The slow-growing and the Graegorczyk hierarchies

1983 ◽  
Vol 48 (2) ◽  
pp. 399-408 ◽  
Author(s):  
E.A. Cichon ◽  
S.S. Wainer
Keyword(s):  
Initial Segment ◽  
Elementary Proof ◽  
The Other ◽  
Precise Description ◽  
Limit Ordinal ◽  
Rates Of Growth ◽  
Positive Integers ◽  
Slow Growing ◽  
Initial Choice ◽  
Crucial Parameter

We give here an elementary proof of a recent result of Girard [4] comparing the rates of growth of the two principal (and extreme) examples of a spectrum of “majorization hierarchies”—i.e. hierarchies of increasing number-theoretic functions, indexed by (systems of notations for) initial segments I of the countable ordinals so that if α < β ∈ I then the βth function dominates the αth one at all but finitely-many positive integers x.Hardy [5] was perhaps the first to make use of a majorization hierarchy—the Hα's below—in “exhibiting” a set of reals with cardinality ℵ1. More recently such hierarchies have played important roles in mathematical logic because they provide natural classifications of recursive functions according to their computational complexity. (All the functions considered here are “honest” in the sense that the size of their values gives a measure of the number of steps needed to compute them.)The hierarchies we are concerned with fall into three main classes depending on their mode of generation at successor stages, the other crucial parameter being the initial choice of a particular (standard) fundamental sequence λ0 < λ1 < λ2 < … to each limit ordinal λ under consideration which, by a suitable diagonalization, will then determine the generation at stage λ.Our later comparisons will require the use of a “large” initial segment I of proof-theoretic ordinals, extending as far as the “Howard ordinal”. However we will postpone a precise description of these ordinals and their associated fundamental sequences until later.

α-degrees of α-theories

1972 ◽  
Vol 37 (4) ◽  
pp. 677-682 ◽  
Author(s):  
George Metakides
Keyword(s):  
Initial Segment ◽  
Recursion Theory ◽  
The Other ◽  
Infinite Length ◽  
Infinitary Logic ◽  
Admissible Set ◽  
Recursively Enumerable ◽  
Limit Ordinal ◽  
The Subject ◽  
Further Development

Let α be a limit ordinal with the property that any “recursive” function whose domain is a proper initial segment of α has its range bounded by α. α is then called admissible (in a sense to be made precise later) and a recursion theory can be developed on it (α-recursion theory) by providing the generalized notions of α-recursively enumerable, α-recursive and α-finite. Takeuti [12] was the first to study recursive functions of ordinals, the subject owing its further development to Kripke [7], Platek [8], Kreisel [6], and Sacks [9].Infinitary logic on the other hand (i.e., the study of languages which allow expressions of infinite length) was quite extensively studied by Scott [11], Tarski, Kreisel, Karp [5] and others. Kreisel suggested in the late '50's that these languages (even which allows countable expressions but only finite quantification) were too large and that one should only allow expressions which are, in some generalized sense, finite. This made the application of generalized recursion theory to the logic of infinitary languages appear natural. In 1967 Barwise [1] was the first to present a complete formalization of the restriction of to an admissible fragment (A a countable admissible set) and to prove that completeness and compactness hold for it. [2] is an excellent reference for a detailed exposition of admissible languages.

Floral development of Potamogeton densus

1973 ◽  
Vol 51 (3) ◽  
pp. 647-656 ◽  
Author(s):  
U. Posluszny ◽  
R. Sattler
Keyword(s):  
Floral Development ◽  
The Other ◽  
Floral Meristem ◽  
Developmental Sequence ◽  
Initial Activity ◽  
The Third ◽  
Stamen Primordia ◽  
Procambial Strand ◽  
Inflorescence Axis ◽  
Rates Of Growth

The floral appendages of Potamogeton densus are initiated in an acropetal sequence. The first primordia to be seen externally are those of the lateral tepals, though sectioning young floral buds (longitudinally, parallel to the inflorescence axis) reveals initial activity in the region of the lower median (abaxial) tepal and stamen at a time when the floral meristem is not yet clearly demarcated. The lateral (transversal) stamens are initiated simultaneously and unlike the median stamens each arises as two separate primordia. The upper median (adaxial) tepal and stamen develop late in relation to the other floral appendages, and in some specimens are completely absent. Rates of growth of the primordia vary greatly. Though the lower median tepal and stamen are initiated first, they grow slowly up to gynoecial inception, while the upper median tepal appears late in the developmental sequence but grows rapidly, soon overtaking the other tepal primordia. The four gynoecial primordia arise almost simultaneously, although variation in their sequence of inception occurs. The two-layered tunica of the floral apices gives rise to all floral appendages through periclinal divisions in the second layer. The third layer (corpus) is involved as well in the initiation of the stamen primordia. Procambial strands develop acropetally, lagging behind primordial initiation. The lateral stamens though initiating as two primordia each form a single, central procambial strand, which differentiates after growth between the two primordia of the thecae has occurred. A great amount of deviation from the normal tetramerous flower is found, including completely trimerous flowers, trimerous gynoecia with tetramerous perianth and androecium, and organs differentiating partially as tepals and partially as stamens.

scholarly journals On the Turán Properties of Infinite Graphs

10.37236/771 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Andrzej Dudek ◽  
Vojtěch Rödl
Keyword(s):  
Infinite Graph ◽  
Infinite Graphs ◽  
The Other ◽  
Other Hand ◽  
Turan's Theorem ◽  
Vertex Set ◽  
Turán’S Theorem ◽  
Positive Integers

Let $G^{(\infty)}$ be an infinite graph with the vertex set corresponding to the set of positive integers ${\Bbb N}$. Denote by $G^{(l)}$ a subgraph of $G^{(\infty)}$ which is spanned by the vertices $\{1,\dots,l\}$. As a possible extension of Turán's theorem to infinite graphs, in this paper we will examine how large $\liminf_{l\rightarrow \infty} {|E(G^{(l)})|\over l^2}$ can be for an infinite graph $G^{(\infty)}$, which does not contain an increasing path $I_k$ with $k+1$ vertices. We will show that for sufficiently large $k$ there are $I_k$–free infinite graphs with ${1\over 4}+{1\over 200} < \liminf_{l\rightarrow \infty} {|E(G^{(l)})|\over l^2}$. This disproves a conjecture of J. Czipszer, P. Erdős and A. Hajnal. On the other hand, we will show that $\liminf_{l\rightarrow \infty} {|E(G^{(l)})|\over l^2}\le{1\over 3}$ for any $k$ and such $G^{(\infty)}$.

scholarly journals Orthologic Desargues' figure

1979 ◽  
Vol 28 (3) ◽  
pp. 295-302 ◽  
Author(s):  
Sahib Ram Mandan
Keyword(s):  
Elementary Proof ◽  
The Other

AbstractIn 1894 Sondat published a theorem that the centre of perspectivity and the 2 orthologic centres of any 2 bilogic (perspective as well as orthologic) triangles lie on a line perpendicular to their axis of perspectivity. Thébault (1952) gave an elementary proof of this theorem. Here we give two new proofs, one synthetic and the other analytic, and then deduce the existence of an orthologic Desargues' figure where all the 10 pairs of perspective triangles in it are orthologic. Consequently we arrive at an orthologic Veronese configuration of 15 points and 10 pairs of perpendicular lines studied in 5 different ways.

scholarly journals Equations with powers of singular moduli

2019 ◽  
Vol 15 (03) ◽  
pp. 445-468 ◽  
Author(s):  
Antonin Riffaut
Keyword(s):  
Rational Number ◽  
The Other ◽  
Other Hand ◽  
Singular Moduli ◽  
Cm Points ◽  
The One ◽  
Straight Lines ◽  
Positive Integers

We treat two different equations involving powers of singular moduli. On the one hand, we show that, with two possible (explicitly specified) exceptions, two distinct singular moduli [Formula: see text] such that the numbers [Formula: see text], [Formula: see text] and [Formula: see text] are linearly dependent over [Formula: see text] for some positive integers [Formula: see text], must be of degree at most [Formula: see text]. This partially generalizes a result of Allombert, Bilu and Pizarro-Madariaga, who studied CM-points belonging to straight lines in [Formula: see text] defined over [Formula: see text]. On the other hand, we show that, with obvious exceptions, the product of any two powers of singular moduli cannot be a non-zero rational number. This generalizes a result of Bilu, Luca and Pizarro-Madariaga, who studied CM-points belonging to a hyperbola [Formula: see text], where [Formula: see text].

scholarly journals On modification of the q-L-series and its applications

2001 ◽  
Vol 164 ◽  
pp. 185-197 ◽  
Author(s):  
Hirofumi Tsumura
Keyword(s):  
Elementary Proof ◽  
Positive Integers

We slightly modify the definitions of q-Hurwitz ζ-functions and q-L-series constructed by J. Satoh. By using these modified functions, we give some relations for the ordinary Dirichlet L-series. Especially we give an elementary proof of Katsurada’s formula on the values of Dirichlet L-series at positive integers.

Problems of Growth of the African Migratory Locust

1938 ◽  
Vol 29 (4) ◽  
pp. 425-456 ◽  
Author(s):  
A. J. Duarte
Keyword(s):  
Dry Weight ◽  
Hind Femur ◽  
The Other ◽  
The Third ◽  
Cell Divisions ◽  
First Instar ◽  
Wet Weight ◽  
Solitary Males ◽  
Rates Of Growth ◽  
Control Batch

General Growth.Generally, females have higher rates of growth than males. The phases, however, do not show appreciable differences in the rate. The pronotum has for increase in length the highest values which decrease throughout the instars (whereas the constants for the other parts remain fairly stable up to the fifth instar).Dyar's rule was applied for the growth in length of the middle femur and the width of the head, and it was found that the rule holds good for these parts.Przibram's rule, as modified by Bodenheimer, holds true for the growth in length of the different parts and shows the occurrence of latent cell-divisions varying from one (width of head and of pronotum) to four (length of pronotum). The number of latent cell-divisions keeps fairly constant in both phases.For wet weight Przibram's principle is inapplicable, owing to the large percentage of differences between the actual and calculated values.Gregarious males are heavier than solitary males up to the third stadium ; gregarious females are heavier than solitary females up to the third stadium ; fourth, fifth and adult stadia being characterized by higher values in wet weight for solitary females than for gregarious females. Females have higher rates of increase in wet weight than males. No significant differences exist in the rates of increase between gregarious and solitary individuals. In the fifth-adult stadium all the rates decrease except in gregarious females, which show a rise.Gregarious insects have higher values in dry weight than solitary insects, except solitary females in the adult stadium. The coefficients are higher for females than for males.The rates of increase reach the highest values in the second-third stadium of gregarious insects and solitary females, whereas solitary males have their highest value in the fourth-fifth stadium.With the exception of solitary females, all the rates of growth in dry weight decline in the fifth-adult stadium.The rates of growth of the hind legs obtained from the cube-roots of their wet weights are compared with the rates of linear growth of the hind femora. Their variation throughout the instars seems to be in opposite directions. Therefore it is suggested that the rates of growth in wet weight of the hind legs and the rates of growth in length of the respective hind femora are independent of each other.Growth of the parts.The application of the exponential allometry formula y=bxα to the data on dimensions of the parts of Locusta shows the existence of negative, positive and almost isometric growth.The pronotum has the highest value for the growth in length relatively to the growth in length of the middle femur ; the lowest value pertains to the growth in width of the head.Males have higher values than females ; phase gregaria exhibits higher growth-ratios than phase solitaria.With the exception of the hind femur the growth-ratios decline throughout the instars. The greatest decline pertains to the growth in length of the pronotum.A growth-gradient exists in Locusta with the highest value in the pronotum. The middle femur divides the growth-gradient into two parts : an anterior part with values decreasing with the growth of the insect, and a posterior part whose values increase with its growth.Effects of the amputation of the hind tibiae on crowded locusts.The effects obtained by mutilating both hind tibiae of three hundred first instar hoppers of Locusta migratoria migratorioides and placing them in a crowded condition are compared with the effects obtained by crowding a batch of the same number of first instar unoperated insects.The insects with their hind tibiae cut off did not develop as far as those of the control batch ; the differences in dimensions are greater for the hind femur than for the other parts of the body.In the experimental batch the hind femur, as a consequence of its useless condition, became extremely short as compared with the elytra, bringing the ratio E/F to a high value (over 1·950), thus leading to a false interpretation.The occurrence of the black-orange coloration in both batches suggested that both developed towards phase gregaria. This coloration was stronger and more uniform in the control batch than it was in the experimental batch. Thus the control animals developed into a better gregarious type.

Universal Grammar According to Some 12th Century Grammarians

1980 ◽  
Vol 7 (1-2) ◽  
pp. 69-84 ◽  
Author(s):  
Karin Margareta Fredborg
Keyword(s):  
Liberal Arts ◽  
Theoretical Foundation ◽  
Universal Grammar ◽  
Good Deal ◽  
Linguistic Analysis ◽  
Linguistic Competence ◽  
The Other ◽  
Precise Description ◽  
12Th Century ◽  
The Individual

Summary As early as the 12th century the concept of universal grammar became a commonly discussed and accepted doctrine among the Latin grammarians. Universal grammar is discussed within the context of whether grammar (and the other Liberal Arts) could be diversified into species, i. e., the grammar of the individual languages. Some grammarians accepted the existence of ‘species grammaticae’ but only with the proviso that there were to be two kinds of grammarians: the teacher of grammar expounding the universal grammar and the person exercising his linguistic competence in the individual languages. Along with the interest in the ‘species grammaticae’ grew a continuous interest in crosslinguistc analysis by appeal to the vernacular on matters of pronunciation, semantics and syntax. By the end of the century more determined efforts were made to solve the questions of the identity of words in different languages. These attempts proved abortive with respect to the precise description of pronunciation, orthography and morphosyntactical features, whereas a more dialectically orientated analysis of requirements for sentence-constituents is handled successfully. Further, a good deal of the upsurge of cross-linguistic analysis is hampered by the stricter adherence to the formal features as found in the established theoretical framework of Latin grammar, to the detriment of linguistic description of the vernacular, to which no theoretical foundation is conceded.

scholarly journals ON ISOTOPISMS OF COMMUTATIVE PRESEMIFIELDS AND CCZ-EQUIVALENCE OF FUNCTIONS

2011 ◽  
Vol 22 (06) ◽  
pp. 1243-1258 ◽  
Author(s):  
LILYA BUDAGHYAN ◽  
TOR HELLESETH
Keyword(s):  
Equivalence Relation ◽  
The Other ◽  
Ccz Equivalence ◽  
Positive Divisor ◽  
Positive Integers

A function F from Fpnto itself is planar if for any [Formula: see text] the function F(x+a)-F(x) is a permutation. CCZ-equivalence is the most general known equivalence relation of functions preserving planar property. This paper considers two possible extensions of CCZ-equivalence for functions over fields of odd characteristics, one proposed by Coulter and Henderson and the other by Budaghyan and Carlet. We show that the second one in fact coincides with CCZ-equivalence, while using the first one we generalize one of the known families of PN functions. In particular, we prove that, for any odd prime p and any positive integers n and m, the indicators of the graphs of functions F and F' from Fpnto Fpmare CCZ-equivalent if and only if F and F′ are CCZ-equivalent.We also prove that, for any odd prime p, CCZ-equivalence of functions from Fpnto Fpm, is strictly more general than EA-equivalence when n ≥ 3 and m is greater or equal to the smallest positive divisor of n different from 1.

Quantitative anatomical studies of the composition of the pig at 50, 68 and 92 kg. carcass weight II. Gross composition and skeletal composition

1962 ◽  
Vol 59 (2) ◽  
pp. 215-223 ◽  
Author(s):  
A. Cuthbertson ◽  
R. W. Pomeroy
Keyword(s):  
Growth Rate ◽  
Cervical Vertebrae ◽  
The Other ◽  
Carcass Weight ◽  
Anatomical Studies ◽  
Pig Carcasses ◽  
Rates Of Growth ◽  
Second Period ◽  
Different Parts

1. Results are presented concerning the gross composition of pig carcasses at 50, 68 and 92 kg. carcass weight in ten litters. They show that during the periods under consideration the proportion of muscle in the carcass exceeded that of the other tissues. However, while the rates of growth of muscle and bone were similar the growth rate of fat was markedly greater. The result was that at 92 kg. the proportions of muscle and fat in the carcass were 43·53 and 41·37%, respectively.2. Results are also presented dealing with the relative development of the different parts of the skeleton. Of the five major anatomical regions of the skeleton the sacrum grew relatively fastest between 50 and 68 kg. carcass weight while the cervical vertebrae grew fastest in the second period. In both periods the bones of the thorax and loin grew at rates intermediate between the fastest and slowest growing regions.

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