Symbolic Topology Research Articles

Uniform Symbolic Topologies and Hypersurfaces

We study the question of which rings, and which families of ideals, have uniform symbolic topologies. In particular, we show that the uniform symbolic topology property holds for all dimension one primes in any normal complete local domain, provided dimension one primes in hypersurfaces have the uniform symbolic topology property. We also discuss bootstrapping techniques and provide a strong bootstrapping statement in positive characteristic. We apply these techniques to give families of primes in hypersurfaces of positive characteristic which have uniform symbolic topologies.

The uniform symbolic topology property for diagonally F-regular algebras

Let k be a field of positive characteristic. Building on the work of the second named author, we define a new class of k-algebras, called diagonally F-regular algebras, for which the so-called Uniform Symbolic Topology Property (USTP) holds effectively. We show that this class contains all essentially smooth k-algebras. We also show that this class contains certain singular algebras, such as the affine cone over Pkr×Pks, when k is perfect. By reduction to positive characteristic, it follows that USTP holds effectively for the affine cone over PCr×PCs and more generally for complex varieties of diagonal F-regular type.

Equivalence of symbolic and adic topologies

Let [Formula: see text] denote an ideal of a Noetherian ring [Formula: see text] and [Formula: see text] a nonzero finitely generated [Formula: see text]-module. It is shown that if the [Formula: see text]-symbolic topology is equivalent to the [Formula: see text]-adic topology on [Formula: see text], for all [Formula: see text], then the [Formula: see text]-symbolic topology on [Formula: see text] is equivalent to the [Formula: see text]-adic topology on [Formula: see text]. Moreover, we show that if [Formula: see text] consists of a single prime ideal, for all [Formula: see text], then the [Formula: see text]-adic and the [Formula: see text]-symbolic topologies on [Formula: see text] are equivalent. Finally, it is shown that if for every one-dimensional prime ideal [Formula: see text] in [Formula: see text], the [Formula: see text]-adic and the [Formula: see text]-symbolic topologies are equivalent on [Formula: see text], then [Formula: see text] is unmixed and [Formula: see text] has only one element.

Uniform symbolic topologies in abelian extensions

In this paper we prove that, under mild conditions, an equicharacteristic integrally closed domain which is a finite abelian extension of a regular domain has the uniform symbolic topology property.

Uniform symbolic topologies via multinomial expansions

When does a Noetherian commutative ring R R have uniform symbolic topologies on primes–read, when does there exist an integer D > 0 D>0 such that the symbolic power P ( D r ) ⊆ P r P^{(Dr)} \subseteq P^r for all prime ideals P ⊆ R P \subseteq R and all r > 0 r >0 ? Groundbreaking work of Ein-Lazarsfeld-Smith, as extended by Hochster and Huneke, and by Ma and Schwede in turn, provides a beautiful answer in the setting of finite-dimensional excellent regular rings. It is natural to then search for analogues where the ring R R is non-regular, or where the above ideal containments can be improved using a linear function whose growth rate is slower. This manuscript falls under the overlap of these research directions. Working with a prescribed type of prime ideal Q Q inside of tensor products of domains of finite type over an algebraically closed field F \mathbb {F} , we present binomial and multinomial expansion criteria for containments of type Q ( E r ) ⊆ Q r Q^{(E r)} \subseteq Q^r , or even better, of type Q ( E ( r − 1 ) + 1 ) ⊆ Q r Q^{(E (r-1)+1)} \subseteq Q^r for all r > 0 r>0 . The final section consolidates remarks on how often we can utilize these criteria, presenting an example.

Rational singularities and uniform symbolic topologies

Take $(R,\mathfrak{m})$ any normal Noetherian domain, either local or $\mathbb{N}$-graded over a field. We study the question of when $R$ satisfies the uniform symbolic topology property (USTP) of Huneke, Katz, and Validashti: namely, that there exists an integer $D>0$ such that for all prime ideals $P\subseteq R$, the symbolic power $P^{(Da)}\subseteq P^{a}$ for all $a>0$. Reinterpreting results of Lipman, we deduce that when $R$ is a two-dimensional rational singularity, then it satisfies the USTP. Emphasizing the non-regular setting, we produce explicit, effective multipliers $D$, working in two classes of surface singularities in equal characteristic over an algebraically closed field, using: (1) the volume of a parallelogram in $\mathbb{R}^{2}$ when $R$ is the coordinate ring of a simplicial toric surface; or (2) known invariants of du Val isolated singularities in characteristic zero due to Lipman.

Uniform symbolic topologies and finite extensions

We study the behavior of rings with uniform symbolic topologies with respect to finite extensions.

Reading North Korea: An Ethnological Inquiry by Sonia Ryang (review)

Sonia Ryang, Reading North Korea: An Ethnological Inquiry. Cambridge: Harvard University Press, 2012. 244 pp.As title Reading North Korea implies, Sonia Ryang invites readers to approach North Korea as if it were a text. Yet, as Ryang boldly pos- its, less effort has been made to recognize North Korea as a land of human beings, and more effort has been made to advocate for, accuse, or ridicule country (4). Anthropologists, in general, pride themselves on their sci- entific and ethical commitment to years of ethnographic fieldwork in order to interpret societies and people a native viewpoint. Therefore, why not direct these approaches toward North Korea? It is in this sense that Ryang calls for general readers, anthropological scholars, and students in particular to humanize or, in author's term, (re)anthropologize the faceless Koreans (8) by employing ethnological approaches, just as Ruth Benedict did among Japanese during World War II. The purpose of book is to comprehend North Korean logic, structure, and operative mode of everyday life (3). Ryang's methods include reading and analyzing select novels and films that develop themes of love, war, and a sense of self that, she suggests, ground North Korean human relations.The book consists of five interconnected chapters that develop proj- ect of humanizing what is often infamously described as world's most isolated country. In introduction, A Journey into Abyss, Ryang stresses how her stance diverges existing popular opinions that fig- ure North Korea as either a neo-Confucian or a guerilla state (cf. Lee 1976, Kwon and Chung 2012, Wada 2002, Cumings 2004). Instead of labeling country with concepts already familiar to outsiders, Ryang stresses critical transformation of Kim Il Sung, Supreme of North Korea, in symbolic topology of 1970s and 1980s. The shift from a theoretico-ideological leader into an ethico-spiritual one (25) is essen- tial to understanding relationship between leaders and people. The leaders, Kim Il Sung to Kim Jong Il and to Kim Jong Un, cannot be compared or interchanged. In particular, Kim Il Sung's position as Supreme is permanent, while position of his successor Kim Jong Il as is not. This point is reminiscent of that raised by Heonik Kwon and Byung-Ho Chung (2012) on resilience of charismatic leadership in North Korea. Namely, Kwon and Chung investigate not characteristics of charismatic leadership per se, but ways by which charisma carries over decades and is further inherited by next genera- tions. Using Clifford Geertz's concept of theater state, Kwon and Chung pay attention to varied yet coherent state-run cultural performances, products, and practices that are ritualized and serve to routinize Kim Il Sung's personal charismatic power in form of hereditary succession. Returning to Ryang's standpoint, however, Kim Il Sung's supreme power is not transferred to his successors but remains solely in him and with people eternally. In other words, Ryang asserts that Kim Jong Il and current leader Kim Jong Un cannot replace Kim Il Sung's image and power but can only replicate it (or pretend to). Sanctification of Kim Il Sung is thus seen as a politico-cultural project and historical process accentuated in 1970s and throughout 1980s by which the Great Leader came to be embed- ded in, and thus determine, people's everyday human relations. The subsequent chapters continue to explore love, war, and a sense of self that Ryang claims are essential aspects of North Korean society.Chapter 1, Love, examines how North Korean exemplary love is ani- mated in an eternal triangle between men and women (i.e., people) and Great Kim Il Sung, whose position is at apex of triangle. Readers will enjoy main storylines of selected novels that Ryang suc- cinctly describes and then further analyzes to show how they demonstrate nature of North Korean love, which includes a core revolutionary spirit and sensitivity for a greater self and nation-state. …

The set of stable switching sequences for discrete-time linear switched systems

In this paper we study the characterization of the asymptotical stability for discrete-time switched linear systems. We first translate the system dynamics into a symbolic setting under the framework of symbolic topology. Then by using the ergodic measure theory, a lower bound estimate of Hausdorff dimension of the set of asymptotically stable sequences is obtained. We show that the Hausdorff dimension of the set of asymptotically stable switching sequences is positive if and only if the corresponding switched linear system has at least one asymptotically stable switching sequence. The obtained result reveals an underlying fundamental principle: a switched linear system either possesses uncountable numbers of asymptotically stable switching sequences or has none of them, provided that the switching is arbitrary. We also develop frequency and density indexes to identify those asymptotically stable switching sequences of the system.

Almost Sure Stability of Discrete-Time Switched Linear Systems: A Topological Point of View

In this paper, we study the stability of discrete-time switched linear systems via symbolic topology formulation and the multiplicative ergodic theorem. A sufficient and necessary condition for $\mu_A$-almost sure stability is derived, where $\mu_A$ is the Parry measure of the topological Markov chain with a prescribed transition (0,1)-matrix A. The obtained $\mu_A$-almost sure stability is invariant under small perturbations of the system. The topological description of stable processes of switched linear systems in terms of Hausdorff dimension is given, and it is shown that our approach captures the maximal set of stable processes for linear switched systems. The obtained results cover the stochastic Markov jump linear systems, where the measure is the natural Markov measure defined by the transition probability matrix. Two examples are provided to illustrate the theoretical outcomes of the paper.

Generating Feminine Discourse in Boccaccio's Decameron : The "Valle delle donne" as Julia Kristeva's chora

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY GENERATING FEMININE DISCOURSE IN BOCCACCIO’S DECAMERON: THE “VALLE DELLE DONNE” AS JULIA KRISTEVA’S CHORA NEAL MCTIGHE AT the end of Day VI, the seven ladies of the brigata depart for the uniquely feminine “Valle delle donne,” which is perhaps the most beautiful and harmonious locus amoenus in all of the Decameron. The “Valle delle donne,” however, is more than a mere ornamental space, and more than solely a locus amoenus.1 It is, I sustain, a pre-symbolic space that generates and empowers the feminine discourse that defines Day VII. The “Valle delle donne” lends Day VII a certain symbolic topology inside of which “si ragiona delle beffe, le quali o per amore o per salvamento di loro le donne hanno già fatte a’ suoi mariti, senza essersene avveduti o sì” (VII). In an effort to expose this locus amoenus as a presymbolic space, this paper explores the “Valle delle donne” through the lens offered by Julia Kristeva’s notion of the chora – an approach that has not yet been applied to a reading of Boccaccio’s “Valle delle donne.” The “Valle delle donne,” I believe, resembles Kristeva’s concept because both are pre-symbolic spaces in which energy and meaning are generated . Just as the semiotic chora is linked to the world of the symbolic, the “Valle delle donne” is likewise linked to the ten novellas that make up Day VII. That is, although the valley is not present in any of the stories of Day VII, it serves, however, to frame and determine the nature of the symbolic space in which the stories are told. In each of the ten novellas of Day VII, therefore, is recognizable the presence of an empowered 1 For a detailed analysis of the “Valle delle donne” as a model of the tradition of the locus amoenus, see Raja. For a discussion on the valley as an ornamental space, see Cerisola. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY 41 feminine discourse – akin to the day’s theme – conceived in the Kristevan chora that is the “Valle delle donne.” What exactly is the chora? In the section entitled “The Semiotic Chora Ordering the Drives” of Revolution in Poetic Language (1974), a profound meditation as to how poetic language is constituted in speaking subjects, Julia Kristeva proposes an answer to this question. To begin, Kristeva borrows the term chora from Plato’s Timaeus. In this work Plato likens the chora to the figure of a mother – one who nourishes and cares for another living being. As commentator Miglena Nikolchina stresses, however, Kristeva’s use of the concept “is a matter not of a simple transfer from Plato, but of a radical translation” (164). In this light, Kristeva’s “radical translation” is built around two points of agreement with Plato: 1) the chora is generative; and 2) it is an indeterminable and mutable space. Building upon this, Kristeva defines the chora as a space that “precedes evidence, verisimilitude, spatiality, and temporality” (26). Moreover, she claims that the chora is “a nonexpressive totality formed by the drives and their stases in a motility that is as full of movement as it is regulated” (25). As such, “one can situate the chora and, if necessary, lend it a topology, but one can never give it axiomatic form” (26).2 Kristeva thereby claims that although one must use signifiers to express the notion of the chora, it is of great importance to note that the chora precedes the very linguistic sign that it determines and shapes. This is achieved because the chora is, as Kristeva elaborates, the “kinetic functional stage of the semiotic” (27; her emph.). The semiotic chora transmits to the symbolic sphere its own “semiotic articulations ” through “biological code or physiological ‘memory’” (29). One must keep in mind that although the symbolic sphere and its antecedent chora are distinct, once a speaking subject has been constituted, the two establish a mutual dependency. This dependency is formed because “[o]ur discourse – all discourse – moves with and against the chora in the sense that it simultaneously depends upon and refuses it” (26). The 42 ROMANCE NOTES 2 It is important to note Kristeva’s use of the term topology instead of topography...

On the symbolic topology of an ideal

For an ideal I in a Noetherian ring, it is proved that the I-adic and the I-symbolic topologies are equivalent if and only if the only essential prime divisors of I are its minimal associated primes. We investigate rings where these two topologies are linearly equivalent for all ideals. Locally unmixed rings are characterized as those Noetherian rings in which these two topologies are equivalent for every ideal of the principal class. An example is given to show that two results given by O. Zariski and P. Samuel (“Commutative Algebra,” Vol. II, p. 274, Van Nostrand, Princeton, NJ. 1960) in this area are not true as stated.

On ideals whose adic and symbolic topologies are linearly equivalent

Ideals whose adic and symbolic topologies are linearly equivalent are characterized in terms of analytic spread and u-essential prime divisors. Using this characterization, under certain conditions on a Noetherian ring R and an ideal I of R it is shown that the I-adic and the I-symbolic topologies are linearly equivalent iff gr( I,R) red is a domain, and locally unmixed rings are characterized as those rings in which the adic and the symbolic topologies of every ideal of the principal class are linearly equivalent.

On linearity equivalent ideal topologies

A new ring transpormation T ( bA ) (for each non-nilpotent b in a Noetherian ring A ) is introduced and it is shown that T ( bA ) is always a finite A -module and is the largest transformation of its type that has this property. Then, using this transformation on the element u in the Rees ringR( A, I ) of A with respect to an ideal I in A , it is shown that the I-adic topology on A is linearly equivalent to the u -symbolic topology on A ; that is, there exists k ⩾0 such that I n ⊆I [n] ⊆ n − k for all n ≥ k , where I [ n ] = u n T ( u R( A,I ))∩ A . Several corollaries of this are proved, and Cohen-Macaulay rings are characterized in terms of the filtrations I { [ n ] ; n ≥0}, where I is an ideal of the principal class in A .

Symbolic Powers of Prime Ideals and Their Topology

For a prime ideal $P$ of a commutative Noetherian ring $R$ a necessary and sufficient condition is given to determine when the $P$-adic topology is equivalent, resp. linearly equivalent, to the $P$-symbolic topology. The last means that the symbolic Rees ring is a finitely generated module over the ordinary Rees ring of $P$. Then it is considered when the integral closure of all the powers of $P$ are primary.