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Using tools from algebraic topology and algebraic geometry (fundamental polygons and blowups, respectively), we prove that cutting along the center of a Möbius strip yields a cylinder. [ABSTRACT FROM AUTHOR]

We investigate the notions of weak annihilator and nilpotent associated prime defined by Ouyang and Birkenmeier (Weak annihilator over extension rings, 2012) in the setting of skew PBW extensions. We extend several results formulated in the literature concerning annihilators and associated primes of commutative rings and skew polynomial rings to a more general setting of algebras not considered before. We exemplify our results with families of algebras appearing in the theory of enveloping algebras, noncommutative algebraic geometry, and theoretical physics. Finally, we present some ideas for future research. [ABSTRACT FROM AUTHOR]

Abstract: The topic of inequalities is one of the fundamental subject areas for the development of mathematical knowledge. Although there is a considerable body of research focused on one-variable inequalities, studies that investigate students' performance on two-variable inequalities are scarce. With this in mind, the aim of this study is twofold: first, it provides an account of the difficulties that students face in interpreting and constructing different representations of two-variable inequalities; and second, it investigates the effectiveness of a computer-supported teaching sequence aimed at developing students' understanding of different representations of two-variable inequalities. The participants in the study were 33 elementary mathematics teacher candidates. The data were gathered via a test implemented as a pre- and post-test. The main finding of the study was that students had difficulty in establishing the link between the coordinates of the points in a solution set and the algebraic representation of a two-variable inequality. Moreover, it was found that the students were not able to correctly use and interpret the logical connectives in compound inequalities. The comparison of the pre- and post-test scores indicated that the designed instructional sequence was beneficial for supporting students in understanding the different representations of two-variable inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we study the identifiability of specific forms (symmetric tensors), with the target of extending recent methods for the case of 3 variables to more general cases. In particular, we focus on forms of degree 4 in 5 variables. By means of tools coming from classical algebraic geometry, such as Hilbert function, liaison procedure and Serre's construction, we give a complete geometric description and criteria of identifiability for ranks ≥ 9 , filling the gap between rank ≤ 8 , covered by Kruskal's criterion, and 15, the rank of a general quartic in 5 variables. For the case r = 12, we construct an effective algorithm that guarantees that a given decomposition is unique. [ABSTRACT FROM AUTHOR]

One way to study a hypergraph is to attach to it a tensor. Tensors are a generalization of matrices, and they are an efficient way to encode information in a compact form. In this paper, we study how properties of weighted hypergraphs are reflected on eigenvalues and eigenvectors of their associated tensors. We also show how to efficiently compute eigenvalues with some techniques from numerical algebraic geometry. [ABSTRACT FROM AUTHOR]

Technology of Ancient Greece As Nicos pointed out "automaton" (Graph HT ht ) is a Homeric word that first appeared in the Iliad (Vasileiadou et al., [14]). Assigning the roots of Graph HT ht is much more difficult. Likewise the Greek word Graph HT ht "system" comes from the ancient verb Graph HT ht which occurs frequently in ancient Greek written sources in many contexts with a basic meaning that the whole is made from parts and that the whole may have properties not always attributed to the individual parts. [Extracted from the article]

In this paper, we give a counterexample to Zariski's cancellation problem for principal ${\mathbb{G}}_{a}$ G a -bundles over certain open subvarieties of non- $\mathbb{A}^{1}$ A 1 -uniruled affine varieties. [ABSTRACT FROM AUTHOR]

The commuting variety of matrices over a given field is a well-studied object in linear algebra and algebraic geometry. As a set, it consists of all pairs of square matrices with entries in that field that commute with one another. In this paper, we generalize the commuting variety by using the commuting distance of matrices. We show that over an algebraically closed field, each of our sets does indeed form a variety. We compute the dimension of the distance-2 commuting variety and characterize its irreducible components. We also work over other fields, showing that the distance-2 commuting set is a variety but that the higher distance commuting sets may or may not be varieties, depending on the field and on the size of the matrices. [ABSTRACT FROM AUTHOR]

The wild McKay correspondence, a variant of the McKay correspondence in positive characteristics, shows that stringy motives of quotient varieties equal some motivic integrals on the moduli space of the Galois covers of a formal disk. In this paper, we determine when the integral converges for the case of cyclic groups of prime power order. As an application, we give a criterion for the quotient variety being canonical or log canonical. [ABSTRACT FROM AUTHOR]

Polstra showed that the cardinality of the torsion subgroup of the divisor class group of a local strongly F-regular ring is finite. We expand upon this result and prove that the reciprocal of the F-signature of a local strongly F-regular ring R bounds the cardinality of the torsion subgroup of the divisor class group of R. [ABSTRACT FROM AUTHOR]

In 1878, Gaston Darboux published a method for finding first integrals of differential systems with polynomial coefficients. Combined with tools from algebraic geometry and computer algebra, Darboux's method has opened up a vast field of work that is still being actively pursued. In this article, we introduce some of the key ideas and problems in this area. [ABSTRACT FROM AUTHOR]

Let C be a nodal curve, and let E be a union of semistable subcurves of C. We consider the problem of contracting the connected components of E to singularities in a way that preserves the genus of C and makes sense in families. In order to do this, we introduce the notion of mesa curve, a nodal curve C with a logarithmic structure and a piecewise linear function λ ¯ on the tropicalization of C. We then construct a contraction of the support of λ ¯ inside of C for families of mesa curves. [ABSTRACT FROM AUTHOR]

In this article, we check that Fano schemes of lines on certain rational cubic fourfolds are birational to Hilbert schemes of two points on K3 surfaces. [ABSTRACT FROM AUTHOR]

In this paper, we reconsider the topological characterization of gestures in a convenient category of spaces mentioned by Mazzola in 2009, recovering Arias's 2018 result that the relevant equivalence is a homeomorphism. We also show the topological characterization of gestures extends to an adjunction between the category of gestures and the category of continuous maps whose domain is a one-dimensional CW complex. Our arguments utilize only basic tools from category theory and almost no point-set topology. With little generality lost from our restriction of spaces considered, the ease of conceptualization in this setting provides an advantageous entry point for researchers interested in studying or applying gestures but who may not have advanced knowledge of category theory, algebraic geometry, or point-set topology. Thus one may view this work, in part, as an advertisement for the topological interpretation of gestures developed within. [ABSTRACT FROM AUTHOR]

The study of invariants of group actions on commutative polynomial rings has motivated many developments in commutative algebra and algebraic geometry. It has been of particular interest to understand what conditions on the group result in an invariant ring satisfying useful properties. In particular, Watanabe's Theorem states that the invariant subring of k [ x 1 , ... , x n ] under the natural action of a finite subgroup of S L n (k) is always Gorenstein. In this paper, we study this question in the more general setting of group actions on noncommutative non-connected algebras A. We develop the notion of a homological determinant of an automorphism of A, then use the homological determinant to study actions of finite groups G on A. We give a sufficient condition so that the invariant ring AG has finite injective dimension and satisfies the generalized Gorenstein condition. More precisely, let A be a noetherian N -graded generalized Gorenstein algebra with finite global dimension. Suppose all elements of G fix the idempotents of A and act with trivial homological determinant. Then the invariant ring AG is generalized Gorenstein. [ABSTRACT FROM AUTHOR]

The Isbell duality tells us a dual equivalence between spatial frames (aka. locales) and sober spaces; it is induced by the prime spectrum functor on frames. In the present paper, we give another dual equivalence induced by the maximal spectrum functor. The Isbell duality subsumes all sober spaces, but not all T 1 spaces; the duality shown in this paper subsumes all T 1 spaces, but not all sober spaces. Non-sober T 1 spaces are particularly important in classical algebraic geometry; they include, inter alia, algebraic varieties in the traditional sense, the points of which can be recovered from their open set frames via the maximal spectrum functor (and cannot via the prime spectrum functor). The duality in this paper is particularly useful for those spaces in algebraic geometry. In addition to the duality induced by maximal spectra, we give a dual adjunction lurking behind it, and an algebraic characterization of having enough points in terms of maximal spectra. Communicated by Jason P. Bell [ABSTRACT FROM AUTHOR]

We classify the conics passing through six points on the sides of a triangle equidistant from the corresponding midpoints. We use tools from synthetic geometry and real algebraic geometry. [ABSTRACT FROM AUTHOR]

Let Gr (k , n) be the Grassmannian. The quantum multiplication by the first Chern class c 1 (Gr (k , n)) induces an endomorphism c ̂ 1 of the finite-dimensional vector space QH * (Gr (k , n)) | q = 1 specialized at q = 1. Our main result is a case that a conjecture by Galkin holds. It states that the largest real eigenvalue of c ̂ 1 is greater than or equal to dim Gr (k , n) +1 with equality if and only if Gr (k , n) = P n − 1. [ABSTRACT FROM AUTHOR]

The characteristics of tensor decomposition in certain desirable forms are its uniqueness which appears to be essential in many applications. Extensive work under the framework of algebraic geometry has provided many fundamental results associated with tensor rank and dimension to ensure the generic identifiability. Different from most current approaches in the literature, in this paper, by using algebraic geometry theory, we study the rank L r and rank L r ⊗ rank-1 block term tensor decomposition for higher order tensors. In particular, we establish the sufficient and necessary conditions for a general tensor to have finitely many block term decompositions, and sufficient conditions to guarantee a unique decomposition. [ABSTRACT FROM AUTHOR]

In this paper, we consider the wide class of subvarieties of the variety of all representation of Lie algebras over a field k of characteristic 0. We study the relation between the geometric equivalence and automorphic equivalence of the representations from these subvarieties. [ABSTRACT FROM AUTHOR]

In this paper, an algorithm based on linear algebra tools is proposed to compute a weighted sum of squares decomposition of a given form whose length is lower than the number of variables. Such an objective is pursued by using linear algebra techniques to perform tasks that are usually carried out through computational algebraic geometry tools. Several examples are reported to show that the use of linear algebra rather than algebraic geometry leads to a reduction of the execution times, without affecting the effectiveness of the algorithm. Applications of the given procedure to system analysis and to control design problems are reported as well as a detailed complexity analysis. [ABSTRACT FROM AUTHOR]

In this article, we define the notion of Brauer-Clifford group for H-locally finite (S, H)-Azumaya algebras, when H is a cocommutative Hopf algebra and S is an H-locally finite commutative H-module algebra over a commutative noetherian ring k. This is the situation that arises in applications with connections to algebraic geometry. This Brauer-Clifford group turns out to be an example of a Brauer group of a symmetric monoidal category. [ABSTRACT FROM AUTHOR]

In this article, we investigate, for varieties over with trivial group of 0-cycles, the gap between essential -dimension 2 and essential -dimension 0. In particular, we present sufficient (and necessary) conditions for a variety with trivial group of 0-cycles and essential -dimension to have, in fact, essential -dimension 0. [ABSTRACT FROM AUTHOR]

In this paper, we construct new sequences of asymptotically good convolutional codes (AGCC's). These sequences are obtained from sequences of transitive, self-orthogonal and self-dual algebraic geometry (AG) codes attaining the Tsfasman–Vladut–Zink bound. Furthermore, by applying the techniques of expanding, extending, puncturing, direct sum, the construction and the product code construction to these block codes, we construct more new sequences of asymptotically good convolutional codes. Additionally, we show that the new constructions presented here also hold when applied to al l sequences of asymptotically good block codes where and exist. [ABSTRACT FROM AUTHOR]

An exploratory study was conducted to investigate the use of magic activities in a math course for prospective middle-school math teachers. This research report focuses on a lesson using two versions of math magic: (1) the 5-4-3-2-1-½ Magic involves having students choose a secret number and apply six arithmetic operations in sequence to arrive at a resultant number, and the teacher-magician can spontaneously reveal a student's secret number from the resultant number; and (2) the Everyone-Got-9 Magic also involves choosing a secret number and applying arithmetic operations in sequence, but everyone will end up with the same resultant number of 9. These magic activities were implemented to reinforce students' understanding of foundational algebra concepts like variables, expressions, and inverse functions. Analysis of students' written responses revealed that (1) all students who figured out the trick in the first magic activity did not used algebra, (2) most students could apply what they learned in one trick to a similar trick but not to a different trick, and (3) many students were weak in symbolic representations and manipulations. Responses from a survey and a focus group indicate that students found the magic activities to be fun and intellectually engaging. [ABSTRACT FROM AUTHOR]

For a formal scheme over a complete discrete valuation ring with a good action of a finite group, we define equivariant motivic integration, and we prove a change of variable formula for that. To do so, we construct and examine an induced group action on the Greenberg scheme of such a formal scheme. Using this equivariant motivic integration, we define an equivariant volume Poincaré series, from which we deduce Denef and Loeser's motivic zeta function including the action of the profinite group of roots of unity. [ABSTRACT FROM AUTHOR]

Proposed problems should be submitted online at americanmathematicalmonthly.submittable.com/submit. Proposed solutions to the problems below should be submitted by May 31, 2019 via the same link. More detailed instructions are available online. Proposed problems must not be under consideration concurrently at any other journal nor be posted to the internet before the deadline date for solutions. An asterisk (*) after the number of a problem or a part of a problem indicates that no solution is currently available. [ABSTRACT FROM AUTHOR]

A submodule N of a module M is δ-small in M if N+X≠M for any proper submodule X of M with M∕X singular. A projective δ-cover of a module M is a projective module P with an epimorphism to M whose kernel is δ-small in P. A module M is called δ-semiperfect if every factor module of M has a projective δ-cover. In this paper, we prove various properties, including a structure theorem and several characterizations, for δ-semiperfect modules. Our proofs can be adapted to generalize several results of Mares [8] and Nicholson [11] from projective semiperfect modules to arbitrary semiperfect modules. [ABSTRACT FROM AUTHOR]

Let M be a hypermodule over a hyperring R such that the intersection of any two subhypermodules of M, is a subhypermodule of M. We introduce the concept of an essential subhypermodule in M relative to an arbitrary subhypermodule T of M, which is called a T-essential subhypermodule of M. Our main goal in this work is to investigate properties of (relative) essential subhypermodules. We apply this concept to introduce extending hypermodules. Examples are provided to illustrate different concepts. [ABSTRACT FROM AUTHOR]

Nikolai Durov introduced the theory of generalized rings and schemes to study Arakelov geometry in an alternative algebraic framework, and introduced the residue field at the infinite place, 픽∞. We show an elementary algebraic approach to modules and algebras over this object, define prime congruences, show that the polynomial ring of n variables is of Krull dimension n, and derive a prime decomposition theorem for these primes. [ABSTRACT FROM AUTHOR]

In this paper, we will classify normal Gorenstein stable log surfaces with . [ABSTRACT FROM AUTHOR]

We study the problem of lifting of polynomial symplectomorphisms in characteristic zero to automorphisms of the Weyl algebra by means of approximation by tame automorphisms. In 1983, Anick proved the fundamental result on approximation of polynomial automorphisms. We obtain similar approximation theorems for symplectomorphisms and Weyl algebra authomorphisms. We then formulate the lifting problem. More precisely, we prove the possibility of lifting of a symplectomorphism to an automorphism of the power series completion of the Weyl algebra of the corresponding rank. The lifting problem has its origins in the context of deformation quantization of the affine space and is closely related to several major open problems in algebraic geometry and ring theory. This paper is a continuation of the study [19]. [ABSTRACT FROM AUTHOR]

In [4], Mezzetti and Miró-Roig proved that the minimal number of generators μ(I) of a minimal (smooth) monomial Togliatti system satisfies and they classify all smooth minimal monomial Togliatti systems with 2n+1≤μ(I)≤2n+2. In this paper, we address the first open case. We classify all smooth monomial Togliatti systems of forms of degree d≥4 with μ(I) = 2n+3 and n≥2 and all monomial Togliatti systems of forms of degree d≥6 with μ(I) = 7. [ABSTRACT FROM AUTHOR]

A depiction of a nonnoetherian integral domain R is a special coordinate ring that provides a framework for describing the geometry of R. We show that if R is noetherian in codimension 1, then R has a unique maximal depiction T. In this case, the geometric dimensions of the points of Spec R may be computed directly from T. If in addition R has a normal depiction S, then S is the unique maximal depiction of R. [ABSTRACT FROM AUTHOR]

The matrix Sturm-Liouville equation on a finite interval with a Bessel-type singularity in the end of the interval is studied. We consider inverse problems by the Weyl matrix and by the spectral data for this equation. Constructive solutions, based on the method of spectral mappings, are obtained for these inverse problems. [ABSTRACT FROM AUTHOR]

In this paper, we use Block’s results to classify irreducible modules over the diamond Lie algebra𝔇. As a corollary, we also give a classification of irreducible modules over the Euclidean algebra𝔢(2). [ABSTRACT FROM PUBLISHER]

Letbe a connected reductive algebraic group defined over an algebraically closed field. Letbe a maximal torus ofand letbe a Borel subgroup ofcontaining. LetWdenote the Weyl group ofand letẇdenote an element in the preimage of the natural map fromtoW. We investigate what conditions are necessary and sufficient for the double cosets(called Bruhat cells) to contain only regular elements. An application is provided on how this information about Bruhat cells can be used to determine character values for certain Hecke algebras. [ABSTRACT FROM PUBLISHER]

We study multiple-class classification problems. Both ordinal and categorical labeled cases are discussed. The common approaches for multiple-class classification are built on binary classifiers, in which one-versus-one and one-versus-rest are typical approaches. When the number of classes is large, then these binary-classifier-based methods may suffer from either computational costs or the highly imbalanced sample sizes in their training stage. In order to alleviate the computational burden and the imbalanced training data issue in multiple-class classification problems, we propose a method that has competitive performance and retains the ease of model interpretation, which is essential for a prognostic/predictive model. [ABSTRACT FROM AUTHOR]

Let 𝒳 be an irreducible algebraic curve defined over a finite field 𝔽q of characteristic p>2. Assume that the 𝔽q-automorphism group of 𝒳 admits a subgroup isomorphic to the direct product of two cyclic groups Cm and Cn of orders m and n prime to p, such that both quotient curves 𝒳∕Cn and 𝒳∕Cm are rational. In this paper, we provide a complete classification of such curves as well as a characterization of their full automorphism groups. [ABSTRACT FROM AUTHOR]

We provide an explicit algorithm for computing intersection numbers between basis elements of complementary codimension in the Hilbert scheme ofNpoints in the projective plane. [ABSTRACT FROM AUTHOR]

LetGbe a finite group,Ea normal subgroup ofGandpa fixed prime. We say thatEisp-hypercyclically embeddedinGif everyp-G-chief factor ofEis cyclic. A subgroupHofGis said to satisfy Π-propertyinGif |G∕K:NG∕K((H∩L)K∕K)| is aπ((H∩L)K∕K)-number for any chief factorL∕KinG; we say thatHhas Π -property inGifH∩Oπ(H)(G) has Π-property inG. In this paper, we prove thatEisp-hypercyclically embedded inGif and only if some classes ofp-subgroups ofEhave Π -property inG. Some recent results are extended. [ABSTRACT FROM AUTHOR]

Differential algebraic geometry seeks to extend the results of its algebraic counterpart to objects defined by differential equations. Many notions, such as that of a projective algebraic variety, have close differential analogues but their behavior can vary in interesting ways. Workers in both differential algebra and model theory have investigated the property of completeness of differential varieties. After reviewing their results, we extend that work by proving several versions of a “differential valuative criterion" and using them to give new examples of complete differential varieties. We conclude by analyzing the first examples of incomplete, finite-rank projective differential varieties, demonstrating a clear difference from projective algebraic varieties. [ABSTRACT FROM PUBLISHER]

Let Λ be a non semisimple symmetric artin algebra over a commutative artinian ring R and let repdim(Λ) be its representation dimension. Assume also that Λ is a finite quiver whose connected components are amalgamations of single cycles possibly with multiple arrows, and that ℐ is an admissible ideal in the generalized path algebra ΛΩ. We prove that repdim... ≤ (Λ)+1. [ABSTRACT FROM AUTHOR]

We consider how a standard theorem in algebraic geometry relating properties of a curve with a (Z/2Z)²-action to the properties of its quotients generalizes to results about sets and graphs that admit (Z/2Z)²-actions. [ABSTRACT FROM AUTHOR]

Similarity between the objects has close association with the geometrical structure and details of the objects. Therefore, in this study, a framework was presented based on the geometric criteria, the fuzzy Membership Functions (MF) and human spatial cognition. To increase the efficiency of MF and for the problem to get closer to reality, 1 K, 2 K, 5 K, 10 K, 25 K, 50 K, and 100 K data were used in this framework. The results showed that the degree of spatial similarity is specific to the objects themselves and their geometric structure. No linear or non-linear relationship could be found between scale changes and degree of spatial similarity of the objects, because the geometry of any object independently affects its degree of similarity to other objects in different scales and sources. [ABSTRACT FROM AUTHOR]

The present survey deals with the notion of topological inverse-representation for spaces and groups, a concept introduced by Valentin Poénaru in the 80’s, and largely used in successive works in geometric topology. We recall the origin of this concept as a low-dimensional topology tool for studying compact 3-manifolds and their universal covering spaces. We also illustrate some recent developments in geometric group theory. [ABSTRACT FROM PUBLISHER]

A famous theorem of Hopf [14] states that: Every finite-dimensional real commutative division algebraAis at most two-dimensional. To this day, no elementary proof of Hopf’s theorem is known. In the present paper, we extend this result to more general situation. Indeed, we show that ifAis a real commutative algebraic algebra without divisors of zero, thenAis finite dimensional. Moreover, we prove by a simple algebraic method, there is no four-dimensional real commutative division algebra. Finally, we prove that every third-power associative real algebra, with unit element, without divisors of zero, and algebraic of degree ≠8, is quadratic. This last generalizes previously known results of Diankha et al. [6]. [ABSTRACT FROM PUBLISHER]

For any skewed-gentle algebra, we characterize its indecomposable Gorenstein projective modules explicitly and describe its Cohen–Macaulay Auslander algebra. We prove that skewed-gentle algebras are always Gorenstein, which is independent of the characteristic of the ground field, and the Cohen–Macaulay Auslander algebras of skewed-gentle algebras are also skewed-gentle algebras. [ABSTRACT FROM PUBLISHER]