Natural Factors of the Medvedev Lattice Capturing IPC

Skvortsova showed that there is a factor of the Medvedev lattice which captures intuitionistic propositional logic (IPC). However, her factor is unnatural in the sense that it is constructed in an ad hoc manner. We present a more natural example of such a factor. We also show that for every non-trivial factor of the Medvedev lattice its theory is contained in Jankov's logic, the deductive closure of IPC plus the weak law of the excluded middle. This answers a question by Sorbi and Terwijn

Intuitionistic logic and Muchnik degrees

We prove that there is a factor of the Muchnik lattice that captures intuitionistic propositional logic. This complements a now classic result of Skvortsova for the Medvedev lattice

Cartan-Geometric Approaches to Submaximally Symmetric Ordinary Differential Equations

This thesis is concerned with a symmetry classification problem for ordinary differential equations (ODEs) that dates back to Sophus Lie. We focus on higher order ODEs, i.e. scalar ODEs of order greater than or equal to 4 or vector ODEs of order greater than or equal to 3, up to contact transformations. The maximal contact symmetry algebra dimensions for these ODEs are known. We determine for all higher order ODEs the submaximal (i.e. next largest realizable) contact symmetry dimensions. Using the known contact fundamental (generalized Wilczynski or C-class) invariants for higher order ODEs, we also determine submaximal symmetry dimensions for several classes of the ODEs that are contact invariant. Moreover, we give a complete local classification of all submaximally symmetric vector ODEs of C-class, i.e. ODEs with symmetry dimensions realizing submaximal symmetry dimensions that are characterized by the vanishing of all generalized Wilczynski invariants. Our results refine the classical results for scalar ODEs, and also provide generalizations of those results to vector ODEs

Decomposition and Structure theorems for Garside-like groups with modular lattice structure

Despite being a vast generalization of Garside groups, right $\ell$-groups with noetherian lattice structure and strong order unit share a lot of the properties of Garside groups. In the present work, we prove that every modular noetherian right $\ell$-group with strong order unit decomposes as a direct product of beams, which are sublattices that correspond to the directly indecomposable factors of the strong order interval. Furthermore, we show that the beams of dimension $\delta \geq 4$ can be coordinatized by the $R$-lattices in $Q^{\delta}$, where $Q$ is a noncommutative discrete valuation field with valuation ring $R$. In particular, this gives a precise description of a very big family of modular Garside groups.Comment: 39 pages, Comments welcome

Reset'i katsumus: Barack Obama ja Dmitri Medvedevi Venemaa ja USA viidete diskursusanalüüs aastatel 2008–2012

Laclau ja Mouffe’i (1985) teooria kohaselt peab poliitiline jõud tänapäeval võimu saavutamiseks osaliselt siduma tühja tähistaja, mis on hetkel ühiskondlikult ja poliitiliselt oluline (nt demokraatia), teatud sõlmpunktidega (nt õigusriik ja inimõigused) nii, et tunduks, et sellel hegemoonilisel artikulatsioonil puuduvad alternatiivid. Selleks aga, et hegemoonia püsiks, peab hegemoonilist artikulatsiooni üha uuesti ja erineval viisil taasesitama, et konkureerida teiste poliitiliste jõududega, kes pakuvad välja selle ühiskondlikult tähtsa tühja tähistaja alternatiivseid artikulatsioone. Poliitiline retoorika keskendubki hegemooniliste artikulatsioonide edendamisele, kasutades erinevate kuulajaskondade veenmiseks erinevaid käike. Poliitilist retoorikat aga piiravad riigi ajaloo käigus kujunenud diskursused, mis mõjutavad riigi identiteeti ja suhteid teiste riikidega. Näiteks on Venemaa ja USA poliitikutel raske edendada stabiilset partnerlusretoorikat teise riigi kohta, kuna nii Venemaa kui ka USA kultuuris on teist riiki kaua konstrueeritud oma riigi identiteedi jaoks tähtsa Teisena. Lisaks ajaloolistele diskursustele teevad poliitilise retoorika ebastabiilseks ka tühja tähistaja muud artikulatsioonid, millega poliitik on varjatult dialoogis ja millest ta püüdleb vabaneda, kuid mis siiski tihtipeale hägustavad tema retoorikat. Käesolev väitekiri oligi suunatud selliste ebastabiilsuste ja diskursiivsete võitluste tuvastamisele Barack Obama ja Dmitri Medvedevi lausungites USA ja Venemaa vaheliste suhete kohta aastatel 2008–2012, perioodil, mil riigid proovisid omavahelisi suhteid normaliseerida. Diskursusanalüüs, mis ühendas Laclau ja Mouffe’i diskursusteooria kriitilise diskursusanalüüsi meetoditega (deiksis ja oletused), näitas muuhulgas seda, kuidas Obama saavutas oma retoorikaga hegemoonia USA ja Venemaa suhetes tollel ajal olulise tähenduse fikseerimise sõlmpunkti, suhete taaskäivitamise (reset), üle. Samuti näitas analüüs, et suhete taaskäivitamise diskursus oli lõppkokkuvõttes nihestatud, sest Obama ei suutnud vastu võtta Medvedevi hegemoonilist artikulatsiooni, mille keskmes oli USA ja Venemaa võrdne koostöö julgeoleku küsimustes, kuna USA Kesk- ja Ida-Euroopa liitlased artikuleerisid sellist koostööd USA nõrgenenud pühendumusega nende julgeoleku tagamisele. Medvedev omakorda ei suutnud nõustuda Obama hegemoonilise artikulatsiooniga, mis rõhutas Venemaa avatust läänele, kuna Medvedevi Euroopa julgeolekulepingu projekt, mis andnuks Venemaale keskse rolli Euroopa julgeolekus, lükati tagasi, samal ajal kui raketitõrjekilp, mida Venemaa fikseeris ohuga oma riiklikule julgeolekule, loodi hoolimata Venemaa tugevast vastuseisust.According to Laclau and Mouffe (1985), hegemony is achieved when a political force succeeds in fixing a socially important empty signifier (e.g., democracy) with specific nodal points (e.g., rule of law and human rights), so that there appear to be no alternatives to this hegemonic articulation. However, such a fixation can only be temporary. Since hegemony is inherently unstable, the hegemonic articulation needs to be constantly reproduced in various ways, to compete with other political forces articulating this empty signifier differently, in a struggle for political power. This is one of the core principles of political rhetoric. Political rhetoric is, however, limited by discourses that have evolved throughout the history of a state and that influence its relations with other states. For instance, for politicians in Russia and the U.S, it is hard to maintain a stable partnership rhetoric about the other country because in both the other has been an important Other in the country’s identity construction for a long time. Political rhetoric is often undermined from within by other articulations, with which a speaker is in an implicit dialog, and which he/she seeks to suppress, but which often loom large at the background of what is being said. This dissertation detects exactly such instabilities and discursive struggles in the utterances of Barack Obama and Dmitry Medvedev on U.S.–Russian relations from 2008–2012, in the period when the two countries attempted to improve their dialog. Discourse analysis, combining analytical concepts from Laclau and Mouffe’s theory with the linguistic tools from critical discourse analysis (indexicality and assumptions) performed in the dissertation demonstrated, among other things, how Obama, through his rhetoric, achieved hegemony over reset, a significant nodal point in U.S.–Russian relations at that time. The analysis also showed that the reset discourse, although successful to some degree, was ultimately dislocated because Obama could not accept Medvedev’s hegemonic articulation of reset with Russia’s and U.S.’s equal cooperation on security, since the U.S.’s CEE allies articulated such cooperation with the U.S.’s weakened commitment to protecting their security. Medvedev, in turn, could not accept Obama’s hegemonic articulation of reset with Russia’s openness to the West, since Medvedev’s project of the European Security Treaty, through which Russia would play a central role in European security, was rejected, while missile defense shield, fixed with a threat to Russia’s national security, was being, in a reviewed form, implemented despite Russia’s strong opposition.https://www.ester.ee/record=b529246

Quantum chaos, thermalization, and entanglement generation in real-time simulations of the Banks-Fischler-Shenker-Susskind matrix model

We study numerically the onset of chaos and thermalization in the Banks-Fischler-Shenker-Susskind (BFSS) matrix model with and without fermions, considering Lyapunov exponents, entanglement generation, and quasinormal ringing. We approximate the real-time dynamics in terms of the most general Gaussian density matrices with parameters which obey self-consistent equations of motion, thus extending the applicability of real-time simulations beyond the classical limit. Initial values of these Gaussian density matrices are optimized to be as close as possible to the thermal equilibrium state of the system. Thus attempting to bridge between the low-energy regime with a calculable holographic description and the classical regime at high energies, we find that quantum corrections to classical dynamics tend to decrease the Lyapunov exponents, which is essential for consistency with the Maldacena-Shenker-Stanford bound at low temperatures. The entanglement entropy is found to exhibit an expected "scrambling" behavior-rapid initial growth followed by saturation. At least at high temperatures the entanglement saturation time appears to be governed by classical Lyapunov exponents. Decay of quasinormal modes is found to be characterized by the shortest timescale of all. We also find that while the bosonic matrix model becomes nonchaotic in the low-temperature regime, for the full BFSS model with fermions the leading Lyapunov exponent, entanglement saturation time, and decay rate of quasinormal modes all remain finite and nonzero down to the lowest temperatures