Arithmetic Geometric Mean Research Articles

Tighter sum unitary uncertainty relation

Abstract Unitary uncertainty relations provide a theoretical framework that enhances our understanding of the principles underlying quantum mechanics and its applications in quantum information science. In this study, we derive both the unitary uncertainty relation and the weighted unitary uncertainty relation based on the sum variance for arbitrary pairs of unitary operators. By applying the arithmetic geometric mean inequality, we obtain a lower bound that is tighter than the one provided by Bagchi and Pati. [Physical Review A 94,042104] for two unitary operators. To illustrate our results, we include examples of both the unitary uncertainty relation and the weighted uncertainty relation based on sum variance.

Fractional Hermite–Hadamard, Newton–Milne, and Convexity Involving Arithmetic–Geometric Mean-Type Inequalities in Hilbert and Mixed-Norm Morrey Spaces ℓq(·)(Mp(·),v(·)) with Variable Exponents

Function spaces play a crucial role in the study and application of mathematical inequalities. They provide a structured framework within which inequalities can be formulated, analyzed, and applied. They allow for the extension of inequalities from finite-dimensional spaces to infinite-dimensional contexts, which is crucial in mathematical analysis. In this note, we develop various new bounds and refinements of different well-known inequalities involving Hilbert spaces in a tensor framework as well as mixed Moore norm spaces with variable exponents. The article begins with Newton–Milne-type inequalities for differentiable convex mappings. Our next step is to take advantage of convexity involving arithmetic–geometric means and build various new bounds by utilizing self-adjoint operators of Hilbert spaces in tensorial frameworks for different types of generalized convex mappings. To obtain all these results, we use Riemann–Liouville fractional integrals and develop several new identities for these operator inequalities. Furthermore, we present some examples and consequences for transcendental functions. Moreover, we developed the Hermite–Hadamard inequality in a new and significant way by using mixed-norm Moore spaces with variable exponent functions that have not been developed previously with any other type of function space apart from classical Lebesgue space. Mathematical inequalities supporting tensor Hilbert spaces are rarely examined in the literature, so we believe that this work opens up a whole new avenue in mathematical inequality theory.

Notes on the arithmetic–geometric mean inequality

Notes on the arithmetic–geometric mean inequality

The Complete Squares Method and the Arithmetic-Geometric Mean Inequality to Solve and Analyze A Two-Level Supply Chain Problem

Bu çalışmada, farklı ülkelerde bulunan tek bir üretici ve tek bir perakendeciden oluşan iki aşamalı tedarik zinciri problemi için bir bütünleşik stok kontrol modeli geliştirilmiştir. Bu çalışmanın amacı, bütünleşik toplam maliyeti minimum yapacak şekilde üreticinin parti sayısının ve perakendecinin parti büyüklüğünün, yani bütünleşik üretim-stok kontrol politikası parametrelerinin birlikte hesaplanmasıdır. Tek kalem ürünün siparişi, eşit büyüklükte partiler halinde teslim alınmaktadır. Perakendecinin teslim aldığı her parti iyi kaliteli ürünlerle birlikte kusurlu ürünler de içermektedir. Kusurlu ürünler, kalite kontrol işleminin ardından indirimli fiyattan satılmak üzere tek parti halinde stoktan çıkarılmaktadır; kusurlu ürün sayısı kadar iyi kaliteli fakat daha yüksek fiyatlı ürünler yerel bir tedarikçiden satın alınmaktadır. Üretici ve perakendecinin toplam stok maliyeti fonksiyonları elde edilmiş ve bütünleşik toplam stok maliyeti fonksiyonu türetilmiştir. Optimum çözüm diferensiyel hesabı kullanmadan aritmetik-geometrik ortalama eşitsizliği ve kareye tamamlama yöntemiyle elde edilmiştir. Sayısal bir örnek yardımıyla teorik sonuçlar elde edilmiş ve duyarlılık analizleri verilmiştir.

Probabilistic Interval Ordering Prioritized Averaging Operator and Its Application in Bank Investment Decision Making

Probabilistic interval ordering, as a helpful tool for expressing positive and negative information, can effectively address multi-attribute decision-making (MADM) problems in reality. However, when dealing with a significant number of decision-makers and decision attributes, the priority relationships between different attributes and their relative importance are often neglected, resulting in deviations in decision outcomes. Therefore, this paper combines probability interval ordering, the prioritized aggregation (PA) operator, and the Gauss–Legendre algorithm to address the MADM problem with prioritized attributes. First, considering the significance of interval priority ordering and the distribution characteristics of attribute priority, the paper introduces probability interval ordering elements that incorporate attribute priority, and it proposes the probabilistic interval ordering prioritized averaging (PIOPA) operator. Then, the probabilistic interval ordering Gauss–Legendre prioritized averaging operator (PIOGPA) is defined based on the Gauss–Legendre algorithm, and various excellent properties of this operator are explored. This operator considers the priority relationships between attributes and their importance level, making it more capable of handling uncertainty. Finally, a new MADM method is constructed based on the PIOGPA operator using probability intervals and employs the arithmetic–geometric mean (AGM) algorithm to compute the weight of each attribute. The feasibility and soundness of the proposed method are confirmed through a numerical example and comparative analysis. The MADM method introduced in this paper assigns higher weights to higher-priority attributes to establish fixed attribute weights, and it reduces the impact of other attributes on decision-making results. It also utilizes the Gauss AGM algorithm to streamline the computational complexity and enhance the decision-making effectiveness.

Mean Inequalities for the Numerical Radius

Extending certain scalar and norm inequalities, we present new inequalities for the numerical radius, which generalize and refine some known results. Applications of the obtained inequalities include a new original proof of the matrix arithmetic-geometric mean inequality and certain extensions of some well-established results from the literature for products of matrices.

A Deep Reinforcement Learning Scheme for Sum Rate and Fairness Maximization Among D2D Pairs Underlaying Cellular Network With NOMA

Device-to-device (D2D) communication is an emerging technology in 5G and upcoming 6G networks due to its properties to enhanced spectral efficiency (SE), energy-efficiency (EE), and sum rate. Despite these advantages, co-channel and cross-channel interference, and ultra-massive connectivity are major issues which can deteriorate performance of any implemented solution in this environment. To address these issues, in this paper, we integrated the power domain non-orthogonal multiple access techniques (PD-NOMA) on the base station (BS). NOMA serves more than one user using the same resource block (RB) and reduces the effect of interference at CUs due to the presence of successive interference cancellation (SIC). The problem is formulated as a mixed-integer non-linear programming (MINLP) with associated resources and power constraints of the BS and DDPs with an aim to maximize the sum rate and fairness among the NOMA-enabled CUs and D2D pairs (DDPs). We firstly used the centralized deep deterministic policy gradient (DDPG) and arithmetic-geometric mean approximation (AGMA) technique to reduce cross-channel interference (CR-CI) and control the power. Then, to provide fairness to all the users, we transformed the proposed solution into distributed deep deterministic policy gradient (D3PG). Also, the successive convex approximation technique is then integrated into the D3PG to mitigate the effect of co-channel (CO-CI) interference among DDPs. The experimental results show that the proposed scheme has superior performance with respect to sum rate and fairness. Also, the results reveal that the proposed scheme has 21.05%, 34.21%, and 49.8% higher sum rate in comparison to DDPG, Deep dueling, and DQN scheduling.

Weighted Sum-Rate Maximization in Multi-IRS-Aided Multi-Cell mmWave Communication Systems for Suppressing ICI

Intelligent reflecting surface (IRS) has been envisioned as a disruptive technology with the capability of reconfiguring the wireless transmission environment. Hence, the IRS is used to assist the communication system, however, the impacts of inter-cell interference (ICI) on cell-edge users are severe in multi-cell communications but have been ignored in most literature about IRS-assisted communication systems. In this paper, we consider the downlink transmission of cell-edge users in multi-cell millimeter wave (mmWave) communication systems with the aid of multiple IRSs deployed at the cell boundary. To suppress the ICI, we maximize the weighted sum-rate (WSR) by jointly optimizing the IRS-user association matrix, the transmit beamforming of mmWave base stations (MBSs), and the phase shifts of IRSs, subject to the power limits of MBSs, the reflection constraints of IRSs, and the association constraints. Then, a joint IRS-user association and beamforming algorithm with alternating optimization (AO) framework is developed to tackle the mixed-integer non-convex problem. Specifically, a simplified objective function with respect to the association variables is derived and the non-convex binary constraints are handled by the relaxation operation and the first-order Taylor expansion, while the arithmetic-geometric mean inequality (AGMI)-based rank-one relaxation algorithm and the ring-based penalty convex-concave procedure (CCP) method are proposed for obtaining the transmit beamforming matrix of MBSs and the phase shifts of IRSs, respectively. Simulation results demonstrate the superiority of the proposed algorithm over the benchmark schemes. The useful insights into the optimization of IRS-user association and design of the number of elements of IRSs are also presented for future networks.

Norm inequalities for the spectral spread of Hermitian operators

AbstractIn this work, we introduce a new measure for the dispersion of the spectral scale of a Hermitian (self‐adjoint) operator acting on a separable infinite‐dimensional Hilbert space that we call spectral spread. Then, we obtain some submajorization inequalities involving the spectral spread of self‐adjoint operators, that are related to Tao's inequalities for anti‐diagonal blocks of positive operators, Kittaneh's commutator inequalities for positive operators and also related to the arithmetic–geometric mean inequality. In turn, these submajorization relations imply inequalities for unitarily invariant norms (in the compact case).

Two Methods Used for Radiative Micropolar Nanofluid Passed Through a Porous Channel

In this study, radiative micropolar nanofluid passed through a porous channel was analyzed. Because of the considerable amount of difference, a Finite element technique with the approval of the AGM method was used. Different parameter and dimensionless numbers like Reynolds number Re, Hartman number M, the Thermophoretic number N_t with a variety of velocity, temperature, and concentration profile are shown graphically. There was no change in velocity or microrotation profile while there was a change in Reynolds number. Although the concentration profile decreased with increasing Reynolds number

On convex functions and related inequalities

The main result of this paper is to give refinement and reverse the celebrated Jensen inequality. We directly apply our results to establish several weighted arithmetic-geometric mean inequality. We also present a stronger estimate for the first inequality in the Hermite-Hadamard inequality.

Investigation for brownian motion of nonlinear thermal bioconvective SPF in a nanofluid utilizing AGM method

An analysis of the bioconvective slump point flow of nanofluid is addressed. The studied nanofluid, which contains gyrotactic microorganisms, was investigated in a porous medium on a nonlinear tensile plate in which it was placed. The scaling group transformation approach converted the analyzed PDEs to ODEs. ODEs obtained are solved using the numerical method (rkf-45) and AGM, and the results are compared with DTM and RKF methods. The accuracy of the obtained answers is remarkable compared to the mentioned methods. An increase in the density of motile microorganism distribution caused by the increment in values of bioconvection Peclet number is desired. In addition, a rapid increase in heat transfer rate and mass transfer rate happens when there is an increase in the thermophoresis parameter, heat source parameter, chemical reaction parameter, and Brownian motion parameter in a sequence. In this study, we investigated the SPF of bioconvective nanofluids containing GM. It is observed that the flow velocity increases as Da−1 decreases. As Rb increases, bioconvection increases. These studies may be used for biotechnological applications, such as the design of bioconjugates or the increase of mass transfer in microfluidics.

Optimal Designs of SVC-Based Content Placement and Delivery in Wireless Caching Networks.

To allieviate the heavy traffic burden over backhaul links and improve the user's quality of service (QoS), edge caching plays an important role in wireless networks. This paper investigated the optimal designs of content placement and transmission in wireless caching networks. The contents to be cached and requested were encoded into individual layers by scalable video coding (SVC), and different sets of layers can provide different viewing qualities to end users. The demanded contents were provided by helpers caching the requested layers, or by the macro-cell base station (MBS) otherwise. In the content placement phase, this work formulated and solved the delay minimization problem. In the content transmission phase, the sum rate optimization problem was established. To effectively solve the nonconvex problem, the methods of semi-definite relaxation (SDR), successive convex approximation (SCA), and arithmetic-geometric mean (AGM) inequality were adopted, after which the original problem was transformed into the convex form. The numerical results show that the transmission delay is reduced by caching contents at helpers. Moreover, the fast convergence of the proposed algorithm for solving the sum rate maximization problem is presented, and the sum rate gain of edge caching is also revealed, as compared to the benchmark scheme without content caching.

RIS-Assisted Cooperative NOMA With SWIPT

This letter proposes a two-stage reconfigurable intelligent surface (RIS)-assisted transmission scheme for cooperative non-orthogonal multiple access networks with simultaneous wireless information and power transfer. We focus on improving the achievable rate of the strong user with guaranteed weak user’s quality of service by jointly optimizing power splitting factors, beamforming coefficients, and RIS reflection coefficients in two transmission stages. To tackle this challenging problem, we first use the alternating optimization framework to transform it into three subproblems, and then apply the penalty-based arithmetic-geometric mean approximation algorithm and the successive convex approximation-based method to solve them. Numerical results verify the superiority of the proposed algorithm over the baseline schemes.

Diagnostic Tree Model을 활용한 수학 서술형 문항 인지진단 평가 적용 연구

This study aimed to analyze mathematics assessments with constructed-response items using diagnostic tree model (DTM), which accommodates polytomous responses with multiple strategies and obtain diagnostic information about examinees’ attribute mastery. A cognitive diagnostic assessment evaluating the “absolute inequality” domain with constructed-response items was designed based on the DTM and administered to students attending a science high school for gifted children. The findings from cognitive diagnosis of examinees indicate that students face more challenges in mastering the Cauchy-Schwarz inequality compared with the arithmetic-geometric mean inequality, with the greatest difficulty being in the application of the substitution method. The analysis revealed that for certain test items, the estimated guess and slip parameters vary depending on the nodes of the path selected by the chosen strategy, thus indicating that the difficulty to solve the problem may depend on the strategy that the students choose. Additionally, this study suggests several methods to visualize the item response of each examinee using a tree structure and estimated parameters for the DTM and to utilize it for individualized diagnostic feedback.

Norm inequalities for hyperaccretive quasinormal operators, with extensions of the arithmetic-geometric means inequality

Let Φ , Ψ be symmetrically norming (s.n.) functions, let C Ψ ( H ) be the ideal of compact Hilbert space operators, associated with the s.n. function Ψ, p ⩾ 2 and let A , B , X ∈ B ( H ) be such that A, B are accretive and AX + XB ∈ C Ψ ( H ) . Then A ∗ + A X B + B ∗ ∈ C Ψ ( H ) as well, and | | A ∗ + A X B + B ∗ | | Ψ ⩽ | | AX + XB | | Ψ , under any of the following conditions: Ψ := Φ ( p ) ∗ and A (resp. B ∗ ) is quasinormal operator with its adjoint operator being 2-hyperaccretive and having the injective real part; if both A and B ∗ are quasinormal operators with its adjoint operators being 2-hyperaccretive operators and having injective real parts. Also, for M , N ∈ N which are such that K := ( M + N ) / 2 ∈ N , for I , J ∈ N ∪ { 0 } , for s.n. functions Φ , Ψ , for bounded Hilbert space operators A , B , X , such that A is ( M + I ) -hyperaccretive and B ∗ is ( N + J ) -hyperaccretive, satisfying Δ A , B K ( X ) = def ∑ k = 0 K ( K k ) A K − k X B k ∈ C Ψ ( H ) , then there exists V A , B M + I , N + J , − X = def lim w lim ⁡ T → + ∞ ⁡ ( Δ A ∗ , A M + I ( I ) ) 1 / 2 e − T A X e − T B ( Δ B , B ∗ N + J ( I ) ) 1 / 2 and | | ( ∑ m = 0 M + I ( M + I m ) A ∗ m A M + I − m ) 1 2 X ( ∑ n = 0 N + J ( N + J n ) B N + J − n B ∗ n ) 1 2 − V A , B M + I , N + J , − X | | Ψ ⩽ ( M − 1 ) ! ( N − 1 ) ! ( M + N 2 − 1 ) ! | | ( ∑ i = 0 I ( I i ) A ∗ i A I − i ) 1 2 ∑ k = 0 K ( K k ) A K − k X B k ( ∑ j = 0 J ( J j ) B J − j B ∗ j ) 1 2 | | Ψ , hold under any of the following conditions: if p ⩾ 2 , Ψ := Φ ( p ) ∗ and A or B ∗ is normal (in which case V A , B M + I , N + J , − X = 0 ), if both A and B ∗ are normal (in which case V A , B M + I , N + J , − X = 0 ), if | | ⋅ | | Ψ := | | ⋅ | | 1 and AX + XB ∈ C 1 ( H ) .

An Inequality Indicator for High-Resistance Connection Fault Diagnosis in Marine Current Turbine

Marine current energy is an abundant renewable energy resource. Marine current turbines (MCTs) can convert the kinetic energy of marine currents into electrical energy. However, the variations in the marine currents are violent and complex. These characteristics will be reflected in the variation in the operating condition of MCTs, thus interfering with normal diagnosis for the high-resistance connection (HRC). The HRC can be caused by damaged connections between device components that are easily made due to the harsh marine environment. To diagnose HRC in MCT, an inequality indicator is proposed to quantify the current imbalance caused by HRC. The inequality indicator is defined based on the arithmetic–geometric mean inequality and can identify slight current imbalances in the early stages of HRC. The inequality indicator is robust to the variable operating conditions of the permanent magnet synchronous machine (PMSM) in the MCT. Experimental results show that the inequality indicator can be used to effectively diagnose HRC in the MCT with 100% accuracy. This research will help maintain the health condition of the MCTs and provide some ideas for diagnosis in MCTs. Moreover, the inequality indicator may provide a different approach to the analysis for other faults that can lead to a current imbalance.

On the irregularity of graphs based on the arithmetic-geometric mean inequality

On the irregularity of graphs based on the arithmetic-geometric mean inequality

Operator inequalities related to the arithmetic–geometric mean inequality and characterizations

Operator inequalities related to the arithmetic–geometric mean inequality and characterizations

MULTIPLE-TERM REFINEMENTS OF ALZER–FONSECA–KOVAČEC INEQUALITIES

By the weighted arithmetic-geometric mean inequality we present multiple-term refinements of one of the most important extensions to Young’s inequalities due to Alzer, Fonseca and Kovačec (Linear Multilinear Algebra 63(3) (2015), 622–635). As applications, we give some related inequalities for operators and matrices.