Narrow Results

A signed graph $Ĝ=(G,\gamma )$ is defined on a graph $G$ (known as the underlying graph of $Ĝ$), where the edges are assigned $+$ or − sign by a sign function $\gamma$. In this paper, we introduce the Laplacian permanental polynomial of a signed graph. The matrix $L(Ĝ)=D(G)-A(Ĝ)$ is defined as the Laplacian matrix of $Ĝ$, where $D(G)$ is the degree diagonal matrix of the underlying graph $G$ and $A(Ĝ)$ is the adjacency matrix of $Ĝ$. The polynomial $\psi (L(Ĝ);x)=\mathrm{\text{per}}(xI-L(Ĝ))$ is called the Laplacian permanental polynomial of $Ĝ$, where $I$ is the identity matrix and $\mathrm{\text{per}}$ denotes the permanent of a matrix. In this paper, we develop some recursive formulae for computing $\psi (L(Ĝ);x)$. We study the polynomial $\psi (L(Ĝ);x)$, when $Ĝ$ is a balanced signed graph, a signed tree, a balanced signed bipartite graph, and a signed graph with only odd-length negative cycles. Moreover, for any signed graph $Ĝ$, we compute the general expressions for the first five coefficients and the constant term of $\psi (L(Ĝ);x)$.

Valiant defined #P-completeness using Turing reductions. We propose a more restrictive definition, a variant of many-one reductions, and prove that the (0, 1)-permanent remains #P-complete under this definition.

This paper presents new generators for the center of the universal enveloping algebra of the symplectic Lie algebra. These generators are expressed in terms of the column-permanent and it is easy to calculate their eigenvalues on irreducible representations. We can regard these generators as the counterpart of central elements of the universal enveloping algebra of the orthogonal Lie algebra given in terms of the column-determinant by Wachi. The earliest prototype of all these central elements is the Capelli determinants in the universal enveloping algebra of the general linear Lie algebra.

We study two kinds of buy-it-now options, temporary and permanent, under a theoretical model of Stackelberg game. In this two-stage game, the bidders, as the followers, use a two-threshold strategy to determine whether to bid or directly buy the item at the posted price, given an auction configuration featured by the seller in the first stage and other common knowledge. Under the uniform distribution assumption for the bidders' valuation, we derive the optimal necessary conditions of the starting price and the buy-it-now price for maximizing the seller's expected revenue. Then, we use two numerical experiments to find some interesting insights, which include that under identical bidders' participation costs, the temporary buy-it-now option can acquire a higher expected revenue for the seller than the permanent option, and a buy-it-now price auction always nontrivially dominates a regular auction in terms of the achieved expected revenue, no matter whether the seller or the bidders are risk-averse.

We review the fundamental ideas of free complement (FC) method through its application on both ground and first excited states of helium atom. We have found that lower energies can be obtained with fewer number of terms in the FC expansion of the ground state wavefunction. In this case, the optimization of orbital exponents was not necessary for achieving spectroscopic accuracy, especially at higher orders where the structure of the FC wavefunction converges to that of the exact one. We have discovered that permanents naturally appear in the FC expansion of the first triplet excited state wavefunction. Including permanents in the FC expansion is shown to be energetically important for the first triplet excited state of helium atom whereas it is not computationally favorable at higher orders. Finally, considering the group theoretical properties of the symmetric group $S_2$ and using immanants, a compact and more elegant form for the FC expansion of the first triplet excited state of the helium atom is achieved.

In order to extend the results of $F_p$ in [P. Das, The number of permutation polynomials of a given degree over a finite field, Finite Fields Appl.8(4) (2002) 478–490], where $p$ is a prime, to arbitrary finite fields $F_{p^r}$, we find a formula for the number of permutation polynomials of degree $q-2$ over a finite field $F_q$, which has $q$ elements, in terms of the permanent of a matrix. We write down an expression for the number of permutation polynomials of degree $q-2$ over a finite field $F_q$, using the permanent of a matrix whose entries are $p$th roots of unity and using this we obtain a nontrivial bound for the number. Finally, we provide a formula for the number of permutation polynomials of degree $d$ less than $q-2$.

A delayed SEIR epidemic model with vertical transmission and non-monotonic incidence is formulated. The equilibria and the threshold of the model have been determined on the bases of the basic reproduction number. The local stability of disease-free equilibrium and endemic equilibrium is established by analyzing the corresponding characteristic equations. By comparison arguments, it is proved that, if R₀ < 1, the disease-free equilibrium is globally asymptotically stable. Whereas, the disease-free equilibrium is unstable if R₀ > 1. Moreover, we show that the disease is permanent if the basic reproduction number is greater than one. Furthermore, the sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium when R₀ > 1.

Let $H_d[x_1,\dots ,x_m]$ be the complex vector space of homogeneous polynomials of degree $d$ with the independent variables $x_1,\dots ,x_m$. Let $V$ be the complex vector space of homogeneous linear polynomials in the variables $x_1,\dots ,x_m$. For any linear operator $T$ acting on $V$, there is a (unique) induced operator $P(T)$ acting on $H_d[x_1,\dots ,x_m]$ satisfying