scholarly journals Los infinitos de algunas series divergentes

2020 ◽  
Vol 20 (2) ◽  
Author(s):  
Diego Miramontes de León ◽  
Gerardo Miramontes de León
Keyword(s):  
Infinite Number ◽  
The Other ◽  
Divergent Series ◽  
Harmonic Series ◽  
Natural Numbers

En este trabajo interesa mostrar que dos series divergentes, aunque ambas tienen un número infinito de términos, al tener términos diferentes, su valor al infinito también difiere. En el documento se muestra que la serie armónica, dada por la suma del inverso de los números naturales, puede descomponerse en dos series. Una de ellas dada por la suma del inverso de los naturales de la forma 1/np con p > 1 y la otra, que será llamada subarmónica, formada por el resto de los términos que completan la serie armónica original. Se muestra que cada una de estas series es, una convergente y la otra divergente, obteniendo así la serie original divergente. Se incluye la demostración de la divergencia de las nuevas series, y como extensión de esta descomposición de la serie armónica, se hace una comparación de dos series subarmónicas las cuales, a pesar de ser ambas divergentes, difieren en su valor al infinito.   Abstract This work aims to show that two divergent series, although both have an infinite number of terms, if they have different terms, their value to infinity also differs. In this document, it is shown that the harmonic series, given by the sum of the inverse of natural numbers, can be decomposed into two series; one of them is given by the sum of the inverse of the naturals in the form 1/np where p > 1 and the other, which will be called subharmonic, formed by the rest of the terms that complete the originalharmonic series. It is shown that each of these series is one convergent and the other divergent, thus obtaining the original divergent series. It is included the demonstration of the divergence of the new series, and as an extension of this decomposition of the harmonic series, a comparison is made of two subharmonic series which, despite being both divergent, differ in their value to infinity.

Non-constructive rules of inference

2018 ◽  
Author(s):  
A.P. Hazen
Keyword(s):  
Natural Number ◽  
Infinite Number ◽  
Deductive System ◽  
Higher Order ◽  
The Other ◽  
Natural Numbers ◽  
Substitutional Quantification ◽  
Deductive Systems ◽  
Formal Systems ◽  
Order Arithmetic

For some theoretical purposes, generalized deductive systems (or, ‘semi-formal’ systems) are considered, having rules with an infinite number of premises. The best-known of these rules is the ‘ω-rule’, or rule of infinite induction. This rule allows the inference of ∀nΦ(n) from the infinitely many premises Φ(0), Φ(1),… that result from replacing the numerical variable n in Φ(n) with the numeral for each natural number. About 1930, in part as a response to Gödel’s demonstration that no formal deductive system had as theorems all and only the true formulas of arithmetic, several writers (most notably, Carnap) suggested considering the semi-formal systems obtained, from some formulation of arithmetic, by adding this rule. Since no finite notation can provide terms for all sets of natural numbers, no comparable rule can be formulated for higher-order arithmetic. In effect, the ω-rule is valid just in case the relevant quantifier can be interpreted substitutionally; looked at from the other side, the validity of some analogue of the ω-rule is the essential mathematical characteristic of substitutional quantification.

Mathematical Knowledge and the Origin of Phenomenology

2021 ◽  
Vol 21 ◽  
pp. 273-294
Author(s):  
Gabriele Baratelli ◽  
Keyword(s):  
Mathematical Knowledge ◽  
Cardinal Number ◽  
The Other ◽  
Mathematical Science ◽  
Natural Numbers ◽  
Symbolic Thinking ◽  
The One

The paper is divided into two parts. In the first one, I set forth a hypothesis to explain the failure of Husserl’s project presented in the Philosophie der Arithmetik based on the principle that the entire mathematical science is grounded in the concept of cardinal number. It is argued that Husserl’s analysis of the nature of the symbols used in the decadal system forces the rejection of this principle. In the second part, I take into account Husserl’s explanation of why, albeit independent of natural numbers, the system is nonetheless correct. It is shown that its justification involves, on the one hand, a new conception of symbols and symbolic thinking, and on the other, the recognition of the question of “the formal” and formalization as pivotal to understand “the mathematical” overall.

Orthonormal basis of wavelets with customizable frequency bands

2016 ◽  
Vol 14 (06) ◽  
pp. 1650050 ◽  
Author(s):  
Hiroshi Toda ◽  
Zhong Zhang
Keyword(s):  
Frequency Domain ◽  
Infinite Number ◽  
Orthonormal Basis ◽  
The Other ◽  
Just Intonation ◽  
Frequency Bands ◽  
Orthonormal Bases ◽  
Major Scale ◽  
Dilation Factor ◽  
New Type

We already proved the existence of an orthonormal basis of wavelets having an irrational dilation factor with an infinite number of wavelet shapes, and based on its theory, we proposed an orthonormal basis of wavelets with an arbitrary real dilation factor. In this paper, with the development of these fundamentals, we propose a new type of orthonormal basis of wavelets with customizable frequency bands. Its frequency bands can be freely designed with arbitrary bounds in the frequency domain. For example, we show two types of orthonormal bases of wavelets. One of them has an irrational dilation factor, and the other is designed based on the major scale in just intonation.

Determination of Stresses in Cemented Lap Joints

1953 ◽  
Vol 20 (3) ◽  
pp. 355-364
Author(s):  
R. W. Cornell
Keyword(s):  
Differential Equations ◽  
Infinite Number ◽  
Lap Joints ◽  
The Other ◽  
Complete Analysis ◽  
Stress Distributions ◽  
Lap Joint ◽  
Analogy Method ◽  
Set Up

Abstract A variation and extension of Goland and Reissner’s (1) method of approach is presented for determining the stresses in cemented lap joints by assuming that the two lap-joint plates act like simple beams and the more elastic cement layer is an infinite number of shear and tension springs. Differential equations are set up which describe the transfer of the load in one beam through the springs to the other beam. From the solution of these differential equations a fairly complete analysis of the stresses in the lap joint is obtained. The spring-beam analogy method is applied to a particular type of lap joint, and an analysis of the stresses at the discontinuity, stress distributions, and the effects of variables on these stresses are presented. In order to check the analytical results, they are compared to photoelastic and brittle lacquer experimental results. The spring-beam analogy solution was found to give a fairly accurate presentation of the stresses in the lap joint investigated and should be useful in analyzing other cemented lap-joint structures.

A Characterization of the Topological Dimension

1982 ◽  
Vol 25 (4) ◽  
pp. 487-490
Author(s):  
Gerd Rodé
Keyword(s):  
Hausdorff Space ◽  
The Other ◽  
Natural Numbers ◽  
Topological Dimension ◽  
Other Hand ◽  
The One

AbstractThis paper gives a new characterization of the dimension of a normal Hausdorff space, which joins together the Eilenberg-Otto characterization and the characterization by finite coverings. The link is furnished by the notion of a system of faces of a certain type (N1,..., NK), where N1,..., NK, K are natural numbers. It is shown that a space X contains a system of faces of type (N1,..., NK) if and only if dim(X) ≥ N1 + … + NK. The two limit cases of the theorem, namely Nk = 1 for 1 ≤ k ≤ K on the one hand, and K = 1 on the other hand, give the two known results mentioned above.

Vibration of an Infinitely Extending Plate Resting on an Elastic Foundation

1963 ◽  
Vol 30 (3) ◽  
pp. 355-362 ◽  
Author(s):  
Kazuyosi Ono
Keyword(s):  
Free Vibration ◽  
Elastic Foundation ◽  
Infinite Number ◽  
Free Vibrations ◽  
Forced Vibrations ◽  
The Other ◽  
Damped Oscillation

Free vibrations and forced vibrations of an infinitely extending plate resting on an elastic foundation and carrying a mass are solved. Then the amplitudes of the free vibrations produced by an impulse applied to the mass on the plate are determined, and it is found that two kinds of vibration are produced in the plate: One is a free vibration and the other is a special vibration, which consists of an infinite number of free vibrations and resembles a damped oscillation.

Every analytic set is Ramsey

1970 ◽  
Vol 35 (1) ◽  
pp. 60-64 ◽  
Author(s):  
Jack Silver
Keyword(s):  
Measurable Cardinal ◽  
Infinite Subset ◽  
The Other ◽  
Natural Numbers ◽  
Ramsey Property ◽  
Analytic Subset ◽  
Borel Set ◽  
Analytic Set ◽  
Principal Theorem ◽  
Countably Infinite

If X is a set, [Χ]ω will denote the set of countably infinite subsets of X. ω is the set of natural numbers. If S is a subset of [ω]ω, we shall say that S is Ramsey if there is some infinite subset X of ω such that either [Χ]ω ⊆ S or [Χ]ω ∩ S = 0. Dana Scott (unpublished) has asked which sets, in terms of logical complexity, are Ramsey.The principal theorem of this paper is: Every Σ11 (i.e., analytic) subset of [ω]ω is Ramsey (for the Σ, Π notations, see Addison [1]). This improves a result of Galvin-Prikry [2] to the effect that every Borel set is Ramsey. Our theorem is essentially optimal because, if the axiom of constructibility is true, then Gödel's Σ21 Π21 well-ordering of the set of reals [3], having the convenient property that the set of ω-sequences of reals enumerating initial segments is also Σ21 ∩ Π21, rather directly gives a Σ21 ∩ Π21 set which is not Ramsey. On the other hand, from the assumption that there is a measurable cardinal we shall derive the conclusion that every Σ21 (i.e., PCA) is Ramsey. Also, we shall explore the connection between Martin's axiom and the Ramsey property.

Theories incomparable with respect to relative interpretability

1962 ◽  
Vol 27 (2) ◽  
pp. 195-211 ◽  
Author(s):  
Richard Montague
Keyword(s):  
Positive Integer ◽  
The Other ◽  
Natural Numbers ◽  
Infinite Set

The present paper concerns the relation of relative interpretability introduced in [8], and arises from a question posed by Tarski: are there two finitely axiomatizable subtheories of the arithmetic of natural numbers neither of which is relatively interpretable in the other? The question was answered affirmatively (without proof) in [3], and the answer was generalized in [4]: for any positive integer n, there exist n finitely axiomatizable subtheories of arithmetic such that no one of them is relatively interpretable in the union of the remainder. A further generalization was announced in [5] and is proved here: there is an infinite set of finitely axiomatizable subtheories of arithmetic such that no one of them is relatively interpretable in the union of the remainder. Several lemmas concerning the existence of self-referential and mutually referential formulas are given in Section 1, and will perhaps be of interest on their own account.

Diophantine quadruples and near-Diophantine quintuples from P3,K sequences

2017 ◽  
Vol 10 (01) ◽  
pp. 1750010
Author(s):  
A. M. S. Ramasamy
Keyword(s):  
Infinite Number ◽  
Fibonacci Numbers ◽  
Natural Numbers ◽  
Text Type ◽  
Type Formula ◽  
Fourth Term ◽  
Unexplored Area ◽  
Diophantine Quintuples ◽  
Diophantine Quadruples

The question of a non-[Formula: see text]-type [Formula: see text] sequence wherein the fourth term shares the property [Formula: see text] with the first term has not been investigated so far. The present paper seeks to fill up the gap in this unexplored area. Let [Formula: see text] denote the set of all natural numbers and [Formula: see text] the sequence of Fibonacci numbers. Choose two integers [Formula: see text] and [Formula: see text] with [Formula: see text] such that their product increased by [Formula: see text] is a square [Formula: see text]. Certain properties of the sequence [Formula: see text] defined by the relation [Formula: see text] are established in this paper and polynomial expressions for Diophantine quadruples from the [Formula: see text] sequence [Formula: see text] are derived. The concept of a near-Diophantine quintuple is introduced and it is proved that there exist an infinite number of near-Diophantine quintuples.

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