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In this paper, the Sphere-packing bound, Wang-Xing-Safavi-Naini bound, Johnson bound and Gilbert-Varshamov bound on the subspace code of length $2\nu +\delta$, size $M$, minimum subspace distance $2j$ based on $m$-dimensional totally singular subspace in the $(2\nu +\delta )$-dimensional orthogonal space ${{𝔽}_q}^{(2\nu +\delta )}$ over finite fields ${𝔽}_q$ of characteristic 2, denoted by ${(2\nu +\delta ,M,2j,m)}_q$, are presented, where $\nu$ is a positive integer, $\delta =0,1,2$, $0\le m\le \nu$, $0\le j\le m$. Then, we prove that ${(2\nu +\delta ,M,2j,m)}_q$ codes attain the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures in ${ℳ}(m,0,0;2\nu +\delta )$, where ${ℳ}(m,0,0;2\nu +\delta )$ denotes the collection of all the $m$-dimensional totally singular subspaces in the $(2\nu +\delta )$-dimensional orthogonal space ${{𝔽}_q}^{(2\nu +\delta )}$ over ${𝔽}_q$ of characteristic 2. Finally, Gilbert-Varshamov bound and linear programming bound on the subspace code ${(2\nu +\delta ,M,d)}_q$ in ${ℳ}(2\nu +\delta )$ are provided, where ${ℳ}(2\nu +\delta )$ denotes the collection of all the totally singular subspaces in the $(2\nu +\delta )$-dimensional orthogonal space ${{𝔽}_q}^{(2\nu +\delta )}$ over ${𝔽}_q$ of characteristic 2.

The disadvantage of a nondimensionalized model of a permanent-magnet synchronous Motor (PMSM) is identified. The original PMSM model is transformed into a Kolmogorov system to aid dynamic force analysis. The vector field of the PMSM is analogous to the force field including four types of torque — inertial, internal, dissipative, and generalized external. Using the feedback thought, the error torque between external torque and dissipative torque is identified. The pitchfork bifurcation of the PMSM is performed. Four forms of energy are identified for the system — kinetic, potential, dissipative, and supplied. The physical interpretations of the decomposition of force and energy exchange are given. Casimir energy is stored energy, and its rate of change is the error power between the dissipative energy and the energy supplied to the motor. Error torque and error power influence the different types of dynamic modes. The Hamiltonian energy and Casimir energy are compared to find the function of each in producing the dynamic modes. A supremum bound for the chaotic attractor is proposed using the error power and Lagrange multiplier.

The generalized Hamiltonian function is proposed for the brushless DC motor (BLDCM) chaotic system. The Hamiltonian and Casimir functions are derived from the generalized Hamiltonian function. In this way the Casimir energy is proven to be a special type of the generalized Hamiltonian function. The derivative of the Hamiltonian function is used for analyzing the various dynamical behaviors under different combination of energy components. An analytical optimal bound of the BLDCM is simply proposed from the Hamiltonian power. Along the study, the comparison between the Hamiltonian and Casimir powers is conducted, and physical interpretations and mechanism revealing the onset of chaos are provided for the BLDCM chaotic system. Bifurcation analysis through the Hamiltonian power and Casimir power identifies the different dynamic patterns.

In this paper, we give a bound for the number of boundary slopes of orientable immersed proper π₁-injective surfaces of given genus g in an orientable Haken 3-manifold M with a torus boundary, where the bound is independent of M, and a function of g and m, the number of the Jaco–Shalen–Johannson decomposition tori of M.

Fractal dimensions of Weyl–Marchaud fractional derivative of certain continuous functions are investigated in this paper. Upper Box dimension of Weyl–Marchaud fractional derivative of certain continuous functions with Box dimension one has been proved to be no more than the sum of one and its order.

As more and more organizations collect, store, and release large amounts of personal information, it is increasingly important for the organizations to conduct privacy risk assessment so as to comply with various emerging privacy laws and meet information providers' demands. Existing statistical database security and inference control solutions may not be appropriate for protecting privacy in many new uses of data as these methods tend to be either less or over-restrictive in disclosure limitation or are prohibitively complex in practice. We address a fundamental question in privacy risk assessment which asks: how to accurately derive bounds for protected information from inaccurate released information or, more particularly, from bounds of released information. We give an explicit formula for calculating such bounds from bounds, which we call square bounds or S-bounds. Classic F-bounds in statistics become a special case of S-bounds when all released bounds retrograde to exact values. We propose a recursive algorithm to extend our S-bounds results from two dimensions to high dimensions. To assess privacy risk for a protected database of personal information given some bounds of released information, we define typical privacy disclosure measures. For each type of disclosure, we investigate the distribution patterns of privacy breaches as well as effective and efficient controls that can be used to eliminate privacy risk, both based on our S-bounds results.

In this work, we introduce a new class of optimal tensor codes related to the Ravagnani-type anticodes defined in 2023. We show that it extends the family of $j$-maximum rank distance codes and contains the $j$-binomial moment determined codes (with respect to the Ravagnani-type anticodes) as a proper subclass. We define and study the zeta function for tensor codes. We establish connections between this object and the weight enumerator of a tensor code with respect to the Ravagnani-type anticodes. We introduce a new refinement of the invariants of tensor codes exploiting the structure of product lattices of some classes of anticodes classified in 2023 and we derive the corresponding MacWilliams identities. In this framework, we also define a multivariate version of the tensor weight enumerator and we establish relations with the corresponding zeta function. As an application, we derive connections on the tensor weights.

Entropy is a key concept of quantum information theory. The entropy of a quantum system is a measure of its randomness and has many applications in quantum communication protocols, quantum coherence, and so on. In this paper, based on the Rényi entropy and Tsallis entropy, we derive the bounds of the expectation value and variance of a quantum observable. By the maximal value of Rényi entropy, we show an upper bound on the product of variance and entropy. Furthermore, we obtain the reverse uncertainty relation for the product and sum of the variances for $n$ observables respectively.

A graph invariant is any function on a graph that does not depend on a labeling of its vertices. One of the best known graph invariants successfully applied in chemical graph theory over the last decade is the harmonic index. It is defined for a graph $G$ as the sum of the terms $\frac{2}{d_G(u)+d_G(v)}$ over all edges $uv$ of $G$, where $d_G(u)$ and $d_G(v)$ denote the degrees of the vertices $u$ and $v$ in $G$, respectively. The eccentric version of harmonic index has recently been proposed in an analogous way by replacing the vertex degrees with the vertex eccentricities. One of the main topics in chemical graph theory is to study how certain invariants of product graphs are related to the corresponding invariants of their components. Due to this, we investigate here the behavior of the eccentric version of harmonic index under various families of graph products and apply the derived results on some graphs of chemical and general interest.

Subspace code is very useful in the error correction for random network coding. In this paper, subspace code based on flats in affine space over finite fields is constructed. Sphere-packing bound, Wang–Xing–Safavi-Naini bound, anticode bound, Ahlswede–Aydinian bound and Gilbert–Varshamov bound for the size of code based on flats in affine space over finite fields are provided. Finally, we introduce another method of obtaining the anticode bound for the size of code based on flats in affine space from Erdős–Ko–Rado theorem.

The eccentric connectivity index and second Zagreb eccentricity index are well-known graph invariants defined as the sums of contributions dependent on the eccentricities of adjacent vertices over all edges of a connected graph. The coindices of these invariants have recently been proposed by considering analogous contributions from the pairs of non-adjacent vertices. Here, we obtain several lower and upper bounds on the eccentric connectivity coindex and second Zagreb eccentricity coindex in terms of some graph parameters such as order, size, number of non-adjacent vertex pairs, radius, and diameter, and relate these invariants to some well-known graph invariants such as Zagreb indices and coindices, status connectivity indices and coindices, ordinary and multiplicative Zagreb eccentricity indices, Wiener index, degree distance, total eccentricity, eccentric connectivity index, second eccentric connectivity index, and eccentric-distance sum. Moreover, we compute the values of these coindices for two graph constructions, namely, double graphs and extended double graphs.