basic effects responsible for the operation of a transformer. Coil $1$ consists of a conducting wire wound in the form of a long solenoid. Around this coil—and insulated from it—is wound coil $2$, consisting of a few turns of wire. If now a current is passed through coil $1$, we know that a magnetic field will appear inside it. This magnetic field also passes through coil $2$. As the current in coil $1$ is varied, the magnetic flux will also vary, and there will be an induced emf in coil $2$. We will now calculate this induced emf.
We have seen in Section 13–5 that the magnetic field inside a long solenoid is uniform and has the magnitude $$\label{Eq:II:17:23} B=\frac{1}{\epsO c^2}\,\frac{N_1I_1}{l},$$ where $N_1$ is the number of turns in coil $1$, $I_1$ is the current through it, and $l$ is its length. Let’s say that the cross-sectional area of coil $1$ is $S$; then the flux of $\FLPB$ is its magnitude times $S$. If coil $2$ has $N_2$ turns, this flux links the coil $N_2$ times. Therefore the emf in coil $2$ is given by $$\label{Eq:II:17:24} \emf_2=-N_2S\,\ddt{B}{t}.$$ The only quantity in Eq. (17.23) which varies with time is $I_1$. The emf is therefore given by $$\label{Eq:II:17:25} \emf_2=-\frac{N_1N_2S}{\epsO c^2l}\,\ddt{I_1}{t}.$$
We see that the emf in coil $2$ is proportional to the rate of change of the current in coil $1$. The constant of proportionality, which is basically a geometric factor of the two coils, is called the mutual inductance, and is usually designated $\mutualInd_{21}$. Equation (17.25) is then written $$\label{Eq:II:17:26} \emf_2=\mutualInd_{21}\,\ddt{I_1}{t}.$$
Suppose now that we were to pass a current through coil $2$ and ask about the emf in coil $1$. We would compute the magnetic field, which is everywhere proportional to the current $I_2$. The flux linkage through coil $1$ would depend on the geometry, but would be proportional to the current $I_2$. The emf in coil $1$ would, therefore, again be proportional to $dI_2/dt$: We can write $$\label{Eq:II:17:27} \emf_1=\mutualInd_{12}\,\ddt{I_2}{t}.$$ The computation of $\mutualInd_{12}$ would be more difficult than the computation we have just done for $\mutualInd_{21}$. We will not carry through that computation now, because we will show later in this chapter that $\mutualInd_{12}$ is necessarily equal to $\mutualInd_{21}$.
Since for any coil its field is proportional to its current, the same kind of result would be obtained for any two coils of wire. The equations (17.26) and (17.27) would have the same form; only the constants $\mutualInd_{21}$ and $\mutualInd_{12}$ would be different. Their values would depend on the shapes of the coils and their relative positions.
Suppose that we wish to find the mutual inductance between any two arbitrary coils—for example, those shown in Fig. 17–9. We know that the general expression for the emf in coil $1$ can be written as \begin{equation*} \emf_1=-\ddt{}{t}\int_{(1)}\FLPB\cdot\FLPn\,da, \end{equation*} where $\FLPB$ is the magnetic field and the integral is to be taken over a surface bounded by circuit $1$. We have seen in Section 14–1 that such a surface integral of $\FLPB$ can be related to a line integral of the vector potential. In particular, \begin{equation*} \int_{(1)}\FLPB\cdot\FLPn\,da=\oint_{(1)}\FLPA\cdot d\FLPs_1, \end{equation*} where $\FLPA$ represents the vector potential and $d\FLPs_1$ is an element of circuit $1$. The line integral is to be taken around circuit $1$. The emf in coil $1$ can therefore be written as $$\label{Eq:II:17:28} \emf_1=-\ddt{}{t}\oint_{(1)}\FLPA\cdot d\FLPs_1.$$
Now let’s assume that the vector potential at circuit $1$ comes from currents in circuit $2$. Then it can be written as a line integral around circuit $2$: $$\label{Eq:II:17:29} \FLPA=\frac{1}{4\pi\epsO c^2}\oint_{(2)}\frac{I_2\,d\FLPs_2}{r_{12}},$$ where $I_2$ is the current in circuit $2$, and $r_{12}$ is the distance from the element of the circuit $d\FLPs_2$ to the point on circuit $1$ at which we are evaluating the vector potential. (See Fig. 17–9.) Combining Eqs. (17.28) and (17.29), we can express the emf in circuit $1$ as a double line integral: \begin{equation*} \emf_1=-\frac{1}{4\pi\epsO c^2}\,\ddt{}{t}\oint_{(1)}\oint_{(2)} \frac{I_2\,d\FLPs_2}{r_{12}}\cdot d\FLPs_1. \end{equation*} In this equation the integrals are all taken with respect to stationary circuits. The only variable quantity is the current $I_2$, which does not depend on the variables of integration. We may therefore take it out of the integrals. The emf can then be written as \begin{equation*} \emf_1=\mutualInd_{12}\,\ddt{I_2}{t}, \end{equation*} where the coefficient $\mutualInd_{12}$ is $$\label{Eq:II:17:30} \mutualInd_{12}=-\frac{1}{4\pi\epsO c^2}\oint_{(1)}\oint_{(2)} \frac{d\FLPs_2\cdot d\FLPs_1}{r_{12}}.$$ We see from this integral that $\mutualInd_{12}$ depends only on the circuit geometry. It depends on a kind of average separation of the two circuits, with the average weighted most for parallel segments of the two coils. Our equation can be used for calculating the mutual inductance of any two circuits of arbitrary shape. Also, it shows that the integral for $\mutualInd_{12}$ is identical to the integral for $\mutualInd_{21}$. We have therefore shown that the two coefficients are identical. For a system with only two coils, the coefficients $\mutualInd_{12}$ and $\mutualInd_{21}$ are often represented by the symbol $\mutualInd$ without subscripts, called simply the mutual inductance: \begin{equation*} \mutualInd_{12}=\mutualInd_{21}=\mutualInd. \end{equation*}
17–7Self-inductance
In discussing the induced electromotive forces in the two coils of Figs. 17–8 or 17–9, we have considered only the case in which there was a current in one coil or the other. If there are currents in the two coils simultaneously, the magnetic flux linking either coil will be the sum of the two fluxes which would exist separately, because the law of superposition applies for magnetic fields. The emf in either coil will therefore be proportional not only to the change of the current in the other coil, but also to the change in the current of the coil itself. Thus the total emf in coil $2$ should be written2 $$\label{Eq:II:17:31} \emf_2=\mutualInd_{21}\,\ddt{I_1}{t}+\mutualInd_{22}\,\ddt{I_2}{t}.$$ Similarly, the emf in coil $1$ will depend not only on the changing current in coil $2$, but also on the changing current in itself: $$\label{Eq:II:17:32} \emf_1=\mutualInd_{12}\,\ddt{I_2}{t}+\mutualInd_{11}\,\ddt{I_1}{t}.$$ The coefficients $\mutualInd_{22}$ and $\mutualInd_{11}$ are always negative numbers. It is usual to write $$\label{Eq:II:17:33} \mutualInd_{11}=-\selfInd_1,\quad \mutualInd_{22}=-\selfInd_2,$$ where $\selfInd_1$ and $\selfInd_2$ are called the self-inductances of the two coils.
The self-induced emf will, of course, exist even if we have only one coil. Any coil by itself will have a self-inductance $\selfInd$. The emf will be proportional to the rate of change of the current in it. For a single coil, it is usual to adopt the convention that the emf and the current are considered positive if they are in the same direction. With this convention, we may write for the emf of a single coil $$\label{Eq:II:17:34} \emf=-\selfInd\,\ddt{I}{t}.$$ The negative sign indicates that the emf opposes the change in current—it is often called a “back emf.”
Since any coil has a self-inductance which opposes the change in current, the current in the coil has a kind of inertia. In fact, if we wish to change the current in a coil we must overcome this inertia by connecting the coil to some external voltage source such as a battery or a generator, as shown in the schematic diagram of Fig. 17–10(a). In such a circuit, the current $I$ depends on the voltage $\voltage$ according to the relation $$\label{Eq:II:17:35} \voltage=\selfInd\,\ddt{I}{t}.$$
Fig. 17–10.(a) A circuit with a voltage source and an inductance. (b) An analogous mechanical system.
This equation has the same form as Newton’s law of motion for a particle in one dimension. We can therefore study it by the principle that “the same equations have the same solutions.” Thus, if we make the externally applied voltage $\voltage$ correspond to an externally applied force $F$, and the current $I$ in a coil correspond to the velocity $v$ of a particle, the inductance $\selfInd$ of the coil corresponds to the mass $m$ of the particle.3 See Fig. 17–10(b). We can make the following table of corresponding quantities.
Particle Coil \begin{align*} &F\text{ (force)}\\[.5ex] &v\text{ (velocity)}\\[.5ex] &x\text{ (displacement)} \end{align*} \begin{align*} &\voltage\text{ (potential difference)}\\[.5ex] &I\text{ (current)}\\[.5ex] &q\text{ (charge)} \end{align*} $\displaystyle F=m\,\ddt{v}{t}$ $\displaystyle \voltage=\selfInd\,\ddt{I}{t}$ $mv\text{ (momentum)}$ $\selfInd I$ $\tfrac{1}{2}mv^2\text{ (kinetic energy)}$ $\tfrac{1}{2}\selfInd I^2\text{ (magnetic energy)}$
17–8Inductance and magnetic energy
Continuing with the analogy of the preceding section, we would expect that corresponding to the mechanical momentum $p=mv$, whose rate of change is the applied force, there should be an analogous quantity equal to $\selfInd I$, whose rate of change is $\voltage$. We have no right, of course, to say that $\selfInd I$ is the real momentum of the circuit; in fact, it isn’t. The whole circuit may be standing still and have no momentum. It is only that $\selfInd I$ is analogous to the momentum $mv$ in the sense of satisfying corresponding equations. In the same way, to the kinetic energy $\tfrac{1}{2}mv^2$, there corresponds an analogous quantity $\tfrac{1}{2}\selfInd I^2$. But there we have a surprise. This $\tfrac{1}{2}\selfInd I^2$ is really the energy in the electrical case also. This is because the rate of doing work on the inductance is $\voltage I$, and in the mechanical system it is $Fv$, the corresponding quantity. Therefore, in the case of the energy, the quantities not only correspond mathematically, but also have the same physical meaning as well.
We may see this in more detail as follows. As we found in Eq. (17.16), the rate of electrical work by induced forces is the product of the electromotive force and the current: \begin{equation*} \ddt{W}{t}=\emf I. \end{equation*} Replacing $\emf$ by its expression in terms of the current from Eq. (17.34), we have $$\label{Eq:II:17:36} \ddt{W}{t}=-\selfInd I\,\ddt{I}{t}.$$ Integrating this equation, we find that the energy required from an external source to overcome the emf in the self-inductance while building up the current4 (which must equal the energy stored, $U$) is $$\label{Eq:II:17:37} -W=U=\tfrac{1}{2}\selfInd I^2.$$ Therefore the energy stored in an inductance is $\tfrac{1}{2}\selfInd I^2$.
Applying the same arguments to a pair of coils such as those in Figs. 17–8 or 17–9, we can show that the total electrical energy of the system is given by $$\label{Eq:II:17:38} U=\tfrac{1}{2}\selfInd_1I_1^2+\tfrac{1}{2}\selfInd_2I_2^2+\mutualInd I_1I_2.$$ For, starting with $I=0$ in both coils, we could first turn on the current $I_1$ in coil $1$, with $I_2=0$. The work done is just $\tfrac{1}{2}\selfInd_1I_1^2$. But now, on turning up $I_2$, we not only do the work $\tfrac{1}{2}\selfInd_2I_2^2$ against the emf in circuit $2$, but also an additional amount $\mutualInd I_1I_2$, which is the integral of the emf [$\mutualInd(dI_2/dt)$] in circuit $1$ times the now constant current $I_1$ in that circuit.
Suppose we now wish to find the force between any two coils carrying the currents $I_1$ and $I_2$. We might at first expect that we could use the principle of virtual work, by taking the change in the energy of Eq. (17.38). We must remember, of course, that as we change the relative positions of the coils the only quantity which varies is the mutual inductance $\mutualInd$. We might then write the equation of virtual work as \begin{equation*} -F\,\Delta x=\Delta U=I_1I_2\,\Delta\mutualInd\quad(\text{wrong}). \end{equation*} But this equation is wrong because, as we have seen earlier, it includes only the change in the energy of the two coils and not the change in the energy of the sources which are maintaining the currents $I_1$ and $I_2$ at their constant values. We can now understand that these sources must supply energy against the induced emf’s in the coils as they are moved. If we wish to apply the principle of virtual work correctly, we must also include these energies. As we have seen, however, we may take a short cut and use the principle of virtual work by remembering that the total energy is the negative of what we have called $U_{\text{mech}}$, the “mechanical energy.” We can therefore write for the force $$\label{Eq:II:17:39} -F\,\Delta x=\Delta U_{\text{mech}}=-\Delta U.$$ The force between two coils is then given by \begin{equation*} F\,\Delta x=I_1I_2\,\Delta\mutualInd. \end{equation*}
Equation (17.38) for the energy of a system of two coils can be used to show that an interesting inequality exists between mutual inductance $\mutualInd$ and the self-inductances $\selfInd_1$ and $\selfInd_2$ of the two coils. It is clear that the energy of two coils must be positive. If we begin with zero currents in the coils and increase these currents to some values, we have been adding energy to the system. If not, the currents would spontaneously increase with release of energy to the rest of the world—an unlikely thing to happen! Now our energy equation, Eq. (17.38), can equally well be written in the following form: $$\label{Eq:II:17:40} U=\frac{1}{2}\,\selfInd_1\biggl(I_1+\frac{\mutualInd}{\selfInd_1}\,I_2\biggr)^2+ \frac{1}{2}\biggl(\selfInd_2-\frac{\mutualInd^2}{\selfInd_1}\biggr)I_2^2.$$ \begin{align} U=\;\;\frac{1}{2}\,\selfInd_1&\biggl(I_1+\frac{\mutualInd}{\selfInd_1}\,I_2\biggr)^2\notag\\[.75ex] \label{Eq:II:17:40} +\;\frac{1}{2}&\biggl(\selfInd_2-\frac{\mutualInd^2}{\selfInd_1}\biggr)I_2^2. \end{align} That is just an algebraic transformation. This quantity must always be positive for any values of $I_1$ and $I_2$. In particular, it must be positive if $I_2$ should happen to have the special value $$\label{Eq:II:17:41} I_2=-\frac{\selfInd_1}{\mutualInd}\,I_1.$$ But with this current for $I_2$, the first term in Eq. (17.40) is zero. If the energy is to be positive, the last term in (17.40) must be greater than zero. We have the requirement that \begin{equation*} \selfInd_1\selfInd_2>\mutualInd^2. \end{equation*} We have thus proved the general result that the magnitude of the mutual inductance $\mutualInd$ of any two coils is necessarily less than or equal to the geometric mean of the two self-inductances. ($\mutualInd$ itself may be positive or negative, depending on the sign conventions for the currents $I_1$ and $I_2$.) $$\label{Eq:II:17:42} \abs{\mutualInd}<\sqrt{\selfInd_1\selfInd_2}.$$
The relation between $\mutualInd$ and the self-inductances is usually written as $$\label{Eq:II:17:43} \mutualInd=k\sqrt{\selfInd_1\selfInd_2}.$$ The constant $k$ is called the coefficient of coupling. If most of the flux from one coil links the other coil, the coefficient of coupling is near one; we say the coils are “tightly coupled.” If the coils are far apart or otherwise arranged so that there is very little mutual flux linkage, the coefficient of coupling is near zero and the mutual inductance is very small.
For calculating the mutual inductance of two coils, we have given in Eq. (17.30) a formula which is a double line integral around the two circuits. We might think that the same formula could be used to get the self-inductance of a single coil by carrying out both line integrals around the same coil. This, however, will not work, because the denominator $r_{12}$ of the integrand will go to zero when the two line elements $d\FLPs_1$ and $d\FLPs_2$ are at the same point on the coil. The self-inductance obtained from this formula is infinite. The reason is that this formula is an approximation that is valid only when the cross sections of the wires of the two circuits are small compared with the distance from one circuit to the other. Clearly, this approximation doesn’t hold for a single coil. It is, in fact, true that the inductance of a single coil tends logarithmically to infinity as the diameter of its wire is made smaller and smaller.
We must, then, look for a different way of calculating the self-inductance of a single coil. It is necessary to take into account the distribution of the currents within the wires because the size of the wire is an important parameter. We should therefore ask not what is the inductance of a “circuit,” but what is the inductance of a distribution of conductors. Perhaps the easiest way to find this inductance is to make use of the magnetic energy. We found earlier, in Section 15–3, an expression for the magnetic energy of a distribution of stationary currents: $$\label{Eq:II:17:44} U=\tfrac{1}{2}\int\FLPj\cdot\FLPA\,dV.$$ If we know the distribution of current density $\FLPj$, we can compute the vector potential $\FLPA$ and then evaluate the integral of Eq. (17.44) to get the energy. This energy is equal to the magnetic energy of the self-inductance, $\tfrac{1}{2}\selfInd I^2$. Equating the two gives us a formula for the inductance: $$\label{Eq:II:17:45} \selfInd=\frac{1}{I^2}\int\FLPj\cdot\FLPA\,dV.$$ We expect, of course, that the inductance is a number depending only on the geometry of the circuit and not on the current $I$ in the circuit. The formula of Eq. (17.45) will indeed give such a result, because the integral in this equation is proportional to the square of the current—the current appears once through $\FLPj$ and again through the vector potential $\FLPA$. The integral divided by $I^2$ will depend on the geometry of the circuit but not on the current $I$.
Equation (17.44) for the energy of a current distribution can be put in a quite different form which is sometimes more convenient for calculation. Also, as we will see later, it is a form that is important because it is more generally valid. In the energy equation, Eq. (17.44), both $\FLPA$ and $\FLPj$ can be related to $\FLPB$, so we can hope to express the energy in terms of the magnetic field—just as we were able to relate the electrostatic energy to the electric field. We begin by replacing $\FLPj$ by $\epsO c^2\FLPcurl{\FLPB}$. We cannot replace $\FLPA$ so easily, since $\FLPB=\FLPcurl{\FLPA}$ cannot be reversed to give $\FLPA$ in terms of $\FLPB$. Anyway, we can write $$\label{Eq:II:17:46} U=\frac{\epsO c^2}{2}\int(\FLPcurl{\FLPB})\cdot\FLPA\,dV.$$
The interesting thing is that—with some restrictions—this integral can be written as $$\label{Eq:II:17:47} U=\frac{\epsO c^2}{2}\int\FLPB\cdot(\FLPcurl{\FLPA})\,dV.$$ To see this, we write out in detail a typical term. Suppose that we take the term $(\FLPcurl{\FLPB})_zA_z$ which occurs in the integral of Eq. (17.46). Writing out the components, we get \begin{equation*} \int\biggl(\ddp{B_y}{x}-\ddp{B_x}{y}\biggr)A_z\,dx\,dy\,dz. \end{equation*} (There are, of course, two more integrals of the same kind.) We now integrate the first term with respect to $x$—integrating by parts. That is, we can say \begin{equation*} \int\ddp{B_y}{x}\,A_z\,dx=B_yA_z-\int B_y\,\ddp{A_z}{x}\,dx. \end{equation*} Now suppose that our system—meaning the sources and fields—is finite, so that as we go to large distances all fields go to zero. Then if the integrals are carried out over all space, evaluating the term $B_yA_z$ at the limits will give zero. We have left only the term with $B_y(\ddpl{A_z}{x})$, which is evidently one part of $B_y(\FLPcurl{\FLPA})_y$ and, therefore, of $\FLPB\cdot(\FLPcurl{\FLPA})$. If you work out the other five terms, you will see that Eq. (17.47) is indeed equivalent to Eq. (17.46).
But now we can replace $(\FLPcurl{\FLPA})$ by $\FLPB$, to get $$\label{Eq:II:17:48} U=\frac{\epsO c^2}{2}\int\FLPB\cdot\FLPB\,dV.$$ We have expressed the energy of a magnetostatic situation in terms of the magnetic field only. The expression corresponds closely to the formula we found for the electrostatic energy: $$\label{Eq:II:17:49} U=\frac{\epsO}{2}\int\FLPE\cdot\FLPE\,dV.$$
One reason for emphasizing these two energy formulas is that sometimes they are more convenient to use. More important, it turns out that for dynamic fields (when $\FLPE$ and $\FLPB$ are changing with time) the two expressions (17.48) and (17.49) remain true, whereas the other formulas we have given for electric or magnetic energies are no longer correct—they hold only for static fields.
If we know the magnetic field $\FLPB$ of a single coil, we can find the self-inductance by equating the energy expression (17.48) to $\tfrac{1}{2}\selfInd I^2$. Let’s see how this works by finding the self-inductance of a long solenoid. We have seen earlier that the magnetic field inside a solenoid is uniform and $\FLPB$ outside is zero. The magnitude of the field inside is $B=nI/\epsO c^2$, where $n$ is the number of turns per unit length in the winding and $I$ is the current. If the radius of the coil is $r$ and its length is $L$ (we take $L$ very long, so that we can neglect end effects, i.e., $L\gg r$), the volume inside is $\pi r^2L$. The magnetic energy is therefore \begin{equation*} U=\frac{\epsO c^2}{2}\,B^2\cdot(\text{Vol})=\frac{n^2I^2}{2\epsO c^2}\, \pi r^2L, \end{equation*} which is equal to $\tfrac{1}{2}\selfInd I^2$. Or, $$\label{Eq:II:17:50} \selfInd=\frac{\pi r^2n^2}{\epsO c^2}\,L.$$
1. Now that we are using the letter $A$ for the vector potential, we prefer to let $S$ stand for a surface area.
2. The sign of $\mutualInd_{12}$ and $\mutualInd_{21}$ in Eqs. (17.31) and (17.32) depends on the arbitrary choices for the sense of a positive current in the two coils.
3. This is, incidentally, not the only way a correspondence can be set up between mechanical and electrical quantities.
4. We are neglecting any energy loss to heat from the current in the resistance of the coil. Such losses require additional energy from the source but do not change the energy which goes into the inductance.
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# 8.5: Appendix 4 - Hybrid Orbitals
## Linear alignment with two neighbors (sp hybridization)
Consider three atoms in a line, as shown in Figure $$\PageIndex{1}$$. Arbitrarily we align the atoms with the x-axis.
We wish to determine the contribution of the central atom's orbitals to $$\sigma$$ bonds. Recall that $$\sigma$$ bonds have electron density on the axis between the atoms.
Now, if the basis set consists of s and p orbitals, only $$s$$ and $$p_{x}$$ orbitals can contribute to $$\sigma$$ bonds on the x-axis. $$p_{y}$$ and $$p_{z}$$ orbitals have zero density on the x-axis and therefore cannot contribute to the $$\sigma$$ bonds. They may contribute to $$\pi$$ bonds however.
Let's define the symmetry adapted atomic orbitals that contribute to $$\sigma$$ bonds generally as:
$\phi_{\sigma}=c_{s}\phi_{s}+c_{p_{z}}\phi_{p_{z}} \label{8.5.1}$
where $$c_{s}$$ and $$c_{p_{x}}$$ are the weighting coefficients for the $$s$$ and $$p_{x}$$ orbitals respectively. There are two $$\sigma$$ bonds: one to the left, and one to the right. We'll define the two symmetry adapted atomic orbitals that contribute to these $$\sigma$$ bonds as $$\phi_{\sigma}^{1}$$ and $$\phi_{\sigma}^{2}$$ respectively.
The s orbital contributes equally to both symmetry adapted atomic orbitals. i.e.
$|c_{s}|^{2} = \frac{1}{2},\ \ \ c_{s} =\frac{1}{\sqrt{2}} \label{8.5.2}$
Since the $$p_{x}$$ orbital is aligned with the x-axis, we can weight the $$p_{x}$$ orbital components by the coordinates of the two neighboring atoms at x = +1 and x = -1,
$\phi_{\sigma}^{1} = c_{s}\phi_{s} + c_{p}\phi_{p_{x}} \\ \phi_{\sigma}^{2} = c_{s}\phi_{s} - c_{p}\phi_{p_{x}} \label{8.5.3}$
In the first orbital, we are adding the s and $$p_{x}$$ orbitals in-phase. Consequently, we have maximum electron density in the positive x-direction. In the second, we are adding the s and $$p_{x}$$ orbitals out of phase, yielding a maximum electron density in the negative x-direction.
Normalizing each orbital gives
$|c_{p}|^{2} = \frac{1}{2},\ \ \ c_{p} =\sqrt{\frac{1}{2}} \label{8.5.4}$
Thus, the first symmetry adapted atomic orbital is
$\phi_{\sigma}^{1} = \frac{1}{\sqrt{2}}(\phi_{s}+\phi_{p_{x}}) \label{8.5.5}$
Similarly, the second symmetry adapted atomic orbital is
$\phi_{\sigma}^{2} = \frac{1}{\sqrt{2}}(\phi_{s}-\phi_{p_{x}}) \label{8.5.6}.$
Thus, based purely on symmetry arguments, in a linear chain of atoms it is convenient to re-express the four atomic orbitals s, $$p_{x}$$, $$p_{y}$$ and $$p_{z}$$, as
$\phi_{\sigma} = \frac{1}{\sqrt{2}}(\phi_{s} \pm\phi_{p_{x}}) \label{8.5.7},$
where $$p_{y}$$ and $$p_{z}$$ remain unaffected. This is known as sp hybridization since we have combined one s atomic orbital, and one p atomic orbital to create two atomic orbitals that contribute to $$\sigma$$ bonds.
The remaining $$p_{y}$$ and $$p_{z}$$ atomic orbitals may combine in molecular orbitals with higher energy. The highest occupied molecular orbital (HOMO) is also known as the frontier molecular orbital. In an sp-hybridized material, the frontier molecular orbital will be a linear combination of $$p_{y}$$ and $$p_{z}$$ atomic orbitals. The frontier molecular orbital is relevant to us, because it more likely than deeper levels to be partially filled. Consequently, conduction is more likely to occur through the HOMO than deeper orbitals.
Now, $$\sigma$$ bonds possess electron densities localized between atoms. But $$\pi$$ bonds composed of linear combinations of p orbitals can be delocalized along a chain or sheet of atoms. Thus, if the HOMO is a $$\pi$$ bond, it is much easier to push an electron through it; we'll see some examples of this in the next section.
## Planar alignment with three neighbors ($$sp^{2}$$ hybridization)
Consider a central atom with three equispaced neighbors at the points of a triangle; as shown in Figure $$\PageIndex{5}$$. Arbitrarily we align the atoms on the x-y plane.
Once again, we wish to determine the contribution of the central atom's orbitals to $$\sigma$$ bonds. If the basis set consists of s and p orbitals, only s, $$p_{x}$$ and $$p_{y}$$ atomic orbitals can contribute to $$\sigma$$ bonds in the x-y plane. $$p_{z}$$ orbitals can only contribute to $$\pi$$ bonds.
Let's define the symmetry adapted atomic orbitals that contribute individually to $$\sigma$$ bonds generally as:
$\phi_{\sigma} = c_{s}\phi_{s} + c_{p_{x}}\phi_{p_{x}} + c_{p_{y}}\phi_{p_{y}} \label{8.5.8}$
The s orbital contributes equally to all three symmetry adapted atomic orbitals. i.e.
$|c_{s}|^{2} = \frac{1}{3},\ \ \ c_{s} = \frac{1}{\sqrt{3}} \label{8.5.9}$
Since the $$p_{x}$$ orbital is aligned with the x-axis, and $$p_{y}$$ with the y-axis, we can weight the p orbital components by the coordinates of the triangle of neighboring atoms
$\phi_{\sigma}^{1} = c_{s}\phi_{s} + c_{p} \left( +1\phi_{p_{x}} + 0\phi_{p_{y}} \right) \\ \phi_{\sigma}^{2} = c_{s}\phi_{s} + c_{p} \left( -\frac{1}{2} \phi_{p_{x}} + \frac{\sqrt{3}}{2} \phi_{p_{y}} \right) \\ \phi_{\sigma}^{3} = c_{s}\phi_{s} + c_{p} \left( -\frac{1}{2} \phi_{p_{x}} - \frac{\sqrt{3}}{2} \phi_{p_{y}} \right) \label{8.5.10}$
Normalizing each orbital gives
$|c_{p}|^{2} = \frac{2}{3},\ \ \ c_{p} = \sqrt{\frac{2}{3}} \label{8.5.11}$
Thus,
$\phi_{\sigma}^{1} = \frac{1}{\sqrt{3}} \phi_{s} + \sqrt{\frac{2}{3}} \phi_{p_{x}} + 0 \phi_{p_{y}} \\ \phi_{\sigma}^{2} = \frac{1}{\sqrt{3}} \phi_{s} - \frac{1}{\sqrt{6}} \phi_{p_{x}} + \frac{1}{\sqrt{2}} \phi_{p_{y}} \\ \phi_{\sigma}^{2} = \frac{1}{\sqrt{3}} \phi_{s} - \frac{1}{\sqrt{6}} \phi_{p_{x}} - \frac{1}{\sqrt{2}} \phi_{p_{y}} \nonumber$
This is known as $$sp^{2}$$ hybridization since we have combined one s atomic orbital, and two p atomic orbitals to create three atomic orbitals that contribute individually to $$\sigma$$ bonds. The bond angle is 120º.
The remaining $$p_{z}$$ atomic orbitals will contribute to the frontier molecular orbitals of an $$sp^{2}$$ hybridized material; see for example ethene in Figure $$\PageIndex{7}$$.
As in the sp hybridized case, electrons in these $$\pi$$ molecular orbitals may be delocalized. If electrons are delocalized over several neighboring atoms, then the molecule is said to be conjugated. Another $$sp^{2}$$ hybridized material was shown in Figure 6.3.1. This is 1,3- butadiene, a chain of $$4 \times sp^{2}$$ hybridized carbon atoms. Note the extensive electron delocalization in the $$\pi$$ bonds.
Some archetypal conjugated carbon-based molecules are shown in Figure $$\PageIndex{8}$$. In each material the carbon atoms are $$sp^{2}$$ hybridized (surrounded by three neighbors at points of an equilateral triangle). Note that another typical characteristic of $$sp^{2}$$ hybridized materials is alternating single and double bonds.
## Tetrahedral alignment with four neighbors ($$sp^{3}$$ hybridization)
Consider a central atom with four equispaced neighbors. Repulsion between these atoms will push them to the points of a tetrahedron; see Figure $$\PageIndex{9}$$.
Now, all atomic orbitals will contribute to $$\sigma$$ bonds. There are no $$\pi$$ bonds.
Let's define the symmetry adapted atomic orbitals that contribute individually to $$\sigma$$ bonds generally as:
$\phi_{\sigma} = c_{s}\phi_{s} + c_{p_{x}}\phi_{p_{x}} + c_{p_{y}}\phi_{p_{y}} + c_{p_{z}}\phi_{p_{z}} \label{8.5.13}$
Once again, the s orbital contributes equally to all four symmetry adapted atomic orbitals. i.e.
$|c_{s}|^{2} = \frac{1}{4},\ \ \ c_{s} = \frac{1}{2} \label{8.5.14}$
Since the $$p_{x}$$ orbital is aligned with the x-axis, $$p_{y}$$ with the y-axis and $$p_{z}$$ with the z-axis, we can weight the p orbital components by the coordinates of the triangle of neighboring atoms
$\phi_{\sigma}^{1} = c_{s}\phi_{s} + (+1\phi_{p_{x}}\ +1\phi_{p_{y}}\ +1\phi_{p_{x}}) \\ \phi_{\sigma}^{2} = c_{s}\phi_{s} + (-1\phi_{p_{x}}\ +1\phi_{p_{y}}\ -1\phi_{p_{x}}) \\ \phi_{\sigma}^{3} = c_{s}\phi_{s} + (+1\phi_{p_{x}}\ -1\phi_{p_{y}}\ -1\phi_{p_{x}}) \\ \phi_{\sigma}^{1} = c_{s}\phi_{s} + (-1\phi_{p_{x}}\ -1\phi_{p_{y}}\ -1\phi_{p_{x}}) \label{8.5.15}$
Normalizing each orbital gives
$|c_{p}|^{2} = \frac{1}{4},\ \ \ c_{p} = \frac{1}{2} \label{8.5.16}$
Thus,
$\phi_{\sigma}^{1} = \frac{1}{2}(\phi_{s} + \phi_{p_{x}} + \phi_{p_{y}} + \phi_{p_{z}}) \\ \phi_{\sigma}^{2} = \frac{1}{2}(\phi_{s} - \phi_{p_{x}} + \phi_{p_{y}} - \phi_{p_{z}}) \\ \phi_{\sigma}^{3} = \frac{1}{2}(\phi_{s} + \phi_{p_{x}} - \phi_{p_{y}} - \phi_{p_{z}}) \\ \phi_{\sigma}^{4} = \frac{1}{2}(\phi_{s} - \phi_{p_{x}} - \phi_{p_{y}} - \phi_{p_{z}}) \label{8.5.17}$
This is known as $$sp^{3}$$ hybridization since we have combined one s atomic orbital, and three p atomic orbitals to create four possible atomic orbitals that contribute individually to $$\sigma$$ bonds. The bond angle is 109.5º.
8.5: Appendix 4 - Hybrid Orbitals is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.
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# Math Help - Limit wit compliacated (looking) intergrals
1. ## Limit wit compliacated (looking) intergrals
$\lim_{\displaystyle x \to 0+} \frac{\displaystyle\int_{0}^{x}\left(\int_{0}^{t}\ sqrt{\displaystyle1+z^4}\mathrm{d}z\right) \mathrm{d}t}{\displaystyle\int_{0}^{x}\left(\int_{ 0}^{t}\sqrt{\displaystyle1+z^6}\mathrm{d}z\right) \mathrm{d}t}$
Any help would be appreciated!
2. You could try several approaches, I think. One is to use l"Hospital's rule. Another is to reverse the order of integration in both numerator and denominator. You should be able to get traction one way or the other. How does that strike you?
3. Thank you for your help.
I am afraid, I still can't solve the problem
I uderstand that $\lim_{\displaystyle x \to 0+}\displaystyle\int_{0}^{x}\left(\int_{0}^{t}\sqr t{\displaystyle1+z^4}\mathrm{d}z\right) \mathrm{d}t=\lim_{\displaystyle x \to 0+}\displaystyle\int_{0}^{x}\left(\int_{0}^{t}\sqr t{\displaystyle1+z^6}\mathrm{d}z\right) \mathrm{d}t=0$
so we can use L'Hospital's rule. But I don't know how to derive these expressions (with respect to z?).
So I need a little more help!
Thank you!
4. $F(x) = \int_0^x {\int_0^t {\sqrt {1 + z^4 } dz} dt} \; \Rightarrow \;F'(x) = \int_0^x {\sqrt {1 + z^4 } dz} \; \Rightarrow \;F''(x) = \sqrt {1 + x^4 }$
5. Originally Posted by doug
$\lim_{\displaystyle x \to 0+} \frac{\displaystyle\int_{0}^{x}\left(\int_{0}^{t}\ sqrt{\displaystyle1+z^4}\mathrm{d}z\right) \mathrm{d}t}{\displaystyle\int_{0}^{x}\left(\int_{ 0}^{t}\sqrt{\displaystyle1+z^6}\mathrm{d}z\right) \mathrm{d}t}$ Any help would be appreciated!
$\displaystyle \lim_{x \rightarrow 0+} \frac{\int_{0}^{x} \int_{0}^{t} \sqrt{1+z^{4}}\ dz\ dt}{\int_{0}^{x} \int_{0}^{t} \sqrt{1+z^{6}}\ dz\ dt} = \lim_{x \rightarrow 0+} \frac{\int_{0}^{x} \sqrt{1+z^{4}}\ dz} {\int_{0}^{x} \sqrt{1+z^{6}}\ dz}=$
$\displaystyle = \lim_{x \rightarrow 0+} \frac{\sqrt{1+x^{4}}}{\sqrt{1+x^{6}}} = 1$
$\chi$ $\sigma$
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# Math Help - Word Problem
1. ## Word Problem
Jack, Kay, and Lynn deliver flyers. If alone Jack takes 4 hours, and Lynn takes 1 hour longer than Kay. Together they can deliver in 40% of the time it takes Kay working alone. *How long does it take Kay to deliver all the flyers alone?
2. Originally Posted by Bsauer05
Jack, Kay, and Lynn deliver flyers. If alone Jack takes 4 hours, and Lynn takes 1 hour longer than Kay. Together they can deliver in 40% of the time it takes Kay working alone. *How long does it take Kay to deliver all the flyers alone?
jack's rate = (1 job)/(4 hrs)
kay's rate = (1 job)/(k hrs)
lynn's rate = (1 job)/(k+1 hrs)
$\left(\frac{1}{4} + \frac{1}{k} + \frac{1}{k+1}\right) = \frac{2}{5k}$
solve for k
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# Math Help - Equivalence of terms
1. ## Equivalence of terms
We have
A = ∀x¬P(x) ⊻ ∀yQ(y)
B = ∃xP(x) ⇔ ∀yQ(y)
Are A and ¬B equal? I get that they are.
Is it correct? I have solved it like this:
First we negate the B:
∃xP(x) ⇔ ∀yQ(y)
¬∃xP(x) ⇔ ¬∀yQ(y)
then we get
∀x¬P(x) ⇔ ∃y¬Q(y)
And then we insert something. (for example):
P(x) x is divisible by 2
Q(y) y is not divisible by 2
For both A and not B i get zero. So they are equal.
Is my thinking correct?
2. First, A and ~B are not equal, since they are not the same formulas. A formula is a string of symbols and formulas are equal if and only if they are the same string of symbols. However, they might be logically EQUIVALENT.
So the question is whether A and ~B are logically equivalent. (Should I surmise that the connective in your A is a typo and is supposed to be the "if and only if" sign just as is the connective in B?)
Second, you did not properly negate B. Indeed, what you thought was negating B is just an equivalent (so very much not a negation) of B.
Third, you don't show equivalence by using a particular interpretation of the predicates, such as you interpreted P and Q. You can use particular interpretations to prove the invalidy of a formula (here the formula in question is A <-> ~B), but not of the validty of a formula, since validity depends on the formula being true with ANY interpretation of the predicates.
Those three points suggest to me that you need a very thorough review of the fundamentals of symbolic logic.
3. OK, thank you for this introduction.
How would I properly negate B?
So now I know what do I have to proof:
∀x¬P(x) ⊻ ∀yQ(y) ⇔ ¬(∃xP(x) ⇔ ∀yQ(y))
Sorry if I don't know, but I don't have any book for predicates. Do you know a website with examples and some introdoction for predicate.
4. The negation of any formula P is ~P, so since B is ExPx <-> AyQy, the negation of B is ~(ExPx <-> AyQy).
What does the symbol
stand for?
/
For a good book, get "Logic: Techniques Of Formal Reasoning" by Kalish, Montague, and Mar?
Are you taking a course that has no textbook specified?
5. Originally Posted by MoeBlee
What does the symbol
⊻ stand for?
Usually $\underline \vee$ is use for the exclusive or.
6. Ah, I hadn't see that one before. Thanks.
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numerically approximate the real roots of an expression using Newton's method - Maple Help
Home : Support : Online Help : Education : Student Package : Numerical Analysis : Visualization : Student/NumericalAnalysis/Newton
Student[NumericalAnalysis][Newton] - numerically approximate the real roots of an expression using Newton's method
Calling Sequence Newton(f, x=a, opts) Newton(f, a, opts)
Parameters
f - algebraic; expression in the variable x representing a continuous function x - name; the independent variable of f a - numeric; the initial approximate root opts - (optional) equation(s) of the form keyword=value, where keyword is one of fixedpointiterator, functionoptions, lineoptions, maxiterations, output, pointoptions, showfunction, showlines, showpoints, showverticalline, stoppingcriterion, tickmarks, caption, tolerance, verticallineoptions, view; the options for approximating the roots of f
Description
• The Newton command numerically approximates the roots of an algebraic function, f, using the classical Newton-Raphson method.
• Given an expression f and an initial approximate a, the Newton command computes a sequence ${p}_{k}$, $k$=$0..n$, of approximations to a root of f, where $n$ is the number of iterations taken to reach a stopping criterion. For sufficiently well-behaved functions and sufficiently good initial approximations, the convergence of ${p}_{k}$ toward the exact root is quadratic.
• The Newton command is a shortcut for calling the Roots command with the method=newton option.
Notes
• Newton's method will fail if $\frac{\partial }{\partial x}f\left({p}_{k-1}\right)$=0.$\frac{ⅆ}{ⅆx}f\left({p}_{k-1}\right)=0$
Examples
> $\mathrm{with}\left(\mathrm{Student}[\mathrm{NumericalAnalysis}]\right):$
> $f:={ⅇ}^{x}+{2}^{-x}+2\mathrm{cos}\left(x\right)-6:$
> $\mathrm{Newton}\left(f,x=2.0,\mathrm{tolerance}={10}^{-2}\right)$
${1.829383715}$ (1)
> $\mathrm{Newton}\left(f,x=2.0,\mathrm{tolerance}={10}^{-2},\mathrm{output}=\mathrm{sequence}\right)$
${2.0}{,}{1.850521336}{,}{1.829751202}{,}{1.829383715}$ (2)
> $\mathrm{Newton}\left(f,x=2,\mathrm{output}=\mathrm{plot},\mathrm{stoppingcriterion}=\mathrm{function_value}\right)$
To play the following animation in this help page, right-click (Control-click, on Macintosh) the plot to display the context menu. Select Animation > Play.
> $\mathrm{Newton}\left(f,x=1.3,\mathrm{output}=\mathrm{animation},\mathrm{stoppingcriterion}=\mathrm{absolute}\right)$
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# Suppose a Firm Uses Labor and Capital to Produce Output
Question 314
Multiple Choice
Question 314
Multiple Choice
## Suppose a firm uses labor and capital to produce output.The last unit of labor hired has a marginal product of 12 units of output, and the last unit of capital employed has a marginal product of 20 units.Use the optimal combination of inputs rule to calculate the price of capital if the price of labor is $6 per unit.The price of capital is A)$2.
B)$10. C)$20.
D)impossible to determine with the information given.
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I Minimum deviation in prism spectroscope
Tags:
1. May 6, 2017
crick
Suppose that I use a prism (vertex angle $\alpha$) spectroscope to analyze a beam of visible light from a mercury lamp (different wavelenghts) and I want the determine the refraction index of the prism using the minmum deviation angles $D_{min}$
$$n(\lambda)=\frac{\mathrm{sin}(\frac{D_{min}(\lambda)+\alpha }{2})}{\mathrm{sin}\frac{\alpha}{2}}$$
$D_{min}$ depends on $\lambda$ and so does $n$.
Nevertheless suppose that, even if I want to find experimentally $D_{min}$ for all the spectral lines that I see in the spectroscope, I want to do the setting of the prism for minmum deviation condition only for the two extreme ones, say a violet and a red and then try to extimate the values of $D_{min}$ for other wavelenghts from these two. (I know that this is not theoretically correct).
Therefore I set the prism in the condition of minimum deviation for red and then I measure the angle of deviation $D$ for all the other $\lambda$s (i'll take, for example, the yellow one) and then I repeat the same procedure but setting the condition of minimum deviation for violet.
My question is: should I expect that $$D_{yellow_{\mathrm{minmum \, deviation \, condition \,for \,RED}}}<D_{yellow_{\mathrm{minmum \, deviation \, condition \,for \,VIOLET}}}$$
or the opposite? Or nothing can be said?
More generally how does $D_{yellow}$ vary as a function of the wavelenght (or frequency) for which the minimum deviation condition is set in the spectroscope?
Alternatevely, called $i$ the angle of incidence on the prism of the beam, what should I expect for the relation between $D_{min}$, $\lambda$ and $i$?
I know that $$D(i)=i -\alpha +\mathrm{arcsin}(\mathrm{sin}\sqrt{n^2+\mathrm{sin }^2 i}-\mathrm{cos} \alpha \,\,\,\mathrm{sin}i)$$
Also, under minimum deviation condition, $D_{min}=2i-\alpha$.
But what is the relation $D_{min}(\lambda)$? As far as I understood it should be decreasing (therefore $D_{min}(f)$ is increasing), but i do not think that it is a direct proportionality (correct?)
If what I said is correct, then suppose to plot $D_{yellow_{\mathrm{minmum \, deviation \, condition \,for \,RED}}}$ and $D_{yellow_{\mathrm{minmum \, deviation \, condition \,for \,VIOLET}}}$ as a function of the frequencies of red and violet. Should I expect to have a curve of tendency as the one in picture?
If this is the case then I cannot say which between red or violet would give the higher deviation for yellow, unless there is a way to get the function that describes the behaviour in the above graph, is there one?
Any suggestion on the topic or reference on where to find information about this is highly appreciated
2. May 7, 2017
sophiecentaur
What are you actually wanting to find? What frequencies are you choosing for your R,Y and V? and why did you want to minimise the deviation for yellow? Does it have some special significance for you?
A reference isn't really required here because the theory is pretty straightforward (ideal case). You could be lucky and find something published about your particular problem but I presume you have already does some searching.
Snell's Law and a bit of simple ray tracing will give you the path for any given frequency, numerically. If you want a minimum deviation for yellow then just calculate it over a range of incidence angles to find the minimum. You could do it analytically, I guess but you would end up with a long equation to solve.
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Introduction
What is Structured Prediction?
The ultimate objective of discriminative learning is to train a system to optimize a desired measure of performance. In binary classification we are interested in finding a function that assigns a binary label to a single object, and minimizes the error rate (correct or incorrect) on unseen data.
In structured prediction the input is a complex object (a spoken utterance, an image, a phrase, a graph, etc.), and we are interested in the prediction of a structured label (a sequence of words, a bounding box, a parsing tree, etc.). Typically, each structured prediction task has its own measure of performance or evaluation metric, such as word error rate in speech recognition, the BLEU score in machine translation or the intersection-over-union score in object segmentation.
Example In the problem of vowel duration measurement, our goal is to predict the start and end times of a vowel phoneme in a spoken word. One approach is to simply train a binary classifier at the frame level to predict for each frame weather it is a vowel or not. This approach does not take into consideration the internal structure of the desire label and relation between its components, such as typical vowel duration, or its temporal and spectral characteristics. In contrary, the structured prediction approach can look at the desired label as one piece and define a set of features maps that captures the relations between target internal components. A typical feature can be the presume duration of the vowel relative to the average vowel duration. We provide very detailed tutorial on the vowel duration measurement which can be found here.
We begin by posing the structured learning setting. We consider a supervised learning setting with input objects $x \in \mathcal{X}$ and target labels $y \in \mathcal{Y}$. The labels may be sequences, trees, grids, or other high-dimensional objects with internal structure. We assumed a fixed mapping $\phi: \mathcal{X} \times \mathcal{Y} \rightarrow \mathbb{R}^d$ from the set of input objects and target labels to a real vector of length $d$. We call the elements of this mapping feature functions/feature maps.
Example In the problem of pronunciation modeling, the input $x$ is a pronounced sequence of phones, and the output $y$ is the pronounced word. Most often pronunciation variations cannot be found in a lexicon of canonical pronunciation, and have to be modeled statistically. In this task, the feature functions map a pronunciation from a set of pronunciations of different lengths along with a proposed word to a vector of fixed dimension in $\mathbb{R}^d$. One such feature function might measure the edit distance between the pronunciation $x$ and the canonical pronunciation of the word $y$ in the lexicon. This feature function counts the minimum number of edit operations (insertions, deletions, and substitutions) that are needed to convert the actual pronunciation to the lexicon pronunciation; it is low if the actual pronunciation is close to the lexicon and high otherwise. See Hao, Keshet, and Livescu (2012) and the references therein for full description of this task and other examples of feature functions.
More Formally
Consider a supervised learning setting with input instances $x \in \mathcal{X}$ and target labels $y \in \mathcal{Y}$, which refers to a set of complex objects with an internal structure. We assumed a fixed mapping called feature functions (or feature maps) $\phi: \mathcal{X} \times \mathcal{Y} \rightarrow \mathbb{R}^d$ from the set of input objects and target labels to a real vector of length $d$. Also consider a linear decoder with parameters $w \in \mathbb{R}^d$, such that $\hat{y}$ is a good approximation to the true label of $x$, as follows:
$$\label{eq:decoding} \hat{y} = argmax_{y \in \mathcal{y}} ~ w^\top \phi(x, y)$$
Ideally, the learning algorithm finds $w$ such that the prediction rule optimizes the expected desired $\textit{measure of preference}$ or $\textit{evaluation metric}$ on unseen data. We define a $\textit{cost}$ function, $\ell(y, \hat{y})$, to be a non-negative measure of error when predicting $\hat{y}$ instead of $y$ as the label of $x$. Our goal is to find the parameters vector $w$ that minimizes this function. Often the desired evaluation metric is a utility function that needs to be maximized (like BLEU or NDCG) and then we define the cost to be 1 minus the evaluation metric. We assume that exists some unknown probability distribution $\rho$ over pairs $(x,y)$ where $y$ is the desired output for input $x$. We would like to set $w$ so as to minimize the expected cost, or the $\textit{risk}$, for predicting $\hat{y}$,
$$\label{eq:w*} w^* = argmin_{w} ~ \mathbb{E}_{(x,y) \sim \rho} [\ell(y,\hat{y}(x))].$$
This objective function is hard to minimize directly (Keshet, 2014). Given a training set of examples $\mathcal{S} = \{(x_i,y_i)\}_{i=1}^{m}$, where each pair $(x_i, y_i)$ is drawn i.i.d from $\rho$, a common practice is to find the model parameters that minimize the regularized mean surrogate loss,
$$\label{eq:reg-loss} w^* = argmin_{w} ~ \frac{1}{m}\sum_{i=1}^{m} \bar{\ell}(w,x_i,y_i) + \frac{\lambda}{2} \|w\|^2,$$
where $\bar{\ell}(w,x,y)$ is a surrogate loss function, and $\lambda$ is a trade-off parameter between the loss term and the regularization. Each algorithm has its own definition of the surrogate loss, e.g., the surrogate loss in max-margin Markov model (Taskar et al., 2003) is the structured hinge loss with the Hamming cost, whereas the surrogate loss in conditional random fields (Lafferty et al., 2001) is the log loss function. A general survey on structured prediction algorithms and their prediction rules is given in (Keshet, 2014).
Modules
Let us first define the loss augmented function as follows:
$$\label{eq:loss-augmented} \hat{y}^{\epsilon} = argmax_{\hat{y} \in \mathcal{Y}} ~ w^\top \phi(x, \hat{y}) + \epsilon \, \ell (y,\hat{y})$$
where epsilon can be 0 if we wish to use a standard linear decoder, 1 if we wish to use a loss-augmented or any other $\epsilon$ value (as used in DLM).
We implemented six structured prediction algorithms in StructED, all the loss functions and update rules are summarized in the following table:
Loss Update Rule
Structured Perceptron (Collins, 2002).
- $w_{t+1} = w_t + \phi(x_t, y_t) - \phi(x_t,\hat{y}^0_t)$
Structured SVM (Tsochantaridis et al., 2005).
$max_{\hat{y}} ~[ \ell(y,\hat{y}) - w^\top\phi(x,y) + w^\top\phi(x,\hat{y})]$ $w_{t+1} = (1-\eta_t \lambda)w_t + \eta_t( \phi(x_t, y_t) - \phi(x_t,\hat{y}^1_t))$
Passive Aggressive (Crammer et al., 2006).
- $w_{t+1} = w_{t} + \tau_{t}(\phi(x_t,y_t)) - \phi(x_t,\hat{y}^1_t))$
CRF (Lafferty et al., 2001).
$-\ln P_{w}(y \,|\, x)$, where $P_{w}(y \,|\, x) = \frac{1}{Z_{w}(x)} \exp\{w^\top \phi(x, y)\}$ $w_{t+1} = (1-\eta_t \lambda)\,w_{t} + \eta_t\Big( \phi(x_{j_t},y_{j_t}) - \mathbb{E}_{y'\sim P_{w}(y' \,|\, x)}[ \phi(x_{j_t},y')] \Big)$
Direct Loss Minimization (McAllester et al., 2010).
$\ell (y,\hat{y})$ $w_{t+1} = w_{t} + \frac{\eta_t}{\epsilon}(\phi(x_t,\hat{y}^{-\epsilon}_t) - \phi(x_t,\hat{y}^0_{t}))$
Structured Ramp Loss (McAllester and Keshet, 2011).
$\max_{\hat{y}} [ \ell(y,\hat{y}) + w^\top \phi(x,\hat{y})] - \max_{\tilde{y}} [w^\top \phi(x,\tilde{y})]$ $w_{t+1} = (1-\eta_t\lambda)\,w_{t} + \eta_t(\phi(x_t,\hat{y}^0_t) - \phi(x_t,\hat{y}^1_t))$
Probit Loss (Keshet et al., 2011).
$\mathbb{E}_{\gamma \sim \mathcal{N}(0,I)}[\ell (y,\hat{y}_{w+\gamma})]$ $w_{t+1} = (1-\eta_t\lambda)\,w_t + \eta_t \mathbb{E}_{\gamma \sim \mathcal{N}(0,I)}[ \gamma \, \ell (y_t,\hat{y}^0_{w+\gamma})]$
Although it is not so popular in structured prediction, we also implemented two kernel expansion functions:
1. Polynomial.
2. RBF using 2nd and 3rd Taylor approximation (Cotter, Keshet and Srebro, 2011)
Moreover, in order to support multi-class, we also implemented the feature functions, inference, loss-augmented inference and and use zero-one loss as a cost function. Hence, using StructED as a multi-class classifier is straightforward. More details about using StructED for multi-class tasks can be found here.
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Solved
# Rename Files to Include Full Path in File Name
Posted on 2013-11-27
301 Views
I am using Windows 7.
I have a series of files named file1,file2,file3 in the folder \folder1\folder2. So for example the full path of file1 would be \folder1\folder2\file1. How can I rename these files to include the full path in their file name.
For example I want to rename file1 to folder1-folder2-file1.
0
Question by:zabac
LVL 57
Expert Comment
ID: 39682626
Do you want them to stay in the original location? So the full name would be:
\folder1\folder2\folder1-folder2-file1
0
Author Comment
ID: 39682692
Yes they should stay in the original folders,
0
LVL 34
Accepted Solution
Dan Craciun earned 500 total points
ID: 39682769
Try this (in Powershell 3):
Param (
[string]$inputPath = "X:\path\to\files" )$fileList = Get-ChildItem -Path $inputPath -Recurse:1 -file foreach ($i in $fileList) {$newName = $i.FullName.split("\")$newName = $newName[1..($newName.count - 1)] ## remove the X: part
$newName =$newName -join("-")
$newNameWithPath =$i.DirectoryName + "\" + $newName Rename-Item$i.FullName \$newNameWithPath
}
It will rename all files inside a folder and subfolders.
HTH,
Dan
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# Sleight of Hand
Keeping up with all the hoopla regarding MVNOs, Portugal is now at the forefront of that burgeoning market, sporting a ratio of roughly one MVNO per mobile operator (whether or not MVNOs can be considered as such when wholly or partly owned by their parent operators or their conglomerates is something I leave to the legal eagles).
I've been holding on tightly to my Disclaimer regarding this sort of thing (and there is plenty of commentary on Portuguese sites regarding the current offerings), but I couldn't avoid pointing out their somewhat shady web marketing tactics.
Doing a "view source" on Rede4, I found their meta tags to have a rather interesting choice of keywords:
<meta name="KEYWORDS" content="rede4; telemóvel; SMS; grátis; Nokia;
tarifários; toques; logos; minuto; barato; móvel; rede; uzo;" >
Makes sense, and their choice of keywords is rather restrained (I would even say "politically correct"), with only a single mention to their competitors.
Doing a "view source" on UZO, however, turned up a much more interesting choice of keywords:
<META NAME="KEYWORDS" CONTENT="descomplicado,desbloqueado,grátis,desbloquear
telemoveis,gratis sms,imagens telemovel,motorola,nokia,sms,siemens,
telemoveis imagens,toques telemovel,tmn,vodafone,optimus,rede4,uzo,uso,use,uze" />
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Select Page
a be a positive integer from set {2, 3, 4, … 9999}. Show that there are exactly two positive integers in that set such that 10000 divides a*a-1.
1. Put $n^2 -1$ in place of 9999. How many positive integers a exists such that $n^2$divides a(a-1)
Discussion:
Important Note: If you are reading this, then please understand the full use of this discussion. A discussion typically begins with a comment from Teacher which contain hint for solution. After reading the first comment from teacher you should try to go back to the problem and solve it on your own. If you are unable to make any progress come back to this dialog. With each passing comment the solution unfolds. So it is more useful NOT to read the entire dialog in one shot. After reading each comment try the problem again and again. The only way to learn mathematics is by doing and the discussions in this blog are created to help you do exactly that.
Teacher: This is a clever problem. Especially the second part. For the first part can you find at least one value of a by trial and error? Use the prime factorization of 10000 and also the fact a and a-1 are coprime numbers (why?)
Student: Sure. $10000 = 2^4 5^4$ . As a and a-1 are coprime (they are consecutive numbers), hence we all the 2’s must divide either a or a-1 and all the 5’s must divide the other one.
Teacher: Absolutely
Student: So one of the numbers is a multiple of 625 and by adding or subtracting 1 to it we hope to find a multiple of 16.
Teacher: Right! You can easily get one by trial and error. You want $625k \pm 1 \equiv 0$ mod 16 .
Student: 625 works! If we subtract 1 from it we get 624 and that is a multiple of 16.
Teacher: Good. So 625 is one such number. Any other number?
Student: Should I check all multiples of 625 (and 16) and add (subtract) 1 from them. That will be tedious but will surely give the answer.
Teacher: That will definitely give the answer. But it will be very tedious. Instead observe that $625 \equiv 1$ mod 16.
Student: Oh! That will help. We want $625k \pm 1 \equiv 0$ mod 16 . Since $625 \equiv 1$ mod 16 , we want $k \pm 1 \equiv 0$ mod 16. So possible values of k are 1, 15, 17, 31 (or $6t \pm 1$ numbers). However if we put k = 17 , the 625*17 exceeds 9999 (hence is not from the set). So k=15 is the other choice. Hence a-1 = 9375 (625*15) and a= 9376 is the other choice.
Teacher: Very nice. Now lets try the second part. For starters can you tell me for what kind of n no such awill exist?
Student: Obviously if n is a prime. For if n is a prime (say p) then it is impossible to have two consecutive numbers from the set {2, … $p^2 -1$ } whose product will be divisible by $p^2$ (one of the two consecutive numbers needs to have 2 p’s but that is impossible because the first number with two p’s in it’s prime factorization is $p^2$ but we are taking numbers from the set {2 , 3, … , $p^2 - 1$ }
Teacher: Is it only primes that has this problem?
Student: Actually if $n = p^k$ where p is a prime then we have this problem (that is there will be no such a for which a(a-1) is divisible by $n^2$ )
Teacher: Okay. Now let us consider numbers which have more that one prime factor. Suppose n = xy such that x and y are coprime and both greater than 1, We split n into two co-prime factors because we want $n^2$ to divide a(a-1) and we need $x^2$ to divide a (or (a-1)) and $y^2$ to divide the other one.
Student: I see. We can proceed as before. The idea is this: if we add or subtract 1 from a multiple of $x^2$ we need that to be divisible by $y^2$ given that $gcd (x^2 , y^2 ) = 1$ .
So $x^2 k \pm 1 \equiv y^2$
I cannot think what to do from here?
Teacher: We are allowed to take numbers from the set {2,3 ,4 ..$x^2 y^2 - 1$ } , so think about the permissible values for k.
Student: Oh okay! So k can be 1, 2, 3, .. $y^2 - 1$ (if k = $y^2$ or more then $x^2\times k$ will exceed $x^2 y^2 -1$ ) .
Now lets consider the set { $x^2 , 2x^2 , 3x^2 , ... , (y^2 -1) x^2$ } . There are $y^2 -1$ terms in this set. If we reduce each term modulo $y^2$ we will get all the numbers $1, 2, 3, ... y^2 -1$ (we get all residues except 0 as $x^2 \times k$ is not divisible by $y^2$ for any value of k from 1, 2, … $y^2 -1$ as gcd $( x^2 , y^2 )$ = 1 .
Teacher: How can you say that we get $1, 2, 3, ... y^2 -1$ as residues when we reduce{ $x^2 , 2x^2 , 3x^2 , ... , (y^2 -1) x^2$ } modulo$y^2$?
Student: Firstly we won’t get 0 (that I have mentioned in my argument before). Secondly we get $y^2 -1$ values . So if we can show that all residues that we get are distinct we are done. Suppose not.
Then $x^2 k_1 \equiv x^2 k_2 \implies x^2 (k_1 - k_2) \equiv 0$ mod $y^2$ . But this is not possible as $x^2$ and $y^2$ has no common factor and $k_1 - k_2$ is smaller than $y^2$ (hence not completely divisible by it)
Thus all residues we get are distinct.
Teacher: Very well. Proceed.
Student: So there will exactly two values of k for which the residues will be +1 and -1. Hence we will get exactly two values of a for that x, y split of n.
Teacher: Right! So what is the final answer.
Student: If n has k (> 1)prime factors then there are $2^k$ ways to split it into two co prime parts and for each of them there are exactly two values of a. If n is a prime (or a power of a single prime) there are no such a .
CMI PAPER 2015
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# Week 7 - Dynamic Programming
## Notes
### Matrix-chain multiplication
Given a chain of $n$ matrices $\langle A_1, ..., A_i, ..., A_n \rangle$ , where $A_i$ has dimension $p_{i-1} \times p_i$, parenthesize it in a way that minimizes the number of scalar multiplications.
#### Optimal structure
Within optimal parentheses, two splitted sub-chains must be optimal as well.
Prove: alternative 'better' sub-chains will yield a solution 'better' than the optimum: a contradiction.
#### Recursively defining
1. Definition: for $1 \leq i \leq j$, let $m[i, j]$ be optimal solution to chain $[i, j]$
2. The optimal solution is made of optimal solutions of sub-chains
• When the problem is trivial: $i = j$, single matrix, $m[i, j] = m[i, i] = 0$
• When the problem is not trivial: $i < j$, let $A_{i...k}$ and $A_{k+1...j}$ be the optimal parentheses, the cost will be sum of two sub-product plus product of two sub-matrix, which is
m[i, j] = m[i, k] + m[k + 1, j] + p_{i-1} p_k p_j
3. All possible splitting must be considered to yield optimal solutions of sub-chains, so we review all $i \leq k < j$ and find the optimal (minimum here):
\min_{i \leq k < j} \{m[i, j] = m[i, k] + m[k + 1, j] + p_{i-1} p_k p_j\}
In conclusion, we recursively define the minimum cost of parenthesizing product as
\begin{align*} m[i, j] = \begin{cases} 0 &\text{if }i = j\\\\ \min_{i \leq k < j} \{m[i, j] = m[i, k] + m[k + 1, j] + p_{i-1} p_k p_j\} &\text{if }i < j\end{cases} \end{align*}
It's trivial.
#### Computing Optimal
Running time $O(n^3)$
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 class ch15: """Algorithms in chapter 15 """ def matrix_up_np(self, p): """matrix-chain product Given a chain of n matrices, where A_i has dimension p_{i-1} \times p_i, fully parenthesize in a way that minimizes the number of scalar multiplications. Arguments: p {list} -- list of matrix shape consecutively memo {dict} -- dict of memoization """ import numpy as np n = len(p) - 1 m = np.zeros((n+1, n+1)) # m -> optimal number s = np.zeros((n+1, n+1)) # s -> optimal k for i in range(1, n+1): m[i, i] = 0 for l in range(2, n+1): # l -> chain length for i in range(1, n-l+2): # head position j = i + l - 1 # tail position m[i, j] = float('inf') for k in range(i, j): q = m[i, k] + m[k + 1, j] + p[i - 1]*p[k]*p[j] if q < m[i, j]: m[i, j] = q s[i, j] = k print(m[1:, 1:]) # start from 1 print(s[1:, 1:]) return m, s def matrix_print(self, s, i, j): """print parentheses Arguments: s {np.array} -- chart of 'k' i {int} -- from j {int} -- to """ i, j = int(i), int(j) if i == j: print('A_'+str(i), end=' ') else: print('(', end='') # print(i, j) self.matrix_print(s, i, s[i, j]) self.matrix_print(s, s[i, j] + 1, j) print(')', end='') >>> quiz = ch15() >>> p = [30, 35, 15, 5, 10, 20, 25] # same to the textbook sample >>> _, s = quiz.matrix_up_np(p) [[ 0. 15750. 7875. 9375. 11875. 15125.] [ 0. 0. 2625. 4375. 7125. 10500.] [ 0. 0. 0. 750. 2500. 5375.] [ 0. 0. 0. 0. 1000. 3500.] [ 0. 0. 0. 0. 0. 5000.] [ 0. 0. 0. 0. 0. 0.]] [[0. 1. 1. 3. 3. 3.] [0. 0. 2. 3. 3. 3.] [0. 0. 0. 3. 3. 3.] [0. 0. 0. 0. 4. 5.] [0. 0. 0. 0. 0. 5.] [0. 0. 0. 0. 0. 0.]] >>> quiz.matrix_print(s, 1, 6) ((A_1 (A_2 A_3 ))((A_4 A_5 )A_6 ))
### Longest common subsequence
#### Recursively Defining
\begin{align*} z[i, j] = \begin{cases} 0 &\text{ if }i = 0 \text{ or }j = 0\\\\ z[i-1, j-1] + 1 &\text{ if }X_i = Y_j\\\\ \max(z(i-1, j), z(i, j-1)) &\text{ if }X_i != Y_j\end{cases} \end{align*}
### Optimal Binary Search Tree
\begin{align} E &= \sum_{i=1}^n (\text{depth}_T(k_i) + 1)\times p_i + \sum_{i=0}^n (\text{depth}_T(d_i) + 1)\times q_i \\ &= 1 + \sum_{i=1}^n \text{depth}_T(k_i)\times p_i + \sum_{i=0}^n \text{depth}_T(d_i)\times q_i \end{align}
#### Optimal Structure
If an optimal BST has a subtree, the latter must be optimal as well.
Prove: If not, replace the subtree with the optimal one, results in a 'better than optimal' BST: contradiction to optimality.
#### Recursively Defining
1. Definition
• $e[i, j]$ is cost optimal BST containing keys $\langle k_i, \dots, k_j\rangle$ and dummy keys $\langle d_{i-1}, \dots, d_j\rangle$.
2. Push down cost: $e[i, j]$ is the cost when we access the tree directly. When it becomes a subtree of a node, the original subtree is pushed down one level. This increases the cost of each node by 1 level, and the sum of which will be
w(i, j) = \sum_{r=i}^j p_r + \sum_{l=i-1}^j q_r
Note that if $key_r$ if the root, then
w(i, j) = w(i, r-1) + p_r + w(r+1, j)
3. Thus, if there exist a root, we choose $k_r, i \leq r \leq j$ as the root
• The cost of root node is $p_r$
• The cost of $left$ subtree = direct access cost + push down 1 level
• The cost of $right$ subtree = direct access cost + push down 1 level
\begin{align} e[i, j] &= \text{cost of root node} + \text{cost of left subtree} + \text{cost of right subtree} \\ &= p_r + (e[i, r-1] + w(i, r-1)) + (e[r+1, j] + w(r+1, j))\\ &= e[i, r-1] + e[r+1, j] + (w(i, r-1) +p_r + w(r+1, j))\\ &= e[i, r-1] + e[r+1, j] + w(i, j) \end{align}
4. In order to find the optimal (minimum) cost above, we shall consider all possible root:
e[i, j] = \min_{i\leq r\leq j} \{e[i, r-1] + e[r+1, j] + w(i, j)\}
5. The trivial case is there's no root but only dummy key, which happens when $j = i-1$, and the cost of which is the dummy key itself $q_{i-1}$.
6. Combine the trivial case with none trivial ones, we have:
\begin{align} e[i, j] =\begin{cases} q_{i-1} &\text{ if }j = i - 1 \\\\ \min_{i\leq r\leq j} \{e[i, r-1] + e[r+1, j] + w(i, j)\} &\text{ if }i\leq j \\ \end{cases} \end{align}
## Assignment - Due 11/10/2018
Credit: source code of questions are from walkccc.
### 15.1-2
Show, by means of a counterexample, that the following ''greedy'' strategy does not always determine an optimal way to cut rods. Define the density of a rod of length $i$ to be $p_i / i$, that is, its value per inch. The greedy strategy for a rod of length $n$ cuts off a first piece of length $i$, where $1 \le i \le n$, having maximum density. It then continues by applying the greedy strategy to the remaining piece of length $n - i$.
\begin{array}{l|ccccc} i & 1 & 2 & 3 & 4 & 5 \\ \hline p_i & 1 & 10 & 12 & 8 & 10 \\ p_i/i & 1 & 5 & 4 & 2 & 2 \\ \end{array}
Given the price table above, suppose ord length is 3, according to $p_i/i$ greedy strategy, it will be cut to $\{2, 1\}$ with income of $11$. However the optimal way is without cutting, having income of $12$.
### 15.1-3
Consider a modification of the rod-cutting problem in which, in addition to a price $p_i$ for each rod, each cut incurs a fixed cost of $c$. The revenue associated with a solution is now the sum of the prices of the pieces minus the costs of making the cuts. Give a dynamic-programming algorithm to solve this modified problem.
#### Top-down method
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 def rod_dynamic_down(price={1:1, 2:5, 3:8, 4:9, 5:10, 6:17, 7:17, 8:30}, avail=21, pool={}, c=0): """Get maximum value of rod to the avail, top-down implementation Keyword Arguments: price {dict} -- dict of length and price {length:price, } avail {int} -- [description] (default: {21}) """ price_len = sorted(price.keys()) if avail < min(price_len): # less than min-cutting return 0 if pool.get(avail): # dynamic - found return pool.get(avail) max_r = 0 # in case rod <= avail for rod in [i for i in price_len if i <= avail]: max_r = max(max_r, price[rod] + rod_dynamic_down(price, avail-rod, pool) - c) pool[avail] = max_r # dynamic - not found return max_r
#### Bottom-up method
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 def rod_dynamic_up(price={1:1, 2:5, 3:8, 4:9, 5:10, 6:17, 7:17, 8:30}, avail=21, c=0): """Bottom-up Keyword Arguments: price {dict} -- dict of length and price {length:price, } avail {int} -- [description] (default: {100}) """ record = {0:0, } for total in range(1, avail+1): max_r = 0 for cut in range(1, min(total, max(price)) + 1): max_r = max(max_r, price[cut] + record[total - cut] - c) record[total] = max_r return max_r
### 15.2-1
Find an optimal parenthesization of a matrix-chain product whose sequence of dimensions is $\langle 5, 10, 3, 12, 5, 50, 6 \rangle$.
### 15.2-3
Use the substitution method to show that the solution to the recurrence $\text{(15.6)}$ is $\Omega(2^n)$.
### 15.4-1
Determine an $\text{LCS}$ of $\langle 1, 0, 0, 1, 0, 1, 0, 1 \rangle$ and $\langle 0, 1, 0, 1, 1, 0, 1, 1, 0 \rangle$.
### 15.4-2
Give pseudocode to reconstruct an $\text{LCS}$ from the completed $c$ table and the original sequences $X = \langle x_1, x_2, \ldots, x_m \rangle$ and $Y = \langle y_1, y_2, \ldots, y_n \rangle$ in $O(m + n)$ time, without using the $b$ table.
### 15.4-3
Give a memoized version of $\text{LCS-LENGTH}$ that runs in $O(mn)$ time.
### 15.4-4
Show how to compute the length of an $\text{LCS}$ using only $2 \cdot \min(m, n)$ entries in the $c$ table plus $O(1)$ additional space. Then show how to do the same thing, but using $\min(m, n)$ entries plus $O(1)$ additional space.
### 15.5-1
Write pseudocode for the procedure $\text{CONSTRUCT-OPTIMAL-BST}(root)$ which, given the table $root$, outputs the structure of an optimal binary search tree. For the example in Figure 15.10, your procedure should print out the structure
\begin{align} & \text{$k_2$ is the root} \\ & \text{$k_1$ is the left child of $k_2$} \\ & \text{$d_0$ is the left child of $k_1$} \\ & \text{$d_1$ is the right child of $k_1$} \\ & \text{$k_5$ is the right child of $k_2$} \\ & \text{$k_4$ is the left child of $k_5$} \\ & \text{$k_3$ is the left child of $k_4$} \\ & \text{$d_2$ is the left child of $k_3$} \\ & \text{$d_3$ is the right child of $k_3$} \\ & \text{$d_4$ is the right child of $k_4$} \\ & \text{$d_5$ is the right child of $k_5$} \end{align}
corresponding to the optimal binary search tree shown in Figure 15.9(b).
### 15.5-2
Determine the cost and structure of an optimal binary search tree for a set of $n = 7$ keys with the following probabilities
\begin{array}{c|cccccccc} i & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline p_i & & 0.04 & 0.06 & 0.08 & 0.02 & 0.10 & 0.12 & 0.14 \\ q_i & 0.06 & 0.06 & 0.06 & 0.06 & 0.05 & 0.05 & 0.05 & 0.05 \end{array}
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# Why does the Hubble constant have a subscript zero?
The title says it all. Why is the Hubble constant written $H_0$ and not just $H$? What is the purpose of that subscript $_0$?
The Hubble parameter is defined as $\frac{\dot{a}}{a}$, where $a$ is the scale factor in the Friedmann-Robertson-Walker metric: $$g_{\mu\nu} \, \mathrm{d}x^{\mu} \, \mathrm{d}x^{\nu}=c^2 \, \mathrm{d}t^2-a^2 \left(\frac{\mathrm{d}r^2}{1-kr^2}+r^2\mathrm{d}\Omega ^2 \right) \,.$$
When you put this metric into Einstein Field Equation, you will get an equation that describes how $a$ evolves with time $t$. Thus in general the Hubble parameter is a function of time.
$H_0$ is used to indicate the actual value of the Hubble parameter, measured at our time.
The value of the Hubble parameter at the present epoch is represented with $H_0$.
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# Finding the eccentric anamoly
by smmSTV
Tags: anamoly, eccentric
P: 366 The correct equation is $$M = E - e*sin(E)$$ Solve using the Newton Method: $$E_{n+1} = E_{n} - \frac{f(E)}{f'(E)}$$ where f(E) = E - e * sin(E) - M and f'(E) = 1 - e * cos(E) Loop the above equations until: $$\frac{f(E)}{f'(E)} < 0.00001$$Or some substantially low number not zero. Also,$$r = \frac{a * (1 - e ^ 2)}{(1 + e * cos(TA))}$$ where TA - True Anomaly and a - Semi-Major Axis of Mars
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# Computing mod inverse?
How might one compute $4^{-1} \mod 17$ I know the answer is 13. I'm just not sure how to arrive at that number, and can't find any good explanations. Any help would be great
• – user12859 Oct 3 '15 at 23:50
• What you linked me to is the Extended Euclidean Algorithm which gives the GCD (Greatest Common Divisor) of two numbers, no? It isn't helpful. I'm not looking for the GCD of these two numbers. I'm looking for an example similar to solve $x^{-1} \mod y$ – Shammy Oct 4 '15 at 0:09
• What I linked you "to is the Extended Euclidean Algorithm which gives the" $\hspace{1.5 in}$ "coefficients of Bézout's identity". $\;$ – user12859 Oct 4 '15 at 0:14
• I'm very confused. I thought it was rather simple to solve the problem I have. Are you saying I need to find the GCD(4,17) first? I just want to be clear – Shammy Oct 4 '15 at 0:17
• No, you'll need either x or y, depending on which of {a,b} is to be inverted. $\;$ – user12859 Oct 4 '15 at 0:19
## 1 Answer
In order to compute the inverse of $a$ modulo $n$, use the extended Euclidean algorithm to find the GCD of $a$ and $n$ (which should be 1), together with coefficients $x,y$ such that $ax + ny = 1$. The inverse of $a$ modulo $n$ is thus $x$.
The extended Euclidean algorithm gives a constructive proof of Bézout's identity, which states that for all integers $a,b$ there exist integers $x,y$ such that $ax+by = \mathrm{gcd}(a,b)$. A different proof shows that the minimal positive value of $ax+by$ (over all $x,y$) is $\mathrm{gcd}(a,b)$.
The extended Euclidean algorithm works in greater generality, for any Euclidean domain. An important example is the ring of polynomials over a field.
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Controlled stochastic networks in heavy traffic: Convergence of value functions
# Controlled stochastic networks in heavy traffic: Convergence of value functions
[ [ [ [ University of North Carolina and Iowa State University Department of Statistics
and Operations Research
University of North Carolina
Chapel Hill, North Carolina 27599-3260
USA
Department of Statistics
3216 Snedecor Hall
Iowa State University
Ames, Iowa 50011-1210
USA
\smonth3 \syear2010\smonth12 \syear2010
\smonth3 \syear2010\smonth12 \syear2010
\smonth3 \syear2010\smonth12 \syear2010
###### Abstract
Scheduling control problems for a family of unitary networks under heavy traffic with general interarrival and service times, probabilistic routing and an infinite horizon discounted linear holding cost are studied. Diffusion control problems, that have been proposed as approximate models for the study of these critically loaded controlled stochastic networks, can be regarded as formal scaling limits of such stochastic systems. However, to date, a rigorous limit theory that justifies the use of such approximations for a general family of controlled networks has been lacking. It is shown that, under broad conditions, the value function of the suitably scaled network control problem converges to that of the associated diffusion control problem. This scaling limit result, in addition to giving a precise mathematical basis for the above approximation approach, suggests a general strategy for constructing near optimal controls for the physical stochastic networks by solving the associated diffusion control problem.
[
\kwd
\doi
10.1214/11-AAP784 \volume22 \issue2 2012 \firstpage734 \lastpage791 \newproclaimassu[theorem]Assumption \newproclaimremark[theorem]Remark \newproclaimdefinition[theorem]Definition \newproclaimdefn[theorem]Definition \newproclaimcond[theorem]Condition \newproclaimpr[theorem]Property \newproclaimexampleExample
\runtitle
Convergence of value functions
{aug}
A]\fnmsAmarjit \snmBudhiraja\thanksreft1label=e1]budhiraj@email.unc.edu and B]\fnmsArka P. \snmGhosh\corref\thanksreft2label=e2]apghosh@iastate.edu \thankstextt1Supported in part by the NSF (DMS-10-04418), Army Research Office (W911NF-0-1-0080, W911NF-10-1-0158) and the US-Israel Binational Science Foundation (2008466). \thankstextt2Supported in part by NSF Grant DMS-06-08634.
class=AMS] \kwd[Primary ]60K25 \kwd68M20 \kwd90B22 \kwd90B36 \kwd[; secondary ]60J70. Heavy traffic \kwdstochastic control \kwdscaling limits \kwddiffusion approximations \kwdunitary networks \kwdcontrolled stochastic processing networks \kwdasymptotic optimality \kwdsingular control with state constraints \kwdBrownian control problem (BCP).
## 1 Introduction
As an approximation to control problems for critically-loaded stochastic networks, Harrison (in harri2 (), see also harri1 (), harri-canon ()) has formulated a stochastic control problem in which the state process is driven by a multidimensional Brownian motion along with an additive control that satisfies certain feasibility and nonnegativity constraints. This control problem, that is, usually referred to as the Brownian Control Problem (BCP) has been one of the key developments in the heavy traffic theory of controlled stochastic processing networks (SPN). BCPs can be regarded as formal scaling limits for a broad range of scheduling and sequencing control problems for multiclass queuing networks. Finding optimal (or even near-optimal) control policies for such networks—which may have quite general non-Markovian primitives, multiple server capabilities and rather complex routing geometry—is in general prohibitive. In that regard, BCPs that provide significantly more tractable approximate models are very useful. In this diffusion approximation approach to policy synthesis, one first finds an optimal (or near-optimal) control for the BCP which is then suitably interpreted to construct a scheduling policy for the underlying physical network. In recent years there have been many works ata-kumar (), bellwill (), bellwill2 (), BudGho (), meyn (), ward-kumar (), chen-yao (), dai-lin () that consider specific network models for which the associated BCP is explicitly solvable (i.e., an optimal control process can be written as a known function of the driving Brownian motions) and, by suitably adapting the solution to the underlying network, construct control policies that are asymptotically (in the heavy traffic limit) optimal. The paper KuMa () also carries out a similar program for the crisscross network where the state–space is three dimensional, although an explicit solution for the BCP here is not available.
Although now there are several papers which establish a rigorous connection between a network control problem and its associated BCP by exploiting the explicit form of the solution of the latter, a systematic theory which justifies the use of BCPs as approximate models has been missing. In a recent work BudGho2 () it was shown that for a large family of Unitary Networks (following terminology of Will-Bram-2work (), these are networks with a structure as described in Section 2), with general interarrival and service times, probabilistic routing and an infinite horizon discounted linear holding cost, the cost associated with any admissible control policy for the network is asymptotically, in the heavy traffic limit, bounded below by the value function of the BCP. This inequality, which provides a useful bound on the best achievable asymptotic performance for an admissible control policy, was a key step in developing a rigorous general theory relating BCPs with SPN in heavy traffic.
The current paper is devoted to the proof of the reverse inequality. The network model is required to satisfy assumptions made in BudGho2 () (these are summarized above Theorem 2.1). In addition, we impose a nondegeneracy condition (Assumption 2), a condition on the underlying renewal processes regarding probabilities of deviations from the mean (Assumption 2) and regularity of a certain Skorohod map (Assumption 2) (see next paragraph for a discussion of these conditions). Under these assumptions we prove that the value function of the BCP is bounded below by the heavy traffic limit (limsup) of the value functions of the network control problem (Theorem 2.2). Combining this with the result obtained in BudGho2 () (see Theorem 2.1), we obtain the main result of the paper (Theorem 2.4). This theorem says that, under broad conditions, the value function of the network control problem converges to that of the BCP. This result provides, under general conditions, a rigorous basis for regarding BCPs as approximate models for critically loaded stochastic networks.
Conditions imposed in this paper allow for a wide range of SPN models. Some such models, whose description is taken from Will-Bram-2work (), are discussed in detail in Examples 2(a)–(c). We note that our approach does not require the BCP to be explicitly solvable and the result covers many settings where explicit solutions are unavailable. Most of the conditions that we impose are quite standard and we only comment here on three of them: Assumptions 2, 2 and 2. Assumption 2 says that each buffer is processed by at least one basic activity (see Remark 2). This condition, which was introduced in Will-Bram-2work (), is fundamental for our analysis. In fact, Will-Bram-2work () has shown that without this assumption even the existence of a nonnegative workload matrix may fail. Assumption 2 is a natural condition on the geometry of the underlying network. Roughly speaking, it says that a nonzero control action leads to a nonzero state displacement. Assumption 2 is the third key requirement in this work. It says that the Skorohod problem associated with a certain reflection matrix [see equation (43) for the definition of ] is well posed and the associated Skorohod map is Lipschitz continuous. As Example 2 discusses, this condition holds for a broad family of networks (including all multiclass open queuing networks, as well as a large family of parallel server networks and job-shop networks).
The papers ata-kumar (), bellwill (), bellwill2 (), BudGho (), ward-kumar (), dai-lin () noted earlier, that treat the setting of explicitly solvable BCP, do much more than establish convergence of value functions. In particular, these works give an explicit implementable control policy for the underlying network that is asymptotically optimal in the heavy traffic limit. In the generality treated in the current work, giving explicit recipes (e.g., threshold type policies) is unfeasible, however, the policy sequence constructed in Section 4.1 suggests a general approach for building near asymptotically optimal policies for the network given a near optimal control for the BCP. Obtaining near optimal controls for the BCP in general requires numerical approaches (see, e.g., DuKu (), Kushbook (), meyn-book ()), discussion of which is beyond the scope of the current work.
Using a near-optimal control of the form given in Section 3 (cf. Theorem 3.5), we then proceed to construct a sequence of policies for the underlying network. The key relation that enables translation of into is (16) using which one can loosely interpret as the asymptotic deviation, with suitable scaling, of from the nominal allocation (see Definition 2 for the definition of nominal allocation vector). Recall that is constructed by modifying, through a Skorohod constraining mechanism, a pure jump process (say, ). In particular, has sample paths that are, in general, discontinuous. On the other hand, note that an admissible policy is required to be a Lipschitz function (see Remark 2). This suggests the following construction for . Over time periods (say, ) of constancy of one should use the nominal allocation (i.e., ), while jump-instants should be stretched into periods of length of order (note that in the scaled network, time is accelerated by a factor of and so such periods translate to intervals of length in the scaled evolution and thus are negligible) over which a nontrivial control action is employed as dictated by the jump vector (see Figure 4 for a more complete description). This is analogous to the idea of a discrete review policy proposed by Harrison bigstep () (see also ata-kumar () and references therein). There are some obvious difficulties with the above prescription, for example, a nominal allocation corresponds to the average behavior of the system and for a given realization is feasible only when the buffers are nonempty. Thus, one needs to modify the above construction to incorporate idleness, that is, caused due to empty buffers. The effect of such a modification is, of course, very similar to that of a Skorohod constraining mechanism and it is tempting to hope that the deviation process corresponding to this modified policy converges to (in an appropriate sense), as . However, without further modifications, it is not obvious that the reflection terms that are produced from the idling periods under this policy are asymptotically consistent with those obtained from the Skorohod constraining mechanism applied to (the state process corresponding to) . The additional modification [see (100)] that we make roughly says that jobs are processed from a given buffer over a small interval , only if at the beginning of this interval there are a “sufficient” number of jobs in the buffer. This idea of safety stocks is not new and has been used in previous works (see, e.g., bellwill (), bellwill2 (), ata-kumar (), BudGho (), meyn-book ()). The modification, of course, introduces a somewhat nonintuitive idleness even when there are jobs that require processing. However, the analysis of Section 4 shows that this idleness does not significantly affect the asymptotic cost. The above very rough sketch of construction of is made precise in Section 4.1.
The rest of the paper is devoted to showing that the cost associated with converges to that associated with . It is unreasonable to expect convergence of controls (e.g., with the usual Skorohod topology)—in particular, note that has Lipschitz paths for every while is a (modification of) a pure jump process – however, one finds that the convergence of costs holds. This convergence proof, and the related weak convergence analysis, is carried out in Sections 4.2 and 4.3.
The paper is organized as follows. Section 2 describes the network structure, all the associated stochastic processes and the heavy-traffic assumptions as well as the other assumptions of the paper. The section also presents the SPN control problem, that is, considered here, along with the main result of the paper (Theorem 2.4). Section 3 constructs (see Theorem 3.5) a near-optimal control policy for the BCP which can be suitably adapted to the network control problem. In Section 4 the near-optimal control policy from Section 3 is used to obtain a sequence of admissible control policies for the scaled SPN. The main result of the section is Theorem 4.5, which establishes weak convergence of various scaled processes. Convergence of costs (i.e., Theorem 2.3) is an immediate consequence of this weak convergence result. Theorem 2.4 then follows on combining Theorem 2.3 with results of BudGho2 () (stated as Theorem 2.1 in the current work). Finally, the Appendix collects proofs of some auxiliary results.
The following notation will be used. The space of reals (nonnegative reals), positive (nonnegative) integers will be denoted by (), (), respectively. For and will denote the space of continuous functions from (resp. ) to with the topology of uniform convergence on compacts (resp. uniform convergence). Also, will denote the space of right continuous functions with left limits, from (resp. ) to with the usual Skorohod topology. For and , we write and , where for , All vector inequalities are to be interpreted component-wise. We will call a function nonnegative if for all . A function is called nondecreasing if it is nondecreasing in each component. All (stochastic) processes in this work will have sample paths that are right continuous and have left limits, and thus can be regarded as -valued random variables with a suitable . For a Polish space , will denote the corresponding Borel sigma-field. Weak convergence of valued random variables to will be denoted as . Sequence of processes is tight if and only if the measures induced by ’s on form a tight sequence. A sequence of processes with paths in () is called -tight if it is tight in and any weak limit point of the sequence has paths in almost surely (a.s.). For processes , defined on a common probability space, we say that converge to , uniformly on compact time intervals (u.o.c.), in probability (a.s.) if for all , converges to zero in probability (resp. a.s.). To ease the notational burden, standard notation (that follow bram-will-1 (), Will-Bram-2work ()) for different processes are used (e.g., for queue-length, for idle time, for workload process etc.). We also use standard notation, for example, , to denote fluid scaled, respectively, diffusion scaled, versions of various processes of interest [see (2) and (2)]. All vectors will be column vectors. An -dimensional vector with all entries will be denoted by . For a vector , will denote the diagonal matrix such that the vector of its diagonal entries is . will denote the transpose of a matrix . Also, will denote generic constants whose values may change from one proof to the next.
## 2 Multiclass queueing networks and the control problem
Let be a probability space. All the random variables associated with the network model described below are assumed to be defined on this probability space. The expectation operation under will be denoted by .
### Network structure
We begin by introducing the family of stochastic processing network models that will be considered in this work. We closely follow the terminology and notation used in harri2 (), harri1 (), harri-canon (), bellwill2 (), Will-Bram-2work (), bram-will-1 (). The network has infinite capacity buffers (to store many different classes of jobs) and nonidentical servers for processing jobs. Arrivals of jobs, given in terms of suitable renewal processes, can be from outside the system and/or from the internal rerouting of jobs that have already been processed by some server. Several different servers may process jobs from a particular buffer. Service from a given buffer by a given server is called an activity. Once a job starts being processed by an activity, it must complete its service with that activity, even if its service is interrupted for some time (e.g., for preemption by a job from another buffer). When service of a partially completed job is resumed, it is resumed from the point of preemption—that is, the job needs only the remaining service time from the server to get completed (preemptive-resume policy). Also, an activity must complete service of any job that it started before starting another job from the same buffer. An activity always selects the oldest job in the buffer that has not yet been served, when starting a new service [i.e., First In First Out (FIFO) within class]. There are activities [at most one activity for a server-buffer pair , so that ]. Here the integers are strictly positive. Figure 1 gives a schematic for such a model.
Let , and . The correspondence between the activities and buffers, and activities and servers are described by two matrices and respectively. is an matrix with if the th activity processes jobs from buffer , and otherwise. The matrix is with if the th server is associated with the th activity, and otherwise. Each activity associates one buffer and one server, and so each column of has exactly one 1 (and similarly, every column of has exactly one 1). We will further assume that each row of (and ) has at least one 1, that is, each buffer is processed by (server is processing, resp.) at least one activity. For , let , if activity corresponds to the th server processing class jobs. Let, for , and . Thus, for the th server, denotes the set of activities that the server can perform, and represents the corresponding buffers from which the jobs can be processed.
### Stochastic primitives
We are interested in the study of networks that are nearly critically loaded. Mathematically, this is modeled by considering a sequence of networks that “approach heavy traffic,” as , in the sense of Definition 2 below. Each network in the sequence has identical structure, except for the rate parameters that may depend on . Here , where is a countable set: with and , as . One thinks of the physical network of interest as the th network embedded in this sequence, for a fixed large value of . For notational simplicity, throughout the paper, we will write the limit along the sequence as simply as “.” Also, will always be taken to be an element of and, thus, hereafter the qualifier will not be stated explicitly.
The th network is described as follows. If the th class () has exogenous job arrivals, the interarrival times of such jobs are given by a sequence of nonnegative random variables that are i.i.d with mean and standard deviation respectively. Let, by relabeling if needed, the buffers with exogenous arrivals correspond to , where . We set and , for . Service times for the th type of activity (for ) are given by a sequence of nonnegative random variables that are i.i.d. with mean and standard deviation respectively. We will assume that the above random variables are in fact strictly positive, that is,
(1)
We will further impose the following uniform integrability condition:
the collection {(uri(1))2,(vrj(1))2;r≥1,j∈J,i∈I′} is uniformly integrable.
(2)
Rerouting of jobs completed by the th activity is specified by a sequence of -dimensional vector , where . For each and , if the th completed job by activity gets rerouted to buffer , and takes the value zero otherwise, where represents jobs leaving the system. It is assumed that for each fixed , , , are (mutually) independent sequences of i.i.d , where . That, in particular, means, for , . Furthermore, for fixed ,
Cov(ϕj,ri1(n),ϕj,ri2(n))=σϕji1i2=−pji1pji2+pji1δi1,i2, (3)
where is if and otherwise. We also assume that, for each , the random variables
{uri(n),vrj(n),ϕj,r0(n),ϕj,r(n),n≥1,i∈I,j∈J} are mutually independent.
(4)
Next we introduce the primitive renewal processes, , that describe the state dynamics. The process is the -dimensional exogenous arrival process, that is, for each , is a renewal process which denotes the number of jobs that have arrived to buffer from outside the system over the interval . For class to which there are no exogenous arrivals (i.e., ), we set for all . We will denote the process by . For each activity , denotes the number of complete jobs that could be processed by activity in if the associated server worked continuously and exclusively on jobs from the associated buffer in and the buffer had an infinite reservoir of jobs. The vector is denoted by . More precisely, for , let
ξri(m)≐m∑n=1uri(n),ηrj(m)≐m∑n=1vrj(n). (5)
We set . Then , are renewal processes given as follows. For ,
Eri(t)=max{m≥0\dvtxξri(m)≤t},Srj(t)=max{m≥1\dvtxηrj(m)≤t}. (6)
Finally, we introduce the routing sequences. Let denote the number of jobs that are routed to the th buffer, among the first jobs completed by activity . Thus, for ,
\boldsΦj,ri(n)=n∑m=1ϕj,ri(m),n=1,2,…. (7)
We will denote the -dimensional sequence corresponding to routing of jobs completed by the th activity by . Also, will denote the matrix .
### Control
A Scheduling policy or control for the th SPN is specified by a nonnegative, nondecreasing -dimensional process . For any , represents the cumulative amount of time spent on the th activity up to time . For a control to be admissible, it must satisfy additional properties which are specified below in Definition 2.
### State processes
For a given scheduling policy , the state processes of the network are the associated -dimensional queue length process and the -dimensional idle time process . For each , , represents the queue-length at the th buffer at time (including the jobs that are in service at that time), and for , is the total amount of time the th server has idled up to time . Let be the -dimensional vector of queue-lengths at time . Note that, for , is the total number of services completed by the th activity up to time . The total number of completed jobs (by activity ) up to time that get rerouted to buffer equals . Recalling the definition of matrices and , the state of the system at time can be described by the following equations:
Qri(t) = qr+Eri(t)−J∑j=1CijSrj(Trj(t))+J∑j=1\boldsΦj,ri(Srj(Trj(t))),i∈I, (8) Irk(t) = (9)
### Heavy traffic
We now describe the main heavy traffic assumptionharri1 (), harri-canon (). We begin with a condition on the convergence of various parameters in the sequence of networks .
{assu}
There are , , such that , if and only if , and, as ,
θr1 ≐ r(αr−α)→θ1,θr2≐r(βr−β)→θ2, (10) σu,r → σu,σv,r→σv,^qr≐qrr→q.
The definition of heavy traffic, for the sequence , as introduced in harri1 () (also see Will-Bram-2work (), bram-will-1 (), harri-canon ()), is as follows.
{defn}
[[Heavy traffic]] Define matrices , such that , for , and
R≐(C−P′)diag(β). (11)
We say that the sequence approaches heavy traffic as if, in addition to Assumption 2, the following two conditions hold: {longlist}[(ii)]
There is a unique optimal solution to the following linear program (LP):
minimize ρ such that Rx=α and Ax≤ρ1Kfor all x≥0. (12)
The pair satisfies
ρ∗=1andAx∗=1K. (13)
{assu}
The sequence of networks approaches heavy traffic as .
{remark}
From Assumption 2, given in (i) of Definition 2 is the unique -dimensional nonnegative vector satisfying
Rx∗=α,Ax∗=1K. (14)
Following harri1 (), assume without loss of generality (by relabeling activities, if necessary), that the first components of are strictly positive (corresponding activities are referred to as basic) and the rest are zero (nonbasic activities). For later use, we partition the following matrices and vectors in terms of basic and nonbasic components:
x∗=[x∗b0],Tr=[TrbTrn],A=[B:N],R=[H:M], (15)
where is some control policy, is a -dimensional vector of zeros, are , , and matrices, respectively. The following assumption (see Will-Bram-2work ()) says that for each buffer there is an associated basic activity.
{assu}
For every , there is a such that and .
### Other processes
Components of the vector defined above can be interpreted as the nominal allocation rates for the activities. Given a control policy , define the deviation process as the difference between and the nominal allocation:
Yr(t)≐x∗t−Tr(t),t≥0. (16)
It follows from (9) and (14) that the idle-time process has the following representation:
Ir(t)=AYr(t),t≥0.
Let . Next we define a matrix and -dimensional process as follows:
K≐[BN0−I],Ur(t)≐KYr(t),t≥0, (17)
where denotes a identity matrix. Note that, with as in (15),
Ur(t)=[Ir(t)Trn(t)],t≥0. (18)
Finally, we introduce the workload process which is defined as a certain linear transformation of the queue-length process and is of dimension no greater than of the latter. More precisely, is an -dimensional process (, see Will-Bram-2work ()) defined as
Wr(t)=ΛQr(t),t≥0, (19)
where is a -dimensional matrix with rank and nonnegative entries, called the workload matrix. We will not give a complete description of since that requires additional notation; and we refer the reader to Will-Bram-2work (), harri-canon () for details. The key fact that will be used in our analysis is that there is a matrix with nonnegative entries (see (3.11) and (3.12) in harri-canon ()) such that
ΛR=GK. (20)
We will impose the following additional assumption on which says that each of its columns has at least one strictly positive entry. The assumption is needed in the proof of Lemma 3.7 [see (3.1)]. {assu} There exists a such that for every , .
### Rescaled processes
We now introduce two types of scalings. The first is the so-called fluid scaling, corresponding to a law of large numbers, and the second is the standard diffusion scaling, corresponding to a central limit theorem.
Fluid Scaled Process: This is obtained from the original process by accelerating time by a factor of and scaling down space by the same factor. The following fluid scaled processes will play a role in our analysis. For ,
¯Er(t) ≐ r−2Er(r2t),¯Sr(t)≐r−2Sr(r2t), ¯\boldsΦr(t) ≐ r−2\boldsΦr(⌊r2t⌋),¯Tr(t)≐r−2Tr(r2t), (21) ¯Ir(t) ≐ r−2Ir(r2t),¯Qr(t)≐r−2Qr(r2t).
Here for , denotes its integer part, that is, the greatest integer bounded by .
Diffusion Scaled Process: This is obtained from the original process by accelerating time by a factor of and, after appropriate centering, scaling down space by . Some diffusion scaled processes that will be used are as follows. For ,
^Er(t) ≐ (Er(r2t)−αrr2t)r,^Sr(t)≐(Sr(r2t)−βrr2t)r, ^\boldsΦr(t) ≐ (\boldsΦr(⌊r2t⌋])−⌊r2t⌋P′)r, (22) ^Ur(t) ≐ r−1Ur(r2t),^Qr(t)≐r−1Qr(r2t), ^Wr(t) ≐ r−1Wr(r2t),^Yr(t)≐r−1Yr(r2t).
The processes are not centered, as one finds (see Lemma 3.3 of BudGho2 ()) that, with any reasonable control policy, their fluid scaled versions converge to zero as . Define for ,
^Xri(t) ≐ ^Eri(t)−J∑j=1(Cij−pji)^Srj(¯Trj(t)) −J∑j=1^\boldsΦj,ri(¯Srj(¯Trj(t))),i∈I.
Recall and from Assumption 2. Using (8), (9), (14) and (17), one has the following relationships between the various scaled quantities defined above. For all ,
^Qr(t)=^ζr(t)+R^Yr(t),^Ur(t)=K^Yr(t),
where
^ζr(t)=^qr+^Xr(t)+[θr1t−(C−P′)diag(θr2)¯Tr(t)]. (24)
Also, using (19), (20) and (24), for all ,
^Wr(t)=Λ^qr+Λ^Xr(t)+Λ[θr1t−(C−P′)diag(θr2)¯Tr(t)]+G^Ur(t). (25)
The definition of admissible policies (Definition 2), given below, incorporates appropriate nonanticipativity requirements and ensures feasibility by requiring that the associated queue-length and idle-time processes () are nonnegative.
For we define the multiparameter filtration generated by interarrival and service times and routing variables as
¯Fr((m,n)) (26) =σ{uri(m′i),vrj(n′j),ϕj,ri(n′j)\dvtxm′i≤mi,n′j≤nj;i∈I,j∈J}.
Then is a multiparameter filtration with the following (partial) ordering:
(m1,n1)≤(m2,n2)if and only ifm1i≤m2i,n1j≤n2j;i∈I,j∈J.
We refer the reader to Section 2.8 of Kurtz-redbook () for basic definitions and properties of multiparameter filtrations, stopping times and martingales. Let
¯Fr≐⋁(m,n)∈NI+J¯Fr((m,n)). (27)
For all , we define where denotes the vector of 1’s. It will be convenient to allow for extra randomness, than that captured by , in formulating the class of admissible policies. Let be a -field independent of . For , let .
{defn}
For a fixed and , a scheduling policy is called admissible for with initial condition if for some independent of , the following conditions hold: {longlist}[(iii)]
is nondecreasing, nonnegative and satisfies for .
defined by (9) is nondecreasing, nonnegative and satisfies for .
defined in (8) is nonnegative for .
Define for each ,
σr0(t) = (σr,E0(t),σr,S0(t)) ≐ (Eri(r2t)+1\dvtxi∈I;Srj(Trj(r2t))+1\dvtxj∈J).
Then, for each ,
σr0(t) is a {Fr((m,n))\dvtxm∈NI,n∈NJ} stopping time. (29)
Define the filtration as
Fr1(t) ≐ Fr(σr0(t)) = σ{A∈Fr\dvtxA∩{σr0(t)≤(m,n)}∈Fr((m,n)),m∈NI,n∈NJ}.
Then
Denote by the collection of all admissible policies for with initial condition .
{remark}
(i) and (ii) in Definition 2 imply, in view of (9) and properties of the matrix , that
0≤Trj(t)−Trj(s)≤t−s,j∈J for% all 0≤s≤t<∞. (32)
In particular, is a process with Lipschitz continuous paths. Condition (iv) in Definition 2 can be interpreted as a nonanticipativity condition. Proposition 2.8 and Theorem 5.4 of BudGho2 () give general sufficient conditions under which this property holds (see also Proposition 4.1 of the current work).
### Cost function
For the network , we consider an expected infinite horizon discounted (linear) holding cost associated with a scheduling policy and initial queue length vector :
Jr(qr,Tr)≐E(∫∞0e−γth⋅^Qr(t)dt)+E(∫∞0e−γtp⋅d^Ur(t)). (33)
Here, is the “discount factor” and , an -dimensional vector with each component , is the vector of “holding costs” for the buffers. In the second term, is an -dimensional vector. The first block of corresponds to the idleness process , and, thus, the second term in the cost, in particular, captures the idleness cost. The last components of correspond to the time spent on nonbasic activities. Thus, this formulation of the cost allows, in addition to the idleness cost, the user to put a penalty for using nonbasic activities.
The formulation of the cost function considered in our work goes back to the original work of Harrison et al. harri1 (), harri2 ().
The scheduling control problem for is to find an admissible control policy that minimizes the cost . The value function for this control problem is defined as
Vr(qr)≐infTr∈Ar(qr)Jr(qr,Tr),qr∈NI0. (34)
### Brownian control problem
The goal of this work is to characterize the limit of value functions as , as the value function of a suitable diffusion control problem. In order to see the form of the diffusion control problem, we will like to send in (24). Using the functional central limit theorem for renewal processes, it is easily seen that, for all reasonable control policies (see again Lemma 3.3 of BudGho2 ()), when converges to some , defined in (24) converges weakly to
~ζ=q+~X+θ\mathpzci, (35)
where
θ≐θ1−(C−P′)diag(θ2)x∗. (36)
Here is the identity map and is a Brownian motion with drift 0 and covariance matrix
Σ≐Σu+(C−P′)Σvdiag(x∗)(C−P′)′+J∑j=1βjx∗jΣϕj, (37)
where is a diagonal matrix with diagonal entries , is a diagonal matrix with diagonal entries and s are matrices with entries [see (3)]. Although the process in (24), for a general policy sequence , need not converge, upon formally taking limit as , one is led to the following diffusion control problem.
{defn}
[[Brownian Control Problem (BCP)]] A -dimensional adapted process , defined on some filtered probability space which supports an -dimensional -Brownian motion with drift 0 and covariance matrix given by (37), is called an admissible control for the Brownian control problem with the initial condition iff the following two properties hold -a.s.:
~Q(t) ≐ ~ζ(t)+R~Y(t)≥0where ~ζ(t)=q+~X(t)+θt,t≥0, (38) ~U ≐ K~Y is nondecreasing and ~U(0)≥0, (39)
where and are as in (35) and (36) respectively. We refer to as a system. We denote the class of all such admissible controls by . The Brownian control problem is to
infimize ~J(q,~Y)≐~E[∫∞0e−γth⋅~Q(t)dt+∫[0,∞)e−γtp⋅d~U(t)], (40)
over all admissible controls . Define the value function
~J∗(q)=inf~Y∈~A(q)~J(q,~Y). (41)
Recall our standing assumptions (1), (2), (4), Assumptions 2, 2, 2 and 2. The following is the main result of BudGho2 ().
###### Theorem 2.1 ((Budhiraja and Ghosh BudGho2 (), Theorem 3.1, Corollary 3.2))
Fix and for , such that as . Then
liminfr→∞Vr(qr)≥~J∗(q).
{remark}
The proof in BudGho2 () is presented for the case where in the definition of [see (34)], is replaced by the smaller family which consists of all that satisfy (iv) of Definition 2 with replaced by . Proof for the slightly more general setting considered in the current paper requires only minor modifications and, thus, we omit the details.
For the main result of this work, we will need additional assumptions.
{assu}
The matrix is positive definite. We will make the following assumption on the probabilities of deviations from the mean for the underlying renewal processes. Similar conditions have been used in previous works on construction of asymptotically optimal control policies bellwill (), bellwill2 (), BudGho (), ata-kumar (), dai-lin ().
{assu}
There exists and, for each , some such that, for
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Calculus Volume 3
Key Equations
Calculus Volume 3Key Equations
Key Equations
Double integral $∬Rf(x,y)dA=limm,n→∞∑i=1m∑j=1nf(xij*,yij*)ΔA∬Rf(x,y)dA=limm,n→∞∑i=1m∑j=1nf(xij*,yij*)ΔA$ Iterated integral $∫ab∫cdf(x,y)dxdy=∫ab[∫cdf(x,y)dy]dx∫ab∫cdf(x,y)dxdy=∫ab[∫cdf(x,y)dy]dx$ or $∫cd∫baf(x,y)dxdy=∫cd[∫abf(x,y)dx]dy∫cd∫baf(x,y)dxdy=∫cd[∫abf(x,y)dx]dy$ Average value of a function of two variables $fave=1AreaR∬Rf(x,y)dxdyfave=1AreaR∬Rf(x,y)dxdy$
Iterated integral over a Type I region $∬Df(x,y)dA=∬Df(x,y)dydx=∫ab[∫g1(x)g2(x)f(x,y)dy]dx∬Df(x,y)dA=∬Df(x,y)dydx=∫ab[∫g1(x)g2(x)f(x,y)dy]dx$ Iterated integral over a Type II region $∬Df(x,y)dA=∬Df(x,y)dxdy=∫cd[∫h1(y)h2(y)f(x,y)dx]dy∬Df(x,y)dA=∬Df(x,y)dxdy=∫cd[∫h1(y)h2(y)f(x,y)dx]dy$
Double integral over a polar rectangular region $RR$ $∬Rf(r,θ)dA=limm,n→∞∑i=1m∑j=1nf(rij*,θij*)ΔA=limm,n→∞∑i=1m∑j=1nf(rij*,θij*)rij*ΔrΔθ∬Rf(r,θ)dA=limm,n→∞∑i=1m∑j=1nf(rij*,θij*)ΔA=limm,n→∞∑i=1m∑j=1nf(rij*,θij*)rij*ΔrΔθ$ Double integral over a general polar region $∬Df(r,θ)rdrdθ=∫θ=αθ=β∫r=h1(θ)r=h2(θ)f(r,θ)rdrdθ∬Df(r,θ)rdrdθ=∫θ=αθ=β∫r=h1(θ)r=h2(θ)f(r,θ)rdrdθ$
Triple integral $liml,m,n→∞∑i=1l∑j=1m∑k=1nf(xijk*,yijk*,zijk*)ΔxΔyΔz=∭Bf(x,y,z)dVliml,m,n→∞∑i=1l∑j=1m∑k=1nf(xijk*,yijk*,zijk*)ΔxΔyΔz=∭Bf(x,y,z)dV$
Triple integral in cylindrical coordinates $∭Bg(x,y,z)dV=∭Bg(rcosθ,rsinθ,z)rdrdθdz=∭Bf(r,θ,z)rdrdθdz∭Bg(x,y,z)dV=∭Bg(rcosθ,rsinθ,z)rdrdθdz=∭Bf(r,θ,z)rdrdθdz$ Triple integral in spherical coordinates $∭Bf(ρ,θ,φ)ρ2sinφdρdφdθ=∫φ=γφ=ψ∫θ=αθ=β∫ρ=aρ=bf(ρ,θ,φ)ρ2sinφdρdφdθ∭Bf(ρ,θ,φ)ρ2sinφdρdφdθ=∫φ=γφ=ψ∫θ=αθ=β∫ρ=aρ=bf(ρ,θ,φ)ρ2sinφdρdφdθ$
Mass of a lamina $m=limk,l→∞∑i=1k∑j=1lmij=limk,l→∞∑i=1k∑j=1lρ(xij*,yij*)ΔA=∬Rρ(x,y)dAm=limk,l→∞∑i=1k∑j=1lmij=limk,l→∞∑i=1k∑j=1lρ(xij*,yij*)ΔA=∬Rρ(x,y)dA$ Moment about the x-axis $Mx=limk,l→∞∑i=1k∑j=1l(yij*)mij=limk,l→∞∑i=1k∑j=1l(yij*)ρ(xij*,yij*)ΔA=∬Ryρ(x,y)dAMx=limk,l→∞∑i=1k∑j=1l(yij*)mij=limk,l→∞∑i=1k∑j=1l(yij*)ρ(xij*,yij*)ΔA=∬Ryρ(x,y)dA$ Moment about the y-axis $My=limk,l→∞∑i=1k∑j=1l(xij*)mij=limk,l→∞∑i=1k∑j=1l(xij*)ρ(xij*,yij*)ΔA=∬Rxρ(x,y)dAMy=limk,l→∞∑i=1k∑j=1l(xij*)mij=limk,l→∞∑i=1k∑j=1l(xij*)ρ(xij*,yij*)ΔA=∬Rxρ(x,y)dA$ Center of mass of a lamina $x−=Mym=∬Rxρ(x,y)dA∬Rρ(x,y)dAx−=Mym=∬Rxρ(x,y)dA∬Rρ(x,y)dA$ and $y−=Mxm=∬Ryρ(x,y)dA∬Rρ(x,y)dAy−=Mxm=∬Ryρ(x,y)dA∬Rρ(x,y)dA$
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# Leibniz
## 3.1.3 Concave and convex functions
The concept of diminishing marginal product corresponds to the mathematical property of concavity. By looking at the mathematical idea of concave and convex functions, we can gain some further insights into the economic properties of production functions.
concave function
A function of two variables for which the line segment between any two points on the function lies entirely below the curve representing the function (the function is convex when the line segment lies above the function).
We saw in Leibniz 3.1.2 that in the case of the production function $y = Ah^\alpha$, with $A \gt 0$ and $0 \lt \alpha \lt 1$, the marginal product of labour is diminishing. This means that when we move to the right along the graph of the production function, the slope of the curve decreases. A function with this property is said to be concave.
An implication of concavity (and its algebraic definition) is that ‘the function of the average is greater than the average of the function’. To illustrate what this statement means, suppose that for a function $f(h)$, we take any two values $h_0$ and $h_1$. Then:
The left-hand side is the function of the average of the two values, and the right-hand side is the average of the function of the two values. (To see why the inequality holds, try drawing a concave production function, choosing two values on the horizontal axis, and finding the points on the diagram that correspond to the two averages.)
We can give this property a very neat economic interpretation. To understand what it means, consider the following example.
Suppose that Alexei has a production function like the one above, with $A = 20$ and $\alpha = 0.5$; that is:
He has just started at university and is considering two different ways of organizing his time. Because he does not yet know anyone, he thinks the first term might be better spent socializing, so that his average daily hours of study for the end-of-semester exam would be $0$. Having established his position on the social scene, he would return to study with full fervour in the second semester and study $9$ hours a day, every day. From his production function, we find that his grades under this regime would be $20\sqrt{0} =0$ for the first semester exam, and $20\sqrt{9} =60$ for the second semester. His average exam result would thus be $\frac{1}{2}\times 0 + \frac{1}{2}\times 60=30$.
Alternatively, he could work on his social life and academic results more steadily, studying $4.5$ hours a day every day in both semesters. Notice that under this strategy, he gives up the same total hours of free time as under the previous approach—the total inputs are the same. What grades can he then expect? He will get $20\sqrt{4.5} = 42.4$ in each semester, which gives him $42.4$ on average.
Comparing these two possible strategies, Alexei realizes that in his case, slow and steady indeed wins the race, because his total output is higher when hours are constant rather than fluctuating. This is the economic implication of concavity.
By contrast, had we assumed that $\alpha \gt 1$, we would have found that total output is higher when hours fluctuate: in this case, Alexei learns more when he studies more intensely for a shorter period. When $\alpha \gt 1$, the slope of the graph of the production function increases as hours increase, so the marginal product of labour is increasing rather than diminishing, as in Figure 3 in which $\alpha =1.6$. We would then describe the production function as convex rather than concave. A special case is the function $y=Ah^2$: you can check by differentiating that for this production function, the graph of the marginal product of labour is an upward-sloping straight line.
The production function y = 1.5h1.6 and the corresponding marginal product.
Figure 3 The production function y = 1.5h1.6 and the corresponding marginal product.
Read more: Section 8.4 of Malcolm Pemberton and Nicholas Rau. 2015. Mathematics for economists: An introductory textbook, 4th ed. Manchester: Manchester University Press.
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## Parametrization of a regular planar polygon with an arbitrary number of sides
I was wondering if anyone knew of a common technique for parametrizing a regular polygon with an arbitrary number of sides. I figured such a problem would be easy or at least be well documented online, but that doesn't seem to be the case.
I started by assuming that the polygon was centered at the origin of the Polar Plane and the sides were of length R. Then since the polygon has n vertices, we can draw n line segments, each starting at the center and protruding to a vertex. Now, we know that the sides are all equal, as well as the interior angles, and that the center is equidistant from each vertex, then the angles between the circumradii are multiples of 2pi/n. These contraints also require that the circumradii bisect the interior angles, and therefore they partition the polygon into n isosceles triangles. Then the magnitude of the interior angle situated at the ith vertex is then given by
$$\left| {\angle {V_i}\left( {{P_n}} \right)} \right| = \pi - \frac{{2\pi }}{n}$$
we now use the law of sines to determine the length of the ith radius.
$$\left| {{R_i}\left( {{P_n}} \right)} \right| = \frac{{\sin \left( {\frac{\pi }{2} - \frac{\pi }{n}} \right)R}}{{\sin \left( {\frac{{2\pi }}{n}} \right)}} = \frac{R}{{2\sin \left( {\frac{\pi }{n}} \right)}}$$
Next, I transformed these pairs into their Cartesian representation because parametrizing straight line segments in polar form seemed a little inefficient. Which is simple enough but then, in attempting to construct a unit vector, computation became cumbersome very quickly. I assumed that there was a more compact form since the magnitude must be independent of i, but after attempting to use the multiple angle formulae I gave up.
So, I went back to polar form and realized that the the radius vector oscillates back and forth between the circumradius and inradius as the polar angle varies, but I can't seem to put that statement into a parametrization of the polygon.
Anyone have any suggestions? I'm thinking this has to be something simple, and hopefully is somewhat elegant. My brain is just still in full reboot mode from midterms. :(
PhysOrg.com science news on PhysOrg.com >> Galaxies fed by funnels of fuel>> The better to see you with: Scientists build record-setting metamaterial flat lens>> Google eyes emerging markets networks
Btw sorry if you don't like my notation, I tend to make it up as I go along and I'm extremely OCD when it comes to notation, neglecting brevity for the sake of organization.
Could you give a precise definition of what you mean by a parametrization of the polygon? Are you referring to the interior region or the boundary? If you are referring to the boundary "curve", then you could do something like this: For 0
## Parametrization of a regular planar polygon with an arbitrary number of sides
My bad, I meant the boundary. And I was looking for a more of a vector representation
If you use Euler's identity and multiplication of complex numbers, then the the expression I wrote works as a constant speed, vector valued function of t that starts at (1,0) and goes counterclockwise around the polygon. http://en.wikipedia.org/wiki/Euler%27s_formula This assumes that one of the polygon's vertices is (1,0). Then each successive vertex will be given by exp(i2pi*k/n). For example, the next vertex would be (cos(2pi/n), sin(2pi/n)). So the parametrization of the first side would be: (1,0) + t*( cos(2pi/n) - 1, sin(2pi/n)) for 0 < t < 1.
Yes, I know I could take the real and imaginary parts of that as the x and y vectors but I meant in terms of x and y or r and theta without introducing a parameter t. I guess it's easier for me to see what's going on this way For instance, for n= 4 $$\begin{array}{l} {{\vec P}_4}\left( {x,y} \right) = \left( {x\hat x - R\hat y} \right) - \left( { - R\hat x - R\hat y} \right) = \left( {x + R} \right)\hat x\\ x \in \left[ { - R,R} \right]\\ {{\vec P}_4}\left( {x,y} \right) = \left( {R\hat x + y\hat y} \right) - \left( {R\hat x - R\hat y} \right) = \left( {y + R} \right)\hat y\\ y \in \left[ { - R,R} \right]\\ {{\vec P}_4}\left( {x,y} \right) = \left( { - x\left( { - \hat x} \right) + R\hat y} \right) - \left( {R\hat x + R\hat y} \right) = \left( { - x + R} \right)\left( { - \hat x} \right)\\ x \in \left[ {R, - R} \right]\\ {{\vec P}_4}\left( {x,y} \right) = \left( {R( - \hat x) - y( - \hat y)} \right) - \left( {R( - \hat x) - R( - \hat y)} \right) = \left( { - y - R} \right)( - \hat y)\\ y \in \left[ {R, - R} \right] \end{array}$$ I'm a weirdo but Idk like I said it's just easier for me to see whats going on this way. I was looking for kind of an analogous parametrization for an arbitrary value of n.
Oh wait you're talking roots of unity! Duh! Wow my bad lol
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## Capture without multiple histograms
Discussions on extending SharpCap using the built in Python scripting functionality
SteveJP
Posts: 27
Joined: Sat Apr 28, 2018 8:35 am
Location: Melbourne, Australia
### Capture without multiple histograms
Hi,
I'm using SharpCap.SelectedCamera.CaptureSingleFrame() to capture a series of frames, but I get an additional ..CameraSettings.txt file and a histogram file each time. I'd like to limit this to one of each of these files per capture session like the standard capture dialog provides for. Is there a way to optionally suppress the generation of the settings and histogram files?
Thanks
Steve
Posts: 2522
Joined: Sat Feb 11, 2017 3:52 pm
Location: Vale of the White Horse, UK
Contact:
### Re: Capture without multiple histograms
Hi Steve,
you can turn the creation of these files on and off from scripting by assigning the appropriate value to
Code: Select all
SharpCap.Settings.CreateCameraSettingsFile
Cheers, Robin
SteveJP
Posts: 27
Joined: Sat Apr 28, 2018 8:35 am
Location: Melbourne, Australia
### Re: Capture without multiple histograms
Thu Dec 13, 2018 6:36 pm
Hi Steve,
you can turn the creation of these files on and off from scripting by assigning the appropriate value to
Code: Select all
SharpCap.Settings.CreateCameraSettingsFile
Cheers, Robin
Thanks Robin, that's great.
### Who is online
Users browsing this forum: No registered users and 0 guests
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# Lower bound of generating a biased coin?
If we have an unbiased coin and we want to generate a biased coin with probability $p$ of getting a head and $1-p$ of getting a tail. What is the lower bound of the expected number of flips that generating this biased coin?
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As a function of $p$? – deinst Mar 26 '12 at 20:41
For which simulation algorithm? – Did Mar 26 '12 at 20:58
DidierPiau: Not a specific kind, just to find out a lower bound – Mathematics Lover Mar 26 '12 at 21:22
$1$ is a lower bound; $2$ is an upper bound for the expectation. What do you mean by the lower bound? – Henry Mar 26 '12 at 21:42
I guess you are looking for entropy, in such case you need $p\log_2 (p) + (1-p)\log_2(1-p)$ coin flips to get your result (note that for $p\in [0,1]$ the result is in $[0,1]$ too, i.e. for $p \neq 0.5$ you will need less than one coin flip). (In other cases, what happens if $p$ is transcendental?) – dtldarek Mar 26 '12 at 23:14
This answer concerns the maximal (rather than expected) number of tosses; which is not what is asked for.
After flipping a fair coin $n$ times, you have $2^n$ equally likely outcomes. Every event defined in terms of these outcomes has probability $k/2^n$ for some $k\in\{0,\dots,2^n\}$. And conversely, for every $k$ there is such an event. Conclusion:
Required number of flips is $\inf\{n: 2^np\in\mathbb Z\}$.
Which is infinite when $p$ is not a dyadic rational; meaning that for such $p$ you can't simulate the biased coin at all.
Example: if $p=0.375$, you need $3$ flips.
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This is not correct: note that the problem says "expected number of coin flips". E.g. you can simulate a biased coin with $p = \frac13$ (which is not a dyadic rational) by the process of flipping a coin twice, saying "heads" if you see "HH", "tails" if you see "HT" or "TH", and repeating the experiment if you see "TT". This has probability $\frac14$ of requiring yet another pair of flips, but the expected number of coin flips to terminate is $\frac43 (2) = 8/3$, not infinite. See e.g. this question (there are probably others too on math.SE). – ShreevatsaR Jul 16 '13 at 4:18
@ShreevatsaR Good point; would you like to give a correct answer to this question? – 40 votes Jul 16 '13 at 4:21
If you expand $p$ in binary, then take head of your flips as a $1$, tail as a $0$, you can state head or tail as soon as you disagree with $p$. There is no lower bound, as the sequence of flips could match the expansion of $p$ as long as you want. Say $p=\frac 13=0.01010101\overline{01}_2$. As long as you alternate tails and heads you can't tell. With probability $1$ that will stop sometime. The expected number of flips is $\sum_{i=1}^{\infty} \frac i{2^i}=2$
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Lower bound on the expected number of the flips is what the problem asks for. Did you mean "there is no upper bound"? (Which also should exist (in terms of $p$), I guess?) And are you saying that that the expected number of flips is $2$ irrespective of the value of $p$? Something seems wrong with that... I haven't thought through this clearly, but e.g. if $p = \frac1{100}$ I suspect we'll need at least 7 flips (and probably more). – ShreevatsaR Jul 16 '13 at 5:27
Actually now I understand why you say there is no lower bound: in the case of $p=\frac1{100}$ for instance, there is less information than in the $p=\frac12$ case, because most of the time we can just say "tails". (I was earlier thinking of the $(\frac1{100}, \frac1{100}, \dots, \frac1{100})$ distribution rather than the $(\frac1{100}, \frac{99}{100})$ distribution we care about here.) However, there is still a lower bound in terms of $p$, I think it's $H(p)/H(\frac12)$ i.e. $p\lg\frac1p+(1-p)\lg\frac1{1-p}$. This is $< 1$ though. – ShreevatsaR Jul 16 '13 at 6:56
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# AEM Dispatcher Custom Invalidation Scripts
Dec. 5, 2019, 6:31 p.m. bryce AEM Dispatcher AEM Tips and Tricks
When Adobe made the change to begin using etc.clientlibs as a directory, we encountered issues with some existing Dispatcher cache invalidation configurations. For certain cache files, Dispatcher may delete the file on disk when a flush is called. Directories should not get deleted and Dispatcher will drop a .stat file within them. This .stat file's time stamp is what Dispatcher checks when deciding whether to update the requested cache file or not.
However, because of that pesky . character within the name of the client library cache, the Dispatcher was fooled into thinking the cache (etc.clientlibs) should be deleted on invalidation of any /etc files. This caused further issues when it would attempt to recreate the directory. On the next request it could be unable to create the etc.clientlibs directory at the file system level due to a race condition. Thus, no cached client libraries would be created, and until this was manually fixed (recreate the directory by hand), all requests would hit the Publishers.
To fix this we wrote a shell script which was triggered by the Dispatcher during invalidation. To call the script we added the following to our Dispatcher configuration:
/invalidateHandler "/opt/dispatcher/scripts/invalidate.sh" [1]
This ran our invalidate.sh [2] shell script every time our Dispatcher's cache was invalidated from a Flush Agent, thus triggering a recreation of the etc.clientlibs directory and preventing our race condition.
script tips dispatcher
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# Entropy Regularised Deterministic Optimal Control: From Path Integral Solution to Sample-Based Trajectory Optimisation
Sample-based trajectory optimisers are a promising tool for the control of robotics with non-differentiable dynamics and cost functions. Contemporary approaches derive from a restricted subclass of stochastic optimal control where the optimal policy can be expressed in terms of an expectation over stochastic paths. By estimating the expectation with Monte Carlo sampling and reinterpreting the process as exploration noise, a stochastic search algorithm is obtained tailored to (deterministic) trajectory optimisation. For the purpose of future algorithmic development, it is essential to properly understand the underlying theoretical foundations that allow for a principled derivation of such methods. In this paper we make a connection between entropy regularisation in optimisation and deterministic optimal control. We then show that the optimal policy is given by a belief function rather than a deterministic function. The policy belief is governed by a Bayesian-type update where the likelihood can be expressed in terms of a conditional expectation over paths induced by a prior policy. Our theoretical investigation firmly roots sample based trajectory optimisation in the larger family of control as inference. It allows us to justify a number of heuristics that are common in the literature and motivate a number of new improvements that benefit convergence.
• 6 publications
• 7 publications
05/06/2022
10/07/2019
### Stochastic Optimal Control as Approximate Input Inference
Optimal control of stochastic nonlinear dynamical systems is a major cha...
09/13/2019
### HJB Optimal Feedback Control with Deep Differential Value Functions and Action Constraints
Learning optimal feedback control laws capable of executing optimal traj...
03/23/2021
### Smoothing-Averse Control: Covertness and Privacy from Smoothers
In this paper we investigate the problem of controlling a partially obse...
10/16/2020
### Direct Policy Optimization using Deterministic Sampling and Collocation
We present an approach for approximately solving discrete-time stochasti...
03/01/2019
### GuSTO: Guaranteed Sequential Trajectory Optimization via Sequential Convex Programming
Sequential Convex Programming (SCP) has recently seen a surge of interes...
03/29/2021
### Distributionally Robust Trajectory Optimization Under Uncertain Dynamics via Relative-Entropy Trust Regions
Trajectory optimization and model predictive control are essential techn...
## I Introduction
Trajectory optimisation is ubiquitous in robotics. It is used to synthesise complex dynamic behaviour [mordatch2012trajopt] as well as to compute real-time feedback control [dantec2021icra]. Currently there are two classes of algorithms that address trajectory optimisation: (1) gradient based algorithms deriving from iLQR [todorov2005ilqr] and DDP [mayne1966ddp] that rely on linear-quadratic approximations about a nominal trajectory, and (2) sample-based approaches akin to Model Predictive Path Integral (MPPI) control [williams2016agressive, williams2017model, williams2018information]. As opposed to the first category, the second class of methods relies on sampled trajectories to probe the local optimisation landscape and use approximate inference techniques to update the solution. These methods ought to be less prone to failure when the models are non-differentiable or many local minima exist [ha2016path]. In addition, they are highly parallelisable [williams2018information]. For this reason sample-based trajectory optimisation has gained the interest of many researchers in robotics[ha2018path, kahn2021badgr, nagabandi2020deep, bhardwaj2021fast].
MPPI derives from a restricted subclass of stochastic optimal control where inference emerges naturally; see [kappen2005linear] and sec. II. This class is known as Linearly Solvable Optimal Control (LSOC) or path integral control. Essentially, the optimal policy can be expressed as a conditional expectation of an exponential cost-to-go. Because the expectation is taken over the passive stochastic dynamics, it is possible to compute the control from passive sampled trajectories.
A body of work has proposed adaptations to MPPI by pursuing similarities with well-known gradient-based and stochastic optimisation algorithms[stulp2012path, bhardwaj2021fast, rajamaki2016sampled, ghandi2021ieeeral, lefebvre2019path]. These adjustments outperform the original algorithm but lack justification. The main objective of this article is to root the derivation of sample-based trajectory optimisers in a generic theoretical framework. In fact, MPPI belongs to a larger family of ideas collectively known as control as inference. The idea is to reformulate the optimal control problem as a probabilistic inference problem so one can draw from the computational machinery that addresses inference. There are a number of ways to do this, and there is still ongoing research to determine how the different approaches are connected to each other [rawlik2013stochastic, Watson2021cai, levine2018reinforcement]
. The main classes are path integral control, entropy regularised reinforcement learning
[ziebart2010modeling, rawlik2013stochastic] and message passing [toussaint2009robot, watson2020stochastic]. Message passing algorithms reformulate stochastic optimal control as Bayesian input estimation by reinterpreting the likelihood of an observation as the desirability of a state-action pair. This problem can be solved using the Bayesian filtering and smoothing equations. As far as we are aware of, no sample-based algorithms are related to this idea. Entropy regularisation in reinforcement learning addresses stochastic optimal control with additional entropic terms in the control objective. This leads to a so-called soft Bellman recursive equation, which can be solved for Linear Gaussian Quadratic regulators but not for arbitrary non-linear problems.
Regardless, entropy regularisation turns out to be a fruitful direction for our purpose. Recent research illustrates how the principle of entropic inference can be put forth as a principled motivation for entropy regularisation in deterministic optimisation [lefebvre2020elsoc, luo2019minima]. In this paper we make a direct connection between entropy regularised optimisation and deterministic optimal control. To this end, we answer two questions
1. Why does entropic regularisation work in stochastic optimisation, and what is the underlying principle?
2. Why do heuristic adjustments of MPPI, not justified by theory, work better than the original algorithm?
The first step towards answering these questions was taken in [lefebvre2020elsoc], where the underlying problem statement was still closely related to LSOC. As a result, the dynamics had to be invertible and the optimisation variable was a state transition distribution rather than a policy. To extract a locally linear feedback policy, the dynamics also needed to be control affine.
In this paper, we essentially show that we can drop these assumptions. We specifically demonstrate that entropic deterministic control gives rise to a path integral expression for optimal control and how a method akin to MPPI can be derived from this result. Therewith, we establish a theoretical foundation that allows the derivation of generic sample-based search algorithms tailored to deterministic trajectory optimisation.
### I-a Notation
Discrete time is denoted with subscript . The iteration number in algorithms is denoted with subscript . Entities in sample sets are labelled with subscript . We use bold font to denote sequences, e.g. . We use time subscript to denote a subsequence starting from time instant , e.g. . A sequence can also be expressed as . We rely on context to imply the range. We combine both notations so that e.g. refers to iterates of a sequence . Expression implies that the uncertain variable is distributed according to distribution , denotes the expected value of .
denotes the multivariate Gaussian or Normal distribution with mean
and positive definite covariance matrix . is often shortened to .
### I-B Problem description
We consider finite horizon discrete time deterministic optimal control. Variables and denote states and controls, respectively. We assume dynamics are governed by a non-linear time variant difference equation given by . State-action couples are denoted as , except for . A trajectory is defined as . A feasible trajectory satisfies the dynamic difference equation. Functions and denote the running and terminal costs, respectively. The cost-to-go from is defined as
Rn(τn)=rN(sN)+∑N−1n′=nrn′(sn′,an′)
We consider the trajectory optimisation problem defined below. The problem solves for open-loop control sequence .
minaR0(τ0) s.t. sn+1=fn(sn,an), s0=s∗0 (1)
The optimal cost-to-go or value function is defined as , subject to the dynamics and initial state . Relying on Bellman’s principle of optimality, (1) can be cast into a recursive problem. This gives rise to Bellman’s backward recursion equation with boundary condition . Sequence contains state-dependent closed-loop policy functions .
Vn(s) =minarn(s,a)+Vn+1(fn(s,a)) (2) an(s) =argminarn(s,a)+Vn+1(fn(s,a))
## Ii Control as inference through LSOC
LSOC refers to an interesting but restrictive subclass of continuous-time stochastic optimal control problems. Here, we give a brief overview of the main ideas [kappen2005linear]. We consider control affine stochastic dynamics where denotes a Wiener process, so that
A state dependent cost-to-go is defined as
C(t)=cT(s(T))+∫Ttc(τ,s(τ))dτ
We look for a continuous time optimal control . The cost is appended with an input-dependent cost. Note that the rate is inversely proportional to the noise covariance.
defines the continuous-time value function. The expectation is taken over the path probability
induced by the stochastic dynamics and conditioning on .
V(t,s(t))=mina(t→T)EP(a|s(t))[C(t)+∫Ttλ2∥a(τ)∥2Σ−1dτ] (3)
Defining the desirability function , it can be shown that the solution of (3
) is governed by a linear partial differential equation. Remarkably, the solution can then be expressed as a path integral according to the Feynman–Kac formula. Note that the expectation is taken over
passive paths.
Z(t,s(t))=EP(0|s(t)[exp(−1λC(t))]
Second it can also be shown that the optimal control satisfies
a∗(t,s(t))=1Z(t,s(t))EP(0|s(t)[exp(−1λC(t))dξ]
On account of Girsanov’s theorem, the measure of the expectation can be changed to the system dynamics induced by any arbitrary control [ha2016path]. This basically amounts to importance sampling in continuous time. Here, represents the Radon-Nikodym derivative of with respect to .
a∗(t,s(t))∝EP(ag|s(t)[dP(0)% dP(ag)exp(−1λC(t))dξ]
This summarises the main concepts from LSOC.
From this theory, one can derive a sample-based trajectory optimisation algorithm known as MPPI. Because this (and related) method(s) rely on the calculation of a path integral, they are also referred to as path integral control. The MPPI algorithm is summarised in Alg. 1. Due to Girsanov, sampled trajectories are obtained about a reference trajectory induced by with simulated control perturbations . An updated control is inferred according to the theory of LSOC.
A few other remarks are in place:
1. The algorithm solves stochastic optimal control problem (3). The policy itself is (thus) deterministic and inference is established by the inherent process noise.
2. The control penalty in (3) is quadratic, i.e., , with . Hence, when choosing two out of the three parameters , and , we fix the problem that we solve.
3. In practice, the covariance is often updated comparable to CMA-ES [stulp2012path, bhardwaj2021fast], .
4. Also, the theoretical running cost in line 8 is replaced by a generalised running cost, .
Although the two adjustments proposed above lead to better performance, they are not supported by any theory. Furthermore, the combination of remarks 2 and 3 implies that we are solving different problems with every update. By consequence, we argue they are also not properly understood.
## Iii Entropic optimisation
In this section we answer question 1) mentioned in the introduction. It serves as a stepping stone to question 2).
Algorithms for numerical problems, such as optimisation, proceed iteratively, with each iteration providing information that improves a running estimate of the correct solution. Probabilistic numerics [oates2019probnum] pursues methods that, in place of such estimates, update beliefs111The manifestation and interpretation of probability is epistemic and arises from missing information in a computation that is otherwise deterministic. over the solution space. In brief, in this section, we aim to rephrase the generic problem of optimisation as a problem of inference. By embedding deterministic optimisation into the framework of entropic inference, we provide a theoretical argument for the use of entropy regularisation in optimisation in addition to the vast empirical validation documented in previous work.
### Iii-a Entropic inference
In probability theory, inference refers to the rational processing of incomplete information
[jaynes2003]. In the present context, we make use of probabilities to encode our uncertainty about an underlying deterministic quantity and refer to them as beliefs. We seek a posterior, , which encodes new information into a prior, , which encodes information that we already have.
In Bayesian inference, new information is contained in data or experiments. To the contrary, we are interested in information that is contained in an expectation. The principle that allows us to address this type of information is that of minimum relative entropy or discrimination information
[kullback1951information, jaynes1986background, jaynes1982rationale]. The principle states that the unique posterior,
, living in the space of probability distributions,
, constrained by an expectation of the form , is the one that is hardest to discriminate from the prior, . Equivalently, it is the one that minimises their relative entropy. Mathematically, this gives rise to a variational optimisation problem of the following form
minπ∈P D[π∥ρ]relative entropy s.t. μ=Eπ[g]new information (4)
where . The solution is a Boltzmann distribution with so that .
### Iii-B Entropic inference for optimisation
Mathematical optimisation addresses problems of the following form where represents the feasible subset.
x∗=argminx∈Xq(x) (5)
Classic numerical optimisation strategies iterate a running estimate of the solution , assimilating new information with each iteration.
Instead of focussing on a single estimate, we wish to model this search with a belief sequence . The more information that is assimilated, the more certain we get about the solution. To suit the purpose of optimisation, such a sequence should exhibit the following property
limg→∞Eπg[x]=argminx∈Xq=x∗ (6)
Second, we need an inference procedure that facilitates an update operation based on some form of new information
πg+1←πg
Our approach to arrive at such an inference procedure tailored to optimisation is straightforward. A priori no information is available. We can represent this situation mathematically by encoding our uncertainty about the solution in a prior probability density function,
222If no information is available, we can choose .
. In order to gradually decrease our uncertainty, we then look for a posterior probability density function,
, that discriminates the least from our initial guess but has an expectation over the objective function that produces a lower estimate than the prior expectation.
minπ∈P D[π∥ρ] (7) s.t. Eπ[q]≤Eρ[q]−Δ, Δ>0
Whether a solution exists depends on the parameter . If it exists, the solution is a Boltzmann distribution similar to (4). It can be shown that if we choose , there also exists some [luo2019minima]. For now, this is sufficient.
π(x)∝ρ(x)⋅e−λq(x) (8)
Clearly, these expressions bear close correspondence with the classical Bayesian update, generating a posterior, , by multiplying a prior, , with an expression encoding a likelihood, which is the likelihood of optimality in this case. By applying substitutions and , we can establish the sequence . This sequence has exactly property (6) [luo2019minima].
This property basically implies that the sequence converges to the Dirac delta distribution centred at the optimum. Since the next idea is to emulate the behaviour of this distribution numerically, the property also implies that the sample set will become more and more localised. Although this is exactly what we want from a theoretical point of view, this is troublesome from an algorithmic perspective. Therefore, we desire to encode a second prior into the sequence that stimulates exploration. We introduce the augmented entropic optimisation problem. Here represents the uniform on and is a scaling factor that allows us to attribute more importance to either prior.
minπ∈P αD[π∥πg]+(1−α)D[π∥UX],0<α<1 (9) s.t. Eπ[q]≤Eπg[q]−Δ, Δ>0
For some , the solution is given by333
For notational convenience, we absorb the uniform distribution
into the proportionality. Note that is therefore zero outside the set . [lefebvre2020elsoc]
πg+1∝παg⋅U1−αX⋅e−λq∝παg⋅e−λq (10)
and it can be verified that
limα→1limg→∞πg∝limα→1e−λ1−αq∝δ(x−x∗) (11)
By emulating this sequence numerically, we can construct a stochastic search algorithm tailored to the problem (5).
### Iii-C Stochastic search methods
The idea is to update a parametrised belief and match its empirical distribution features with those of the theoretical distribution . In practice, we use a parametric density function with (e.g., ) to approximate the entities in . We derive the associated parameter sequence, , from samples. Therefore, we project the theoretical entity onto the density space generated by , minimising their relative entropy. The objective is then manipulated into an expectation over , which we estimate by sampling .
θg+1 =argminθ∈ΘD[πg+1∥πθ] ≈argmaxθ∈Θ^EDg[e−λ(q+(1−α)logπθg)logπθ]
where . For details, see appx. A.
A basic implementation is presented in Alg. 2. Clearly, the computation bears close correspondence with Alg. 1, though it remains unclear what underlying principle is mutual. For further exploration of similar ideas, we refer to [abdolmaleki2015model, abdolmaleki2017deriving].
## Iv Entropic deterministic optimal control
In this section we answer question 2). We provide an original derivation for a sample based trajectory optimiser that demonstrates similar heuristics as were described in sec. II. Contrary to MPPI contrary our derivation follows directly from of the ideas in sec. III applied to the problem in sec. I-B.
Although entropy regularisation is a well-known concept in reinforcement learning, it has only been applied to stochastic problems that are naturally embedded in a probabilistic framework. As we will show for deterministic optimal control, entropy regularisation gives rise to explicit expressions for the posterior policies in terms of a conditional expectation taken over trajectories induced by prior policies similar to the setting of LSOC. The crucial difference with Entropic Deterministic Optimal Control (EDOC) is that LSOC inherently solves a stochastic optimal control problem whilst EDOC still solves a deterministic optimal control problem. The inference in LSOC is facilitated by the input noise inherent to the stochastic problem. Contrarily, in EDOC, the inference is put in place intentionally. The major consequence of this difference is that it disentangles the inference from the control.
### Iv-a Entropic Bellman equation
We consider the following EDOC problem, where and .
minπ∈P (12) s.t.
Starting from problem (1), we reason as follows. The optimisation variables are given by the control sequence . This problem is no different than problem (5), so we could address it with Alg. 2444 Note that in this case, the weights would not be time dependent.. However, we would instead like to exploit the dynamic structure of the problem. Although the solution of (1) is indeed a trajectory, from (2), we know that a sequence of optimal policies, , underpins this trajectory, which we only happen to know evaluated over deterministic dynamics. Therefore, we introduce the conditional policy beliefs, . These beliefs express our uncertainty about control given state at time for the -th iteration. Second, we define a trajectory probability density function conditioned on and .
pn(πn) =p(τn|sn,πn) =∏N−1n=nδ(sn′+1−fn′(sn′,an′))πn(an|sn)
This definition allows us to cast (1) in a probabilistic manner.
Vn(sn)=minanEpn(πn)[Rn(τn)]
In the case that the dynamics are deterministic, this probabilistic problem is completely equivalent to the original problem and solves for a Dirac distribution or deterministic policy. In turn, this probabilistic model motivates problem (12).
We will address the solution to this problem in the next paragraph. First, we further motivate our problem definition by demonstrating that we would have arrived at the same result if we had regularised the Bellman equation (2) instead. The following lemma certifies the consistency of the proposed regularisation. It also emphasises that we only regularise over the optimisation variables, , but not the dynamics. Finally, it introduces the entropic Bellman equation.
###### Lemma 1.
The solution of problem (12) is governed by the following entropic Bellman equation
Vg+1,n =minπ∈PEπ[Qg+1,n]+D[π∥παg,n⋅U1−αA] (13) πg+1,n =argminπ∈PEπ[Qg+1,n]+D[π∥παg,n⋅U1−αA]
where .
###### Proof.
First, by introducing multiplier , we can recast (12). Note that the dynamics cancel out in the fractions.
minπn∈PEpn(πn)[λRn+αlogπnπg,n+(1−α)logπnUA,n]
Then, writing out the objective, we retrieve an optimisation problem that adheres to Bellman’s principle of optimality
minπn∈PEpn(πn)[λrN+…∑N−1n′=n(λrn′+αlogπn′πg,n′+(1−α)logπn′UA)]=minπn∈PEπn[λrn+αlogπnπg,n+(1−α)logπnUA+…minπn+1∈PEpn+1(πn+1)[…λRn+1+αlogπn+1πg,n+1+(1−α)πn+1UA,n+1]]=minπn∈PEπn[λrn+αlogπnπg,n+(1−α)logπnUA+Vg+1,n+1]
where we have defined
Vg+1,n+1=minπn+1∈PEpn+1(πn+1)[λRn+1+logπn+1παg,n+1U1−αA,n+1]
The lemma follows. ∎
Henceforth, we will treat as a hyper-parameter.
### Iv-B Path integral solution
In this section, we address the tractability of the EDOC problems in (12) and (13). First, we establish a recurrence relation for the optimal policy belief sequence similar to that done for problem (9) with equation (10). Second, we illustrate that this solution gives rise to an explicit path integral expression for the optimal posterior policy belief.
The first result is summarised by the following theorem. For the proof, we refer to appendix B. This result is known as the soft Bellman equation and has been studied in combination with stochastic system dynamics, e.g., in [rawlik2013stochastic] for and in [levine2018reinforcement] for . The second, which is novel, is summarised in the corollary beneath it.
###### Theorem 1.
The solution of problem (13) is given by
πg+1,n ∝παg,nexp(−Qg+1,n+Vg+1,n) Vg+1,n =−log∫παg,nexp(−Qg+1,n)da
Theorem 1 points out that, similarly to equation (10), the relationship between the posterior and prior optimal policy belief functions ( and , respectively) is governed by a Bayesian type recurrence relation. However, in this case, the recursion still depends on the functions and as a consequence of the problem’s sequential nature. This is in contrast to (10), which only depends on the objective .
Fortunately, the results can be further developed into an explicit expression for the posterior optimal policy belief function that depends solely on prior information contained in . This result is a direct consequence of the deterministic system dynamics and is summarised by the following corollary.
###### Corollary 1.
The posterior optimal policy belief function can be expressed as
πg+1,n(a|s)=πg,n(a|s)e−rg,nZg+1,n+1(fn(s,a))Zg+1,n(s)
where
rg,n =λrn+(1−α)logπg,n Zg+1,n =exp(−Vg+1,n)
and
Zg+1,n(s) =Epn(πg,n)[exp(−Rg,n)] pn(πg+1,n) ∝pn(πg,n)exp(−Rg,n) Rg,n =λrN+∑N−1n′=nrg,n′
###### Proof.
First, we define the function as in the theorem. Substituting the transformation into the solution of Theorem 1 yields the following recurrence relation for (this is only because is evaluated deterministically in ).
Zg+1,n =∫παg,ne−λrnZg+1,n+1da =Eπg,n[e−rg,nZg+1,n+1]
We can then easily develop this recursion into an explicit expression for as a conditional expectation over the path probability initialised at , that is, . Recall that this probability is conditioned on , so is indeed a function of it, which proves the expression for .
Zg+1,n=Epn(πg,n)[exp(−Rg,n)]
Now we can substitute this result into the recursive expression for and recover the main result.
Finally, one can substitute the explicit expression for into the trajectory probability, .
pn(πg+1,n)pn(πg,n)=Zg+1,n+1Zg+1,nZg+1,n+2Zg+1,n+1⋯Zg+1,N−1Zg+1,N−21Zg+1,N−1e−Rg,n
It follows that the quotients cancel out when we multiply over the entire trajectory expect for which in fact normalizes the trajectory density function. ∎
The corollary above implies that the function, , and hence the posterior policy belief, , can be quantified explicitly by evaluating conditional expectations over the prior path probabilities, . Although these results follow quite naturally from Theorem 1, they have important consequences in terms of the tractability and computability that are unique to the entropic regularisation of deterministic (rather than stochastic) optimal control. As noted, the framework collapses when aside from the purposeful uncertainty, stochasticity is introduced on account of the dynamics. In this case, an expectation over the stochastic dynamics emerges in the definition of , i.e., instead of . As a result, it is impossible to develop the recursion from Theorem 1 into explicit expressions because we cannot get rid of the expectation in the exponent.
Clearly, these expressions also bear a close correspondence with LSOC. We note that the Feynman-Kac formula establishes a relation between certain partial differential equations and stochastic processes. In particular, it expresses the solution of a partial differential equation as a conditional expectation. The association with stochastic processes emerges, as they constitute a framework where such conditional expectations arise naturally. In our work, the conditional expectation is not associated to any stochastic process, but rather to the Bayesian policy beliefs. Put differently we compute what we might refer to as a Bayesian path integral instead of a stochastic path integral. The difference lies in that the conditional expectation is taken over purposeful uncertainty introduced to construct a consistent inference procedure about the underlying deterministic optimal control problem; it is not inherent to the problem.
### Iv-C Entropic MPPI
From corollary (1), we can derive a sample-based trajectory optimiser akin to MPPI. The algorithm demonstrates similar heuristics to those discussed in sec. II, amongst novel attributes. Although the algorithm’s structure closely relates to Alg. 1, its derivation stems from an entirely different theoretical context. As a disclaimer, we note that our derivation seeks similarities with Alg. 1 intentionally and therefore uses a restricted class of parametric policy beliefs. However, we note that corollary (1) does not in fact exclude the use of any policy, so one could also consider using a Gaussian mixture to address certain problems, e.g., those with many local optima.
In particular, we approximate the posterior policy beliefs, , with a parametric distribution and infer the associated parameters from samples. In this case, we apply the prior policy belief sequence, , for a given and fixed initial state, , and collect data in the form of sample paths or trajectories, . This means that we will sample from the policies as if they were probability density functions rather than Bayesian belief functions. Averaging over these sample trajectories then allows us to evaluate the expressions in corollary 1. Given the clear resemblance with Alg. 1, we can reasonably refer to it as Entropic MPPI (EMPPI); Alg. 3.
#### Iv-C1 Policy belief parametrisation
Pursuing general tractability, we are interested in locally linear Gaussian policies with temporal distribution parameters, , where . With the exception of the covariance, this is similar to imposing a piecewise linear controller. Since the normal is unimodal, this approximation renders the stochastic search method locally similar to gradient-based approaches.
πg,n(a|s)≈πn(a|s;θg)=N(a|kg,n+Kg,ns,Σg,n)
#### Iv-C2 Projection strategy
Since the beliefs are conditioned on , we extend the projection idea with an expectation over the available samples for that time instant, . This expression denotes the probability of state conditioned on the initial state and the prior policy belief sequence .
θg,n =argmaxθ∑jwn,j∑jwn,jlogπ(an,j|sn,j,θ)
with
−logwn,j=Rg,n,j=λRn,j+(1−α)∑N−1n′=nlogπg,n′,j (14)
Apart from the cost, , the weights also contain a second term . Given that if was less likely than , this auxiliary cost encourages unlikely whilst penalizing likely actions. Note that if we had not augmented the entropic optimisation problem with a second prior this term would vanish (
). This optimisation problem is solved most efficiently by calculating a likelihood weighted estimate of the joint Gaussian distribution,
. Parameters are then found by conditioning on . Further note that we solve this problem for every
Kg+1,n =^Σas,g+1,n^Σ−1ss,g+1,n (15) kg+1,n =^μa,g+1,n−Kg+1,n^μs,g+1,n Σg+1,n =^Σaa,g+1,n−Kg+1,n^Σss,g+1,nK⊤g+1,n
where
^μτ,g+1,n =⟨τg,n,j⟩ ^Σττ,g+1,n =⟨(τg,n,j−^μτ,g+1,n)(τg,n,j−^μτ,g+1,n)⊤⟩
with .
Similar algorithms are described in references [williams2016agressive, williams2017model, williams2018information, bhardwaj2021fast, rajamaki2016sampled, ghandi2021ieeeral, lefebvre2019path]. Amongst these MPPI implementations, the presence of feedback, the presence of the term in the likelihood weights and the applicability to general deterministic optimal control problems are, as far as we are aware, unique to Alg. 3, and are now theoretically justified.
#### Iv-C3 Numerical implementation
To increase the overall numerical stability, we first use exponential smoothing
θg+1,n ←βθg+1,n+(1−β)θg,n (16)
Second the use of Monte Carlo estimates implies that . To estimate the covariance matrix of the trajectory, , we need at least a multiple of
samples. It is clear that the updates are prone to high variance for finite
. To remedy this issue we project the time signals on a polynomial space of order spanned by the basis . Despite this measure the update for remained too unstable. Hence instead we substitute a fixed gain matrix for in the updates in (15). Finally, since , an auxiliary procedure is used that changes until .
#### Iv-C4 Relation to other algorithms
Standard gradient-based trajectory optimisation algorithms iterate between a forward and a backward pass to probe the local problem geometry about the iterate trajectory and to optimise it, respectively. Similarly, Alg. 1 and 3 iterate between a forward Monte Carlo step and an inference step. As opposed to the backward step in gradient-based algorithms, the inference step does not possess a causal structure and is thus amenable to parallelisation . In Alg. 1 and 3, the weights are time dependent, so in some way, the recursive nature of the problem emerges. This is opposed to Alg. 2 and other stochastic search algorithms such as CMA-ES[abdolmaleki2017deriving], which could be used to solve (1) as well. The use of conditional policies is not straightforward, nor are the weights time dependent. So taking into account the blue print architecture of Alg. 2, EMPPI can be understood as a temporal rollout of an Evolutionary Strategy or a stochastic implementation of gradient-based trajectory optimisation algorithms where gradient information is inferred from the sampled trajectories. Finally, we emphasise again that Alg. 1 solves specific stochastic optimal control problem (3) whilst Alg. 3 solves general deterministic optimal control problem (1).
### Iv-D Numerical example
We present results for numerical experiments with a 4 dimensional planar robot arm operating in an environment with a single obstacle. This is not only to demonstrate the practical implications of our theoretical investigation (corollary 1 specifically), but also to investigate the effect of the different MPPI algorithms and changing their parameter settings (Alg. 3) on the exploration versus exploitation behaviour of the resulting distributions.
#### Iv-D1 Environment
The environment consists of a planar pendulum with links of length . Masses are concentrated at the end of each link. We use torque inputs directly instead of generating kinematic trajectories and relying on low-level controllers. An OCP is formulated with horizon . We use a relatively coarse time discretisation . The cost rate function is defined as , penalising the energy consumption and aggressive moves. We intentionally do not encode information about the non-linearity of the dynamics nor the obstacles. The system is drawn towards a final end-effector configuration using the final cost term , where
represents the distance vector between the end-effector and goal configuration. We do not represent the obstacles in the cost function. Interactions with the obstacles are strictly through the dynamics; gradient-based algorithms are not in favour here. The contact dynamics are modelled through forces
, where is the Heaviside function, is the distance vector between the obstacle and the near contact point and is the Jacobian matrix computed at the nearest contact point. Described contact dynamics are inelastic.
#### Iv-D2 Experiments
We compare three versions of Alg. 3. In version A, we set , only update , set and . Version A is the most closely related to the MPPI implementation in [williams2018information] and therefore serves as a baseline. An important improvement of MPPI comes from updating the covariance (deviating from the theory of LSOC) [stulp2012path, bhardwaj2021fast]. As a result, it can be observed empirically that the covariance collapses prematurely. Therefore, in version B, we set but update . According to (11), the policy belief functions will converge to a Dirac delta and it is anticipated that the search will converge prematurely. Version C implements the full algorithm. Versions A, B and C are initialised with the feedforward and with covariances . Unless specified, we set and then run 200 generations.
#### Iv-D3 Results
The solution after generations is visualised in Fig. 1. Clearly, only C completes the tasks successfully. As anticipated, we can observe the presence of premature distribution collapse for version B. The entropy of the distribution evaporates and the search stalls. The performance of A is superior to that of B yet also it fails to execute the final reach of the complete manoeuvre within 200 generations. The main benefit of C over A is that it can automatically adapt the covariance of the policy. This allows the policy to discover interesting directions more rapidly. One can also observe that some of the alternative histories still collide with the obstacle; howevers the bulk effectively reaches the goal. These observations are confirmed by Fig. 3 and 3. In particular, for version C, Fig. 3 clearly illustrates how the covariance of the policy self-adapts to the progress made on the problem.
## V Conclusion
Sample- based trajectory optimisation is a promising tool for robotics with complex and non-smooth dynamics and cost functions, both to synthesise complex behaviour and to compute real-time feedback controls. In this contribution, we have proposed an alternative derivation of the popular MPPI algorithm. Our derivation is founded on the framework of EDOC, an entropy-regularised version of the standard deterministic optimal control problem, for which we have shown that the optimal policy can be given by a policy belief function and expressed as a Bayesian path integral similarly to LSOC. We argue that our derivation allows for more principled future algorithmic development of sample-based trajectory optimisers and will become incorporated into other challenges and applications.
## Acknowledgements
This research received funding from the “Flemisch Artificial Intelligence Research (FlAIR)” programme, Belgium.
## Appendix A Stochastic search algorithms
We return to the derivation in section III-C. Now, we wish to manipulate it into an expectation over the prior , which would allow us to estimate it using Monte Carlo sampling. To arrive at the second line, we substitute the definition for the relative entropy. Since we optimise for , we can further neglect the first term. Then, to arrive at the third line, we make use of the recurrence relation. This expectation is then approximated using a sample where .
θg+1 =argminθ∈ΘD[πg+1∥πθ] =argmaxθ∈ΘEπg[π1−αge−λqlogπθ] ≈argmaxθ∈Θ^EDg[e−(λq+(1−α)logπθg)logπθ]
When we approximate beliefs using the Normal distribution ; this approach generates the updates shown in algorithm 2. We refer to [lefebvre2020elsoc] for further details.
## Appendix B Proof of Theorem 1
###### Proof.
Consider , where and are multipliers associated to the normalisation and inequality constraints.
L =
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# Site Reliability Engineering
Site Reliability Engineering (Platform Engineering, Production Engineering) is an engineering discipline enabling organisations to sustainably achieve the appropriate level of reliability in their platforms.
It applies software (Private) engineering principles to IT operations and service management. It can be considered a narrow implementation of DevOps, and is aimed at giving operators agency over their work.
Founding principles:
1. Operations is a software problem.
2. Manage by Service Level Objectives.
3. Work to minimise toil.
4. Automate this year's job away.
5. Move fast by reducing the cost of failure.
6. Share ownership with developers.
7. Use the same tooling, regardless of function or job title (but APIs will outlive tools).
Primary responsibilities:
• Monitor everything
• Reduce toil through automation and problem reduction
• Manage risk through SLIs, SLOs and an error budget
• Documenting and sharing knowledge, encouraging best practice
• Building resilient-enough services, early in the design phase
• Remediating escalations
• Carrying a pager and being on-call
• Learn from outages using meaningful postmortems
# Focuses
Nuances differ, but key focuses are commonly:
• Availability
• Latency
• Performance
• Efficiency
• Change management
• Monitoring
• Emergency response
• Capacity planning
# Core tenets
• Ensuring a durable focus on engineering by capping toil at 50% and diverting excess work (on-call rota, bugs) at the product team, producing postmortems for all incidents.
• Maximising change velocity without violating SLOs through use of an error budget to address the reliability vs innovation conflict. SRE recognises that there are many obstacles to 100% availability and that aiming for such is rarely valuable.
• Monitoring should alert only at the point action needs to be taken. Less critical notifications should be ticketed, and background noise should be relegated to logs.
• Change management, acknowledging that 70% of operational incidents are caused by changes, and reducing impact using progressive rollouts (see Continuous delivery), improving detection of problems and rolling back safely.
• Demand forecasting and capacity planning: ensuring there's sufficient capacity for user traffic through regular load tests based on accurate organic demand forecasts and inorganic event sources.
• Provisioning of instances based on capacity planning exercises.
• Efficiency and performance of the provisioned system must be maintained through monitoring and assessment of cost and performance.
## Embrace risk
Extreme reliability is costly; costs trend exponentially toward infinity for each additional nine. Often unconsidered is the opportunity cost of lost sales, caused by missed opportunities for product innovation.
Risk is measured against uptime (Nines of reliability). In a single region uptime can be measured in time:
availability = uptime / total time
Uptime for a multi-region service might be based on aggregate transactions:
availability = successful requests / total requests
Set quarterly targets and measure performance on a daily or weekly basis. Targets might consider:
• Expected level of service.
• Revenue generating?
• Paid or free?
• Competitor level of service
• The audience: consumers or enterprise
Risk tolerance should differ across failure modes: exposing users' data to the wrong audiences would be more harmful than a partial service outage.
The failure cases differ by workload too: throughput vs latency vs reliability.
## Automate this year's job away
Toil is operational work of little lasting value that can be automated away or removed entirely through reworking of software.
• Software engineering involves writing or modifying source code, either for automation or making robustness improvements.
• Systems engineering is system configuration or documentation for the purpose of making lasting improvements.
• Toil is work tied to operating a service that is manual, repetitive, automatable, has no lasting value, and scales proportionally to the service's growth.
• Overhead might be ticketing system hygiene, process improvement or HR activities like training.
Toil can be cathartic in lower volume but, as it scales, can drive low morale and cause career stagnation. Note the different tolerances for toil amongst different SREs.
## Release engineering
Philosophy:
• Self-service, enabling teams to be self-sufficient and determine their own release pace.
• High velocity teams want to reduce the lag time between features being completed and being available in production, e.g. push-on-green.
• Hermetic builds provide consistency and repeatability, allowing building historical versions in the event we need to troubleshoot a failure mode and cherry-picking fixes from newer branches onto existing deployed branches.
• Enforcement of policies and procedures, allowing gating operations that need review; e.g. source code and configuration changes that require review can't be merged to master prior to receipt of an approval.
Configuration management approaches differ by the change frequency and how it aligns with deployments. Prefer building static values in to the binary or as part of packaging where possible.
## Simplicity
Software is inherently dynamic and unstable; total stability is possible only inside a vacuum. Our job is to maintain the balance between agility and stability.
Boring won't wake you up at 3am. Avoid over-engineering, and don't be afraid of purging old code; it'll still be in the source code management history anyway and removing code reduces risk, maintenance burden and complexity.
Focused releases are easier to troubleshoot: change will happen, so minimising the scope of a release will help with isolation of a problem later.
Minimal APIs are the hallmark of a well understood problem. Modularity can introduce complexity, but can be used to demarcate different responsibilities between teams
# Practices
The Service Reliability Hierarchy applies Maslow's Hierarchy of Needs to service delivery. In order to deliver higher levels of the hierarchy the baser levels of the hierarchy must be met.
• Product
• Development
• Capacity planning
• Testing and release procedures
• Postmortem/Root cause analysis
• Incident response
• Monitoring
*[IT]: Information Technology
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2022-09-25 08:42:09 +0200 commented answer Batch processing Since you insist on a text file, the Python documentation should answer your question. A remark : you are not solvng eq 2022-09-25 08:41:28 +0200 commented answer Batch processing Since you insist on a text file, the Python documentation should answer your question. Noe in passing : you are not sol 2022-09-25 08:40:13 +0200 commented answer Batch processing Since you insist on a text file, the Python documentation should answer your question. 2022-09-25 08:38:56 +0200 commented answer Batch processing The Python documentation should answer your question. 2022-09-23 17:20:05 +0200 commented question Limit comparison test Your problem seems to be the use of arguments in a Python function rather than Sage-specific. A few notes : Your code 2022-09-20 08:31:40 +0200 answered a question Batch processing This tutorial and the relevant part of the manual should give you some inspiration... The very short version : load r 2022-09-20 05:16:11 +0200 received badge ● Nice Answer (source) 2022-09-19 13:45:48 +0200 edited answer How to write integrals symbolically in Sagemath similar to Mathematica ∫ is Unicode's U+222B. Depending on your platform, this can be input in various ways. For example : On most Linux's te 2022-09-17 08:54:17 +0200 commented question definite_integral of max function @FrédéricC : Would you mind making a proper answer for the benefit of future ask.sagemath.org perusers ? 2022-09-17 08:54:01 +0200 commented question definite_integral of max function @FrédéricC : Would you mind making a roper answer for the benefit of future ask.sagemath.org perusers ? 2022-09-17 08:43:29 +0200 answered a question How to write integrals symbolically in Sagemath similar to Mathematica ∫ is Unicode's U+222B. Depending on your platform, this can be input in various ways. For example : On most Linux's te 2022-09-14 08:21:16 +0200 commented question Simplifying a simple rational expression with indeterminate exponent @Juanjo : you should make an answer of your comment, for the benefit of future ask.sagemath.org perusers. 2022-09-11 10:45:13 +0200 answered a question How to extract list of vertices of Hamiltonian path What's wrong with : sage: G = graphs.Grid2dGraph(5,5) sage: %time hp=G.hamiltonian_path(algorithm="backtrack") CPU time 2022-09-04 10:00:53 +0200 edited answer How to name new libraries/modules properly with conflictions with 3rd party code you may import controlunder a name convenient for you. Example : sage: import sympy as sy now you can use "sy" to acc 2022-09-03 09:40:08 +0200 edited answer How to name new libraries/modules properly with conflictions with 3rd party code you may import controlunder a name convenient for you. Example : sage: import sympy as sy now you can use "sy" to acc 2022-09-03 09:38:51 +0200 answered a question How to name new libraries/modules properly with conflictions with 3rd party code you may import controlunder a name convenient for you. Example : sage: import sympy as sy now you can use "sy" to acc 2022-08-22 11:02:05 +0200 answered a question Subsets() with same element multiple times? Well... Let's try to be lazy, and reuse work already done : sage: subsets? Docstring: Iterator over the *list* 2022-08-20 13:19:34 +0200 commented question How to serially number the output ? @max-alekseyev/ : could you turn your comment into an answer, in order to help future perusers of ask.sagemath.org with 2022-08-20 13:14:21 +0200 commented question Effective resistance matrix and nonedgesonly=True This interesting question regards the implementation choices made by the pakage's author(s), is indeed a mathematical qu 2022-08-20 13:13:15 +0200 commented question Effective resistance matrix and nonedgesonly=True This interesting question regards the implementation choices made by the pakage's author(s), is a math question and coul 2022-08-20 13:12:59 +0200 commented question Effective resistance matrix and nonedgesonly=True This interesting question regards the implementation choices made by the pakage's author(s), is a math question and coul 2022-08-12 07:41:03 +0200 commented question How to copy huge output in another code? What is the problem with saving your "huge output" in a variable ? 2022-08-04 18:38:36 +0200 commented answer Solve fails to identify the maximum of the entropy function Alternatives : sage: Ex=-(x*log(x)+(1-x)*log(1-x)) sage: solve(Ex.diff(x),x) [log(x) == log(-x + 1)] Fails indeed. bu 2022-08-04 18:37:00 +0200 commented answer Solve fails to identify the maximum of the entropy function Alternatives : sage: Ex=-(x*log(x)+(1-x)*log(1-x)) sage: solve(Ex.diff(x),x) [log(x) == log(-x + 1)] Fails indeed. bu 2022-08-04 18:32:40 +0200 commented answer Solve fails to identify the maximum of the entropy function Alternatives : sage: Ex=-(x*log(x)+(1-x)*log(1-x)) sage: solve(Ex.diff(x),x) [log(x) == log(-x + 1)] sage: solve(Ex.dif 2022-08-01 07:02:12 +0200 answered a question Writing a matrix as a sum of matrices Well... This problem (decomposing a matrix as the sum of $p$ matrices with your constraints) can be turned in the search 2022-07-29 00:19:12 +0200 commented answer In a system of 5 nonlinear equations with 5 unknowns using the solve command, Sage gives 5 lists of 5 solutions. Is it possible to print only one list with the 5 third solutions? The .variety() method of ideals returns someting only if its .dimension() is 0 ; in other cases, you have to search the 2022-07-29 00:09:27 +0200 commented answer Matrix dimensions being symbolic Nope. After : sage: var("n, p") (n, p) the Python variable n is indeed bound to a symbolic variable (element of SR) : 2022-07-29 00:09:00 +0200 commented answer Matrix dimensions being symbolic Nope. After : sage: var("n, p") (n, p) the Python variable n is indeed bound to a symbolic variable (element of SR) : 2022-07-29 00:07:30 +0200 commented answer Matrix dimensions being symbolic Nope. After : sage: var("n, p") (n, p) Here, n is bound too a symbolic variable : sage: n.parent() Symbolic Ring B 2022-07-24 20:39:39 +0200 commented answer Writing a matrix as a sum of matrices I took the liberty to edit your "answer" in order to get somethng barely legible. I hope that I haven't introduced error 2022-07-24 20:37:38 +0200 edited answer Writing a matrix as a sum of matrices I asked this thing because a friend of mine made code for this but it is very time taking. For 2nd order matrices outp 2022-07-24 20:26:52 +0200 answered a question error interact in 9.6 on my computer but not on sagecell Same problem here (9.7.beta5 sompiled from Git on Deban testing). But this : @interact def _(v=input_grid(1, 3, 2022-07-23 22:48:16 +0200 commented question Why, in any system of equations, does Sage give repeated solutions only once? Solving via SR's solve isn't exact : given : var('a,b,c,d,e,f,g,x') Sys=[(a - g) + 2, - (a*g - b) - 1, - (b*g - c), - 2022-07-23 22:47:27 +0200 commented question Why, in any system of equations, does Sage give repeated solutions only once? @georgeafr : @rburing is right. He was faster|smarter than me... My remark was not (even tentatively) an answer to your 2022-07-23 22:47:00 +0200 commented question Why, in any system of equations, does Sage give repeated solutions only once? @georgeafr : @rburning is right. He was faster|smarter than me... My remark was not (even tentatively) an answer to your 2022-07-23 22:44:54 +0200 commented question Why, in any system of equations, does Sage give repeated solutions only once? A better (checkable) solution is to use the ring of polynomials of QQbar : Vars=[str(u).upper() for u in vars] # My 2022-07-23 21:34:17 +0200 commented question Why, in any system of equations, does Sage give repeated solutions only once? A better (checkable) solution is to use the ring of polynomials of QQbar : Vars=[str(u).upper() for u in vars] # My 2022-07-23 21:33:40 +0200 commented question Why, in any system of equations, does Sage give repeated solutions only once? A better (checkable) solution is to use the ring of polynomials of QQbar : Vars=[str(u).upper() for u in vars] # My 2022-07-23 21:31:45 +0200 commented question Why, in any system of equations, does Sage give repeated solutions only once? A better (checkable) solution is to use the ring of polynomials of QQbar : Vars=[str(u).upper() for u in vars] # My 2022-07-23 21:27:40 +0200 commented question Why, in any system of equations, does Sage give repeated solutions only once? Solving via SR's solve isn't *exact` : given : var('a,b,c,d,e,f,g,x') Sys=[(a - g) + 2, - (a*g - b) - 1, - (b*g - c), 2022-07-23 17:52:24 +0200 commented answer Mittag-Leffler In the same vein : sage: mathematica.MittagLefflerE? Signature: mathematica.MittagLefflerE(*args, **kwds) Type: 2022-07-19 21:18:53 +0200 commented question how to write an expression in latex without it being transformed by sage/sagetex First embryo of an idea : work by generating strings representing your "raw" polynomials, format this to LaTeX (possibly 2022-07-19 21:11:53 +0200 commented question A sageCell question Try this Google group... 2022-07-16 16:54:16 +0200 commented answer In a system of 5 nonlinear equations with 5 unknowns using the solve command, Sage gives 5 lists of 5 solutions. Is it possible to print only one list with the 5 third solutions? A possibly cleaner solution (avoiding to define a truckload of unused symbolic variables and needlessly scratch predefi 2022-07-16 16:53:18 +0200 commented answer In a system of 5 nonlinear equations with 5 unknowns using the solve command, Sage gives 5 lists of 5 solutions. Is it possible to print only one list with the 5 third solutions? A possibly cleaner solution (avoiding to define a truckload of unused symbolic variables and scratch needlessly predefin 2022-07-16 16:39:53 +0200 edited question In a system of 5 nonlinear equations with 5 unknowns using the solve command, Sage gives 5 lists of 5 solutions. Is it possible to print only one list with the 5 third solutions? In a system of 5 nonlinear equations with 5 unknowns using the solve command, Sage gives 5 lists of 5 solutions. Is it p 2022-07-08 22:44:13 +0200 commented question Ubuntu on Windows cannot open notebook @ortollj : for the benefit of future ask(per)users, could you turn your comments in a structured answer ? 2022-07-06 20:51:53 +0200 edited answer why there is no exact result with this fractions sum ? from srange? : Docstring: Return a list of numbers "start, start+step, ..., start+k*step", where "start+k*st 2022-07-06 20:50:20 +0200 answered a question why there is no exact result with this fractions sum ? from srange? : Docstring: Return a list of numbers "start, start+step, ..., start+k*step", where "start+k*st
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## WeBWorK Main Forum
### Feedback emails
by Maria Nogin -
Number of replies: 9
Is it possible to set it up so that every instructor in a course receives feedback emails only from students in their own section (rather than all instructors receive all emails)?
### Re: Feedback emails
by Maria Nogin -
Or may be at least indicate the section in the subject of the email? Then it would be easy for each instructor to see which emails are from their students.
### Re: Feedback emails
by Raghu Gompa -
Dear Maria,
When you choose E-mail option under course configuration, you will be able to specify what you would like in the Format for the subject line in feedback e-mails. For example:
[WWfeedback] course:%c user:%u set:%s prob:%p sec:%x rec:%r
Hope this helps.
I would like to know the answer to your original question: Is it possible to set it up so that every instructor in a course receives feedback emails only from students in their own section (rather than all instructors receive all emails)?
I wait for the experts to help.
Raghu
### Re: Feedback emails
by Gavin LaRose -
Hi Raghu and Maria,
Prior to release 2.4.5 it was not possible to restrict feedback e-mail by class section. I believe this is possible as of the 2.4.5 release, however. In the global.conf or course.conf file, I believe one can set $feedback_by_section = 1 to make feedback be sent only to the instructor of a section. Gavin In reply to Maria Nogin ### Re: Feedback emails by Arnold Pizer - Hi Maria, In global.conf you will see the lines: # If this value is true, feedback will only be sent to users with the same # section as the user initiating the feedback.$feedback_by_section = 0;
You should set this to 1 and then make sure that the professors and TA's are in the same section as their students.
If you just wanted this to apply to a single course, copy this to course.conf and of course set $feedback_by_section = 1; Arnie In reply to Arnold Pizer ### Re: Feedback emails by Jason Aubrey - Hi All, Is it possible to have the email from two or more sections sent to the same instructor? E.g. suppose I have T. Cher teaching sections 3 and 5, and I want all email from both sections to go to him? In that case, maybe I could count section 3 as a "Section" in WW and section 5 as a recitation? (Does feedback by section work for recitations too?) But even if that works, what if T. Cher has to teach more than 2 sections? Thanks, Jason In reply to Arnold Pizer ### Re: Feedback emails by Arnold Pizer - Following is an exchange of emails (slightly edited) that I had with Dan Margalit <dan.margalit@tufts.edu> that others might find useful. Hi Arnie, I found this on the WebWork forum. Am I correct that this is a setting for the whole server (and not for a particular class)? If so, what is the setting on the server at Rochester where I have our Tufts U course? Thanks! Dan ------------- Hi Maria, In global.conf you will see the lines: # If this value is true, feedback will only be sent to users with the same # section as the user initiating the feedback.$feedback_by_section = 0;
You should set this to 1 and then make sure that the professors and TA's are in the same section as their students.
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# Electric Field inside a Conductor— not Gauss's Law Explanation Please
I am trying to figure out why in a conductor, lets for example say a uniform sheet of metal (infinitely large to neglect edge effects) that has a net positive charge, at all points inside the conductor the electric field is zero. I understand the Gauss's law explanation that there is no net free charge enclosed by a Gaussian surface inside the conductor because all the free charge is on the surface. This makes sense, but then I get confused if I think further. If you look at a point inside the slab that is closer to one side of the sheet of charged metal, shouldn't there be a stronger field as you approach that side due to closer proximity to the charges on that surface? I appreciate the help
This example illustrates the answer for spherical geometry but the argument is the same in the planar geometry of your question.
On the near side of the sphere, closer to the point inside the sphere, the amount of charge captured in the angular opening is small, but the charges are closer. On the far side, the amount of charge captured by the angular opening is large but the charges are farther away. Because the area of the angular opening grows like $r^2$ while the electric field $\vert E\vert\sim 1/r^2$, the effect of the larger distance is exactly balanced by the effect of the larger area.
This other example shows how this works for an arbitrary surface.
By inscribing a sphere centred at any point inside, one shows that the same must hold for the opposing areas of the surface, so that the contribution from the larger number of farther charges exactly cancels the contribution from the nearer number of fewer charges.
To be a little more explicit, the right dashed lines define a cone (in 3d) that would intercept the sphere and so contain on the intercepted surface a certain amount of charge $dq_R\sim \sigma \ell_R^2$, where $\ell_R$ is the distance from the point to this patch of surface. The contribution to the field from this patch of charge is $dE_R\sim \sigma \ell_R^2/\ell_R^2\sim\sigma$. The same goes for the dashed lines on the left, where $dq_L\sim \sigma \ell_L^2$ and $dE_L\sim \sigma \ell_L^2/\ell_L^2\sim\sigma$. The magnitude of $dE_L$ is thus the same as that of $dE_R$ but their directions are opposite, so they exactly cancel at the point $P$ inside your surface.
Obviously in the case of planar geometry, this can only hold if the surfaces are infinite in extent.
I should add that strictly speaking this argument only works for infinitesimal openings, so that all the points "on the right" are at the same distance from the point inside, and all the points "on the left" are also at the same distance from the point inside.
• I am not sure I understand the setup for the explanation using the angular openings? Could you explain this logic further for me? Thanks – Joe Jul 17 '17 at 23:11
• @Joe I added a paragraph which hopefully helps. – ZeroTheHero Jul 17 '17 at 23:24
• @ZeroTheHero Does the demonstration for the arbitrarily shaped conductor assume a uniform surface charge density $\sigma$ ? – Hilbert Aug 14 at 22:40
• @Hilbert Good question. I think yes inasmuch as charges would distribute themselves uniformly on a sphere or on an infinite slab, as the example of the OP. – ZeroTheHero Aug 15 at 0:23
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# Inverse of Sparse Diagonal Array not Sparse
Kind of bummed that when I take the inverse of a matrix that is a diagonal sparse array, the result in not a sparse array. Further, the time to compute the inverse is the same as the time to compute the inverse of a normal array.
Just checking to see if I am missing some obvious switch or setting or method that will let MMa speed up the process. I know I can invert the elements of the list that I am building the diagonal matrix from prior, and get the result I want.
v = DiagonalMatrix@SparseArray[Range[5000] // N]
res = Inverse@v; // Timing
(* {3.6875, Null} *)
res is not a sparse array. The time to create it is the same as if it had never been sparse in the first place.
v = DiagonalMatrix@Range[5000] // N
res = Inverse@v; // Timing
(* {3.625, Null} *)
Manually inverting the list I am building the diagonal matrix from. This is the performance I was hoping to see. Also, was hoping to get a sparse matrix on the output.
res = SparseArray@(1/Range[50000]); // Timing
(* {0.125, Null} *)
res is by creation a sparse array.
• This seems relevant: Efficient way of sparse matrix inversion – Lukas Lang Jun 10 at 17:35
• Hm. I don't see why Inverse should check for diagonal matrices first: It is a rather seldom case in which we may expect the user to know that they are about to invert a diagonal matrix. In any case, I suggest to use LinearSolve instead of Inverse in almost all applications, where the matrix size is bigger than, say $5 \times 5$. – Henrik Schumacher Jun 10 at 18:18
• I've a problem that involves inverting a matrix (my diagonal one), adding it to another, inverting their sum, and then summing up the elements of that inverted matrix. So there's not really a place for LinearSolve in the problem. Not a great bother to work around it, was just mildly surprised that there was no way to signal that the Inverse was sparse too, and to tell Mathematica to attempt to keep it sparse. – MikeY Jun 10 at 18:33
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# Firefox – How to force Firefox to always show the URL and navigation toolbar in popup windows
firefoxpopups
How do I prevent websites from disabling the URL bar navigation bar when they open popup windows using javascript. Here is an example of such a window: http://jsfiddle.net/nLmw8t5q/1/
Basically, I always want to see the URLbar/address bar and navigation controls. I am using Firefox.
Thanks!
dom.disable_window_open_feature.toolbar
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On Skew Polycyclic Codes over $\mathbb{Z}_4[u]/\langle u^2-2\rangle$
Received:April 01, 2022 Revised:October 04, 2022
Key Words: skew polycyclic code polycyclic code cyclic code generator polynomial Gray map
Fund Project:Supported by the National Natural Science Foundation of China (Grant No.12201361).
Author Name Affiliation Wei QI School of Mathematics and Statistics, Shandong University of Technology, Shandong 255000, P. R. China Xiaolei ZHANG School of Mathematics and Statistics, Shandong University of Technology, Shandong 255000, P. R. China
Hits: 31
In this paper, we investigate some classes of skew polycyclic codes and polycyclic codes over $R=\mathbb{Z}_4[u]/\langle u^2-2\rangle$. We first obtain the generator polynomials of all $(1,2u)$-polycyclic codes over $R$. Then, by defining some Gray maps, we show that the images of (skew) $(1,2u)$-polycyclic codes over $R$ are cyclic or quasi-cyclic with index 2 over $\mathbb{Z}_4$. Finally, an example of some $(1,2u)$-polycyclic codes over $R$ is given to exhibit the main results of the paper.
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## College Algebra 7th Edition
(a) $m(0)=13$ (b) $m(45)=6.62$ kg
We are given: $m(t)=13e^{-0.015t}$ (a) We plug in $t=0$: $m(0)=13e^{-0.015*0}=13*1=13$ (b) We plug in $t=45$: $m(45)=13e^{-0.015*45}=13e^{-0.675}=6.62$ kg
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Vol. 10, No. 2, 2017
Recent Issues
The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-4184 (e-only) ISSN: 1944-4176 (print) Author Index Coming Soon Other MSP Journals
Combinatorial curve neighborhoods for the affine flag manifold of type $A_1^1$
Leonardo C. Mihalcea and Trevor Norton
Vol. 10 (2017), No. 2, 317–325
Abstract
Let $X$ be the affine flag manifold of Lie type ${A}_{1}^{1}$. Its moment graph encodes the torus fixed points (which are elements of the infinite dihedral group ${D}_{\infty }$) and the torus stable curves in $X$. Given a fixed point $u\in {D}_{\infty }$ and a degree $d=\left({d}_{0},{d}_{1}\right)\in {ℤ}_{\ge 0}^{2}$, the combinatorial curve neighborhood is the set of maximal elements in the moment graph of $X$ which can be reached from $u$ using a chain of curves of total degree $\le d$. In this paper we give a formula for these elements, using combinatorics of the affine root system of type ${A}_{1}^{1}$.
Keywords
affine flag manifolds, moment graph, curve neighborhood
Mathematical Subject Classification 2010
Primary: 05E15
Secondary: 17B67, 14M15
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Looking for Algebra worksheets?
Check out our pre-made Algebra worksheets!
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##### Browse Questions
You can create printable tests and worksheets from these Grade 8 Algebraic Expressions questions! Select one or more questions using the checkboxes above each question. Then click the add selected questions to a test button before moving to another page.
1 2
Simplify the expression.
(-8)(-8-d)
1. -64-8d
2. -64+8d
3. 64-8d
4. 64+8d
If $a = -5$ and $b=6$, which expression has a value of 150?
1. $5ab$
2. $a^2 b$
3. $-ab^2$
4. $5a^2 b$
expressions that have the same variables raised to the same exponents
1. absolute value
2. variable
3. coefficient
4. like terms
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Note
This documents the development version of NetworkX. Documentation for the current release can be found here.
# networkx.generators.nonisomorphic_trees.number_of_nonisomorphic_trees¶
number_of_nonisomorphic_trees(order)[source]
Returns the number of nonisomorphic trees
Parameters
order (int) – order of the desired tree(s)
Returns
length
Return type
Number of nonisomorphic graphs for the given order
References
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# Electrostatic speaker
1. Mar 20, 2013
### Number2Pencil
1. The problem statement, all variables and given/known data
An electrostatic speaker is constructed using two conductive plates (stators) and an electrostatically charged diaphragm in the middle which vibrates. One of the conductors is grounded, and the other has an amplified voltage applied to it to drive the force.
A uniform charge density, ps (C/m^2) is maintained on the diaphragm. The stator conductors are separated by distance d.
Find the symbol equation for Pressure, P, as a function of Voltage, V(t). Use only the symbols Q, ps, A, E, V (where V=V(t)), d, and F.
2. Relevant equations
$$P=\frac{F}{A}$$
$$F=Q(E+v \times B)$$
$$Q=\rho_s A$$
$$V = -\int{E \cdot dl}$$
3. The attempt at a solution
I am having a hard time conceptualizing how to find voltage or electric field. Here is what I've done:
------------------
METHOD 1:
------------------
Since there are no magnetic fields in this problem...
$$F=QE$$
Since all electric fields are parallel (angle = 0)
$$V = -\int{Edl}$$
$$V = -Ed$$
$$E = -\frac{V}{d}$$
So now I have Q, and I have E, let's solve for F:
$$F = - \rho_s A \frac{V}{d}$$
Solving for pressure:
$$P = \frac{F}{A} = -\rho_s \frac{V}{d}$$
But alas, it says this is incorrect. I am guessing that the E-Field at the plate is more complicated than just -V/d
-----------------------------
--METHOD 2: Point Charges
-----------------------------
So next I thought, "If the charges are uniformly distributed, maybe I can look at just single point charges on the
conductors."
Coulombs Law:
$$F = \frac{k Q_{plate} Q}{r^2}$$
We can leave Q as Q since that is a symbol in the answer, but we can use Gauss' Law to come up with a relationship
between electric fields and charge:
$$Q = \epsilon_0 E A$$
I took the liberty of getting the results from someone doing a parallel plate capacitor example.
Similarily:
$$E = -\frac{V}{d}$$
Giving us:
$$Q_{plate} = -\frac{\epsilon_0 V A}{d}$$
and we know that the distance between the positive plate and the diaphragm is d/2, putting all this together:
$$F = \frac{4 k \epsilon_0 V A Q}{d^3}$$
Change to pressure by dividing by Area:
$$P = \frac{F}{A} = \frac{4 k \epsilon_0 V Q}{d^3}$$
Since our gap medium is air, we know that k is:
$$k = \frac{1}{4 \pi \epsilon_0}$$
Plugging and canceling out common terms, I wind up with this answer:
$$P = -\frac{VQ}{\pi d^3}$$
But the answer guide said it was incorrect. At this point I will just admit that I am unclear on how to approach this
problem...can someone do some nudging or hand-holding and explain the flaws in my shoddy physics?
2. Mar 20, 2013
### rude man
I see nothing wrong with your method 1. Maybe the minus sign?
Anyone else?
3. Mar 20, 2013
### Number2Pencil
A quick negative sign swap did not appease the homework software...
4. Mar 21, 2013
### Number2Pencil
Turns out my method 1 answer was correct, and it was simply the homework answer analysis-algorithm that was faulty.
Yay...
I guess now that I know the answer, I can explain this in (hopefully) understanding terms for my own sanity. There was a constant electric field established by the voltage conductor and ground conductor throughout the gap between them, the same way there is an electric field inside a capacitor. It is true that the charged diaphragm also contained charge, and thus electric fields, but this didn't "counteract" the other electric fields.....they just do what electric fields do. If the conductors were not rigidly attached to a frame, they would also experience a force.
The equation F=QE talks of a force on a charge which is produced when that charge is exposed to an external electric field. Q was the charge of the Diaphragm, and E was the field it was exposed to (created by the two conductors).
Sound good?
5. Mar 21, 2013
### rude man
Sure does.
Even without the diaphragm, the two plates do experience a force. One nifty way to show this is via the principle of virtual work:
Consider 1 m2 of plates separated by d. The E field is V/d and the energy of this part of the field is εE2d/2 = εV2/2d. Now if d were increased a small amount δd, the new energy per m2 would be as above but with d → d + δd. The difference in energy under 1 m2 of plates would be ≈ -εV2/2 (δd/d2) = F δd, so F = - εV2/2d2 per sq. meter or call it pressure P.
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In my previous post I illustrated why it is not possible to compute the Jordan canonical form numerically (i.e. in floating point numbers). The simple reason: For every matrix ${A}$ and every ${\epsilon>0}$ there is a matrix ${A_{\epsilon}}$ which differs from ${A}$ by at most ${\epsilon}$ (e.g. in every entry – but all norms for matrices are equivalent, so this does not really play a role) such that ${A_{\epsilon}}$ is diagonalizable. So why should you bother about computing the Jordan canonical form anyway? Or even learning or teaching it? Well, the prime application of the Jordan canonical form is to calculate solutions of linear systems of ODEs. The equation
$\displaystyle y'(t) = Ay(t),\quad y(0) = y_{0}$
with matrix ${A\in {\mathbb R}^{n\times n}}$ and initial value ${y_{0}\in{\mathbb R}^{n}}$ (both could also be complex). This system has a unique solution which can be given explicitly with the help of the matrix exponential as
$\displaystyle y(t) = \exp(At)y_{0}$
where the matrix exponential is
$\displaystyle \exp(At) = \sum_{k=0}^{\infty}\frac{A^{k}t^{k}}{k!}.$
It is not always simple to work out the matrix exponential by hand. The straightforward way would be to calculate all the powers of ${A}$, weight them by ${1/k!}$ and sum the series. This may be a challenge, even for simple matrices. My favorite example is the matrix
$\displaystyle A = \begin{bmatrix} 0 & 1\\ 1 & 1 \end{bmatrix}.$
Its first powers are
$\displaystyle A^{2} = \begin{bmatrix} 1 & 1\\ 1 & 2 \end{bmatrix},\quad A^{3} = \begin{bmatrix} 1 & 2\\ 2 & 3 \end{bmatrix}$
$\displaystyle A^{4} = \begin{bmatrix} 2 & 3\\ 3 & 5 \end{bmatrix},\quad A^{5} = \begin{bmatrix} 3 & 5\\ 5 & 8 \end{bmatrix}.$
You may notice that the Fibonicci numbers appear (and this is pretty clear on a second thought). So, finding a explicit form for ${\exp(A)}$ leads us to finding an explicit form for the ${k}$-th Fibonacci number (which is possible, but I will not treat this here).
Another way is diagonalization: If ${A}$ is diagonalizable, i.e. there is an invertible matrix ${S}$ and a diagonal matrix ${D}$ such that
$\displaystyle S^{-1}AS = D\quad\text{or, equivalently}\quad A = SDS^{-1},$
you see that
$\displaystyle \exp(At) = S\exp(Dt)S^{-1}$
and the matrix exponential of a diagonal matrix is simply the exponential function applied to the diagonal entries.
But not all matrices are diagonalizable! The solution that is usually presented in the classroom is to use the Jordan canonical form instead and to compute the matrix exponential of Jordan blocks (using that you can split a Jordan block ${J = D+N}$ into the sum of a diagonal matrix ${D}$ and a nil-potent matrix ${N}$ and since ${D}$ and ${N}$ commute one can calculate ${\exp(J) = \exp(D)\exp(N)}$ and both matrix exponentials are quite easy to compute).
But in light of the fact that there are a diagonalizable matrices arbitrarily close to any matrix, on may ask: What about replacing a non-diagonalizable matrix ${A}$ with a diagonalizable one (with a small error) and then use this one?
Let’s try this on a simple example:
We consider
$\displaystyle A = \begin{bmatrix} -1 & 1\\ 0 & -1 \end{bmatrix}$
which is not diagonalizable. The linear initial value problem
$\displaystyle y' = Ay,\quad y(0) = y_{0}$
has the solution
$\displaystyle y(t) = \exp( \begin{bmatrix} -t & t\\ 0 & -t \end{bmatrix}) y_{0}$
and the matrix exponential is
$\displaystyle \begin{array}{rcl} \exp( \begin{bmatrix} -t & t\\ 0 & -t \end{bmatrix}) & = &\exp(\begin{bmatrix} -t & 0\\ 0 & -t \end{bmatrix})\exp(\begin{bmatrix} 0 & t\\ 0 & 0 \end{bmatrix})\\& = &\begin{bmatrix} \exp(-t) & 0\\ 0 & \exp(-t) \end{bmatrix}\begin{bmatrix} 1 & t\\ 0 & 1 \end{bmatrix}\\ &=& \begin{bmatrix} \exp(-t) & t\exp(-t)\\ 0 & \exp(-t) \end{bmatrix}. \end{array}$
So we get the solution
$\displaystyle y(t) = \begin{bmatrix} e^{-t}(y^{0}_{1} + ty^{0}_{2})\\ e^{-t}y^{0}_{2} \end{bmatrix}.$
Let us take a close-by matrix which is diagonalizable. For some small ${\epsilon}$ we choose
$\displaystyle A_{\epsilon} = \begin{bmatrix} -1 & 1\\ 0 & -1+\epsilon \end{bmatrix}.$
Since ${A_{\epsilon}}$ is upper triangular, it has its eigenvalues on the diagonal. Since ${\epsilon\neq 0}$, there are two distinct eigenvalues and hence, ${A_{\epsilon}}$ is diagonalizable. Indeed, with
$\displaystyle S = \begin{bmatrix} 1 & 1\\ 0 & \epsilon \end{bmatrix},\quad S^{-1}= \begin{bmatrix} 1 & -\tfrac1\epsilon\\ 0 & \tfrac1\epsilon \end{bmatrix}$
we get
$\displaystyle A = S \begin{bmatrix} -1 & 0 \\ 0 & -1+\epsilon \end{bmatrix}S^{-1}.$
The matrix exponential of ${A_{\epsilon}t}$ is
$\displaystyle \begin{array}{rcl} \exp(A_{\epsilon}t) &=& S\exp( \begin{bmatrix} -t & 0\\ 0 & -t(1-\epsilon) \end{bmatrix} )S^{-1}\\ &=& \begin{bmatrix} e^{-t} & \tfrac{e^{-(1-\epsilon)t} - e^{-t}}{\epsilon}\\ 0 & e^{-(1-\epsilon)t} \end{bmatrix}. \end{array}$
Hence, the solution of ${y' = Ay}$, ${y(0) = y_{0}}$ is
$\displaystyle y(t) = \begin{bmatrix} e^{-t}y^{0}_{1} + \tfrac{e^{-(1-\epsilon)t} - e^{-t}}{\epsilon}y^{0}_{2}\\ e^{-(1-\epsilon)t}y^{0}_{2} \end{bmatrix}.$
How is this related to the solution of ${y'=Ay}$? How far is it away?
Of course, the lower right entry of ${\exp(A_{\epsilon}t)}$ converges to ${e^{-t}}$ for ${\epsilon \rightarrow 0}$, but what about the upper right entry? Note that the entry
$\displaystyle \tfrac{e^{-(1-\epsilon)t} - e^{-t}}{\epsilon}$
is nothing else that the (negative) difference quotient for the derivative of the function ${f(a) = e^{-at}}$ at ${a=1}$. Hence
$\displaystyle \tfrac{e^{-(1-\epsilon)t} - e^{-t}}{\epsilon} \stackrel{\epsilon\rightarrow 0}{\longrightarrow} -f'(1) = te^{-t}$
and we get
$\displaystyle \exp(A_{\epsilon}t) \stackrel{\epsilon\rightarrow 0}{\longrightarrow} \begin{bmatrix} e^{-t} & te^{-t}\\ 0 & e^{-t} \end{bmatrix} = \exp(At)$
as expected.
It turns out that a fairly big $\epsilon$ is already enough to get a quite good approximation and even the correct asymptotics: The blue curve it first component of the exact solution (initialized with the second standard basis vector), the red one corresponds $\epsilon = 0.1$ and the yellow on (pretty close to the blue on) is for $\epsilon = 0.01$.
to \$\e
I remember from my introductory class in linear algebra that my instructor said
It is impossible to calculate the Jordan canonical form of a matrix numerically.
Another quote I remember is
The Jordan canonical form does not depend continuously on the matrix.
For both quotes I did not remember the underlying reasons and since I do teach an introductory class on linear algebra this year, I got thinking about these issues again.
Here is a very simple example for the fact in the second quote:
Consider the matrix
$\displaystyle A_{\varepsilon} = \begin{pmatrix}1 & \varepsilon\\ 0 & 1\end{pmatrix}$
for ${\varepsilon>0}$. This matrix has ${1}$ as a double eigenvalue, but the corresponding eigenspace is one-dimensional and spanned by ${v_{1} = e_{1}}$. To extend this vector to a basis we calculate a principle vector by solving
$\displaystyle (A_{\varepsilon}-I)v_{2} = v_{1}$
$\displaystyle v_{2} = \begin{pmatrix} 0\\\varepsilon^{-1} \end{pmatrix}.$
Defining ${S = [v_{1}\, v_{2}]}$ we get the Jordan canonical form of ${A_{\varepsilon}}$ as
$\displaystyle J_{\varepsilon} = S^{-1}A_{\varepsilon}S = \begin{pmatrix} 1 & 1\\ 0 & 1 \end{pmatrix}$
So we have
$\displaystyle A_{\varepsilon}\rightarrow A = I\quad\text{and}\quad J_{\varepsilon} \rightarrow J = \begin{pmatrix} 1 & 1\\ 0 & 1 \end{pmatrix},$
but ${J}$ is not the Jordan canonical form of ${A}$. So, in short: The Jordan canonical form of the limit of ${A_{\varepsilon}}$ is not the limit of the Jordan canonical form of ${A_{\varepsilon}}$. In other words: Taking limits does not commute with forming the Jordan canonical form.
A side note: Of course, the Jordan canonical form is not even unique in general, so speaking of “dependence on the matrix” is an issue. What we have shown is, that there is no way to get continuous dependence on the matrix even if non-uniqueness is not an issue (like in the example above).
What about the first quote? Why is it impossible to compute the Jordan canonical form numerically? Let’s just try! We start with the simplest non-diagonalizable matrix
$\displaystyle A = \begin{pmatrix} 1 & 1\\ 0 & 1 \end{pmatrix}$
If we ask MATLAB or Octave to do the eigenvalue decomposition we get
>> [V,D] = diag(A)
V =
1.00000 -1.00000
0.00000 0.00000
D =
1 0
0 1
We see that ${V}$ does not seem to be invertible and indeed we get
>> rank(V)
ans = 1
What is happening? MATLAB did not promise the produce an invertible ${V}$ and it does not promise that the putcome would fulfill ${V^{-1}AV = D}$ (which is my definition of diagonalizability). It does promise that ${AV = VD}$ and in fact
>> A*V
ans =
1.00000 1.00000
0.00000 -0.00000
>> V*D
ans =
1.00000 -1.00000
0.00000 0.00000
>> A*V-V*D
ans =
0.0000e+00 2.2204e-16
0.0000e+00 0.0000e+00
so the promised identity is fulfilled up to machine precision (which is actually equal $\texttt{2.2204e-16}$ and we denote it by ${\epsilon}$ from here on).
How did MATLAB diagonalize this matrix? Here is the thing: The diagonalizable matrices are dense in ${{\mathbb C}^{n\times n}}$! (You probably have heard that before…) What does that mean numerically? Any matrix that you represent in floating point numbers is actually a representative of a whole bunch of matrices. Each entry is only known up to a certain precision. But this bunch of matrices does contain some matrix which is diagonalizable! This is exactly, what it means to be a dense set! So it is impossible to say if a matrix given in floating point numbers is actually diagonalizable or not. So, what matrix was diagonalized by MATLAB? Let us have closer look at the matrix ${V}$: The entries in the first row are in fact ${1}$ and ${-1}$:
>> V(1,:)
ans =
1 -1
In the second row we have
>> V(2,1)
ans =
0
>> V(2,2)
ans = 2.2204e-16
and there we have it. The inverse of ${V}$ does exist (although the matrix has numerical rank ${1}$) and it is
>> inv(V) warning: matrix singular to machine precision, rcond = 1.11022e-16
ans =
1.0000e+00 4.5036e+15
0.0000e+00 4.5036e+15
and note that $\texttt{4.5036e+15}$ is indeed just the inverse of the machine precision, so this inverse is actually 100% accurate. Recombining gives
>> inv(V)*D*V warning: matrix singular to machine precision, rcond = 1.11022e-16
ans =
1 0
0 1
which is not even close to our original ${A}$.
How is that? Here is a solution. The matrix
$\displaystyle \tilde A = \begin{pmatrix} 1 & 1\\ 0 & 1+\epsilon^{-2} \end{pmatrix}$
is indistinguishable from ${A}$ numerically. However, it has two distinct eigenvalues, so it is diagonalizable. Indeed, a basis of eigenvectors is
$\displaystyle \tilde V = \begin{pmatrix} 1 & -1\\ 0 & \epsilon^{-2} \end{pmatrix}$
which is indistinguishable from ${V}$ above and it holds that
$\displaystyle \tilde V^{-1}\tilde A\tilde V = \begin{pmatrix} 1 & 0\\ 0 & 1+\epsilon^{-2}. \end{pmatrix}$
which is indistinguishable from $D$.
Im Wintersemster 2018/19 habe ich die Vorlesung “Lineare Algebra 1” gehalten. Hier die lecture notes dazu:
Taking the derivative of the loss function of a neural network can be quite cumbersome. Even taking the derivative of a single layer in a neural network often results in expressions cluttered with indices. In this post I’d like to show an index-free way to do it.
Consider the map ${\sigma(Wx+b)}$ where ${W\in{\mathbb R}^{m\times n}}$ is the weight matrix, ${b\in{\mathbb R}^{m}}$ is the bias, ${x\in{\mathbb R}^{n}}$ is the input, and ${\sigma}$ is the activation function. Usually ${\sigma}$ represents both a scalar function (i.e. mapping ${{\mathbb R}\mapsto {\mathbb R}}$) and the function mapping ${{\mathbb R}^{m}\rightarrow{\mathbb R}^{m}}$ which applies ${\sigma}$ in each coordinate. In training neural networks, we would try to optimize for best parameters ${W}$ and ${b}$. So we need to take the derivative with respect to ${W}$ and ${b}$. So we consider the map
$\displaystyle \begin{array}{rcl} G(W,b) = \sigma(Wx+b). \end{array}$
This map ${G}$ is a concatenation of the map ${(W,b)\mapsto Wx+b}$ and ${\sigma}$ and since the former map is linear in the joint variable ${(W,b)}$, the derivative of ${G}$ should be pretty simple. What makes the computation a little less straightforward is the fact the we are usually not used to view matrix-vector products ${Wx}$ as linear maps in ${W}$ but in ${x}$. So let’s rewrite the thing:
There are two particular notions which come in handy here: The Kronecker product of matrices and the vectorization of matrices. Vectorization takes some ${W\in{\mathbb R}^{m\times n}}$ given columnwise ${W = [w_{1}\ \cdots\ w_{n}]}$ and maps it by
$\displaystyle \begin{array}{rcl} \mathrm{Vec}:{\mathbb R}^{m\times n}\rightarrow{\mathbb R}^{mn},\quad \mathrm{Vec}(W) = \begin{bmatrix} w_{1}\\\vdots\\w_{n} \end{bmatrix}. \end{array}$
The Kronecker product of matrices ${A\in{\mathbb R}^{m\times n}}$ and ${B\in{\mathbb R}^{k\times l}}$ is a matrix in ${{\mathbb R}^{mk\times nl}}$
$\displaystyle \begin{array}{rcl} A\otimes B = \begin{bmatrix} a_{11}B & \cdots &a_{1n}B\\ \vdots & & \vdots\\ a_{m1}B & \cdots & a_{mn}B \end{bmatrix}. \end{array}$
We will build on the following marvelous identity: For matrices ${A}$, ${B}$, ${C}$ of compatible size we have that
$\displaystyle \begin{array}{rcl} \mathrm{Vec}(ABC) = (C^{T}\otimes A)\mathrm{Vec}(B). \end{array}$
Why is this helpful? It allows us to rewrite
$\displaystyle \begin{array}{rcl} Wx & = & \mathrm{Vec}(Wx)\\ & = & \mathrm{Vec}(I_{m}Wx)\\ & = & \underbrace{(x^{T}\otimes I_{m})}_{\in{\mathbb R}^{m\times mn}}\underbrace{\mathrm{Vec}(W)}_{\in{\mathbb R}^{mn}}. \end{array}$
So we can also rewrite
$\displaystyle \begin{array}{rcl} Wx +b & = & \mathrm{Vec}(Wx+b )\\ & = & \mathrm{Vec}(I_{m}Wx + b)\\ & = & \underbrace{ \begin{bmatrix} x^{T}\otimes I_{m} & I_{m} \end{bmatrix} }_{\in{\mathbb R}^{m\times (mn+m)}}\underbrace{ \begin{bmatrix} \mathrm{Vec}(W)\\b \end{bmatrix} }_{\in{\mathbb R}^{mn+m}}\\ &=& ( \underbrace{\begin{bmatrix} x^{T} & 1 \end{bmatrix}}_{\in{\mathbb R}^{1\times(n+1)}}\otimes I_{m}) \begin{bmatrix} \mathrm{Vec}(W)\\b \end{bmatrix}. \end{array}$
So our map ${G(W,b) = \sigma(Wx+b)}$ mapping ${{\mathbb R}^{m\times n}\times {\mathbb R}^{m}\rightarrow{\mathbb R}^{m}}$ can be rewritten as
$\displaystyle \begin{array}{rcl} \bar G( \begin{pmatrix} \mathrm{Vec}(W)\\b \end{pmatrix}) = \sigma( ( \begin{bmatrix} x^{T} & 1 \end{bmatrix}\otimes I_{M}) \begin{bmatrix} \mathrm{Vec}(W)\\b \end{bmatrix}) \end{array}$
mapping ${{\mathbb R}^{mn+m}\rightarrow{\mathbb R}^{m}}$. Since ${\bar G}$ is just a concatenation of ${\sigma}$ applied coordinate wise and a linear map, now given as a matrix, the derivative of ${\bar G}$ (i.e. the Jacobian, a matrix in ${{\mathbb R}^{m\times (mn+m)}}$) is calculated simply as
$\displaystyle \begin{array}{rcl} D\bar G( \begin{pmatrix} \mathrm{Vec}(W)\\b \end{pmatrix}) & = & D\sigma(Wx+b)( \begin{bmatrix} x^{T} & 1 \end{bmatrix}\otimes I_{M})\\ &=& \underbrace{\mathrm{diag}(\sigma'(Wx+b))}_{\in{\mathbb R}^{m\times m}}\underbrace{( \begin{bmatrix} x^{T} & 1 \end{bmatrix}\otimes I_{M})}_{\in{\mathbb R}^{m\times(mn+m)}}\in{\mathbb R}^{m\times(mn+m)}. \end{array}$
While this representation of the derivative of a single layer of a neural network with respect to its parameters is not particularly simple, it is still index free and moreover, straightforward to implement in languages which provide functions for the Kronecker product and vectorization. If you do this, make sure to take advantage of sparse matrices for the identity matrix and the diagonal matrix as otherwise the memory of your computer will be flooded with zeros.
Now let’s add a scalar function ${L}$ (e.g. to produce a scalar loss that we can minimize), i.e. we consider the map
$\displaystyle \begin{array}{rcl} F( \begin{pmatrix} \mathrm{Vec}(W)\\b \end{pmatrix}) = L(G(Wx+b)) = L(\bar G( \begin{pmatrix} \mathrm{Vec}(W)\\b \end{pmatrix}). \end{array}$
The derivative is obtained by just another application of the chain rule:
$\displaystyle \begin{array}{rcl} DF( \begin{pmatrix} \mathrm{Vec}(W)\\b \end{pmatrix}) = DL(G(Wx+b))D\bar G( \begin{pmatrix} \mathrm{Vec}(W)\\b \end{pmatrix}). \end{array}$
If we want to take gradients, we just transpose the expression and get
$\displaystyle \begin{array}{rcl} \nabla F( \begin{pmatrix} \mathrm{Vec}(W)\\b \end{pmatrix}) &=& D\bar G( \begin{pmatrix} \mathrm{Vec}(W)\\b \end{pmatrix})^{T} DL(G(Wx+b))^{T}\\ &=& ([x^{T}\ 1]\otimes I_{m})^{T}\mathrm{diag}(\sigma'(Wx+b))\nabla L(G(Wx+b))\\ &=& \underbrace{( \begin{bmatrix} x\\ 1 \end{bmatrix} \otimes I_{m})}_{\in{\mathbb R}^{(mn+m)\times m}}\underbrace{\mathrm{diag}(\sigma'(Wx+b))}_{\in{\mathbb R}^{m\times m}}\underbrace{\nabla L(G(Wx+b))}_{\in{\mathbb R}^{m}}. \end{array}$
Note that the right hand side is indeed vector in ${{\mathbb R}^{mn+m}}$ and hence, can be reshaped to a tupel ${(W,b)}$ of an ${m\times n}$ matrix and an ${m}$ vector.
A final remark: the Kronecker product is related to tensor products. If ${A}$ and ${B}$ represent linear maps ${X_{1}\rightarrow Y_{1}}$ and ${X_{2}\rightarrow Y_{2}}$, respectively, then ${A\otimes B}$ represents the tensor product of the maps, ${X_{1}\otimes X_{2}\rightarrow Y_{1}\otimes Y_{2}}$. This relation to tensor products and tensors explains where the tensor in TensorFlow comes from.
The problem of optimal transport of mass from one distribution to another can be stated in many forms. Here is the formulation going back to Kantorovich: We have two measurable sets ${\Omega_{1}}$ and ${\Omega_{2}}$, coming with two measures ${\mu_{1}}$ and ${\mu_{2}}$. We also have a function ${c:\Omega_{1}\times \Omega_{2}\rightarrow {\mathbb R}}$ which assigns a transport cost, i.e. ${c(x_{1},x_{2})}$ is the cost that it takes to carry one unit of mass from point ${x_{1}\in\Omega_{2}}$ to ${x_{2}\in\Omega_{2}}$. What we want is a plan that says where the mass in ${\mu_{1}}$ should be placed in ${\Omega_{2}}$ (or vice versa). There are different ways to formulate this mathematically.
A simple way is to look for a map ${T:\Omega_{1}\rightarrow\Omega_{2}}$ which says that thet mass in ${x_{1}}$ should be moved to ${T(x_{1})}$. While natural, there is a serious problem with this: What if not all mass at ${x_{1}}$ should go to the same point in ${\Omega_{2}}$? This happens in simple situations where all mass in ${\Omega_{1}}$ sits in just one point, but there are at least two different points in ${\Omega_{2}}$ where mass should end up. This is not going to work with a map ${T}$ as above. So, the map ${T}$ is not flexible enough to model all kinds of transport we may need.
What we want is a way to distribute mass from one point in ${\Omega_{1}}$ to the whole set ${\Omega_{2}}$. This looks like we want maps ${\mathcal{T}}$ which map points in ${\Omega_{1}}$ to functions on ${\Omega_{2}}$, i.e. something like ${\mathcal{T}:\Omega_{1}\rightarrow (\Omega_{2}\rightarrow{\mathbb R})}$ where ${(\Omega_{2}\rightarrow{\mathbb R})}$ stands for some set of functions on ${\Omega_{2}}$. We can de-curry this function to some ${\tau:\Omega_{1}\times\Omega_{2}\rightarrow{\mathbb R}}$ by ${\tau(x_{1},x_{2}) = \mathcal{T}(x_{1})(x_{2})}$. That’s good in principle, be we still run into problems when the target mass distribution ${\mu_{2}}$ is singular in the sense that ${\Omega_{2}}$ is a “continuous” set and there are single points in ${\Omega_{2}}$ that carry some mass according to ${\mu_{2}}$. Since we are in the world of measure theory already, the way out suggests itself: Instead of a function ${\tau}$ on ${\Omega_{1}\times\Omega_{2}}$ we look for a measure ${\pi}$ on ${\Omega_{1}\times \Omega_{2}}$ as a transport plan.
The demand that we should carry all of the mass in ${\Omega_{1}}$ to reach all of ${\mu_{2}}$ is formulated by marginals. For simplicity we just write these constraints as
$\displaystyle \int_{\Omega_{2}}\pi\, d x_{2} = \mu_{1},\qquad \int_{\Omega_{1}}\pi\, d x_{1} = \mu_{2}$
(with the understanding that the first equation really means that for all continuous function ${f:\Omega_{1}\rightarrow {\mathbb R}}$ it holds that ${\int_{\Omega_{1}\times \Omega_{2}} f(x_{1})\,d\pi(x_{1},x_{2}) = \int_{\Omega_{1}}f(x_{1})\,d\mu_{1}(x_{1})}$).
This leads us to the full transport problem
$\displaystyle \min_{\pi}\int_{\Omega_{1}\times \Omega_{2}}c(x,y)\,d\pi(x_{1}x_{2})\quad \text{s.t.}\quad \int_{\Omega_{2}}\pi\, d x_{2} = \mu_{1},\quad \int_{\Omega_{1}}\pi\, d x_{1} = \mu_{2}.$
There is the following theorem which characterizes optimality of a plan and which is the topic of this post:
Theorem 1 (Fundamental theorem of optimal transport) Under some technicalities we can say that a plan ${\pi}$ which fulfills the marginal constraints is optimal if and only if one of the following equivalent conditions is satisfied:
1. The support ${\mathrm{supp}(\pi)}$ of ${\pi}$ is ${c}$-cyclically monotone.
2. There exists a ${c}$-concave function ${\phi}$ such that its ${c}$-superdifferential contains the support of ${\pi}$, i.e. ${\mathrm{supp}(\pi)\subset \partial^{c}\phi}$.
A few clarifications: The technicalities involve continuity, integrability, and boundedness conditions of ${c}$ and integrability conditions on the marginals. The full theorem can be found as Theorem 1.13 in A user’s guide to optimal transport by Ambrosio and Gigli. Also the notions ${c}$-cyclically monotone, ${c}$-concave and ${c}$-superdifferential probably need explanation. We start with a simpler notion: ${c}$-monotonicity:
Definition 2 A set ${\Gamma\subset\Omega_{1}\times\Omega_{2}}$ is ${c}$-monotone, if for all ${(x_{1}x_{2}),(x_{1}',x_{2}')\in\Gamma}$ it holds that
$\displaystyle c(x_{1},x_{2}) + c(x_{1}',x_{2}')\leq c(x_{1},x_{2}') + c(x_{1}',x_{2}).$
If you find it unclear what this has to do with monotonicity, look at this example:
Example 1 Let ${\Omega_{1/2}\in{\mathbb R}^{d}}$ and let ${c(x_{1},x_{2}) = \langle x_{1},x_{2}\rangle}$ be the usual scalar product. Then ${c}$-monotonicity is the condition that for all ${(x_{1}x_{2}),(x_{1}',x_{2}')\in\Gamma\subset\Omega_{1}\times\Omega_{2}}$ it holds that
$\displaystyle 0\leq \langle x_{1}-x_{1}',x_{2}-x_{2}'\rangle$
which may look more familiar. Indeed, when ${\Omega_{1}}$ and ${\Omega_{2}}$ are subset of the real line, the above conditions means that the set ${\Gamma}$ somehow “moves up in ${\Omega_{2}}$” if we “move right in ${\Omega_{1}}$”. So ${c}$-monotonicity for ${c(x_{1},x_{2}) = \langle x_{1},x_{2}\rangle}$ is something like “monotonically increasing”. Similarly, for ${c(x_{1},x_{2}) = -\langle x_{1},x_{2}\rangle}$, ${c}$-monotonicity means “monotonically decreasing”.
You may say that both ${c(x_{1},x_{2}) = \langle x_{1},x_{2}\rangle}$ and ${c(x_{1},x_{2}) = -\langle x_{1},x_{2}\rangle}$ are strange cost functions and I can’t argue with that. But here comes: ${c(x_{1},x_{2}) = |x_{1}-x_{2}|^{2}}$ (${|\,\cdot\,|}$ being the euclidean norm) seems more natural, right? But if we have a transport plan ${\pi}$ for this ${c}$ for some marginals ${\mu_{1}}$ and ${\mu_{2}}$ we also have
$\displaystyle \begin{array}{rcl} \int_{\Omega_{1}\times \Omega_{2}}c(x_{1},x_{2})d\pi(x_{1},x_{2}) & = & \int_{\Omega_{1}\times \Omega_{2}}|x_{1}|^{2} d\pi(x_{1},x_{2})\\ &&\quad- \int_{\Omega_{1}\times \Omega_{2}}\langle x_{1},x_{2}\rangle d\pi(x_{1},x_{2})\\ && \qquad+ \int_{\Omega_{1}\times \Omega_{2}} |x_{2}|^{2}d\pi(x_{1},x_{2})\\ & = &\int_{\Omega_{1}}|x_{1}|^{2}d\mu_{1}(x_{1}) - \int_{\Omega_{1}\times \Omega_{2}}\langle x_{1},x_{2}\rangle d\pi(x_{1},x_{2}) + \int_{\Omega_{2}}|x_{2}|^{2}d\mu_{2}(x_{2}) \end{array}$
i.e., the transport cost for ${c(x_{1},x_{2}) = |x_{1}-x_{2}|^{2}}$ differs from the one for ${c(x_{1},x_{2}) = -\langle x_{1},x_{2}\rangle}$ only by a constant independent of ${\pi}$, so may well use the latter.
The fundamental theorem of optimal transport uses the notion of ${c}$-cyclical monotonicity which is stronger that just ${c}$-monotonicity:
Definition 3 A set ${\Gamma\subset \Omega_{1}\times \Omega_{2}}$ is ${c}$-cyclically monotone, if for all ${(x_{1}^{i},x_{2}^{i})\in\Gamma}$, ${i=1,\dots n}$ and all permutations ${\sigma}$ of ${\{1,\dots,n\}}$ it holds that
$\displaystyle \sum_{i=1}^{n}c(x_{1}^{i},x_{2}^{i}) \leq \sum_{i=1}^{n}c(x_{1}^{i},x_{2}^{\sigma(i)}).$
For ${n=2}$ we get back the notion of ${c}$-monotonicity.
Definition 4 A function ${\phi:\Omega_{1}\rightarrow {\mathbb R}}$ is ${c}$-concave if there exists some function ${\psi:\Omega_{2}\rightarrow{\mathbb R}}$ such that
$\displaystyle \phi(x_{1}) = \inf_{x_{2}\in\Omega_{2}}c(x_{1},x_{2}) - \psi(x_{2}).$
This definition of ${c}$-concavity resembles the notion of convex conjugate:
Example 2 Again using ${c(x_{1},x_{2}) = -\langle x_{1},x_{2}\rangle}$ we get that a function ${\phi}$ is ${c}$-concave if
$\displaystyle \phi(x_{1}) = \inf_{x_{2}}-\langle x_{1},x_{2}\rangle - \psi(x_{2}),$
and, as an infimum over linear functions, ${\phi}$ is clearly concave in the usual way.
Definition 5 The ${c}$-superdifferential of a ${c}$-concave function is
$\displaystyle \partial^{c}\phi = \{(x_{1},x_{2})\mid \phi(x_{1}) + \phi^{c}(x_{2}) = c(x,y)\},$
where ${\phi^{c}}$ is the ${c}$-conjugate of ${\phi}$ defined by
$\displaystyle \phi^{c}(x_{2}) = \inf_{x_{1}\in\Omega_{1}}c(x_{1},x_{2}) -\phi(x_{1}).$
Again one may look at ${c(x_{1},x_{2}) = -\langle x_{1},x_{2}\rangle}$ and observe that the ${c}$-superdifferential is the usual superdifferential related to the supergradient of concave functions (there is a Wikipedia page for subgradient only, but the concept is the same with reversed signs in some sense).
Now let us sketch the proof of the fundamental theorem of optimal transport: \medskip
Proof (of the fundamental theorem of optimal transport). Let ${\pi}$ be an optimal transport plan. We aim to show that ${\mathrm{supp}(\pi)}$ is ${c}$-cyclically monotone and assume the contrary. That is, we assume that there are points ${(x_{1}^{i},x_{2}^{i})\in\mathrm{supp}(\pi)}$ and a permutation ${\sigma}$ such that
$\displaystyle \sum_{i=1}^{n}c(x_{1}^{i},x_{2}^{i}) > \sum_{i=1}^{n}c(x_{1}^{i},x_{2}^{\sigma(i)}).$
We aim to construct a ${\tilde\pi}$ such that ${\tilde\pi}$ is still feasible but has a smaller transport cost. To do so, we note that continuity of ${c}$ implies that there are neighborhoods ${U_{i}}$ of ${x_{1}^{i}}$ and ${V_{i}}$ of ${x_{2}^{i}}$ such that for all ${u_{i}\in U_{1}}$ and ${v_{i}\in V_{i}}$ it holds that
$\displaystyle \sum_{i=1}^{n}c(u_{i},v_{\sigma(i)}) - c(u_{i},v_{i})<0.$
We use this to construct a better plan ${\tilde \pi}$: Take the mass of ${\pi}$ in the sets ${U_{i}\times V_{i}}$ and shift it around. The full construction is a little messy to write down: Define a probability measure ${\nu}$ on the product ${X = \bigotimes_{i=1}^{N}U_{i}\times V_{i}}$ as the product of the measures ${\tfrac{1}{\pi(U_{i}\times V_{i})}\pi|_{U_{i}\times V_{i}}}$. Now let ${P^{U_{1}}}$ and ${P^{V_{i}}}$ be the projections of ${X}$ onto ${U_{i}}$ and ${V_{i}}$, respectively, and set
$\displaystyle \nu = \tfrac{\min_{i}\pi(U_{i}\times V_{i})}{n}\sum_{i=1}^{n}(P^{U_{i}},P^{V_{\sigma(i)}})_{\#}\nu - (P^{U_{i}},P^{V_{i}})_{\#}\nu$
where ${_{\#}}$ denotes the pushforward of measures. Note that the new measure ${\nu}$ is signed and that ${\tilde\pi = \pi + \nu}$ fulfills
1. ${\tilde\pi}$ is a non-negative measure
2. ${\tilde\pi}$ is feasible, i.e. has the correct marginals
3. ${\int c\,d\tilde \pi<\int c\,d\pi}$
which, all together, gives a contradiction to optimality of ${\pi}$. The implication of item 1 to item 2 of the theorem is not really related to optimal transport but a general fact about ${c}$-concavity and ${c}$-cyclical monotonicity (c.f.~this previous blog post of mine where I wrote a similar statement for convexity). So let us just prove the implication from item 2 to optimality of ${\pi}$: Let ${\pi}$ fulfill item 2, i.e. ${\pi}$ is feasible and ${\mathrm{supp}(\pi)}$ is contained in the ${c}$-superdifferential of some ${c}$-concave function ${\phi}$. Moreover let ${\tilde\pi}$ be any feasible transport plan. We aim to show that ${\int c\,d\pi\leq \int c\,d\tilde\pi}$. By definition of the ${c}$-superdifferential and the ${c}$-conjugate we have
$\displaystyle \begin{array}{rcl} \phi(x_{1}) + \phi^{c}(x_{2}) &=& c(x_{1},x_{2})\ \forall (x_{1},x_{2})\in\partial^{c}\phi\\ \phi(x_{1}) + \phi^{c}(x_{2}) & \leq& c(x_{1},x_{2})\ \forall (x_{1},x_{2})\in\Omega_{1}\times \Omega_{2}. \end{array}$
Since ${\mathrm{supp}(\pi)\subset\partial^{c}\phi}$ by assumption, this gives
$\displaystyle \begin{array}{rcl} \int_{\Omega_{1}\times \Omega_{2}}c(x_{1},x_{2})\,d\pi(x_{1},x_{2}) & =& \int_{\Omega_{1}\times \Omega_{2}}\phi(x_{1}) + \phi^{c}(x_{1})\,d\pi(x_{1},x_{2})\\ &=& \int_{\Omega_{1}}\phi(x_{1})\,d\mu_{1}(x_{1}) + \int_{\Omega_{1}}\phi^{c}(x_{2})\,d\mu_{2}(x_{2})\\ &=& \int_{\Omega_{1}\times \Omega_{2}}\phi(x_{1}) + \phi^{c}(x_{1})\,d\tilde\pi(x_{1},x_{2})\\ &\leq& \int_{\Omega_{1}\times \Omega_{2}}c(x_{1},x_{2})\,d\tilde\pi(x_{1},x_{2}) \end{array}$
which shows the claim.
${\Box}$
Corollary 6 If ${\pi}$ is a measure on ${\Omega_{1}\times \Omega_{2}}$ which is supported on a ${c}$-superdifferential of a ${c}$-concave function, then ${\pi}$ is an optimal transport plan for its marginals with respect to the transport cost ${c}$.
This is a short follow up on my last post where I wrote about the sweet spot of the stepsize of the Douglas-Rachford iteration. For the case $\beta$-Lipschitz + $\mu$-strongly monotone, the iteration with stepsize $t$ converges linear with rate
$\displaystyle r(t) = \tfrac{1}{2(1+t\mu)}\left(\sqrt{2t^{2}\mu^{2}+2t\mu + 1 +2(1 - \tfrac{1}{(1+t\beta)^{2}} - \tfrac1{1+t^{2}\beta^{2}})t\mu(1+t\mu)} + 1\right)$
Here is animated plot of this contraction factor depending on $\beta$ and $\mu$ and $t$ acts as time variable:
What is interesting is, that this factor has increasing or decreasing in $t$ depending on the values of $\beta$ and $\mu$.
For each pair $(\beta,\mu)$ there is a best $t^*$ and also a smallest contraction factor $r(t^*)$. Here are plots of these quantities:
Comparing the plot of te optimal contraction factor to the animated plot above, you see that the right choice of the stepsize matters a lot.
I blogged about the Douglas-Rachford method before here and here. It’s a method to solve monotone inclusions in the form
$\displaystyle 0 \in Ax + Bx$
with monotone multivalued operators ${A,B}$ from a Hilbert space into itself. Using the resolvent ${J_{A} = (I+A)^{-1}}$ and the reflector ${R_{A} = 2J_{A} - I}$, the Douglas-Rachford iteration is concisely written as
$\displaystyle u^{n+1} = \tfrac12(I + R_{B}R_{A})u_{n}.$
The convergence of the method has been clarified is a number of papers, see, e.g.
Lions, Pierre-Louis, and Bertrand Mercier. “Splitting algorithms for the sum of two nonlinear operators.” SIAM Journal on Numerical Analysis 16.6 (1979): 964-979.
for the first treatment in the context of monotone operators and
Svaiter, Benar Fux. “On weak convergence of the Douglas–Rachford method.” SIAM Journal on Control and Optimization 49.1 (2011): 280-287.
for a recent very general convergence result.
Since ${tA}$ is monotone if ${A}$ is monotone and ${t>0}$, we can introduce a stepsize for the Douglas-Rachford iteration
$\displaystyle u^{n+1} = \tfrac12(I + R_{tB}R_{tA})u^{n}.$
It turns out, that this stepsize matters a lot in practice; too small and too large stepsizes lead to slow convergence. It is a kind of folk wisdom, that there is “sweet spot” for the stepsize. In a recent preprint Quoc Tran-Dinh and I investigated this sweet spot in the simple case of linear operators ${A}$ and ${B}$ and this tweet has a visualization.
A few days ago Walaa Moursi and Lieven Vandenberghe published the preprint “Douglas-Rachford splitting for a Lipschitz continuous and a strongly monotone operator” and derived some linear convergence rates in the special case they mention in the title. One result (Theorem 4.3) goes as follows: If ${A}$ is monotone and Lipschitz continuous with constant ${\beta}$ and ${B}$ is maximally monotone and ${\mu}$-strongly monotone, than the Douglas-Rachford iterates converge strongly to a solution with a linear rate
$\displaystyle r = \tfrac{1}{2(1+\mu)}\left(\sqrt{2\mu^{2}+2\mu + 1 +2(1 - \tfrac{1}{(1+\beta)^{2}} - \tfrac1{1+\beta^{2}})\mu(1+\mu)} + 1\right).$
This is a surprisingly complicated expression, but there is a nice thing about it: It allows to optimize for the stepsize! The rate depends on the stepsize as
$\displaystyle r(t) = \tfrac{1}{2(1+t\mu)}\left(\sqrt{2t^{2}\mu^{2}+2t\mu + 1 +2(1 - \tfrac{1}{(1+t\beta)^{2}} - \tfrac1{1+t^{2}\beta^{2}})t\mu(1+t\mu)} + 1\right)$
and the two plots of this function below show the sweet spot clearly.
If one knows the Lipschitz constant of ${A}$ and the constant of strong monotonicity of ${B}$, one can minimize ${r(t)}$ to get on optimal stepsize (in the sense that the guaranteed contraction factor is as small as possible). As Moursi and Vandenberghe explain in their Remark 5.4, this optimization involves finding the root of a polynomial of degree 5, so it is possible but cumbersome.
Now I wonder if there is any hope to show that the adaptive stepsize Quoc and I proposed here (which basically amounts to ${t_{n} = \|u^{n}\|/\|Au^{n}\|}$ in the case of single valued ${A}$ – note that the role of ${A}$ and ${B}$ is swapped in our paper) is able to find the sweet spot (experimentally it does).
<p
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# Finding the “top” or “bottom” vertex of a simplex
A vertex $v$ of a simplex in $\mathbb{R}^n$ is a "top" vertex if there exists a point $p \neq v$ in the simplex with $v \ge p$ (i.e. $v_1 \ge p_1$, ... , $v_n \ge p_n$). Similarly, $v$ is a "bottom" vertex if there exists $p \neq v$ with $v \le p$.
It's not hard to show that any simplex has at least one top or bottom vertex. I'm looking for a test I can run (preferably in polynomial time) that identifies at least one such vertex, and whether it's top or bottom.
-
Checking that a vertex $u$ is a top one can be done by solving a linear program, as follows: write $p$ in baricentric coordinates, i.e. $p=p_x=\sum_{v\in V} x_v v$, $x_v\geq 0$ for any $v\in V$, and $\sum_{v\in V} x_v=1$ (I denote by $V$ the set of vertices of the simplex). To check $u$ is top is equivalent to checking that there exists $x=(x_{v_1},\dots,x_{v_{|V|}})$ satisfying $x\geq 0$, $\sum_{v\in V} x_v=1$, $u\geq p_x$, and $x_u<1$.
So you solve the linear program $$\min x_u \text{ subject to x\geq 0, \sum_{v\in V} x_v=1, u\geq p_x},$$ and it can be done in polynomial time. If the value of the objective is strictly less than 1 then $u$ is top.
Testing for $u$ to be bottom is essentially the same, just replace the inequalities $u\geq p_x$ by $u\leq p_x$.
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# Sequence homework
If (bn) is an unbounded sequence then it has a sequence (b(kn)) such that lim (1/(b(kn)))=0
(where kn is a subsequence of bn )
Related Calculus and Beyond Homework Help News on Phys.org
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Homework Helper
Welcome to PF!
Hi gankutsuou! Welcome to PF!
(try using the X2 tag just above the Reply box )
Do the obvious …
start "If {bn} is unbounded, then for any n … "
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# Iverson brackets and SwiftUI modifiers
⇐ Notes archive
(This is an entry in my technical notebook. There will likely be typos, mistakes, or wider logical leaps — the intent here is to “let others look over my shoulder while I figure things out.”)
I love noticing when an everyday engineering practice has a named analog in mathematics. This time around, it was Iverson brackets. The Wikipedia page is…a lot, so no frets if it’s intimidating — the non-mathematician’s reading of it is the expression $[P]$ is equal to $1$ if $P$ is true or $0$, if false, where $P$ is some predicatetrue-false statement.
In Swift speak, a function from Bool to Int1.
In SwiftUI speak, conditionally setting a modifier’s value2. Most commonly with opacity(_:),
someView.opacity(isShown ? 1 : 0).
And implicitly with others like rotationEffect(_:anchor:),
someView
.rotationEffect(.degrees(isRotated ? 90 : 0))
// which expands out to,
someView
.rotationEffect(.degrees(90 * (isRotated ? 1 : 0)))
The isShown ? 1 : 0 and isRotated ? 1 : 0 ternaries are Iverson brackets in disguise. Kinda nifty to see another domain’s language around this type of expression. I came across the notation in an answer to the question of “What is the sum of number of digits of the numbers $2^{2001}$ and $5^{2001}$?” asked over on Math Stack Exchange.
The next note will likely pencil in the intermediary steps of that solution.
1. Or, a BinaryInteger conformance for the nerds.
2. Harlan posted a thread on why this approach is preferred over if-elseing in ViewBuilders.
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# American Institute of Mathematical Sciences
doi: 10.3934/mcrf.2022021
Online First
Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.
Readers can access Online First articles via the “Online First” tab for the selected journal.
## A geometric approach of gradient descent algorithms in linear neural networks
1 Laboratoire des Signaux et Systèmes, CentraleSupélec, Université Paris-Saclay, France 2 EIC, Huazhong University of Science and Technology, Wuhan, China 3 University Grenoble Alpes, Inria, CNRS, Grenoble INP, LIG, 38000 Grenoble, France
* Corresponding author: Yacine Chitour
Received April 2021 Revised March 2022 Early access April 2022
In this paper, we propose a geometric framework to analyze the convergence properties of gradient descent trajectories in the context of linear neural networks. We translate a well-known empirical observation of linear neural nets into a conjecture that we call the overfitting conjecture which states that, for almost all training data and initial conditions, the trajectory of the corresponding gradient descent system converges to a global minimum. This would imply that the solution achieved by vanilla gradient descent algorithms is equivalent to that of the least-squares estimation, for linear neural networks of an arbitrary number of hidden layers. Built upon a key invariance property induced by the network structure, we first establish convergence of gradient descent trajectories to critical points of the square loss function in the case of linear networks of arbitrary depth. Our second result is the proof of the overfitting conjecture in the case of single-hidden-layer linear networks with an argument based on the notion of normal hyperbolicity and under a generic property on the training data (i.e., holding for almost all training data).
Citation: Yacine Chitour, Zhenyu Liao, Romain Couillet. A geometric approach of gradient descent algorithms in linear neural networks. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2022021
##### References:
show all references
##### References:
Illustration of a $H$-hidden-layer linear neural network
A geometric "vision" of the loss landscape
Visual representation of normal hyperbolicity
Visual representation of normal hyperbolicity and trajectories samples (in green)
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2021 Impact Factor: 1.141
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# Bayes' Theorem
1. Sep 23, 2008
### sampahmel
Dear all,
1.) The difference of conditional probability and Bayes' formula.
2.) Is Bayes' formula a "all weather condition" formula for all conditional probabilities problem?
Thank you,
S
2. Sep 25, 2008
### sampahmel
3. Sep 27, 2008
### Focus
1) Conditional probability is $$\mathbb{P}(A|B):=\frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)}$$.
Bayes Theorem is used to swap the condition around,
$$\mathbb{P}(A|B):=\frac{\mathbb{P}(B|A) \mathbb{P}(A)}{\mathbb{P}(B)}$$
2) You can use Bayes formula for any conditional probabilities such that $$\mathbb{P}(B)>0$$
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1. ## simple integration problem
$\int_4^9 \frac{1}{(2x)(1+ \sqrt x)}$
2. ## Re: simple integration problem
Originally Posted by earthboy
$\int_4^9 \frac{1}{(2x)(1+ \sqrt x)}$
Assuming you mean \displaystyle \begin{align*} \int_4^9{\frac{1}{2x\left(1 + \sqrt{x}\right)}\,dx} &= \int_4^9{\frac{1}{2\sqrt{x}\sqrt{x}\left(1 + \sqrt{x}\right)}\,dx} \end{align*}
Now make the substitution \displaystyle \begin{align*} u = \sqrt{x} \implies du = \frac{1}{2\sqrt{x}}\,dx \end{align*} and also note that this means \displaystyle \begin{align*} 1 + \sqrt{x} = 1 + u \end{align*} and the limits will change to 2 and 3, and the integral becomes
\displaystyle \begin{align*} \int_4^9{\frac{1}{2\sqrt{x}\sqrt{x}\left(1 + \sqrt{x}\right)}\,dx} = \int_2^3{\frac{1}{u\left(1 + u\right)}\,du} \end{align*}
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# Probability: Permutations
Consider the experiment of picking a random permutation $\pi$ on $\{1,2,...,n\}$, and define the associated random variable $f(\pi)$ as the number of fixed points of $\pi$, i.e, the number of $i$ such that $f(i)=i$.
I know that a permutation of $X=\{1,2,\ldots ,n\}$ is a one-to-one function $\pi: X \rightarrow X$. Any two functions, $\pi_1, \pi_2$ can be composed and the resulting function is also a permutation. How can I find $E(F)$ and what do we know about $E(f(\pi^2))$ and $E(f( \pi^k))$?
Hint: what is the probability that $1$ is a fixed point? Now use the linearity of expectation.
These can be done using generating functions. First, consider $E[F].$ The exponential generating function of the set of permutations by sets of cycles where fixed points are marked is $$G(z, u) = \exp\left(uz - z + \log \frac{1}{1-z}\right) = \frac{1}{1-z} \exp(uz-z).$$ Now to get $E[F]$ compute $$\left.\frac{d}{du} G(z, u)\right|_{u=1} = \left. \frac{z}{1-z} \exp(uz-z) \right|_{u=1} = \frac{z}{1-z}.$$ This means that $E[F] = 1,$ there is one fixed point on average.
Let $E[F_2]$ denote the expectation of the number of fixed points in $\sigma^2$, where $\sigma$ is a random permutation. Now we need to mark both fixed points and two-cycles, since the latter turn into two fixed points under squaring. We get $$G(z, u) = \exp\left(uz - z + u^2 \frac{z^2}{2} - \frac{z^2}{2} + \log \frac{1}{1-z}\right) = \frac{1}{1-z} \exp\left(uz - z + u^2 \frac{z^2}{2} - \frac{z^2}{2}\right).$$ Continuing as before, we find $$\left.\frac{d}{du} G(z, u)\right|_{u=1} = \left. \frac{z+z^2}{1-z} \exp\left(uz - z + u^2 \frac{z^2}{2} - \frac{z^2}{2}\right) \right|_{u=1} = \frac{z+z^2}{1-z}.$$ The conclusion is that $E[F_2] = 2$ for $n\ge 2$ and there are two fixed points on average.
The pattern should now be readily apparent. For every divisor $d$ of $k$ a cycle of length $d$ splits into $d$ fixed points when raised to the power $k.$ Hence we need to mark these cycles with $u^d.$ To illustrate this consider $E[F_6].$
We get $$G(z, u) = \exp\left(uz - z + u^2 \frac{z^2}{2} - \frac{z^2}{2} + u^3 \frac{z^3}{3} - \frac{z^3}{3} + u^6 \frac{z^6}{6} - \frac{z^6}{6} + \log \frac{1}{1-z}\right) \\ = \frac{1}{1-z} \exp\left(uz - z + u^2 \frac{z^2}{2} - \frac{z^2}{2} + u^3 \frac{z^3}{3} - \frac{z^3}{3} + u^6 \frac{z^6}{6} - \frac{z^6}{6}\right).$$ Once more continuing as before, we find $$\left.\frac{d}{du} G(z, u)\right|_{u=1} \\ = \left. \frac{z+z^2+z^3+z^6}{1-z} \exp\left(uz - z + u^2 \frac{z^2}{2} - \frac{z^2}{2} + u^3 \frac{z^3}{3} - \frac{z^3}{3} + u^6 \frac{z^6}{6} - \frac{z^6}{6}\right) \right|_{u=1} = \frac{z+z^2+z^3+z^6}{1-z}.$$ The conclusion is that $E[F_6] = 4$ for $n\ge 6$ and there are four fixed points on average.
The general procedure is $$G(z, u) = \exp\left(\sum_{d\mid k} \left(u^d \frac{z^d}{d} - \frac{z^d}{d}\right) + \log \frac{1}{1-z} \right)= \frac{1}{1-z} \exp\left(\sum_{d\mid k} \left(u^d \frac{z^d}{d} - \frac{z^d}{d}\right)\right).$$ Once more continuing as before, we find $$\left.\frac{d}{du} G(z, u)\right|_{u=1} = \left. \frac{\sum_{d\mid k} z^d}{1-z} \exp\left(\sum_{d\mid k} \left(u^d \frac{z^d}{d} - \frac{z^d}{d}\right)\right) \right|_{u=1} = \frac{\sum_{d\mid k} z^d}{1-z}.$$ We have shown that the value of $E[F_k]$ is equal to $\tau(k)$ (the number of divisors of $k$) as soon as $n\ge k.$ It starts out at $1$ for $n=1$ and increases by one every time $n$ hits a divisor of $k$ up to and including $k$ itself.
In reference to the calculation above I can offer some Maple code that confirms the correctness of the result (answer is $\tau(k)$). This algorithm is very fast and does not iterate over all permutations but only over cycle decompositions, which are treated according to their multiplicity in $S_n$. This makes it possible to compute with values like $n=24,$ which would otherwise be out of reach ($24!$ has $23$ digits).
Here is some sample output:
> seq(v(n, 1), n=1..24);
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
> seq(v(n, 6), n=1..24);
1, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4
> seq(v(n, 12), n=1..24);
1, 2, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6
> seq(v(n, 16), n=1..24);
1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5
> seq(v(n, 24), n=1..24);
1, 2, 3, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8
This is the code:
with(group);
pet_cycleind_vrec :=
proc(n)
local p, s;
option remember;
if n=0 then return 1; fi;
end;
fp :=
proc(n, p, k)
local nonfixed, q, j, cyc;
q := p;
for j to k-1 do
q := mulperms(q, p);
od;
nonfixed := 0;
for cyc in q do
nonfixed := nonfixed + nops(cyc);
od;
n - nonfixed;
end;
v :=
proc(n, k)
option remember;
local t, el, v, q, cf, d, len, deg, res, cycs;
res := 0;
for t in pet_cycleind_vrec(n) do
el := 1;
cf := t; cycs := [];
for v in indets(t) do
deg := degree(t, v);
cf := cf/v^deg;
len := op(1, v);
if len>1 then
for d to deg do
cycs :=
[op(cycs), [seq(el+q, q=0..(len-1))]];
el := el + len;
od;
else
el := el + deg;
fi;
od;
res := res + cf*fp(n, cycs, k);
od;
res;
end;
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Citation
## Material Information
Title:
On some new examples of Cameron-Liebler line classes
Creator:
Rodgers, Morgan J.
Place of Publication:
Denver, CO
Publisher:
Publication Date:
Language:
English
## Thesis/Dissertation Information
Degree:
Doctorate ( Doctor of philosophy)
Degree Grantor:
Degree Divisions:
Department of Mathematical and Statistical Sciences, CU Denver
Degree Disciplines:
Applied mathematics
Committee Chair:
Payne, Stanley E.
Committee Members:
Cherowitzo, William
Penttila, Timothy
White, Diana
Williford, Jason
## Record Information
Source Institution:
Holding Location:
Auraria Library
Rights Management:
Copyright Morgan J. Rodgers. Permission granted to University of Colorado Denver to digitize and display this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Full Text
ON SOME NEW EXAMPLES OF CAMERON-LIEBLER LINE CLASSES
by
Morgan J. Rodgers
B.S., Humboldt State University, 2005 M.S., University of Colorado Denver, 2011
A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Applied Mathematics
2012
This thesis for the Doctor of Philosophy degree by Morgan J. Rodgers has been approved by
Stanley E. Payne, Advisor and Chair William Cherowitzo Timothy Penttila Diana White Jason Williford
Date
n
Rodgers, Morgan J. (Ph.D., Applied Mathematics)
On some new examples of Cameron-Liebler line classes Thesis directed by Professor Stanley E. Payne
ABSTRACT
Cameron-Liebler line classes are sets of lines in PG(3,q) having many nice combinatorial properties; among them, a Cameron-Liebler line class £ shares precisely x lines with any spread of the space for some non-negative integer x, called the parameter of the set. These objects were originally studied as generalizations of symmetric tactical decompositions of PG(3,q), as well as of subgroups of PTL(4,g) having equally many orbits on points and lines of PG(3, q). They have connections to many other combinatorial objects, including blocking sets in PG(2,g), certain error-correcting codes, and strongly regular graphs.
We construct many new examples of Cameron-Liebler line classes, each stabilized by a cyclic group of order q2 + q + 1 having a semi-regular action on the lines. In particular, new examples are constructed in PG(3,q) having parameter |(g2 — 1) for all values of q = 5 or 9 mod 12 with q < 200; with parameter |(q + l)2 (found in collaboration with Jan de Beule, Klaus Metsch, and Jeroen Schillewaert) for all values of q = 2 mod 3 with 2 < q < 128; with parameter 336 in PG(3, 27); and with parameter 495 in PG(3,32). The new examples with parameter |(q2 — 1) when q = 9 mod 12 are of particular interest. These induce a symmetric tactical decomposition of PG(3, q) having four classes on points and lines, one of the line classes being the set of lines in a common plane. This decomposition can be used to construct examples of two-intersection sets in the affine plane AG(2, q). Since the only previously known examples of two-intersection sets in affine planes of odd order are in planes of order 9, our example in AG(2,81) is new.
m
The form and content of this abstract are approved. I recommend its publication.
Approved: Stanley E. Payne
IV
ACKNOWLEDGMENTS
This thesis would not have been possible without the help of many people. I owe part of my success as a graduate student to every member of my committee, and I would like to thank them all for their time and support. In particular, I would like to thank my advisor Stan Payne, whose dedication to mathematics has truly been inspiring, and Tim Penttila, who generously suggested this problem to work on. Tim’s help in learning to program with Magma was truly indispensible in conducting this research, and he also brought a few key articles to my attention that were instrumental in filling in some details of this work. Bill Cherowitzo has also offered constant support and assistance; most notably, he convinced the department to give me an office to work in when I was a first year student with no other form of departmental support. Much of the computational work involved in this thesis was done on the University of Wyoming cluster, and I thank Jason Williford for going out of his way to set me up with this access. My survival in academia would have been much more difficult without the help and advice of these people and many others, especially Diana White, Mike Ferrara, and Oscar Vega.
I also would like to thank the Bateman family for their financial support of the mathematics department; I have been fortunate to receive three semesters of support from the Lynn Bateman Memorial Fellowship, and the completion of this research would have been much more difficult without the relief from teaching duties this provided. I would like to extend thanks to Jan de Beule and the Universiteit Gent as well for supporting me as a visiting researcher. Part of the work of this thesis was conducted during that trip, in collaboration with Jan, Klaus Metsch, and Jeroen Schillewaert.
Most importantly, I thank my loving wife who has never lost faith in my ability to succeed, even when I questioned it myself. She helped more than she knows; I definitely would not have finished this thesis without her support.
v
Tables.................................................................... viii
Chapter
1. Introduction............................................................... 1
1.1 Overview............................................................. 1
1.2 Finite fields........................................................ 2
1.3 The projective geometry PG(n, q) ................................... 4
1.4 Collineations and dualities ......................................... 5
1.5 Combinatorics of PG(n, q) ........................................... 7
1.6 Bilinear and quadratic forms......................................... 9
1.7 Orthogonality and totally isotropic subspaces....................... 10
1.8 Orthogonal polar spaces in PG(n, q)................................. 11
1-9 Q+(5,q) and the Klein correspondence................................ 14
2. Cameron-Liebler line classes.............................................. 18
2.1 Definitions and history ............................................ 18
2.2 Tight sets of Q+ (5, q)............................................. 19
2.3 Two-intersection sets, two-weight codes, and strongly regular graphs 21
2.4 Trivial examples.................................................... 23
2.5 Non-existence results .............................................. 24
2.6 Known examples...................................................... 26
2.6.1 Bruen and Drudge examples................................. 26
2.6.2 Penttila and Govaerts example in PG(3,4).................. 27
3. Methodology............................................................... 30
3.1 An eigenvector method for tight sets................................ 30
3.2 Tactical Decompositions............................................. 31
3.3 A model of Q+ (5, q)................................................ 33
3.4 The general method.................................................. 34
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4. New examples......................................................... 36
4.1 New examples with parameter |(g2 — 1)................... 37
4.1.1 The construction......................................... 37
4.1.2 Some details of these examples........................... 41
4.2 New examples with parameter |(g + l)2................. 43
4.3 Some other new examples....................................... 44
5. Planar two-intersection sets ........................................ 46
5.1 Projective examples........................................... 46
5.2 Affine examples............................................... 48
5.3 Constructions from Cameron-Liebler line classes............. 50
5.3.1 A two-intersection set in AG(2, 9)...................... 50
5.3.2 A new two-intersection set in AG(2, 81) ................ 52
5.3.3 A family of examples in AG(2, 32e)? 53
Appendix
A. Algorithms.......................................................... 56
A. l CLaut Matrix.................................................. 56
B. Programs ........................................................... 58
B. l CLshell.mgm................................................... 58
B.2 CLpreamble.mgm................................................ 60
B.3 CLaut.mgm .................................................... 62
B.4 CLbcirc.mgm................................................... 65
B.5 CLvspace.mgm.................................................. 68
B.6 CLpg3q.mgm.................................................... 70
B.7 CLint.mgm..................................................... 71
B.8 CL81int.mgm .................................................. 72
B.9 MNset.mgm..................................................... 73
References.............................................................. 76
vii
TABLES
Table
4.1 Parameters and automorphism groups of the new examples of Cameron-
Liebler line classes constructed....................................... 36
4.2 Intersection numbers of line classes with parameter |(g2 — 1) with the
planes of PG(3, q)..................................................... 42
4.3 Intersection numbers of line classes with parameter + l)2 with the
planes ofPG(3,q)....................................................... 44
4.4 Intersection numbers of some other new line classes.................. 45
5.1 Lines per point for the symmetric tactical decomposition induced on
PG(3,9) by a Cameron-Liebler line class of parameter AO............. 51
5.2 Points per line for the symmetric tactical decomposition induced on PG(3, 9)
by a Cameron-Liebler line class of parameter AO..................... 51
5.3 Lines per point for the symmetric tactical decomposition induced on
PG(3,81) by a Cameron-Liebler line class of parameter 3280.......... 53
5.4 Points per line for the symmetric tactical decomposition induced on PG(3, 81)
by a Cameron-Liebler line class of parameter 3280................... 53
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1. Introduction
1.1 Overview
The focus of this dissertation is to construct new examples of Cameron-Liebler line classes admitting a certain cyclic automorphism group. These line classes have many different characterizations. Most notably, a Cameron-Liebler line class C, has the property that, for some integer x called the parameter, C, shares precisely x lines with every spread of the space. Cameron-Liebler line classes are also of interest to other areas of mathematics including group theory and coding theory. These sets of lines were originally studied in relation to a group theory problem regarding collineation groups of PG(3,q) having the same number of orbits on points and on lines. They also serve as generalizations of the notion of a symmetric tactical decomposition of PG(3,q); i.e., a tactical decomposition having the same number of point classes and line classes. Through the Klein correspondence, a Cameron-Liebler line class is equivalent to a set of points of the hyperbolic quadric Q+(5, q) called a tight set. Tight sets of this quadric often determine two-intersection sets of the underlying projective space PG(5,g), that is, sets of points having two intersection numbers with respect to hyperplanes. Two-intersection sets can then be used to construct error correcting codes with codewords having precisely two nonzero weights which, in turn, give rise to examples of strongly regular graphs.
After reviewing the geometry of PG(3, q) and Q+(5, q), as well as their relationship through the Klein correspondence, we survey the known results on Cameron-Liebler line classes, including the known examples as well as some non-existence results. We show the equivalence of these objects with tight sets of Q+(5, q) and give results on when these sets determine two-intersection sets of the underlying PG(5, q). We also look at the construction of two-weight codes and strongly regular graphs from these two-intersection sets. Once we have developed this background material, we develop tools which are used to construct new examples. The main tools are
1
an eigenvector method for finding tight sets and results on tactical decompositions which facilitate this method. Since we are primarily working from the point of view of Q+ifi,q), an algebraic model for this space is introduced. This allows us to give a concise notation for a cyclic group of order q2 + q + 1 acting semi-regularly on the space, which is contained in the stabilizer of each new example constructed.
We use the tools we develop to construct several new examples of Cameron-Liebler line classes in PG(3,q), including many having parameters \{q2 — 1) and |(g+l)2. We also describe other examples in PG(3, 27) and PG(3, 32). For all of these new examples, we detail structural information such as automorphism groups and intersection numbers with planes of PG(3,q), as well as the related two-intersection sets of PG(5,g), two-weight codes, and strongly regular graphs. Furthermore, when q = 9 or 81, the new examples with parameter |(g2 — 1) are line partitions of a symmetric tactical decomposition of PG(3, q) having four parts on points and on lines; we give a construction from this decomposition of two-intersection sets in AG(2,g). Few examples of two-intersection sets of odd order affine planes are known; in fact, the only previously known examples are in planes of order 9. Thus, our example in AG(2, 81) is new.
1.2 Finite fields
A finite held always has order q = pe, where p is a prime. This held, which is unique up to isomorphism, will be denoted Fg and has characteristic p; i.e., Ym=i x = 0 for every xeF, (and p is the smallest integer for which this is true). The multiplicative group F* of nonzero elements of Fg is a cyclic group of order q — 1; an element a G F* having order g — 1 is called a primitive element, and (a) = F* for such an element.
Let K = Fq, where q = ph. A subset F of K which is also a held under the
same operations is called a subfield of K; we write F < K. K contains a unique
subheld isomorphic to Fpe for each e dividing h, consisting of {a E K : ap£ = a}. The
2
intersection of all subfields of K is called the prime subfield of K and is isomorphic to Fp. We can construct a larger held E = from K by considering an irreducible polynomial f(x) in K[x\ of degree d; in this case
E = K[x\/(f(x)) = {a0 + a\x + ... + ad-ixd~l\ cp e K, f(x) = 0}
is a hnite held of order qd containing K as a subheld. We say that E is an extension field of K.
A map a : Fg —>• Fg is called an automorphism of Fg if a is a permutation of the elements such that (x + y)a = xa + ya, and (xy)a = {xa){ya) for all x, y in Fg. We write Aut(Fg) for the group of automorphisms of Fg. If q = pe, p prime, then Aut(Fg) is cyclic, isomorphic to Ze, and is generated by the Frobenius automorphism (p : x i y .
Let q = pe, F = Fg, and K = Fg/> with F < K; then Au.t(K/F), the group of automorphisms of K hxing every element of F, has order h and is generated by (f)e : x i y xp£ = xq. We dehne the relatiue trace map from K to F, TK/F : K i—>• F, by
Tk/f(o) = aa = a + aq + af + ... + aq(h X).
a£Aut(K / F)
This map has the following properties:
1. TK/F(a + b) = Tk/f(o) + TK/F(b) for all a, b E K.
2. TK/F(ca) = cTK/F(a) for all c E F, a E K.
3. Tk/f(o) = ha for all a E F.
4. Tk/f{(F) = T^/j?(a) for all a E K and for all a E Aut(K/F).
Notice that the hrst two items imply that, if K is viewed as a vector space over F, then TK/F is a linear transformation from K to F. We actually have more; the map TK/F maps K onto F, and in fact, every linear map from K into F takes the form Lb(a) = TK/F(ba) for some b E K.
3
1.3 The projective geometry PG(n,g)
Much of this material is treated thoroughly in [25] or in [18]. There are also
examples of projective planes which are not of the form PG(2, q); for details on these examples, see [19].
Definition 1.1 Let F = Fq be a finite field of order q and V be a vector space of dimension n+ 1 over F. We define the geometry PG(n,q) as follows:
• The one-dimensional vector subspaces of V are the points of PG(n, q).
• The (d+ l)-dimensional vector subspaces ofV are the d-dimensional subspaces ofPG(n,q).
• Incidence is defined in terms of containment of the corresponding vector subspaces.
The word “dimension†is used in two ways with a different meaning; when it is not clear which we mean from context, we will specify projective dimension when we are talking about the dimension in PG(n,q), or vector space dimension when we are talking about a subspace of V. This definition of a projective space allows us to associate vectors in V with points of PG(n, 5); namely, a nonzero vector v e V represents a point of PG(n, q), with nonzero vectors v, w representing the same point if and only if v = cw for some c G F*.
We will call a subspace of PG(n, 5) having dimension n — la hyperplane; the set of points on a hyperplane can be described as those satisfying a homogeneous linear equation. We write the coefficients the equation describing a hyperplane H as h = [xo, • • • , xfi, with the convention that a point u is in H if and only if uhT = 0. We will speak of a set of points or of hyperplanes as being linearly independent if the corresponding vectors are linearly independent in the vector space.
It is frequently useful to describe a subspace U of PG(n, q) as either the intersection or the span of other subspaces. Given two subspaces U\ and U2 of PG(n, q), the
4
intersection U\ fl U2 is again a subspace of PG(n, q). We define the span of U\ and U2 to be the smallest subspace of PG(n, 5) containing both U\ and U2; we denote this by (U\, U2). In general, the projective dimension of (U\, U2) is given by
dim {Ui, U2) = dim LA + dim U2 — dim (LA fl U2).
The span of two distinct points, for example, is the line containing both of them. Given a subspace U of dimension d and a hyperplane id, we have that U fl H is either equal to U, or has dimension d — 1. Thus we can describe a d-dimensional subspace U of PG(n, q) as either the span of d + 1 linearly independent points, or as the intersection of n — d linearly independent hyperplanes. If hi,..., \vn-d are the vectors containing the coefficients of the equations for these hyperplanes, then we can associate U with the left null space of the matrix
hT hT
nl • • • nn-d
1.4 Collineations and dualities
A bijection on the points of PG(n,g) which preserves the lines is called a collineation; that is, a map 6 : PG(n, 5) —> PG(n,q) such that for all lines £ of PG(n,q), the image 8(£) is also a line of PG(n, 5). This necessarily implies that any d-dimensional subspace of PG(n, q) gets mapped by 6 to another d-dimensional subspace. Since we view the subspaces of PG(n, 5) as corresponding to subspaces of an (n +1)-dimensional vector space V over Fg, we can describe a collineation of PG(n, q) in terms of its action on V. In particular, any matrix A E GL(n + 1, q) can be used to define a collineation : x 1—> xA of PG(n, q). Collineations of this type are called
homographies. Note that, for any A G F*, the matrices A and A A define the same map on PG(n, 5); we say these two maps are projectively equivalent. The group
PGL(n+ 1, q) = GL(n+ 1, q)/Z(GL(n + l,q))
5
is called the projective linear group, and acts faithfully on PG(n, q). It is worth noting that, for a hyperplane H represented by h and a matrix A E PGL(n + 1, q), we have that x G If if and only if (xA)(A_1hT) = 0, so the hyperplane H gets mapped by La to a new hyperplane defined by A-1 hT. Automorphisms of Fg also give examples of collineations of PG(n, q). Given a E Aut(Fg), the map on PG(n, q) induced by
a : V ->â– V : (x0, â– â– â– , xn) ^ (x%,..., xjj)
is called an automorphic collineation.
The Fundamental Theorem of Projective Geometry tells us that any collineation of PG(n,g) can be obtained by composing an automorphic collineation with a ho-mography. Such a map is of the form
La o a : x 1—> La{?G) = xCTA,
where a E Aut(Fg) and A E PGL(n+ 1 ,q), and is called a projective semilinear map. The group of these maps is denoted
PTL(n +1, q) = PGL(n + 1, q) x Aut(Fg).
Associated with a projective geometry S is the so-called dual geometry S*-, this geometry’s points and hyperplanes are, respectively, the hyperplanes and points of
S. A projective geometry of the form PG(n, 5) is isomorphic to its dual geometry, and we call an isomorphism from the points of PG(n, 5) onto the hyperplanes a reciprocity. One important example is the map sending a point x to the hyperplane determined by xT. By our earlier comments about the Fundamental Theorem of Projective Geometry, any reciprocity can be written in the form x —> (x.aA)T, where a E Aut(Fg) and A E PGL(n + 1, q). If we have a reciprocity 6 that is an involution, that is, if 92 = 1, then we call 6 a polarity.
1.5 Combinatorics of PG(n, q)
6
Theorem 1.2 [18] In PG (n,q), there are
{q"q-i)1) Points, lines, (
d-dimensional subspaces.
(gn+1-1)(g"-1) iines an(i
nGb'-i)
Given d < s, the number of s dimensional subspaces containing a fixed d-dimensional subspace is given by
n—d
n tf-1)
i=s—d-\-1
i= 1
It is useful to note that, since PG(u,g) is self dual, the number of d-dimensional subspaces is the same as the number of (n — d)-dimensional subspaces. In particular, there are the same number of hyperplanes as there are points. Also, the number of hyperplanes through a fixed point is the same as the number of points in a fixed hyperplane.
We now give some consideration to the orders of various groups associated with
PGM [16].
Theorem 1.3 If q = pe, then
ra+l
GL(n+l,q) has order q\nG+l) — 1);
k=1
ra+l
PGL(n+l,q) has order qhnG+l) — 1), and
k=2
ra+l
PTL(n+ 1 ,q) has order eq^n(-n+ ^ — !)•
k=2
The projective space we will be primarily interested in will be PG(3, q). Specializing the above results to this specific case, we see that PG(3, q) contains q3 + q2 + q-f 1 points as well as planes, and (q2 + q + 1 )(q2 + 1) lines. Through each point there are q2 + q + 1 lines and cj2 + q + 1 planes; also, given a line, there are q + 1 planes containing that line.
7
The groups acting on PG(3, q) are PGL(4, q) and PTL(4, q); if q = pe, the orders of these are
PGL{A,q) has order Q6(q2 — l)(g3 — l)(g4 — 1) and PTL(A,q) has order eq6(q2 — l)(g3 — f){qA — f) .
A set 7Z of q + f mutually skew lines in PG(3, q) is called a regulus provided
1. through every point of every line of 7Z there is a transversal of the lines of 7Z (that is, a line meeting each of the lines of 1Z); and,
2. through every point of every transversal there is a line of 1Z.
It is clear that the set of transversals of 7Z is also a regulus which we call 7£0pp> the opposite regulus of 1Z. Any three skew lines in PG(3,q) determine a unique regulus. A spread of PG(3, q) is a set of cj2 + 1 lines of the space that partitions the points. A spread S is called regular if, given any three skew lines in S, the regulus determined by those three lines is also contained in S. Spreads can be defined in higher dimensional spaces as well, and are of considerable interest, as they can be used to construct examples of projective planes. Their classification is an important problem in finite geometry that is beyond the scope of this thesis.
A k-arc K of PG(2, q) (or any projective plane of order q) is a set of k points such that no three are collinear. Thus any line £ of PG(2, q) meets K in 0, 1, or 2 points; we call these lines external, tangent, or secant to K respectively. A k-arc must have k < q + 2. A k-arc K which is not contained in any (k + l)-arc is called maximal. A (q + l)-arc is called an oval, and if q is odd, any (q + l)-arc is maximal. However, if q is even, every tangent line to an oval K passes through a common point N which we call the nucleus of 1C. In this situation, K U {N} is a (q + 2)-arc, which we call a hyperoval. Given a plane tt embedded in PG(3,q) containing an oval or hyperoval O, and a point p not in tt, we define a cone over O to be the set of points on the
8
lines joining p to points of O. The lines are called the generators of the cone, and p is called the vertex of the cone.
Let V be a vector space of dimension n + 1 over F = ¥q A bilinear form on V is a function B : V x V F that is linear in each argument; that is,
B(au + v, w) = aB(u, w) + B(v, w) and B(u, av + w) = aB(u, v) + B(u, w)
for all u, v, w G V and all a E F.
A bilinear form B is said to be symmetric if B(u, v) = B(v, u) for all u, v e V, and alternating if B(u,u) = 0 for all u e V. We are strictly interested in reflexive bilinear forms, that is, those for which B(u,v) = 0 implies B(v, u) = 0. Every reflexive bilinear form is either symmetric or alternating.
A quadratic form on a vector space V is a map Q : V -E F defined by a homogeneous degree 2 polynomial in the coordinates of V relative to some basis. Equivalently, we call Q : V F a quadratic form if
Q(au) = a2Q( u) for all a E F and uef and B : (u, v) i y Q(u + v) — Q(u) — Q(v) gives a bilinear form on V.
It is clear that B is symmetric if q is odd and alternating if q is even. This associated bilinear form, B, is called the polar form of Q. A vector space equipped with a quadratic form is called an orthogonal space.
Given a bilinear form B and an ordered basis B = {v0,..., vra} for V, we put bij = B(vj,Vj). The Gram matrix relative to B is then defined by B = [bifl. This matrix has the property that, if [u]b and [v]g are coordinate vectors of u and v relative to B, then B(u,v) = [u]sB[v]g. Any two Gram matrices of a bilinear form B have the same rank, which we define to be the rank of B. If Q is a quadratic form
9
on V, we define the upper triangular matrix
A = (a,ij), where
B(vi, vj), i < j; < Q(vi), i = j;
0 * > j.
V
We then have that Q(u) = [u]ed[u]g, and the Gram matrix for the polar form of Q with respect to B is then B = A + AT.
1.7 Orthogonality and totally isotropic subspaces
Let B be a reflexive bilinear form on PG(n,g). We define an orthogonality relationship on the points of PG(n, 5) by u _L v if B(u,v) = 0. If S C V, we define S “perp†to be
S1 = {v G P| v 1 s Vs e S}.
A point v of PG(n, q) is called singular with respect to a bilinear form B if v1 = V; it is called singular with respect to a quadratic form Q if it is singular with respect to the associated bilinear form and Q(v) = 0. We say B or Q is degenerate if there is a singular point, and nondegenerate otherwise.
We place a special significance on points v for which B(v,v) = 0 or Q(v) = 0. Such a point v is called isotropic with respect to to bilinear form B or the quadratic form Q, respectively. We call a subspace W of V isotropic if it contains an isotropic point, anisotropic otherwise, and totally isotropic if B(u, v) = 0 for all u,v G W (for a bilinear form), or if Q(v) = 0 for all v e W (for a quadratic form). If a subspace is totally isotropic with respect to a quadratic form Q, then it is also totally isotropic with respect to the associated bilinear form, though the converse only holds if q is odd. The set of isotropic points in PG(n, q) with respect to a nondegenerate quadratic form is called a quadric, and has the property that any line of PG(n, 5) containing more than two points of a quadric must be completely contained in the quadric.
If B is nondegenerate form, the orthogonality relation can be used to define a
polarity a : U 1—> U1- of PG(n, 5). In this case, if U and W are subspaces of V with
10
U < W, then W1- < U-1; furthermore, for any subspace U of V, dim 17 + dimf/-1 = dimVh A point x is said to be isotropic with respect to the polarity if x C x-1, and a subspace U is said to be totally isotropic with respect to the polarity if U < UA This is in agreement with the notions of being isotropic or totally isotropic with respect to the bilinear form. When q is odd, the polar form of a nondegenerate quadratic form is necessarily nondegenerate. Thus in this case the notions of being (totally) isotropic with respect to the quadratic form, the bilinear form, and the associated polarity all agree.
The situation is more complicated when q is even, since it is possible for the polar form of Q to be degenerate even when Q is nondegenerate. In this case, we do not have a polarity associated with the quadratic form. Even if the polar form B of Q is nondegenerate, we have B(u,u) = 0 for every u e PG(n,q), so every point of PG(n,g) is incident with its image under the induced polarity (such a polarity is called a null polarity). Thus the set of points which are isotropic with respect to this polarity does not agree with the set of points which are isotropic with respect to the quadratic form.
1.8 Orthogonal polar spaces in PG(n,q)
Definition 1.4 A polar space of rank r is an incidence geometry consisting of a set of points, lines, projective planes, ..., (r — l)-dimensional projective spaces called subspaces such that
1. Any two subspaces intersect in a subspace.
2. IfU is a subspace of dimension r — 1 and p is a point not in U, there is a unique subspace W containing p with U fl W having dimension r — 2; it consists of all points of U which are joined to p by some line.
3. There are two disjoint subspaces of dimension r — 1.
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The (r — l)-dimensional subspaces are called maximals of the polar space.
The finite classical polar spaces are the examples naturally embedded in a projective space PG(n,g); they are defined by a nondegenerate quadratic or sesquilinear form on the space. A result of Tits [33] proved that any polar space with rank at least 3 is classical. Rank 2 polar spaces are a special case. They are called generalized quadrangles, and there are nonclassical examples of these; see [26] for a detailed treatment.
Let F = Fg, and k be a (n + 1)-dimensional vector space over F. Take Q to be a nondegenerate quadratic form on V with polar form B. The geometry consisting of the totally isotropic subspaces of PG(n,g) with respect to Q is an example of a classical polar space; we call an example arising in this way an orthogonal polar space.
Note: We have now introduced three very closely related terms, an orthogonal space, a quadric, and an orthogonal polar space.
• The vector space V along with a nondegenerate quadratic form is an orthogonal space.
• The set of isotropic points of PG(n,g) with respect to the quadratic form is called a quadric.
• The geometry of totally isotropic subspaces with respect to the quadratic form is called an orthogonal polar space, in this context the polar space can either be considered as embedded in PG(n, q) or as a geometry in its own right.
The most general collineation of PG(n,g) preserving a quadric is called a semisimilarity, this is a map a such that, for some a G F* and some r G Aut(Fg),
Q((?(x)) = a(Q(x))T-
We call a a similarity if r = 1, and we call a an isometry if a = 1 and r = 1. The following important theorem is known as Witt’s Extension Theorem:
12
Theorem 1.5 IfU,W< V, and a : U —>• W an isometry, then there is an isometry r : V —> V such that r\u = o.
Corollary 1.6 Any two maximals ofV have the same dimension.
The vector space dimension of a maximal is called the Witt index of the polar space. The Witt index of a nondegenerate form is less than or equal to \ dim V, since a totally isotropic subspace W is contained in W^.
We define two distinct points u, v of the quadric to be a hyperbolic pair if B(u, v) = 1. We then call (u, v) a hyperbolic line. Note that this is a line of PG(n, q) containing precisely two points of the quadric.
Theorem 1.7 Any nondegenerate orthogonal space of Witt index r over Fg is isometric to one of the following:
1. A hyperbolic quadric Q+{2r — 1 ,q) is the orthogonal direct sum of r hyperbolic lines.
2. A parabolic quadric Q(2r, q) is the orthogonal direct sum of r hyperbolic lines and a one-dimensional anisotropic space. These fall into two isometry classes and one similarity class when q is odd, and one isometry class when q is even.
3. An elliptic quadric Q~(2r + 1 ,q) is the orthogonal direct sum of r hyperbolic lines and a two-dimensional anisotropic space.
The group of isometries of Q+(2r—1, q), Q(2r, q), or Q_(2r+1, q) is denoted 0+(2r, q), 0(2r + l,q), or 0~{2r + 2,q), respectively. For the projective versions of these groups, we prefix this with P, PG, or PT depending on whether we want the group of isometries, similarities, or semi-similarities, respectively.
If we have a set of points O in a polar space such that every maximal of the polar space meets O in a unique point, then we call O an ovoid of the polar space. The
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classification of ovoids in classical polar spaces is an important open problem in finite geometry; we are primarily interested in these objects because of how they interact with other objects in the space.
1.9 Q+(5,q) and the Klein correspondence
The 5-dimensional hyperbolic orthogonal space Q+(5, q) plays an important role, as this geometry is closely related to PG(3,q). This quadric is made up of the orthogonal direct sum of three hyperbolic lines, and the standard associated quadratic form is given by
Q ■{%0, Xi,X2, Xs, X4, X5) 1—> X0X1 + X2X3 + X4X5, which is described by the matrix
0 1 0 0 0 0
0 0 0 0 0 0
0 0 0 1 0 0
0 0 0 0 0 0
0 0 0 0 0 1
0 0 0 0 0 0
The Gram matrix for the polar form B with respect to the standard basis is then
B = A + At.
Another way to think of the structure of this polar space is given by taking one point from each of the three hyperbolic pairs. Since the hyperbolic lines they determine are pairwise orthogonal, these three points are also pairwise orthogonal and so span a totally isotropic plane tti, necessarily a maximal of the polar space. The three remaining points from the hyperbolic pairs then span a totally isotropic plane 7r2 which is disjoint from tti-
The geometries PG(3,q) and Q+(5,q) are closely related through a mapping
known as the Klein correspondence. This refers to a bijection from the lines of PG(3, q)
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to the points of Q+(5,q) such that two lines of PG(3,q) intersect if and only if their images are collinear in Q+(5,q). To define the bijection, we will first establish a way to describe lines of PG(3,q) using Plucker coordinates. Let x = (To, an, x2, x3) and y = (yo, Vi, y2, 2/3) be distinct points on a line £ of PG(3,q). Dehne G(£) = (P01 > P23, P02, P31, P03, P12), where
Pa
Xf ^ Xj
for 0 < i < j < 3.
Vi Vj
If we consider G(£) E PG(5,g), any choice of two points on £ determine the same point. Furthermore, it can be seen that P01P23 +P02P31 +P03P12 = 0. Thus, G takes lines of PG(3, q) to points of Q+(5, q).
Now, given a point a = (a0, «i, a2) a3, a^, a5) G Q+(5, q), so a0«i +^2^3 + «4«5 = 0, the matrix
0 a\ a3 0>5
—a\ 0 a4 —a2
-a3 —a 4 0 a0
— 0-5 a 2 —ao 0
has rank two. Therefore the map
H (a) = row
0 a\ a.3 (15
—a\ 0 a4 —a 2
—CL3 —Op 0 ao
— 0-5 a2 —ao 0
takes the point a to a line of PG(3, q). It can be verified that H(G(£)) = l. Thus G and H give a bijection between the lines of PG(3, q) and the points of Q+(5, q). We refer to this bijection as the Klein correspondence.
Here we detail some important properties of the Klein correspondence.
Theorem 1.8 Two lines £ and £! of PG(3,q) are concurrent if and only if their corresponding points L and L' are collinear in Q+(5,g).
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Corollary 1.9 The set of lines in a spread of PG(3, q) correspond to an ovoid of
Q+(5,g).
Corollary 1.10 Let £ and £' be two concurrent lines in PG(3, q) with corresponding points L and L' in Q+(5, q). Then the lines of the fiat pencil of lines in PG(3, q) determined by £ and £' correspond to the line of Q+(5, q) through L and L'. Conversely, each line of Q+(5, q) corresponds to a set of lines in PG(3, q) lying in a fiat pencil.
Corollary 1.11 The set of points in a totally isotropic plane of Q+ (5, q) corresponds to a set of cj2 + q + 1 lines in PG(3, q), any two of which are concurrent. Thus, they correspond to either the set of lines through a common point p, denoted star(p), or the set of lines in a common plane ir, denoted lineijr).
We call a totally isotropic plane of Q+(5, q) a Latin plane if it corresponds to star(p) for some point p E PG(3, q), and a Greek plane if it corresponds to line(7r) for some plane tt of PG(3, q).
Corollary 1.12 Any two distinct planes of the same type in Q+(5,q) intersect in a 0-dimensional subspace (a single point). Any two planes of different types are either disjoint, or meet in a line of Q+(5,q). Thus two planes are of the same type if and only if their intersection has even dimension.
These correspondences allow us to count the following:
Corollary 1.13 Q+(5,q) contains
1. (q2 + 1 )(q2 + q + 1) points;
2. (q3 + q2 + q + 1 )(q2 + q + 1) lines;
3. 2(q3 + q2 + q + 1) planes;
4â– q(q + l)2 points collinear to a given point;
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5. (q + l)2 lines containing a given point;
6. 2(q + 1) planes containing a given point;
7. 2 planes containing a given line.
The Klein correspondence also gives us a connection between the groups PTL(4, q) acting on PG(3,q) and PTO+(6,q) acting on Q+(5,q). Specihcally, any element of PTL(4:,q) induces an action on the points of Q+(5,q) preserving collinearity, and so PTO+(6,q) has a subgroup isomorphic to PTL(4:,q). Any map on Q+(5,q) arising in this fashion maps Greek planes to Greek planes and Latin planes to Latin planes. Any correlation of PG(3, q) sends lines to lines, and so also induces an action on the points of Q+(5,q) preserving collinearity. A map arising in this fashion interchanges the Greek and Latin planes. These are known to be the only automorphisms of
Q+(5,g).
Theorem 1.14 The structure of the projective similarity and semi-similarity groups ofQ+(5,q) are as follows:
• PGO+(6,q) ~ PGL(4,q) x Z2.
• PTO+(6,q) ~ PTL(4,q) x Z2.
Using this connection between the lines of PG(3,q) and the points of Q+(5,q) can be helpful, especially when dealing with combinatorics of sets of lines in PG(3, q). In addition to having many theoretical results to apply, it is more computationally convenient to deal with sets of points. For this reason, much of our work in this thesis is done in the context of Q+(5, q).
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2. Cameron-Liebler line classes
In this chapter, we will survey many of the known results on Cameron-Liebler line classes. This includes non-existence results, known constructions, and a discussion of the images of these line sets in Q+(5,q) under the Klein correspondence.
2.1 Definitions and history
Here we detail sets of lines in PG(3,q) having some special combinatorial properties. These sets of lines were originally studied by Cameron and Liebler [8], who called them “special†line classes, in connection with the study of collineation groups of PG(3,q) having the same number of orbits on points and lines. Such a group induces a symmetric tactical decomposition of the incidence structure of points and lines in PG(3,q), and they showed that a line class from such a decomposition has nice intersection properties with respect to reguli and spreads of the space. They abstracted the concept of sets of lines with these properties, hoping it would lead to the classification of symmetric tactical decompositions and collineation groups of PG(3,q) with this orbit structure. However, this problem proved interesting in a more general setting than originally envisioned.
Definition 2.1 Let A be the point-line incidence matrix ofPG(3,q) with respect to some ordering of the points and lines, and let £ be a set of lines in PG(3,q) with characteristic vector x = Xc- We will write (x)e for the entry of x corresponding to the line l. The following statements are all equivalent; if they hold, £ is called a Cameron-Liebler line class [8], [27].
1. xc G row(A).
2. xc £ {null{AT))-L
3. \R fl £\ = \Ropp H £| for every regulus 7Z and its opposite Ropp-
4• There exists x G TA such that \£ fl S\ = x for every spread S.
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5. There exists x G Z+ such that \C fl S\ = x for every regular spread S.
6. There exists x G Z+ such that, for every incident point-plane pair (p, tt),
|star{p) fl C\ + \ line(7r) fl C\ = x + (q + l)\pencil(p,7r) fl C\.
7. There exists x G Z+ such that, for every line l in PG(3, q),
| {lines m G C meeting | = + 1) + (q2 + l)(x)r-
5. T/iere exists x G Z+ skc/i that, for every pair l, m of skew lines in PG(3, q), \{n G £ : n is a transversal to l, m}\ = x + q[(x)t + (x)m\ â–
The value x must satisfy 0 < x < q2 + 1, and will necessarily be the same in each instance; we call x the parameter of the line class. If £ is a Cameron-Liebler line class with parameter x, then \C\ = x(q2 + q+ 1). The complement £ of a Cameron-Liebler line class C, with parameter x is also a Cameron-Liebler line class having parameter cj2 + 1 — x, and the union of two disjoint Cameron-Liebler line classes with parameters x\ and X2 is a Cameron-Liebler line class with parameter x\ + X2■A Cameron-Liebler line class is said to be irreducible if it does not properly contain any other line class as a subset.
2.2 Tight sets of Q+(5,q)
To investigate the existence of Cameron-Liebler line classes, it is frequently useful to translate their definition to the setting of Q+(5, q) using the Klein correspondence. In this context, part 7 of Definition 2.1 has an especially interesting interpretation; C, is a Cameron-Liebler line class if and only if its image M in Q+(5, q) has the following property:
There exists x G Z+ such that, for every point p in Q+(5, q),
Ip^nMl = x(q+ 1) + q2(x)p, where x = Xm is the characteristic vector of M.
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Definition 2.2 Let S be a polar space of rank r > 3 over Fg. Then a set T of points in S is an x-tight set if for all points p E S
n T |
+ qr 1 if p eT
ifp i r
Adapting this definition for the rank 3 polar space Q+(5, q), we see that a Cameron-Liebler line class with parameter x is equivalent to an x-tight set of Q+(5, q).
Point sets in polar spaces having precisely two intersection numbers with respect to perps of points are called intriguing by Bamberg, Kelly, Law and Penttila [2]. There are two types of intriguing sets in finite polar spaces, and they can be characterized in terms of their intersection numbers. If X is an intriguing set of a polar space having intersection numbers h\ for perps of points inside X and h2 for perps of points outside X, then X is a tight set if h\ > h2. A tight set of points in a finite polar space can also be defined as a set of points T such that each point of the space is, on average, collinear with as many points in T as possible. These sets were originally studied in generalized quadrangles by Payne [24] and their definition was later extended to more general polar spaces by Drudge [12], An intriguing set with h\ < h2 is called an m-ovoid for some m. These are generalizations of the concept of an ovoid in a polar space. The study of intriguing sets in finite polar spaces is an active area of research with many open problems; for details, see [2],
In addition to the equivalent characterizations of Cameron-Liebler line classes carrying over under the Klein correspondence, we have a few additional properties in this context that do not have a good interpretation in PG(3, q).
Theorem 2.3 Let M be a set of points in Q+(5,q) C PG(5,g). The following are equivalent [22].
1. M corresponds to a Cameron-Liebler line class ofPG(3,q) with parameter x.
2. M is an x-tight set ofQ+(5,q).
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3. There exists x G Z+ such that \M\ = x(q2 + q + 1), every tangent hyperplane to <2+(5, q) at a point of M meets M in q2 + x(q + 1) points, and every other hyperplane ofPG(5,q) meets M in x(q + 1) points.
4. There exists x G Z+ such that \£L fl M\ = q\£ fl M\ + x for every line £ of
PGM.
5. There exists x G Z+ such that l-G1 fl M\ = q\£ fl M\ + x for every line £ of one of the four line types in PG(3,q) (external, tangent, secant, totally isotropic).
It is important to note that the last three characterizations are stronger than their related versions in PG(3,q). Part 3 in particular states that, in addition to knowing the intersection numbers for tangent hyperplanes of Q+(5, q), we also know that every nontangent hyperplane section of Q+(5,q) meets an x-tight set in x(q + I) points. This property is important enough that we state it on its own, as it will be used in the next section to construct related combinatorial objects.
Theorem 2.4 Let T be a proper x-tight set of Q+(5, q) that spans the ambient projective space. Then the set of points covered by T has two intersection numbers with respect to hyperplanes o/PG(5,g). These numbers are
hi = (q2 + 1) + x(q + I) and h2 = x(q + I).
2.3 Two-intersection sets, two-weight codes, and strongly regular graphs
Tight sets of Q+(5,q) are related to many other combinatorial objects; here we investigate some properties of these objects.
Definition 2.5 A set of points S of PG(n, q) is called a two-intersection set with intersection numbers h\ and h2 if every hyperplane of PG(n, q) intersects S in either h\ or h2 points. Such a set is also sometimes called a set of type (hi, h2).
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From the previous theorem, an x-tight set of Q+(5,q) whose points span PG(5,g) is a two-intersection set of PG(5,g). These sets are related to a wide range of other combinatorial objects. We begin by detailing results on an important class of linear codes.
An [n, k]q code C is a fc-dimensional subspace of the vector space V = Fâ„¢. Vectors in C are called codewords, and the vjeight wt(v) of a codeword v is the number of nonzero entries of v. A two-weight code C is a code whose codewords have precisely two nonzero weights. Given a code C, we define the dual code
C1- = {v G V\ vcT = 0 Vc G C}.
We have that C1- is an [n, n — k\q code.
Let C be an [n, k]q code; there exist linear functionals f : —>• Fg such that
C = {(/i(v), • • •, /n(v)) : v ^ F^}. Since (u, v) uvT is a nondegenerate bilinear form, there exist u1; ..., ura G F^ such that fiiy) = vuf for all v G F^. Thus, we have that C = {(vu[, ..., vu^)| v G V}, and since dim(G) = k, the ip span F^. We say C is projective if no two of the ip represent the same point in PG(k — 1, q).
Let G C V \ {0}. We say G is a {Ai, A2} difference set if, for every v G V \ {0}, the number of pairs (x, y) G G2 such that x — y = v is Ai if v G G, and A2 if v ^ G. If —G = G, we dehne a graph G(Q) whose vertices are the vectors in V, with u and v adjacent if and only if u — v G Q.
Definition 2.6 A strongly regular graph vnth parameters (v, k, A, p) is a connected k-regular (simple, undirected) graph on v vertices, not null or complete, such that any two adjacent vertices share A common neighbors, and any two nonadjacent vertices share p common neighbors.
We now give a result connecting the concepts of two intersection sets, two-weight codes, {Ai,A2} difference sets, and strongly regular graphs due to Calderbank and Kantor [7].
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Theorem 2.7 Let V = Fâ„¢+1; O = {yi \ 1 < i < r} be a set of vectors which span V (so r > n + 1) and are pairwise independent, and let Q = {cyi \ c E F*} be the set of nonzero scalar multiples of the yi; then the following statements are equivalent:
1. O is a set of type (r — w\,r — u>2) in PG(n, q) for some w\, W2;
2. C = {(x â– yi, ..., x â– yk)\ x E V} is a projective two-weight [r, n+ l]q code with nonzero weights W\ and u>2;
3. Q is a {Ai, A2} difference set for some X\, A2;
4- G(Q) is a strongly regular graph with parameters (qn+1, r(q —l),X,y), where for some w\, W2 we have
X = r2(q — l)2 + 3r(q — 1) — qw\W2 — r(q — l)(wi + w2) + q2(w\ + u>2) and
_ q2wi«i2 t1 qn+1 '
This means that if £ = {yi,..., yx(q2+q+i)} is an x-tight set of Q+{3,q) which spans PG(5,g), then
f. C is a set of type {{q2 + 1) + x(q + 1 ),x(q + f)) in PG(5, q);
2. the points of £ define a projective two-weight [x(q2+q+ f), 6\q code with weights (q — 1 ) 3. Q = {cyi | c E F*} is a {Ai, A2} difference set for some Ai, A2; and
4. G(Q) is strongly regular with parameters
(q6, x(q3 — 1), x(x — 3) + q3, x(x — 1)).
2.4 Trivial examples
There are a few examples which trivially satisfy the necessary requirements to be a Cameron-Liebler line class.
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1. The empty set 0 is a Cameron-Liebler line class with parameter 0.
2. The set star(p) of lines through a common point p of PG(3,q) is a Cameron-Liebler line class with parameter 1 corresponding to a 1-tight set of Q+(5,q) consisting of the set of points in a (Latin) plane.
3. The set line(7r) of lines in a plane tt of PG(3, q) is a Cameron-Liebler line class with parameter 1 corresponding to a 1-tight set of Q+(5,q) consisting of the set of points in a (Greek) plane (this is equivalent to the previous example in
Q+M).
4. The set star(p) Uline(7r), where tt is a plane of PG(3, q) and p is a point not in tt, is a Cameron-Liebler line class with parameter 2 corresponding to a 2-tight set of Q+(5, q) which is a union of two disjoint planes (one Latin and one Greek).
5. The complements of the above sets are Cameron-Liebler line classes with parameters q2 + 1, q2, q2, q2 — 1 respectively.
We call the Cameron-Liebler line classes in this list trivial.
2.5 Non-existence results
Cameron and Liebler conjectured that there were no nontrivial examples of these line classes, and proved this conjecture for classes with parameter < 2. Many other results followed, leading to some interesting connections with various geometric objects.
Many of the early non-existence results relied strictly on counting arguments;
specifically, we can think of sets of the type star(p) or line(7r) for a point p or a
line 7r as being essentially the same, and refer to these as cliques. The equivalent
definitions for a Cameron-Liebler line class allow us to perform some analysis on the
potential intersection numbers with respect to cliques of a hypothetical line class
with a given parameter x. Using these arguments, Penttila [27] was able to rule out
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several parameters in specific cases, and Bruen and Drudge [5] were able to rule out the existence of line classes with parameter 2 < x < y/q.
These methods were greatly improved in 1999 by Drudge [13] when he showed that if the intersection of an indecomposable Cameron-Liebler line class £ with parameter x > 2 and some clique C has x < |£ fl C| < x + q then £ n C forms a blocking set in C (in this context, a set of lines not containing any pencil, such that every point is on at least one of the lines; the normal definition is dual to this). Blocking sets are well studied, and there are many results on their minimum possible size. This gives a powerful tool for investigating the feasibility of certain parameters for Cameron-Liebler line classes. Drudge used this method to rule out the case where 2 < x < \{q + 1) when q is prime, and also gave the first nontrivial example of a Cameron-Liebler line class having parameter 5 in PG(3, 3). Soon after, he and Bruen [6] constructed an infinite family of examples of line classes having parameter \{ In 2004, Govaerts and Storme [15] used these blocking set techniques to improve the nonexistence result, eliminating the possibility of 2 < x < q when q was an odd prime. Soon after, Govaerts and Penttila [14] were able to rule out a few parameters in PG(3,4) by considering intersections with multiple blocking sets. In this same paper, they constructed the first example of a Cameron-Liebler line class for even q, an example with parameter 7 in PG(3, q). Multiple blocking sets can also be viewed as a special case of a more general class of combinatorial objects called minihypers. De Beule, Hallez, and Storme [10] used known results on these objects in Q+(5, q) to show that, when q is not prime, we cannot have 2 < x < |.
While the previous results are of considerable interest, in that they relate Cameron-Liebler line classes to other well-studied objects, the most recent and strongest nonexistence result is notable in that it uses primarily geometric arguments. In 2010, Metsch [22] looked at how an x-tight set of Q+(5,q) could potentially inter-
25
sect a parabolic Q(4, q) embedded in the quadric. He was able to use this technique to show the following:
Theorem 2.8 A Cameron-Liebler line class C, with parameter x < q exists only for x <2, and corresponds in Q+(5,q) to the union of x skew planes.
This shows that any nontrivial example must have parameter q < x < q2 — q.
2.6 Known examples
We now give constructions for the known examples of Cameron-Liebler line classes.
2.6.1 Bruen and Drudge examples
Drudge constructed the first known nontrivial example of a Cameron-Liebler line class in his 1998 doctoral dissertation [12]. His original example was in PG(3, 3) and had parameter x = 5. Not long after, he and Bruen [6] generalized this construction to an infinite family of examples in PG(3, q) having parameter x = \{q2 + 1) for every odd q. Here we describe their construction.
Let q be odd, and O = Q~(3, q) C PG(3, q) be an elliptic quadric with quadratic form Q. The rank 1 geometry O has q2 + 1 isotropic points and no totally isotropic lines, so no three points of the quadric are collinear; therefore every line of PG(3,q) contains 0, 1, or 2 points of O. These lines are called external, tangent, or secant, respectively. Each point p G O lies on q2 secants to O, and so lies on q + 1 tangent lines. Let Cp be a set of \{q + 1) of the tangent lines to O through p, and let S be the set of secant lines to O; then
£ = (UpeoTp) U S
is a set of
l(q2 + i)(? + i) + l(q2 +1 )(q2) = \(q2 +1 )(q2 + q +1)
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lines of PG(3,q), which is the number of lines in a Cameron-Liebler line class with parameter \{q2 + 1). The goal is to select the sets £p in such a way that C, is in fact a Cameron-Liebler line class.
Every plane of PG(3, q) is either tangent to O, and so contains a unique point of O, or else intersects O in a conic. The nontangent plane sections of O can be used to associate the points and nontangent plane sections of O with points and circles of the inversive plane IP (5) [23], so that each circle of IP (5) corresponds to a section of O by a nontangent plane 7r containing q + 1 tangent lines to O. An intersecting pencil of circles is the set of (q + 1) circles through two common points of IP (q), and a tangent pencil of circles is a (maximal) set of q mutually tangent circles on a given point of IP (q).
An equivalence relation ~ can be defined on the circles of IP (q) by
Ci ~ C2 -<=/- 3C such that C is tangent to both C\ and C2.
The circles of IP (5) fall into precisely two equivalence classes under this relation according to whether is a square or nonsquare, where 7r is the plane containing
the circle in question. Let A be one of these equivalence classes; A contains exactly \ of the circles in each intersecting pencil and either all or none of the circles in each tangent pencil. Thus if we define £p to be the set of tangent lines at p contained in a plane section which corresponds to a circle in A, £p contains \(q + 1) of the tangent lines to O at p.
Bruen and Drudge show that C, is a Cameron-Liebler line class with parameter + 1) by showing the set of lines in C, has a certain “matching†property with respect to the external lines to O which are the intersection of two tangent planes.
2.6.2 Penttila and Govaerts example in PG(3,4)
Another known example of a nontrivial Cameron-Liebler was constructed by Penttila and Govaerts [14]. This is an example in PG(3,4) with parameter x = 7, and
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was the first known nontrivial example when q is even. So far there has not been a generalization of this construction.
Let 7T be a plane in PG(3, 4) containing a hyperoval O and let p be a point not in 7r. Define C to be the cone with base O and vertex p, with G the set of generators of C, S' the set of secants to C which do not contain a point of O, and E the set of lines in n which are external to O.
Theorem 2.9 The set C = GUSUE is a Garner on-Liebler line class with parameter 7.
Proof: There are seven types of lines in PG(3,4) with respect to the cone C and the distinguished plane tt containing O.
1. Generators of C; this is the set G C C.
2. Secants to C which are skew to O; this is the set S C C.
3. Lines in tt which are skew to O; this is the set E C C.
4. Lines through p not contained in C.
5. Secants to C which meet a single point of O.
6. Secants to O.
7. Lines skew to C which are not contained in tt.
The points are of 5 types. Here we count the number of lines of each type through a point of each type.
1. {p}; of the 21 lines through p, 6 are of type 1, and 15 are of type 4.
2. Points on C \ ({p} U 7r); of the 21 lines through such a point, 1 is of type 1, 15 are of type 2, and 5 are of type 5.
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3. Points on 0\ of the 21 lines through such a point, 1 is of type 1, 15 are of type 5, and 5 are of type 6.
4. Points on tt \ 0\ of the 21 lines through such a point, 9 are of type 2, 2 are of
type 3, 1 is of type 4, 3 are of type 6, and 6 are of type 7.
5. Points on PG(3,4) \ (C U tt); of the 21 lines on such a point, 9 are of type 2, 1
is of type 4, 6 are of type 5, and 5 are of type 7.
From this, we can count that a line in £ meets 50 other lines of £, and a line not in
£ meets 35 lines of £. Thus £ is a Cameron-Liebler line class with parameter 7. â–
Unfortunately this construction does not generalize to other values of q in any obvious way, as we do not get the correct number of lines for a Cameron-Liebler line class unless q = 4.
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3. Methodology
Here we describe some algebraic techniques which we will use to search for new examples of Cameron-Liebler line classes of PG(3,q). We will search for these as tight sets of Q+(5, q); as such, we will develop a model of this quadric which will be convenient for our computational work.
3.1 An eigenvector method for tight sets
Our search for new Cameron-Liebler line classes will be conducted in the context of searching for new x-tight sets of Q+(5, q). An eigenvector method will be used to search for these objects, which is due to the following result of Bamberg, Kelly, Law and Penttila [2].
Theorem 3.1 Let £ be a set of points in Q+(5,q) with characteristic vector x and let A be the collinearity matrix of Q+(5,q). Then £ is an x-tight set if and only if
(x ~
x
q2 + 1
where j is the all-ones vector.
S)A = (q2-l)(x~
x
q2 + 1
j),
Proof: By definition, £ is an x-tight set if and only if, for p G £, p is collinear with (q2 — 1) + (q + l)x other points of Q+(5, q) and, for p ^ £, p is collinear with (q + l)x points of Q+(5, q). Thus £ is an x-tight set if and only if
XA= (q2 - l)x + x(q + l)j.
Since j A = q(q + l)2j, the above formula follows immediately. â–
In Q+(5,q), there exist two disjoint totally isotropic planes tv\ and 7^; our goal is to find tight sets which are disjoint from [jii U 772). The above method will be slightly modified to account for this. We will let A1 be the submatrix of A obtained by throwing away the rows and columns corresponding to points in (771 U 772).
Theorem 3.2 Let £ be a set of points of Q+(5,q) disjoint from 7Ti and 1r2 and let
xf be the vector obtained from the characteristic vector of £ by removing entries
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corresponding to points of tt\ and 7t2. Then £ is an x-tight set of Q+(5, q) if and only
if
(x'~
rJM = {q - i)(x -
rJ)-
q2 — 1 q2 — r
Proof: Denote the eigenspace of A corresponding to the eigenvalue (q2 — 1) by E, and the eigenspace of A1 corresponding to the eigenvalue (q2 — 1) by E'. Since U 7r2
is a 2-tight set,
2 . ^
T — X(7riU7T2) — ^2 _|_ ^ E-
Let C be a set of points of Q+(5, q) disjoint from 7Ti and tt2 with characteristic vector X; then £ is a x-tight set if and only if
x
X
rj e E
V = (x - -^rj) + ~Y~-T e E.
q2 — q
q2 + 1 q2 + f
The entries of v corresponding to points of (7riU7r2) are 0, and the entry corresponding to a point p ^ fni U 7t2) is given by
2
(x)P -
x
x
(x)P -
X
q2 + 1 q2 — 1 q2 + 1 q2 — 1
Thus if we obtain a new vector v' from v by throwing away entries corresponding to points in [jii U 7t2), and a new vector yf from x in the same manner,
v' = xf
x
q2 — 1
j eE'
X
x
q2 + 1
j eE
£ is an x-tight set.
3.2 Tactical Decompositions
For any incidence structure, a tactical decomposition is a partition of the points into point classes and the blocks into block classes such that the number of points in a point class which lie on a block depends only on the class in which the block lies, and similarly with points and blocks interchanged. Examples can be obtained by taking as point and block classes the orbits of some collineation group acting on the structure.
31
The idea of a tactical decomposition can also be extended to matrices. Let A = [a,ij\ be a matrix, along with a partition of the row indices into subsets R\, ..., Rt and a partition of the column indices into subsets C\, ..., Ctj. We will call this a tactical decomposition of A if for every (i, j), 1 < i < t, 1 < j < t', the submatrix [ah,e] (h G Ri, £ £ Cj) has constant column sums and row sums r^. A tactical decomposition of an incidence structure corresponds to a tactical decomposition of its incidence matrix. The row and column sum matrices of A are defined to to be Ra = Vij\ and Ca = [cp] respectively.
Utilizing a tactical decomposition makes finding eigenvectors corresponding to x-tight sets easier, as an eigenvector of the column sum matrix Ca obtained from the decomposition can be used to recover an eigenvector of A. The following result comes from the theory of the interlacing of eigenvalues, which was introduced by Higman and Sims, used by Payne in the study of generalized quadrangles, and further developed by Haemers; see [17] for a detailed survey.
Theorem 3.3 Suppose the matrix A can be partitioned as
A
An â– â– â– An
(3.1)
Aki â– â– â– Akk
vnth each An square, 1 < i < k, and each Aij having constant column sum Cij; then any eigenvalue of the column sum matrix Ca = [cp] is also an eigenvalue of A.
Proof: An eigenvector of Ca can be expanded according to the partition of A (by duplicating the entries corresponding to each part) to construct an eigenvector of A.
To apply this theorem to the task of finding eigenvectors of the collinearity matrix of Q+(5, q), we define an incidence structure R with both “points†and “blocks†being given by the points of Q+(5,q), and incidence being given by collinearity. Thus the
32
incidence matrix A of LL is given by the collinearity matrix of Q+(5, q). Furthermore, any automorphism of Q+(5, q) determines an automorphism of % in an obvious way. The matrix A is symmetric, and any tactical decomposition arising from an automorphism group of Q+(5, q) will induce the same partition on the rows of A and the columns of A. The following theorem gives us a nice relationship between the row and column sums in arising from such a tactical decomposition.
Theorem 3.4 Let A be a symmetric matrix and let Oi, Ok be the parts of a tactical decomposition of A (so the row and column partition is the same) with \Oi\ = Oi; then rij = Cji, and Oirij = OjCij.
Proof: If Aij is the submatrix associated with the row part corresponding to Oi and the column part corresponding to Oj, then Aiq = Aj^ thus for all i, j. Also,
each of the o* rows of Aiq has row sum r^, and each of the Oj columns has column sum dj. Summing over all entries of A^ in two ways gives tyry,- = OjC^. m
Corollary 3.5 Let A be a symmetric matrix with a tactical decomposition having the same parts for rows and columns, with part i containing Oi rows/columns; then we have the following relationship between the row and column sum matrices:
Rl = M = M = [%«} = cA.
°j
3.3 A model of Q+(5,q)
We now describe a model for Q+(5,q) which gives us a range of algebraic tools to use in searching for tight sets. Let F = Fg, E = ¥q3, and
T = Te/f : x i—> xq2 + xq + x.
We consider Q+(5, q) to have V = E2 as its underlying vector space, considered over F and equipped with the quadratic form
Q â– (x,y) ^ T(xy).
33
The polar form B of Q is then given by
B((ui,u2), (vi,v2)) = T(uiv2) + T(u2vi).
This form is nondegenerate, since if (vi,v2) E V has B((ui,w2), (x,y)) = 0 for all (x,y) E V, then T(v\y) + T(v2x) = 0 for all (x,y) E V. Setting x = 0 forces
T(v\y) = v\y + v\yq + v\ yq2 = 0 for all y E E*,
thus v\ = 0. Likewise, setting y = 0 can be seen to force v2 = 0, and so (wi,w2) =
(0,0).
It can be also be seen that
7Ti = {(x} 0) : x E E*} and vr2 = {(0,y) : y E E*}
are totally isotropic planes with respect to this form. This shows that the quadric defined by Q has Witt index 3, and so is hyperbolic.
3.4 The general method
Theorem 3.6 Let q ^ 1 mod 3. Take y E E* with \y\ = q2 + q + 1, and define the map g on Q+(5,q) by
9 â– (x,y) ^ {iax,ia~ly)-
then the group C = (g) < PGO+(6, q) and has \C\ = q2 + q + 1. This group acts semi-regularly on the points of Q+(5,q) and stabilizes the totally isotropic planes and 7t2 •
Proof: It is clear that g is an isometry of Q+(5, q) having order cj2 + q + 1. To see that C acts semi-regularly on the points of Q+(5,q), notice that gl((x,y)) = (x,y) implies that jT E F. But
(q2 + q + l,q- 1) = (q - 1,3) = 1,
34
since q ^ 1 mod 3. Thus this can only happen when fil = 1, and so the identity is the only element of this group fixing a point. â–
If a is a primitive element of if, we can without loss of generality assume that fj, = aq~l. The semi-regular action of C on Q+(5, q) gives us the following result.
Theorem 3.7 Let A be the collinearity matrix of Q+(5, q), q ^ 1 mod 3, vrith a tactical decomposition induced by the action of the cyclic group C defined aboue; then the row of each submatrix of the decomposition is the same as the column sum. Thus the decomposition matrix (which is the same for row sums and column sums) is symmetric.
Proof: This follows directly from 3.4 since all orbits have the same size. â–
Since each orbit has size q2 + q + 1, a union of x orbits contains the right number of points to be an x-tight set of Q+(5, q). Our goal will be to find ways of combining these orbits which will result in an x-tight set. We accomplish this by considering large subgroups G < NPro+(6,q)(C) having relatively few orbits on the points of Q+(5,q). The orbits of such a group are unions of orbits of C. We use such a group G to induce a tactical decomposition on the points of Q+(5, q), and then use this decomposition to form the column sum matrix B of the collinearity matrix A, after throwing away the entries corresponding to points in 7Ti and 7r2. The eigenspace of B for the eigenvalue g2 — 1 is then searched for eigenvectors having a form corresponding to an x-tight set of Q+(5,q). Whenever new examples show a pattern, e.g. a common formula for x in terms of q or a similar stabilizing group, algebraic, geometric, and combinatorial details are analyzed in an attempt to find a construction for a new infinite family of tight sets.
35
4. New examples
Throughout this chapter, we let q ^ 1 mod 3, E = ¥q3 with E* = (a), and F = Fq < E with F* = (uj) where uj = aq2+q+1. The hyperbolic quadric Q+(5,q) is defined over the vector space V = E2 (considered over F) and has quadratic form Q((x,y)) = T(xy), where T = Tf/e, and polar form B(u,v) = T(uiu2) + T(u2ui) as described in Chapter 3. Put y = cC_1, and define the cyclic group C = (g), where
9 â– (x,y) ^ {yx,y~ly)-
then C acts semi-regularly on the points of Q+(5, q) and stabilizes the disjoint totally isotropic planes tv\ = {(aqO) : x E E*} and 7r2 = {(0,y) : y E E*}.
Below is a summary of the new examples of Cameron-Liebler line classes which are described in this chapter. Notice that here we consider the parameter of the line class to be smaller than that of its complement, and we take the line class to be disjoint from [fi U 7t2); thus a new example with parameter x described below also gives new line classes with parameter x + 1, x + 2, (g2 +1) — x, q2 — x, and (q2 — 1) — x.
X 9 Aut(£)
W-i) q = 5 or 9 mod 12 and q < 200 (Zg2+g+i x Zi(ff_1}) xi Z3
W + if q = 2 mod 3 and q < 150 ^ 336 q = 27 (Zg2_|_g_|_i X Z2) X Zg
495 q = 32 ^q2-\-q-\-1 ^ ^15
Table 4.1: Parameters and automorphism groups of the new examples of Cameron-Liebler line classes constructed.
36
4.1 New examples with parameter |(g2 — 1)
Here we describe a construction giving many new examples of tight sets in Q+(5,q) having parameter |(g2 — 1). This construction requires us to have q = 5 or 9 mod 12, and has resulted in new tight sets for all such q < 200.
4.1.1 The construction
Let S = {x : x G E*\ T(x) = 0}; then the orbits of C on the points of Q+(5,q) are tv\ = (l,0)c, tt2 = (0, l)c, and (l,x)c for each x E S. We also let the group H = (h), where
h : (x,y) ^ (x,u4y), act on the space, and put G = (C,H).
Lemma 4.1 The group H defined above centralizes C, and intersects C trivially.
Proof: To show that H centralizes C, we only need to show that g and h commute; we have that
%((T V))) = H(/ax, ia~ly)) = (yx, y~lu4y) and g(h((x, y))) = g((x, u4y)) = (yx, y~lu4y).
Since the powers of y are pairwise independent over F, it is clear that HC\C contains only the identity. â–
Corollary 4.2 The group G defined above is equal to C x H, and so |G| = \{q— l)(q2 + q+l).
Furthermore, it can be seen that G acts semiregularly on the points of Q+(5, q) \ ijii U 7t2), as H acts semi-regularly on those points, and induces a semi-regular action on those orbits of C as well.
37
Let Sk be the subset of S containing the elements with loga(x) = k mod 4 for 0 < k < 3. For x G S, put x = {u)Atx : 0 < t < (q — l)/4}; for shorthand we will write (1,4:) = {{l,x') : x' G x}. Now we define
«(z,y) := |(f,^)± n (i,y)c\.
In terms of the tactical decomposition induced by G on Q+(5, q), n(x, y) is the number of points in (l,t/)c collinear with any given point of {l,x)G. Thus k(x' ,y') = n(x,y) for all i'gi and y' G y.
Let A be the matrix obtained from the tactical decomposition induced by G on the collinearity matrix of Q+(5,q) after throwing away the entries corresponding to points in 7Ti U 7t2. We use a specific ordering of the orbits of G to define A. Notice that S0 contains \( (1,X0)C, (l,Xg)C, (1,UJX0)C, . . . , (l,lVXg)C,
(l,UJ2Xo)C, (1 ,U2Xg)C, (1,U3X0)C, , (1 ,U)3Xg)C.
Now A can be described as follows:
Aq Ai A2 As
As Aq Ai A2
41 =
A2 As Aq Ai A\ A2 As Aq
where Ak = (n(xi, ujkXj))0<. for 0 < k < 3. This matrix is block-circulant, which allows us to apply the following result on eigenvectors of block-circulant matrices due to Garry Tee [30].
Theorem 4.3 Let ( be any fourth root of unity, and A be a block-circulant matrix as defined aboue, with blocks Aq, A\, A2, As each hauing size nj4. Take a uector v G IR?. Then the uector
w = [v(v(2v(3v]
38
is an eigenvector of A for X if and only i/v is an eigenvector of A0 + (Ai + (2A2 + C3«3 for X.
We now investigate some properties of k in order to better understand the structure of A.
Lemma 4.4 For x,y E S (not necessarily distinct), n(x,y) = n(y,x).
Proof: This follows directly from Theorem 3.4, along with the fact that (l,x)c and (1 ,y)c are the same size. â–
Corollary 4.5 A is symmetric; thus A0 and A2 are symmetric, and Ai = Aff.
Lemma 4.6 For x,y E So (not necessarily distinct), n(x,ujky) = n(y,ojkx) for 0 < k< 3.
Proof: First we notice that (y) contains q2 + q + 1 distinct elements of E*, no
two differing by a multiple in F. Thus, for any z E E*, there exists an integer 0 log« Asz) = loga (y3) = loga (aj{q~1)) = 0 mod 4.
Now take x, y in So] we have that
K(x,ujky) = |{z : 0 < i < cf2 + q + 1| T(ylx + y~lA1 ujky) = 0}|.
0 There exists (a unique) j with 0 yi+jx + y-{l+3)oj4tojky = ylAsy +
39
From this we can see (by relabeling indices) that
n(x,u}ky) = |{z : 0 < i < q2 + q + 1| T(fAx + y loj4tojky) = 0}|
0 = | {z : 0 0 = n{u)4sy,u)kx) = n(y,u}kx).
Lemma 4.7 For x,y E S0 (not necessarily distinct), n(x,ujy) = /i(x,uj3y).
Proof: We have that n(x,u}3y) = n{u)x,u)4y) = n{u)x,y) = n(y,ujx) = n(x,ujy). â–
Corollary 4.8 Ai = A3, and so all blocks of A are symmetric.
The following two lemmas have been verified computationally. It seems reasonable to attempt to prove that they will hold true for all values of q = 5 or 9 mod 12, although it is not clear why this would occur.
Lemma 4.9 If q < 200, then for all x E So, n(x,x) — k(x,ou2x) = — 1.
Proof: Computed with Magma, see Appendix B. â–
Lemma 4.10 If x < 200, then for all x,y E S0 vnth x/y, n(x, y) — n(x, u2y) = ±q. Proof: Computed with Magma, see Appendix B. â–
Lemma 4.11 If x < 200, then for x,y,z E So, distinct, n(x,y) — n{x,uj2y) = k(x, z) — k(x, oj2z) if and only if n(y, z) — n(y, oj2z) = q.
Proof: Computed with Magma, see Appendix B. â–
Theorem 4.12 If q < 200 is a prime povjer congruent to either 5 or 9 mod 12, then there exists an x-tight set C, in Q+(5,q) vnth x = \{q2 — 1) in PG(3,q) stabilized by a cyclic group of order q2 + q + 1 acting semi-regularly on the points. We also have
40
that C is disjoint from a trivial 2-tight set consisting of a union of two skew totally isotropic planes.
Proof: We have that i is a fourth root of unity. The matrix
H = Aq + iA\ + ff A2 + i3A3 = Aq + iA\ — A2 — iA3 = Aq — A2
has a nice form; all of the diagonal entries are —1, and all other entries are +q. Furthermore, there is some partition of {xo, ..., xq}, into parts L\ and L2 such that Hij = q if and only if i yt j and ay, ay are in the same part; say \Lf = a and |L2| = b, with a + b = q + 1. If we take K to be the adjacency matrix of the graph KLl © KL2 (where KLl and KL2 are the complete graphs on the sets L1 and L2, respectively) and K' be the adjacency matrix of the complement of this graph, then H = qK — qK' — I. We will form the vector v = xm — ~ \xl2- Notice that = \ if we let
x = Now we have that
(X© - Xl2)H=(xl1 ~ XL2){qK - qK1 - I)
= ((a - l)qxia ~ aqxL2 ~ X©) - ((& - 1 )qXL2 ~ bqxr2 ~ Xl2)
= {{a + b — 1 )q - 1) x+ - {{a + b- 1 )q - 1) xl2
= (q2 - 1 )(x© - x+),
thus v is an eigenvector of H having the desired form. We can use v to construct an eigenvector of A in four different ways, namely
±[ v —v —v V ]
±[ V v —v —v ]
each of which in turn can be used to construct an eigenvector of the collinearity matrix M of Q+{5,q) corresponding to a \{q2 — l)-tight set. â–
4.1.2 Some details of these examples
Our examples with parameter |(g2 — 1) have a group isomorphic to Zg2+g+1 x
acting on them, by construction. By observing details about the orbits used
41
in the construction, we notice that these examples are also stabilized by Aut(Fg3), thus they are stabilized by a group isomorphic to (Zq2+q+i x Zi(^q_1~)) x Z3. For those examples small enough to compute their full stabilizer in PrL(4, q), which are those with q < 41, this is in fact the full group.
We can also compute intersection numbers of these line classes in PG(3,q) with respect to planes and point stars of PG(3,q); this becomes prohibitively expensive, computationally, when q > 32. Plere we include details for some small values of q, as well as q = 81; this special case was of particular interest, see Chapter 5, so a considerable amount of time was dedicated to computing these values.
In this table, we include the intersection numbers with respect to planes of PG(3,q)-, the examples considered here are all isomorphic to their dual, and so have the same intersection numbers with the same multiplicities with respect to the point stars of PG(3, q).
q X Intersection numbers, with multiplicity; we have r = q2 + q + 1
5 12 0(b, 6^, 12(2r), 18^, 24^
9 40 0(0, 30(4r),40(r), 60(4r)
17 144 0(0, 108^, 126(4r), 144^, 180(4r), 198(4r)
29 420 0(0, 330(7r)) 39o(7b; 420^, 480(7r), 540^
81 3280 0W, 29 52(40r), 3280(r), 369Q(40r)
Table 4.2: Intersection numbers of line classes vrith parameter — 1) vrith the planes of PG(3, q).
The numbers in bold represent the intersection numbers for the plane corresponding to 7Ti in Q+(5,q), which is disjoint from our line class, and the planes through the point which corresponds to 7T2 in Q+(5, q), which shares \{q2 — 1) lines with our line class. It is worth noting that the number of lines shared by each plane with the line
42
class is divisible by (q + 1). The multiplicities being divisible by q2 + q + 1 is a side effect of C acting semi-regularly on the planes of PG(3, q) not corresponding to 7Ti.
The new examples of tight sets in Q+(5, q) give examples of two-intersection sets, two-weight codes, and strongly regular graphs, as detailed in Chapter 2. While there are tables of known strongly regular graphs, these examples on q6 vertices are too large to appear. Furthermore, if there were known examples, checking isomorphism would most likely be unreasonable.
4.2 New examples with parameter |(g + l)2
Here we give details on many new examples which have been constructed in joint work with Jan De Beule, Klaus Metsch, and Jeroen Schillewaert, having parameter |(q + l)2. These examples have been constructed for all values of q = 2 mod 3 which are computationally feasible. Unfortunately, in general these examples do not exhibit much symmetry; all of the examples found have C x Aut(Fg3) as their full stabilizing group. When q is prime, this does not give much to work with. These are found through more of a search than a construction; first we put together the tactical decomposition matrix for Q+(5, q) \ [jii U^) with respect to the group, then we search over the eigenspace for appropriate eigenvectors (see Appendix A for details about the algorithms used). With a small stabilizer, there are lots of orbits; for example, if q is prime, there are |(g2 — 1) orbits to consider. As such, forming the matrix for the tactical decomposition is a large task. Furthermore, we do not currently have a good method for reducing the size of the eigenspace to search over, so finding these examples is computationally infeasible if the eigenspace of the tactical decomposition matrix is too large (12 dimensions starts to push the limits of our computing power).
An important subcase of these examples occurs when q = 2e, where e > 1 and
odd. In this case, we have a slightly larger stabilizing group to work with. These
examples are also of particular interest since there is only one previously known
Cameron-Liebler line class in PG(3,q) for q even (see Chapter 2 for this construc-
43
tion). New examples with this parameter have been found for q e {8,32,128}, as well as for odd primes q < 100 which are congruent to 2 mod 3, and for q = 125. In all of the cases where it is feasible to compute the stabilizer group (q < 32), we have that C xi Aut(Fg3) is the full group. Below, we describe how some of these line classes intersect planes of PG(3,q). All of the examples considered below are isomorphic to their dual, and so have the same intersection numbers with the same multiplicities with respect to point stars of PG(3, q).
Q X Intersection numbers, with multiplicity; we have r = q2 + q + 1
5 12 0(b, 6^, 12Gb, 18 8 27 0(0, 18Gb, 27w, 36Gb, 54Gb
11 48 0«, 24G), 36Gb, 48Gb, 60GG, 72 17 108 OG), 72Gb, 90GG, 108(4G, 126Gb, 144Gb, 216
23 192 0(0, 120G), 144Gb, 168Gb, 192Gb; 216Gb, 240Gb, 264 32 363 OG), 264Gb, 33QG°b, 363(r), 396G°b, 462Gb, 726G)
Table 4.3: Intersection numbers of line classes with parameter |(g + l)2 with the planes of PG(3, q).
4.3 Some other new examples
We also have a couple of other new examples which do not currently seem to fit into a nice grouping. These examples have been found by assuming a group acting on the points of Q+(5,q) (usually a subgroup of N0+(etq)(C)), forming the orbits of Q+(5,q) \ ("7Ti U 7r2) and the associated matrix for the tactical decomposition, and searching over all possible parameters. The number of possible parameters can be very large, especially if the orbits are not all the same size. It was computationally feasible when q < 23 to assume that the examples we were looking for admitted the
44
group C x Aut(Fg3/Fg) as a stabilizer, though these searches did not yield any new examples.
For q = 27, the variation in the orbit sizes gives a large number of possible parameters, and there is a relatively large eigenspace to consider. Thus, considering a small stabilizing group was not feasible. By assuming the group C x Aut(Fg3) stabilized the examples, we were able to find a new tight set with parameter 336. This example is also stabilized by the map (x, y) i—> (x, —y), and so has full stabilizer isomorphic to (Zq2+q+1 x Z2) x Z9. Restricting our first search with C x Aut(Fg3/Fg) by assuming our parameter was divisible by q + 1, we found no other new examples.
With q = 32, all of the point orbits of the group C x Aut(Fg3) on Q+(5, q) have the same size, so the search is feasible assuming this stabilizer. We are able to find two new examples having parameter 495, each having C x Aut(F323/F2) as their full stabilizer. In this case, these two examples are isomorphic as tight sets, but not as Cameron-Liebler line classes. In PG(3,q), they are dual to one another.
Below we detail how these examples intersect the planes of PG(3,q). Note that only the first example is self-dual; in this case, the intersection numbers and multiplicities for point stars of PG(3, q) are the same as for planes. For the two examples with q = 32, the plane intersection numbers of one example are the point star intersection numbers for the other, and vice-versa.
Q X Intersection numbers, with multiplicity; we have r = cj2 + q + 1
27 336 Od), 252(-6r\ 336(13b; 420^ 504<2r)
32 495 Od), 330d), 396dd, 462d°d; 495w, 528^ 594W)
32 495 Qd), 396d°d, 495w, 528d5r), 660<6r>
Table 4.4: Intersection numbers of some other new line classes.
45
5. Planar two-intersection sets
A set of type (m, n) in a projective or affine plane is a set K of points such that every line of the plane contains either morn points of /C; we require that m < n, and we want both values to occur. For projective planes, there are many examples of these types of sets with q both even and odd, however the situation is quite different for affine planes. When q is even, we obtain a set of type (0,2) in AG(2,g) from a hyperoval of the corresponding projective plane, and similarly a set of type (0, n) from a maximal arc of degree n. Examples of sets of type (m, n) in affine planes of odd order, on the other hand, are extremely scarce; the only previously known examples exist in affine planes of order 9 [28].
Here we examine some combinatorial properties of these point sets in both affine and projective planes, and review the current state of the art for the affine situation. We then look at how affine sets of type (m, n) can be constructed from some of the Cameron-Liebler line classes found in Chapter 3.
5.1 Projective examples
We can deduce some information about sets of type (m, n) in projective planes through elementary counting.
Lemma 5.1 Let K be a set of type (m, n) in a projective plane ir of order q. Let tm and tn be the number of m-secants and n-secants to 1C, and let k = \1C\; then
tm + tn = q2 + Q + 1, (5-1)
mtm + ntn = k(q + 1), and (5.2)
m(m — 1 )tm + n(n — 1 )tn = k{k — 1). (5.3)
Proof: Each of the q2 + q + 1 lines in tt is either a m-secant or an n-secant, giving us (5.1). Counting over all secants, tm meet m points of /C, and tn meet n points of
/C; this counts each of the k points of K q + 1 times, giving (5.2). To obtain (5.3),
we notice that each of the tm m-secants contains m(m — 1) ordered pairs of points in
46
/C, and each of the tn n-secants contains n(n — 1) such pairs. Each of the k(k — f) ordered pairs of points in /C is counted once in this manner. â–
Corollary 5.2 If we have a set K of type (m,n) in a projective plane of order q, then k = \1C\ must satisfy
k2 — k(q(n + m — l) + n + m) + mn(q2 + q + 1) = 0. (5.4)
If we take a fixed point p G 1C, and let pm and pn be the number of m-secants and n-secants through p, respectively, we see that
Pm + Pn = q + 1 and (m - 1 )pm + (n - 1 )pn = k - 1.
From this, we see that
Pm = (n(q + 1) — k — q)/(n — m) and Pn = (k + q - m(q + 1 ))/{n - m),
and so pm and pn do not depend on our choice of p. Likewise, if we take a fixed point q ^ /C, and let am and an be the number of m-secants and n-secants through q, we see that
Cm + Again, these values can be seen to be independent of our choice of q; we have that
Vm = (n(q + 1) — k)/(n — m) and crn= {k - m(q + 1))/(n — m).
From these numbers we see that, given a set of type (m,n) in a projective plane of order q, we can construct three other related sets with two intersection numbers (for a proof, see [18]).
47
Theorem 5.3 Let K be a set of type (m, n) in a projective plane ir of order q, with \1C\ = k.
1. The complement of K, is a set of type (q + 1 — n, q + 1 — m) in tt containing 2. The set of m-secants to 1C is a set of type (pm, am) in the dual plane to ir containing tm points.
3. The set of n-secants to 1C is a set of type (pm,crm) in the dual plane to ir containing tn points.
Notice that pn — an = q/(n — m) and so it is necessary for n — m to divide q. If n — m = q, then /C can be seen to be either the set of points on a common line, or the complement of this; the examples having n — m = 1 are dual to these, and we consider the examples in either of these situations to be trivial.
One major class of examples are the sets of type (0,n). These examples are also known as maximal arcs of degree n, or as (qn — q + n, n)-arcs (as they necessarily contain qn — q + n points). The prototypical examples are given by hyperovals, which are sets of type (0,2); other families of maximal arcs of degree larger than 2 have been described by Denniston [11], Thas [31] [32], and Mathon [21]. Maximal arcs of degree 2“ are known to exist in PG(2,2e) for all pairs (a,e) with 0 < a < e, and it was proven by Ball, Blokhuis and Mazzoca [1] that there are no examples in PG(2, q) for odd q.
5.2 Affine examples
We will be concerned with sets of type (m, n) in affine planes. Through elementary counting, we get formulas very similar to those in Lemma 5.1, but giving different results.
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Lemma 5.4 Let K be a set of type (m,n) in an affine plane ir of order q. Let tm and tn be the number or m-secants and n-secants to 1C, and let k = \1C \; then
tm + tn = q2 + q, (5.5)
mtm + ntn = k(q + 1), and (5.6)
m(m — 1 )tm + n(n — 1 )tn = k(k — 1). (5.7)
These modified formulas lead to the following alternate version of Corollary 5.2.
Corollary 5.5 If we have a set K of type (m,n) in an affine plane of order q, then k = \1C\ must satisfy
k2 — k(q(n + m — 1) + n + m) + mnq(q + 1) = 0.
We again get constant values pm and pn for the number of m-secants and n-secants through a point in /C, and am and an for the number of m-secants and n-secants through a point not in /C, given by the formulas
pm = (n(q + 1) - k - q)/(n - m),
Pn = (k + q - m(q + l))/(n - m),
= (n(q + 1) — k)/(n — m), and an= {k - m(q + 1))/(n - m).
This tells us that, as for the projective case, we must have n — m dividing q. However, since the dual of an affine plane is not again an affine plane, we do not have results about the m-secants or n-secants forming another planar set with two intersection numbers.
There are very few known examples of sets of type (m, n) in affine planes. For
planes of even order, we can obtain an example from a set of type (0, n) in a projective
plane, by choosing an external line to the set as the line at infinity to form the affine
plane. However, sets of this type do not exist in projective planes of odd order.
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In affine planes of odd order, the only previously known examples of sets of type (m, n) are sets of type (3, 6) in planes of order 9. These sets were found through an exhaustive computer search, see [28], and examples were found in each of the four projective planes of order 9.
The size k of a set of type (3, 6) in a plane of order 9 must satisfy
k2 - 81 A: + 1620 = 0
which has solutions k\ = 36 and fc2 = 45. The complement of a set of type (3, 6) in a plane of order 9 will again be a set of type (3, 6), and the complement of a set of size k\ will contain fc2 points. The 45-sets of type (3, 6) have p3 = 2, p6 = 8, 5.3 Constructions from Cameron-Liebler line classes
We now describe a method of constructing some of the known sets of type (3, 6) in AG(2,9) starting with a Cameron-Liebler line class with parameter 40 in PG(3, 9). We then generalize this method to give a new example in AG(2, 81).
5.3.1 A two-intersection set in AG(2,9)
Take a Cameron-Liebler line class C\ of parameter 40 in PG(3, 9), as constructed in the Chapter 4. This set of lines is disjoint from a trivial Cameron-Liebler line class with parameter 2, which we will consider to be star(p) U line(7r), where p is a point in PG(3,q) and tt is a plane not containing p. This line class induces a symmetric tactical decomposition on PG(3, 9) having four classes of points and lines, as follows: the four line classes are
1. star(p),
2. line(7r),
3. Ci, and
4. £2 = line(PG(3, q)) \ (line(7r) U star(p) U C\).
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Each point of PG(3, q) \ ({p} U tt) lies on either 30 or 60 lines of C\ (see Table 4.2) and so the four point classes of the tactical decomposition are
!• {p}>
2. TT,
3. V\ = {u G PG(3,q) : star(u) fl C\ = 30}, and
4. V2 = {v € PG(3, q) : star(v) fl C\ = 60}.
The numbers of lines in each given line class through a fixed point in each given point class can be found in Table 5.1, and the numbers of points in each given point class on a fixed line in each given line class can be found in Table 5.2.
star(p) line(7r) Ci c2
{p} 91 0 0 0
7r 1 10 40 40
Vi 1 0 30 60
v2 1 0 60 30
Table 5.1: Lines per point for the symmetric tactical decomposition induced on PG(3,9) by a Cameron-Liebler line class of parameter 40.
star(p) line(7r) A c2
{p} 1 0 0 0
7r 1 10 l 1
v1 4 0 3 6
v2 4 0 6 3
Table 5.2: Points per line for the symmetric tactical decomposition induced on PG(3,9) by a Cameron-Liebler line class of parameter 40.
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Now, if we take a plane tt' of PG(3,9) not equal to tt and not containing p, tt' contains precisely one line of line(7r) and no lines of star(p). Furthermore, tt' will contain either 30 or 60 lines of £\, and so 60 or 30 lines of £2 (see Table 4.2). Without loss of generality, we may assume that 7b contains 30 lines of £\ and 60 lines of £2. As for the various point classes, 7b does not contain p, and contains 10 points of tt, all on a common line. Under our assumptions, 7b also contains 30 points of V\ and 60 points of TV In fact, we have a symmetric tactical decomposition of 7b having 3 classes on points and lines induced by our tactical decomposition of the larger space. By taking 7b fl tt to be the line at infinity and removing it (along with all of its points) from 7b, we obtain an affine plane AG(2, 9). All of the points of this affine plane are in Vi or TV and all of the lines are in £\ or £2. It can be easily verified that tt fl V\ is a set of type (3, 6) in this plane containing 30 points. This set admits a stabilizer isomorphic to Z3. As the sets of type (m, n) in AG(2, 9) were completely classified in [28], this set is not new.
5.3.2 A new two-intersection set in AG(2,81)
We are also able to follow the above procedure with a Cameron-Liebler line class £\ of parameter 3280 in PG(3,81) constructed as in Chapter 4. In this case, the line classes are formed as before. As for the point classes, each point of PG(3,81) is on either 2952 lines of £1, or on 3690 points of £\. We define V\ to be the set of points on 2952 lines of £\. These point and line classes give a symmetric tactical decomposition of PG(3,81); the numbers of lines in each given line class through a fixed point in each given point class can be found in Table 5.3, and the numbers of points in each given point class on a fixed line in each given line class can be found in Table 5.4.
We let 7b be a plane of PG(3,81) not equal to 7r, and not containing p. Then 7b
contains one line of line(7r) and no lines of star(p). Also, 7b will contain either 2952
lines of £\ or 3690 lines of £\ (see Table 4.2); without loss of generality, assume 7b
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star(p) line(7r) Cl c2
{p} 6643 0 0 0
7r 1 82 3280 3280
Vi 1 0 2952 3690
v2 1 0 3690 2952
Table 5.3: Lines per point for the symmetric tactical decomposition induced on PG(3,81) by a Cameron-Liebler line class of parameter 3280.
star(p) line (77) Ci c2
{P} 1 0 0 0
7r 1 82 1 1
v1 40 0 36 45
v2 40 0 45 36
Table 5.4: Points per line for the symmetric tactical decomposition induced on PG(3,81) by a Cameron-Liebler line class of parameter 3280.
contains 2952 lines of C\. The point set of tt' is again disjoint from {p}, and contains 82 points of tt, all on a common line. By taking this line, which is tt' fl tt, to be the line at infinity and removing it along with all of its points from tt', we obtain an affine plane AG(2,81). All of the points of this affine plane are in V\ or V2, and all of the lines are in or C2. If is clear that V\ is a set of type (36,45) containing 2952 points. There are no previously known examples of sets of type (m,n) in AG(2,81), so this example is new. Using Magma, the stabilizer of this set is computed and is isomorphic to Z6.
5.3.3 A family of examples in AG(2,32e)?
The combinatorics of our Cameron-Liebler line classes of parameter |(g2 — 1) seem to be especially nice over fields of order 32e, inducing a symmetric tactical decomposition on the space having four classes of lines and of points. A future
53
direction of research related to this observation is to focus on proving the existence of an infinite family of Cameron-Liebler line classes having this parameter in the specific case where q = 32e, and examining the intersection numbers with respect to the planes and point stars of PG(3,q). If these line classes always induce such a tactical decomposition, with one of the classes being a plane and another a point star, then we will be able to construct an infinite family of sets of type (m, n) in AG(2, 32e).
Assume we have a Cameron-Liebler line class C\ with parameter ^(36e — 1) in PG(3, 32e) which is disjoint from a trivial Cameron-Liebler line class star(p) Uline(7r) with parameter 2 (so p ^ tt), and that this line class induces a symmetric tactical decomposition of PG(3,q) as above having point classes {p}, tt, V\, V2, and line
classes star(p), line(7r), £1, £2. Take a plane tt' distinct from 7r and not containing p.
The points in the affine plane tt' \ (7b fl 7r) are all in V\ or V2, and the lines are all in £1 or £2- If we let K = V\ fl tt', then the lines of the affine plane have precisely two intersection numbers with K depending on whether they are in C\ or £2. Without loss of generality we will assume that each line of C\ fl star(7r/) meets m points of 1C. Let A and B be such that each point in V\ is on A lines of C\ and B lines of £2; thus each point in V2 is on B lines of C\ and A lines of £2.
The most likely possibility for n — m, and the situation for our earlier examples, is that n — m = 3e. Assume that this is the case, so that n = m + 3e. By Definition 2.1, we have
^(36e - 1) + (32e + 1 )pm = |£i n tt'I + A and
-(36e — 1) + (32e + l)crm = (£2 fl n'\ + A,
by applying the result to C\ using the incident point-plane pair (u^') with u e V\, and to £2 using the incident point-plane pair (v, 7b) with v e TV Since am — pm = 3e, we see that
|£2 n tt'I - |£i n tt'I = (32e + 1)( 54
which, along with the fact that
|C2 n tt'I + |£i n tt'I = 34e + 32e = (32e + l)32e
, tells us that
|£i n tt'I = ^(32e - 3e)(32e + 1) and IAHtt'I = ^(32e + 3e)(32e + l).
This allows us to solve for
m = ^(32e — 3e) and n = t(32' + 3').
Conjecture 5.6 For any e > l, there exist sets of type (§(32e — 3e), |(32e + 3e)) in AG(2, 32e).
Our hope is that, in the future, we will be able to prove that we have an infinite family of Cameron-Liebler line classes in PG(3, 32e which induce tactical decompositions of the space, allowing us to show the existence of these two-intersection sets.
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APPENDIX A. Algorithms
Here we detail some of the algorithms that facilitate our findings.
A.l CLaut Matrix
• We have
— C is our cyclic group of order q2 + q + 1 whose orbits on Q+(5, q) \ {jii U 7t2) are represented by elements of ORep= {x : x G F*3 | T(x) = 0} (considered as an ordered set for consistency).
— Z\ = Aut(Fg3/Fp) and Z2 = F*; these groups, considered on Q+(5,q), normalize C, so we only consider their action on ORep). Z\ is assumed to stabilize our tight set but Z2 is not.
— XBLOCK and XO are the orbits of Z\ and Z2, respectively; the elements of each orbit are also ordered.
• Want to store, for each x gORep, indexed pairs and so that X =Xblock[/1] [/1]=XO[/2] [y2].
• We now form a set R such that, for each x gORep, we have a unique r eR such
k
that uolrp = x for some i, k.
• Form the structure S= [sr], where sr = [|(1, r)-1 fl (l,x)c| : x gORep].
• Form the array 01P, where 07P[/][y] =k, where k is such that gXblock[/c].
• Each row and column of our matrix A corresponds to an orbit on Q+(5,q) \ (iti U 7r2) under G =. We consider them to be ordered according to the ordering of the orbits XBLOCK of Z\. We find Aij as follows:
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— Let a, b be such that (Wry EZi[i].
— Then Aij is formed by summing over sn[k\ as k ranges over the values satisfying 01P[a][k\=j.
In other words, for each Zi[i], we can find an x in Zi[i], an r in R, and an integer a such that x = u)ar. We know how many elements of (1, y)G are collinear with (1, r), so we consider how the map y i—> ujay on ORep permutes the associated orbits of C, and consider which of these orbits have their representatives in Zi\j\, and add them up.
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APPENDIX B. Programs
We have structured the code to examine tight sets of Q+(5,q) in a modular fashion. That is, we have a shell program that contains the parameters we may wish to modify, and from there we load the hies containing specific methods, and then call the specific functions we are interested in. Here is the code for our basic shell program; we comment out any parts we are not interested in before submitting the job to the cluster. Any restrictions for the parameters, either from the requirements of the algorithms or for the sake of computational speed, will be mentioned when we discuss the individual pieces of the code.
B.l CLshell.mgm
We have variables p, h, and t which can be modified; p does not need to be prime. The idea is that we will have q = ph, and / Au{Fpah/Fp) will be assumed to act on a f-tight set found by the search. We usually define t in terms of q, but we must take care that t is an integer, p := 81 ;
h ;= 1;
CLpreamble .mgm sets up our basic infrastructure, load “CLpreamble .mgmâ€; t := RationalsQ ! (1/2)*(g2 -1);
CLaut. mgm holds the basic search algorithm; when h = 1, it will find any of our examples having parameter t, although it is very slow. Using larger values of h (while leaving q fixed) makes things much faster, but will only find examples stabilized by this larger group.
Findl_(f, ~Z_);
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CLbcirc.mgm is specialized for h = 1, p = 1 mod 4, and t = (1/2)(g2 — 1). It is very fast and memory efficient.
We have that L contains the set of orbit representatives for each tight set found, print “There wereâ€, #L,
“line classes found with parameterâ€, f;
CLvspace.mgm contains definitions for the vector spaces V = E2, W = F6, and the map (f) : V —> W. This map uses a basis for W which gives the standard orthogonal form. The vector space U = FA is also defined, along with maps 5 : U —> W and 7 : W —> U which map a point to a line of PG(3, q) via the Klein correspondence, load “CLvspace.mgmâ€;
CLpg3q.mgm defines the function LU(), which maps a set of trace zero elements of E* to the set of lines of PG(3, q) corresponding to their orbits under C. Also defines the groups GL = PrL(4, q) and CU ~ C acting on lines of PG(3, q). load “CLpg3q.mgmâ€;
S := Stabilizer(GZ-, LU(L[1]));
print “The stabilizer of L is as follows :\nâ€, S;
CLint.mgm is used to compute intersection numbers. The orbit representatives are expanded to the full pointset throuth LW(), and intersection numbers with stars and planes are computed using intPlane() and intStar(), respectively, load “CLint.mgmâ€;
LL := LW(L[1]);
print “The intersection numbers (with multiplicity) of L with point stars of the space are as follows :\nâ€, intStar (Z_Z_);
print “The intersection numbers with respect to
59
planes of the space are as follows :\nâ€, intPlane(LL);
CLint81.mgm gives an alternate way to compute intersection numbers. It is much slower, but more memory efficient. We use it for the case where p = 81, since the computations are impossible otherwise, load “CL8lint .mgmâ€;
print “The intersection numbers (with multiplicity) of L with point stars of the space are as follows :\nâ€, i ntStar (LW (Z_ [1 ]));
print “The intersection numbers with respect to planes of the space are as follows :\nâ€, intStar(LWd(L[1]));
MNset.mgm will give the intersection of the set in PG(3,q) with a plane piPrime as described in Chapter 5. The set K will also be given, which should be a two-intersection set of AG(2, q) when q = 32e. load “MNsetTH2 .mgmâ€;
K := MN(L[1]);
S := KStab(K');
print “The stabilizer of K is as follows :\nâ€, S;
B.2 CLpreamble.mgm
This code is required for everything that follows.
We let F = ¥q and E = Fg3, with primitive elements oj and a, respectively, and view V as E2, with bilinear form (x,y) T(xy), where T = Tr(E/F).
q := Ph\
A := (q2-1);
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F := FiniteField(q) ; if not IsPrimitive(u;) then
uj := PrimitiveElement(F) ;
end if;
E := exf;
T := func;
It is useful to have these ordered. This ordering makes Fstar[i]= likewise
Estar[i]= aA+b
Fstar := {@ uj1 : i in [0.. q-2]@};
Estar := {@ a1 : i in [0. . q3-2]@};
H is a element of E with order r = q2 + q + I. jj, := a 1); r := q2 +q +1 ;
ORep is the set of nonzero elements with trace 0; fills the role of S in the thesis. ORep := {@ x: x in Estar\ T(x) eq 0
L is a placeholder sequence; it will contain subsets of ORep corresponding to f-tight sets of the quadric.
L := [POWERSET(OFep)|];
Instead of defining our cyclic group acting onk = E2, we define C to be a permutation group on Estar, generated by C.1: x e-> p * x.
C := PermutationGroup;
We define the group Z\= Aut(£,/Fp); to save memory, we define this group acting on just the trace zero elements ORep C Estar. This requires some consideration to the order in which we apply groups to look at orbits.
Gr := Sym(ORep);
Zi := sub;
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0 := Zi.1;
The group Z2 generated by z : x e-> uj * x permutes the orbits of C; the orbits of this group on the trace zero elements of Estar are used to efficiently form the tactical decomposition.
Z2 := sub-, z := Z2.1 ;
B.3 CLaut.mgm
The method used here is the most general search technique. Forming the matrix for the tactical decomposition can be very computationally expensive.
Xblock â– = Orb its (Zi);
XO := Orbits(Z2) ; n := #Xblock]
The following record format is used to keep track of important information about the trace zero elements of Estar, such as their location in the cycles induced by specific group elements on representative trace zero elements. See Appendix A for details. TZero := recformat
01 : car<{i: i in [1 ..#Xblock]}, {/: j in [1.. (3*/))]}>,
02 : car<{i: i in [1..#XO]}, {/ : j in [1.. (qf—1 )]}>>;
TZ := [rec : / in [1 ..#ORep]\;
for / in \t..#Xblock\ do
for j in \1 ,.#Xblock\i\\ do
7Z[lNDEx(0/:?ep, Xblock[i]\j])]'Oi := ; end for; end for;
for / in [1..#XO] do
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for j in [1 ,.#XO[i]\ do
TZ[\NDEx(ORep, X0[i][j])]'02 := \ end for; end for;
It is efficient to sort the records according to the size of the orbit of the trace zero element under Zq we maintain separate sequences of the trace zero elements, and the records associated with them, each in the same order. The memory used to store this redundant information is made up for in the time saved accessing the information, forward m;
Sort(~7Z, func, ~m);
ORep := {@ ORep[im] : i in [1 ..#ORep\@};
For details on the formation of A, see Appendix A.
02block := {@{@ 7Z[/]'Oi[1]
: j in [Index(ORep, x) : x in XO[TZ[i\02[ 1]]]@}
: / in [1..#ORep] @};
R := {@ Index(ORep, Xblock]02block\i11111111) : / in [1 ..#02block]@};
Cy := [[Trace(E ! x, F) : x in Cycle(C.1, XO[i][ 1])] : / in [1..#XO]]; yC := [RoTATE(REVERSE(Cy[/]), 1): / in [1..#Cy]];
002 := [[Fsfar[x[2]]*yC[x[1]][/] : j in [1..#Cy[x[1]]]] where x := TZ[R[i]]'02 : / in [1..#/?]];
M2 := func | 002[i][k] + Fstar{x[2]\*Cy{x[\}\{k) eq 0 where x := TZ[y]'02}>;
S := [PowerSequence(Rationals())| ]; for / in [1 ..#R] do
s := [M2(/,_/) : j in [1 ..#ORep]]\
Append(~S, s);
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end for;
01P := [ [TZ[\HDEx(ORep, x*0f?ep[/])]'0i[1]: j in [1 ..#ORep]]: x in Fstar]]
M := function(/,y)
d := exists(/c,s)
{ : m in [1..#R], n in [1 ..#Fstar]
| TZ[\NDEx(ORep, ORep[R[m]]*Fstar[n])]'Oi[1] eq i }; return &+[ S[/c][x] : x in [1 ..#01P[s]] \ 01P[s][x] eq j]; end function;
A := SymmetricMatrix(Rationals(), &caf[[M(/',y) :/' in [1..y]] : y in [1.. /?]])
- ScalarMatrix(Rationals(), n, 1);
This loop adjusts the values of the matrix if the orbits are not all the same size, which occurs when q = 0 mod 3 or when 3 divides h. s := [#Xblock[i] : i in \1..#Xblock\ \; for / in [1.. (n— 1)] do
for y in [(/+1).. n\ do
A[i,j\ := A[i,j\/(s\j\/s[i})] end for; end for;
We now find the eigenspace for A, and define a function FindL() which will search for a tight set with a given parameter t and return the sets of orbit representatives (trace zero elements of Estar) corresponding to any tight sets found.
Ba := Basis(Eigenspace(/\, A));
FindL := procedure^, ~Z.)
p := RationalsQ ! t/(q2 -1); e := (1 —p); f := -p;
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Since the basis vectors for the eigenspace for A are normalized, a vector with all entries equal to e or / must be a linear combination of these basis vectors with all weights equal to e or /; we search over all such linear combinations for tight sets, for c in CartesianPower({0, 1}, #Ba) do
s := [ (c[i] eq 1) select e else f . i in [1..#6a]]; v := &+[s[/'] * Ba[i] : i in [1..#6a]]; if forall{/:/ in [1.. n\ \ v[i] in {e, f}} then
Append(~Z_, &join{Xblock[i]: i in [1.. n] \ v[i] eq e}); print v\
end if; end for; end procedure;
B.4 CLbcirc.mgm
This code searches for tight sets of Q+(5,q) as described in Chapter 4 when q = 1 mod 4, by forming a block circulant matrix for the tactical decomposition. The requirements for the parameters in the shell are as follows:
p= 1 mod 4,
e = 1, and
t = Rationals((l/2) * (q2 — 1)).
We must also have loaded the CLpreamble.mgm hie.
We define the subgroup Z3= (z4) Z3 := sub;
Xblock contains the orbits on the trace zero elements under ( Z\, Z3 ). We order these orbits according to whether loga x = 0,1, 2, 3 mod 4, as described in Chapter 4.
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This way, XO[/][/c] is the orbit given by u/fc ^ * X0[/][1] for 1 < k < 4, where X0[/][1] is an orbit containing elements with loga x = 0 mod 4.
Xblock := ORB\Js(sub); n := #Xblock]
Sort Xblock, func);
SoRT(~Xb/oc/c, func);
XO := {@ Xblock[i]Z2 : i in [1.. n]@}; n := n div 4;
The following algorithm is used to generate the matrix corresponding to the tactical decomposition of Q+(5,q) induced by the group ( C, Zl5 Z3 ); see the details in Appendix A.
Cy := [[[Trace(E ! x, F) : x in Cycle(C.1, y)\ : y in XO[/'][1]] : / in [1..#XO]];
002 := [R0TATE(REVERSE(Cy[/'][1]), 1) : / in [1../?]];
M := func | 002[/] [z] + Fsfar[/f]*Cy[/][x][z] eq 0}
: x in [l..#Cy[/]] ] >;
Ablock â– = [ SymmetricMatrix
(RationalsQ, &caf[[M(/,y,/c): / in [1..y]] : y in [1.. /?]])
: k in [1.. 3] ]; if ((q mod 3) eq 0) then
s := 3;
for / in [1.. 3] do
for j in [2.. n\ do
Abiock[i)[\, j) := Ab/oc/c[/][1,y']/s; end for; end for;
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end if;
Ab!ock[\] := Ab!ock[\] - ScalarMatrix(Rationals(), n, 1);
Ablock[4] := Ablock[2];
The eigenvectors of the block symmetric matrix are now constructed over the cy-clotomic held R[(\ :=Q[z]. We are only interested in real valued eigenvectors. For these examples, for all values of q which are feasible computationally, we get a onedimensional eigenspace of H[1] = Ablock[]]—Ablock[3], giving eigenvectors which correspond to (1/2)(g2 — l)-tight sets (in essentially four equivalent ways).
R<(> := CyclotomicField(4) ;
H := [&+[( h[ 1];
v := Basis(Eigenspace(H[1], (q2 —1 )))[1];
A := HORIZONTALjOIN([VERTICALjOIN([/4b/OC/c[1 + ((/-_/) mod 4)]
: / in [0.. 3]]) : j in [0.. 3]]);
This loop puts together a set of the orbit representatives for each of the four f-tight sets. (Need some explanation of the method, which is based on conjecture.) The sequence L consists of a sequence of sets of orbit representatives, each corresponding to a f-tight set.
p ■= RationalsQ ! t/(q2 -1); e := (1 —p); f := -p;
Sp := &.cat([[ : k in [1../7]]: j in [1..4]]); v := Vector([Rationals() | (IntegersQ ! Ablock[2] [/, 1 ] mod 2) - p
: / in [1.. n]]);
for c in CartesianPower({1,2}, 2) do
u := Vector(&caf([ Eltseq(((-1)c[1])*i/),
67
ELTSEQ(((-1)c[2])*l/),
ELTSEQ(((-1) (c[1]+1))*i/),
Eltseq(((-1 ) (c[2]+1))*i/)]));
if (u*A eq A*u) then
Append(~Z_, &join{XO[Sp[i] [2]] [Sp[/] [1 ]]
: i in \1..#Xblock\ \ u[i] eq e});
print u\
end if; end for;
B.5 CLvspace.mgm
This code includes definitions of various vector spaces, as well as maps to implement the Klein correspondence. It is required for all of the code in the proceeding sections.
We begin by defining our vector spaces V and W, and our bilinear form.
V := VectorSpace(E, 2);
1/1/, (p := VectorSpace(V, F);
B := func;
Q := func;
BW := func\
QW := func;
OForm := Matrix(F, [[BW(Basis(I/I/)[/], Basis(I/I/)[/])
: / in [1 • • 6]]: j in [1..6]]);
We now find a basis for W under which we can use the standard Plucker coordinates for the Klein correspondence.
Ba := [Basis(l/l/)[1 ]];
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Ba := AppEND(Ba,a) where a := rep{x. x in W
| QW(x) eq 0 and BW(Ba[1], x) ne 0};
for / in [1.. 3] do
Ba[2*/'—1] := (BW(Ba[2*/—1], Ba[2*/]))(_1)*8a[2*/-1];
8a[2*/] := QW(Ba[2*/'])*Ba[2*/'-1] + Ba[2*/']; if (/' ne 3) then
Ba := AppEND(Ba, a)
where a := rep{x : x in Nullspace
(OForm*TRANSPOSE(MATRix([Ba[/c] : k in [1--2 */]])))
| x ne 0 and QW(x) eq 0};
Ba := AppEND(Ba, a)
where a := rep{x : x in Nullspace
(OForm*TRANSPOSE(MATRix([Ba[/c] : k in [1--2 */]])))
| x ne 0 and QW(x) eq 0
and BW(Ba[2*/'+1],x) ne 0};
end if; end for;
Ba := [Ba[1], Ba[3], Ba[5], Ba[6], Ba[4], Ba[2]];
WBa := (Matrix(F, 6, 6, Ba))(“1);
We redefine to map V —> W in such a way that we can use the standard orthogonal form for the quadric.
(f) := map(v)*WBa, w ((f)~ 1)(w*l/l/Ba—1)>;
WForm â– = Matrix(F, [[BW(Ba[/], Ba[/]): / in [1..6]] : y in [1.. 6]]);
BW := func;
QW := func;
We now define the vector space U which will underlie PG(3,q); 5 : U —>■W and 7 : W U give the Klein correspondence.
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U := VectorSpace(F, 4);
8 := func [—x[1], 0, x[4], —x[5]],
[—x[2], —x[4], 0, x[6]],
[—x[3], x[5], —x[6], 0]]))>;
pK := func [[Basis (//ne) [1 ] [/+1 ], BASis(//ne) [1 ] [/c+1 ]], [Basis (//ne) [2] [y+1 ], Basis (line) [2] [/c+1 ]]]))>;
7 := func [pK(line, 0, 1), pK(line, 0, 2), pK(line, 0, 3), pK(line, 1, 2), pK(line, 3, 1), pK(line, 2, 3)]>)[1]>;
B.6 CLpg3q.mgm
This code requires CLvspace.mgm in order to run.
We set up the group acting on PG(3, q), and it’s action on the lines.
G, P := PGammaL(L/) ;
PointsU := func {Index(P, N0RMALiZE(BASis(//'ne)[2]))} join {Index(P, N0RMALiZE(BASis(//'ne)[1] + x* Basis (line) [2])) : x in F}>;
Lines := PointsU(sub)G;
Lines := GSet(G, Lines)-, p, GL := Action(G, Lines)]
Here we define the action of our cyclic group on PG(3, q).
CU := (p~'[)(sub [ y[1] ne 0 select i/[1]c-1 else 0, v[2] ne 0 select i/[2]c-_1 else 0])))
70
where v is (4>~'[){rj{sub)) where / is rep{ : a, b in Lines[i] \ a ne b}
: i in [1 ..#Lines]]>)m,
The function LU() maps our orbit representatives to a set of lines in PG(3,q).
LU := func; B.7 CLint.mgm
This code requires CLvspace.mgm in order to run.
H, N := PG0Plus(I/1/) ;
Hprime := Subgroups(H : IndexEqual := 2)[1 ]'subgroup-,
CW := sub where z is (4>~ 1)(x) : x in A/]>;
LW := func< L | &y'o//7{ Index(/V, Normalize(0(1/ ! [1,x])))cw : x in L} intStar := function(l/1/CZ.)
7ti := { Index(/V, Normalize(0(1/ ! [x,0]))) : x in Estar };
Stars := TTi HPrime ;
return {* #(WCL meet tt) : ir in Stars*}-, end function;
intPlane := function(l/l/C/.)
7t2 := { Index(/V, Normalize(0(1/ ! [0,y]))) : y in Estar };
Planes := 7r2HPrime;
return {* #(WCL meet ir) : ir in Planes*}-, end function;
B.8 CL81int.mgm
This code requires CLvspace.mgm in order to run.
G, P := PGammaL(LZ) ;
aC := \pLk : k in [1.. r]];
Ca := FtEVERSE(aC);
ROTATE(~aC, 1);
CW := func;
ULines := Parent(^(0(1/ ! [1, ORep[\]])*WBa))m,
LW := function(/./:?ep)
/./.:=[ ULines |]; for x in LRep do
for v in CW(x) do
Append(~LL, 5(4>(v)))m, end for; end for; return /_/_;
end function;
LWd := function(LRep)
/./.:=[ ULines |]; for x in LRep do
for v in CW(x) do
Append(~LL, \[v[2], i^[1 ]])));
end for; end for; return /_/_;
end function;
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intStar := function(UCL)
INT := {* IntegersQ| *}; for x in P do
xINT := #{v : v in UCL \ x in v}]
Include(~/A/7, xINT) ; end for; return !NT\ end function;
intPlane := function (UCL)
INT := {* Integers()| *};
Nv := [Nullspace(Transpose(Matrix([Basis(i/)[1], Basis(ix)[2]]))) : v in UCL];
for x in P do
xINT := #{/' : / in [1..#Nv] | x in Nv[/']};
Include(~/A/7, xINT) ; end for; return INT] end function;
B.9 MNset.mgm
This code requires CLvspace.mgm in order to run. aC := \p,k : k in [1.. r]];
Ca := FtEVERSE(aC);
RoTATE(~aC, 1);
CW := func;
ULines := Parent(5((V ! [1, ORep[\]])*WBa))i
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7r := Nullspace(Transpose(Matrix(L/ ! [1,0,0,0]))); piW â– = sub),
â€/(sub),
â€/(sub)>;
LWpi := function(Z-Bep)
/./.:=[ ULines |]; for x in LRep do
for v in CW(x) do
if v in piW then
Append(~Z_Z_, 8(v))m,
end if; end for; end for; return /_/_; end function;
MN := function(UCL)
UCLpi := LWpi (UCL)]
I := {x : x in 7T |
#{v : v in UCLpi \ x in v} eq (#UCLpi div (g+1))};
/ := sub< 7T | / >;
Bp := ExtendBasis(/, 7r);
Bp := [Bp[3], Bp[1], Bp[2]];
H,N := PGL(3, q);
mn â– = {* #{v : v in UCLpi \ x in v} : x in ir*}; m := MiN(mn);
74
K := {NORMALIZE(
Vector(F, Coordinates(VectorSpaceWith Basis (Bp), x))) : x in 7r \#{v : v in UCLpi \ x in v} eq m};
K := {[x [2], x[3]] : x in K}\ return K ; end function;
KStab := function(mnSef)
H, J := AGammaL(2,q) ;
K := {Index(J, x) : x in mnSet}]
S := Stabilizer(/-/, K); return S; end function;
75
REFERENCES
[1] S. Ball, A. Blokhuis, and F. Mazzocca. Maximal arcs in desarguesian planes of odd order do not exist. Combinatorica, 17:31-41, 1997. 10.1007/BF01196129.
[2] J. Bamberg, S. Kelly, M. Law, and T. Penttila. Tight sets and m-ovoids of finite polar spaces. Journal of Combinatorial Theory, Series A, 114(7): 1293—1314, 2007.
[3] J. Bamberg, M. Law, and T. Penttila. Tight sets and m-ovoids of generalised quadrangles. Combinatorica, 29(1):1—17, 2009.
[4] W. Bosma, J. Cannon, and C. Playoust. The Magma algebra system. I. The user language. J. Symbolic Comput., 24(3-4):235-265, 1997. Computational algebra and number theory (London, 1993).
[5] A.A. Bruen and K. Drudge. On the non-existence of certain Cameron-Liebler line classes in PG(3,q). Designs, Codes and Cryptography, 14(2): 127—132, 1998.
[6] A.A. Bruen and K. Drudge. The construction of Cameron-Liebler line classes in PG(3,q). Finite Fields and Their Applications, 5(1):35—45, 1999.
[7] R. Calderbank and W. M. Kantor. The geometry of two-weight codes. Bulletin of the London Mathematical Society, 18(2):97—122, 1986.
[8] P.J. Cameron and R.A. Liebler. Tactical decompositions and orbits of projective groups. Linear Algebra and its Applications, 46:91-102, 1982.
[9] J. De Beule, P. Govaerts, A. Hallez, and L. Storme. Tight sets, weighted recovers, weighted m-ovoids, and minihypers. Designs, Codes and Cryptography, 50(2): 187—201, 2009.
[10] J. De Beule, A. Hallez, and L. Storme. A non-existence result on Cameron-Liebler line classes. Journal of Combinatorial Designs, 16(4):342 349, 2008.
[11] R.H.F. Denniston. Some maximal arcs in finite projective planes. Journal of Combinatorial Theory, 6(3):317 — 319, 1969.
[12] K. Drudge. Extremal sets in projective and polar spaces. PhD thesis, University of Western Ontario, 1998.
[13] K. Drudge. On a conjecture of Cameron and Liebler. European Journal of Combinatorics, 20(4):263-269, 1999.
[14] P. Govaerts and T. Penttila. Cameron-Liebler line classes in PG(3,4). Bulletin of the Belgian Mathematical Society - Simon Stevin, 12(5):793—804, 2005.
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[15] P. Govaerts and L. Storme. On Cameron-Liebler line classes. Advances in Geometry, 4(3):279-286, 2004.
[16] L.C. Grove. Classical Groups and Geometric Algebra. American Mathematical Society, 2002.
[17] W.H. Haemers. Interlacing eigenvalues and graphs. Linear Algebra and its Applications, 226-228(0):593 - 616, 1995.
[18] J. Hirschfeld. Projective Geometries over Finite Fields (Oxford Mathematical Monographs). Oxford University Press, USA, 2 edition, 1998.
[19] D. R. Hughes and F. C. Piper. Projective Planes. Springer-Verlag New York,, 1973.
[20] R. Lidl and H. Niederreiter. Finite Fields (Encyclopedia of Mathematics and its Applications). Cambridge University Press, 1996.
[21] R. Mathon. New maximal arcs in desarguesian planes. Journal of Combinatorial Theory, Series A, 97(2):353 - 368, 2002.
[22] K. Metsch. The non-existence of Cameron-Liebler line classes with parameter 2 < x < q. Bulletin of the London Mathematical Society, 42(6):991-996, 2010.
[23] W.F. Orr. The Miquelian inversive plane IP(g) and the associated projective planes. PhD thesis, University of Wisconsin, 1973.
[24] S. Payne. Tight pointsets in finite generalized quadrangles. Congressus Numer-antiurn, 60:243-260, 1987.
[25] S. Payne. Topics in Finite Geometry: Ovals, Ovoids, and Generalized Quadrangles. UC Denver Course Notes, 2009. For a draft version, see http://math.ucdenver.edu/~spayne/classnotes/topics.pdf.
[26] S. Payne and J. Thas. Finite Generalized Quadrangles. European Mathematical Society, 2nd edition, 2009.
[27] T. Penttila. Cameron-Liebler line classes in PG(3,q). Geometriae Dedicata, 37(3):245-252, 1991.
[28] T. Penttila and G.F. Royle. Sets of type (m, n) in the affine and projective planes of order nine. Designs, Codes and Cryptography, 6(3) :229—245, 1995.
[29] T. Penttila and B. Williams. Regular packings of PG(3,g). European Journal of Combinatorics, 19(6):713 - 720, 1998.
[30] G. Tee. Eigenvectors of block circulant and alternating circulant matrices. New Zealand Journal of Mathematics, 36:195-211, 2007.
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[31] J. Thas. Construction of maximal arcs and partial geometries. Geornetriae Dedicata, 3:61-64, 1974.
[32] J. Thas. Construction of maximal arcs and dual ovals in translation planes. European Journal of Combinatorics, 1:189-192, 1980.
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Full Text
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ThisthesisfortheDoctorofPhilosophydegreeby MorganJ.Rodgers hasbeenapproved by StanleyE.Payne,AdvisorandChair WilliamCherowitzo TimothyPenttila DianaWhite JasonWilliford Date ii
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Rodgers,MorganJ.Ph.D.,AppliedMathematics OnsomenewexamplesofCameron-Lieblerlineclasses ThesisdirectedbyProfessorStanleyE.Payne ABSTRACT Cameron-LieblerlineclassesaresetsoflinesinPG ;q havingmanynicecombinatorialproperties;amongthem,aCameron-Lieblerlineclass L sharesprecisely x lineswithanyspreadofthespaceforsomenon-negativeinteger x ,calledthe parameteroftheset.Theseobjectswereoriginallystudiedasgeneralizationsof symmetrictacticaldecompositionsofPG ;q ,aswellasofsubgroupsofPL ;q havingequallymanyorbitsonpointsandlinesofPG ;q .Theyhaveconnectionsto manyothercombinatorialobjects,includingblockingsetsinPG ;q ,certainerrorcorrectingcodes,andstronglyregulargraphs. WeconstructmanynewexamplesofCameron-Lieblerlineclasses,eachstabilized byacyclicgroupoforder q 2 + q +1havingasemi-regularactiononthelines.In particular,newexamplesareconstructedinPG ;q havingparameter 1 2 q 2 )]TJ/F15 11.9552 Tf 13.037 0 Td [(1 forallvaluesof q 5or9mod12with q< 200;withparameter 1 3 q +1 2 found incollaborationwithJandeBeule,KlausMetsch,andJeroenSchillewaertforall valuesof q 2mod3with2
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Theformandcontentofthisabstractareapproved.Irecommenditspublication. Approved:StanleyE.Payne iv
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ACKNOWLEDGMENTS Thisthesiswouldnothavebeenpossiblewithoutthehelpofmanypeople.I owepartofmysuccessasagraduatestudenttoeverymemberofmycommittee, andIwouldliketothankthemallfortheirtimeandsupport.Inparticular,I wouldliketothankmyadvisorStanPayne,whosededicationtomathematicshas trulybeeninspiring,andTimPenttila,whogenerouslysuggestedthisproblemto workon.Tim'shelpinlearningtoprogramwithMagmawastrulyindispensiblein conductingthisresearch,andhealsobroughtafewkeyarticlestomyattentionthat wereinstrumentalinllinginsomedetailsofthiswork.BillCherowitzohasalso oeredconstantsupportandassistance;mostnotably,heconvincedthedepartment togivemeanocetoworkinwhenIwasarstyearstudentwithnootherformof departmentalsupport.Muchofthecomputationalworkinvolvedinthisthesiswas doneontheUniversityofWyomingcluster,andIthankJasonWillifordforgoingout ofhiswaytosetmeupwiththisaccess.Mysurvivalinacademiawouldhavebeen muchmoredicultwithoutthehelpandadviceofthesepeopleandmanyothers, especiallyDianaWhite,MikeFerrara,andOscarVega. IalsowouldliketothanktheBatemanfamilyfortheirnancialsupportofthe mathematicsdepartment;Ihavebeenfortunatetoreceivethreesemestersofsupport fromtheLynnBatemanMemorialFellowship,andthecompletionofthisresearch wouldhavebeenmuchmoredicultwithouttherelieffromteachingdutiesthis provided.IwouldliketoextendthankstoJandeBeuleandtheUniversiteitGent aswellforsupportingmeasavisitingresearcher.Partoftheworkofthisthesis wasconductedduringthattrip,incollaborationwithJan,KlausMetsch,andJeroen Schillewaert. Mostimportantly,Ithankmylovingwifewhohasneverlostfaithinmyability tosucceed,evenwhenIquestioneditmyself.Shehelpedmorethansheknows;I denitelywouldnothavenishedthisthesiswithouthersupport. v
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TABLEOFCONTENTS Tables........................................viii Chapter 1.Introduction...................................1 1.1Overview.................................1 1.2Finiteelds................................2 1.3TheprojectivegeometryPG n;q ...................4 1.4Collineationsanddualities.......................5 1.5CombinatoricsofPG n;q .......................7 1.6Bilinearandquadraticforms......................9 1.7Orthogonalityandtotallyisotropicsubspaces.............10 1.8OrthogonalpolarspacesinPG n;q ..................11 1.9 Q + ;q andtheKleincorrespondence.................14 2.Cameron-Lieblerlineclasses..........................18 2.1Denitionsandhistory.........................18 2.2Tightsetsof Q + ;q ..........................19 2.3Two-intersectionsets,two-weightcodes,andstronglyregulargraphs21 2.4Trivialexamples.............................23 2.5Non-existenceresults..........................24 2.6Knownexamples.............................26 2.6.1BruenandDrudgeexamples...................26 2.6.2PenttilaandGovaertsexampleinPG ; 4...........27 3.Methodology...................................30 3.1Aneigenvectormethodfortightsets..................30 3.2TacticalDecompositions.........................31 3.3Amodelof Q + ;q ...........................33 3.4Thegeneralmethod...........................34 vi
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4.Newexamples..................................36 4.1Newexampleswithparameter 1 2 q 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1................37 4.1.1Theconstruction.........................37 4.1.2Somedetailsoftheseexamples.................41 4.2Newexampleswithparameter 1 3 q +1 2 ................43 4.3Someothernewexamples........................44 5.Planartwo-intersectionsets..........................46 5.1Projectiveexamples...........................46 5.2Aneexamples.............................48 5.3ConstructionsfromCameron-Lieblerlineclasses...........50 5.3.1Atwo-intersectionsetinAG ; 9................50 5.3.2Anewtwo-intersectionsetinAG ; 81............52 5.3.3AfamilyofexamplesinAG ; 3 2 e ?..............53 Appendix A.Algorithms....................................56 A.1CLautMatrix..............................56 B.Programs....................................58 B.1CLshell.mgm...............................58 B.2CLpreamble.mgm............................60 B.3CLaut.mgm...............................62 B.4CLbcirc.mgm...............................65 B.5CLvspace.mgm..............................68 B.6CLpg3q.mgm...............................70 B.7CLint.mgm................................71 B.8CL81int.mgm..............................72 B.9MNset.mgm...............................73 References ......................................76 vii
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TABLES Table 4.1 ParametersandautomorphismgroupsofthenewexamplesofCameronLieblerlineclassesconstructed. .......................36 4.2 Intersectionnumbersoflineclasseswithparameter 1 2 q 2 )]TJ/F15 11.9552 Tf 13.132 0 Td [(1 withthe planesof PG ;q . ..............................42 4.3 Intersectionnumbersoflineclasseswithparameter 1 3 q +1 2 withthe planesof PG ;q . ..............................44 4.4 Intersectionnumbersofsomeothernewlineclasses. ...........45 5.1 Linesperpointforthesymmetrictacticaldecompositioninducedon PG ; 9 byaCameron-Lieblerlineclassofparameter 40 . ........51 5.2 Pointsperlineforthesymmetrictacticaldecompositioninducedon PG ; 9 byaCameron-Lieblerlineclassofparameter 40 . .............51 5.3 Linesperpointforthesymmetrictacticaldecompositioninducedon PG ; 81 byaCameron-Lieblerlineclassofparameter 3280 . ......53 5.4 Pointsperlineforthesymmetrictacticaldecompositioninducedon PG ; 81 byaCameron-Lieblerlineclassofparameter 3280 . ............53 viii
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1.Introduction 1.1Overview ThefocusofthisdissertationistoconstructnewexamplesofCameron-Liebler lineclassesadmittingacertaincyclicautomorphismgroup.Theselineclasseshave manydierentcharacterizations.Mostnotably,aCameron-Lieblerlineclass L hasthe propertythat,forsomeinteger x calledthe parameter , L sharesprecisely x lineswith everyspreadofthespace.Cameron-Lieblerlineclassesarealsoofinteresttoother areasofmathematicsincludinggrouptheoryandcodingtheory.Thesesetsoflines wereoriginallystudiedinrelationtoagrouptheoryproblemregardingcollineation groupsofPG ;q havingthesamenumberoforbitsonpointsandonlines.They alsoserveasgeneralizationsofthenotionofasymmetrictacticaldecomposition ofPG ;q ;i.e.,atacticaldecompositionhavingthesamenumberofpointclasses andlineclasses.ThroughtheKleincorrespondence,aCameron-Lieblerlineclassis equivalenttoasetofpointsofthehyperbolicquadric Q + ;q calleda tightset .Tight setsofthisquadricoftendeterminetwo-intersectionsetsoftheunderlyingprojective spacePG ;q ,thatis,setsofpointshavingtwointersectionnumberswithrespect tohyperplanes.Two-intersectionsetscanthenbeusedtoconstructerrorcorrecting codeswithcodewordshavingpreciselytwononzeroweightswhich,inturn,giverise toexamplesofstronglyregulargraphs. AfterreviewingthegeometryofPG ;q and Q + ;q ,aswellastheirrelationshipthroughtheKleincorrespondence,wesurveytheknownresultsonCameronLieblerlineclasses,includingtheknownexamplesaswellassomenon-existence results.Weshowtheequivalenceoftheseobjectswithtightsetsof Q + ;q andgive resultsonwhenthesesetsdeterminetwo-intersectionsetsoftheunderlyingPG ;q . Wealsolookattheconstructionoftwo-weightcodesandstronglyregulargraphs fromthesetwo-intersectionsets.Oncewehavedevelopedthisbackgroundmaterial, wedeveloptoolswhichareusedtoconstructnewexamples.Themaintoolsare 1
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aneigenvectormethodforndingtightsetsandresultsontacticaldecompositions whichfacilitatethismethod.Sinceweareprimarilyworkingfromthepointofview of Q + ;q ,analgebraicmodelforthisspaceisintroduced.Thisallowsustogive aconcisenotationforacyclicgroupoforder q 2 + q +1actingsemi-regularlyonthe space,whichiscontainedinthestabilizerofeachnewexampleconstructed. WeusethetoolswedeveloptoconstructseveralnewexamplesofCameronLieblerlineclassesinPG ;q ,includingmanyhavingparameters 1 2 q 2 )]TJ/F15 11.9552 Tf 13.099 0 Td [(1and 1 3 q +1 2 .WealsodescribeotherexamplesinPG ; 27andPG ; 32.Forallofthese newexamples,wedetailstructuralinformationsuchasautomorphismgroupsand intersectionnumberswithplanesofPG ;q ,aswellastherelatedtwo-intersection setsofPG ;q ,two-weightcodes,andstronglyregulargraphs.Furthermore,when q =9or81,thenewexampleswithparameter 1 2 q 2 )]TJ/F15 11.9552 Tf 12.93 0 Td [(1arelinepartitionsofa symmetrictacticaldecompositionofPG ;q havingfourpartsonpointsandonlines; wegiveaconstructionfromthisdecompositionoftwo-intersectionsetsinAG ;q . Fewexamplesoftwo-intersectionsetsofoddorderaneplanesareknown;infact, theonlypreviouslyknownexamplesareinplanesoforder9.Thus,ourexamplein AG ; 81isnew. 1.2Finiteelds Aniteeldalwayshasorder q = p e ,where p isaprime.Thiseld,whichis uniqueuptoisomorphism,willbedenoted F q andhas characteristic p ;i.e., P p i =1 x =0 forevery x 2 F q and p isthesmallestintegerforwhichthisistrue.Themultiplicativegroup F q ofnonzeroelementsof F q isacyclicgroupoforder q )]TJ/F15 11.9552 Tf 12.208 0 Td [(1;anelement 2 F q havingorder q )]TJ/F15 11.9552 Tf 12.487 0 Td [(1iscalleda primitiveelement ,and h i = F q forsuchan element. Let K = F q ,where q = p h .Asubset F of K whichisalsoaeldunderthe sameoperationsiscalleda subeld of K ;wewrite F K . K containsaunique subeldisomorphicto F p e foreach e dividing h ,consistingof f a 2 K : a p e = a g .The 2
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intersectionofallsubeldsof K iscalledthe primesubeld of K andisisomorphic to F p .Wecanconstructalargereld E = F q d from K byconsideringanirreducible polynomial f x in K [ x ]ofdegree d ;inthiscase E = K [ x ] = f x = f a 0 + a 1 x + ::: + a d )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 x d )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 j a i 2 K;f x =0 g isaniteeldoforder q d containing K asasubeld.Wesaythat E isan extension eld of K . Amap : F q ! F q iscalledan automorphism of F q if isapermutationof theelementssuchthat x + y = x + y ,and xy = x y forall x , y in F q . WewriteAut F q forthegroupofautomorphismsof F q .If q = p e , p prime,then Aut F q iscyclic,isomorphicto Z e ,andisgeneratedbythe Frobeniusautomorphism : x 7! x p . Let q = p e , F = F q ,and K = F q h with F K ;thenAut K=F ,thegroup ofautomorphismsof K xingeveryelementof F ,hasorder h andisgeneratedby e : x 7! x p e = x q .Wedenethe relativetracemapfrom K to F ,T K=F : K 7! F ,by T K=F a = X 2 Aut K=F a = a + a q + a q 2 + ::: + a q h )]TJ/F18 5.9776 Tf 5.756 0 Td [(1 . Thismaphasthefollowingproperties: 1.T K=F a + b =T K=F a +T K=F b forall a;b 2 K . 2.T K=F ca = c T K=F a forall c 2 F , a 2 K . 3.T K=F a = ha forall a 2 F . 4.T K=F a =T K=F a forall a 2 K andforall 2 Aut K=F . Noticethatthersttwoitemsimplythat,if K isviewedasavectorspaceover F ,thenT K=F isalineartransformationfrom K to F .Weactuallyhavemore;the mapT K=F maps K onto F ,andinfact,everylinearmapfrom K into F takesthe form L b a =T K=F ba forsome b 2 K . 3
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1.3Theprojectivegeometry PG n;q Muchofthismaterialistreatedthoroughlyin[25]orin[18].Therearealso examplesofprojectiveplaneswhicharenotoftheformPG ;q ;fordetailsonthese examples,see[19]. Denition1.1 Let F = F q beaniteeldoforder q and V beavectorspaceof dimension n +1 over F .Wedenethegeometry PG n;q asfollows: Theone-dimensionalvectorsubspacesof V arethepointsof PG n;q . The d +1 -dimensionalvectorsubspacesof V arethe d -dimensionalsubspaces of PG n;q . Incidenceisdenedintermsofcontainmentofthecorrespondingvectorsubspaces. Theworddimension"isusedintwowayswithadierentmeaning;whenitisnot clearwhichwemeanfromcontext,wewillspecify projective dimensionwhenwe aretalkingaboutthedimensioninPG n;q ,or vectorspace dimensionwhenwe aretalkingaboutasubspaceof V .Thisdenitionofaprojectivespaceallowsus toassociatevectorsin V withpointsofPG n;q ;namely,anonzerovector v 2 V representsapointofPG n;q ,withnonzerovectors v , w representingthesamepoint ifandonlyif v = c w forsome c 2 F . WewillcallasubspaceofPG n;q havingdimension n )]TJ/F15 11.9552 Tf 12.69 0 Td [(1a hyperplane ;the setofpointsonahyperplanecanbedescribedasthosesatisfyingahomogeneous linearequation.Wewritethecoecientstheequationdescribingahyperplane H as h =[ x 0 ;:::;x n ],withtheconventionthatapoint u isin H ifandonlyif uh T =0. Wewillspeakofasetofpointsorofhyperplanesasbeing linearlyindependent ifthe correspondingvectorsarelinearlyindependentinthevectorspace. Itisfrequentlyusefultodescribeasubspace U ofPG n;q aseithertheintersectionorthespanofothersubspaces.Giventwosubspaces U 1 and U 2 ofPG n;q ,the 4
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intersection U 1 U 2 isagainasubspaceofPG n;q .Wedenethe span of U 1 and U 2 tobethesmallestsubspaceofPG n;q containingboth U 1 and U 2 ;wedenotethis by h U 1 ;U 2 i .Ingeneral,theprojectivedimensionof h U 1 ;U 2 i isgivenby dim h U 1 ;U 2 i =dim U 1 +dim U 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(dim U 1 U 2 . Thespanoftwodistinctpoints,forexample,isthelinecontainingbothofthem. Givenasubspace U ofdimension d andahyperplane H ,wehavethat U H is eitherequalto U ,orhasdimension d )]TJ/F15 11.9552 Tf 12.687 0 Td [(1.Thuswecandescribea d -dimensional subspace U ofPG n;q aseitherthespanof d +1linearlyindependentpoints,oras theintersectionof n )]TJ/F19 11.9552 Tf 12.369 0 Td [(d linearlyindependenthyperplanes.If h 1 ;:::; h n )]TJ/F20 7.9701 Tf 6.587 0 Td [(d arethe vectorscontainingthecoecientsoftheequationsforthesehyperplanes,thenwecan associate U withtheleftnullspaceofthematrix h T 1 ::: h T n )]TJ/F20 7.9701 Tf 6.586 0 Td [(d . 1.4Collineationsanddualities AbijectiononthepointsofPG n;q whichpreservesthelinesiscalleda collineation ;thatis,amap :PG n;q ! PG n;q suchthatforalllines ` of PG n;q ,theimage ` isalsoalineofPG n;q .Thisnecessarilyimpliesthatany d -dimensionalsubspaceofPG n;q getsmappedby toanother d -dimensionalsubspace.SinceweviewthesubspacesofPG n;q ascorrespondingtosubspacesofan n +1-dimensionalvectorspace V over F q ,wecandescribeacollineationofPG n;q intermsofitsactionon V .Inparticular,anymatrix A 2 GL n +1 ;q canbeused todeneacollineation L A : x 7! x A ofPG n;q .Collineationsofthistypearecalled homographies .Notethat,forany 2 F q ,thematrices A and A denethesame maponPG n;q ;wesaythesetwomapsare projectivelyequivalent .Thegroup PGL n +1 ;q = GL n +1 ;q =Z GL n +1 ;q 5
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iscalledthe projectivelineargroup ,andactsfaithfullyonPG n;q .Itisworthnoting that,forahyperplane H representedby h andamatrix A 2 PGL n +1 ;q ,wehave that x 2 H ifandonlyif x A A )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 h T =0 ; sothehyperplane H getsmappedby L A toanewhyperplanedenedby A )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 h T .Automorphismsof F q alsogiveexamplesof collineationsofPG n;q .Given 2 Aut F q ,themaponPG n;q inducedby : V ! V : x 0 ;:::;x n 7! x 0 ;:::;x n iscalledan automorphiccollineation . TheFundamentalTheoremofProjectiveGeometrytellsusthatanycollineation ofPG n;q canbeobtainedbycomposinganautomorphiccollineationwithahomography.Suchamapisoftheform L A : x 7! L A x = x A; where 2 Aut F q and A 2 PGL n +1 ;q ,andiscalleda projectivesemilinear map. Thegroupofthesemapsisdenoted P )]TJ/F19 11.9552 Tf 7.314 0 Td [(L n +1 ;q = PGL n +1 ;q o Aut F q : Associatedwithaprojectivegeometry S istheso-called dualgeometry S ;this geometry'spointsandhyperplanesare,respectively,thehyperplanesandpointsof S .AprojectivegeometryoftheformPG n;q isisomorphictoitsdualgeometry, andwecallanisomorphismfromthepointsofPG n;q ontothehyperplanesa reciprocity .Oneimportantexampleisthemapsendingapoint x tothehyperplane determinedby x T .ByourearliercommentsabouttheFundamentalTheoremof ProjectiveGeometry,anyreciprocitycanbewrittenintheform x ! x A T ,where 2 Aut F q and A 2 PGL n +1 ;q .Ifwehaveareciprocity thatisaninvolution, thatis,if 2 =1,thenwecall a polarity . 1.5Combinatoricsof PG n;q 6
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Theorem1.2 [18]In PG n;q ,thereare q n +1 )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 q )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 points, q n +1 )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 q n )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 q )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 2 q +1 lines,and Q n +1 i = n )]TJ/F21 5.9776 Tf 5.756 0 Td [(d +1 q i )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 Q r +1 i =1 q i )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 d -dimensionalsubspaces. Given d
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ThegroupsactingonPG ;q are PGL ;q and P )]TJ/F19 11.9552 Tf 7.315 0 Td [(L ;q ;if q = p e ,theorders oftheseare PGL ;q hasorder q 6 q 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 q 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 q 4 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1and P )]TJ/F19 11.9552 Tf 7.314 0 Td [(L ;q hasorder eq 6 q 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 q 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 q 4 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1. Aset R of q +1mutuallyskewlinesinPG ;q iscalleda regulus provided 1.througheverypointofeverylineof R thereisatransversalofthelinesof R thatis,alinemeetingeachofthelinesof R ;and, 2.througheverypointofeverytransversalthereisalineof R . Itisclearthatthesetoftransversalsof R isalsoareguluswhichwecall R opp ,the opposite regulusof R .AnythreeskewlinesinPG ;q determineauniqueregulus. A spread ofPG ;q isasetof q 2 +1linesofthespacethatpartitionsthepoints.A spread S iscalled regular if,givenanythreeskewlinesin S ,theregulusdeterminedby thosethreelinesisalsocontainedin S .Spreadscanbedenedinhigherdimensional spacesaswell,andareofconsiderableinterest,astheycanbeusedtoconstruct examplesofprojectiveplanes.Theirclassicationisanimportantprobleminnite geometrythatisbeyondthescopeofthisthesis. A k -arc K ofPG ;q oranyprojectiveplaneoforder q isasetof k pointssuch thatnothreearecollinear.Thusanyline ` ofPG ;q meets K in0,1,or2points; wecalltheselines external , tangent ,or secant to K respectively.A k -arcmusthave k q +2.A k -arc K whichisnotcontainedinany k +1-arciscalled maximal .A q +1-arciscalledan oval ,andif q isodd,any q +1-arcismaximal.However,if q iseven,everytangentlinetoanoval K passesthroughacommonpoint N which wecallthe nucleus of K .Inthissituation, K[f N g isa q +2-arc,whichwecall a hyperoval .Givenaplane embeddedinPG ;q containinganovalorhyperoval O ,andapoint p notin ,wedenea cone over O tobethesetofpointsonthe 8
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linesjoining p topointsof O .Thelinesarecalledthe generators ofthecone,and p iscalledthe vertex ofthecone. 1.6Bilinearandquadraticforms Let V beavectorspaceofdimension n +1over F = F q A bilinearform on V is afunctionB: V V ! F thatislinearineachargument;thatis, B a u + v ; w = a B u ; w +B v ; w and B u ;a v + w = a B u ; v +B u ; w forall u ; v ; w 2 V andall a 2 F . AbilinearformBissaidtobe symmetric ifB u ; v =B v ; u forall u ; v 2 V , and alternating ifB u ; u =0forall u 2 V .Wearestrictlyinterestedin reexive bilinearforms,thatis,thoseforwhichB u ; v =0impliesB v ; u =0.Every reexivebilinearformiseithersymmetricoralternating. A quadraticform onavectorspace V isamap Q : V ! F denedbyahomogeneousdegree2polynomialinthecoordinatesof V relativetosomebasis.Equivalently, wecall Q : V ! F aquadraticformif Q a u = a 2 Q u forall a 2 F and u 2 V and B: u ; v 7! Q u + v )]TJ/F19 11.9552 Tf 11.955 0 Td [(Q u )]TJ/F19 11.9552 Tf 11.955 0 Td [(Q v givesabilinearformon V . ItisclearthatBissymmetricif q isoddandalternatingif q iseven.Thisassociated bilinearform,B,iscalledthe polarform of Q .Avectorspaceequippedwitha quadraticformiscalledan orthogonalspace . GivenabilinearformBandanorderedbasis B = f v 0 ;:::; v n g for V ,weput b ij =B v i ; v j .The Grammatrixrelativeto B isthendenedby ^ B =[ b ij ].This matrixhasthepropertythat,if[ u ] B and[ v ] B arecoordinatevectorsof u and v relativeto B ,thenB u ; v =[ u ] B ^ B[ v ] T B .AnytwoGrammatricesofabilinearform Bhavethesamerank,whichwedenetobethe rank ofB.If Q isaquadraticform 9
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on V ,wedenetheuppertriangularmatrix A = a ij ,where a ij = 8 > > > > < > > > > : B v i ; v j , ij . Wethenhavethat Q u =[ u ] B A [ u ] T B ,andtheGrammatrixforthepolarformof Q withrespectto B isthen ^ B = A + A T . 1.7Orthogonalityandtotallyisotropicsubspaces LetBbeareexivebilinearformonPG n;q .Wedenean orthogonality relationshiponthepointsofPG n;q by u ? v ifB u ; v =0.If S V ,wedene S perp"tobe S ? = f v 2 V j v ? s 8 s 2 S g : Apoint v ofPG n;q iscalled singular withrespecttoabilinearformBif v ? = V ; itiscalled singular withrespecttoaquadraticform Q ifitissingularwithrespect totheassociatedbilinearformand Q v =0.WesayBor Q is degenerate ifthereis asingularpoint,and nondegenerate otherwise. Weplaceaspecialsignicanceonpoints v forwhichB v ; v = 0 or Q v =0. Suchapoint v iscalled isotropic withrespecttotobilinearformBorthequadratic form Q ,respectively.Wecallasubspace W of V isotropicifitcontainsanisotropic point, anisotropic otherwise,and totallyisotropic ifB u ; v =0forall u ; v 2 W for abilinearform,orif Q v =0forall v 2 W foraquadraticform.Ifasubspaceis totallyisotropicwithrespecttoaquadraticform Q ,thenitisalsototallyisotropic withrespecttotheassociatedbilinearform,thoughtheconverseonlyholdsif q is odd.ThesetofisotropicpointsinPG n;q withrespecttoanondegeneratequadratic formiscalleda quadric ,andhasthepropertythatanylineofPG n;q containing morethantwopointsofaquadricmustbecompletelycontainedinthequadric. IfBisnondegenerateform,theorthogonalityrelationcanbeusedtodenea polarity : U 7! U ? ofPG n;q .Inthiscase,if U and W aresubspacesof V with 10
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U W ,then W ? U ? ;furthermore,foranysubspace U of V ,dim U +dim U ? = dim V .Apoint x issaidtobe isotropic withrespecttothepolarityif x x ? ,anda subspace U issaidtobe totallyisotropic withrespecttothepolarityif U U ? .This isinagreementwiththenotionsofbeingisotropicortotallyisotropicwithrespectto thebilinearform.When q isodd,thepolarformofanondegeneratequadraticform isnecessarilynondegenerate.Thusinthiscasethenotionsofbeingtotallyisotropic withrespecttothequadraticform,thebilinearform,andtheassociatedpolarityall agree. Thesituationismorecomplicatedwhen q iseven,sinceitispossibleforthe polarformof Q tobedegenerateevenwhen Q isnondegenerate.Inthiscase,we donothaveapolarityassociatedwiththequadraticform.EvenifthepolarformB of Q isnondegenerate,wehaveB u ; u =0forevery u 2 PG n;q ,so every point ofPG n;q isincidentwithitsimageundertheinducedpolaritysuchapolarityis calleda nullpolarity .Thusthesetofpointswhichareisotropicwithrespecttothis polaritydoesnotagreewiththesetofpointswhichareisotropicwithrespecttothe quadraticform. 1.8Orthogonalpolarspacesin PG n;q Denition1.4 A polarspaceofrank r isanincidencegeometryconsistingofaset ofpoints,lines,projectiveplanes,..., r )]TJ/F15 11.9552 Tf 12.798 0 Td [(1 -dimensionalprojectivespacescalled subspaces suchthat 1.Anytwosubspacesintersectinasubspace. 2.If U isasubspaceofdimension r )]TJ/F15 11.9552 Tf 10.359 0 Td [(1 and p isapointnotin U ,thereisaunique subspace W containing p with U W havingdimension r )]TJ/F15 11.9552 Tf 11.639 0 Td [(2 ;itconsistsofall pointsof U whicharejoinedto p bysomeline. 3.Therearetwodisjointsubspacesofdimension r )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 . 11
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The r )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 -dimensionalsubspacesarecalled maximals ofthepolarspace. The niteclassicalpolarspaces aretheexamplesnaturallyembeddedinaprojective spacePG n;q ;theyaredenedbyanondegeneratequadraticorsesquilinearform onthespace.AresultofTits[33]provedthatanypolarspacewithrankatleast 3isclassical.Rank2polarspacesareaspecialcase.Theyarecalledgeneralized quadrangles,andtherearenonclassicalexamplesofthese;see[26]foradetailed treatment. Let F = F q ,and V bea n +1-dimensionalvectorspaceover F .Take Q tobe anondegeneratequadraticformon V withpolarformB.Thegeometryconsisting ofthetotallyisotropicsubspacesofPG n;q withrespectto Q isanexampleofa classicalpolarspace;wecallanexamplearisinginthiswayan orthogonalpolarspace . Note: Wehavenowintroducedthreeverycloselyrelatedterms,an orthogonal space ,a quadric ,andan orthogonalpolarspace . Thevectorspace V alongwithanondegeneratequadraticformisan orthogonalspace . ThesetofisotropicpointsofPG n;q withrespecttothequadraticformis calleda quadric . Thegeometryoftotallyisotropicsubspaceswithrespecttothequadraticform iscalledan orthogonalpolarspace ,inthiscontextthepolarspacecaneither beconsideredasembeddedinPG n;q orasageometryinitsownright. ThemostgeneralcollineationofPG n;q preservingaquadriciscalleda semisimilarity ;thisisamap suchthat,forsome a 2 F q andsome 2 Aut F q , Q x = a Q x . Wecall a similarity if =1,andwecall an isometry if a =1and =1.The followingimportanttheoremisknownasWitt'sExtensionTheorem: 12
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Theorem1.5 If U , W V ,and : U ! W anisometry,thenthereisanisometry : V ! V suchthat j U = . Corollary1.6 Anytwomaximalsof V havethesamedimension. Thevectorspacedimensionofamaximaliscalledthe Wittindex ofthepolarspace. TheWittindexofanondegenerateformislessthanorequalto 1 2 dim V ,sincea totallyisotropicsubspace W iscontainedin W ? . Wedenetwodistinctpoints u , v ofthequadrictobea hyperbolicpair if B u ; v =1.Wethencall h u ; v i a hyperbolicline .NotethatthisisalineofPG n;q containingpreciselytwopointsofthequadric. Theorem1.7 AnynondegenerateorthogonalspaceofWittindex r over F q isisometrictooneofthefollowing: 1.Ahyperbolicquadric Q + r )]TJ/F15 11.9552 Tf 12.029 0 Td [(1 ;q istheorthogonaldirectsumof r hyperbolic lines. 2.Aparabolicquadric Q r;q istheorthogonaldirectsumof r hyperboliclines andaone-dimensionalanisotropicspace.Thesefallintotwoisometryclasses andonesimilarityclasswhen q isodd,andoneisometryclasswhen q iseven. 3.Anellipticquadric Q )]TJ/F15 11.9552 Tf 7.085 -4.339 Td [( r +1 ;q istheorthogonaldirectsumof r hyperbolic linesandatwo-dimensionalanisotropicspace. Thegroupofisometriesof Q + r )]TJ/F15 11.9552 Tf 9.414 0 Td [(1 ;q , Q r;q ,or Q )]TJ/F15 11.9552 Tf 7.085 -4.339 Td [( r +1 ;q isdenoted O + r;q , O r +1 ;q ,or O )]TJ/F15 11.9552 Tf 7.085 -4.339 Td [( r +2 ;q ,respectively.Fortheprojectiveversionsofthesegroups, weprexthiswith P , PG ,or P )-488(dependingonwhetherwewantthegroupof isometries,similarities,orsemi-similarities,respectively. Ifwehaveasetofpoints O inapolarspacesuchthateverymaximalofthepolar spacemeets O inauniquepoint,thenwecall O an ovoid ofthepolarspace.The 13
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classicationofovoidsinclassicalpolarspacesisanimportantopenprobleminnite geometry;weareprimarilyinterestedintheseobjectsbecauseofhowtheyinteract withotherobjectsinthespace. 1.9 Q + ;q andtheKleincorrespondence The5-dimensionalhyperbolicorthogonalspace Q + ;q playsanimportantrole, asthisgeometryiscloselyrelatedtoPG ;q .Thisquadricismadeupoftheorthogonaldirectsumofthreehyperboliclines,andthestandardassociatedquadratic formisgivenby Q : x 0 ;x 1 ;x 2 ;x 3 ;x 4 ;x 5 7! x 0 x 1 + x 2 x 3 + x 4 x 5 ; whichisdescribedbythematrix A = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 010000 000000 000100 000000 000001 000000 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 . TheGrammatrixforthepolarformBwithrespecttothestandardbasisisthen ^ B = A + A T . Anotherwaytothinkofthestructureofthispolarspaceisgivenbytaking onepointfromeachofthethreehyperbolicpairs.Sincethehyperboliclinesthey determinearepairwiseorthogonal,thesethreepointsarealsopairwiseorthogonal andsospanatotallyisotropicplane 1 ,necessarilyamaximalofthepolarspace. Thethreeremainingpointsfromthehyperbolicpairsthenspanatotallyisotropic plane 2 whichisdisjointfrom 1 . ThegeometriesPG ;q and Q + ;q arecloselyrelatedthroughamapping knownasthe Kleincorrespondence .ThisreferstoabijectionfromthelinesofPG ;q 14
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tothepointsof Q + ;q suchthattwolinesofPG ;q intersectifandonlyiftheir imagesarecollinearin Q + ;q .Todenethebijection,wewillrstestablishaway todescribelinesofPG ;q using Pluckercoordinates .Let x = x 0 ;x 1 ;x 2 ;x 3 and y = y 0 ;y 1 ;y 2 ;y 3 bedistinctpointsonaline ` ofPG ;q .Dene G ` = p 01 ;p 23 ;p 02 ;p 31 ;p 03 ;p 12 ,where p ij = x i x j y i y j for0 i
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Corollary1.9 Thesetoflinesinaspreadof PG ;q correspondtoanovoidof Q + ;q . Corollary1.10 Let ` and ` 0 betwoconcurrentlinesin PG ;q withcorresponding points L and L 0 in Q + ;q .Thenthelinesoftheatpenciloflinesin PG ;q determinedby ` and ` 0 correspondtothelineof Q + ;q through L and L 0 .Conversely, eachlineof Q + ;q correspondstoasetoflinesin PG ;q lyinginaatpencil. Corollary1.11 Thesetofpointsinatotallyisotropicplaneof Q + ;q corresponds toasetof q 2 + q +1 linesin PG ;q ,anytwoofwhichareconcurrent.Thus,they correspondtoeitherthesetoflinesthroughacommonpoint p ,denotedstar p ,or thesetoflinesinacommonplane ,denotedline . Wecallatotallyisotropicplaneof Q + ;q a Latinplane ifitcorrespondstostar p forsomepoint p 2 PG ;q ,anda Greekplane ifitcorrespondstoline forsome plane ofPG ;q . Corollary1.12 Anytwodistinctplanesofthesametypein Q + ;q intersectina 0 -dimensionalsubspaceasinglepoint.Anytwoplanesofdierenttypesareeither disjoint,ormeetinalineof Q + ;q .Thustwoplanesareofthesametypeifand onlyiftheirintersectionhasevendimension. Thesecorrespondencesallowustocountthefollowing: Corollary1.13 Q + ;q contains 1. q 2 +1 q 2 + q +1 points; 2. q 3 + q 2 + q +1 q 2 + q +1 lines; 3. 2 q 3 + q 2 + q +1 planes; 4. q q +1 2 pointscollineartoagivenpoint; 16
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5. q +1 2 linescontainingagivenpoint; 6. 2 q +1 planescontainingagivenpoint; 7. 2 planescontainingagivenline. TheKleincorrespondencealsogivesusaconnectionbetweenthegroups P )]TJ/F19 11.9552 Tf 7.314 0 Td [(L ;q actingonPG ;q and P )]TJ/F19 11.9552 Tf 7.314 0 Td [(O + ;q actingon Q + ;q .Specically,anyelementof P )]TJ/F19 11.9552 Tf 7.314 0 Td [(L ;q inducesanactiononthepointsof Q + ;q preservingcollinearity,andso P )]TJ/F19 11.9552 Tf 7.314 0 Td [(O + ;q hasasubgroupisomorphicto P )]TJ/F19 11.9552 Tf 7.314 0 Td [(L ;q .Anymapon Q + ;q arising inthisfashionmapsGreekplanestoGreekplanesandLatinplanestoLatinplanes. AnycorrelationofPG ;q sendslinestolines,andsoalsoinducesanactiononthe pointsof Q + ;q preservingcollinearity.Amaparisinginthisfashioninterchanges theGreekandLatinplanes.Theseareknowntobetheonlyautomorphismsof Q + ;q . Theorem1.14 Thestructureoftheprojectivesimilarityandsemi-similaritygroups of Q + ;q areasfollows: PGO + ;q ' PGL ;q oZ 2 . P )]TJ/F19 11.9552 Tf 7.314 0 Td [(O + ;q ' P )]TJ/F19 11.9552 Tf 7.314 0 Td [(L ;q oZ 2 . UsingthisconnectionbetweenthelinesofPG ;q andthepointsof Q + ;q canbehelpful,especiallywhendealingwithcombinatoricsofsetsoflinesinPG ;q . Inadditiontohavingmanytheoreticalresultstoapply,itismorecomputationally convenienttodealwithsetsofpoints.Forthisreason,muchofourworkinthisthesis isdoneinthecontextof Q + ;q . 17
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2.Cameron-Lieblerlineclasses Inthischapter,wewillsurveymanyoftheknownresultsonCameron-Lieblerline classes.Thisincludesnon-existenceresults,knownconstructions,andadiscussionof theimagesoftheselinesetsin Q + ;q undertheKleincorrespondence. 2.1Denitionsandhistory HerewedetailsetsoflinesinPG ;q havingsomespecialcombinatorialproperties.ThesesetsoflineswereoriginallystudiedbyCameronandLiebler[8],who calledthemspecial"lineclasses,inconnectionwiththestudyofcollineationgroups ofPG ;q havingthesamenumberoforbitsonpointsandlines.Suchagroup inducesasymmetrictacticaldecompositionoftheincidencestructureofpointsand linesinPG ;q ,andtheyshowedthatalineclassfromsuchadecompositionhas niceintersectionpropertieswithrespecttoreguliandspreadsofthespace.They abstractedtheconceptofsetsoflineswiththeseproperties,hopingitwouldlead totheclassicationofsymmetrictacticaldecompositionsandcollineationgroupsof PG ;q withthisorbitstructure.However,thisproblemprovedinterestingina moregeneralsettingthanoriginallyenvisioned. Denition2.1 Let A bethepoint-lineincidencematrixof PG ;q withrespectto someorderingofthepointsandlines,andlet L beasetoflinesin PG ;q with characteristicvector = L .Wewillwrite ` fortheentryof corresponding totheline ` .Thefollowingstatementsareallequivalent;iftheyhold, L iscalleda Cameron-Lieblerlineclass[8],[27]. 1. L 2 row A . 2. L 2 null A T ? 3. jRLj = jR opp Lj foreveryregulus R anditsopposite R opp . 4.Thereexists x 2 Z + suchthat jLSj = x foreveryspread S . 18
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5.Thereexists x 2 Z + suchthat jLSj = x forevery regular spread S . 6.Thereexists x 2 Z + suchthat,foreveryincidentpoint-planepair p; , j star p Lj + j line Lj = x + q +1 j pencil p; Lj . 7.Thereexists x 2 Z + suchthat,foreveryline ` in PG ;q , jf lines m 2L meeting `;m 6 = ` gj = x q +1+ q 2 +1 ` . 8.Thereexists x 2 Z + suchthat,foreverypair ` , m ofskewlinesin PG ;q , jf n 2L : n isatransversalto `;m gj = x + q [ ` + m ] . Thevalue x mustsatisfy0 x q 2 +1,andwillnecessarilybethesameineach instance;wecall x the parameter ofthelineclass.If L isaCameron-Lieblerlineclass withparameter x ,then jLj = x q 2 + q +1.Thecomplement L 0 ofaCameron-Liebler lineclass L withparameter x isalsoaCameron-Lieblerlineclasshavingparameter q 2 +1 )]TJ/F19 11.9552 Tf 10.711 0 Td [(x ,andtheunionoftwodisjointCameron-Lieblerlineclasseswithparameters x 1 and x 2 isaCameron-Lieblerlineclasswithparameter x 1 + x 2 .ACameron-Liebler lineclassissaidtobe irreducible ifitdoesnotproperlycontainanyotherlineclass asasubset. 2.2Tightsetsof Q + ;q ToinvestigatetheexistenceofCameron-Lieblerlineclasses,itisfrequentlyuseful totranslatetheirdenitiontothesettingof Q + ;q usingtheKleincorrespondence. Inthiscontext,part7ofDenition2.1hasanespeciallyinterestinginterpretation; L isaCameron-Lieblerlineclassifandonlyifitsimage M in Q + ;q hasthefollowing property: Thereexists x 2 Z + suchthat,foreverypoint p in Q + ;q , j p ? M j = x q +1+ q 2 p , where = M isthecharacteristicvectorof M . 19
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Denition2.2 Let S beapolarspaceofrank r 3 over F q .Thenaset T ofpoints in S isan x -tightset ifforallpoints p 2S j p ? Tj = 8 > < > : x q r )]TJ/F18 5.9776 Tf 5.756 0 Td [(1 )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 q )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 + q r )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 if p 2T x q r )]TJ/F18 5.9776 Tf 5.756 0 Td [(1 )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 q )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 if p 62T Adaptingthisdenitionfortherank3polarspace Q + ;q ,weseethataCameronLieblerlineclasswithparameter x isequivalenttoan x -tightsetof Q + ;q . Pointsetsinpolarspaceshavingpreciselytwointersectionnumberswithrespect toperpsofpointsarecalled intriguing byBamberg,Kelly,LawandPenttila[2].There aretwotypesofintriguingsetsinnitepolarspaces,andtheycanbecharacterizedin termsoftheirintersectionnumbers.If I isanintriguingsetofapolarspacehaving intersectionnumbers h 1 forperpsofpointsinside I and h 2 forperpsofpointsoutside I ,then I isatightsetif h 1 >h 2 .Atightsetofpointsinanitepolarspacecan alsobedenedasasetofpoints T suchthateachpointofthespaceis,onaverage, collinearwithasmanypointsin T aspossible.Thesesetswereoriginallystudied ingeneralizedquadranglesbyPayne[24]andtheirdenitionwaslaterextendedto moregeneralpolarspacesbyDrudge[12].Anintriguingsetwith h 1
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3.Thereexists x 2 Z + suchthat j M j = x q 2 + q +1 ,everytangenthyperplane to Q + ;q atapointof M meets M in q 2 + x q +1 points,andeveryother hyperplaneof PG ;q meets M in x q +1 points. 4.Thereexists x 2 Z + suchthat j ` ? M j = q j ` M j + x foreveryline ` of PG ;q . 5.Thereexists x 2 Z + suchthat j ` ? M j = q j ` M j + x foreveryline ` ofone ofthefourlinetypesin PG ;q external,tangent,secant,totallyisotropic. Itisimportanttonotethatthelastthreecharacterizationsarestrongerthantheir relatedversionsinPG ;q .Part3inparticularstatesthat,inadditiontoknowing theintersectionnumbersfortangenthyperplanesof Q + ;q ,wealsoknowthatevery nontangenthyperplanesectionof Q + ;q meetsan x -tightsetin x q +1points. Thispropertyisimportantenoughthatwestateitonitsown,asitwillbeusedin thenextsectiontoconstructrelatedcombinatorialobjects. Theorem2.4 Let T beaproper x -tightsetof Q + ;q thatspanstheambientprojectivespace.Thenthesetofpointscoveredby T hastwointersectionnumberswith respecttohyperplanesof PG ;q .Thesenumbersare h 1 = q 2 +1+ x q +1 and h 2 = x q +1 : 2.3Two-intersectionsets,two-weightcodes,andstronglyregulargraphs Tightsetsof Q + ;q arerelatedtomanyothercombinatorialobjects;herewe investigatesomepropertiesoftheseobjects. Denition2.5 Asetofpoints S of PG n;q iscalleda two-intersectionset with intersectionnumbers h 1 and h 2 ifeveryhyperplaneof PG n;q intersects S ineither h 1 or h 2 points.Suchasetisalsosometimescalleda setoftype h 1 ;h 2 . 21
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Fromtheprevioustheorem,an x -tightsetof Q + ;q whosepointsspanPG ;q is atwo-intersectionsetofPG ;q .Thesesetsarerelatedtoawiderangeofother combinatorialobjects.Webeginbydetailingresultsonanimportantclassoflinear codes. An[ n;k ] q code C isa k -dimensionalsubspaceofthevectorspace V = F n q .Vectors in C arecalled codewords ,andthe weight wt v ofacodeword v isthenumberof nonzeroentriesof v .A two-weightcode C isacodewhosecodewordshaveprecisely twononzeroweights.Givenacode C ,wedenethe dualcode C ? = f v 2 V j vc T =0 8 c 2 C g : Wehavethat C ? isan[ n;n )]TJ/F19 11.9552 Tf 11.955 0 Td [(k ] q code. Let C bean[ n;k ] q code;thereexistlinearfunctionals f i : F k q ! F q suchthat C = f f 1 v ;:::;f n v : v 2 F k q g .Since u ; v 7! uv T isanondegeneratebilinear form,thereexist u 1 , ::: , u n 2 F k q suchthat f i v = vu T i forall v 2 F k q .Thus,we havethat C = f vu T 1 ;:::; vu T n j v 2 V g ,andsincedim C = k ,the u i span F k q . Wesay C is projective ifnotwoofthe u i representthesamepointinPG k )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 ;q . Let V nf 0 g .Wesayisa f 1 ; 2 g dierenceset if,forevery v 2 V nf 0 g , thenumberofpairs x ; y 2 2 suchthat x )]TJ/F34 11.9552 Tf 11.979 0 Td [(y = v is 1 if v 2 ,and 2 if v 62 . If )]TJ/F15 11.9552 Tf 9.299 0 Td [(=,wedeneagraph G whoseverticesarethevectorsin V ,with u and v adjacentifandonlyif u )]TJ/F34 11.9552 Tf 11.955 0 Td [(v 2 . Denition2.6 A stronglyregulargraph withparameters v;k;; isaconnected k -regularsimple,undirectedgraphon v vertices,notnullorcomplete,suchthatany twoadjacentverticesshare commonneighbors,andanytwononadjacentvertices share commonneighbors. Wenowgivearesultconnectingtheconceptsoftwointersectionsets,two-weight codes, f 1 ; 2 g dierencesets,andstronglyregulargraphsduetoCalderbankand Kantor[7]. 22
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Theorem2.7 Let V = F n +1 q , O = f y i j 1 i r g beasetofvectorswhichspan V so r n +1 andarepairwiseindependent,andlet = f cy i j c 2 F q g bethesetof nonzeroscalarmultiplesofthe y i ;thenthefollowingstatementsareequivalent: 1. O isasetoftype r )]TJ/F19 11.9552 Tf 11.955 0 Td [(w 1 ;r )]TJ/F19 11.9552 Tf 11.955 0 Td [(w 2 in PG n;q forsome w 1 , w 2 ; 2. C = f x y 1 ;:::;x y k j x 2 V g isaprojectivetwo-weight [ r;n +1] q codewith nonzeroweights w 1 and w 2 ; 3. isa f 1 ; 2 g dierencesetforsome 1 , 2 ; 4. G isastronglyregulargraphwithparameters q n +1 ;r q )]TJ/F15 11.9552 Tf 10.788 0 Td [(1 ;; ,wherefor some w 1 , w 2 wehave = r 2 q )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 +3 r q )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(qw 1 w 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(r q )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 w 1 + w 2 + q 2 w 1 + w 2 and = q 2 w 1 w 2 q n +1 : Thismeansthatif L = f y 1 ;:::;y x q 2 + q +1 g isan x -tightsetof Q + ;q which spansPG ;q ,then 1. L isasetoftype q 2 +1+ x q +1 ;x q +1inPG ;q ; 2.thepointsof L deneaprojectivetwo-weight[ x q 2 + q +1 ; 6] q codewithweights q )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 q 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1and xq 2 ; 3.= f cy i j c 2 F q g isa f 1 ; 2 g dierencesetforsome 1 , 2 ;and 4. G isstronglyregularwithparameters q 6 ;x q 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ;x x )]TJ/F15 11.9552 Tf 11.955 0 Td [(3+ q 3 ;x x )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 : 2.4Trivialexamples Thereareafewexampleswhichtriviallysatisfythenecessaryrequirementstobe aCameron-Lieblerlineclass. 23
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1.Theemptyset ; isaCameron-Lieblerlineclasswithparameter0. 2.Thesetstar p oflinesthroughacommonpoint p ofPG ;q isaCameronLieblerlineclasswithparameter1correspondingtoa1-tightsetof Q + ;q consistingofthesetofpointsinaLatinplane. 3.Thesetline oflinesinaplane ofPG ;q isaCameron-Lieblerlineclass withparameter1correspondingtoa1-tightsetof Q + ;q consistingofthe setofpointsinaGreekplanethisisequivalenttothepreviousexamplein Q + ;q . 4.Thesetstar p [ line ,where isaplaneofPG ;q and p isapointnotin , isaCameron-Lieblerlineclasswithparameter2correspondingtoa2-tightset of Q + ;q whichisaunionoftwodisjointplanesoneLatinandoneGreek. 5.ThecomplementsoftheabovesetsareCameron-Lieblerlineclasseswithparameters q 2 +1, q 2 , q 2 , q 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1respectively. WecalltheCameron-Lieblerlineclassesinthislist trivial . 2.5Non-existenceresults CameronandLieblerconjecturedthattherewerenonontrivialexamplesofthese lineclasses,andprovedthisconjectureforclasseswithparameter 2.Manyother resultsfollowed,leadingtosomeinterestingconnectionswithvariousgeometricobjects. Manyoftheearlynon-existenceresultsreliedstrictlyoncountingarguments; specically,wecanthinkofsetsofthetypestar p orline forapoint p ora line asbeingessentiallythesame,andrefertotheseas cliques .Theequivalent denitionsforaCameron-Lieblerlineclassallowustoperformsomeanalysisonthe potentialintersectionnumberswithrespecttocliquesofahypotheticallineclass withagivenparameter x .Usingthesearguments,Penttila[27]wasabletoruleout 24
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severalparametersinspeciccases,andBruenandDrudge[5]wereabletoruleout theexistenceoflineclasseswithparameter2 2andsomeclique C has x< jLCj x + q then LC formsa blocking set in C inthiscontext,asetoflinesnotcontaininganypencil,suchthatevery pointisonatleastoneofthelines;thenormaldenitionisdualtothis.Blocking setsarewellstudied,andtherearemanyresultsontheirminimumpossiblesize. Thisgivesapowerfultoolforinvestigatingthefeasibilityofcertainparametersfor Cameron-Lieblerlineclasses.Drudgeusedthismethodtoruleoutthecasewhere 2
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sectaparabolic Q ;q embeddedinthequadric.Hewasabletousethistechnique toshowthefollowing: Theorem2.8 ACameron-Lieblerlineclass L withparameter x q existsonlyfor x 2 ,andcorrespondsin Q + ;q totheunionof x skewplanes. Thisshowsthatanynontrivialexamplemusthaveparameter q
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linesofPG ;q ,whichisthenumberoflinesinaCameron-Lieblerlineclasswith parameter 1 2 q 2 +1.Thegoalistoselectthesets L p insuchawaythat L isinfact aCameron-Lieblerlineclass. EveryplaneofPG ;q iseithertangentto O ,andsocontainsauniquepointof O ,orelseintersects O inaconic.Thenontangentplanesectionsof O canbeused toassociatethepointsandnontangentplanesectionsof O withpointsandcirclesof theinversiveplaneIP q [23],sothateachcircleofIP q correspondstoasection of O byanontangentplane containing q +1tangentlinesto O .An intersecting pencil ofcirclesisthesetof q +1circlesthroughtwocommonpointsofIP q ,and a tangentpencil ofcirclesisamaximalsetof q mutuallytangentcirclesonagiven pointofIP q . Anequivalencerelation canbedenedonthecirclesofIP q by C 1 C 2 9C suchthat C istangenttoboth C 1 and C 2 . ThecirclesofIP q fallintopreciselytwoequivalenceclassesunderthisrelation accordingtowhether Q ? isasquareornonsquare,where istheplanecontaining thecircleinquestion.Let A beoneoftheseequivalenceclasses; A containsexactly 1 2 ofthecirclesineachintersectingpencilandeitherallornoneofthecirclesineach tangentpencil.Thusifwedene L p tobethesetoftangentlinesat p containedina planesectionwhichcorrespondstoacirclein A , L p contains 1 2 q +1ofthetangent linesto O at p . BruenandDrudgeshowthat L isaCameron-Lieblerlineclasswithparameter 1 2 q 2 +1byshowingthesetoflinesin L hasacertainmatching"propertywith respecttotheexternallinesto O whicharetheintersectionoftwotangentplanes. 2.6.2PenttilaandGovaertsexamplein PG ; 4 AnotherknownexampleofanontrivialCameron-LieblerwasconstructedbyPenttilaandGovaerts[14].ThisisanexampleinPG ; 4withparameter x =7,and 27
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wastherstknownnontrivialexamplewhen q iseven.Sofartherehasnotbeena generalizationofthisconstruction. Let beaplaneinPG ; 4containingahyperoval O andlet p beapointnot in .Dene C tobetheconewithbase O andvertex p ,with G thesetofgenerators of C , S thesetofsecantsto C whichdonotcontainapointof O ,and E thesetof linesin whichareexternalto O . Theorem2.9 Theset L = G [ S [ E isaCameron-Lieblerlineclasswithparameter 7 . Proof: ThereareseventypesoflinesinPG ; 4withrespecttothecone C and thedistinguishedplane containing O . 1.Generatorsof C ;thisistheset G L . 2.Secantsto C whichareskewto O ;thisistheset S L . 3.Linesin whichareskewto O ;thisistheset E L . 4.Linesthrough p notcontainedin C . 5.Secantsto C whichmeetasinglepointof O . 6.Secantsto O . 7.Linesskewto C whicharenotcontainedin . Thepointsareof5types.Herewecountthenumberoflinesofeachtypethrougha pointofeachtype. 1. f p g ;ofthe21linesthrough p ,6areoftype1,and15areoftype4. 2.Pointson Cn f p g[ ;ofthe21linesthroughsuchapoint,1isoftype1,15 areoftype2,and5areoftype5. 28
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3.Pointson O ;ofthe21linesthroughsuchapoint,1isoftype1,15areoftype 5,and5areoftype6. 4.Pointson nO ;ofthe21linesthroughsuchapoint,9areoftype2,2areof type3,1isoftype4,3areoftype6,and6areoftype7. 5.PointsonPG ; 4 n C[ ;ofthe21linesonsuchapoint,9areoftype2,1 isoftype4,6areoftype5,and5areoftype7. Fromthis,wecancountthatalinein L meets50otherlinesof L ,andalinenotin L meets35linesof L .Thus L isaCameron-Lieblerlineclasswithparameter7. Unfortunatelythisconstructiondoesnotgeneralizetoothervaluesof q inany obviousway,aswedonotgetthecorrectnumberoflinesforaCameron-Lieblerline classunless q =4. 29
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3.Methodology Herewedescribesomealgebraictechniqueswhichwewillusetosearchfornew examplesofCameron-LieblerlineclassesofPG ;q .Wewillsearchfortheseas tightsetsof Q + ;q ;assuch,wewilldevelopamodelofthisquadricwhichwillbe convenientforourcomputationalwork. 3.1Aneigenvectormethodfortightsets OursearchfornewCameron-Lieblerlineclasseswillbeconductedinthecontext ofsearchingfornew x -tightsetsof Q + ;q .Aneigenvectormethodwillbeusedto searchfortheseobjects,whichisduetothefollowingresultofBamberg,Kelly,Law andPenttila[2]. Theorem3.1 Let L beasetofpointsin Q + ;q withcharacteristicvector and let A bethecollinearitymatrixof Q + ;q .Then L isan x -tightsetifandonlyif )]TJ/F19 11.9552 Tf 25.136 8.088 Td [(x q 2 +1 j A = q 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F19 11.9552 Tf 25.135 8.088 Td [(x q 2 +1 j , where j istheall-onesvector. Proof: Bydenition, L isan x -tightsetifandonlyif,for p 2L , p iscollinearwith q 2 )]TJ/F15 11.9552 Tf 10.99 0 Td [(1+ q +1 x otherpointsof Q + ;q and,for p 62L , p iscollinearwith q +1 x pointsof Q + ;q .Thus L isan x -tightsetifandonlyif A = q 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 + x q +1 j : Since j A = q q +1 2 j ,theaboveformulafollowsimmediately. In Q + ;q ,thereexisttwodisjointtotallyisotropicplanes 1 and 2 ;ourgoal istondtightsetswhicharedisjointfrom 1 [ 2 .Theabovemethodwillbe slightlymodiedtoaccountforthis.Wewilllet A 0 bethesubmatrixof A obtained bythrowingawaytherowsandcolumnscorrespondingtopointsin 1 [ 2 . Theorem3.2 Let L beasetofpointsof Q + ;q disjointfrom 1 and 2 andlet 0 bethevectorobtainedfromthecharacteristicvectorof L byremovingentries 30
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correspondingtopointsof 1 and 2 .Then L isan x -tightsetof Q + ;q ifandonly if 0 )]TJ/F19 11.9552 Tf 25.233 8.087 Td [(x q 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 j A 0 = q 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F19 11.9552 Tf 25.232 8.087 Td [(x q 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 j . Proof: Denotetheeigenspaceof A correspondingtotheeigenvalue q 2 )]TJ/F15 11.9552 Tf 12.095 0 Td [(1by E , andtheeigenspaceof A 0 correspondingtotheeigenvalue q 2 )]TJ/F15 11.9552 Tf 10.97 0 Td [(1by E 0 .Since 1 [ 2 isa2-tightset, = 1 [ 2 )]TJ/F15 11.9552 Tf 25.535 8.088 Td [(2 q 2 +1 j 2 E: Let L beasetofpointsof Q + ;q disjointfrom 1 and 2 withcharacteristicvector ;then L isa x -tightsetifandonlyif )]TJ/F19 11.9552 Tf 25.136 8.088 Td [(x q 2 +1 j 2 E v = )]TJ/F19 11.9552 Tf 25.135 8.088 Td [(x q 2 +1 j + x q 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(q 2 E: Theentriesof v correspondingtopointsof 1 [ 2 are0,andtheentrycorresponding toapoint p 62 1 [ 2 isgivenby p )]TJ/F19 11.9552 Tf 25.136 8.087 Td [(x q 2 +1 )]TJ/F19 11.9552 Tf 25.233 8.087 Td [(x q 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 q 2 +1 = p )]TJ/F19 11.9552 Tf 25.233 8.087 Td [(x q 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 : Thusifweobtainanewvector v 0 from v bythrowingawayentriescorrespondingto pointsin 1 [ 2 ,andanewvector 0 from inthesamemanner, v 0 = 0 )]TJ/F19 11.9552 Tf 25.232 8.088 Td [(x q 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 j 2 E 0 )]TJ/F19 11.9552 Tf 25.136 8.088 Td [(x q 2 +1 j 2 E L isan x -tightset. 3.2TacticalDecompositions Foranyincidencestructure,a tacticaldecomposition isapartitionofthepoints intopointclassesandtheblocksintoblockclassessuchthatthenumberofpoints inapointclasswhichlieonablockdependsonlyontheclassinwhichtheblock lies,andsimilarlywithpointsandblocksinterchanged.Examplescanbeobtained bytakingaspointandblockclassestheorbitsofsomecollineationgroupactingon thestructure. 31
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Theideaofatacticaldecompositioncanalsobeextendedtomatrices.Let A =[ a ij ]beamatrix,alongwithapartitionoftherowindicesintosubsets R 1 , ::: , R t andapartitionofthecolumnindicesintosubsets C 1 , ::: , C t 0 .Wewillcallthis a tacticaldecomposition of A ifforevery i;j ,1 i t ,1 j t 0 ,thesubmatrix [ a h;` ] h 2 R i , ` 2 C j hasconstantcolumnsums c ij androwsums r ij .Atactical decompositionofanincidencestructurecorrespondstoatacticaldecompositionof itsincidencematrix.Therowandcolumnsummatricesof A aredenedtotobe R A =[ r ij ]and C A =[ c ij ]respectively. Utilizingatacticaldecompositionmakesndingeigenvectorscorrespondingto x -tightsetseasier,asaneigenvectorofthecolumnsummatrix C A obtainedfromthe decompositioncanbeusedtorecoveraneigenvectorof A .Thefollowingresultcomes fromthetheoryoftheinterlacingofeigenvalues,whichwasintroducedbyHigmanand Sims,usedbyPayneinthestudyofgeneralizedquadrangles,andfurtherdeveloped byHaemers;see[17]foradetailedsurvey. Theorem3.3 Supposethematrix A canbepartitionedas A = 2 6 6 6 6 4 A 11 A 1 k . . . . . . . . . A k 1 A kk 3 7 7 7 7 5 .1 witheach A ii square, 1 i k ,andeach A ij havingconstantcolumnsum c ij ;then anyeigenvalueofthecolumnsummatrix C A =[ c ij ] isalsoaneigenvalueof A . Proof: Aneigenvectorof C A canbeexpandedaccordingtothepartitionof A by duplicatingtheentriescorrespondingtoeachparttoconstructaneigenvectorof A . Toapplythistheoremtothetaskofndingeigenvectorsofthecollinearitymatrix of Q + ;q ,wedeneanincidencestructure H withbothpoints"andblocks"being givenbythepointsof Q + ;q ,andincidencebeinggivenbycollinearity.Thusthe 32
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incidencematrix A of H isgivenbythecollinearitymatrixof Q + ;q .Furthermore, anyautomorphismof Q + ;q determinesanautomorphismof H inanobviousway. Thematrix A issymmetric,andanytacticaldecompositionarisingfromanautomorphismgroupof Q + ;q willinducethesamepartitionontherowsof A andthe columnsof A .Thefollowingtheoremgivesusanicerelationshipbetweentherow andcolumnsumsinarisingfromsuchatacticaldecomposition. Theorem3.4 Let A beasymmetricmatrixandlet O 1 , ::: , O k bethepartsofa tacticaldecompositionof A sotherowandcolumnpartitionisthesamewith j O i j = o i ;then r ij = c ji ,and o i r ij = o j c ij . Proof: If A ij isthesubmatrixassociatedwiththerowpartcorrespondingto O i and thecolumnpartcorrespondingto O j ,then A ij = A T ji ,thus r ij = c ji forall i , j .Also, eachofthe o i rowsof A ij hasrowsum r ij ,andeachofthe o j columnshascolumn sum c ij .Summingoverallentriesof A ij intwowaysgives o i r ij = o j c ij . Corollary3.5 Let A beasymmetricmatrixwithatacticaldecompositionhavingthe samepartsforrowsandcolumns,withpart i containing o i rows/columns;thenwe havethefollowingrelationshipbetweentherowandcolumnsummatrices: R T A =[ r ji ]=[ c ij ]=[ o i o j r ij ]= C A : 3.3Amodelof Q + ;q Wenowdescribeamodelfor Q + ;q whichgivesusarangeofalgebraictools touseinsearchingfortightsets.Let F = F q , E = F q 3 ,and T=T E=F : x 7! x q 2 + x q + x: Weconsider Q + ;q tohave V = E 2 asitsunderlyingvectorspace,consideredover F andequippedwiththequadraticform Q : x;y 7! T xy : 33
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ThepolarformBof Q isthengivenby B u 1 ;u 2 ; v 1 ;v 2 =T u 1 v 2 +T u 2 v 1 : Thisformisnondegenerate,sinceif v 1 ;v 2 2 V hasB v 1 ;v 2 ; x;y =0forall x;y 2 V ,thenT v 1 y +T v 2 x =0forall x;y 2 V .Setting x =0forces T v 1 y = v 1 y + v q 1 y q + v q 2 1 y q 2 =0forall y 2 E , thus v 1 =0.Likewise,setting y =0canbeseentoforce v 2 =0,andso v 1 ;v 2 = ; 0. Itcanbealsobeseenthat 1 = f x; 0: x 2 E g and 2 = f ;y : y 2 E g aretotallyisotropicplaneswithrespecttothisform.Thisshowsthatthequadric denedby Q hasWittindex3,andsoishyperbolic. 3.4Thegeneralmethod Theorem3.6 Let q 6 1mod3 .Take 2 E with j j = q 2 + q +1 ,anddenethe map g on Q + ;q by g : x;y 7! x; )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 y ; thenthegroup C = h g i PGO + ;q andhas j C j = q 2 + q +1 .Thisgroupacts semi-regularlyonthepointsof Q + ;q andstabilizesthetotallyisotropicplanes 1 and 2 . Proof: Itisclearthat g isanisometryof Q + ;q havingorder q 2 + q +1.Tosee that C actssemi-regularlyonthepointsof Q + ;q ,noticethat g i x;y = x;y impliesthat i 2 F .But q 2 + q +1 ;q )]TJ/F15 11.9552 Tf 11.955 0 Td [(1= q )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ; 3=1 ; 34
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since q 6 1mod3.Thusthiscanonlyhappenwhen i =1,andsotheidentityis theonlyelementofthisgroupxingapoint. If isaprimitiveelementof E ,wecanwithoutlossofgeneralityassumethat = q )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 .Thesemi-regularactionof C on Q + ;q givesusthefollowingresult. Theorem3.7 Let A bethecollinearitymatrixof Q + ;q , q 6 1mod3 ,withatacticaldecompositioninducedbytheactionofthecyclicgroup C denedabove;then therowofeachsubmatrixofthedecompositionisthesameasthecolumnsum.Thus thedecompositionmatrixwhichisthesameforrowsumsandcolumnsumsissymmetric. Proof: Thisfollowsdirectlyfrom3.4sinceallorbitshavethesamesize. Sinceeachorbithassize q 2 + q +1,aunionof x orbitscontainstherightnumber ofpointstobean x -tightsetof Q + ;q .Ourgoalwillbetondwaysofcombining theseorbitswhichwillresultinan x -tightset.Weaccomplishthisbyconsideringlarge subgroups G N P )]TJ/F20 7.9701 Tf 5.289 0 Td [(O + ;q C havingrelativelyfeworbitsonthepointsof Q + ;q . Theorbitsofsuchagroupareunionsoforbitsof C .Weusesuchagroup G toinduce atacticaldecompositiononthepointsof Q + ;q ,andthenusethisdecompositionto formthecolumnsummatrix B ofthecollinearitymatrix A ,afterthrowingawaythe entriescorrespondingtopointsin 1 and 2 .Theeigenspaceof B fortheeigenvalue q 2 )]TJ/F15 11.9552 Tf 11.268 0 Td [(1isthensearchedforeigenvectorshavingaformcorrespondingtoan x -tightset of Q + ;q .Whenevernewexamplesshowapattern,e.g.acommonformulafor x intermsof q orasimilarstabilizinggroup,algebraic,geometric,andcombinatorial detailsareanalyzedinanattempttondaconstructionforanewinnitefamilyof tightsets. 35
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4.Newexamples Throughoutthischapter,welet q 6 1mod3, E = F q 3 with E = h i ,and F = F q E with F = h ! i where ! = q 2 + q +1 .Thehyperbolicquadric Q + ;q is denedoverthevectorspace V = E 2 consideredover F andhasquadraticform Q x;y = T xy ,where T = T F=E ,andpolarformB u ; v = T u 1 v 2 + T u 2 v 1 as describedinChapter3.Put = q )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 ,anddenethecyclicgroup C = h g i ,where g : x;y 7! x; )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 y ; then C actssemi-regularlyonthepointsof Q + ;q andstabilizesthedisjointtotally isotropicplanes 1 = f x; 0: x 2 E g and 2 = f ;y : y 2 E g . BelowisasummaryofthenewexamplesofCameron-Lieblerlineclasseswhich aredescribedinthischapter.Noticethathereweconsidertheparameteroftheline classtobesmallerthanthatofitscomplement,andwetakethelineclasstobe disjointfrom 1 [ 2 ;thusanewexamplewithparameter x describedbelowalso givesnewlineclasseswithparameter x +1, x +2, q 2 +1 )]TJ/F19 11.9552 Tf 10.64 0 Td [(x , q 2 )]TJ/F19 11.9552 Tf 10.64 0 Td [(x ,and q 2 )]TJ/F15 11.9552 Tf 10.641 0 Td [(1 )]TJ/F19 11.9552 Tf 10.64 0 Td [(x . x q Aut L 1 2 q 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 q 5or9mod12 Z q 2 + q +1 Z 1 4 q )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 oZ 3 and q< 200 1 3 q +1 2 q 2mod3 Z q 2 + q +1 oZ 3 and q< 150 336 q =27 Z q 2 + q +1 Z 2 oZ 9 495 q =32 Z q 2 + q +1 oZ 15 Table4.1: ParametersandautomorphismgroupsofthenewexamplesofCameronLieblerlineclassesconstructed. 36
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4.1Newexampleswithparameter 1 2 q 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 Herewedescribeaconstructiongivingmanynewexamplesoftightsetsin Q + ;q havingparameter 1 2 q 2 )]TJ/F15 11.9552 Tf 13.062 0 Td [(1.Thisconstructionrequiresustohave q 5or9mod12,andhasresultedinnewtightsetsforallsuch q< 200. 4.1.1Theconstruction Let S = f x : x 2 E j T x =0 g ;thentheorbitsof C onthepointsof Q + ;q are 1 = ; 0 C , 2 = ; 1 C ,and ;x C foreach x 2S .Wealsoletthegroup H = h h i ,where h : x;y 7! x;! 4 y ; actonthespace,andput G = h C;H i . Lemma4.1 Thegroup H denedabovecentralizes C ,andintersects C trivially. Proof: Toshowthat H centralizes C ,weonlyneedtoshowthat g and h commute; wehavethat h g x;y = h x; )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 y = x; )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 ! 4 y and g h x;y = g x;! 4 y = x; )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 ! 4 y : Sincethepowersof arepairwiseindependentover F ,itisclearthat H C contains onlytheidentity. Corollary4.2 Thegroup G denedaboveisequalto C H ,andso j G j = 1 4 q )]TJ/F15 11.9552 Tf -422.702 -23.98 Td [(1 q 2 + q +1 . Furthermore,itcanbeseenthat G actssemiregularlyonthepointsof Q + ;q n 1 [ 2 ,as H actssemi-regularlyonthosepoints,andinducesasemi-regularactionon thoseorbitsof C aswell. 37
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Let S k bethesubsetof S containingtheelementswithlog x k mod4for 0 k 3.For x 2S ,put x = f ! 4 t x :0 t< q )]TJ/F15 11.9552 Tf 12.318 0 Td [(1 = 4 g ;forshorthandwewill write ; x = f ;x 0 : x 0 2 x g .Nowwedene x;y := ;x ? ; y C : Intermsofthetacticaldecompositioninducedby G on Q + ;q , x;y isthenumber ofpointsin ; y C collinearwithanygivenpointof ; x C .Thus x 0 ;y 0 = x;y forall x 0 2 x and y 0 2 y . Let A bethematrixobtainedfromthetacticaldecompositioninducedby G on thecollinearitymatrixof Q + ;q afterthrowingawaytheentriescorrespondingto pointsin 1 [ 2 .Weuseaspecicorderingoftheorbitsof G todene A .Notice that S 0 contains 1 4 q 2 )]TJ/F15 11.9552 Tf 12.137 0 Td [(1elementsof E ,andsocontains q +1equivalenceclasses oftheform x .Let x 0 , ::: , x q berepresentativesfromthese q +1orbits.Weorderthe orbitsas ; x 0 C ;:::; ; x q C ; ;! x 0 C ;:::; ;! x q C ; ;! 2 x 0 C ;:::; ;! 2 x q C ; ;! 3 x 0 C ;:::; ;! 3 x q C : Now A canbedescribedasfollows: A = 2 6 6 6 6 6 6 6 4 A 0 A 1 A 2 A 3 A 3 A 0 A 1 A 2 A 2 A 3 A 0 A 1 A 1 A 2 A 3 A 0 3 7 7 7 7 7 7 7 5 , where A k = )]TJ/F19 11.9552 Tf 5.48 -9.683 Td [( x i ;! k x j 0 i;j q for0 k 3.Thismatrixisblock-circulant,which allowsustoapplythefollowingresultoneigenvectorsofblock-circulantmatricesdue toGarryTee[30]. Theorem4.3 Let beanyfourthrootofunity,and A beablock-circulantmatrix asdenedabove,withblocks A 0 , A 1 , A 2 , A 3 eachhavingsize n= 4 .Takeavector v 2 R n 4 .Thenthevector w =[ v v 2 v 3 v ] 38
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isaneigenvectorof A for ifandonlyif v isaneigenvectorof A 0 + A 1 + 2 A 2 + 3 a 3 for . Wenowinvestigatesomepropertiesof inordertobetterunderstandthestructureof A . Lemma4.4 For x;y 2S notnecessarilydistinct, x;y = y;x . Proof: ThisfollowsdirectlyfromTheorem3.4,alongwiththefactthat ; x C and ; y C arethesamesize. Corollary4.5 A issymmetric;thus A 0 and A 2 aresymmetric,and A 1 = A T 3 . Lemma4.6 For x;y 2 S 0 notnecessarilydistinct, x;! k y = y;! k x for 0 k 3 . Proof: Firstwenoticethat h i contains q 2 + q +1distinctelementsof E ,no twodieringbyamultiplein F .Thus,forany z 2 E ,thereexistsaninteger 0 j
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Fromthiswecanseebyrelabelingindicesthat x;! k y = X 0 t< q )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 = 4 jf i :0 i
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that L isdisjointfromatrivial 2 -tightsetconsistingofaunionoftwoskewtotally isotropicplanes. Proof: Wehavethat i isafourthrootofunity.Thematrix H = A 0 + iA 1 + i 2 A 2 + i 3 A 3 = A 0 + iA 1 )]TJ/F19 11.9552 Tf 11.956 0 Td [(A 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(iA 3 = A 0 )]TJ/F19 11.9552 Tf 11.955 0 Td [(A 2 hasaniceform;allofthediagonalentriesare )]TJ/F15 11.9552 Tf 9.298 0 Td [(1,andallotherentriesare q . Furthermore,thereissomepartitionof f x 0 ;:::;x q g ,intoparts L 1 and L 2 suchthat H ij = q ifandonlyif i 6 = j and x i , x j areinthesamepart;say j L 1 j = a and j L 2 j = b , with a + b = q +1.Ifwetake K tobetheadjacencymatrixofthegraph K L 1 K L 2 where K L 1 and K L 2 arethecompletegraphsonthesets L 1 and L 2 ,respectivelyand K 0 betheadjacencymatrixofthecomplementofthisgraph,then H = qK )]TJ/F19 11.9552 Tf 10.694 0 Td [(qK 0 )]TJ/F19 11.9552 Tf 10.694 0 Td [(I . Wewillformthevector v = L 1 )]TJ/F17 7.9701 Tf 12.685 4.707 Td [(1 2 j = 1 2 L 1 )]TJ/F17 7.9701 Tf 12.686 4.707 Td [(1 2 L 2 .Noticethat x q 2 )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 = 1 2 ifwelet x = q 2 )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 2 .Nowwehavethat L 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [( L 2 H = L 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [( L 2 qK )]TJ/F19 11.9552 Tf 11.955 0 Td [(qK 0 )]TJ/F19 11.9552 Tf 11.955 0 Td [(I = a )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 q L 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(aq L 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [( L 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( b )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 q L 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(bq L 1 )]TJ/F19 11.9552 Tf 11.956 0 Td [( L 2 = a + b )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 q )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 L 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( a + b )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 q )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 L 2 = q 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 L 1 )]TJ/F19 11.9552 Tf 11.955 0 Td [( L 2 , thus v isaneigenvectorof H havingthedesiredform.Wecanuse v toconstructan eigenvectorof A infourdierentways,namely [ v )]TJ/F34 11.9552 Tf 9.299 0 Td [(v )]TJ/F34 11.9552 Tf 9.299 0 Td [(vv ] [ vv )]TJ/F34 11.9552 Tf 9.299 0 Td [(v )]TJ/F34 11.9552 Tf 9.298 0 Td [(v ] eachofwhichinturncanbeusedtoconstructaneigenvectorofthecollinearitymatrix M of Q + ;q correspondingtoa 1 2 q 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1-tightset. 4.1.2Somedetailsoftheseexamples Ourexampleswithparameter 1 2 q 2 )]TJ/F15 11.9552 Tf 12.535 0 Td [(1haveagroupisomorphicto Z q 2 + q +1 Z 1 4 q )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 actingonthem,byconstruction.Byobservingdetailsabouttheorbitsused 41
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intheconstruction,wenoticethattheseexamplesarealsostabilizedbyAut F q 3 , thustheyarestabilizedbyagroupisomorphicto Z q 2 + q +1 Z 1 4 q )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 oZ 3 .Forthose examplessmallenoughtocomputetheirfullstabilizerin P )]TJ/F19 11.9552 Tf 7.314 0 Td [(L ;q ,whicharethose with q 41,thisisinfactthefullgroup. WecanalsocomputeintersectionnumbersoftheselineclassesinPG ;q with respecttoplanesandpointstarsofPG ;q ;thisbecomesprohibitivelyexpensive, computationally,when q> 32.Hereweincludedetailsforsomesmallvaluesof q , aswellas q =81;thisspecialcasewasofparticularinterest,seeChapter5,soa considerableamountoftimewasdedicatedtocomputingthesevalues. Inthistable,weincludetheintersectionnumberswithrespecttoplanesof PG ;q ;theexamplesconsideredhereareallisomorphictotheirdual,andsohave thesameintersectionnumberswiththesamemultiplicitieswithrespecttothepoint starsofPG ;q . q x Intersectionnumbers,withmultiplicity;wehave r = q 2 + q +1 5 12 0 ; 6 r ; 12 r ; 18 r ; 24 r 9 40 0 ; 30 r ; 40 r ; 60 r 17 144 0 ; 108 r ; 126 r ; 144 r ; 180 r ; 198 r 29 420 0 ; 330 r ; 390 r ; 420 r ; 480 r ; 540 r 81 3280 0 ; 2952 r ; 3280 r ; 3690 r Table4.2: Intersectionnumbersoflineclasseswithparameter 1 2 q 2 )]TJ/F15 11.9552 Tf 12.377 0 Td [(1 withthe planesof PG ;q . Thenumbersinboldrepresenttheintersectionnumbersfortheplanecorresponding to 1 in Q + ;q ,whichisdisjointfromourlineclass,andtheplanesthroughthe pointwhichcorrespondsto 2 in Q + ;q ,whichshares 1 2 q 2 )]TJ/F15 11.9552 Tf 11.845 0 Td [(1lineswithourline class.Itisworthnotingthatthenumberoflinessharedbyeachplanewiththeline 42
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classisdivisibleby q +1.Themultiplicitiesbeingdivisibleby q 2 + q +1isaside eectof C actingsemi-regularlyontheplanesofPG ;q notcorrespondingto 1 . Thenewexamplesoftightsetsin Q + ;q giveexamplesoftwo-intersectionsets, two-weightcodes,andstronglyregulargraphs,asdetailedinChapter2.Whilethere aretablesofknownstronglyregulargraphs,theseexampleson q 6 verticesaretoo largetoappear.Furthermore,iftherewereknownexamples,checkingisomorphism wouldmostlikelybeunreasonable. 4.2Newexampleswithparameter 1 3 q +1 2 Herewegivedetailsonmanynewexampleswhichhavebeenconstructedinjoint workwithJanDeBeule,KlausMetsch,andJeroenSchillewaert,havingparameter 1 3 q +1 2 .Theseexampleshavebeenconstructedforallvaluesof q 2mod3 whicharecomputationallyfeasible.Unfortunately,ingeneraltheseexamplesdonot exhibitmuchsymmetry;alloftheexamplesfoundhave C o Aut F q 3 astheirfull stabilizinggroup.When q isprime,thisdoesnotgivemuchtoworkwith.Theseare foundthroughmoreofasearchthanaconstruction;rstweputtogetherthetactical decompositionmatrixfor Q + ;q n 1 [ 2 withrespecttothegroup,thenwesearch overtheeigenspaceforappropriateeigenvectorsseeAppendixAfordetailsabout thealgorithmsused.Withasmallstabilizer,therearelotsoforbits;forexample, if q isprime,thereare 1 3 q 2 )]TJ/F15 11.9552 Tf 12.086 0 Td [(1orbitstoconsider.Assuch,formingthematrixfor thetacticaldecompositionisalargetask.Furthermore,wedonotcurrentlyhavea goodmethodforreducingthesizeoftheeigenspacetosearchover,sondingthese examplesiscomputationallyinfeasibleiftheeigenspaceofthetacticaldecomposition matrixistoolargedimensionsstartstopushthelimitsofourcomputingpower. Animportantsubcaseoftheseexamplesoccurswhen q =2 e ,where e> 1and odd.Inthiscase,wehaveaslightlylargerstabilizinggrouptoworkwith.These examplesarealsoofparticularinterestsincethereisonlyonepreviouslyknown Cameron-LieblerlineclassinPG ;q for q evenseeChapter2forthisconstruc43
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tion.Newexampleswiththisparameterhavebeenfoundfor q 2f 8 ; 32 ; 128 g ,as wellasforoddprimes q 100whicharecongruentto2mod3,andfor q =125.In allofthecaseswhereitisfeasibletocomputethestabilizergroup q 32,wehave that C o Aut F q 3 isthefullgroup.Below,wedescribehowsomeoftheselineclasses intersectplanesofPG ;q .Alloftheexamplesconsideredbelowareisomorphicto theirdual,andsohavethesameintersectionnumberswiththesamemultiplicities withrespecttopointstarsofPG ;q . q x Intersectionnumbers,withmultiplicity;wehave r = q 2 + q +1 5 12 0 ; 6 r ; 12 r ; 18 r ; 24 r 8 27 0 ; 18 r ; 27 r ; 36 r ; 54 r 11 48 0 ; 24 r ; 36 r ; 48 r ; 60 r ; 72 r ; 96 r 17 108 0 ; 72 r ; 90 r ; 108 r ; 126 r ; 144 r ; 216 r 23 192 0 ; 120 r ; 144 r ; 168 r ; 192 r ; 216 r ; 240 r ; 264 r ; 384 r 32 363 0 ; 264 r ; 330 r ; 363 r ; 396 r ; 462 r ; 726 r Table4.3: Intersectionnumbersoflineclasseswithparameter 1 3 q +1 2 withthe planesof PG ;q . 4.3Someothernewexamples Wealsohaveacoupleofothernewexampleswhichdonotcurrentlyseemtot intoanicegrouping.Theseexampleshavebeenfoundbyassumingagroupacting onthepointsof Q + ;q usuallyasubgroupof N O + ;q C ,formingtheorbitsof Q + ;q n 1 [ 2 andtheassociatedmatrixforthetacticaldecomposition,and searchingoverallpossibleparameters.Thenumberofpossibleparameterscanbe verylarge,especiallyiftheorbitsarenotallthesamesize.Itwascomputationally feasiblewhen q 23toassumethattheexampleswewerelookingforadmittedthe 44
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group C o Aut F q 3 = F q asastabilizer,thoughthesesearchesdidnotyieldanynew examples. For q =27,thevariationintheorbitsizesgivesalargenumberofpossible parameters,andthereisarelativelylargeeigenspacetoconsider.Thus,considering asmallstabilizinggroupwasnotfeasible.Byassumingthegroup C o Aut F q 3 stabilizedtheexamples,wewereabletondanewtightsetwithparameter336. Thisexampleisalsostabilizedbythemap x;y 7! x; )]TJ/F19 11.9552 Tf 9.298 0 Td [(y ,andsohasfullstabilizer isomorphicto Z q 2 + q +1 Z 2 oZ 9 .Restrictingourrstsearchwith C o Aut F q 3 = F q byassumingourparameterwasdivisibleby q +1,wefoundnoothernewexamples. With q =32,allofthepointorbitsofthegroup C o Aut F q 3 on Q + ;q have thesamesize,sothesearchisfeasibleassumingthisstabilizer.Weareabletond twonewexampleshavingparameter495,eachhaving C o Aut F 32 3 = F 2 astheirfull stabilizer.Inthiscase,thesetwoexamplesareisomorphicastightsets,butnotas Cameron-Lieblerlineclasses.InPG ;q ,theyaredualtooneanother. BelowwedetailhowtheseexamplesintersecttheplanesofPG ;q .Notethat onlytherstexampleisself-dual;inthiscase,theintersectionnumbersandmultiplicitiesforpointstarsofPG ;q arethesameasforplanes.Forthetwoexamples with q =32,theplaneintersectionnumbersofoneexamplearethepointstarintersectionnumbersfortheother,andvice-versa. q x Intersectionnumbers,withmultiplicity;wehave r = q 2 + q +1 27 336 0 ; 252 r ; 336 r ; 420 r ; 504 r 32 495 0 ; 330 r ; 396 r ; 462 r ; 495 r ; 528 r ; 594 r 32 495 0 ; 396 r ; 495 r ; 528 r ; 660 r Table4.4: Intersectionnumbersofsomeothernewlineclasses. 45
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5.Planartwo-intersectionsets Asetoftype m;n inaprojectiveoraneplaneisaset K ofpointssuchthat everylineoftheplanecontainseither m or n pointsof K ;werequirethat m
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K ,andeachofthe t n n -secantscontains n n )]TJ/F15 11.9552 Tf 12.316 0 Td [(1suchpairs.Eachofthe k k )]TJ/F15 11.9552 Tf 12.316 0 Td [(1 orderedpairsofpointsin K iscountedonceinthismanner. Corollary5.2 Ifwehaveaset K oftype m;n inaprojectiveplaneoforder q , then k = jKj mustsatisfy k 2 )]TJ/F19 11.9552 Tf 11.956 0 Td [(k q n + m )]TJ/F15 11.9552 Tf 11.956 0 Td [(1+ n + m + mn q 2 + q +1=0 : .4 Ifwetakeaxedpoint p 2K ,andlet m and n bethenumberof m -secants and n -secantsthrough p ,respectively,weseethat m + n = q +1and m )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 m + n )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 n = k )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 : Fromthis,weseethat m = n q +1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(k )]TJ/F19 11.9552 Tf 11.956 0 Td [(q = n )]TJ/F19 11.9552 Tf 11.955 0 Td [(m and n = k + q )]TJ/F19 11.9552 Tf 11.955 0 Td [(m q +1 = n )]TJ/F19 11.9552 Tf 11.956 0 Td [(m ; andso m and n donotdependonourchoiceof p .Likewise,ifwetakeaxedpoint q 62K ,andlet m and n bethenumberof m -secantsand n -secantsthrough q ,we seethat m + n = q +1and m m + n n = k: Again,thesevaluescanbeseentobeindependentofourchoiceof q ;wehavethat m = n q +1 )]TJ/F19 11.9552 Tf 11.956 0 Td [(k = n )]TJ/F19 11.9552 Tf 11.955 0 Td [(m and n = k )]TJ/F19 11.9552 Tf 11.956 0 Td [(m q +1 = n )]TJ/F19 11.9552 Tf 11.955 0 Td [(m : Fromthesenumbersweseethat,givenasetoftype m;n inaprojectiveplaneof order q ,wecanconstructthreeotherrelatedsetswithtwointersectionnumbersfor aproof,see[18]. 47
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Theorem5.3 Let K beasetoftype m;n inaprojectiveplane oforder q ,with jKj = k . 1.Thecomplementof K isasetoftype q +1 )]TJ/F19 11.9552 Tf 12.488 0 Td [(n;q +1 )]TJ/F19 11.9552 Tf 12.487 0 Td [(m in containing q 2 + q +1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(k points. 2.Thesetof m -secantsto K isasetoftype m ; m inthedualplaneto containing t m points. 3.Thesetof n -secantsto K isasetoftype m ; m inthedualplaneto containing t n points. Noticethat n )]TJ/F19 11.9552 Tf 12.004 0 Td [( n = q= n )]TJ/F19 11.9552 Tf 12.004 0 Td [(m andsoitisnecessaryfor n )]TJ/F19 11.9552 Tf 12.004 0 Td [(m todivide q .If n )]TJ/F19 11.9552 Tf 12.262 0 Td [(m = q ,then K canbeseentobeeitherthesetofpointsonacommonline,or thecomplementofthis;theexampleshaving n )]TJ/F19 11.9552 Tf 12.405 0 Td [(m =1aredualtothese,andwe considertheexamplesineitherofthesesituationstobetrivial. Onemajorclassofexamplesarethesetsoftype ;n .Theseexamplesarealso knownasmaximalarcsofdegree n ,oras qn )]TJ/F19 11.9552 Tf 12.535 0 Td [(q + n;n -arcsastheynecessarily contain qn )]TJ/F19 11.9552 Tf 10.924 0 Td [(q + n points.Theprototypicalexamplesaregivenbyhyperovals,which aresetsoftype ; 2;otherfamiliesofmaximalarcsofdegreelargerthan2have beendescribedbyDenniston[11],Thas[31][32],andMathon[21].Maximalarcsof degree2 a areknowntoexistinPG ; 2 e forallpairs a;e with0
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Lemma5.4 Let K beasetoftype m;n inananeplane oforder q .Let t m and t n bethenumberor m -secantsand n -secantsto K ,andlet k = jKj ;then t m + t n = q 2 + q; .5 mt m + nt n = k q +1 ; and .6 m m )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 t m + n n )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 t n = k k )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 : .7 ThesemodiedformulasleadtothefollowingalternateversionofCorollary5.2. Corollary5.5 Ifwehaveaset K oftype m;n inananeplaneoforder q ,then k = jKj mustsatisfy k 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(k q n + m )]TJ/F15 11.9552 Tf 11.955 0 Td [(1+ n + m + mnq q +1=0 : Weagaingetconstantvalues m and n forthenumberof m -secantsand n -secants throughapointin K ,and m and n forthenumberof m -secantsand n -secants throughapointnotin K ,givenbytheformulas m = n q +1 )]TJ/F19 11.9552 Tf 11.956 0 Td [(k )]TJ/F19 11.9552 Tf 11.955 0 Td [(q = n )]TJ/F19 11.9552 Tf 11.955 0 Td [(m ; n = k + q )]TJ/F19 11.9552 Tf 11.955 0 Td [(m q +1 = n )]TJ/F19 11.9552 Tf 11.955 0 Td [(m ; m = n q +1 )]TJ/F19 11.9552 Tf 11.956 0 Td [(k = n )]TJ/F19 11.9552 Tf 11.955 0 Td [(m ; and n = k )]TJ/F19 11.9552 Tf 11.955 0 Td [(m q +1 = n )]TJ/F19 11.9552 Tf 11.955 0 Td [(m : Thistellsusthat,asfortheprojectivecase,wemusthave n )]TJ/F19 11.9552 Tf 10.524 0 Td [(m dividing q .However, sincethedualofananeplaneisnotagainananeplane,wedonothaveresults aboutthe m -secantsor n -secantsforminganotherplanarsetwithtwointersection numbers. Thereareveryfewknownexamplesofsetsoftype m;n inaneplanes.For planesofevenorder,wecanobtainanexamplefromasetoftype ;n inaprojective plane,bychoosinganexternallinetothesetasthelineatinnitytoformtheane plane.However,setsofthistypedonotexistinprojectiveplanesofoddorder. 49
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Inaneplanesofoddorder,theonlypreviouslyknownexamplesofsetsoftype m;n aresetsoftype ; 6inplanesoforder9.Thesesetswerefoundthroughan exhaustivecomputersearch,see[28],andexampleswerefoundineachofthefour projectiveplanesoforder9. Thesize k ofasetoftype ; 6inaplaneoforder9mustsatisfy k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(81 k +1620=0 whichhassolutions k 1 =36and k 2 =45.Thecomplementofasetoftype ; 6ina planeoforder9willagainbeasetoftype ; 6,andthecomplementofasetofsize k 1 willcontain k 2 points.The45-setsoftype ; 6have 3 =2, 6 =8, 3 =5,and 6 =5. 5.3ConstructionsfromCameron-Lieblerlineclasses Wenowdescribeamethodofconstructingsomeoftheknownsetsoftype ; 6 inAG ; 9startingwithaCameron-Lieblerlineclasswithparameter40inPG ; 9. WethengeneralizethismethodtogiveanewexampleinAG ; 81. 5.3.1Atwo-intersectionsetin AG ; 9 TakeaCameron-Lieblerlineclass L 1 ofparameter40inPG ; 9,asconstructed intheChapter4.ThissetoflinesisdisjointfromatrivialCameron-Lieblerlineclass withparameter2,whichwewillconsidertobestar p [ line ,where p isapoint inPG ;q and isaplanenotcontaining p .Thislineclassinducesasymmetric tacticaldecompositiononPG ; 9havingfourclassesofpointsandlines,asfollows: thefourlineclassesare 1.star p , 2.line , 3. L 1 ,and 4. L 2 =linePG ;q n line [ star p [L 1 . 50
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EachpointofPG ;q n f p g[ liesoneither30or60linesof L 1 seeTable4.2, andsothefourpointclassesofthetacticaldecompositionare 1. f p g , 2. , 3. P 1 = f u 2 PG ;q :star u L 1 =30 g ,and 4. P 2 = f v 2 PG ;q :star v L 1 =60 g . Thenumbersoflinesineachgivenlineclassthroughaxedpointineachgivenpoint classcanbefoundinTable5.1,andthenumbersofpointsineachgivenpointclass onaxedlineineachgivenlineclasscanbefoundinTable5.2. star p line L 1 L 2 f p g 91000 1104040 P 1 103060 P 2 106030 Table5.1: Linesperpointforthesymmetrictacticaldecompositioninducedon PG ; 9 byaCameron-Lieblerlineclassofparameter 40 . star p line L 1 L 2 f p g 1000 11011 P 1 4036 P 2 4063 Table5.2: Pointsperlineforthesymmetrictacticaldecompositioninducedon PG ; 9 byaCameron-Lieblerlineclassofparameter 40 . 51
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Now,ifwetakeaplane 0 ofPG ; 9notequalto andnotcontaining p , 0 containspreciselyonelineofline andnolinesofstar p .Furthermore, 0 will containeither30or60linesof L 1 ,andso60or30linesof L 2 seeTable4.2.Without lossofgenerality,wemayassumethat 0 contains30linesof L 1 and60linesof L 2 . Asforthevariouspointclasses, 0 doesnotcontain p ,andcontains10pointsof , allonacommonline.Underourassumptions, 0 alsocontains30pointsof P 1 and 60pointsof P 2 .Infact,wehaveasymmetrictacticaldecompositionof 0 having3 classesonpointsandlinesinducedbyourtacticaldecompositionofthelargerspace. Bytaking 0 tobethelineatinnityandremovingitalongwithallofitspoints from 0 ,weobtainananeplaneAG ; 9.Allofthepointsofthisaneplaneare in P 1 or P 2 ,andallofthelinesarein L 1 or L 2 .Itcanbeeasilyveriedthat P 1 isasetoftype ; 6inthisplanecontaining30points.Thissetadmitsastabilizer isomorphicto Z 3 .Asthesetsoftype m;n inAG ; 9werecompletelyclassiedin [28],thissetisnotnew. 5.3.2Anewtwo-intersectionsetin AG ; 81 WearealsoabletofollowtheaboveprocedurewithaCameron-Lieblerlineclass L 1 ofparameter3280inPG ; 81constructedasinChapter4.Inthiscase,the lineclassesareformedasbefore.Asforthepointclasses,eachpointofPG ; 81 isoneither2952linesof L 1 ,oron3690pointsof L 1 .Wedene P 1 tobetheset ofpointson2952linesof L 1 .Thesepointandlineclassesgiveasymmetrictactical decompositionofPG ; 81;thenumbersoflinesineachgivenlineclassthrougha xedpointineachgivenpointclasscanbefoundinTable5.3,andthenumbersof pointsineachgivenpointclassonaxedlineineachgivenlineclasscanbefound inTable5.4. Welet 0 beaplaneofPG ; 81notequalto ,andnotcontaining p .Then 0 containsonelineofline andnolinesofstar p .Also, 0 willcontaineither2952 linesof L 1 or3690linesof L 1 seeTable4.2;withoutlossofgenerality,assume 0 52
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star p line L 1 L 2 f p g 6643000 18232803280 P 1 1029523690 P 2 1036902952 Table5.3: Linesperpointforthesymmetrictacticaldecompositioninducedon PG ; 81 byaCameron-Lieblerlineclassofparameter 3280 . star p line L 1 L 2 f p g 1000 18211 P 1 4003645 P 2 4004536 Table5.4: Pointsperlineforthesymmetrictacticaldecompositioninducedon PG ; 81 byaCameron-Lieblerlineclassofparameter 3280 . contains2952linesof L 1 .Thepointsetof 0 isagaindisjointfrom f p g ,andcontains 82pointsof ,allonacommonline.Bytakingthisline,whichis 0 ,tobethe lineatinnityandremovingitalongwithallofitspointsfrom 0 ,weobtainanane planeAG ; 81.Allofthepointsofthisaneplanearein P 1 or P 2 ,andallofthe linesarein L 1 or L 2 .Itisclearthat P 1 isasetoftype ; 45containing2952 points.Therearenopreviouslyknownexamplesofsetsoftype m;n inAG ; 81, sothisexampleisnew.UsingMagma,thestabilizerofthissetiscomputedandis isomorphicto Z 6 . 5.3.3Afamilyofexamplesin AG ; 3 2 e ? ThecombinatoricsofourCameron-Lieblerlineclassesofparameter 1 2 q 2 )]TJ/F15 11.9552 Tf 12.891 0 Td [(1 seemtobeespeciallyniceovereldsoforder3 2 e ,inducingasymmetrictactical decompositiononthespacehavingfourclassesoflinesandofpoints.Afuture 53
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directionofresearchrelatedtothisobservationistofocusonprovingtheexistence ofaninnitefamilyofCameron-Lieblerlineclasseshavingthisparameterinthe speciccasewhere q =3 2 e ,andexaminingtheintersectionnumberswithrespect totheplanesandpointstarsofPG ;q .Iftheselineclassesalwaysinducesucha tacticaldecomposition,withoneoftheclassesbeingaplaneandanotherapointstar, thenwewillbeabletoconstructaninnitefamilyofsetsoftype m;n inAG ; 3 2 e . AssumewehaveaCameron-Lieblerlineclass L 1 withparameter 1 2 6 e )]TJ/F15 11.9552 Tf 12.584 0 Td [(1in PG ; 3 2 e whichisdisjointfromatrivialCameron-Lieblerlineclassstar p [ line withparameter2so p 62 ,andthatthislineclassinducesasymmetrictactical decompositionofPG ;q asabovehavingpointclasses f p g , , P 1 , P 2 ,andline classesstar p ,line , L 1 , L 2 .Takeaplane 0 distinctfrom andnotcontaining p . Thepointsintheaneplane 0 n 0 areallin P 1 or P 2 ,andthelinesareallin L 1 or L 2 .Ifwelet K = P 1 0 ,thenthelinesoftheaneplanehavepreciselytwo intersectionnumberswith K dependingonwhethertheyarein L 1 or L 2 .Without lossofgeneralitywewillassumethateachlineof L 1 star 0 meets m pointsof K . Let A and B besuchthateachpointin P 1 ison A linesof L 1 and B linesof L 2 ;thus eachpointin P 2 ison B linesof L 1 and A linesof L 2 . Themostlikelypossibilityfor n )]TJ/F19 11.9552 Tf 11.843 0 Td [(m ,andthesituationforourearlierexamples, isthat n )]TJ/F19 11.9552 Tf 12.012 0 Td [(m =3 e .Assumethatthisisthecase,sothat n = m +3 e .ByDenition 2.1,wehave 1 2 6 e )]TJ/F15 11.9552 Tf 11.955 0 Td [(1+ 2 e +1 m = jL 1 0 j + A and 1 2 6 e )]TJ/F15 11.9552 Tf 11.955 0 Td [(1+ 2 e +1 m = jL 2 0 j + A; byapplyingtheresultto L 1 usingtheincidentpoint-planepair u ; 0 with u 2P 1 , andto L 2 usingtheincidentpoint-planepair v ; 0 with v 2P 2 .Since m )]TJ/F19 11.9552 Tf 10.403 0 Td [( m =3 e , weseethat jL 2 0 j)-222(jL 1 0 j = 2 e +1 )]TJ/F19 11.9552 Tf 11.955 0 Td [( = 2 e +1 e 54
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which,alongwiththefactthat jL 2 0 j + jL 1 0 j =3 4 e +3 2 e = 2 e +1 2 e ,tellsusthat jL 1 0 j = 1 2 2 e )]TJ/F15 11.9552 Tf 11.956 0 Td [(3 e 2 e +1and jL 2 0 j = 1 2 2 e +3 e 2 e +1 : Thisallowsustosolvefor m = 1 2 2 e )]TJ/F15 11.9552 Tf 11.955 0 Td [(3 e and n = 1 2 2 e +3 e : Conjecture5.6 Forany e 1 ,thereexistsetsoftype 1 2 2 e )]TJ/F15 11.9552 Tf 12.16 0 Td [(3 e ; 1 2 2 e +3 e in AG ; 3 2 e . Ourhopeisthat,inthefuture,wewillbeabletoprovethatwehaveaninnite familyofCameron-LieblerlineclassesinPG ; 3 2 e whichinducetacticaldecompositionsofthespace,allowingustoshowtheexistenceofthesetwo-intersectionsets. 55
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APPENDIXA.Algorithms Herewedetailsomeofthealgorithmsthatfacilitateourndings. A.1CLautMatrix Wehave { C isourcyclicgroupoforder q 2 + q +1whoseorbitson Q + ;q n 1 [ 2 arerepresentedbyelementsof OR EP = f x : x 2 F q 3 j T x =0 g considered asanorderedsetforconsistency. { Z 1 =Aut F q 3 = F p and Z 2 = F q ;thesegroups,consideredon Q + ;q , normalize C ,soweonlyconsidertheiractionon OR EP . Z 1 isassumedto stabilizeourtightsetbut Z 2 isnot. { X BLOCK and XO aretheorbitsof Z 1 and Z 2 ,respectively;theelementsof eachorbitarealsoordered. Wanttostore,foreach x 2 OR EP ,indexedpairs < i 1 , j 1 > and < i 2 , j 2 > sothat x = X BLOCK [ i 1 ][ j 1 ]= XO [ i 2 ][ j 2 ]. Wenowformaset R suchthat,foreach x 2 OR EP ,wehaveaunique r 2 R such that ! i r p k = x forsome i;k . Formthestructure S =[ s r ],where s r =[ j ;r ? ;x C j : x 2 OR EP ]. Formthearray O1P ,where O1P [ i ][ j ]= k ,where k issuchthat ! j r i 2 X BLOCK [ k ]. Eachrowandcolumnofourmatrix A correspondstoanorbiton Q + ;q n 1 [ 2 under G = < C , Z 1 > .Weconsiderthemtobeorderedaccordingto theorderingoftheorbits X BLOCK of Z 1 .Wend A ij asfollows: 56
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{ Let a;b besuchthat ! a r b 2 Z 1 [ i ]. { Then A ij isformedbysummingover s r b [ k ]as k rangesoverthevalues satisfying O1P [ a ][ k ]= j . Inotherwords,foreach Z 1 [ i ],wecanndan x in Z 1 [ i ],an r in R ,andaninteger a suchthat x = ! a r .Weknowhowmanyelementsof ;y C arecollinearwith ;r , soweconsiderhowthemap y 7! ! a y on OR EP permutestheassociatedorbitsof C , andconsiderwhichoftheseorbitshavetheirrepresentativesin Z 1 [ j ],andaddthem up. 57
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APPENDIXB.Programs Wehavestructuredthecodetoexaminetightsetsof Q + ;q inamodular fashion.Thatis,wehaveashellprogramthatcontainstheparameterswemaywish tomodify,andfromthereweloadthelescontainingspecicmethods,andthen callthespecicfunctionsweareinterestedin.Hereisthecodeforourbasicshell program;wecommentoutanypartswearenotinterestedinbeforesubmittingthe jobtothecluster.Anyrestrictionsfortheparameters,eitherfromtherequirements ofthealgorithmsorforthesakeofcomputationalspeed,willbementionedwhenwe discusstheindividualpiecesofthecode. B.1CLshell.mgm Wehavevariables p , h ,and t whichcanbemodied; p doesnotneedtobeprime. Theideaisthatwewillhave q = p h ,and =Au F p 3 h =F p willbeassumedtoactona t -tightsetfoundbythesearch.Weusuallydene t intermsof q ,butwemusttake carethat t isaninteger. p := 81; h := 1; CLpreamble.mgm setsupourbasicinfrastructure. load “ CLpreamble.mgm â€; t := R ATIONALS ! 1 / 2 q 2 )]TJ/F41 10.8792 Tf 9.299 0 Td [(1 ; CLaut.mgm holdsthebasicsearchalgorithm;when h =1,itwillndanyofour exampleshavingparameter t ,althoughitisveryslow.Usinglargervaluesof h while leaving q xedmakesthingsmuchfaster,butwillonlyndexamplesstabilizedby thislargergroup. load “ CLaut.mgm â€; FindL t , L ; 58
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CLbcirc.mgm isspecializedfor h =1, p 1mod4,and t = = 2 q 2 )]TJ/F15 11.9552 Tf 12.047 0 Td [(1.Itisvery fastandmemoryecient. load “ CLbcirc.mgm â€; WehavethatLcontainsthesetoforbitrepresentativesforeachtightsetfound. print “ Therewere â€,# L , “ lineclassesfoundwithparameter â€, t ; CLvspace.mgm containsdenitionsforthevectorspaces V = E 2 , W = F 6 ,andthe map : V ! W .Thismapusesabasisfor W whichgivesthestandardorthogonal form.Thevectorspace U = F 4 isalsodened,alongwithmaps : U ! W and : W ! U whichmapapointtoalineofPG ;q viatheKleincorrespondence. load “ CLvspace.mgm â€; CLpg3q.mgm denesthefunction LU ,whichmapsasetoftracezeroelementsof E tothesetoflinesofPG ;q correspondingtotheirorbitsunder C .Alsodenesthe groups GL = P )]TJ/F19 11.9552 Tf 7.314 0 Td [(L ;q and CU ' C actingonlinesofPG ;q . load “ CLpg3q.mgm â€; S := S TABILIZER GL ,LU L [ 1 ] ; print “ ThestabilizerofLisasfollows:n â€, S ; CLint.mgm isusedtocomputeintersectionnumbers.Theorbitrepresentativesare expandedtothefullpointsetthrouth LW ,andintersectionnumberswithstarsand planesarecomputedusing intPlane and intStar ,respectively. load “ CLint.mgm â€; LL := LW L [ 1 ] ; print “ TheintersectionnumberswithmultiplicityofL withpointstarsofthespaceareasfollows:n â€, intStar LL ; print “ Theintersectionnumberswithrespectto 59
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planesofthespaceareasfollows:n â€, intPlane LL ; CLint81.mgm givesanalternatewaytocomputeintersectionnumbers.Itismuch slower,butmorememoryecient.Weuseitforthecasewhere p =81,sincethe computationsareimpossibleotherwise. load “ CL81int.mgm â€; print “ TheintersectionnumberswithmultiplicityofL withpointstarsofthespaceareasfollows:n â€, intStar LW L [ 1 ] ; print “ Theintersectionnumberswithrespectto planesofthespaceareasfollows:n â€, intStar LWd L [ 1 ] ; MNset.mgm willgivetheintersectionofthesetinPG ;q withaplanepiPrime asdescribedinChapter5.Theset K willalsobegiven,whichshouldbeatwointersectionsetofAG2 ;q when q =3 2 e . load “ MNsetTH2.mgm â€; K := MN L [ 1 ] ; S := KStab K ; print “ ThestabilizerofKisasfollows:n â€, S ; B.2CLpreamble.mgm Thiscodeisrequiredforeverythingthatfollows. Welet F = F q and E = F q 3 ,withprimitiveelements ! and ,respectively,and view V as E 2 ,withbilinearform x;y 7! T xy ,where T =Tr E=F . q := p h ; := q 2 )]TJ/F41 10.8792 Tf 9.298 0 Td [(1 ; 60
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F := F INITE F IELD q ; if not I S P RIMITIVE ! then ! := P RIMITIVE E LEMENT F ; endif ; E <> := ext < F j 3 > ; T := func < x j T RACE E ! x , F > ; Itisusefultohavetheseordered.Thisorderingmakes Fstar [ i ]= ! i +1 ,likewise Estar [ i ]= i +1 Fstar := f @ ! i : i in [ 0 :: q )]TJ/F41 10.8792 Tf 9.298 0 Td [(2 ] @ g ; Estar := f @ i : i in [ 0 :: q 3 )]TJ/F41 10.8792 Tf 9.298 0 Td [(2 ] @ g ; isaelementof E withorder r = q 2 + q +1. := q )]TJ/F41 8.7034 Tf 9.299 0 Td [(1 ; r := q 2 + q + 1; ORep isthesetofnonzeroelementswithtrace0;llstheroleof S inthethesis. ORep := f @ x : x in Estar j T x eq 0@ g ; L isaplaceholdersequence;itwillcontainsubsetsof ORep correspondingto t -tight setsofthequadric. L :=[ P OWER S ET ORep j ] ; Insteadofdeningourcyclicgroupactingon V = E 2 ,wedene C tobeapermutation groupon Estar ,generatedby C : 1 : x 7! x . C := P ERMUTATION G ROUP < Estar j [ x : x in Estar ] > ; Wedenethegroup Z 1 =Aut E= F p ;tosavememory,wedenethisgroupactingon justthetracezeroelements ORep Estar .Thisrequiressomeconsiderationtothe orderinwhichweapplygroupstolookatorbits. Gr := S YM ORep ; Z 1 := sub < Gr j [ x p : x in ORep ] > ; 61
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o := Z 1 : 1; Thegroup Z 2 generatedby z : x 7! ! x permutestheorbitsof C ;theorbitsof thisgrouponthetracezeroelementsof Estar areusedtoecientlyformthetactical decomposition. Z 2 := sub < Gr j [ ! x : x in ORep ] > ; z := Z 2 : 1; B.3CLaut.mgm Themethodusedhereisthemostgeneralsearchtechnique.Formingthematrix forthetacticaldecompositioncanbeverycomputationallyexpensive. Xblock := O RBITS Z 1 ; XO := O RBITS Z 2 ; n := # Xblock ; Thefollowingrecordformatisusedtokeeptrackofimportantinformationaboutthe tracezeroelementsof Estar ,suchastheirlocationinthecyclesinducedbyspecic groupelementsonrepresentativetracezeroelements.SeeAppendixAfordetails. TZ ERO := recformat < rep : ORep , O 1 : car < f i : i in [ 1 :: # Xblock ] g , f j : j in [ 1 :: 3 h ] g > , O 2 : car < f i : i in [ 1 :: # XO ] g , f j : j in [ 1 :: q )]TJ/F41 10.8792 Tf 9.298 0 Td [(1 ] g >> ; TZ :=[ rec < TZ ERO j rep := ORep [ i ] > : i in [ 1 :: # ORep ]] ; for i in [ 1 :: # Xblock ] do for j in [ 1 :: # Xblock [ i ]] do TZ [ I NDEX ORep , Xblock [ i ][ j ]] O 1 := < i , j > ; endfor ; endfor ; for i in [ 1 :: # XO ] do 62
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for j in [ 1 :: # XO [ i ]] do TZ [ I NDEX ORep , XO [ i ][ j ]] O 2 := < i , j > ; endfor ; endfor ; Itisecienttosorttherecordsaccordingtothesizeoftheorbitofthetracezero elementunder Z 1 ;wemaintainseparatesequencesofthetracezeroelements,andthe recordsassociatedwiththem,eachinthesameorder.Thememoryusedtostorethis redundantinformationismadeupforinthetimesavedaccessingtheinformation. forward m ; S ORT TZ , func < x , y j # Xblock [ x O 1 [ 1 ]] )]TJ/F41 10.8792 Tf 13.686 0 Td [(# Xblock [ y O 1 [ 1 ]] > , m ; ORep := f @ ORep [ i m ]: i in [ 1 :: # ORep ] @ g ; Fordetailsontheformationof A ,seeAppendixA. O2block := f @ f @ TZ [ j ] O 1 [ 1 ] : j in [ I NDEX ORep , x : x in XO [ TZ [ i ] O 2 [ 1 ]]] @ g : i in [ 1 :: # ORep ] @ g ; R := f @I NDEX ORep , Xblock [ O2block [ i ][ 1 ]][ 1 ]: i in [ 1 :: # O2block ] @ g ; Cy :=[[ T RACE E ! x , F : x in C YCLE C : 1, XO [ i ][ 1 ]]: i in [ 1 :: # XO ]] ; yC :=[ R OTATE R EVERSE Cy [ i ] ,1 : i in [ 1 :: # Cy ]] ; OO2 :=[[ Fstar [ x [ 2 ]] yC [ x [ 1 ]][ j ]: j in [ 1 :: # Cy [ x [ 1 ]]]] where x := TZ [ R [ i ]] O 2 : i in [ 1 :: # R ]] ; M 2 := func < i , j j # f k : k in [ 1 :: r ] j OO2 [ i ][ k ]+ Fstar [ x [ 2 ]] Cy [ x [ 1 ]][ k ] eq 0 where x := TZ [ j ] O 2 g > ; S :=[ P OWER S EQUENCE R ATIONALS j ] ; for i in [ 1 :: # R ] do s :=[ M 2 i , j : j in [ 1 :: # ORep ]] ; A PPEND S , s ; 63
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endfor ; O1P :=[[ TZ [ I NDEX ORep , x ORep [ j ]] O 1 [ 1 ]: j in [ 1 :: # ORep ]]: x in Fstar ] ; M := function i , j d := exists k , s f < m , n > : m in [ 1 :: # R ] , n in [ 1 :: # Fstar ] j TZ [ I NDEX ORep , ORep [ R [ m ]] Fstar [ n ]] O 1 [ 1 ] eq i g ; return & +[ S [ k ][ x ]: x in [ 1 :: # O1P [ s ]] j O1P [ s ][ x ] eq j ] ; endfunction ; A := S YMMETRIC M ATRIX R ATIONALS ,& cat [[ M i , j : i in [ 1 :: j ]]: j in [ 1 :: n ]] )]TJ/F41 10.8792 Tf 13.686 0 Td [(S CALAR M ATRIX R ATIONALS , n ,1 ; Thisloopadjuststhevaluesofthematrixiftheorbitsarenotallthesamesize,which occurswhen q 0mod3orwhen3divides h . s :=[ # Xblock [ i ]: i in [ 1 :: # Xblock ]] ; for i in [ 1 :: n )]TJ/F41 10.8792 Tf 9.299 0 Td [(1 ] do for j in [ i + 1 :: n ] do A [ i , j ]:= A [ i , j ]/ s [ j ]/ s [ i ] ; endfor ; endfor ; Wenowndtheeigenspacefor ,anddeneafunction FindL whichwillsearchfor atightsetwithagivenparameter t andreturnthesetsoforbitrepresentativestrace zeroelementsof Estar correspondingtoanytightsetsfound. Ba := B ASIS E IGENSPACE A , ; FindL := procedure t , L := R ATIONALS ! t / q 2 )]TJ/F41 10.8792 Tf 9.299 0 Td [(1 ; e := 1 )]TJ/F19 11.9552 Tf 9.298 0 Td [( ; f := )]TJ/F19 11.9552 Tf 9.298 0 Td [( ; 64
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Sincethebasisvectorsfortheeigenspacefor arenormalized,avectorwithallentries equalto e or f mustbealinearcombinationofthesebasisvectorswithallweights equalto e or f ;wesearchoverallsuchlinearcombinationsfortightsets. for c in C ARTESIAN P OWER f 0,1 g ,# Ba do s :=[ c [ i ] eq 1 select e else f : i in [ 1 :: # Ba ]] ; v := & +[ s [ i ] Ba [ i ]: i in [ 1 :: # Ba ]] ; ifforall f i : i in [ 1 :: n ] j v [ i ] in f e , f gg then A PPEND L ,& join f Xblock [ i ]: i in [ 1 :: n ] j v [ i ] eq e g ; print v ; endif ; endfor ; endprocedure ; B.4CLbcirc.mgm Thiscodesearchesfortightsetsof Q + ;q asdescribedinChapter4when q 1mod4,byformingablockcirculantmatrixforthetacticaldecomposition.The requirementsfortheparametersintheshellareasfollows: p 1mod4 ; e =1 ; and t =Rationals = 2 q 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 : WemustalsohaveloadedtheCLpreamble.mgmle. Wedenethesubgroup Z 3 = h z 4 i Z 2 ;groupsaredenedtoactonthetrace zeroelementsof E tosavememory. Z 3 := sub < Z 2 j z 4 > ; Xblock containstheorbitsonthetracezeroelementsunder h Z 1 , Z 3 i .Weorder theseorbitsaccordingtowhetherlog x 0 ; 1 ; 2 ; 3mod4,asdescribedinChapter4. 65
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Thisway, XO [ i ][ k ]istheorbitgivenby ! k )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 XO [ i ][ 1 ]for1 k 4,where XO [ i ][ 1 ] isanorbitcontainingelementswithlog x 0mod4. Xblock := O RBITS sub < Gr j Z 1 , Z 3 > ; n := # Xblock ; S ORT Xblock , func < x , y j # x )]TJ/F41 10.8792 Tf 13.686 0 Td [(# y > ; S ORT Xblock , func < x , y j L OG x [ 1 ] mod 4 )]TJ/F15 11.9552 Tf 13.686 0 Td [( L OG y [ 1 ] mod 4 > ; XO := f @ Xblock [ i ] Z 2 : i in [ 1 :: n ] @ g ; n := n div 4; Thefollowingalgorithmisusedtogeneratethematrixcorrespondingtothetactical decompositionof Q + ;q inducedbythegroup h C , Z 1 , Z 3 i ;seethedetailsin AppendixA. Cy :=[[[ T RACE E ! x , F : x in C YCLE C : 1, y ]: y in XO [ i ][ 1 ]]: i in [ 1 :: # XO ]] ; OO2 :=[ R OTATE R EVERSE Cy [ i ][ 1 ] ,1 : i in [ 1 :: n ]] ; M := func < i , j , k j & +[ # f z : z in [ 1 :: r ] j OO2 [ i ][ z ]+ Fstar [ k ] Cy [ j ][ x ][ z ] eq 0 g : x in [ 1 :: # Cy [ j ]]] > ; Ablock :=[ S YMMETRIC M ATRIX R ATIONALS ,& cat [[ M i , j , k : i in [ 1 :: j ]]: j in [ 1 :: n ]] : k in [ 1 :: 3 ]] ; if q mod 3 eq 0 then s := 3; for i in [ 1 :: 3 ] do for j in [ 2 :: n ] do Ablock [ i ][ 1, j ]:= Ablock [ i ][ 1, j ]/ s ; endfor ; endfor ; 66
PAGE 75
endif ; Ablock [ 1 ]:= Ablock [ 1 ] )]TJ/F41 10.8792 Tf 13.686 0 Td [(S CALAR M ATRIX R ATIONALS , n ,1 ; Ablock [ 4 ]:= Ablock [ 2 ] ; Theeigenvectorsoftheblocksymmetricmatrixarenowconstructedoverthecyclotomiceld R [ ]:= Q [ i ].Weareonlyinterestedinrealvaluedeigenvectors.For theseexamples,forallvaluesof q whicharefeasiblecomputationally,wegetaonedimensionaleigenspaceof H [ 1 ]= Ablock [ 1 ] )]TJ/F40 10.8792 Tf 9.299 0 Td [(Ablock [ 3 ],givingeigenvectorswhichcorrespondto = 2 q 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1-tightsetsinessentiallyfourequivalentways. R <> := C YCLOTOMIC F IELD 4 ; H :=[ & +[ j i )]TJ/F41 8.7034 Tf 9.299 0 Td [(1 Ablock [ i ]: i in [ 1 :: 4 ]]: j in [ 1 :: 4 ]] ; H [ 1 ] ; v := B ASIS E IGENSPACE H [ 1 ] , q 2 )]TJ/F41 10.8792 Tf 9.299 0 Td [(1 [ 1 ] ; A := H ORIZONTAL J OIN [ V ERTICAL J OIN [ Ablock [ 1 + i )]TJ/F40 10.8792 Tf 9.299 0 Td [(j mod 4 ] : i in [ 0 :: 3 ]]: j in [ 0 :: 3 ]] ; Thisloopputstogetherasetoftheorbitrepresentativesforeachofthefour t -tight sets.Needsomeexplanationofthemethod,whichisbasedonconjecture.The sequence L consistsofasequenceofsetsoforbitrepresentatives,eachcorresponding toa t -tightset. := R ATIONALS ! t / q 2 )]TJ/F41 10.8792 Tf 9.299 0 Td [(1 ; e := 1 )]TJ/F19 11.9552 Tf 9.298 0 Td [( ; f := )]TJ/F19 11.9552 Tf 9.298 0 Td [( ; Sp := & cat [[ < j , k > : k in [ 1 :: n ]]: j in [ 1 :: 4 ]] ; v := V ECTOR [ R ATIONALS j I NTEGERS ! Ablock [ 2 ][ i ,1 ] mod 2 )]TJ/F19 11.9552 Tf 13.686 0 Td [( : i in [ 1 :: n ]] ; for c in C ARTESIAN P OWER f 1,2 g ,2 do u := V ECTOR & cat [ E LTSEQ )]TJ/F41 10.8792 Tf 9.299 0 Td [(1 c [ 1 ] v , 67
PAGE 76
E LTSEQ )]TJ/F41 10.8792 Tf 9.298 0 Td [(1 c [ 2 ] v , E LTSEQ )]TJ/F41 10.8792 Tf 9.298 0 Td [(1 c [ 1 ]+ 1 v , E LTSEQ )]TJ/F41 10.8792 Tf 9.298 0 Td [(1 c [ 2 ]+ 1 v ] ; if u A eq u then A PPEND L ,& join f XO [ Sp [ i ][ 2 ]][ Sp [ i ][ 1 ]] : i in [ 1 :: # Xblock ] j u [ i ] eq e g ; print u ; endif ; endfor ; B.5CLvspace.mgm Thiscodeincludesdenitionsofvariousvectorspaces,aswellasmapstoimplementtheKleincorrespondence.Itisrequiredforallofthecodeintheproceeding sections. Webeginbydeningourvectorspaces V and W ,andourbilinearform. V := V ECTOR S PACE E ,2 ; W , := V ECTOR S PACE V , F ; B := func < u , v j T u [ 1 ] v [ 2 ]+ T u [ 2 ] v [ 1 ] > ; Q := func < v j T v [ 1 ] v [ 2 ] > ; BW := func < u , v j B )]TJ/F41 8.7034 Tf 9.298 0 Td [(1 u , )]TJ/F41 8.7034 Tf 9.298 0 Td [(1 v > ; QW := func < v j Q )]TJ/F41 8.7034 Tf 9.299 0 Td [(1 v > ; OForm := M ATRIX F , [[ BW B ASIS W [ i ] ,B ASIS W [ j ] : i in [ 1 :: 6 ]]: j in [ 1 :: 6 ]] ; Wenowndabasisfor W underwhichwecanusethestandardPluckercoordinates fortheKleincorrespondence. Ba :=[ B ASIS W [ 1 ]] ; 68
PAGE 77
Ba := A PPEND Ba , a where a := rep f x : x in W j QW x eq 0 and BW Ba [ 1 ] , x ne 0 g ; for i in [ 1 :: 3 ] do Ba [ 2 i )]TJ/F41 10.8792 Tf 9.299 0 Td [(1 ]:= BW Ba [ 2 i )]TJ/F41 10.8792 Tf 9.299 0 Td [(1 ] , Ba [ 2 i ] )]TJ/F41 8.7034 Tf 9.299 0 Td [(1 Ba [ 2 i )]TJ/F41 10.8792 Tf 9.298 0 Td [(1 ] ; Ba [ 2 i ]:= QW Ba [ 2 i ] Ba [ 2 i )]TJ/F41 10.8792 Tf 9.298 0 Td [(1 ]+ Ba [ 2 i ] ; if i ne 3 then Ba := A PPEND Ba , a where a := rep f x : x in N ULLSPACE OForm T RANSPOSE M ATRIX [ Ba [ k ]: k in [ 1 :: 2 i ]] j x ne 0 and QW x eq 0 g ; Ba := A PPEND Ba , a where a := rep f x : x in N ULLSPACE OForm T RANSPOSE M ATRIX [ Ba [ k ]: k in [ 1 :: 2 i ]] j x ne 0 and QW x eq 0 and BW Ba [ 2 i + 1 ] , x ne 0 g ; endif ; endfor ; Ba :=[ Ba [ 1 ] , Ba [ 3 ] , Ba [ 5 ] , Ba [ 6 ] , Ba [ 4 ] , Ba [ 2 ]] ; WBa := M ATRIX F ,6,6, Ba )]TJ/F41 8.7034 Tf 9.299 0 Td [(1 ; Weredene tomap V ! W insuchawaythatwecanusethestandardorthogonal formforthequadric. := map < V ! W j v 7! v WBa , w 7! )]TJ/F41 8.7034 Tf 9.298 0 Td [(1 w WBa )]TJ/F41 8.7034 Tf 9.298 0 Td [(1 > ; WForm := M ATRIX F , [[ BW Ba [ i ] , Ba [ j ]: i in [ 1 :: 6 ]]: j in [ 1 :: 6 ]] ; BW := func < u , v j B )]TJ/F41 8.7034 Tf 9.299 0 Td [(1 u , )]TJ/F41 8.7034 Tf 9.298 0 Td [(1 v > ; QW := func < v j Q )]TJ/F41 8.7034 Tf 9.299 0 Td [(1 v > ; Wenowdenethevectorspace U whichwillunderliePG3 ;q ; : U ! W and : W ! U givetheKleincorrespondence. 69
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U := V ECTOR S PACE F ,4 ; := func < x j R OWSPACE M ATRIX F ,4,4, [[ 0, x [ 1 ] , x [ 2 ] , x [ 3 ]] , [ )]TJ/F40 10.8792 Tf 9.299 0 Td [(x [ 1 ] ,0, x [ 4 ] , )]TJ/F40 10.8792 Tf 9.299 0 Td [(x [ 5 ]] , [ )]TJ/F40 10.8792 Tf 9.299 0 Td [(x [ 2 ] , )]TJ/F40 10.8792 Tf 9.298 0 Td [(x [ 4 ] ,0, x [ 6 ]] , [ )]TJ/F40 10.8792 Tf 9.299 0 Td [(x [ 3 ] , x [ 5 ] , )]TJ/F40 10.8792 Tf 9.298 0 Td [(x [ 6 ] ,0 ]] > ; p K := func < line , j , k j D ETERMINANT M ATRIX F ,2,2, [[ B ASIS line [ 1 ][ j + 1 ] ,B ASIS line [ 1 ][ k + 1 ]] , [ B ASIS line [ 2 ][ j + 1 ] ,B ASIS line [ 2 ][ k + 1 ]]] > ; := func < line j B ASIS sub < W j [ p K line ,0,1 , p K line ,0,2 , p K line ,0,3 , p K line ,1,2 , p K line ,3,1 , p K line ,2,3 ] > [ 1 ] > ; B.6CLpg3q.mgm Thiscoderequires CLvspace.mgm inordertorun. WesetupthegroupactingonPG ;q ,andit'sactiononthelines. G , P := PG AMMA L U ; PointsU := func < line j f I NDEX P ,N ORMALIZE B ASIS line [ 2 ] g join f I NDEX P ,N ORMALIZE B ASIS line [ 1 ]+ x B ASIS line [ 2 ] : x in F g > ; Lines := PointsU sub < U j P [ 1 ] , P [ 2 ] > G ; Lines := GS ET G , Lines ; , GL := A CTION G , Lines ; HerewedenetheactionofourcyclicgrouponPG ;q . CU := )]TJ/F41 8.7034 Tf 9.298 0 Td [(1 sub < GL j [ PointsU V ! [ v [ 1 ] ne 0 select v [ 1 ] C : 1 else 0, v [ 2 ] ne 0 select v [ 2 ] C : )]TJ/F41 8.7034 Tf 9.299 0 Td [(1 else 0 ] 70
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where v is )]TJ/F41 8.7034 Tf 9.298 0 Td [(1 sub < U j P [ l [ 1 ]] , P [ l [ 2 ]] > where l isrep f < a , b > : a , b in Lines [ i ] j a ne b g : i in [ 1 :: # Lines ]] > ; Thefunction LU mapsourorbitrepresentativestoasetoflinesinPG ;q . LU := func < L j & join f @ PointsU V ! [ 1, x ] CU : x in L @ g > ; B.7CLint.mgm Thiscoderequires CLvspace.mgm inordertorun. H , N := PGOP LUS W ; Hprime := S UBGROUPS H : I NDEX E QUAL := 2 [ 1 ] subgroup ; CW := sub < H j [ I NDEX N ,N ORMALIZE V ! [ z [ 1 ] , )]TJ/F41 8.7034 Tf 9.299 0 Td [(1 z [ 2 ]] where z is )]TJ/F41 8.7034 Tf 9.299 0 Td [(1 x : x in N ] > ; LW := func < L j & join f I NDEX N ,N ORMALIZE V ! [ 1, x ] CW : x in L g > ; intStar := function WCL 1 := f I NDEX N ,N ORMALIZE V ! [ x ,0 ]: x in Estar g ; Stars := 1 Hprime ; return f # WCL meet : in Stars g ; endfunction ; intPlane := function WCL 2 := f I NDEX N ,N ORMALIZE V ! [ 0, y ]: y in Estar g ; Planes := 2 Hprime ; return f # WCL meet : in Planes g ; endfunction ; B.8CL81int.mgm 71
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Thiscoderequires CLvspace.mgm inordertorun. G , P := PG AMMA L U ; aC :=[ k : k in [ 1 :: r ]] ; Ca := R EVERSE aC ; R OTATE aC ,1 ; CW := func < x jf @ V ! [ aC [ i ] , Ca [ i ] x ]: i in [ 1 :: r ] @ g > ; ULines := P ARENT V ! [ 1, ORep [ 1 ]] WBa ; LW := function LRep LL :=[ ULines j ] ; for x in LRep do for v in CW x do A PPEND LL , v ; endfor ; endfor ; return LL ; endfunction ; LWd := function LRep LL :=[ ULines j ] ; for x in LRep do for v in CW x do A PPEND LL , V ! [ v [ 2 ] , v [ 1 ]] ; endfor ; endfor ; return LL ; endfunction ; 72
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intStar := function UCL INT := f I NTEGERS jg ; for x in P do xINT := # f v : v in UCL j x in v g ; I NCLUDE INT , xINT ; endfor ; return INT ; endfunction ; intPlane := function UCL INT := f I NTEGERS jg ; N V :=[ N ULLSPACE T RANSPOSE M ATRIX [ B ASIS v [ 1 ] ,B ASIS v [ 2 ]] : v in UCL ] ; for x in P do xINT := # f i : i in [ 1 :: #N V ] j x in N V [ i ] g ; I NCLUDE INT , xINT ; endfor ; return INT ; endfunction ; B.9MNset.mgm Thiscoderequires CLvspace.mgm inordertorun. aC :=[ k : k in [ 1 :: r ]] ; Ca := R EVERSE aC ; R OTATE aC ,1 ; CW := func < x jf @ V ! [ aC [ i ] , Ca [ i ] x ]: i in [ 1 :: r ] @ g > ; ULines := P ARENT V ! [ 1, ORep [ 1 ]] WBa ; 73
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:= N ULLSPACE T RANSPOSE M ATRIX U ! [ 1,0,0,0 ] ; piW := sub < W j sub < U j B ASIS [ 1 ] ,B ASIS [ 2 ] > , sub < U j B ASIS [ 1 ] ,B ASIS [ 3 ] > , sub < U j B ASIS [ 2 ] ,B ASIS [ 3 ] > > ; LWpi := function LRep LL :=[ ULines j ] ; for x in LRep do for v in CW x do if v in piW then A PPEND LL , v ; endif ; endfor ; endfor ; return LL ; endfunction ; MN := function UCL UCLpi := LWpi UCL ; l := f x : x in j # f v : v in UCLpi j x in v g eq # UCLpi div q + 1 g ; l := sub < j l > ; Bp := E XTEND B ASIS l , ; Bp :=[ Bp [ 3 ] , Bp [ 1 ] , Bp [ 2 ]] ; H , N := PGL 3, q ; mn := f # f v : v in UCLpi j x in v g : x in g ; m := M IN mn ; 74
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K := f N ORMALIZE V ECTOR F ,C OORDINATES V ECTOR S PACE W ITH B ASIS Bp , x : x in j # f v : v in UCLpi j x in v g eq m g ; K := f [ x [ 2 ] , x [ 3 ]]: x in K g ; return K ; endfunction ; KStab := function mnSet H , J := AG AMMA L 2, q ; K := f I NDEX J , x : x in mnSet g ; S := S TABILIZER H , K ; return S ; endfunction ; 75
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REFERENCES [1]S.Ball,A.Blokhuis,andF.Mazzocca.Maximalarcsindesarguesianplanesof oddorderdonotexist. Combinatorica ,17:31{41,1997.10.1007/BF01196129. [2]J.Bamberg,S.Kelly,M.Law,andT.Penttila.Tightsetsand m -ovoidsofnite polarspaces. JournalofCombinatorialTheory,SeriesA ,114:1293{1314, 2007. [3]J.Bamberg,M.Law,andT.Penttila.Tightsetsand m -ovoidsofgeneralised quadrangles. Combinatorica ,29:1{17,2009. [4]W.Bosma,J.Cannon,andC.Playoust.TheMagmaalgebrasystem.I.Theuser language. J.SymbolicComput. ,24-4:235{265,1997.Computationalalgebra andnumbertheoryLondon,1993. [5]A.A.BruenandK.Drudge.Onthenon-existenceofcertainCameron{Liebler lineclassesinPG ;q . Designs,CodesandCryptography ,14:127{132,1998. [6]A.A.BruenandK.Drudge.TheconstructionofCameron{Lieblerlineclassesin PG ;q . FiniteFieldsandTheirApplications ,5:35{45,1999. [7]R.CalderbankandW.M.Kantor.Thegeometryoftwo-weightcodes. Bulletin oftheLondonMathematicalSociety ,18:97{122,1986. [8]P.J.CameronandR.A.Liebler.Tacticaldecompositionsandorbitsofprojective groups. LinearAlgebraanditsApplications ,46:91{102,1982. [9]J.DeBeule,P.Govaerts,A.Hallez,andL.Storme.Tightsets,weighted m covers,weighted m -ovoids,andminihypers. Designs,CodesandCryptography , 50:187{201,2009. [10]J.DeBeule,A.Hallez,andL.Storme.Anon-existenceresultonCameron{ Lieblerlineclasses. JournalofCombinatorialDesigns ,16:342{349,2008. [11]R.H.F.Denniston.Somemaximalarcsinniteprojectiveplanes. Journalof CombinatorialTheory ,6:317{319,1969. [12]K.Drudge. Extremalsetsinprojectiveandpolarspaces .PhDthesis,University ofWesternOntario,1998. [13]K.Drudge.OnaconjectureofCameronandLiebler. EuropeanJournalof Combinatorics ,20:263{269,1999. [14]P.GovaertsandT.Penttila.Cameron{LieblerlineclassesinPG ; 4. Bulletin oftheBelgianMathematicalSociety{SimonStevin ,12:793{804,2005. 76
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[15]P.GovaertsandL.Storme.OnCameron{Lieblerlineclasses. Advancesin Geometry ,4:279{286,2004. [16]L.C.Grove. ClassicalGroupsandGeometricAlgebra .AmericanMathematical Society,2002. [17]W.H.Haemers.Interlacingeigenvaluesandgraphs. LinearAlgebraanditsApplications ,226{228:593{616,1995. [18]J.Hirschfeld. ProjectiveGeometriesoverFiniteFieldsOxfordMathematical Monographs .OxfordUniversityPress,USA,2edition,1998. [19]D.R.HughesandF.C.Piper. ProjectivePlanes .Springer-VerlagNewYork,, 1973. [20]R.LidlandH.Niederreiter. FiniteFieldsEncyclopediaofMathematicsandits Applications .CambridgeUniversityPress,1996. [21]R.Mathon.Newmaximalarcsindesarguesianplanes. JournalofCombinatorial Theory,SeriesA ,97:353{368,2002. [22]K.Metsch.Thenon-existenceofCameron{Lieblerlineclasseswithparameter 2
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[31]J.Thas.Constructionofmaximalarcsandpartialgeometries. Geometriae Dedicata ,3:61{64,1974. [32]J.Thas.Constructionofmaximalarcsanddualovalsintranslationplanes. EuropeanJournalofCombinatorics ,1:189{192,1980. [33]J.Tits. BuildingsofSphericalTypeandFinite BN -pairs .Springer-Verlag,1974. 78
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# How to draw a simple cone with height and radius with TikZ?
I need to make a very simple cone with h and r (like the picture I've uploaded as example) and I can't find it in previous questions.
With arc
\documentclass[tikz,border=10pt]{standalone}
\usetikzlibrary{calc}
\begin{document}
\begin{tikzpicture}
\draw[dashed] (0,0) arc (170:10:2cm and 0.4cm)coordinate[pos=0] (a);
\draw (0,0) arc (-170:-10:2cm and 0.4cm)coordinate (b);
\draw[densely dashed] ([yshift=4cm]$(a)!0.5!(b)$) -- node[right,font=\footnotesize] {$h$}coordinate[pos=0.95] (aa)($(a)!0.5!(b)$)
-- node[above,font=\footnotesize] {$r$}coordinate[pos=0.1] (bb) (b);
\draw (aa) -| (bb);
\draw (a) -- ([yshift=4cm]$(a)!0.5!(b)$) -- (b);
\end{tikzpicture}
\end{document}
With ellipse
\documentclass[tikz,border=10pt]{standalone}
\usetikzlibrary{calc}
\begin{document}
\begin{tikzpicture}
\begin{scope}
\clip (-2,0) rectangle (2,1cm);
\draw[dashed] (0,0) circle(2cm and 0.35cm);
\end{scope}
\begin{scope}
\clip (-2,0) rectangle (2,-1cm);
\draw (0,0) circle(2cm and 0.35cm);
\end{scope}
\draw[densely dashed] (0,4) -- node[right,font=\footnotesize] {$h$}coordinate[pos=0.95] (aa)(0,0)
-- node[above,font=\footnotesize] {$r$}coordinate[pos=0.1] (bb) (2,0);
\draw (aa) -| (bb);
\draw (-2,0) -- (0,4) -- (2,0);
\end{tikzpicture}
\end{document}
Gonzalo has kindly provided the shadings to the cylinder and I am reproducing his code (with thanks):
\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
\fill[
top color=gray!50,
bottom color=gray!10,
opacity=0.25
]
(0,0) circle (2cm and 0.5cm);
\fill[
left color=gray!50!black,
right color=gray!50!black,
middle color=gray!50,
opacity=0.25
]
(2,0) -- (0,6) -- (-2,0) arc (180:360:2cm and 0.5cm);
\draw
(-2,0) arc (180:360:2cm and 0.5cm) -- (0,6) -- cycle;
\draw[dashed]
(-2,0) arc (180:0:2cm and 0.5cm);
\draw[dashed]
(2,0) -- node[below] {$r$} (0,0) -- node[left] {h} (0,6) ;
\draw
(0,8pt) -- ++(8pt,0) -- (8pt,0);
\end{tikzpicture}
\end{document}
• Since you answered providing two options, perhaps you could add a new option with shadings as in my answer? I see no point in giving almost the same code in two different answers. – Gonzalo Medina Apr 13 '14 at 1:47
• @GonzaloMedina Thanks and I used your code as such as I felt that it is good. thank you. – user11232 Apr 13 '14 at 15:12
• There is an unpleasant mistake at the base since the sides should be tangent to the bottom ellipse and in this case they are not. Drawing a larger the problem gets bigger. – ThePunisher Sep 29 '16 at 16:31
This requires a mathematician. You need to calculate the points of tangency and then connect the vertex of the cone to those points, not to the ends of the major axis. If the ellipse has its major axis from (-a,0) to (a,0), its minor axis from (0,-b) to (b,0), and its vertex at (0,h) (with h>b), then one point of tangency is (a*sqrt(1-(b/h)^2), b*(b/h)) and the other is the same but with its x-coordinate negated.
Here is MetaPost code to draw the cone correctly:
beginfig(1)
a:=2in; b:=.5in; h:= 3in; % for example
draw fullcircle xscaled 2a yscaled 2b; % a x b ellipse
pair Z[];
Z2 := (0,h); % vertex
Z1 := (a*sqrt(1 - (b/h)*(b/h)),b*(b/h)); % right tangency
Z3 := (-xpart Z1, ypart Z1); % left tangency
draw Z1--Z2--Z3;
endfig;
end
I don't use TikZ, so I will let others provide a translation if necessary. (And provide for the dashed portions.)
Dan's solution translated to TikZ:
\documentclass[tikz,border=2mm]{standalone}
\usetikzlibrary{positioning, calc}
\begin{document}
\begin{tikzpicture}
\newcommand{\height}{6}
\draw[fill=gray!30] (a)--(0,\height)--(b)--cycle;
\begin{scope}
\clip ([xshift=-2mm]a) rectangle ($(b)+(1mm,-2*\radiusy)$);
\end{scope}
\begin{scope}
\clip ([xshift=-2mm]a) rectangle ($(b)+(1mm,2*\radiusy)$);
\end{scope}
\draw[dashed] (0,\height)|-(\radiusx,0) node[right, pos=.25]{$h$} node[above,pos=.75]{$r$};
\draw (0,.15)-|(.15,0);
\end{tikzpicture}
\end{document}
• I tried \newcommand{\height}{1}, the circle of base very bad. – minhthien_2016 Feb 5 '19 at 8:51
Just for fun with PSTricks.
\documentclass[pstricks,border=12pt,12pt]{standalone}
\usepackage{pst-node}
\begin{document}
\begin{pspicture}[dimen=m](8,10)
\psellipticarc[linestyle=dashed](4,1)(4,.65){0}{180}
\psellipticarcn(4,1)(4,.65){0}{180}
\psline[linecap=0](0,1)(4,10)(8,1)
\pcline[linestyle=dashed](4,10)(4,1)\naput{$h$}
\pcline[linestyle=dashed](4,1)(8,1)\naput{$r$}
\rput(4,1){\psline(0,9pt)(9pt,9pt)(9pt,0)}
\end{pspicture}
\end{document}
• There is an unpleasant mistake at the basis since the sides should be tangent to the bottom ellipse and in this case they are not. Drawing a larger the problem gets bigger. – ThePunisher Sep 29 '16 at 16:30
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Outlook: Kimco Realty Corporation Class L Depositary Shares each of which represents a one-one thousandth fractional interest in a share of 5.125% Class L Cumulative Redeemable Preferred Stock liquidation preference \$25000.00 per share is assigned short-term Ba1 & long-term Ba1 estimated rating.
Dominant Strategy : Hold
Time series to forecast n: 09 Mar 2023 for (n+3 month)
Methodology : Reinforcement Machine Learning (ML)
## Abstract
Kimco Realty Corporation Class L Depositary Shares each of which represents a one-one thousandth fractional interest in a share of 5.125% Class L Cumulative Redeemable Preferred Stock liquidation preference \$25000.00 per share prediction model is evaluated with Reinforcement Machine Learning (ML) and Sign Test1,2,3,4 and it is concluded that the KIM^L stock is predictable in the short/long term. According to price forecasts for (n+3 month) period, the dominant strategy among neural network is: Hold
## Key Points
3. Why do we need predictive models?
## KIM^L Target Price Prediction Modeling Methodology
We consider Kimco Realty Corporation Class L Depositary Shares each of which represents a one-one thousandth fractional interest in a share of 5.125% Class L Cumulative Redeemable Preferred Stock liquidation preference \$25000.00 per share Decision Process with Reinforcement Machine Learning (ML) where A is the set of discrete actions of KIM^L stock holders, F is the set of discrete states, P : S × F × S → R is the transition probability distribution, R : S × F → R is the reaction function, and γ ∈ [0, 1] is a move factor for expectation.1,2,3,4
F(Sign Test)5,6,7= $\begin{array}{cccc}{p}_{a1}& {p}_{a2}& \dots & {p}_{1n}\\ & ⋮\\ {p}_{j1}& {p}_{j2}& \dots & {p}_{jn}\\ & ⋮\\ {p}_{k1}& {p}_{k2}& \dots & {p}_{kn}\\ & ⋮\\ {p}_{n1}& {p}_{n2}& \dots & {p}_{nn}\end{array}$ X R(Reinforcement Machine Learning (ML)) X S(n):→ (n+3 month) $∑ i = 1 n s i$
n:Time series to forecast
p:Price signals of KIM^L stock
j:Nash equilibria (Neural Network)
k:Dominated move
a:Best response for target price
For further technical information as per how our model work we invite you to visit the article below:
How do AC Investment Research machine learning (predictive) algorithms actually work?
## KIM^L Stock Forecast (Buy or Sell) for (n+3 month)
Sample Set: Neural Network
Stock/Index: KIM^L Kimco Realty Corporation Class L Depositary Shares each of which represents a one-one thousandth fractional interest in a share of 5.125% Class L Cumulative Redeemable Preferred Stock liquidation preference \$25000.00 per share
Time series to forecast n: 09 Mar 2023 for (n+3 month)
According to price forecasts for (n+3 month) period, the dominant strategy among neural network is: Hold
X axis: *Likelihood% (The higher the percentage value, the more likely the event will occur.)
Y axis: *Potential Impact% (The higher the percentage value, the more likely the price will deviate.)
Z axis (Grey to Black): *Technical Analysis%
## IFRS Reconciliation Adjustments for Kimco Realty Corporation Class L Depositary Shares each of which represents a one-one thousandth fractional interest in a share of 5.125% Class L Cumulative Redeemable Preferred Stock liquidation preference \$25000.00 per share
1. A single hedging instrument may be designated as a hedging instrument of more than one type of risk, provided that there is a specific designation of the hedging instrument and of the different risk positions as hedged items. Those hedged items can be in different hedging relationships.
2. Because the hedge accounting model is based on a general notion of offset between gains and losses on the hedging instrument and the hedged item, hedge effectiveness is determined not only by the economic relationship between those items (ie the changes in their underlyings) but also by the effect of credit risk on the value of both the hedging instrument and the hedged item. The effect of credit risk means that even if there is an economic relationship between the hedging instrument and the hedged item, the level of offset might become erratic. This can result from a change in the credit risk of either the hedging instrument or the hedged item that is of such a magnitude that the credit risk dominates the value changes that result from the economic relationship (ie the effect of the changes in the underlyings). A level of magnitude that gives rise to dominance is one that would result in the loss (or gain) from credit risk frustrating the effect of changes in the underlyings on the value of the hedging instrument or the hedged item, even if those changes were significant.
3. At the date of initial application, an entity shall determine whether the treatment in paragraph 5.7.7 would create or enlarge an accounting mismatch in profit or loss on the basis of the facts and circumstances that exist at the date of initial application. This Standard shall be applied retrospectively on the basis of that determination.
4. An embedded prepayment option in an interest-only or principal-only strip is closely related to the host contract provided the host contract (i) initially resulted from separating the right to receive contractual cash flows of a financial instrument that, in and of itself, did not contain an embedded derivative, and (ii) does not contain any terms not present in the original host debt contract.
*International Financial Reporting Standards (IFRS) adjustment process involves reviewing the company's financial statements and identifying any differences between the company's current accounting practices and the requirements of the IFRS. If there are any such differences, neural network makes adjustments to financial statements to bring them into compliance with the IFRS.
## Conclusions
Kimco Realty Corporation Class L Depositary Shares each of which represents a one-one thousandth fractional interest in a share of 5.125% Class L Cumulative Redeemable Preferred Stock liquidation preference \$25000.00 per share is assigned short-term Ba1 & long-term Ba1 estimated rating. Kimco Realty Corporation Class L Depositary Shares each of which represents a one-one thousandth fractional interest in a share of 5.125% Class L Cumulative Redeemable Preferred Stock liquidation preference \$25000.00 per share prediction model is evaluated with Reinforcement Machine Learning (ML) and Sign Test1,2,3,4 and it is concluded that the KIM^L stock is predictable in the short/long term. According to price forecasts for (n+3 month) period, the dominant strategy among neural network is: Hold
### KIM^L Kimco Realty Corporation Class L Depositary Shares each of which represents a one-one thousandth fractional interest in a share of 5.125% Class L Cumulative Redeemable Preferred Stock liquidation preference \$25000.00 per share Financial Analysis*
Rating Short-Term Long-Term Senior
Outlook*Ba1Ba1
Income StatementCBaa2
Balance SheetCB3
Leverage RatiosCaa2Ba1
Cash FlowCB1
Rates of Return and ProfitabilityBaa2Ba3
*Financial analysis is the process of evaluating a company's financial performance and position by neural network. It involves reviewing the company's financial statements, including the balance sheet, income statement, and cash flow statement, as well as other financial reports and documents.
How does neural network examine financial reports and understand financial state of the company?
### Prediction Confidence Score
Trust metric by Neural Network: 76 out of 100 with 582 signals.
## References
1. Mnih A, Kavukcuoglu K. 2013. Learning word embeddings efficiently with noise-contrastive estimation. In Advances in Neural Information Processing Systems, Vol. 26, ed. Z Ghahramani, M Welling, C Cortes, ND Lawrence, KQ Weinberger, pp. 2265–73. San Diego, CA: Neural Inf. Process. Syst. Found.
2. Chernozhukov V, Chetverikov D, Demirer M, Duflo E, Hansen C, Newey W. 2017. Double/debiased/ Neyman machine learning of treatment effects. Am. Econ. Rev. 107:261–65
3. Knox SW. 2018. Machine Learning: A Concise Introduction. Hoboken, NJ: Wiley
4. L. Prashanth and M. Ghavamzadeh. Actor-critic algorithms for risk-sensitive MDPs. In Proceedings of Advances in Neural Information Processing Systems 26, pages 252–260, 2013.
5. Athey S, Imbens GW. 2017a. The econometrics of randomized experiments. In Handbook of Economic Field Experiments, Vol. 1, ed. E Duflo, A Banerjee, pp. 73–140. Amsterdam: Elsevier
6. N. B ̈auerle and A. Mundt. Dynamic mean-risk optimization in a binomial model. Mathematical Methods of Operations Research, 70(2):219–239, 2009.
7. Swaminathan A, Joachims T. 2015. Batch learning from logged bandit feedback through counterfactual risk minimization. J. Mach. Learn. Res. 16:1731–55
Frequently Asked QuestionsQ: What is the prediction methodology for KIM^L stock?
A: KIM^L stock prediction methodology: We evaluate the prediction models Reinforcement Machine Learning (ML) and Sign Test
Q: Is KIM^L stock a buy or sell?
A: The dominant strategy among neural network is to Hold KIM^L Stock.
Q: Is Kimco Realty Corporation Class L Depositary Shares each of which represents a one-one thousandth fractional interest in a share of 5.125% Class L Cumulative Redeemable Preferred Stock liquidation preference \$25000.00 per share stock a good investment?
A: The consensus rating for Kimco Realty Corporation Class L Depositary Shares each of which represents a one-one thousandth fractional interest in a share of 5.125% Class L Cumulative Redeemable Preferred Stock liquidation preference \$25000.00 per share is Hold and is assigned short-term Ba1 & long-term Ba1 estimated rating.
Q: What is the consensus rating of KIM^L stock?
A: The consensus rating for KIM^L is Hold.
Q: What is the prediction period for KIM^L stock?
A: The prediction period for KIM^L is (n+3 month)
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# What is the least $n$ such that it is possible to embed $\operatorname{GL}_2(\mathbb{F}_5)$ into $S_n$?
Let $\operatorname{GL}_2(\mathbb{F}_5)$ be the group of invertible $2\times 2$ matrices over $\mathbb{F}_5$, and $S_n$ be the group of permutations of $n$ objects.
What is the least $n\in\mathbb{N}$ such that there is an embedding (injective homomorphism) of $\operatorname{GL}_2(\mathbb{F}_5)$ into $S_n$?
Such a question has been asked today during an exam; it striked me as quite difficult. There is an obvious embedding with $n=24$, and since $|\operatorname{GL}_2(\mathbb{F}_5)|=480$ and in $\operatorname{GL}_2(\mathbb{F}_5)$ there are many elements with order $20$, we have $n\geq 9$. However, "filling the gap" between $9$ and $24$ looks hard, at least to me. Can someone shed light on the topic? I would bet that representation theory and Cayley graphs may help, but I am not so much confident to state something non-trivial. I think that proving that $\operatorname{GL}_2(\mathbb{F}_5)$ is generated by three elements (is this true?) may help, too.
I would be interested also in having a proof of something sharper than $9\leq n\leq 24$.
Update. The following Wikipedia page claims, in paragraph Exceptional isomorphisms, that $\operatorname{PGL}(2,5)$ is isomorphic to $S_5$. This seems to suggest that $\operatorname{GL}_2(\mathbb{F}_5)$ embeds in $\mathbb{Z}_4\times S_5$ that embeds in $S_9$. Am I right?
Second Update. No, I am wrong, since $\mathbb{F}_{25}^*$ embeds in $\operatorname{GL}(2,\mathbb{F}_5)$, so there is an element with order $24$, so $n\geq 11$.
• I wonder if we can feed GAP with this sort of question. – Martin Brandenburg Sep 10 '14 at 22:23
• $\mathrm{GL}_2(\mathbb F_5)$ contains a subgroup isomorphic to $\mathbb{F}_{25}^{\times}$, so it should have an element of order 24, right? – Thomas Andrews Sep 10 '14 at 22:23
• @Thomas: right. That bumps up the lower bound to $3 + 8 = 11$, I think. – Qiaochu Yuan Sep 10 '14 at 22:30
• @Jack: no, that doesn't follow. You need to know that the short exact sequence $\mathbb{Z}_4 \to \text{GL}_2(\mathbb{F}_5) \to \text{PGL}_2(\mathbb{F}_5)$ is trivial to conclude that, and as far as I know that's not the case. – Qiaochu Yuan Sep 10 '14 at 22:31
• Note that $24$ is the highest order of an element. If the minimal polynomial for a matrix $A$ is prime or a product of distinct linear factors, that polynomial is a divisor of $X^{24}-1$ (since $X$ is never a factor.) So the only case we care about are of the form $(X-u)^2$. In that case, it is a factor of $X^{20}-1=(X^4-1)^{5}$. – Thomas Andrews Sep 11 '14 at 0:01
The answer is $24$. The natural action on ${\mathbb F}_5 \setminus \{0\}$ shows that ${\rm GL}_2(5) < S_{24}$.
To show that this is the smallest possible we prove the stronger result that $24$ is the smallest $n$ with $G:={\rm SL}_2(5) \le S_n$. The centre $Z = \{ \pm I_2 \}$ of $G$ has order $2$ and, since $G/Z \cong {\rm PSL}_2(5) \cong A_5$ is simple, $Z$ is the only nontrivial proper normal subgroup of $G$. So any non-faithful permutation action of $G$ has $Z$ in its kernel. It follows that the smallest degree faithful representation is transitive, and so it is equivalent to an action on the cosets of a subgroup $H < G$ with $H \cap Z = 1$. Hence we are looking for the largest subgroup $H$ of $G$ with $H \cap Z = 1$.
Since $-I_2$ is the only element of order $2$ in $G$, all subgroups of $G$ of even order contain $Z$. There is no subgroup of order $15$, so the largest odd order subgroup has order $5$, and the permutation action on its cosets has degree $120/5 = 24$.
In general, for a finite group $G$ with a complicated structure, the problem of finding the least $n$ with $G \le S_n$ seems to be very difficult, and I have not come across any computer algorithms that solve it efficiently. The difficulty comes from the fact that the the smallest $n$ does not generally come from a transitive action, so you have to look at all possibilities of combining transitive actions to get trivial intersectino of kernels. In this particular case, we are lucky in that we can reduce the problem to ${\rm SL}_2(5)$, where we are guaranteed that the minimal action is transitive, so then it just becomes a search for the largest core-free subgroup, which can be done computationally if the group is not too huge.
• Very nice! I was having thoughts along these lines (mostly the third paragraph, not the second) but I failed to realize that restricting my attention to $\text{SL}_2(\mathbb{F}_5)$ would make the argument easier. – Qiaochu Yuan Sep 11 '14 at 8:24
Here are some thoughts. Every group action of a group $G$ is a disjoint union of transitive group actions $G/G_i$ for various subgroups $G_i$. A group action is faithful iff for every $g \in G$ not equal to the identity there is some $i$ such that $g$ does not act by the identity on $G/G_i$. Hence there must be some coset $h G_i$ such that $gh G_i \neq h G_i$, or equivalently such that $hgh^{-1} \not \in G_i$. So the condition is that the conjugacy class of $g$ is not entirely contained in $G_i$. Finally, for a group action to be small we want the $G_i$ to be large.
This condition is hardest to satisfy when $g$ is central; in that case, the condition is that there must exist some $G_i$ such that $g \not \in G_i$. But it seems hard to me to find a large subgroup of $\text{GL}_2(\mathbb{F}_5)$ that doesn't contain a nontrivial central element. The action of $\text{GL}_2(\mathbb{F}_5)$ on $\mathbb{F}_5^2 \setminus \{ (0, 0) \}$ comes from the subgroup of matrices of the form
$$\left[ \begin{array}{cc} 1 & \ast \\ 0 & \ast \\ \end{array} \right]$$
which has order $20$, and that's the largest subgroup I can think of that doesn't have a nontrivial central element.
If we allow ourselves to ignore the center we can do much better. $\text{GL}_2(\mathbb{F}_5)$ acts on $\mathbb{P}^1(\mathbb{F}_5)$, which has $6$ elements, with kernel precisely the center. We can capture half of the center by using the determinant $\det : \text{GL}_2(\mathbb{F}_5) \to \mathbb{F}_5^{\times}$, which gives a group action with $4$ elements. The disjoint union of these two group actions has $10$ elements and has kernel just $-1$.
• The argument I had previously written down about a lower bound was nonsense. – Qiaochu Yuan Sep 11 '14 at 0:00
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10. The location of a pointP in a Cartesian plane can be expressed in eitherthe rectangular coordinates (x,y) or the polar coordinates(r,?), as shown in the figure below. Therelationships among these two sets of coordinates are given by thefollowing equation:
Write two functionsrectTOpolar andpolarTOrect that convert coordinates fromrectangular to polar form, and vice versa, where the angle?is expressed indegrees.
Note that MATLAB’s (and also C’s)trigonometric functions work in radians, so we must convert fromdegrees to radians and vice versa when solving this problem. Therelationship between degrees and radians is 180 degrees =?radians
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# How to break up function definitions across several lines?
if i write a function definition and i break it up across several lines, the sage prepocessor complains:
phi(epsilon, Q, r) = 1/(4*pi*
epsilon
)*Q/r
show(phi(1,2,3))
the sage prepocessor complains:
sage demo.sage
File "demo.sage.py", line 6
__tmp__=var("epsilon,Q,r"); phi = symbolic_expression(_sage_const_1 /(_sage_const_4 *pi*).function(epsilon,Q,r)
^
SyntaxError: invalid syntax
While this is a very short function that is easily kept on one line, I do have rather long and intricate ones which I would like to break up (and potentially comment) over several lines.
Linebreaks with backslash as the last char of the line don't work:
cat demo.sage
phi(epsilon, Q, r) = 1/(4*pi*\
epsilon\
)*Q/r
show(phi(1,2,3))
% sage demo.sage
File "demo.sage.py", line 6
__tmp__=var("epsilon,Q,r"); phi = symbolic_expression(_sage_const_1 /(_sage_const_4 *pi* * BackslashOperator() * ).function(epsilon,Q,r)
^
SyntaxError: invalid syntax
How do I do that?
edit retag close merge delete
Sort by » oldest newest most voted
Hello, @stockh0lm. This seems to be a problem with the preprocessor, since it is assuming all \ to be a matrix operation (i.e., the Matlab-style operator for solving systems of linear equations).
As a workaround you could use consistent indentation and wrap the code with parenthesis, so that Sage's preparser understands the complete extension of your code. The following worked for me:
phi(epsilon, Q, r) = (1 / (4 * pi
* epsilon)
* Q / r)
show(phi(1,2,3))
Notice all lines defining phi are equally indented, and I started and finished its definition with parenthesis. The preparser seems to assume the code defining the function ends on the line with the last parenthesis, so we are forcing it to consider our complete definition.
Also, I am using the PEP 8 convention of breaking a line BEFORE an operator, i.e., the asterisk doesn't end a line, but it starts one. (This last convention has nothing to do with the working of the code; it's just a convention.)
more
how weird, this does not work for me, either:
% sage demo.sage
File "demo.sage.py", line 8
* Q/r
^
IndentationError: unexpected indent
% cat demo.sage
phi(epsilon, Q, r) = 1/(4*pi
* epsilon)
* Q/r
show(phi(1,2,3))
( 2019-04-27 00:45:42 +0200 )edit
Yes, sorry. You're right! This is weird, but Sage's preparser thinks the expression extends until the line that closes the last parenthesis (in your code until the end of the second line). Thus, the third line is not considered to be part of the definition, and Sage doesn't know why you indented.
I was lucky! Notice I closed the parenthesis on the last line (I think this is not good style), so in my case, it worked. Funny result!
Try wrapping all the expression in parenthesis, like:
phi(epsilon, Q, r) = (1/(4*pi
* epsilon)
* Q/r)
show(phi(1,2,3))
( 2019-04-27 01:42:10 +0200 )edit
this works! Thank you! - can you please put it in a proper answer, instead of a comment, so i can upvote and accept it?
( 2019-04-27 16:40:29 +0200 )edit
Hello, @stockh0lm. I think this webpage doesn't allow multiple answers from the same user, so I have edited my original answer to include my comment that gave you the correct answer. Therefore, you can mark that as an answer. Thank you!
( 2019-04-27 20:39:39 +0200 )edit
This does also work without indentation
phi(epsilon, Q, r) = (1 / (4 * pi
* epsilon)
* Q / r)
( 2019-04-27 22:02:52 +0200 )edit
This is indeed a bug of the preparse function. Your syntax is perfectly valid in Python
>>> A = 1 + (3*
... 4
... ) * 2
25
However Sage preparser function is not smart in matching the start and end of an instruction.
sage: s = """phi(epsilon, Q, r) = 1/(4*pi*
....: epsilon
....: )*Q/r"""
sage: print(s)
phi(epsilon, Q, r) = 1/(4*pi*
epsilon
)*Q/r
sage: print(preparse(s))
__tmp__=var("epsilon,Q,r"); phi = symbolic_expression(Integer(1)/(Integer(4)*pi*).function(epsilon,Q,r)
epsilon
)*Q/r
Somehow, only the first line went transformed into a function. If you call directly preparse_calculus (which is the part that perform the preparsing of the instruction defining the function) it works
sage: from sage.repl.preparse import preparse_calculus
sage: print(preparse_calculus(";" + s + ";"))
;__tmp__=var("epsilon,Q,r"); phi = symbolic_expression(1/(4*pi*
epsilon
)*Q/r).function(epsilon,Q,r);
(Don't ask me why, but the function preparse_calculus needs a ";" at the begining and the end of the instruction). This can then be executed manually
sage: exec(preparse_calculus(";" + s + ";")[1:-1])
sage: phi
(epsilon, Q, r) |--> 1/4*Q/(pi*epsilon*r)
Note that this problem has been reported 8 years ago in trac ticket #11621.
more
Hi,
you can use line breaks line breaks
phi(epsilon, Q, r) = 1/(4*pi* \
epsilon \
)*Q/r
show(phi(1,2,3))
Worked for me in the sage command line v8.2
more
No.
File "demo.sage.py", line 6
__tmp__=var("epsilon,Q,r"); phi = symbolic_expression(_sage_const_1 /(_sage_const_4 *pi* * BackslashOperator() * ).function(epsilon,Q,r)
^
SyntaxError: invalid syntax
actually i had tried that, and i cant. see error above.
( 2019-04-26 17:06:58 +0200 )edit
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# Circular Motion and Tension in a string
1. Jul 31, 2008
### Sdarcy
Okay, I have given this a go but its been 2 years since I've done any dynamics so I think I've done something stupid...
A ball is attached horizontally by a string of length L to a central point C. The mass, m, of the ball is 4.775kg. It is released from rest and allowed to swing downwards. What is the tension in the string (in N) when the ball has fallen through 45 degrees.
This is what I've done so far:
$$\sum$$F$$_{}n$$ = ma$$_{}n$$
T- m sin$$\alpha$$ = m/g (v$$^{}2$$/L)
T = m(sin$$\alpha$$ + v$$^{}2$$/gL)
$$\sum$$F$$_{}t$$ = ma$$_{}t$$
m cos$$\alpha$$ = m/g a$$_{}t$$
a$$_{}t$$ = g cos$$\alpha$$
vdv = a$$_{}$$t ds
ds = L d$$\alpha$$
vdv = a$$_{}$$t L $$\alpha$$
vdv = g L cos$$\alpha$$ d$$\alpha$$
then integrate that equation
v = $$\sqrt{}$$2gl(sin$$\alpha$$ - sin$$\alpha$$0)
T = m(3 sin $$\alpha$$ - 2 sin $$\alpha$$0)
= 4.775[3sin(45) - 2sin(0)]
= 10.13N
Which is apparently very, very wrong.
Not sure what I've stuffed up, but help would REALLY be appreciated. Thanks.
2. Aug 1, 2008
### Hootenanny
Staff Emeritus
I think that you have over complicated this question somewhat. Start by writing down the net horizontal and vertical forces acting on the ball.
3. Aug 3, 2008
### Sdarcy
Lets pretend I'm stupid (which I am) and that I don't know anything about dynamics (which I currently don't), can you explain more clearly what you mean? :D
Thanks
4. Aug 3, 2008
### Hootenanny
Staff Emeritus
Whilst you seem to heave already put a lot of work in I don't mind doing a bit of work for you ... just this once .
I'll admit to making a mistake in my previous post, I should have said radial and tangential components rather that horizontal and vertical. So for the radial components:
$$\sum F_r = T - mg\sin\theta$$
As you correctly have. Now since the ball is following a circular path and applying Newton's second law we obtain:
$$T - mg\sin\theta = m\frac{v^2}{L}$$
Now from here, rather than attempting to solve a differential equation, it would be much more straight forward to apply conservation of energy to determine the velocity as a function of theta. Do you follow?
On a related point, you should note the mistake in going from the first line to the second line in your OP:
The first term on the LHS should also be divided by a factor of g.
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# Jones figures that the total number of thousands of miles that an auto
Jones figures that the total number of thousands of miles that an auto can be driven before it would need to be junked is an exponential random variable with parameter $\frac{1}{20}$. Smith has a used car that he claims has been driven only 10,000 miles. If Jones purchases the car, what is the probability that she would get at least 20,000 additional miles out of it? Repeat under the assumption that the lifetime mileage of the car is not exponentially distributed, but rather is (in thousands of miles) uniformly distributed over (0,40).
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Step 1
Let X be the number of miles the car can be driven before being junked.
a) First, let $X\sim Exp\left(\frac{1}{2}\right)$.
We know that the car has been driven 10,000 miles and we want to find the probability that it will last an additional 20,000 miles. We use the memoryless property of exponential distribution, where .
Then, $P\left(X\ge 10000+20000\mid X>10000\right)=P\left(X\ge 20000\right)$.
$P\left(X\ge 30000\mid X>10000\right)=P\left(X\ge 20000\right)$
$={\int }_{20}^{\mathrm{\infty }}{e}^{-\frac{1}{20}x}dx$
$={e}^{-1}$
Step 2
b) Let $X\sim U\left(0,40\right)$. We want to find $P\left(X>30\mid X>10\right)$. By Bayes
###### Not exactly what you’re looking for?
Bertha Jordan
Let X denote the exponential random variable with parameter 1.20, U denote the uniform random variable with parameters 0 and 40. The probability function of an exponential random variable with parameter 1/20 is
$F\left(x\right)=1-{e}^{-\frac{x}{20}},x\ge 0$
The desired probability is $P\left(X>30\mid X>10\right)$. Since exponential random variable have memoryless property, we have
$P\left(X>30\mid X>10\right)=P\left(X>20\right)=1-P\left(X\le 20\right)={e}^{-1}\approx 0.3679$
The probability function of an uniform random variable with parameter 0 and 40 is
$F\left(x\right)=\frac{1}{40-0}x=\frac{x}{40},0\le x\le 40$
Hence $P\left(U>30\mid U>10\right)=\frac{P\left(U>30\right)}{P\left(U>10\right)}=\frac{1-P\left(U\le 30\right)}{1-P\left(U\le 10\right)}=\frac{1-\frac{30}{40}}{1-\frac{10}{40}}=\frac{1}{3}$ is the desired probability.
###### Not exactly what you’re looking for?
nick1337
Let X denote the total number of thousands of miles that the auto is driven before it needs to be junked. We want to compute $P\left(\left\{X\ge 30\right\}|\left\{X\ge 10\right\}\right)$. Assuming X is an exponential random variable with parameter $1/20$, we have
$P\left\{X\ge 30\right\}={\int }_{30}^{\mathrm{\infty }}\frac{1}{20}{e}^{-x/20}dx={\int }_{3/2}^{\mathrm{\infty }}{e}^{-t}dt={e}^{-3/2},$
$P\left\{X\ge 10\right\}={\int }_{10}^{\mathrm{\infty }}\frac{1}{20}{e}^{-x/20}dx={\int }_{1/2}^{\mathrm{\infty }}{e}^{-t}dt={e}^{-1/2},$
$P\left(\left\{X\ge 30\right\}|\left\{X\ge 10\right\}\right)=P\left\{X\ge 30\right\}/P\left\{X\ge 10\right\}={e}^{-3/2}/{e}^{-1/2}={e}^{-1}\approx .3678.$
Note that we could have saved some work here by using the memoryless property of the exponential random variable.
Assuming X is uniformly distributed over (0,40), we have
$P\left\{X\ge 30\right\}={\int }_{30}^{40}\frac{1}{40}dx=1/4,$
$P\left\{X\ge 10\right\}={\int }_{10}^{40}\frac{1}{40}dx=3/4,$
$P\left(\left\{X\ge 30\right\}|\left\{X\ge 10\right\}\right)=P\left\{X\ge 30\right\}/P\left\{X\ge 10\right\}=\left(1/4\right)/\left(3/4\right)=1/3.$
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demo_map
# Demonstration of the BuTools MAP package¶
Set the precision and initialize butools (load all packages)
In [1]:
%precision %g
%run "~/github/butools/Python/BuToolsInit.py"
Butools V2.0
Packages loaded: utils, mc, moments, reptrans, trace, ph, dph, map, dmap, fitting, mam, queues
Global variables:
butools.verbose = False , butools.checkInput = True , butools.checkPrecision = 1e-12
First the global butools.verbose flag is set to True to obtain more messages from the functions.
In [2]:
butools.verbose = True
## MAPs and RAPs, MMAPs and MRAPs¶
RAPs (Rational Arrival Processes) are the generalizations of MAPs without the Markovian restrictions for the rate matrices. Some functions need a MAP, some others a RAP input, which is enforced if the global flag called butools.checkInput is set to True, with tolerance given by the global butools.checkPrecision variable.
The corresponding functions to check if a matrix pair is a MAP or a RAP are called CheckMAPRepresentation and CheckRAPRepresentation.
In [3]:
H0 = ml.matrix([[-1., 0, 0],[0, -2., 1.],[0, -1., -2.]])
H1 = ml.matrix([[1., 0, 0],[0, 1., 0],[1., 1., 1.]])
In [4]:
CheckMAPRepresentation(H0, H1)
CheckGenerator: The generator has negative off-diagonal element (precision: 1e-12)!
Out[4]:
False
In [5]:
CheckRAPRepresentation(H0, H1)
Out[5]:
True
In [6]:
H0 = ml.matrix([[-2., 0, 0],[0, -1., 1.],[0, -1., -1.]])
H1 = ml.matrix([[1., 0, 1.],[0, 1., -1.],[1., 0, 1.]])
CheckRAPRepresentation(H0, H1)
CheckRAPRepresentation: The dominant eigenvalue of D0 is not real!
Out[6]:
False
BuTools also supports the multi-type, marked variant of these arrival processes, which are referred to as MMAPs and MRAPs. Instead of a matrix pair, they are characterized by a list of matrices. If the arrival process is able to generate K different types of arrivals, K+1 matrices are necessary to define the process. The list of these K+1 matrices is the input of the corresponding functions of BuTools.
In [7]:
H0 = ml.matrix([[-5., 0.28, 0.9, 1.],[1., -8., 0.9, 0.1],[0.9, 0.1, -4., 1.],[1., 2., 3., -9.]])
H1 = ml.matrix([[-0.08, 0.7, 0.1, 0.1],[0.1, 1., 1.8, 0.1],[0.1, 0.1, 0.1, 0.7],[0.7, 0.1, 0.1, 0.1]])
H2 = ml.matrix([[0.1, 0.1, 0.1, 1.7],[1.8, 0.1, 1., 0.1],[0.1, 0.1, 0.7, 0.1],[0.1, 1., 0.1, 0.8]])
In [8]:
CheckMMAPRepresentation([H0, H1, H2])
CheckMMAPRepresentation: Some of the matrices H1 ... HM have a negative element!
Out[8]:
False
In [9]:
CheckMRAPRepresentation([H0, H1, H2])
Out[9]:
True
## Simple functions to compute various properties of MAPs and RAPs¶
Let us define a simple MAP given by matrices $D_0$ and $D_1$
In [10]:
D0 = ml.matrix ([[-5, 1, 0], [3, -3, 0], [1, 1, -5]])
D1 = ml.matrix ([[0, 0, 4], [0, 0, 0], [1, 1, 1]])
If the AT&T graphviz tool is installed, the ImageFromMAP and ImageFromMMAP functions are able to visualize the MAPs and MMAPs.
In [11]:
ImageFromMAP(D0, D1)
Out[11]:
It is possible to obtain the phase type distributed marginal distribution.
In [12]:
alpha, A = MarginalDistributionFromMAP (D0, D1)
print("alpha = ", alpha)
print("A = ", A)
alpha = [[ 0.14285714 0.14285714 0.71428571]]
A = [[-5 1 0]
[ 3 -3 0]
[ 1 1 -5]]
From this distribution it is easy to compute the marginal moments by the appropriate functions of the PH package. However, there is a convenience function MarginalMomentsFromMAP that does the same.
In [13]:
print(MarginalMomentsFromMAP (D0, D1, 3))
print(MomentsFromPH(alpha, A, 3))
[0.4285714285714286, 0.40000000000000002, 0.58285714285714296]
[0.4285714285714286, 0.40000000000000002, 0.58285714285714296]
Now we compute the autocorrelation function up to lag 10.
In [14]:
LagCorrelationsFromMAP (D0, D1, 10)
Out[14]:
array([ -3.01886792e-02, 1.20754717e-02, -4.83018868e-03,
1.93207547e-03, -7.72830189e-04, 3.09132075e-04,
-1.23652830e-04, 4.94611321e-05, -1.97844528e-05,
7.91378113e-06])
The marginal moments and the lag-1 joint moments are known to characterize the MAP uniquely:
In [15]:
m = MarginalMomentsFromMAP (D0, D1)
Nm = LagkJointMomentsFromMAP (D0, D1)
print("marginal moments = ", m)
print("lag-1 joint moments = ", Nm)
marginal moments = [0.4285714285714286, 0.40000000000000002, 0.58285714285714296, 1.1520000000000004, 2.8662857142857159]
lag-1 joint moments = [[ 1. 0.42857143 0.4 ]
[ 0.42857143 0.17714286 0.16228571]
[ 0.4 0.16228571 0.1472 ]]
## Matching methods¶
BuTools has several functions for the inverse characterization of MAPs and RAPs. For example, we can obtain a simple 2-state MAP based on 3 moments and the lag-1 autocorrelation. (Note however, that the moments and the autocorrelation must fall into a given interval, othervise MAP2FromMoments does not return a valid result)
In [16]:
D0,D1 = MAP2FromMoments([1,3,20], 0.2)
print("D0 = ", D0)
print("D1 = ", D1)
print("Checking moments and correlation:")
moms = MarginalMomentsFromMAP (D0, D1)
ro = LagCorrelationsFromMAP (D0, D1, 1)
print("Moments = ", moms)
print("Lag-1 autocorrelation = ", ro)
D0 = [[-0.3417355 0.06209041]
[ 0. -1.35057219]]
D1 = [[ 0.27964509 0. ]
[ 0.03021729 1.32035491]]
Checking moments and correlation:
Moments = [0.999999999999998, 2.999999999999984, 19.999999999999851]
Lag-1 autocorrelation = [ 0.2]
Based on $2N-1$ marginal moments and $N\times N$ lag-1 joint moments we can obtain an order-$N$ RAP as well. First we define a new MAP, and compute its marginal and joint moments...
In [17]:
D0 = ml.matrix ([[-6, 2, 1], [3, -4, 0], [2, 0, -5]])
D1 = ml.matrix ([[0, 1, 2], [0, 0, 1], [1, 1, 1]])
m = MarginalMomentsFromMAP (D0, D1)
Nm = LagkJointMomentsFromRAP (D0, D1)
print("marginal moments = ", m)
print("lag-1 joint moments = ", Nm)
marginal moments = [0.42857142857142855, 0.38327526132404183, 0.52825698988697201, 0.98482942167740695, 2.311135705693844]
lag-1 joint moments = [[ 1. 0.42857143 0.38327526]
[ 0.42857143 0.1820345 0.16181674]
[ 0.38327526 0.16198671 0.14352756]]
... and solve the inverse characterization problem.
In [18]:
H0, H1 = RAPFromMoments (m, Nm)
print("H0 = ", H0)
print("H1 = ", H1)
H0 = [[ -3.06944444 22.81703515 -22.36663832]
[ -1.06307165 -5.8404154 4.5345846 ]
[ -0.49248391 4.28485984 -6.09014016]]
H1 = [[ 0.06944444 2.78092404 -0.23132086]
[ 0.81408791 0.46490727 1.08990727]
[ 0.74146765 1.09064829 0.46564829]]
We can check if the moments are really the same:
In [19]:
m = MarginalMomentsFromRAP (H0, H1)
Nm = LagkJointMomentsFromRAP (H0, H1)
print("marginal moments = ", m)
print("lag-1 joint moments = ", Nm)
marginal moments = [0.42857142857142866, 0.3832752613240421, 0.52825698988697256, 0.98482942167740828, 2.311135705693848]
lag-1 joint moments = [[ 1. 0.42857143 0.38327526]
[ 0.42857143 0.1820345 0.16181674]
[ 0.38327526 0.16198671 0.14352756]]
It is easy to see that $(H_0,H_1)$ is a non-Markovian representation. However, it is not at all easy to check if it is a valid process (with non-negative joint densities), or not. Something like MonocyclicPHFromME for RAPs is not available yet.
## Representation transformation methods¶
Now we try to transform $(H_0,H_1)$ into a Markovian representation.
In [20]:
D0, D1 = MAPFromRAP (H0, H1)
print("D0=", D0)
print("D1=", D1)
D0= [[-5.36050875 1.8098798 0.19876097]
[ 1.0793193 -7.12929544 1.68636471]
[ 0.04875948 0.95197382 -2.51019581]]
D1= [[ 0.28136405 0.74482106 2.32568287]
[ 1.57362442 0.08452984 2.70545717]
[ 0.74915196 0.12620444 0.63410611]]
Altough we managed to find a Markovian representation, keep in mind that it is not always possible.
If a MAP or a RAP is redundant, we have functions to obtain the minimal representation as well. The next $(D_0,D_1)$ matrices define a redundant MAP.
In [21]:
D0 = ml.matrix ([[-5, 1, 0], [3, -3, 0], [1, 1, -5]])
D1 = ml.matrix ([[0, 0, 4], [0, 0, 0], [1, 1, 1]])
Let us compute the minimal representation:
In [22]:
H0, H1 = MinimalRepFromRAP(D0, D1)
print("H0=", H0)
print("H1=", H1)
H0= [[-4.40740741 1.69312169]
[ 0.84259259 -2.59259259]]
H1= [[ 2.03703704 0.67724868]
[ 2.78703704 -1.03703704]]
Thus it is an order-2 rational arrival process. We can check that $(D_0,D_1)$ and $(H_0,H_1)$ are equal indeed by comparing their moments.
In [23]:
m = MarginalMomentsFromMAP (D0, D1)
N = LagkJointMomentsFromMAP (D0, D1, 2)
print("marginal moments = ", m)
print("lag-1 joint moments = ", N)
m = MarginalMomentsFromRAP (H0, H1)
N = LagkJointMomentsFromRAP (H0, H1, 2)
print("marginal moments = ", m)
print("lag-1 joint moments = ", N)
marginal moments = [0.4285714285714286, 0.40000000000000002, 0.58285714285714296, 1.1520000000000004, 2.8662857142857159]
lag-1 joint moments = [[ 1. 0.42857143 0.4 ]
[ 0.42857143 0.17714286 0.16228571]
[ 0.4 0.16228571 0.1472 ]]
marginal moments = [0.42857142857142855, 0.40000000000000002, 0.58285714285714285]
lag-1 joint moments = [[ 1. 0.42857143 0.4 ]
[ 0.42857143 0.17714286 0.16228571]
[ 0.4 0.16228571 0.1472 ]]
...or, by finding an appropriate similarity transformation matrix (with the SimilarityMatrix function of the RepTrans package) that transforms $(D_0,D_1)$ to $(H_0,H_1)$.
In [24]:
T = SimilarityMatrix(H0,D0)
Check if matrix $T$ transforms $(D_0,D_1)$ to $(H_0,H_1)$ indeed:
In [25]:
la.norm (T*D0*T.I - H0), la.norm (T*D1*T.I - H1)
Out[25]:
(1.1065e-14, 7.36773e-15)
Most MAP/RAP related functions in butools work with marked processes as well. Let us try out the representation minimization method with the following MMAP.
In [26]:
D0 = ml.matrix ([[-6, 0, 0], [0, -5, 2], [0, 0, -3]])
D1 = ml.matrix ([[0, 1, 5], [1, 1, 0], [1, 0, 1]])
D2 = ml.matrix ([[0, 0, 0], [0, 1, 0], [0, 0, 1]])
H0, H1, H2 = MinimalRepFromMRAP ([D0, D1, D2])
print("H0=", H0)
print("H1=", H1)
print("H2=", H2)
H0= [[-2.58578644 -1.41421356]
[ 1. -6.41421356]]
H1= [[ 4.27614237 -0.94280904]
[ 8.49509379 -3.27614237]]
H2= [[ 1.13807119 -0.47140452]
[ 0.33333333 -0.13807119]]
Thus it is an order-2 representation again. To confirm it, we compare the marginal and the lag-1 joint moments.
In [27]:
m = MarginalMomentsFromMMAP ([D0, D1, D2], 5)
N = LagkJointMomentsFromMMAP ([D0, D1, D2], 2)
print("marginal moments = ", m)
print("lag-1 joint moments = ", N)
m = MarginalMomentsFromMRAP ([H0, H1, H2], 5)
N = LagkJointMomentsFromMRAP ([H0, H1, H2], 2)
print("marginal moments = ", m)
print("lag-1 joint moments = ", N)
marginal moments = [0.29166666666666663, 0.18055555555555558, 0.1736111111111111, 0.22685185185185186, 0.37422839506172839]
lag-1 joint moments = [matrix([[ 0.75 , 0.20833333, 0.125 ],
[ 0.20833333, 0.05555556, 0.03240741],
[ 0.125 , 0.03240741, 0.01851852]]), matrix([[ 0.25 , 0.08333333, 0.05555556],
[ 0.08333333, 0.02777778, 0.01851852],
[ 0.05555556, 0.01851852, 0.01234568]])]
marginal moments = [0.29166666666666652, 0.18055555555555547, 0.17361111111111099, 0.22685185185185169, 0.374228395061728]
lag-1 joint moments = [matrix([[ 0.75 , 0.20833333, 0.125 ],
[ 0.20833333, 0.05555556, 0.03240741],
[ 0.125 , 0.03240741, 0.01851852]]), matrix([[ 0.25 , 0.08333333, 0.05555556],
[ 0.08333333, 0.02777778, 0.01851852],
[ 0.05555556, 0.01851852, 0.01234568]])]
The tool called RAPFromMomentsAndCorrelations does what its name suggests. The input is a set of moments and lag-k correlations, and the output is a RAP.
For testing purposes we feed it with the moments and correlations of an other RAP, $(H_0,H_1)$:
In [28]:
H0 = ml.matrix([[-6.2, 2., 0],[2., -9., 1.],[1., 0, -3.]])
H1 = ml.matrix([[2.2, 0, 2.],[0, 4., 2.],[0, 1., 1.]])
mom = MarginalMomentsFromRAP(H0, H1)
corr = LagCorrelationsFromRAP(H0, H1, 3)
G0, G1 = RAPFromMomentsAndCorrelations(mom, corr)
print("G0=",G0)
print("G1=",G1)
G0= [[ -8.96289388 22.25257011 -18.54409809]
[ -0.99178156 -4.66699225 2.33103341]
[ -1.2472989 2.42791179 -4.57011387]]
G1= [[ 2.20274746 -1.3172514 4.36892581]
[ 1.2179263 1.82172664 0.28808746]
[ 1.02115416 0.41735382 1.950993 ]]
The next code shows that the procedure was successfull.
In [29]:
rmom = MarginalMomentsFromRAP(G0, G1)
rcorr = LagCorrelationsFromRAP(G0, G1, 3)
print("mom=",mom)
print("rmom=",rmom)
print("corr=",corr)
print("rcorr=",rcorr)
mom= [0.29774127310061604, 0.19283643304803644, 0.19448147792730755, 0.26597325539245531, 0.45833053059627116]
rmom= [0.29774127310061604, 0.19283643304803638, 0.19448147792730741, 0.26597325539245492, 0.45833053059627044]
corr= [ 0.01239357 0.0027412 0.00072384]
rcorr= [ 0.01239357 0.0027412 0.00072384]
However, the we did not get back the original RAP! Let us find the similarity matrix that transforms $H_0$ to $G_0$:
In [30]:
T = SimilarityMatrix (H0, G0)
This matrix does the job: transforms $H_0$ to $G_0$:
In [31]:
T.I*H0*T - G0
Out[31]:
matrix([[ -8.02913291e-12, -1.76925141e-12, -1.78346227e-12],
[ -1.52489132e-12, -4.89386309e-13, -3.93463040e-13],
[ -1.73239201e-12, -4.48974191e-13, -4.06785716e-13]])
...but the same similarity transformation does not transform $H_1$ to $G_1$!
In [32]:
T.I*H1*T - G1
Out[32]:
matrix([[ 0.95226605, -8.83362948, 7.88136343],
[-0.16334624, 0.21780655, -0.05446031],
[ 0.16334624, -0.21780655, 0.05446031]])
The last inverse characterization too is MAPFromFewMomentsAndCorrelations. Contrary to all previous procedures, this one always gives a valid MAP representation based on 2 or 3 moments and the lag-1 autocorrelation.
In [33]:
D0, D1 = MAPFromFewMomentsAndCorrelations([1.2, 4.32, 20.], -0.4)
print("D0=",D0)
print("D1=",D1)
D0= [[ -0.33604187 0.33604187 0. 0. 0. 0. ]
[ 0. -36.02667283 0. 0. 0. 0. ]
[ 0. 0. -1.28286281 1.28286281 0. 0. ]
[ 0. 0. 0. -1.28286281 1.28286281 0. ]
[ 0. 0. 0. 0. -1.28286281
1.28286281]
[ 0. 0. 0. 0. 0. -1.37471852]]
D1= [[ 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
0.00000000e+00 0.00000000e+00]
[ 2.47407456e-02 1.77659290e+00 2.34745531e+01 0.00000000e+00
0.00000000e+00 1.07507861e+01]
[ 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
0.00000000e+00 0.00000000e+00]
[ 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
0.00000000e+00 0.00000000e+00]
[ 0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00
0.00000000e+00 0.00000000e+00]
[ 1.79372562e-02 1.28804533e+00 4.71447523e-02 0.00000000e+00
0.00000000e+00 2.15911735e-02]]
The result is sometimes large, but at least Markovian. Let us check the target parameters.
In [34]:
rmoms = MarginalMomentsFromMAP(D0, D1, 3)
rcorr1 = LagCorrelationsFromMAP(D0, D1, 1)
print("rmoms=",rmoms)
print("rcorr1=",rcorr1)
rmoms= [1.2000000000000002, 4.3200000000000021, 19.999999999999915]
rcorr1= [-0.4]
## Randomness¶
The RandomMAP and RandomMMAP procedures provide a random MAP and MMAP, respectively.The input parameters are the number of phases, the number of arrival types (in case of MMAP), the mean inter-arrival time and the number of zero entries.
In [35]:
D0, D1 = RandomMAP(4, 1.62, 10)
print("D0=",D0)
print("D1=",D1)
D0= [[-2.58668134 0.23931909 0. 0.63351354]
[ 0. -1.75671785 0.09718683 0.01550621]
[ 0.04462193 0.80081749 -1.5411998 0. ]
[ 0. 0. 0. -0.40711131]]
D1= [[ 0.13948285 0.14126085 0.6699959 0.76310911]
[ 0.29832463 0.44959652 0.79414025 0.10196341]
[ 0. 0.07989564 0. 0.61586473]
[ 0. 0. 0.19373849 0.21337281]]
In [36]:
D0, D1, D2, D3 = RandomMMAP(4, 3, 1.62, 10)
print("D0=",D0)
print("D1=",D1)
print("D2=",D2)
print("D3=",D3)
D0= [[-0.82684732 0.10509885 0.01955105 0. ]
[ 0.0665087 -1.04377322 0.02035592 0.05023649]
[ 0.066475 0.03017514 -0.53943718 0.02570979]
[ 0.07044526 0.0906659 0.0396139 -0.74180676]]
D1= [[ 0.0676032 0.03704518 0.04165858 0.04829198]
[ 0.10830712 0.11452429 0.11203451 0.04137123]
[ 0.08150456 0. 0.03539041 0.04125597]
[ 0.00866007 0.10553133 0. 0.03193119]]
D2= [[ 0.09247225 0.12156338 0.05404064 0.0231576 ]
[ 0.09804396 0.06668159 0.09061084 0.01751566]
[ 0.02552027 0.08998726 0.03548746 0. ]
[ 0.10850257 0.09873466 0.07408211 0. ]]
D3= [[ 0.02775122 0.00348846 0.11137723 0.0737477 ]
[ 0.07607181 0.12185031 0.05966079 0. ]
[ 0. 0. 0.07814517 0.02978613]
[ 0. 0. 0.02466867 0.0889711 ]]
The SamplesFromMAP / SamplesFromMMAP functions return a vector of random samples from the given arrival processes. These samples can be used in simulations, for instance.
In [37]:
D0,D1 = MAP2FromMoments([1,3,20], 0.2)
x = SamplesFromMAP(D0, D1, 100000)
print("mean=",np.mean(x))
mean= 1.0071132382
In the MMAP case the result is two dimensional, the output is the list of sample-type pairs.
In [38]:
D = RandomMMAP(4, 3, 1.62, 10)
x = SamplesFromMMAP(D, 10000)
print(x[0:10])
[[ 3.16164714 1. ]
[ 0.37266944 3. ]
[ 1.41439525 2. ]
[ 0.830808 3. ]
[ 0.00969904 1. ]
[ 0.21160724 2. ]
[ 2.96377196 1. ]
[ 3.04674254 2. ]
[ 2.57087294 1. ]
[ 2.29580264 2. ]]
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Problem with ListContourPlot feeding data format and WeatherData
With following code I am trying to show the mean wind speed patterns for a few geographic locations.
cords = Select[
Reverse[CityData[#, "Coordinates"]] & /@ {"Oslo", "Vadso","Hammerfest"},
];
ListContourPlot[
Join @@ {#,List@WeatherData[#,
"MeanWindSpeed",
{{2013, 1, 25}, {2013, 5, 1}, "Day"}
][[1,2]]} & /@ cords
]
How can I improve the code to solve following problems:
1. Even if the requested date window is wider (a few days), it only takes one day data from WeatherData and when I am changing the partitioning part to [[All,2]] the plot doesn't work and says the format it wrong.
2. For even wider data range (a few years), there are some missing data for some days. How can I also filter that bad data in the code?
Remember, I need them exactly on the geographic location no matter how wide my data range is.
-
This kind of questions don't show a real interest in learning the language. You're throwing in a whole problem's coding and asking for a solution. Perhaps you should break it up in pieces, and ask about your doubts. Neither of your questions are related to a Plotting problem. – belisarius Jun 1 '13 at 21:57
1) Your [[1,2]] takes the first windspeed from every location. So, it doesn't help to increase the data range. 2. [[All,2]] results in passing a list of windspeeds for every location to ListContourPlot. However, ListContourPlot expects a single dependent value for each point not a list. How would you interpret that? You need to apply Mean on that list to get a single, average value. 3) You can delete Missing[...] data points using DeleteCases. – Sjoerd C. de Vries Jun 1 '13 at 22:26
@belisarius with many thanks for your previous hint; but frankly I am struggling with this language for around two days. I have to admit that it was very constructive hint of you and made me to look a the Join and List functionality thoroughly.I advanced the code considerably and knew theListContourPlot better now but those problems are really hindering me of any progress. – Alex Jun 1 '13 at 22:32
Using the Mean is making use of all recorded data. I'm afraid you are being rather unclear here. What do you want ListContourPlot to show on each location based on that set of windspeeds? You need a single color there. So, what should that depict? – Sjoerd C. de Vries Jun 1 '13 at 22:45
That's because you used Reverse to get your coordinates. That's OK for plotting, but not for getting information from WeatherData, which expects latlong, not longlat. – Sjoerd C. de Vries Jun 1 '13 at 23:53
1. Your [[1,2]] takes the first windspeed from every location. So, it doesn't help to increase the data range.
2. [[All,2]] results in passing a list of windspeeds for every location to ListContourPlot. However, ListContourPlot expects a single dependent value for each point not a list. How would you interpret that? You need to apply Mean on that list to get a single, average value.
3. You can delete Missing[...] data points using DeleteCases.
Something like the following should work:
ListContourPlot[
Append[#,
Mean[
DeleteCases[
WeatherData[
#,
"MeanWindSpeed",
{{2013, 1, 25}, {2013, 5, 1}, "Day"}
][[All, 2]],
Missing["NotAvailable"]
]
]
] & /@ cords
]
If I were you, I'd increase the number of locations in the plot. As it is, it looks rather boring.
-
manay thanks for advices but I don't want to make mean. It is very easy to make mean of those data.But I need the ListContourPlot to make the plot based on the all recorded data?I need the intensity of colors based on the analysis of previous data. Is it possible? – Alex Jun 1 '13 at 22:45
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Environ Eng Res > Volume 26(2); 2021 > Article
Min, Kim, Oh, Ryu, and Park: Flow velocity and cell pair number effect on current efficiency in plating wastewater treatment through electrodialysis
### Abstract
Electrodialysis has been used for treating toxic substances such as heavy metals and minimizing secondary environmental pollution problems effectively. However, electrodialysis depends on the operating parameters as well as fluid dynamics and electrical properties. This study provides design elements for the treatment of heavy metal-containing wastewater by electrodialysis. We found that the limiting current density (LCD) is proportional but not completely linear to the diluate concentration over a threshold value. In contrast, it is linear to the linear flow velocity for the whole range. As the number of cell pairs increases, because linear flow velocity and LCD increase, the removal efficiency of heavy metals also increases. Therefore, for highly concentrated wastewater, increasing the linear flow velocity, the applied voltage, and the number of cell pairs can effectively improve removal efficiency. It was found that the current efficiency is as low as 17% when the removal efficiency of heavy metals exceeds 95%. Thus, it is necessary to select an operating range that optimizes the operating and initial investment costs for the effective removal of heavy metals using electrodialysis.
### 1. Introduction
Climate change and excessive use of water resources have caused water shortage. To solve this problem, research has been focused on practical solutions with regard to treatment and reuse of wastewater [13]. In particular, electrodialysis has been used as a technique for minimizing secondary environmental pollution problems and treating toxic substances such as heavy metals effectively. Electrodialysis is a compact process that reduces the consumption of chemicals, energy, water, and land. Additionally, its operation cost can be reduced by minimizing the occurrence of chemical sludge and enabling automatic operation. Furthermore, the electrodialysis process can concentrate heavy metals. Electrodialysis, when combined with heavy metal extraction technologies, can thus efficiently recover heavy metals at a minimal cost [46].
Electrodialysis is a membrane separation technology that extracts salt ions from a solution by using an electric field and an ion exchange membrane, producing concentrated water containing salt ion and treated water with almost no salt ions.. The process occurs in a stack composed of the ion (cation/anion) exchange membrane, a spacer, a diluate and a concentrate channel. The system formed by a cation exchange membrane, a concentrate containing cell, an anion exchange membrane, and a diluate containing cell is referred to as a cell pair. The factors affecting the efficiency of electrodialysis include influent, hydrodynamic, and electrical properties [2, 711].
Additionally, the performance of electrodialysis is determined by a set of fixed and variable process parameters, such as feed and product concentration, stack construction, ion exchange membrane permselectivity, flow velocity, current density, rates, etc. Therefore, to carry out electrodialysis efficiently, not only the operating parameters but also the components and characteristics should be optimized in terms of the overall cost [8, 1214]. In particular, the process parameters directly affect the current efficiency and concentration polarization. In general, concentration polarization refers to the dissociation of water into H+ and OH when the ion concentration on the surface of the cation and/or anion exchange membrane in the diluate cell is 0. Here, because the anions (cations) are transported through the anion (cation) exchange membrane to the concentrate cell, the cations (anions) are concentrated in the boundary layer of the membrane surface to the diluate cell and increase at the surface facing the concentrate cell [810, 1517]. Therefore, scaling may occur due to the deposition of metal hydroxides on the surface of the ion exchange membrane [8, 1819].
In general, when the set current density is reached, the cell resistance increases rapidly, and the voltage does not increase until the current density increases at the applied voltage. When the applied voltage exceeds a certain threshold called the limiting current density (LCD), the current density increases again. In electrodialysis, when the LCD is exceeded, the electrical resistance of the diluate rapidly increases because of the depletion of ions in the boundary layer of the membrane surface. In this situation, the current efficiency decreases [23]. Thus, the LCD is one of the most important design parameters to determine the efficiency of plant operation.
Therefore, to treat wastewater efficiently using electrodialysis, it is important to study the dependence of the LCD on the operating parameters [23, 16]. Most studies have focused on the use of electrodialysis as a seawater desalination process. In the case of studies on the separation of heavy metals such as copper and nickel, the separation efficiency and the ion LCD have been reported for wastewater of high concentration, containing single ions, or mixed with NaCl [1921]. However, these values are difficult to use as a design factor because of the great differences from actual wastewater. In this context, this study aims to provide design factors for the treatment of plating wastewater containing low concentrations of heavy metals using an electrodialysis process.
The specific objectives of this study were: (a) to investigate the correlation between heavy metal concentration and LCD at constant linear flow velocity; (b) to correlate the LCD with the linear flow velocity at constant heavy metal concentration; (c) to measure the separation and current efficiency of heavy metals according to the number of cell pairs.
### 2.1. Wastewater
The properties of wastewater from electroplating facilities were analyzed to determine copper and nickel concentrations. The analysis revealed that the copper concentration ranged from 19 to 250 mg/L and the nickel one from 18 to 150 mg/L. The electrical conductivity ranged from 2,500 to 7,000 μS/cm and pH from 2.0 to 2.4. Nickel of synthetic plating wastewater was prepared using nickel (II) sulfate hexahydrate (NiSO4·6H2O, Sigma-Aldrich, USA), and copper was prepared using copper (II) sulfate pentahydrate (CuSO4·5H2O, Sigma-Aldrich, USA). The pH was measured after preparing 10% sulfuric acid (H2SO4, Sigma-Aldrich, USA) using distilled water. Table 1 summarizes the concentrations of synthetic plating wastewater.
### 2.2. Materials
The most important factor in determining the performance of an electrodialysis process is the ion exchange membrane. The ion exchange membrane can be regarded as a film of ion exchange resin. The properties of ion exchange membranes are determined by different parameters, such as the density of the polymer network, the hydrophobic or hydrophilic character of the matrix polymer, the type and concentration of the fixed charges in the polymer, and the morphology of the membrane itself. The membranes should have low electrical resistance, good mechanical, high chemical, thermal stability, high permselectivity, and low production costs [9].
In this study, we used a Neosepta CMX-SB and AMX-SB from ASTOM (Tokyo, Japan) as cationic and anionic ion exchange membrane, respectively. Their physical properties are reported in Table 2. The ion exchange membrane was immersed in distilled water for 1 h at room temperature before use. At the end of the experiment, to remove the salt attached to the ion exchange membrane, it was immersed in 0.36% H2SO4 and 1.22% Na2SO4 (Sigma-Aldrich, USA) for 1 h. The ion exchange membrane used in the experiment was used after immersion for 24 h in the solution to be used in the experiment at 25±1°C.
### 2.3. Experimental Setup
The electrodialysis system used in this experiment (CJ-S3, ChangJotechno Co. Ltd., Korea) consists of a dilution/concentration/electrolyte solution tank, ion exchange stack, diluate/concentrate/electrolyte feed pump, and power supply (Fig. 1). The capacity of the solution tanks, made of PVC, is 0.5 L.
The ion exchange stack is 115 mm (W) × 225 mm (H), and the distance between the cation and anion exchange membrane is 0.73 mm, filled by a planar diluate/concentrate/net-like spacer. The effective surface area of the anion/cation exchange membrane was 55.5 cm2. The flow paths in the diluate and concentrate cell of the ion exchange stack are independent and are maximized by design. In particular, the net-like spacers induce a zigzag flow, providing better turbulent flows and reducing concentration polarization phenomena [9].
At both ends of the stack are the cathode and anode. Their electrode size was 45 mm (W) × 105 mm (H) × 2 mm (D), and their titanium base was platted with a solution containing platinum and ruthenium to prevent corrosion.
To prevent leakage in each cell, a silicone rubber seal was inserted, and a steel block was installed at the top and bottom of the cell to press the ion exchange membrane and the spacer together, and then screwed. The power supply used in the experiment (P3030, Advantek Co. Ltd., Korea) can supply potential and direct current up to 30 V and 3 A, respectively.
### 2.4. Operating Conditions
A laboratory-scale electrodialysis system was used to measure the LCD as a function of the Cu and Ni concentrations. Using an ion exchange stack consisting of 5 pairs, the LCD of synthetic wastewater, prepared with concentrations of CuSO4 and NiSO4 of 20, 100, and 200 mg/L, was measured. Current and resistance were measured at each potential. The potential was increased by 1 V from 1 to 10 V, and by 2 V from 10 to 30 V. The current and resistance were measured after 5 minutes of stabilization at each step [22]. By changing the number of the pairs of membranes to 1, 3, 5, and 9, the effect of the linear flow velocity of the membrane on the LCD was measured. Here, the linear flow velocity of both the diluate and concentrate was controlled. Additionally, using synthetic wastewater with a 20 mg/L concentration of CuSO4 and NiSO4 and according to the LCD of each pair, the copper and nickel concentrations, electrical conductivity, and current over time were measured. The electrolyte used was a 4% (w/w) Na2SO4 solution [18, 23].
### 2.5. Analytical Methods
Copper and nickel concentrations were measured using inductively coupled plasma optical emission spectroscopy (ICP-OES, ICP-6000, Thermo Fisher Scientific Inc., USA) and according to method EN ISO 11885:2007. The samples were filtered by a syringe equipped with polyethersulfone filters with an average pore diameter of 0.45 μm and then acidified to 2% with concentrated HCl. Dilutions of the multielement standards solution 6 for ICP (Sigma-Aldrich Co., LLC., USA) were used for calibration, using a six-point function with blank, 0.1, 1.0, 5.0, 10.0 and 20.0 mg/L. Samples were diluted with 2% HCl if they contained higher concentrations than the endpoint of the calibration function. The multielement standards solution was initially acidified to 1 % with concentrated HCl and dilutions were made with a 2 % HCl solution. The analytical wavelengths of metal ions in this study was Ni (231.60 nm) and Cu (324.75 nm). The ion separation efficiencies were obtained using the changes in the concentration of copper and nickel and the conductivity of the diluate during electrodialysis. The conductivity of all samples was measured at regular intervals using an Orion 5 Star instrument (Thermo Fisher Scientific Inc., USA).
The decrease in conductivity over time in the dilution tank was determined using
##### (1)
$CtCo=e-k1t$
where Co and Ct are the conductivity (μS/cm) of the diluate at the beginning of the experiment and at time t, respectively, and k1 is a first order constant [24].
In general, the LCD is known to be proportional to the mass transfer coefficient and ion concentration of the diluate as
##### (2)
$LCD=kCn$
which can be rewritten as [8, 25]
##### (3)
$lnLCD=lnk+nlnC$
Here, k is the mass transfer coefficient (m/s) and C is the ion molar concentration (mol/m3) in the diluate.
The flow rate in an electrodialysis system can be expressed as [25]
##### (4)
$Q=u·Δ·w·N$
where Q is the volumetric flow rate (m3/s),is the theoretical linear flow velocity (m/s), Δ is the thickness of unit cell (m), w is the width of the diluate cell (m), and N is the total number of cell pairs.
In this experiment, the linear flow velocity was changed by supplying the maximum flow rate to each cell pair. The mass transfer coefficientcan generally be expressed as a nonlinear function of the linear flow velocity:
##### (5)
$k=a(vd)b$
Here, vd is the linear flow velocity (cm/s) of the diluate, and a and b are empirical parameters.
Using Eq. (5), a simple empirical equation that gives reasonable results for the LCD as a function of concentration and linear flow velocity can be found, as seen by equation (6) and (7):
##### (6)
$LCD=kC=a(vd)bCn$
##### (7)
$a(vd)b=aebln(vd)$
Because the charge passes through all the cell pairs connected in series, the current efficiency can be expressed as [9, 15, 25]
##### (8)
$I=zFQCsΔζN$
where I is the total electric current passing through the stack (A), $CsΔ$ is the concentration difference between the feed and product solutions obtained during the operation (mol/m3), ζ is the current utilization, z the valence number, F the Faraday constant (C/mol), and Q and N are the same as in Eq. (4).
### 3.1. Dependence of the LCD on Metal Concentration
In physical models, the LCD is empirically correlated to the flow velocity and bulk concentration of the diluate [16]. Some studies have reported LCD equations that depend on the hydrodynamic conditions, including flow velocity, and that include coefficients that are constant or functions of the flow velocity [15, 17]. However, most of them are the result of experiments performed on electrochemical cells under very specific conditions, and therefore require correction. Moreover, it is very difficult to determine the LCD in industrial electrodialysis systems because of the varying membrane thicknesses [26]. Therefore, the LCD must be approximated or measured using a reproducible electrodialysis system [16].
To analyze the dependence of the LCD on the diluate concentration, we measured the LCD at copper and nickel concentrations of 20, 100 and 200 mg/L. In the electrodialysis process, the maximum current per unit area of the membrane must be maintained as high as possible, though the operational level is limited by polarization [2]. The LCD is determined from the graph of the inverse of the current as a function of the cell resistance (Fig. 2(a)). Because the LCD is a current with zero ion concentration at the surface of the ion exchange membrane, the current required for ion depletion increases as the electrolyte concentration in the bulk layer increases [8]. Therefore, the LCD is the point where the slope of the potential/current changes because of water dissociation on the surface of the ion exchange membrane (Fig. 2(b)).
In this study, for copper and nickel concentration of 20, 100, and 200 mg/L in the diluate, the LCD increased from 14.1 to 16.4 and 25.0 A/m2, respectively, under the same linear flow velocity conditions (Fig. 3(a)). However, a linear relationship cannot be assumed at all concentrations. In fact, at low electrolyte concentrations, the LCD and the coefficients can be affected by high electrical resistivity and by the diffusion coefficient in solution [8]. According to equation (3), it is reasonable to measure only at concentrations above 1 mol/m3, because below this value the LCD is zero. Because the measured LCD depends on the ion concentration in the diluate, a proper choice of the concentration range is necessary to determine the LCD.
### 3.2. Dependence of the LCD on the Llinear Flow Velocity
The relationship between the LCD and linear flow velocity was investigated by changing the number of membrane pairs under the same metal concentration.
The LCD increased with increasing linear flow velocity (Fig. 3(b)). The coefficients a and b were measured 7.645 and 1.318, respectively, in the 1–2 cm/s linear velocity range.
Some researchers measured coefficients a and b at different linear flow velocities and NaCl concentrations. They report that coefficient b is affected by the fluid properties, including the linear feed flow rate, whereas coefficient a is related to the cell composition, including the ion transport through the ion exchange [2728]. In contrast, for well-mixed solutions, the linear flow velocity did not significantly affect the transport of electrolyte through the membrane [8, 29].
### 3.3. Dependence of the Separation Efficiency on the Number of Cell Pairs
The separation efficiency of copper and nickel was measured as a function of the number of cell pairs to evaluate the applicability of electrodialysis to heavy metals. The change in conductivity in the electrodialysis process was used to indirectly identify desalination. Electrodialysis has been reported to be more economical in the 2,000–10,000 μS/cm conductivity range and to have better ion removal efficiency than nanofiltration [1, 30]. Thus, electrodialysis should efficiently separate heavy metals contained in plating wastewater, because the wastewater has a conductivity of approximately 2,500 μS/cm.
In the electrodialysis process, when an electric potential is applied, the anions and cations are accumulated in a concentrate cell by the ion exchange membrane. The applied voltage is not only crucial for the process to occur but also determines the separation efficiency. Therefore, the ion separation efficiency in electrodialysis can be measured by regression analysis of the experimental data as a first-order constant.
Fig. 4 shows the conductivity as a function of the applied potential for each cell pair. As the number of cell pairs increases, the linear flow velocity and the LCD increase. Therefore, the ion transfer rate also increases, resulting in an increase in reaction rate and, in turn, in a reduction of the time required for separation.
However, the separation rates of nickel and copper were nearly the same under all conditions (Fig. 5). This is because both copper and nickel are divalent ions [31]. The change in conductivity is almost the same as the change in copper and nickel concentration, which can be indirectly used to evaluate the degree of desalination and separation efficiency.
### 3.4. Dependence of the Current Efficiency on the Number of Cell Pairs
The total cost of electrodialysis is the sum of the initial investment and the energy and maintenance costs. The energy costs increase linearly as the current density increases, whereas the investment costs decrease. Therefore, there is a value of the current density that optimizes the total cost.
Measuring the current efficiency under different experimental conditions, we found that the current efficiency decreased with increases in the number of cell pairs and operating time. However, in the case of 5 pairs and 9 pairs, the current efficiency was approximately 16% and 17%, respectively, when the efficiency of heavy metals separation exceeded 95% (Fig. 6). Therefore, the fewer the cell pairs, the higher the efficiency of the current, because the ion separation performance decreases at each incremental step (pair). Thus, the efficiency of the current is a measure of the electric power used for ion separation.
The loss of current efficiency is largely determined by the selectivity of the membrane and its scaling, e.g. in water movement, current leakage, and failing because of osmosis and electroosmosis. The selectivity of the ion-selective membrane strongly depends on the concentration of salt in the external solution. When this increases, the intensity of the diffusion flux from the external solution into the membrane increases. The amount of water transported because of electroosmosis is proportional to the amount of salt transported from the diluate to concentration cell, i.e. It is directly proportional to the electric current (charge) passing through the membranes.
Generally, exceeding the LCD causes concentration polarization at the surface of the ion exchange membrane [2, 32]. In electrodialysis, exceeding the LCD results in the electrical resistance of the solution increasing, leading to a drastic drop in the efficiency of the process. Additionally, the dissolution of water can cause a rapid change in pH in the solution on the ion exchange membrane surface. This change in pH can lead to the deposition of metal hydroxides at the surface of the ion exchange membrane, resulting in scaling [8]. If the range of the operating pH of the membrane is small, the membrane may be damaged, resulting in lower removal and current efficiency [2]. Additionally, the current efficiency can be reduced by the formation of additional electrically conductive paths between the manifold, the channels inside the stack, and the external electrodes [89].
### 4. Conclusions and Summary
In electrodialysis, the number of cell pairs and the applied voltage are crucial operating factors that determine the initial investment cost, energy consumption, and efficiency of metal ions separation. This study provided design elements to treat wastewater containing copper and nickel using electrodialysis.
The LCD was found to be proportional to the dilute concentration, though not in the whole range. In contrast, a linear relationship was found in the whole range of the linear flow velocity. In the 1–2 cm/s linear velocity range, coefficientsandwere measured 7.645 and 1.318, respectively. The optimum flow velocity and number of cell pairs in this experiment were 2.0 cm/s and 9, respectively. There was no significant difference in the separation rate of copper and nickel in the wastewater containing them both. Therefore, for highly concentrated wastewater, increasing the linear flow rate and applied voltage can effectively reduce the treatment time.
Regarding the efficiency of the current, because the ion transport depends on the charge supplied, according to Faraday’s law, as the processing efficiency increased, the current efficiency decreased. When the separation efficiency was over 95%, the current efficiency was as low as 16–17%. This may be because of the selectivity of the membrane, the movement of water caused by osmosis and electroosmosis, current leakage, scaling of the membrane or failure. Because the current density is the most important factor in determining operating and initial investment costs, a proper operating LCD range is of high utility for applications.
### Acknowledgment
This subject is supported by Korea Ministry of Environment (MOE) as “Technologies for the Risk Assessment & Management Program (2017000140006)” and the “Human Resource Program (Grant No. 20194010201790)” of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Ministry of Trade, Industry & Energy (MOTIE) of the Republic of Korea. This article was presented at the 2019 International Desalination Workshop (IDW2019) held on 28–30 August 2019, Jeju, Korea.
### Notes
Author Contributions
K.J.M. (Ph.D.) checked all experimental results and wrote a manuscript. J.H.K. (M.D student) conducted all the experiments. E.J.O. (M.D student) conducted all the experiments. J.H.R. (Ph.D.) supported writing manuscript. K.Y.P. (Ph.D.) approved all experimental results and modified the manuscript.
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##### Fig. 1
Photograph and schematic diagram of the electrodialysis system.
##### Fig. 2
(a) Resistance as a function of the inverse of the current and (b) Experimental and theoretical limiting current density.
##### Fig. 3
(a) LCD as a function of the molar concentration and (b) The linear flow velocity of the diluate.
##### Fig. 4
Conductivity as a function of time for different numbers of ion exchange membrane pairs.
##### Fig. 5
Concentration of (a) Copper and (b) Nickel as a function of time for different numbers of ion-exchange membrane pairs.
##### Fig. 6
Current efficiency for different numbers of ion exchange membrane pairs.
##### Table 1
Properties of the Synthetic Wastewater
Salt solution (mg/L) Main cation Main anion Linear flow velocity (cm/s)
CuSO4(20) + NiSO4(20) Cu2+, Ni2+ SO42− 1.1, 1.6, 1.7, 2.0
CuSO4(100) + NiSO4(100) Cu2+, Ni2+ SO42− 1.1, 1.6, 1.7, 2.0
CuSO4(200) + NiSO4(200) Cu2+, Ni2+ SO42− 1.1, 1.6, 1.7, 2.0
##### Table 2
Properties of the ASTOM ion Exchange Membranes Used [12]
Ion Membrane Exchange Type Thickness (mm) Electrical resistance (Ω cm2) Burst strength (kPa) Counterion transport number
AMX-SB Strongly acidic 0.14 2.4 30 > 0.98
CMX-SB Strongly basic 0.17 3.0 40 > 0.98
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The power loss due to current $I(t)=A\, \sin\omega t$ for resistor R is
$P=I^2R=A^2\, \sin^2\omega t \, R=A^2 \frac{1-\cos 2\omega t}{2} \, R$
oic, the blue line is to show [color=blue]$Power loss in R =A^2 \frac{1-\cos 2\omega t}{2} \, R$[/color]
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#### 9.2.9 NewLabelDot3D
• NewLabelDot3D( <coordinate>, <"name">, <direction> [, DrawDot, distance] ).
• Description: the argument <coordinate> is a space point, that can be in the form $M\left(x,y,z\right)$ or $\left[x+iy,z\right]$. The macro create a global variable called <"name"> with the value <coordinate>. It also create a graphical element displaying the variable name besides the point <coordinate>. The direction (in the screen plane) can be "N" for north, "NO" for north-west, "SO" for south-west, "SE" for south-east...etc, or a list in the form [longueur, direction] where direction is a complex, In that second case, the optional parameter <distance> is ignored. The point is also displayed when <DrawDot> is $1$ (default value) and the <distance> (in cm) between the point and the text can be redefined (0.25cm by default). The graphical element calls the macro LabelDot.
• That macro is associated to a button in the toolbar: Supplément 3D (more 3D tools).
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## Subintuitionistic Logics
Once the Kripke semantics for normal modal logics were introduced, a whole family of modal logics other than the Lewis systems S1 to S5 were discovered. These logics were obtained by changing the semantics in natural ways. The same can be said of the Kripke-style semantics for relevant logics: a whole range of logics other than the standard systems R, E and T were unearthed once a semantics was given. In a similar way, weakening the structural rules of the Gentzen formulation of classical logic gives rise to other “substructural” logics such as linear logic. This process of “strategic weakening” is becoming popular today, with the discovery of applications of these logics to areas such as linguistics and the theory of computation. This paper examines what the process of weakening does to the Kripke-style semantics of intuitionistic logic, introducing the family of subintuitionistic logics.
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## Solution to Measure, Integration & Real Analysis by Axler
### Chapter 1 Riemann Integration
• §1A Review: Riemann Integral
• §1B Riemann Integral Is Not Good Enough
### Chapter 2 Measures
• §2A Outer Measure on R
(#1) (#2) (#3) (#4) (#5) (#6)
• §2B Measurable Spaces and Functions
• §2C Measures and Their Properties
• §2D Lebesgue Measure
• §2E Convergence of Measurable Functions
### Chapter 3 Integration
• §3A Integration with Respect to a Measure
• §3B Limits of Integrals & Integrals of Limits
#### Linearity
This website is supposed to help you study Linear Algebras. Please only read these solutions after thinking about the problems carefully. Do not just copy these solutions.
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Question
# The binding energy per nucleon of $$^7_3 Li$$ and $$^4_2 He$$ nuclei are 5.60 MeV and 7.06 Me V, respectively. In the nuclear reaction $$^7_3 Li + ^1_1 He \longrightarrow ^4_2 H + ^4_2 He + Q$$ the value of energy Q released is
A
-2.4 MeV
B
8.4 MeV
C
17.3 MeV
D
19.6 MeV
Solution
## The correct option is C 17.3 MeVThe binding energy for $$_1H^1$$ is around zero and also not give in the question so we can ignore it,$$Q=2(4\times 7.06)-7\times (5.60)$$$$=(56.48-39.2)MeV$$$$=17.28\ MeV$$$$=17.3\ MeV$$Physics
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# PKCS#1 v1.5 encryption (RSA)¶
Warning
Use PKCS#1 OAEP (RSA) instead. This module is provided only for legacy purposes.
See RFC8017 or the original RSA Labs specification .
This scheme is more properly called RSAES-PKCS1-v1_5.
As an example, a sender may encrypt a message in this way:
>>> from Crypto.Cipher import PKCS1_v1_5
>>> from Crypto.PublicKey import RSA
>>> from Crypto.Hash import SHA
>>>
>>> message = b'To be encrypted'
>>> h = SHA.new(message)
>>>
>>> cipher = PKCS1_v1_5.new(key)
>>> ciphertext = cipher.encrypt(message+h.digest())
At the receiver side, decryption can be done using the private part of the RSA key:
>>> From Crypto.Hash import SHA
>>> from Crypto import Random
>>>
>>>
>>> dsize = SHA.digest_size
>>> sentinel = Random.new().read(15+dsize) # Let's assume that average data length is 15
>>>
>>> cipher = PKCS1_v1_5.new(key)
>>> message = cipher.decrypt(ciphertext, sentinel)
>>>
>>> digest = SHA.new(message[:-dsize]).digest()
>>> if digest==message[-dsize:]: # Note how we DO NOT look for the sentinel
>>> print "Encryption was correct."
>>> else:
>>> print "Encryption was not correct."
Crypto.Cipher.PKCS1_v1_5.new(key, randfunc=None)
Create a cipher for performing PKCS#1 v1.5 encryption or decryption.
Parameters: key (RSA key object) – The key to use to encrypt or decrypt the message. This is a Crypto.PublicKey.RSA object. Decryption is only possible if key is a private RSA key. randfunc (callable) – Function that return random bytes. The default is Crypto.Random.get_random_bytes(). A cipher object PKCS115_Cipher.
class Crypto.Cipher.PKCS1_v1_5.PKCS115_Cipher(key, randfunc)
This cipher can perform PKCS#1 v1.5 RSA encryption or decryption. Do not instantiate directly. Use Crypto.Cipher.PKCS1_v1_5.new() instead.
can_decrypt()
Return True if this cipher object can be used for decryption.
can_encrypt()
Return True if this cipher object can be used for encryption.
decrypt(ciphertext, sentinel)
Decrypt a PKCS#1 v1.5 ciphertext.
This function is named RSAES-PKCS1-V1_5-DECRYPT, and is specified in section 7.2.2 of RFC8017.
Parameters: Returns: ciphertext (bytes/bytearray/memoryview) – The ciphertext that contains the message to recover. sentinel (any type) – The object to return whenever an error is detected. A byte string. It is either the original message or the sentinel (in case of an error). If the ciphertext length is incorrect If the RSA key has no private half (i.e. it cannot be used for decyption).
Warning
You should never let the party who submitted the ciphertext know that this function returned the sentinel value. Armed with such knowledge (for a fair amount of carefully crafted but invalid ciphertexts), an attacker is able to recontruct the plaintext of any other encryption that were carried out with the same RSA public key (see Bleichenbacher’s attack).
In general, it should not be possible for the other party to distinguish whether processing at the server side failed because the value returned was a sentinel as opposed to a random, invalid message.
In fact, the second option is not that unlikely: encryption done according to PKCS#1 v1.5 embeds no good integrity check. There is roughly one chance in 216 for a random ciphertext to be returned as a valid message (although random looking).
1. Select as sentinel a value that resembles a plausable random, invalid message.
2. Not report back an error as soon as you detect a sentinel value. Put differently, you should not explicitly check if the returned value is the sentinel or not.
3. Cover all possible errors with a single, generic error indicator.
4. Embed into the definition of message (at the protocol level) a digest (e.g. SHA-1). It is recommended for it to be the rightmost part message.
5. Where possible, monitor the number of errors due to ciphertexts originating from the same party, and slow down the rate of the requests from such party (or even blacklist it altogether).
If you are designing a new protocol, consider using the more robust PKCS#1 OAEP.
encrypt(message)
Produce the PKCS#1 v1.5 encryption of a message.
This function is named RSAES-PKCS1-V1_5-ENCRYPT, and it is specified in section 7.2.1 of RFC8017.
Parameters: Returns: message (bytes/bytearray/memoryview) – The message to encrypt, also known as plaintext. It can be of variable length, but not longer than the RSA modulus (in bytes) minus 11. A byte string, the ciphertext in which the message is encrypted. It is as long as the RSA modulus (in bytes). If the RSA key length is not sufficiently long to deal with the given message.
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# A gas has a volume of 350 mL at 45°C. If the volume changes to 400 mL, what is the new temperature?
May 19, 2016
28°C.
#### Explanation:
The correct equation to use with these kinds of questions is the Combined Gas Law.
$\frac{{P}_{1} \cdot {V}_{1}}{T} _ 1 = \frac{{P}_{2} \cdot {V}_{2}}{T} _ 2$
This equation, though, is a little unnecessary at times because of the possibility of extra constants.
In this case you have a pressure constant which enables us to shorten your longer equation down to Charles' Law because pressure remains the same and doesn't affect the calculation.
$\frac{{V}_{1}}{T} _ 1 = \frac{{V}_{2}}{T} _ 2$
The next step is to isolate the variable that you want, which is, in this case, ${T}_{2}$.
${T}_{2} = \frac{{V}_{1} {T}_{1}}{V} _ 2$
Now it is time to replace your known variables with numbers
T_2 = (350mL*45°C)/(400mL)
And calculate...
T_2 = 28.125°C
Significant digits...
T_2 = 28°C
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Teletraffic engineering/What is the Engset calculation?
Summary
The Engset traffic model explores the relationship between offered traffic usually during the busy hour, the blocking that will occur in that traffic and the number of circuits provided where there number of sources from which the traffic is generated is known. It is used in place of the Erlang B traffic model in cases where the ratio of the number of sources to the number of circuits is less than 10, as the Erlang B overestimates blocking for a finite number of sources [1]. The Engset formula assumes that calls, when blocked, are cleared ( only valid if the calls are overflowed to another trunk group). It is used in applications such as small telephone systems or PBX systems, where a finite number of users have dial access [2].
Definition
The Engset formula is used to determine the blocking probability or probability of congestion occuriing within a circuit group. It is similar to the Erlang B formula but specifies a finite number of sources.It also assumes that blocked calls are cleared or overflowed to another circuit group.
Engset Calculation
The Engset calculation, was developed by Tore Olaus Engset to determine the probability of congestion occurring within a circuit group. The level of congestion can be used to determine a network's perfomance as it is measured by the grade of service. The Engset formula requires that the user knows the expected peak traffic, the number of sources and the number of circuits in the network [3].
Engset's equation is similar to the Erlang B formula except that the Erlang B assumes an infinite number to sources. In situations where you have a limited number of sources generating calls to a call centre, Erlang B can result in too high a number of circuits required [4]. Engset specifies a finite number of callers and thus would produce a more accurate result for traffic generated by a finite source [3, 4]. For a large user population, however, the Engset and the Erlang B give the same result [3].
Solving the Engset formula involves iteration in that to obtain the answer to this probability, the calculation must first determine an initial estimate. The user makes an initial guess of probability Pb, and runs the Engset formula using that guess. The process is repeated using the answer found as the new guess until it (the guess at that time) converges with the answer [3,2].
Engset formula [2]:
${\displaystyle P(b)={\frac {\left[{\frac {\left(S-1\right)!}{N!\cdot \left(S-1-N\right)!}}\right]\cdot M^{N}}{\sum _{X=1}^{N}\left[{\frac {\left(S-1\right)!}{X!\cdot \left(S-1-X\right)!}}\right]\cdot M^{X}}}}$
${\displaystyle M={\frac {A}{S-A\cdot \left(1-P(b)\right)}}}$
where
A = offered traffic in erlangs, from all sources
S = number of sources of traffic
N = number of circuits
P(b) = probability of blocking
Example
Suppose that company X has 10 telephone extensions, generating 5 erlangs of traffic for outgoing calls during the busy hour. Using the Erlang B formula for a grade of service of 1%, 11 circuits (lines) would be needed by company X. Calculations for Erlang B can be carried out using the Erlang B calculator. There are only 10 extensions so 11 lines is an overestimate.
Using the Engset formula for the same example, 9 lines would be specified which is fine as not all extensions will be used simultaneously all the time. Calculations done using the Engset calculator.
Exercises
Exercise 1
Using the Erlang B formula and the Engset formula, compare the number of lines specified in each of the calculations for the same number of sources and traffic offered for a grade of service of 0.01.
Number of sources: 10, 20, and 100.
Offered traffic = 5 erlangs.
Exercise 2
Finite source formulas such as the Engset formula have fewer applications than infinite formulas (Erlang). Another finite source formula is the Binomial formula. It differs from the Engset formula in that it uses traffic per source rather than the total traffic from all sources. It also assumes that traffic is queued rather cleared when blocked as does the Engset formula [2].
The Binomial finite source formula is given by [2]:
${\displaystyle {\begin{matrix}\mathbf {Pb=\sum _{x=N}^{S-1}~{\frac {(S-1)!}{x!(S-1-X)!}}~A^{x}(1-A)^{(S-1-X)}} &{\begin{array}{l}\mathbf {Where:} \\\mathbf {\quad A=Offered~Erlangs~per~source} \\\mathbf {\quad S=Number~of~sources} \\\mathbf {\quad N=Number~of~servers} \\\mathbf {\quad Pb=Probability~of~blcoking} \end{array}}\end{matrix}}}$
Compare the probabilities for the Binomial and Engset calculations for the same number of sources for traffic offered equal to 2 erlangs, with 5 available lines and a grade of service of 0.02.
Number of sources: 10, 20, and 30.
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# Find the characteristic of $Z_n \times Z_m$:
so I was given the problem: find the characteristic of $Z_3\times Z_4$ and I got $\operatorname{char}(Z_3\times Z_4)=12$, is it true that for any $Z_n \times Z_m$, $\operatorname{char}(Z_n \times Z_m)=n*m$???
so for instance, does $\operatorname{char}(Z_2\times Z_3)=6$?
For $\operatorname{char}(Z_{10}\times Z_{20})$ would you take 20*10, or 10 because thats the GCF, or 2 because thats the lowest common factor????
Thanks!
• A tip: When you hover your mouse on top of a tag, you get a brief description of what it means. Doing that here and reading it would reveal that [tag: characteristic-function] is not appropriate for your question. – Jyrki Lahtonen Dec 15 '14 at 6:14
• But have you tried applying the definition of characteristic? What problems did you encounter while doing that? – Jyrki Lahtonen Dec 15 '14 at 6:24
Hint: I'd say $char(\mathbb{Z}_m\times \mathbb{Z}_n)=lcm(m,n)$. Can you see why?
Note $\mathbb{Z}_m\times \mathbb{Z}_n$ is a ring with unity $(1,1)$, so if we can find the characteristic of $(1,1)$, then we are done.
Let $k$ be its characteristic.
Then, $k(1,1)=(0\pmod m,0\pmod n)$
$(k,k)=(0\pmod m,0\pmod n)$
$m|k$ and $n|k\Rightarrow lcm(m,n)|k$
It is fairly simple to show that $k|lcm(m,n)$
• So for Z10*Z20 it would be 10? – Kaitlyn Dec 15 '14 at 6:22
• $lcm(10,20)=20$ – Swapnil Tripathi Dec 15 '14 at 6:23
• Sorry, I misread the beginning of the post... that makes sense! – Kaitlyn Dec 15 '14 at 6:26
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Integrated rate law
1. Ukitake Jyuushirou
123
this seems a fairly straight forward qn but the ans at the back of the book does not agree with my ans :(
we have a rate constant 5e-2 mol/L
inital concentration of 1e-3 M
calculate concentration after the time elapsed is 5e-3 s
my ans is 7.5e-4 but the ans at the back claims it is 2.5e-4
formula i using is the integrated rate law [A] = -kt + [A initial]
did i miss something?
2. siddharth
1,191
The units of your rate constant aren't correct. It should be mole L-1 sec-1.
Since it's a zero order reaction, your rate law is correct. Check your calculations.
3. Ukitake Jyuushirou
123
yea, thanks i manage to fig this one out :D
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# Problem
Say you have N stacks named S1 through SN, where each Sk (k = 1 to N) contains N copies of the number k.
For example, when N = 3 the stacks looks like this:
1 2 3 <- top of stack
1 2 3
1 2 3 <- bottom of stack
=======
1 2 3 <- stack index
Here there are 3 stacks indexed as 1, 2, and 3, and each one contains N instances of its own index.
The goal is to rearrange the N stacks such that each of them identically contains the numbers 1 through N in order from top to bottom.
e.g. for N = 3 the goal is to rearrange the stacks into:
1 1 1
2 2 2
3 3 3
=======
1 2 3
The only action you can perform with the stacks is taking the top number from one of the stacks (popping) then immediately placing it on top of a different stack (pushing). This is subject to these stipulations:
• A number can only be pushed onto a stack if it is less than or equal to the top number on that stack.
• e.g. a 1 can be pushed onto a stack with a 1, 2, or 3 at the top, but a 2 can only be pushed onto a stack with a 2 or 3 (or higher) at the top.
• This has the effect that stacks are always monotonically increasing from top to bottom.
• Any nonempty stack may be popped from, and, assuming the previous bullet is satisfied, any stack may be pushed to.
• Any number may be pushed onto an empty stack.
• Stacks have no maximum height limit.
• Stacks cannot be created or destroyed, there are always N of them.
This challenge is about deciding which pops and pushes to do in order to complete the stack exchange, not necessarily in the fewest moves, but in a surefire way.
(Practicing with a deck of cards is a good way to get a feel for the problem.)
# Challenge
Write a program or function that takes in a positive integer N, guaranteed to be 3 or above. Print or return a string that denotes all the pop-push actions required to rearrange the stacks from the initial state:
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
=============
1 2 3 4 5
(N = 5 case)
To the final state:
1 1 1 1 1
2 2 2 2 2
3 3 3 3 3
4 4 4 4 4
5 5 5 5 5
=============
1 2 3 4 5
Every line in your output must contain two numbers separated by a space. The first number is the index of the stack to pop from and the second number is the index of the stack to push to. Performing the actions of all the lines in order should arrange the stacks correctly without breaking any rules.
For example, here is a potential valid output for the N = 3 case:
1 2 [move the top number on stack 1 to the top of stack 2]
1 2 [repeat]
1 2 [repeat]
3 1 [move the top number on stack 3 to the top of stack 1]
2 3 [etc.]
2 3
2 3
2 1
2 1
2 1
3 1
3 1
3 1
3 2
1 2
1 2
1 2
1 3
2 3
2 3
2 3
1 2
3 2
3 1
### Notes
• Your output does not need to be optimal, only correct. i.e. you do not need to minimize the number of pops and pushes.
• So it would be alright if, say, some move were repeatedly made and immediately reversed.
• Popping and pushing to the same stack in one move, e.g. 2 2, is allowed as well (though of course pointless).
• Your output does need to be deterministic and finite.
• Remember that stacks have 1-based indexing. 0-based indexing is not allowed.
• N greater than 9 should of course work just as well as single digit N.
• If desired you may use any two distinct, non-digit printable ASCII characters in place of spaces and newlines. A trailing newline (or newline substitute) in the output is fine.
# Scoring
The shortest code in bytes wins. Tiebreaker is higher voted answer.
Valueless brownie points if you can show your algorithm is optimal.
• Stop with the "extra points for small things" nonsense >_> May 30 '16 at 17:02
• @zyabin101 You just lost any chance at brownies. May 30 '16 at 17:04
• You always come up with such wonderful titles! May 30 '16 at 17:27
• @HelkaHomba -._(._.)_.- May 30 '16 at 18:18
• Is the possible output you include for the case of N=3 optimal? Jun 5 '16 at 23:17
# Pyth 96 94 bytes
Mt*Q+++bGdHM|%+y_GHQQg1 2++Qd1g2 3g2 1g3 1++Qd2Vr3QgNtN++QdN;g1QVStQVStQI<NHgnNHnNtH)++nN0dnNH
Try it here
## How does it work?
This explanation will be using N=5.
### Part 1: Create the bottom layer on every stack
The reason why this is needs a separate piece of code is because every stack needs to be used: the first 4 need a 5 to be put beneath them, and the last stack must provide the 5s. This means that we can't just move all the 4s somewhere, put a 5 there, and move the 4s back.
Visualization: (parentheses mean what will be moved)
_
11111 |
22222 |_ Can't move 4s here, not monotonically increasing
33333_|
(44444)------------??? Where to put the 4s?
55555 <- Must supply the 5 that will be moved
Instead, to do this first exchange, we will first move all the 1s over to to the second stack, move a 5 to the first stack (which is now empty), move the 1s to the third stack, move the 2s to the first stack, move the 1s back to the first stack, and finally move a 5 to the second stack.
(11111)-----.
2222211111<-'
===============================
5<---------.
2222211111 : (from stack 5)
===============================
5
22222(11111)-.
3333311111<--'
===============================
522222<-.
(22222)-'
3333311111
===============================
52222211111<-.
|
33333(11111)-'
===============================
52222211111
5<-----.
33333 |
44444 |
555(5)-'
Now that we have a free space to move stacks into (stack 2, which only contains a 5 that is placed in the right spot), we can move all the 3s to stack 2 and place a 5 in stack 3. We can then repeat the same thing for stack 4, and now we get all the 5s in the right place! And just one more thing: we will move all the 1s to stack 5 so that we get a nice setup for the next stack exchange.
522222(11111)-.
533333 |
544444 |
5 |
511111<-------'
## Part 2: Do everything else :)
This is much easier now, because now we will always have a free stack to move other numbers we need to juggle around into. So, first we figure out where the 4 is. A bit of examination will show that it will always be 1 up from where it started, or 2 above the last stack. Now, we just keep going down the stacks, placing a 4 in the stack if it's free, or moving the other numbers up 1 stack if it's not. Now we have all the 4s in place.
522222<------.
533333<----. |
544444-.-.-'-'
5<-----' |
511111<--'
===============================
5433333
54
54
5411111
5422222
Now, we realize that the 3s are 2 stacks above where the 4s where. This means that we can do the exact same thing we did with the 4s! And as it turns out, we can keep doing this as long as we wrap the stack index around to the other side.
5433333-'wrap around 543
54 543
54 54311111
5411111 .----------->54322222
5422222 |2 stacks up 543
And so, we can keep doing this until we have exchanged all the stacks.
## Code explanation:
### First of all: The (important) predefined variables.
Q: Evaluated input.
b: The newline character, '\n'
d: A space, ' '
### There are 2 lambda definitions.
M | g(G)(H), used for moving Q numbers at a time.
| We will call these Q numbers a "(number) block"
t | Tail, used to remove beginning newline
*Q | Repeat the following Q times
+++bGdH | '\n' + G + ' ' + H. Just a whole bunch of concatenating.
|
M | n(G)(H), used for figuring out which stacks to move from
| Q | If the following code is 0 (false), then use Q instead
% Q | Mod Q
y | Multiply by 2
_G | Negate (remember in the explanation part 2? Always 2 stacks above?)
### The stack exchanging: part 1
g1 2 | Move the 1 block to stack 2
++Qd1 | Move a Q to stack 1
g2 3 | Move the 1 block to stack 3
g2 1 | Move the 2 block to stack 1
g3 1 | Move the 1 block back to stack 1
++Qd2 | Move a Q to stack 2
v---Code-continuation---' |I don't have enough room!!!
Vr3Q | For N in range(3, Q)
gNtN | Move the number block in stack N up 1
++QdN | Move a Q to stack N
;g1Q | End for loop; move the 1 block to the last stack
### The stack exchanging: part 2
VStQ | For N in [1, 2, ..., Q - 1]
VStQ | For H in [1, 2, ..., Q - 1]
I<NH | If N < H
g | Number block move
nNH | (find number block)
nNtH | (find the previous stack)
) | End "For H"
++nN0dnNH | Find start, move number to next location down
I already know I'm not getting brownie points, cuz I can see many more efficient and more complicated methods :(
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# S/PHI/nX Binary Distribution (*.deb)
Written by sixten on . Posted in All Posts, S/PHI/nX
1st S/PHI/nX Binary Distribution released.
The object-oriented Density-Functional-Theory program package S/PHI/nX and its underlying high-performance general purpose C++ framework SxAccelerate have been released recently as open-source. Although building the libraries is straightforward, Gemmantics creates and maintains binary packages of both S/PHI/nX and SxAccelerate. In the upcoming weeks we will deploy native installers for various operating systems. By using the binary packages you can start working with S/PHI/nX or developing with SxAccelerate within one or two minutes.
The first operating system which we support for shipping S/PHI/nX binaries is Debian (others will follow). Debian’s native package management system is apt-get, packages can be installed with the dpkg program.
We provide two versions:
1. The complete S/PHI/nX program package and
2. a single bundle containing only the SxAccelerate framework.
The complete S/PHI/nX program package provides
• SxAccelerate,
• S/PHI/nX libraries,
• Intel MKL runtime environment,
• S/PHI/nX executable, and
The second package provides only the SxAccelerate framework in both Debug and Release mode. The download of this package is mainly interesting for C++ developers.
Notes about installing Debian along with building instructions for S/PHI/nX can be found in the previous blog.
# Installing
You can either install the full S/PHI/nX program package or only the programming environment SxAccelerate. Either way, just execute the following commands under root privileges:
# install netCD apt-get install libgfortran3 libnetcdf6 # download package wget http://download.gemmantics.com\ /sx/Debian/6.0/x86_64\ /sphinx_2.0.1-2_amd64.deb # install S/PHI/nX package dpkg -i sphinx_2.0.1-2_amd64.deb # install netCDF apt-get install libgfortran3 libnetcdf6 # download package wget http://download.gemmantics.com\ /sx/Debian/6.0/x86_64\ /sxaccelerate_1.0.1-1_amd64.deb # install S/PHI/nX package dpkg -i sxaccelerate_1.0.1-1_amd64.deb
# First Steps with S/PHI/nX
After installing the complete S/PHI/nX the package you can start to perform the first “hello world” calculation. The S/PHI/nX distribution comes with a few example files which can be found in:
/opt/sphinx/VERSION/share/examples
/opt/sxaccelerate/VERSION/share/examples
In order to perform the first test calculation with S/PHI/nX please copy one or all of the example folders to your home directory:
# prepare some working environment
cd ~
mkdir sxwork
cd sxwork
# copy examples to ~/sxwork
cp -r /opt/sphinx/VERSION/share/examples/* .
# Input
Let’s begin with a very simple example, namely the computation of the total energy of gallium arsenide bulk (GaAs). Please copy the folder GaAs2-bulk-CG to your work directory.
cd GaAs2-bulk-CG
vim input.sx
S/PHI/nX reads typically from the input.sx located in the current working directory.
format sphinx;
include ;
project = "Test: GaAs bulk";
aLat = 10.2;
pseudoPot {
species { include "ga-lda-ham.sx"; }
species { include "as-lda-ham.sx"; }
}
structure {
include <structures/fcc.sx>;
species {
atom { coords = [0,0,0]; relative; }
}
species {
atom { coords = [1/4,1/4,1/4]; relative; }
}
}
basis {
eCut = 20; // Ry
kPoint { coords = [1/2, 1/2, 1/2]; weight = 1; relative; }
folding = 4* [1, 1, 1];
}
PWHamiltonian {
nEmptyStates = 0;
ekt = 0;
xc = LDA;
}
initialGuess {
waves { lcao { maxSteps=1; rhoMixing=0; } }
rho { atomicOrbitals; }
}
main {
CCG {
dEnergy=1e-7;
finalDiag;
}
}
Every S/PHI/nX program begins with the header statement “format sphinx;” (line 1). The S/PHI/nX file format is a markup-language defined as a set of hierarchal groups. In this example there are groups to specify the atomic structure (line 12) and the corresponding potentials (line 7). GaAs bulk crystalizes in the Zincblende structure. The plane-wave basis configuration can be found in lines 23-27. The Hamiltonian (line 29-33) will use LDA as exchange-correlation functional. In the main loop (lines 40-45) the conjugate-gradient method will be applied to compute the total energy.
# Launch
After inspecting the input file S/PHI/nX can be started by running the following command:
# compute the total energy of GaAs bulk
/opt/sphinx/VERSION/bin/sphinx --log
The command line argument “--log” redirects all output to stdout to the log file “sphinx.log”. The computation takes only a few seconds.
# Output
During the computation S/PHI/nX generates a variety of output files:
The main log file is sphinx.log. It contains all relevant information of the entire run.
Furthermore, S/PHI/nX computes the self-consistent electronic charge density rho.sxb and corresponding wave functions waves.sxb.
The computed one-particle spectra can be found in eps.dat.
The energy convergence can be inspected from the file energy.dat as can be seen from this figure.
A S/PHI/nX run may generate additional output files depending on the configuration provided in the input file.
# Next steps
The S/PHI/nX Wiki contains more detailed information about the input file and usage of S/PHI/nX.
Please also consider to subscribe to the S/PHI/nX mailing lists where all questions concerning S/PHI/nX can be posted. You can join the S/PHI/nX discussion forum by sending a blank email to
sxusers-subscribe at mpie dot de.
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ale we lengths of the sides of a ma...
Question
ale we lengths of the sides of a mangley, um 4. A, B, C be three sets such that n(A) = 2. n(B) = 3, n(C) = 4. If n(P(P(C))) n(PIP(A))) n(P(P(B))). Sum of digits of Kis M(A) = 2, n(B) = 3, n(C) = 4. If PIX) denotes power set of X
JEE/Engineering Exams
Maths
Solution
122
4.0 (1 ratings)
( n(c)=4 ) ( n(P(c))=2^{4}=16 ) ( n(P(P(C)))=2^{16} ) ( n(A)=2 ) ( n(P(A))=2^{2}=4 ) ( n(P(P(A)))=2^{4} ) ( n(B)=3 ) ( n(P(B))=2^{3}=8 ) ( k=nleft(P(P(B))=2^{8}right. ) ( n(P(P(A))) n(P(P(B)))_{16}=frac{2^{16}}{2^{4} cdot 2^{8}}=frac{2^{16}}{2^{12}} )
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## Test de Connaissance du Français (TCF) Practice Tests, Manual & Exam Preparation Resources
Posted: 21st April 2011 by admin in Other
Tags: , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,
Free Sample Questions, Test Manual & Materials for Test de Connaissance du Français (TCF)
Every year, many of medical students and graduates, as well as those from other fields tend to continue their studies or immigrate to francophone countries, including immigration to Quebec, Canada. That’s why, the TCF exam might be a requirement for them. We are going to provide them with the opportunity to download free materials for their preparation for TEF exam.
All applicants who want to sit for the Test de Connaissance du Français (TCF) or TEFAQ, and look for the Sample TCF Questions, can download TCF questions for Free easily here and also here.
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We hope this helps you with your TCF exam preparation! Good luck!
Donc, familiarisez-vous avec les questions proposées dans les sessions du TCF ici! Bonne Chance!
1. kulwinder says:
It is stated that I have passed DELF A1 and A2 now I intend to appear in TCF exame, inorder to get immigration visa for Qubec, Canada.
It is therefore requested to please provide me free sample of test material and Practice guide / Book.
Thanks
kulwinder
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Introduction
2019-03-22
This package tries to smooth over some of the differences in encryption approaches (symmetric vs. asymmetric, sodium vs. openssl) to provide a simple interface for users who just want to encrypt or decrypt things.
The scope of the package is to protect data that has been saved to disk. It is not designed to stop an attacker targeting the R process itself to determine the contents of sensitive data. The package does try to prevent you accidentally saving to disk the contents of sensitive information, including the keys that could decrypt such information.
This vignette works through the basic functionality of the package. It does not offer much in the way of an introduction to encryption itself; for that see the excellent vignettes in the openssl and sodium packages (see vignette("crypto101") and vignette("bignum") for information about how encryption works). This package is a wrapper around those packages in order to make them more accessible.
Keys and the like
To encrypt anything we need a key. There are two sorts of key “types” we will concern ourselves with here “symmetric” and “asymmetric”.
• “symmetric” keys are used for storing secrets that multiple people need to access. Everyone has the same key (which is just a bunch of bytes) and with that we can either encrypt data or decrypt it.
• a “key pair” is a public and a private key; this is used in communication. You hold a private key that nobody else ever sees and a public key that you can copy around all over the show. These can be used for a couple of different patterns of communication (see below).
We support symmetric keys and asymmetric key pairs from the openssl and sodium packages (which wrap around industry-standard cryptographic libraries) - this vignette will show how to create and load keys of different types as they’re used.
The openssl keys have the advantage of a standard key format, and that many people (especially on Linux and macOS) have a keypair already (see below if you’re not sure if you do). The sodium keys have the advantage of being a new library, starting from a clean slate rather than carrying with it accumulated ideas from the last 20 years of development.
The idea in cyphr is that we can abstract away some differences in the types of keys and the functions that go with them to create a standardised interface to encrypting and decrypting strings, R objects, files and raw vectors. With that, we can then create wrappers around functions that create files and simplify the process of adding encryption into a data workflow.
Below, I’ll describe the sorts of keys that cyphr supports and in the sections following describe how these can be used to actually do some encryption.
Symmetric encryption
This is the simplest form of encryption because everyone has the same key (like a key to your house or a single password). This raises issues (like how do you store the key without other people reading it) but we can deal with that below.
openssl
To generate a key with openssl, you can use:
k <- openssl::aes_keygen()
which generates a raw vector
k
## aes raw 45:31:02:6f:2b:50:2a:3b:57:6e:f6:1a:a1:da:cb:cf
(this prints nicely but it really is stored as a 16 byte raw vector).
The encryption functions that this key supports are openssl::aes_cbc_encrypt, openssl::aes_ctr_encrypt and openssl::aes_gcm_encrypt (along with the corresponding decryption functions). The cyphr package tries to abstract this away by using a wrapper cyphr::key_openssl
key <- cyphr::key_openssl(k)
key
## <cyphr_key: openssl>
With this key, one can encrypt a string with cyphr::encrypt_string:
secret <- cyphr::encrypt_string("my secret string", key)
and decrypt it again with cyphr::decrypt_string:
cyphr::decrypt_string(secret, key)
## [1] "my secret string"
See below for more functions that use these key objects.
sodium
The interface is almost identical using sodium symmetric keys. To generate a symmetric key with libsodium you would use sodium::keygen
k <- sodium::keygen()
This is really just a raw vector of length 32, without even any class attribute!
The encryption functions that this key supports are sodium::data_encrypt and sodium::data_decrypt. To create a key for use with cyphr that knows this, use:
key <- cyphr::key_sodium(k)
key
## <cyphr_key: sodium>
This key can then be used with the high-level cyphr encryption functions described below.
Asymmetric encryption (“key pairs”)
With asymmetric encryption everybody has two keys that differ from everyone else’s key. One key is public and can be shared freely with anyone you would like to communicate with and the other is private and must never be disclosed.
In the sodium package there is a vignette (vignette("crypto101")) that gives a gentle introduction to how this all works. In practice, you end up creating a pair of keys for yourself. Then to encrypt or decrypt something you encrypt messages with the recipient’s public key and they (and only they) can decrypt it with their private key.
One use for asymmetric encryption is to encrypt a shared secret (such as a symmetric key) - with this you can then safely store or communicate a symmetric key without disclosing it.
openssl
Let’s suppose that we have two parties “Alice” and “Bob” who want to talk with one another. For demonstration purposes we need to generate SSH keys (with no password) in temporary directories (to comply with CRAN policies). In a real situation these would be on different machines (Alice has no access to Bob’s key!) and these keys would be password protected.
path_key_alice <- cyphr::ssh_keygen(password = FALSE)
path_key_bob <- cyphr::ssh_keygen(password = FALSE)
Note that each directory contains a public key (id_rsa.pub) and a private key (id_rsa).
dir(path_key_alice)
## [1] "id_rsa" "id_rsa.pub"
dir(path_key_bob)
## [1] "id_rsa" "id_rsa.pub"
Below, the full path to the key (e.g., .../id_rsa) could be used in place of the directory name if you prefer.
If Alice wants to send a message to Bob she needs to use her private key and his public key
pair_a <- cyphr::keypair_openssl(path_key_bob, path_key_alice)
pair_a
## <cyphr_keypair: openssl>
with this pair she can write a message to “bob”:
secret <- cyphr::encrypt_string("secret message", pair_a)
The secret is now just a big pile of bytes
secret
## [1] 58 0a 00 00 00 02 00 03 05 02 00 02 03 00 00 00 02 13 00 00 00 04 00
## [24] 00 00 18 00 00 00 10 63 6b d1 89 27 49 74 84 ed 7b 19 ce 74 4a 43 c2
## [47] 00 00 00 18 00 00 01 00 7c 1b f9 17 f8 2c 08 75 9a bf e0 8e c4 9a 0b
## [70] 37 9d f1 c1 9f c8 98 a8 21 b0 2c c6 23 5e 64 4b 8d df b5 59 40 ba c2
## [93] 75 f9 8a 35 2b a9 af d4 5d f1 c9 73 2f 13 c1 34 a4 7f af 38 e9 b8 bc
## [116] 90 83 a1 cb 7f 29 32 c4 92 0c 0d 37 2c 2d a3 a9 0f fd 54 80 b2 71 eb
## [139] 8c 0d c5 8a d5 d3 96 42 41 d4 c6 d1 27 91 10 96 70 e9 48 25 04 97 78
## [162] 4b bb f2 a9 e2 5e 9d a5 ef 2e d4 f6 d1 2e 2b e3 98 f8 4f 3b 4b ac e5
## [185] 25 02 45 43 91 92 19 fc 29 51 4f a1 66 64 65 cd bc 8e dc d6 3d f2 d3
## [208] 5e d1 34 8f 25 8f b1 5f 0c b7 0b 6c 41 74 71 71 9c e8 b0 45 42 d0 d8
## [231] de ae 47 6d 35 ea ac a5 82 69 dc 38 1a e1 0b 68 04 71 94 94 e6 cd 8e
## [254] 84 df 35 0f 0d 64 ef 10 b3 cb 25 f7 b1 fc 5b cb f0 0b e2 76 27 e1 b7
## [277] a3 25 0d 28 7d 98 56 fe 19 e2 f9 e9 19 b4 24 78 02 fe 29 4a 04 d6 79
## [300] 7d ab 79 d1 56 1f be e5 e3 2a 38 00 00 00 18 00 00 00 10 5b 53 fa c5
## [323] 7c 23 8b 13 d8 b3 98 f4 59 20 86 e3 00 00 00 18 00 00 01 00 68 ba 10
## [346] 21 9c e9 67 05 dc b9 5e bc b2 87 0c 3a 1b 59 2d 5c a9 06 56 ef 68 d6
## [369] 97 94 65 ca 1a 39 bf 1c b8 fc 53 e7 de 2a 11 2e b5 04 a5 8e 24 45 a3
## [392] 32 02 46 b7 10 0a 2b 4f 44 f1 2b 0b 55 77 d8 ac b7 a1 71 15 51 e7 cc
## [415] 1f b9 ae 07 43 a4 78 2c 9b 10 a5 97 0d fe 02 9e 02 25 6f 4e 63 53 85
## [438] b6 8f 1e 91 60 be ad d6 fd 7b 40 75 aa f6 fd fb 0e 9f 89 13 96 3d c9
## [461] e4 c6 3c 35 e7 ad 27 b6 6b 82 55 b5 11 ad 98 5c d1 bc 27 04 5a 99 cb
## [484] 35 94 51 58 4b 51 6c fe 4f 2c 1f 19 8a 7a 66 8c 0f 92 3e b9 f3 59 09
## [507] fc a5 7d 49 bf de a5 83 7e a3 21 b0 e7 1a 5f cf b8 e0 a9 7c 68 7b 1d
## [530] 45 70 77 34 c6 1f d5 d3 00 60 a8 86 6a 52 4d a0 14 94 3f 00 13 08 ea
## [553] 32 6c 06 2c f2 a6 af b3 93 04 ed 4b 17 87 b8 70 8d f7 e4 41 37 78 2f
## [576] 9a 74 6b d9 5d 70 f2 6e 50 cd 6d 7b 98 27 1a 5e 30 70 c9 d4 ac d8 a4
## [599] 00 00 04 02 00 00 00 01 00 04 00 09 00 00 00 05 6e 61 6d 65 73 00 00
## [622] 00 10 00 00 00 04 00 04 00 09 00 00 00 02 69 76 00 04 00 09 00 00 00
## [645] 07 73 65 73 73 69 6f 6e 00 04 00 09 00 00 00 04 64 61 74 61 00 04 00
## [668] 09 00 00 00 09 73 69 67 6e 61 74 75 72 65 00 00 00 fe
Note that unlike symmetric encryption above, Alice cannot decrypt her own message:
cyphr::decrypt_string(secret, pair_a)
## Error: OpenSSL error in RSA_padding_check_PKCS1_type_2: pkcs decoding error
For Bob to read the message, he uses his private key and Alice’s public key (which she has transmitted to him previously).
pair_b <- cyphr::keypair_openssl(path_key_alice, path_key_bob)
With this keypair, Bob can decrypt Alice’s message
cyphr::decrypt_string(secret, pair_b)
## [1] "secret message"
And send one back of his own:
secret2 <- cyphr::encrypt_string("another message", pair_b)
secret2
## [1] 58 0a 00 00 00 02 00 03 05 02 00 02 03 00 00 00 02 13 00 00 00 04 00
## [24] 00 00 18 00 00 00 10 9d bb cf c7 59 6a 7a 44 d6 c1 1d db 07 1f c1 d5
## [47] 00 00 00 18 00 00 01 00 1a 44 92 13 4f 25 6d 89 a0 60 2b 16 d9 f5 a8
## [70] 1d a4 d3 f8 90 94 93 fd b8 b2 59 f9 e1 80 86 61 77 73 0c b7 43 e0 61
## [93] de 10 9f ad ac f5 ee 8e 4c 61 67 e0 b4 3e 91 51 e0 6b d2 32 ff a9 33
## [116] f5 52 2c 32 9d 75 34 56 bf e5 2b d2 df d3 09 d2 05 bd 70 75 28 14 00
## [139] 9f 27 3d 6a bf 26 88 d4 bc a3 61 2f 24 b6 0d ba da 71 e7 f2 af f9 c9
## [162] dd 07 79 8f 63 3e 5c f1 f2 62 4e ce 25 c6 24 b3 77 d4 1d 62 92 db 5b
## [185] 7e b0 75 1a 5e 2f 5c 2a 33 46 64 f4 79 f2 36 2e 9e 8d 76 c9 9c bf 63
## [208] 94 3e 05 3e 1f 90 77 7c 7d ab c5 28 16 d9 f9 3b 22 3a 7a 48 f8 21 0d
## [231] 52 2d 18 47 ff 76 1d 67 56 e9 e5 f2 66 94 91 b7 18 89 56 ce 35 41 84
## [254] 96 b5 21 ad 83 59 a4 f3 16 25 d6 17 6b ed 72 6c 23 af 51 97 3a 70 e1
## [277] ee c5 82 41 ca 36 7c f2 5f a7 0a 27 d2 2e 99 e6 8e 03 e3 f4 84 27 64
## [300] 1b 30 87 a3 4e 5d 27 38 01 f5 1f 00 00 00 18 00 00 00 10 56 5e fc 6a
## [323] 14 3b 63 3e 04 c2 62 c5 b1 9e 4c 9a 00 00 00 18 00 00 01 00 9b 1b 11
## [346] 23 fb 62 67 68 2f 26 7b e1 45 90 63 a4 77 d8 6f 87 16 8b c1 11 b3 77
## [369] c6 db d0 3b 85 21 19 b5 18 e4 56 5a 62 82 7a 09 15 d5 17 78 b1 a9 34
## [392] 33 1d e7 b1 23 95 b0 62 f6 62 64 6b df 85 ea 75 dd de 9d 60 db e6 d4
## [415] c6 fb b3 1f 76 60 44 d8 4a cb 28 47 35 7c 01 fb 69 88 b6 9a 6d f7 f5
## [438] ce c7 40 9c 6b 4a e5 3f 2a 0c 12 76 f3 e8 52 81 9f 9d a7 fb e9 2c 7d
## [461] 1a b4 50 5e 29 3e d2 9f 9b 9a ec 47 64 28 95 6e be c5 f6 d7 d4 04 02
## [484] 96 b5 df 4e 48 da 6c 77 85 ea f7 f1 c9 10 75 36 23 91 5b bb 7f 31 49
## [507] b0 24 c8 b7 f3 9a a6 fb 01 73 fc 08 08 ca bc 24 fd ea bc 9b a2 a2 ce
## [530] a3 d4 3c 4d f4 d3 58 d0 a2 16 96 fc e7 2b 19 7f fd c0 41 40 93 0d 12
## [553] 05 3a 49 4b 5b d9 fd b2 11 00 08 a1 f5 f1 58 c2 a3 7a bf f0 7c f5 ae
## [576] cb c7 13 48 75 5f af d1 01 03 4e 05 79 92 36 12 27 6c 0d bd 1a 62 8d
## [599] 00 00 04 02 00 00 00 01 00 04 00 09 00 00 00 05 6e 61 6d 65 73 00 00
## [622] 00 10 00 00 00 04 00 04 00 09 00 00 00 02 69 76 00 04 00 09 00 00 00
## [645] 07 73 65 73 73 69 6f 6e 00 04 00 09 00 00 00 04 64 61 74 61 00 04 00
## [668] 09 00 00 00 09 73 69 67 6e 61 74 75 72 65 00 00 00 fe
which she can decrypt
cyphr::decrypt_string(secret2, pair_a)
## [1] "another message"
Chances are, you have an openssl keypair in your .ssh/ directory. If so, you would pass NULL as the path for the private (or less usefully, the public) key pair part. So to send a message to Bob, we’d include the path to Bob’s public key.
pair_us <- cyphr::keypair_openssl(path_key_bob, NULL)
This all skips over how Alice and Bob will exchange this secret information. Because the secret is bytes, it’s a bit odd to work with. Alice could save the secret to disk with
secret <- cyphr::encrypt_string("secret message", pair_a)
path_for_bob <- file.path(tempdir(), "for_bob_only")
writeBin(secret, path_for_bob)
And then send Bob the file for_bob_only (over email or any other insecure medium).
and bob could read the secret in with:
secret <- readBin(path_for_bob, raw(), file.size(path_for_bob))
cyphr::decrypt_string(secret, pair_b)
## [1] "secret message"
As an alternative, you can “base64 encode” the bytes into something that you can just email around:
secret_base64 <- openssl::base64_encode(secret)
secret_base64
## [1] "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"
This can be converted back with openssl::base64_decode:
identical(openssl::base64_decode(secret_base64), secret)
## [1] TRUE
Or, less compactly but also suitable for email, you might just convert the bytes into their hex representation:
secret_hex <- sodium::bin2hex(secret)
secret_hex
## [1] "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"
and the reverse with sodium::hex2bin:
identical(sodium::hex2bin(secret_hex), secret)
## [1] TRUE
(this is somewhat less space efficient than base64 encoding.
As a final option, you can just save the secret with saveRDS and read it in with readRDS like any other option. This will be the best route if the secret is saved into a more complicated R object (e.g., a list or data.frame).
See the other cyphr vignette (vignette("data", package = "cyphr")) for a suggested workflow for exchanging secrets within a team, and the wrapper functions below for more convenient ways of working with encrypted data.
Do you already have an ssh keypair? To find out, run
cyphr::keypair_openssl(NULL, NULL)
One of three things will happen:
1. you will be prompted for your password to decrypt your private key, and then after entering it an object <cyphr_keypair: openssl> will be returned - you’re good to go!
2. you were not prompted for your password, but got a <cyphr_keypair: openssl> object. You should consider whether this is appropriate and consider generating a new keypair with the private key encrypted. If you don’t then anyone who can read your private key can decrypt any message intended for you.
3. you get an error like Did not find default ssh public key at ~/.ssh/id_rsa.pub. You need to create a keypair.
To create a keypair, you can use the cyphr::ssh_keygen() function as
cyphr::ssh_keygen("~/.ssh")
This will create the keypair as ~/.ssh/id_rsa and ~/.ssh/id_rsa.pub, which is where cyphr will look for your keys by default. See ?ssh_keygen for more information. (On Linux and macOS you might use the ssh-keygen command line utility. On windows, PuTTY has a utility for creating keys.)
sodium
With sodium, things are largely the same with the exception that there is no standard format for saving sodium keys. The bits below use an in-memory key (which is just a collection of bytes) but these can also be filenames, each of which contains the contents of the key written out with writeBin.
First, generate keys for Alice:
key_a <- sodium::keygen()
pub_a <- sodium::pubkey(key_a)
the public key is derived from the private key, and Alice can share that with Bob. We next generate Bob’s keys
key_b <- sodium::keygen()
pub_b <- sodium::pubkey(key_b)
Bob would now share is public key with Alice.
If Alice wants to send a message to Bob she again uses her private key and Bob’s public key:
pair_a <- cyphr::keypair_sodium(pub_b, key_a)
As above, she can now send a message:
secret <- cyphr::encrypt_string("secret message", pair_a)
secret
## [1] f3 ac 15 fa f7 d1 cc 3a 39 35 88 a0 ff 5a be 2a c2 38 08 22 f8 7e b1
## [24] cf 32 9e 71 68 38 f9 7c 85 93 bc 5e 4e 84 8e 7c 9e 2c 41 1e 42 18 2b
## [47] 7b 37 cf 85 8d 84 b0 cb
Note how this line is identical to the one in the openssl section.
To decrypt this message, Bob would use Alice’s public key and his private key:
pair_b <- cyphr::keypair_sodium(pub_a, key_b)
cyphr::decrypt_string(secret, pair_b)
## [1] "secret message"
Encrypting things
Above, we used cyphr::encrypt_string and cyphr::decrypt_string to encrypt and decrypt a string. There are several such functions in the package that encrypt and decrypt
• R objects encrypt_object / decrypt_object (using serialization and deserialization)
• strings: encrypt_string / decrypt_string
• raw vectors: encrypt_data / decrypt_data
• files: encrypt_file / decrypt_file
For this section we will just use a sodium symmetric encryption key
key <- cyphr::key_sodium(sodium::keygen())
For the examples below, in the case of asymmetric encryption (using either cyphr::keypair_openssl or cyphr::keypair_sodium) the sender would use their private key and the recipient’s public key and the recipient would use the complementary key pair.
Objects
Here’s an object to encrypt:
obj <- list(x = 1:10, y = "secret")
This creates a bunch of raw bytes corresponding to the data (it’s not really possible to print this as anything nicer than bytes).
secret <- cyphr::encrypt_object(obj, key)
secret
## [1] 59 57 c0 17 ec 9f de e9 01 43 6f 06 6d 7d 76 66 3a a7 d6 1e a8 87 bc
## [24] 6e d3 78 b3 53 f1 af f3 38 c3 27 62 50 8c d8 f9 35 34 60 65 ef ac c4
## [47] 9d a5 ab bc a2 87 7e 8a bf 8a c8 29 62 43 a2 71 3e 7c 84 2f fd fe 47
## [70] cf 67 8a c5 b3 fc c2 db 5f 05 83 e9 fc 9e 07 41 95 68 bf 0d c9 c1 4a
## [93] e2 69 6f f0 68 cf 48 59 11 57 45 12 22 7b cf 1f 61 14 41 ac be 56 33
## [116] a4 49 0d ef 47 b3 7b 78 ce d2 f9 1f ae f8 f7 68 13 1c c2 dc 2e f7 16
## [139] 1a e7 64 a1 ef 10 6d cc 0d 0d 09 72 b6 01 f0 ae 6d 82 75 af da 85 60
## [162] 3a fe 8d 2f 1c 0a de 0e ba cf d0 b4 ca b7 41 17 23 3e 43 31 44 0d
The data can be decrypted with the decrypt_object function:
cyphr::decrypt_object(secret, key)
## $x ## [1] 1 2 3 4 5 6 7 8 9 10 ## ##$y
## [1] "secret"
Optionally, this process can go via a file, using a third argument to the functions (note that temporary files are used here for compliance with CRAN policies - any path may be used in practice).
path_secret <- file.path(tempdir(), "secret.rds")
cyphr::encrypt_object(obj, key, path_secret)
There is now a file called secret.rds in the temporary directory:
file.exists(path_secret)
## [1] TRUE
though it is not actually an rds file:
readRDS(path_secret)
## Error in readRDS(path_secret): unknown input format
When passed a filename (as opposed to a raw vector), cyphr::decrypt_object will read the object in before decrypting it
cyphr::decrypt_object(path_secret, key)
## $x ## [1] 1 2 3 4 5 6 7 8 9 10 ## ##$y
## [1] "secret"
Strings
For the case of strings we can do this in a slightly more lightweight way (the above function routes through serialize / deserialize which can be slow and will create larger objects than using charToRaw / rawToChar)
secret <- cyphr::encrypt_string("secret", key)
secret
## [1] 6e 1b 54 23 f0 df fd 2c 64 7d f1 25 0b 74 09 7e 23 89 af 30 26 99 88
## [24] 75 a9 5e 85 70 39 ec 4d 14 13 99 82 30 f6 cd d1 04 64 e1 da d5 43 70
and decrypt:
cyphr::decrypt_string(secret, key)
## [1] "secret"
Plain raw data
If these are not enough for you, you can work directly with raw objects (bunches of bytes) by using encrypt_data:
dat <- sodium::random(100)
dat # some random bytes
## [1] 69 29 8f 8b f3 0f c2 eb 3f 3f 3a 20 2b 91 6e ad 2c 81 e7 f8 7a 70 e8
## [24] db 9a 78 82 56 db cd e9 03 ef 29 14 1c cf 64 0d 65 7c aa a2 b4 e3 e4
## [47] 6b a3 8c 1f 3a 50 68 cc 1e e3 9b 3a c8 0b 63 c1 75 b7 dd 87 e5 2b b2
## [70] 5f fa 10 4e fe ad e4 78 b5 a3 5b bd 34 80 5e 80 ee 9e de 23 bf f2 97
## [93] 0e 10 2c 15 f8 5e 51 0f
secret <- cyphr::encrypt_data(dat, key)
secret
## [1] 0a 7e 0a 9c 45 b8 4c f7 86 c3 06 5c e4 f6 8a c4 6b cb fb 69 4e c7 3f
## [24] f0 2a 3d 39 97 a8 43 e2 85 7e 74 4c 60 86 a2 1b 94 b4 74 66 c2 9f 8c
## [47] 63 37 46 77 f6 0f 94 24 0c d5 eb 93 2f c2 e5 8b 9d e0 74 b1 bc f8 ac
## [70] ab 9c f5 13 30 9c 05 6f eb 47 8a ac 65 4f 71 0f 20 ab ff fe 42 6c bd
## [93] 17 fb ac b4 9a 1e 28 10 92 7e bb 97 c2 59 d4 45 06 82 14 53 c8 6f e9
## [116] d0 3c 73 5e 5d 2f e0 98 98 2c d2 d1 91 d9 22 3a 81 48 bc a0 44 7f b8
## [139] 1e ac
Decrypted data is the same as a the original data
identical(cyphr::decrypt_data(secret, key), dat)
## [1] TRUE
Files
Suppose we have written a file that we want to encrypt to send to someone (in a temporary directory for compliance with CRAN policies)
path_data_csv <- file.path(tempdir(), "iris.csv")
write.csv(iris, path_data_csv, row.names = FALSE)
You can encrypt that file with
path_data_enc <- file.path(tempdir(), "iris.csv.enc")
cyphr::encrypt_file(path_data_csv, key, path_data_enc)
This encrypted file can then be decrypted with
path_data_decrypted <- file.path(tempdir(), "idis2.csv")
cyphr::decrypt_file(path_data_enc, key, path_data_decrypted)
Which is identical to the original:
tools::md5sum(c(path_data_csv, path_data_decrypted))
## /var/folders/z7/c2kx_kt96zn2tt_6179bkc4m0000gp/T//Rtmp4OHgUv/iris.csv
## "5fe92fe6a2c1928ef5a67b8939fdaf8d"
## /var/folders/z7/c2kx_kt96zn2tt_6179bkc4m0000gp/T//Rtmp4OHgUv/idis2.csv
## "5fe92fe6a2c1928ef5a67b8939fdaf8d"
An even higher level interface for files
This is the most user-friendly way of using the package when the aim is to encrypt and decrypt files. The package provides a pair of functions cyphr::encrypt and cyphr::decrypt that wrap file writing and file reading functions. In general you would use encrypt when writing a file and decrypt when reading one. They’re designed to be used like so:
Suppose you have a super-secret object that you want to share privately
key <- cyphr::key_sodium(sodium::keygen())
x <- list(a = 1:10, b = "don't tell anyone else")
If you save x to disk with saveRDS it will be readable by everyone until it is deleted. But if you encrypted the file that saveRDS produced it would be protected and only people with the key can read it:
path_object <- file.path(tempdir(), "secret.rds")
cyphr::encrypt(saveRDS(x, path_object), key)
(see below for some more details on how this works).
This file cannot be read with readRDS:
readRDS(path_object)
## Error in readRDS(path_object): unknown input format
but if we wrap the call with decrypt and pass in the config object it can be decrypted and read:
cyphr::decrypt(readRDS(path_object), key)
## $a ## [1] 1 2 3 4 5 6 7 8 9 10 ## ##$b
## [1] "don't tell anyone else"
What happens in the call above is cyphr uses “non standard evaluation” to rewrite the call above so that it becomes (approximately)
1. use cyphr::decrypt_file to decrypt “secret.rds” as a temporary file
2. call readRDS on that temporary file
3. delete the temporary file (even if there is an error in the above calls)
This non-standard evaluation breaks referential integrity (so may not be suitable for programming). You can always do this manually with encrypt_file / decrypt_file so long as you make sure to clean up after yourself.
The encrypt function inspects the call in the first argument passed to it and works out for the function provided (saveRDS) which argument corresponds to the filename (here "secret.rds"). It then rewrites the call to write out to a temporary file (using tempfile()). Then it calls encrypt_file (see below) on this temporary file to create the file asked for ("secret.rds"). Then it deletes the temporary file, though this will also happen in case of an error in any of the above.
The decrypt function works similarly. It inspects the call and detects that the first argument represents the filename. It decrypts that file to create a temporary file, and then runs readRDS on that file. Again it will delete the temporary file on exit.
The functions supported via this interface are:
• readLines / writeLines
• readRDS / writeRDS
• read / save
• read.table / write.table
• read.csv / read.csv2 / write.csv
• read.delim / read.delim2
But new functions can be added with the rewrite_register function. For example, to support the excellent rio package, whose import and export functions take the filename file you could use:
cyphr::rewrite_register("rio", "import", "file")
cyphr::rewrite_register("rio", "export", "file")
now you can read and write tabular data into and out of a great many different file formats with encryption with calls like
cyphr::encrypt(rio::export(mtcars, "file.json"), key)
cyphr::decrypt(rio::import("file.json"), key)
The functions above use non standard evaluation and so may not be suitable for programming or use in packages. An “escape hatch” is provided via encrypt_ and decrypt_ where the first argument is a quoted expression.
cyphr::encrypt_(quote(saveRDS(x, path_object)), key)
cyphr::decrypt_(quote(readRDS(path_object)), key)
## $a ## [1] 1 2 3 4 5 6 7 8 9 10 ## ##$b
## [1] "don't tell anyone else"
Session keys
When using key_openssl, keypair_openssl, key_sodium, or keypair_sodium we generate something that can decrypt data. The objects that are returned by these functions can encrypt and decrypt data and so it is reasonable to be concerned that if these objects were themselves saved to disk your data would be compromised.
To avoid this, cyphr does not store private or symmetric keys directly in these objects but instead encrypts the sensitive keys with a cyphr-specific session key that is regenerated each time the package is loaded. This means that the objects are practically only useful within one session, and if saved with save.image (perhaps automatically at the end of a session) the keys cannot be used to decrypt data.
To manually invalidate all keys you can use the cyphr::session_key_refresh function. For example, here is a symmetric key:
key <- cyphr::key_sodium(sodium::keygen())
which we can use to encrypt a secret string
secret <- cyphr::encrypt_string("my secret", key)
and decrypt it:
cyphr::decrypt_string(secret, key)
## [1] "my secret"
If we refresh the session key we invalidate the key object
cyphr::session_key_refresh()
and after this point the key cannot be used any further
cyphr::decrypt_string(secret, key)
## Error in is.raw(key): Failed to decrypt key as session key has changed
This approach works because the package holds the session key within its environment (in cyphr:::session\$key) which R will not serialize. As noted above - this approach does not prevent an attacker with the ability to snoop on your R session from discovering your private keys or sensitive data but it does prevent accidentally saving keys in a way that would be useful for an attacker to use in a subsequent session.
• The vignettes in the openssl (vignette(package = "openssl")) and sodium (vignette(package = "openssl")) packages have explanations of how the tools used in cyphr work and interface with R.
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研究論文
2017年
1. Isao Nakazawa and Ken Umeno, "Fundamental study on almost periodic frequency arrangements for super-multi-access radio communication systems", to appear in IEICE Communications Express (ComEx) (2017).
2, Hirofumi Tsuda and Ken Umeno, "Weyl Spreading Sequence Optimizing CDMA", to
appear in IEICE Transactions on Communications (2017).
3. Hirofumi Tsuda and Ken Umeno, "Non-Linear Programming: Maximize SINR for Designing Spreading Sequence", to
appear in IEEE Transactions on Communications (2017).
4. Atsushi Iwasaki and Ken Umeno, "Further Improving Security of Vector Stream Cipher", Nonlinear Theory and Its Applications, IEICE, vol. 8, No.3, (2017) pp. 215-223.
5. Hiroki Okada and Ken Umeno, "Randomness Evaluation with the Discrete Fourier Transform Test Based on Exact Analysis of the Reference Distribution", IEEE Transactions on Information Forensics and Security, vol. 12, No.5, (2017) pp. 1218-1226.
6. Ken-ichi Okubo and Ken Umeno, "New Chaos Indicators for Systems with Extremely Small Lyapunov Exponents", in Chaos, Complexity and Transport,Edited by Xavier Leoncini, Christophe Eloy and Gwenn Boedec, (World Scientific,
2017), pp. 185-203.
7. Takuya Iwata and Ken Umeno,"Preseismic ionospheric anomalies detected before the 2016 Kumamoto earthquake",Journal of Geophysical Research, vol. 122 (2017) pp.3602-3616, doi:10.1002/2017JA023921
8. Atsushi Iwasaki and Ken Umeno,"One-stroke polynomials over a ring of modulo 2^w", JSIAM Letters, Vol. 9 (2017) pp.5-9, doi:10.14495/jsiaml.9.5
2016年
1. Hirofumi Tsuda and Ken Umeno, "Orthogonal basis spreading sequence for optimal CDMA", JSIAM Letters, vol. 8 (2016) pp. 77-80, doi:10.14495/jsiaml.8.77
2. Iwata Takuya and Ken Umeno, "Correlation Analysis for Preseismic Total Electron Content Anomalies around the 2011 Tohoku-Oki Earthquake",Journal of Geophysical Research, ,vol. 121(2016) pp.8969-8984, doi: 10.1002/2016JA023036
3. Ken Umeno and Ken-ichi Okubo,Exact Lyapunov exponents of the generalized Boole transformations, Prog. Theor. Exp. Phys. 021A01, pp.1-10, doi: 10.1093/ptep/ptv195
4. Atsushi Iwasaki and Ken Umeno, "Improving security of Vector Stream Cipher", Nonlinear Theory and Its Applications, IEICE, vol. 7 (2016) pp. 30-37, doi: 10.1587/nolta.7.30
5. Ken Umeno, "Ergodic transformatons on R preserving Cauchy laws",Nonlinear Theory and Its Applications, IEICE, vol. 7 (Invited Paper) (2016) pp.14-20, doi:10.1587/nolta.7.14
6. Shin-itiro Goto, "Contact geometric descriptions of vector fields on dually flat spaces and their applications in electric circuit models and nonequilibrium statistical mechanics",Journal of Mathematical Physics, vol. 57, (2016)102702, doi: http://dx.doi.org/10.1063/1.4964751
2015年
1. H. Okada, K. Umeno, D. Handoko, M. Ihsan, R. Maharani, P. Nursetia, H. Rosyad, Warsito, "Brain Electrical Capacitance Volume Tomography Signals Analysis with Moving Maximum Lyapunov Exponents", Advanced Science, Engineering and Medicine, vol. 7, No. 10 (2015), pp. 897-899.
2. 大久保 健一, 梅野 健,弱カオス系のカオス性判定について, 日本応用数理学会論文誌, vol.25, No.3(2015), pp. 165-190.
3. 岡田 大樹、梅野 健、"新たな非線形時系列解析の手法 —移動最大リアプノフ指数線によるカオス解析ー", レーザー研究, vol. 43, No. 6(2015), pp.
4. Chen-An Yang, Kung Yao, Ken Umeno, and Ezio Biglieri, "Superefficient Monte Carlo Simulations" in Simulation Technologies in Networking and Communications Selecting the Best Tool for Test, Edited by Al-Sakib Khan Pathan, Mohammad Mostafa Monowar Shafiullah Khan (CRC Press, 2015) pp.69-91.
2014年
1. Ryo Takahashi and Ken Umeno, "Performance Evaluation of CDMA Using Chaotic Spreading Sequence with Constant Power in Indoor Power Line Fading Channels", IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences,E67-A,No. 7 (2014), pp.1619-1622, doi: 10.1587/transfun.E97.A.1619
2013年
1. K. Umeno and M. H. Kao, "Chaos Theory as the answer to limited spectrum?", ITU News, No.10, 2013.
2. Chen-An Yang, Kung Yao, Ken Umeno and Ezio Biglieri, "Using Deterministic Chaos for Superefficient Monte Carlo Simulations", IEEE Circuits and Systems Magazine, 13 (2013), pp.26 - 35.
3. Shun Ogawa, Spectral and formal stability criteria of spatially inhomogeneous solutions to the Vlasov equation for the Hamiltonian mean-field model, Phys. Rev. E 87, 062107 (2013), arXiv:1301.1130
4. 梅野健 情報の統計力学 数理科学 No. 600 (2013), pp. 35-41
5. K. Umeno and A.-H. Sato, "Chaotic Method for Generating q-Gaussian Random Variables", IEEE Transactions on Information Theory, 59 (2013), pp.3199 - 3209.
修士論文
＊修士課程学生は修了のために、修士論文を作成します。
ここ数年の修士論文の題目および摘要は、
以下のリストの氏名をクリックしてご覧下さい。
特別研究報告書
＊４回生は卒業のために、特別研究報告書を作成します。
ここ数年間の特別研究報告書の題目および摘要は、
以下のリストの氏名をクリックしてご覧下さい。
有井 伴樹
Study on CDMA Systems with Primitive Root Codes (原始根符号を用いたCDMAシステムの研究)
In this thesis, we study on CDMA systems with primitive root codes. First, we explain primitive roots modulo a prime number $p$, and introduce the concept of safe prime numbers. Second, using the properties of primitive roots, we construct synchronous CDMA systems with primitive root codes which are complex number sequences. In this paper, we construct the system with the following situations: (1) The same primitive root $q_1$ is assigned to every user. (2) Different primitive roots are assigned to each user. (3) Some users are assigned by the same primitive root $q_1$, and other users are assigned by another primitive root $q_2$. Third, we analyse theoretically Bit Error Rate (BER) of the systems and effects of a safe prime number. Fourth, we simulate numerically BER of the systems. Finally, we compare the theoretical and numerical analysis of BER and evaluate the performance of the systems.
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入江 哲史
Study on the high-precision Monte-Carlo computation using random numbers with nonuniform density (非一様乱数を使用する高精度モンテカルロ計算についての研究)
In this paper, a generalized method and several practical methods of the high-precision Monte-Carlo computation using random numbers with nonuniform density which decrease are proposed. Also the results of comparison between the proposed methods and previous methods, which use random or quasi-random number sequences with uniform density, are shown for applications to several multidimensional integral computation. As a result, the superiority of this methods is demonstrated.
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久世 友博
Robustness of interdependent networks with degree-correlated inter-connections (次数相関を持って複数のネットワークが相互につながっている場合のロバスト性)
Modern systems are constructed with multiple networks that are connected to each other. For example, electrical systems are constructed with the power grids and their communication support systems. In such a interdependent network, failure of nodes in one constituent network leads nodes in the other network to fail. This happens recursively and leads to a cascade of failures. It is known that the interdependent networks with random inter-connections have weaker robustness, tolerance to failures, than the individual networks. However, if the interdependent networks have degree correlations between the networks constructing them, the robustness of the interdependent networks may be changed. Since actual interdependent networks have some correlations, we investigate the effects of the correlations on the networks.
A group that nodes connected with several links is called a cluster and if the number of nodes in the cluster is large, the network is robust. We perform simulations for various ratios of the initial failure of nodes and evaluate the cluster sizes after the cascade of failures. We show that when a interdependent network has a positive degree correlation between two networks that construct it, it has the stronger robustness than that for the networks with no degree correlation. Moreover, as the result of the numerical simulation this system shows a percolation phase transition and the threshold is approximately a linear function of the correlation coefficient. Then, we show not only the numerical simulation results but theoretical ones for the robustness of the interdependent networks. The theory can be applied to the interdependent networks with any degree distributions and any inter-correlations. The theoretical results correspond to the numerical simulation results mainly at any case.
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家治川 博
Linear Regression Analysis of Foreign Exchanges with a Method of Segmenting Time Series Based on the Likelihood-Ratio Test
There has been a lot of researches to analyze time series data. On the other hand, a variation of information which we can obtain is getting wider and wider these days because of the development of information technology. Therefore, it is important to deal with various information for a purpose to analyze financial time series data more accurately.
In this paper, we conduct a linear regression analysis and set Google search queries as explanatory variables of the model in order to analyze financial time series data with taking various information into consideration. Additionally, we also set the interbank exchange frequencies and the volatility calculated by GARCH(1,1) model as explanatory variables.
Since financial time series are not generally modeled as a stationary process, but modeled as a non-stationary process, we assume that the non-stationary time series consists of several stationary segments with different properties. In order to discriminate the joint of these distributions, we conduct a likelihood- ratio test. According to this test, the point which maximizes the likelihood- ratio between the null model (homogeneous disturbance distribution) and the alternative model (a mixture of two different normal distributions) is regarded as the most possible combination point. Since likelihood usually includes sample errors, we evaluate the significance level of it by using the bootstrap method. We employ the bootstrap distribution as a discriminant measure to divide the time series into two segments at an adequate point, recursively.
We apply this method to the data of foreign exchange market and make an analysis of them in terms of segmenting points and regression coefficients. We compare our proposed method with the ordinary linear regression method visually and numerically. We conclude that Google search term which segments the time series at a point different from other search term has an additional relation with the explained variable. Our proposed method can detect the data point at which the tendency of the time series changes and can analyze the time series better than the ordinary method.
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守田 悠三
Analysis of Foreign Exchange Rates Based on Parametric Risk Assessment Procedures withq-Gaussian and Pearson type IV Distributions
Recently, it has been much easier for individuals to buy and sell foreign currencies in the market and it is becoming more important to understand the foreign currency risks to hold foreign currencies safely. Exchange rates sometimes fluctuate unpredictably and it could cause loss of the deposit. Especially, it is well-known that the volatilities of the price fluctuate de- pending on the time period, which seems to cause fat-tailedness observed in actual data. Therefore, it is important to regard its fat-tailedness when we estimate the risk from historical data.
In this paper, we introduce parametric risk assessment procedures in the foreign exchange market. In order to consider the ruin probability when we have some deposit, we assume two types of distributions, the q-Gaussian and the Pearson type IV distributions that log-returns in foreign exchange rates obey.
We perform parameter estimation of q-Gaussian distribution for 30 currency pairs with a maximum likelihood method. The parameter q is estimated in the range of from 1.3 to 1.7, and we confirm that the empirical distributions of the market data have fat-tails. To check whether the estimated parameters are statistically significant, we calculate p-values of two types of statistical test, Kolmogorov-Smirnov (KS) test and Anderson-Darling (AD) test. We reveal that all p-values in KS test are larger than 0.1, although p-values of 9 currency pairs are less than 0.1. This means that the log-returns obey the q-Gaussian as a whole, but do not when we focus on tails of the distributions.
We also perform parameter estimation of the Pearson type IV distribution, which has skewness, for 30 currency pairs. In this case, the average p-values in both KS and AD test are better than those of the q-Gaussian and the p-values in AD test for only 5 pairs are less than 0.1. This means that the Pearson type IV is better-fitted to the market data. Therefore, we reveal that the model with skewness is preferred for risk estimation. We calculate 1% Value-at-Risk (VaR) in both cases. The difference of the VaR between the q-Gaussian and the Pearson type IV is about 10%. This indicates that VaR with the q-Gaussian could cause underestimation of the risk.
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吉村 玄太
Capacity-Approaching LDPC Codes Constructed from Extended Protographs
Low-density parity-check (LDPC) codes are in a class of the most powerful error correcting codes available today. In this paper, to improve the decoding performance we introduce a new class of LDPC codes constructed from a template called an extended protograph, which belongs to the superset of the popular protographs. We exploit the extrinsic information transfer (EXIT) chart and asymptotic ensemble weight enumerator techniques to predict the capacity-approaching performance of extended protograph code ensembles. EXIT charts compute the iterative decoding threshold of a code ensemble, which contributes to the waterfall region performance for the decoding error rate. Asymptotic ensemble weight enumerators estimate whether or not the minimum distance of code ensemble increases linearly with code length, which affects the error floor performance for the decoding error rate. Taking account of these conflicting performance indices, we apply the simulated annealing (SA) algorithm to find extended protograph based codes which have low iterative decoding threshold and the linear minimum distance growth property. Finally, the performance of optimized extended protograph based codes over the binary-input additive white Gaussian noise (BIAWGN) channel are compared with that of existing protograph based codes and the quasi-cyclic (QC) code adopted by IEEE 802.16e (WiMAX) standard.
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池田 和雄
Opinion propagation using partisan voter models on several networks
The partisan voter model is the one of models which treat the social opinion dynamics. We use them in order to treat opinion dynamics on several networks. This model is a modified version of the voter model which describes the evolution to consensus in the networks of nodes, voters, that possess respective opinions within a discrete set. In the partisan voter model, each node has also an innate and fixed liking for one opinion. This liking determines the probability that each node changes its opinion. We calculate the convergence time toward consensus of opinions in several networks by using the partisan voter model. The convergence time in the scale-free networks, the BA model, is smaller than that in the complete graph. With the modified BA model we investigate the dependence on the exponent of the degree distribution, that is, on the number and size of hubs on the network. If the exponent is small, the number of hubs is small and the size of hubs is large. From our simulation, for the small exponent, the convergence time is shorter than that for the large exponent. The correlation of degrees is controlled by rewires of the links in the network initially created by the BA model. For assortative networks, we confirm with our simulation that convergence time is, in general, longer than that for uncorrelated networks. On the other hand, for disassortative networks, it is shorter than that for uncorrelated one in general.
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野田 実
Japanese hotel statistics in terms of regional room capacities
In this paper, in order to understand the dependence of stay capacity on regionality, we propose a method to determine districts depending on the number of rooms and to classify its districts. We empirically analyze the geographical positions and the number of rooms about 2,881 Japanese hotels which have 582,898 rooms in total. Firstly, we conduct a clustering analysis of the regional stay capacity by centroid method. Secondly, we introduce the maximum entropy principle in order to divide regional areas into some levels. It may be concluded that the rank size distribution for the number of rooms in the cluster is fitted with a power-law function and that the scaling exponent is dependent on the number of clusters.
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森岡 篤
Optimization of routing strategies for data transfer in peer-to-peer networks
Recently, peer-to-peer file-sharing systems have become familiar and the information traffic in the networks is increasing. Therefore it causes various traffic problems in peer-to-peer networks. In this paper, we model some features of the peer-to-peer networks, and investigate the traffic problems. Peer-to-peer networks have two notable characters. One is that each peer frequently searches for a file and download it from a peer who has the requested file. To decide whether a peer has the requested file or not in modeling of the search and download process, we introduce a file-parameter Pj , which expresses the normalized amount of files stored in peer j. It is assumed that if Pj is large, peer j has many files and can meet other peers' requests with high probability. The other character is that peers leave and join into the network repeatedly. Many researchers address traffic problems of data transfer in computer communication networks. To our knowledge, however, no reports focus on those in peer-to-peer networks whose topology changes with time. For routing paths of data transfer, generally, the shortest paths are used in usual computer networks. In this paper, we introduce a new optimal routing strategy which uses weights of peers to avoid traffic congestion. We find that the new routing strategy is superior to the shortest path strategy in terms of data traveling time when many peers join in the data transfer.
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永田 啓悟
Analysis of image encryption schemes using chaotic maps
Along with the development of the information and telecommunications networks, various kinds of encryption schemes have been developed by many researchers. Some of them pay attention to chaotic encryption. They utilize chaotic properties such as initial sensitivity, synchronization, and so on, for enhancing security and effect of encryption. In this paper, we consider image encryption schemes using baker's map, which is one of chaotic maps. Image encryption scheme is divided into two phases, "permutation" and "diffusion". Permutation is encryption on positions of pixels, and diffusion is that on gray values. Both of them are necessary for secure encryption. In some previous studies, baker's map is used for permutation. With repeated application of this map, the pixels are permutated, and the original image comes to look featureless and random like noise. We can decrypt it easily if and only if the widths of the rectangles, called keys, are known because baker's map is reversible. First, we analyze the properties of baker's map and improve it to surmount the weakness. Secondly, we apply baker's map to diffusion. Finally, we evaluate the security of the proposed encryption scheme, and prove the usefulness of it.
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中本 武志
Estimating the tail index of distributions: Case study on the foreign exchange market
We study an unconditional distribution derived from Alfarano-Lux model which has the two parameters: the herding propensity and the autonomous switching tendency. One of the parameters, the herding propensity characterizes the tail shape of its distribution, but does not accord with the well-known Hill's estimator in case of foreign exchange market data. In this paper, we transform the probability density function of Alfarano-Lux model to the expanded form and obtain the analytical form of its cumulative distribution function. Additionally, we explain the reason why the difference between the estimates of Alfarano-Lux model and Hill's estimator is generated, and conduct Kolmogorov-Smirnov test to measure the goodness-of-fit. As an application of Alfarano-Lux model to the foreign exchange market data, we measure the fluctuation risk for some currency pairs.
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西岡 謙太
Log returns of stock values and q-Gaussian distributions: Application to the risk-assessment
We assess a risk of financial time series with the distribution estimated from them. Because it is not easy to infer its tail shape due to a lack of data in a practical manner.(1) We introduce Value at Risk(VaR) to a risk measure and compare it with variance under the q-Gaussian assumption.(2) We examine performance of the maximum likelihood estimator with the q-Gaussian log-likelihood function.(3) By using the distribution estimates, we verify errors of the VaR to estimated one. Finally we conduct an empirical analysis on log-returns of a stock traded in the Tokyo Stock Exchange.
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大谷 卓也
Sequential associative memories on complex networks
The models of associative memories in neural networks, such as Hopfield model, are quite tractable mathematically, and applied to many areas. One of them is a model of sequential associative memories. Most of the models, however, including sequential associative memories, are usually applied to all-to-all networks. Since the studies of neural networks originate in brain science, it is natural to apply them to more realistic networks. Many researchers have been interested in complex networks, which is said to describe properties of real networks in society, computers, and neurons, that is, the small-world and scale-free properties. For example, random graph, the Watts-Strogatz model, the Barabási-Albert model, are the most famous complex networks which have some interesting properties. These days, the studies of complex networks are developed drastically. In this paper, the model of sequential associative memories is applied to these complex networks. The main subject is how the network topology affects the temperature dependence of the retrieval performance. Computer simulation reveals that the temperature dependence varies with the network topology, and it also comes out that the local performance on each node is increasing monotonically with its degree. Modifying the existing theory to be suitable for complex networks, we have been obtained a new approximated equation for overlap using a mean-field approximation and the the central limit theorem. This approximated equation can describe the temperature dependence of performance with any networks whose degree distributions are known. Numerical solution of the equation on the Barabási-Albert model is compatible with simulation results. On the other networks, the numerical solutions do not agree well with the results. Loops of networks are considered as one of the possible causes for the inagreement because the loops could disturb the independence of node states which is essential to apply the central limit theorem. As the Barabási-Albert model is known to have a very small number of loops than the other complex networks, the above compatibility between simulation and theoretical results is consistent with the loop effects. In order to determine whether the loops are the cause of the error, in the last of this paper, we present new approximated simultaneous equations which evaluate the effects of loops. As far as the numerical solution on random graph, it is likely that the loops have little effect on the performance of sequential associative memories on complex networks.
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砂川 敦志
Active random walkers: a simple model fro catalyst dynamics
Some problems related to active random walkers are studied with use of a model for a chemical system which consists of catalysts(C), products(P) and reactants(R). For numerical experiments we use a Monte Carlo method to follow motion of C- and P-particles. For analytical considerations a Langevin model for particle dynamics is introduced, which is converted to a coupled diffusion equation, whose linear stability is studied in relation to numerical experiments. We assume either attractive or repulsive interactions between C-particles and P-particles and the catalysts C are able to change environment locally by producing P-particles from R-particles. P-particles and C-particles can diffuse in the system and P-particles are characterized by a decay constant, which prevents the number of P-particles from increasing indefinitely. Our main results include
◦1) In case of attractive interaction between C- and P-particles, the density fields of C-particles and P-particles tend to a stationary bound state, which consists of a bump of C-particles and a bump of P-particles occupying the same small region (in a long time limit).
◦2) In case of repulsive interaction between C- and P-particles, we observe (irregular) density waves for both C-particles and P-particles, in which a region of high P-particles density corresponds to a region of low C-particles density, and vice versa.
◦3) When the amount of R-particles, available for the chemical reaction, is finite we observe that a ring, inside of which R-particles are consumed completely, expands outward.
◦4) Linear stability analyses give us information on the characteristic length of the stationary state and this turned out to be consistent with our findings (1) and (2) above.
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君塚 誠
Properties of coupled double well systems with delay and noise
The interplay between noise and delay in physical as well as biological systems gives rise to a lot of interesting phenomena and is gathering interest of many researchers. An especially hot area in this connection is the semiconductor laser emitter (VCSEL in short), which is modeled by a Brownian particle in a double-well potential under delayed force. If there were no delayed force, the system would show a simple barrier-crossing or hopping, which means for VCSEL the transition from the vertically polarized state of light to the horizontally polarized one and vice versa. In the presence of delay, which represents the delayed feedback for VCSEL, this hopping is modified due to strong correlation between (x(t)) and (x(t-τ)) , with (x(t)) and (τ) denoting position of the Brownian particle at time t and the delay time, respectively. In this work we consider a model which consists of (N)-Brownian particles put in a double-well potential and investigate some theoretical problems such as the positional stationary state distribution function (Pss) and the time correlation function(TCF). This model may be considered as a model for VCSEL in which (N) laser emitters are connected in cascade. At the moment we have no experimental results available for comparison with our theory. Thus we performed ourselves computer experiments to obtain some physical quantities of interest. For theoretical analysis of Brownian dynamics in a double well potential, we employ a two-state approximation, which replaces the continuous position x(t) with s(t), which takes only two values +1 ( if (x(t)>0) ) and -1 ( if (x(t)<0) ). This greatly simplifies our problem and at the same time gives much insight into our problem. Some results obtained in this work include, 1) the positional stationary state distribution function (Pss) for the Brownian particle model is calculated (computer) experimentally and its (τ) and feedback strength ε dependences are clarified. These dependences are theoretically studied for the case (N=2) based on the two-state approximation, which results in (Pss) well correlated with our experiments. It is remarked that this (τ) dependence is absent for the case (N=1, 2) the time correlation function is also calculated numerically by solving the Langevin equation and theoretically based on the two-state approximation.
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近藤 健夫
Signal response in scale-free network of bistable units
Information processing in biological systems, such as brains or cell membranes, has several features in common: (i) it is an analog information processing; (ii) scale-free networks play an important role; and (iii) each information processing unit possesses strong nonlinearity. We studied the efficiency of information processing in a scale-free network, consisting of many interacting nonlinear units with double-well potential, both theoretically and (computer) experimentally. We introduced the gain (G) to quantify the efficiency, which is defined as the ratio of the output strength aL of the maximum response unit L to the periodic input signal strength (A), i.e. (G≡aL/A) . In the previous work, Acebron et al. calculated (G) as a function of coupling (or interaction) strength (λ) i.e. &math (G(λ)) , by simulations and they found that (G(λ)) showed plateau behavior in some range of (λ) They tried to understand their numerical results with emphasis put on a hub, which is a typical feature of the scale-free network. That is, they analyzed dynamics based on a simplified model (starlike network). However, the analysis contains some problems: (i) it is limited to a very small (λ) region; and (ii) the overall role of the scale-free network for information processing is not touched upon. In this study, we develop a formalism, in which the gain (G) is studied based on a one-body problem, which turns out to be a good approximation to replace the dynamics of the complex network. We analyzed this model, and obtained four theoretical predictions: (i) the value (Gplateau) of the plateau height of (G(λ)) (ii) the initial transient behavior of (G(λ)) in a small (λ) region, (Gtrans(λ)) with finiteness of the system taken into account, (iii) the asymptotic (G) value, (Gsync) after full synchronization is achieved for large (λ) and (iv) the critical value sync) , beyond which all units are fully synchronized and the gain (G); becomes (Gsync) We compared these theoretical predictions with computer experiments and confirmed that our theory reproduces experimental results at least semi-quantitatively. From this we may say that we could understand (G(λ)) in the whole coupling strength range by revealing the role of the complex network in information processing.
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西村 麻衣子
Scaling analysis on quotation activities in the foreign exchange market: Empirical investigation and stochastic modeling
We investigate quotation activities in the foreign exchange market both empirically and theoretically. We found the scaling relationship between means of the number of quotations during window lengths and their standard deviations. We confirm that the scaling exponent temporally changes from 0.8 to 0.9 depending on observation days and that it tends to be unity as the time window length increases. We also estimate a cross-correlation coefficients and relative frequencies of the quotation activities. As a result, it is found that the scaling exponent and the cross-correlation coefficients show a significant correspondence relationship. Therefore, it is concluded that scaling analysis would be one of adequate ways to grasp the total market state. Besides, we propose a stochastic model to understand the market participants' activities and conduct a theoretical analysis for the proposed model with parameter fitting by empirical data. Comparing empirical results with theoretical ones, we examine the adequency of the model. Consequently, we found that the fluctuation of the probability with which market participants decide their attitudes and the weight of each currency pair have an important role to determine the value of a scaling exponent under the condition that the probability is homogeneous for every market participant.
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岩間 心平
Effects of time delay on nonlinear stochastic systems
Considerable attention is paid to some stochastic systems, whose dynamics is determined by the present state x(t) and the state x(t-T) in the past with r(> 0) denoting the delay time. Usually this delay is ascribed to the finite speed of information transmission and it is rather natural that effects of delay are intensively studied mainly for biological systems, e.g. as models to describe postural sway, visual feedback, and brain activity, to mention a few. Effects of delay are also studied for chemical, physical, and engineering systems. So long as we know, there have been no systematic studies on nonlinear systems where effects of delay play essential roles to determine system properties. Also from the viewpoint of methodology we have no reliable method, which can be applied for large delay time and strong nonlinearity. From these points stochastic systems with delay are offering many problems, interesting both from mathematical and physical viewpoints. Recent progress in understanding the delayed system may be partly due to the advent of the Fokker-Planck equation for the (one-body) distribution function p(x, t). However this equation is not a closed one for the distribution function p(x, t) in the sense that it contains additional 'collision' term, expressed in terms of two-body conditional distribution function p(y, t-T | x, t) for x(t-T) = y when x(t) = x. Developing some approximation schemes to make it a closed one, we give detailed discussions on the range of validity of each approximation scheme for the delay Fokker-Planck equation. Our strategy is as follows: First we note that the delay Fokker-Planck equation has two important parameters, the delay time T and the strength E of a delay term. We propose three approximation schemes, namely, which are supposed to be applicable for (i) small T, (ii) small E and (iii) large T regions. By applying these schemes to a double-well potential system and comparing theoretical predictions with (numerical) experiments, we estimate how our proposed schemes work to this problem.
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川本 大樹
Efficient packet routing strategy in complex networks
We investigate new packet routing strategies which mitigate traffic congestion on complex networks. Instead of using shortest paths, we propose efficient paths which avoid hubs on scale-free networks with a weight of each node. Firstly, we compare the routing strategy using degree-based weights with that using betweenness-based weights on two types of scale-free networks. The strategy using degree-based weights is more efficient than that using betweenness-based weights on scale-free networks generated by a preferential attachment. On the other hand, the ascendancy of the strategy using degree-based weights over that using betweenness-based weights is reversed on scale-free networks composed by taking into account the distance between nodes. Next, we consider the heuristic algorithm which improves step by step routing properties on congestion by using the information of betweenness of each node in every step. We propose new heuristic algorithm which balances traffic on networks by achieving minimization of the maximum betweenness in the much smaller number of iteration steps.
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酒井 洋
Discrimination and Cluster methods for multi-dimensional time series of the Foreign Exchange Market based on Spectral Distances
In this thesis, the similarities between multi-dimensional time series extracted from high frequency financial data of the foreign exchange market are measured and the hierarchical clustering is performed based on them. We introduce two methods to calculate their similarities based on spectral distances. One is the sum of the Kullback-Leibler divergence between two normalized power spectra of time series of each elements for different observations, and the other is that between two largest eigenvalues of cross spectral matrix computed from multiple time series. In order to verify adequateness of these methods, we introduce the agent-based model of the foreign exchange market in which N market participants exchange M currency pairs, and perform numerical simulation and the hierarchical clustering with pseudo price movements obtained from it. As the results, it is found that the tendency that time series are unified sequentially as the difference of parameters became large, and that results obtained by means of two methods are almost similar. Finally applying this procedure to actual tick data we confirm that this can extract meaningful information from a large amount of data. The day when the rate movements of several currency pairs are greatly abnormal has a tendency that that on the day belongs to a separated cluster from those on other days. Therefore, it is concluded that this procedure is applicable to automatic extraction of meaningful information about markets from enomarous amounts of financial data.
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多羅尾 光記
Information Flow and Causality in Coupled Systems
In studies of a complex system, one of the major concerns is the detection and quantification of "causal interdependencies" among many dynamical subsystem, which constitute the complex system. For systems studied via physics or chemistry, this interdependency may be called "coupling" or "interaction". The purpose of this thesis is to study both physical and man-made systems on equal footing, using the concept of entropy transfer (or information flow) between subsystems. From physics we know that heat flows from a high to a low temperature region and sound wave propagates through a system with a constant velocity. These may be regarded as a kind of information transfer and there have been many works to understand these phenomena from the viewpoint of information flow. Based on the recent advance in methodology to quantify entropy transfer, as developed by Schreiber, we consider three systems, (1) a linearly coupled Langevin system, (2) FitzHugh?-Nagumo neural networks and (3) foreign exchange market. For each system we quantitatively analyze the system dynamics, especially the interrelation among subsystems, based on entropy transfer. The characteristics of these systems is that they have many units that are coupled with each other, unidirectionally or bidirectionally. In the linear system composed of two Brownian particles, we derive the transfer entropy rate in a theoretical way by taking the limit dt→0 where dt represents the sampling time. And we compare it with the entropy production rate. In the non-linear system that consists of many neurons, the information flow in three types of network, (i) linear array type, (ii) triangular type and (iii) a star-like one is analyzed. For a foreign exchange market, we treat the thirteen currency pairs. By calculating the entropy transfer among currency pairs, we observed some rules, which are rather vague but suggestive and seem to be consistent with our intuition.
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小崎 元也
Application of the Beck model to stock markets: Value-at-Risk and portfolio risk assessment
We have applied the Beck model, developed for mechanical systems that exhibit scaling properties, to stock markets. Our study revealed that the Beck model elucidates properties of stock market returns and is applicable to practical use such as the Value-at-Risk estimation and the portfolio analysis. We have performed empirical analysis with daily/intraday data of the S&P 500 index returns and found that the volatility fluctuation of real markets iw well consistent with the assumptions of the Beck model: the volatility fluctuates in much larger time scale than return itself and the inverse of variance, or "inverse temperature," beta obeys Gamma-distribution. As predicted by the Beck model. the method of Value-at-Risk (VaR), one of the most significant indicator in risk management, is studied for q-Gaussian distribution. Our proposed method enables the VaR estimation in consideration of tail risk, which is underestimated by the variance-covariance method. A framework of portfolio risk assessment under the existence of tail risk is considered. We have proposed a multi-asset model with a single volatility fluctuation shared by all assets, named the single beta model, and empirically examined the agreement between the model an and an imaginary portfolio with Dow Jones indices. It turns out that the single beta model gives good approximation to portfolios composed of the assets with non-Gaussian and correlated returns.
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新谷 幸平
Empirical analysis and numerical simulation of foreign exchange market dynamics
In this thesis, the dynamics of the foreign currency markets is analyzed with the tick frequency, which is the trace of market participatnts' activities. With the double-threshold agent model, the relation between the tick frequency time series of two agent groups is discussed. Empirical analysis with actual tick data indicates that the similarity of the tick frequency time series is getting similar as time passes and the similarity network among currency pairs changes by the areas where the markets are active. It means that in the present financial markets, market participants become more homogeneous, and the risks in the financial markets must be calibrated depending on areas, respectively. Alternative analysis indicates that the similarity between the best ask price and the similarity between the tick frequency time series influence each other constantly and weakly, and that the co-movement of them quickly and largely changes. Finally, a way of constructing the real-time market monitoring system is discussed.
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徐 普永
Self-tuning of activation energy in a two-state system
Many researchers have observed, in various biological and physical systems, that noise can enhance responses(i.e. output signals) of nonlinear systems to a weak periodic driving force(i.e. input signals) in a positive way, thus transfering more information than a case without noise. The phenomenon is closely related to stochastic resonance(SR), which states that there exists optimal noise strength T* where best information transfer is achieved. However it seems that not much attention has been paid to a situation where noise is rather weak and information transfer is very limited. In order to solve this problem, we attempt to apply an idea of self-tuning (ST), first proposed to explain high sensitivity of auditory systems of animals. That is, we consider a simple adaptation process of an activation energy(to be regarded as a threshold), which turns out to work well for a two-state system (TSS) in a weak noise region. Improvement of information processing ability of the TSS in a strong noise region was made possible by analytically studying the adaptation equation. We apply our ST method to a double-well potential system (DWPS), showing that the ST method works well also for the DWPS. Some quantities, not directly obtainable theoretically such as the first-passage-time distribution function, are calculated based on Monte-Carlo simulations. Finally we consider ST and SR from the viewpoint of 'energy transfer' from input signals to reservoirs. It is shown from numerical experiments that this energy transfer shows similar behavior as the information transfer.
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長沼 佑樹
Packet routing strategy using neural networks on complex networks
We investigate routing strategies on complex networks. Firstly, using neural networks, we introduce a routing strategy where path lengths and queue lenghts are taken into account within a framework of statistical physics. The performance in this strategy becomes more efficient from improvement of the distance term. At the same time, we analyze how the properties of networks influence the performance of this strategy. Secondly, we propose a routing strategy where connection weights in neural networks are adjusted by local information. We also confirm how the distance term and the properties of networks influence the performance of this adjustive strategy.
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松田 雄馬
Synchronization of a randomly coupled map model for neural networks
Neurons are known to oscillate synchronously with each other to achieve various functions. Studies on randomly coupled Hodgkin-Huxley neuron models have show that neurons synchronize their activity even in noisy environments. Synchronization is reported to occur when the size of networks is large enough even for sparsely connected neurons. Because fluctuation is neglected in large size networks, synchronization is controlled only by average states of neurons. Although it is widely observed in nature, especially in nervous systems, most of the mechanism is still unknown. In this paper, the condition for synchronization on neural networks is investigated with numerical simulations of a one-dimensional map neuron model. Then, the synchronization in noisy fields is theoretically understood throughout the distribution of neuron states formulated by inductive statistics and Gaussian mixture models.
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岡田 大樹
離散フーリエ変換検定法とその改良についての研究
また，DFT検定の問題点の考察を通して，DFT検定とは検定対象が多少異なるが，周期的特徴を検出する独自の検定法を考案した．この検定法について，少数のフーリエ係数のみが大きくなるような乱数の検定においては離散フーリエ変換検定法より周期的特徴の検出能力が優れることが確認できた．
それの他に，離散フーリエ変換をより一般化した変換について考察し，それをもとにして作る新たな離散変換を離散フーリエ変換と置き換えるという変更を離散フーリエ変換検定法に加えることで，チェビシェフ写像などを生成法の基にするカオスな擬似乱数系列等を検定可能とする，擬似乱数系列の持つカオス性（＝非乱数性）を棄却検定する検定法を構成した．
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長谷川 史晃
嗅覚系モデルにみられる神経集団システムの研究
また，力学系が振動するためのパラメータの条件を求め，これが正しいことを示すために，力学系が振動する時のリミットサイクルの内部の面積を解析的に求め，求めたパラメータの条件のもとで，面積が存在することを導き出した．
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亀山 慎吾
原子時系発生システムの安定化と高精度化に関する研究
そして、最後に精度の観点から現在米国のNISTで用いられている標準時の時系アルゴリズムにおいて、指数フィルタと呼ばれるものについて議論した。また、これにLSFを導入することで負の相関をもたせ、より高精度なシステムになる可能性があることを提案した。この提案は今後の研究の課題の1つとしてこれから調査していかなければならないことである。
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猶原 僚也
パワー一定カオス拡散符号を用いたCDMA通信システムの性能評価：ルベーグスペクトラムフィルタの適用
また、自己相関特性を制御し、他のユーザからの干渉雑音を抑制するルベーグスペクトラムフィルタ（Lebesgue-Spectrum-Filter）という線形フィルタを導入することによって、ビット誤り率やSNRが低下したり、SIR（信号対干渉比）が上昇することをシミュレーションによって示す。具体的には、従来のシステムより同じビット誤り率でユーザが15\%増加できてそれが非同期CDMAシステムでは解析的に最適である。
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吉村 玄太
適応型ネットワークにおける潜伏期を持つ感染症のダイナミクス
ネットワークのダイナミクス（トポロジーつまり接続状態の変化）とネットワーク上のダイナミクス（ノー ド状態の変化）が相互作用するようなネットワークを適応型ネットワークと呼ぶ。本論文では潜伏期を持つ感染症が適応型ネットワークにおいてどのような定常状態に達するかを明らかにする。まず適応型ネットワークにおける潜伏期を持つ感染症のダイナミクスを表現するため、Susceptible-Infected-Susceptibleモデル（SISモデル）を発展させたSusceptible-Latent-Infected-Susceptibleモデル（SLISモデル）を導入する。次にネットワークのノード状態及びトポロジーを表す平均場量を用いて近似し、導入した SLISモデルを支配する連立常微分方程式を導出する。この方程式を解くことで、SLISモデルの定常状態における感染ノード密度を理論的に求める。また SLISモデルを計算機上で実装し、シミュレーションを行うことで定常状態における感染ノード密度を数値的に求める。感染確率pに対して、無病状態から流行状態への転移が起きる侵入閾値 pinv 、及び流行状態から無病状態への転移が起きる持続閾値 pper が定義できる。従来のSISモデルでは、リンクの張り替えにより感染症との接触を避ける防衛の効果によって、多くの場合に pper < pinv となり、2つの閾値の間では履歴に応じて無病状態と流行状態の2つの安定な状態をとる、いわゆる双安定性を有する。これに対して本研究で導入したSLISモデルでは、潜伏期が長くなるほど防衛の効果が薄れ、 pinv, pper が減少していくこと、双安定性が失われることを、得られた解析及びシミュレーションの結果から示す。た pinv, pper の、潜伏期の長さを特徴付けるxに対する依存性を示す。さらに潜伏期における自覚症状の有無により定常状態、特に pper や流行状態の感染ノード密度に変化が生じることを示す。
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木津 幸子
時間周期外力の働く系における揺らぎ定理
エントロピー生成あるいはエントロピー生成率は、非平衡な過程を特徴付ける最も基本的かつ一般的な物理 量である。1990年代初頭に見出された「揺らぎ定理」の出現によりエントロピー生成は益々その重要性を増しつつある。観測時間τでのエントロピー生成を Δ とすると、「揺らぎ定理」はその確率分布 p(Δ) p(Δ)/p(-Δ)=exp(Δ) を満足することを主張する。小さな体系や短い観測時間では揺らぎが大きく、熱力学第2法則に反してエントロピー生成が負になることもありうるが、揺らぎ定理はその確率を定量的に議論することを可能にした。
ただし、確率分布 p(Δ) から直ちに熱力学第2法則 ⟨Δ⟩= ∫ dΔp(Δ)Δ > 0 が導かれるので、揺らぎ定理は熱力学第2法則を、揺らぎを含めた形に一般化（拡張）したものであるといえ る。揺らぎ定理研究の初期には、主として決定論的な力学系が扱われたが、次第に確率系も研究の対象になり、現在では非決定論的な、すなわちストキヤス ティックな力学系も多くの研究者の関心を集めている。また、「揺らぎ定理」の多くのものでは観測時間Tが十分大きいときに成立することを主張するが、現在、この制限を緩める方向でいくつかの研究がなされている。本研究においては、ポテンシャルV(x)の中を運動する（過減衰）ブラウン粒子に周期外の時間周期外力が作用する系をランジェバン方程式で記述し、 i）観測時間丁において、上に述べた揺らぎ定理が厳密に成立することを示し、 ii）これを以下のポテンシャルをもつ3つの系に適用した。
a）調和振動子系 V(x)=x2/2
b）2次一4次系 V(x)=(x+1)2(x-1)2
c）区分線形系 V(x)=x(2f) (0 ≤ x < f) , V(x)=(1-x)/2(1-f) (f ≤ x < 1)
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武田 隆之
社会ネットワークにおけるランダム性を含んだ意見形成モデルについて
あるネットワークを構成する者の間で周囲の意見を取り入れ、周囲の意見の平均値と自己の意見の積に雑音を加えたものを元に自己の意見の更新がなされるモデルを考える。一定時間後にそのネットワーク全体の意見がどのような結論状態 に達するかはネットワークの構造と雑音の強度に依存する。いま初期状態をランダムに決定し、雑音の最大値を指定した上で、十分なステップ数だけ意見の更新を行った状態を結論状態とみなす。リンク数とリンク先が固定された規則的なネットワーク、リンク数は固定でありながら単位時間ごとにリンク先がランダムに決定されるネットワーク、リンク数の分布がべき乗になりリンク先が固定されたネットワークに対して結論状態を求める。平均場近似を用いて理論値を導出し、各ネットワークの結論状態と比較する。また、一定の割合で自己の意見を固持するようなモデルを導入し、固持する割合による結論状態の差異を調べ、ネットーク内で固持する割合が一定であるモデルとそうでないモデルの間での比較を行い、ネットワークの構造ごとの結論状態の差異について考察する。
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平松 将
多変量自己回帰モデルを用いたスペクトル距離の分析
データに基づき体系の背後に横たわるダイナミックスを定量化することによって、 現象の理解を深める目的で、本研究では、時系列の緩和の程度を定量化するスペ クトラル・エントロピーに基づいたスペクトル距離の評価方法を提案する。そして、 多変量自己回帰モデルによって人工的に生成した時系列に対して、提案するスペクトル距離の定量化手法が妥当であることを示す。
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# Authors on the same line, change after authblk is loaded
Without loading the authblk package, if a document has 2 authors they are put next to each other (unless wide) on a titlepage. After loading the authblk package, the 2 authors are above each other. Is there a way to retain the next-to-each-other arrangement even with the authblk package loaded.
\documentclass[a4paper,11pt]{article}
%\usepackage{authblk}
\title{title}%
\date{\today}%
\author{%
author 1 \\ author 1 university%
\and
author 2 \\ author 2 university}%
\date{ ~ \\ \today }%
\begin{document}
\maketitle%
\end{document}
Thank you
According to the authblk documentation, you have to use the \author{...}\affil{...} notation, and set the noblocks package option. This will give you the footnote style of affiliation notation, which isn't quite the same as the original, but the only way I can see to keep them on one line.
\documentclass[a4paper,11pt]{article}
\usepackage[noblocks]{authblk}
\title{title}%
\date{\today}%
\author{%
author 1}
\affil{author 1 university}%
\author{author 2}
\affil{author 2 university}%
\date{ ~ \\ \today }%
\begin{document}
\maketitle%
\end{document}
Here is another way to do this,
\documentclass{article}
\usepackage{authblk}
\makeatletter
\renewcommand\AB@affilsepx{, \protect\Affilfont}
\makeatother
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# Calculate borrow/loan or repo rate
I was given this question on interview and couldn't find an answer in time (it is a software developer job in a place that deals with options). Can someone explain how to do this or point me to a good source of material?
Given two European options - a call $$C$$ and a put $$P$$ - struck at $$K$$, expiring at time $$T$$, on an underlying asset priced at $$S_t$$ today, derive the formula for the implied asset financing rate, i.e., the asset borrow/loan or repo rate. Assume the discount rate is $$r$$ and the dividend yield on the asset is $$q$$, both annual continuously compounded.
My answer was: Interest rate = [(future value/present value) – 1] x year/number of days, that's what I was able to find online at the time.
I believe, they are testing two things here:
1. That you know the Put-Call Parity (with dividends)
2. That you can successfully rearrange an equation
The Put-Call Parity with continuously compounded dividends is:
$$C-P=Se^{-qT}-Ke^{-rT}$$
The second part of the question is to rearrange the above for r.
Which gives:
$$r = \frac{-1}{T}\ln\left( \frac{Se^{-qT}-C+P}{K}\right)$$
I hope that this helps.
Let me know if you would like me to break down the rearranging?
• They also mentioned the rate $r$. Maybe the original interviewer distinguished between a treasury (i.e. unsecured) rate $r$, and a repo rate $h$. But I agree with the gist of your answer. In an interview I think something along these lines should probably be enough. Jul 8 '20 at 17:42
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# Liouville's theorem
1. Oct 4, 2009
### Petar Mali
Phase volume is constant.
$$\int_{G_0}dx^0=\int_{G_t}dx^t$$
$$x=(x_1,...,x_{6N})$$
$$\int_{G_0}dx^0=\int_{G_t}dx^t=\int_{G_0}Jdx^0$$
We must prove that $$J=1$$
$$J=\frac{\partial (x_1^t,...,x_{6N}^t)}{\partial (x_1^0,...,x_{6N}^0)}$$
$$J$$ is determinant with elements
$$a_{ik}=\frac{\partial x_i^t}{\partial x_k^0}$$
Minor for $$a_{ik}$$ is
$$D_{ik}=\frac{\partial J}{\partial a_{ik}}$$
And now $$J$$ is define like
$$J=\sum_{k}D_{ik}a_{ik}$$
Why is define like this? Why not
$$J=\sum_{i,k}D_{ik}a_{ik}$$ ?
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# Nonvariational systems
Nonvariational systems
A typical nonvariational elliptic system has the form
$(NV)\;\;\left\{\begin{array}[]{ll}-\Delta u=f(x;u,v),&\,x\in\Omega\\ -\Delta v=g(x;u,v),&\,x\in\Omega\\ u=v=0\;\mbox{or}\;\frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=% 0&\,x\in\partial\Omega\end{array}\right.$
where $\Omega\subset{\mathbb{R}}^{N}(N\geq 1)$ is an open bounded domain, $f(x;u,v),g(x;u,v)\in\mathcal{C}^{1}(\overline{\Omega}\times\mathbb{R}^{2};% \mathbb{R})$ in the variables $(u,v)\in\mathbb{R}^{2}$. Here, we further assume that there exists no function $G(x;u,v)$ with $\nabla G=(f,\pm g)$ or $\nabla G=(g,f)$. Under this assumption, it is easy to see that problem (NV) is nonvariational.
Title Nonvariational systems NonvariationalSystems1 2013-03-11 19:28:55 2013-03-11 19:28:55 linor (11198) (0) 1 linor (0) Definition
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?
Free Version
Moderate
# Finding the Volume of a Cone
GEOM-4RS@ZH
Find the volume of a right circular cone in which the height (H), the radius (R), and the slant height (S) of the cone form a 30-60-90 triangle. The radius of the cone and its slant height form the 60 degree angle, and the diameter of the base of the cone is 6 inches. (See picture below.)
A
16.324 $in^3$
B
48.973 $in^3$
C
84.823 $in^3$
D
195.890 $in^3$
E
4.141 $in^3$
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Copyright © University of Cambridge. All rights reserved.
## 'Series Sums' printed from http://nrich.maths.org/
### Show menu
As each sum develops it should become clear that the last number in each sum is triangular. So for $S_n$, the last number in the sum is the $n^{th}$ triangular number $= n(n + 1)/2$. Bearing this in mind and the fact that the first number in the sum is the $(n - 1)^{th}$ triangular number plus $1$, then,
\begin{eqnarray}S_n &=& \frac{n(n - 1)} {2} + 1 + \frac{n(n - 1)} {2} + 2 + \frac{n(n - 1)} {2} + 3 + \frac{n(n - 1)} {2} + \cdots + \frac{n(n - 1)} {2} + n \\ \; &=& \frac{n^2(n - 1)}{2} + (1 + 2 + 3 + 4 + \cdots + n) \\ \; &=& \frac{n^2(n - 1)}{2} + \frac{n(n + 1)}{2} \\ \; &=& \frac{n(n^2 - n)}{2} + \frac{n(n + 1)}{2} \\ S_n &=& \frac{n(n^2 + 1)}{2}\end{eqnarray}
Therefore $S_{17} = 17 \times\frac{17^2+1}{2} = 17 \times{290\over2} = 2465$
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0
Research Papers: Forced Convection
# Flow and Heat Transfer Over a Stretched Microsurface
[+] Author and Article Information
Suhil Kiwan1
Department of Mechanical Engineering, Jordan University of Science and Technology, P.O. Box 3030, Irbid, 22110, Jordankiwan@just.edu.jo
M. A. Al-Nimr
Department of Mechanical Engineering, Jordan University of Science and Technology, P.O. Box 3030, Irbid, 22110, Jordan
1
Corresponding author.
J. Heat Transfer 131(6), 061703 (Apr 09, 2009) (8 pages) doi:10.1115/1.3090811 History: Received May 24, 2008; Revised October 15, 2008; Published April 09, 2009
## Abstract
The convection heat transfer induced by a stretching flat plate has been studied. Similarity conditions are obtained for the boundary layer equations for a flat plate subjected to a power law temperature and velocity variations. It is found that a similarity solution exists only for a linearly stretching plate and only when the plate is isothermal. The analysis shows that three parameters control the flow and heat transfer characteristics of the problem. These parameters are the velocity slip parameter $K1$, the temperature slip parameter $K2$, and the Prandtl number. The effect of these parameters on the flow and heat transfer of the problem has been studied and presented. It is found that the slip velocity parameter affect both the flow and heat transfer characteristics of the problem. It is found that the skin friction coefficient decreases with increasing $K1$ and most of the changes in the skin friction takes place in the range $0. A correlation between the skin friction coefficient and $K1$ and $Rex$ has been found and presented. It is found that $cf=23Rex−0.5(K1+0.64)−0.884$ for $0 with an error of ±0.8%. Other correlations between Nu and $K1$ and $K2$ has been found and presented in Eq. 28.
<>
## Figures
Figure 13
Variation in the thermal boundary layer thickness with the variation of jump parameter K2 for different Prandtl numbers and K1=1
Figure 14
Variation in skin friction parameter with the variation of slip parameter K1 for all values of Pr and K2
Figure 15
Variation in Nusselt number with the variation of slip parameter K1 for different values of Pr and K2=1
Figure 16
Variation in Nusselt number with the variation of jump parameter K2 for different values of Pr and K1=1
Figure 1
Schematic for the problem under consideration
Figure 2
Variation in the dimensionless transverse velocity distribution with the similarity parameter η at different slip parameter K1
Figure 3
Variation in the dimensionless axial velocity distribution with the similarity parameter η at different slip parameter K1
Figure 4
Variation in the dimensionless shear parameter distribution with the similarity parameter η at different slip parameter K1
Figure 5
Variation in the dimensionless temperature distribution with the similarity parameter η at different slip parameter K1
Figure 6
Variation in the dimensionless temperature gradient distribution with the similarity parameter η at different slip parameter K1
Figure 7
Variation in the dimensionless temperature distribution with the similarity parameter η at different jump parameters K2, K1=1, Pr=1
Figure 8
Variation in the dimensionless temperature gradient distribution with the similarity parameter η at different jump parameters K2, K1=1, Pr=1
Figure 9
Variation in the dimensionless temperature distribution with the similarity parameter η at different Prandtl numbers for K1=1, K2=0.5
Figure 10
Variation in the dimensionless temperature gradient with the similarity parameter η at different Prandtl numbers for K1=1, and K2=0.5
Figure 11
Variation in the displacement thickness with the variation in the slip parameter K1 for all values of K2 and Pr
Figure 12
Variation in the thermal boundary layer thickness with the variation of slip parameter K1 for different Prandtl numbers and K2=1
## Discussions
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# Coprime
## Definition 1
Two integers are coprime if and only if the only integer that divides both of them is $$1$$.
## Definition 2
$$\forall a,b \in \mathbb{Z} : a \perp b \Leftrightarrow gcd(a, b) = 1$$
THEOREM $$\forall a \in \mathbb{Z} : 1 \perp a$$
coprime_1_left: coprime 1 ?a
Proof unavailable
THEOREM $$\forall a \in \mathbb{Z} : a \perp 1$$
coprime_1_left: coprime 1 ?a
Proof unavailable
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## Applied Statistics and Probability for Engineers, 6th Edition
The range is $[ 0 , 99999 ]$
Possible Values are integers from $0$ to $99999$
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# Variational Inference¶
Variational Inference is a powerful algorithm for fitting Bayesian networks. In this blog, you will learn about maths and intuition behind Variational Inference, Mean Field approximation and its implementation in Tensorflow Probability.
## Intro to Bayesian Networks¶
### Random Variables¶
Random Variables are simply variables whose values are uncertain. Eg -
1. In case of flipping a coin $n$ times, a random variable $X$ can be number of heads shown up.
2. In COVID-19 pandemic situation, random variable can be number of patients found positive with virus daily.
### Probability Distributions¶
Probability Distributions governs the amount of uncertainty of random variables. They have a math function with which they assign probabilities to different values taken by random variables. The associated math function is called probability density function (pdf). For simplicity, let's denote any random variable as $X$ and its corresponding pdf as $P\left (X\right )$. Eg - Following figure shows the probability distribution for number of heads when an unbiased coin is flipped 5 times.
### Bayesian Networks¶
Bayesian Networks are graph based representations to acccount for randomness while modelling our data. The nodes of the graph are random variables and the connections between nodes denote the direct influence from parent to child.
### Bayesian Network Example¶
Let's say a student is taking a class during school. The difficulty of the class and the intelligence of the student together directly influence student's grades. And the grades affects his/her acceptance to the university. Also, the intelligence factor influences student's SAT score. Keep this example in mind.
More formally, Bayesian Networks represent joint probability distribution over all the nodes of graph - $P\left (X_1, X_2, X_3, ..., X_n\right )$ or $P\left (\bigcap_{i=1}^{n}X_i\right )$ where $X_i$ is a random variable. Also Bayesian Networks follow local Markov property by which every node in the graph is independent on its non-descendants given its parents. In this way, the joint probability distribution can be decomposed as -
$$P\left (X_1, X_2, X_3, ..., X_n\right ) = \prod_{i=1}^{n} P\left (X_i | Par\left (X_i\right )\right )$$
Extra: Proof of decomposition
First, let's recall conditional probability,
$$P\left (A|B\right ) = \frac{P\left (A, B\right )}{P\left (B\right )}$$ The above equation is so derived because of reduction of sample space of $A$ when $B$ has already occured. Now, adjusting terms -
$$P\left (A, B\right ) = P\left (A|B\right )*P\left (B\right )$$ This equation is called chain rule of probability. Let's generalize this rule for Bayesian Networks. The ordering of names of nodes is such that parent(s) of nodes lie above them (Breadth First Ordering).
$$P\left (X_1, X_2, X_3, ..., X_n\right ) = P\left (X_n, X_{n-1}, X_{n-2}, ..., X_1\right )\\ = P\left (X_n|X_{n-1}, X_{n-2}, X_{n-3}, ..., X_1\right ) * P \left (X_{n-1}, X_{n-2}, X_{n-3}, ..., X_1\right ) \left (Chain Rule\right )\\ = P\left (X_n|X_{n-1}, X_{n-2}, X_{n-3}, ..., X_1\right ) * P \left (X_{n-1}|X_{n-2}, X_{n-3}, X_{n-4}, ..., X_1\right ) * P \left (X_{n-2}, X_{n-3}, X_{n-4}, ..., X_1\right )$$ Applying chain rule repeatedly, we get the following equation -
$$P\left (\bigcap_{i=1}^{n}X_i\right ) = \prod_{i=1}^{n} P\left (X_i | P\left (\bigcap_{j=1}^{i-1}X_j\right )\right )$$ Keep the above equation in mind. Let's bring back Markov property. To bring some intuition behind Markov property, let's reuse Bayesian Network Example. If we say, the student scored very good grades, then it is highly likely the student gets acceptance letter to university. No matter how difficult the class was, how much intelligent the student was, and no matter what his/her SAT score was. The key thing to note here is by observing the node's parent, the influence by non-descendants towards the node gets eliminated. Now, the equation becomes -
$$P\left (\bigcap_{i=1}^{n}X_i\right ) = \prod_{i=1}^{n} P\left (X_i | Par\left (X_i\right )\right )$$ Bingo, with the above equation, we have proved Factorization Theorem in Probability.
The decomposition of running Bayesian Network Example can be written as -
$$P\left (Difficulty, Intelligence, Grade, SAT, Acceptance Letter\right ) = P\left (Difficulty\right )*P\left (Intelligence\right )*\left (Grade|Difficulty, Intelligence\right )*P\left (SAT|Intelligence\right )*P\left (Acceptance Letter|Grade\right )$$
### Why care about Bayesian Networks¶
Bayesian Networks allow us to determine the distribution of parameters given the data (Posterior Distribution). The whole idea is to model the underlying data generative process and estimate unobservable quantities. Regarding this, Bayes formula can be written as -
$$P\left (\theta | D\right ) = \frac{P\left (D|\theta\right ) * P\left (\theta\right )}{P\left (D\right )}$$
$\theta$ = Parameters of the model
$P\left (\theta\right )$ = Prior Distribution over the parameters
$P\left (D|\theta\right )$ = Likelihood of the data
$P\left (\theta|D\right )$ = Posterior Distribution
$P\left (D\right )$ = Probability of Data. This term is calculated by marginalising out the effect of parameters.
$$P\left (D\right ) = \int P\left (D, \theta\right ) d\left (\theta\right )\\ P\left (D\right ) = \int P\left (D|\theta\right ) P\left (\theta\right ) d\left (\theta\right )$$
So, the Bayes formula becomes -
$$P\left (\theta | D\right ) = \frac{P\left (D|\theta\right ) * P\left (\theta\right )}{\int P\left (D|\theta\right ) P\left (\theta\right ) d\left (\theta\right )}$$
The devil is in the denominator. The integration over all the parameters is intractable. So we resort to sampling and optimization techniques.
## Intro to Variational Inference¶
### Information¶
Variational Inference has its origin in Information Theory. So first, let's understand the basic terms - Information and Entropy . Simply, Information quantifies how much useful the data is. It is related to Probability Distributions as -
$$I = -\log \left (P\left (X\right )\right )$$
The negative sign in the formula has high intuitive meaning. In words, it signifies whenever the probability of certain events is high, the related information is less and vica versa. For example -
1. Consider the statement - It never snows in deserts. The probability of this statement being true is significantly high because we already know that it is hardly possible to snow in deserts. So, the related information is very small.
2. Now consider - There was a snowfall in Sahara Desert in late December 2019. Wow, thats a great news because some unlikely event occured (probability was less). In turn, the information is high.
### Entropy¶
Entropy quantifies how much average Information is present in occurence of events. It is denoted by $H$. It is named Differential Entropy in case of Real Continuous Domain.
$$H = E_{P\left (X\right )} \left [-\log\left (P\left (X\right )\right )\right ]\\ H = -\int_X P_X\left (x\right ) \log\left (P_X\left (x\right )\right ) dx$$
### Entropy of Normal Distribution¶
As an exercise, let's calculate entropy of Normal Distribution. Let's denote $\mu$ as mean nd $\sigma$ as standard deviation of Normal Distribution. Remember the results, we will need them further.
$$X \sim Normal\left (\mu, \sigma^2\right )\\ P_X\left (x\right ) = \frac{1}{\sigma \sqrt{2 \pi}} e^{ - \frac{1}{2} \left ({\frac{x- \mu}{ \sigma}}\right )^2}\\ H = -\int_X P_X\left (x\right ) \log\left (P_X\left (x\right )\right ) dx$$
Only expanding $\log\left (P_X\left (x\right )\right )$ -
$$H = -\int_X P_X\left (x\right ) \log\left (\frac{1}{\sigma \sqrt{2 \pi}} e^{ - \frac{1}{2} \left ({\frac{x- \mu}{ \sigma}}\right )^2}\right ) dx\\ H = -\frac{1}{2}\int_X P_X\left (x\right ) \log\left (\frac{1}{2 \pi {\sigma}^2}\right )dx - \int_X P_X\left (x\right ) \log\left (e^{ - \frac{1}{2} \left ({\frac{x- \mu}{ \sigma}}\right )^2}\right ) dx\\ H = \frac{1}{2}\log \left ( 2 \pi {\sigma}^2 \right)\int_X P_X\left (x\right ) dx + \frac{1}{2{\sigma}^2} \int_X \left ( x-\mu \right)^2 P_X\left (x\right ) dx$$
Identifying terms -
$$\int_X P_X\left (x\right ) dx = 1\\ \int_X \left ( x-\mu \right)^2 P_X\left (x\right ) dx = \sigma^2$$
Substituting back, the entropy becomes -
$$H = \frac{1}{2}\log \left ( 2 \pi {\sigma}^2 \right) + \frac{1}{2\sigma^2} \sigma^2\\ H = \frac{1}{2}\left ( \log \left ( 2 \pi {\sigma}^2 \right) + 1 \right )$$
### KL divergence¶
This mathematical tool serves as the backbone of Variational Inference. Kullback–Leibler (KL) divergence measures the mutual information between two probability distributions. Let's say, we have two probability distributions $P$ and $Q$, then KL divergence quantifies how much similar these distributions are. Mathematically, it is just the difference between entropies of probabilities distributions. In terms of notation, $KL(Q||P)$ represents KL divergence with respect to $Q$ against $P$.
$$KL(Q||P) = H_P - H_Q\\ = -\int_X P_X\left (x\right ) \log\left (P_X\left (x\right )\right ) dx + \int_X Q_X\left (x\right ) \log\left (Q_X\left (x\right )\right ) dx$$
Changing $-\int_X P_X\left (x\right ) \log\left (P_X\left (x\right )\right ) dx$ to $-\int_X Q_X\left (x\right ) \log\left (P_X\left (x\right )\right ) dx$ as the KL divergence is with respect to $Q$.
$$= -\int_X Q_X\left (x\right ) \log\left (P_X\left (x\right )\right ) dx + \int_X Q_X\left (x\right ) \log\left (Q_X\left (x\right )\right ) dx\\ = \int_X Q_X\left (x \right) \log \left( \frac{Q_X\left (x \right)}{P_X\left (x \right)} \right) dx$$
Remember? We were stuck upon Bayesian Equation because of denominator term but now, we can estimate the posterior distribution $p(\theta|D)$ by another distribution $q(\theta)$ over all the parameters of the model.
$$KL(q(\theta)||p(\theta|D)) = \int q(\theta) \log \left( \frac{q(\theta)}{p(\theta|D)} \right) d\theta\\$$
Note
If two distributions are similar, then their entropies are similar, implies the KL divergence with respect to two distributions will be smaller. And vica versa. In Variational Inference, the whole idea is to minimize KL divergence so that our approximating distribution $q(\theta)$ can be made similar to $p(\theta|D)$.
Extra: What are latent variables?
If you go about exploring any paper talking about Variational Inference, then most certainly, the papers mention about latent variables instead of parameters. The parameters are fixed quantities for the model whereas latent variables are unobserved quantities of the model conditioned on parameters. Also, we model parameters by probability distributions. For simplicity, let's consider the running terminology of parameters only.
### Evidence Lower Bound¶
There is again an issue with KL divergence formula as it still involves posterior term i.e. $p(\theta|D)$. Let's get rid of it -
$$KL(q(\theta)||p(\theta|D)) = \int q(\theta) \log \left( \frac{q(\theta)}{p(\theta|D)} \right) d\theta\\ KL = \int q(\theta) \log \left( \frac{q(\theta) p(D)}{p(\theta, D)} \right) d\theta\\ KL = \int q(\theta) \log \left( \frac{q(\theta)}{p(\theta, D)} \right) d\theta + \int q(\theta) \log \left(p(D) \right) d\theta\\ KL + \int q(\theta) \log \left( \frac{p(\theta, D)}{q(\theta)} \right) d\theta = \log \left(p(D) \right) \int q(\theta) d\theta\\$$
Identifying terms -
$$\int q(\theta) d\theta = 1$$
So, substituting back, our running equation becomes -
$$KL + \int q(\theta) \log \left( \frac{p(\theta, D)}{q(\theta)} \right) d\theta = \log \left(p(D) \right)$$
The term $\int q(\theta) \log \left( \frac{p(\theta, D)}{q(\theta)} \right) d\theta$ is called Evidence Lower Bound (ELBO). The right side of the equation $\log \left(p(D) \right)$ is constant.
Observe
Minimizing the KL divergence is equivalent to maximizing the ELBO. Also, the ELBO does not depend on posterior distribution.
Also,
$$ELBO = \int q(\theta) \log \left( \frac{p(\theta, D)}{q(\theta)} \right) d\theta\\ ELBO = E_{q(\theta)}\left [\log \left( \frac{p(\theta, D)}{q(\theta)} \right) \right]\\ ELBO = E_{q(\theta)}\left [\log \left(p(\theta, D) \right) \right] + E_{q(\theta)} \left [-\log(q(\theta)) \right]$$
The term $E_{q(\theta)} \left [-\log(q(\theta)) \right]$ is entropy of $q(\theta)$. Our running equation becomes -
$$ELBO = E_{q(\theta)}\left [\log \left(p(\theta, D) \right) \right] + H_{q(\theta)}$$
So far, the whole crux of the story is - To approximate the posterior, maximize the ELBO term. ADVI = Automatic Differentiation Variational Inference. I think the term Automatic Differentiation deals with maximizing the ELBO (or minimizing the negative ELBO) using any autograd differentiation library. Coming to Mean Field ADVI (MF ADVI), we simply assume that the parameters of approximating distribution $q(\theta)$ are independent and posit Normal distributions over all parameters in transformed space to maximize ELBO.
### Transformed Space¶
To freely optimize ELBO, without caring about matching the support of model parameters, we transform the support of parameters to Real Coordinate Space. In other words, we optimize ELBO in transformed/unconstrained/unbounded space which automatically maps to minimization of KL divergence in original space. In terms of notation, let's denote a transformation over parameters $\theta$ as $T$ and the transformed parameters as $\zeta$. Mathematically, $\zeta=T(\theta)$. Also, since we are approximating by Normal Distributions, $q(\zeta)$ can be written as -
$$q(\zeta) = \prod_{i=1}^{k} N(\zeta_k; \mu_k, \sigma^2_k)$$
Now, the transformed joint probability distribution of the model becomes -
$$p\left (D, \zeta \right) = p\left (D, T^{-1}\left (\zeta \right) \right) \left | det J_{T^{-1}}(\zeta) \right |\\$$
Extra: Proof of transformation equation
To simplify notations, let's use $Y=T(X)$ instead of $\zeta=T(\theta)$. After reaching the results, we will put the values back. Also, let's denote cummulative distribution function (cdf) as $F$. There are two cases which respect to properties of function $T$.
Case 1 - When $T$ is an increasing function $$F_Y(y) = P(Y <= y) = P(T(X) <= y)\\ = P\left(X <= T^{-1}(y) \right) = F_X\left(T^{-1}(y) \right)\\ F_Y(y) = F_X\left(T^{-1}(y) \right)$$Let's differentiate with respect to $y$ both sides - $$\frac{\mathrm{d} (F_Y(y))}{\mathrm{d} y} = \frac{\mathrm{d} (F_X\left(T^{-1}(y) \right))}{\mathrm{d} y}\\ P_Y(y) = P_X\left(T^{-1}(y) \right) \frac{\mathrm{d} (T^{-1}(y))}{\mathrm{d} y}$$Case 2 - When $T$ is a descreasing function $$F_Y(y) = P(Y <= y) = P(T(X) <= y) = P\left(X >= T^{-1}(y) \right)\\ = 1-P\left(X < T^{-1}(y) \right) = 1-P\left(X <= T^{-1}(y) \right) = 1-F_X\left(T^{-1}(y) \right)\\ F_Y(y) = 1-F_X\left(T^{-1}(y) \right)$$Let's differentiate with respect to $y$ both sides - $$\frac{\mathrm{d} (F_Y(y))}{\mathrm{d} y} = \frac{\mathrm{d} (1-F_X\left(T^{-1}(y) \right))}{\mathrm{d} y}\\ P_Y(y) = (-1) P_X\left(T^{-1}(y) \right) (-1) \frac{\mathrm{d} (T^{-1}(y))}{\mathrm{d} y}\\ P_Y(y) = P_X\left(T^{-1}(y) \right) \frac{\mathrm{d} (T^{-1}(y))}{\mathrm{d} y}$$Combining both results - $$P_Y(y) = P_X\left(T^{-1}(y) \right) \left | \frac{\mathrm{d} (T^{-1}(y))}{\mathrm{d} y} \right |$$Now comes the role of Jacobians to deal with multivariate parameters $X$ and $Y$. $$J_{T^{-1}}(Y) = \begin{vmatrix} \frac{\partial (T_1^{-1})}{\partial y_1} & ... & \frac{\partial (T_1^{-1})}{\partial y_k}\\ . & & .\\ . & & .\\ \frac{\partial (T_k^{-1})}{\partial y_1} & ... &\frac{\partial (T_k^{-1})}{\partial y_k} \end{vmatrix}$$Concluding - $$P(Y) = P(T^{-1}(Y)) |det J_{T^{-1}}(Y)|\\P(Y) = P(X) |det J_{T^{-1}}(Y)|$$Substitute $X$ as $\theta$ and $Y$ as $\zeta$, we will get - $$P(\zeta) = P(T^{-1}(\zeta)) |det J_{T^{-1}}(\zeta)|\\$$
### ELBO in transformed Space¶
Let's bring back the equation formed at ELBO. Expressing ELBO in terms of $\zeta$ -
$$ELBO = E_{q(\theta)}\left [\log \left(p(\theta, D) \right) \right] + H_{q(\theta)}\\ ELBO = E_{q(\zeta)}\left [\log \left(p\left (D, T^{-1}\left (\zeta \right) \right) \left | det J_{T^{-1}}(\zeta) \right | \right) \right] + H_{q(\zeta)}$$
Since, we are optimizing ELBO by factorized Normal Distributions, let's bring back the results of Entropy of Normal Distribution. Our running equation becomes -
$$ELBO = E_{q(\zeta)}\left [\log \left(p\left (D, T^{-1}\left (\zeta \right) \right) \left | det J_{T^{-1}}(\zeta) \right | \right) \right] + H_{q(\zeta)}\\ ELBO = E_{q(\zeta)}\left [\log \left(p\left (D, T^{-1}\left (\zeta \right) \right) \left | det J_{T^{-1}}(\zeta) \right | \right) \right] + \frac{1}{2}\left ( \log \left ( 2 \pi {\sigma}^2 \right) + 1 \right )$$
Success
The above ELBO equation is the final one which needs to be optimized.
### Let's Code¶
In [1]:
# Imports
%matplotlib inline
import numpy as np
import scipy as sp
import pandas as pd
import tensorflow as tf
from scipy.stats import expon, uniform
import arviz as az
import pymc3 as pm
import matplotlib.pyplot as plt
import tensorflow_probability as tfp
from pprint import pprint
plt.style.use("arviz-darkgrid")
from tensorflow_probability.python.mcmc.transformed_kernel import (
make_transform_fn, make_transformed_log_prob)
tfb = tfp.bijectors
tfd = tfp.distributions
dtype = tf.float32
In [2]:
# Plot functions
def plot_transformation(theta, zeta, p_theta, p_zeta):
fig, (const, trans) = plt.subplots(nrows=2, ncols=1, figsize=(6.5, 12))
const.plot(theta, p_theta, color='blue', lw=2)
const.set_xlabel(r"$\theta$")
const.set_ylabel(r"$P(\theta)$")
const.set_title("Constrained Space")
trans.plot(zeta, p_zeta, color='blue', lw=2)
trans.set_xlabel(r"$\zeta$")
trans.set_ylabel(r"$P(\zeta)$")
trans.set_title("Transfomed Space");
### Transformed Space Example-1¶
Transformation of Standard Exponential Distribution
$$P_X(x) = e^{-x}$$
The support of Exponential Distribution is $x>=0$. Let's use log transformation to map the support to real number line. Mathematically, $\zeta=\log(\theta)$. Now, let's bring back our transformed joint probability distribution equation -
$$P(\zeta) = P(T^{-1}(\zeta)) |det J_{T^{-1}}(\zeta)|\\ P(\zeta) = P(e^{\zeta}) * e^{\zeta}$$
Converting this directly into Python code -
In [3]:
theta = np.linspace(0, 5, 100)
zeta = np.linspace(-5, 5, 100)
dist = expon()
p_theta = dist.pdf(theta)
p_zeta = dist.pdf(np.exp(zeta)) * np.exp(zeta)
plot_transformation(theta, zeta, p_theta, p_zeta)
### Transformed Space Example-2¶
Transformation of Uniform Distribution (with support $0<=x<=1$)
$$P_X(x) = 1$$
Let's use logit or inverse sigmoid transformation to map the support to real number line. Mathematically, $\zeta=logit(\theta)$.
$$P(\zeta) = P(T^{-1}(\zeta)) |det J_{T^{-1}}(\zeta)|\\ P(\zeta) = P(sig(\zeta)) * sig(\zeta) * (1-sig(\zeta))$$
where $sig$ is the sigmoid function.
Converting this directly into Python code -
In [4]:
theta = np.linspace(0, 1, 100)
zeta = np.linspace(-5, 5, 100)
dist = uniform()
p_theta = dist.pdf(theta)
sigmoid = sp.special.expit
p_zeta = dist.pdf(sigmoid(zeta)) * sigmoid(zeta) * (1-sigmoid(zeta))
plot_transformation(theta, zeta, p_theta, p_zeta)
Infer $\mu$ and $\sigma$ for Normal distribution.
In [5]:
# Generating data
mu = 12
sigma = 2.2
data = np.random.normal(mu, sigma, size=200)
In [6]:
# Defining the model
model = tfd.JointDistributionSequential([
# sigma_prior
tfd.Exponential(1, name='sigma'),
# mu_prior
tfd.Normal(loc=0, scale=10, name='mu'),
# likelihood
lambda mu, sigma: tfd.Normal(loc=mu, scale=sigma)
])
In [7]:
print(model.resolve_graph())
(('sigma', ()), ('mu', ()), ('x', ('mu', 'sigma')))
In [8]:
# Let's generate joint log probability
joint_log_prob = lambda *x: model.log_prob(x + (data,))
In [9]:
# Build Mean Field ADVI
parameters = model.sample(1)
parameters.pop()
dists = []
for i, parameter in enumerate(parameters):
shape = parameter[0].shape
loc = tf.Variable(
tf.random.normal(shape, dtype=dtype),
name=f'meanfield_{i}_loc',
dtype=dtype
)
scale = tfp.util.TransformedVariable(
tf.fill(shape, value=tf.constant(0.02, dtype=dtype)),
tfb.Softplus(), # For positive values of scale
name=f'meanfield_{i}_scale'
)
approx_parameter = tfd.Normal(loc=loc, scale=scale)
dists.append(approx_parameter)
return tfd.JointDistributionSequential(dists)
TFP handles transformations differently as it transforms unconstrained space to match the support of distributions.
In [10]:
unconstraining_bijectors = [
tfb.Exp(),
tfb.Identity()
]
posterior = make_transformed_log_prob(
joint_log_prob,
unconstraining_bijectors,
direction='forward',
enable_bijector_caching=False
)
In [11]:
opt = tf.optimizers.Adam(learning_rate=.1)
@tf.function(autograph=False)
def run_approximation():
elbo_loss = tfp.vi.fit_surrogate_posterior(
posterior,
optimizer=opt,
sample_size=200,
num_steps=10000)
return elbo_loss
elbo_loss = run_approximation()
WARNING:tensorflow:From /usr/local/lib/python3.8/site-packages/tensorflow_probability/python/math/minimize.py:74: calling <lambda> (from tensorflow_probability.python.vi.optimization) with loss is deprecated and will be removed after 2020-07-01.
Instructions for updating:
The signature for trace_fns passed to minimize has changed. Trace functions now take a single traceable_quantities argument, which is a tfp.math.MinimizeTraceableQuantities namedtuple containing traceable_quantities.loss, traceable_quantities.gradients, etc. Please update your trace_fn definition.
In [12]:
plt.plot(elbo_loss, color='blue')
plt.yscale("log")
plt.xlabel("No of iterations")
plt.ylabel("Negative ELBO")
plt.show()
In [13]:
graph_info = model.resolve_graph()
approx_param = dict()
for i, (rvname, param) in enumerate(graph_info[:-1]):
approx_param[rvname] = {"mu": free_param[i*2].numpy(),
"sd": free_param[i*2+1].numpy()}
In [14]:
print(approx_param)
{'sigma': {'mu': 0.7740287, 'sd': -0.7494337}, 'mu': {'mu': 11.233825, 'sd': 1.7977774}}
We got pretty good estimates of sigma and mu. We need to transform sigma via exp and I believe it will be something close to 2.2
## Drawbacks of this blog post¶
1. I have not used consistent notation for probability density functions (pdfs). Because I like equations handled this way.
2. Coming up with more good examples using minibatches.
3. The ADVI papers also mention Elliptical standardization and Adaptive step size for optimizers. I have not understood those sections well and thus, haven't tried to implement them.
## Special Thanks¶
• Website: codecogs.com to help me generate LaTeX equations.
• Comments: #1 and #2 by Luciano Paz that cleared my all doubts regarding transformations.
Last update: September 11, 2020
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# Step-by-step Solution
Go!
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8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
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(◻)
+
-
×
◻/◻
/
÷
2
e
π
ln
log
log
lim
d/dx
Dx
|◻|
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc
asin
acos
atan
acot
asec
acsc
sinh
cosh
tanh
coth
sech
csch
asinh
acosh
atanh
acoth
asech
acsch
## Step-by-step explanation
Problem to solve:
$\int_1^{\infty}\left(\frac{x-2}{x\cdot\left(x^2+1\right)}\right)dx$
Learn how to solve definite integrals problems step by step online.
$\frac{x-2}{x\left(x^2+1\right)}=\frac{A}{x}+\frac{Bx+C}{x^2+1}$
Learn how to solve definite integrals problems step by step online. Integrate (x-2)/(x(x^2+1)) from 1 to \infty. Rewrite the fraction \frac{x-2}{x\left(x^2+1\right)} in 2 simpler fractions using partial fraction decomposition. Find the values of the unknown coefficients. The first step is to multiply both sides of the equation by x\left(x^2+1\right). Multiplying polynomials. Simplifying.
$0.0923-2\ln\left|\infty \right|+\ln\left|\infty \right|$
### Problem Analysis
$\int_1^{\infty}\left(\frac{x-2}{x\cdot\left(x^2+1\right)}\right)dx$
### Main topic:
Definite integrals
~ 0.35 seconds
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Writing firewall rules
Congratulations, you're now a network administrator! As your first task you will need to configure a firewall to stop any traffic to some blacklisted addresses.
Unfortunately, the only firewall at your disposal only takes in rules in this format:
x.y.z.t/m ALLOW
There's no DENY keyword. If you want to exclude a specific address, you'll have to write a lot of rules to allow everything except that address. If you want to exclude more than one address or a whole range, then you're in luck because these ALLOW rules will overlap to some degree.
The challenge
You are given a list of blacklisted IPs. Write a program that will print out rules for our firewall. Also, be sensible when writing rules: don't just enumerate all 256^4 and don't print out duplicates.
Code golf rules apply: shortest code wins
*this problem is inspired by a real life assignment; a friend actually needed this done
Example
To block address 0.1.2.3 you could write the following ALLOW rules (this is not the only solution possible, and it's certainly not the shortest):
0.1.2.0 ALLOW
0.1.2.1 ALLOW
0.1.2.2 ALLOW
0.1.2.4 ALLOW
...
0.1.2.255 ALLOW
0.1.0.0/24 ALLOW
0.1.1.0/24 ALLOW
0.1.4.0/24 ALLOW
...
0.1.255.0/24 ALLOW
0.0.0.0/16 ALLOW
0.2.0.0/16 ALLOW
...
0.255.0.0/16 ALLOW
1.0.0.0/8 ALLOW
...
255.0.0.0/8 ALLOW
• Could you add some test cases? – grc May 25 '13 at 7:39
• "shortest code" conflicts with "be sensible". Are there any hard requirements for "being sensible"? Note there is always exactly one optimal solution and it's fairly easy to find and verify optimality. You could require that. – John Dvorak May 25 '13 at 8:14
Python3 - 325 chars
import sys
r=[(0,32)]
for b in (int('%02x'*4%tuple(int(x) for x in y.split('.')),16) for y in sys.stdin):
i=0
while i<len(r):l,q=r[i];o=2**q//2;m=l+o;h=m+o;k=l<=b<h;r[i:i+1]=[(l,q-k),(m,q-k)][0:k+1*(not l==h==b)];i+=1-k
for n,m in r:print('%s/%d ALLOW'%('.'.join(str(int(('%08x'%n)[i:i+2],16)) for i in range(0,8,2)),32-m))
Python 3: 253 characters
Yay, Python!
from ipaddress import *
import sys
def e(x,a):
b=[]
for i in a:
except ValueError: b+=[i]
return b
a=[ip_network('0.0.0.0/0')]
for h in sys.stdin:
for i in h.split(): a=e(i,a)
for i in a: print(i,'ALLOW')
It accepts a list of addresses on stdin, separated by any (ASCII) whitespace you like (no commas, though).
You can save 12 characters by replacing the stdin loop with for i in sys.stdin: a=e(i.strip(),a). The strip() is necessary to get rid of a trailing newline anyway, though, so I figured I'd use split() and another for loop and allow any whitespace.
You can probably save a few more characters by moving the code out of that pesky function. I didn't because I figured I was already confusing myself enough.
You can also make it work for IPv6 instead by changing only one line.
Ungolfed version:
#!/usr/bin/env python3
import sys
new_networks = []
for network in networks:
try:
except ValueError:
new_networks.append(network)
except TypeError:
pass
return new_networks
for line in sys.stdin:
networks = exclude(line, networks)
for network in networks:
print(network)
This one has the added benefit of working for IPv4 and IPv6, no tweaking necessary. :)
• You could still remove unnecessary spaces, such as the ones after :s and before the first *. Great first answer on this site! :) +1 – Doorknob Mar 1 '14 at 22:53
PHP 5.3, 465 characters
$r=array_map(function($v){return explode('.',$v);},$w);$l=256;$n=" ALLOW\r\n";$o='';$a=$b=$c=$d=range(0,$l-1);foreach($r as$i){unset($a[$i[0]],$b[$i[1]],$c[$i[2]],$d[$i[3]]);}for($x=0;$x<$l;$x++){for($y=0;$y<$l;$y++){for($z=0;$z<$l;$z++){for($t=0;$t<$l;$t++){$k="$x.$y.$z.$t";if(isset($a[$x])){$o.="$k/8$n";break 3;}elseif(isset($b[$y])){$o.="$k/16$n";break 2;}elseif(isset($c[$z])){$o.="$k/24$n";break;}else$o.=isset($d[$t])?"$k$n":'';}}}}file_put_contents($p,$o); Usage: $w = array('2.5.5.4', '1.2.3.4');
$p = 'data.txt';$r=array_map(function($v){return explode('.',$v);},$w);$l=256;$n=" ALLOW\r\n";$o='';$a=$b=$c=$d=range(0,$l-1);foreach($r as$i){unset($a[$i[0]],$b[$i[1]],$c[$i[2]],$d[$i[3]]);}for($x=0;$x<$l;$x++){for($y=0;$y<$l;$y++){for($z=0;$z<$l;$z++){for($t=0;$t<$l;$t++){$k="$x.$y.$z.$t";if(isset($a[$x])){$o.="$k/8$n";break 3;}elseif(isset($b[$y])){$o.="$k/16$n";break 2;}elseif(isset($c[$z])){$o.="$k/24$n";break;}else$o.=isset($d[$t])?"$k$n":'';}}}}file_put_contents($p,$o);
This will write to data.txt
Uncompressed:
$deny = array('2.5.5.4', '1.2.3.4');$file = 'data.txt';
$list = array_map(function($v){return explode('.', $v);},$deny);
$l = 256;$n = " ALLOW\r\n";$o = '';$first = $second =$third = $fourth = range(0,$l-1);
foreach($list as$ip){unset($first[$ip[0]], $second[$ip[1]], $third[$ip[2]], $fourth[$ip[3]]);}
for($x=0;$x<$l;$x++){
for($y=0;$y<$l;$y++){
for($z=0;$z<$l;$z++){
for($t=0;$t<$l;$t++){
if(isset($first[$x])){
$o .= "$x.0.0.0/8$n";break 3; }else{ if(isset($second[$y])){$o.="$x.$y.0.0/16$n";break 2; }else{ if(isset($third[$z])){$o.="$x.$y.$z.0/24$n";break 1;
}else{
if(isset($fourth[$t])){
$o.="$x.$y.$z.$t$n";
}
}
}
}
}
}
}
}
file_put_contents($file,$o);
• Won't this solution only consider the subnets /8, /16 and /24? – jarnbjo May 27 '13 at 14:56
• @jarnbjo correct, but it will write the rest without /x. For examlpe in this case it will write 1.2.3.0 ALLOW\r\n1.2.3.1 ALLOW\r\n 1.2.3.2 ALLOW\r\n1.2.3.3 ALLOW\r\n1.2.3.5 ALLOW .... – HamZa May 27 '13 at 14:59
• The shortest rule list to block only 0.1.2.3 would require the rule "0.1.2.0/31 ALLOW" instead of allowing the IP addresses 0.1.2.0 and 0.1.2.1 separately. – jarnbjo May 27 '13 at 15:59
• @jarnbjo I know, but if I would take that into consideration the code will be too huge, I'm already using PHP so I'm suffering enough from the code length. Besides it isn't specified by the OP :) – HamZa May 27 '13 at 16:30
• I assumed that returning the shortest rule list is part of the "being sensible" requirement. Otherwise you could shorten your code a lot and simply output each allowed IP address, but that's obviously not what's asked for either. – jarnbjo May 27 '13 at 23:07
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# How Much Water Should Be Added to 15 Grams of Salt to Obtain 15 per Cent Salt Solution ? - Science
Short Note
Sum
How much water should be added to 15 grams of salt to obtain 15 per cent salt solution ?
#### Solution
To make the salt solution of 15% with 15 gm as the mass of the solute, the mass of the solvent must be 85 gm.
"concentration" = "mass of solute"/"mass of solute + mass of solvent"xx 100
Mass of solution = mass of solute + mass of solvent
Mass of the solvent = mass of the solution – mass of solvent
= 100 – 15 = 85 gm
Therefore, 85 gm of water should be added.
Concept: Matter
Is there an error in this question or solution?
#### APPEARS IN
Lakhmir Singh Class 9 Chemistry - Science Part 2
Chapter 2 Is Matter Around Us Pure
Very Short Answers | Q 6 | Page 79
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#### Archived
This topic is now archived and is closed to further replies.
# if ( FAILED (...)
This topic is 5696 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
Professional game programs are full of : if FAILED(lpD3DRM->CreateMesh...) goto generic_error; I never use if FAILED ... being a lazy amateur I do not care if my code FAIL , from time to time, but just as a matter of curiosity. Does it fail so often? If so, who is the guilty? The hardware, Windows, the third law of thermodinamics ? may be direct x? thanks and regards to all GameDev members
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quote:
Original post by AlbertoT
Professional game programs are full of :
if FAILED(lpD3DRM->CreateMesh...)
goto generic_error;
this is the standard way of checking for errors in COM world, to which DX belongs.
quote:
I never use if FAILED ... being a lazy amateur I do not care if my code FAIL , from time to time, but just as a matter of curiosity.
you better start checking for errors, because they are inevitable and error checking saves on debugging time.
quote:
Does it fail so often?
it depends on how sloppy you write your programs.
quote:
If so, who is the guilty?
The hardware, Windows, the third law of thermodinamics ? may be direct x?
i think it would be safe to say that in over 99% of the cases the programmer is, so add error checking to your programs.
---
Come to #directxdev IRC channel on AfterNET
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Ideally, it never fails at all. But when it does, it''s nice to know EXACTLY when that is. You say you''re lazy; I say you''re inexperienced. Once you''ve done a lot of debugging, you''ll appreciate how essential assiduous error checking is.
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Not so much as the API failing, but more, if you try to do something that something on the system does not support, or if the system is busy doing something else. For example, if you try to set a display adapter resolution that the adapter doesn''t support (i.e. 2048x1536x24bpp), which might exist on some high-end graphics cards) then you need to be able to tell whether or not that attempt fails in software, to properly compensate and notify the user that it''s not possible. Likewise, if you try to page flip on a device that has only a primary buffer, you would get a return value (not necessarily an error) indicating the action is not possible (or the screen update should be blitted rather than page-flipped).
As for whether or not testing with FAILED() is professional is a matter of personal opinion.
MatrixCubed
http://MatrixCubed.cjb.net
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Well, to add my two pence - even if you don''t really mind if FAILED is true, I can guess the users of your programs do Of course, if you don''t have any - it doesn''t matter! Using FAILED will **at the very least** enable you to present a message to the user explaining why all around them is about to come crashing down - apart from that, yep, its useful for debugging.
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If you don''t bother checking for errors and you''re actually getting anywhere in what your doing you''ll have a rude awakening when you want your stuff to run on other computers.
------------
- outRider -
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Two types of things which will return failure SCODEs/HRESULTs:
1. Things which could feasibly happen in a bug free app in the real world. For example DX device creation failing due to lack of memory.
2. Things which shouldn''t happen in a properly written, bug free app. In DX for example you have caps flags and ValidateDevice, so a SetRenderState for example should never fail because code earlier on should have prevented invalid values being passed.
For Type #1, we always use the FAILED() and SUCCEEDED() macros, and often also act on the actual code returned after it''s been trapped by one of those.
For Type #2, we use a macro which performs the test and a report in debug builds and compiles away to nothing in release builds:
#ifdef _DEBUG #define TRYDX(fn) do { \ HRESULT __dbgHResult__ = fn; \ if (FAILED(__dbgHResult__)) { \ _dbgLogDirectXError( __FILE__, __LINE__, __dbgHResult__, #fn ); \ } \ } while(0);#else #define TRYDX(fn) (fn)#endif
_dbgLogDirectXError is a global function only present in debug builds which produces a report including the line number and source file of where the error occured along with the actual line itself and a translation of the error code into English.
Usually this report is sent to the debug stream (OutputDebugString), but can be logged to a file or even cause an error box.
--
Simon O''Connor
Creative Asylum Ltd
www.creative-asylum.com
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nice macro btw, I kinda like it
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quote:
Original post by S1CA
2. Things which shouldn''t happen in a properly written, bug free app. In DX for example you have caps flags and ValidateDevice, so a SetRenderState for example should never fail because code earlier on should have prevented invalid values being passed.
I used to check all dx return values, but gave up, and here''s my reason why. The docs state that retail runtime doesn''t perform parameter validation in many cases, so an error return from SetRenderState or similar or very unlikely (from the retail runtime). Therefore, checking for error returns is pointless. At the same time, with debug runtime I use ''break on error'' feature and there''s no need to add error handling because dx will break anyway, along wth something more meaningful than D3DERR_INVALIDCALL. I do check all functions that might fail in normal conditions, like Create... ones and Present.
Reasonable?
---
Come to #directxdev IRC channel on AfterNET
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quote:
I used to check all dx return values, but gave up, and here''s my reason why. The docs state that retail runtime doesn''t perform parameter validation in many cases, so an error return from SetRenderState or similar or very unlikely (from the retail runtime).
Yep it is entirely reasonable to not check calls which fall into category 2 of my original reply.
"Break on D3D error" is indeed a good way to trap those for D3D errors.
However, there isn''t a "break on error" for other DirectX components, plus its use implies you have a development environment and the full source code available.
When we''re testing stuff which is still in development, people within the company without the full source code or even dev environments will sometimes report problems. The function called by the macro will log to a file (when set to do so) so we get to the problem much quicker on any machine.
It''s much easier to install the debug DX runtime than it is to install dev tools and the codebase!
IMO you should do as much as possible at code level (asserts etc) to provide an early warning system - e.g. someone changes something wrongly - an assert pops up as soon as they test it - they can fix the issue immediately while they can still remember what they changed .
--
Simon O''Connor
Creative Asylum Ltd
www.creative-asylum.com
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quote:
Original post by S1CA
However, there isn''t a "break on error" for other DirectX components, plus its use implies you have a development environment and the full source code available.
Good point, thanks. I''ll consider asserting everything.
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Come to #directxdev IRC channel on AfterNET
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# The Mechanics of Omitted Variable Bias: Bias Amplification and Cancellation of Offsetting Biases
Peter M. Steiner and Yongnam Kim
# Abstract
Causal inference with observational data frequently requires researchers to estimate treatment effects conditional on a set of observed covariates, hoping that they remove or at least reduce the confounding bias. Using a simple linear (regression) setting with two confounders – one observed (X), the other unobserved (U) – we demonstrate that conditioning on the observed confounder X does not necessarily imply that the confounding bias decreases, even if X is highly correlated with U. That is, adjusting for X may increase instead of reduce the omitted variable bias (OVB). Two phenomena can cause an increasing OVB: (i) bias amplification and (ii) cancellation of offsetting biases. Bias amplification occurs because conditioning on X amplifies any remaining bias due to the omitted confounder U. Cancellation of offsetting biases is an issue whenever X and U induce biases in opposite directions such that they perfectly or partially offset each other, in which case adjusting for X inadvertently cancels the bias-offsetting effect. In this article we discuss the conditions under which adjusting for X increases OVB, and demonstrate that conditioning on X increases the imbalance in U, which turns U into an even stronger confounder. We also show that conditioning on an unreliably measured confounder can remove more bias than the corresponding reliable measure. Practical implications for causal inference will be discussed.
## Introduction
Causal inference with observational studies frequently requires researchers to estimate treatment effects conditional on a set of observed baseline covariates in order to remove confounding bias. Covariate-adjusted effect estimates can be obtained by controlling for the observed covariates in a regression analysis, or by matching cases on the observed covariates or the corresponding propensity score. It is well known that the confounding bias can be removed if all the confounding covariates that simultaneously determine treatment selection and the outcome are observed. This condition is frequently referred to as the conditional independence assumption, selection on observables, strong ignorability assumption, unconfoundedness, or the backdoor or adjustment criterion [1, 2, 34]. If one fails to reliably measure all the confounding covariates, the causal effect is not identified and the covariate-adjusted treatment effect will usually remain biased. In the linear regression context, the bias due an omitted variable is formalized in the omitted variable bias (OVB) formula [2, 5, 6, 7]. [1]
Though OVB is well known and has been discussed for decades, the mechanics of OVB are not yet fully understood which regularly leads to misguided advice regarding the reduction of confounding bias in practice. Applied and methodological articles and textbooks regularly suggest that including more variables in a regression model will more likely establish the conditional independence assumption and thus reduce or at least not increase confounding bias (e. g., [8, 9, 10], see [7], for a brief discussion of this ill-advised rationale for including more rather than less covariates). Similarly, there is a strong belief that adjusting for an observed variable that is correlated with unobserved confounders necessarily removes a part of the bias induced by the unobserved confounders and, thus, further reduces bias. Particularly the matching literature suggests that matching on variables that are correlated with unobserved confounders reduces the imbalance in and the bias due to unobserved confounders (e. g., [11, 12, 13]). We will show that even a high correlation neither guarantees a decrease in imbalance in the unobserved confounders nor a decreasing bias. We will also show that measurement error in covariates (unreliability) does not imply that less bias is removed.
Recently, researchers started looking at the mechanics of OVB in more detail. In particular, they have been investigating what happens if one conditions on covariates that have the potential to induce or amplify bias. Such covariates are collider variables that induce their own bias in addition to any OVB [14, 15, 16], or instrumental variables (IVs) that amplify any bias left after conditioning on a set of observed covariates [17, 18]. Another class of bias-amplifying covariates are near-IVs that strongly determine treatment selection but affect the outcome only weakly (the weak instead of absent association with the outcome turns them into a near-IV). Pearl [17, 19], see also [20, 21], formally showed that adjusting for a near-IV removes the near-IV’s own confounding bias but also amplifies any bias left due to omitted confounders. Also simulation studies have been used to demonstrate that the inclusion of additional variables can actually increase OVB [7, 21, 22].
In this article we give a thorough formal characterization of the mechanics that lead to OVB. In particular, we discuss conditions under which adjusting for a confounder actually increases instead of reduces OVB. We use a linear setting with only two continuous confounders, X and U, that confound the relationship between a continuous treatment Z and a continuous outcome variable Y. This allows us to keep the complexity of the OVB formulas low, and thus to better understand the OVB mechanics.
In the following we first review and explain the phenomenon of bias amplification when one conditions on an IV in the presence of an omitted variable. Then we focus on the case of two uncorrelated confounders (one observed, the other unobserved), followed by the more general case with two correlated confounders. Slowly increasing the complexity of the confounding structure – from the IV case to two correlated confounders – allows us to clearly disentangle the effects of bias amplification, cancellation of offsetting biases, correlated confounders, and unreliable covariate measurement. We conclude with a discussion of practical implications. The appendices contain (a) an explanation of bias amplification in the context of matching or stratifying on an IV (Appendix A), (b) OVB formulas for a dichotomous treatment variable (Appendix B), and (c) proofs of results discussed in this article (Appendix C).
## Amplification of bias and imbalance: the instrumental variable case
Several publications [17, 18, 19, 20] demonstrated that conditioning on an instrumental variable (IV) amplifies any remaining bias due to an omitted variable. [2] The causal graph in Figure 1 represents a simple data generating model (DGM) for the outcome Y and treatment Z with one confounder U and an instrumental variable IV (which is a variable that has no effect on the outcome Y except for the indirect effect via treatment Z). The corresponding linear structural causal model (SCM) is given by
I V = ε I V , U = ε U , Z = α I V I V + α U U + ε Z , Y = τ Z + β U U + ε Y ,
### Figure 1:
Causal graph with an instrumental variable (IV). Z is the treatment, Y the outcome, and U an unobserved confounder (represented by the vacant node).
where α U , β U , and τ are standardized parameters and ε I V , ε U , ε Z , and ε Y are mutually independent error terms (representing unknown factors or measurement error) with variances that ensure that
V a r I V = V a r U = V a r Z = V a r Y = 1.
Conducting a linear regression analysis that neither conditions on U nor on IV, Y ˆ = γ ˆ + τ ˆ Z , results in a biased regression estimator τ ˆ for the treatment effect with E ( τ ˆ ) = τ + α U β U . Thus, the initial OVB, that is, the bias before conditioning on IV, is given by O V B ( τ ˆ | { } ) = E ( τ ˆ ) τ = α U β U . The empty set in O V B ( τ ˆ | { } ) indicates that we did not adjust for any covariates. Note that the initial OVB, α U β U , represents the confounding bias due to the unblocked (open) backdoor path ZUY. [3]
### Bias amplification
Omitting U but including IV in the regression model, Y ˆ = γ ˆ + τ ˆ Z + α ˆ I V I V , also results in bias [17]:
(1) O V B ( τ ˆ | I V ) = α U β U 1 α I V 2 .
However, conditioning on IV amplifies any bias left due to an unblocked backdoor path because 0 < 1 α I V 2 < 1 . Thus, the absolute OVB after adjusting for IV is always greater than the absolute initial OVB: α U β U 1 α I V 2 > α U β U . If we were to condition on U in addition to IV (in case U would be observed), no OVB would be left because U blocks the backdoor path ZUY. Thus, if all confounders (or at least a set of variables that blocks all backdoor paths) are reliably measured, conditioning on an IV does not result in any OVB because there is no bias left to be amplified (provided the functional form of the regression is correctly specified). However, adjusting for the IV still reduces the efficiency of the treatment effect estimate [21, 23].
### Imbalance in the unobserved confounder U
Bias amplification occurs because conditioning on the IV increases the imbalance in the unobserved confounder U. For our linear framework, we define imbalance as the difference in the expected value of U for subpopulations with Z = z and Z = z + 1 (if Z would be dichotomous the imbalance would measure the mean difference between the two groups). That is, without conditioning on the IV or any other covariate the imbalance in U is obtained by regressing U on Z: I m b a l a n c e ( U | { } ) = E ( U | Z = z + 1 ) E ( U | Z = z ) = α U . After conditioning on IV, we get I m b a l a n c e ( U | I V ) = E ( U | Z = z + 1 , I V ) E ( U | Z = z , I V ) = α U 1 α I V 2 (Proof 1 in Appendix C). The comparison of the two imbalance formulas reveals that conditioning on the IV amplifies U’s imbalance by the factor 1 / ( 1 α I V 2 ) . Thus, we can write the OVB as the product of the amplified imbalance in U and U’s direct effect on the outcome: O V B ( τ ˆ | I V ) = α U 1 α I V 2 × β U . This formula highlights that conditioning on IV turns U into a relatively stronger confounder.
The increased imbalance in U can be explained as follows (similar explanations can be found in [21], and [24]): Since Z = α I V I V + α U U + ε Z is a function of IV, U, and the error term ε Z , conditioning on the IV removes IV’s effect on Z such that the remaining variation in Z is determined by U and the error term alone. With only two sources of variation left (U and ε Z ), U now explains a larger portion of variance in Z. Hence, the association between U and Z for a given IV = v is necessarily greater than before conditioning on IV. The increased association between U and Z implies an increase in U’s absolute imbalance: I m b a l a n c e ( U | I V ) = α U 1 α I V 2 > I m b a l a n c e ( U | { } ) = α U . Appendix A contains a more intuitive explanation within the context of matching or stratifying treatment and control cases on an IV.
## OVB and imbalance due to conditioning on an uncorrelated confounder
Bias-amplification occurs not only when one conditions on an IV but also when one conditions on a confounder. For an unobserved confounder U and an uncorrelated confounder X that both induce bias in the same direction (i. e., either positive or negative selection bias), prior studies have shown that conditioning on a confounder, where X is a near-IV that is highly predictive of treatment Z but only weakly predictive of the outcome, has two effects: it removes X’s own confounding bias and amplifies any remaining bias due the omitted confounder [17, 19, 21]. The bias-amplifying effect may actually dominate the bias-reducing effect such that conditioning on a confounder X may increase instead of reduce OVB in the treatment effect. In order to fully characterize the mechanics of OVB, we discuss the more general case where X and U (a) are (un)correlated, (b) induce biases in different directions, and (c) where X is unreliably measured. We first discuss the case of uncorrelated confounders and then the case where X and U are correlated.
The left graph in Figure 2 shows the DGM with two uncorrelated confounders, an observed confounder X and an unobserved confounder U. The corresponding linear SCM is given by
(2) X = ε X , U = ε U , Z = α X X + α U U + ε Z , Y = τ Z + β X X + β U U + ε Y ,
### Figure 2:
Causal graphs with two uncorrelated confounders X and U, with X reliably measured in the left graph, and X measured with error in the right graph.
with the same constraints as before such that the parameters represent standardized coefficients. For this linear SCM, the initial OVB due to omitted confounders X and U is O V B ( τ ˆ | { } ) = α X β X + α U β U , which represents the biases induced by the two open backdoor paths ZXY and ZUY. It is important to note that the two bias terms add up if both terms are either positive or negative, but partially or fully offset each other if one term is positive and the other negative.
### Reliably measured confounder X
Adjusting for a reliably measured confounder X results in a biased regression estimator with
(3) O V B ( τ ˆ | X ) = α U β U × 1 1 α X 2 .
A comparison of this bias formula (Proof 3 in Appendix C) with the initial OVB indicates that conditioning on X has two effects: First, a bias-reducing effect because X blocks the backdoor path ZXY and thus eliminates its own confounding bias ( α X β X ). Second, a bias-increasing effect because the bias due to the unblocked backdoor path ZUY ( α U β U ) is amplified by the factor 1 / ( 1 α X 2 ) .
If the bias-increasing effect dominates the bias-reducing effect then conditioning on X leads to an increase in the absolute OVB, that is, the OVB after conditioning on the confounder X is greater than without conditioning on X: α U β U 1 α X 2 > α X β X + α U β U . The discussion of the conditions under which the absolute OVB actually increases requires a distinction between the case where X and U induce bias in the same direction (no offsetting biases) and where they induce bias in different directions such that their respective confounding biases partially or fully offset each other.
Biases in the Same Direction. If both confounders induce bias in the same direction, s g n ( α X β X ) = s g n ( α U β U ) , then conditioning on X results in an increasing OVB only if the bias-amplifying effect dominates the bias-reducing effect, which is the case if [4]
(4) α U β U α X β X > 1 α X 2 α X 2 . 4
Conditioning on X very likely increases the absolute OVB in two situations. First, if the bias induced by U ( α U β U ) is much larger than the bias induced by X ( α X β X ), implying that the bias ratio on the left-hand side in (4) is large. And second, if X strongly determines Z ( α X close to 1) such that the right-hand side in (4) is close to zero. Thus, adjusting for a confounder with α X close to 1 and β X close to zero (i. e., a near-IV) very likely increases the absolute bias.
In the upper left plot of Figure 3 the two dark grey areas show combinations of α X values and bias ratios α U β U α X β X for which the absolute OVB increases. The two light grey areas indicate areas of decreasing OVB. The line separating the dark and light grey areas represents the 100% bias contour line where conditioning on X neither reduces nor increases OVB (i. e., 100% of the initial OVB is left). The darker shade of the two dark grey areas indicates the region where conditioning on X leads to a bias that is at least twice as large as the initial bias. Thus, the contour line that separates the two dark grey areas represents the 200% bias contour line. Similarly, the very light grey area indicates that less than 50% of the initial bias is remaining. The contour line separating the two light grey areas represents the 50% bias contour line. For example, conditioning on a confounder with α X = . 1 results in an increasing OVB only if the bias ratio α U β U α X β X is greater than 1 . 1 2 . 1 2 = 101 , that is, if the bias induced by the unobserved confounder U is at least 101 times greater than the bias induced by X. However, if X is strongly related to treatment, α X = . 9 , conditioning on X results in an increasing OVB if the bias induced by U is at least one fourth ( 1 . 9 2 . 9 2 = . 23 ) of X’s bias. In this case, bias amplification dominates bias reduction: though conditioning on X removes its own bias α X β X which amounts to 81% (= 1/(1 + .23)) of the total confounding bias, [5] the amplification of the remaining 19% (= .23/(1 + .23)) due to omitting U ( α U β U ) is strong enough to offset the bias-reducing effect because the bias amplification factor is 1/(1 − .92) = 5.26.
### Figure 3:
Increasing and decreasing OVB due to conditioning on an uncorrelated confounder X. The two dark grey areas indicate an increasing OVB, with 100%-200% (lighter shade) and 200% or more (darker shade) remaining bias. The two light grey areas indicate a decreasing OVB, with 50%-100% (darker shade) and 50% or less (lighter shade) remaining bias.
Offsetting Biases. For sgn ( α X β X ) sgn ( α U β U ) , the confounding biases induced by X and U partially or even completely offset each other such that α X β X + α U β U < max ( α X β X , α U β U ) . If U induces less bias than X, α U β U α X β X , adjusting for the observed confounder X increases rather than reduces OVB only if
(5) α U β U α X β X 1 α X 2 2 α X 2 ( P r o o f 4 i n A p p e n d i x C ) .
But if U induces more bias than X, α U β U > α X β X , then conditioning on X always increases OVB because the remaining bias due to the unblocked backdoor path ZUY is necessarily greater than the initial bias: α U β U > α X β X + α U β U .
The upper right plot in Figure 3 shows areas of increasing and decreasing absolute OVB when biases (partially) offset each other. For α X 0 , OVB increases as long as the bias induced by U is at least half of X’s bias: lim α X 0 1 α X 2 2 α X 2 = 1 2 . For α X = . 5 , OVB increases if the bias ratio exceeds 1 . 5 2 2 . 5 2 = . 43 . If α X is close to 1, say .95, then OVB increases as long as the bias induced by U is at least about one tenth of X’s bias ( 1 . 95 2 2 . 95 2 = . 09 ).
To summarize, for offsetting biases, the absolute OVB increases in two situations: First, if the confounding biases induced by X and U nearly offset each other ( α X β X α U β U ). In fact, independent of the value of α X , OVB always increases if the bias induced by the unobserved confounder U is at least half of X’s bias ( α U β U > α X β X / 2 ). And second, if X strongly determines Z such that α X is close to 1, then the absolute OVB increases even when α X β X >> α U β U . The increase in the absolute OVB is mostly a result of the cancellation of the bias-offsetting effect, but the amplification of the remaining bias adds to the increase. Also note that the sign of the initial and adjusted OVB may differ. For instance, the initial OVB might be positive, but adjusting for X might turn the positive OVB into a negative OVB.
### Unreliably measured confounder X
The OVB formula in (3) only holds for a reliably measured uncorrelated confounder X. The right graph in Figure 2 shows the case with a fallibly measured X. The node of X now turns into a vacant node (open circle) indicating that X is not directly observed. Instead, we only have an unreliable measure X* which is given by X = X + e , where e is an independent error with mean zero and variance σ e 2 . [6] Since Var(X) = 1, the reliability of X* is given by γ = 1 / ( 1 + σ e 2 ) . Measurement error in X* has no influence on the initial OVB, O V B ( τ ˆ | { } ) = α X β X + α U β U , but affects OVB after adjusting for the fallible X* (Proof 3 in Appendix C):
(6) O V B ( τ ˆ | X ) = { α U β U + α X β X ( 1 γ ) } × 1 1 α X 2 γ .
In comparison to the OVB for a reliably measured confounder X in (3), measurement error has two effects: First, the bias left due to (partially) unblocked backdoor paths now consists of two components, α U β U and α X β X ( 1 γ ) . Besides the open backdoor path ZUY (due to omitting U), adjusting for X* no longer fully blocks the backdoor path ZXY such that ( 1 γ ) % of X’s bias is left. That is, X* removes the bias induced by X only to the degree of its reliability ( γ ). The less reliable the measurement, the more of X’s bias will remain. Second, measurement error attenuates the bias amplification factor since 1 / ( 1 α X 2 γ ) is always less than 1 / ( 1 α X 2 ) because 0 γ 1 . A completely unreliable measure X* with γ 0 neither removes nor amplifies any bias such that the initial OVB remains: lim γ 0 O V B ( τ ˆ | X ) = α X β X + α U β U (also see [25]). On the other extreme, with a perfectly reliable measure X ( γ = 1 ), the OVB formula in (6) reduces to the OVB formula in (3).
Biases in the Same Direction. The second and third row of plots in Figure 3 show the areas of increasing OVB (the two dark grey areas) and decreasing OVB (the two light grey areas) for an unreliably measured confounder X ( γ = . 75 in the second row and γ = . 5 in the third row). In the left columns of plots for sgn ( α X β X ) = sgn ( α U β U ) , the 100% bias contour lines are the same as for the reliably measured confounder (upper left plot), but the 200% and 50% bias contour lines change. Unreliability in X does not change the 100 % contour line because measurement error always results in an attenuation of OVB toward the initial OVB [26] (see Proof 5 in Appendix C which also contains a more detailed discussion). Since the 100% contour line represents situations where conditioning on X does not alter the initial OVB (i. e., bias reduction is exactly offset by bias amplification), measurement error has no effect. But if adjusting for the reliable X increases OVB then measurement error attenuates the increase as shown by the retreating 200% contour line (as one moves from the plot in the first row to the plots in the second and third row). If conditioning on the reliable X reduces OVB then measurement error attenuates bias reduction as indicated by the retreating 50% contour line.
Offsetting Biases. For offsetting biases ( sgn ( α X β X ) sgn ( α U β U ) , shown in the right column of Figure 3), all bias contour lines depend on the extent of measurement error. In comparison to the reliably measured confounder (upper right plot), more measurement error in X* results in an expansion of the light grey areas of diminishing OVB, that is, measurement error makes an increasing OVB less likely because the cancellation of the offsetting biases is attenuated. Though unreliability decreases the chances of an increasing OVB, it does not imply that the fallible X* necessarily removes more bias than the corresponding reliable measure. A comparison of the 50% bias contour lines (or the very light grey area) across the three plots reveals that the fallibly X* can remove less OVB than the reliably X.
### Imbalance in confounders U and X
For both reliably and unreliably measured confounders X, bias amplification operates via increasing the imbalance in U and X. For an unreliably measured confounder X*, the initial imbalance in U ( α U ) and remaining imbalance in X ( α X ( 1 γ ) ) are inflated by the factor 1 / ( 1 α X 2 γ ) : I m b a l a n c e ( U | X ) = α U 1 α X 2 γ and I m b a l a n c e ( X | X ) = α X ( 1 γ ) 1 α X 2 γ (Proof 1 in Appendix C). The imbalance formula for U indicates that adjusting for X* always increases the absolute imbalance in U because the amplification factor 1 / ( 1 α X 2 γ ) is less than one (but note that measurement error attenuates bias amplification and thus the decrease in U’s absolute imbalance). Regarding the imbalance in X, conditioning on X* cannot fully balance X because the unreliable X* fails to completely remove the association between Z and X. However, the unreliable measure X* will be balanced, I m b a l a n c e ( X | X ) = 0 . Thus, balance in a fallible covariate X* does not imply that the underlying data-generating confounder X will be balanced. Particularly if α X >> 0 or γ < . 75 , then the absolute imbalance in X after adjusting for X* may still be large but it will never exceed the absolute initial imbalance, I m b a l a n c e ( X | { } ) = α X (Proof 2 in Appendix C). This result does not generalize to the more general case with multiple observed confounders. If one conditions not only on a single unreliable confounder but on multiple, possibly uncorrelated confounders simultaneously, the resulting imbalance in the latent X might exceed the initial imbalance. This is so because the remaining imbalance in X after conditioning on X*, I m b a l a n c e ( X | X ) , is further amplified by any other confounder we condition on (just like the imbalance in U).
## OVB and imbalance due to conditioning on a correlated confounder
The mechanics of OVB become slightly more complex when confounders are correlated. Intuitively, one might think that the correlation between an observed (X) and unobserved confounder (U) always helps in reducing OVB when conditioning on X. But this is not necessarily true because the correlation also triggers the bias-amplifying potential of the hidden confounder or might result in a cancellation of offsetting biases (e. g., if both X and U induce positive bias on their own, a negative correlation would partially offset their biases). These bias-increasing effects can actually dominate the bias-reducing effects. Since bias amplification, cancellation of offsetting biases, and measurement error operate as before, we only highlight the changes due to the correlation of confounders.
The left graph in Figure 4 shows the DGM with correlated confounders X and U. The linear SCM is the same as for the uncorrelated case in Eq. (1), except that X and U are correlated with C o r ( X , U ) = ρ . The correlation between X and U might be due to a common cause C (XCU), a causal effect of X on U (XU), or a causal effect of U on X (UX). The initial OVB is then given by O V B ( τ ˆ | { } ) = α X β X + α U β U + α X ρ β U + α U ρ β X , which reflects the biases due to all four backdoor paths between Z and Y in Figure 4: ZXY, ZUY, ZX UY, and ZU XY.
### Figure 4:
Causal graphs with two correlated confounders X and U, with X reliably measured in the left graph and X measured with error in the right graph.
### Reliably measured confounder X
Adjusting for the reliably measured confounder X but omitting U results in [7]
(7) O V B ( τ ˆ | X ) = α U β U ( 1 ρ 2 ) × 1 1 ( α X + α U ρ ) 2 . 7
The OVB formula indicates that conditioning on a correlated confounder X has three effects. First, it eliminates its own confounding bias ( α X β X ) but also the entire confounding bias induced by X’s correlation with U ( α X ρ β U + α U ρ β X ). That is, conditioning on X blocks all backdoor paths going through X (i. e., ZXY, ZX UY, and ZU XY). Second, because of X and U’s correlation, X partially blocks the backdoor path ZUY to the extent of the squared correlation ρ 2 , thus the bias due to the unobserved U reduces to α U β U ( 1 ρ 2 ) . And third, the correlation also affects the bias amplification factor 1 / ( 1 ( α X + α U ρ ) 2 ) because conditioning on X triggers U’s bias-amplifying potential to the extent of their correlation as reflected by the additional term α U ρ in the denominator.
Depending on the sign of α U ρ , the correlation can strengthen, weaken, or even neutralize the bias amplification factor. If sgn ( α U ρ ) = sgn ( α X ) then the correlation boosts bias amplification in comparison to the uncorrelated case because α X + α U ρ > α X . The stronger the correlation and the larger α U , the stronger the bias-amplifying effect. If sgn ( α U ρ ) sgn ( α X ) , the correlation can strengthen (if α X + α U ρ > α X ), weaken (if α X + α U ρ < α X ) or completely cancel bias amplification (if α X = α U ρ ). Thus, even with highly correlated confounders X and U, there is no guarantee that conditioning on a correlated X reduces OVB (examples are briefly discussed at the end of the following subsection).
### Unreliably measured confounder X
The right graph in Figure 4 shows the same causal diagram as before but with the fallible covariate X*. In this case, one can show (Proof 3 in Appendix C) that conditioning on X* results in an OVB of
(8) O V B ( τ ˆ | X ) = { α U β U ( 1 ρ ˜ 2 ) + ( α X β X + α X ρ β U + α U ρ β X ) ( 1 γ ) } × 1 1 ( α X + α U ρ ) 2 γ .
All four terms of the initial bias appear in the OVB formula, but the biases induced by the four backdoor paths are not fully effective. First, the correlation of the unreliable X* with the unobserved confounder U, C o r ( X , U ) = ρ ˜ = ρ γ , reduces the bias induced by U to the extent of the squared correlation ρ ˜ 2 , leaving a bias of α U β U ( 1 ρ ˜ 2 ) . Second, the unreliable X* blocks the three backdoor paths via X only to the extent of its reliability ( γ ) and thus leaves a bias of ( α X β X + α X ρ β U + α U ρ β X ) ( 1 γ ) . Finally, the remaining bias due to the four partially unblocked backdoor paths is amplified but the bias amplification factor is attenuated by the reliability γ .
Due to the increased complexity of the OVB formulas, an easily interpretable inequality as for the uncorrelated confounder case is not derivable. Thus, we illustrate the effect of correlated confounders with two examples. The first row of plots in Figure 5 shows for two different parameter settings the areas of increasing (dark grey) and decreasing OVB (light grey) as a function of the correlation ρ (abscissa) and the unobserved confounder’s coefficient α U (ordinate). For both plots we set β X = β U = . 1 , but α X = . 3 in the left plot and α X = . 9 in the right plot (making X a near-IV in the latter case). In each plot, quadrant I (with ρ 0 and α U 0 ) represents the situation where all biases induced by X and U go into the same direction because all five data-generating parameters are positive. Quadrants II, III, and IV show the results for partial or completely offsetting biases (because the signs of the parameters differ).
### Figure 5:
Increasing and decreasing OVB due to conditioning on a correlated confounder X. The two dark grey areas indicate an increasing OVB, with 100%-200% (lighter shade) and 200% or more (darker shade) remaining bias. The two light grey areas indicate a decreasing OVB, whith 50%-100% (darker shade) and 50% or less (lighter shade) remaining bias. The white areas indicate parameter combinations that are impossible for standardized path coefficients.
Consider quadrant I of the top right plot in Figure 3, where the confounder X strongly affects Z ( α X = . 9 ): OVB can exceed the initial bias even if one conditions on a confounder X that is almost perfectly correlated with U. In general, it is hard to derive a generalizable pattern from the two example plots. Without knowing the sign and magnitude of the five parameters it is impossible to predict whether conditioning on a correlated X or X* reduces or increases OVB even if X is highly correlated with U. The second and third row in Figure 3 shows the effect of measurement error, which is the same as for the uncorrelated case (i. e., attenuation to the initial bias; Proof 5 in Appendix C).
### Imbalance in confounders U and X
As for the case of uncorrelated confounders, the bias-amplifying effect of conditioning on a reliably or unreliably measured confounder X can be explained by the amplified imbalance in U and X. The absolute initial imbalance in U, I m b a l a n c e ( U | { } ) = α U + α X ρ , might increase or decrease once one conditions on X*, even when U is correlated with X*. Adjusting for the correlated X* changes the initial imbalance in U to α U ( 1 ρ ˜ 2 ) + α X ρ ( 1 γ ) , which then is amplified by the factor 1 / ( 1 ( α X + α U ρ ) 2 γ ) such that we obtain I m b a l a n c e ( U | X ) = α U ( 1 ρ ˜ 2 ) + α X ρ ( 1 γ ) 1 ( α X + α U ρ ) 2 γ . Compared to the absolute value of the initial imbalance (before adjusting for X*), the absolute imbalance in U after adjusting for X* might be smaller or larger (Proof 2 in Appendix C). Despite the correlation, conditioning on X* can increase the imbalance in U because the term α U ρ may strengthen the bias amplification factor.
Correspondingly, conditioning on X* first reduces the absolute initial imbalance in X from I m b a l a n c e ( X | { } ) = α X + α U ρ to ( α X + α U ρ ) ( 1 γ ) , which again is amplified such that I m b a l a n c e ( X | X ) = ( α X + α U ρ ) ( 1 γ ) 1 ( α X + α U ρ ) 2 γ . Multiplying U’s imbalance by β U and X’s imbalance by β X , and then adding the two terms results in the OVB formula (8). As for the uncorrelated confounder case, the absolute imbalance in X after adjusting for X* will always be smaller than before the adjustment, I m b a l a n c e ( X | X ) I m b a l a n c e ( X | { } ) (Proof 2 in Appendix C). Again, this only holds for the case with a single observed confounder X. Conditioning on multiple confounders, including X*, can actually increase the imbalance in X (but as for the imbalance in U, whether the imbalance in X decreases or increases depends on the correlation among the observed confounders).
With a perfectly reliably measured X ( γ = 1 ), X will be fully balanced but U remains imbalanced with I m b a l a n c e ( U | X ) = α U ( 1 ρ 2 ) 1 ( α X + α U ρ ) 2 . Note that neither the imbalance in U nor in X (given it is unreliably measured) can be tested empirically since both are unobserved.
## Discussion
The investigation of the OVB mechanics revealed that conditioning on a confounder provokes two opposing effects, a bias-removing effect and a bias-increasing effect. If the bias-increasing effect dominates the bias-removing effect, then OVB increases. The increase in OVB can be caused by the amplification of any bias left due to unblocked backdoor paths, the cancellation of offsetting biases, or by both together. The overall extent of bias amplification is driven by two factors: (i) the bias left due to unblocked backdoor paths and (ii) the size of the multiplicative bias amplification factor. Both factors depend on the strength of the correlation between the observed and unobserved confounder and the degree of measurement error in the observed confounder. Though the correlation helps in partially removing the bias induced by the unobserved confounder, it also picks up the bias-amplifying potential of the unobserved confounder, and thus can further boost bias amplification. Therefore, even a high correlation between the observed and unobserved confounder does not guarantee that OVB will decrease. Though measurement error attenuates the bias amplification factor it also attenuates the confounder’s potential to remove bias such that measurement error may have a positive or negative effect on OVB. Bias amplification is not an issue if conditioning on a set of confounders removes all the bias (i. e., no bias is left to be amplified) or if the amplification factor is one (i. e., α X = α U ρ ). Table 1 and Table 2 summarize the formulas and results for uncorrelated and correlated confounders, respectively. Appendix B shows that the very same OVB mechanics operate with dichotomous instead of continous treatment variables (though the formulas a slightly different).
Table 1:
Uncorrelated confounders X and U: Omitted variable bias (OVB) and imbalance before and after adjusting for X*.
Initial OVB and Imbalance OVB and Imbalance after adjusting for X*
Omitted variable bias O V B ( τ ˆ | { } ) = α X β X + α U β U O V B ( τ ˆ | X ) = α U β U + α X β X ( 1 γ ) 1 α X 2 γ
Imbalance in U I m b a l a n c e ( U | { } ) = α U I m b a l a n c e ( U | X ) = α U 1 α X 2 γ
Imbalance in X I m b a l a n c e ( X | { } ) = α X I m b a l a n c e ( X | X ) = α X ( 1 γ ) 1 α X 2 γ
Effect of conditioning on X* when …
biases are in the same direction biases offset each other
Absolute omitted variable bias Increase in OVB is most likely if (a) the bias induced by the unobserved confounder U is much larger than the bias induced by confounder X or (b) confounder X strongly affects Z. If the bias induced by the unobserved confounder U exceeds half of the bias induced by X, OVB always increases (this case also includes almost perfectly offsetting biases). If the bias induced by the unobserved confounder U is less than half of the bias induced by X, OVB most likely increases if X strongly affects Z (provided X is reliably measured).
Absolute imbalance Imbalance in U always increases. Imbalance in X always decreases. Imbalance in U always increases. Imbalance in X always decreases.
Effect of measurement error Attenuates any increase in OVB and attenuates any decrease in OVB. If the bias induced by the unobserved confounder U exceeds half of the bias induced by X, measurement error attenuates any increase in OVB. If the bias induced by the unobserved confounder U is less than half of the bias induced by X, measurement error attenuates any increase in OVB (and might even turn an increase into a decrease) but may attenuate or strengthen any decrease in OVB.
Table 2:
Correlated confounders X and U: Omitted variable bias (OVB) and imbalance before and after adjusting for X*.
Initial OVB and Imbalance OVB and Imbalance after adjusting for X*
Omitted variable bias O V B ( τ ˆ | { } ) = α X β X + α U β U + α X ρ β U + α U ρ β X O V B ( τ ˆ | X ) = α U β U ( 1 ρ ˜ 2 ) + ( α X β X + α X ρ β U + α U ρ β X ) ( 1 γ ) 1 ( α X + α U ρ ) 2 γ
Imbalance in U I m b a l a n c e ( U | { } ) = α U + α X ρ I m b a l a n c e ( U | X ) = α U ( 1 ρ ˜ 2 ) + α X ρ ( 1 γ ) 1 ( α X + α U ρ ) 2 γ
Imbalance in X I m b a l a n c e ( X | { } ) = α X + α U ρ I m b a l a n c e ( X | X ) = ( α X + α U ρ ) ( 1 γ ) 1 ( α X + α U ρ ) 2 γ
Effect of conditioning on X* when …
biases are in same the direction biases offset each other
Absolute omitted variable bias Increase in OVB is most likely if(a) the bias induced by the unobserved confounder U is much larger than the bias induced by confounder X and the correlation between X and U is low, or (b) confounder X strongly affects Z –a high correlation between X and U strongly boosts bias amplification. Whether OVB increases strongly depends. on the signs and magnitudes of all five parameters. If the biases induced by X and U strongly offset each other, an increase in OVB almost surely results – unless the correlation between X and U is close to 1.
Absolute imbalance Imbalance in U may increase or decrease. Imbalance in X always decreases. Imbalance in U may increase or decrease. Imbalance in X always decreases.
Effect of measurement error Attenuates any increase in OVB and attenuates any decrease in OVB. Attenuates any increase in OVB (and might even turn an increase into a decrease) but may attenuate or strengthen any decrease in OVB.
Though we restricted our discussion of OVB to the case with a single observed and unobserved confounder, the principles of the OVB mechanics also apply to the multiple confounder case where X and U represent sets of observed and unobserved confounders. However, the OVB formulas would be by far more complex because the correlation structure within and between the two sets of confounders also needs to be considered (for an OVB formula in matrix notation, see [27]). Moreover, cancellation of offsetting biases and bias amplification are not restricted to the linear case, they also occur in nonlinear settings [17]; but it is much harder to derive closed OVB formulas that are informative about the OVB mechanics.
We also showed that bias amplification operates via increasing the imbalance in unobserved confounders. That is, conditioning on an observed confounder can significantly increase the unobserved confounders’ imbalance and, thus, turn them into even stronger confounders. If the observed and unobserved confounders are uncorrelated, the imbalance in the unobserved confounders always increases. Thus, balancing a large set of observed covariates via matching or regression adjustment does not imply that the imbalance in unobserved confounders decreases.
In the presence of omitted or unobserved variables, is it possible to select a subset of observed covariates that minimizes OVB? Or, is it at least possible to make sure that the selected covariates do not increase OVB? With almost perfect knowledge about the data-generating selection and outcome models one could actually select the set of covariates that minimizes OVB. But such knowledge is rarely available. Without reliable knowledge about the true DGM it seems impossible to know whether conditioning on a set of covariates minimizes or even reduces the confounding bias. While empirical covariate selection strategies, that rely on observed relations between the covariates and the treatment or outcome, can be very successful when all confounding covariates are reliably measured, it is not clear how good or bad these strategies perform in the presence of unobserved or unreliably measured confounders. However, partial knowledge might occasionally allow an informed assessment of whether adjusting for a set of covariates brings us at least closer to a causal effect estimate (for instance, we might know that only positive selection took place and that the observed covariates cover the most important confounders but no near-IVs).
The OVB mechanics discussed in this article have far-reaching implications for practice. Given unobserved confounders, neither conditioning on all or a large set of observed pre-treatment covariates (as publicized in [28], or [9]), nor conditioning on a small set of covariates that has been selected on subject-matter or empirical grounds [21] can guarantee that OVB will decrease. For matching designs like propensity score matching this means that achieving balance on all observed pre-treatment covariates neither implies that the confounding bias has been minimized or even reduced nor that the imbalance in unobserved covariates, including the latent constructs of fallible measures, diminished. The same holds for all methods dealing with bias due to nonresponse or attrition – conditioning on a large set of covariates does not imply that nonresponse or attrition bias in the statistic of interest is successfully addressed [22]; Or for two-stage least-squares analyses (2SLS) of conditional IV designs, conditioning on a set of observed covariates does not guarantee that the bias due to a potential violation of the exclusion restriction is minimized. Whenever covariate adjustments are made in the hope to reduce different types of confounding bias, a thoughtless or automated selection of covariates may increase instead of reduce the bias.
Since we used a very simple data-generating model to explain the mechanics of OVB, one needs to be careful in deriving practical guidelines about when to condition on an observed covariate and when not. The decision about adjusting for a given covariate strongly depends on the presumed real-world data-generating model. For instance, if there would be only a single confounder X but which has been unreliably measured, then conditioning on X* would always reduce selection bias. But when there is one or multiple unobserved confounders, then it is already less clear whether conditioning on X* actually reduces OVB. In practice, the situation is usually even more complex because a confounding path might be blocked in more than one way. For instance, if we observed an intermediate covariate W on U’s confounding path, ZWUY, then conditioning on W would not result in any OVB despite the omission of confounder U (provided there are no other unobserved confounders). But if one conditions neither on U nor on W the OVB mechanics are in place again.
Sometimes it is also possible to circumvent unobserved confounding by using designs that exploit other observed covariates. For instance, if the observed set of covariates contains an instrumental variable then we could use an instrumental variable design to identify the complier average treatment effect. Or, if data contain a pretest measure of the outcome then a gain score or difference-in-differences design can deal with unobserved time-invariant confounding [29]. However, the assumptions underlying these designs might be less credible than the conditional independence assumption such that covariate adjustments via regression or matching methods might be preferable. But given the uncertainty about the magnitude of OVB left after adjusting for a set of covariates, it is important to conduct sensitivity analyses that assess the estimated treatment effect’s sensitivity to unobserved confounders [30, 31, 32]. Or, with partial knowledge about the data-generating process, one can pursue a partial identification strategy and compute bounds on the treatment effect [33]. In any case, lacking strong subject-matter theory, researchers should abstain from making strong causal claims from a single observational study. Causal claims are much more credible when built on multiple independent replications with different study designs.
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## Appendix A: Bias amplification when matching or stratifying on an IV
Bias amplification can also be intuitively explained within the context of matching or stratifying treatment and control cases on the IV (i. e., with a dichotomous treatment Z). Consider the case of exact full matching on IV, that is, all treatment and control cases with IV=v are matched together (this is equivalent to exact stratification because the set of matched cases forms a unique stratum with IV=v). For simplicity, we first assume that the dichotomous treatment Z is a deterministic function of IV and U: Z = f ( I V , U ) = 1 I V + U > c , where Z = 1 if the sum IV + U exceeds a threshold c, otherwise Z = 0 (indicating the control condition). Now assume that we match on the observed IV in the hope to remove potential confounding bias. Then, for a given stratum with IV = v, the treatment status Z = f ( U | I V = v ) = 1 U > c v is now exclusively determined by U: Cases with U > c v received the treatment and cases with U c v received the control condition. Thus, all treatment cases with IV = v must have strictly larger values in U than the control cases, that is, the treatment and control cases distribution of U no longer overlap. But without matching on IV, the distributions of U would have overlapped, enabling exact matches on U. Thus, matching on the IV increases the treatment and control group’s heterogeneity in U which is reflected by the increased imbalance.
The same argument holds for a treatment function with an independent error term (i. e., unobserved factors determining Z): Z = f ( I V , U , ε ) = 1 I V + U + ε > c . Matching on IV then restricts the pool of potential matches with regard to U – if one were to match on the unobserved U. Due to the error term, we still could find exact matches on U but, nonetheless, the difference in the treatment and control cases distribution of U is larger than before matching on IV. Note that the imbalance in U does not necessarily have to increase within each stratum, but it will necessarily increase on average across strata.
## Appendix B: Bias amplification and cancellation of offsetting biases for a dichotomous treatment
All the bias formulas we discussed so far referred to regression estimators for a continuous treatment variable. Since treatment variables are frequently dichotomous, we briefly characterize the bias for a dichotomous treatment indicator Z* (this section follows the formalization used by [14]). Figure 6 shows the DGM with two correlated confounders, one measured with error and the other one unobserved. The corresponding SCM we used for the following derivations is given by
X = ε X X = X + e U = ε U Z = α X X + α U U + ε Z Z = 1 i f Z c a n d Z = 0 i f Z < c Y = τ Z + β X X + β U U + ε Y
### Figure 6:
Causal graph for two correlated confounders X and U. The vacant nodes for X, Z and U indicate that they are unobserved. Z* is dichotomous.
In order to derive corresponding OVB formulas, we assume that X and U are distributed according to a bivariate normal distribution with zero expectation, unit variances, and a correlation ρ . Consequently, also Z is normally distributed with zero expectation. We further assume that the treatment effect is zero which considerably simplifies the derivation of the OVB formulas. As before, coefficients of α X , α U , β X , and β U represent standardized coefficients, and the normally distributed error terms ε Z and ε U were chosen such that Var(Z) = 1 and Var(Y) = 1. The dichotomous treatment Z* is obtained from the continuous Z and the cutoff c. The cutoff value c refers to the quantiles of a standard normal distribution ϕ ( c ) because Z N ( 0 , 1 ) . The unreliable measure X* is given by X* = X + e with e N ( 0 , σ e 2 ) .
Under these assumptions the standardized effect of X on Z* is given by α X = α X ϕ ( c ) Φ ( c ) Φ ( c ) and the standardized effect of U on Z* is given by α U = α U ϕ ( c ) Φ ( c ) Φ ( c ) where ϕ ( c ) and Φ ( c ) denote the standard normal probability density and cumulative distribution function, respectively (the Proof is given at the end of the section). Then, the regression estimator’s initial bias before any conditioning (i. e., Y ˆ = γ ˆ + τ ˆ Z Z ) is
(9) O V B ( τ ˆ Z | { } ) = ( α X β X + α U β U + α X ρ β U + α U ρ β X ) × 1 Φ ( c ) Φ ( c ) .
After conditioning on X*, we obtain
(10) O V B ( τ ˆ Z | X ) = { α U β U ( 1 ρ ˜ 2 ) + ( α X β X + α X ρ β U + α U ρ β X ) ( 1 γ ) } × 1 1 ( α X + ρ α U ) 2 γ × 1 Φ ( c ) Φ ( c ) .
Both OVB formulas are identical to the OVB formulas for a continuous treatment variable, except for the constant 1 / Φ ( c ) Φ ( c ) = 1 / V a r ( Z ) which ensures that OVB refers to the change in Z* from 0 to 1 (without the constant the OVB formula would refer to a change in Z* by one standard deviation, just as in the continuous case). Thus, we have the same OVB mechanics and conditions under which conditioning on X* increases OVB as for the continuous treatment case. However, since α X < α X and α U < α U the bias-amplifying effects will always be weaker for a dichotomous treatment than for a corresponding continuous treatment (because the dichotomized version of the continuous treatment will always be less strongly correlated with the continuous confounders). But this does not imply that bias amplification and an increasing OVB is less of an issue with a dichotomous treatment. Just assume that the dichotomous Z* is directly affected by dichotomous confounders X and U (i. e., with respect to Figure 6, X and U are dichotomous and there is no continuous Z on the causal pathway from the dichotomous confounders to Z*; instead X and U directly affect Z*: XZ* and UZ*). In this case, the dichotomous confounders can affect Z* at least as strongly as continuous confounders can affect a continuous Z ( α X and α U are no longer attenuated and the correlation between the confounder and the treatment can theoretically be one as in the continuous treatment and confounder case).
Proof.
OVB with a Dichotomous Treatment
Using the data generating model in Figure 6 with a treatment effect of zero ( τ =0), we derive the OVB formula for the treatment effect from the regression of Y on Z* and X*. We assume that X and U are bivariate normally distributed with zero means, unit variances, and a correlation ρ . This implies that also Z is normally distributed. The unstandardized OLS estimator for the treatment effect can be written in terms of observed correlations as b Z = r Y Z r Y X r Z X 1 r Z X 2 × 1 S D ( Z ) . To obtain the three correlation coefficients, we use the corresponding covariances:
C o v ( Y , Z ) = ϕ ( c ) ( α X β X + α U β U + ρ α X β U + ρ α U β X ) C o v ( Y , X ) = C o v ( X + e , Y ) = C o v ( X , Y ) = C o v ( X , β X X + β U U + e Y ) = β X + ρ β U , C o v ( Z , X ) = C o v ( Z , X + e ) = C o v ( Z , X ) = ϕ ( c ) ( α X + ρ α U ) ,
where ϕ ( x ) denotes the standard normal density function. While C o v ( Y , X ) directly follows from the structural equations, C o v ( Y , Z ) and C o v ( Z , X ) need some further explanations which we exemplify for C o v ( Y , Z ) .
Assuming a constant treatment effect of zero, the treatment effect’s regression estimator from the regression of Y on Z* can be written as the expected difference in the outcome Y for Z*=1 and Z*=0, that is, E ( Y | Z = 1 ) E ( Y | Z = 0 ) . Since the OLS estimator is given by C o v ( Y , Z ) / V a r ( Z ) , we obtain C o v ( Y , Z ) = V a r ( Z ) E ( Y | Z = 1 ) E ( Y | Z = 0 ) . Then, using V a r ( Z ) = Φ ( c ) Φ ( c ) and E ( Y | Z = 1 ) E ( Y | Z = 0 ) = E ( Y | Z c ) E ( Y | Z < c ) = r Z Y ϕ ( c ) Φ ( c ) Φ ( c ) from Lemma 1 and Lemma 2 (see below), and r Z Y = C o v ( α X X + α U U + e Z , β X X + β U U + e Y ) = α X β X + α U β U + ρ α X β U + ρ α U β X , we get C o v ( Y , Z ) = ϕ ( c ) ( α X β X + α U β U + ρ α X β U + ρ α U β X ) .
The covariances and Lemma 1 are then used to obtain expressions for the correlations:
r Y Z = C o v ( Y , Z ) / S D ( Z ) = ϕ ( c ) ( α X β X + α U β U + ρ α X β U + ρ α U β X ) / Φ ( c ) Φ ( c ) r Y X = C o v ( Y , X ) / S D ( X ) = ( β X + ρ β U ) γ , r Z X = C o v ( Z , X ) / S D ( Z ) S D ( X ) = ϕ ( c ) ( α X + ρ α U ) γ Φ ( c ) Φ ( c )
Plugging the correlations into the formula for the treatment effect’s regression estimator results in b Z = O V B ( τ ˆ Z | X ) = ϕ ( c ) α U β U ( 1 ρ ˜ 2 ) + ( α X β X + ρ α X β U + ρ α U β X ) ( 1 γ ) Φ ( c ) Φ ( c ) ϕ ( c ) 2 ( α X + ρ α U ) 2 γ which is equivalent to the OVB since the derivations are based on a treatment effect of zero. The initial bias in the treatment effect of Z* on Y can be obtained by regressing Y onto Z*, that is,
O V B ( τ ˆ Z | { } ) = C o v ( Z , Y ) / V a r ( Z ) = φ ( c ) ( α X β X + α U β U + ρ α X β U + ρ α U β X ) / Φ ( c ) Φ ( c ) .
The two OVBs can be rewritten as
O V B ( τ ˆ Z | { } ) = ( α X β X + α U β U + α X ρ β U + α U ρ β X ) × 1 Φ ( c ) Φ ( c ) a n d O V B ( τ ˆ Z | X ) = { α U β U ( 1 ρ ˜ 2 ) + ( α X β X + α X ρ β U + α U ρ β X ) ( 1 γ ) } × 1 1 ( α X + ρ α U ) 2 γ × 1 Φ ( c ) Φ ( c ) ,
where α X = α X ϕ ( c ) Φ ( c ) Φ ( c ) is the standardized effect of X on Z* and α U = α U ϕ ( c ) Φ ( c ) Φ ( c ) is the standardized effect of U on Z*. α X is the product of the effect of X on Z ( α X ) and the standardized effect of Z on Z* ( ϕ ( c ) Φ ( c ) Φ ( c ) ). The latter is obtained from the regression of Z* on Z together with Lemmas 1 and 2, that is,
C o v ( Z , Z ) V a r ( Z ) × S D ( Z ) S D ( Z ) = C o v ( Z , Z ) V a r ( Z ) × S D ( Z ) S D ( Z ) = { E ( Z | Z = 1 ) E ( Z | Z = 0 ) } × S D ( Z ) = ϕ ( c ) Φ ( c ) ϕ ( c ) Φ ( c ) × Φ ( c ) Φ ( c ) = ϕ ( c ) Φ ( c ) Φ ( c ) .
The first equality follows from inverting the regression, that is, regressing Z on Z* (using the fact the standardized coefficients of the original and inverted regression are equivalent), the second equality rewrites the effect of Z* on Z in terms of conditional expectations and uses SD(Z)=1, and the third equality directly follows from Lemma 1 and 2.
Lemma 1.
Assume Z is distributed according to a standard normal distribution and a binary variable Z* is determined from Z using a cutoff c such that Z = 1 if Z c and Z = 0 otherwise. Then the new random variable Z* follows a Bernoulli distribution with Pr ( Z = 1 ) = p . Since p = Pr ( Z = 1 ) = Pr ( Z c ) = 1 Φ ( c ) = Φ ( c ) , we get V a r ( Z ) = p ( 1 p ) = Φ ( c ) Φ ( c ) .
Lemma 2.
[14]. Assume X and Y follow a bivariate normal distribution with zero means, unit variances, and correlation coefficient ρ . Under these assumptions we have E ( Y | X < c ) = ρ E ( X | X < c ) . Since E ( X | X < c ) = 1 Φ ( c ) c x ϕ ( x ) d x = 1 Φ ( c ) c d ϕ ( x ) = ϕ ( c ) Φ ( c ) , we obtain E ( Y | X < c ) = ρ ϕ ( c ) Φ ( c ) . Similarly, we obtain E ( Y | X c ) = ρ ϕ ( c ) Φ ( c ) = ρ ϕ ( c ) Φ ( c ) since E ( Y | X c ) = E ( Y | X < c ) .
## Appendix C: Proofs
### Proof 1 Imbalance in confounders U and X
For the linear structural model formulated in Eq. (2) and represented by the right causal diagram in Figure 4, we prove for the general case with a correlated and unreliably measured confounder X* the imbalance formula
I m b a l a n c e ( U | X ) = E X E ( U | Z = z + 1 , X ) E ( U | Z = z , X ) = α U ( 1 ρ ˜ 2 ) + α X ρ ( 1 γ ) 1 ( α X + α U ρ ) 2 γ ,
where X, U, Z, and Y are unit-variance variables and X* is a fallible measure of X with reliability γ = 1 / ( 1 + σ e 2 ) (i. e., X = X + e with e N ( 0 , σ e 2 ) ). The correlation between X and U is given by c o r ( X , U ) = ρ < 1 , and the corresponding correlation with X* is c o r ( X , U ) = ρ γ = ρ ˜ . Due to the linearity of the structural model, the difference in expectations of the above imbalance formula is given by the partial regression coefficient for Z of the regression of U on Z and X*: b Z = r U Z r U X r Z X 1 r Z X 2 , where r A B is the correlation coefficient between A and B (note that the difference in expectations represents the change due to a one-unit increase in Z). Then, using correlations
r U Z = C o v ( U , Z ) = C o v ( U , α X X + α U U + ε Z ) = α X ρ + α U r U X = C o v ( U , X ) / S D ( X ) = C o v ( U , X + e ) γ = ρ γ , a n d r Z X = C o v ( Z , X ) / S D ( X ) = C o v ( α X X + α U U + ε Z , X + e ) γ = ( α X + α U ρ ) γ
we obtain
I m b a l a n c e ( U | X ) = r U Z r U X r Z X 1 r Z X 2 = { α U ( 1 ρ ˜ 2 ) + α X ρ ( 1 γ ) } 1 1 ( α X + α U ρ ) 2 γ .
In setting ρ = 0 or γ = 1 all other imbalance formulas presented in this article can be directly derived.
Analogously, the imbalance formula for X is given by the partial regression coefficient for Z from the regression of X on Z and X*. Using
r X Z = C o v ( X , Z ) = C o v ( X , α X X + α U U + ε Z ) = α X + α U ρ a n d r X X = C o v ( X , X ) / S D ( X ) = C o v ( X , X + e ) γ = γ
we get
I m b a l a n c e ( X | X ) = r X Z r X X r Z X 1 r Z X 2 = ( α X + α U ρ ) ( 1 γ ) 1 ( α X + α U ρ ) 2 γ .
### Proof 2 Imbalance inequalities
We prove the following three results: (i) Conditioning on a fallible X* does not fully balance the latent X, and the imbalance can never exceed the initial imbalance (i. e., without conditioning on X or X*): I m b a l a n c e ( X | X ) I m b a l a n c e ( X | { } ) . (ii) If X and U are uncorrelated, conditioning on a fallible X* increases the imbalance in U: I m b a l a n c e ( U | X ) > I m b a l a n c e ( U | { } ) . (iii) For correlated X and U, conditioning on a fallible X* may increase or decrease the imbalance in U.
1. (i)
We show that I m b a l a n c e ( X | X ) I m b a l a n c e ( X | { } ) , that is, ( α X + α U ρ ) ( 1 γ ) 1 ( α X + α U ρ ) 2 γ α X + α U ρ . For ease of notation, we use a = α X + ρ α U such that the inequality simplifies to a ( 1 γ ) 1 a 2 γ a which is identical to writing ( 1 γ ) 1 a 2 γ a a since 0 < γ 1 and a 2 1 (because the path coefficients refer to variables with unit variances). Because of the constraints on γ and a we know that ( 1 γ ) 1 a 2 γ 1 , proving our result. Note that conditioning on X* does not reduce the imbalance in X if a = α X + ρ α U = 0 (another setting would be a = α X + ρ α U = 1 but this is not possible due to the parameter constraints).
2. (ii)
For uncorrelated X and U we show that I m b a l a n c e ( U | X ) > I m b a l a n c e ( U | { } ) , that is, α U 1 α X 2 γ > α U . Using 0 < γ 1 and α X 2 < 1 , we get α U 1 α X 2 γ > α U . And knowing that 1 α X 2 γ < 1 verifies the inequality.
3. (iii)
For correlated X and U conditioning on X* can increase or decrease the imbalance in U, that is, I m b a l a n c e ( U | X ) > I m b a l a n c e ( U | { } ) or I m b a l a n c e ( U | X ) I m b a l a n c e ( U | { } ) . Using two different restrictions on α U , we show that the difference in absolute imbalances, I m b a l a n c e ( U | { } ) I m b a l a n c e ( U | X ) = α U + α X ρ α U ( 1 ρ ˜ 2 ) + α X ρ ( 1 γ ) 1 ( α X + α U ρ ) 2 γ , can be negative or positive. Using α U = α X ρ with α X ρ > 0 as first restriction results in a negative difference. Since I m b a l a n c e ( U | { } ) = 0 and I m b a l a n c e ( U | X ) = γ ( 1 ρ 2 ) 1 ( α X + α U ρ ) 2 γ α X ρ > 0 we obtain I m b a l a n c e ( U | { } ) I m b a l a n c e ( U | X ) < 0 . Using α U = α X ρ ( 1 γ ) 1 ρ ˜ 2 with α X ρ > 0 as second restriction results in a positive difference. Since I m b a l a n c e ( U | { } ) = γ ( 1 ρ 2 ) 1 ρ 2 γ α X ρ > 0 and I m b a l a n c e ( U | X ) = 0 we get I m b a l a n c e ( U | { } ) I m b a l a n c e ( U | X ) > 0 .
### Proof 3 Bias in the linear regression estimator τ ˆ
Using the same linear setting as in Proof 1, we show that, after conditioning on X*, the bias in the linear regression estimator τ ˆ is given by
O V B ( τ ˆ | X ) = { α U β U ( 1 ρ ˜ 2 ) + ( α X β X + α X ρ β U + α U ρ β X ) ( 1 γ ) } × 1 1 ( α X + α U ρ ) 2 γ .
The estimator τ ˆ for the effect of treatment Z is obtained from regressing Y onto Z and X*: τ ˆ = r Y Z r Y X r Z X 1 r Z X 2 . Plugging the population correlations
r Y Z = C o v ( Y , Z ) = C o v ( β X X + β U U + τ Z + ε Y , α X X + α U U + ε Z ) = τ + α X β X + α U β U + α X β U ρ + α U β X ρ , r Y X = C o v ( Y , X ) / S D ( X ) = C o v ( β X X + β U U + τ Z + ε Y , X + e ) γ = ( β X + τ α X + β U ρ + τ α U ρ ) γ , r Z X = C o v ( Z , X ) / S D ( X ) = C o v ( α X X + α U U + ε Z , X + e ) γ = ( α X + α U ρ ) γ
into the above OVB formula we get τ ˆ τ = r Y Z r Y X r Z X 1 r Z X 2 τ . In setting ρ = 0 or γ = 1 all other bias formulas contained in this article directly follow from this general formula.
### Proof 4 Inequalities for increasing bias when conditioning on an uncorrelated and reliably measured confounder X
For uncorrelated confounders X and U (with standardized coefficients), we prove the inequalities (i) α U β U α X β X > 1 α X 2 α X 2 if sgn ( α X β X ) = sgn ( α U β U ) , (ii) α U β U α X β X > 1 α X 2 2 α X 2 if sgn ( α X β X ) sgn ( α U β U ) and α X β X > α U β U , and (iii) α U β U α X β X > 1 1 α X 2 if sgn ( α X β X ) sgn ( α U β U ) and α X β X < α U β U . Given the biases before and after conditioning on X, O V B ( τ ˆ | { } ) = α X β X + α U β U and O V B ( τ ˆ | X ) = α U β U 1 α X 2 , adjusting for X increases the absolute bias if
(C1) α U β U 1 α X 2 > α X β X + α U β U
First, if sgn ( α X β X ) = sgn ( α U β U ) , (C1) is equivalent to α U β U 1 α X 2 > α X β X + α U β U . In dividing both sides by α U β U we obtain 1 1 α X 2 > α X β X α U β U + 1 and, finally, α U β U α X β X > 1 α X 2 α X 2 .
Second, if sgn ( α X β X ) sgn ( α U β U ) and α X β X > α U β U , then (C1) can be written as α U β U 1 α X 2 > α X β X α U β U . Dividing both sides by α U β U we obtain 1 1 α X 2 > α X β X α U β U 1 , and thus α U β U α X β X > 1 α X 2 2 α X 2 .
Third, if sgn ( α X β X ) sgn ( α U β U ) and α X β X < α U β U , then (C1) is equivalent to α U β U 1 α X 2 > α U β U α X β X . Then, dividing both sides by α U β U we obtain 1 1 α X 2 > 1 α X β X α U β U and finally α U β U α X β X > 1 1 α X 2 . For α X β X < α U β U this inequality is always true because the left-hand side is always greater than one while the right-hand side is always less than one. That is, for sgn ( α X β X ) sgn ( α U β U ) and α X β X < α U β U , conditioning on X always increases rather than reduces the bias.
### Proof 5 Inequalities among absolute biases
It is important to note that measurement error in X* always attenuates OVB towards the initial bias [26], that is,
O V B ( τ ˆ | X ) < O V B ( τ ˆ | X ) < O V B ( τ ˆ | { } ) i f O V B ( τ ˆ | X ) < O V B ( τ ˆ | { } ) a n d O V B ( τ ˆ | X ) > O V B ( τ ˆ | X ) > O V B ( τ ˆ | { } ) i f O V B ( τ ˆ | X ) > O V B ( τ ˆ | { } ) .
Since the initial OVB and the OVB after adjusting for X can be of opposite signs, the two inequalities do not imply that measurement error necessarily increases the absolute OVB. Thus, the corresponding inequalities with absolute OVBs,
O V B ( τ ˆ | X ) < O V B ( τ ˆ | X ) < O V B ( τ ˆ | { } ) i f O V B ( τ ˆ | X ) < O V B ( τ ˆ | { } ) a n d O V B ( τ ˆ | X ) > O V B ( τ ˆ | X ) > O V B ( τ ˆ | { } ) i f O V B ( τ ˆ | X ) > O V B ( τ ˆ | { } )
do not hold in general. They only hold if X and U induce bias in the same direction, that is, all four terms in the initial bias formula have the same sign. To show the impact of measurement error on the bias, we prove the following four inequalities:
1. (i)
O V B ( τ ˆ | X ) O V B ( τ ˆ | X ) O V B ( τ ˆ | { } ) h o l d s i f O V B ( τ ˆ | X ) O V B ( τ ˆ | { } ) 1 a n d sgn O V B ( τ ˆ | { } ) = sgn O V B ( τ ˆ | X ) ,
2. (ii)
O V B ( τ ˆ | { } ) < O V B ( τ ˆ | X ) < O V B ( τ ˆ | X ) h o l d s i f O V B ( τ ˆ | X ) O V B ( τ ˆ | { } ) > 1 a n d sgn O V B ( τ ˆ | { } ) = sgn O V B ( τ ˆ | X ) ,
3. (iii)
O V B ( τ ˆ | X ) O V B ( τ ˆ | X ) O V B ( τ ˆ | { } ) h o l d s i f k < O V B ( τ ˆ | X ) O V B ( τ ˆ | { } ) 1 a n d sgn O V B ( τ ˆ | { } ) sgn O V B ( τ ˆ | X ) ,
4. (iv)
O V B ( τ ˆ | X ) O V B ( τ ˆ | { } ) < O V B ( τ ˆ | X ) h o l d s i f 1 < O V B ( τ ˆ | X ) O V B ( τ ˆ | { } ) k a n d sgn O V B ( τ ˆ | { } ) sgn O V B ( τ ˆ | X ) ,
where k = 1 γ γ 1 ( α X + ρ α U ) 2 and γ is the reliability of X*.
For ease of notation we use a = α X + ρ α U , u = 1 ρ 2 α U β U and i n i = O V B ( τ ˆ | { } ) = α X β X + α U β U + ρ α X β U + ρ α U β X . Then, we can write the absolute OVB differences as
B 0 = O V B ( τ ˆ | X ) O V B ( τ ˆ | { } ) = u 1 a 2 + σ 2 + i n i 1 a 2 + σ 2 σ 2 i n i , B X = O V B ( τ ˆ | X ) O V B ( τ ˆ | X ) = u 1 a 2 + σ 2 + i n i 1 a 2 + σ 2 σ 2 u 1 a 2 .
We first prove that a 2 < 1 . Due to the constraints of our parameters (unit variance of variables) we have α X 2 + α U 2 + 2 ρ α X α U < 1 . Adding ( 1 ρ 2 ) α U 2 to both sides we get 1 ( α X + ρ α X ) 2 > ( 1 ρ 2 ) α U 2 . Since 1 < ρ < 1 we obtain the true inequality ( α X + ρ α U ) 2 < 1 . Consequently, 1 a 2 + σ 2 > 0 in both B 0 and B X .
Now consider the situation where sgn O V B ( τ ˆ | { } ) = sgn O V B ( τ ˆ | X ) holds (inequalities (i) and (ii)). The equality of signs directly implies sgn ( u ) = sgn ( i n i ) such that
B 0 = 1 1 a 2 + σ 2 u 1 a 2 i n i a n d B X = σ 2 1 a 2 + σ 2 u 1 a 2 i n i .
Then, inequality (i) holds if O V B ( τ ˆ | X ) O V B ( τ ˆ | { } ) 1 because B 0 0 and B X 0 . Inequality (ii) holds if O V B ( τ ˆ | X ) O V B ( τ ˆ | { } ) > 1 because B 0 > 0 and B X < 0 .
Now consider the situation where sgn O V B ( τ ˆ | { } ) sgn O V B ( τ ˆ | X ) and u > i n i σ 2 (inequality (iii)). The two absolute OVB differences are given by
B 0 = 1 1 a 2 + σ 2 u 1 a 2 + 2 σ 2 i n i a n d B X = σ 2 1 a 2 + σ 2 u 1 a 2 + i n i .
Then, inequality (iii) holds if O V B ( τ ˆ | X ) O V B ( τ ˆ | { } ) 1 and O V B ( τ ˆ | X ) O V B ( τ ˆ | { } ) > 1 γ γ 1 ( α X + ρ α U ) 2 because B 0 0 and B X 0 . Note that B 0 0 holds because u 1 a 2 + 2 σ 2 i n i u 1 a 2 i n i 0 .
Finally consider the situation where sgn O V B ( τ ˆ | { } ) sgn O V B ( τ ˆ | X ) and u i n i σ 2 (inequality (iv)). The two absolute OVB differences are given by
B 0 = 1 1 a 2 + σ 2 u + 1 a 2 i n i a n d B X = σ 2 1 a 2 + σ 2 i n i u 1 a 2 2 u σ 2 .
Inequality (iv) holds if O V B ( τ ˆ | X ) O V B ( τ ˆ | { } ) > 1 and O V B ( τ ˆ | X ) O V B ( τ ˆ | { } ) 1 γ γ 1 ( α X + ρ α U ) 2 because B 0 0 and B X 0 . Note that B X 0 hold because i n i u 1 a 2 2 u σ 2 i n i u 1 a 2 < 0 .
Published Online: 2016-11-8
Published in Print: 2016-9-1
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## Foliations and the Bott Vanishing Theorem
Let $M$ be a closed manifold and $TM$ its tangent vecto […]
## The Chern-Simons Transgressed Form
In the end of the proof of theorem1.9, we have \[ tr \l […]
## K-groups and the Chern Character
Let $E$ be a complex vector bundle over a compact smoot […]
## Some Examples
Now, we take $f$ to be some special function to obtain […]
## Chern-Weil Theorem
Recall that given a vector bundle on $M$, there exists […]
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# Homework Help: Simple p-series question
1. Jul 31, 2012
### e^(i Pi)+1=0
I just wanted to check that this was legal.
$\sum_5^\infty \frac{1}{(n-4)^2} = \sum_1^\infty \frac{1}{n^2}$ ?
2. Jul 31, 2012
### Dick
Sure it is. They are the same series, aren't they?
3. Jul 31, 2012
Thanks
4. Aug 1, 2012
### HallsofIvy
A slightly more formal derivation would be:
$$\sum_{n=5}^\infty \frac{1}{(n-4)^2}$$
Let i= n- 4. Then $(n-4)^2= i^3$ and when n= 5, i= 5- 4= 1. Of course, i= n-4 goes to infinity as n goes to infinity so
$$\sum_{n=5}^\infty \frac{1}{(n-4)^2}= \sum_{i=1}^\infty \frac{1}{i^2}$$
But both "n" and "i" are "dummy" variables- the final sum does not involve either- so we can change them at will. Changing "i" to "n" in the last sum,
$$\sum_{n=5}^\infty \frac{1}{(n-4)^2}= \sum_{i=n}^\infty \frac{1}{n^2}$$
The crucial point is that, as Dick said, "they are the same sequence":
$$\sum_{n=5}^\infty \frac{1}{(n-4)^2}+ \frac{1}{(5-4)^2}+ \frac{1}{(6-4)^2}+ \frac{1}{(7- 4)^2}+ \cdot\cdot\cdot= 1+ \frac{1}{4}+ \frac{1}{9}+ \cdot\cdot\cdot$$
$$\sum_{n= 1}^\infty \frac{1}{n^2}= \frac{1}{1^2}+ \frac{1}{2^2}+ \frac{1}{3^2}+ \cdot\cdot\cdot= 1+ \frac{1}{4}+ \frac{1}{9}+ \cdot\cdot\cdot$$
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# What do parameters like Torque (Nm) mean for ebikes?
Motor Torque is the characteristic which reflects the traction (thrust) and is important for climbing up hills.
• The traction force equals Torque divided by wheel radius (F=T/R).
• The more torque at the wheel gives you a higher grade-ability.
• The smaller wheel diameter gives a higher traction force or climb-ability with same motor torque.
The torque at the wheel for mid-drive motors should be calculated like this:
• Max Torque at Wheel for crank motors = Max Motor Torque x Number of rear cog teeth / Number of Chain wheel teeth
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# Electron shell
Possible Number of Electrons in shells 1-7
Shell Electrons
K or 1st 2
L or 2nd 8
M or 3rd 8
N or 4th 18
O or 5th 18
P or 6th 32
Q or 7th 32
Articles WikiDoc Resources for Electron shell
## Overview
An electron shell, also known as a main energy level, is a group of atomic orbitals with the same value of the principal quantum number n. Electron shells are made up of one or more electron subshells, or sublevels, which have two or more orbitals with the same angular momentum quantum number l. Electron shells make up the electron configuration of an atom. It can be shown that the number of electrons that can reside in a shell is equal to ${\displaystyle 2n^{2}}$ [1].
## History
The existence of electron shells was first observed experimentally in Charles Barkla's and Henry Moseley's X-ray absorption studies. Barkla labelled them with the letters K, L, M, etc. (The origin of this terminology was alphabetic. K for hypothetical spectral lines that were never discovered.) These letters were later found to correspond to the n-values 1, 2, 3, etc. They are used in the spectroscopic Siegbahn notation.
The name for electron shells originates from the Bohr model, in which groups of electrons were believed to orbit the nucleus at certain distances, so that their orbits formed "shells" around the nucleus.
## Valence shell
The valence shell is the outermost shell of an atom in its uncombined state, which contains the electrons most likely to account for the nature of any reactions involving the atom and of the bonding interactions it has with other atoms. The outermost shell of an ion is not commonly termed valence shell. Electrons in the valence shell are referred to as valence electrons. The physical chemist Gilbert Lewis was responsible for much of the early development of the theory of the participation of valence shell electrons in chemical bonding. Linus Pauling later generalized and extended the theory while applying insights from quantum mechanics.
In a noble gas, an atom tends to have 8 electrons in its outer shell (except helium, which is only able to fill its shell with 2 electrons). This serves as the model for the octet rule which is mostly applicable to main group elements of the second and third periods. In terms of atomic orbitals, the electrons in the valence shell are distributed 2 in the single s orbital and 2 each in the three p orbitals.
For coordination complexes containing transition metals, the valence shell consists of electrons in these s and p orbitals, as well as up to 10 additional electrons, distributed as 2 into each of 5 d orbitals, to make a total of 18 electrons in a complete valence shell for such a compound. This is referred to as the eighteen electron rule.
Each shell (n=1, 2, 3, 4) can hold 2, 8, 18, or 32 electrons, or ${\displaystyle 2n^{2}}$ electrons. It is important to note that the number of valence electrons is not necessarily equal to the total number of electrons in a given electron shell. For example, because the 3d subshell has a higher energy than the 4s subshell, the 3d electrons are considered to be part of the 4th valence shell. So, while the 3rd electron shell can contain a total of 18 electrons (2 in the 3s orbital, 6 in the 3p orbitals, and 10 in the 3d orbitals), the 3rd valence shell contains only 8 electrons as the 3d electrons are typically not part of the 3rd valence shell.
## References
1. Tipler, Paul & Ralph Llewellyn (2003). Modern Physics (4th ed.). New York: W. H. Freeman and Company. ISBN 0-7167-4345-0
ar:غلاف إلكتروني
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Estimating and testing a structured covariance matrix for three-level multivariate data.(English)Zbl 1216.62095
Summary: This article considers an approach to estimating and testing a new Kronecker product covariance structure for three-level (multiple time points $$(p)$$, multiple sites $$(u)$$, and multiple response variables $$(q))$$ multivariate data. Testing of such covariance structure is potentially important for high dimensional multi-level multivariate data. The hypothesis testing procedure developed in this article can not only test the hypothesis for three-level multivariate data, but also can test many different hypotheses, such as blocked compound symmetry, for two-level multivariate data as special cases. The tests are implemented with two real data sets.
MSC:
62H15 Hypothesis testing in multivariate analysis 62H12 Estimation in multivariate analysis 15A99 Basic linear algebra 65C60 Computational problems in statistics (MSC2010)
Full Text:
References:
[1] DOI: 10.1214/aoms/1177729698 · Zbl 0042.38203 [2] DOI: 10.1080/03610929108830562 · Zbl 0751.62027 [3] DOI: 10.1016/S0378-3758(01)00235-X · Zbl 1044.62059 [4] DOI: 10.1080/00949659908811970 · Zbl 0960.62056 [5] DOI: 10.1080/03610929408831436 · Zbl 0875.62274 [6] Johnson R. A., Applied Multivariate Statistical Analysis., 6. ed. (2007) · Zbl 1269.62044 [7] DOI: 10.1016/j.jmva.2006.06.002 · Zbl 1112.62057 [8] DOI: 10.1016/j.jspi.2008.11.013 · Zbl 1162.62059 [9] DOI: 10.1016/j.spl.2005.04.020 · Zbl 1071.62052 [10] DOI: 10.1080/02664760120011626 · Zbl 0991.62038 [11] Rao C. R., Curr. Sci. 14 pp 66– (1945) [12] Rao C. R., Sankhy 12 pp 229– (1953) [13] DOI: 10.1006/jmva.2001.2009 · Zbl 1011.62011 [14] DOI: 10.1002/sim.2320 [15] DOI: 10.1002/bimj.200510192 [16] Roy A., Proc. Amer. Statist. Assoc. Statist. Comput. Sec. pp 2157– (2007) [17] Roy A., J. Appl. Statist. Sci. 12 pp 91– (2003) [18] DOI: 10.1016/j.stamet.2005.07.003 · Zbl 1248.62092 [19] Roy A., Proc. Statist. Data Anal. Sect., Paper 199 pp 1– (2005) [20] DOI: 10.1201/9781420010923.ch11 [21] DOI: 10.1007/s11634-007-0013-0 · Zbl 1182.62142 [22] DOI: 10.1016/j.spl.2008.01.066 · Zbl 1147.62343 [23] DOI: 10.1111/j.0006-341X.2002.00521.x · Zbl 1210.62074 [24] DOI: 10.1002/sim.1887
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