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\"Introduces the reader to units and measurements.\"-- Provided by publisher.
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Quantum theory based on real numbers can be experimentally falsified
Although complex numbers are essential in mathematics, they are not needed to describe physical experiments, as those are expressed in terms of probabilities, hence real numbers. Physics, however, aims to explain, rather than describe, experiments through theories. Although most theories of physics are based on real numbers, quantum theory was the first to be formulated in terms of operators acting on complex Hilbert spaces
1
,
2
. This has puzzled countless physicists, including the fathers of the theory, for whom a real version of quantum theory, in terms of real operators, seemed much more natural
3
. In fact, previous studies have shown that such a ‘real quantum theory’ can reproduce the outcomes of any multipartite experiment, as long as the parts share arbitrary real quantum states
4
. Here we investigate whether complex numbers are actually needed in the quantum formalism. We show this to be case by proving that real and complex Hilbert-space formulations of quantum theory make different predictions in network scenarios comprising independent states and measurements. This allows us to devise a Bell-like experiment, the successful realization of which would disprove real quantum theory, in the same way as standard Bell experiments disproved local physics.
A Bell-like experiment that discriminates between real-number and complex-number multipartite quantum systems could disprove real quantum theory.
Journal Article
Finding Minimum Production Time and Optimal Procedure for a Project When Step Times are Given by Constructive Real Numbers
In the production process, some jobs can only be done sequentially, while others can be done synchronously. Given the time needed to complete each job, an oriented graph with a source vertex and a sink vertex can be generated. Each route starting from the source vertex and ending at the sink vertex represents a way of completing the project. It is possible to find the minimum production time and the optimal production path when time for each job is given by a rational number. However, this paper proves that there does not exist a computer program that can always generate the minimum time and optimal path when the time needed for each job is a constructive real number.
Journal Article
THE LIFTING PROBLEM FOR UNIVERSAL QUADRATIC FORMS OVER SIMPLEST CUBIC FIELDS
The lifting problem for universal quadratic forms over a totally real number field K consists of determining the existence or otherwise of a quadratic form with integer coefficients (or
$\\mathbb {Z}$
-form) that is universal over K. We prove the nonexistence of universal
$\\mathbb {Z}$
-forms over simplest cubic fields for which the integer parameter is big enough. The monogenic case is already known. We prove the nonexistence in the nonmonogenic case by using the existence of a totally positive nonunit algebraic integer in K with minimal (codifferent) trace equal to one.
Journal Article
Calculus of Linear Fuzzy Numerical Functions Based on Non-Increasing Fuzzy Real Numbers
Fuzzy calculus is the basis of researching fuzzy differential equations. This article mainly investigate the differentiability and integrability of linear fuzzy numerical functions constructed from non-negative and increasing real functions and decreasing fuzzy real numbers. Firstly, we introduced the definitions of limit and continuity on fuzzy numerical functions, which are the cornerstone of researching the differentiation and integration of fuzzy numerical functions. What is more, the concept of differentiability of fuzzy numerical functions is discussed. To be specific, we also introduce some theorems and several examples of solving differential of fuzzy numerical functions. Finally, we proposed the concept and some properties of integrability of linear fuzzy numerical functions. These concepts improve the fuzzy numerical function calculus theory.
Journal Article
On Signifiable Computability: Part I: Signification of Real Numbers, Sequences, and Types
Signifiable computability aims to separate what is theoretically computable from what is computable through performable processes on computers with finite amounts of memory. Real numbers and sequences thereof, data types, and instances are treated as finite texts, and memory limitations are made explicit through a requirement that the texts be stored in the available memory on the devices that manipulate them. In Part I of our investigation, we define the concepts of signification and reference of real numbers. We extend signification to number tuples, data types, and data instances and show that data structures representable as tuples of discretely finite numbers are signifiable. From the signification of real tuples, we proceed to the constructive signification of multidimensional matrices and show that any data structure representable as a multidimensional matrix of discretely finite numbers is signifiable.
Journal Article
A Novel Definition of Fuzzy Difference on Non-increasing Fuzzy Real Numbers
In this paper, firstly, a novel definition of a fuzzy difference based on non-increasing fuzzy real numbers is introduced, which is different from the previous definitions by using fuzzy interval-numbers and Zadeh’s extension principle. Then we give some important conclusions of fuzzy difference from the view of two cuts of fuzzy sets. Moreover, a definition of a opposite fuzzy real number is given so that we show the connection between the fuzzy difference and fuzzy addition on fuzzy real numbers. Finally, we provide several examples for the sake of illustrating the fuzzy difference on non-increasing fuzzy real numbers is reasonable generalization of classical difference. In addition, we give the prospect that we want to use this difference to research fuzzy derivatives.
Journal Article
The shape of cyclic number fields
Let
$m>1$
and
$\\mathfrak {d} \\neq 0$
be integers such that
$v_{p}(\\mathfrak {d}) \\neq m$
for any prime p. We construct a matrix
$A(\\mathfrak {d})$
of size
$(m-1) \\times (m-1)$
depending on only of
$\\mathfrak {d}$
with the following property: For any tame
$ \\mathbb {Z}/m \\mathbb {Z}$
-number field K of discriminant
$\\mathfrak {d}$
, the matrix
$A(\\mathfrak {d})$
represents the Gram matrix of the integral trace-zero form of K. In particular, we have that the integral trace-zero form of tame cyclic number fields is determined by the degree and discriminant of the field. Furthermore, if in addition to the above hypotheses, we consider real number fields, then the shape is also determined by the degree and the discriminant.
Journal Article
Substitutions and Cantor real numeration systems
We consider Cantor real numeration system as a frame in which every non-negative real number has a positional representation. The system is defined using a bi-infinite sequence B=(βn)n∈Z of real numbers greater than one. We introduce the set of B-integers and code the sequence of gaps between consecutive B-integers by a symbolic sequence in general over the alphabet N. We show that this sequence is S-adic. We focus on alternate base systems, where the sequence B of bases is periodic, and characterize alternate bases B in which B-integers can be coded by using a symbolic sequence vB over a finite alphabet. With these so-called Parry alternate bases we associate some morphisms and show that vB is a fixed point of their composition. We then provide two classes of Parry alternate bases B generating sturmian fixed points. The paper generalizes results of Fabre and Burdík et al. obtained for the Rényi numerations systems, i.e., in the case when the Cantor base B is a constant sequence.
Journal Article
Exploring Hardware Fault Impacts on Different Real Number Representations of the Structural Resilience of TCUs in GPUs
The most recent generations of graphics processing units (GPUs) boost the execution of convolutional operations required by machine learning applications by resorting to specialized and efficient in-chip accelerators (Tensor Core Units or TCUs) that operate on matrix multiplication tiles. Unfortunately, modern cutting-edge semiconductor technologies are increasingly prone to hardware defects, and the trend to highly stress TCUs during the execution of safety-critical and high-performance computing (HPC) applications increases the likelihood of TCUs producing different kinds of failures. In fact, the intrinsic resiliency to hardware faults of arithmetic units plays a crucial role in safety-critical applications using GPUs (e.g., in automotive, space, and autonomous robotics). Recently, new arithmetic formats have been proposed, particularly those suited to neural network execution. However, the reliability characterization of TCUs supporting different arithmetic formats was still lacking. In this work, we quantitatively assessed the impact of hardware faults in TCU structures while employing two distinct formats (floating-point and posit) and using two different configurations (16 and 32 bits) to represent real numbers. For the experimental evaluation, we resorted to an architectural description of a TCU core (PyOpenTCU) and performed 120 fault simulation campaigns, injecting around 200,000 faults per campaign and requiring around 32 days of computation. Our results demonstrate that the posit format of TCUs is less affected by faults than the floating-point one (by up to three orders of magnitude for 16 bits and up to twenty orders for 32 bits). We also identified the most sensible fault locations (i.e., those that produce the largest errors), thus paving the way to adopting smart hardening solutions.
Journal Article