Abstract
In this paper, we discuss the unicity problem of certain shift polynomials. Suppose that
c
j
(
j
= 1, …,
s
) be distinct complex numbers,
n, m, s
and
μ
j
(
j
= 1, …,
s
) are integers ...satisfying
n
+
m
> 4σ + 14, where σ =
μ
1
+
μ
2
+ …
μ
s
. We prove that if
p
n
(
γ
)
(
p
(
γ
)
−
1
)
m
∏
j
=
1
s
p
(
γ
+
c
j
)
μ
j
and
q
n
(
γ
)
(
q
(
γ
)
−
1
)
m
∏
j
=
1
s
q
(
γ
+
c
j
)
μ
j
share ″(
α
(
γ
),0)″, then either
p
(
γ
)
≡
q
(
γ
)
or
p
n
(
p
−
1
)
m
∏
j
=
1
s
p
(
γ
+
c
j
)
μ
j
−
q
n
(
q
−
1
)
m
∏
j
=
1
s
q
(
γ
+
c
j
)
μ
j
. The results obtained greatly improve the results of Saha (Korean J. Math. 28(4)(2020)) and C. Meng (Mathematica Bohemica 139(2014)).
Lagrange's Trigonometric Identity is usually proven analytically by summing a geometric series in the complex numbers. This Proof Without Words is purely geometric.
Wave-particle duality is one of the basic features of quantum mechanics, giving rise to the use of complex numbers in describing states of quantum systems and their dynamics and interaction. Since ...the inception of quantum theory, it has been debated whether complex numbers are essential or whether an alternative consistent formulation is possible using real numbers only. Here, we attack this long-standing problem theoretically and experimentally, using the powerful tools of quantum resource theories. We show that, under reasonable assumptions, quantum states are easier to create and manipulate if they only have real elements. This gives an operational meaning to the resource theory of imaginarity. We identify and answer several important questions, which include the state-conversion problem for all qubit states and all pure states of any dimension and the approximate imaginarity distillation for all quantum states. As an application, we show that imaginarity plays a crucial role in state discrimination, that is, there exist real quantum states that can be perfectly distinguished via local operations and classical communication but that cannot be distinguished with any nonzero probability if one of the parties has no access to imaginarity. We confirm this phenomenon experimentally with linear optics, discriminating different two-photon quantum states by local projective measurements. Our results prove that complex numbers are an indispensable part of quantum mechanics.
In complex analysis courses, it is common to use physical interpretations as a didactic tool for teaching complex numbers. In the case of operations between complex numbers, the geometric ...interpretation of addition and subtraction is well known; however, many authors avoid the interpretation of the multiplication of complex numbers. In this paper, using the physical concepts of rotation and scaling, we will explain the multiplication of complex numbers through visualization in the Argand plane. In addition, we use visual representations in order to obtain proofs without words for some identities.
Abstract
In industrial automation, there is always increasing demand of generating desired motion with accuracy and precision. It is possible when the mechanism generates motion for large number of ...precision points. Therefore, in present work, the dimensional synthesis of a 6-bar, Watt-I mechanism has been carried for twenty coupler positions. The single degree of freedom mechanism has revolute pairs at each of its joint and is able to generate required motion. All the links of the mechanism are made of metal. The dyad-triad design equations are formed in terms of complex number algebra. To solve these equations, a MATLAB code is generated and utilized to yield the dimensional parameters i.e. lengths and angular orientations of all links of the mechanism. The complete synthesis methodology is modeled and demonstrated on a numerical problem. To verify the outcomes, SAM 7.0 application software has been incorporated and applied on the given example.
Gaussian modulo integer is a complex number a + ib where a,b ∊ ℤ n. Some of the characteristics of the prime ideal on Gaussian integer are a trivial ideal {0} is a prime ideal, and I ideal prime if ...and only if I almost prime ideal. These characteristics do not necessarily apply to modulo Gaussian integers. In this paper, we give some characteristics of the prime ideal on modulo Gaussian integer.
Lagrange's Trigonometric Identity is usually proven analytically by summing a geometric series in the complex numbers. This Proof Without Words is purely geometric.
ERDŐS–LIOUVILLE SETS CHALEBGWA, TABOKA PRINCE; MORRIS, SIDNEY A.
Bulletin of the Australian Mathematical Society,
04/2023, Volume:
107, Issue:
2
Journal Article
Peer reviewed
Open access
In 1844, Joseph Liouville proved the existence of transcendental numbers. He introduced the set
$\mathcal L$
of numbers, now known as Liouville numbers, and showed that they are all transcendental. ...It is known that
$\mathcal L$
has cardinality
$\mathfrak {c}$
, the cardinality of the continuum, and is a dense
$G_{\delta }$
subset of the set
$\mathbb {R}$
of all real numbers. In 1962, Erdős proved that every real number is the sum of two Liouville numbers. In this paper, a set W of complex numbers is said to have the Erdős property if every real number is the sum of two numbers in W. The set W is said to be an Erdős–Liouville set if it is a dense subset of
$\mathcal {L}$
and has the Erdős property. Each subset of
$\mathbb {R}$
is assigned its subspace topology, where
$\mathbb {R}$
has the euclidean topology. It is proved here that: (i) there exist
$2^{\mathfrak {c}}$
Erdős–Liouville sets no two of which are homeomorphic; (ii) there exist
$\mathfrak {c}$
Erdős–Liouville sets each of which is homeomorphic to
$\mathcal {L}$
with its subspace topology and homeomorphic to the space of all irrational numbers; (iii) each Erdős–Liouville set L homeomorphic to
$\mathcal {L}$
contains another Erdős–Liouville set
$L'$
homeomorphic to
$\mathcal {L}$
. Therefore, there is no minimal Erdős–Liouville set homeomorphic to
$\mathcal {L}$
.
We introduce a complex-plane generalization of the consecutive level-spacing ratio distribution used to distinguish regular from chaotic quantum spectra. Our approach features the distribution of ...complex-valued ratios between nearest- and next-to-nearest-neighbor spacings. We show that this quantity can successfully detect the chaotic or regular nature of complex-valued spectra, which is done in two steps. First, we show that, if eigenvalues are uncorrelated, the distribution of complex spacing ratios is flat within the unit circle, whereas random matrices show a strong angular dependence in addition to the usual level repulsion. The universal fluctuations of Gaussian unitary and Ginibre unitary universality classes in the large-matrix-size limit are shown to be well described by Wigner-like surmises for small-size matrices with eigenvalues on the circle and on the two-torus, respectively. To study the latter case, we introduce the toric unitary ensemble, characterized by a flat joint eigenvalue distribution on the two-torus. Second, we study different physical situations where non-Hermitian matrices arise: dissipative quantum systems described by a Lindbladian, nonunitary quantum dynamics described by non-Hermitian Hamiltonians, and classical stochastic processes. We show that known integrable models have a flat distribution of complex spacing ratios, whereas generic cases, expected to be chaotic, conform to random matrix theory predictions. Specifically, we are able to clearly distinguish chaotic from integrable dynamics in boundary-driven dissipative spin-chain Liouvillians and in the classical asymmetric simple exclusion process and to differentiate localized from delocalized regimes in a non-Hermitian disordered many-body system.
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