A divergence formula for randomness and dimension

This paper examines the constructive Hausdorff and packing dimensions of weak truth-table degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dimH(S) and constructive packing dimension dimP(S) is weak truth-table equivalent to a sequence R with \({\rm dim_H}({\it R}) \geq {\rm dim_H}(S) / {\rm dim_P}(S) - \epsilon\), for arbitrary ε> 0. Furthermore, if dimP(S) > 0, then dimP(R) ≥ 1 − ε. The reduction thus serves as a randomness extractor that increases the algorithmic randomness of S, as measured by constructive dimension.A number of applications of this result shed new light on the constructive dimensions of wtt degrees (and, by extension, Turing degrees). A lower bound of dimH(S) / dimP(S) is shown to hold for the wtt degree of any sequence S. A new proof is given of a previously-known zero-one law for the constructive packing dimension of wtt degrees. It is also shown that, for any regular sequence S (that is, dimH(S) = dimP(S)) such that dimH(S) > 0, the wtt degree of S has constructive Hausdorff and packing dimension equal to 1.Finally, it is shown that no single Turing reduction can be a universal constructive Hausdorff dimension extractor.

The variance conjecture on projections of the cube

Fix an odd integer p≥5. Let M n be a uniform p-angulation with n vertices, endowed with the uniform probability measure on its vertices. We prove that there exists C p ∈ℝ + such that, after rescaling distances by C p /n 1/4 , M n converges in distribution for the Gromov–Hausdorff–Prokhorov topology towards the Brownian map. To prove the preceding fact, we introduce a bootstrapping principle for distributional convergence of random labelled plane trees. In particular, the latter allows to obtain an invariance principle for labeled multitype Galton–Watson trees, with only a weak assumption on the centering of label displacements.

Let (M,g) be a compact Riemannian surface. Consider a family of L2 normalized Laplace–Beltrami eigenfunctions, written in the semiclassical form −h 2 j Δgφhj=φhj, whose eigenvalues satisfy hh −1 j ∈(1,1+hD] for D>0 a large enough constant. Let Ph be a uniform probability measure on the L2 unit-sphere Sh of this cluster of eigenfunctions and take u∈Sh. Given a closed curve γ⊂M, there exists C1(γ,M),C2(γ,M)>0 and h0>0 such that for all h∈(0,h0], C1h1/2≤Eh[|∫γudσ|]≤C2h1/2. This result contrasts the previous deterministic O(1) upperbounds obtained by Chen–Sogge, Reznikov, and Zelditch. Furthermore, we treat the higher dimensional cases and compute large deviation estimates. Under a measure zero assumption on the periodic geodesics in S∗M, we can consider windows of small width D=1 and establish a O(h1/2) estimate. Lastly, we treat probabilistic Lq restriction bounds along curves.

Let $M_{n}$ be a simple triangulation of the sphere $\mathbb{S}^{2}$, drawn uniformly at random from all such triangulations with $n$ vertices. Endow $M_{n}$ with the uniform probability measure on its vertices. After rescaling graph distance by $(3/(4n))^{1/4}$, the resulting random measured metric space converges in distribution, in the Gromov–Hausdorff–Prokhorov sense, to the Brownian map. In proving the preceding fact, we introduce a labelling function for the vertices of $M_{n}$. Under this labelling, distances to a distinguished point are essentially given by vertex labels, with an error given by the winding number of an associated closed loop in the map. We establish similar results for simple quadrangulations.

It is known that when the random variables being compared have a uniform probability measure, as is the case when using financial market data, the computation of efficient sets for first- and second-degree dominance can be greatly simplified if majorization conditions are used instead of the dominance definitions. This paper presents an equally efficient technique for third-degree dominance.

Facial Expressions play major role in interpersonal communication and imparting intelligence to computer for identifying facial expressions is a crucial task. In this paper we present an efficient preprocessing algorithm combined with feature extraction using Local Binary Patterns (LBP) followed by classification using Kullback Leibler (KL) divergence. Firstly Viola Jones algorithm is used to detect pair of eyes using which effective part of face is obtained which is further processed to eliminate illumination effect. LBP operator is then applied on the preprocessed image to extract local features represented by histogram. Template histograms for seven basic expressions using training images are formed which are compared with the test histogram distribution using an efficient KL divergence for dissimilarity measure. This algorithm is implemented on JAFFE database resulting in a high classification accuracy of 95.24%.

The Shannon entropy and the associated Kullback—Leibler divergence measure (relative entropy) between probability measures are fundamental from the applications point of view, and arise naturally from statistical concepts. While restricted to information theory and statistics in the work of Shannon [38] and Kullback—Leibler [31] in the early 1950s, the concept of entropy started to be used in optimization modeling for various problems of engineering and management science. An early work on entropy optimization problems over linear constraints sets (equality or inequality) was studied by Chames and Cooper [16] via convex programming techniques. Many other useful applications in a diversity of problems such as traffic engineering, game theory, information theory, and marketing were developed by Chames et al. (see, e.g., [15,17-19] and the references therein). For further general information and applications, we refer the reader to Frieden [23] and Kay and Marple [30] for engineering problems and to Lev and Theil [32] and more recently to Theil and Feibig [39] for economic and finance models.

The paper studies the Hausdorff dimension of harmonic measures on various boundaries of a relatively hyperbolic group which are associated with random walks driven by a probability measure with finite first moment. With respect to the Floyd metric and the shortcut metric, we prove that the Hausdorff dimension of the harmonic measure equals the ratio of the entropy and the drift of the random walk.If the group is infinitely-ended, the same dimension formula is obtained for the end boundary endowed with a visual metric. In addition, the Hausdorff dimension of the visual metric is identified with the growth rate of the word metric. These results are complemented by a characterization of doubling visual metrics for accessible infinitely-ended groups: the visual metrics on the end boundary is doubling if and only if the group is virtually free. Consequently, there are at least two different bi-Hölder classes (and thus quasi-symmetric classes) of visual metrics on the end boundary.

which is intended to show that the concept of continuity is more subtle than one expects, because f is continuous exactly the irrational points (see [1, p. 86] or [2, p. 94]). And for the average student, this is the only property of the function f, that is, it is not interesting otherwise because there seems to be no reasonable context in which this function arises naturally. Its strange continuity behavior is only just punishment for having created a function by an artificial f-definition! But there is an interesting interpretation of this function: We consider only positive arguments, and we interpret t as the slope of a ray in the first quadrant of the x-y plane, starting the origin. Let us intercept the ray x = n, where n is any natural number chosen at random. Then f(t) is the that the y-coordinate of the interception point is also an integer, because this is the case if and only if n is divisible by q, and thus in making a great number of random guesses of the integer x-coordinate n we shall have also an integer y-coordinate in one of q cases. (Strictly speaking, there is no probability measure on N that makes the choice of each natural number equally probable! But if you choose your number n among the natural numbers 1, 2,. . ., N, randomly with respect to the uniform probability measure, your chance to get one which is divisible by q is [N/q]/N, which tends to l/q for N -> oo. So f(t) = l/q is the asymptotic density that n chosen in N is divisible by q.) Of course, this interpretation of the function f may be reformulated for a torus instead of the x-y plane. A slight modification of the function f doesn't change its continuity behavior but admits another nice interpretation: Place a square billiard table in the first quadrant of the x-y plane, with its lower-left corner the origin, and let a ball start this corner and proceed along the line with slope t > 0. Then

The limit frequency properties of trajectories of the simplest dynamical system generated by the left shift on the space of sequences of letters from a finite alphabet are studied. The following modification of the Shannon-McMillan-Breiman theorem is proved: for any invariant (not necessarily ergodic) probability measure μ on the sequence space, the logarithm of the cardinality of the set of all μ-typical sequences of length n is equivalent to nh(μ), where h(μ) is the entropy of the measure μ. Here a typical finite sequence of letters is understood as a sequence such that the empirical measure generated by it is close to μ (in the weak topology).

Embeddings of metric Boolean algebras in [formula omitted

We study the inter-dinucleotide distance distributions in the human genome, both in the whole-genome and protein-coding regions. The inter-dinucleotide distance is defined as the distance to the next occurrence of the same dinucleotide. We consider the 16 sequences of inter-dinucleotide distances and two reading frames. Our results show a period-3 oscillation in the protein-coding inter-dinucleotide distance distributions that is absent from the whole-genome distributions. We also compare the distance distribution of each dinucleotide to a reference distribution, that of a random sequence generated with the same dinucleotide abundances, revealing the CG dinucleotide as the one with the highest cumulative relative error for the first 60 distances. Moreover, the distance distribution of each dinucleotide is compared to the distance distribution of all other dinucleotides using the Kullback-Leibler divergence. We find that the distance distribution of a dinucleotide and that of its reversed complement are very similar, hence, the divergence between them is very small. This is an interesting finding that may give evidence of a stronger parity rule than Chargaff's second parity rule.

On hyperbolic once-punctured-torus bundles IV: Automata for lightning curves

Voice conversion with SI-DNN and KL divergence based mapping without parallel training data