scholarly journals A NEW UNCONSCIOUS PARADIGM AT THE HEART OF MATHEMATICS AND PHYSICS

Author(s):  
Maxim G. Godarev-Lozovsky ◽  

It is already the case that philosophical foundations of mathematics and physics, need a serious critical analysis and revision of a number of generally accepted assumptions. In the future, this work may lead to a shift in the paradigm related to mathematics and physics. The article deals with the problem of the ideas of actual and potential infinity being not distinguished in the «fragmented thinking» of many mathematicians. We consider it necessary to differentiate between the concept of «representation of a number by an infinite decimal fraction» and the concept of «writing a numeral». A real number can be written in different ways, but every number must be uniquely represented using an infinite decimal fraction. First of all, we overcome the ambiguity of the representation of number 1 by assuming a potentially infinite set of signs of a periodic fraction and an actually infinite set of signs of a non-periodic fraction. This leads to the following harmonious scientific and philosophical system required by broad-minded scientists. 1. Every real number, including 0, (9) is represented by a single point of a continuous number line. 2. Every irrational number in decimal representation, unlike a rational number, does not have the last digit. 3. Real space, as well as past and future time are not mathematically equal and are referents of potentially and actually infinite, countable and uncountable sets. 4. The motion of a quantum micro-object, as a fundamental particle, is mathematically imaginary because a quantum particle has a countable set of points in time that is insufficient to move temporally and an uncountable set of points in space that is excessive for moving along the trajectory. Therefore, the motion of a quantum particle can be described as the path of a point in the plane of a complex variable.

2013 ◽  
Vol 53 (5) ◽  
pp. 416-426 ◽  
Author(s):  
Pavel Exner ◽  
Jiří Lipovský

We discuss resonances for a nonrelativistic and spinless quantum particle confined to a two- or three-dimensional Riemannian manifold to which a finite number of semiinfinite leads is attached. Resolvent and scattering resonances are shown to coincide in this situation. Next we consider the resonances together with embedded eigenvalues and ask about the high-energy asymptotics of such a family. For the case when all the halflines are attached at a single point we prove that all resonances are in the momentum plane confined to a strip parallel to the real axis, in contrast to the analogous asymptotics in some metric quantum graphs; we illustrate this on several simple examples. On the other hand, the resonance behaviour can be influenced by a magnetic field. We provide an example of such a ‘hedgehog’ manifold at which a suitable Aharonov-Bohm flux leads to absence of any true resonance, i.e. that corresponding to a pole outside the real axis.


1957 ◽  
Vol 9 ◽  
pp. 277-290 ◽  
Author(s):  
R. A. Rankin

1. Introduction. Let ω be an irrational number. It is well known that there exists a positive real number h such that the inequality(1)has infinitely many solutions in coprime integers a and c. A theorem of Hurwitz asserts that the set of all such numbers h is a closed set with supremum √5. Various proofs of these results are known, among them one by Ford (1), in which he makes use of properties of the modular group. This approach suggests the following generalization.


2003 ◽  
Vol 13 (08) ◽  
pp. 2253-2277 ◽  
Author(s):  
Gian-Italo Bischi ◽  
Laura Gardini ◽  
Christian Mira

This paper is the second part of an earlier work devoted to the properties specific to maps of the plane characterized by the presence of a vanishing denominator, which gives rise to the generation of new types of singularities, called set of nondefinition, focal points and prefocal curves. A prefocal curve is a set of points which are mapped (or "focalized") into a single point, called focal point, by the inverse map when it is invertible, or by at least one of the inverses when it is noninvertible. In the case of noninvertible maps, the previous text dealt with the simplest geometrical situation, which is nongeneric. To be more precise this situation occurs when several focal points are associated with a given prefocal curve. The present paper defines the generic case for which only one focal point is associated with a given prefocal curve. This is due to the fact that only one inverse of the map has the property of focalization, but with properties different from those of invertible maps. Then the noninvertible maps of the previous Part I appear as resulting from a bifurcation leading to the merging of two prefocal curves, without merging of two focal points.


2012 ◽  
Vol 98 (3) ◽  
pp. 384-391 ◽  
Author(s):  
H. A. Bonhoff ◽  
A. Eslami

The concept of source descriptor and coupling function is commonly recognized to form a rigorous basis for structure-borne sound source characterization. While this concept initially is valid for the single-point case only, it can be extended to sources with multi-point coupling by including the interface mobility approach. By considering a continuous interface that passes all contact points, the velocities, forces and mobilities are series expanded into interface orders by means of a spatial Fourier decomposition. The use of a continuous formulation for the multi-point case, however, can be problematic from a practical point of view. This paper discusses a reformulation of the interface mobility approach for a simplified calculation and clarified interpretation of the interface orders. With a discrete Fourier series as a basis for the interface mobility approach, the interface is reduced to a set of points and the interface orders are shown to describe the interplay of the data at the contact points. A discrete formulation furthermore yields simplified equations and a strict upper bound for the number of orders that have to be included, thus enhancing the practicability of interface mobilities for source characterization.


2018 ◽  
Vol 48 (2) ◽  
pp. 27-29
Author(s):  
Virginija Garbaliauskienė ◽  
Antanas Garbaliauskas

In the paper, we prove the discrete limit theorem in the sense of the weak convergence of probability measures in the space of analytic on DV = {s ∈ C : 1 < σ < 3/2, |t| <  V} functions for L-functions of elliptic curves LE(s). The main statement of the paper is as follows. Let h > 0 be a fixed real number such that exp {2πk/h} is an irrational number for all k∈Z\{0}. Then the probability measure μN(LE(s + imh)∈A), A ∈ B(H(DV)), converges weakly to the measure PLE as N→∞.


Author(s):  
DANIELA MAFTULEAC

CAT(0) metric spaces and hyperbolic spaces play an important role in combinatorial and geometric group theory. In this paper, we present efficient algorithms for distance problems in CAT(0) planar complexes. First of all, we present an algorithm for answering single-point distance queries in a CAT(0) planar complex. Namely, we show that for a CAT(0) planar complex [Formula: see text] with n vertices, one can construct in O(n2 log n) time a data structure [Formula: see text] of size O(n2) so that, given a point [Formula: see text], the shortest path γ(x, y) between x and the query point y can be computed in linear time. Our second algorithm computes the convex hull of a finite set of points in a CAT(0) planar complex. This algorithm is based on Toussaint's algorithm for computing the convex hull of a finite set of points in a simple polygon and it constructs the convex hull of a set of k points in O(n2 log n + nk log k) time, using a data structure of size O(n2 + k).


Psichologija ◽  
2004 ◽  
Vol 29 ◽  
Author(s):  
Kristina Vanagaitė

Straipsnyje nagrinėjami T. Valentine’o veidų erdvės ir M. B. Lewiso bei R. A. Johnstono Voronoi modeliai, kuriuose pagrindinis dėmesys skiriamas veidų reprezentacijos atmintyje ypatumams. Pažymima, jog veidai reprezentuojami hipotetinėje daugiamatėje veidų erdvėje. Veidų erdvės matmenys (arba geometrinės ašys) atspindi koduojamos veido informacijos požymių įverčius. Tuomet pats veidas apibūdinamas kaip unikalus įvairių matmenų įverčių derinys, kuris gali būti reprezentuotas veidų erdvėje kaip vienintelis abstraktus taškas arba vektorius (pagal tai, ar kodavimo metu atsižvelgiama į veidą-prototipą). Remiantis veidų išsidėstymo erdvėje principais (nuotoliu nuo veido-prototipo ir atstumu tarp reprezentuojamų veidų) aiškinami kai kurie veidų informacijos apdorojimo dėsningumai (pvz., informacijos atgaminimo tikslumas, veidų tarpusavio panašumo, savitumo / tipiškumo nustatymas ir pan.). Pagrindinė problema – netikslus ir nedetalus modeliuose vartojamų sąvokų bei aprašomų veido informacijos apdorojimo procesų apibūdinimas. Tai lemia atliekamų tyrimų rezultatų prieštaringumą ir pasunkina empirinį modelių prielaidų patvirtinimą. MODELS OF FACES REPRESENTATIONKristina Vanagaitė SummaryConsidering that information alteration of faces and other visual stimulus differ qualitative, it is important to establish models, which would enable to explain peculiar consistent patterns of information processing of faces.One of the most important stages of informatikon processing – representation in memory. Recently, seeking to discover the peculiarities of faces representation in memory, the biggest attention is given to face space of T. Valentine and patterns of M. B. Lewis and R. A. Johnston Voronoi.In T. Valentine’s multidimensional face-space framework, faces are encoded according to a series of visual or physical parameter particular to each face. Each of these parameters is represented along a space dimension. The dimensions of the space will be those that serve to discriminate between faces and they represent the perceptual dimensions or features of faces that distinguish the faces. A face’s representation will be made up of many values taken along many different dimensions. This set of values will be unique for every individual face and will be represented as a single point (or vector) in an N-dimensional face-space. The main problem – because of non-comprehensive description of face’s dimension and encoded information of faces it is not clear the way of distinguishing physical and psychological faces’ dimensions, the possibilities to range the encoded face dimensions according to their importance and significance.Two different models based on this framework, norm-based and exemplar-based models are described and contrasted. Models that incorporate the facial prototype are referred to as norm-based models. On such models, faces are represented by face vectors taken relative to the facial prototype at the center of the face-space, which represent the average value of the population on each dimension. Although the existence of face-prototype is certified by the research of faces’ caricature, it is unclear why the faces of representatives of other races are reproduced inaccurate, how are the valuation mediums of encoded faces’ dimension calculated, does the face-prototype is kept in memory as the real face?As an alternative to these – exemplar-based models have face representations that are stored as absolutes, relative only to other exemplars. The present study describes Voronoi diagram, developed from the exemplar-based model, which accounts for the caricature advantage without reference to an explicitly encoded facial prototype. Voronoi diagram involves interpreting the face-space as a space of regions of identity (also called „Voronoi cells“) rather than a set of points or vectors. The advantage of this pattern – the identity of observed face is connected with appropriate „cell“. It is important, because such structure, differently to the pattern oriented to standard, enables to explain the fact, that faces are recognized despite their dimensions or change of circumstances of observation.Distinguishing between these two models remains difficult because they make similar predictions. For example, both models predict that faces that are judged to be distinctive will be recognized faster and more accurately than those judged to be typical. Also indicate that typical faces are generally located near the center of the space in more densely populated facespace while distinctive faces will tend to be present in the outer regions. However, with help of patterns oriented to standards, as well as to examples, it is possible to explain only some of peculiarities of informatikon alteration of faces and the results of empirical researches are quite contradictory. Further analysis of pattern of faces’ expansion is necessary – it would not only specify the hypothetical structure of faces’ expansion, but also elaborate the proceeding processes.


2018 ◽  
Vol 2019 (24) ◽  
pp. 7691-7732 ◽  
Author(s):  
Dong Han Kim ◽  
Lingmin Liao

Abstract Fix an irrational number θ. For a real number τ &gt; 0, consider the numbers y satisfying that for all large number Q, there exists an integer 1 ≤ n ≤ Q, such that ∥nθ − y∥ &lt; Q−τ, where ∥⋅∥ is the distance of a real number to its nearest integer. These numbers are called Dirichlet uniformly well-approximated numbers. For any τ &gt; 0, the Haussdorff dimension of the set of these numbers is obtained and is shown to depend on the Diophantine property of θ. It is also proved that with respect to τ, the only possible discontinuous point of the Hausdorff dimension is τ = 1.


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