Positive definite solution of two kinds of nonlinear matrix equations

Abstract. Based on the elegant properties of the Thompson metric, we prove that the following two kinds of nonlinear matrix equations always have a unique positive definite solution. Iterative methods are proposed to compute the unique positive definite solution. We show that the iterative methods are more effective as δ = max{|δi|, i = 1, 2, · · · , m} decreases. Perturbation bounds for the unique positive definite solution are derived in the end

A Class of Weighted Low Rank Approximation of the Positive Semidefinite Hankel Matrix

We consider the weighted low rank approximation of the positive semidefinite Hankel matrix problem arising in signal processing. By using the Vandermonde representation, we firstly transform the problem into an unconstrained optimization problem and then use the nonlinear conjugate gradient algorithm with the Armijo line search to solve the equivalent unconstrained optimization problem. Numerical examples illustrate that the new method is feasible and effective

Diagonal Sums of Doubly Substochastic Matrices

Let Ωn denote the convex polytope of all n x n doubly stochastic matrices, and ωn denote the convex polytope of all n x n doubly substochastic matrices. For a matrix A ϵ ωn, define the sub-defect of A to be the smallest integer k such that there exists an (n + k) x (n + k) doubly stochastic matrix containing A as a submatrix. Let ωn,k denote the subset of ωn which contains all doubly substochastic matrices with sub-defect k. For π a permutation of symmetric group of degree n, the sequence of elements a1π(1); a2π(2), ..., anπ(n) is called the diagonal of A corresponding to π. Let h(A) and l(A) denote the maximum and minimum diagonal sums of A ϵ ωn,k, respectively. In this paper, existing results of h and l functions are extended from Ωn to ωn,k. In addition, an analogue of Sylvesters law of the h function on ωn,k is proved

Specifying cycles of minimal length for commonly used linear layers in block ciphers

With the advances of Internet-of-Things (IoT) applications in smart cities and the pervasiveness of network devices with limited resources, lightweight block ciphers have achieved rapid development recently. Due to their relatively simple key schedule, nonlinear invariant attacks have been successfully applied to several families of lightweight block ciphers. This attack relies on the existence of a nonlinear invariant g:\F_2^n \rightarrow \F_2 for the round function $F_k$ so that $g(x) + g(F_k(x))$ is constant for any input value $x$. Whereas invariants of the entire $S$-box layer has been studied in terms of the corresponding cycle structure [TLS16,WRP20] (assuming the use of bijective S-boxes), a similar analysis for the linear layer has not been performed yet. In this article, we provide a theoretical analysis for specifying the minimal length of cycles for commonly used linear permutations (implementing linear layers) in lightweight block ciphers. Namely, using a suitable matrix representation, we exactly specify the minimal cycle lengths for those (efficiently implemented) linear layers that employ ShiftRows, Rotational-XOR and circular Boolean matrix operations which can be found in many well-known families of block ciphers. These results are practically useful for the purpose of finding nonlinear invariants of the entire encryption rounds since these can be specified using the intersection of cycles corresponding to the linear and S-box layer. We also apply our theoretical analysis practically and specify minimal cycle lengths of linear layers for certain families of block ciphers including some NIST candidates

Adopting a Theophylline-Responsive Riboswitch for Flexible Regulation and Understanding of Glycogen Metabolism in Synechococcus elongatus PCC7942

Cyanobacteria are supposed to be promising photosynthetic microbial platforms that recycle carbon dioxide driven into biomass and bioproducts by solar energy. Glycogen synthesis serves as an essential natural carbon sink mechanism, storing a large portion of energy and organic carbon source of photosynthesis. Engineering glycogen metabolism to harness and rewire carbon flow is an important strategy to optimize efficacy of cyanobacteria platforms. ADP-glucose pyrophosphorylase (GlgC) catalyzes the rate-limiting step for glycogen synthesis. However, knockout of glgC fails to promote cell growth or photosynthetic production in cyanobacteria, on the contrary, glgC deficiency impairs cellular fitness and robustness. In this work, we adopted a theophylline-responsive riboswitch to engineer and control glgC expression in Synechococcus elongatus PCC7942 and achieved flexible regulation of intracellular GlgC abundance and glycogen storage. With this approach, glycogen synthesis and glycogen contents in PCC7942 cells could be regulated in a range from about 40 to 300% of wild type levels. In addition, the results supported a positive role of glycogen metabolism in cyanobacteria cellular robustness. When glycogen storage was reduced, cellular physiology and growth under standard conditions was not impaired, while cellular tolerance toward environmental stresses was weakened. While when glycogen synthesis was enhanced, cells of PCC7942 displayed optimized cellular robustness. Our findings emphasize the significance of glycogen metabolism for cyanobacterial physiology and the importance of flexible approaches for engineering and understanding cellular physiology and metabolism

Analysis of Time Series Gene Expression and DNA Methylation Reveals the Molecular Features of Myocardial Infarction Progression

Myocardial infarction (MI) is one of the deadliest diseases in the world, and the changes at the molecular level after MI and the DNA methylation features are not clear. Understanding the molecular characteristics of the early stages of MI is of significance for the treatment of the disease. In this study, RNA-seq and MeDIP-seq were performed on heart tissue from mouse models at multiple time points (0 h, 10 min, 1, 6, 24, and 72 h) to explore genetic and epigenetic features that influence MI progression. Analysis based on a single point in time, the number of differentially expressed genes (DEGs) and differentially methylated regions (DMRs) increased with the time of myocardial infarction, using 0 h as a control group. Moreover, within 10 min of MI onset, the cells are mainly in immune response, and as the duration of MI increases, apoptosis begins to occur. Analysis based on time series data, the expression of 1012 genes was specifically downregulated, and these genes were associated with energy metabolism. The expression of 5806 genes was specifically upregulated, and these genes were associated with immune regulation, inflammation and apoptosis. Fourteen transcription factors were identified in the genes involved in apoptosis and inflammation, which may be potential drug targets. Analysis based on MeDIP-seq combined with RNA-seq methodology, focused on methylation at the promoter region. GO revealed that the downregulated genes with hypermethylation at 72 h were enriched in biological processes such as cardiac muscle contraction. In addition, the upregulated genes with hypomethylation at 72 h were enriched in biological processes, such as cell-cell adhesion, regulation of the apoptotic signaling pathway and regulation of angiogenesis. Among these genes, the Tnni3 gene was also present in the downregulated model. Hypermethylation of Tnni3 at 72 h after MI may be an important cause of exacerbation of MI