Related papers: Imaginary Number Bases
We show how some orthonormal bases can be generated by representations of the Cuntz algebra; these include Fourier bases on fractal measures, generalized Walsh bases on the unit interval and piecewise exponential bases on the middle third…
This is a leisurely introductory account addressed to non-experts and based on previous work by the authors, on how methods borrowed from physics can be used to "count" an infinite number of points. We begin with the classical case of…
Bases of finite-dimensional Hilbert spaces (in dimension d) of relevance for quantum information and quantum computation are constructed from angular momentum theory and su(2) Lie algebraic methods. We report on a formula for deriving in…
In this paper some new ways of generalizing perfect numbers are investigated, numerical results are presented and some conjectures are established.
There are many complementary approaches to the construction of solutions to the field equations of general relativity. Among these, numerical approximation offers the only possibility to compute a variety of dynamical spacetimes, and so has…
Let $D<0$ be a fundamental discriminant and denote by $E(D)$ the exponent of the ideal class group $\text{Cl}(D)$ of $K={\mathbb Q}(\sqrt{D})$. Under the assumption that no Siegel zeros exist we compute all such $D$ with $E(D)$ is a divisor…
Eugene Wigner's much-discussed notion of the "unreasonable effectiveness of mathematics" as applied to describing the physics of empirical reality is simultaneously both trivial and profound. After all, the relevant mathematics was (in the…
In this paper we consider certain quaternary quadratic forms and octonary quadratic forms and by using the theory of modular forms, we find formulae for the number of representations of a positive integer by these quadratic forms.
We give a~detailed construction of the complete ordered field of real numbers by means of infinite decimal expansions. We prove that in the canonical encoding of decimals neither addition nor multiplication is {\em computable}, but that…
Let $A$ be a real quadratic order of discriminant $p$ or $4p$ with a prime $p$. In this paper we classify all proper totally imaginary quadratic $A$-orders $B$ with index $w(B)=[B^\times: A^\times]>1$. We also calculate numerical invariants…
Quantum computers promise a highly efficient approach to investigate quantum phase transitions, which describe abrupt changes between different ground states of many-body systems. At quantum critical points, the divergent correlation length…
We use entropy numbers in combination with the polynomial method to derive a new general lower bound for the n-th minimal error in the quantum setting of information-based complexity. As an application, we improve some lower bounds on…
The status of multifractional theories is reviewed using comparative tables. Theoretical foundations, classical matter and gravity dynamics, cosmology and experimental constraints are summarized and the application of the multifractional…
we derive new, improved lower bounds for the block complexity of an irrational algebraic number and for the number of digit changes in the b-ary expansion of an irrational algebraic number. To this end, we apply a quantitative version of…
Complex potential transformations which add imaginary parts to chosen energy levels are given and qualitatively explained. Unexpected shape similarity of potential perturbations for real and imaginary E-shifts of bound states are exhibited.…
We give upper and lower bounds on the Chevalley-Bass number of a field of characteristic zero, whenever this quantity is well-defined. We also describe an algorithm which computes the Chevalley-Bass number of a field, provided its maximal…
An enormous variety of quantum nanoobjects and nanosystems calls for the development of new approaches to their description and parametrization. Corresponding methods should be simple, universal enough and ensuring the retention of…
Quantum computation is a promising emerging technology which, compared to conventional computation, allows for substantial speed-ups e.g. for integer factorization or database search. However, since physical realizations of quantum…
We show how to represent an interval of real numbers in an abstract numeration system built on a language that is not necessarily regular. As an application, we consider representations of real numbers using the Dyck language. We also show…
Inspired by computer assisted proofs in analysis, we present an interval approach to real-number computations.