Zero Element Research Articles

Analytical Bounds on the Local Lipschitz Constants of ReLU Networks.

In this article, we determine analytical upper bounds on the local Lipschitz constants of feedforward neural networks with rectified linear unit (ReLU) activation functions. We do so by deriving Lipschitz constants and bounds for ReLU, affine-ReLU, and max pooling functions, and combining the results to determine a network-wide bound. Our method uses several insights to obtain tight bounds, such as keeping track of the zero elements of each layer, and analyzing the composition of affine and ReLU functions. Furthermore, we employ a careful computational approach which allows us to apply our method to large networks such as AlexNet and VGG-16. We present several examples using different networks, which show how our local Lipschitz bounds are tighter than the global Lipschitz bounds. We also show how our method can be applied to provide adversarial bounds for classification networks. These results show that our method produces the largest known bounds on minimum adversarial perturbations for large networks such as AlexNet and VGG-16.

Semilattices of stratified extensions

Grillet (Semigroup Forum 50:25–36, 1995) introduced the concept of stratified semigroups as a kind of generalisation of finite nilsemigroups. We extend Grillet’s ideas by introducing the notion of the base of a semigroup and show that a semigroup is stratified if and only if its base is either empty or consists of only the zero element. The general structure of semigroups with non-trivial bases is studied and we show that these can be described in terms of ideal extensions of semigroups by stratified semigroups. We consider certain types of group-bound semigroups and also ideal extensions of Clifford semigroups, and show how to describe them as semilattices of ideal extensions by stratified semigroups and provide a number of interesting examples.

CONSTRUCT THE TRIPLE ZERO GRAPH OF RING Z_n USING PYTHON

Let be a commutative ring with nonzero identity and there exists such that , , , denotes the set of all triple zero elements of . The triple zero graph of , denoted by , is an undirected graph with vertex set where two distinct vertices and are adjacent if and only if , and there exists a nonzero element of such that , , and . Python is a programming language with simple and easy-to-learn code that can be used to solve problems in algebra and graphs. In this paper, we construct the triple zero graph of ring using Python. Based on the output of the program, several properties of are obtained, such as if and , then is a planar graph, if with is prime numbers, then is a complete graph , and if with is prime numbers and , then is a connected graph.

A Symmetry of Boundary Functions Method for Solving the Backward Time-Fractional Diffusion Problems

In the paper, we develop three new methods for estimating unknown initial temperature in a backward time-fractional diffusion problem, which is transformed to a space-dependent inverse source problem for a new variable in the first method. Then, the initial temperature can be recovered by solving a second-order boundary value problem. The boundary functions and a unique zero element constitute a group symmetry. We derive energetic boundary functions in the symmetry group as the bases to retrieve the source term as an unknown function of space and time. In the second method, the solution bases are energetic boundary functions, and then by collocating the governing equation we obtain the expansion coefficients for retrieving the entire solution and initial temperature. For the first two methods, boundary fluxes are over-specified to retrieve the initial condition. In the third method, we give two boundary conditions and a final time temperature to construct the bases in another symmetry group; the governing equation is collocated to a linear system to obtain the whole solution (initial temperature involved). These three methods are assessed and compared by numerical experiments.

Diffuse orthogonally additive operators

A regular orthogonally additive operator is a diffuse operator if it is disjoint from all operators in the band generated by the disjointness preserving operators. We present a criterion for principal lateral projections in an order complete vector lattice $E$ to be disjoint. We also state a criterion for a regular orthogonally additive operator to be diffuse. A criterion for the regularity of an integral Urysohn operator acting on ideal spaces of measurable functions is also presented. This criterion is used to show that an integral operator is diffuse. Examples of vector lattices are considered in which the sets of diffuse operators consist only of the zero element. The general form of an order projection operator onto the band generated by the disjointness preserving operators is found. Bibliography: 47 titles.

Multi-Label Weighted Contrastive Cross-Modal Hashing

Due to the low storage cost and high computation efficiency of hashing, cross-modal hashing has been attracting widespread attention in recent years. In this paper, we investigate how supervised cross-modal hashing (CMH) benefits from multi-label and contrastive learning (CL) by overcoming the following two challenges: (i) how to combine multi-label and supervised contrastive learning to consider diverse relationships among cross-modal instances, and (ii) how to reduce the sparsity of multi-label representation so as to improve the similarity measurement accuracy. To this end, we propose a novel cross-modal hashing framework, dubbed Multi-Label Weighted Contrastive Hashing (MLWCH). This framework involves compact consistent similarity representation, a new designed multi-label similarity calculation method that efficiently reduces the sparsity of multi-label by reducing redundant zero elements. Furthermore, a novel multi-label weighted contrastive learning strategy is developed to significantly improve hashing learning by assigning similarity weight to positive samples under both linear and non-linear similarities. Extensive experiments and ablation analysis over three benchmark datasets validate the superiority of our MLWCH method, especially over several outstanding baselines.

FEATURES OF THE CONSTRUCTION OF UNIFORM ASYMPTOTIC SOLUTION OF A SYSTEM OF DIFFERENTIAL EQUATIONS WITH A TURNING POINT WITH POSITIVE COEFFICIENTS OF THE MATRIX

The work is devoted to the analysis of the coefficients of the singular operator of the Orr-Sommerfeld type in vector form including the turning point. The system of singularly perturbed differential equations with a small parameter at the highest derivative is investigated. We consider the case when the spectrum of the limit operator contains multiple and identically equal zero elements. Using the method of essentially singular functions, the uniform asymptotic solution of the system is constructed. For the case of a stable turning point, the asymptotics of the solutions of the system are constructed in the sector that contains the turning point. The asymptotics of the first two solutions for the homogeneous problem are constructed using the Airy functions and their derivatives. The third formal solution of a homogeneous system for this case presents certain difficulties. Therefore, taking into account the specified conditions, in order to construct the uniform asymptotics of the solution for the given system, we used the partial solution of the heterogeneous system as the third solution of the homogeneous system.

A Category of Ordered Algebras Equivalent to the Category of Multialgebras

It is well known that there is a correspondence between sets and complete, atomic Boolean algebras (\(\textit{CABA}\)s) taking a set to its power-set and, conversely, a complete, atomic Boolean algebra to its set of atomic elements. Of course, such a correspondence induces an equivalence between the opposite category of \(\textbf{Set}\) and the category of \(\textit{CABA}\)s. We modify this result by taking multialgebras over a signature \(\Sigma\), specifically those whose non-deterministic operations cannot return the empty-set, to \(\textit{CABA}\)s with their zero element removed (which we call a \(\textit{bottomless Boolean algebra}\)) equipped with a structure of \(\Sigma\)-algebra compatible with its order (that we call \(\textit{ord-algebras}\)). Conversely, an ord-algebra over \(\Sigma\) is taken to its set of atomic elements equipped with a structure of multialgebra over \(\Sigma\). This leads to an equivalence between the category of \(\Sigma\)-multialgebras and the category of ord-algebras over \(\Sigma\). The intuition, here, is that if one wishes to do so, non-determinism may be replaced by a sufficiently rich ordering of the underlying structures.

Machine learning classification can significantly reduce the cost of calculating the Hamiltonian matrix in CI calculations.

Hamiltonian matrices in electronic and nuclear contexts are highly computation intensive to calculate, mainly due to the cost for the potential matrix. Typically, these matrices contain many off-diagonal elements that are orders of magnitude smaller than diagonal elements. We illustrate that here for vibrational H-matrices of H2O, C2H3 (vinyl), and C2H5NO2 (glycine) using full-dimensional abinitio-based potential surfaces. We then show that many of these small elements can be replaced by zero with small errors of the resulting full set of eigenvalues, depending on the threshold value for this replacement. As a result of this empirical evidence, we investigate three machine learning approaches to predict the zero elements. This is shown to be successful for these H-matrices after training on a small set of calculated elements. For H-matrices of vinyl and glycine, of order 15 552 and 8828, respectively, training on a percent or so of elements is sufficient to obtain all eigenvalues with a mean absolute error of roughly 2 cm-1.

Automated discovery of angle theorems

We consider geometry theorems whose premises and statement comprise a set of bisector conditions. Each premise and the statement can be represented as the rows of a “bisector matrix”: one with three non zero elements per row, one element with value -2 and the others with value 1. The existence of a theorem corresponds to rank deficiency in this matrix. Our method of theorem discovery starts with identification of rank deficient bisector matrices. Some such matrices can be represented as graphs whose vertices correspond to matrix rows and whose edges correspond to matrix columns. We show that if a bisector matrix which can be represented as a graph is rank deficient, then the graph is bicubic. We give an algorithm for finding the rank deficient matrices for a Hamiltonian bicubic graph, and report on the results for graphs with 6,8,10 and 12 vertices. We discuss a method of deriving rank deficient bisector matrices with more than 2 non-zero elements. We extend the work to matrices containing rows corresponding to angle trisectors.

Zonal graphs of small cycle rank

A zonal labeling of a plane graph G is an assignment of the two nonzero elements of the ring Z 3 of integers modulo 3 to the vertices of G such that the sum of the labels of the vertices on the boundary of each region of G is the zero element of Z 3 . A plane graph possessing such a labeling is a zonal graph. There is a connection between zonal labelings of connected bridgeless cubic plane graphs and the Four Color Theorem. Zonal labelings of cycles play a role in this connection. The cycle rank of a connected graph of order n and size m is m − n + 1 . Thus, cycles have cycle rank 1. All zonal connected graphs of cycle rank at most 2 are determined.

Cooperative Nearest-Neighbor Control of Multi-Agent Systems: Consensus and Formation Control Problems

This letter studies the problem of cooperative nearest-neighbor control of multi-agent systems where each agent can only realize a finite set of control points. Under the assumption that the underlying graph representing the communication network between agents is connected and the interior of the convex hull of all finite actions of each agent contains the zero element, consensus or distance-based formation problems can practically be stabilized by means of nearest-neighbor control approach combined with the well-known consensus control or distributed formation control laws, respectively. Furthermore, we provide the convergence bound for each corresponding error vector which can be computed based on the information of individual agent’s finite control points. Finally, we show Monte Carlo numerical simulations that confirm our analysis.

Two families of circulant nut graphs

A circulant nut graph is a non-trivial simple graph whose adjacency matrix is a circulant matrix of nullity one such that its non-zero null space vectors have no zero elements. The study of circulant nut graphs was originally initiated by Basic et al. [Art Discrete Appl. Math. 5(2) (2021) #P2.01], where a conjecture was made regarding the existence of all the possible pairs (n, d) for which there exists a d-regular circulant nut graph of order n. Later on, it was proved by Damnjanovic and Stevanovic [Linear Algebra Appl. 633 (2022) 127-151] that for each odd t ? 3 such that t ?/10 1 and t ?/18 15, the 4t-regular circulant graph of order n with the generator set {1, 2, 3,..., 2t+1}\{t}) must necessarily be a nut graph for each even n ? 4t + 4. In this paper, we extend these results by constructing two families of circulant nut graphs. The first family comprises the 4t-regular circulant graphs of order n which correspond to the generator sets {1, 2,..., t?1} ? {n/4, n/4 + 1} ? {n/2?(t?1),..., n/2 ? 2, n/2 ? 1}, for each odd t ? N and n ? 4t + 4 divisible by four. The second family consists of the 4t-regular circulant graphs of order n which correspond to the generator sets {1, 2,..., t?1} ? {n+2/4, n+6/4} ? {n/2 ?(t?1),..., n/2?2, n/2?1}, for each t ? N and n ? 4t + 6 such that n ?4 2. We prove that all of the graphs which belong to these families are indeed nut graphs, thereby fully resolving the 4t-regular circulant nut graph order-degree existence problem whenever t is odd and partially solving this problem for even values of t as well.

On employing pythagorean fuzzy processing time to minimize machine rental cost

<abstract> <p>The aim of this paper is to obtain the minimal rental cost of the three-phases flow shop scheduling problems. A novel strategy to tackle this issue using Pythagorean fuzzy processing time is introduced. It depends on converting the three stages machine into two stages when the minimum value of processing time of the first machine is greater than the maximum value of processing time of the second machine. The vague processing time does not convert to its crisp form. The jobs sequencing in machines is obtained using Johnson procedure. The zero element of the Pythagorean set is defined as, $ {\widetilde{\mathrm{O}}}^{\mathrm{p}} = \left(\mathrm{0, 1}\right) $ i.e., it has zero membership and one nonmembership values. A numerical example include Pythagorean rental cost is delivered to demonstrate the reliability of the suggested strategy. The idle time, utilization time, and the overall cost are calculated. The idle time of all machines is zero, which minimize the required time and hence, minimize the total rental cost.</p> </abstract>

Calculation of $\Sigma$-function for commutative non-cyclic semigroup of order 3 without identity and zero elements

Calculation of $\Sigma$-function for commutative non-cyclic semigroup of order 3 without identity and zero elements

Fixed product preserving mappings on Banach algebras

In this paper, we describe linear maps between complex Banach algebras that preserve products equal to fixed elements. This generalizes some important special cases where the fixed elements are the zero or identity element. First we show that if such map preserves products equal to a finite-rank operator, then it must also preserve the zero product. In several instances, this is enough to show that a product preserving map must be a scalar multiple of an algebra homomorphism. Second, we explore a more general problem concerning the existence of product preserving maps and the relationship between the fixed elements. Lastly, motivated by Kaplansky's problem on invertibility preservers, we show that maps preserving products equal to fixed invertible elements are either homomorphisms or antihomomorphisms multiplied on the left by a fixed element.

Inner zonality in graphs

A zonal labelling of a plane graph G is an assignment of the two nonzero elements of the ring of integers modulo 3 to the vertices of G such that the sum of the labels of the vertices on the boundary of each region of G is the zero element of . A plane graph possessing such a labelling is a zonal graph. A cubic map is a connected 3-regular bridgeless plane graph. It is known that if an independent proof could be given that every cubic map is zonal, then the Four Color Theorem would follow as a corollary. As a step in this direction, it is shown that certain subgraphs of cubic maps are nearly zonal.

ANTI FUZZY SUBSEMIRING

When the first operation’s inverse axiom is deleted from the ring, an algebraic structure, the semiring, is generated. Subsemiring is one of the subjects covered in semiring. The concepts of fuzzy subsemiring, anti subsemiring fuzzy semiring, and complement are introduced in this paper. In addition, the anti-subsemiring fuzzy semiring, a wedge, or a combination of two or more fuzzy anti-subsemiring associated with a non-empty subset of the semiring whose membership criteria are defined by the membership value of the zero elements will be discussed.

Zero-Divisor Graphs of Order-Decreasing Full Transformation Semigroups

Let 𝑛 ∈ ℤ+ and 𝑋𝑛 = {1,2, … , 𝑛} be a finite set. Let 𝐷𝑛 be the order-decreasing full transformation semigroup on 𝑋𝑛. In this paper, we find the left zero-divisors, the right zero-divisors and two sided zerodivisors of 𝐷𝑛. Moreover, for 𝑛 ≥ 4 we define an undirected graph Γ(𝐷𝑛) whose vertices are two-sided zero divisors of 𝐷𝑛 excluding the zero element 𝜃 of 𝐷𝑛. In the graph, distinct two vertices 𝛼 and 𝛽 are joined if and only if 𝛼𝛽 = 𝜃 = 𝛽𝛼. In this paper, we prove that Γ(𝐷𝑛) is a connected graph, and we find diameter, girth, the degrees of all vertices, the maximum degree and the minimum degree in Γ(𝐷𝑛). Moreover, we give lower bounds for clique number and choromatic number of Γ(𝐷𝑛).

Tensor theory for higher-dimensional Chern insulators with large Chern numbers

Recent advances in topological artificial systems open the door to realizing topological states in dimensions higher than the usual three-dimensional space. Here, we present a "tensor product" theory, which offers a method to construct Chern insulators with arbitrarily high dimensions and Chern numbers. Particularly, we show that the tensor product of a $d_A$D Chern insulator $\langle \mathcal{H}_A^{(\kappa_{A})}, C_A\rangle$ with a $d_B$D Chern insulator $\langle \mathcal{H}_B^{(\kappa_B)}, C_B\rangle$ leads to a $(d_A+d_B)$D Chern insulator $\langle \mathcal{H}_{A B}^{(\kappa_A\star \kappa_B)},-2C_AC_B\rangle $, where in the brackets, $\mathcal{H}^{(\kappa)}$ is the $d$D Hamiltonian with $d$ even, $C$ is the corresponding $(d/2)$th Chern number, and $\kappa$ labels the five non-chiral Altland-Zirnbauer symmetry classes A, AI, D, AII and C. The four real classes AI, D, AII and C form a Klein four-group under the multiplication `$\star$' with class AI the identity, and class A is the zero element. Our theory leads to novel higher-dimensional topological physics. (i) The construction can generate large higher-order Chern numbers, e.g., for some cases the resultant classification is $8\mathbb{Z}$. (ii) Fascinatingly, the boundary states feature flat nodal hypersurfaces with nontrivial Chern charges. For the constructed $(d_A+d_B)$D Chern insulator, a boundary perpendicular to a direction of $\mathcal{H}_A$ generically hosts $|C_A|$ $d_B$D nodal hypersurfaces, each of which has topological charge $\pm 2C_B$. Under perturbations, each nodal hypersurface bursts into stable unit nodal points, with the total Chern charge conserved. Examples are given to demonstrate our theory, which can be experimentally realized in artificial systems such as acoustic crystals, electric circuit arrays, ultracold atoms, or mechanical networks.