Max Function Research Articles

Enhancing River Flood Prediction in Early Warning Systems Using Fuzzy Logic-Based Learning

Previous studies show that the fuzzy-based approach predicts incoming floods better than machine learning (ML). However, with numerous observation points, difficulties in manually determining fuzzy rules and membership values increase. This research proposes a novel fuzzy logic-based learning (FLBL) that embeds missing data imputations and a fuzzy rule optimization strategy to enhance ML performance while still benefiting from fuzzy theory. The simple moving average handles sensors’ missing data. The logical mapping is used for fuzzification automation and fuzzy rule generation. The join function between the Szymkiewicz–Simpson coefficient similarity and max function is applied to optimize a fuzzy rules model. The case study uses observation data from three rivers traversing three districts in Semarang City. As a result, FLBL achieves 97.87% accuracy in predicting flood, outperforming the decision tree (96%) and the neural network (73.07%). This work is significant as a part of preventive flood-related disaster plans.

Fuzzy Logic-Based Quantitative Development Model for Job Satisfaction in College Graduates

College graduates’ job satisfaction is validated via their feedback, experiences and personal developments during their career progression. Validation is accomplished to ensure the students’ job satisfaction and retain them for a prolonged time. The traditional validation models have difficulties in analyzing the individual satisfaction level. The research issue is addressed by introducing the Quantitative Assessment Method (QAM) using Fuzzy Logic to validate the job satisfaction level of different newly placed students. The QAM approach assesses the student experience and personal development across various quarters. The fuzzy optimization uses two factors differentially using partial derivatives. The partial derivatives are extracted using the min–max functions of the fuzzification process such that the derivatives are halted after the maximum factors. The proposed method optimizes the validation using individual satisfaction levels and cumulative experience shared by the students. The available derivatives identify the best-afford job satisfaction level for different progression levels. This best-fit feature is handled using multiple min and max derivatives to extract optimal outputs. This proposed method is valid for improving satisfaction levels and experience analysis.

The ultrametric backbone is the union of all minimum spanning forests

Minimum spanning trees and forests are powerful sparsification techniques that remove cycles from weighted graphs to minimize total edge weight while preserving node reachability, with applications in computer science, network science, and graph theory. Despite their utility and ubiquity, they have several limitations, including that they are only defined for undirected networks, they significantly alter dynamics on networks, and they do not generally preserve important network features such as shortest distances, shortest path distribution, and community structure. In contrast, distance backbones, which are subgraphs formed by all edges that obey a generalized triangle inequality, are well defined in directed and undirected graphs and preserve those and other important network features. The backbone of a graph is defined with respect to a specified path-length operator that aggregates weights along a path to define its length, thereby associating a cost to indirect connections. The backbone is the union of all shortest paths between each pair of nodes according to the specified operator. One such operator, the max function, computes the length of a path as the largest weight of the edges that compose it (a weakest link criterion). It is the only operator that yields an algebraic structure for computing shortest paths that is consistent with De Morgan's laws. Applying this operator yields the ultrametric backbone of a graph in that (semi-triangular) edges whose weights are larger than the length of an indirect path connecting the same nodes (i.e. those that break the generalized triangle inequality based on max as a path-length operator) are removed. We show that the ultrametric backbone is the union of minimum spanning forests in undirected graphs and provides a new generalization of minimum spanning trees to directed graphs that, unlike minimum equivalent graphs and minimum spanning arborescences, preserves all shortest paths and De Morgan's law consistency.

An offset-transformer hierarchical model for point cloud-based resistance spot welding quality classification

Resistance spot welding (RSW) is a widely used welding technology in automotive manufacturing, and weld nugget quality is closely related to the quality of the vehicle body. Offline random checks are largely relied on the quality inspection of weld nuggets, but they have low efficiency and high cost. To address this issue, this paper proposes a deep learning model for RSW weld nugget classification, named the offset-transformer hierarchical model (OFTFHC), which is based on the point cloud data of its appearance shape. OFTFHC uses a hierarchical network structure to gradually expand the receptive field. A local feature module is introduced to extract local features from the point cloud, effectively enabling the recognition of the fine structural features of the resistance spot weld point cloud. A residual ratio module, which is based on MLP_MA and uses max and average functions for feature enhancement, is designed to adapt to the complex spatial structure of the point cloud. The offset-transformer structure is used to learn global context features, thereby enhancing the global feature extraction capability. Through classification experiments on RSW weld nuggets across 5 categories with a total of 1050 samples, OFTFHC achieved an average accuracy of 80.6 %, outperforming existing models. This demonstrates the effectiveness and superiority of the method, making it highly suitable for weld nugget quality control in automotive automation production lines.

Noisy Computing of the OR and MAX Functions

Noisy Computing of the OR and MAX Functions

The Effect of Basic Fusion Techniques in Deep Ensemble Learning-Based Models For Covid-19 Diagnosis

The coronavirus disease (COVID-19), declared as a global epidemic disease (pandemic), is a new viral respiratory disease. The disease is transmitted from person to person through droplets or contact. İt is very important to detect the disease early with rapid diagnosis rates to prevent the spread of the disease. However, long-term pathological laboratory tests and low diagnosis rates in test results led researchers to apply different techniques. Radiological imaging has begun to be used to monitor COVID-19 disease as well as being useful in detecting various lung diseases. The application of deep learning techniques together with radiological imaging has a very important place in the correct detection of this disease. İn this study, the effect of basic fusion functions on classification performance on ensemble learning algorithms was investigated using the COVİD-19 X-ray dataset. Two different ensemble models were created to combine different deep learning models; Ensemble-1 (Ens-1) ve Ensemble-2 (Ens-2). The basic fusion rules of Max, Mode, Sum, Average, and Product were tested in these ensemble models. When the obtained values are examined, it is seen that the Max and Product basic fusion functions have a positive effect on the classification performance. İn multi-classification, the Max function for both Ens-1 and Ens-2 becomes prominent with an accuracy rate of 85% and 86%, respectively. The Product function achieved the highest performance with 99% in binary classification. The results show that the fusion methods can achieve better classification performance in binary classification.

Efficient Automatic Subdifferentiation for Programs with Linear Branches

Computing an element of the Clarke subdifferential of a function represented by a program is an important problem in modern non-smooth optimization. Existing algorithms either are computationally inefficient in the sense that the computational cost depends on the input dimension or can only cover simple programs such as polynomial functions with branches. In this work, we show that a generalization of the latter algorithm can efficiently compute an element of the Clarke subdifferential for programs consisting of analytic functions and linear branches, which can represent various non-smooth functions such as max, absolute values, and piecewise analytic functions with linear boundaries, as well as any program consisting of these functions such as neural networks with non-smooth activation functions. Our algorithm first finds a sequence of branches used for computing the function value at a random perturbation of the input; then, it returns an element of the Clarke subdifferential by running the backward pass of the reverse-mode automatic differentiation following those branches. The computational cost of our algorithm is at most that of the function evaluation multiplied by some constant independent of the input dimension n, if a program consists of piecewise analytic functions defined by linear branches, whose arities and maximum depths of branches are independent of n.

Strong Stationarity for Optimal Control Problems with Non-smooth Integral Equation Constraints: Application to a Continuous DNN

Motivated by the residual type neural networks (ResNet), this paper studies optimal control problems constrained by a non-smooth integral equation associated to a fractional differential equation. Such non-smooth equations, for instance, arise in the continuous representation of fractional deep neural networks (DNNs). Here the underlying non-differentiable function is the ReLU or max function. The control enters in a nonlinear and multiplicative manner and we additionally impose control constraints. Because of the presence of the non-differentiable mapping, the application of standard adjoint calculus is excluded. We derive strong stationary conditions by relying on the limited differentiability properties of the non-smooth map. While traditional approaches smoothen the non-differentiable function, no such smoothness is retained in our final strong stationarity system. Thus, this work also closes a gap which currently exists in continuous neural networks with ReLU type activation function.

Pura Vida Neutrosophic Algebra

We introduce Pura Vida Neutrosophic Algebra, an algebraic structure consisting of neutrosophic numbers equipped with two binary operations namely addition and multiplication. The addition can be calculated sometimes with the function min and other times with the max function. The multiplication operation is the usual sum between numbers. Pura Vida Neutrosophic Algebra is an extension of both Tropical Algebra (also known as Min-Plus, or Min-Algebra) and Max-Plus Algebra (also known as Max-algebra). Tropical and Max-Plus algebras are algebraic structures included in semirings and their operations can be used in matrices and vectors. Pura Vida Neutrosophic Algebra is included in Neutrosophic semirings and can be used in Neutrosophic matrices and vectors.

Nonsymmetric examples for Gaussian correlation inequalities

In this paper, we compare two variances of maxima of N standard Gaussian random variables. One is a sequence of N i.i.d. standard Gaussians, and the other one is N standard Gaussians with covariances σ1,2=ρ∈(0,1) and σi,j=0, for other i≠j. It turns out that we need to discuss the covariance of two functions with respect to multivariate Gaussian distributions. Gaussian correlation inequalities hold for many symmetric (with respect to the origin) cases. However, in our case, the max function and its derivatives are not symmetric about the origin. We have two main results in this paper. First, we prove a specific case for a convex/log-concave correlation inequality for the standard multivariate Gaussian distribution. The other result is that the variance of maxima of standard Gaussians with σ1,2=ρ∈(0,1), while σi,j=0, for other i≠j, is larger than the variance of maxima of independent standard Gaussians. This implies that the variance of maxima of N i.i.d. standard Gaussians is decreasing in N.

Energy reduction of stochastic time-constrained robot stations

This paper looks at the problem of reducing the energy use of robot movements in a robot station with stochastic execution times, while keeping the productivity of the station. The problem is formulated as a stochastic optimization problem, that constrains the makespan of the station to meet a deadline with a high probability. The energy use of the station is a function of the execution times of the robot operations, and the goal is to reduce this energy use by finding the optimal execution times and operation order. A theoretical motivation to why the stochastic variables in the problem, under some conditions, can be approximated as independent and normally distributed is presented, together with a derivation of the max function of stochastic variables. This allows the stochastic optimization problem to be approximated with a deterministic version, that can be solved with a commercial solver. The accuracy of the deterministic approximation is evaluated on multiple numerical examples, which show that the method successfully reduces the energy use, while the deadlines of the stations are met with high probabilities.

Approximating Max Function in Fully Homomorphic Encryption

This study focuses on efficiently finding the location of the maximum value for large-scale values encrypted by the CKKS (Cheon—Kim—Kim–Song) method. To find the maximum value, logM+1 comparison operations and logM rotation operations, and 2logM+3 additions and 2logM+1 multiplications are required. However, there is no known way to find a k-approximate maximum value, i.e., a value with the same most significant k-bits as the maximum value. In this study, when the value range of all data in each slot in the ciphertext is [0, 1], we propose a method for finding all slot positions of values whose most significant k-bits match the maximum value. The proposed method can find all slots from the input ciphertexts where their values have the same most significant k-bits as the maximum value by performing 2k comparison operations, (4k+2) multiplications, (6k+2klogM+3) additions, and 2klogM rotation operations. Through experiments and complexity analysis, we show that the proposed method is more efficient than the existing method of finding all locations where the k MSB is equal to the maximum value. The result of this can be applied to various privacy-preserving applications in various environments, such as IoT devices.

MP59-02 INITIAL EXPERIENCE WITH SUPERPULSED THULIUM LASER IN PROSTATE ENUCLEATION: PROSPECTIVE RANDOMIZED MULTICENTER COMPARATIVE STUDY BETWEEN HOLEP AND THULEP IN TERMS OF EFFICACY AND SAFETY

You have accessJournal of UrologyCME1 Apr 2023MP59-02 INITIAL EXPERIENCE WITH SUPERPULSED THULIUM LASER IN PROSTATE ENUCLEATION: PROSPECTIVE RANDOMIZED MULTICENTER COMPARATIVE STUDY BETWEEN HOLEP AND THULEP IN TERMS OF EFFICACY AND SAFETY Javier Romero Otero, Raquel Sopeña Sutil, Borja García Gómez, Alfredo Rodriguez Antolin, Juan Justo Quintas, Ricardo Brime Menéndez, Javier Amalio Feltes Ochoa, David Saenz Calzada, Laura Avilleira García, Elena Peña Vallejo, Cristina Calzas Montalvo, Diana María Vallejo Arzayus, and Camila Martinez Otero Javier Romero OteroJavier Romero Otero More articles by this author , Raquel Sopeña SutilRaquel Sopeña Sutil More articles by this author , Borja García GómezBorja García Gómez More articles by this author , Alfredo Rodriguez AntolinAlfredo Rodriguez Antolin More articles by this author , Juan Justo QuintasJuan Justo Quintas More articles by this author , Ricardo Brime MenéndezRicardo Brime Menéndez More articles by this author , Javier Amalio Feltes OchoaJavier Amalio Feltes Ochoa More articles by this author , David Saenz CalzadaDavid Saenz Calzada More articles by this author , Laura Avilleira GarcíaLaura Avilleira García More articles by this author , Elena Peña VallejoElena Peña Vallejo More articles by this author , Cristina Calzas MontalvoCristina Calzas Montalvo More articles by this author , Diana María Vallejo ArzayusDiana María Vallejo Arzayus More articles by this author , and Camila Martinez OteroCamila Martinez Otero More articles by this author View All Author Informationhttps://doi.org/10.1097/JU.0000000000003312.02AboutPDF ToolsAdd to favoritesDownload CitationsTrack CitationsPermissionsReprints ShareFacebookLinked InTwitterEmail Abstract INTRODUCTION AND OBJECTIVE: Holmium laser enucleation of the prostate (HoLEP) is considered as the new “gold standard” to treat surgically BPH. Other types of lasers have been employed to enucleate, but with worse results or less evidence. The new Superpulsed Thulium laser available in the market offers new features that could be a good alternative. Our objective is to compare for the first time in the literature the efficacy and safety of HoLEP 100 Wat versus Superpulsed Thulium laser (TFL) 60 Wat enucleation of the prostate (ThuLEP) in the surgical treatment of BPH. METHODS: Multicenter prospective comparative study of 100 patients randomized 1:1 to be treated with HoLEP vs. superpulsed ThuLEP (TFL), performed by 3 experts surgeons in laser enucleation. Inclusion criteria: patients with moderate/severe symptoms due to BPH. Exclusion criteria: Patients who have received other previous surgical treatment for BPH, urethral stenosis, bladder tumor, or lithiasis. A descriptive analysis of the groups was performed to compare: intraoperative, perioperative, and postoperative results at 3 and 6 months. Statistical significance was established at <0.005. RESULTS: Both arms were similar in preoperative prostate volume, Q max, PSA, bladder catheterization, IPSS, and sexual function (measured by IIEF15). Total surgical time was longer in the HoLEP (mean of 52.90 min) vs. the ThuLEP group (45.55 min), (p=0.075). However, enucleation time was longer with the TFL (26.77 min vs. 23.90 min, p=0.09). No intraoperative complications were recorded. At the perioperative features: no differences between both groups were noted: bladder catheterization: 30.6 vs 32.4 hours, hospitalization 34.2 vs 36.8 hours. Drop in hemoglobin was similar in both groups (no transfusion was required). No differences were noted in: rehospitalization, emergency room, recatheterization, or infection. The resected tissue was 28gr vs 31gr. At three-month follow-up, the improvement in Q max was 14.00 ml/s and 10.75 ml/s for HoLEP and ThuLEP, respectively, but statistical significance was not reached. No differences were found in the IIEF-15 questionnaire and only one patient in each arm reported preserving ejaculatory function. The incontinence rate was slightly higher in the ThuLEP group but was equalized at 6-month follow-up. CONCLUSIONS: In expert hands, the first cases of prostatic enucleation with superpulsed Thulium laser (TFL) are comparable to Holmium laser in terms of functional results and post-surgical complications. Source of Funding: No © 2023 by American Urological Association Education and Research, Inc.FiguresReferencesRelatedDetails Volume 209Issue Supplement 4April 2023Page: e806 Advertisement Copyright & Permissions© 2023 by American Urological Association Education and Research, Inc.MetricsAuthor Information Javier Romero Otero More articles by this author Raquel Sopeña Sutil More articles by this author Borja García Gómez More articles by this author Alfredo Rodriguez Antolin More articles by this author Juan Justo Quintas More articles by this author Ricardo Brime Menéndez More articles by this author Javier Amalio Feltes Ochoa More articles by this author David Saenz Calzada More articles by this author Laura Avilleira García More articles by this author Elena Peña Vallejo More articles by this author Cristina Calzas Montalvo More articles by this author Diana María Vallejo Arzayus More articles by this author Camila Martinez Otero More articles by this author Expand All Advertisement PDF downloadLoading ...

An Inverse Optimal Value Approach for Synchronously Optimizing Activity Durations and Worker Assignments with a Project Ideal Cost

Most companies survive the pain of cost and schedule overruns because of inaccurate project activity time settings. In order to deliver a project with a target cost and on schedule, this research proposes an inverse optimal value approach to optimize activity durations and the corresponding worker assignments synchronously to make the optimal project cost infinitely close to an ideal cost. The leader model reflects cost orientation and adjustability of activity durations, the follower model reflects the complexity of activity sequence, critical path completion time, cost pressure, skill matching, energy consumption, and other costs. Through upper-level and lower-level feedback and interaction of activity durations and worker assignments it is possible to deliver a project with an ideal cost. With considerations of the mathematical model characteristics of bi-level programming, nonlinearity, NP hard, and MAX functions, an improved genetic algorithm combining adaptive artificial fish swarms is designed. From the comparison results of random examples and an actual example, the error rate of the optimal value of the improved algorithm is acceptable. Numerical experiments show that the inverse optimal approach can deliver a project without delay and with an ideal cost. The inverse optimization method is more in line with the idea of target management, and can help managers achieve the purpose of cost control.

WILLIAM GOLDING’İN PİRAMİT ADLI ROMANINDA TOPLUMSAL TABAKALAŞMANIN GÖSTERGELERİ OLARAK SINIF, STATÜ VE GÜÇ

Sociologically, it is understood that the concepts of power and inequality refer to social stratification. According to Max Weber, the importance and function of power in the main structure of societies is undeniable and every society has a power-based structure. In this sense, in the Weberian approach, the concepts of class, status and party constitute the contents of power. While Karl Marx polarized the concept of social stratification as rich and poor, Weber explains it as the triad of class, status and party. In this article, based on the views of Marx and Weber, the concept of social stratification in William Golding's The Pyramid has been examined. As a distinguished English writer Golding lived from 1911 to 1993 and was awarded the 1983 Nobel Prize in literature. In The Pyramid, Golding invites the reader to the time before the Second World War and emphasizes the sociological concept of class. Moreover, Golding reveals the social stratification and inequality problems of the British people, as well as their perception of class and power through the experiences of the people from the lower and upper classes living in Stilbourne in the begining of the twentieth century. Thus, in the study, power inequalities and social blockages emphasized by Golding in the Pyramid are analyzed in the light of Weber’s and Marx's theories of social stratification. As a result, this literary study aims to make an interdisciplinary contribution to the field with references to sociology in terms of Weber’s and Marx’s ideas..

Selection of Aggregation Function in Fuzzy Inference System for Metabolic Syndrome

Metabolic syndrome (MetS) has long-term, very detrimental effects, including chronic kidney disease, cardiovascular disease, stroke, and diabetes mellitus. Therefore, early detection of MetS is very important. Numerous global health organizations have made some Metabolic Syndrome (MetS) diagnosis criteria, but they are still mostly in a dichotomous form. On the other hand, a continuous MetS risk score has been proven to be more sensitive and with less risk of error. This study aims to build a Fuzzy Inference System (FIS) model. MetS diagnostic criteria issued by NCEP-APT III are used as a reference for generating rules. This model uses max, probor, and additive functions to obtain membership values as a result of rules aggregation in seven steps: 1) Identification of variables; 2) Determination of fuzzy sets and their membership functions; 3) Knowledge base generation; 4) Implementation of the implication functions; 5) Fuzzy rules aggregation; 6) Defuzzification; 7) Performance testing of the model and selecting the best aggregation function. The findings show the max function as the most suitable function for the aggregation process with an accuracy, sensitivity, specificity, and precision value of 100% according to the measurement results with NCEP-ATP III. A continuous risk score between 0% and 99.99% is considered a non-high risk, whereas a score of 100% indicates a high risk. This function also has an ideal risk value distribution according to the neighborhood level of the NCEP-ATP III diagnostic criteria.

Intra-frame Based Video Compression Using Deep Convolutional Neural Network (DCNN)

In principle, a video codec is built by implementing various algorithms and their development. The next generation of codecs involves more artificial intelligence applications and their development. DCNN (Deep Convolutional Neural Network) is a multi-layer NN concept with a deep learning approach in the field of artificial intelligence development. This study has proposed a DCNN with three hidden layers for intra-frame-based video compression. DCT and fractal methods were used to compare the performance of the proposed method. The training image (obtained from the average of all down-sampled frames) is divided into several square blocks using the square block shift operation until all parts of the image are fulfilled. All pixels in each block act as input data patterns. After the training process, the trained proposed DCNN was then used to construct the feature and sub-feature image obtained through the max function operation in the feature bank and sub-feature bank. These feature and sub-feature images were then a spatial redundancy minimizer with specific manipulation techniques and simultaneously a quantizer without converting the frame's pixels to a bit-stream. The result of this process is a compressed image. Experiments on the entire dataset resulted in AAPR (Average Approximate Performance Ratio) of 147.71%, or an average of 1.5 times better than other methods. For further studies, the performance improvement of the proposed DCNN is performed by modifying its structure so that the output is direct in the form of feature and sub-feature images. Another way is to combine it with the DCT or fractal method to improve the performance of the result.

Anderson Acceleration for Nonsmooth Fixed Point Problems

We give new convergence results of Anderson acceleration for the composite $\max$ fixed point problem. We prove that Anderson(1) and EDIIS(1) are q-linear convergent with a smaller q-factor than existing q-factors. Moreover, we propose a smoothing approximation of the composite max function in the contractive fixed point problem. We show that the smoothing approximation is a contraction mapping with the same fixed point as the composite $\max$ fixed point problem. Our results rigorously confirm that the nonsmoothness does not affect the convergence rate of the Anderson acceleration method when we use the proposed smoothing approximation for the composite $\max$ fixed point problem. Numerical results for constrained minimax problems, complementarity problems, and nonsmooth differential equations are presented to show the efficiency and good performance of the proposed Anderson acceleration method with smoothing approximation.

Cardiorespiratory Fitness and Neuromuscular Function of Mechanically Ventilated ICU COVID-19 Patients.

The aim of the current study was to investigate the level of cardiorespiratory fitness and neuromuscular function of ICU survivors after COVID-19 and to examine whether these outcomes are related to ICU stay/mechanical ventilation duration. Prospective nonrandomized study. Patients hospitalized in ICU for COVID-19 infection. Sixty patients hospitalized in ICU (mean duration: 31.9 ± 18.2 d) were recruited 4-8 weeks post discharge from ICU. None. Patients visited the laboratory on two separate occasions. The first visit was dedicated to quality of life questionnaire, cardiopulmonary exercise testing, whereas measurements of the knee extensors neuromuscular function were performed in the second visit. Maximal oxygen uptake (V o2 max) was 18.3 ± 4.5 mL·min -1 ·kg -1 , representing 49% ± 12% of predicted value, and was significantly correlated with ICU stay/mechanical ventilation (MV) duration ( R = -0.337 to -0.446; p < 0.01 to 0.001), as were maximal voluntary contraction and electrically evoked peak twitch. V o2 max (either predicted or in mL· min -1 ·kg -1 ) was also significantly correlated with key indices of pulmonary function such as predicted forced vital capacity or predicted forced expiratory volume in 1 second ( R = 0.430-0.465; p ≤ 0.001) and neuromuscular function. Both cardiorespiratory fitness and neuromuscular function were correlated with self-reported physical functioning and general health status. V o2 max was on average only slightly above the 18 mL·min -1 ·kg -1 , that is, the cut-off value known to induce difficulty in performing daily tasks. Overall, although low physical capacities at admission in ICU COVID-19 patients cannot be ruled out to explain the association between V o2 max or neuromuscular function and ICU stay/MV duration, altered cardiorespiratory fitness and neuromuscular function observed in the present study may not be specific to COVID-19 disease but seem applicable to all ICU/MV patients of similar duration.

Exponential stability and hybrid [formula omitted]-gain analysis of hybrid systems via max functions

This paper investigates exponential stability and hybrid L2h-gain performance for a class of hybrid systems. A novel Lyapunov function composed of two max functions, one describing the flow-storage function and the other describing the jump-storage function, is presented to analyze these problems. Due to the effect of flow set, jump set and a timer, the novel Lyapunov function is allowed to increase along flows with a bounded increasing rate and is required to decrease to compensate the possible increasing energy, which relax the constraint on non-increasing requirement of existing common quadratic functions in literature. Then, a stability criterion of hybrid systems is obtained, which can also be used to judge the stability of hybrid systems with unstable flow behaviors. Furthermore, this paper studies the effect of two external disturbances in flows and jumps of hybrid systems, while most literature only study external disturbances existing in flows. The concept of hybrid L2h-gain is first presented to hybrid systems. By analyzing hybrid L2h-gain performance of hybrid systems, one can obtain that L2-gain performance for flows or l2-gain performance for jumps can be improved by degrading the other one. Finally, two numerical examples are provided to illustrate the effectiveness and reliability of the results.