Patterns of projecta

1981 ◽  
Vol 46 (2) ◽  
pp. 287-295
Author(s):  
Adam Krawczyk
Keyword(s):  
Finite Sequence ◽  
Natural Numbers ◽  
Main Technique

AbstractRoughly speaking, a pattern is a finite sequence coding the set of natural numbers n for which the Σn+1 projectum is less than the Σn projectum for a given admissible ordinal. We prove that for each pattern there exists an ordinal realizing it. Several results on the orderings of patterns are given. We conclude the paper with remarks on Δn projecta. The main technique, used throughout the paper, is Jensen's Uniformisation Theorem.

Note on an idea of Fitch

1949 ◽  
Vol 14 (3) ◽  
pp. 175-176 ◽  
Author(s):  
John R. Myhill
Keyword(s):  
Finite Sequence ◽  
Basic Logic ◽  
Natural Numbers ◽  
Recursive Predicate

The concept of a recursively definite predicate of natural numbers was introduced by F. B. Fitch in his An extension of basic logic as follows:Every recursive predicate is recursively definite. If R(x1, …, xn) is recursively definite so is (Ey)R(x1, …, xn−1, y) and (y)R(x1, …, xn−1, y). If R is recursively definite and S is the proper ancestral of R, then S is recursively definite, where the proper ancestral of a relation is defined as follows: if R is of even degree, say 2m, then the proper ancestral of R is the relation S such that for all x1, …, xm, y1, …, ym, S(x1, …, xm, y1, …, ym) is true if and only if there is a finite sequence of sequences (z11, …, zm1), (z12, …, Zm2), …, (z1k, …, Zmk) such that R(Z11, …, Zm1, z12, …, zm2), R(z12, …, zm2, z13, …, zm3), …, R(z1,k−1, z1k, …, Zmk) are all true, where (z11, …, zm1) is (x1, …, xm) and (z1k, …, zmk) is (y1, …, ym).An arithmetic predicate is one which is definable in terms of the operations ‘+’ and ‘·’ of elementary arithmetic, the connectives of the classical prepositional calculus, and quantifiers.

scholarly journals On the General Erdős-Turán Conjecture

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Georges Grekos ◽  
Labib Haddad ◽  
Charles Helou ◽  
Jukka Pihko
Keyword(s):  
Finite Sequence ◽  
General Term ◽  
Natural Numbers ◽  
Intrinsic Interest ◽  
Positive Integers ◽  
Increasing Function

The general Erdős-Turán conjecture states that if A is an infinite, strictly increasing sequence of natural numbers whose general term satisfies an≤cn2, for some constant c>0 and for all n, then the number of representations functions of A is unbounded. Here, we introduce the function ψ(n), giving the minimum of the maximal number of representations of a finite sequence A={ak:1≤k≤n} of n natural numbers satisfying ak≤k2 for all k. We show that ψ(n) is an increasing function of n and that the general Erdős-Turán conjecture is equivalent to limn→∞ψ(n)=∞. We also compute some values of ψ(n). We further introduce and study the notion of capacity, which is related to the ψ function by the fact that limn→∞ψ(n) is the capacity of the set of squares of positive integers, but which is also of intrinsic interest.

The Argument from (Natural) Numbers

2018 ◽  
Author(s):  
Tyron Goldschmidt
Keyword(s):  
Natural Numbers ◽  
Existence Of God ◽  
Divine Attribute ◽  
Rule Out

This chapter considers Plantinga’s argument from numbers for the existence of God. Plantinga sees divine psychologism as having advantages over both human psychologism and Platonism. The chapter begins with Plantinga’s description of the argument, including the relation of numbers to any divine attribute. It then argues that human psychologism can be ruled out completely. However, what rules it out might rule out divine psychologism too. It also argues that the main problem with Platonism might also be a problem with divine psychologism. However, it will, at the least, be less of a problem. In any case, there are alternative, possibly viable views about the nature of numbers that have not been touched by Plantinga’s argument. In addition, the chapter touches on the argument from properties, and its relation to the argument from numbers.

To Discriminate in Order to Fight Discrimination

2017 ◽  
Author(s):  
Dolores Morondo Taramundi
Keyword(s):  
Human Rights ◽  
United Kingdom ◽  
The Other ◽  
The United Kingdom ◽  
European Court ◽  
Main Technique ◽  
Antidiscrimination Law ◽  
The One ◽  
Paradoxical Results ◽  
Conflicts Of Rights

This chapter analyses arguments regarding conflicts of rights in the field of antidiscrimination law, which is a troublesome and less studied area of the growing literature on conflicts of rights. Through discussion of Ladele and McFarlane v. The United Kingdom, a case before the European Court of Human Rights, the chapter examines how the construction of this kind of controversy in terms of ‘competing rights’ or ‘conflicts of rights’ seems to produce paradoxical results. Assessment of these apparent difficulties leads the discussion in two different directions. On the one hand, some troubles come to light regarding the use of the conflict of rights frame itself in the field of antidiscrimination law, particularly in relation to the main technique (‘balancing of rights’) to solve them. On the other hand, some serious consequences of the conflict of rights frame on the development of the antidiscrimination theory of the ECtHR are unearthed.

The Natural Numbers

2018 ◽  
Author(s):  
Øystein Linnebo
Keyword(s):  
Natural Number ◽  
Peano Arithmetic ◽  
Number Sequence ◽  
Natural Numbers ◽  
Ordinal Numbers ◽  
Cardinal Numbers

How are the natural numbers individuated? That is, what is our most basic way of singling out a natural number for reference in language or in thought? According to Frege and many of his followers, the natural numbers are cardinal numbers, individuated by the cardinalities of the collections that they number. Another answer regards the natural numbers as ordinal numbers, individuated by their positions in the natural number sequence. Some reasons to favor the second answer are presented. This answer is therefore developed in more detail, involving a form of abstraction on numerals. Based on this answer, a justification for the axioms of Dedekind–Peano arithmetic is developed.

Asymptotics with remainder term for moments of the total cycle number of random A-permutation

2021 ◽  
Vol 31 (1) ◽  
pp. 51-60
Author(s):  
Arsen L. Yakymiv
Keyword(s):  
Remainder Term ◽  
Random Permutation ◽  
Natural Numbers ◽  
Cycle Number ◽  
Fixed Set ◽  
Cycle Lengths

Abstract Dedicated to the memory of Alexander Ivanovich Pavlov. We consider the set of n-permutations with cycle lengths belonging to some fixed set A of natural numbers (so-called A-permutations). Let random permutation τ n be uniformly distributed on this set. For some class of sets A we find the asymptotics with remainder term for moments of total cycle number of τ n .

scholarly journals Efficient Private Conjunctive Query Protocol Over Encrypted Data

Cryptography ◽  
2021 ◽  
Vol 5 (1) ◽  
pp. 2
Author(s):  
Tushar Kanti Saha ◽  
Takeshi Koshiba
Keyword(s):  
Homomorphic Encryption ◽  
Data Access ◽  
Conjunctive Query ◽  
Conjunctive Queries ◽  
Private Data ◽  
Binary Format ◽  
Main Technique ◽  
Batch Technique ◽  
Outsourced Database ◽  
Packing Method

Conjunctive queries play a key role in retrieving data from a database. In a database, a query containing many conditions in its predicate, connected by an “and/&/∧” operator, is called a conjunctive query. Retrieving the outcome of a conjunctive query from thousands of records is a heavy computational task. Private data access to an outsourced database is required to keep the database secure from adversaries; thus, private conjunctive queries (PCQs) are indispensable. Cheon, Kim, and Kim (CKK) proposed a PCQ protocol using search-and-compute circuits in which they used somewhat homomorphic encryption (SwHE) for their protocol security. As their protocol is far from being able to be used practically, we propose a practical batch private conjunctive query (BPCQ) protocol by applying a batch technique for processing conjunctive queries over an outsourced database, in which both database and queries are encoded in binary format. As a main technique in our protocol, we develop a new data-packing method to pack many data into a single polynomial with the batch technique. We further enhance the performances of the binary-encoded BPCQ protocol by replacing the binary encoding with N-ary encoding. Finally, we compare the performance to assess the results obtained by the binary-encoded BPCQ protocol and the N-ary-encoded BPCQ protocol.

scholarly journals Endoscopic Study of the Oral and Pharyngeal Cavities in the Common Dolphin, Striped Dolphin, Risso’s Dolphin, Harbour Porpoise and Pilot Whale: Reinforced with Other Diagnostic and Anatomic Techniques

Animals ◽  
2021 ◽  
Vol 11 (6) ◽  
pp. 1507
Author(s):  
Álvaro García de los Ríos y Loshuertos ◽  
Marta Soler Laguía ◽  
Alberto Arencibia Espinosa ◽  
Francisco Martínez Gomariz ◽  
Cayetano Sánchez Collado ◽  
...  
Keyword(s):  
Oral Cavity ◽  
Common Dolphin ◽  
Resonance Imaging ◽  
Pilot Whale ◽  
Anatomical Structures ◽  
Vascular Plexus ◽  
Endoscopic Study ◽  
Main Technique ◽  
The Common ◽  
Magnetic Resonance Imaging Mri

In this work, the fetal and newborn anatomical structures of the dolphin oropharyngeal cavities were studied. The main technique used was endoscopy, as these cavities are narrow tubular spaces and the oral cavity is difficult to photograph without moving the specimen. The endoscope was used to study the mucosal features of the oral and pharyngeal cavities. Two pharyngeal diverticula of the auditory tubes were discovered on either side of the choanae and larynx. These spaces begin close to the musculotubaric channel of the middle ear, are linked to the pterygopalatine recesses (pterygoid sinus) and they extend to the maxillopalatine fossa. Magnetic Resonance Imaging (MRI), osteological analysis, sectional anatomy, dissections, and histology were also used to better understand the function of the pharyngeal diverticula of the auditory tubes. These data were then compared with the horse’s pharyngeal diverticula of the auditory tubes. The histology revealed that a vascular plexus inside these diverticula could help to expel the air from this space to the nasopharynx. In the oral cavity, teeth remain inside the alveolus and covered by gums. The marginal papillae of the tongue differ in extension depending on the fetal specimen studied. The histology reveals that the incisive papilla is vestigial and contain abundant innervation. No ducts were observed inside lateral sublingual folds in the oral cavity proper and caruncles were not seen in the prefrenular space.

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