Digit Set Research Articles

UNIQUE EXPANSION OF POINTS OF A CLASS OF SELF‐SIMILAR SETS WITH OVERLAPS

For q >1, the set F q of real numbers which can be expanded in base q with respect to the digit set {0,1, q } is just a self-similar set with overlaps. We consider the subset of F q whose elements have a unique expansion and calculate its Hausdorff dimension for the case where .

Spectrality of a class of self-affine measures with decomposable digit sets

The self-affine measure associated with an expanding matrix and a finite digit set is uniquely determined by the self-affine identity with equal weight. The spectral and non-spectral problems on the selfaffine measures have some surprising connections with a number of areas in mathematics, and have been received much attention in recent years. In the present paper, we shall determine the spectrality and non-spectrality of a class of self-affine measures with decomposable digit sets. We present a method to deal with such case, and clarify the spectrality and non-spectrality of a class of self-affine measures by applying this method.

INTERSECTIONS OF CERTAIN DELETED DIGITS SETS

We consider some properties of the intersection of deleted digits Cantor sets with their translates. We investigate conditions on the set of digits such that, for any t between zero and the dimension of the deleted digits Cantor set itself, the set of translations such that the intersection has that Hausdorff dimension equal to t is dense in the set F of translations such that the intersection is non-empty. We make some simple observations regarding properties of the set F, in particular, we characterize when F is an interval, in terms of conditions on the digit set.

A genetic algorithm based region sampling for selection of local features in handwritten digit recognition application

Identification of local regions from where optimal discriminating features can be extracted is one of the major tasks in the area of pattern recognition. To locate such regions different kind of region sampling techniques are used in the literature. There is no standard methodology to identify exactly such regions. Here we have proposed a methodology where local regions of varying heights and widths are created dynamically. Genetic algorithm (GA) is then applied on these local regions to sample the optimal set of local regions from where an optimal feature set can be extracted that has the best discriminating features. We have evaluated the proposed methodology on a data set of handwritten Bangla digits. In the present work, we have randomly generated seven sets of local regions and from every set, GA selects an optimal group of local regions which produces best recognition performance with a support vector machine (SVM) based classifier. Other popular optimization techniques like simulated annealing (SA) and hill climbing (HC) have also been evaluated with the same data set and maximum recognition accuracies were found to be 97%, 96.7% and 96.7% for GA, SA and HC, respectively. We have also compared the performance of the present technique with those of other zone based techniques on the same database.

The Euclid Algorithm is totally gaussian

We consider Euclid’s gcd algorithm for two integers $(p, q)$ with $1 \leq p \leq q \leq N$, with the uniform distribution on input pairs. We study the distribution of the total cost of execution of the algorithm for an additive cost function $d$ on the set of possible digits, asymptotically for $N \to \infty$. For any additive cost of moderate growth $d$, Baladi and Vallée obtained a central limit theorem, and –in the case when the cost $d$ is lattice– a local limit theorem. In both cases, the optimal speed was attained. When the cost is non lattice, the problem was later considered by Baladi and Hachemi, who obtained a local limit theorem under an intertwined diophantine condition which involves the cost $d$ together with the “canonical” cost $c$ of the underlying dynamical system. The speed depends on the irrationality exponent that intervenes in the diophantine condition. We show here how to replace this diophantine condition by another diophantine condition, much more natural, which already intervenes in simpler problems of the same vein, and only involves the cost $d$. This “replacement” is made possible by using the additivity of cost $d$, together with a strong property satisfied by the Euclidean Dynamical System, which states that the cost $c$ is both “strongly” non additive and diophantine in a precise sense. We thus obtain a local limit theorem, whose speed is related to the irrationality exponent which intervenes in the new diophantine condition. We mainly use the previous proof of Baladi and Hachemi, and “just” explain how their diophantine condition may be replaced by our condition. Our result also provides a precise comparison between the rational trajectories of the Euclid dynamical system and the real trajectories.

Model selection for Gaussian kernel PCA denoising.

We propose kernel parallel analysis (kPA) for automatic kernel scale and model order selection in Gaussian kernel principal component analysis (KPCA). Parallel analysis is based on a permutation test for covariance and has previously been applied for model order selection in linear PCA, we here augment the procedure to also tune the Gaussian kernel scale of radial basis function based KPCA. We evaluate kPA for denoising of simulated data and the U.S. postal data set of handwritten digits. We find that kPA outperforms other heuristics to choose the model order and kernel scale in terms of signal-to-noise ratio of the denoised data.

ERROR RECOVERY IN CONTINUOUS VALUED NUMBER SYSTEM

Continuous valued number system is a novel number system which can be employed for developing analog signal processing units. It is a continuous number system which is represented by a set of continuous digits. One of the main features of this system is that digits share information, and have a digit-level redundancy. This redundancy is used to protect the digits against environment imperfections when implemented by analog circuits. In this paper, integrity of this number system in representing real values is explored. The study is required to show the effect of implementation imperfections which is an indicative of its feasibility. Effects of possible errors and error threshold for implementing this system are studied in this paper. An error recovery method is proposed, which enhances this number system representation. An efficient error recovery method extends the application of this number system in high density memory and storage devices for hardware implementations of neural networks.

Structure of planar integral self‐affine tilings

AbstractFor a self‐affine tile in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf {R}^2$\end{document} generated by an expanding matrix \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$A\in M_2(\mathbf {Z})$\end{document} and an integral consecutive collinear digit set \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal D}$\end{document}, Leung and Lau [Trans. Amer. Math. Soc. 359, 3337–3355 (2007).] provided a necessary and sufficient algebraic condition for it to be disklike. They also characterized the neighborhood structure of all disklike tiles in terms of the algebraic data A and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal D}$\end{document}. In this paper, we completely characterize the neighborhood structure of those non‐disklike tiles. While disklike tiles can only have either six or eight edge or vertex neighbors, non‐disklike tiles have much richer neighborhood structure. In particular, other than a finite set, a Cantor set, or a set containing a nontrivial continuum, neighbors can intersect in a union of a Cantor set and a countable set.

Spectrality of planar self-affine measures with two-element digit set

The iterated function system with two-element digit set is the simplest case and the most important case in the study of self-affine measures. The one-dimensional case corresponds to the Bernoulli convolution whose spectral property is understandable. The higher dimensional analogue is not known, for which two conjectures about the spectrality and the non-spectrality remain open. In the present paper, we consider the spectrality and non-spectrality of planar self-affine measures with two-element digit set. We give a method to deal with the two-dimensional case, and clarify the spectrality and non-spectrality of a class of planar self-affine measures. The result here provides some supportive evidence to the two related conjectures.

Analysis of a class of spectral pair conditions

For an expanding integer matrix M ∈ M n (ℤ) and two finite digit sets D, S ⊂ ℤ n with 0 ∈ D ∩ S, we shall investigate and study the possible conditions on the spectral pair (µ M, D , Λ(M, S)) associated with the iterated function systems {ϕ d (x) = M −1(x + d)} d∈D and {ψ s (x) = M*x + s} s∈S in the case when |D| = |S| = |det(M)|. Under the condition that (M −1 D, S) is a compatible pair, we obtain a series of necessary and sufficient conditions for (µ M, D , Λ(M, S)) to be a spectral pair. These conditions include how to characterize the invariant sets Λ(M, S) and T(M, D) such that Λ(M, S) = ℤ n and µ L (T(M, D)) = 1 which play an important role in the number system research and in the construction of Haar-type orthogonal wavelet basis respectively.

Transfer of Learned Manipulation following Changes in Degrees of Freedom

The present study was designed to determine whether manipulation learned with a set of digits can be transferred to grips involving a different number of digits, and possible mechanisms underlying such transfer. The goal of the task was to exert a torque and vertical forces on a visually symmetrical object at object lift onset to balance the external torque caused by asymmetrical mass distribution. Subjects learned this manipulation through consecutive practice using one grip type (two or three digits), after which they performed the same task but with another grip type (e.g., after adding or removing one digit, respectively). Subjects were able to switch grip type without compromising the behavioral outcome (i.e., the direction, timing, and magnitude of the torque exerted on the object was unchanged), despite the use of significantly different digit force-position coordination patterns in the two grip types. Our results support the transfer of learning for anticipatory control of manipulation and indicate that the CNS forms an internal model of the manipulation task independent of the effectors that are used to learn it. We propose that sensory information about the new digit placement--resulting from adding or removing a digit immediately after the switch in grip type--plays an important role in the accurate modulation of new digit force distributions. We discuss our results in relation to studies of manipulation reporting lack of learning transfer and propose a theoretical framework that accounts for failure or success of motor learning generalization.

Continuous dependence on parameters of certain self-affine measures, and their singularity

In this paper, we first prove that the self-affine sets depend continuously on the expanding matrix and the digit set, and the corresponding self-affine measures with respect to the probability weight behave in much the same way. Moreover, we obtain some sufficient conditions for certain self-affine measures to be singular.

Rapid parallel semantic processing of numbers without awareness

In this study, we investigate whether multiple digits can be processed at a semantic level without awareness, either serially or in parallel. In two experiments, we presented participants with two successive sets of four simultaneous Arabic digits. The first set was masked and served as a subliminal prime for the second, visible target set. According to the instructions, participants had to extract from the target set either the mean or the sum of the digits, and to compare it with a reference value. Results showed that participants applied the requested instruction to the entire set of digits that was presented below the threshold of conscious perception, because their magnitudes jointly affected the participant’s decision. Indeed, response decision could be accurately modeled as a sigmoid logistic function that pooled together the evidence provided by the four targets and, with lower weights, the four primes. In less than 800ms, participants successfully approximated the addition and mean tasks, although they tended to overweight the large numbers, particularly in the sum task. These findings extend previous observations on ensemble coding by showing that set statistics can be extracted from abstract symbolic stimuli rather than low-level perceptual stimuli, and that an ensemble code can be represented without awareness.

Connectedness of a class of planar self-affine tiles

We consider a class of planar self-affine tiles T that are generated by the lower triangular expanding matrices and the product-form digit sets. We give necessary and sufficient conditions for T to be connected and disk-like. Also for the disconnect case, we give a condition that enumerates the number of connected components of T.

Improving the speed of decimal division

The authors study previous major contributions to digit recurrence decimal division hardware and focus on techniques for improving the performance of quotient digit selection (QDS) as the most complex part. In particular, Design D1 uses the digit set [-5, 5] for quotient digits. Another design (D2) uses mixed binary/decimal carry-save manipulation of the few most significant digits of partial remainders. Motivated by successful combined arithmetic algorithms such as hybrid adders, the authors offer a decimal division scheme that takes advantage of the best design options of D1 and D2 with due modifications that significantly enhance the division speed. In particular, they configure the architectures of QDS and partial remainder computation paths in favour of reduced balanced latencies of both. Furthermore, they remove the rounding cycle by cost-free auto-rounding, which is an exclusive advantage of the digit set [-5, 5]. The authors of D1 and D2 have used logical effort (LE) and circuit synthesis to evaluate their dividers, respectively. Therefore for a fair comparison, the authors evaluate the proposed design (D3) with both methods. The LE-based D3/D1 comparison shows 21- more speed at the cost of 6- more area, whereas the synthesis-based D3/D2 comparison results in 46- less latency and 23- less area.

Assessing the impact of verbal and visuospatial working memory load on eye-gaze cueing

Observers tend to respond more quickly to peripheral stimuli that are being gazed at by a centrally presented face, than to stimuli that are not being gazed at. While this gaze-cueing effect was initially seen as reflexive, there have also been some indications that top-down control processes may be involved. Therefore, the present investigation employed a dual-task paradigm to attempt to disrupt the putative control processes involved in gaze cueing. Two experiments examined the impact of working memory load on gaze cueing. In Experiment 1, participants were required to hold a set of digits in working memory during each gaze trial. In Experiment 2, the gaze task was combined with an auditory task that required the manipulation and maintenance of visuospatial information. Gaze cueing effects were observed, but they were not modulated by dual-task load in either experiment. These results are consistent with traditional accounts of gaze cueing as a highly reflexive process.

REMARKS ON SELF-AFFINE FRACTALS WITH POLYTOPE CONVEX HULLS

Suppose that the set [Formula: see text] of real n × n matrices has joint spectral radius less than 1. Then for any digit set D = {d1, …, dq} ⊂ ℝn, there exists a unique non-empty compact set [Formula: see text] satisfying [Formula: see text], which is typically a fractal set. We use the infinite digit expansions of the points of F to give simple necessary and sufficient conditions for the convex hull of F to be a polytope. Additionally, we present a technique to determine the vertices of such polytopes. These answer some of the related questions of Strichartz and Wang, and also enable us to approximate the Lebesgue measure of such self-affine sets. To show the use of our results, we also give several examples including the Levy dragon and the Heighway dragon.

Boundary parametrization of planar self-affine tiles with collinear digit set

We consider a class of planar self-affine tiles \( T = M^{ - 1} \cup _{a \in \mathcal{D}} (T + a) \) generated by an expanding integral matrix M and a collinear digit set \( \mathcal{D} \) as follows: $$ M = \left( \begin{gathered} 0 - B \hfill \\ 1 - A \hfill \\ \end{gathered} \right),\mathcal{D} = \left\{ {\left( \begin{gathered} 0 \hfill \\ 0 \hfill \\ \end{gathered} \right),...,\left( \begin{gathered} |B| - 1 \hfill \\ 0 \hfill \\ \end{gathered} \right)} \right\} $$ . We give a parametrization \( \mathbb{S}^1 \to \partial T \) of the boundary of T with the following standard properties. It is Holder continuous and associated with a sequence of simple closed polygonal approximations whose vertices lie on ∂T and have algebraic preimages. We derive a new proof that T is homeomorphic to a disk if and only if 2|A| ⩽ |B + 2|.

Fraction Sets for Basic Digit Sets

A finite set of integers $D$ with $0\in D$ is basic for the base $b\in \mathbb Z$ if every $z\in\mathbb Z$ can be written uniquely as an integer in the base $b$ with digits from $D$. Since such a base representation is unsigned, basic sets must have a negative base $b$ or some negative integers in $D$. The fraction set for $(b,D)$ is the set of all representable numbers with integer part zero. We show that if $(b,D)$ is basic and $D$ consists of consecutive digits, then the fraction set is an interval of length one whose endpoints have redundant representations. Furthermore, we show that if $D$ does not consist of consecutive integers, then $F$ is disconnected.

Redundant τ-adic expansions I: non-adjacent digit sets and their applications to scalar multiplication

This paper investigates some properties of τ-adic expansions of scalars. Such expansions are widely used in the design of scalar multiplication algorithms on Koblitz curves, but at the same time they are much less understood than their binary counterparts. Solinas introduced the width-wτ-adic non-adjacent form for use with Koblitz curves. This is an expansion of integers $${z = \sum_{i=0}^\ell z_i \tau^i}$$, where τ is a quadratic integer depending on the curve, such that z i ? 0 implies z w+i-1 = . . . = z i+1 = 0, like the sliding window binary recodings of integers. It uses a redundant digit set, i.e., an expansion of an integer using this digit set need not be uniquely determined if the syntactical constraints are not enforced. We show that the digit sets described by Solinas, formed by elements of minimal norm in their residue classes, are uniquely determined. Apart from this digit set of minimal norm representatives, other digit sets can be chosen such that all integers can be represented by a width-w non-adjacent form using those digits. We describe an algorithm recognizing admissible digit sets. Results by Solinas and by Blake, Murty, and Xu are generalized. In particular, we introduce two new useful families of digit sets. The first set is syntactically defined. As a consequence of its adoption we can also present improved and streamlined algorithms to perform the precomputations in τ-adic scalar multiplication methods. The latter use an improvement of the computation of sums and differences of points on elliptic curves with mixed affine and Lopez---Dahab coordinates. The second set is suitable for low-memory applications, generalizing an approach started by Avanzi, Ciet, and Sica. It permits to devise a scalar multiplication algorithm that dispenses with the initial precomputation stage and its associated memory space. A suitable choice of the parameters of the method leads to a scalar multiplication algorithm on Koblitz Curves that achieves sublinear complexity in the number of expensive curve operations.