Arithmetic-geometric Mean Inequality Research Articles

On Rotfel'd type inequalities for sectorial matrices

Using a matrix decomposition for sectorial matrices and an AM-GM inequality for singular values, we present a new Rotfel'd type inequality for sectorial matrices, which generalizes and partially improves some results by Zhang. We also give alternative simpler proofs of a main result of a recent paper by Yang, Lu and Chen as well as that by Fu and Liu.

Geometric-arithmetic mean inequality for q-rung orthopair fuzzy Hamacher aggregations

The geometric-arithmetic mean inequality is undoubtedly the most important one in the area of information aggregation. Recently, some q-rung orthopair fuzzy aggregation operators were proposed based on the Hamacher operations. In this paper, we give a detailed theoretical and practical analysis of the developed Hamacher arithmetic and geometric operators for q-rung orthopair fuzzy values. First, we investigate the monotonicity of these Hamacher aggregation operators on q-rung orthopair fuzzy values with respect to the parameter within Hamacher operations. Then, we discuss the limiting cases of these q-rung orthopair fuzzy Hamacher aggregation operators as the parameter therein approaches to zero or infinity and give a new characterization of the boundedness of these aggregation operators. Subsequently, we establish the geometric-arithmetic mean inequality for q-rung orthopair fuzzy information based on Hamacher operations. Finally, we present a decision making method by use of these aggregation operators and apply it to the problem of enterprise resource planning system selection.

Notes on the arithmetic–geometric mean inequality

Notes on the arithmetic–geometric mean inequality

OPTIMAL POLICY OF THE INTERVAL EPQ MODEL USING C-L INTERVAL INEQUALITY

The objective of this work is to study the optimal policy of the classical economic production quantity (EPQ) model under interval uncertainty using interval inequality. To serve this purpose existing arithmetic mean-geometric mean (AM-GM) inequality is extended for interval numbers using c-L interval order relation. Then, using the said AM-GM interval inequality, the optimal policy of the classical EPQ model in the interval environment is developed. Thereafter, the optimality policy of the classical EPQ model in a crisp environment is obtained as a special case of that of the interval environment. Finally, all the optimality results are illustrated with the help of some numerical examples.

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.

Arithmetic Mean-Geometric Mean Inequality for Convex Fuzzy Sets

Convex analysis is a discipline of mathematics dedicated to the explication of the properties of convex sets and convex functions. Convex functions are extremely useful in proving many famous inequalities in mathematics. It is widely known that inequalities defined by convex functions originated with the works done by Holder and Jensen among others and applied to modeling a variety of problems both in hard and soft sciences. It is said that the arsenal of an analyst is heavily stocked with inequalities. Studies related to convexity have kept occupying a central position in almost all areas of mathematics, especially in functional analysis and operations research. Following the emergence of fuzzy set theory, fuzzy convexity alongside a number of related concepts has been explicated. However, not much has been done regarding fuzzy inequalities defined by fuzzy convexity. In this paper, a novel attempt to study arithmetic mean-geometric mean is proposed. In this short note, we provide two proofs of the arithmetic mean-geometric mean inequality for convex fuzzy sets. Mathematics Subject Classification 2020: 03E72, and 94D05.

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.

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).

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.

Fairness and Welfare Quantification for Regret in Multi-Armed Bandits

We extend the notion of regret with a welfarist perspective. Focussing on the classic multi-armed bandit (MAB) framework, the current work quantifies the performance of bandit algorithms by applying a fundamental welfare function, namely the Nash social welfare (NSW) function. This corresponds to equating algorithm's performance to the geometric mean of its expected rewards and leads us to the study of Nash regret, defined as the difference between the - a priori unknown - optimal mean (among the arms) and the algorithm's performance. Since NSW is known to satisfy fairness axioms, our approach complements the utilitarian considerations of average (cumulative) regret, wherein the algorithm is evaluated via the arithmetic mean of its expected rewards. This work develops an algorithm that, given the horizon of play T, achieves a Nash regret of O ( sqrt{(k log T)/T} ), here k denotes the number of arms in the MAB instance. Since, for any algorithm, the Nash regret is at least as much as its average regret (the AM-GM inequality), the known lower bound on average regret holds for Nash regret as well. Therefore, our Nash regret guarantee is essentially tight. In addition, we develop an anytime algorithm with a Nash regret guarantee of O( sqrt{(k log T)/T} log T ).

AM–GM Algorithm for Evaluating, Analyzing, and Correcting the Spatial Scaling Bias of the Leaf Area Index

The leaf area index (LAI) is a crucial variable in climate, ecological, and land surface modeling. However, the estimation of the LAI from coarse-resolution remote sensing data can be affected by the spatial scaling bias, which arises from the nonlinearity of retrieval models and the heterogeneity of the land surface. This study provides an algorithm named Arithmetic Mean and Geometric Mean (AM–GM) to correct the spatial scaling bias. It is established based on negative logarithmic functions and avoids second-order stationarity. In this algorithm, relationships are derived between the scaling bias of LAI and the arithmetic and geometric means of directional gap probability for two commonly used remote sensing models, the Beer–Lambert law and a semi-empirical transfer function, respectively. According to the AM–GM algorithm, the expression representing the model nonlinearity is derived and utilized for the analysis of LAI scaling bias. Furthermore, the AM–GM algorithm is simplified by a linear relationship, which is constructed between two quantities related to the directional gap probability between two specific resolutions. Two scenes simulated by the LargE-Scale remote sensing data and image Simulation framework (LESS) model and three sites are used to evaluate the proposed algorithm and analyze the scaling bias of LAI. The validation results show that the AM–GM algorithm provides accurate correction of LAI scaling bias. The analyses based on the AM–GM algorithm demonstrate that the scaling bias of LAI increases with the increase in the LAI value, with stronger surface heterogeneity and coarser spatial resolution. The validation results of the simplified AM–GM algorithm demonstrate that at the Sud-Ouest site, the absolute value of the bias for the estimated LAI decreases from 0.10, 0.22, 0.29, and 0.31 to 0.04, 0.01, 0.04, and 0.05 at 200 m, 500 m, 1000 m, and 1500 m resolutions, respectively. In conclusion, the proposed algorithm is effective in the analysis and correction of the scaling bias for coarse-resolution LAI.

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.

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.

Energy efficient transceiver design for SWIPT systems with non-orthogonal multiple access and power splitting

This paper studies the energy efficient (EE) optimization for the simultaneous wireless information and power transfer (SWIPT) systems combined with non-orthogonal multiple access for multi-user support. Power splitting at receiver is adopted with nonlinear energy harvesting (EH) model. Our goal is to pursue high EE for information transmission by jointly design the transmit beamforming and power splitting coefficients while satisfying quality of service requirements, such as maximum transmit power budget, minimum data rate, and minimum harvested energy per terminal. The coupling of variables make it the non-convex even with linear EH models, which is difficult to solve. Firstly, after approximating the nonlinear EH model in the optimization objective by linear one, the nonlinear fractional function is transformed into a linear subtractive form with two groups of coupled variables. After transformations, the original problem is approximately solved in an alternate way. Through the iterative procedure, semi-definite relaxation and arithmetic–geometric mean inequality are adopted to deal with the non-convex constraints. Simulation results reveal that our proposed method has a faster convergence speed and a higher energy efficiency than the benchmarks.