Uncertainty Inequality Research Articles

Free Metaplectic K-Wigner Distribution: Uncertainty Principles and Applications

This study focuses on a novel parameterized Wigner distribution, which is an organic integration of the free metaplectic Wigner distribution and the K-Wigner distribution. We style this as the free metaplectic K-Wigner distribution (FMKWD) and investigate its uncertainty principles and related applications. We establish a crucial equivalence relation between the uncertainty product in time-FMKWD and free metaplectic transformation (FMT)-FMKWD domains and those in two FMT domains, from which we derive two types of orthogonality conditions: an orthonormality condition; and two sub-types of minimum or maximum eigenvalue commutativity conditions on the FMKWD. Finally we separately formulate an uncertainty inequality in FMKWD domains for real-valued functions, three kinds of uncertainty inequalities in orthogonal FMKWD domains, an uncertainty inequality in orthonormal FMKWD domains, and four kinds of uncertainty inequalities in the minimum or maximum eigenvalue commutative FMKWD domains for complex-valued functions. The time-frequency resolution of the FMKWD is compared with those of the free metaplectic Wigner distribution, K-Wigner distribution, and N-dimensional Wigner distribution to demonstrate its superiority in super-resolution analysis. For applications, the uncertainty inequalities derived are used to estimate the bandwidth in FMKWD domains, and the FMKWD is applied to detect noisy linear frequency-modulated signals.

Analysis of National Financial Policy in the 2020-2024 RPJMN

The 2020-2024 RPJMN is a strategic planning document that guides national development policies, including state financial management, to support economic growth and societal well-being. The state financial policy within the RPJMN encompasses revenue, expenditure, and financing aspects that must be optimally managed to achieve sustainable and inclusive development targets. This study aims to evaluate the implementation of state financial policies in the 2020-2024, identify challenges in fiscal management, and provide policy recommendations to enhance the effectiveness and efficiency of state financial management in the future. The research employs a descriptive method with a qualitative approach. Data were obtained from official government documents, academic journals, and financial institution publications, which were then analyzed using data reduction, data presentation, and conclusion-drawing techniques. The findings indicate that the analysis of state financial policies in the 2020-2024 RPJMN emphasizes its strategic role in optimizing resources to support national development priorities. Although various strategies have been implemented, their execution faces challenges such as global economic uncertainty, climate change, and social inequality, necessitating policy adjustments. Therefore, more adaptive and sustainable financial policies are required to maintain economic stability and ensure the achievement of national development targets. This study provides empirical insights that can inform policymakers in designing more resilient and forward-looking fiscal policies. The findings contribute to the academic discourse on public financial management by highlighting the importance of policy adaptability in response to dynamic socio-economic and environmental factors.

Novel Gabor-Type Transform and Weighted Uncertainty Principles

The linear canonical Fourier transform is one of the most celebrated time-frequency tools for analyzing non-transient signals. In this paper, we will introduce and study the deformed Gabor transform associated with the linear canonical Dunkl transform (LCDT). Then, we will formulate several weighted uncertainty principles for the resulting integral transform, called the linear canonical Dunkl-Gabor transform (LCDGT). More precisely, we will prove some variations in Heisenberg’s uncertainty inequality. Then, we will show an analog of Pitt’s inequality for the LCDGT and formulate a Beckner-type uncertainty inequality via two approaches. Finally, we will derive a Benedicks-type uncertainty principle for the LCDGT, which shows the impossibility of a non-trivial function and its LCDGT to both be supported on sets of finite measure. As a side result, we will prove local uncertainty principles for the LCDGT.

Biquaternion Windowed Linear Canonical Transform and Associated Uncertainty Inequalities

Biquaternion Windowed Linear Canonical Transform and Associated Uncertainty Inequalities

Two-Wavelet Multipliers and Applications

This paper delves into the Dunkl-Bessel operator on and its corresponding harmonic analysis. A generalized form of Heisenberg-type uncertainty inequality is established. Schatten-von Neumann properties for the two-wavelet multiplier within the Dunkl-Bessel theory framework are elucidated. Additionally, the trace formula for a two-wavelet Dunkl-Bessel multiplier is proven as a bounded linear operator in the trace class from into . Furthermore, subject to appropriate conditions, the boundedness and compactness of these Dunkl-Bessel two-wavelet multipliers are proven, applicable to , 1 ≤ p ≤ ∞. Finally, using a class of concentration operators for the Dunkl-Bessel two-wavelet, we show that the eigenfunctions of the Dunkl-Bessel two-wavelet are maximally concentrated in the time-frequency domain. Leveraging this result, we derive approximation inequalities for functions that exhibit significant concentration within specific regions of the time-frequency plane.

Offset quaternion linear canonical transform: Properties, uncertainty inequalities and application

Offset quaternion linear canonical transform: Properties, uncertainty inequalities and application

Uncertainty Principles for the Weinstein Wavelet Transform

In the present paper we explore the localization properties of the Weinstein continuous wavelet transform via entropy and we introduce a version of Lp local uncertainty inequalities.

Estimation of function's supports under arithmetic constraints

The well-known inequality |supp(f)||supp(f^)|≥|G| gives a lower estimation for each support. In this paper we consider the case where there exists a slowly increasing function F such that |supp(f)|≤F(|supp(f^)|). We will show that this can be done under some arithmetic constraint.The two links that help us come from additive combinatorics and theoretical computer science. The first is the additive energy which plays a central role in additive combinatorics. The second is the influence of Boolean functions. Our main tool is the spectral analysis of Boolean functions. We prove an uncertainty inequality in which the influence of a function and its Fourier spectrum play a role.

State-dependent and state-independent uncertainty relations for skew information and standard deviation

In this work, we derive state-dependent uncertainty relations (uncertainty equalities) in which commutators of incompatible operators (not necessarily Hermitian) are explicitly present and state-independent uncertainty relations based on the Wigner-Yanase (-Dyson) skew information. We derive uncertainty equality based on standard deviation for incompatible operators with mixed states, a generalization of previous works in which only pure states were considered. We show that for pure states, the Wigner-Yanase skew information based state-independent uncertainty relations become standard deviation based state-independent uncertainty relations which turn out to be tighter uncertainty relations for some cases than the ones given in previous works, and we generalize the previous works for arbitrary operators. As the Wigner-Yanase skew information of a quantum channel can be considered as a measure of quantum coherence of a density operator with respect to that channel, we show that there exists a state-independent uncertainty relation for the coherence measures of the density operator with respect to a collection of different channels. We show that state-dependent and state-independent uncertainty relations based on a more general version of skew information called generalized skew information which includes the Wigner-Yanase (-Dyson) skew information and the Fisher information as special cases hold. In qubits, we derive tighter state-independent uncertainty inequalities for different form of generalized skew informations and standard deviations, and state-independent uncertainty equalities involving generalized skew informations and standard deviations of spin operators along three orthogonal directions. Finally, we provide a scheme to determine the Wigner-Yanase (-Dyson) skew information of an unknown observable which can be performed in experiment using the notion of weak values.

Uncertainty principles for short‐time free metaplectic transformation

This study devotes to extend Heisenberg's uncertainty inequalities in free metaplectic transformation (FMT) domains into short‐time free metaplectic transformation (STFMT) domains. We disclose an equivalence relation between spreads in time‐STFMT and time domains, as well as FMT‐STFMT and FMT domains. We use them to set up an inequality relation between the uncertainty product in time‐STFMT and FMT‐STFMT domains and that in time and FMT domains and an inequality relation between the uncertainty product in two FMT‐STFMT domains and that in two FMT domains. We deduce uncertainty inequalities of real‐valued functions and complex‐valued window functions for the STFMT and uncertainty inequalities of complex‐valued (window) functions for the orthogonal STFMT, the orthonormal STFMT, and the STFMT without the assumption of orthogonality, respectively. To formulate the attainable lower bounds, we also propose some novel uncertainty inequalities of complex‐valued (window) functions for the orthogonal FMT and the FMT without the assumption of orthogonality, respectively.

Biquaternion ambiguity function and associated uncertainty principles

A mathematical tool used to analyze signals in high-dimensional areas is the biquaternion ambiguity function. While typical ambiguity functions deal with time delay and Doppler frequency, biquaternion ambiguity functions extend the concept to rotations and scaling factors. Biquaternion ambiguity functions offer a more thorough representation of signal ambiguity in these high-dimensional spaces, which can help develop better signal processing algorithms for a range of uses. In this paper, we introduce the notion of biquaternion ambiguity function which is formed from conventional ambiguity function by invoking biquaternion algebra. We first establish the various properties, namely dilation, translation, shift, nonlinearity, boundedness, reconstruction formula, Moyal’s formula, and inversion formula. An interesting relationship between the biquaternion Fourier transform and the biquaternion ambiguity function is also established. Furthermore, we establish the various versions of uncertainty inequalities namely, Pitt’s inequality, sharp Hausdorff–Young inequality, sharp Local uncertainty principle and logarithmic uncertainty principle for biquaternion ambiguity function. Some potential applications are also presented at the end.

The logarithmic Sobolev inequality on the Heisenberg group and applications to the uncertainty inequality and heat equation

We consider the logarithmic Sobolev inequality on the Heisenberg group. One can derive the logarithmic Sobolev inequality from the Sobolev inequality, and we consider an application to the uncertainty inequality on the Heisenberg group. Moreover, one can also obtain a dissipative estimate of a solution of the heat equation on the Heisenberg group from the logarithmic Sobolev inequality.

Maximally localized Gabor orthonormal bases on locally compact Abelian groups

Maximally localized Gabor orthonormal bases on locally compact Abelian groups

Uncertainty principles for coupled fractional Wigner-Ville distribution.

The coupled fractional Wigner-Ville distribution is a more general version of the fractional Wigner-Ville distribution. Main properties including boundedness, Moyal's formula and inversion formula are studied in detail for the transformation. Additionally, the relation of the coupled fractional Wigner-Ville distribution with the two-dimensional Fourier transform is studied. We also present the relationship between the coupled fractional Wigner-Ville distribution with the two-dimensional Wigner-Ville distribution. We show how the properties and relations allow us to derive several versions of the uncertainty inequalities related to the coupled fractional Wigner-Ville distribution.

Hypoelliptic functional inequalities

In this paper we derive a variety of functional inequalities for general homogeneous invariant hypoelliptic differential operators on nilpotent Lie groups. The obtained inequalities include Hardy, Sobolev, Rellich, Hardy–Littllewood–Sobolev, Gagliardo–Nirenberg, Caffarelli–Kohn–Nirenberg and Heisenberg–Pauli–Weyl type uncertainty inequalities. Some of these estimates have been known in the case of the sub-Laplacians, however, for more general hypoelliptic operators almost all of them appear to be new as no approaches for obtaining such estimates have been available. The approach developed in this paper relies on establishing integral versions of Hardy inequalities on homogeneous Lie groups, for which we also find necessary and sufficient conditions for the weights for such inequalities to be true. Consequently, we link such integral Hardy inequalities to different hypoelliptic inequalities by using the Riesz and Bessel kernels associated to the described hypoelliptic operators.

Uncertainty inequality for weighted Fock spaces

Uncertainty inequality for weighted Fock spaces

Do world uncertainty and income inequality moderate the efficacy of environmental innovation in driving global renewable energy generation?

ABSTRACT Environmental innovation is pivotal in tackling climate change by offering diverse and practical solutions to reduce carbon footprints, promote resource conservation, and enhance ecological sustainability. As anthropogenic activities continue to accelerate global environmental degradation, innovative solutions are imperative for mitigating the impacts of climate change and fostering sustainability. However, the effectiveness of these innovations is threatened by global uncertainties and rising income inequalities. Uncertainties in geopolitical landscape and economic disparities may impede the development and widespread adoption of eco-friendly technologies and practices. Given the above background, this study empirically investigates the influences of environmental innovation, world uncertainty and income inequality on renewable energy generation for a balanced panel of 58 countries between 1990 and 2020. The estimated models also control for the effects of globalisation and economic growth. After diagnosing for necessary panel diagnostic tests including cross-sectional dependence, heterogeneity, and panel unit-root, the Westerlund cointegration test confirms a significant long-run relationship between the variables of the study. Using panel Augmented Mean Group (AMG) estimation, the results reveal that while environmental innovation continues to promote the renewable energy generation, global uncertainty and income inequality hamper it. More importantly, the moderating effects of world uncertainty and income inequality reduce the favourable impact of environmental innovation on renewable energy generation. Besides, economic growth and globalisation continue to play critical roles in speeding up the quest for renewable energy. The PCSE estimation technique confirms the significance of these baseline findings. Potential policy implications are also discussed.

The nexus of climate policy uncertainty, renewable energy and energy inequality under income disparities

This study has made a significant contribution to the literature on energy inequality by utilizing an economic-environmental framework to investigate this phenomenon across different income groups of countries worldwide. The study employed the Theil index to calculate intra-regional and inter-regional energy inequality among high-income, upper-middle-income, and lower-middle-income countries during the period 2001–2020. The study further examined the impact of key economic and environmental factors, including climate policy uncertainty (CPU), renewable energy consumption (RE), financial development (FD), income, and urbanization (UR), on energy inequality using a Two-Stage Least Square Method (2SLS) model. The findings indicated a negative impact of CPU, FD, and income on energy inequality, while UR was found to exacerbate energy inequality. RE was found to reduce intra-regional inequality in high-income and upper-middle-income countries, but it was found to worsen inter-regional inequality. These findings highlight the importance of addressing key economic and environmental factors to achieve equitable distribution of energy consumption and access to energy services.

A covariant tapestry of linear GUP, metric-affine gravity, their Poincaré algebra and entropy bound

Motivated by the potential connection between metric-affine gravity and linear generalized uncertainty principle (GUP) in the phase space, we develop a covariant form of linear GUP and an associated modified Poincaré algebra, which exhibits distinctive behavior, nearing nullity at the minimal length scale proposed by linear GUP. We use three-torus geometry to visually represent linear GUP within a covariant framework. The three-torus area provides an exact geometric representation of Bekenstein’s universal bound. We depart from Bousso’s approach, which adapts Bekenstein’s bound by substituting the Schwarzschild radius ( rs ) with the radius (R) of the smallest sphere enclosing the physical system, thereby basing the covariant entropy bound on the sphere’s area. Instead, our revised covariant entropy bound is described by the area of a three-torus, determined by both the inner radius rs and outer radius R where rs⩽R due to gravitational stability. This approach results in a more precise geometric representation of Bekenstein’s bound, notably for larger systems where Bousso’s bound is typically much larger than Bekensetin’s universal bound. Furthermore, we derive an equation that turns the standard uncertainty inequality into an equation when considering the contribution of the three-torus covariant entropy bound, suggesting a new avenue of quantum gravity.

A variation of Lp local uncertainty principles for Weinstein transform

The main crux of this paper is to introduce $L^p$ local uncertainty inequalities for the Weinstein transform, and we study $L^p$ version of the Heisenberg-Pauli-Weyl uncertainty inequalities for this transform. Then, by using the $L^p$ local uncertainty inequalities for the Weinstein transform and the tools of Donoho-Stark, we obtain uncertainty principles of concentration in the $L^p$ theory, for all $1 p \leq 2$.