Minimal Dimension Research Articles

Minimal Nonsolvable Bieberbach Groups

Several authors have shown that there exist nonsolvable Bieberbach groups of dimension 15. In this paper, we show that this is, in fact, a minimal dimension for such groups.

Interval observers for hybrid continuous-time stationary systems

The problem of interval observer design for hybrid continuous-time stationary systems under external disturbances and measurement noise is studied. It is assumed that continuous-time dynamic of such systems is described by linear or nonlinear differential state equations with linear function of output. System parameters depend on system states and are switched based on the control system with finite set of states. The relations allowing designing hybrid interval observer of minimal dimension estimating the set of admissible values of the prescribed linear vector function of the system states are derived. To solve the problem, algebra of partitions and linear algebra are used. Theoretical results are illustrated by example.

Fault detection in linear systems via interval parity relations

The problem of fault detection via interval parity relations designed for systems described by linear models under the external disturbances and measurement noise is considered. To solve the problem, the reduced-order model of minimal dimension insensitive or having minimal sensitivity to the external disturbances is used. Based on this model, the interval parity relations are designed. The relations that allow to construct two residuals are obtained and theorem showing that zero is between these residuals for healthy system is proved. Theoretical results are illustrated by example.

Fully Automated Localization and Measurement of Levator Hiatus Dimensions Using 3-D Pelvic Floor Ultrasound

ObjectiveTo develop an algorithm for the automated localization and measurement of levator hiatus (LH) dimensions (AI-LH) using 3-D pelvic floor ultrasound. MethodsThe AI-LH included a 3-D plane regression model and a 2-D segmentation model, which first achieved automated localization of the minimal LH dimension plane (C-plane) and measurement of the hiatal area (HA) on maximum Valsalva on the rendered LH images, but not on the C-plane. The dataset included 600 volumetric data. We compared AI-LH with sonographer difference (ASD) as well as the inter-sonographer differences (IESD) in the testing dataset (n = 240). The assessment encompassed the mean absolute error (MAE) for the angle and center point distance of the C-plane, along with the Dice coefficient, MAE, and intra-class correlation coefficient (ICC) for HA, and included the time consumption. ResultsThe MAE of the C-plane of ASD was 4.81 ± 2.47° with 1.92 ± 1.54 mm. AI-LH achieved a mean Dice coefficient of 0.93 for LH segmentation. The MAE on HA of ASD (1.44 ± 1.12 mm²) was lower than that of IESD (1.63 ± 1.58 mm²). The ICC on HA of ASD (0.964) was higher than that of IESD (0.949). The average time costs of AI-LH and manual measurement were 2.00 ± 0.22 s and 59.60 ± 2.63 s (t = 18.87, p < 0.01), respectively. ConclusionAI-LH is accurate, reliable, and robust in the localization and measurement of LH dimensions, which can shorten the time cost, simplify the operation process, and have good value in clinical applications.

Surfkit: An atomic toolkit for surface modelling with molecular adsorption

The surface model with molecular adsorption is critical to study various surface related functions by means of first principles approaches such as catalytic activity, energy storage, corrosion, charge transfer, ion adsorption, etc. However, as the atomic configurations in surface modelling become progressively complex, the public programs exhibit insufficient automation or even deficiency in the construction of specific surface-related models, e.g. the high-index surface slab, the bilayer/heterostructures and the adsorption configuration with specified symmetry. All these issues bring necessity to develop a user-friendly program that is dedicated to automatic surface modeling with the guidance of symmetry. In this work, we present an open-source program, termed as Surfkit, that is specifically developed for automatically constructing the complicated surface-related models for first-principles calculations. This program features three major functionalities, including the construction of complex surface model with minimal lattice dimension, the twisted bilayers or heterostructures with minimized misfit strain, and the molecular adsorption models with specified symmetry. Several options can be applied consecutively, allowing for a comprehensive workflow from a unit cell to complex surface model. We provide not only a throughout overview on various numerical methods and their implementation into Surfkit program, but some critical evaluations, specific tests and applications, demonstrating the capability of the program on each functionality.

Geometry of certain foliations on the complex projective plane

Let $d\geq2$ be an integer. The set $\mathbf{F}(d)$ of foliations of degree $d$ on the complex projective plane can be identified with a Zariski's open set of a projective space of dimension $d^2+4d+2$ on which $\mathrm{Aut}(\mathbb P^2_{\mathbb C})$ acts. We show that there are exactly two orbits $\mathcal{O}(\mathcal{F}_{1}^{d})$ and $\mathcal{O}(\mathcal{F}_{2}^{d})$ of minimal dimension $6$, necessarily closed in $\mathbf{F}(d)$. This generalizes known results in degrees $2$ and $3.$ We deduce that an orbit $\mathcal{O}(\mathcal{F})$ of an element $\mathcal{F}\in\mathbf{F}(d)$ of dimension $7$ is closed in $\mathbf{F}(d)$ if and only if $\mathcal{F}_{i}^{d}\not\in\overline{\mathcal{O}(\mathcal{F})}$ for $i=1,2.$ This allows us to show that in any degree $d\geq3$ there are closed orbits in $\mathbf F(d)$ other than the orbits $\mathcal{O}(\mathcal{F}_{1}^{d})$ and $\mathcal{O}(\mathcal{F}_{2}^{d}),$ unlike the situation in degree $2.$ On the other hand, we introduce the notion of the basin of attraction $\mathbf{B}(\mathcal{F})$ of a foliation $\mathcal{F}\in\mathbf{F}(d)$ as the set of $\mathcal{G}\in\mathbf{F}(d)$ such that $\mathcal{F}\in\overline{\mathcal{O}(\mathcal{G})}.$ We show that the basin of attraction $\mathbf{B}(\mathcal{F}_{1}^{d})$, resp. $\mathbf{B}(\mathcal{F}_{2}^{d})$, contains a quasi-projective subvariety of $\mathbf{F}(d)$ of dimension greater than or equal to $\dim\mathbf{F}(d)-(d-1)$, resp. $\dim \mathbf{F}(d)-(d-3)$. In particular, we obtain that the basin $\mathbf{B}(\mathcal{F}_{2}^{3})$ contains a non-empty Zariski open subset of $\mathbf{F}(3)$. This is an analog in degree $3$ of a result on foliations of degree $2$ due to Cerveau, D\'eserti, Garba Belko and Meziani.

Interval estimation in discrete-time linear systems with parametric uncertainties

The problem of interval observer design for discrete-time linear systems under the external disturbances, measurement noise, and parametric uncertainties is studied. The relation allowing designing the interval observer of minimal dimension estimating the set of admissible values of the specified linear vector function of the system state are derived. Theoretical results are illustrated by the example.

On linear codes with random multiplier vectors and the maximum trace dimension property

Abstract Let C C be a linear code of length n n and dimension k k over the finite field F q m {{\mathbb{F}}}_{{q}^{m}} . The trace code Tr ( C ) {\rm{Tr}}\left(C) is a linear code of the same length n n over the subfield F q {{\mathbb{F}}}_{q} . The obvious upper bound for the dimension of the trace code over F q {{\mathbb{F}}}_{q} is m k mk . If equality holds, then we say that C C has maximum trace dimension. The problem of finding the true dimension of trace codes and their duals is relevant for the size of the public key of various code-based cryptographic protocols. Let C a {C}_{{\boldsymbol{a}}} denote the code obtained from C C and a multiplier vector a ∈ ( F q m ) n {\boldsymbol{a}}\in {\left({{\mathbb{F}}}_{{q}^{m}})}^{n} . In this study, we give a lower bound for the probability that a random multiplier vector produces a code C a {C}_{{\boldsymbol{a}}} of maximum trace dimension. We give an interpretation of the bound for the class of algebraic geometry codes in terms of the degree of the defining divisor. The bound explains the experimental fact that random alternant codes have minimal dimension. Our bound holds whenever n ≥ m ( k + h ) n\ge m\left(k+h) , where h ≥ 0 h\ge 0 is the Singleton defect of C C . For the extremal case n = m ( h + k ) n=m\left(h+k) , numerical experiments reveal a closed connection between the probability of having maximum trace dimension and the probability that a random matrix has full rank.

Classifying spaces for families of abelian subgroups of braid groups, RAAGs and graphs of abelian groups

Abstract Given a group $G$ and an integer $n\geq 0$ , we consider the family ${\mathcal F}_n$ of all virtually abelian subgroups of $G$ of $\textrm{rank}$ at most $n$ . In this article, we prove that for each $n\ge 2$ the Bredon cohomology, with respect to the family ${\mathcal F}_n$ , of a free abelian group with $\textrm{rank}$ $k \gt n$ is nontrivial in dimension $k+n$ ; this answers a question of Corob Cook et al. (Homology Homotopy Appl. 19(2) (2017), 83–87, Question 2.7). As an application, we compute the minimal dimension of a classifying space for the family ${\mathcal F}_n$ for braid groups, right-angled Artin groups, and graphs of groups whose vertex groups are infinite finitely generated virtually abelian groups, for all $n\ge 2$ . The main tools that we use are the Mayer–Vietoris sequence for Bredon cohomology, Bass–Serre theory, and the Lück–Weiermann construction.

Interval Observer Design for Fault Diagnosis

The problem of interval observer design for systems described by linear models under the external disturbances is considered. The problem is solved based on the reduced-order model of the original system of minimal dimension. The relations allowing designing interval observer insensitive or having minimal sensitivity to the disturbance are obtained. The designed interval observers are used to solve the problem of fault diagnosis. Theoretical results are illustrated by example.

Failure of the well-rounded retract for outer space and Teichmüller space

The well-rounded retract for S L n ( Z ) SL_n(\mathbb {Z}) is defined as the set of flat tori of unit volume and dimension n n whose systoles generate a finite-index subgroup in homology. This set forms an equivariant spine of minimal dimension for the space of flat tori. For both the Outer space X n X_n of metric graphs of rank n n and the Teichmüller space T g \mathcal {T}_g of closed hyperbolic surfaces of genus g g , we show that the literal analogue of the well-rounded retract does not contain an equivariant spine. We also prove that the sets of graphs whose systoles fill either topologically or geometrically (two analogues of a set proposed as a spine for T g \mathcal {T}_g by Thurston) are spines for X n X_n but that their dimension is larger than the virtual cohomological dimension of O u t ( F n ) Out(F_n) in general.

On minimal epimorphic subgroups in simple algebraic groups of rank 2

The category of linear algebraic groups admits non-surjective epimorphisms. For simple algebraic groups of rank 2 defined over algebraically closed fields, we show that the minimal dimension of a closed epimorphic subgroup is 3.

Optimization of System of Early Detection of Oncological Diseases in Outpatient Medical Organizations

INTRODUCTION: Increased attention to the early detection and prevention of malignant neoplasms (MNP) is conditioned by their high medical and social significance.&#x0D; AIM: To develop a methodology of early defection and stratification of risk for development of MNP at the outpatient stage using modern information technologies.&#x0D; MATERIALS AND METHODS: The data of the official statistics of Rosstat and Health Ministry of Russia for the Voronezh region (VR) were used. To collect the primary information about the level of oncological alertness (OA) of primary care workers, a survey was conducted using a specially designed questionnaire (Oncological Alertness on an Outpatient Visit) which contained 10 questions reflecting the frequency of MNP and precancer detection, doctors’ knowledge, methods of early diagnosis and patient routing in case this pathology is identified. In the survey, 112 medical workers participated. To identify MNP at early stages, methods of evaluation and stratification of the risk of MNP development were elaborated on the basis of a multi-stage analysis of significance of the identified risk factors (‘danger signals’) with the use of artificial intelligence. The method was tried on a test sample (100 patients, MNP in 55).&#x0D; RESULTS: A ‘rough’ increase in the incidence rate for 2013–2022 was 11.4%. The mortality rate from MNPs in 2022 was 170.5 per 100 thousand cases, which is 0.2 higher than in the previous year. About 60% of the newly identified MNPs were diagnosed at III–IV stage. A comprehensive study of the causative factors of advanced cases permitted to identify the most significant ones: late seeking medical care, latent asymptomatic course of MNP and insufficient OA level of primary care physicians. When testing the developed technique, the probability for the development of the disease was estimated as high in 41 (82%) patients with MNP, medium in 7 (14%), and in 2 (4%) patients the prognosis was erroneous — a low probability was predicted. Of the 50 patients who did not have MNP at the time of examination, 23 (46%) were referred to the group of low, 21 (42%) — of medium, 6 (12%) — of high risk of having a MNP. After the introduction of the developed technique, the detectability of gastric cancer increased by 3%, of colon cancer — by 2%, of tracheal, bronchial and lung cancer — by 6%, of breast, cervix and prostate cancer — by 1%, 8%, and 2%, respectively.&#x0D; CONCLUSION: The developed method permits to identify and exclude unreliable data, to select the optimal feature space characterized by the minimal dimension with sufficient informational value. This permits identification of precancerous conditions at the preclinical stage and facilitates timely detection of MNP at early stages.

Minimal Realizations of Nonlinear Time-Delay Systems

The problem of finding a minimal realization of a single-input single-output nonlinear retarded time-delay input-output equation with constant commensurable delays is addressed. First, an algorithm is given to find an observable realization of dimension <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> of a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> -th order input-output equation. Then, this algorithm is modified under two technical assumptions to find a realization of lower dimension, which is observable and accessible. It is shown that under the same technical assumptions an observable and accessible realization has minimal possible state-space dimension among all possible realizations. An algebraic approach based on polynomial tools and modules of differential 1-forms is used to derive the results.

Desirability function analysis approach for optimization of fused deposition modelling process parameters

Abstract This study uses desirability function analysis to optimize the process parameters namely layer thickness, fill pattern, fill density, and build orientation for fused deposition modelling on polyethylene terephthalate glycol material. The Taguchi-desirability function analysis is one of the optimization techniques for the multi-objective decision-making process. Taguchi L9 orthogonal array is employed in the test trials during the fused deposition modelling process. The output responses measured are dimension error, ductility, and tensile strength. The goal of process parameter optimization is to attain the best of the trials for minimal dimension error and maximum tensile strength and ductility. The best feasible amalgamation of input variables is discovered for the investigated and assessed response features utilizing desirability function analysis. As per the results obtained from experiments, it is concluded that the optimal combination of parameters is layer thickness of 0.1 mm, build orientation of 0°, fill density of 90 % and hexa fill pattern. By following this combination, the error in the dimensions was slightly increased which needed to be reduced, but both tensile strength and ductility were increased which are more desirable characteristics.

A post-quantum associative memory

Associative memories are devices storing information that can be fully retrieved given partial disclosure of it. We examine a toy model of associative memory and the ultimate limitations to which it is subjected within the framework of general probabilistic theories (GPTs), which represent the most general class of physical theories satisfying some basic operational axioms. We ask ourselves how large the dimension of a GPT should be so that it can accommodate 2 m states with the property that any N of them are perfectly distinguishable. Call the minimal such dimension. Invoking an old result by Danzer and Grünbaum, we prove that , to be compared with when the GPT is required to be either classical or quantum. This yields an example of a task where GPTs outperform both classical and quantum theory exponentially. More generally, we resolve the case of fixed N and asymptotically large m, proving that (as ) for every , which yields again an exponential improvement over classical and quantum theories. Finally, we develop a numerical approach to the general problem of finding the largest N-wise mutually distinguishable set for a given GPT, which can be seen as an instance of the maximum clique problem on N-regular hypergraphs.

A Rosenbrock framework for tangential interpolation of port-Hamiltonian descriptor systems

ABSTRACT We present a new structure-preserving model order reduction (MOR) framework for large-scale port-Hamiltonian descriptor systems (pH-DAEs). Our method exploits the structural properties of the Rosenbrock system matrix for this system class and utilizes condensed forms which often arise in applications and reveal the solution behaviour of a system. Provided that the original system has such a form, our method produces reduced-order models (ROMs) of minimal dimension, which tangentially interpolate the original model’s transfer function and are guaranteed to be again in pH-DAE form. This allows the ROM to be safely coupled with other dynamical systems when modelling large system networks, which is useful, for instance, in electric circuit simulation.

Tests of the Charge Convexity Conjecture in Caswell-Banks-Zaks theory

The Charge Convexity Conjecture (CCC) states that in a unitary conformal field theory in d ≥ 3 dimensions with a global symmetry, the minimal dimension of operators in certain representations of the symmetry, as a function of the charge q of the representation (or a generalized notion of it), should be convex. More precisely, this was conjectured to be true when q is restricted to positive integer multiples of some integer q0. The CCC was tested on a number of examples, most of which are in d < 4 dimensions, and its version in which q0 is taken to be the charge of the lowest-dimension positively-charged operator was shown to hold in all of them.In this paper we test the conjecture in a non-trivial example of a d = 4 theory, which is the family of Caswell-Banks-Zaks IR fixed points of SU(Nc) gauge theory coupled to Nf massless fermions and Ns massless scalars. In these theories, the lowest-dimension gauge-invariant operators that transform non-trivially under the global symmetry are mesons. These may consist of two scalars, two fermions or one of each.We find that the CCC holds in all applicable cases, providing significant new evidence for its validity, and suggesting a stronger version for non-simple global symmetry groups.

Robust virtual sensors design for linear and nonlinear systems under external disturbances

SummaryThe problem of robust virtual sensors design in linear and nonlinear dynamic systems is studied. To design such sensors, two different canonical forms are used: identification and Jordan canonical forms. The problems generally are solved in three steps: at the first step, the linear model invariant with respect to the disturbance is designed; at the second step, a possibility to take into account the nonlinear term and to estimate the given variable is checked; finally, stability of the observer is obtained. The relations allowing to design virtual sensor of minimal dimension estimating the prescribed component of the state vector of the system are obtained. The problem of virtual sensor location is studied and solved. The theoretical results are illustrated by the example of three tank system.

Miniaturization of 2 × 4 90-Degree Hybrid Optical Couplers

In this work we study the limits of miniaturization of a 90-degree hybrid coupler working in the L, C and S bands, with respect to a number of performance parameters aimed at its application for balanced detection. We investigate the main effects responsible for the degradation of the performance of the devices during miniaturization, and establish the minimal dimension that such devices can have without significant degradation for photonic applications such as balanced detection. The miniaturized device in InP generic technology has a footprint of only 2200μm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , more than 5 times smaller than the conventional device used as reference. The scaling approach is based on the use of the number of propagating modes which are sustained in both the miniaturized MMI and port waveguides as scaling parameters. This approach allows us to generalize the miniaturization problem from a specific platform and offers a methodology which is flexible and transferable to multiple platforms. We tested the scaling methodology based on the number of modes in other platforms commonly used in integrated photonics, such as Si/SiO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (SOI), TriPleX or polymer platforms, obtaining comparable results and proving the universality of our approach, finally we performed a fabrication tolerance analysis of the miniaturized devices.