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The demand for deploying deep convolutional neural networks (DCNNs) on resource-constrained devices for real-time applications remains substantial. However, existing state-of-the-art structured pruning methods often involve intricate implementations, require modifications to the original network architectures, and necessitate an extensive fine-tuning phase. To overcome these challenges, we propose a novel method that, for the first time, incorporates the concepts of charge and electrostatic force from physics into the training process of DCNNs. The magnitude of this force is directly proportional to the product of the charges of the convolution filter and the source filter, and inversely proportional to the square of the distance between them. We applied this electrostatic-like force to the convolution filters, either attracting filters with opposite charges toward non-zero weights or repelling filters with like charges toward zero weights. Consequently, filters subject to repulsive forces have their weights reduced to zero, enabling their removal, while the attractive forces preserve filters with significant weights that retain information. Unlike conventional methods, our approach is straightforward to implement, does not require any architectural modifications, and simultaneously optimizes weights and ranks filter importance, all without the need for extensive fine-tuning. We validated the efficacy of our method on modern DCNN architectures using the MNIST, CIFAR, and ImageNet datasets, achieving competitive performance compared to existing structured pruning approaches.

Let $S_n$ denote the symmetric group. We consider \begin{equation*} N_{\ell}(n) := \frac{\left\vert Hom\left( \mathbb{Z}^{\ell},S_n\right) \right\vert}{n!} \end{equation*} which also counts the number of $\ell$-tuples $\pi=\left( \pi_1, \ldots, \pi_{\ell}\right) \in S_n^{\ell}$ with $\pi_i \pi_j = \pi_j \pi_i$ for $1 \leq i,j \leq \ell$ scaled by $n!$. A recursion formula, generating function, and Euler product have been discovered by Dey, Wohlfahrt, Bryman and Fulman, and White. Let $a,b, \ell \geq 2$. It is known by Bringman, Franke, and Heim, that the Bessenrodt--Ono inequality \begin{equation*} \Delta_{a,b}^{\ell}:= N_{\ell}(a) \, N_{\ell}(b) - N_{\ell}(a+b) >0 \end{equation*} is valid for $a,b \gg 1$ and by Bessenrodt and Ono that it is valid for $\ell =2$ and $a+b >9$. In this paper we prove that for each pair $(a,b)$ the sign of $\{\Delta_{a,b}^{\ell} \}_{\ell}$ is getting stable. In each case we provide an explicit bound. The numbers $N_{\ell}\left( n\right) $ had been identified by Bryan and Fulman as the $n$-th orbifold characteristics, generalizing work by Macdonald and Hirzebruch--H\"{o}fer concerning the ordinary and string-theoretic Euler characteristics of symmetric products, where $N_2(n)=p(n) $ represents the partition function.

The classical multidimensional resultant can be defined as the, suitably normalized, generator of a projective elimination ideal in the ring of universal coefficients. This is the approach via the so-called inertia forms or Tr\"{a}gheitsformen. Using clever substitutions, Mertens and Hurwitz gave a criterion, for recognizing such inertia forms, which amounts to a linear system for their numerical coefficients. In this article we explicitly solve this linear system. We do so by identifying a subset of the available equations which forms a unitriangular system. The key notion we use is that of transversal, i.e., a selection of a monomial term in each of the homogeneous polynomials at hand. We need two such transversals which are disjoint and extremal, in the sense that they relate to extremizers of a, possibly new, determinantal inequality for differences of two substochastic matrices. Thanks to this notion of extremal pair of transversals, we derive an explicit formula for the coefficients of general multidimensional resultants, as a sum of terms made of a sign times a product of multinomial coefficients, thereby explicitly showing they are integer-valued., Comment: 36 pages

Let $S_n$ denote the symmetric group of permutations acting on $n$ elements. We investigate the double sequence $\{N_{\ell}(n)\}$ counting the number of $\ell$ tuples of elements of the symmetric group $S_n$, where the components commute, normalized by the order of $S_n$. Our focus lies on exploring log-concavity with respect to $n$: $$ N_{\ell}(n)^2 - N_{\ell}(n-1) \,\, N_{\ell}(n+1) \geq 0.$$ We establish that this depends on $n \pmod{3}$ for sufficiently large $\ell$. These numbers are studied by Bryan and Fulman as the $n$th orbifold characteristics, generalizing work of Macdonald and Hirzebruch--Hofer concerning the ordinary and string-theoretic Euler characteristics of symmetric products. Notably, $N_2(n)$ represents the partition numbers $p(n)$, while $N_{3}(n)$ represents the number of non-equivalent $n$-sheeted coverings of a torus studied by Liskovets and Medynkh. The numbers also appear in algebra since $ \vert S_n \vert \,\, N_{\ell}(n) = \left\vert Hom \left( \mathbb{Z}^{\ell},S_n\right) \right\vert $.

Optical flow is the pattern of apparent motion of objects in a scene. The computation of optical flow is a critical component in numerous computer vision tasks such as object detection, visual object tracking, and activity recognition. Despite a lot of research, efficiently managing abrupt changes in motion remains a challenge in motion estimation. This paper proposes novel variational regularization methods to address this problem since they allow combining different mathematical concepts into a joint energy minimization framework. In this work, we incorporate concepts from signal sparsity into variational regularization for motion estimation. The proposed regularization uses a robust l1 norm, which promotes sparsity and handles motion discontinuities. By using this regularization, we promote the sparsity of the optical flow gradient. This sparsity helps recover a signal even with just a few measurements. We explore recovering optical flow from a limited set of linear measurements using this regularizer. Our findings show that leveraging the sparsity of the derivatives of optical flow reduces computational complexity and memory needs., Comment: 12 pages, 9 figures, and 3 tables

Distributed Computing in Blockchain Technology (BCT) hinges on a trust assumption among independent nodes. Without a third-party interface or what is known as a Blockchain Oracle, it can not interact with the external world. This Oracle plays a crucial role by feeding extrinsic data into the Blockchain, ensuring that Smart Contracts operate accurately in real time. The Oracle problem arises from the inherent difficulty in verifying the truthfulness of the data sourced by these Oracles. The genuineness of a Blockchain Oracle is paramount, as it directly influences the Blockchain's reliability, credibility, and scalability. To tackle these challenges, a strategy rooted in Byzantine fault tolerance {\phi} is introduced. Furthermore, an autonomous system for sustainability and audibility, built on heuristic detection, is put forth. The effectiveness and precision of the proposed strategy outperformed existing methods using two real-world datasets, aimed to meet the authenticity standards for Blockchain Oracles., Comment: 12 pages, 16 figures

Let $A(\ell,n,k)$ denote the number of $\ell$-tuples of commuting permutations of $n$ elements whose permutation action results in exactly $k$ orbits or connected components. We provide a new proof of an explicit formula for $A(\ell,n,k)$ which is essentially due to Bryan and Fulman, in their work on orbifold higher equivariant Euler characteristics. Our proof is self-contained, elementary, and relies on the construction of an explicit bijection, in order to perform the $\ell+1\rightarrow \ell$ reduction. We also investigate a conjecture by the first author, regarding the log-concavity of $A(\ell,n,k)$ with respect to $k$. The conjecture generalizes a previous one by Heim and Neuhauser related to the Nekrasov-Okounkov formula., Comment: 11 pages

For a simple complex Lie algebra $\mathfrak{g}$, fixing a principal $\mathfrak{sl}_2$-triple and highest weight vectors induces a basis of $\mathfrak{g}$ as vector space. For $\mathfrak{sl}_n$, we describe how to compute the Lie bracket in this basis using transvectants. This generalizes a well-known rule for $\mathfrak{sl}_2$ using Poisson brackets and degree 2 monomials in two variables. Our proof method uses a graphical calculus for classical invariant theory. Other Lie algebra types are discussed., Comment: 36 pages, 55 figures

Let $A(p,n,k)$ be the number of $p$-tuples of commuting permutations of $n$ elements whose permutation action results in exactly $k$ orbits or connected components. We formulate the conjecture that, for every fixed $p$ and $n$, the $A(p,n,k)$ form a log-concave sequence with respect to $k$. For $p=1$ this is a well known property of unsigned Stirling numbers of the first kind. As the $p=2$ case, our conjecture includes a previous one by Heim and Neuhauser, which strengthens a unimodality conjecture for the Nekrasov-Okounkov hook length polynomials. In this article, we prove the $p=\infty$ case of our conjecture. We start from an expression for the $A(p,n,k)$ which follows from an identity by Bryan and Fulman, obtained in the their study of orbifold higher equivariant Euler characteristics. We then derive the $p\rightarrow\infty$ asymptotics. The last step essentially amounts to the log-concavity in $k$ of a generalized Tur\'an number, namely, the maximum product of $k$ positive integers whose sum is $n$., Comment: 9 pages

Various Non-negative Matrix factorization (NMF) based methods add new terms to the cost function to adapt the model to specific tasks, such as clustering, or to preserve some structural properties in the reduced space (e.g., local invariance). The added term is mainly weighted by a hyper-parameter to control the balance of the overall formula to guide the optimization process towards the objective. The result is a parameterized NMF method. However, NMF method adopts unsupervised approaches to estimate the factorizing matrices. Thus, the ability to perform prediction (e.g. classification) using the new obtained features is not guaranteed. The objective of this work is to design an evolutionary framework to learn the hyper-parameter of the parameterized NMF and estimate the factorizing matrices in a supervised way to be more suitable for classification problems. Moreover, we claim that applying NMF-based algorithms separately to different class-pairs instead of applying it once to the whole dataset improves the effectiveness of the matrix factorization process. This results in training multiple parameterized NMF algorithms with different balancing parameter values. A cross-validation combination learning framework is adopted and a Genetic Algorithm is used to identify the optimal set of hyper-parameter values. The experiments we conducted on both real and synthetic datasets demonstrated the effectiveness of the proposed approach.

The model described in this paper belongs to the family of non-negative matrix factorization methods designed for data representation and dimension reduction. In addition to preserving the data positivity property, it aims also to preserve the structure of data during matrix factorization. The idea is to add, to the NMF cost function, a penalty term to impose a scale relationship between the pairwise similarity matrices of the original and transformed data points. The solution of the new model involves deriving a new parametrized update scheme for the coefficient matrix, which makes it possible to improve the quality of reduced data when used for clustering and classification. The proposed clustering algorithm is compared to some existing NMF-based algorithms and to some manifold learning-based algorithms when applied to some real-life datasets. The obtained results show the effectiveness of the proposed algorithm.

We consider the correlations of invariant observables for the $O(N)$ and $\mathbb{C}\mathbb{P}^{N-1}$ models at zero coupling, namely, with respect to the natural group-invariant measure. In the limit where one takes a large power of the integrand, we show that these correlations become inverse powers of the Kirchhoff polynomial. The latter therefore provide a simplified toy model for the investigation of inequalities between products of correlations. Properties such as ferromagnetic behavior for spin model correlations correspond, in this asymptotic limit, to log-ultramodularity which is a consequence of the Rayleigh property of the Kirchhoff polynomial. In addition to the above rigorous asymptotics, the main result of this article is a general theorem which shows that inverse half-integer powers of certain determinantal stable polynomials, such as the Kirchhoff polynomials, satisfy generalizations of the GKS 2 inequalities and the Ginibre inequalities. We conclude with some open problems, e.g., the question of whether the last statement holds for powers which are not half-integers. This leads to a Hirota-bilinear analogue of the complete monotonicity property recently investigated by Scott and Sokal., Comment: 17 pages, fixed minor errors and typos

5G network operators consider the dense deployment of small base-stations (SBSs) to increase network coverage and capacity. Hence, operators face the challenge of X-hauling, i.e., backhauling or fronthauling, their traffic to the core network. Also, SBSs densification will increase the possibility of failure of these X-haul links. To cope with this problem, an X-haul outage compensation scheme with the assistance of Reconfigurable Intelligent Surfaces (RIS) is proposed to mitigate or at least alleviate the effect of X-haul failure. The RIS is a newly adopted technology that is able to improve the performance of wireless networks. In this paper, we present and evaluate an X-haul outage compensation scheme based on placing a number of RIS panels in pre-planned locations to mitigate the effect of X-haul failure. This evaluation is done using frequencies below and beyond 6 GHz. Based on our analytical results, the proposed RIS scheme shows that placing a sufficient number of RIS elements in proximity to the failed SBS under certain conditions can help acquire the same X-haul rate before the occurrence of the failure. Also, we show that for high X-haul spectral density, the RIS-assisted transmission with a certain number of elements can be more energy-efficient than line-of-sight and non-line-of-sight transmissions. Finally, the system's energy efficiency is addressed with and without RIS, and the optimal number of RIS reflecting elements is derived., Comment: This paper was submitted to a conference May 2022