The Essential Dimension of Stacks of Parabolic Vector Bundles over Curves

AbstractWe characterize all fields of definition for a given coherent sheaf over a projective scheme in terms of projective modules over a finite-dimensional endomorphism algebra. This yields general results on the essential dimension of such sheaves. Applying them to vector bundles over a smooth projective curveC, we obtain an upper bound for the essential dimension of their moduli stack. The upper bound is sharp if the conjecture of Colliot-Thélène, Karpenko and Merkurjev holds. We find that the genericity property proved for Deligne–Mumford stacks by Brosnan, Reichstein and Vistoli still holds for this Artin stack, unless the curveCis elliptic.

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Upper bounds for the essential dimension of the moduli stack of SL n -bundles over a curve

Transformation Groups ◽

10.1007/s00031-009-9069-6 ◽

2009 ◽

Vol 14 (4) ◽

pp. 747-770 ◽

Cited By ~ 1

Author(s):

A. Dhillon ◽

N. Lemire

Keyword(s):

Upper Bounds ◽

Essential Dimension ◽

Moduli Stack

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The Heine-Borel theorem in extended basic logic

Journal of Symbolic Logic ◽

10.2307/2268972 ◽

1949 ◽

Vol 14 (1) ◽

pp. 9-15 ◽

Cited By ~ 5

Author(s):

Frederic B. Fitch

Keyword(s):

Lower Bound ◽

Lower Bounds ◽

Upper Bound ◽

Fundamental Theorem ◽

Upper Bounds ◽

Basic Logic ◽

Real Numbers ◽

Consistent Theory

A demonstrably consistent theory of real numbers has been outlined by the writer in An extension of basic logic1 (hereafter referred to as EBL). This theory deals with non-negative real numbers, but it could be easily modified to deal with negative real numbers also. It was shown that the theory was adequate for proving a form of the fundamental theorem on least upper bounds and greatest lower bounds. More precisely, the following results were obtained in the terminology of EBL: If С is a class of U-reals and is completely represented in Κ′ and if some U-real is an upper bound of С, then there is a U-real which is a least upper bound of С. If D is a class of (U-reals and is completely represented in Κ′, then there is a U-real which is a greatest lower bound of D.

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Class numbers of real cyclotomic fields of composite conductor

LMS Journal of Computation and Mathematics ◽

10.1112/s1461157014000382 ◽

2014 ◽

Vol 17 (A) ◽

pp. 404-417 ◽

Cited By ~ 4

Author(s):

John C. Miller

Keyword(s):

Lower Bounds ◽

Upper Bound ◽

Upper Bounds ◽

Class Numbers ◽

Prime Ideals ◽

Cyclotomic Fields ◽

Class Field ◽

Divisibility Properties ◽

Composite Conductor ◽

Hilbert Class Field

AbstractUntil recently, the ‘plus part’ of the class numbers of cyclotomic fields had only been determined for fields of root discriminant small enough to be treated by Odlyzko’s discriminant bounds.However, by finding lower bounds for sums over prime ideals of the Hilbert class field, we can now establish upper bounds for class numbers of fields of larger discriminant. This new analytic upper bound, together with algebraic arguments concerning the divisibility properties of class numbers, allows us to unconditionally determine the class numbers of many cyclotomic fields that had previously been untreatable by any known method.In this paper, we study in particular the cyclotomic fields of composite conductor.

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Sharp upper bounds on perfect retrieval in the Hopfield model

Journal of Applied Probability ◽

10.1017/s0021900200017708 ◽

1999 ◽

Vol 36 (03) ◽

pp. 941-950 ◽

Cited By ~ 1

Author(s):

Anton Bovier

Keyword(s):

Fixed Points ◽

Lower Bounds ◽

Upper Bound ◽

Random Variables ◽

Upper Bounds ◽

Negative Association ◽

Hopfield Model ◽

Gradient Dynamics ◽

One Step ◽

The One

We prove a sharp upper bound on the number of patterns that can be stored in the Hopfield model if the stored patterns are required to be fixed points of the gradient dynamics. We also show corresponding bounds on the one-step convergence of the sequential gradient dynamics. The bounds coincide with the known lower bounds and confirm the heuristic expectations. The proof is based on a crucial idea of Loukianova (1997) using the negative association properties of some random variables arising in the analysis.

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Upper Bounds for Online Ramsey Games in Random Graphs

Combinatorics Probability Computing ◽

10.1017/s0963548308009620 ◽

2009 ◽

Vol 18 (1-2) ◽

pp. 259-270 ◽

Cited By ~ 12

Author(s):

MARTIN MARCINISZYN ◽

RETO SPÖHEL ◽

ANGELIKA STEGER

Keyword(s):

Large Class ◽

Lower Bounds ◽

Random Graphs ◽

Upper Bound ◽

Upper Bounds ◽

Arbitrary Size ◽

Threshold Functions ◽

Current Graph ◽

Player Game ◽

Monochromatic Copy

Consider the following one-player game. Starting with the empty graph onnvertices, in every step a new edge is drawn uniformly at random and inserted into the current graph. This edge has to be coloured immediately with one ofravailable colours. The player's goal is to avoid creating a monochromatic copy of some fixed graphFfor as long as possible. We prove an upper bound on the typical duration of this game ifFis from a large class of graphs including cliques and cycles of arbitrary size. Together with lower bounds published elsewhere, explicit threshold functions follow.

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Quantum universality by state distillation

Quantum Information and Computation ◽

10.26421/qic9.11-12-7 ◽

2009 ◽

Vol 9 (11&12) ◽

pp. 1030-1052

Author(s):

B.W. Reichardt

Keyword(s):

Lower Bounds ◽

Upper Bound ◽

Fault Tolerant ◽

Upper Bounds ◽

Mixed States ◽

Single Qubit ◽

Noise Rate ◽

Pure States ◽

Rate Threshold ◽

Stabilizer States

Quantum universality can be achieved using classically controlled stabilizer operations and repeated preparation of certain ancilla states. Which ancilla states suffice for universality? This ``magic states distillation" question is closely related to quantum fault tolerance. Lower bounds on the noise tolerable on the ancilla help give lower bounds on the tolerable noise rate threshold for fault-tolerant computation. Upper bounds show the limits of threshold upper-bound arguments based on the Gottesman-Knill theorem. We extend the range of single-qubit mixed states that are known to give universality, by using a simple parity-checking operation. For applications to proving threshold lower bounds, certain practical stability characteristics are often required, and we also show a stable distillation procedure.}{No distillation upper bounds are known beyond those given by the Gottesman-Knill theorem. One might ask whether distillation upper bounds reduce to upper bounds for single-qubit ancilla states. For multi-qubit pure states and previously considered two-qubit ancilla states, the answer is yes. However, we exhibit two-qubit mixed states that are not mixtures of stabilizer states, but for which every postselected stabilizer reduction from two qubits to one outputs a mixture of stabilizer states. Distilling such states would require true multi-qubit state distillation methods.

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Upper Bounds for the Essential Dimension of E7

Canadian Mathematical Bulletin ◽

10.4153/cmb-2013-006-x ◽

2013 ◽

Vol 56 (4) ◽

pp. 795-800 ◽

Cited By ~ 3

Author(s):

Mark L. MacDonald

Keyword(s):

Upper Bound ◽

Upper Bounds ◽

Essential Dimension ◽

Connected Group ◽

Simply Connected

Abstract.This paper gives a new upper bound for the essential dimension and the essential 2-dimension of the split simply connected group of type E7 over a field of characteristic not 2 or 3. In particular, ed(E7) ≤ 29, and ed(E7, 2) ≤ 27.

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Lower Bounds for the Cop Number when the Robber is Fast

Combinatorics Probability Computing ◽

10.1017/s0963548311000101 ◽

2011 ◽

Vol 20 (4) ◽

pp. 617-621 ◽

Cited By ~ 6

Author(s):

ABBAS MEHRABIAN

Keyword(s):

Graph Theory ◽

Lower Bound ◽

Lower Bounds ◽

Regular Graph ◽

Upper Bound ◽

Upper Bounds ◽

Cops And Robbers ◽

Vertex Graph

We consider a variant of the Cops and Robbers game where the robber can movetedges at a time, and show that in this variant, the cop number of ad-regular graph with girth larger than 2t+2 is Ω(dt). By the known upper bounds on the order of cages, this implies that the cop number of a connectedn-vertex graph can be as large as Ω(n2/3) ift≥ 2, and Ω(n4/5) ift≥ 4. This improves the Ω($n^{\frac{t-3}{t-2}}$) lower bound of Frieze, Krivelevich and Loh (Variations on cops and robbers,J. Graph Theory, to appear) when 2 ≤t≤ 6. We also conjecture a general upper boundO(nt/t+1) for the cop number in this variant, generalizing Meyniel's conjecture.

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Sharp upper bounds on perfect retrieval in the Hopfield model

Journal of Applied Probability ◽

10.1239/jap/1032374647 ◽

1999 ◽

Vol 36 (3) ◽

pp. 941-950 ◽

Cited By ~ 11

Author(s):

Anton Bovier

Keyword(s):

Fixed Points ◽

Lower Bounds ◽

Upper Bound ◽

Random Variables ◽

Upper Bounds ◽

Negative Association ◽

Hopfield Model ◽

Gradient Dynamics ◽

One Step ◽

The One

We prove a sharp upper bound on the number of patterns that can be stored in the Hopfield model if the stored patterns are required to be fixed points of the gradient dynamics. We also show corresponding bounds on the one-step convergence of the sequential gradient dynamics. The bounds coincide with the known lower bounds and confirm the heuristic expectations. The proof is based on a crucial idea of Loukianova (1997) using the negative association properties of some random variables arising in the analysis.

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