Nonlinear Layer Research Articles

Nonlinear Deep Learning Framework for Precise Sentiment Classification in Textual Data

The study presents the Nonlinear Deep Learning Framework for Precise Sentiment Classification (NPSC), focusing on improving sentiment analysis in text data. The model addresses the challenge of capturing complex and hidden sentiment patterns using a combination of BERT embeddings, Bidirectional LSTM (BiLSTM), Convolutional Neural Networks (CNN), and an attention mechanism. This setup helps in identifying subtle relationships within text, enabling more precise sentiment predictions. NPSC tackles common issues faced by existing models that struggle with detailed text analysis. Using the IMDB movie review dataset, the approach relies on BERT to convert words into dense vectors, BiLSTM for capturing context from both directions, and CNN to extract key local features. An attention layer highlights important words, refining the sentiment detection process. The integration of these components in a nonlinear fusion layer allows for enhanced feature interaction, leading to accurate classification through a final softmax layer. The findings show that NPSC performs better across metrics like accuracy, precision, recall, and F1-score. Ablation studies reveal the impact of each component, demonstrating their role in enhancing the model’s effectiveness. Error analysis shows how NPSC manages mixed sentiments and challenging text patterns. The study highlights NPSC’s potential as a reliable method for sentiment analysis, contributing to better handling of complex textual data.

Mathematical Nonlinear Graph Theory Topology Layer Model for Photoelectric Tracking System

The performance of the photoelectric tracking system mainly depends on the tracking accuracy. In order to achieve the purpose of high precision tracking, controlling the power dragging device, the main component of the photoelectric tracking system, is the main means to achieve this purpose. In order to improve the tracking speed and accuracy of photoelectric tracking systems, the author proposed a mathematical nonlinear graph theory topology layer model for photoelectric tracking systems. The topology layer model and the motion node servo mechanism model of the photoelectric tracking system are studied, and the two-stage disturbance sources that affect the tracking stability are analyzed. The results show that the interference estimation error range designed by the author reaches 0.003 through comparison, it can be seen that the estimation error designed by the author is significantly smaller than the ESO estimation error, and the buffeting is small. Through comparison, it can be concluded that the estimation error of the algorithm and the disturbance observer designed by the author are significantly smaller than the estimation error of ESO, and the buffeting is small, and they have strong compensation ability for the disturbance of different frequencies. The interference observer can quickly and accurately estimate the interference and prove its stability. The effectiveness of the proposed observer and disturbance observer is fully demonstrated. The finite time integrated site selection mode disturbance observer (F-ISMDOB) is designed to quickly estimate and compensate for equivalent interference, and effectively improves the anti-interference performance of the system structure layer.

On azimuthal wave-induced distortion of variable density swirling flows across non-linear critical layers

On azimuthal wave-induced distortion of variable density swirling flows across non-linear critical layers

Monolith: Circuit-Friendly Hash Functions with New Nonlinear Layers for Fast and Constant-Time Implementations

Hash functions are a crucial component in incrementally verifiable computation (IVC) protocols and applications. Among those, recursive SNARKs and folding schemes require hash functions to be both fast in native CPU computations and compact in algebraic descriptions (constraints). However, neither SHA-2/3 nor newer algebraic constructions, such as Poseidon, achieve both requirements. In this work we overcome this problem in several steps. First, for certain prime field domains we propose a new design strategy called Kintsugi, which explains how to construct nonlinear layers of high algebraic degree which allow fast native implementations and at the same time also an efficient circuit description for zeroknowledge applications. Then we suggest another layer, based on the Feistel Type-3 scheme, and prove wide trail bounds for its combination with an MDS matrix. We propose a new permutation design named Monolith to be used as a sponge or compression function. It is the first arithmetization-oriented function with a native performance comparable to SHA3-256. At the same time, it outperforms Poseidon in a circuit using the Merkle tree prover in the Plonky2 framework. Contrary to previously proposed designs, Monolith also allows for efficient constant-time native implementations which mitigates the risk of side-channel attacks.

Non-similar solution of Casson fluid flow over a curved stretching surface with viscous dissipation; Artificial neural network analysis

PurposeThe main focus is to provide a non-similar solution for the magnetohydrodynamic (MHD) flow of Casson fluid over a curved stretching surface through the novel technique of the artificial intelligence (AI)-based Lavenberg–Marquardt scheme of an artificial neural network (ANN). The effects of joule heating, viscous dissipation and non-linear thermal radiation are discussed in relation to the thermal behavior of Casson fluid.Design/methodology/approachThe non-linear coupled boundary layer equations are transformed into a non-linear dimensionless Partial Differential Equation (PDE) by using a non-similar transformation. The local non-similar technique is utilized to truncate the non-similar dimensionless system up to 2nd order, which is treated as coupled ordinary differential equations (ODEs). The coupled system of ODEs is solved numerically via bvp4c. The data sets are constructed numerically and then implemented by the ANN.FindingsThe results indicate that the non-linear radiation parameter increases the fluid temperature. The Casson parameter reduces the fluid velocity as well as the temperature. The mean squared error (MSE), regression plot, error histogram, error analysis of skin friction, and local Nusselt number are presented. Furthermore, the regression values of skin friction and local Nusselt number are obtained as 0.99993 and 0.99997, respectively. The ANN predicted values of skin friction and the local Nusselt number show stability and convergence with high accuracy.Originality/valueAI-based ANNs have not been applied to non-similar solutions of curved stretching surfaces with Casson fluid model, with viscous dissipation. Moreover, the authors of this study employed Levenberg–Marquardt supervised learning to investigate the non-similar solution of the MHD Casson fluid model over a curved stretching surface with non-linear thermal radiation and joule heating. The governing boundary layer equations are transformed into a non-linear, dimensionless PDE by using a non-similar transformation. The local non-similar technique is utilized to truncate the non-similar dimensionless system up to 2nd order, which is treated as coupled ODEs. The coupled system of ODEs is solved numerically via bvp4c. The data sets are constructed numerically and then implemented by the ANN.

Layered heterogeneous structures integrated device for multiplication, division arithmetic unit and multiple-physical sensing

A layered heterogeneous structure (LHS), consisting of silver, liquid crystal, and nonlinear dielectric layers, is proposed to realize functions of computing and sensing. By leveraging the optical Tamm state, the intrinsic absorption principle of liquid crystal, and nonlinear effects, the design of an integrated device capable of passive multiplication and division operations, along with high-performance multi-physical quantity sensing functionalities, is achieved. The given LHS exhibits Janus properties, with different physical functions manifested depending on the direction of electromagnetic wave (EW) propagation. During forward propagation of EWs, the LHS displays high and sharp absorptivity peaks at 774.8 and 1517.6 nm. The relationship between the two peaks approximates a frequency multiplication factor of 1.960, enabling signal multiplication. Furthermore, the two absorptivity peaks at different wavelengths facilitate the sensing of serum creatinine solution concentration and external pressure, with sensitivity (S), quality factors (Q), and figure of merit (FOM) of 266.76 μmol L−1/nm and 213.33 GPa/nm, as well as 248.76 and 348.22, 84.1 L (μmol)−1 and 49.06 GPa−1, respectively. During backward propagation of EWs, absorptivity peaks with distinct resolutions are observed at 1423 and 2809 nm, with a multiple relationship between them of 1.974, enabling frequency doubling for signal division. Additionally, the absorptivity curve facilitates temperature sensing over a wide range from 257 to 347 K. Owing to the unique temperature S of liquid crystal, different sensitivities and resolutions are observed at 257 to 297 K and 307 to 347 K, with S of 1.015 and 0.686 K/nm, and corresponding Q and FOM of 21.57 and 12.576, 0.076 K−1 and 0.003 04 K−1, respectively.

One-Dimensional-Mixed Convolution Neural Network and Covariance Pooling Model for Mineral Mapping of Porphyry Copper Deposit Using PRISMA Hyperspectral Data

Mapping distribution of alterations around porphyry copper deposits (PCDs) greatly affects mineral exploration. Diverse geological processes generate irregular alteration patterns with diverse spectral characteristics in mineral deposits. Applying remotely sensed hyperspectral images (HSIs) is an appealing technology for geologic surveyors to generate alteration maps. Conventional methods mainly use shallow spectral absorption features to discriminate minerals and cannot extract their important spectral information. Deep neural networks with nonlinear layers can evoke the deep spectral and spatial information of HSIs. Deep learning???based methods include fully connected neural networks, convolutional neural networks, and hybrid convolutional networks like mixed convolution neural network and covariance pooling (MCNN‐CP) algorithms. However, each has its advantages and limitations. To significantly avoid losing important spectral features, we proposed a new method by fusing a one‐dimensional convolutional neural network (1D‐CNN) with MCNN‐CP (1D‐MCNN‐CP), achieving an overall accuracy (97.44%) of mineral mapping from PRISMA HSIs. This research deduced that 1D‐MCNN‐CP improved performance and reduced misclassification errors among minerals sharing similar spectral features.

Precision improvement of robotic bioprinting via vision-based tool path compensation

Robotic 3D bioprinting is a rapidly advancing technology with applications in organ fabrication, tissue restoration, and pharmaceutical testing. While the stepwise generation of organs characterizes bioprinting, challenges such as non-linear material behavior, layer shifting, and trajectory tracking are common in freeform reversible embedding of suspended hydrogels (FRESH) bioprinting, leading to imperfections in complex organ construction. To overcome these limitations, we propose a computer vision-based strategy to identify discrepancies between printed filaments and the reference robot path. Employing error compensation techniques, we generate an adjusted reference path, enhancing robotic 3D bioprinting by adapting the robot path based on vision system data. Experimental assessments confirm the reliability and agility of our vision-based robotic 3D bioprinting approach, showcasing precision in fabricating human blood vessel segments through case studies. Significantly, it minimizes the printing layer width disparity to just 0.15 mm compared to the 0.6 mm in traditional methods, and it decreases the average error for curved filaments to 7.0 mm2 from the previous 12.7 mm2 in conventional printing. While these results underscore the significant potential of our innovation in creating precise biomimetic constructs, further investigation is necessary to tackle challenges such as accurately distinguishing closely stacked layers using a vision system, especially under varying lighting conditions. These limitations, coupled with issues of computational complexity and scalability in larger-scale bioprinting, emphasize the importance of enhancing the reliability of the vision-based approach across various conditions. Nonetheless, our innovation demonstrates substantial promise in creating precise biomimetic constructs and paves the way for future advancements in vision-guided robotic bioprinting, including the integration of multi-material printing techniques to enhance versatility.

Pre-sensor computing with compact multilayer optical neural network.

Moving computation units closer to sensors is becoming a promising approach to addressing bottlenecks in computing speed, power consumption, and data storage. Pre-sensor computing with optical neural networks (ONNs) allows extensive processing. However, the lack of nonlinear activation and dependence on laser input limits the computational capacity, practicality, and scalability. A compact and passive multilayer ONN (MONN) is proposed, which has two convolution layers and an inserted nonlinear layer, performing pre-sensor computations with designed passive masks and a quantum dot film for incoherent light. MONN has an optical length as short as 5 millimeters, two orders of magnitude smaller than state-of-the-art lens-based ONNs. MONN outperforms linear single-layer ONN across various vision tasks, off-loading up to 95% of computationally expensive operations into optics from electronics. Motivated by MONN, a paradigm is emerging for mobile vision, fulfilling the demands for practicality, miniaturization, and low power consumption.

Extraction of nonlinearity in neural networks with Koopman operator

Nonlinearity plays a crucial role in deep neural networks. In this paper, we investigate the degree to which the nonlinearity of the neural network is essential. For this purpose, we employ the Koopman operator, extended dynamic mode decomposition, and the tensor-train format. The Koopman operator approach has been recently developed in physics and nonlinear sciences; the Koopman operator deals with the time evolution in the observable space instead of the state space. Since we can replace the nonlinearity in the state space with the linearity in the observable space, it is a hopeful candidate for understanding complex behavior in nonlinear systems. Here, we analyze learned neural networks for the classification problems. As a result, the replacement of the nonlinear middle layers with the Koopman matrix yields enough accuracy in numerical experiments. In addition, we confirm that the pruning of the Koopman matrix gives sufficient accuracy even at high compression ratios. These results indicate the possibility of extracting some features in the neural networks with the Koopman operator approach.

Saturation and absorption effects on the transmission properties of Kerr-type nonlinear layers

The effects of absorption and saturation on soliton states in Kerr-type nonlinear layers are theoretically investigated. In addition to the usual gray and bright soliton structures observed in nonlinear slabs, a flat soliton, i.e., a particular soliton excitation with electric field amplitude independent of the position within the layer, is researched in cases of self-defocusing and self-focusing nonlinearities. Effects caused by the combination of absorption and saturation, such as the shift and extinction of the flat soliton peak, the decrease in the amplitude of the electric field within the nonlinear layer, and the suppression of the multistable behavior of the transmission coefficient in the vicinity of the soliton peaks, are discussed. The present theoretical results are compared and found in good quantitative agreement with previous experimental measurements.Graphical

Multi-task learning for hand heat trace time estimation and identity recognition

The thermal imager can capture the invisible thermal trace, and the potential information contained in the thermal trace can be extracted through the identification and analysis of the infrared thermal trace, such as the identity information and time information of the trace. This technology has great application value in criminal investigation, military and other fields. However, with the passage of time, the heat traces will gradually fade, and the intra-class distances will gradually increase. For hand heat trace, the trace may exhibit significant variations at different time points or in different states. This will pose challenges for extracting potential information and present significant challenges for model recognition. Therefore, we propose a multi task framework using deep convolutional neural networks (MTLHand) to jointly handle the two tasks of heat trace identification and heat trace departure time estimation. It can learn invariant representations of thermal trace identities, better capturing the dynamic changes in palmprint data, to enhance the accuracy of identity recognition. Specifically, soft threshold is inserted as nonlinear transformation layers into an improved ResNet to extract deep mixed hand trace features. The mixed features are decomposed into identity related features and time related features through spatial attention and channel attention. Use the multi task training method to carry out the corresponding learning tasks. In addition, we collected and constructed an infrared hand heat trace dataset containing labels such as departure time, gender, hand posture, identity category, etc., to promote the development of heat trace research. Experiments on this dataset show that our method is superior to other deep learning methods for the specific task of infrared heat trace recognition, and the identification accuracy can reach 83.48 %. In the trace leaving time estimation task, the recognition error rate is 18 % if the residual value between the estimated time and the real time is less than 60 s, and 3.55 % if the residual value is less than 120 s.

Non-linear radiative second-grade nano fluid with sinusoidal magnetic force and arrhenius activation energy: A computational exploration

The goal of this work is to further improve our knowledge of the nonlinear radiative second-grade nano fluid flow boundary layer phenomena which is associated with an Arrhenius activation energy, a sinusoidal magnetic field, and a stretched peripheral with a heat source. The unsteady governing equations are transformed into a proper dimensionless arrangement, and then the explicit finite difference (EFD) method is applied to numerically calculate the equations. However, precise stability and convergence criteria have been developed to make the solution convergent. Along with the typical profile of other flow fields, the oscillatory forms of the velocity are shown. Tabular research has even demonstrated a relationship between the Nusselt number and other parameters, and graphical depiction has been used for regression and data prediction. The novel conclusions drawn from this research indicate that, in comparison to linear patterns, nonlinear radiative heat flux significantly raises (30.35 %) flow profiles with second-grade characteristics. Moreover, the heat transfer rates of second-grade Nano fluids are seen to be significantly influenced (35.14 %) by the sinusoidal magnetic component. When considering nonlinear thermal radiation, activation energy principles cause a major change (34.19 % more) in mass transmission, as high-temperature processes become an essential part of chemical reactions.

High energy storage properties for dielectric composite by asymmetric three-layer films design

Polymer dielectrics are extensively studied for applications in electronics pulsed power systems owing to their excellent processability, high breakdown strength and low energy loss. To synergistically achieve high energy density and high discharge efficiency, we report an asymmetric three-layer all-polymer composite, which consists of a mixture of P(VDF-HFP) and P(St-MMA) as an intermediate layer sandwiched between a linear polymer P(St-MMA) dielectric layer and a nonlinear polymer P(VDF-HFP) dielectric layer. The results of the study showed that a high efficiency of 94.14 % and a high energy discharge density of 14.86 J/cm3 were achieved concurrently. In this case, the linear dielectric layer provides high energy efficiency, and the high energy density is provided by the nonlinear layer dielectric layer. In particular, the middle layer can effectively balance the electric field distribution, which improves the breakdown strength and increases the energy density. The structural design of asymmetric three-layer all-polymer films provides an effective way to realize high-performance dielectric energy storage materials.

Non-reciprocal optical bistability of sandwiched structure containing magnetic Weyl semimetals

We investigate non-reciprocal optical bistability (OB) of sandwiched structure comprising of two magnetic Weyl semimetal (WSM) layers and a nonlinear layer. The resonant tunneling mode in the sandwiched structure with a nonlinear layer creates favorable conditions for the emergence of OB. Besides, the inherent non-reciprocal nature of WSM enables non-reciprocal transmission of OB in the absence of an external magnetic field. By employing the transfer matrix method, we simulate the OB curves for forward and backward propagations, as well as analyze the non-reciprocity of OB. Furthermore, the impacts of various parameters on non-reciprocal OB are comprehensively discussed. The results demonstrate that the non-reciprocal OB curve and the width of the isolation of the non-reciprocal OB can be tuned by adjusting the thickness and the axial vector of the WSM, the temperature as well as the incident angle. This work would provide crucial insights for the design of high-performance non-reciprocal devices.

Numerical Study of Natural Convective Flow of a Nanofluid Over Rotating Truncated Cone with Convective Heating

In this paper, we consider the natural convection flow of a nanofluid simultaneously heat and mass transfer over a truncated rotating porous cone with convective boundary condition. Buongiorno’s model is used to describe the present nanofluid flow problem which is represented by Brownian motion and thermophoresis effects. Governing set of nonlinear boundary layer equations for the nanofluid over a rotating truncated cone is converted into a set of non-similar forms through non-similarity transformations. The obtained reduced nonlinear PDE system is linearized locally and solved via an accurate Chebyshev Spectral Collocation Method. The accuracy of the obtained numerical solutions is tested against existing results in the literature for specific cases and demonstrates good agreement. In addition, the impacts of active parameters such as the rotational parameter (0 ≤ χ ≤ 3), Biot number (0.1 ≤ Bi ≤ 1), and suction/injection parameter (−0.5 ≤ fw ≤ 0.5) on velocities, temperature, and nanoparticle volume fraction profiles along with heat and mass transfer rates are examined. It is found that the local surface drag and heat transfer rate increase, and the nanoparticle mass transfer rate decreases, with the increase in both the spinning parameter and Biot number.

The Impact of Volume Fraction and Nanoparticles in the Flow of Nanofluids along the Boundary Layer Over a Flat Plate

Introduction: The forced convective, steady, viscous, nonlinear boundary layer flow of a two-dimensional problem over a flat plate with heat transfer nanofluids. The flow is laminar. Two different types of nanoparticles, namely, Titanium dioxide () and Silver are used to prepare nanofluids with base fluid is chosen as water. Nanofluids has a large number of applications in the area of biomedical, cooling and tribological applications. Objectives: To study about the effect different nanoparticles and their volume fraction in the flow of nanofluids along the boundary layer over a flat plate. Analyse the changes in the viscosity, density, specific heat capacity, thermal conductivity, Prandtl number, skin friction coefficient and Nusselt number. Methods: The flow of the fluid that are governed by the PDEs that are nonlinear are transformed into the ODEs that are nonlinear by employing similarity transformations, which then solved by the use of MATLAB bvp5c solution procedure. Results: Both thermal conductivity and heat transfer coefficient increases with the increase in the volume fraction of the corresponding nanoparticle. Also, Silver-water nanofluid has the high thermal conductivity and heat transfer rate when compared with Titanium dioxide-water nanofluid. Conclusions: Silver water nanofluid surpasses titanium dioxide water nanofluid in terms of thermal conductivity and heat transfer rate. Moreover, the physical attributes like viscosity, density, and thermal conductivity of nanofluids escalate proportionally with the volume fraction of nanoparticles. Conversely, specific heat capacity and Prandtl number exhibit an inverse relationship with the volume fraction of nanoparticles, decreasing as it increases.

Tunable and narrowband shortwave infrared light sensing enabled by dual-Fano resonance enhanced sum-frequency generation

Tunable and narrowband light detection provides a means of selectively detecting optical signals at a specific wavelength, enabling a precise tool for object identification, machine vision, spectroscopy, and so on. Simultaneous tunable and narrowband response in shortwave infrared (SWIR) detectors is critical yet still challenging. This work utilizes dual-Fano resonance enhanced sum-frequency generation in a two-layer structure comprising a silicon metasurface and a two-dimensional (2D) nonlinear GaSe layer to realize tunable and narrowband light detection in the SWIR range. The silicon metasurface affords a high-quality-factor dual-Fano resonance in the SWIR regime, which enhances the near-field optical density of the two resonant wavelengths (pump and signal) when passing through the 2D nonlinear layer, leading to drastically enhanced sum-frequency generation. The sum-frequency light at a visible wavelength that contains the information of the SWIR signal light, can then be detected by a low-noise visible detector. The tunability and selectivity in the response spectrum stem from the geometry-dependent dual-Fano resonance in the silicon metasurface, covering the 1200–1550 nm range. The upconversion detector exhibits a sub-nanometer narrowband detection with a full width at half maximum of down to ∼0.1 nm, owing to the high quality factor of the Fano resonances. This SWIR narrowband detection is one of the best performances reported so far, much narrower than commercial filter products. The peak value of the specific detectivity of 1.5 × 1012 Jones at 1256.3 nm is achieved, comparable to broadband commercial InGaAs detectors. The detector designs in this work open up the opportunity of upconversion sensors for delicate spectroscopic applications.

An information entropy-based fuzzy stochastic configuration network for robust data modeling

In data-driven modeling, the generalization performance of a neural network based on the minimum mean square error criterion may be affected when the samples contain noise. To handle uncertain data modeling problems, this study contributes a novel robust data modeling method called information entropy-based fuzzy stochastic configuration networks (IEFSCNs), which incorporates the maximum correntropy criterion (MCC) to reflect the squared error of the training sample set and uses the quantized minimum error entropy (QMEE) criterion to mine the distribution structure of the modeling data and suppress outlier data. Fuzzy systems are designed to generate nodes in the fuzzy transformation layer. A nonlinear enhancement layer is established as a hidden layer to enhance the representational capacity of the target system. The fuzzy transformation and nonlinear enhancement layers are fully connected to determine the outputs of the IEFSCNs. In addition, an incremental parameter learning algorithm is developed based on a supervision mechanism for optimizing the cost function, and its convergence property is demonstrated. Owing to the superiorities of the cost function by synthesizing MCC and QMEE, IEFSCNs can improve the modeling accuracy and reduce noise suppression. The performances are evaluated using various regression and classification tasks. Compared with existing methods, including SCNs, RVFL, F-SCNs, RSC-KDE, RSC-MCC, and SM-RSC, the proposed method demonstrates an effective robust modeling performance by reducing network structure redundancy while enhancing its ability to suppress noisy data.

Propagation properties of a non-linear mapping based on squaring in odd characteristic

Many modern cryptographic primitives for hashing and (authenticated) encryption make use of constructions that are instantiated with an iterated cryptographic permutation that operates on a fixed-width state consisting of an array of bits. Often, such permutations are the repeated application of a relatively simple round function consisting of a linear layer and a non-linear layer. These constructions do not require that the underlying function is a permutation and they can plausibly be based on a non-invertible transformation. Recently, Grassi proposed the use of non-invertible mappings operating on arrays of digits that are elements of a finite field of odd characteristic for so-called MPC-/FHE-/ZK-friendly symmetric cryptographic primitives. In this work, we consider a mapping that we call γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document} that has a simple expression and is based on squaring. We discuss, for the first time, the differential and linear propagation properties of γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document} and observe that these follow the same rules up to a relabeling of the digits. This is an intriguing property that, as far as we know, only exists for γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document} and the binary mapping χ3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi _{_{3}}$$\\end{document} that is used in the cryptographic permutation Xoodoo. Moreover, we study the implications of its non-invertibility on differentials with zero output difference and on biases at the output of the γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document} mapping and show that they are as small as they can possibly be.