On $m$-th roots of nilpotent matrices

A new necessary and sufficient condition for the existence of an $m$-th root of a nilpotent matrix in terms of the multiplicities of Jordan blocks is obtained and expressed as a system of linear equations with nonnegative integer entries which is suitable for computer programming. Thus, computation of the Jordan form of the $m$-th power of a nilpotent matrix is reduced to a single matrix multiplication; conversely, the existence of an $m$-th root of a nilpotent matrix is reduced to the existence of a nonnegative integer solution to the corresponding system of linear equations. Further, an erroneous result in the literature on the total number of Jordan blocks of a nilpotent matrix having an $m$-th root is corrected and generalized. Moreover, for a singular matrix having an $m$-th root with a pair of nilpotent Jordan blocks of sizes $s$ and $l$, a new $m$-th root is constructed by replacing that pair by another one of sizes $s+i$ and $l-i$, for special $s,l,i$. This method applies to solutions of a system of linear equations having a special matrix of coefficients. In addition, for a matrix $A$ over an arbitrary field that is a sum of two commuting matrices, several results for the existence of $m$-th roots of $A^k$ are obtained.

PRINCIPAL RADICAL SYSTEMS, LEFSCHETZ PROPERTIES, AND PERFECTION OF SPECHT IDEALS OF TWO-ROWED PARTITIONS

We show that the Specht ideal of a two-rowed partition is perfect over an arbitrary field, provided that the characteristic is either zero or bounded below by the size of the second row of the partition, and we show this lower bound is tight. We also establish perfection and other properties of certain variants of Specht ideals, and find a surprising connection to the weak Lefschetz property. Our results, in particular, give a self-contained proof of Cohen–Macaulayness of certain h-equals sets, a result previously obtained by Etingof–Gorsky–Losev over the complex numbers using rational Cherednik algebras.

ADS Using NER Systems

Data accessible over the net is generally unstructured. Offers distributed by different sources like banks, digital wallets, merchants, etc., are one of the foremost gotten to advertising data in today’s world. This information gets gotten to by millions of people on a every day premise and is effortlessly deciphered by people, but since it is generally unstructured and differing, utilizing an algorithmic way to extricate significant data out of these offers is hard. Distinguishing the basic offer substances (for occasion, its amount, the item on which the offer is pertinent, the merchant giving the offer, etc.) from these offers plays a vital role in focusing on the proper clients to make strides deals.This work presents and assesses different existing Named Substance Recognizer (NER) models which can distinguish the desired substances from offer feeds. We moreover propose a novel NER demonstration constructed by two-level stacking of Conditional Arbitrary Field, Bidirectional LSTM and Spacy models at the primary level and an SVM classifier at the moment. The proposed cross breed demonstrate has been tried on offer feeds collected from different sources and has appeared better performance within the offered space when compared to the existing models. Index Terms—Named Substance Acknowledgment, Information Mining, Machine Learning, Stanford NER, Bidirectional LSTM, Spacy, Bolster Vector Machines.

Математична модель антени з нелінійними характеристиками для розрахунку елект-ромагнітного поля у зоні Френеля

The paper presents a mathematical model of radio-electronic systems (RES), which include antennas and their excitation paths with nonlinear characteristics. The model provides acceptable accuracy of RES quality indicator analysis and electromagnetic compatibility (EMC) for further practical design. General purpose: the development of a mathematical model of a transmitting multi-input radiating structure with nonlinear characteristics under the Fresnel zone. Objective: choice justification of a structural schema of a radiating multi-input system with a radiator that has a distributed nonlinear surface impedance; obtaining the nonlinear integral equations (NIE) related to the current density for radiators with distributed nonlinearity, excited by an arbitrary field distribution for solving the general analysis problem; obtaining a ratio for calculating focused electromagnetic fields (EMF) created by multi-input radiating structures with nonlinear characteristics in the Fresnel zone. The methods used in the paper are mathematical methods of electrodynamics and antennas theory with nonlinear elements (ANE), theory of microwave circuits, and multipoles. The following results were obtained. An electrodynamics approach is proposed to analyze the entire set of nonlinear effects arising in transmitting multi-input radiating structures with nonlinear characteristics. It allows considering the mutual influence of the transmitting and receiving antennas with nonlinear characteristics in the system itself and the electrodynamics interaction of the transmitting antenna with nonlinear characteristics with RES for other purposes. Component equations (NIE) of multi-input radiating structures that establish the relationship of amplitude-phase distribution at the inputs of radiators with distributed nonlinearity and amplitude-phase distribution on their surfaces are obtained. A mathematical model of multi-input radiator structures with nonlinear characteristics in the Fresnel zone for analysis purposes has been produced. Conclusions. The scientific novelty of the obtained results is as follows: a generalized theory of transmitting antennas of arbitrary configuration with nonlinear characteristics in the Fresnel zone, which makes it possible to analyze the characteristics of these antennas considering the positive and negative (beneficial and adverse) nonlinear effects that arise in them.

An infinite-dimensional 2-generated primitive axial algebra of Monster type

Rehren proved in Axial algebras. Ph.D. thesis, University of Birmingham (2015), Trans Am Math Soc 369:6953–6986 (2017) that a primitive 2-generated axial algebra of Monster type $$(\alpha ,\beta )$$ ( α , β ) , over a field of characteristic other than 2, has dimension at most 8 if $$\alpha \notin \{2\beta ,4\beta \}$$ α ∉ { 2 β , 4 β } . In this note, we show that Rehren’s bound does not hold in the case $$\alpha =4\beta $$ α = 4 β by providing an example (essentially the unique one) of an infinite-dimensional 2-generated primitive axial algebra of Monster type $$(2,\frac{1}{2})$$ ( 2 , 1 2 ) over an arbitrary field $${{\mathbb {F}}}$$ F of characteristic other than 2 and 3. We further determine its group of automorphisms and describe some of its relevant features.

Surface Susceptibilities as Compact Full-Wave Simulation Models of Fully-Reflective Volumetric Metasurfaces

<p>While metasurfaces (MSs) are constructed from deeply-subwavelength unit cells, they are generally electrically-large and full-wave simulations of the complete structure are computationally expensive. Thus, to reduce this high computational cost, non-uniform MSs can be modeled as zero-thickness boundaries, with sheets of electric and magnetic polarizations related to the fields by surface susceptibilities and the generalized sheet transition conditions (GSTCs). While these two-sided boundary conditions have been extensively studied for single sheets of resonant particles, it has not been shown if they can correctly model structures where the two sides are electrically isolated, such as a fully-reflective surface. In particular, we consider in this work whether the fields scattered from a fully reflective metasurface can be correctly predicted for arbitrary field illuminations, with the source placed on either side of the surface. In the process, we also show the mapping of a PEC sheet with a dielectric cover layer to bi-anisotropic susceptibilities. Finally, we demonstrate the use of the susceptibilities as compact models for use in various simulation techniques, with an illustrative example of a parabolic reflector, for which the scattered fields are correctly computed using a integral equation (IE) based solver.<br></p>

Surface Susceptibilities as Compact Full-Wave Simulation Models of Fully-Reflective Volumetric Metasurfaces

<p>While metasurfaces (MSs) are constructed from deeply-subwavelength unit cells, they are generally electrically-large and full-wave simulations of the complete structure are computationally expensive. Thus, to reduce this high computational cost, non-uniform MSs can be modeled as zero-thickness boundaries, with sheets of electric and magnetic polarizations related to the fields by surface susceptibilities and the generalized sheet transition conditions (GSTCs). While these two-sided boundary conditions have been extensively studied for single sheets of resonant particles, it has not been shown if they can correctly model structures where the two sides are electrically isolated, such as a fully-reflective surface. In particular, we consider in this work whether the fields scattered from a fully reflective metasurface can be correctly predicted for arbitrary field illuminations, with the source placed on either side of the surface. In the process, we also show the mapping of a PEC sheet with a dielectric cover layer to bi-anisotropic susceptibilities. Finally, we demonstrate the use of the susceptibilities as compact models for use in various simulation techniques, with an illustrative example of a parabolic reflector, for which the scattered fields are correctly computed using a integral equation (IE) based solver.<br></p>

A STUDY OF SOLVING SYSTEM OF LINEAR EQUATION USING DIFFERENT METHODS AND ITS REAL LIFE APPLICATIONS:

Solving a system of linear equations (or linear systems or, also simultaneous equations) is a common situation in many scientific and technological problems. Many methods either analytical or numerical, have been developed to solve them so, in this paper, I will explain how to solve any arbitrary field using the different – different methods of the system of linear equation for this we need to define some concepts. Like a general method most used in linear algebra is the Gauss Elimination or variation of this sometimes they are referred as “direct methods “Basically it is an algorithm that transforms the system into an equivalent one but with a triangular matrix, thus allowing a simpler resolution, Other methods can be more effective in solving system of the linear equation like Gauss Elimination or Row Reduction, Gauss Jordan and Crammer’s rule, etc. So, in this paper I will explain this method by taking an example also, in this paper I will explain the Researcher’ works that how they explain different –different methods by taking different examples. And I worked on using these different methods in solving a single example, i.e. I will use these methods in an example. In this paper, I will explain the real-life application that how a System of Linear Equation is used in our daily life.