Measure Zero Research Articles

On effectively closed sets of effective strong measure zero

The strong measure zero sets of reals have been widely studied in the context of set theory of the real line. The notion of strong measure zero is straightforwardly effectivized. A set of reals is said to be of effective strong measure zero if for any computable sequence {εn}n∈N of positive rationals, a sequence of intervals In of diameter εn covers the set. We observe that a set is of effective strong measure zero if and only if it is of measure zero with respect to any outer measure constructed by Monroe's Method from a computable atomless outer premeasure defined on all open balls. This measure-theoretic restatement permits many characterizations of strong measure zero in terms of semimeasures as well as martingales. We show that for closed subsets of Cantor space, effective strong nullness is equivalent to another well-studied notion called diminutiveness, the property of not having a computably perfect subset. Further, we prove that if P is a nonempty effective strong measure zero Π10 set consisting only of noncomputable elements, then some Martin-Löf random reals compute no element in P, and P has an element that computes no autocomplex real. Finally, we construct two different special Π10 sets, one of which is not of effective strong measure zero, but consists only of infinitely-often K-trivial reals, and the other is perfect and of effective strong measure zero, but contains no anti-complex reals.

Zero-Sum Magic Labelings and Null Sets of Regular Graphs

For every $h\in \mathbb{N}$, a graph $G$ with the vertex set $V(G)$ and the edge set $E(G)$ is said to be $h$-magic if there exists a labeling $l : E(G) \rightarrow\mathbb{Z}_h \setminus \{0\}$ such that the induced vertex labeling $s : V (G) \rightarrow \mathbb{Z}_h$, defined by $s(v) =\sum_{uv \in E(G)} l(uv)$ is a constant map. When this constant is zero, we say that $G$ admits a zero-sum $h$-magic labeling. The null set of a graph $G$, denoted by $N(G)$, is the set of all natural numbers $h \in \mathbb{ N} $ such that $G$ admits a zero-sum $h$-magic labeling. In 2012, the null sets of 3-regular graphs were determined. In this paper we show that if $G$ is an $r$-regular graph, then for even $r$ ($r > 2$), $N(G)=\mathbb{N}$ and for odd $r$ ($r\neq5$), $\mathbb{N} \setminus \{2,4\}\subseteq N(G)$. Moreover, we prove that if $r$ is odd and $G$ is a $2$-edge connected $r$-regular graph ($r\neq 5$), then $ N(G)=\mathbb{N} \setminus \{2\}$. Also, we show that if $G$ is a $2$-edge connected bipartite graph, then $\mathbb{N} \setminus \{2,3,4,5\}\subseteq N(G)$.

A novel string field theory solving string theory by liberating left and right movers

We put forward ideas to a novel string field theory based on making some “objects” that essentially describe “liberated” left- and right- mover fields $ X_L^{\mu } $ (τ + σ) and $ X_R^{\mu } $ (τ − σ) on the string. Our novel string field theory is completely definitely different from any other string theory in as far as a “null set” of information in the string field theory Fock space has been removed relatively, to the usual string field theories. So our theory is definitely new. The main progress is that we manage to make our novel string field theory provide the correct mass square spectrum for the string. We finally suggest how to obtain the Veneziano amplitude in our model.

Differentiability of Lipschitz functions in Lebesgue null sets

We show that if $$n>1$$ then there exists a Lebesgue null set in $${\mathbb {R}}^{n}$$ containing a point of differentiability of each Lipschitz function $$f:{\mathbb {R}}^{n} \rightarrow {\mathbb {R}}^{n-1}$$ ; in combination with the work of others, this completes the investigation of when the classical Rademacher theorem admits a converse. Avoidance of $$\sigma $$ -porous sets, arising as irregular points of Lipschitz functions, plays a key role in the proof.

The effects of time-dependent dissipation on the basins of attraction for the pendulum with oscillating support

We consider a pendulum with vertically oscillating support and time-dependent damping coefficient which varies until reaching a finite final value. Although it is the final value which determines which attractors eventually exist, the sizes of the corresponding basins of attraction are found to depend strongly on the full evolution of the dissipation. In particular, we investigate numerically how dissipation monotonically varying in time changes the sizes of the basins of attraction. It turns out that, in order to predict the behaviour of the system, it is essential to understand how the sizes of the basins of attraction for constant dissipation depend on the damping coefficient. For values of the parameters where the systems can be considered as a perturbation of the simple pendulum, which is integrable, we characterise analytically the conditions under which the attractors exist and study numerically how the sizes of their basins of attraction depend on the damping coefficient. Away from the perturbation regime, a numerical study of the attractors and the corresponding basins of attraction for different constant values of the damping coefficient produces a much more involved scenario: changing the magnitude of the dissipation causes some attractors to disappear either leaving no trace or producing new attractors by bifurcation, such as period doubling and saddle-node bifurcation. Finally, we pass to the case of an initially non-constant damping coefficient, both increasing and decreasing to some finite final value, and we numerically observe the resulting effects on the sizes of the basins of attraction: when the damping coefficient varies slowly from a finite initial value to a different final value, without changing the set of attractors, the slower the variation the closer the sizes of the basins of attraction are to those they have for constant damping coefficient fixed at the initial value. Furthermore, if during the variation of the damping coefficient attractors appear or disappear, remarkable additional phenomena may occur. For instance, it can happen that, in the limit of very large variation time, a fixed point asymptotically attracts the entire phase space, up to a zero-measure set, even though no attractor with such a property exists for any value of the damping coefficient between the extreme values.

One-dimensional Schrödinger operators with [formula omitted]-interactions on Cantor-type sets

We introduce a novel approach for defining a δ′-interaction on a subset of the real line of Lebesgue measure zero which is based on Sturm–Liouville differential expression with measure coefficients. This enables us to establish basic spectral properties (e.g., self-adjointness, lower semiboundedness and spectral asymptotics) of Hamiltonians with δ′-interactions concentrated on sets of complicated structures.

Lattice Trace Operators

A bounded linear operator T on a Hilbert space ℋ is trace class if its singular values are summable. The trace class operators on ℋ form an operator ideal and in the case that ℋ is finite-dimensional, the trace tr(T) of T is given by ∑jajj for any matrix representation {aij} of T. In applications of trace class operators to scattering theory and representation theory, the subject is complicated by the fact that if k is an integral kernel of the operator T on the Hilbert space L2(μ) with μ a σ-finite measure, then k(x,x) may not be defined, because the diagonal {(x,x)} may be a set of (μ⊗μ)-measure zero. The present note describes a class of linear operators acting on a Banach function space X which forms a lattice ideal of operators on X, rather than an operator ideal, but coincides with the collection of hermitian positive trace class operators in the case of X=L2(μ).

Ideals of [formula omitted] and bimodules over maximal abelian selfadjoint algebras

This paper is concerned with weak⁎ closed masa-bimodules generated by A(G)-invariant subspaces of VN(G). An annihilator formula is established, which is used to characterise the weak⁎ closed subspaces of B(L2(G)) which are invariant under both Schur multipliers and a canonical action of M(G) on B(L2(G)) via completely bounded maps. We study the special cases of extremal ideals with a given null set and, for a large class of groups, we establish a link between relative spectral synthesis and relative operator synthesis.

Standard decomposition of expansive ergodically supported dynamics

In this work we introduce the notion of weak quasigroups, that are quasigroup operations defined almost everywhere on some set. Then we prove that the topological entropy and the ergodic period of an invertible expansive ergodically supported dynamical system $(X,T)$ with the shadowing property establishes a sufficient criterion for the existence of quasigroup operations defined almost everywhere outside of universally null sets and for which $T$ is an automorphism. Furthermore, we find a decomposition of the dynamics of $T$ in terms of $T$-invariant weak topological subquasigroups.

Fourier Transforms of Distributions and Hausdorff Measures

Consider a distribution whose support has Hausdorff \(h\)-measure zero. How fast can its Fourier coefficients decay to zero? If \(h\) is close to linear, we can improve on known results by a factor of about \((\log n)^{-1/2}\).

Haar Null Sets and the Consistent Reflection of Non-meagreness

AbstractA subset X of a Polish group G is called Haar null if there exist a Borel set B ⊃ X and Borel probability measure μ on G such that μ(gBh) = 0 for every g; h ∊ G. We prove that there exist a set X ⊂ R that is not Lebesgue null and a Borel probability measure μ such that μ (X + t) = 0 for every t ∊ R. This answers a question from David Fremlin’s problem list by showing that one cannot simplify the definition of a Haar null set by leaving out the Borel set B. (The answer was already known assuming the Continuum Hypothesis.)This result motivates the following Baire category analogue. It is consistent with ZFC that there exist an abelian Polish group G and a Cantor set C ⊂ G such that for every non-meagre set X ⊂ G there exists a t ∊ G such that C ∩ (X + t) is relatively non-meagre in C. This essentially generalizes results of Bartoszyński and Burke–Miller.

On the Unique Identification of Continuous-Time Autoregressive Models From Sampled Data

In this work, we investigate the relationship between continuous-time autoregressive (AR) models and their sampled version. We consider uniform sampling and derive criteria for uniquely determining the continuous-time parameters from sampled data; the model order is assumed to be known. We achieve this by removing a set of measure zero from the collection of all AR models and by investigating the asymptotic behavior of the remaining set of autocorrelation functions. We provide necessary and sufficient conditions for uniqueness of general AR models, and we demonstrate the usefulness of this result by considering particular examples. We further exploit our theory and introduce an estimation algorithm that recovers continuous-time AR parameters from sampled data, regardless of the sampling interval. We demonstrate the usefulness of our algorithm for various Gaussian and non-Gaussian AR processes.

A Feasibility Test for Linear Interference Alignment in MIMO Channels With Constant Coefficients

In this paper, we consider the feasibility of linear interference alignment (IA) for multiple-input-multiple-output (MIMO) channels with constant coefficients for any number of users, antennas, and streams per user, and propose a polynomial-time test for this problem. Combining algebraic geometry techniques with differential topology ones, we first prove a result that generalizes those previously published on this topic. In particular, we consider the input set (complex projective space of MIMO interference channels), the output set (precoder and decoder Grassmannians), and the solution set (channels, decoders, and precoders satisfying the IA polynomial equations), not only as algebraic sets, but also as smooth compact manifolds. Using this mathematical framework, we prove that the linear alignment problem is feasible when the algebraic dimension of the solution variety is larger than or equal to the dimension of the input space and the linear mapping between the tangent spaces of both smooth manifolds given by the first projection is generically surjective. If that mapping is not surjective, then the solution variety projects into the input space in a singular way and the projection is a zero-measure set. This result naturally yields a simple feasibility test, which amounts to checking the rank of a matrix. We also provide an exact arithmetic version of the test, which proves that testing the feasibility of IA for generic MIMO channels belongs to the bounded-error probabilistic polynomial complexity class.

On Quadrilateral Orbits in Complex Algebraic Planar Billiards

The famous conjecture of V.Ya.Ivrii (1978) says that in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero. In the present paper we study the complex algebraic version of Ivrii’s conjecture for quadrilateral orbits in two dimensions, with reflections from complex algebraic curves. We present the complete classification of 4-reflective algebraic counterexamples: billiards formed by four complex algebraic curves in the projective plane that have open set of quadrilateral orbits. As a corollary, we provide classification of the so-called real algebraic pseudo-billiards with open set of quadrilateral orbits: billiards formed by four real algebraic curves; the reflections allow to change the side with respect to the reflecting tangent line.

Complete Multipartite Graphs and their Null Set

For every natural number h, a graph G is said to be h-magic if there exists a labelling l:E(G)→Zh\\{0} such that the induced vertex set labelling l+:V(G)→Zh defined byl+(v)=∑uv∈E(G)l(uv), is a constant map. When this constant is zero, it is said that G admits a zero-sum h-magic labelling. The null set of a graph G, denoted by N(G), is the set of all natural numbers h∈N such that G admits an h-zero-sum magic labelling. In 2007, E. Salehi determined the null set of complete bipartite graphs. In this paper we generalize this result by obtaining the null set of complete multipartite graphs.

Addendum to ``On Meager Additive and Null Additive Sets in the Cantor space 2ωand in R'' (Bull. Polish Acad. Sci. Math. 57 (2009), 91–99)

We prove in ZFC that there is a set $A\subseteq 2^{\omega}$ and a surjective function $H : A\to \langle0,1\rangle $ such that for every null additive set $X\subseteq \langle0,1)$, $H^{-1}(X)$ is null additive in $2^\omega$. This settles in the affirmati

On Generic Nonexistence of the Schmidt--Eckart--Young Decomposition for Complex Tensors

The Schmidt--Eckart--Young theorem for matrices states that the optimal rank-$r$ approximation of a matrix is obtained by retaining the first $r$ terms from the singular value decomposition of that matrix. This paper considers a generalization of this optimal truncation property to the rank decomposition (Candecomp/Parafac) of tensors and establishes a necessary orthogonality condition. We prove that this condition is not satisfied at least by an open set of positive Lebesgue measure in complex tensor spaces. It is proved, moreover, that for complex tensors of small rank this condition can be satisfied only by a set of tensors of Lebesgue measure zero. Finally, we demonstrate that generic tensors in cubic tensor spaces are not optimally truncatable.

Topological Properties of the Catastrophe Map of a General Equilibrium Production Model with Uncertain States of Nature

This paper shows existence and efficiency of equilibria of a production model with uncertainty, where production is modeled in the demand function of the consumer. Existence and efficiency of equilibria are a direct consequence of the catastrophe map being smooth and proper. Topological properties of the equilibrium set are studied. It is shown that the equilibrium set has the structure of a smooth submanifold of the Euclidean space which is diffeomorphic to the sphere implying connectedness, simple connectedness, and contractibility. The set of economies with discontinuous price systems is shown to be of Lebesgue measure zero.

J-holomorphic discs and real analytic hypersurfaces

We give in R 6 a real analytic almost complex structure J, a real analytic hypersurface M and a vector v in the Levi null set at 0 of M, such that there is no germ of J-holomorphic disc included in M with (0) = 0 and @ (0) = v, although the Levi form of M has constant rank. Then for any hypersurface M and any complex structure J, we give necessary conditions under which there exists such a germ of disc.

Zero measure Cantor spectra for continuum one-dimensional quasicrystals

We study Schrödinger operators on R with measures as potentials. Choosing a suitable subset of measures we can work with a dynamical system consisting of measures. We then relate properties of this dynamical system with spectral properties of the associated operators. The constant spectrum in the strictly ergodic case coincides with the union of the zeros of the Lyapunov exponent and the set of non-uniformities of the transfer matrices. This result enables us to prove Cantor spectra of zero Lebesgue measure for a large class of operator families, including many operator families generated by aperiodic subshifts.