The dimensions of the graded quotients of the cohomology of a plane curve complement U = P-2/C with respect to the Hodge filtration are described in terms of simple geometrical invariants. The case of curves with ordinary singularities is discussed in detail. We also give a precise numerical estimate for the difference between the Hodge filtration and the pole order filtration on H-2(U, C).

We study Hilbert functions, Lefschetz properties, and Jordan type of Artinian Gorenstein algebras associated to Perazzo hypersurfaces in projective space. The main focus lies on Perazzo threefolds, for which we prove that the Hilbert functions are always unimodal. Further we prove that the Hilbert function determines whether the algebra is weak Lefschetz, and we characterize those Hilbert functions for which the weak Lefschetz property holds. By example, we verify that the Hilbert functions of Perazzo fourfolds are not always unimodal. In the particular case of Perazzo threefolds with the smallest possible Hilbert function, we give a description of the possible Jordan types for multiplication by any linear form. 

Codimension two Artinian algebras have the strong and weak Lefschetz propertiesprovided the characteristic is zero or greater than the socle degree. It is open to whatextent such results might extend to codimension three Artinian Gorenstein algebras. De-spite much work, the strong Lefschetz property for codimension three Artinian Gorensteinalgebra has remained largely mysterious; our results build on and strengthen some of theprevious results. We here show that every standard-graded codimension three ArtinianGorenstein algebra A having maximum value of the Hilbert function at most six has thestrong Lefschetz property, provided that the characteristic is zero. When the characteris-tic is greater than the socle degree of A, we show that A is almost strong Lefschetz, theyare strong Lefschetz except in the extremal pair of degrees.

We classify the orbits of nets of conics under the action of the projective linear group and we determine the specializations of these orbits, using geometric and algebraic methods. We study related geometric questions, as the parametrization of planar cubics. We show that Artinian algebras of Hilbert function H=(1,3,3,0) determined by nets, can be smoothed—deformed to a direct sum of fields; and that algebras of Hilbert function H=(1,r,2,0), determined by pencils of quadrics, can also be smoothed. This portion is a translation and update of a 1977 version, a typescript by the second two authors that was distributed as a preprint of University of Paris VII. In a new Historical Appendix A we describe related work prior to 1977. In an Update Appendix B we survey some developments since 1977 concerning nets of conics, related geometry, and deformations of Artinian algebras of small length.

Let R= k [x, y, z], the polynomial ring over a field k. Several of the authors previously classified nets of ternary conics and their specializations over an algebraically closed field, Abdallah et al. (Eur J Math 9(2), Art. No. 22, 2023). We here show that when k is algebraically closed, and considering the Hilbert function sequence T =(1,3(k),1), k >= 2 (i.e. T = (1, 3, 3, ... , 3, 1) where k is the multiplicity of 3), then the family GT parametrizing graded Artinian algebra quotients A = R/I of R having Hilbert function T is irreducible, and G(T) is the closure of the family Gor(T) of Artinian Gorenstein algebras of Hilbert function T. We then classify up to isomorphism the elements of these families Gor(T) and of G(T). Finally, we give examples of codimension 3 Gorenstein sequences, such as (1, 3, 5, 3, 1), for which G(T) has several irreducible components, one being the Zariski closure of Gor(T).

Special partial matchings (SPMs) are a generalisation of Brenti's special matchings. Let a pircon be a poset in which every non-trivial principal order ideal is finite and admits an SPM. Thus pircons generalise Marietti's zircons. We prove that every open interval in a pircon is a PL ball or a PL sphere. It is then demonstrated that Bruhat orders on certain twisted identities and quasiparabolic W-sets constitute pircons. Together, these results extend a result of Can, Cherniaysky, and Twelbeck, prove a conjecture of Hultman, and confirm a claim of Rains and Vazirani.

In (Stanley, 1978), Stanley constructs an example of an Artinian Gorenstein (AG) ring A with non-unimodal H-vector (1,13,12,13,1). Migliore-Zanello show in (Migliore and Zanello, 2017) that for regularity r=4, Stanley's example has the smallest possible codimension c for an AG ring with non-unimodal H-vector.The weak Lefschetz property (WLP) has been much studied for AG rings; it is easy to show that an AG ring with non-unimodal H-vector fails to have WLP. In codimension c=3 it is conjectured that all AG rings have WLP. For c=4, Gondim shows in (Gondim, 2017) that WLP always holds for r≤4 and gives a family where WLP fails for any r≥7, building on Ikeda's example (Ikeda, 1996) of failure for r=5. In this note we study the minimal free resolution of A and relation to Lefschetz properties (both weak and strong) and Jordan type for c=4 and r≤6.

A net in P^2 is a configuration of lines A and points X satisfying certain incidence properties. Nets appear in a variety of settings, ranging from quasigroups to combinatorial design to classification of Kac–Moody algebras to cohomology jump loci of hyperplane arrangements. For a matroid M and rank r, we associate a monomial ideal (a monomial variant of the Orlik–Solomon ideal) to the set of flats of M of rank ≤r. In the context of line arrangements in P^2, applying Alexander duality to the resulting ideal yields insight into the combinatorial structure of nets.

How fast can you comfortably travel between two points A and B? This question is formulated as a minimization problem of a functional where the discomfort is quantified in terms of the integral of the square of the acceleration between A and B. The problem is solved in terms of the corresponding Euler-Lagrange equation and approximately using a direct variational approach based on trial functions and Ritz optimization. The main purpose of the analysis is to introduce undergraduate students to variational calculus in an interesting and pedagogical way.

A short introduction is given of direct variational methods and its relation to Galerkin and moment methods, all flexible and powerful approaches for finding approximate solutions of difficult physical equations. A pedagogical application of moment methods is given to the physically and technically important Child–Langmuir law in electron physics. The analysis is shown to provide simple, yet accurate, approximate solutions of the two-dimensional problem (a problem which does not allow an exact analytical solution) and illustrates the usefulness and the power of moment methods.

An analysis based on the Galerkin method is given of some nonlinear oscillator equations that have been analyzed by several other methods, including harmonic balance and direct variational methods. The present analysis is shown to provide simple yet accurate approximate solutions of these nonlinear equations and illustrates the usefulness and the power of the Galerkin method. (C) 2010 American Association of Physics Teachers.

Direct variational methods are used to find simple approximate solutions of the Thomas–Fermi equations describing the properties of self-gravitating radially symmetric stellar objects both in the non-relativistic and ultra-relativistic cases. The approximate solutions are compared and shown to be in good agreement with exact and numerically obtained solutions.

We construct a variational formulation for the problem of interpolating seismic data in the case of missing traces. We assume that we have derivative information available at the traces. The variational problem is essentially the minimization of the integral over the smallest eigenvalue of the structure tensor associated with the interpolated data. This has the physical meaning of penalizing the local presence of more than one direction in the interpolation. The variational problem is used to justify the solutions of a non-standard anisotropic diffusion problem as reasonable interpolated images. We show existence and uniqueness for this type of anisotropic diffusion. In particular, the uniqueness property is important as it guarantees that the solution can be obtained by the numerical schemes we propose.

Rat models are frequently used for finding genes contributing to the arthritis phenotype. In most studies, however, limitations in the number of animals result in a low resolution. As a result, the linkage between the autoimmune experimental arthritis phenotype and the genomic region, that is, the quantitative trait locus, can cover several hundred genes. The purpose of this work was to facilitate the search for candidate genes in such regions by introducing a web tool called Candidate Gene Capture (CGC) that takes advantage of free text data on gene function. The CGC tool was developed by combining genomic regions in the rat, associated with the autoimmune experimental arthritis phenotype, with rat/human gene homology data, and with descriptions of phenotypic gene effects and selected keywords. Each keyword was assigned a value, which was used for ranking genes based on their description of phenotypic gene effects. The application was implemented as a web-based tool and made public at http://ratmap.org/cgc. The CGC application ranks gene candidates for 37 rat genomic regions associated with autoimmune experimental arthritis phenotypes. To evaluate the CGC tool, the gene ranking in four regions was compared with an independent manual evaluation. In these sample tests, there was a full agreement between the manual ranking and the CGC ranking for the four highest-ranked genes in each test, except for one single gene. This indicates that the CGC tool creates a ranking very similar to that made by human inspection. The exceptional gene, which was ranked as a gene candidate by the CGC tool but not in the manual evaluation, was found to be closely associated with rheumatoid arthritis in additional literature studies. Genes ranked by the CGC tools as less likely gene candidates, as well as genes ranked low, were generally rated in a similar manner to those done manually. Thus, to find genes contributing to experimentally induced arthritis, we consider the CGC application to be a helpful tool in facilitating the evaluation of large amounts of textual information.

Today, textiles and fiber science in US, Europe and Japan from its once lofty perch in the global economy, stands in stark contrast to its preeminent position of few decades ago. Its influence on the society as a whole has eroded enormously. Many of the synthetic fiber products that once fuelled the rapid growth of the industry have become mature commodity products now characterized by low growth and lower profit margins. To add to the current dilemma, organizational ‘health’ and growth processes are constantly threatened in this era of turbulence. Thus the drive for survival and success has translated, in recent times, to quest for resiliency – to survive and thrive in turbulences. On the other hand, most managers and academicians agree that innovation ensures superior organizational performance while recent research has shown that most resilient companies can dynamically orchestrate diverse innovation strategies. Resiliency in such a context has become a prerequisite for a sustained long term business prosperity fuelled by diverse technological innovations. This has intensified the organization’s search for differentiated products and services, processes, business models, technology, strategies etc. pushing firms to gain competitive advantage and also to develop new knowledge and innovation performance to drive sustainable growth. Organizations now follow multiple innovation strategies to pragmatically devise their innovation repertoire for delivering growth, hence, success in turbulent times while emphasizing resiliency. What does the future hold and how can we reverse the trend to achieve and sustain the impressive credentials of the past? To understand the significance of what the future may hold, and to reverse the downward spiral of the industry, we must evaluate the successes and failures of the past and come to grips with rapid global changes and turbulences currently underway. The present article seeks to explore such an inexorable phenomenon of quantifying and correlating innovation and business resiliency over a time line, from the annual financial data of 35 healthy and unhealthy companies along with 5 textile companies over a span of few decades. These are then extrapolated with certain predictive capabilities to suggest future trends and strategies for the textile companies. Learning from these companies, if adopted, will yield capacity to transform the scenario. Assessments and classification of the economic health of a company is typically made based on some quantity derived from selected indices, such as Altman’s Z-score. These methods can describe an instantaneous status, or a “time snap” of an economical subject but lack information about the time-dynamics of the assessment, which is important for investors, shareholders and the management. We suggest using historical data to estimate current trends in the form of the first and second time-derivative of the appropriate quantity in the time domain. This new information is independent on the quantity itself and beside more precise description can be used as new predictor to improve effectiveness of classification of successful and unsuccessful subjects. This approach is further discussed in this paper.

In this paper we use a Choquet type theorem on adapted spaces to obtain or reobtain some results in harmonic analysis on semigroups. Thus we give a L ́evy]Khinchine formula for some negative definite functions defined on a com- mutative semigroup with neutral element, we prove that completely monotonic Žresp. alternating. functions are completely positive Žresp. negative. definite, we characterize the completely monotonic and the completely alternating func- tions defined on N* [ 1, 2, 3, . . . 4, and we consider a Stieltjes’ moment problem.

In this paper we obtain integral representations, independent of a L ́evy function, for negative definite functions with real part bounded below defined on a commu- tative involutive semigroup and for continuous negative definite functions defined on the group Rⁿ

The book is an introduction to linear algebra intended as a textbook for the first course in linear algebra. In the first six chapters we present the core topics: matrices, the vector space ℝn, orthogonality in ℝn, determinants, eigenvalues and eigenvectors, and linear transformations. The book gives students an opportunity to better understand linear algebra in the next three chapters: Jordan forms by examples, singular value decomposition, and quadratic forms and positive definite matrices.

In the first nine chapters everything is formulated in terms of ℝn. This makes the ideas of linear algebra easier to understand. The general vector spaces are introduced in Chapter 10. The last chapter presents problems solved with a computer algebra system. At the end of the book we have results or solutions for odd numbered exercises.

In this paper we obtain the quadratic form in the Lévy-Khinchin formula on a commutative involutive semigroup, with a neutral element, as a sum of two simpler quadratic forms.

Dans cet article on obtient un théorème du type Bochner-Godement pour des algèbres commutatives involutives et avec unité et puis on applique ce théorème pour obtenir des résolutions pour le problème des moments sur les polydisques et les polyrectangles de S (où S est un semi-groupe abélien avec involution et élément neutre) et sur certains de leurs sous-ensembles.

The book makes a first course in linear algebra more accessible to the majority of students and it assumes no prior knowledge of the subject. It provides a careful presentation of particular cases of all core topics. Students will find that the explanations are clear and detailed in manner. It is considered as a bridge over the obstacles in linear algebra and can be used with or without the help of an instructor.While many linear algebra texts neglect geometry, this book includes numerous geometrical applications. For example, the book presents classical analytic geometry using concepts and methods from linear algebra, discusses rotations from a geometric viewpoint, gives a rigorous interpretation of the right-hand rule for the cross product using rotations and applies linear algebra to solve some nontrivial plane geometry problems.Many students studying mathematics, physics, engineering and economics find learning introductory linear algebra difficult as it has high elements of abstraction that are not easy to grasp. This book will come in handy to facilitate the understanding of linear algebra whereby it gives a comprehensive, concrete treatment of linear algebra in R² and R³. This method has been shown to improve, sometimes dramatically, a student's view of the subject.

A space of generalized functions is constructed that allows us to generalize Bochner's theorem so that all Radon measures on a locally compact group are in a one-to-one correspondence with elements of that space of generalized functions. This defines a Fourier transform for all Radon measures on a locally compact group.

We introduce some spaces of generalized functions that are defined as generalized quotients and Boehmians. The spaces provide simple and natural frameworks for extensions of the Fourier transform.

A space of pseudoquotients is introduced that is shown to be isomorphic to the space of tempered distributions on RN. The Fourier transform is defined as a map from the space of pseudoquotients to the space of tempered distributions and as a transformation on pseudoquotients.

We consider pseudoquotient extensions of positive linear functionals on a commutative Banach algebra A and give conditions under which the constructed space of pseudoquotients can be identified with all Radon measures on the structure space A.

This is a book for the second course in linear algebra whereby students are assumed to be familiar with calculations using real matrices. To facilitate a smooth transition into rigorous proofs, it combines abstract theory with matrix calculations. This book presents numerous examples and proofs of particular cases of important results before the general versions are formulated and proved. The knowledge gained from a particular case, that encapsulates the main idea of a general theorem, can be easily extended to prove another particular case or a general case. For some theorems, there are two or even three proofs provided. In this way, students stand to gain and study important results from different angles and, at the same time, see connections between different results presented in the book.

Two related ethnographic research projects on mathematics teacher education in Sweden are presented in this paper. They represent a response to recent policy developments that reaffirm the value of authoritative subject studies content as the central and most important component in the professional knowledge base of would-be teachers and concomitant increases in the amount of subject studies in teacher education. These policy changes, in Sweden at least, lack scientific research support and the article argues that these policies need to be seriously rethought, as the increased emphasis on subject content may undermine the development of key professional skills.

We provide a mathematical account of the recent letter by Tarnopolsky, Kruchkov and Vishwanath (Phys. Rev. Lett.122:10 (2019), art. id. 106405). The new contributions are a spectral characterization of magic angles, its accurate numerical implementation and an exponential estimate on the squeezing of all bands as the angle decreases. Pseudospectral phenomena due to the nonhermitian nature of operators appearing in the model considered in the letter of Tarnopolsky et al. play a crucial role in our analysis.

We consider a tight-binding model recently introduced by Timmel and Mele (Phys Rev Lett 125:166803, 2020) for strained moiré heterostructures. We consider two honeycomb lattices to which layer antisymmetric shear strain is applied to periodically modulate the tunneling between the lattices in one distinguished direction. This effectively reduces the model to one spatial dimension and makes it amenable to the theory of matrix-valued quasi-periodic operators. We then study the charge transport and spectral properties of this system, explaining the appearance of a Hofstadter-type butterfly and the occurrence of metal/insulator transitions that have recently been experimentally verified for non-interacting moiré systems (Wang et al. in Nature 577:42–46, 2020). For sufficiently incommensurable moiré lengths, described by a diophantine condition, as well as strong coupling between the lattices, which can be tuned by applying physical pressure, this leads to the occurrence of localization phenomena. 

In this article we generalize the Bohr–Sommerfeld rule for scalar symbols at a potential well to matrix-valued symbols having eigenvalues that may coalesce precisely at the bottom of the well. As an application, we study the existence of approximately flat bands in moiré heterostructures such as strained two-dimensional honeycomb lattices in a model recently introduced by Timmel and Mele. 

The reformulation, using Fock space vertex operators, of the original light-front cubic interaction terms for higher spin gauge fields is reviewed with comments on quantum higher spin gravity. The formalism is generalized to all orders in the interaction. The ensuing recursive equations for the higher order vertices, if they can be explicitly solved, will encode all interaction data into a denumerable set of rational functions of p+, the overall transverse momentum structure being fixed already at the kinematical level. A more thorough exposition can be found in the archive.

The original cubic interaction terms for higher spin gauge fields in four dimensions and their reformulation using Fock space vertex operators is reviewed. As a new result, the complete list of all cubic vertex functions in D=4 is derived. It is observed, contrary to what would have been expected, that the non-linear dynamical Poincar\'e transformations do not restrict the cubic interactions beyond what is required by kinematics. The role of the SU(1,1) algebra of tracelessness constraints is clarified. It is shown that higher spin fields couple non-minimally to gravity at the cubic level in D=4 light-front dynamics. Based on a detailed analysis of the structure of the light-front Poincar\'e algebra, the formalism is then generalized to all orders in the interaction. The interacting theory, being a deformation of the free theory, takes the form of a strongly homotopy Lie algebra. It is conjectured that the ensuing recursive equations, if they can be explicitly solved, will encode all interaction data into a denumerable set of functions of p+, the overall transverse momentum structure being fixed already at the kinematical level.

In this paper we discuss gravity in the light-front formulation (light-cone gauge) and show how possible counterterms arise. We find that Poincare invariance is not enough to find the three-point counterterms uniquely. Higher-spin fields can intrude and mimic three-point higher derivative gravity terms. To select the correct term we have to use the remaining reparametrization invariance that exists after the gauge choice. We finally sketch how the corresponding programme for N=8 Supergravity should work.

Research in warehouse optimization has gotten increased attention in the last few years due to e-commerce. The warehouse contains a waste range of different products. Due to the nature of the individual order, it is challenging to plan the picking list to optimize the material flow in the process. There are also challenges in minimizing costs and increasing production capacity, and this complexity can be defined as a multidisciplinary optimization problem with an IDF nature. In recent years the use of parallel computing using GPGPUs has become increasingly popular due to the introduction of CUDA C and accompanying applications in, e.g., Python. 

In the case study at the company in the field of retail, a case study including a system design optimization (SDO) resulted in an increase in throughput with well over 20% just by clustering different categories and suggesting in which sequence the orders should be picked during a given time frame. 

The options provided by implementing a distributed high-performance computing network based on GPUs for subsystem optimization have shown to be fruitful in developing a functioning SDO for warehouse optimization. The toolchain can be used for designing new warehouses or evaluating and tuning existing ones.