Minimal Zero Research Articles

Minimal zero entropy subshifts can be unrestricted along any sparse set

Abstract We present a streamlined proof of a result essentially presented by the author in [Some counterexamples in topological dynamics. Ergod. Th. & Dynam. Sys.28(4) (2008), 1291–1322], namely that for every set $S = \{s_1, s_2, \ldots \} \subset \mathbb {N}$ of zero Banach density and finite set A, there exists a minimal zero-entropy subshift $(X, \sigma )$ so that for every sequence $u \in A^{\mathbb {Z}}$ , there is $x_u \in X$ with $x_u(s_n) = u(n)$ for all $n \in \mathbb {N}$ . Informally, minimal deterministic sequences can achieve completely arbitrary behavior upon restriction to a set of zero Banach density. As a corollary, this provides counterexamples to the polynomial Sarnak conjecture reported by Eisner [A polynomial version of Sarnak’s conjecture. C. R. Math. Acad. Sci. Paris353(7) (2015), 569–572] which are significantly more general than some recently provided by Kanigowski, Lemańczyk and Radziwiłł [Prime number theorem for analytic skew products. Ann. of Math. (2)199 (2024), 591–705] and by Lian and Shi [A counter-example for polynomial version of Sarnak’s conjecture. Adv. Math.384 (2021), Paper no. 107765] and shows that no similar result can hold under only the assumptions of minimality and zero entropy.

On the structure of the 6 × 6 copositive cone

In this work we complement the description of the extreme rays of the 6×6 copositive cone with some topological structure. In a previous paper we decomposed the set of extreme elements of this cone into a disjoint union of subsets of algebraic varieties of different dimension. In this paper we link this classification to the recently introduced combinatorial characteristic called extended minimal zero support set. We determine those subsets which are essential, i.e., which are not embedded in the boundary of other subsets. This allows to drastically decrease the number of cases one has to consider when investigating different properties of the 6×6 copositive cone. As an application, we construct an example of a copositive 6×6 matrix with all ones on the diagonal which does not belong to the Parrilo inner sum of squares relaxation K6(1).

Minimal Zero Sequences of Finite Cyclic Groups

Minimal Zero Sequences of Finite Cyclic Groups

On equivalent representations and properties of faces of the cone of copositive matrices

The paper is devoted to the study of the cone of copositive matrices. Based on the concept of immobile indices known from semi-infinite optimization, we define zero and minimal zero vectors of a subset of the cone and use them to obtain different representations of the faces of and the corresponding dual cones. The minimal face of containing a given convex subset of this cone is described, and some propositions are proved that allow obtaining equivalent descriptions of the feasible sets of copositive problems.

Structural Properties of Faces of the Cone of Copositive Matrices

In this paper, we study the properties of faces and exposed faces of the cone of copositive matrices (copositive cone), paying special attention to issues related to their geometric structure. Based on the concepts of zero and minimal zero vectors, we obtain several explicit representations of faces of the copositive cone and compare them. Given a face of the cone of copositive matrices, we describe the subspace generated by that face and the minimal exposed face containing it. Summarizing the results obtained in the paper, we systematically show what information can be extracted about the given copositive face in the case of incomplete data. Several examples for illustrating the main findings of the paper and also for justifying the usefulness of the developed approach to the study of the facial structure of the copositive cone are discussed.

Characteristic measures of symbolic dynamical systems

AbstractA probability measure is a characteristic measure of a topological dynamical system if it is invariant to the automorphism group of the system. We show that zero entropy shifts always admit characteristic measures. We use similar techniques to show that automorphism groups of minimal zero entropy shifts are sofic.

The extreme rays of the $$6\times 6$$ copositive cone

We provide a complete classification of the extreme rays of the $$6 \times 6$$ copositive cone $$\mathcal {COP}^{6}$$ . We proceed via a coarse intermediate classification of the possible minimal zero support set of an exceptional extremal matrix $$A \in \mathcal {COP}^{6}$$ . To each such minimal zero support set we construct a stratified semi-algebraic manifold in the space of real symmetric $$6 \times 6$$ matrices $${\mathcal {S}}^{6}$$ , parameterized in a semi-trigonometric way, which consists of all exceptional extremal matrices $$A \in \mathcal {COP}^{6}$$ having this minimal zero support set. Each semi-algebraic stratum is characterized by the supports of the minimal zeros u as well as the supports of the corresponding matrix-vector products Au. The analysis uses recently and newly developed methods that are applicable to copositive matrices of arbitrary order.

On the algebraic structure of the copositive cone

We decompose the copositive cone $$\mathcal {COP}^{n}$$ into a disjoint union of a finite number of open subsets $$S_{{\mathcal {E}}}$$ of algebraic sets $$Z_{{\mathcal {E}}}$$ . Each set $$S_{{\mathcal {E}}}$$ consists of interiors of faces of $$\mathcal {COP}^{n}$$ . On each irreducible component of $$Z_{{\mathcal {E}}}$$ these faces generically have the same dimension. Each algebraic set $$Z_{{\mathcal {E}}}$$ is characterized by a finite collection $${{\mathcal {E}}} = \{(I_{\alpha },J_{\alpha })\}_{\alpha = 1,\dots ,|\mathcal{E}|}$$ of pairs of index sets. Namely, $$Z_{{\mathcal {E}}}$$ is the set of symmetric matrices A such that the submatrices $$A_{J_{\alpha } \times I_{\alpha }}$$ are rank-deficient for all $$\alpha $$ . For every copositive matrix $$A \in S_{{\mathcal {E}}}$$ , the index sets $$I_{\alpha }$$ are the minimal zero supports of A. If $$u^{\alpha }$$ is a corresponding minimal zero, then $$J_{\alpha }$$ is the set of indices j such that $$(Au^{\alpha })_j = 0$$ . We call the pair $$(I_{\alpha },J_{\alpha })$$ the extended support of the zero $$u^{\alpha }$$ , and $${{\mathcal {E}}}$$ the extended minimal zero support set of A. We provide some necessary conditions on $${{\mathcal {E}}}$$ for $$S_{{\mathcal {E}}}$$ to be non-empty, and for a subset $$S_{{{\mathcal {E}}}'}$$ to intersect the boundary of another subset $$S_{{\mathcal {E}}}$$ .

Zeros of recursively defined polynomials

Let g, h be two arithmetic functions. In this paper, we study zero-free intervals of polynomials of the type We use a difference equation approach. Let be the divisor sum function and the identity function, then the zeros of the polynomials dictate the vanishing properties of the coefficients of all powers of the Dedekind eta function. This is directly related to the Lehmer conjecture on the non-vanishing of the Ramanujan τ-function. Let be the identity function. Then, the polynomials are related to generalized Laguerre polynomials for and to Chebyshev polynomials of the second kind for . The paper is furnished with several interesting examples. We found polynomials with decreasing minimal real zeros. We provide a simple arithmetic function g to approximate the minimal real zeros attached to σ.

Fuzzification of Zero Forcing Process

In this paper, we investigate the fuzzification of zero forcing process. For this, first we introduce a new embedding of a graph [Formula: see text] by considering a minimal zero forcing set of [Formula: see text] and an arbitrary list of maximal forcing chains of this zero forcing set. Then we get a comparison between zero forcing sets of a graph by using fuzzy concepts. Finally, we give an application for this procedure.

Variable <sc>on</sc>-Time Control Scheme for the Secondary-Side Controlled Flyback Converter

This paper presents an analysis of a novel control approach for the secondary side controlled flyback concept, along with improved drain–source voltage sensing for more precise gate signals and reduced losses. In contrast to the existing control scheme for this concept, the approach presented here sustains constant switching frequency throughout the load range without any additional hardware, boosting efficiency and simplifying coupled inductor design optimization. A 65-W demonstrator shows minimal output ripple during load changes, peak efficiency of 89.90%, natural output current limiting in overload conditions, and utilizes a novel lossless synchronous-rectification sensing subcircuit with minimal zero current crossing delay.

A new certificate for copositivity

In this article, we introduce a new method of certifying any copositive matrix to be copositive. This is done through the use of a theorem by Hadeler and the Farkas Lemma. For a given copositive matrix this certificate is constructed by solving finitely many linear systems, and can be subsequently checked by checking finitely many linear inequalities. In some cases, this certificate can be relatively small, even when the matrix generates an extreme ray of the copositive cone which is not positive semidefinite plus nonnegative. This certificate can also be used to generate the set of minimal zeros of a copositive matrix. In the final section of this paper we introduce a set of newly discovered extremal copositive matrices.

Extremal Copositive Matrices with Zero Supports of Cardinality n-2

Let $A \in {\cal C}^n$ be an exceptional extremal copositive $n \times n$ matrix with positive diagonal. A zero $u$ of $A$ is a non-zero nonnegative vector such that $u^TAu = 0$. The support of a zero $u$ is the index set of the positive elements of $u$. A zero $u$ is minimal if there is no other zero $v$ such that $\Supp v \subset \Supp u$ strictly. Let $G$ be the graph on $n$ vertices which has an edge $(i,j)$ if and only if $A$ has a zero with support $\{1,\dots,n\} \setminus \{i,j\}$. In this paper, it is shown that $G$ cannot contain a cycle of length strictly smaller than $n$. As a consequence, if all minimal zeros of $A$ have support of cardinality $n - 2$, then $G$ must be the cycle graph $C_n$.

Extremal Copositive Matrices with Zero Supports of Cardinality n-2

Let $A \in {\cal C}^n$ be an exceptional extremal copositive $n \times n$ matrix with positive diagonal. A zero $u$ of $A$ is a non-zero nonnegative vector such that $u^TAu = 0$. The support of a zero $u$ is the index set of the positive elements of $u$. A zero $u$ is minimal if there is no other zero $v$ such that $\Supp v \subset \Supp u$ strictly. Let $G$ be the graph on $n$ vertices which has an edge $(i,j)$ if and only if $A$ has a zero with support $\{1,\dots,n\} \setminus \{i,j\}$. In this paper, it is shown that $G$ cannot contain a cycle of length strictly smaller than $n$. As a consequence, if all minimal zeros of $A$ have support of cardinality $n - 2$, then $G$ must be the cycle graph $C_n$.

Extremal copositive matrices with minimal zero supports of cardinality two

Let be an extremal copositive matrix with unit diagonal. Then the minimal zeros of A all have supports of cardinality two if and only if the elements of A are all from the set . Thus the extremal copositive matrices with minimal zero supports of cardinality two are exactly those matrices which can be obtained by diagonal scaling from the extremal unit diagonal matrices characterized by [Hoffman, Pereira, 1973].

The best constants in the Wirtinger inequality

This study aimed to determine the best constants in the Wirtinger inequality [Formula: see text] where [Formula: see text] is defined on [Formula: see text] with [Formula: see text]. First, we referred the computation of [Formula: see text] to the maximal eigenvalue of an integral type operator [Formula: see text]. Second, we proved that the computation of the eigenvalues of [Formula: see text] is equivalent to the solution of a Strum–Liouville problem with some boundary conditions and hence we referred the computation of [Formula: see text] to finding the minimal zero of a function with one variable. Third, by comparing with a result of Lifshits, Papageorgiou, Woźniakowski, we obtained that the strong asymptotic order of [Formula: see text]: [Formula: see text].

Minimal zeros of copositive matrices

Let A be an element of the copositive cone Cn. A zero u of A is a nonzero nonnegative vector such that uTAu=0. The support of u is the index set suppu⊂{1,…,n} corresponding to the positive entries of u. A zero u of A is called minimal if there does not exist another zero v of A such that its support suppv is a strict subset of suppu. We investigate the properties of minimal zeros of copositive matrices and their supports. Special attention is devoted to copositive matrices which are irreducible with respect to the cone S+(n) of positive semi-definite matrices, i.e., matrices which cannot be written as a sum of a copositive and a nonzero positive semi-definite matrix. We give a necessary and sufficient condition for irreducibility of a matrix A with respect to S+(n) in terms of its minimal zeros. A similar condition is given for the irreducibility with respect to the cone Nn of entry-wise nonnegative matrices. For n=5 matrices which are irreducible with respect to both S+(5) and N5 are extremal. For n=6 a list of candidate combinations of supports of minimal zeros which an exceptional extremal matrix can have is provided.

Minimal Zero Norm Solutions of Linear Complementarity Problems

In this paper, we study minimal zero norm solutions of the linear complementarity problems, defined as the solutions with smallest cardinality. Minimal zero norm solutions are often desired in some real applications such as bimatrix game and portfolio selection. We first show the uniqueness of the minimal zero norm solution for Z-matrix linear complementarity problems. To find minimal zero norm solutions is equivalent to solve a difficult zero norm minimization problem with linear complementarity constraints. We then propose a p norm regularized minimization model with p in the open interval from zero to one, and show that it can approximate minimal zero norm solutions very well by sequentially decreasing the regularization parameter. We establish a threshold lower bound for any nonzero entry in its local minimizers, that can be used to identify zero entries precisely in computed solutions. We also consider the choice of regularization parameter to get desired sparsity. Based on the theoretical results, we design a sequential smoothing gradient method to solve the model. Numerical results demonstrate that the sequential smoothing gradient method can effectively solve the regularized model and get minimal zero norm solutions of linear complementarity problems.

Preliminary evaluation of a novel intraparenchymal capacitive intracranial pressure monitor

Intracranial pressure (ICP) monitors are currently based on fluid-filled, strain gauge, or fiberoptic technology. Capacitive sensors have minimal zero drift and energy requirements, allowing long-term implantation and telemetric interrogation; their application to neurosurgery has only occasionally been reported. The aim of this study was to undertake a preliminary in vitro and in vivo evaluation of a capacitive telemetric implantable ICP monitor. Four devices were tested in air- and saline-filled pressure chambers; long-term capacitance-pressure curves were obtained. Devices implanted in a gel phantom and in a piglet were placed in a 3-T MR unit to evaluate MR compatibility. Four devices were implanted in a piglet neonatal hydrocephalus model; output was compared with ICP obtained through fluid-filled transduction and a strain-gauge ICP monitor. The capacitance-pressure relationship was constant over 4 weeks, suggesting minimal zero drift during this period. There were no temperature changes around the monitor. Signal loss at the sensor was minimal in both the phantom and the piglet. Over 114,000 measurements were obtained; the difference between mean capacitive ICP and fluid-transduced ICP was 1.8 ± 1.42 mm Hg. The correlation between ICP from the capacitive sensor and fluid-filled transducer (r = 0.97, p < 0.0001) or strain-gauge monitor (r = 0.99, p < 0.0001) was excellent. In vivo monitoring was restricted to 48 hours due to problems with robustness in the clinical environment. This preliminary study demonstrates minimal long-term zero drift in vitro, good MR compatibility, and good correlation with other methods of ICP monitoring in vivo in the short term. Further long-term in vivo study is required.

Minimal zero sum sequences of length four over finite cyclic groups

Text Let G be a finite cyclic group. Every sequence S over G can be written in the form S = ( n 1 g ) ⋅ … ⋅ ( n l g ) where g ∈ G and n 1 , … , n l ∈ [ 1 , ord ( g ) ] , and the index ind ( S ) of S is defined to be the minimum of ( n 1 + ⋯ + n l ) / ord ( g ) over all possible g ∈ G such that 〈 g 〉 = 〈 supp ( S ) 〉 . The problem regarding the index of sequences has been studied in a series of papers, and a main focus is to determine sequences of index 1. In the present paper, we show that if G is a cyclic of prime power order such that gcd ( | G | , 6 ) = 1 , then every minimal zero-sum sequence of length 4 has index 1. Video For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=BC7josX_xVs.