Search

For positive integer [Formula: see text], a graph [Formula: see text] is said to be [Formula: see text]-magic if the edges of [Formula: see text] can be labeled with the nonzero elements of Abelian group [Formula: see text], where [Formula: see text] (the set of integers) and [Formula: see text] is the group of integers mod [Formula: see text], so that the sum of the labels of the edges incident to any vertex of [Formula: see text] is the same. When this constant sum is [Formula: see text], we say that [Formula: see text] is a zero-sum [Formula: see text]-magic graph. The set of all [Formula: see text] for which [Formula: see text] is a zero-sum [Formula: see text]-magic graph is the null set of [Formula: see text]. In this paper, we will completely determine the null set of the join of a finite number of paths.

The concept of soft sets and its application in decision making is introduced by Molodtsov. Later it is generalized to soft fuzzy set. In our study we introduce the definition of soft Generalized vague set and define operations on it namely subset, equality, null set union, intersection, OR operator, AND operator on soft generalized vague sets with illustrative example.

The notion of Haar null set was introduced by J.P.R. Christensen in 1973 and reintroduced in 1992 in the context of dynamical systems by Hunt, Sauer and Yorke. During the last twenty years this notion has been useful in studying exceptional sets in diverse areas. These include analysis, dynamical systems, group theory, and descriptive set theory. Inspired by these various results, we introduce the topological analogue of the notion of Haar null set. We call it Haar meager set. We prove some basic properties of this notion, state some open problems and suggest a possible line of investigation which may lead to the unification of these two notions in certain context.

We develop a Fourier-Feynman theory for Fourier-type func-tionals k F and [F on Wiener space. We show that Fourier-Feynmantransform and convolution of Fourier-type functionals exist. We alsoshow that the Fourier-Feynman transform of the convolution product ofFourier-type functionals is a product of Fourier-Feynman transforms ofeach functionals. 1. Introduction and preliminariesLet C 0 [0;T] denote the Wiener space, that is, the space of real valuedcontinuous functions xon [0;T] with x(0) = 0. Let Mdenote the class ofall Wiener measurable subsets of C 0 [0;T] and let mdenote Wiener measure.(C 0 [0;T];M;m) is a complete measure space and we denote the Wiener integralof a functional FbyZ C 0 [0;T] F(x)dm(x):A subset Eof C 0 [0;T] is said to be scale-invariant measurable provided ˆEis Wiener measurable for every ˆ>0, and a scale-invariant measurable setN is said to be scale-invariant null provided m(ˆN) = 0 for every ˆ>0. Aproperty that holds except on a scale-invariant null set is said to hold scale-invariant almost everywhere (s-a.e.). Given two complex-valued functions Fand Gon C

In this paper, we prove the Hyers–Ulam stability theorem when $${f, g, h : \mathbb{R} \to \mathbb{R}}$$ satisfy $$|f(x + y) - g(x) - h(y)| \leq \epsilon$$ in a set $${\Gamma \subset \mathbb{R}^{2}}$$ of measure $${m(\Gamma) = 0}$$ , which refines a previous result in Chung (Aequat Math 83:313–320, 2012) and gives an affirmative answer to the question in the paper. As a direct consequence we obtain that if $${f, g, h : \mathbb{R} \to \mathbb{R}}$$ satisfy the Pexider equation $$f(x + y) - g(x) - h(y) = 0$$ in $${\Gamma}$$ , then the equation holds for all $${x, y \in \mathbb{R}}$$ . Using our method of construction of the set, we can find a set $${\Gamma \subset \mathbb{R}^{2n}}$$ of 2n-dimensional measure 0 and obtain the above result for the functions $${f, g, h : \mathbb{R}^{n} \to \mathbb{C}}$$ .

Let \(\Omega \subset {\mathbb {R}}^{n}\) be a domain, \(n \ge 2\). We show that a continuous, open and discrete mapping \(f \in W_{\mathrm{loc }}^{1,n-1}(\Omega , {\mathbb {R}}^{n})\) with integrable inner distortion is differentiable almost everywhere on \(\Omega \). As a corollary we get that the branch set of such a mapping has measure zero.

We consider the following variant of the Monge-Kantorovich transportation problem. Let S be a finite set of point sites in d dimensions. A bounded set C in d-dimensional space is to be distributed among the sites p in S such that (i) each p receives a subset C_p of prescribed volume, and (ii) the average distance of all points of C from their respective sites p is minimized. In our model, volume is quantified by some measure, and the distance between a site p and a point z is given by a function d_p(z). Under quite liberal technical assumptions on C and on the functions d_p we show that a solution of minimum total cost can be obtained by intersecting with C the Voronoi diagram of the sites in S, based on the functions d_p with suitable additive weights. Moreover, this optimum partition is unique up to sets of measure zero. Unlike the deep analytic methods of classical transportation theory, our proof is based directly on geometric arguments., 15 pages, 2 figures. Submitted to the journal Computational Geometry, Theory and Applications, Special issue for the 28th European Workshop on Computational Geometry (EuroCG'12) in Assisi, March 2012

A colouring of a graphGis called distinguishing if its stabilizer in AutGis trivial. It has been conjectured that, if every automorphism of a locally finite graph moves infinitely many vertices, then there is a distinguishing 2-colouring. We study properties of random 2-colourings of locally finite graphs and show that the stabilizer of such a colouring is almost surely nowhere dense in AutGand a null set with respect to the Haar measure on the automorphism group. We also investigate random 2-colourings in several classes of locally finite graphs where the existence of a distinguishing 2-colouring has already been established. It turns out that in all of these cases a random 2-colouring is almost surely distinguishing.

In the first section, a new proof is given that along the radius \([0,1)\) a Blaschke product cannot tend to zero as fast as \(\exp (-\frac{1+x}{1-x})\) but can tend to zero at any slower rate. The second section is devoted to analyzing the subsets \(Z\) of the circle which are of the form \(\{e^{i\theta }:B(re^{i\theta })\rightarrow 0 \text{ as } r\rightarrow 1^-\}\), where \(B\) is a Blascke product. A complete characterization is not found, but every \(F_\sigma \) of measure zero is such a set \(Z\), and every such \(Z\) is the intersection of a countable number of such \(F_\sigma \). Of independent interest is the theorem that for every Cantor-like subset \(E\) of the unit interval there is a monotone continuous function which is constant on the complementary intervals and has infinity as the limit of the symmetric derivates at every point of \(E\), just as the original Cantor function behaves for the original Cantor set.

In this paper we present a new construction of the ternary Cantor set within the context of Gromov hyperbolic geometry. Unlike the standard construction, where one proceeds by removing middle-third intervals, our construction uses the collection of the removed intervals. More precisely, we first hyperbolize (in the sense of Gromov) the collection of the removed middle-third open intervals, then we define a visual metric on its boundary at infinity and then we show that the resulting metric space is isometric to the Cantor set. 1. The ternary Cantor set The ternary Cantor set C is one of the most familiar fractals in mathematics. Recall its standard construction, which is based on the Euclidean notion of length. Begin with the closed unit interval C0DT0; 1U R, then remove the open middle-third interval, constructing C1D 0; 1 [ 2 ; 1 . We then remove the middle-third of

We consider parametric classes ( T r ) r ∈ ( 0 , 1 / 2 ) of so-called transformation matrices and their induced families ( A r ) r ∈ ( 0 , 1 / 2 ) and ( μ r ) r ∈ ( 0 , 1 / 2 ) of two-dimensional copulas and doubly stochastic measures with fractal support respectively. By using tools from Symbolic Dynamics we show that for each pair r , r ′ ∈ ( 0 , 1 / 2 ) with r ≠ r ′ there exists a homeomorphism H r r ′ between the supports of μ r and μ r ′ mapping a Borel set of μ r -measure one to a set of μ r ′ -measure zero. Differentiability properties of these homeomorphisms are studied and Hausdorff dimensions of related sets are calculated. Several examples and graphics illustrate the main results.

A linear Cantor setC with zero Lebesgue measure is associated with the countable collection of the bounded complementary open intervals. A rearrangment of C has the same lengths of its complementary intervals, but with different locations. We study the Hausdorff and packing h-measures and dimensional properties of the set of all rearrangments of some given C for general dimension functions h. For each set of complementary lengths, we construct a Cantor set rearrangement which has the maximal Hausdorff and the minimal packing h-premeasure, up to a constant. We also show that if the packing measure of this Cantor set is positive, then there is a rearrangement which has infinite packing measure.

We consider a dynamical system on the semi-infinite cylinder which models the high energy dynamics of a family of mechanical models. We provide conditions under which we ensure that the set of orbits undergoing Fermi acceleration has measure zero.

We prove that the stable classes for continuous maps on compact metric spaces have measure zero with respect to any ergodic invariant measure with positive entropy. Then, every continuous map with positive topological entropy on a compact metric space has uncountably many stable classes. We also prove that every continuous map with positive topological entropy of a compact metric space cannot be Lyapunov stable on its recurrent set. For homeomorphisms on compact metric spaces we prove that the sets of heteroclinic points, and sinks in the canonical coordinates case, have zero measure with respect to any ergodic invariant measure with positive entropy. These results generalize those of Fedorenko and Smital [Maps of the interval Ljapunov stable on the set of nonwandering points. Acta Math. Univ. Comenian. (N.S.)60 (1) (1991), 11–14], Huang and Ye [Devaney’s chaos or 2-scattering implies Li–Yorke’s chaos. Topology Appl.117 (3) (2002), 259–272], Reddy [The existence of expansive homeomorphisms on manifolds. Duke Math. J.32 (1965), 627–632], Reddy and Robertson [Sources, sinks and saddles for expansive homeomorphisms with canonical coordinates. Rocky Mountain J. Math.17 (4) (1987), 673–681], Sindelarova [A counterexample to a statement concerning Lyapunov stability. Acta Math. Univ. Comenian. 70 (2001), 265–268], and Zhou [Some equivalent conditions for self-mappings of a circle. Chinese Ann. Math. Ser. A12(suppl.) (1991), 22–27].

We provide several explicit examples of 3D quasiperiodic linear skew-products with simple Lyapunov spectrum, that is with $3$ different Lyapunov multipliers, for which the corresponding Oseledets bundles are measurable but not continuous, colliding in a measure zero dense set.

In this paper we consider all possible period sets $P(f)$ for self-maps of the Cantor set, $f:C\to C$. We prove that the possible period sets are completely unrestricted provided that, in addition, one allows points that are not periodic. However, if every point is periodic, we show that a surprising finiteness condition is imposed on $P(f)$: namely, there is a finite subset $B$ of $P(f)$ such that every element of $P(f)$ is divisible by at least one element of $B$.

We introduce the concept of hereditarily non uniformly perfect sets, compact sets for which no compact subset is uniformly perfect, and compare them with the following: Hausdorff dimension zero sets, logarithmic capacity zero sets, Lebesgue 2-dimensional measure zero sets, and porous sets. In particular, we give an example of a compact set in the plane of Hausdorff dimension 2 (and positive logarithmic capacity) which is hereditarily non uniformly perfect., 14 pages. See also http://rstankewitz.iweb.bsu.edu/, http://sugawa.cajpn.org/index_E.html, http://www.math.sci.osaka-u.ac.jp/~sumi/

International audience; We study Kakeya sets over local non-archimedean fields with a probabilistic point of view: we define a probability measure on the set of Kakeya sets as above and prove that, according to this measure, almost all non-archimedean Kakeya sets are neglectable according to the Haar measure. We also discuss possible relations with the non-archimedean Kakeya conjecture.; Nous étudions les ensembles de Kakeya sur les corps locaux non-archimédiens avec un point de vue probabiliste: nous définissons une mesure de probabilité sur l'ensemble de ces ensembles de Kakeya et démontrons que presque tous les ensembles de Kakeya non-archimédiens ont une mesure de Haar nulle. Nous discutons également d'un analogue non-archimédien de la conjecture de Kakeya.

Let A , B be two classes of paths so that if X ∈ A and Y ∈ B , then the integral ∫ X d Y is appropriately defined. Then, despite the fact that ∫ X d Y exists with probability one whenever X , Y are independent brownian motions, A ∩ B has Weiner measure zero.

We study level statistics in ensembles of integrable $N\times N$ matrices linear in a real parameter $x$. The matrix $H(x)$ is considered integrable if it has a prescribed number $n>1$ of linearly independent commuting partners $H^i(x)$ (integrals of motion) $\left[H(x),H^i(x)\right] = 0$, $\left[H^i(x), H^j(x)\right]$ = 0, for all $x$. In a recent work, we developed a basis-independent construction of $H(x)$ for any $n$ from which we derived the probability density function, thereby determining how to choose a typical integrable matrix from the ensemble. Here, we find that typical integrable matrices have Poisson statistics in the $N\to\infty$ limit provided $n$ scales at least as $\log{N}$; otherwise, they exhibit level repulsion. Exceptions to the Poisson case occur at isolated coupling values $x=x_0$ or when correlations are introduced between typically independent matrix parameters. However, level statistics cross over to Poisson at $ \mathcal{O}(N^{-0.5})$ deviations from these exceptions, indicating that non-Poissonian statistics characterize only subsets of measure zero in the parameter space. Furthermore, we present strong numerical evidence that ensembles of integrable matrices are stationary and ergodic with respect to nearest neighbor level statistics., 18 pages, 26 figures, discussion on number variance added; published version

Quantum Darwinism recognizes the role of the environment as a communication channel: Decoherence can selectively amplify information about the pointer states of a system of interest (preventing access to complementary information about their superpositions) and can make records of this information accessible to many observers. This redundancy explains the emergence of objective, classical reality in our quantum Universe. Here, we demonstrate that the amplification of information in realistic spin environments can be quantified by the quantum Chernoff information, which characterizes the distinguishability of partial records in individual environment subsystems. We show that, except for a set of initial states of measure zero, the environment always acquires redundant information. Moreover, the Chernoff information captures the rich behavior of amplification in both finite and infinite spin environments, from quadratic growth of the redundancy to oscillatory behavior. These results will considerably simplify experimental testing of quantum Darwinism, e.g., using nitrogen vacancies in diamond., Comment: 10 pages, 5 figures

We add to the literature the following observation. If µ is a singular measure on the real line which assigns measure zero to every porous set and f : R ! R is a Lipschitz function which is non-differentiable µ-a.e. then for every C 1 function g : R ! R there holds µ{x 2 R : f(x) = g(x)} = 0. In other words the Lusin type approximation property of Lipschitz functions with C 1 functions does not hold with respect to a general Radon measure. Moreover we discuss, only in the one-dimensional setting, how the result contained in (Alberti, A Lusin type theorem for gradients, J. Funct. Anal., 100 (1991)) could be extended and improved when the Lebesgue measure is replaced by an arbitrary Radon measure.

In this chapter, we extend results obtained for the real line to more general spaces supplied with a measure.

Summary. We introduce two new classes of special subsets of the real line: the class of perfectly null sets and the class of sets which are perfectly null in the transitive sense. These classes may play the role of duals to the corresponding classes on the category side. We investigate their properties and, in particular, we prove that every strongly null set is perfectly null in the transitive sense, and that it is consistent with ZFC that there exists a universally null set which is not perfectly null in the transitive sense. Finally, we state some open questions concerning the above classes. Although the main problem of whether the classes of perfectly null sets and universally null sets are consistently dierent remains open, we prove some results related to this question.

We give a precise description of spectra of the Maryland model $ (h_{\lambda,\alpha,\theta}u)_n=u_{n+1}+u_{n-1}+ \lambda \tan \pi(\theta+n\alpha)u_n$ for all values of parameters. We introduce an arithmetically defined index $\delta (\alpha, \theta)$ and show that for $\alpha\notin\mathbb{Q},\,$ $\sigma_{sc}(h_{\lambda,\alpha,\theta})=\overline{\{e:\gamma_{\lambda}(e), Comment: CPAM to appear

In this paper, the author further reveals some intrinsic properties of the Cantor set. By the properties, the author gives a new method for calculating the exact value of the Hausdorff measure of the Cantor set, and shows the facts that each covering which consists of basic intervals, in which any basic interval cannot completely contain the other, is the best covering of the Cantor set, and the Hausdorff measure of the Cantor set can be determined by coverings which only consist of basic intervals.

In the context of arbitrary Polish groups, we investigate the Galvin–Mycielski–Solovay characterization of strong measure zero sets as those sets for which a meager collection of right translates cannot cover the whole group.

Kechris showed in Kechris (Trans. Am. Math. Soc. 202, 259---297, 1975) that there exists a largest ź11${\Pi ^{1}_{1}}$ set of measure 0. An explicit construction of this largest ź11${\Pi ^{1}_{1}}$ nullset has later been given in Hjorth and Nies (J. Lond. Math. Soc. 75(2), 495---508, 2007). Due to its universal nature, it was conjectured by many that this nullset has a high Borel rank (the question is explicitely mentioned in Chong and Yu (J. Symb. Log. 80(04), 1131---1148, 2015) and Yu (Fundam. Math. 215, 219---231, 2011)). In this paper, we refute this conjecture and show that this nullset is merely Σ30${\Sigma }^{0}_{3}$. Together with a result of Liang Yu, our result also implies that the exact Borel complexity of this set is Σ30${\Sigma }^{0}_{3}$. To do this proof, we develop the machinery of effective randomness and effective Solovay genericity, investigating the connections between those notions and effective domination properties.

We study a class of Cantor sets in [0, 1]. We show that the complement of the Cantor set is winning for Schmidt game if and only if the Cantor set is not “fat”. This provides some open dense sets of full Lebesgue measure that are not winning for Schmidt game.

LetXbe a real normed space andYa Banach space andf:X→Y. We prove the Ulam-Hyers stability theorem for the quartic functional equationf(2x+y)+f(2x-y)-4f(x+y)-4f(x-y)-24f(x)+6f(y)=0in restricted domains. As a consequence we consider a measure zero stability problem of the above inequality whenf:R→Y.

The process of constructing impulse-response functions (IRFs) and forecast-error variance decompositions (FEVDs) for a structural vector autoregression (SVAR) usually involves a factorization of an estimate of the error-term variance-covariance matrix V. Examining residuals from a monetary VAR, this paper finds evidence suggesting that all of the variances in V are infinite. Specifically, this study estimates alpha-stable distributions for the reduced-form error terms. The ML estimates of the residuals' characteristic exponents "alpha" range from 1.5504 to 1.7734, with the Gaussian case lying outside 95 percent asymptotic confidence intervals for all six equations of the VAR. Variance-stabilized P-P plots show that the estimated distributions fit the residuals well. Results for subsamples are varied, while GARCH(1,1) filtering yields standardized shocks that are also all likely to be non-Gaussian alpha stable. When one or more error terms have infinite variance, V cannot be factored. Moreover, by Proposition 1, the reduced-form DGP cannot be transformed, using the required nonsingular matrix, into an appropriate system of structural equations with orthogonal, or even finite-variance, shocks. This result holds with arbitrary sets of identifying restrictions, including even the null set. Hence, with one or more infinite-variance error terms, structural interpretation of the reduced-form VAR within the standard SVAR model is impossible.

The parameter lower (upper) distribution set corresponds to the cylindrical lower or upper local dimension set for a self-similar measure on a self-similar set satisfying the open set condition.

In this paper we study singular integrals on small (that is, measure zero and lower than full dimensional) subsets of metric groups. The main examples of the groups we have in mind are Euclidean spaces and Heisenberg groups. In addition to obtaining results in a very general setting, the purpose of this work is twofold; we shall extend some results in Euclidean spaces to more general kernels than previously considered, and we shall obtain in Heisenberg groups some applications to harmonic (in the Heisenberg sense) functions of some results known earlier in Euclidean spaces.

It is well known that the difference between the set of all ordinary density points and the set of all strong density points of an arbitrary measurable subset of the plane is a null set. It is of interest to check how large can a difference between sections of these sets be. Both measure and category cases are considered.

Let and K(λ,3), the 3-part Cantor set, be the attractor of an iterated function system {φ1,φ2,φ3}on a line, where φ1(x) = λx, , φ3(x) = (1 − λ) + λx, x ∈ [0,1]. Set μ be the corresponding 3-part Cantor measure and . We will determine by an explicit formula for every x ∈ K(λ,3) the upper and lower s-densities Θ*s(μ,x), of the 3-part Cantor measure at the point x. Copyright © 2012 John Wiley & Sons, Ltd.

We study restriction and extension theory for semibounded Hermitian operators in the Hardy space of analytic functions on the disk D. Starting with the operator zd/dz, we show that, for every choice of a closed subset F in T=bd(D) of measure zero, there is a densely defined Hermitian restriction of zd/dz corresponding to boundary functions vanishing on F. For every such restriction operator, we classify all its selfadjoint extension, and for each we present a complete spectral picture. We prove that different sets F with the same cardinality can lead to quite different boundary-value problems, inequivalent selfadjoint extension operators, and quite different spectral configurations. As a tool in our analysis, we prove that the von Neumann deficiency spaces, for a fixed set F, have a natural presentation as reproducing kernel Hilbert spaces, with a Hurwitz zeta-function, restricted to FxF, as reproducing kernel., 63 pages, 11 figures

For a compact subset K of the boundary of a compact Hausdorff space X, six properties that K may have in relation to the algebra A(X) are considered. It is shown that in relation to the algebra A(Dn), where Dn denotes the n-dimensional polydisc, the property of being totally null is weaker than the other five properties. A general condition is given on the algebra A(X) which implies the existence of a totally null set that is not null, and several conditions are stated for A(X) , each of which is sufficient for a totally null set to be a null set.

J.C. Oxtoby and S. Ulam proved that each set of the first category in r -dimensional Euclidean space can be transformed into a set of Lebesgue measure zero by some automorphism, and such homeomorphisms constitute a residual set in the set of all automorphisms. We have improved this result changing the sets of measure zero by microscopic sets on the real line and on the plane.

Computing the Hausdorff measure of C × C, where C is the classical ternary Cantor set, is a long standing difficult problem. It is well-known that for a self-similar set, calculating the Hausdorff measure is equivalent to determining its optimal sets. This paper studies optimal sets of C × C: their diameters, measures, symmetries and the shapes. For this purpose, we introduce several devices: the repulsive principle, a bipartite graph G and a W-function. We show that the diameter of the optimal set B is between 1.2993 and 1.3082. Two symmetry properties of B are proved. Finally, we show that the shape of B is very close to a disk. We conjecture that an optimal set might be a disk.

In this paper we consider a class of symmetric Cantor sets in R. Under certain separation condition we determine the exact packing measure of such a Cantor set through the computation of the lower density of the uniform probability measure supported on the set. With an additional restriction on the dimension we give also the exact centered Hausdorff measure by computing the upper density. © 2011 Elsevier Inc. Fil:Zuberman, L. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.

We extend the notion of L2–B-discrepancy introduced in [E. Novak, H. Woźniakowski, L2 discrepancy and multivariate integration, in: W.W.L. Chen, W.T. Gowers, H. Halberstam, W.M. Schmidt, and R.C. Vaughan (Eds.), Analytic Number Theory. Essays in Honour of Klaus Roth, Cambridge University Press, Cambridge, 2009, pp. 359–388] to what we shall call weighted geometric L2-discrepancy. This extension enables us to consider weights in order to moderate the importance of different groups of variables, as well as to consider volume measures different from the Lebesgue measure and classes of test sets different from measurable subsets of Euclidean spaces.We relate the weighted geometric L2-discrepancy to numerical integration defined over weighted reproducing kernel Hilbert spaces and settle in this way an open problem posed by Novak and Woźniakowski.Furthermore, we prove an upper bound for the numerical integration error for cubature formulas that use admissible sample points. The set of admissible sample points may actually be a subset of the integration domain of measure zero. We illustrate that particularly in infinite-dimensional numerical integration it is crucial to distinguish between the whole integration domain and the set of those sample points that actually can be used by the algorithms.

We show that given a one-parameter family Fb of strongly dissipative infinitely renormalizable Henon-like maps, parametrized by a quantity called the 'average Jacobian' b, the set of all parameters b such that Fb has a Cantor set with unbounded geometry has full Lebesgue measure.

The article is devoted to a particular case of Ivriĭ's conjecture on periodic orbits of billiards. The general conjecture states that the set of periodic orbits of the billiard in a domain with smooth boundary in the Euclidean space has measure zero. In this article we prove that for any domain with piecewise $C^4$-smooth boundary in the plane the set of quadrilateral trajectories of the corresponding billiard has measure zero.

This paper addresses two fundamental problems in the context of hidden Markov models (HMMs). The first problem is concerned with the characterization and computation of a minimal order HMM that realizes the exact joint densities of an output process based on only finite strings of such densities (known as HMM partial realization problem). The second problem is concerned with learning a HMM from finite output observations of a stochastic process. We review and connect two fields of studies: realization theory of HMMs, and the recent development in spectral methods for learning latent variable models. Our main results in this paper focus on generic situations, namely, statements that will be true for almost all HMMs, excluding a measure zero set in the parameter space. In the main theorem, we show that both the minimal quasi-HMM realization and the minimal HMM realization can be efficiently computed based on the joint probabilities of length $N$ strings, for $N$ in the order of ${\cal O}(\log_{d}(k))$ . In other words, learning a quasi-HMM and an HMM have comparable complexity for almost all HMMs.

For a measure μ on the complex plane μ-regular points play an important role in various polynomial inequalities. In the present work it is shown that every point in the set {μ′>0} (actually of a larger set where μ is strong) with the exception of a set of zero logarithmic capacity is a μ-regular point. Here “set of zero logarithmic capacity” cannot be replaced by “β-logarithmic Hausdorff measure 0” with β=1 (it can be replaced by “β-logarithmic measure 0” with any β>1). On the other hand, for arbitrary μ the set of μ-regular points can be quite small, but never empty.

In this paper, we show that if the volume sum \( \sum\nolimits_{h = 1}^\infty {{h^{n - 1}}{\Psi^t}(h)} \) converges for a function ψ (not necessarily monotonic), then the set of points \( \left( {x,{w_1}, \ldots, {w_{t - 1}}} \right) \in {\mathbb R} \times {{\mathbb Q}_{{p_1}}} \times \ldots \times {{\mathbb Q}_{{p_{t - 1}}}} \) that simultaneously satisfy the inequalities \( \left| {P(x)} \right| < \Psi (H) {\text{and}} {\left| {P\left( {{w_i}} \right)} \right|_{{p_i}}} < \Phi (H), 1 \leqslant i \leqslant t - 1 \), for infinitely many integer polynomials P has measure zero.

A relative Schottky set in a planar domain D is a subset of D obtained by removing from D open geometric discs whose closures are in D and are pairwise disjoint. In this paper we study quasisymmetric and related maps between relative Schottky sets of measure zero. We prove, in particular, that quasisymmetric maps between such sets in Jordan domains are conformal, locally bi-Lipschitz, and that their first derivatives are locally Lipschitz. We also provide a locally bi-Lipschitz uniformization result for relative Schottky sets in Jordan domains and establish rigidity with respect to local quasisymmetric maps for relative Schottky sets in the unit disc., Comment: 43 pages