Orthogonal Matrices in Four-Space

1949 ◽  
Vol 1 (1) ◽  
pp. 69-72
Author(s):  
C. C. MacDuffee

Every proper orthogonal matrix A can be writtenwhere Q is a skew matrix [6], and conversely every such matrix A is orthogonal. It is also known that every proper orthogonal transformation in real Euclidean four-space may be characterized in term of quaternions [1, 3] by the equationdetermines with the origin a vector having the coordinates (XQ, XI, x2, x3). The relationship between these two representations was clearly shown by Murnaghan [5].

Ramus ◽  
2007 ◽  
Vol 36 (1) ◽  
pp. 25-38 ◽  
Author(s):  
Myles Lavan

(BJ6.350)Those who discard their weapons and surrender their persons, I will let live. Like a lenient master in a household, I will punish the incorrigible but preserve the rest for myself.So ends Titus' address to the embattled defenders of Jerusalem in the sixth book of Josephus'Jewish War(6.328-50). It is the most substantial instance of communication between Romans and Jews in the work. Titus compares himself to the master of a household and the Jewish rebels to his slaves. Is this how we expect a Roman to describe empire? If not, what does it mean for our understanding of the politics of Josephus' history? The question is particularly acute given that this is not just any Roman but Titus himself: heir apparent and, if we believe Josephus, the man who read and approved this historical account. It is thus surprising that, while the speeches of Jewish advocates of submission to Rome such as Agrippa II (2.345-401) and Josephus himself (5.362-419) have long fascinated readers, Titus' speech has received little or no attention. Remarkably, it is not mentioned in any of three recent collections of essays on Josephus. This paper aims to highlight the rhetorical choices that Josephus has made in constructing this voice for Titus—particularly his self-presentation as master—and the interpretive questions these raise for his readers. It should go without saying that the relationship of this text to anything that Titus may have said during the siege is highly problematic. (Potentially more significant, but unfortunately no less speculative, is the question of how it might relate to any speech recorded in the commentaries of Vespasian and Titus that Josephus appears to have used as a source.) What we have is a Josephan composition that is embedded in the broader narrative of theJewish War.


1956 ◽  
Vol 8 ◽  
pp. 256-262 ◽  
Author(s):  
J. De Groot

1. Introduction. We consider the group of proper orthogonal transformations (rotations) in three-dimensional Euclidean space, represented by real orthogonal matrices (aik) (i, k = 1,2,3) with determinant + 1 . It is known that this rotation group contains free (non-abelian) subgroups; in fact Hausdorff (5) showed how to find two rotations P and Q generating a group with only two non-trivial relationsP2 = Q3 = I.


1980 ◽  
Vol 31 (2) ◽  
pp. 221-222 ◽  
Author(s):  
R. D. H. Cohen ◽  
D. L. Garden ◽  
J. P. Langlands

ABSTRACTThe incidence of oestrus in 603 Hereford heifers with a mean age of 545 days (range 515 to 575) was related to live weight (range 130 to 376 kg). Observations were ranked in order of ascending live weight and the percentage of heifers showing oestrus was calculated for consecutive 10 kg increments in live weight from 130 to 380 kg. The percentage showing oestrus was subjected to the probit transformation (Y, probit units) and was then related to live weight (X, kg):It was predicted that 50% of the heifers showed oestrus at 231 ± 1 kg, and that 5% and 95% showed oestrus at 187 and 280 kg respectively.


Author(s):  
P. Srestasathiern ◽  
S. Lawawirojwong ◽  
R. Suwantong ◽  
P Phuthong

This paper address the problem of rotation matrix sampling used for multidimensional probability distribution transfer. The distribution transfer has many applications in remote sensing and image processing such as color adjustment for image mosaicing, image classification, and change detection. The sampling begins with generating a set of random orthogonal matrix samples by Householder transformation technique. The advantage of using the Householder transformation for generating the set of orthogonal matrices is the uniform distribution of the orthogonal matrix samples. The obtained orthogonal matrices are then converted to proper rotation matrices. The performance of using the proposed rotation matrix sampling scheme was tested against the uniform rotation angle sampling. The applications of the proposed method were also demonstrated using two applications i.e., image to image probability distribution transfer and data Gaussianization.


1958 ◽  
Vol 1 (3) ◽  
pp. 183-191 ◽  
Author(s):  
Hans Zassenhaus

Under the assumptions of case of theorem 1 we derive from (3.32) the matrix equationso that there corresponds the matrix B to the bilinear form4.1on the linear space4.2and fP,μ, is symmetric if ɛ = (-1)μ+1, anti-symmetric if ɛ = (-1)μ.The last statement remains true in the case a) if P is symmetric irreducible because in that case fP,μ is 0.


2016 ◽  
Vol 22 (2) ◽  
pp. 249-286 ◽  
Author(s):  
LAURENT BIENVENU ◽  
CHRISTOPHER P. PORTER

AbstractA set of infinite binary sequences ${\cal C} \subseteq 2$ℕ is negligible if there is no partial probabilistic algorithm that produces an element of this set with positive probability. The study of negligibility is of particular interest in the context of ${\rm{\Pi }}_1^0 $ classes. In this paper, we introduce the notion of depth for ${\rm{\Pi }}_1^0 $ classes, which is a stronger form of negligibility. Whereas a negligible ${\rm{\Pi }}_1^0 $ class ${\cal C}$ has the property that one cannot probabilistically compute a member of ${\cal C}$ with positive probability, a deep ${\rm{\Pi }}_1^0 $ class ${\cal C}$ has the property that one cannot probabilistically compute an initial segment of a member of ${\cal C}$ with high probability. That is, the probability of computing a length n initial segment of a deep ${\rm{\Pi }}_1^0 $ class converges to 0 effectively in n.We prove a number of basic results about depth, negligibility, and a variant of negligibility that we call tt-negligibility. We provide a number of examples of deep ${\rm{\Pi }}_1^0 $ classes that occur naturally in computability theory and algorithmic randomness. We also study deep classes in the context of mass problems, examine the relationship between deep classes and certain lowness notions in algorithmic randomness, and establish a relationship between members of deep classes and the amount of mutual information with Chaitin’s Ω.


1993 ◽  
Vol 5 (2) ◽  
pp. 189-224 ◽  
Author(s):  
Malcah Yaeger-Dror

ABSTRACTThis article considers language variation within one “ethnic” group: Israelis of Middle Eastern origins. Earlier studies (Yaeger-Dror, 1988, 1991) found that singers from the dominant “koiné” -speaking social group (Blanc, 1968) use [r] in pop songs and [R] in casual interviews. This can be defined as a register distinction. On the other hand, singers from a MidEastern ethnolinguistic background, whose underlying dialect includes [r], use [R] even in songs. Given that singers whose vernacular consonant invetory does not even include [r], and who should find it easire to use it categorically, have such a difficult time maintaing [r] consistently (and appropriately) in the song register? One of the recorded variants for these singers “merges” the [r] and [R] into coarticulated [rR]. Why does this previously unattested sound arise, and what does it tell us about the linguistic and sociolinguistic situation? Data from various registers are analyzed in order to discover the answers to these questions. This analysis is concerned with the quantifiable evidence of systematic patterns in the use of these three pronunciations for [r] and uses this evidence to demonstrate that subconscious sociolinguistic pressures on members of the minority community influence them to assimilate to the dominant social group while still retaining ethnolinguistic proof of a narrower ethnic identity. For example, the use of [rR] is found to be correlated with a wish to affiliate with both an [R]-using group and an [r]-using group, showing that sociolinguistic techniques can reveal social psychological ethinc affiliation. Like Trudgill's (1986) discussions of dialects in contact, the present theoretical discussion takes advantage of proposals advanced by Giles, to explain why the data reveal both convergence (toward the dominant out-group) and divergence (toward the in-group) (Giles & Coupland, 1991). Sociolinguistic methods permit a quantitative analysis of the strength of these conflicting tendencies, both of which are subsumed under the technical term “accommodation.” Methods are proposed to determine if choice of the[R] or [rR] variant is conscious or not, and variable rule analysis reveals that for most of the singers the less cognitive salience, the greater the degree of convergence to the Koiné norm [R]. The linguistic factors that are correlated with the relative degree of salience can be used in future studies when the relationship between convergence toward another dialect or language and relative cognitive salience is also at issue.


1982 ◽  
Vol 34 (3) ◽  
pp. 355-357 ◽  
Author(s):  
R. T. Cowan ◽  
J. J. Robinson ◽  
I. McDonald

ABSTRACTData from three comparative slaughter experiments involving a total of 73 ewes were used to study the influence of body fat content at the start of lactation (X1 kg) and of metabolizable energy intake (X2, MJ/day), on the rate of loss of body fat by lactating ewes over the first 6 weeks of lactation (Y, g/day). The relationship was described by the equation:Thus the rate of fat loss was greater for ewes with higher initial fat contents, but the differential became less as metabolizable energy intake increased. Since increases in body fatness depress food intake it was not possible to prevent loss of body fat during early lactation in fat ewes given high concentrate diets ad libitum. The likely response in milk yield to increase in body fatness at parturition is therefore strongly dependent on the relative levels of body fatness and metabolizable energy intake. The value of any improvement in condition of the ewe at parturition may be considerable when metabolizable energy intake during lactation is low but much less when it is expected to be high.


2014 ◽  
Vol 4 (4) ◽  
pp. 159-170 ◽  
Author(s):  
Talia Gershon ◽  
Tayfun Gokmen ◽  
Oki Gunawan ◽  
Richard Haight ◽  
Supratik Guha ◽  
...  

Abstract


2009 ◽  
Vol 29 (4) ◽  
pp. 1119-1140 ◽  
Author(s):  
KARMA DAJANI ◽  
YUSUF HARTONO ◽  
COR KRAAIKAMP

AbstractLet 0<α<1 andβ>1. We show that everyx∈[0,1] has an expansion of the formwherehi=hi(x)∈{0,α/β}, andpi=pi(x)∈{0,1}. We study the dynamical system underlying this expansion and give the density of the invariant measure that is equivalent to the Lebesgue measure. We prove that the system is weakly Bernoulli, and we give a version of the natural extension. For special values ofα, we give the relationship of this expansion with the greedyβ-expansion.


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