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In this paper, a differential MOEMS accelerometer based on the Fabry-Perot (FP) micro-cavities is presented. The optical system of the device consists of two FP cavities and the mechanical system is composed of a proof mass that is suspended by four springs. The applied acceleration tends to move the PM from its resting position. This mechanical displacement can be measured by the FP interferometer formed between the proof mass cross-section and the optical fiber end face. The proposed sensor is fabricated on a silicon on insulator (SOI) wafer using the bulk micromachining method. The results of the sensor characterization show that the accelerometer has a linear response in the range of 1g. Also, the optical sensitivity and resolution of the sensor in the static characterization are 6.52 nm/g and 153ug. The sensor sensitivity in the power measurement is 49.6 mV/g and its resonant is at 1372 Hz. Using the differential measurement method increases the sensitivity of the accelerometer. Based on experimental data, the sensor sensitivity is two times as high as that of a similar MOEMS accelerometer with one FP cavity., Comment: arXiv admin note: text overlap with arXiv:2202.05153

A word $w$ is said to be closed if it has a proper factor $x$ which occurs exactly twice in $w$, as a prefix and as a suffix of $w$. Based on the concept of Ziv-Lempel factorization, we define the closed $z$-factorization of finite and infinite words. Then we find the closed $z$-factorization of the infinite $m$-bonacci words for all $m \geq 2$. We also classify closed prefixes of the infinite $m$-bonacci words.

In this paper, we study the generalized gapped k-mer filters and derive a closed form solution for their coefficients. We consider nonnegative integers $\ell$ and $k$, with $k\leq \ell$, and an $\ell$-tuple $B=(b_1,\ldots,b_{\ell})$ of integers $b_i\geq 2$, $i=1,\ldots,\ell$. We introduce and study an incidence matrix $A=A_{\ell,k;B}$. We develop a M\"obius-like function $\nu_B$ which helps us to obtain closed forms for a complete set of mutually orthogonal eigenvectors of $A^{\top} A$ as well as a complete set of mutually orthogonal eigenvectors of $AA^{\top}$ corresponding to nonzero eigenvalues. The reduced singular value decomposition of $A$ and combinatorial interpretations for the nullity and rank of $A$, are among the consequences of this approach. We then combine the obtained formulas, some results from linear algebra, and combinatorial identities of elementary symmetric functions and $\nu_B$, to provide the entries of the Moore-Penrose pseudo-inverse matrix $A^{+}$ and the Gapped k-mer filter matrix $A^{+} A$., Comment: arXiv admin note: substantial text overlap with arXiv:1605.06806

The Fibonacci word $W$ on an infinite alphabet was introduced in [Zhang et al., Electronic J. Combinatorics 2017 24(2), 2-52] as a fixed point of the morphism $2i\rightarrow (2i)(2i+1)$, $(2i+1) \rightarrow (2i+2)$, $i\geq 0$. Here, for any integer $k>2$, we define the infinite $k$-bonacci word $W^{(k)}$ on the infinite alphabet as the fixed point of the morphism $\varphi_k$ on the alphabet $\mathbb{N}$ defined for any $i\geq 0$ and any $0\leq j\leq k-1$, as \begin{equation*} \varphi_k(ki+j) = \left\{ \begin{array}{ll} (ki)(ki+j+1) & \text{if } j = 0,\cdots ,k-2,\\ (ki+j+1)& \text{otherwise}. \end{array} \right. \end{equation*} We consider the sequence of finite words $(W^{(k)}_n)_{n\geq 0}$, where $W^{(k)}_n$ is the prefix of $W^{(k)}$ whose length is the $(n+k)$-th $k$-bonacci number. We then provide a recursive formula for the number of palindromes occur in different positions of $W^{(k)}_n$. Finally, we obtain the structure of all palindromes occurring in $W^{(k)}$ and based on this, we compute the palindrome complexity of $W^{(k)}$, for any $k>2$.

We introduce a variation of the Ziv-Lempel and Crochemore factorizations of words by requiring each factor to be a palindrome. We compute these factorizations for the Fibonacci word, and more generally, for all $m$-bonacci words., Comment: 34 pages

We consider the problem of finding nonzero eigenvalues and the corresponding eigenvectors of a matrix $AA^{\top}$, where $A$ is a special incidence matrix; This matrix can equivalently be defined based on a match relation between some sequences. By using a concrete description of the obtained eigenvectors, we show that these are pairwise orthogonal and satisfy nice properties. The combinatorial arguments, in the sequel, lead us to obtain formulas for entries of matrices $W$ and $WA$, where $W$ is the Moore-Penrose pseudo-inverse of $A$. A special case of this problem has previously found applications in computational biology. ., Comment: 37 pages

In this paper we prove that any distance-balanced graph $G$ with $\Delta(G)\geq |V(G)|-3$ is regular. Also we define notion of distance-balanced closure of a graph and we find distance-balanced closures of trees $T$ with $\Delta(T)\geq |V(T)|-3$., Comment: 8 pages, 4 figures

Studying expressions of the form $(f(x)D)^p$, where $D={\displaystyle \frac{d}{dx}}$ is the derivative operator, goes back to Scherk's Ph.D. thesis in 1823. We show that this can be extended as ${\displaystyle\sum \gamma_{p;a} (f^{(0)})^{a(0)+1} (f^{(1)})^{a(1)}...(f^{(p-1)})^{a(p-1)}D^{p-\sum_i i a(i)}}$}, where the summation is taken over the $p$-tuples $(a_0, a_1,..., a_{p-1})$, satisfying $\sum_{i}a(i)=p-1,\, \sum_{i}i a(i)

We study the area distribution of closed walks of length $n$, beginning and ending at the origin. The concept of area of a walk in the square lattice is generalized and the usefulness of the new concept is demonstrated through a simple argument. It is concluded that the number of walks of length $n$ and area $s$ equals to the coefficient of $z^s$ in the expression $(x+x^{-1}+y+y^{-1})^n$, where the calculations are performed in a special group ring $R[x,y,z]$. A polynomial time algorithm for calculating these values, is then concluded. Finally, the provided algorithm and the results of implementation are compared with previous works., Comment: 12 pages, 1 Figure

Several intersection matrices of $s$-subsets vs. $k$-subsets of a $v$-set are introduced in the literature. We study these matrices systematically through counting arguments and generating function techniques. A number of new or known identities appear as natural consequences of this viewpoint; especially, appearance of the derivative operator $d/dz$ and some related operators reveals some connections between intersection matrices and the "combinatorics of creation-annihilation". As application, the eigenvalues of several intersection matrices including some generalizations of the adjacency matrices of the Johnson scheme are derived; two new bases for the Bose--Mesner algebra of the Johnson scheme are introduced and the associated intersection numbers are obtained as well. Finally, we determine the rank of some intersection matrices., Comment: 17 pages, revised version

Given integers $t$, $k$, and $v$ such that $0\leq t\leq k\leq v$, let $W_{tk}(v)$ be the inclusion matrix of $t$-subsets vs. $k$-subsets of a $v$-set. We modify slightly the concept of standard tableau to study the notion of rank of a finite set of positive integers which was introduced by Frankl. Utilizing this, a decomposition of the poset $2^{[v]}$ into symmetric skipless chains is given. Based on this decomposition, we construct an inclusion matrix, denoted by $W_{\bar{t}k}(v)$, which is row-equivalent to $W_{tk}(v)$. Its Smith normal form is determined. As applications, Wilson's diagonal form of $W_{tk}(v)$ is obtained as well as a new proof of the well known theorem on the necessary and sufficient conditions for existence of integral solutions of the system $W_{tk}\bf{x}=\bf{b}$ due to Wilson. Finally we present anotherinclusion matrix with similar properties to those of $W_{\bar{t}k}(v)$ which is in some way equivalent to $W_{tk}(v)$., Comment: Accepted for publication in Journal of Combinatorial Theory, Series A