Environment monitoring system based on optical sensor

Abstract

The invention discloses an environmental monitoring system based on an optical sensor, and relates to the technical field of environmental monitoring, and the system comprises the following components: a quantum entanglement module, an optical sensor integration module, a quantum measurement and data processing module, and a data transmission and sharing module. By integrating the quantum entanglement module, the optical sensor integration module and the quantum measurement and data processing module, high-precision and real-time monitoring of environmental parameters is realized, the quantum entanglement technology provides measurement precision and anti-interference capability superior to those of a traditional optical sensor for the system, the accuracy and reliability of data are ensured, and meanwhile, the quantum entanglement technology can be used for monitoring the environmental parameters in real time. Due to the design of high sensitivity and wide dynamic range of the optical sensor, the system can stably work under various complex environment conditions, and the efficiency and accuracy of environment monitoring are effectively improved.

Description

Technical Field

[0001] The present invention relates to the technical field of environmental monitoring, and in particular to an environmental monitoring system based on optical sensors. Background Art

[0002] With the rapid development of industrialization and urbanization, environmental pollution problems are becoming increasingly serious, and the demand for environmental monitoring technology is becoming increasingly urgent. Traditional environmental monitoring methods mainly rely on chemical analysis and physical sensors, but these methods often have problems such as insufficient monitoring accuracy, poor real-time performance, and susceptibility to environmental interference.

[0003] Traditional technology has its shortcomings. On the one hand, the preparation and maintenance of quantum entangled states require extremely high technical conditions and extremely high requirements for the stability and precision of the equipment. On the other hand, traditional quantum communication technology is easily affected by environmental noise and interference during long-distance transmission, resulting in the degradation of quantum entangled states and an increase in the error rate of information transmission.

[0004] In summary, although quantum entanglement technology has potential advantages in the field of remote monitoring, traditional technology still faces many technical difficulties and challenges in practical applications. Therefore, it is particularly important to develop an environmental monitoring system based on optical sensors. Summary of the invention

[0005] The purpose of the present invention is to make up for the shortcomings of the prior art and to provide an environmental monitoring system based on optical sensors. It can achieve high-precision and real-time monitoring of environmental parameters by integrating a quantum entanglement module, an optical sensor integration module, and a quantum measurement and data processing module. Quantum entanglement technology provides the system with measurement accuracy and anti-interference capabilities that surpass traditional optical sensors, ensuring the accuracy and reliability of the data. At the same time, the high sensitivity and wide dynamic range design of the optical sensor enable the system to work stably under various complex environmental conditions, effectively improving the efficiency and accuracy of environmental monitoring.

[0006] In order to solve the above technical problems, the present invention provides the following technical solutions: an environmental monitoring system based on optical sensors, the system comprising the following components: a quantum entanglement module, an optical sensor integration module, a quantum measurement and data processing module and a data transmission and sharing module; The quantum entanglement module: takes a periodically polarized lithium niobate crystal as the core, generates entangled photon pairs through a nonlinear optical difference frequency process, regulates the crystal temperature and the external electric field, changes the polarization period, and uses a spatial light modulator to load a phase modulation pattern optimized based on a genetic algorithm. The pattern takes the target transmission direction of the entangled photons and the purity of the orbital angular momentum mode as the optimization target, sets the initial population and genetic operation parameters, and is obtained through multiple generations of genetic evolution. An optical collimation system composed of a multi-layer aspheric lens combination is used, combined with adaptive optical technology, to correct the optical path deviation caused by environmental factors in real time; The optical sensor integrated module: deploys high-sensitivity, wide dynamic range, high-resolution imaging spectrometers and multi-angle light scattering sensor optical sensors at each monitoring point, and the automatic calibration device uses a machine learning algorithm to establish a correlation model based on changes in environmental parameters and through measurements of standard light sources and reference samples to achieve dynamic calibration; The quantum measurement and data processing module: The monitoring point adopts quantum weak measurement technology, jointly measures entangled photons and optical sensor signals to obtain high signal-to-noise ratio quantum state data, and uses quantum state fusion analysis algorithm , combining quantum and classical optical measurement data to extract environmental parameters, including It is a variable related to spatial position, which is used to determine the specific position information of the quantum state in space. , These two variables are also used to describe the spatial position information and are related to the optical signal characteristics. Together, they built a three-dimensional spatial coordinate system to more accurately locate the spatial position of optical signal generation or detection. , are variables related to the classical optical measurement parameters, is the comprehensive value of environmental parameters, are the number of quantum and classical measurement dimensions, For the The weight coefficient of the quantum measurement dimension, For spatial position and time The relevant quantum state function, is a function related to the optical signal characteristics, For the The weight coefficients of the classical measurement dimensions, is a function related to the parameters of classical optical measurement; The data transmission and sharing module: uses key distribution technology based on quantum random walk to encrypt data, transmits it to the data center through quantum or encrypted traditional communication network, and generates keys using the evolution of quantum bit states in quantum random walk. The data center uses quantum secure multi-party computing technology, combined with oblivious transfer protocol and homomorphic encryption technology. The quantum communication network includes quantum communication channels, and data transmission is carried out by quantum optical fiber communication and other methods according to actual needs.

[0007] Furthermore, in the nonlinear optical frequency difference process based on periodically poled lithium niobate crystal, the polarization period of the crystal is prepared by focused ion beam writing technology, and the period accuracy reaches the nanometer level. The pump light frequency , the frequency of the generated entangled photon pair and Satisfaction relationship: ,in is the wave vector of the pump light, are the wave vectors of the two photons in the generated entangled photon pair, and by controlling the crystal temperature exist In the range, It is the lowest temperature that the crystal can tolerate while ensuring that the difference frequency process is carried out effectively and that the thermo-optic effect can be used to fine-tune the crystal refractive index to optimize the phase matching conditions. It is the highest temperature that the crystal can withstand under the same conditions without affecting the difference frequency process and related performance. The refractive index of the crystal is fine-tuned using the thermo-optical effect to optimize the phase matching conditions of the difference frequency process.

[0008] Furthermore, the phase modulation pattern loaded by the spatial light modulator is optimized by genetic algorithm, with the target transmission direction of entangled photons and the purity of orbital angular momentum mode as the optimization target, the initial population and genetic operation parameters are set, and after multiple generations of genetic evolution, the optimal phase modulation pattern is obtained, so that the mode fidelity of entangled photons in the transmission process reaches The multi-layer aspheric lens combination in the optical collimation system is made of low-dispersion, high-refractive-index optical glass. The curvature radius, thickness, and spacing of the lens are optimized through ray tracing software.

[0009] Furthermore, the high-resolution imaging spectrometer adopts the principle of Fourier transform interferometric imaging, obtains interference pattern sequence through uniform linear motion of the moving mirror, and the interference pattern data is processed by fast Fourier transform algorithm to obtain spectral information, and the spectral resolution The maximum moving distance of the moving mirror Decide, satisfy ,in The central wavelength of light detected by the high-resolution imaging spectrometer in vacuum. The multi-angle light scattering sensor adopts a multi-detector array design. The angle distribution of the detectors is optimized according to the Mie scattering theory and actual monitoring needs. The scattering angle range that can be measured is .

[0010] Furthermore, in the quantum weak measurement technology, a weak measurement intensity parameter is introduced , by adjusting The value of the quantum state is achieved without destroying the quantum state. The weak information of the quantum state is amplified and measured. During the measurement, quantum gate operations are used to encode and decode quantum bits. The quantum gate operation sequence is designed according to the principle of quantum error correction code. The quantum detector uses a quantum bit detector based on a superconducting Josephson junction, and its energy resolution reaches ,in Represents the minimum energy difference that the quantum bit detector can distinguish.

[0011] Furthermore, in the quantum state fusion analysis algorithm, the weight coefficient and The dynamic determination method is as follows: first, a multi-dimensional feature space of the monitoring environment is constructed, and a deep neural network is used to extract and classify the features of the monitoring environment. The initial weight coefficients of different measurement dimensions are determined according to the classification results. Then, a reinforcement learning algorithm is used to minimize the error between the measurement results and the actual environmental parameters as the reward function, and the weight coefficients are continuously adjusted.

[0012] Furthermore, in the key distribution technology based on quantum random walk, the number of steps of quantum random walk is According to the communication distance and channel noise level Dynamic determination: When the communication distance increases, the number of steps is increased to ensure the security of the key. When the channel noise level is high, the randomness and anti-attack performance of the key are improved by increasing the number of steps. The specific relationship is: ,in is a constant related to the performance of quantum communication equipment and the environmental noise characteristics. In each random walk, the quantum state evolution equation is updated according to the initial state of the quantum bit and the influence of environmental noise.

[0013] Furthermore, when the quantum communication channel adopts quantum optical fiber communication, the optical fiber adopts ultra-low loss, high bandwidth special optical fiber, and its attenuation coefficient is less than In order to compensate for the signal attenuation and dispersion during optical fiber transmission, distributed Raman amplification technology and chirp compensation technology are used. Distributed Raman amplification realizes distributed amplification of signal light through the interaction between pump light and signal light transmitted in the optical fiber. The amplification gain coefficient is Chirp compensation technology uses a chirped fiber Bragg grating. According to the fiber dispersion characteristics and signal light bandwidth, the chirp parameters of the grating are designed to achieve dispersion compensation for the signal light.

[0014] Furthermore, when the data center uses quantum-safe multi-party computing technology for data integration and analysis, it uses the oblivious transfer protocol to achieve secure data interaction. In the multi-party computing process, in order to improve computing efficiency, homomorphic encryption technology is used to directly calculate the encrypted data, reducing the number of data decryption and re-encryption. The homomorphic encryption algorithm uses a fully homomorphic encryption scheme based on lattice cryptography, which can support addition and multiplication homomorphic operations. Ensure complex statistical analysis and model training of data in the ciphertext state, while ensuring data security and privacy.

[0015] Compared with the prior art, this optical sensor-based environmental monitoring system has the following beneficial effects: 1. The system achieves high-precision, real-time monitoring of environmental parameters by integrating quantum entanglement modules, optical sensor integration modules, and quantum measurement and data processing modules. Quantum entanglement technology provides the system with measurement accuracy and anti-interference capabilities that surpass traditional optical sensors, ensuring the accuracy and reliability of the data. At the same time, the high sensitivity and wide dynamic range design of the optical sensor enable the system to work stably under various complex environmental conditions, effectively improving the efficiency and accuracy of environmental monitoring.

[0016] Second, the keys generated by the system through quantum random walk are highly random and resistant to attacks, which can effectively prevent data from being stolen or tampered with during transmission. At the same time, quantum secure multi-party computing technology allows data centers to integrate and analyze data without decrypting the data, further protecting the security and privacy of the data, and providing strong technical support for the sharing and utilization of environmental monitoring data.

[0017] Other advantages, objectives and features of the present invention will be set forth in part in the following description and, in part, will be apparent to those skilled in the art based on an examination of the following or may be taught from the practice of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the prior art descriptions are briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention, and for ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.

[0019] Figure 1 This is a process operation diagram of an environmental monitoring system based on optical sensors. DETAILED DESCRIPTION

[0020] In order to further explain the technical means and effects adopted by the present invention to achieve the predetermined invention purpose, the specific implementation mode, structure, characteristics and effects of the present invention are described in detail below in combination with the accompanying drawings and preferred embodiments.

[0021] Embodiment 1: This embodiment describes selecting multiple representative monitoring points in a city with a serious smog problem, such as a busy intersection, an industrial area boundary, and an urban park.

[0022] The entangled photon pairs are generated with periodically polarized lithium niobate crystals as the core. The crystal polarization period is prepared by focused ion beam writing technology to achieve nanometer-level precision. The pump light frequency , the frequency of the generated entangled photon pair and Satisfaction relationship: , by controlling the crystal temperature exist , using the thermo-optic effect to fine-tune the crystal refractive index, optimizing the phase matching conditions of the difference frequency process, and using space to load the phase modulation pattern optimized by genetic algorithm into the modulator to make the entangled photon mode fidelity , using multi-layer aspheric lens linear combination and adaptive optics technology to correct optical path deviation.

[0023] Optical sensors with high sensitivity and wide dynamic range are deployed at each monitoring point. The high-resolution imaging spectrometer adopts the principle of Fourier transform interferometric imaging. The moving mirror moves in a straight line at a uniform speed to obtain the interference pattern sequence. The spectral information is obtained by fast Fourier transform algorithm processing. The spectral resolution is , used to analyze the chemical composition in haze, the multi-angle light scattering sensor adopts a multi-detector array design, optimizes the detector angle distribution according to Mie scattering theory and actual monitoring needs, and measures the scattering angle range , accurately measures haze particles in different particle size ranges. The automatic calibration device uses a machine learning algorithm to establish a correlation model based on changes in ambient temperature and humidity parameters, and achieves dynamic calibration by measuring standard light sources and reference samples.

[0024] Using quantum weak measurement technology, introducing weak measurement intensity parameters , amplifies and measures weak information of quantum states without destroying the quantum state, uses quantum gate operations to encode and decode quantum bits, and the quantum gate operation sequence is designed according to the principle of quantum error correction code. The quantum detector uses a quantum bit detector based on superconducting Josephson junction, with an energy resolution of , using quantum state fusion analysis algorithm Extract environmental parameters, where the weight coefficient and By constructing a multi-dimensional feature space of the monitoring environment, deep neural networks are used to extract features and classify to determine the initial values, and then a reinforcement learning algorithm is used to make adjustments based on the reward function of minimizing the error between the measurement results and the actual environmental parameters.

[0025] The key distribution technology based on quantum random walk is used to encrypt data. , according to the communication distance between the monitoring point and the data center and channel noise level Dynamically determined, transmitted to the data center through quantum or encrypted traditional communication networks. If quantum optical fiber communication is used, the optical fiber should use ultra-low loss, high bandwidth special optical fiber with an attenuation coefficient less than ,in The maximum allowable attenuation coefficient value is determined based on the requirements of the quantum communication system for signal transmission loss, combined with the specific communication distance, signal strength and system fault tolerance factors. In order to compensate for the signal attenuation and dispersion during optical fiber transmission, the distributed Raman amplification technology (amplification gain coefficient is ) and chirp compensation technology (using chirp parameters designed by chirped fiber Bragg grating) to compensate for signal attenuation and dispersion. The data center adopts quantum secure multi-party computing technology, combined with the oblivious transfer protocol and the fully homomorphic encryption scheme based on lattice cryptography to integrate and analyze data, providing data support for urban haze control.

[0026] Example 2: This example describes an industrial park with multiple chemical and electronic companies. There are multiple pollution sources such as wastewater, waste gas, and waste residue, which pose a potential threat to the surrounding environment. In order to timely grasp the environmental pollution status of the park, monitoring points are set up around each enterprise in the park, sewage treatment plants, and key locations of waste gas emission outlets.

[0027] Similarly, entangled photon pairs are generated with periodically poled lithium niobate crystals as the core. By adjusting the crystal temperature and applying an external electric field to change the polarization period, the spatial light modulator and optical collimation system are used to shape the entangled photon wavefront and correct the optical path deviation to ensure its stable transmission to each monitoring point.

[0028] The deployed optical sensors are selected according to the needs of industrial pollution monitoring. The high-resolution imaging spectrometer can detect the spectral characteristics of specific pollutants in wastewater. The multi-angle light scattering sensor is used to monitor the concentration and particle size distribution of particulate matter in the exhaust gas. The automatic calibration device uses machine learning algorithms to dynamically calibrate the sensors according to the complex and changeable parameters in the industrial environment, such as pH, temperature, and humidity, to ensure the accuracy of the measurement data.

[0029] Quantum weak measurement technology is used to jointly measure entangled photons and optical sensor signals to obtain high signal-to-noise ratio quantum state data. The quantum state fusion analysis algorithm integrates quantum and classical measurement data to extract environmental parameters. When determining the weight coefficient, a multi-dimensional feature space is constructed based on the pollution characteristics of the industrial park. After the deep neural network extracts and classifies the features, the reinforcement learning algorithm optimizes the weight coefficient with the goal of minimizing the measurement error to adapt to different pollution scenarios.

[0030] Taking into account the communication distance and channel noise characteristics within the industrial park, the key distribution technology based on quantum random walk dynamically determines the number of steps to generate keys to encrypt data. The data is transmitted to the data center through quantum or encrypted traditional communication networks. If quantum fiber optic communication is used, distributed Raman amplification technology and chirp compensation technology are used to ensure reliable data transmission. The data center uses quantum secure multi-party computing technology, combined with oblivious transfer protocol and homomorphic encryption technology, to integrate and analyze data from each monitoring point to provide decision support for park environmental management.

[0031] The above description is only a preferred embodiment of the present invention and does not limit the present invention in any form. Although the present invention has been disclosed as a preferred embodiment as above, it is not used to limit the present invention. Any technical personnel in this field can make some changes or modify the technical contents disclosed above into equivalent embodiments without departing from the scope of the technical solution of the present invention. However, any brief modifications, equivalent changes and modifications made to the above embodiments based on the technical essence of the present invention without departing from the content of the technical solution of the present invention are still within the scope of the technical solution of the present invention.

Claims

1. An environmental monitoring system based on optical sensors, characterized in that: The system includes the following components: quantum entanglement module, optical sensor integration module, quantum measurement and data processing module and data transmission and sharing module; The quantum entanglement module: takes a periodically polarized lithium niobate crystal as the core, generates entangled photon pairs through a nonlinear optical difference frequency process, regulates the crystal temperature and the external electric field, changes the polarization period, and uses a spatial light modulator to load a phase modulation pattern optimized based on a genetic algorithm. The pattern takes the target transmission direction of the entangled photons and the purity of the orbital angular momentum mode as the optimization target, sets the initial population and genetic operation parameters, and is obtained through multiple generations of genetic evolution. An optical collimation system composed of a multi-layer aspheric lens combination is used, combined with adaptive optical technology, to correct the optical path deviation caused by environmental factors in real time; The optical sensor integrated module: deploys high-sensitivity, wide dynamic range, high-resolution imaging spectrometers and multi-angle light scattering sensor optical sensors at each monitoring point, and the automatic calibration device uses a machine learning algorithm to establish a correlation model based on changes in environmental parameters and through measurements of standard light sources and reference samples to achieve dynamic calibration; The quantum measurement and data processing module: The monitoring point adopts quantum weak measurement technology, jointly measures entangled photons and optical sensor signals to obtain high signal-to-noise ratio quantum state data, and uses quantum state fusion analysis algorithm , combining quantum and classical optical measurement data to extract environmental parameters, including It is a variable related to spatial position, which is used to determine the specific position information of the quantum state in space. , These two variables are also used to describe the spatial position information and are related to the optical signal characteristics. Together, they built a three-dimensional spatial coordinate system to more accurately locate the spatial position of optical signal generation or detection. , are variables related to the classical optical measurement parameters, is the comprehensive value of environmental parameters, are the number of quantum and classical measurement dimensions, For the The weight coefficient of the quantum measurement dimension, For spatial position and time The relevant quantum state function, is a function related to the optical signal characteristics, For the The weight coefficients of the classical measurement dimensions, is a function related to the parameters of classical optical measurement; The data transmission and sharing module: uses key distribution technology based on quantum random walk to encrypt data, transmits it to the data center through quantum or encrypted traditional communication network, generates keys using the evolution of quantum bit states in quantum random walk, and the data center uses quantum secure multi-party computing technology, combined with oblivious transfer protocol and homomorphic encryption technology. The quantum communication network includes quantum communication channels, and quantum optical fiber communication is used for data transmission according to actual needs.

2. The optical sensor-based environmental monitoring system according to claim 1, characterized in that: In the nonlinear optical frequency difference process based on periodically poled lithium niobate crystal, the polarization period of the crystal is prepared by focused ion beam writing technology, and the period accuracy reaches the nanometer level. The pump light frequency , the frequency of the generated entangled photon pair and Satisfaction relationship: ,in is the wave vector of the pump light, are the wave vectors of the two photons in the generated entangled photon pair, and by controlling the crystal temperature exist In the range, It is the lowest temperature that the crystal can tolerate while ensuring that the difference frequency process is carried out effectively and that the thermo-optic effect can be used to fine-tune the crystal refractive index to optimize the phase matching conditions. It is the highest temperature that the crystal can withstand under the same conditions without affecting the difference frequency process and related performance. The refractive index of the crystal is fine-tuned using the thermo-optical effect to optimize the phase matching conditions of the difference frequency process.

3. The optical sensor-based environmental monitoring system according to claim 1, characterized in that: The phase modulation pattern loaded by the spatial light modulator is optimized by genetic algorithm, with the target transmission direction of entangled photons and the purity of orbital angular momentum mode as the optimization target, the initial population and genetic operation parameters are set, and after multiple generations of genetic evolution, the optimal phase modulation pattern is obtained, so that the mode fidelity of entangled photons in the transmission process reaches The multi-layer aspheric lens combination in the optical collimation system is made of low-dispersion, high-refractive-index optical glass. The curvature radius, thickness, and spacing of the lens are optimized through ray tracing software.

4. The optical sensor-based environmental monitoring system according to claim 1, characterized in that: The high-resolution imaging spectrometer adopts the principle of Fourier transform interferometric imaging, obtains interference pattern sequence through uniform linear motion of the moving mirror, and obtains spectral information through fast Fourier transform algorithm processing. The maximum moving distance of the moving mirror Decide, satisfy ,in The central wavelength of light detected by the high-resolution imaging spectrometer in vacuum. The multi-angle light scattering sensor adopts a multi-detector array design. The angle distribution of the detectors is optimized according to the Mie scattering theory and actual monitoring needs. The scattering angle range that can be measured is .

5. The optical sensor-based environmental monitoring system according to claim 1, characterized in that: In the quantum weak measurement technology, a weak measurement intensity parameter is introduced , by adjusting The value of the quantum state is achieved without destroying the quantum state. The weak information of the quantum state is amplified and measured. During the measurement, quantum gate operations are used to encode and decode quantum bits. The quantum gate operation sequence is designed according to the principle of quantum error correction code. The quantum detector uses a quantum bit detector based on a superconducting Josephson junction, and its energy resolution reaches ,in Represents the minimum energy difference that the quantum bit detector can distinguish.

6. The optical sensor-based environmental monitoring system according to claim 1, characterized in that: In the quantum state fusion analysis algorithm, the weight coefficient and The dynamic determination method is as follows: first, a multi-dimensional feature space of the monitoring environment is constructed, and a deep neural network is used to extract and classify the features of the monitoring environment. The initial weight coefficients of different measurement dimensions are determined according to the classification results. Then, a reinforcement learning algorithm is used to minimize the error between the measurement results and the actual environmental parameters as the reward function, and the weight coefficients are continuously adjusted.

7. The optical sensor-based environmental monitoring system according to claim 1, characterized in that: In the key distribution technology based on quantum random walk, the number of steps of quantum random walk is According to the communication distance and channel noise level Dynamic determination: When the communication distance increases, the number of steps is increased to ensure the security of the key. When the channel noise level is high, the randomness and anti-attack performance of the key are improved by increasing the number of steps. The specific relationship is: ,in is a constant related to the performance of quantum communication equipment and the environmental noise characteristics. In each random walk, the quantum state evolution equation is updated according to the initial state of the quantum bit and the influence of environmental noise.

8. The optical sensor-based environment monitoring system according to claim 1, characterized in that: When the quantum communication channel adopts quantum optical fiber communication, the optical fiber adopts ultra-low loss, high bandwidth special optical fiber, and its attenuation coefficient is less than ,in The maximum allowable attenuation coefficient value is determined based on the requirements of the quantum communication system for signal transmission loss, combined with the specific communication distance, signal strength and system fault tolerance factors. In order to compensate for the signal attenuation and dispersion during optical fiber transmission, distributed Raman amplification technology and chirp compensation technology are used. Distributed Raman amplification achieves distributed amplification of signal light through the interaction between the pump light and the signal light transmitted in the optical fiber. The amplification gain coefficient is Chirp compensation technology uses a chirped fiber Bragg grating. According to the fiber dispersion characteristics and signal light bandwidth, the chirp parameters of the grating are designed to achieve dispersion compensation for the signal light.

9. The optical sensor-based environment monitoring system according to claim 1, characterized in that: When the data center uses quantum-safe multi-party computing technology for data integration and analysis, it adopts the oblivious transfer protocol to achieve secure data interaction. During the multi-party computing process, in order to improve computing efficiency, homomorphic encryption technology is used to directly calculate the encrypted data, reducing the number of data decryption and re-encryption times. The homomorphic encryption algorithm uses a fully homomorphic encryption scheme based on lattice cryptography, which can support homomorphic operations of addition and multiplication.

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