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Let n and t be integers with 1 < t ≤ n . Let A be an n × n (0, 1)-matrix with a positive permanent, that is, for which there exists a permutation matrix P ≤ A (entrywise order). We investigate the minimum number α (n , t) of zeros possible in such a matrix A which avoids a P with a 12 ⋯ t -pattern, that is, for which there does not exist a permutation matrix P ≤ A containing the identity matrix I t as a submatrix. We conjecture that α (n , t) = k + 1 2 where k = n - t + 1 . We prove this conjecture is correct when t = 2 or 3 and we consider for which matrices equality holds. We also prove the conjecture is correct for all t if n ≥ 2 k - 3 . Finally, we investigate which 12 ⋯ t -permutation avoiding matrices have the maximum permanent. [ABSTRACT FROM AUTHOR]

In 2019, Bóna and Smith introduced the notion of strong pattern avoidance for which a permutation and its square both avoid a given pattern. In this paper, we enumerate the set of permutations π which not only strongly avoid the pattern 312 or 231 but also avoid the pattern τ , for all τ ∈ S 3 and some τ ∈ S 4 . One of these results gives a positive answer to a conjecture of Archer and Geary. [ABSTRACT FROM AUTHOR]

In digital communication, the protection of digital images containing user-sensitive information is an emerging area of research. In the literature, several image encryption schemes have been proposed but they might not be appropriate for use in real-world situations. Some of them are computationally expensive, and some do not offer high security. In this article, we introduce an innovative and robust solution to this challenge: a secure multi-index multi-step image encryption scheme specifically designed for greyscale images. The proposed encryption scheme uses a hybrid combination of substitution, permutation and hyperchaotic structure. The substitution operation is achieved through a substitution box that is defined over the finite field. The permutation operation is achieved through a multi-index permutation algorithm. A hyperchaotic sequence is generated from a four-dimensional hyperchaotic map that is used in bitwise XOR operation for providing masking of image pixels. A comprehensive evaluation of our proposed encryption scheme against established security benchmarks reveals its resilience. Our proposed scheme not only passes all well-known security tests but also attains test values closely aligned with optimal benchmarks. Notably, our encryption scheme exhibits noteworthy efficiency, making it well suited for deployment on low-memory devices. This work not only contributes to advancing the field of digital communication but also underscores the practicality and efficiency of the proposed scheme for real-world applications [ABSTRACT FROM AUTHOR]

The paper proposes a novel difference-based method for testing the hypothesis of no relationship between the dependent and independent variables. We construct three test statistics for nonparametric regression with Gaussian and non-Gaussian random errors. These test statistics have the standard normal as the asymptotic null distribution. Furthermore, we show that these tests can detect local alternatives that converge to the null hypothesis at a rate close to n - 1 / 2 previously achieved only by the residual-based tests. We also propose a permutation test as a flexible alternative. Our difference-based method does not require estimating the mean function or its first derivative, making it easy to implement and computationally efficient. Simulation results demonstrate that our new tests are more powerful than existing methods, especially when the sample size is small. The usefulness of the proposed tests is also illustrated using two real data examples. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new method for counting the number of connected components of multi-curves. Our method is based on associating multi-curves with permutations, where we can see that the number of connected components of a multi-curve is directly related to the number of cycles in a cycle decomposition of the corresponding permutation. Our ultimate goal is to obtain a formula about multi-curves that gives the number of connected components of the curves, and obtaining a formula for the number of cycles of a permutation will also accomplish this goal. While we have several naive methods for counting the number of cycles of permutations, none of them gives us such a formula. As an approach, we develop a new combinatorial technique to count the number of cycles of permutations by introducing a new notation of permutation. [ABSTRACT FROM AUTHOR]

Image based screening is now routinely available for early detection of cancer and other diseases. Quantitative analysis for effects of risk factors on digital images is important to extract biological insights for modifiable factors in prevention studies and understand pathways for targets in preventive drugs. However, current approaches are restricted to summary measures within the image with the assumption that all relevant features needed to characterize an image can be identified and appropriately quantified. Motivated by data challenges in breast cancer, we propose a nonparametric statistical framework for risk factor screening that uses the whole mammogram image as outcome. The proposed permutation test allows assessment of whether a set of scalar risk factors is associated with the whole image in the presence of correlated residuals across the spatial domain. We provide extensive simulation studies and illustrate an application to the Joanne Knight Breast Health Cohort using the mammogram imaging data. [ABSTRACT FROM AUTHOR]

The first Adaptive COVID‐19 Treatment Trial (ACTT‐1) showed that remdesivir improved COVID‐19 recovery time compared with placebo in hospitalized adults. The secondary outcome of mortality was almost significant overall (p = 0.07) and highly significant for people receiving supplemental oxygen at enrollment (p = 0.002), suggesting a mortality benefit concentrated in this group. We explore analysis methods that are helpful when a single subgroup benefits from treatment and apply them to ACTT‐1, using baseline oxygen use to define subgroups. We consider two questions: (1) is the remdesivir effect for people receiving supplemental oxygen real, and (2) does this effect differ from the overall effect? For Question 1, we apply a Bonferroni adjustment to subgroup‐specific hypothesis tests and the Westfall and Young permutation test, which is valid when small cell counts preclude normally distributed test statistics (a frequently unexamined condition in subgroup analyses). For Question 2, we introduce Qmax, the largest standardized difference between subgroup‐specific effects and the overall effect. Qmax simultaneously tests whether any subgroup effect differs from the overall effect and identifies the subgroup benefitting most. We demonstrate that Qmax strongly controls the familywise error rate (FWER) when test statistics are normally distributed with no mean–variance relationship. We compare Qmax to a related permutation test, SEAMOS, which was previously proposed but not extensively applied or tested. We show that SEAMOS can have inflated Type 1 error under the global null when control arm event rates differ between subgroups. Our results support a mortality benefit from remdesivir in people receiving supplemental oxygen. [ABSTRACT FROM AUTHOR]

Localising effects in space, time and other dimensions is a fundamental goal of magneto‐ and electroencephalography (EEG) research. A popular exploratory approach applies mass‐univariate statistics followed by cluster‐sum inferences, an effective way to correct for multiple comparisons while preserving high statistical power by pooling together neighbouring effects. Yet, these cluster‐based methods have an important limitation: each cluster is associated with a unique p‐value, such that there is no error control at individual timepoints, and one must be cautious about interpreting when and where effects start and end. Sassenhagen and Draschkow (2019) provided an important reminder of this limitation. They also reported results from a simulation, suggesting that onsets estimated from EEG data are both positively biased and very variable. However, the simulation lacked comparisons to other methods. Here, I report such comparisons in a new simulation, replicating the positive bias of the cluster‐sum method, but also demonstrating that it performs relatively well, in terms of bias and variability, compared to other methods that provide pointwise p‐values: two methods that control the false discovery rate and two methods that control the familywise error rate (cluster‐depth and maximum statistic methods). I also present several strategies to reduce estimation bias, including group calibration, group comparison and using binary segmentation, a simple change point detection algorithm that outperformed mass‐univariate methods in simulations. Finally, I demonstrate how to generate onset hierarchical bootstrap confidence intervals that integrate variability over trials and participants, a substantial improvement over standard group approaches that ignore measurement uncertainty. [ABSTRACT FROM AUTHOR]

We characterize the permutative automorphisms of the Cuntz algebra \mathcal {O}_n (namely, stable permutations) in terms of two sequences of graphs that we associate to any permutation of a discrete hypercube [n]^t. As applications we show that in the limit of large t (resp. n) almost all permutations are not stable, thus proving Conj. 12.5 of Brenti and Conti [Adv. Math. 381 (2021), p. 60], characterize (and enumerate) stable quadratic 4 and 5-cycles, as well as a notable class of stable quadratic r-cycles, i.e. those admitting a compatible cyclic factorization by stable transpositions. Some of our results use new combinatorial concepts that may be of independent interest. [ABSTRACT FROM AUTHOR]

For a natural number n, the permutation (n!) is defined as the left-to-right product of the first n cycles, namely,. In this article, we prove that for any natural number n, 2 is a primitive root of 2n + 1 if and only if 2n + 1 = pk for some odd prime number p and for some natural number k such that the permutation (n!) has exactly k orbits. We also prove that a prime number p is a Sophie Germain prime if and only if the permutation (p!) has at most two orbits. [ABSTRACT FROM AUTHOR]

Smart door locks pose a large number of threats such as network attacks. Its storage area and power of cipher are severely limited because the wireless nodes of smart door locks are mostly battery‐powered. Therefore, effective security solutions are urgently needed. In this paper, a new lightweight block cipher with low power named LPHD is proposed to ensure the security of the master control chip of the smart door lock terminal. We design a scheme of low power S‐box and construct the two‐stage permutation layer (TP structure) suitable for LPHD by filtering the sets of 8‐bit permutations. LPHD proposes a variant of the 8‐branch generalized Feistel structure (GFS) to realize that the bits of all branches are affected in one encryption round. The problem of slow diffusion in the standard Feistel structure is solved. The key schedule adopts the nonlinear design and reuses the encryption process of LPHD. It improves the security of the cipher and reduces hardware overhead. Moreover, we evaluate the hardware implementation and security of LPHD. The results show that LPHD for the unified encryption and decryption circuits requires only 1276 Gate Equivalents (GEs) and 1.914 μW on UMC 0.18 μm, which is better than other lightweight block ciphers such as SKINNY, PRESENT, and IVLBC. In summary, LPHD provides sufficient security for the master control chip of the smart door lock terminal. [ABSTRACT FROM AUTHOR]

Fermat’s two squares theorem asserts that a prime one more than a multiple of 4 is a sum of two squares. There are many proofs of this gem in number theory, including a remarkable one-sentence proof by Don Zagier based on two involutions on a finite set built from such a prime. Applying the two involutions alternatively leads to an iterative algorithm to find the two squares for the prime. Moreover, a detailed analysis of the computation reveals that it is possible to jump through the iteration nodes, leading to a better hopping algorithm. Here is a formalisation of Zagier’s proof, deriving the involutions using windmill patterns. Theories developed for the formal proof are used to establish the correctness of both algorithms. [ABSTRACT FROM AUTHOR]

This study proposes a highly efficient image encryption algorithm by employing a rapid key generation approach and permutation structure. The image is converted to a matrix, and then an encryption algorithm based on factorial decomposition permutation is applied. Two variants of the algorithm have been proposed in this study, where each variant is distinguished by the elements of the matrix to be permutated. The first variant is based on the permutation of the pixels of the image. In the second variant, the permutation is applied to both columns and rows of the matrix. These variants of the algorithm have been tested and compared. To create a permutation of a collection of elements, the factorial decomposition mathematical technique is applied, where the Euclidian division of a given key is obtained by adding the factorials of all the integers. The experimental results indicate that the proposed approach provides sufficient and optimistic results in terms of computational complexity, Keyspace analysis, Statistical analysis, and Sensitivity analysis attacks. The statistical analysis shows the superiority of the algorithm using two permutation methods, where sensitivity analysis indicates that the number of pixels changing rate (NPCR) achieved around 99.7 and the unified average changed intensity (UACI) is around 33.5, which showed better performance than the other approaches in the literature. Moreover, the proposed approach provided less computational complexity compared with the existing approaches. [ABSTRACT FROM AUTHOR]

Let A, B be n × m matrices. The concept of matrix majorization means the jth column of A is majorized by the jth column of B and this is done for all j by a doubly stochastic matrix D. We define rc-majorization that extended matrix majorization to columns and rows of matrices. Also, the linear preservers of rc-majorization will be characterized. [ABSTRACT FROM AUTHOR]

We define a partial order P n on permutations of any given size n, which is the image of a natural partial order on inversion sequences. We call this the "middle order." We demonstrate that the poset P n refines the weak order on permutations and admits the Bruhat order as a refinement, justifying the terminology. These middle orders are distributive lattices and we establish some of their combinatorial properties, including characterization and enumeration of intervals and boolean intervals (in general, or of any given rank), and a combinatorial interpretation of their Euler characteristic. We further study the (not so well-behaved) restriction of this poset to involutions, obtaining a simple formula for the Möbius function of principal order ideals there. Finally, we offer further directions of research, initiating the study of the canonical Heyting algebra associated with P n , and defining a parking function analogue of P n . [ABSTRACT FROM AUTHOR]

With the rapid advancement of multimedia technology, exchange of digital images has significantly increased. This advancement of information exchange in the form of digital images comes with its own challenge of securing them, from being altered during storage or transmission. In health care sector, medical professionals often need to share medical images such as X-ray, MRI, CT-Scan etc., with other specialists or healthcare facilities for correct disease diagnosis and treatment planning. Encryption allows secure sharing of these images over insecure network, by maintaining the integrity of medical data, which is essential as these medical images contain the personal data of patients. For encryption, chaotic maps have become a popular approach due to their significant characteristics such as sensitivity to initial conditions and ergodicity. In this paper, a novel One-dimensional Cosine within Sine (1D-CwS) chaotic map has been proposed. The proposed novel 1D-CwS chaotic map has been evaluated using Bifurcation diagram, Shannon Entropy and Lyapunov Exponent parameters, to prove that the proposed chaotic map is more complex in behaviour, more random in nature and provides larger chaotic range than many existing chaotic maps. Also, the work in this paper discusses a novel method of permutation and diffusion to propose a new medical image encryption (MIE) technique that uses the novel 1D-CwS chaotic map. The MIE scheme is divided into three major phases. In first phase, padding is done, if the given image dimensions are not square. In second phase, pixel level permutation is performed to reposition the pixels by generating unique random numbers, and by applying division and modulus operations on these unique random numbers. In final phase, diffusion process is performed to change the pixel values, which uses chaotic sequence generated by the novel 1D-CwS map. The proposed MIE has been analysed using standard parameters, which demonstrate that the proposed cryptosystem is resistant to various types of attacks. [ABSTRACT FROM AUTHOR]

The internal kink mode is one of the crucial factors affecting the stability of magnetically confined fusion devices. This paper explores the key features influencing the growth rate of internal kink modes using machine learning techniques such as Random Forest, Extreme Gradient Boosting (XGboost), Permutation, and SHapley Additive exPlanations (SHAP). We conduct an in-depth analysis of the significant physical mechanisms by which these key features impact the growth rate of internal kink modes. Numerical simulation data were used to train high-precision machine learning models, namely Random Forest and XGBoost, which achieved coefficients of determination values of 95.07% and 94.57%, respectively, demonstrating their capability to accurately predict the growth rate of internal kink modes. Based on these models, key feature analysis was systematically performed with Permutation and SHAP methods. The results indicate that resistance, pressure at the magnetic axis, viscosity, and plasma rotation are the primary features influencing the growth rate of internal kink modes. Specifically, resistance affects the evolution of internal kink modes by altering current distribution and magnetic field structure; pressure at the magnetic axis impacts the driving force of internal kink modes through the pressure gradient directly related to plasma stability; viscosity modifies the dynamic behavior of internal kink modes by regulating plasma flow; and plasma rotation introduces additional shear forces, affecting the stability and growth rate of internal kink modes. This paper describes the mechanisms by which these four key features influence the growth rate of internal kink modes, providing essential theoretical insights into the behavior of internal kink modes in magnetically confined fusion devices. [ABSTRACT FROM AUTHOR]

In 2016, Y. Kumar et al. in the paper 'Affine equivalence and non-linearity of permutations over Z n ' conjectured that: For n ≥ 3 , the nonlinearity of any permutation on Z n , the ring of integers modulon, cannot exceed n - 2 . For an odd prime p, we settle the above conjecture when n = 2 p and for p ≡ 3 (mod 4) we prove the above conjecture with an improved upper bound. Further, we derive a lower bound on max N L n when n is an odd prime or twice of an odd prime where max N L n denotes the maximum possible nonlinearity of any permutation on Z n . [ABSTRACT FROM AUTHOR]

Digital image security has become a major concern due to open networks and Internet data sharing. Many image encryption algorithms have been introduced recently for securely sharing data, but they have some drawbacks and efficiency issues. Cryptography uses chaotic systems because of their high randomization properties and accuracy. To overcome the constraints inherent in conventional encryption algorithms, a novel and secure image encryption algorithm based on fractional-order (FO) 4D hyperchaotic system (4D-HCS) and Arnold Cat map (ACM) is proposed in this paper to ensure secure transmission of color images. The proposed algorithm is mainly based on pixel rotation, permutation, bit-wise XOR operations, and substitution. ACM is used for pixel rotation in the color image, eliminating the high correlations between neighboring pixels without obscuring the information. Adding permutations after separating the red ( R c ), green ( G c ), and blue ( B c ) color channels can enhance the confusion. The color image pixel values are diffused by using substitution through hyperchaotic sequences of FO 4D-HCS. The algorithm offers an impressive key space of approximately 2 548 for high cryptographic strength and showcases fast encryption. The proposed algorithm for color image encryption is examined in light of various cryptographic characteristics. These tests yield promising results, assuring that the proposed algorithm for color image encryption is secure and resistant to different standard attacks. [ABSTRACT FROM AUTHOR]

A permutation is said to be cycle-alternating if it has no cycle double rises, cycle double falls or fixed points; thus each index i is either a cycle valley ( σ - 1 (i) > i < σ (i) ) or a cycle peak ( σ - 1 (i) < i > σ (i) ). We find Stieltjes-type continued fractions for some multivariate polynomials that enumerate cycle-alternating permutations with respect to a large (sometimes infinite) number of simultaneous statistics that measure cycle status, record status, crossings and nestings along with the parity of the indices. Our continued fractions are specializations of more general continued fractions of Sokal and Zeng. We then introduce alternating Laguerre digraphs, which are generalization of cycle-alternating permutations, and find exponential generating functions for some polynomials enumerating them. We interpret the Stieltjes–Rogers and Jacobi–Rogers matrices associated to some of our continued fractions in terms of alternating Laguerre digraphs. [ABSTRACT FROM AUTHOR]

AbstractWe prove that an element of the symmetric group of degree n>3 can be represented as the commutator of two permutations whose supports intersect at exactly p≤n points if and only if it is the product of p cycles of length 3. As an accompanying result, and also as a step of the main result’s proof, we give a lower bound for p to express an even permutation as the product of p cycles of length 3, in terms of its decomposition into disjoint cycles. [ABSTRACT FROM AUTHOR]

AbstractWe propose a two-step procedure to model and predict high-dimensional functional time series, where the number of function-valued time series p is large in relation to the length of time series n. Our first step performs an eigenanalysis of a positive definite matrix, which leads to a one-to-one linear transformation for the original high-dimensional functional time series, and the transformed curve series can be segmented into several groups such that any two subseries from any two different groups are uncorrelated both contemporaneously and serially. Consequently in our second step those groups are handled separately without the information loss on the overall linear dynamic structure. The second step is devoted to establishing a finite-dimensional dynamical structure for all the transformed functional time series within each group. Furthermore the finite-dimensional structure is represented by that of a vector time series. Modeling and forecasting for the original high-dimensional functional time series are realized via those for the vector time series in all the groups. We investigate the theoretical properties of our proposed methods, and illustrate the finite-sample performance through both extensive simulation and two real datasets. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work. [ABSTRACT FROM AUTHOR]

In the last few years, the volume of internet traffic has increased enormously. A significant portion of the internet traffic is constituted by applications that stream multimedia data from one part of the world to another. The enormity of the video data requires the Content Delivery Network to distribute the content efficiently and securely among the consumers. The secure and authenticated distribution can be achieved by the end-to-end encryption of video. In this paper, one time random key algorithm and the algorithms and methods based on permutation techniques have been proposed to encrypt the video frames selectively and these encrypted frames are transmitted over the network along with the other video frames. The proposed encryption methods extract the colour channels like Red, Green and Blue from the video frames, and they are encrypted by applying row-column permutation cipher, One Time Pad cipher and row permutation cipher. Encryption of the video frames has been performed at two levels by combining the one-time random key algorithm and permutation techniques to provide security to the video. These encryption techniques have been applied on the twenty-five, thirty, and thirty-five, and so on, up to fifty per cent of the total frames available in the video. The experimental results, histogram analysis and parameters like average Peak Signal-to-noise Noise Ratio (PSNR) and Structural Similarity Index (SSIM) values measure the quality of the video obtained after the application of the proposed methods on the video to be transmitted over the network. [ABSTRACT FROM AUTHOR]

Within the context of Indian religions, Jainism has long been recognized for its extensive use of permutations and combinations. However, the application of these principles within Buddhist scriptures has received relatively little scholarly attention. This paper introduces a new example of the specific application of permutations and combinations in Buddhist scriptures. In this paper, we focus on the first saṅghādisesa rule in the Theravāda-vinaya, which lists a series of element sets and arranges these elements according to a certain pattern known as "ten-roots" (mūla), and we discover that these arrangements form a regular numerical sequence, called "oblong numbers". Moreover, similar patterns with different quantities are also found in the fourth Pārājika and the fifth saṅghādisesa rules. This indicates that the compilers of the Theravāda-vinaya did not use this mathematical knowledge without basis. Interestingly, we also found the use of this sequence in the Bakhshālī manuscript. Therefore, in this article, after summarizing and verifying the arrangement rules of the Theravāda-vinaya, we discuss whether the oblong numbers were influenced by Greek mathematics. [ABSTRACT FROM AUTHOR]

We consider representations of an arbitrary permutation π of degree 2n, n ⩾ 3, by products of the so-called (2n)-permutations (any cycle of such a permutation has length 2). We show that any even permutation is represented by the product of four (2n)-permutations. Products of three (2n)-permutations cannot represent all even permutations. Any odd permutation is realized (for odd n) by a product of five (2n)-permutations. [ABSTRACT FROM AUTHOR]

In this paper, an efficient and high-performance image encryption method with high key and plain-image sensitivity. The encryption process is designed by using a low-cost block cipher like the PRESENT cipher. The proposed method has a simple structure with minimum computations. The block size and key size in the cipher are 128-bit. The structure is designed by using different operations such as non-linear substitution box (S-box), permutation, and Pre-processing units for key and plain-image. These operations provide confusion and diffusion properties for a secure image encryption system. To increase the key and plain-image sensitivity, the main key and input image are changed based on the Pre-processing units. The presented method does not depend on the input image and bit pattern of the main key. In other words, the encryption image has no information that is affected by the input data. The security analyses such as correlation coefficient analysis, histogram analysis, entropy, key sensitivity, and differential attack (the number of pixel change rate (NPCR) and the unified average changing intensity (UACI) values) are performed on encrypted images. The achieved results for the proposed structure are close to that of a completely random image with high security. [ABSTRACT FROM AUTHOR]

Given a finite set of points C ⊆ R d , we say that an ordering of C is protrusive if every point lies outside the convex hull of the points preceding it. We give an example of a set C of 5 points in the Euclidean plane possessing a protrusive ordering that cannot be obtained by ranking the points of C according to the sum of their distances to a finite multiset of points. This answers a question of Alon, Defant, Kravitz, and Zhu. [ABSTRACT FROM AUTHOR]

We consider the problem of determining which matrices are permutable to be supmodular. We show that for small dimensions any matrix is permutable by a universal permutation or by a pair of permutations, while for higher dimensions no universal permutation exists. We raise several questions including to determine the dimensions in which every matrix is permutable. [ABSTRACT FROM AUTHOR]

Test of independence is of fundamental importance in modern data analysis, with broad applications in variable selection, graphical models, and causal inference. When the data is high dimensional and the potential dependence signal is sparse, independence testing becomes very challenging without distributional or structural assumptions. In this article, we propose a general framework for independence testing by first fitting a classifier that distinguishes the joint and product distributions, and then testing the significance of the fitted classifier. This framework allows us to borrow the strength of the most advanced classification algorithms developed from the modern machine learning community, making it applicable to high dimensional, complex data. By combining a sample split and a fixed permutation, our test statistic has a universal, fixed Gaussian null distribution that is independent of the underlying data distribution. Extensive simulations demonstrate the advantages of the newly proposed test compared with existing methods. We further apply the new test to a single cell dataset to test the independence between two types of single cell sequencing measurements, whose high dimensionality and sparsity make existing methods hard to apply. for this article are available online. [ABSTRACT FROM AUTHOR]

For a Coxeter system (W, S) and a subset J ⊆ S, fix u ∈ WJ and ξ ∈ WJ, let λ(−, u) : WJ → W and λ(ξ, −) : WJ → W be the mappings defined by λ(x, u) = xu and λ(ξ, y) = ξy, respectively. We show that λ(−, u) is left weak Bruhat order preserving for all u ∈ WJ, and that λ(ξ,−) is left weak Bruhat order-preserving if and only if it is a convex embedding, if and only if `(y) = `(ξyξ−1 ) for all y ∈ WJ. For the Coxeter groups of types A and B, we explicitly describe the ξ that satisfies the previous equivalent conditions in a purely combinatorial manner. [ABSTRACT FROM AUTHOR]

We initiate the investigation of the parameterized complexity of Diameter and Connectivity in the streaming paradigm. On the positive end, we show that knowing a vertex cover of size k allows for algorithms in the Adjacency List (AL) streaming model whose number of passes is constant and memory is O (log n) for any fixed k. Underlying these algorithms is a method to execute a breadth-first search in O (k) passes and O (k log n) bits of memory. On the negative end, we show that many other parameters lead to lower bounds in the AL model, where Ω (n / p) bits of memory is needed for any p-pass algorithm even for constant parameter values. In particular, this holds for graphs with a known modulator (deletion set) of constant size to a graph that has no induced subgraph isomorphic to a fixed graph H, for most H. For some cases, we can also show one-pass, Ω (n log n) bits of memory lower bounds. We also prove a much stronger Ω (n 2 / p) lower bound for Diameter on bipartite graphs. Finally, using the insights we developed into streaming parameterized graph exploration algorithms, we show a new streaming kernelization algorithm for computing a vertex cover of size k. This yields a kernel of 2k vertices (with O (k 2) edges) produced as a stream in poly (k) passes and only O (k log n) bits of memory. [ABSTRACT FROM AUTHOR]

We write $\mathcal {S}_n(A)$ for the set of permutations of a set A with n non-fixed points and $\mathrm {{seq}}^{1-1}_n(A)$ for the set of one-to-one sequences of elements of A with length n where n is a natural number greater than $1$. With the Axiom of Choice, $|\mathcal {S}_n(A)|$ and $|\mathrm {{seq}}^{1-1}_n(A)|$ are equal for all infinite sets A. Among our results, we show, in ZF, that $|\mathcal {S}_n(A)|\leq |\mathrm {{seq}}^{1-1}_n(A)|$ for any infinite set A if ${\mathrm {AC}}_{\leq n}$ is assumed and this assumption cannot be removed. In the other direction, we show that $|\mathrm {{seq}}^{1-1}_n(A)|\leq |\mathcal {S}_{n+1}(A)|$ for any infinite set A and the subscript $n+1$ cannot be reduced to n. Moreover, we also show that " $|\mathcal {S}_n(A)|\leq |\mathcal {S}_{n+1}(A)|$ for any infinite set A " is not provable in ZF. [ABSTRACT FROM AUTHOR]

Permutation techniques have been used extensively in machine learning algorithms for evaluating variable importance. In ordinary regression, however, variables are often removed to gauge their importance. In this paper, we compared the results for permuting variables to removing variables in regression to assess relations between these two methods. We compared permute-and-predict (PaP) methods with leave-one-covariate-out (LOCO) techniques. We also compared these results with conventional metrics such as regression coefficient estimates, t-statistics, and random forest out-of-bag (OOB) PaP importance. Our results indicate that permutation importance metrics are practically equivalent to those obtained from removing variables in a regression setting. We demonstrate a strong association between the PaP metrics, true coefficients, and regression-estimated coefficients. We also show a strong relation between the LOCO metrics and the regression t-statistics. Finally, we illustrate that manual PaP methods are not equivalent to the OOB PaP technique and suggest prioritizing the use of manual PaP methods on validation data.

The substitution box (S-box) is one of the extremely important components in the design of block cipher. An excellent S-box is necessary for the block cipher algorithm, and its cipher strength directly affects the security of the cipher algorithm. The differential uniformity of the S-box generated by the chaotic system is 10 or 12, which cannot effectively resist differential cryptanalysis. Aiming at the high differential uniformity of the S-box constructed by the chaotic system, a novel efficient S-box construction scheme based on a new chaotic map and permutation is proposed in this paper. In this scheme, the chaotic matrix is generated by a new chaotic map, and then is replaced by permutation sequences to generate S-boxes. Comparative analysis shows that the generated S-boxes have high nonlinearity, low differential uniformity, and satisfy SAC and BIC criteria, which can improve the ability of the algorithm to resist differential cipher attacks and linear cryptographic analysis. [ABSTRACT FROM AUTHOR]

Complete non-ambiguous trees (CNATs) are combinatorial objects which appear in various contexts. Recently, Chen and Ohlig studied the notion of permutations associated to these objects, and proposed a series of nice conjectures. Most of themwere proved by Selig and Zhu, through a connection with the abelian sandpile model. But one conjecture remained open, about the distribution of a natural statistic named determinant. We prove this conjecture, in a bijective way. [ABSTRACT FROM AUTHOR]

Affiliative relationships are well documented in a wide range of wild animals. However, there is limited evidence of their existence in social farm animals, despite potential significant animal welfare implications. We aimed to determine whether pigs have preferential partners for affiliative interactions and associations and whether these preferences remain stable over time or are influenced by sex, dominance rank or litter of origin (i.e. kinship or familiarity). Two pens of 12 domestic pigs, Sus scrofa , two males and two females from three litters, were weaned at 28 days of age and observed from 42 to 57 days (postweaning phase) and from 70 to 85 days (finishing phase) following a change of housing at 62 days. Sociopositive behaviours, including nose-to-nose and nose-to-body contacts, allogrooming and associations while lying were scored on video observations over 6 days per phase. Using social network analysis methods, we found nonrandom social preferences for allogrooming in both pens of pigs, which remained stable between the postweaning and finishing phases. We also observed nonrandom associations for proximities while lying, although these patterns were not found in both pens of pigs and did not systematically persist between phases. In contrast, nose-to-nose interactions were expressed randomly in both pens and phases. Depending on the housing conditions, social preferences for grooming interactions or proximity while lying were only weakly influenced by sex, dominance rank or litter of origin. We did not find any correlation between social networks for grooming interactions and associations while lying. In summary, our study highlights the existence of durable affiliative relationships based on grooming interactions and proximity while lying within stable pig groups. These relationships appear to be minimally affected by sex, dominance, kinship or familiarity. Taking these relationships into account, particularly from a young age, could be pivotal in improving the welfare of pigs on farms. • Pigs showed preferential partners for allogrooming and lying together. • Social preferences for allogrooming persisted after a change of housing. • Lying proximity and social interactions were distinct facets of social affiliation. • Nose-to-nose contacts occurred randomly within groups of pigs. • Affiliation in pigs was not primarily driven by sex, dominance or kinship. [ABSTRACT FROM AUTHOR]

Permutation techniques have been used extensively in machine learning algorithms for evaluating variable importance. In ordinary regression, however, variables are often removed to gauge their importance. In this paper, we compared the results for permuting variables to removing variables in regression to assess relations between these two methods. We compared permute-and-predict (PaP) methods with leave-one-covariate-out (LOCO) techniques. We also compared these results with conventional metrics such as regression coefficient estimates, t-statistics, and random forest out-of-bag (OOB) PaP importance. Our results indicate that permutation importance metrics are practically equivalent to those obtained from removing variables in a regression setting. We demonstrate a strong association between the PaP metrics, true coefficients, and regression-estimated coefficients. We also show a strong relation between the LOCO metrics and the regression t-statistics. Finally, we illustrate that manual PaP methods are not equivalent to the OOB PaP technique and suggest prioritizing the use of manual PaP methods on validation data. [ABSTRACT FROM AUTHOR]

Testing zero variance components is of utmost importance in various applications empowered by the use of mixed-effects models. Focusing on generalized linear models, this article proposes a permutation test using an analogue of the ANOVA test statistic that merely requires fitting the null model with independent observations. Monte Carlo simulations reveal that the new test has correct Type-I error rate and that its power compares favorably to an existing bootstrap score test. A real data application illustrates the advantageous capability of the proposed test in detecting the need for random effects. [ABSTRACT FROM AUTHOR]

A permutation is a mathematical technique for determining the number of possible arrangements in any set where the particular sequence of the arrangements matter. Computation of permutations of a string is a complex task, especially for long strings, and it frequently takes a long time. To address this issue, parallelization becomes critical to explore the vast solution space efficiently. In this paper, the parallelized solution for generating string permutations using the Message Passing Interface (MPI) programming model is discussed. The proposed algorithm divides the workload among multiple processors to render a subset of total permutations. Through a comparative analysis with a sequential approach, the performance of the parallel algorithm is evaluated. The findings emphasize the potential advantages of parallelization in reducing the overall computation time for string permutation generation. [ABSTRACT FROM AUTHOR]

Most semi-quantum secret sharing protocols are suitable for the application with three participants. Based on the n-level states, a multiparty semi-quantum secret sharing protocol is proposed. Compared with similar protocols, it has the merits as follows. (1) The dealer only needs to prepare the single photon sequence without using any product state or entangle state. (2) To share a secret, the classical parties need not have the ability of measurement. They only need to reorder the photon sequence and send it to the receiver. (3) The dealer knows nothing about any agent's secret shadow. Therefore, it is infeasible for the dealer to impersonate the classical participant to pool the secret shadow during the secret recovering phase. (4) Theoretically, the qubit efficiency of the proposed protocol is asymptotically 100%. The protocol is secure against famous attacks such as intercept-resend attack, measurement-resend attack and entangle-measure attack. [ABSTRACT FROM AUTHOR]

In the rapid evolving digital landscape, the imperative to ensure data security has never been more crucial. This paper addresses the pressing challenges in data security by introducing a file encryption management system, leveraging a modified advanced encryption standard (AES) algorithm with reduced round iterations and bit permutation. This system aims to comprehensively secure various file types, providing a dependable solution for file exchange. Our findings reveal substantial improvements in both encryption and decryption processes using the reduced-round permutationbased AES (RRPBA). The adapted algorithm demonstrates a significant 38.8% acceleration in encryption time and a remarkable 44.86% improvement in decryption time, positioning it as a pivotal component for efficient file operations within the management system. Moreover, the throughput assessments showcase a remarkable 33.73% improvement in encryption and 23.72% in decryption, outperforming the original AES, emphasizing the algorithm's superior computational effectiveness, signaling positive implications for future high-performance applications. In conclusion, the study not only addresses critical security challenges but also presents a viable solution with tangible speed advantages for file encryption and decryption processes within digital file management systems. [ABSTRACT FROM AUTHOR]

Let n be any natural number such that 2 is a primitive root of 2 n + 1 . In this article, we prove that the permutation (n!) has two orbits if and only if 2 n + 1 = p 2 for some odd prime p, where (n !) = ∏ k = 0 n - 1 (1 , 2 , ⋯ , (n - k)) . [ABSTRACT FROM AUTHOR]

Neyman’s seminal work in 1923 has been a milestone in statistics over the century, which has motivated many fundamental statistical concepts and methodology. In this review, we delve into Neyman’s groundbreaking contribution and offer technical insights into the design and analysis of randomized experiments. We shall review the basic setup of completely randomized experiments and the classical approaches for inferring the average treatment effects. We shall, in particular, review more efficient design and analysis of randomized experiments by utilizing pretreatment covariates, which move beyond Neyman’s original work without involving any covariate. We then summarize several technical ingredients regarding randomizations and permutations that have been developed over the century, such as permutational central limit theorems and Berry–Esseen bounds, and we elaborate on how these technical results facilitate the understanding of randomized experiments. The discussion is also extended to other randomized experiments including rerandomization, stratified randomized experiments, matched pair experiments, and cluster randomized experiments.

The internal kink mode is one of the crucial factors affecting the stability of magnetically confined fusion devices. This paper explores the key features influencing the growth rate of internal kink modes using machine learning techniques such as Random Forest, Extreme Gradient Boosting (XGboost), Permutation, and SHapley Additive exPlanations (SHAP). We conduct an in-depth analysis of the significant physical mechanisms by which these key features impact the growth rate of internal kink modes. Numerical simulation data were used to train high-precision machine learning models, namely Random Forest and XGBoost, which achieved coefficients of determination values of 95.07% and 94.57%, respectively, demonstrating their capability to accurately predict the growth rate of internal kink modes. Based on these models, key feature analysis was systematically performed with Permutation and SHAP methods. The results indicate that resistance, pressure at the magnetic axis, viscosity, and plasma rotation are the primary features influencing the growth rate of internal kink modes. Specifically, resistance affects the evolution of internal kink modes by altering current distribution and magnetic field structure; pressure at the magnetic axis impacts the driving force of internal kink modes through the pressure gradient directly related to plasma stability; viscosity modifies the dynamic behavior of internal kink modes by regulating plasma flow; and plasma rotation introduces additional shear forces, affecting the stability and growth rate of internal kink modes. This paper describes the mechanisms by which these four key features influence the growth rate of internal kink modes, providing essential theoretical insights into the behavior of internal kink modes in magnetically confined fusion devices.