Sign pattern theory examines what can be said about a matrix if one knows the signs of all or some of its entries but not the exact values. Since all we know is the sign of each entry, we can write these sign patterns as matrices whose entries come from the set {+1, -1, 0, #}, where # is used for an unknown sign. Semirings satisfy all properties of rings with unity except the existence of additive inverses. The set {+1, -1, 0, #} can be viewed as a commutative semiring in natural way. In the thesis, we give a semiring version of the Cayley-Dickson construction which allows one to construct the sign pattern semiring from the Boolean semiring. We use tools from Boolean matrices to study sign nonsingular (SNS) matrices. We also investigate different notions of rank of matrices over semirings. For these rank functions we simplify proofs of classical inequalities for the sum and the product of matrices using the semiring versions of the Cauchy-Binet and Laplace theorems. For matrices over the sign pattern semiring, the minimum rank of the sign pattern is compared with the other versions of the rank. We also characterize irreducible powerful sign pattern matrices and investigate the period and base of an SNS matrix.

A sign pattern matrix is a matrix whose entries are from the set {+,-,0}. The minimum rank of a sign pattern matrix A is the minimum of the rank of the real matrices whose entries have signs equal to the corresponding entries of A. It is conjectured that the minimum rank of every sign pattern matrix can be realized by a rational matrix. The equivalence of this conjecture to several seemingly unrelated statements are established. For some special cases, such as when A is entrywise nonzero, or the minimum rank of A is at most 2, or the minimum rank of A is at least n - 1,(where A is mxn), the conjecture is shown to hold.Connections between this conjecture and the existence of positive rational solutions of certain systems of homogeneous quadratic polynomial equations with each coefficient equal to either -1 or 1 are explored. Sign patterns that almost require unique rank are also investigated.

Many models in the physical and life sciences formulated as dynamical systems have a positive steady state, with the local behavior of this steady state determined by the eigenvalues of its Jacobian matrix. The first part of this thesis is concerned with analyzing the linear stability of the steady state by using sign patterns, which are matrices with entries from the set {+,-,0}. The linear stability is related to the allowed refined inertias of the sign pattern of the Jacobian matrix of the system, where the refined inertia of a matrix is a 4-tuple (n+, n_-, ; nz; 2np) with n+ (n_) equal to the number of eigenvalues with positive (negative) real part, nz equal to the number of zero eigenvalues, and 2np equal to the number of nonzero pure imaginary eigenvalues. This type of analysis is useful when the parameters of the model are of known sign but unknown magnitude. The usefulness of sign pattern analysis is illustrated with several biological examples, including biochemical reaction networks, predator{prey models, and an infectious disease model. The refined inertias allowed by sign patterns with specific digraph structures have been studied, for example, for tree sign patterns. In the second part of this thesis, such results on refined inertias are extended by considering sign and zero-nonzero patterns with digraphs isomorphic to strongly connected orientations of the Petersen graph. / Graduate

A sign pattern(matrix) is a matrix whose entries are from the set {+,-,0}. An n x n sign pattern matrix is a spectrally arbitrary pattern(SAP) if for every monic real polynomial p(x) of degree n, there exists a real matrix B whose entries agree in sign with A such that the characteristic polynomial of B is p(x). An n x n pattern A is an inertialy arbitrary pattern(IAP) if (r,s,t) belongs to the inertia set of A for every nonnegative triple (r,s,t) with r+s+t=n. Some elementary results on these two classes of patterns are first exhibited. Tree sign patterns are then investigated, with a special emphasis on 4 x 4 tridiagonal sign patterns. Connections between the SAP(IAP) classes and the classes of potentially nilpotent and potentially stable patterns are explored. Some interesting open questions are also provided.

A sign pattern is a matrix whose entries are from the set $\{+, -, 0\}$. This thesis contains problems about refined inertias and minimum ranks of sign patterns. The refined inertia of a square real matrix $B$, denoted $\ri(B)$, is the ordered $4$-tuple $(n_+(B), \ n_-(B), \ n_z(B), \ 2n_p(B))$, where $n_+(B)$ (resp., $n_-(B)$) is the number of eigenvalues of $B$ with positive (resp., negative) real part, $n_z(B)$ is the number of zero eigenvalues of $B$, and $2n_p(B)$ is the number of pure imaginary eigenvalues of $B$. The minimum rank (resp., rational minimum rank) of a sign pattern matrix $\cal A$ is the minimum of the ranks of the real (resp., rational) matrices whose entries have signs equal to the corresponding entries of $\cal A$. First, we identify all minimal critical sets of inertias and refined inertias for full sign patterns of order 3. Then we characterize the star sign patterns of order $n\ge 5$ that require the set of refined inertias $\mathbb{H}_n=\{(0, n, 0, 0), (0, n-2, 0, 2), (2, n-2, 0, 0)\}$, which is an important set for the onset of Hopf bifurcation in dynamical systems. Finally, we establish a direct connection between condensed $m \times n $ sign patterns and zero-nonzero patterns with minimum rank $r$ and $m$ point-$n$ hyperplane configurations in ${\mathbb R}^{r-1}$. Some results about the rational realizability of the minimum ranks of sign patterns or zero-nonzero patterns are obtained.

Designing smarter vehicles, aiming to minimize the number of driverbased wrong decisions or accidents, which can be faced with during the drive, is one of hot topics of today&rsquo / s automotive technology. In the design of smarter vehicles, several research issues can be addressed / one of which is Traffic Sign Recognition (TSR). In TSR systems, the aim is to remind or warn drivers about the restrictions, dangers or other information imparted by traffic signs, beforehand. Since the existing signs are designed to draw drivers&rsquo / attention by their colors and shapes, processing of these features is one of the crucial parts in these systems. In this thesis, a Traffic Sign Recognition System, having ability of detection and identification of traffic signs even with bad visual artifacts those originate from some weather conditions or other circumstances, is developed. The developed algorithm in this thesis, segments the required color influenced by the illumination of the environment, then reconstructs the shape of partially occluded traffic sign by its remaining segments and finally, identifies it. These three stages are called as &ldquo / Segmentation&rdquo / , &ldquo / Reconstruction&rdquo / and &ldquo / Identification&rdquo / respectively, within this thesis. Due to the difficulty of analyzing partial segments to construct the main frame (a whole sign), the main complexity of the algorithm takes place in the &ldquo / Reconstruction&rdquo / stage.

A sign pattern matrix is a matrix whose entries are from the set {+,-,0}. For a real matrix B, sgn(B) is the sign pattern matrix obtained by replacing each positive(respectively, negative, zero) entry of B by + (respectively, -, 0). For a sign pattern matrix A, the sign pattern class of A, denoted Q(A), is defined as {B: sgn(B) = A}. An n by n sign pattern matrix A requires all distinct eigenvalues if every real matrix whose sign pattern is represented by A has n distinct eigenvalues. In this thesis, a number of sufficient and/or necessary conditions for a sign pattern to reuiqre all distinct eigenvalues are reviewed. In addition, for n=2 and 3, the n by n sign patterns that require all distinct eigenvalues are surveyed. We determine most of the 4 by 4 irreducible sign patterns that require four distinct eigenvalues.

An n × n sign pattern A is potentially stable (PS) if there exists a real matrix A having the sign pattern A and with all its eigenvalues having negative real parts. The identification of non-trivial necessary and sufficient conditions for potential stability remains a long standing open problem. Here we review some of the previous results and give simplified proofs for some of these results. Three techniques are given for the construction of larger order PS sign patterns from given PS sign patterns. These techniques are: construction of a sign pattern that allows a nested sequence of properly signed principal minors (a nest), bordering of a PS sign pattern with additional rows and columns, and use of a similarity transformation of a matrix that is reducible with two diagonal blocks (one of which is a stable matrix and the other a negative scalar). The minimum number of nonzero entries in an irreducible minimally PS sign pattern is determined for n = 2, . . . , 6 and for an arbitrary sign pattern that allows a nest. We also determine lower bounds for the number of nonzero entries in irreducible minimally PS sign patterns having certain sign patterns for their diagonal entries. For irreducible PS sign patterns of order at least four, a bordering construction leads to a new upper bound for the minimum number of nonzero entries.

A sign pattern matrix is a matrix whose entries are from the set {+,-, 0}. For a real matrix B, sgn(B) is the sign pattern matrix obtained by replacing each positive respectively, negative, zero) entry of B by + (respectively, -, 0). For a sign pattern matrixA, the sign pattern class of A, denoted Q(A), is defined as { B : sgn(B)= A }. The minimum rank mr(A)(maximum rank MR(A)) of a sign pattern matrix A is the minimum (maximum) of the ranks of the real matrices in Q(A). Several results concerning sign patterns A that require almost unique rank, that is to say, the sign patterns A such that MR(A)= mr(A)+1 are established. In particular, a complete characterization of these sign patterns is obtained. Further, the results on sign patterns that require almost unique rank are extended to sign patterns A for which the spread is d =MR(A)-mr(A).