Conjugate gradient algorithm for consistent generalized Sylvester-transpose matrix equations

<abstract><p>We develop an effective algorithm to find a well-approximate solution of a generalized Sylvester-transpose matrix equation where all coefficient matrices and an unknown matrix are rectangular. The algorithm aims to construct a finite sequence of approximated solutions from any given initial matrix. It turns out that the associated residual matrices are orthogonal, and thus, the desire solution comes out in the final step with a satisfactory error. We provide numerical experiments to show the capability and performance of the algorithm.</p></abstract>

On the locations of maxima and minima in a sequence of exchangeable random variables

We show for a finite sequence of exchangeable random variables that the locations of the maximum and minimum are independent from every symmetric event. In particular they are uniformly distributed on the grid without the diagonal. Moreover, for an infinite sequence we show that the extrema and their locations are asymptotically independent. Here, in contrast to the classical approach we do not use affine-linear transformations. Moreover it is shown how the new transformations can be used in extreme value statistics.

Parameter estimation and signal reconstruction

In frequency analysis an often appearing problem is the reconstruction of a signal from given samples. Since the samples are usually noised, pure interpolating approaches are not recommended and appropriate approximation methods are more suitable as they can be interpreted as a kind of denoising. Two approaches are widely used. One uses the reflection coefficients of a finite sequence of Szegő polynomials and the other one the zeros of the so called Prony polynomial. We show that both approaches are closely related. As a kind of inverse problem, it’s not surprising that they have in common that both methods depend very sensitive on sampling errors. We use known properties of the signal to estimate the positions of the zeros of the corresponding Szegő- or Prony-like polynomials and construct adaptive algorithms to calculate these ones. Hereby, we get the corresponding parameters in the exponential parts of the signal, too. Then, the coefficients of the signal (as a linear combination of such exponential functions) can be obtained from a system of linear equations by minimizing the residuals with respect to a suitable norm as a kind of denoising.

Sensor fusion approach for shape estimation of a base-excited cantilever beam

In smart structures, achieving a reliable set of measurement signals to monitor the system’s performance is critical. Also, the essence of using the optimum batch of sensors and an efficient algorithm to process these signals is significant for active vibration control of these structures. This paper primarily introduces a method of sensor fusion using the Kalman filter as an observer to gain the proper position signal from both an accelerometer and an ultrasonic sensor mounted on the tip of a cantilever beam. The main goal of this procedure is to eliminate both sensors’ shortcomings. Also, we present a novel approach to estimate the overall shape of the beam, using only the tip position signal. To this end, a high-speed camera is used to capture the motion of three markers on the beam under different excitation frequencies. Then, three long short-term memory networks are trained by deep learning methods, using a finite sequence of beam tip position, to act as observers for estimating the shape of the beam. The proposed methods are simulated and then validated by experiments.

Estimates of objective function minimum for solving linear fractional unconstrained combinatorial optimization problems on arrangements

The paper is devoted to the study of one class of Euclidean combinatorial optimization problems — combinatorial optimization problems on the general set of arrangements with linear fractional objective function and without additional (non-combinatorial) constraints. The paper substantiates the improvement of the polynomial algorithm for solving the specified class of problems. This algorithm foresees solving a finite sequence of linear unconstrained problems of combinatorial optimization on arrangements. The modification of the algorithm is based on the use of estimates of the objective function on the feasible set. This allows to exclude some of the problems from consideration and reduce the number of problems to be solved. The numerical experiments confirm the practical efficiency of the proposed approach.

A Discrete Morse Perspective on Knot Projections and a Generalised Clock Theorem

We obtain a simple and complete characterisation of which matchings on the Tait graph of a knot diagram induce a discrete Morse function (dMf) on the two sphere, extending a construction due to Cohen. We show these dMfs are in bijection with certain rooted spanning forests in the Tait graph. We use this to count the number of such dMfs with a closed formula involving the graph Laplacian. We then simultaneously generalise Kauffman's Clock Theorem and Kenyon-Propp-Wilson's correspondence in two different directions; we first prove that the image of the correspondence induces a bijection on perfect dMfs, then we show that all perfect matchings, subject to an admissibility condition, are related by a finite sequence of click and clock moves. Finally, we study and compare the matching and discrete Morse complexes associated to the Tait graph, in terms of partial Kauffman states, and provide some computations.

SAFE ETERNAL 1-SECURE SETS IN GRAPHS

An eternal $1$-secure set, in a graph $G = (V, E)$ is a set $D \subset V$ having the property that for any finite sequence of vertices $r_1, r_2, \ldots, r_k$ there exists a sequence of vertices $v_1, v_2, \ldots, v_k$ and a sequence $ D = D_0, D_1, D_2, \ldots, D_k$ of dominating sets of $G$, such that for each $i$, $1 \leq i \leq k$, $D_{i} = (D_{i-1} - \{v_i\}) \cup \{r_i\}$, where $v_i \in D_{i-1}$ and $r_i \in N[v_i]$. Here $r_i = v_i$ is possible. The cardinality of the smallest eternal $1$-secure set in a graph $G$ is called the eternal $1$-security number of $G$. In this paper we study a variations of eternal $1$-secure sets named safe eternal $1$-secure sets. A vertex $v$ is safe with respect to an eternal $1$-secure set $S$ if $N[v] \bigcap S =1$. An eternal 1 secure set $S$ is a safe eternal 1 secure set if at least one vertex in $G$ is safe with respect to the set $S$. We characterize the class of graphs having safe eternal $1$-secure sets for which all vertices - excluding those in the safe $1$-secure sets - are safe. Also we introduce a new kind of directed graphs which represent the transformation from one safe 1 - secure set to another safe 1-secure set of a given graph and study its properties.

The impossibility of a satisfactory population prospect axiology (independently of Finite Fine-Grainedness)

Arrhenius’s impossibility theorems purport to demonstrate that no population axiology can satisfy each of a small number of intuitively compelling adequacy conditions. However, it has recently been pointed out that each theorem depends on a dubious assumption: Finite Fine-Grainedness. This assumption states that there exists a finite sequence of slight welfare differences between any two welfare levels. Denying Finite Fine-Grainedness makes room for a lexical population axiology which satisfies all of the compelling adequacy conditions in each theorem. Therefore, Arrhenius’s theorems fail to prove that there is no satisfactory population axiology. In this paper, I argue that Arrhenius’s theorems can be repurposed. Since all of our population-affecting actions have a non-zero probability of bringing about more than one distinct population, it is population prospect axiologies that are of practical relevance, and amended versions of Arrhenius’s theorems demonstrate that there is no satisfactory population prospect axiology. These impossibility theorems do not depend on Finite Fine-Grainedness, so lexical views do not escape them.

Posted Price Mechanisms and Optimal Threshold Strategies for Random Arrivals

The classic prophet inequality states that, when faced with a finite sequence of nonnegative independent random variables, a gambler who knows the distribution and is allowed to stop the sequence at any time, can obtain, in expectation, at least half as much reward as a prophet who knows the values of each random variable and can choose the largest one. In this work, we consider the situation in which the sequence comes in random order. We look at both a nonadaptive and an adaptive version of the problem. In the former case, the gambler sets a threshold for every random variable a priori, whereas, in the latter case, the thresholds are set when a random variable arrives. For the nonadaptive case, we obtain an algorithm achieving an expected reward within at least a 0.632 fraction of the expected maximum and prove that this constant is optimal. For the adaptive case with independent and identically distributed random variables, we obtain a tight 0.745-approximation, solving a problem posed by Hill and Kertz in 1982. We also apply these prophet inequalities to posted price mechanisms, and we prove the same tight bounds for both a nonadaptive and an adaptive posted price mechanism when buyers arrive in random order.

The Maximum Modulus Set of a Polynomial

We study the maximum modulus set, $${{\mathcal {M}}}(p)$$ M ( p ) , of a polynomial p. We are interested in constructing p so that $${{\mathcal {M}}}(p)$$ M ( p ) has certain exceptional features. Jassim and London gave a cubic polynomial p such that $${{\mathcal {M}}}(p)$$ M ( p ) has one discontinuity, and Tyler found a quintic polynomial $${\tilde{p}}$$ p ~ such that $${{\mathcal {M}}}({\tilde{p}})$$ M ( p ~ ) has one singleton component. These are the only results of this type, and we strengthen them considerably. In particular, given a finite sequence $$a_1, a_2, \ldots , a_n$$ a 1 , a 2 , … , a n of distinct positive real numbers, we construct polynomials p and $${\tilde{p}}$$ p ~ such that $${{\mathcal {M}}}(p)$$ M ( p ) has discontinuities of modulus $$a_1, a_2, \ldots , a_n$$ a 1 , a 2 , … , a n , and $${{\mathcal {M}}}({\tilde{p}})$$ M ( p ~ ) has singleton components at the points $$a_1, a_2, \ldots , a_n$$ a 1 , a 2 , … , a n . Finally we show that these results are strong, in the sense that it is not possible for a polynomial to have infinitely many discontinuities in its maximum modulus set.