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In this paper, we study linear forms $\lambda = \beta_1{\mathrm{e}}^{\alpha_1}+\cdots+\beta_m{\mathrm{e}}^{\alpha_m}$, where $\alpha_i$ and $\beta_i$ are algebraic numbers. An explicit lower bound for $|\lambda|$ is proved, which is derived from "th\'eor\`eme de Lindemann--Weierstrass effectif" via constructive methods in algebraic computation. Besides, an explicit upper bound for the minimal $|\lambda|$ is established on systematic results of counting algebraic numbers.

For a fixed type of Petri nets $\tau$, \textsc{$\tau$-Synthesis} is the task of finding for a given transition system $A$ a Petri net $N$ of type $\tau$ ($\tau$-net, for short) whose reachability graph is isomorphic to $A$ if there is one. The decision version of this search problem is called \textsc{$\tau$-Solvability}. If an input $A$ allows a positive decision, then it is called $\tau$-solvable and a sought net $N$ $\tau$-solves $A$. As a well known fact, $A$ is $\tau$-solvable if and only if it has the so-called $\tau$-\emph{event state separation property} ($\tau$-ESSP, for short) and the $\tau$-\emph{state separation property} ($\tau$-SSP, for short). The question whether $A$ has the $\tau$-ESSP or the $\tau$-SSP defines also decision problems. In this paper, for all $b\in \mathbb{N}$, we completely characterize the computational complexity of \textsc{$\tau$-Solvability}, \textsc{$\tau$-ESSP} and \textsc{$\tau$-SSP} for the types of pure $b$-bounded Place/Transition-nets, the $b$-bounded Place/Transition-nets and their corresponding $\mathbb{Z}_{b+1}$-extensions.

We construct a family of (n,k) convolutional codes with degree \delta in {k,n-k} that have a maximum distance profile. The field size required for our construction is of the order n^{2\delta}, which improves upon the known constructions of convolutional codes with a maximum distance profile. Our construction is based on the theory of skew polynomials.

In this paper, the local convergence analysis of the multi-step seventh order method is presented for solving nonlinear equations assuming that the first-order Fr\'echet derivative belongs to the Lipschitz class. The significance of our work is that it avoids the standard practice of Taylor expansion thereby, extends the applicability of the scheme by applying the technique based on the first-order derivative only. Also, this study provides radii of balls of convergence, the error bounds in terms of distances in addition to the uniqueness of the solution. Furthermore, generalization of this analysis satisfying H\"{o}lder continuity condition is provided since it is more relaxed than Lipschitz continuity condition. We have considered some numerical examples and computed the radii of the convergence balls.

The symbol-pair read channel was first proposed by Cassuto and Blaum. Later, Yaakobi et al. generalized it to the $b$-symbol read channel. It is motivated by the limitations of the reading process in high density data storage systems. One main task in $b$-symbol coding theory is to determine the $b$-symbol weight hierarchy of codes. In this paper, we study the $b$-symbol weight hierarchy of the Kasami codes, which are well known for their applications to construct sequences with optimal correlation magnitudes. The complete symbol-pair weight distribution of the Kasami codes is determined.

The mean width of a convex body is the average distance between parallel supporting hyperplanes when the normal direction is chosen uniformly over the sphere. The Simplex Mean Width Conjecture (SMWC) is a longstanding open problem that says the regular simplex has maximum mean width of all simplices contained in the unit ball and is unique up to isometry. We give a self contained proof of the SMWC in $d$ dimensions. The main idea is that when discussing mean width, $d+1$ vertices $v_i\in\mathbb{S}^{d-1}$ naturally divide $\mathbb{S}^{d-1}$ into $d+1$ Voronoi cells and conversely any partition of $\mathbb{S}^{d-1}$ points to selecting the centroids of regions as vertices. We will show that these two conditions are enough to ensure that a simplex with maximum mean width is a regular simplex.

Invariant finite-difference schemes are considered for one-dimensional magnetohydrodynamics (MHD) equations in mass Lagrangian coordinates for the cases of finite and infinite conductivity. For construction these schemes previously obtained results of the group classification of MHD equations are used. On the basis of the classical Samarskiy-Popov scheme new schemes are constructed for the case of finite conductivity. These schemes admit all symmetries of the original differential model and have difference analogues of all of its local differential conservation laws. Among the conservation laws there are previously unknown ones. In the case of infinite conductivity, conservative invariant schemes constructed as well. For isentropic flows of a polytropic gas proposed schemes possess the conservation law of energy and preserve entropy on two time layers. This is achieved by means of specially selected approximations for the equation of state of a polytropic gas. Also, invariant difference schemes with additional conservation laws are proposed.

In this paper, we construct some piecewise defined functions, and study their $c$-differential uniformity. As a by-product, we improve upon several prior results. Further, we look at concatenations of functions with low differential uniformity and show several results. For example, we prove that given $\beta_i$ (a basis of $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$), some functions $f_i$ of $c$-differential uniformities $\delta_i$, and $L_i$ (specific linearized polynomials defined in terms of $\beta_i$), $1\leq i\leq n$, then $F(x)=\sum_{i=1}^n\beta_i f_i(L_i(x))$ has $c$-differential uniformity equal to $\prod_{i=1}^n \delta_i$.

We show that for every fixed $k\geq 3$, the problem whether the termination/counter complexity of a given demonic VASS is $\mathcal{O}(n^k)$, $\Omega(n^{k})$, and $\Theta(n^{k})$ is coNP-complete, NP-complete, and DP-complete, respectively. We also classify the complexity of these problems for $k\leq 2$. This shows that the polynomial-time algorithm designed for strongly connected demonic VASS in previous works cannot be extended to the general case. Then, we prove that the same problems for VASS games are PSPACE-complete. Again, we classify the complexity also for $k\leq 2$. Interestingly, tractable subclasses of demonic VASS and VASS games are obtained by bounding certain structural parameters, which opens the way to applications in program analysis despite the presented lower complexity bounds.

BCH codes are an interesting class of cyclic codes due to their efficient encoding and decoding algorithms. In many cases, BCH codes are the best linear codes. However, the dimension and minimum distance of BCH codes have been seldom solved. Until now, there have been few results on BCH codes over $\gf(q)$ with length $q^m+1$, especially when $q$ is a prime power and $m$ is even. The objective of this paper is to study BCH codes of this type over finite fields and analyse their parameters. The BCH codes presented in this paper have good parameters in general, and contain many optimal linear codes.