In physics,our expectations for system behavior are often guided by intuitive arithmetic.For systems composed of identical units,we anticipate synergy of the contributions from these units,where 1+1=2.Conversely,for s...In physics,our expectations for system behavior are often guided by intuitive arithmetic.For systems composed of identical units,we anticipate synergy of the contributions from these units,where 1+1=2.Conversely,for systems built from opposing units,we expect cancellation of their contributions,where 1-1=0.This intuitive arithmetic has long underpinned our understanding of physical properties of materials,from electronic transport to optical responses.However,scientific breakthroughs often occur when nature reveals ways to circumvent these seemingly fundamental rules,opening new possibilities that challenge our deepest assumptions about material behavior.展开更多
In this paper, we analyse a new chaos-based cryptosystem with an embedded adaptive arithmetic coder, which was proposed by Li Heng-Jian and Zhang J S （Li H J and Zhang J S 2010 Chin. Phys. B 19 050508）. Although thi...In this paper, we analyse a new chaos-based cryptosystem with an embedded adaptive arithmetic coder, which was proposed by Li Heng-Jian and Zhang J S （Li H J and Zhang J S 2010 Chin. Phys. B 19 050508）. Although this new method has a better compression performance than its original version, it is found that there are some problems with its security and decryption processes. In this paper, it is shown how to obtain a great deal of plain text from the cipher text without prior knowledge of the secret key. After discussing the security and decryption problems of the Li Heng-Jian et al. algorithm, we propose an improved chaos-based cryptosystem with an embedded adaptive arithmetic coder that is more secure.展开更多
目的建立不同产地喜树果中9个成分含量同步检测方法,筛选影响其质量的差异标志物,并对其进行质量评价。方法对7省18个批次喜树果样品进行回流提取,提取物采用高效液相色谱法检测;采用正交偏最小二乘判别分析(Orthogonal partial least s...目的建立不同产地喜树果中9个成分含量同步检测方法,筛选影响其质量的差异标志物,并对其进行质量评价。方法对7省18个批次喜树果样品进行回流提取,提取物采用高效液相色谱法检测;采用正交偏最小二乘判别分析(Orthogonal partial least squares discriminant analysis,OPLS-DA)和加权逼近理想解排序(Technique for order preference by similarity to an ideal solution,TOPSIS)法建立喜树果质量优劣评价模型,对其质量差异性进行综合评价。结果3,4′-O-二甲基鞣花酸、丁香酸、10-羟基喜树碱、喜树碱、10-甲氧基喜树碱、三叶豆苷、短小蛇根草苷、金丝桃苷和喜果苷分别在0.51-12.75、0.23-5.75、3.21-80.25、4.45-111.25、1.88-47.00、0.41-10.25、2.05-51.25、0.34-8.50和7.95-198.75μg·mL^(-1)范围内线性关系良好(r>0.999),平均加样回收率96.95%-100.06%(RSD<2.0%);18批样品聚为3类;喜果苷、10-羟基喜树碱、喜树碱、10-甲氧基喜树碱和短小蛇根草苷可能是影响喜树果产品质量主要潜在标志物;加权TOPSIS法分析结果显示18批喜树果质量评价贴近度(Jb)在0.1090-0.7385,其中S14最大(0.7385)。结论建立了同时测定喜树果中9种成分含量的方法,操作简便、结果准确;采用OPLS-DA及加权TOPSIS法进行客观全面评价,可用于喜树果质量差异性评价。展开更多
An approximately optimal adaptive arithmetic coding (AC) system using a forbidden symbol (FS) over noisy channels was proposed which allows one to jointly and adaptively design the source decoding and channel correcti...An approximately optimal adaptive arithmetic coding (AC) system using a forbidden symbol (FS) over noisy channels was proposed which allows one to jointly and adaptively design the source decoding and channel correcting in a single process, with superior performance compared with traditional separated techniques. The concept of adaptiveness is applied not only to the source model but also to the amount of coding redundancy. In addition, an improved branch metric computing algorithm and a faster sequential searching algorithm compared with the system proposed by Grangetto were proposed. The proposed system is tested in the case of image transmission over the AWGN channel, and compared with traditional separated system in terms of packet error rate and complexity. Both hard and soft decoding were taken into account.展开更多
From such actual conditions as the effects of characteristics of miltilayer petroleum geology and permeation fluid mechanics, a new numerical model is put forward and coupling splitting-up implicit interactive scheme ...From such actual conditions as the effects of characteristics of miltilayer petroleum geology and permeation fluid mechanics, a new numerical model is put forward and coupling splitting-up implicit interactive scheme is formed. For the actual situation of Dongying hollow (four-layer) and Tanhai region (three-layer) of Shengli Petroleum Field, the numerical simulation test results and the actual conditions are coincident.展开更多
In this study an adaptive arithmetic coder is embedded in the Baptista-type chaotic cryptosystem for implementing secure data compression. To build the multiple lookup tables of secure data compression, the phase spac...In this study an adaptive arithmetic coder is embedded in the Baptista-type chaotic cryptosystem for implementing secure data compression. To build the multiple lookup tables of secure data compression, the phase space of chaos map with a uniform distribution in the search mode is divided non-uniformly according to the dynamic probability estimation of plaintext symbols. As a result, more probable symbols are selected according to the local statistical characters of plaintext and the required number of iterations is small since the more probable symbols have a higher chance to be visited by the chaotic search trajectory. By exploiting non-uniformity in the probabilities under which a number of iteration to be coded takes on its possible values, the compression capability is achieved by adaptive arithmetic code. Therefore, the system offers both compression and security. Compared with original arithmetic coding, simulation results on Calgary Corpus files show that the proposed scheme suffers from a reduction in compression performance less than 12% and is not susceptible to previously carried out attacks on arithmetic coding algorithms.展开更多
This paper proposes an efficient lossless image compression scheme for still images based on an adaptive arithmetic coding compression algorithm. The algorithm increases the image coding compression rate and ensures t...This paper proposes an efficient lossless image compression scheme for still images based on an adaptive arithmetic coding compression algorithm. The algorithm increases the image coding compression rate and ensures the quality of the decoded image combined with the adaptive probability model and predictive coding. The use of adaptive models for each encoded image block dynamically estimates the probability of the relevant image block. The decoded image block can accurately recover the encoded image according to the code book information. We adopt an adaptive arithmetic coding algorithm for image compression that greatly improves the image compression rate. The results show that it is an effective compression technology.展开更多
Through a series of studies on arithmetic coding and arithmetic encryption, a novel image joint compression- encryption algorithm based on adaptive arithmetic coding is proposed. The contexts produced in the process o...Through a series of studies on arithmetic coding and arithmetic encryption, a novel image joint compression- encryption algorithm based on adaptive arithmetic coding is proposed. The contexts produced in the process of image compression are modified by keys in order to achieve image joint compression encryption. Combined with the bit-plane coding technique, the discrete wavelet transform coefficients in different resolutions can be encrypted respectively with different keys, so that the resolution selective encryption is realized to meet different application needs. Zero-tree coding is improved, and adaptive arithmetic coding is introduced. Then, the proposed joint compression-encryption algorithm is simulated. The simulation results show that as long as the parameters are selected appropriately, the compression efficiency of proposed image joint compression-encryption algorithm is basically identical to that of the original image compression algorithm, and the security of the proposed algorithm is better than the joint encryption algorithm based on interval splitting.展开更多
This research work considers the following inequalities: <i>λ</i><em>A</em>(<i>a</i>,<i>b</i>) + (1-<i>λ</i>)<em>C</em>(<i>a</i>,&l...This research work considers the following inequalities: <i>λ</i><em>A</em>(<i>a</i>,<i>b</i>) + (1-<i>λ</i>)<em>C</em>(<i>a</i>,<i>b</i>) ≤ <span style="text-decoration:overline;">C</span>(<i>a</i>,<i>b</i>) ≤ <i>μ</i><em>A</em>(<i>a</i>,<i>b</i>) + (1-<i>μ</i>)<em>C</em>(<i>a</i>,<i>b</i>) and <em>C</em>[<i>λ</i><em>a</em> + (1-<i>λ</i>)<em>b</em>, <i>λ</i><em>b</em> + (1-<i>λ</i>)<em>a</em>] ≤ <span style="text-decoration:overline;">C</span>(<i>a</i>,<i>b</i>) ≤ <em>C</em>[<i>μ</i><em>a</em> + (1-<i>μ</i>)<em>b</em>, <i>μ</i><em>b</em> + (1-<i>μ</i>)<em>a</em>] with <img src="Edit_ce892b1d-c056-44ea-a929-31dbcd1b0e91.bmp" alt="" /> . The researchers attempt to find an answer as to what are the best possible parameters <i>λ</i>, <i>μ</i> that (1.1) and (1.2) can be hold? The main tool is the optimization of some suitable functions that we seek to find out. By searching the best possible parameters such that (1.1) and (1.2) can be held. Firstly, we insert <em>f</em>(<i>t</i>) = <i>λ</i><em>A</em>(<i>a</i>,<i>b</i>) + (1-<i>λ</i>)<em>C</em>(<i>a</i>,<i>b</i>) - <span style="text-decoration:overline;">C</span>(<i>a</i>,<i>b</i>) without the loss of generality. We assume that <i>a</i>><i>b</i> and let <img src="Edit_efa43881-9a60-44f8-a86f-d4a1057f4378.bmp" alt="" /> to determine the condition for <i>λ</i> and <i>μ</i> to become f (<i>t</i>) ≤ 0. Secondly, we insert g(<i>t</i>) = <i>μ</i><em>A</em>(<i>a</i>,<i>b</i>) + (1-<i>μ</i>)<em>C</em>(<i>a</i>,<i>b</i>) - <span style="text-decoration:overline;">C</span>(<i>a</i>,<i>b</i>) without the loss of generality. We assume that <i>a</i>><i>b</i> and let <img src="Edit_750dddbb-1d71-45d3-be29-6da5c88ba85d.bmp" alt="" /> to determine the condition for <i>λ</i> and <i>μ</i> to become <em>g</em>(<i>t</i>) ≥ 0.展开更多
In this article, we report the derivation of high accuracy finite difference method based on arithmetic average discretization for the solution of Un=F(x,u,u′)+∫K(x,s)ds , 0 x s < 1 subject to natural boundary co...In this article, we report the derivation of high accuracy finite difference method based on arithmetic average discretization for the solution of Un=F(x,u,u′)+∫K(x,s)ds , 0 x s < 1 subject to natural boundary conditions on a non-uniform mesh. The proposed variable mesh approximation is directly applicable to the integro-differential equation with singular coefficients. We need not require any special discretization to obtain the solution near the singular point. The convergence analysis of a difference scheme for the diffusion convection equation is briefly discussed. The presented variable mesh strategy is applicable when the internal grid points of the solution space are both even and odd in number as compared to the method discussed by authors in their previous work in which the internal grid points are strictly odd in number. The advantage of using this new variable mesh strategy is highlighted computationally.展开更多
Lung cancer is among the most frequent cancers in the world,with over one million deaths per year.Classification is required for lung cancer diagnosis and therapy to be effective,accurate,and reliable.Gene expression ...Lung cancer is among the most frequent cancers in the world,with over one million deaths per year.Classification is required for lung cancer diagnosis and therapy to be effective,accurate,and reliable.Gene expression microarrays have made it possible to find genetic biomarkers for cancer diagnosis and prediction in a high-throughput manner.Machine Learning(ML)has been widely used to diagnose and classify lung cancer where the performance of ML methods is evaluated to identify the appropriate technique.Identifying and selecting the gene expression patterns can help in lung cancer diagnoses and classification.Normally,microarrays include several genes and may cause confusion or false prediction.Therefore,the Arithmetic Optimization Algorithm(AOA)is used to identify the optimal gene subset to reduce the number of selected genes.Which can allow the classifiers to yield the best performance for lung cancer classification.In addition,we proposed a modified version of AOA which can work effectively on the high dimensional dataset.In the modified AOA,the features are ranked by their weights and are used to initialize the AOA population.The exploitation process of AOA is then enhanced by developing a local search algorithm based on two neighborhood strategies.Finally,the efficiency of the proposed methods was evaluated on gene expression datasets related to Lung cancer using stratified 4-fold cross-validation.The method’s efficacy in selecting the optimal gene subset is underscored by its ability to maintain feature proportions between 10%to 25%.Moreover,the approach significantly enhances lung cancer prediction accuracy.For instance,Lung_Harvard1 achieved an accuracy of 97.5%,Lung_Harvard2 and Lung_Michigan datasets both achieved 100%,Lung_Adenocarcinoma obtained an accuracy of 88.2%,and Lung_Ontario achieved an accuracy of 87.5%.In conclusion,the results indicate the potential promise of the proposed modified AOA approach in classifying microarray cancer data.展开更多
In this article, we define the arithmetic operations of generalized trapezoidal picture fuzzy numbers by vertex method which is assembled on a combination of the (α, γ, β)-cut concept and standard interval analysis...In this article, we define the arithmetic operations of generalized trapezoidal picture fuzzy numbers by vertex method which is assembled on a combination of the (α, γ, β)-cut concept and standard interval analysis. Various related properties are explored. Finally, some computations of picture fuzzy functions over generalized picture fuzzy variables are illustrated by using our proposed technique.展开更多
基金supported by the National Natural Science Foundation of China (Grant No.12374109)the National Key Research and Development Program of China (Grant No.2023YFA1406600)。
文摘In physics,our expectations for system behavior are often guided by intuitive arithmetic.For systems composed of identical units,we anticipate synergy of the contributions from these units,where 1+1=2.Conversely,for systems built from opposing units,we expect cancellation of their contributions,where 1-1=0.This intuitive arithmetic has long underpinned our understanding of physical properties of materials,from electronic transport to optical responses.However,scientific breakthroughs often occur when nature reveals ways to circumvent these seemingly fundamental rules,opening new possibilities that challenge our deepest assumptions about material behavior.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 60573172 and 60973152)the Doctoral Program Foundation of Institution of Higher Education of China (Grant No. 20070141014)the Natural Science Foundation of Liaoning Province of China (Grant No. 20082165)
文摘In this paper, we analyse a new chaos-based cryptosystem with an embedded adaptive arithmetic coder, which was proposed by Li Heng-Jian and Zhang J S （Li H J and Zhang J S 2010 Chin. Phys. B 19 050508）. Although this new method has a better compression performance than its original version, it is found that there are some problems with its security and decryption processes. In this paper, it is shown how to obtain a great deal of plain text from the cipher text without prior knowledge of the secret key. After discussing the security and decryption problems of the Li Heng-Jian et al. algorithm, we propose an improved chaos-based cryptosystem with an embedded adaptive arithmetic coder that is more secure.
文摘目的建立不同产地喜树果中9个成分含量同步检测方法,筛选影响其质量的差异标志物,并对其进行质量评价。方法对7省18个批次喜树果样品进行回流提取,提取物采用高效液相色谱法检测;采用正交偏最小二乘判别分析(Orthogonal partial least squares discriminant analysis,OPLS-DA)和加权逼近理想解排序(Technique for order preference by similarity to an ideal solution,TOPSIS)法建立喜树果质量优劣评价模型,对其质量差异性进行综合评价。结果3,4′-O-二甲基鞣花酸、丁香酸、10-羟基喜树碱、喜树碱、10-甲氧基喜树碱、三叶豆苷、短小蛇根草苷、金丝桃苷和喜果苷分别在0.51-12.75、0.23-5.75、3.21-80.25、4.45-111.25、1.88-47.00、0.41-10.25、2.05-51.25、0.34-8.50和7.95-198.75μg·mL^(-1)范围内线性关系良好(r>0.999),平均加样回收率96.95%-100.06%(RSD<2.0%);18批样品聚为3类;喜果苷、10-羟基喜树碱、喜树碱、10-甲氧基喜树碱和短小蛇根草苷可能是影响喜树果产品质量主要潜在标志物;加权TOPSIS法分析结果显示18批喜树果质量评价贴近度(Jb)在0.1090-0.7385,其中S14最大(0.7385)。结论建立了同时测定喜树果中9种成分含量的方法,操作简便、结果准确;采用OPLS-DA及加权TOPSIS法进行客观全面评价,可用于喜树果质量差异性评价。
文摘An approximately optimal adaptive arithmetic coding (AC) system using a forbidden symbol (FS) over noisy channels was proposed which allows one to jointly and adaptively design the source decoding and channel correcting in a single process, with superior performance compared with traditional separated techniques. The concept of adaptiveness is applied not only to the source model but also to the amount of coding redundancy. In addition, an improved branch metric computing algorithm and a faster sequential searching algorithm compared with the system proposed by Grangetto were proposed. The proposed system is tested in the case of image transmission over the AWGN channel, and compared with traditional separated system in terms of packet error rate and complexity. Both hard and soft decoding were taken into account.
文摘From such actual conditions as the effects of characteristics of miltilayer petroleum geology and permeation fluid mechanics, a new numerical model is put forward and coupling splitting-up implicit interactive scheme is formed. For the actual situation of Dongying hollow (four-layer) and Tanhai region (three-layer) of Shengli Petroleum Field, the numerical simulation test results and the actual conditions are coincident.
基金Project supported by the National Natural Science Foundation of China (Grant No. 60971104)the Basic Research Foundation of Sichuan Province,China (Grant No. 2006J013-011)the Outstanding Young Researchers Foundation of Sichuan Province,China (Grant No. 09ZQ026-091)
文摘In this study an adaptive arithmetic coder is embedded in the Baptista-type chaotic cryptosystem for implementing secure data compression. To build the multiple lookup tables of secure data compression, the phase space of chaos map with a uniform distribution in the search mode is divided non-uniformly according to the dynamic probability estimation of plaintext symbols. As a result, more probable symbols are selected according to the local statistical characters of plaintext and the required number of iterations is small since the more probable symbols have a higher chance to be visited by the chaotic search trajectory. By exploiting non-uniformity in the probabilities under which a number of iteration to be coded takes on its possible values, the compression capability is achieved by adaptive arithmetic code. Therefore, the system offers both compression and security. Compared with original arithmetic coding, simulation results on Calgary Corpus files show that the proposed scheme suffers from a reduction in compression performance less than 12% and is not susceptible to previously carried out attacks on arithmetic coding algorithms.
基金supported by the National Natural Science Foundation of China (Grant Nos. 60573172 and 60973152)the Superior University Doctor Subject Special Scientific Research Foundation of China (Grant No. 20070141014)the Natural Science Foundation of Liaoning Province of China (Grant No. 20082165)
文摘This paper proposes an efficient lossless image compression scheme for still images based on an adaptive arithmetic coding compression algorithm. The algorithm increases the image coding compression rate and ensures the quality of the decoded image combined with the adaptive probability model and predictive coding. The use of adaptive models for each encoded image block dynamically estimates the probability of the relevant image block. The decoded image block can accurately recover the encoded image according to the code book information. We adopt an adaptive arithmetic coding algorithm for image compression that greatly improves the image compression rate. The results show that it is an effective compression technology.
基金supported by the Natural Science Foundation of Hainan Province, China (Grant No. 613155)
文摘Through a series of studies on arithmetic coding and arithmetic encryption, a novel image joint compression- encryption algorithm based on adaptive arithmetic coding is proposed. The contexts produced in the process of image compression are modified by keys in order to achieve image joint compression encryption. Combined with the bit-plane coding technique, the discrete wavelet transform coefficients in different resolutions can be encrypted respectively with different keys, so that the resolution selective encryption is realized to meet different application needs. Zero-tree coding is improved, and adaptive arithmetic coding is introduced. Then, the proposed joint compression-encryption algorithm is simulated. The simulation results show that as long as the parameters are selected appropriately, the compression efficiency of proposed image joint compression-encryption algorithm is basically identical to that of the original image compression algorithm, and the security of the proposed algorithm is better than the joint encryption algorithm based on interval splitting.
文摘This research work considers the following inequalities: <i>λ</i><em>A</em>(<i>a</i>,<i>b</i>) + (1-<i>λ</i>)<em>C</em>(<i>a</i>,<i>b</i>) ≤ <span style="text-decoration:overline;">C</span>(<i>a</i>,<i>b</i>) ≤ <i>μ</i><em>A</em>(<i>a</i>,<i>b</i>) + (1-<i>μ</i>)<em>C</em>(<i>a</i>,<i>b</i>) and <em>C</em>[<i>λ</i><em>a</em> + (1-<i>λ</i>)<em>b</em>, <i>λ</i><em>b</em> + (1-<i>λ</i>)<em>a</em>] ≤ <span style="text-decoration:overline;">C</span>(<i>a</i>,<i>b</i>) ≤ <em>C</em>[<i>μ</i><em>a</em> + (1-<i>μ</i>)<em>b</em>, <i>μ</i><em>b</em> + (1-<i>μ</i>)<em>a</em>] with <img src="Edit_ce892b1d-c056-44ea-a929-31dbcd1b0e91.bmp" alt="" /> . The researchers attempt to find an answer as to what are the best possible parameters <i>λ</i>, <i>μ</i> that (1.1) and (1.2) can be hold? The main tool is the optimization of some suitable functions that we seek to find out. By searching the best possible parameters such that (1.1) and (1.2) can be held. Firstly, we insert <em>f</em>(<i>t</i>) = <i>λ</i><em>A</em>(<i>a</i>,<i>b</i>) + (1-<i>λ</i>)<em>C</em>(<i>a</i>,<i>b</i>) - <span style="text-decoration:overline;">C</span>(<i>a</i>,<i>b</i>) without the loss of generality. We assume that <i>a</i>><i>b</i> and let <img src="Edit_efa43881-9a60-44f8-a86f-d4a1057f4378.bmp" alt="" /> to determine the condition for <i>λ</i> and <i>μ</i> to become f (<i>t</i>) ≤ 0. Secondly, we insert g(<i>t</i>) = <i>μ</i><em>A</em>(<i>a</i>,<i>b</i>) + (1-<i>μ</i>)<em>C</em>(<i>a</i>,<i>b</i>) - <span style="text-decoration:overline;">C</span>(<i>a</i>,<i>b</i>) without the loss of generality. We assume that <i>a</i>><i>b</i> and let <img src="Edit_750dddbb-1d71-45d3-be29-6da5c88ba85d.bmp" alt="" /> to determine the condition for <i>λ</i> and <i>μ</i> to become <em>g</em>(<i>t</i>) ≥ 0.
文摘In this article, we report the derivation of high accuracy finite difference method based on arithmetic average discretization for the solution of Un=F(x,u,u′)+∫K(x,s)ds , 0 x s < 1 subject to natural boundary conditions on a non-uniform mesh. The proposed variable mesh approximation is directly applicable to the integro-differential equation with singular coefficients. We need not require any special discretization to obtain the solution near the singular point. The convergence analysis of a difference scheme for the diffusion convection equation is briefly discussed. The presented variable mesh strategy is applicable when the internal grid points of the solution space are both even and odd in number as compared to the method discussed by authors in their previous work in which the internal grid points are strictly odd in number. The advantage of using this new variable mesh strategy is highlighted computationally.
基金supported by the Deanship of Scientific Research,at Imam Abdulrahman Bin Faisal University.Grant Number:2019-416-ASCS.
文摘Lung cancer is among the most frequent cancers in the world,with over one million deaths per year.Classification is required for lung cancer diagnosis and therapy to be effective,accurate,and reliable.Gene expression microarrays have made it possible to find genetic biomarkers for cancer diagnosis and prediction in a high-throughput manner.Machine Learning(ML)has been widely used to diagnose and classify lung cancer where the performance of ML methods is evaluated to identify the appropriate technique.Identifying and selecting the gene expression patterns can help in lung cancer diagnoses and classification.Normally,microarrays include several genes and may cause confusion or false prediction.Therefore,the Arithmetic Optimization Algorithm(AOA)is used to identify the optimal gene subset to reduce the number of selected genes.Which can allow the classifiers to yield the best performance for lung cancer classification.In addition,we proposed a modified version of AOA which can work effectively on the high dimensional dataset.In the modified AOA,the features are ranked by their weights and are used to initialize the AOA population.The exploitation process of AOA is then enhanced by developing a local search algorithm based on two neighborhood strategies.Finally,the efficiency of the proposed methods was evaluated on gene expression datasets related to Lung cancer using stratified 4-fold cross-validation.The method’s efficacy in selecting the optimal gene subset is underscored by its ability to maintain feature proportions between 10%to 25%.Moreover,the approach significantly enhances lung cancer prediction accuracy.For instance,Lung_Harvard1 achieved an accuracy of 97.5%,Lung_Harvard2 and Lung_Michigan datasets both achieved 100%,Lung_Adenocarcinoma obtained an accuracy of 88.2%,and Lung_Ontario achieved an accuracy of 87.5%.In conclusion,the results indicate the potential promise of the proposed modified AOA approach in classifying microarray cancer data.
文摘In this article, we define the arithmetic operations of generalized trapezoidal picture fuzzy numbers by vertex method which is assembled on a combination of the (α, γ, β)-cut concept and standard interval analysis. Various related properties are explored. Finally, some computations of picture fuzzy functions over generalized picture fuzzy variables are illustrated by using our proposed technique.
