The Motzkin numbers count the number of lattice paths which go from (0,0) to (n,0) using steps (1,1),(1,0) and (1,−1) and never go below the x-axis. Let Mn,k be the number of such paths with exactly ...k horizontal steps. We investigate the analytic properties of various combinatorial triangles related to the Motzkin triangle Mn,kn,k≥0, including their total positivity, the real-rootedness and interlacing property of the generating functions of their rows, and the asymptotic normality (by central and local limit theorems) of these triangles. We also prove several identities related to these triangles.
The quasilinear equation −(u′(x)1+(u′(x))2)′=λ(1−u)2−λɛ2(1−u)4 with the boundary condition u(−L)=u(L)=0 governs the steady-state solutions from a regularized MEMS model. We prove that for any ...evolution parameters ɛ∈(0,1) and L>0, the global bifurcation curve of positive solutions is strictly increasing or ⊃-like shaped or S-like shaped in the (λ,‖u‖∞)-plane. The bifurcation curves present a variety of shapes and structures, significantly different from those in non-regularized case (i.e., ɛ=0) and in the simplified semilinear case. The main tools are some new time-map techniques, the total positivity theory, and Sturm’s Theorem.
The totally nonnegative Grassmannian is the set of k-dimensional subspaces V of Rn whose nonzero Plücker coordinates all have the same sign. Gantmakher and Krein (1950) and Schoenberg and Whitney ...(1951) independently showed that V is totally nonnegative iff every vector in V, when viewed as a sequence of n numbers and ignoring any zeros, changes sign at most k−1 times. We generalize this result from the totally nonnegative Grassmannian to the entire Grassmannian, showing that if V is generic (i.e. has no zero Plücker coordinates), then the vectors in V change sign at most m times iff certain sequences of Plücker coordinates of V change sign at most m−k+1 times. We also give an algorithm which, given a non-generic V whose vectors change sign at most m times, perturbs V into a generic subspace whose vectors also change sign at most m times. We deduce that among all V whose vectors change sign at most m times, the generic subspaces are dense. These results generalize to oriented matroids. As an application of our results, we characterize when a generalized amplituhedron construction, in the sense of Arkani-Hamed and Trnka (2013), is well defined. We also give two ways of obtaining the positroid cell of each V in the totally nonnegative Grassmannian from the sign patterns of vectors in V.
We consider piecewise Chebyshevian splines, in the sense of splines with pieces taken from any different five-dimensional Extended Chebyshev spaces, and with connection matrices at the knots. In this ...large context we establish necessary and sufficient conditions for the existence of totally positive refinable B-spline bases. These conditions are applied in many important special cases, e.g. symmetric cardinal geometrically continuous quartic B-spline, parametrically continuous mixed L-splines. The great variety of illustrations provided proves the richness of this class of splines for design. This richness can be exploited in various other fields as well.
Abstract
Selecting the optimal Markowitz portfolio depends on estimating the covariance matrix of the returns of N assets from T periods of historical data. Problematically, N is typically of the ...same order as T, which makes the sample covariance matrix estimator perform poorly, both empirically and theoretically. While various other general-purpose covariance matrix estimators have been introduced in the financial economics and statistics literature for dealing with the high dimensionality of this problem, we here propose an estimator that exploits the fact that assets are typically positively dependent. This is achieved by imposing that the joint distribution of returns be multivariate totally positive of order 2 (MTP2). This constraint on the covariance matrix not only enforces positive dependence among the assets but also regularizes the covariance matrix, leading to desirable statistical properties such as sparsity. Based on stock market data spanning 30 years, we show that estimating the covariance matrix under MTP2 outperforms previous state-of-the-art methods including shrinkage estimators and factor models.
This is the first of a series of papers where we develop a theory of total positivity for loop groups. In this paper, we completely describe the totally nonnegative part of the polynomial loop group ...GLn(Rt,t−1), and for the formal loop group GLn(R((t))) we describe the totally nonnegative points which are not totally positive. Furthermore, we make the connection with networks on the cylinder.
Our approach involves the introduction of distinguished generators, called whirls and curls, and we describe the commutation relations amongst them. These matrices play the same role as the poles and zeros of the Edrei–Thoma theorem classifying totally positive functions (corresponding to our case n=1). We give a solution to the “factorization problem” using limits of ratios of minors. This is in a similar spirit to the Berenstein–Fomin–Zelevinsky Chamber Ansatz where ratios of minors are used. A birational symmetric group action arising in the commutation relation of curls appeared previously in Noumi–Yamada’s study of discrete Painlevé dynamical systems and Berenstein–Kazhdan’s study of geometric crystals.
Balanced Truncation of k-Positive Systems Grussler, Christian; Damm, Tobias; Sepulchre, Rodolphe
IEEE transactions on automatic control,
2022-Jan., 2022-1-00, Volume:
67, Issue:
1
Journal Article
Peer reviewed
Open access
This article considers balanced truncation of discrete-time Hankel <inline-formula><tex-math notation="LaTeX">k</tex-math></inline-formula>-positive systems, characterized by Hankel matrices whose ...minors up to order <inline-formula><tex-math notation="LaTeX">k</tex-math></inline-formula> are nonnegative. Our main result shows that if the truncated system has order <inline-formula><tex-math notation="LaTeX">k</tex-math></inline-formula> or less, then it is Hankel totally positive (<inline-formula><tex-math notation="LaTeX">\infty</tex-math></inline-formula>-positive), meaning that it is a sum of first-order lags. This result can be understood as a bridge between two known results: the property that the first-order truncation of a positive system is positive (<inline-formula><tex-math notation="LaTeX">k=1</tex-math></inline-formula>), and the property that balanced truncation preserves state-space symmetry. It provides a broad class of systems where balanced truncation is guaranteed to result in a minimal internally positive system.
Given a sequence
α
=
(
a
k
)
k
≥
0
of nonnegative numbers, define a new sequence
L
(
α
)
=
(
b
k
)
k
≥
0
by
b
k
=
a
k
2
-
a
k
-
1
a
k
+
1
. The sequence
α
is called
r
-
log-concave
if
L
i
(
α
)
=
L
(
...L
i
-
1
(
α
)
)
is a nonnegative sequence for all
1
≤
i
≤
r
. In this paper, we study the
r
-log-concavity and its
q
-analogue for
r
=
2
,
3
using total positivity of matrices. We show the 6-log-concavity of the Taylor coefficients of the Riemann
ξ
-function. We give some criteria for
r
-
q
-log-concavity for
r
=
2
,
3
. As applications, we get 3-
q
-log-concavity of
q
-binomial coefficients and different
q
-Stirling numbers of two kinds, which extends results for
q
-log-concavity. We also present some results for
r
-
q
-log-concavity from the linear transformations. Finally, we pose an interesting question.
In this article, we first study the likelihood ratio ordering of generalized order statistics (GOS) in both one-sample and two-sample problems. Then, we establish the transmission of the increasing ...hazard rate and decreasing reversed hazard rate aging properties of GOS. To do this, we extend Karlin's basic composition theorem for the functions of three variables. Then, we settle certain open problems in this regard by providing some counterexamples. We further investigate similar transmission cases which have not been addressed in the literature so far.
We study nonparametric maximum likelihood estimation for two classes of multivariate distributions that imply strong forms of positive dependence; namely log‐supermodular (MTP2) distributions and ...log‐L♮‐concave (LLC) distributions. In both cases we also assume log‐concavity in order to ensure boundedness of the likelihood function. Given n independent and identically distributed random vectors from one of our distributions, the maximum likelihood estimator (MLE) exists a.s. and is unique a.e. with probability one when n≥3. This holds independently of the ambient dimension d. We conjecture that the MLE is always the exponential of a tent function. We prove this result for samples in {0,1}d or in ℝ2 under MTP2, and for samples in ℚd under LLC. Finally, we provide a conditional gradient algorithm for computing the maximum likelihood estimate.
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