Spectra of elements in the group ring of SU(2)

We present a new method for establishing the “gap” property for finitely generated subgroups of SU(2), providing an elementary solution of Ruziewicz problem on S² as well as giving many new examples of finitely generated subgroups of SU(2) with an explicit gap. The distribution of the eigenvalues of the elements of the group ring R[SU(2)] in the N-th irreducible representation of SU(2) is also studied. Numerical experiments indicate that for a generic (in measure) element of R[SU(2)], the “unfolded” consecutive spacings distribution approaches the GOE spacing law of random matrix theory (for N even) and the GSE spacing law (for N odd) as N→∞; we establish several results in this direction. For certain special “arithmetic” (or Ramanujan) elements of R[SU(2)] the experiments indicate that the unfolded consecutive spacing distribution follows Poisson statistics; we provide a sharp estimate in that direction. Received June 1, 1998 / final version received September 8, 1998     

Going-down pairs of commutative rings

Résumé  D'après D. E. Dobbs, Houston J. Math. 23 (1997), 1–11, nous disons que l'anneau (commutatif)A est un anneau-“going-down” siA/P est un domaine-“going-down” pour chaque idéal premier deA. Etant donné une extension,R⊆T, nous disons que (R, T) est une paire d'anneaux-“going-down” (respectivement, une paire “going-down”) siS est un anneau-“going-down” pour chaque anneau tels queR⊆S⊆T (resp., si “going-down” est satisfait par chaque extension d'anneauxA⊆B tels queR⊆A⊆B⊆T). On montre que siR est un anneau de la dimension 0 (au sens de Krull), alors (R, T) est une paire d'anneaux-“going-down” si et seulement sitr.deg. _R/(P∩R) T/P≤1 pour chaque idéal premier minimalP deT. Des résultats partiels sont obtenus quandR n'est pas de dimension 0. En outre, si (R, T) est une paire d'anneaux-“going-down” tel queT ait un seul idéal premier minimal, alors (R, T) est une paire “going-down”. Des résultats dans l'esprit ci-dessus sont également obtenus pour quelques autres types de paires.

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This paper is taken from the author's doctoral dissertation of May 2000, written under the direction of Professor David E. Dobbs of the University of Tennessee, Knoxville.     

Excursion decompositions for SLE and Watts' crossing formula

It is known that Schramm-Loewner Evolutions (SLEs) have a.s. frontier points if κ>4 and a.s. cutpoints if 4<κ<8. If κ>4, an appropriate version of SLE(κ) has a renewal property: it starts afresh after visiting its frontier. Thus one can give an excursion decomposition for this particular SLE(κ) “away from its frontier”. For 4<κ<8, there is a two-sided analogue of this situation: a particular version of SLE(κ) has a renewal property w.r.t its cutpoints; one studies excursion decompositions of this SLE “away from its cutpoints”. For κ=6, this overlaps Virág's results on “Brownian beads”. As a by-product of this construction, one proves Watts' formula, which describes the probability of a double crossing in a rectangle for critical plane percolation.     

Connected and Disconnected Maps

A new relation between morphisms in a category is introduced—roughly speaking (accurately in the categories Set and Top), f ∥ g iff morphisms w:dom(f)→dom(g) never map subobjects of fibres of f non-constantly to fibres of g. (In the algebraic setting replace fibre with kernel.) This relation and a slight weakening of it are used to define “connectedness” versus “disconnectedness” for morphisms. This parallels and generalises the classical treatment of connectedness versus disconnectedness for objects in a category (in terms of constant morphisms). The central items of study are pairs (F,G)({\mathcal F},{\mathcal G}) of classes of morphisms which are corresponding fixed points of the polarity induced by the ∥-relation. Properties of such pairs are examined and in particular their relation to (pre)factorisation systems is analysed. The main theorems characterise:

Complete Upper Angle Around a Point on the Minkowski Plane

The dependence of the complete upper angle in the sense of A. D. Aleksandrov about a point on the Minkowski plane on the form of the “unit circle” (the centrally symmetric convex curve Φ determining the Minkowski metric ρ_Φ) is studied.The complete upper angle is computed in three cases: if Φ is a square, a “cut circle,” or a “rounded rhombus.” Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 299, 2003, pp. 42–53.     

On duality gap in linear conic problems

In their paper “Duality of linear conic problems” Shapiro and Nemirovski considered two possible properties (A) and (B) for dual linear conic problems (P) and (D). The property (A) is “If either (P) or (D) is feasible, then there is no duality gap between (P) and (D)”, while property (B) is “If both (P) and (D) are feasible, then there is no duality gap between (P) and (D) and the optimal values val(P) and val(D) are finite”. They showed that (A) holds if and only if the cone K is polyhedral, and gave some partial results related to (B). Later Shapiro conjectured that (B) holds if and only if all the nontrivial faces of the cone K are polyhedral. In this note we mainly prove that both the “if” and “only if” parts of this conjecture are not true by providing examples of closed convex cone in \mathbbR⁴{\mathbb{R}^{4}} for which the corresponding implications are not valid. Moreover, we give alternative proofs for the results related to (B) established by Shapiro and Nemirovski.     

On the minimal free resolution of finite sets in P N

Here we construct many possible free resolutions fors points inP ⁿ . In suitable ranges we construct configurations of points with “good” minimal free resolution and other configurations for which the difference with respect to a “good” resolution is prescribed in advance.     

The <Emphasis Type="Italic">R</Emphasis>-observability and <Emphasis Type="Italic">R</Emphasis>-controllability of linear differential-algebraic systems

We study the R-controllability (the controllability within the attainability set) and the R-observability of time-varying linear differential-algebraic equations (DAE). We analyze DAE under assumptions guaranteeing the existence of a structural form (which is called “equivalent”) with separated “differential” and “algebraic” subsystems. We prove that the existence of this form guarantees the solvability of the corresponding conjugate system, and construct the corresponding “equivalent form” for the conjugate DAE. We obtain conditions for the R-controllability and R-observability, in particular, in terms of controllability and observability matrices. We prove theorems that establish certain connections between these properties.     

Ploucquet’s “Refutation” of the Traditional Square of Opposition

In the 18th century, Gottfried Ploucquet developed a new syllogistic logic where the categorical forms are interpreted as set-theoretical identities, or diversities, between the full extension, or a non-empty part of the extension, of the subject and the predicate. With the help of two operators ‘O’ (for “Omne”) and ‘Q’ (for “Quoddam”), the UA and PA are represented as ‘O(S) – Q(P)’ and ‘Q(S) – Q(P)’, respectively, while UN and PN take the form ‘O(S) > O(P)’ and ‘Q(S) > O(P)’, where ‘>’ denotes set-theoretical disjointness. The use of the symmetric operators ‘–’ and ‘>’ gave rise to a new conception of conversion which in turn lead Ploucquet to consider also the unorthodox propositions O(S) – O(P), Q(S) – O(P), O(S) > Q(P), and Q(S) > Q(P). Although Ploucquet’s critique of the traditional theory of opposition turns out to be mistaken, his theory of the “Quantification of the Predicate” is basically sound and involves an interesting “Double Square of Opposition”. My thanks are due to Hanno von Wulfen for helpful discussions and for transforming the word-document into a Latex-file.     

Commutative lmc algebras with discrete spectrum

SupposeE is a topological algebra with non-empty spectrum Gel'fand spaceM(E) andE M(E)) the algebra of allC-valued continuous functions onM(E). EndowE M(E) with the topologies “c”, “e” of compact resp. equicontinuous convergence. ThenE M (E) characterizes all unital (commutative) semisimple complete lmc algebras with discrete spectrum, while all unital uniform complete lmc algebras with dispersed spectrum are of the formE(E)). The first result may fail if completeness is dropped. The second one fails if “e” is replaced by “c” even ifE is complete. The part of the work due to the 2nd author was carried out during her 3-month visit at the Institute of Mathematics, Univ. of Münster (Germany) in Spring of 1995. The warm hospitality from Professor Dr. G. Maltese and the financial support from a DAAD grant are thankfully acknowledged.     

On log canonical divisors that are log quasi-numerically positive

Let (X Δ) be a four-dimensional log variety that is projective over the field of complex numbers. Assume that (X, Δ) is not Kawamata log terminal (klt) but divisorial log terminal (dlt). First we introduce the notion of “log quasi-numerically positive”, by relaxing that of “numerically positive”. Next we prove that, if the log canonical divisorK _X+Δ is log quasi-numerically positive on (X, Δ) then it is semi-ample.     

Iterated forcing and normal ideals onω 1

We prove that suitable iteration does not collapse ℵ₁ [and does not add reals], i.e., that in such iteration, certain sealing of maximal antichains of stationary subsets ofω ₁ is allowed. As an application, e.g., we prove from supercompact hypotheses, mainly, the consistency of: ZFC + GCH + “for some stationary setS ⊆ω ₁, {ie345-1}(ω ₁)/(D ω ₁ +S) is the Levy algebra” (i.e., the complete Boolean Algebra corresponding to the Levy collapse Levy (ℵ₀,<ℵ₂) (and we can add “a variant of PFA”) and the consistency of the same, with “Ulam property” replacing “Levy algebra”). The paper assumes no specialized knowledge (if you agree to believe in the semi-properness iteration theorem and RCS iteration). This research was partially supported by the NSF. This paper was largely written during the author’s visit at Cal Tech around the end of April 1985. The author would like to thank M. Foreman, A. Kekris and H. Woodin for their hospitality.     

Existence of a-fold Perfect （v, {5, 8}, 1）-Mendelsohn Designs

Let v be a positive integer and let K be a set of positive integers. A （v, K, 1）-Mendelsohn design, which we denote briefly by （v, K, 1）-MD, is a pair （X, B） where X is a v-set （of points） and B is a collection of cyclically ordered subsets of X （called blocks） with sizes in the set K such that every ordered pair of points of X are consecutive in exactly one block of B. If for all t =1, 2,..., r, every ordered pair of points of X are t-apart in exactly one block of B, then the （v, K, 1）-MD is called an r-fold perfect design and denoted briefly by an r-fold perfect （v, K, 1）-MD. If K = {k） and r = k - 1, then an r-fold perfect （v, （k）, 1）-MD is essentially the more familiar （v, k, 1）-perfect Mendelsohn design, which is briefly denoted by （v, k, 1）-PMD. In this paper, we investigate the existence of 4-fold perfect （v, （5, 8}, 1）-Mendelsohn designs.     

Signed words and permutations, IV: Fixed and pixed points

The flag-major index “fmaj” and the classical length function “ℓ” are used to construct two q-analogs of the generating polynomial for the hyperoctahedral group B _n by number of positive and negative fixed points (resp., pixed points). Specializations of those q-analogs are also derived dealing with signed derangements and desarrangements, as well as several classical results that were previously proved for the symmetric group. To Volker Strehl, a dedication à la Goethe, on the occasion of his sixtieth birthday.     

Cantor set and interpolation

In 1998, Y. Benyamini published interesting results concerning interpolation of sequences using continuous functions ℝ → ℝ. In particular, he proved that there exists a continuous function ℝ → ℝ which in some sense “interpolates” all sequences (x _n ) _n∈ℤ ∈ [0, 1]^ℤ “simultaneously.” In 2005, M.R. Naulin and C. Uzcátegui unified and generalized Benyamini’s results. In this paper, the case of topological spaces X and Y with an Abelian group acting on X is considered. A similar problem of “simultaneous interpolation” of all “generalized sequences” using continuous mappings X → Y is posed. Further generalizations of Naulin-Uncátegui theorems, in particular, multidimensional analogues of Benyamini’s results are obtained.     

Instability of the wet x soap film

For idealized, infinitely thin (“dry”) soap films, an Xis unstable, while for very thick (“wet”) soap films it is minimizing. We show that for soap films of relatively small but positive wetness, the Xis unstable. Full stability diagrams for the constant liquid fraction case and the constant pressure case are generated. Analogous questions about other singularities remain controversial.     

Analysis of space–time discontinuous Galerkin method for nonlinear convection–diffusion problems

The paper presents the theory of the discontinuous Galerkin finite element method for the space–time discretization of a nonstationary convection–diffusion initial-boundary value problem with nonlinear convection and linear diffusion. The problem is not singularly perturbed with dominating convection. The discontinuous Galerkin method is applied separately in space and time using, in general, different space grids on different time levels and different polynomial degrees p and q in space and time dicretization. In the space discretization the nonsymmetric, symmetric and incomplete interior and boundary penalty (NIPG, SIPG, IIPG) approximation of diffusion terms is used. The paper is concerned with the proof of error estimates in “L ²(L ²)”- and “DG”-norm formed by the “L ²(H ¹)”-seminorm and penalty terms. A special technique based on the use of the Gauss–Radau interpolation and numerical integration has been used for the derivation of an abstract error estimate. In the “DG”-norm the error estimates are optimal with respect to the size of the space grid. They are optimal with respect to the time step, if the Dirichlet boundary condition has behaviour in time as a polynomial of degree ≤ q.     

Spectral representation of generalized transition kernels

Let (X, A) be a set with a countably σ-generated “Borel” field of subsets; letW be a “Borel” subset of the product of (X, A) with the real line ℝ and its Borel fieldB; and for eachx∈X let γ _x be a measure on the “slice”W _x={(w, t)∈W:w=x}. It is shown that, under reasonable conditions, the σ-field A⊗B|W can be generated by a real-valued functiong in such a way that, given any measurablef:W→ℝ,g can be chosen to be arbitrarily close tof and so that its “slice-integrals” coincide with those off. This theorem is the first step in a study of monotonic sequences of countably generated σ-fields.     

An anyon model

We construct an infinite-dimensional dynamical Hamiltonian system that can be interpreted as a localized structure (“quasiparticle”) on the plane E ₂. The model is based on the theory of an infinite string in the Minkowski space E _1,3 formulated in terms of the second fundamental forms of the worldsheet. The model phase space H is parameterized by the coordinates, which are interpreted as “internal” (E(2)-invariant) and “external” (elements of T*E ₂) degrees of freedom. The construction is nontrivial because H contains a finite number of constraints entangling these two groups of coordinates. We obtain the expressions for the energy and for the effective mass of the constructed system and the formula relating the proper angular momentum and the energy. We consider a possible interpretation of the proposed construction as an anyon model.     

Provability interpretations of modal logic

We consider interpretations of modal logic in Peano arithmetic (P) determined by an assignment of a sentencev ^* ofP to each propositional variablev. We put (⊥)*=“0 = 1”, (χ → ψ)* = “χ* → ψ*” and let (□ψ)* be a formalization of “ψ)* is a theorem ofP”. We say that a modal formula, χ, isvalid if ψ* is a theorem ofP in each such interpretation. We provide an axiomitization of the class of valid formulae and prove that this class is recursive.