FROM IRREDUNDANCE TO ANNIHILATION: A BRIEF OVERVIEW OF SOME DOMINATION PARAMETERS OF GRAPHS

Durante los últimos treinta años, el concepto de dominación en grafos ha levantado un interés impresionante. Una bibliografía reciente sobre el tópico contiene más de 1200 referencias y el número de definiciones nuevas está creciendo continuamente. En vez de intentar dar un catálogo de todas ellas, examinamos las nociones más clásicas e importantes (tales como dominación independiente, dominación irredundante, k-cubrimientos, conjuntos k-dominantes, conjuntos Vecindad Perfecta, ...) y algunos de los resultados más significativos.   PALABRAS CLAVES: Teoría de grafos, Dominación.   ABSTRACT During the last thirty years, the concept of domination in graphs has generated an impressive interest. A recent bibliography on the subject contains more than 1200 references and the number of new definitions is continually increasing. Rather than trying to give a catalogue of all of them, we survey the most classical and important notions (as independent domination, irredundant domination, k-coverings, k-dominating sets, Perfect Neighborhood sets, ...) and some of the most significant results.   KEY WORDS: Graph theory, Domination

2D growth processes: SLE and Loewner chains

This review provides an introduction to two dimensional growth processes. Although it covers a variety processes such as diffusion limited aggregation, it is mostly devoted to a detailed presentation of stochastic Schramm-Loewner evolutions (SLE) which are Markov processes describing interfaces in 2D critical systems. It starts with an informal discussion, using numerical simulations, of various examples of 2D growth processes and their connections with statistical mechanics. SLE is then introduced and Schramm's argument mapping conformally invariant interfaces to SLE is explained. A substantial part of the review is devoted to reveal the deep connections between statistical mechanics and processes, and more specifically to the present context, between 2D critical systems and SLE. Some of the SLE remarkable properties are explained, as well as the tools for computing with SLE. This review has been written with the aim of filling the gap between the mathematical and the physical literatures on the subject.Comment: A review on Stochastic Loewner evolutions for Physics Reports, 172 pages, low quality figures, better quality figures upon request to the authors, comments welcom

Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System

Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics

Progress in Commutative Algebra 2

This is the second of two volumes of a state-of-the-art survey article collection which originates from three commutative algebra sessions at the 2009 Fall Southeastern American Mathematical Society Meeting at Florida Atlantic University. The articles reach into diverse areas of commutative algebra and build a bridge between Noetherian and non-Noetherian commutative algebra. These volumes present current trends in two of the most active areas of commutative algebra: non-noetherian rings (factorization, ideal theory, integrality), and noetherian rings (the local theory, graded situation, and interactions with combinatorics and geometry). This volume contains surveys on aspects of closure operations, finiteness conditions and factorization. Closure operations on ideals and modules are a bridge between noetherian and nonnoetherian commutative algebra. It contains a nice guide to closure operations by Epstein, but also contains an article on test ideals by Schwede and Tucker and more

Blow-up algebras in Algebra, Geometry and Combinatorics

[eng] The primary topic of this thesis lies at the crossroads of Commutative Algebra and its interactions with Algebraic Geometry and Combinatorics. It is mainly focused around the following themes: I Defining equations of blow-up algebras. II Study of rational maps via blow-up algebras. III Asymptotic properties of the powers of edge ideals of graphs. We are primarily interested in questions that arise in geometrical or combinatorial contexts and try to understand how their possible answers manifest in various algebraic structures or invariants. There is a particular algebraic object, the Rees algebra (or blow-up algebra), that appears in many constructions of Commutative Algebra, Algebraic Geometry, Geometric Modeling, Computer Aided Geometric Design and Combinatorics. The workhorse and main topic of this doctoral dissertation has been the study of this algebra under various situations. The Rees algebra was introduced in the field of Commutative Algebra in the famous paper [45]. Since then, it has become a central and fundamental object with numerous applications. The study of this algebra has been so fruitful that it is difficult to single out particular results or papers, instead we refer the reader to the books [52] and [53] to wit the “landscape of blow-up algebras”. From a geometrical point of view, the Rees algebra corresponds with the bi-homogeneous coordinate ring of two fundamental objects: the blow-up of a projective variety along a subvariety and the graph of a rational map between projective varieties (see [27, §II.7]). Therefore, the importance of finding the defining equations of the Rees algebra is probably beyond argument. This is a problem of tall order that has occupied commutative algebraists and algebraic geometers, and despite an extensive effort (see [6, 8, 15–19, 30, 37, 39–43, 51]), it remains open even in the case of polynomial rings in two variables. In [10], Chapter 2 of this dissertation, we use the theory of D-modules to describe the defining ideal of the Rees algebra in the case of a parametrization of a plane curve. The study of rational and birational maps is classical in the literature from both an algebraic and geometric point of view, and it goes back to the work of Cremona [20], at least. A relatively new idea, probably first used in [31], is to look at the syzygies of the base ideal of a rational map to determine birationality. This algebraic method for studying rational maps has now become an active research topic (see [7, 22, 23, 28, 29, 38, 44, 46, 47]). In a joint work with Buse´ and D’Andrea [9], Chapter 3 of this dissertation, we introduce a new algebra that we call the saturated special fiber ring, which turns out to be an important tool to analyze the degree of a rational map. Later, in [11], Chapter 4 of this dissertation, we compute the multiplicity of this new algebra in the case of perfect ideals of height two, which, in particular, provides an effective method to determine the degree of a rational map having those ideals as base ideal. Often a good tactic to approach a challenging problem is to go all the way up to a generic case and then find sufficient conditions for the specialization to keep some of the main features of the former. The procedure depends on taking a dramatic number of variables to allow modifying the given data into a generic shape, and usually receives the name of specialization. This method is seemingly due to Kronecker and was quite successful in the hands of Hurwitz ([34]) in establishing a new elegant theory of elimination and resultants. More recent instances where specialization is used are, e.g., [32], [33], [50], [48]. In a joint work with Simis [14], Chapter 5 of this dissertation, we consider the behavior of the degree of a rational map under specialization of the coefficients of the defining linear system. The Rees algebra of the edge ideal of a graph is a well studied object (see [24, 25, 49, 54–57]), that relates combinatorial properties of a graph with algebraic invariants of the powers of its edge ideal. For the Rees algebra of 1 2 YAIRON CID RUIZ the edge ideal of a bipartite graph, in [12], Chapter 6 of this dissertation, we compute the universal Gro¨ bner basis of its defining equations and its total Castelnuovo-Mumford regularity as a bigraded algebra. It is a celebrated result that the regularity of the powers of a homogeneous ideal is asymptotically a linear function (see [21, 36]). Considerable efforts have been put forth to understand the form of this asymptotic linear function in the case of edge ideals (see [1–5, 26, 35]). In a joint work with Jafari, Picone and Nemati [13], Chapter 7 of this dissertation, for bicyclic graphs, i.e. graphs containing exactly two cycles, we characterize the regularity of its edge ideal in terms of the induced matching number and determine the previous asymptotic linear function in special cases. The basic outline of this thesis is as follows. In Chapter 1, we recall some preliminary results and definitions to be used along this work. Then, the thesis is divided in three different parts. The first part corresponds with the theme “ I Defining equations of blow-up algebras” and consists of Chapter 2. The second part corresponds with the theme “ II Study of rational maps via blow-up algebras” and consists of Chapter 3, Chapter 4 and Chapter 5. The third part corresponds with the theme “ III Asymptotic properties of the powers of edge ideals of graphs” and consists of Chapter 6 and Chapter 7. The common thread and main tool in the three parts of this thesis is the use of blow-up algebras. References [1] A. Alilooee and A. Banerjee, Powers of edge ideals of regularity three bipartite graphs, J. Commut. Algebra 9 (2017), no. 4, 441–454. [2] A. Alilooee, S. Beyarslan, and S. Selvaraja, Regularity of powers of unicyclic graphs, Rocky Mountain J. Math. (2018). Advance publication. [3] A. Banerjee, The regularity of powers of edge ideals, J. Algebraic Combin. 41 (2015), no. 2, 303–321. [4] A. Banerjee, S. Beyarslan, and H. T. Ha, Regularity of edge ideals and their powers, arXiv preprint arXiv:1712.00887 (2017). [5] S. Beyarslan, H. T. Ha`, and T. N. Trung, Regularity of powers of forests and cycles, J. Algebraic Combin. 42 (2015), no. 4, 1077–1095. [6] J. A. Boswell and V. Mukundan, Rees algebras and almost linearly presented ideals, J. Algebra 460 (2016), 102–127. [7] N. Botbol, L. Buse´, M. Chardin, S. H. Hassanzadeh, A. Simis, and Q. H. Tran, Effective criteria for bigraded birational maps, J. Symbolic Comput. 81 (2017), 69–87. [8] L. Buse´, On the equations of the moving curve ideal of a rational algebraic plane curve, J. Algebra 321 (2009), no. 8, 2317–2344. [9] L. Buse´, Y. Cid-Ruiz, and C. D’Andrea, Degree and birationality of multi-graded rational maps, ArXiv e-prints (May 2018), available at 1805.05180. [10] Y. Cid-Ruiz, A D-module approach on the equations of the Rees algebra, to appear in J. Commut. Algebra (2017). arXiv:1706.06215. [11] , Multiplicity of the saturated special fiber ring of height two perfect ideals, ArXiv e-prints (July 2018). 1807.03189. [12] , Regularity and Gro¨ bner bases of the Rees algebra of edge ideals of bipartite graphs, Le Matematiche 73 (2018), no. 2, 279–296. [13] Y. Cid-Ruiz, S. Jafari, N. Nemati, and B. Picone, Regularity of bicyclic graphs and their powers, to appear in J. Algebra Appl. (2018). arXiv:1802.07202. [14] Y. Cid-Ruiz and A. Simis, Degree of rational maps via specialization, arXiv preprint arXiv:1901.06599 (2019). [15] T. Cortadellas Ben´ıtez and C. D’Andrea, Rational plane curves parameterizable by conics, J. Algebra 373 (2013), 453–480. [16] , Minimal generators of the defining ideal of the Rees algebra associated with a rational plane parametrization with µ = 2, Canad. J. Math. 66 (2014), no. 6, 1225–1249. [17] , The Rees algebra of a monomial plane parametrization, J. Symbolic Comput. 70 (2015), 71–105. [18] D. Cox, The moving curve ideal and the Rees algebra, Theoret. Comput. Sci. 392 (2008), no. 1-3, 23–36. [19] D. Cox, J. W. Hoffman, and H. Wang, Syzygies and the Rees algebra, J. Pure Appl. Algebra 212 (2008), no. 7, 1787–1796. [20] L. Cremona, Sulle trasformazioni geometriche delle figure piane, Mem. Acad. Bologna 2 (1863), no. 2, 621–630. [21] S. D. Cutkosky, J. Herzog, and N. V. Trung, Asymptotic behaviour of the Castelnuovo-Mumford regularity, Compositio Math. 118 (1999), no. 3, 243–261. MR1711319 [22] A. V. Doria, S. H. Hassanzadeh, and A. Simis, A characteristic-free criterion of birationality, Adv. Math. 230 (2012), no. 1, 390–413. [23] D. Eisenbud and B. Ulrich, Row ideals and fibers of morphisms, Michigan Math. J. 57 (2008), 261–268. Special volume in honor of Melvin Hochster. [24] L. Fouli and K.-N. Lin, Rees algebras of square-free monomial ideals, J. Commut. Algebra 7 (2015), no. 1, 25–54. [25] I. Gitler, C. Valencia, and R. H. Villarreal, A note on the Rees algebra of a bipartite graph, J. Pure Appl. Algebra 201 (2005), no. 1-3, 17–24. [26] H. T. Ha`, Regularity of squarefree monomial ideals, Connections between algebra, combinatorics, and geometry, 2014, pp. 251–276. [27] R. Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. [28] S. H. Hassanzadeh and A. Simis, Plane Cremona maps: saturation and regularity of the base ideal, J. Algebra 371 (2012), 620–652. [29] , Bounds on degrees of birational maps with arithmetically Cohen-Macaulay graphs, J. Algebra 478 (2017), 220–236. [30] J. Hong, A. Simis, and W. V. Vasconcelos, On the homology of two-dimensional elimination, J. Symbolic Comput. 43 (2008), no. 4, 275–292. [31] K. Hulek, S. Katz, and F.-O. Schreyer, Cremona transformations and syzygies., Math. Z. 209 (1992), no. 3, 419–443. [32] C. Huneke and B. Ulrich, Residual intersections, J. Reine Angew. Math. 390 (1988), 1–20. [33] , Generic residual intersections, Commutative algebra (Salvador, 1988), 1990, pp. 47–60. [34] A. Hurwitz, U¨ ber die Tra¨ gheitsformen eines algebraischen Moduls., Annali di Mat. (3) 20 (1913), 113–151 (Italian). [35] A. V. Jayanthan, N. Narayanan, and S. Selvaraja, Regularity of powers of bipartite graphs, Journal of Algebraic Combinatorics (2017May). BRIEF RESUME OF THE PHD THESIS 3 [36] V. Kodiyalam, Asymptotic behaviour of Castelnuovo-Mumford regularity, Proc. Amer. Math. Soc. 128 (2000), no. 2, 407–411. [37] A. Kustin, C. Polini, and B. Ulrich, Rational normal scrolls and the defining equations of Rees algebras, J. Reine Angew. Math. 650 (2011), 23–65. [38] , Blowups and fibers of morphisms, Nagoya Math. J. 224 (2016), no. 1, 168–201. [39] , The bi-graded structure of symmetric algebras with applications to Rees rings, J. Algebra 469 (2017), 188–250. [40] , The equations defining blowup algebras of height three Gorenstein ideals, Algebra Number Theory 11 (2017), no. 7, 1489–1525. [41] K.-N. Lin and C. Polini, Rees algebras of truncations of complete intersections, J. Algebra 410 (2014), 36–52. [42] J. Madsen, Equations of rees algebras of ideals in two variables, ArXiv Mathematics e-prints (2015nov), available at arXiv:1511.04073. [43] S. Morey and B. Ulrich, Rees algebras of ideals with low codimension, Proc. Amer. Math. Soc. 124 (1996), no. 12, 3653–3661. [44] I. Pan and A. Simis, Cremona maps of de Jonquie`res type, Canad. J. Math. 67 (2015), no. 4, 923–941. [45] D. 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Villarreal, Rees algebras of edge ideals, Comm. Algebra 23 (1995), no. 9, 3513–3524. [55] , Rees algebras of complete bipartite graphs, Mat. Contemp. 16 (1999), 281–289. 15th School of Algebra (Portuguese) (Canela, 1998). [56] , Rees algebras and polyhedral cones of ideals of vertex covers of perfect graphs, J. Algebraic Combin. 27 (2008), no. 3, 293–305. [57] , Monomial algebras, Second, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2015

International Congress of Mathematicians: 2022 July 6–14: Proceedings of the ICM 2022

Following the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library

Volume 40, Number 02 (February 1922)

Most Remarkable Pianoforte Recital Ever Given Spirit of Chopin More Reasons Why She Couldn\u27t Hold Her Pupils Remarkable Mind of Camille Saint-Saëns: Passing of the Great French Composer at Advance Age: A Review of his Works as Reflected from Some of His Writings, and From the Writings of His Friends What the Teacher Should Demand Saint-Saëns\u27 Last Public Address Musical Biographical Catechism: Tiny Life Stories of Great Masters What Guido Suffered How I Overcame the Greatest Obstacle in My Career, Symposium Handel\u27s Sensitive Ear Unavoidable Practice Little Lessons from a Master\u27s Workshop Exercises to Prevent Arm Strain Vioin and the Piano in Harmony Study Composite Music Variations on the Pupil\u27s Recital From the Known to the Unknown Difference Between a Sonata and a Symphony Hand and the Keyboard (interview with Arthur Schnabel) Modern Piano Bench Hoary Headed Jazz Mountebank Teachers of the Past8 Why Don\u27t You Count? What Your Musical Success Really Depends Upon: The Pupil\u27s Relation to Success or Failure Secrets of the Success of Great Musicians Musical Sieves and Musical Sponges What is a Monotone? Something Not Generally Included in Music Teaching What Scales Have Done for Me Tirawa\u27s Vengeance Why Study Piano? Music as It is Defined Do You Believe in Preparedness? Changing Musical Perception Sound-Reproducing Machine as a Music Teacher Infallible Nerve Tonichttps://digitalcommons.gardner-webb.edu/etude/1686/thumbnail.jp