Let G be a finite simple graph with edge ideal I (G). Let I (G)∨ denote the Alexander dual of I (G). We show that a description of all induced cycles of odd length in G is encoded in the associated primes of (I (G)∨)2. This result forms the basis for a method to detect odd induced cycles of a graph via ideal operations, e.g., intersections, products and colon operations. Moreover, we get a simp...

Let G denote a finite simple graph with edge ideal I(G). Letting I(G) denote the Alexander dual of I(G), we show that a description of the induced cycles of G of odd length is encoded in the associated primes of (I(G)). This result forms the basis for an algorithm to detect all the odd induced cycles of a graph via ideal operations, e.g., intersections, products, and colon operations. Moreover,...

Book Reviews classical medical training. Following a six year apprenticeship in an English country surgery and a brief period in London, he returned to Berkeley in 1792 to practice variolation, a method that had been popularized to com-The Eradication of Small Pox: Edward bat the smallpox. As the author points out, variolation Jenner and the First and Only Eradication was a highly dangerous pra...

In this paper, we study the nonlinear inverse problem of estimating spectrum a system matrix, that drives finite-dimensional affine dynamical system, from partial observations single trajectory data. noiseless case, prove an annihilating polynomial whose roots are subset spectrum, can be uniquely determined We then which eigenvalues matrix recovered and derive various sufficient necessary condi...

Let V be a vector space over some field k, and let α : V V be a linear map (an ‘endomorphism of V ’). Given any polynomial p with coefficients in k, there is an endomorphism p(α) of V , and we say that p is an annihilating polynomial for α if p(α) = 0. Our first major goal is to see that for any α, the annihilating polynomials can easily be classified: they’re precisely the multiples of a certa...

The major results of Robertson and Seymour on graph well-quasi-ordering establish nonconstructively that many natural graph properties that constitute ideals in the minor or immersion orders are characterized by a nite set of forbidden substructures termed the obstructions for the property. This raises the question of what general kinds of information about an ideal are su cient, or insu cient,...

The major results of Robertson and Seymour on graph well-quasi-ordering establish nonconstructively that many natural graph properties that constitute ideals in the minor or immersion orders are characterized by a nite set of forbidden sub-structures termed the obstructions for the property. This raises the question of what general kinds of information about an ideal are suucient, or insuucient...

The toric Hilbert scheme is a parameter space for all ideals with the same multi-graded Hilbert function as a given toric ideal. Unlike the classical Hilbert scheme, it is unknown whether toric Hilbert schemes are connected. We construct a graph on all the monomial ideals on the scheme, called the flip graph, and prove that the toric Hilbert scheme is connected if and only if the flip graph is ...

The toric Hilbert scheme is a parameter space for all ideals with the same multi-graded Hilbert function as a given toric ideal. Unlike the classical Hilbert scheme, it is unknown whether toric Hilbert schemes are connected. We construct a graph on all the monomial ideals on the scheme, called the flip graph, and prove that the toric Hilbert scheme is connected if and only if the flip graph is ...