Left Zero Research Articles

A Hessian-Based Deep Learning Preprocessing Method for Coronary Angiography Image Analysis

Leveraging its high accuracy and stability, deep-learning-based coronary artery detection technology has been extensively utilized in diagnosing coronary artery diseases. However, traditional algorithms for localizing coronary stenosis often fall short when detecting stenosis in branch vessels, which can pose significant health risks due to factors like imaging angles and uneven contrast agent distribution. To tackle these challenges, we propose a preprocessing method that integrates Hessian-based vascular enhancement and image fusion as prerequisites for deep learning. This approach enhances fuzzy features in coronary angiography images, thereby increasing the neural network’s sensitivity to stenosis characteristics. We assessed the effectiveness of this method using the latest deep learning networks, such as YOLOv10, YOLOv9, and RT-DETR, across various evaluation metrics. Our results show that our method improves AP50 accuracy by 4.84% and 5.07% on RT-DETR R101 and YOLOv10-X, respectively, compared to images without special pre-processing. Furthermore, our analysis of different imaging angles on stenosis localization detection indicates that the left coronary artery zero is the most suitable for detecting stenosis with a AP50(%) value of 90.5. The experimental results have revealed that the proposed method is effective as a preprocessing technique for deep-learning-based coronary angiography image processing and enhances the model’s ability to identify stenosis in small blood vessels.

On Topology Lattices of Acts Over Right or Left Zero Semigroups

On Topology Lattices of Acts Over Right or Left Zero Semigroups

The Vertex Distance on Cayley Digraphs of Rectangular Groups with respect to the Cartesian Product Connection Sets

A rectangular group S is a semigroup isomorphic to the direct product of a group, a left zero semigroup, and a right zero semigroup. The Cayley digraph Cay(S,A) of a finite rectangular group S with respect to a nonempty subset A of S is defined as a digraph with vertex set S and arc set consisting of ordered pairs (u,ua) in SxS for some a in A. The set A is called a connection set of Cay(S,A). Let u and v be in S. The distance from u to v in Cay(S,A), denoted by d(u,v), is the number of arcs in the shortest directed u-v path if one exists and is infinity otherwise. In this paper, we characterize the distances of any two vertices in Cayley digraphs of rectangular groups with respect to the Cartesian product connection sets

Hausdorff series in semigroup rings of rectangular bands

"The Hausdorff series provides a solution to the equation $w=\log(e^ue^v)$ given by a recursive formula which can be expressed as nested commutators of $u$ and $v$. Evolutions of the Haussdorff series in various algebras and rings has been considered in obtaining a closed form of this formula. We consider the rectangular band $L_m\times R_n$ determined by the left zero semigroup $L_m$ and the right zero semigroup $R_n$ of order $m$ and $n$, respectively. Let $\mathbb R\langle L_m\times R_n\rangle$ be the semigroup ring spanned on $L_m\times R_n$ together with the identity element $1$. We provide a closed form of the formula for solving the equation in $\mathbb R\langle L_m\times R_n\rangle$."

Upper Bounds on Mixing Time of Finite Markov Chains

We provide a general framework for computing upper bounds on mixing times of finite Markov chains when its minimal ideal is left zero. Our analysis is based on combining results by Brown and Diaconis with our previous work on stationary distributions of finite Markov chains. Stationary distributions can be computed from the Karnofsky--Rhodes and McCammond expansion of the right Cayley graph of the finite semigroup underlying the Markov chain. Using loop graphs, which are planar graphs consisting of a straight line with attached loops, there are rational expressions for the stationary distribution in the probabilities. From these we obtain bounds on the mixing time. In addition, we provide a new Markov chain on linear extension of a poset with $n$ vertices, inspired by but different from the promotion Markov chain of Ayyer, Klee and the last author. The mixing time of this Markov chain is $O(n \log n)$.

Improved Hoeffding’s Lemma and Hoeffding’s Tail Bounds

The purpose of this article is to improve Hoeffding's lemma and consequently Hoeffding's tail bounds. The improvement pertains to left skewed zero mean random variables X\in[a,b], where a<0 and -a>b. The proof of Hoeffding's improved lemma uses Taylor's expansion, the convexity of \exp(sx), s\in \RR, and an unnoticed observation since Hoeffding's publication in 1963 that for -a>b the maximum of  the intermediate function \tau(1-\tau) appearing in Hoeffding's proof is attained at an endpoint rather than at \tau=0.5 as in the case b>-a. Using Hoeffding's  improved lemma we obtain one sided and two sided  tail bounds  for \PP(S_n\ge t) and \PP(|S_n|\ge t), respectively, where S_n=\sum_{i=1}^nX_i and the X_i\in[a_i,b_i],i=1,...,n are independent zero mean  random variables (not necessarily identically distributed). It is interesting to note that we  could  also improve Hoeffding's two sided bound for all \{X_i:  -a_i\ne b_i,i=1,...,n\}. This is  so because here the one sided bound should be  increased by \PP(-S_n\ge t),  wherein the left skewed intervals become right skewed and vice versa.

Corrections and Extensions in Left and Right Almost Semigroups

In this paper we elaborated the concept that on what conditions left almost semigroup (LA-Semigroup), right almost semigroup (RA-Semigroup) and a groupoid become commutative and further extended these results on medial, LA-Group and RA-Group. We proved that the relation of LA-Semigroup with left double displacement semigroup (LDD-semigroup), RA-Semigroup with left double displacement semigroup (RDD-semigroup) is only commutative property. We highlighted the errors in the recently developed results on LA-Semigroup and semigroup [17, 1, 18] and proved that example discussed in [18] is semigroup with left identity but not paramedial. We extended results on locally associative LA-Semigroup explained in [20, 21] towards LA-Semigroup and RA-Semigroup with left zero and right zero respectively. We also discussed results on n-dimensional LA-Semigroup, n-dimensional RASemigroup, non commutative finite medials with three or more than three left or right identities and finite as well as infinite commutative idempotent medials not studied in literature.

A study of right zero divisor graph of 3 × 3 matrices over ℤ2

In [10] we have computed left zero divisor graph of the ring R, where R is the set of all 3 × 3 matrices over the ring ℤ2 of integers modulo 2 is constructed. We have extended our result for right zero divisor graph of the ring R. Thus we have listed vertices of RZDG by means of lexicographic order. Which enables us to present algorithm that computes the edges of this graph. In addition, it helps to analyze some of its important properties.

INCREASING THE ROUGHNESS OF THE PROCEDURE FOR EVALUATING A VECTOR OF THE CONDITION OF OBJECTS TO THE INFLUENCE OF UNCERTAINTY FACTORS

Algorithms for increasing the roughness of the procedure for assessing the state vector of control objects to the influence of uncertainty factors are given. Expressions are obtained for extended state vectors and observations. Stable inversion algorithms are given for a nondegenerate block matrix with the allocation of its left and right zero divisors of maximum rank. The presented stable computational procedures allow us to regularize the problem of synthesis of algorithms for estimating the parameters of regulators in adaptive control systems with a customizable model and to improve the quality indicators of control processes under conditions of parametric uncertainty

Design of Given Oscillation Index Scalar Controllers: Modal and H∞-Approaches

The algorithms of output controllers design are proposed for linear scalar plants, that ensure the desired or attainable values of oscillation index and of degree of stability, determining the settling time. Both modal control and H∞-approach are used in the design procedurevs. Examples are constructed, demonstrating that striving to provide the degree of stability that is much greater than the distance from the nearest left zero of the plant transfer function to the imaginary axis (even for the minimum phase plants) leads to the quite small gain and phase stability margins, that is unacceptable in practice.

Hausdorff series in a semigroup ring

Let [Formula: see text] and [Formula: see text] be the semigroup rings spanned on the right zero semigroup [Formula: see text], and on the left zero semigroup [Formula: see text], respectively, together with the identity element [Formula: see text]. We suggest a closed formula solving the equation [Formula: see text] which is the evolution of the Campbell–Baker–Hausdorff formula given by the Hausdorff series [Formula: see text] where [Formula: see text], in the algebras [Formula: see text] and [Formula: see text].

The complexity of properties of transformation semigroups

We investigate the computational complexity for determining various properties of a finite transformation semigroup given by generators. We introduce a simple framework to describe transformation semigroup properties that are decidable in [Formula: see text]. This framework is then used to show that the problems of deciding whether a transformation semigroup is a group, commutative or a semilattice are in [Formula: see text]. Deciding whether a semigroup has a left (respectively, right) zero is shown to be [Formula: see text]-complete, as are the problems of testing whether a transformation semigroup is nilpotent, [Formula: see text]-trivial or has central idempotents. We also give [Formula: see text] algorithms for testing whether a transformation semigroup is idempotent, orthodox, completely regular, Clifford or has commuting idempotents. Some of these algorithms are direct consequences of the more general result that arbitrary fixed semigroup equations can be tested in [Formula: see text]. Moreover, we show how to compute left and right identities of a transformation semigroup in polynomial time. Finally, we show that checking whether an element is regular is [Formula: see text]-complete.

Dispersive shock wave, generalized Laguerre polynomials, and asymptotic solitons of the focusing nonlinear Schrödinger equation

We consider dispersive shock waves of the focusing nonlinear Schrödinger equation generated by discontinuous initial conditions which are periodic or quasiperiodic on the left semiaxis and zero on the right semiaxis. As an initial function, we use a finite-gap potential of the Dirac operator given in an explicit form through hyperelliptic theta-functions. The aim of this paper is to study the long-time asymptotics of the solution of this problem in a vicinity of the leading edge, where a train of asymptotic solitons are generated. Such a problem was studied in the work of Kotlyarov and Khruslov [Teor. Mat. Fiz. 68(2), 751–761 (1986)] and Kotlyarov {Mat. Zametki 49(2), 84–94 (1991) [Math. Notes 49(1-2), 172–180 (1991)]} using Marchenko’s inverse scattering techniques. We investigate this problem exceptionally using the Riemann-Hilbert (RH) problem techniques that allow us to obtain explicit formulas for asymptotic solitons themselves in contrast with the cited papers where asymptotic formulas are obtained only for the square of the absolute value of solution. Using transformations of the main RH problems, we arrive at a model problem corresponding to the parametrix at the end points of the continuous spectrum of the Zakharov-Shabat spectral problem. The parametrix problem is effectively solved in terms of the generalized Laguerre polynomials, which naturally appeared after appropriate scaling of the Riemann-Hilbert problem in small neighborhoods of the end points of the continuous spectrum. Further asymptotic analysis gives an explicit formula for solitons at the edge of dispersive waves. Thus, we give the complete description of the train of asymptotic solitons: not only bearing the envelope of each asymptotic soliton, but its oscillating structure is found explicitly. Besides, the second term of asymptotics describing an interaction between these solitons and oscillating background is also found. This gives the fine structure of the edge of dispersive shock waves.

Automorphism groups of semigroups of upfamilies

A family [Formula: see text] of nonempty subsets of a set [Formula: see text] is called an upfamily if for each set [Formula: see text] any set [Formula: see text] belongs to [Formula: see text]. The extension [Formula: see text] of [Formula: see text] consists of all upfamilies on [Formula: see text]. Any associative binary operation [Formula: see text] can be extended to an associative binary operation [Formula: see text]. In the paper, we study automorphisms of extensions of groups, finite monogenic semigroups, null semigroups, right zero semigroups and left zero semigroups. Also, we describe the automorphism groups of extensions of some semigroups of small cardinalities.

Automorphisms of semigroups of k-linked upfamilies

A family $$ \mathcal{A} $$ of non-empty subsets of a set X is called an upfamily, if, for each set $$ A\in \mathcal{A} $$ ; any set B ⊃ A belongs to $$ \mathcal{A} $$ . An upfamily $$ \mathrm{\mathcal{L}} $$ is called k-linked, if $$ \cap \mathrm{\mathcal{F}}\ne \varnothing $$ for any subfamily $$ \mathrm{\mathcal{F}}\subset \mathrm{\mathcal{L}} $$ of cardinality $$ \left|\mathrm{\mathcal{F}}\right|\le k $$ . The extension Nk(X) consists of all k-linked upfamilies on X. Any associative binary operation ∗ : X × X → X can be extended to an associative binary operation ∗ : Nk(X) × Nk(X) → Nk(X). Here, we study automorphisms of the extensions of groups and finite monogenic semigroups. We also describe the automorphism groups of extensions of null semigroups, almost null semigroups, right zero semigroups and left zero semigroups.

Microstrip Bandpass Filter with Left Transmission Zero Controlled by Parasitic Cross Coupling

A third-order microstrip filter is proposed and studied. It is characterized by a left-handed transmission zero fz, caused by the parasitic cross-coupling between non-adjacent resonators. The filter contains a half-wave middle resonator and two quarter-wave resonators located from different sides, near the open ends of the middle resonator. The coupling between all resonators has magnetic character, and the zero of the filter transfer function fz is located to the left of the center frequency of the passband f0. Such filter is described by a modified coupling matrix, where one of the main coupling coefficients is artificially assigned a minus sign. In the proposed filter design, for a given value of the main coupling coefficients, it is possible to provide different values of the cross-coupling coefficient by appropriately selecting the design parameters. This allows adjusting the zero position of the transmission fz for a given bandwidth of the filter, thereby changing the left slope of the amplitude-frequency characteristic. A sequence of steps is proposed for constructing such a filter. The measured and simulated frequency characteristics of the experimental filter are given.

Adaptive Gaussian mixture filter for Markovian jump nonlinear systems with colored measurement noises

This paper considers the state estimation of discrete-time Markovian jump nonlinear systems with colored measurement noises obeying a nonlinear autoregressive process of order n, which is motivated by tracking the maneuvering target under electronic countermeasures with high speed sampling or persistent perturbations. In order to remove the measurement noises correlation, the left zero divisor is explored to reconstruct a new measurement equation via difference approach, with the help of applying statistical linear regression to the colored measurement noise model. Then, a novel hypothesis set constituted of all possible values of multi-step Markov jumping parameters is defined and the posterior probability density of the state is derived recursively. By using Gaussian mixtures to approximate the posterior probability densities, an adaptive Gaussian mixture filter for the considered system is proposed, where the Gaussian components with small weights are pruned adaptively through measuring the Alpha (or Beta) divergence for the original and approximated Gaussian mixtures, to achieve a tradeoff between the estimation accuracy and running time. A maneuvering target tracking accompanied by range gate pull-off with different colored measurement noises cases is simulated to validate the proposed method.

On the Independence Number of Cayley Digraphs of Rectangular Groups

A rectangular group is isomorphic to the direct product of a group, a left zero semigroup, and a right zero semigroup. Some special cases of rectangular groups consisting of left groups and right groups are also considered here. Let $$ \mathrm {Cay}(S,A) $$ denote the Cayley digraph of the rectangular group S with the connection set A. In this paper, we are interested in studying some properties of $$ \mathrm {Cay}(S,A) $$ such as the independence, weakly independence, path independence, and weakly path independence. Furthermore, the independence numbers for those properties of $$ \mathrm {Cay}(S,A) $$ are also determined.

Independent domination number in Cayley digraphs of rectangular groups

Let [Formula: see text] denote the Cayley digraph of the rectangular group [Formula: see text] with respect to the connection set [Formula: see text] in which the rectangular group [Formula: see text] is isomorphic to the direct product of a group, a left zero semigroup, and a right zero semigroup. An independent dominating set of [Formula: see text] is the independent set of elements in [Formula: see text] that can dominate the whole elements. In this paper, we investigate the independent domination number of [Formula: see text] and give more results on Cayley digraphs of left groups and right groups which are specific cases of rectangular groups. Moreover, some results of the path independent domination number of [Formula: see text] are also shown.

SUBTOTAL LEFT MAIN DISEASE IN PATIENT WITH ZERO CALCIUM SCORE: IS IT POSSIBLE?

SUBTOTAL LEFT MAIN DISEASE IN PATIENT WITH ZERO CALCIUM SCORE: IS IT POSSIBLE?