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We determine the best n-term approximation of generalized Wiener model classes in a Hilbert space $H $. This theory is then applied to several special cases.

We show that for any string $w$ of length $n$, $r_B = O(z\log^2 n)$, where $r_B$ and $z$ are respectively the number of character runs in the bijective Burrows-Wheeler transform (BBWT), and the number of Lempel-Ziv 77 factors of $w$. We can further induce a bidirectional macro scheme of size $O(r_B)$ from the BBWT. Finally, there exists a family of strings with $r_B = \Omega(\log n)$ but having only $r=2$ character runs in the standard Burrows--Wheeler transform (BWT). However, a lower bound for $r$ is the minimal run-length of the BBWTs applied to the cyclic shifts of $w$, whose time complexity might be $o(n^2)$ in the light that we show how to compute the Lyndon factorization of all cyclic rotations in $O(n)$ time. Considering also the rotation operation performing cyclic shifts, we conjecture that we can transform two strings having the same Parikh vector to each other by BBWT and rotation operations, and prove this conjecture for the case of binary alphabets and permutations.

We introduce syntactic modal operator $\BOX$ for \textit{being a thesis} into first-order logic. This logic is a modern realization of R. Carnap's old ideas on modality, as logical necessity (J. Symb. Logic, 1946) \cite{Ca46}. We place it within the modern framework of quantified modal logic with a variant of possible world semantics with variable domains. We prove completeness using a kind of normal form and show that in the canonical frame, the relation on all maximal consistent sets, $R = \{\langle \Gamma, \Delta \rangle : \forall A (\BOX A \in \Gamma \Longrightarrow A \in \Delta)\}$, is a universal relation. The strength of the $\BOX$ operator is a proper extension of modal logic $\mathsf{S5}$. Using completeness, we prove that satisfiability in a model of $\BOX A$ under arbitrary valuation implies that $A$ is a thesis of formulated logic. So we can syntactically define logical entailment and consistency. Our semantics differ from S. Kripke's standard one \cite{Kr63} in syntax, semantics, and interpretation of the necessity operator. We also have free variables, contrary to Kripke's and Carnap's approaches, but our notion of substitution is non-standard (variables inside modalities are not free). All $\BOX$-free first-order theses are provable, as well as the Barcan formula and its converse. Our specific theses are \linebreak[4] $\BOX A \to \forall x A$, $\neg \BOX (x = y)$, $\neg \BOX \neg (x = y)$, $\neg \BOX P(x)$, $\neg \BOX \neg P(x)$. We also have $\POS \exists x A(x) \to \POS A(^{y}/_{x})$, and $\forall x \BOX A(x) \to \BOX A(^{y}/_{x})$, if $A$ is a $\BOX$-free formula.

In $2014$, Gupta and Ray proved that the circulant involutory matrices over the finite field $\mathbb{F}_{2^m}$ can not be maximum distance separable (MDS). This non-existence also extends to circulant orthogonal matrices of order $2^d \times 2^d$ over finite fields of characteristic $2$. These findings inspired many authors to generalize the circulant property for constructing lightweight MDS matrices with practical applications in mind. Recently, in $2022,$ Chatterjee and Laha initiated a study of circulant matrices by considering semi-involutory and semi-orthogonal properties. Expanding on their work, this article delves into circulant matrices possessing these characteristics over the finite field $\mathbb{F}_{2^m}.$ Notably, we establish a correlation between the trace of associated diagonal matrices and the MDS property of the matrix. We prove that this correlation holds true for even order semi-orthogonal matrices and semi-involutory matrices of all orders. Additionally, we provide examples that for circulant, semi-orthogonal matrices of odd orders over a finite field with characteristic $2$, the trace of associated diagonal matrices may possess non-zero values.

In various stereological problems an $n$-dimensional convex body is intersected with an $(n-1)$-dimensional Isotropic Uniformly Random (IUR) hyperplane. In this paper the cumulative distribution function associated with the $(n-1)$-dimensional volume of such a random section is studied. This distribution is also known as chord length distribution and cross section area distribution in the planar and spatial case respectively. For various classes of convex bodies it is shown that these distribution functions are absolutely continuous with respect to Lebesgue measure. A Monte Carlo simulation scheme is proposed for approximating the corresponding probability density functions.

In $1998,$ Daemen {\it{ et al.}} introduced a circulant Maximum Distance Separable (MDS) matrix in the diffusion layer of the Rijndael block cipher, drawing significant attention to circulant MDS matrices. This block cipher is now universally acclaimed as the AES block cipher. In $2016,$ Liu and Sim introduced cyclic matrices by modifying the permutation of circulant matrices and established the existence of MDS property for orthogonal left-circulant matrices, a notable subclass within cyclic matrices. While circulant matrices have been well-studied in the literature, the properties of cyclic matrices are not. Back in $1961$, Friedman introduced $g$-circulant matrices which form a subclass of cyclic matrices. In this article, we first establish a permutation equivalence between a cyclic matrix and a circulant matrix. We explore properties of cyclic matrices similar to $g$-circulant matrices. Additionally, we determine the determinant of $g$-circulant matrices of order $2^d \times 2^d$ and prove that they cannot be simultaneously orthogonal and MDS over a finite field of characteristic $2$. Furthermore, we prove that this result holds for any cyclic matrix.

Consider a linear operator equation $x - Kx = f$, where $f$ is given and $K$ is a Fredholm integral operator with a Green's function type kernel defined on $C[0, 1]$. For $r \geq 1$, we employ the interpolatory projection at $2r + 1$ collocation points (not necessarily Gauss points) onto a space of piecewise polynomials of degree $\leq 2r$ with respect to a uniform partition of $[0, 1]$. Previous researchers have established that the iteration in case of the collocation method improves the order of convergence by projection methods and its variants in the case of smooth kernel with piecewise polynomials of even degree only. In this article, we demonstrate the improvement in order of convergence by modified collocation method when the kernel is of Green's function type.

A locally recoverable code of locality $r$ over $\mathbb{F}_{q}$ is a code where every coordinate of a codeword can be recovered using the values of at most $r$ other coordinates of that codeword. Locally recoverable codes are efficient at restoring corrupted messages and data which make them highly applicable to distributed storage systems. Quasi-cyclic codes of length $n=m\ell$ and index $\ell$ are linear codes that are invariant under cyclic shifts by $\ell$ places. %Quasi-cyclic codes are generalizations of cyclic codes and are isomorphic to $\mathbb{F}_{q} [x]/ \langle x^m-1 \rangle$-submodules of $\mathbb{F}_{q^\ell} [x] / \langle x^m-1 \rangle$. In this paper, we decompose quasi-cyclic locally recoverable codes into a sum of constituent codes where each constituent code is a linear code over a field extension of $\mathbb{F}_q$. Using these constituent codes with set parameters, we propose conditions which ensure the existence of almost optimal and optimal quasi-cyclic locally recoverable codes with increased dimension and code length.

Given a finite family $\mathcal{F}$ of graphs, we say that a graph $G$ is "$\mathcal{F}$-free" if $G$ does not contain any graph in $\mathcal{F}$ as a subgraph. A vertex-colored graph $H$ is called "rainbow" if no two vertices of $H$ have the same color. Given an integer $s$ and a finite family of graphs $\mathcal{F}$, let $\ell(s,\mathcal{F})$ denote the smallest integer such that any properly vertex-colored $\mathcal{F}$-free graph $G$ having $\chi(G)\geq\ell(s,\mathcal{F})$ contains an induced rainbow path on $s$ vertices. Scott and Seymour showed that $\ell(s,K)$ exists for every complete graph $K$. A conjecture of N. R. Aravind states that $\ell(s,C_3)=s$. The upper bound on $\ell(s,C_3)$ that can be obtained using the methods of Scott and Seymour setting $K=C_3$ are, however, super-exponential. Gy\'arf\'as and S\'ark\"ozy showed that $\ell(s,\{C_3,C_4\})=\mathcal{O}\big((2s)^{2s}\big)$. For $r\geq 2$, we show that $\ell(s,K_{2,r})\leq (r-1)(s-1)(s-2)/2+s$ and therefore, $\ell(s,C_4)\leq\frac{s^2-s+2}{2}$. This significantly improves Gy\'arf\'as and S\'ark\"ozy's bound and also covers a bigger class of graphs. We adapt our proof to achieve much stronger upper bounds for graphs of higher girth: we prove that $\ell(s,\{C_3,C_4,\ldots,C_{g-1}\})\leq s^{1+\frac{4}{g-4}}$, where $g\geq 5$. Moreover, in each case, our results imply the existence of at least $s!/2$ distinct induced rainbow paths on $s$ vertices. Along the way, we obtain some results on related problems on oriented graphs. For $r\geq 2$, let $\mathcal{B}_r$ denote the orientations of $K_{2,r}$ in which one vertex has out-degree or in-degree $r$. We show that every $\mathcal{B}_r$-free oriented graph $G$ having $\chi(G)\geq (r-1)(s-1)(s-2)+2s+1$ and every bikernel-perfect oriented graph $G$ with girth $g\geq 5$ having $\chi(G)\geq 2s^{1+\frac{4}{g-4}}$ contains every $s$ vertex oriented tree as an induced subgraph.

A toric code, introduced by Hansen to extend the Reed-Solomon code as a $k$-dimensional subspace of $\mathbb{F}_q^n$, is determined by a toric variety or its associated integral convex polytope $P \subseteq [0,q-2]^n$, where $k=|P \cap \mathbb{Z}^n|$ (the number of integer lattice points of $P$). There are two relevant parameters that determine the quality of a code: the information rate, which measures how much information is contained in a single bit of each codeword; and the relative minimum distance, which measures how many errors can be corrected relative to how many bits each codeword has. Soprunov and Soprunova defined a good infinite family of codes to be a sequence of codes of unbounded polytope dimension such that neither the corresponding information rates nor relative minimum distances go to 0 in the limit. We examine different ways of constructing families of codes by considering polytope operations such as the join and direct sum. In doing so, we give conditions under which no good family can exist and strong evidence that there is no such good family of codes.