Set Conjecture Research Articles

Nonexistence of abelian difference sets: Lander’s conjecture for prime power orders

In 1963 Ryser conjectured that there are no circulant Hadamard matrices of order > 4 and no cyclic difference sets whose order is not coprime to the group order. These conjectures are special cases of Lander's conjecture which asserts that there is no abelian group with a cyclic Sylow p-subgroup containing a difference set of order divisible by p. We verify Lander's conjecture for all difference sets whose order is a power of a prime greater than 3.

The Spectral Set Conjecture and Multiplicative Properties of Roots of Polynomials

Fuglede's conjecture states that a set Ω⊂Rn tiles R by translations if and only if L2(Ω) has an orthogonal basis of exponentials. New partial results are obtained supporting the conjecture in dimension 1.

Koszul homology and Lie algebras with application to generic forms and points

We study the Koszul dual for general superalgebras, and apply it to the Koszul homology of a graded algebra. We show that a part of the Koszul homology algebra is related to the homotopy Lie algebra by means of Koszul duality. This is used to study the Minimal Resolution Conjecture and the Ideal Generating Conjecture for sets of generic points in projective space, and for quotients of the polynomial ring (or exterior algebra) modulo generic quadratic forms.

La filtration de Krull de la catégorie 𝒰 et la cohomologie des espaces

Paper written in French -- English abstract: This paper proves a particular case of a conjecture of N. Kuhn. This conjecture is as follows. Consider the Gabriel-Krull filtration of the category U of unstable modules. Let U_n, n>=0, be the n-th step of this filtration. The category U is the smallest thick sub-category that contains all sub-categories U_n and is stable under colimit [L. Schwartz, Unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture, Chicago Lectures in Mathematics Series (1994)]. The category U_0 is the one of locally finite modules, i.e. the modules that are direct limit of finite modules. The conjecture is as follows, let X be a space then : * either H^*X is locally finite, * or H^*X does not belong to U_n, for all n. As an example the cohomology of a finite space, or of the loop space of a finite space are always locally finite. On the other side the cohomology of the classifying space of a finite group whose order is divisible by 2 does belong to any sub-category U_n. One proves this conjecture, modulo the additional hypothesis that all quotients of the nilpotent filtration are finitely generated. This condition is used when applying N. Kuhn's reduction of the problem. It is necessary to do it to be allowed to apply Lannes' theorem on the cohomology of mapping spaces.[N. Kuhn, On topologically realizing modules over the Steenrod algebra, Ann. of Math. 141 (1995) 321-347].

Universal spectra, universal tiling sets and the spectral set conjecture

A subset $\Omega$ of $\mathbf{R}^d$ with finite positive Lebesgue measure is called a spectral set if there exists a subset $\Lambda\subset\mathbf{R}$ such that ${\mathcal E}_\Lambda :=\{e^{i2\pi \langle\lambda, x\rangle}: \lambda\in\Lambda\}$ form an orthogonal basis of $L^2(\Omega)$. The set $\Lambda$ is called a spectrum of the set $\Omega$. The Spectral Set Conjecture states that $\Omega$ is a spectral set if and only if $\Omega$ tiles $\mathbf{R}^d$ by translation. In this paper we prove the Spectral Set Conjecture for a class of sets $\Omega \subset \mathbf{R}$. Specifically we show that a spectral set possessing a spectrum that is a strongly periodic set must tile $\mathbf{R}$ by translates of a strongly periodic set depending only on the spectrum, and vice versa.

Conditional coverage probability of confidence intervals in errors-in-variables and related models

Gleser and Hwang (1987) show that there exists no nontrivial confidence interval with finite expected length for many errors-in-variables and related models. Nonetheless, the validity of using these confidence intervals only when their realizations are of finite length remains uninvestigated. We scrutinize this practice from a conditional frequentist viewpoint and find it not justifiable. It is shown that under some conditions, the minimum coverage probability of the confidence interval is zero if conditioning on the event that the confidence interval has finite length. The result is then applied to confirm Neyman's conjecture for Fieller's confidence sets.

Spectral Sets and Factorizations of Finite Abelian Groups

Aspectral setis a subsetΩofRnwith Lebesgue measure 0<μ(Ω)<∞ such that there exists a setΛof exponential functions which form an orthogonal basis ofL2(Ω). The spectral set conjecture of B. Fuglede states that a set 0 is a spectral set if and only ifΩtilesRnby translation. We study setsΩwhich tileRnusing a rational periodic tile set S=Zn+A, where A⊆(1/N1)Z×…×(1/Nn)Zis finite. We characterize geometrically bounded measurable setsΩthat tileRnwith such a tile set. Certain tile sets S have the property that every bounded measurable setΩwhich tilesRnwith S is a spectral set, with a fixed spectrumΛS. We callΛSa universal spectrum for such S. We give a necessary and sufficient condition for a rational periodic setΛto be a universal spectrum for S, which is expressed in terms of factorizationsA⊕B=GwhereG=ZN1×…×ZNn, andA:=A (modZn). In dimensionn=1 we show that S has a universal spectrum wheneverN1is the order of a “good” group in the sense of Hajós, and for various other sets S.

On the Isomorphism Conjecture for Weak Reducibilities

According to the isomorphism conjecture all NP-complete sets are polynomial?time isomorphic to each other while according to the encrypted complete set conjecture there is ap-one way functionfand an NP-complete setAsuch thatAandf(A) are not polynomial-time isomorphic to each other. In this paper, these two conjectures are translated and investigated for reducibilities weaker than polynomial-time. It is shown that: 1. Relative to reductions computed by one-way logspace DTMs, both the conjectures are false. 2. Relative to reductions computed by one-way logspace NTMs, the isomorphism conjecture is true. 3. Relative to reductions computed by one-way, multi-head, oblivious logspace DTMs, the encrypted complete set conjecture is false. 4. Relative to reductions computed by constant-scan logspace DTMs, the encrypted complete set conjecture is true. It is also shown that the complete degrees of NP under the latter two reducibilities coincide.

The Borsuk dimension of a graph and Borsuk's partition conjecture for finite sets

Borsuk's partition conjecture for finite subsets of euclidean space is placed in a graph theoretic setting and equivalent graph theoretic conjectures are raised. The Borsuk dimension of a graph is defined and the Borsuk dimensions of various graphs are tabulated.

On the direct sum conjecture

We prove the direct sum conjecture for various sets of systems of bilinear forms. Our results depend on a priori knowledge of the complexity of at least one of the direct summands and its underlying algebraic structure. We also briefly survey some previous results concerning the complexity and structure of minimal algorithms for various direct sum systems.

On binary equality sets and a solution to the test set conjecture in the binary case

K. Culik II and J. Karhumäki ( RAIRO Inform. Theor. 14 (1980), 349–369] conjectured that the equality set of two injective morphisms over a binary alphabet is of the form { u, v} ∗ for some (possibly empty) words u and v. Here we show that such an equality set is always either of the above form or of the form (uw ∗v) ∗ for some words u, w, and v. As an application we give a simple proof for the Test Set Conjecture in the binary case; cf. K. Culik II and A. Salomaa [ J. Comput. System Sci. 20 (1980), 379–396]. In fact, we show that a test set can always be chosen to contain no more than three words.

On an inverse problem for nonnegative and eventually nonnegative matrices

Let σ= (λ1,···λn)⊂C. We discuss conditions for which σ is the spectrum of a nonnegative or eventually nonnegative matrix. This brings us to study rational functions with nonnegative Maclaurin coefficients. A conjecture for special sets σ is stated and some evidence in support of this conjecture is given.

On Conway's Conjecture for Integer Sets

Let A be a finite set of integers {ai} and A + A denote {ai + aj} with p different members and A — A denote {ai—aj) with m different members, the interesting conjecture of Conway [1] states that p &lt; m, unless A is symmetric.