Holomorphic Extensions in Smooth Toric Surfaces

SUMMARYAlgebraic methods in connection with classical multidimensional geometry have proven to be very efficient in the computation of direct and inverse kinematics of mechanisms as well as the explanation of strange, pathological behavior. In this paper, we give an overview of the results achieved within the last few years using the algebraic geometric method, geometric preprocessing, and numerical analysis. We provide the mathematical and geometrical background, like Study's parametrization of the Euclidean motion group, the ideals belonging to mechanism constraints, and methods to solve polynomial equations. The methods are explained with different examples from mechanism analysis and synthesis.

Let S be a smooth surface over the complex number field r An a u t o m o r p h i s m q~ of S is cohomologically trivial (resp. numerically trivial) if the induced a u t o m o r p h i s m q~* of the cohomology ring H * (S, 7/) (resp. H * (S, I1~)) is trivial. We denote by A o (S) (resp. A (S)) the quot ient o f the group o fcohomolog ica l ly (resp. numerically) trivial au tomorph i sms of S by its connected component . I t is known tha t A (S) is a finite g roup i f S is Kiihler, [4, 3]. A (S) is even trivial, for example, in the case of ra t ional surfaces, abelian surfaces and K3 surfaces, [9, 5]. In this article, we shall s tudy A (S) for an Enriques surface S. In this case, it is no more true that Ao (S) is always trivial. So far two examples are known.

We investigate one-dimensional ‘generalized convolution behaviors’ (gen. beh.) that comprise differential and delay-differential behaviors in particular. We thus continue work of, for instance, Brethé, van Eijndhoven, Fliess, Gluesing-Luerssen, Habets, Loiseau, Mounier, Rocha, Vettori, Willems, Yamamoto, Zampieri of the last twenty-five years. The signal space for these behaviors is the space E of smooth complex-valued functions on the real line. The ring of operators is the commutative integral domain E′ of distributions with compact support with its convolution product that acts on E by a variant of the convolution product and makes it an E′-module. Both E and E′ carry their standard topologies. Closed E′-submodules of finite powers of E were introduced and studied by Schwartz already in 1947 under the name ‘invariant varieties’ and are called gen. beh. here. A gen. beh. is called a behavior if it can be described by finitely many convolution equations. The ring E′ is not noetherian and therefore the standard algebraic arguments from one-dimensional differential systems theory have to be completed by methods of topological algebra. Standard constructions like elimination or taking (closed) images of behaviors may lead to gen. beh. and therefore the consideration of the latter is mandatory. It is not known whether all gen. beh. are indeed behaviors, but we show that many of them are, in particular all autonomous ones. The E′-module E is neither injective nor a cogenerator and, in particular, does not admit elimination in Willems’ sense. But the signal submodule PE of all polynomial-exponential signals is injective for finitely generated modules and thus admits elimination. This is a useful replacement and approximation of the injectivity of E since the polynomial-exponential part of any gen. beh. is dense in it. We also describe a useful replacement of the cogenerator property and thus establish a strong relation between convolution equations and their solution spaces. Input/output structures of gen. beh. exist and are used to prove that also many nonautonomous generalized behaviors are indeed behaviors. The E′-torsion elements of E, i.e., the smooth functions which satisfy at least one nonzero convolution equation, are called ‘mean-periodic functions’ and were studied by many outstanding analysts. Their results are significant for gen. beh.

In the image analysis via the general theory of moments, some shortcomings were discovered. The major disadvantages of the method of moments are that although the first few moments convey significant information for simple objects, they fail to do so for more complicated objects. The computed invariant moments are expected not to be strictly invariant in the presence of noise, and the computation of moment invariants is time-consuming. In this paper, an effective image analysis method based on the algebraic method is presented. First, a new coordinate transformation from an object image to an invariant matrix is proposed. The invariant matrix representation is independent of image translation, scaling, and rotation. On the basis of invariant matrix, a robust algebraic recognition method based on the projective image is developed. The projective coordinates of projective image on feature images are used as the feature vectors which represent the inherent attributes of objects. In order to test the efficiency of our method, it is used to solve the recognition problem of two-dimensional aircraft models. Theoretical and experimental results show that our method is very reliable for image analysis and the extracted algebraic features are invariant to image translation, scaling, and rotation.

This volume presents an elaborated version of lecture notes for two advanced courses: (Re)Emerging methods in Commutative Algebra and Representation Theory and Building Bridges Between Algebra and Topology, held at the CRM in the spring of 2015. Homological algebra is a rich and ubiquitous area; it is both an active field of research and a widespread toolbox for many mathematicians. Together, these notes introduce recent applications and interactions of homological methods in commutative algebra, representation theory and topology, narrowing the gap between specialists from different areas wishing to acquaint themselves with a rapidly growing field. The covered topics range from a fresh introduction to the growing area of support theory for triangulated categories to the striking consequences of the formulation in the homotopy theory of classical concepts in commutative algebra. Moreover, they also include a higher categories view of Hall algebras and an introduction to the use of idempotent functors in algebra and topology.

Covering an exceptional range of topics, this text provides a unique overview of the Maurer—Cartan methods in algebra, geometry, topology, and mathematical physics. It offers a new conceptual treatment of the twisting procedure, guiding the reader through various versions with the help of plentiful motivating examples for graduate students as well as researchers. Topics covered include a novel approach to the twisting procedure for operads leading to Kontsevich graph homology and a description of the twisting procedure for (homotopy) associative algebras or (homotopy) Lie algebras using the biggest deformation gauge group ever considered. The book concludes with concise surveys of recent applications in areas including higher category theory and deformation theory.

Respiratory motion is a major concern in cone-beam (CB) computed tomography (CT) of the thorax. It causes artifacts such as blur, streaks, and bands, in particular when using slow-rotating scanners mounted on the gantry of linear accelerators. In this paper, we compare two approaches for motion-compensated CBCT reconstruction of the thorax. The first one is analytic; it is heuristically adapted from the method of Feldkamp, Davis, and Kress (FDK). The second one is algebraic: the system of linear equations is generated using a new algorithm for the projection of deformable volumes and solved using the Simultaneous Algebraic Reconstruction Technique (SART). For both methods, we propose to estimate the motion on patient data using a previously acquired 4-D CT image. The methods were tested on two digital and one mechanical motion-controlled phantoms and on a patient dataset. Our results indicate that the two methods correct most motion artifacts. However, the analytic method does not fully correct streaks and bands even if the motion is perfectly estimated due to the underlying approximation. In contrast, the algebraic method allows us full correction of respiratory-induced artifacts.

We give an appropriate extension of the concept of “tower of surjections” to arbitrary inverse systems. We introduce a natural closed model structure (in the sense of D. Quillen) on the category of pro-(Simplicial Abelian Groups) and interpret our condition as the definition of fibrant object.

SAMUEL EILENBERG1. Introduction.The title of this article requires some explanation.The term "abstract algebra" was used to indicate that we shall deal with purely algebraic objects like groups, algebras and Lie algebras rather than topological groups, topological algebras, and so on.The method of study is also purely algebraic but is the replica of an algebraic process which has been widely used in topology, thus the words "toplogical methods" could be replaced by "algebraic methods suggested by algebraic topology."These purely algebraic theories do, however, have several applications in topology.The algebraic process borrowed from topology is the following.Consider a sequence of abelian groups {C q ) and homomorphisms 8,such that ôô = 0.In each group C q two subgroups are distinguished Z q = kernel of 8:C q -* 0 +1 , B q = image of ô:C*~l ~» C q

We discuss the potential use of an algebraic method to compute the value of the solar axial dipole moment at solar minimum, widely considered to be the most reliable precursor of the activity level in the next solar cycle. The method consists of summing up the ultimate contributions of individual active regions to the solar axial dipole moment at the end of the cycle. A potential limitation of the approach is its dependence on the underlying surface flux transport (SFT) model details. We demonstrate by both analytical and numerical methods that the factor relating the initial and ultimate dipole moment contributions of an active region displays a Gaussian dependence on latitude with parameters that only depend on details of the SFT model through the parameter η/Δu where η is supergranular diffusivity and Δu is the divergence of the meridional flow on the equator. In a comparison with cycles simulated in the 2 × 2D dynamo model we further demonstrate that the inaccuracies associated with the algebraic method are minor and the method may be able to reproduce the dipole moment values in a large majority of cycles.

We investigate records in a growing sequence of identical and independently distributed random variables. The record equals the largest value in the sequence, and our focus is on the increment, defined as the difference between two successive records. We investigate sequences in which all increments decrease monotonically, and analyze the case where the random variables are drawn from a uniform distribution with compact support. We find that the fraction IN of sequences that exhibit this property decays algebraically with sequence length N, namely IN ∼ N−ν as N → ∞, and obtain the exponent ν = 0.317 621… using analytic methods. We also study the record distribution and the increment distribution. Whereas the former is a narrow distribution with an exponential tail, the latter is broad and has a power-law tail characterized by the exponent ν. Empirical analysis of records in the sequence of waiting times between successive earthquakes is consistent with the theoretical results.

Stone-type duality theorems, which relate algebraic and relational/topological models, are important tools in logic because -- in addition to elegant abstraction -- they strengthen soundness and completeness to a categorical equivalence, yielding a framework through which both algebraic and topological methods can be brought to bear on a logic. We give a systematic treatment of Stone-type duality for the structures that interpret bunched logics, starting with the weakest systems, recovering the familiar BI and Boolean BI (BBI), and extending to both classical and intuitionistic Separation Logic. We demonstrate the uniformity and modularity of this analysis by additionally capturing the bunched logics obtained by extending BI and BBI with modalities and multiplicative connectives corresponding to disjunction, negation and falsum. This includes the logic of separating modalities (LSM), De Morgan BI (DMBI), Classical BI (CBI), and the sub-classical family of logics extending Bi-intuitionistic (B)BI (Bi(B)BI). We additionally obtain as corollaries soundness and completeness theorems for the specific Kripke-style models of these logics as presented in the literature: for DMBI, the sub-classical logics extending BiBI and a new bunched logic, Concurrent Kleene BI (connecting our work to Concurrent Separation Logic), this is the first time soundness and completeness theorems have been proved. We thus obtain a comprehensive semantic account of the multiplicative variants of all standard propositional connectives in the bunched logic setting. This approach synthesises a variety of techniques from modal, substructural and categorical logic and contextualizes the "resource semantics" interpretation underpinning Separation Logic amongst them.

Dear SIAGA Reader, It is a great pleasure and privilege to welcome you to SIAM Journal on Applied Algebra and Geometry (SIAGA). This journal, with the help of its editorial board, contributors, and readers and the SIAM staff, follows in the outstanding tradition of high-quality SIAM journals. We anticipate that it will become the leading journal dedicated to algebraic, geometric, and topological methods that have strong connection to applications. The journal began accepting submissions in March 2016. By December, it had received 90 manuscripts submitted for publication. Of these, you will find the first nine papers of the initial all-electronic batch published in February 2017. This first group marks the point at which we have established a regular rate of acceptances, building the foundation for a bright future. SIAGA came to exist thanks to the tremendous effort of numerous people. The impetus came from the SIAM Activity Group on Algebraic Geometry, which had its inaugural meeting in Raleigh, North Carolina, in October 2011. Following the second meeting, at Fort Collins, Colorado, in August 2013, a committee was formed to discuss the idea of a new journal. Under the leadership of Frank Sottile and Thorsten Theobald, a proposal was written and then rewritten numerous times. We had strong support from the Journal Committee and SIAM's Board and Council, and the journal received its approval in December 2015. All our invitations to serve on the editorial board were accepted, and SIAGA is now led by a terrific team composed of three Corresponding Editors and 26 Associate Editors. They are doing an amazing job in handling the growing number of submissions in a timely manner. The SIAM staff is a fantastic asset. In particular, Heather Blythe, Mitch Chernoff, and Frank Kunkle have been critical sources of support. The seven cover pictures of SIAGA have been an inspiration for all of us, and we thank Anna Seigal for her SIAM News articles about the mathematics that is expressed in these diagrams. Please enjoy reading the inaugural SIAGA articles. We look forward to receiving your high-quality submissions in the future.

Combinatorial Optimization is an active research area that developed from the rich interaction among many mathematical areas, including combinatorics, graph theory, geometry, optimization, probability, theoretical computer science, and many others. It combines algorithmic and complexity analysis with a mature mathematical foundation and it yields both basic research and applications in manifold areas such as, for example, communications, economics, traffic, network design, VLSI, scheduling, production, computational biology, to name just a few. Through strong inner ties to other mathematical fields it has been contributing to and benefiting from areas such as, for example, discrete and convex geometry, convex and nonlinear optimization, algebraic and topological methods, geometry of numbers, matroids and combinatorics, and mathematical programming. Moreover, with respect to applications and algorithmic complexity, Combinatorial Optimization is an essential link between mathematics, computer science and modern applications in data science, economics, and industry.

Algebraic, Geometric, and Topological Methods in Optimization