搜索结果：Permanente

For a simple graph $G$ with adjacency matrix $A(G)$, let $π(G,x):=\mathrm{per}(xI-A(G))$ be its permanental polynomial with roots $μ_1,\ldots,μ_n \in \mathbb{C}$, and define the permanental energy $E_{\mathrm{per}}(G):=\sum_{i=1}^n |μ_i|$. We prove a sharp universal lower bound: for every $m$-edge graph $G$, $E_{\mathrm{per}}(G) \ge 2\sqrt{m}$, with equality if and only if $G$ is a star together with isolated vertices. We also prove the general upper bound $E_{\mathrm{per}}(G) \le nρ(G)$, where $ρ(G)$ is the spectral radius, and we study $E_{\mathrm{per}}(G)$ on several graph families.

In this article, we compute the $\vv$-number of $2\times 2$ permanental ideals of generic, generic symmetric, and generic Hankel matrices.

Although increasingly used for research, electronic health records (EHR) often lack gold-standard assessment of key data elements. Linking EHRs to other data sources with higher-quality measurements can improve statistical inference, but such analyses must account for selection bias if the linked data source arises from a non-probability sample. We propose a set of novel estimators targeting the average treatment effect (ATE) that combine information from binary outcomes measured with error in a large, population-representative EHR database with gold-standard outcomes obtained from a smaller validation sample subject to selection bias. We evaluate our approach in extensive simulations and an analysis of data from the Adult Changes in Thought (ACT) study, a longitudinal study of incident dementia in a cohort of Kaiser Permanente Washington members with linked EHR data. For a subset of deceased ACT participants who consented to brain autopsy prior to death, gold-standard measures of Alzheimer's disease neuropathology are available. Our proposed estimators reduced bias and improved efficiency for the ATE, facilitating valid inference with EHR data when key data elements are ascertaine

The permanent of an $n \times n$ matrix $M = (m_{ij})$ is defined as $\mathrm{per}(M) = \sum_{σ\in S_n} \prod_{i=1}^n m_{i,σ(i)}$, where $S_n$ denotes the symmetric group on $\{1,2,\ldots,n\}$. The permanental polynomial of $M$, is defined by $ψ(M;x) = \mathrm{per}(xI_n - M)$. We study two fundamental variants: the Laplacian permanental polynomial $ψ(L(G);x)$ and signless Laplacian permanental polynomial $ψ(Q(G);x)$ of a graph $G$. A graph is said to be {determined} by its (signless) Laplacian permanental polynomial if no other non-isomorphic graph shares the same polynomial. A graph is combinedly determined when isomorphism is guaranteed by the equality of both polynomials. Characterizing which graphs are determined by their(signless) Laplacian permanental polynomials is an interesting problem. This paper investigates the permanental characterization problem for several families of starlike graphs, including: spider graphs (tree), coconut tree, perfect binary tree, corona product of $C_m$ and $K_n$, and $\bar K_n$ for various values of $m$ and $n$. We establish which of these graphs are determined by their Laplacian or signless Laplacian permanental polynomials, and which require

Computing the permanent of a non-negative matrix is a computationally challenging, \#P-complete problem with wide-ranging applications. We introduce a novel permanental analogue of Schur's determinant formula, leveraging a newly defined \emph{permanental inverse}. Building on this, we introduce an iterative, deterministic procedure called the \emph{permanent process}, analogous to Gaussian elimination, which yields constructive and algorithmically computable upper bounds on the permanent. Our framework provides particularly strong guarantees for matrices exhibiting approximate diagonal dominance-like properties, thereby offering new theoretical and computational tools for analyzing and bounding permanents.

The rank of an n x n matrix A is equal to the size of its largest square submatrix with a nonzero determinant, and it can be computed in O(n^2.37) time. Analogously, the size of the largest square submatrix with nonzero permanent is defined as the permanental rank. Computing the permanent or the coefficients of the permanental polynomial is #P-complete. The permanental nullity is defined as the multiplicity of zero as a root of the permanental polynomial. We establish a permanental analog of the rank-nullity theorem, showing that the sum of the permanental rank and the permanental nullity equals n for symmetric nonnegative matrices, positive semidefinite matrices, and adjacency matrices of balanced signed graphs. Using this theorem, we can compute the permanental nullity for symmetric nonnegative matrices and adjacency matrices of balanced signed graphs in polynomial time. For symmetric matrices with entries in {0, plus or minus 1}, we also provide a complete characterization of when the permanental rank-nullity identity holds.

This paper is motivated by basic complexity and probability questions about permanents of random matrices over finite fields, and in particular, about properties separating the permanent and the determinant. Fix $q = p^m$ some power of an odd prime, and let $k \leq n$ both be growing. For a uniformly random $n \times k$ matrix $A$ over $\mathbb{F}_q$, we study the probability that all $k \times k$ submatrices of $A$ have zero permanent; namely that $A$ does not have full "permanental rank". When $k = n$, this is simply the probability that a random square matrix over $\mathbb{F}_q$ has zero permanent, which we do not understand. We believe that the probability in this case is $\frac{1}{q} + o(1)$, which would be in contrast to the case of the determinant, where the answer is $\frac{1}{q} + Ω_q(1)$. Our main result is that when $k$ is $O(\sqrt{n})$, the probability that a random $n \times k$ matrix does not have full permanental rank is essentially the same as the probability that the matrix has a $0$ column, namely $(1 +o(1)) \frac{k}{q^n}$. In contrast, for determinantal (standard) rank the analogous probability is $Θ(\frac{q^k}{q^n})$. At the core of our result are some basic lin

In this article, we study permanental varieties, i.e. varieties defined by the vanishing of permanents of fixed size of a generic matrix. Permanents and their varieties play an important, and sometimes poorly understood, role in combinatorics. However, there are essentially no geometric results about them in the literature, in very sharp contrast to the well-behaved and ubiquitous case of determinants and minors. Motivated by the study of the singular locus of the permanental hypersurface, we focus on the codimension of these varieties. We introduce a $\mathbb C^{*}$-action on matrices and prove a number of results. In particular, we improve a lower bound on the codimension of the aforementioned singular locus established by von zur Gathen in 1987.

We characterize ratios of permanents of (generalized) submatrices which are bounded on the set of all totally positive matrices. This provides a permanental analog of results of Fallat, Gekhtman, and Johnson [{\em Adv.\ Appl.\ Math.} {\bf 30} no.\ 3, (2003) pp.\ 442--470] concerning ratios of matrix minors. We also extend work of Drake, Gerrish, and the first author [{\em Electron.\ J.\ Combin.,} {\bf 11} no.\ 1, (2004) Note 6] by characterizing the differences of monomials in $\mathbb{Z}[x_{1,1},x_{1,2},...,x_{n,n}]$ which evaluate positively on the set of all totally positive $n \times n$ matrices.

Let $G$ be a bipartite graph with adjacency matrix $A(G)$. The characteristic polynomial $φ(G,x)=\det(xI-A(G))$ and the permanental polynomial $π(G,x) = \text{per}(xI-A(G))$ are both graph invariants used to distinguish graphs. For bipartite graphs, we define the modified characteristic polynomial, which is obtained by changing the signs of some of the coefficients of $φ(G,x)$. For $4k$-intercyclic bipartite graphs, i.e., those for which the removal of any $4k$-cycle results in a $C_{4k}$-free graph, we provide an expression for $π(G,x)$ in terms of the modified characteristic polynomial of the graph and its subgraphs. Our approach is purely combinatorial in contrast to the Pfaffian orientation method found in the literature to compute the permanental polynomial.

Let $u(s,t)$ be a continuous potential density of a symmetric Lévy process or diffusion with state space $T$ killed at $T_{0}$, the first hitting time of $0$, or at $λ\wedge T_{0}$, where $λ$ is an independent exponential time. Let \[ f(t)=\int_{T} u(t,v)\,dμ(v), \] where $μ$ is a finite positive measure on $T$. Let $X_α=\{X_α(t),t\in T \}$ be an $α-$permanental process with kernel \[ v(s,t)=u(s,t)+f(t). \] Then when $\lim_{t\to 0}u(t,t)=0$, \[ \limsup_{t\downarrow 0}\frac{X_α(t )}{u(t,t)\log \log 1/t }\ge 1 ,\qquad \text{a.s.} \] and \[ \limsup_{t\downarrow 0}\frac{X_α(t )}{u(t,t)\log \log 1/t }\le 1+C_{u,h} ,\qquad \text{a.s.} \] where $C_{u,μ}\le |μ|$ is a constant that depends on both $u$ and $μ$, which is given explicitly, and is different in the different examples.

Existing permanental processes often impose constraints on kernel types or stationarity, limiting the model's expressiveness. To overcome these limitations, we propose a novel approach utilizing the sparse spectral representation of nonstationary kernels. This technique relaxes the constraints on kernel types and stationarity, allowing for more flexible modeling while reducing computational complexity to the linear level. Additionally, we introduce a deep kernel variant by hierarchically stacking multiple spectral feature mappings, further enhancing the model's expressiveness to capture complex patterns in data. Experimental results on both synthetic and real-world datasets demonstrate the effectiveness of our approach, particularly in scenarios with pronounced data nonstationarity. Additionally, ablation studies are conducted to provide insights into the impact of various hyperparameters on model performance.

Let $Y$ be a symmetric Borel right process with locally compact state space $T\subseteq R^{1}$ and potential densities $u(x,y)$ with respect to some $σ$-finite measure on $T$. Let $g$ and $f$ be finite excessive functions for $ Y$. Set $$ u_{g, f}(x,y)= u(x,y)+g(x)f(y),\qquad x,y\in T.$$ In this paper we take $Y$ to be a symmetric Lévy process, or a diffusion, that is killed at the end of an independent exponential time or the first time it hits 0. Under general smoothness conditions on $g$, $f$, $u$ and points $d\in T$, laws of the iterated logarithm are found for $X_{k/2} =\{X_{k/2}(t), t\in T \}$, a $k/2-$permanental process with kernel $ \{u_{g, f}(x,y),x,y\in T \}$, of the following form: For all integers $k\geq 1$, $$\limsup_{x \to 0}\frac{| X_{k/2}( d+x)- X_{k/2} (d)|}{ \left( 2 σ^{2}\left(x\right)\log\log 1/x\right)^{1/2}}= \left( 2 X _{k/2} (d)\right)^{1/2}, \qquad a.s. ,$$ where, $$σ^2(x)=u(d+x,d+x)+u(x,x)-2u(d+x,x).$$ Using these limit theorems and the Eisenbaum Kaspi Isomorphism Theorem, laws of the iterated logarithm are found for the local times of certain Markov processes with potential densities that have the form of $ \{u_{g, f}(x,y),x,y\in T \}$ or are slight modi

Let $G$ be a graph, and let $A(G)$ be the adjacency matrix of $G$. The permanental polynomial of $G$ is defined as $π(G,x)=\mathrm{per}(xI-A(G))$. The permanental sum of $G$ can be defined as the sum of absolute value of coefficients of $π(G,x)$. Computing the permanental sum is $\#$P-complete. Any a bicyclic graph can be generated from three types of induced subgraphs. In this paper, we determine the upper bound of permanental sums of bicyclic graphs generated from each a type of induced subgraph. And we also determine the second maximal permanental sum of all bicyclic graphs.

Let ${\rm Mat}_n(\mathbb{F})$ denote the set of square $n\times n$ matrices over a field $\mathbb{F}$ of characteristic different from two. The permanental rank ${\rm prk}\,(A)$ of a matrix $A \in{\rm Mat}_{n}(\mathbb{F})$ is the size of the maximal square submatrix in $A$ with nonzero permanent. By $Λ^{k}$ and $Λ^{\leq k}$ we denote the subsets of matrices $A \in {\rm Mat}_{n}(\mathbb{F})$ with ${\rm prk}\,(A) = k$ and ${\rm prk}\,(A) \leq k$, respectively. In this paper for each $1 \leq k \leq n-1$ we obtain a complete characterization of linear maps $T: {\rm Mat}_{n}(\mathbb{F}) \to {\rm Mat}_{n}(\mathbb{F})$ satisfying $T(Λ^{\leq k}) = Λ^{\leq k}$ or bijective linear maps satisfying $T(Λ^{\leq k}) \subseteq Λ^{\leq k}$. Moreover, we show that if $\mathbb{F}$ is an infinite field, then $Λ^{k}$ is Zariski dense in $Λ^{\leq k}$ and apply this to describe such bijective linear maps satisfying $T(Λ^{k}) \subseteq Λ^{k}$.

As a variant of the Ulam's vertex reconstruction conjecture and the Harary's edge reconstruction conjecture, Cvetković and Schwenk posed independently the following problem: Can the characteristic polynomial of a simple graph $G$ with vertex set $V$ be reconstructed from the characteristic polynomials of all subgraphs in $\{G-v|v\in V\}$ for $|V|\geq 3$? This problem is still open. A natural problem is: Can the characteristic polynomial of a simple graph $G$ with edge set $E$ be reconstructed from the characteristic polynomials of all subgraphs in $\{G-e|e\in E\}$? In this paper, we prove that if $|V| eq |E|$, then the characteristic polynomial of $G$ can be reconstructed from the characteristic polynomials of all subgraphs in $\{G-uv, G-u-v|uv\in E\}$, and the similar result holds for the permanental polynomial of $G$. We also prove that the Laplacian (resp. signless Laplacian) characteristic polynomial of $G$ can be reconstructed from the Laplacian (resp. signless Laplacian) characteristic polynomials of all subgraphs in $\{G-e|e\in E\}$ (resp. if $|V| eq |E|$).

Let $G$ be a graph with $n$ vertices, and let $L(G)$ and $Q(G)$ be the Laplacian matrix and signless Laplacian matrix of $G$, respectively. The polynomial $π(L(G);x)={\rm per}(xI-L(G))$ (resp. $π(Q(G);x)={\rm per}(xI-Q(G))$) is called {\em Laplacian permanental polynomial} (resp. {\em signless Laplacian permanental polynomial}) of $G$. In this paper, we show that two classes of bicyclic graphs are determined by their (signless) Laplacian permanental polynomials.

Estimating the number of natural disasters benefits the insurance industry in terms of risk management. However, the estimation process is complicated due to the fact that there are many factors affecting the number of such incidents. In this work, we propose a Normal approximation technique for associated point processes for estimating the number of natural disasters under the following two assumptions: 1) the incident counts in any two distinct areas are positively associated and 2) the association between these counts in two distinct areas decays exponentially with respect to distance outside some small local neighborhood. Under the stated assumptions, we extend previous results for the Normal approximation technique for associated point processes, i.e., the establishment of non-asymptotic $L^1$ bounds for the functionals of these processes [Wiroonsri (2019)]. Then we apply this new result to permanental Cox processes that are known to be positively associated. Finally, we apply our Normal approximation results for permanental Cox processes to Thailand's fire data from 2007 to 2020, which was collected by the Geo-Informatics and Space Technology Development Agency of Thailand.

We develop an unsupervised probabilistic model for heterogeneous Electronic Health Record (EHR) data. Utilizing a mixture model formulation, our approach directly models sequences of arbitrary length, such as medications and laboratory results. This allows for subgrouping and incorporation of the dynamics underlying heterogeneous data types. The model consists of a layered set of latent variables that encode underlying structure in the data. These variables represent subject subgroups at the top layer, and unobserved states for sequences in the second layer. We train this model on episodic data from subjects receiving medical care in the Kaiser Permanente Northern California integrated healthcare delivery system. The resulting properties of the trained model generate novel insight from these complex and multifaceted data. In addition, we show how the model can be used to analyze sequences that contribute to assessment of mortality likelihood.