Dyadic Rational Research Articles

On an Example of du Bois-Reymond, Dyadic Rationals, and Thomae’s Function

Summary Paul du Bois-Reymond made several important contributions to the theory of functions. In this paper, we investigate one of these contributions, an example he wrote in passing in a paper without offering further explanation about its inner workings or inspiration. We present a proof he could possibly have envisioned, considering his knowledge at the time, and a reason for what can now be considered an odd choice in his example.

Fast Dimension Spectrum for a Potential with a Logarithmic Singularity

We regard the classic Thue–Morse diffraction measure as an equilibrium measure for a potential function with a logarithmic singularity over the doubling map. Our focus is on unusually fast scaling of the Birkhoff sums (superlinear) and of the local measure decay (superpolynomial). For several scaling functions, we show that points with this behavior are abundant in the sense of full Hausdorff dimension. At the fastest possible scaling, the corresponding rates reveal several remarkable phenomena. There is a gap between level sets for dyadic rationals and non-dyadic points, and beyond dyadic rationals, non-zero accumulation points occur only within intervals of positive length. The dependence between the smallest and the largest accumulation point also manifests itself in a non-trivial joint dimension spectrum.

The Schwartzman group of an affine transformation

We compute the Schwartzman group associated with an ergodic affine automorphism of a compact connected abelian group given by the composition of an automorphism of the group and a translation by an element in the path component of the identity. We show that the Schwartzman group can be characterized by evaluating the invariant characters of the automorphism at the group element by which one translates. As a byproduct, we show that the set of labels associated with the doubling map on the dyadic solenoid is trivial, which in turn allows us to show that any ergodic family of Jacobi matrices defined over the doubling map has connected almost-sure essential spectrum.

Total dual dyadicness and dyadic generating sets

A vector is dyadic if each of its entries is a dyadic rational number, i.e. of the form $$\frac{a}{2^k}$$ for some integers a, k with $$k\ge 0$$ . A linear system $$Ax\le b$$ with integral data is totally dual dyadic if whenever $$\min \{b^\top y:A^\top y=w,y\ge \textbf{0}\}$$ for w integral, has an optimal solution, it has a dyadic optimal solution. In this paper, we study total dual dyadicness, and give a co-NP characterization of it in terms of dyadic generating sets for cones and subspaces, the former being the dyadic analogue of Hilbert bases, and the latter a polynomial-time recognizable relaxation of the former. Along the way, we see some surprising turn of events when compared to total dual integrality, primarily led by the density of the dyadic rationals. Our study ultimately leads to a better understanding of total dual integrality and polyhedral integrality. We see examples from dyadic matroids, T-joins, cycles, and perfect matchings of a graph.

Finitely generated dyadic convex sets

Dyadic rationals are rationals whose denominator is a power of 2. We define dyadic n-dimensional convex sets as the intersections with n-dimensional dyadic space of an n-dimensional real convex set. Such a dyadic convex set is said to be a dyadic n-dimensional polytope if the real convex set is a polytope whose vertices lie in the dyadic space. Dyadic convex sets are described as subreducts (subalgebras of reducts) of certain faithful affine spaces over the ring of dyadic numbers, or equivalently as commutative, entropic and idempotent groupoids under the binary operation of arithmetic mean. The paper contains two main results. First, it is proved that, while all dyadic polytopes are finitely generated, only dyadic simplices are generated by their vertices. This answers a question formulated in an earlier paper. Then, a characterization of finitely generated subgroupoids of dyadic convex sets is provided, and it is shown how to use the characterization to determine the minimal number of generators of certain convex subsets of the dyadic plane.

A Note on Finite Coarse Shape Groups

In this paper, we investigate properties concerning some recently introduced finite coarse shape invariants—the k-th finite coarse shape group of a pointed topological space and the k-th relative finite coarse shape group of a pointed topological pair. We define the notion of finite coarse shape group sequence of a pointed topological pair X,X0,x0 as an analogue of homotopy and (coarse) shape group sequences and show that for any pointed topological pair, the corresponding finite coarse shape group sequence is a chain. On the other hand, we construct an example of a pointed pair of metric continua whose finite coarse shape group sequence fails to be exact. Finally, using the aforementioned pair of metric continua together with a pointed dyadic solenoid, we show that finite coarse-shape groups, in general, differ from both shape and coarse-shape groups.

Synthetic proofs

This is a contribution to the idea that some proofs in first-order logic are synthetic. Syntheticity is understood here in its classical geometrical sense. Starting from Jaakko Hintikka’s original idea and Allen Hazen’s insights, this paper develops a method to define the ‘graphical form’ of formulae in monadic and dyadic fraction of first-order logic. Then a synthetic inferential step in Natural Deduction is defined. A proof is defined as synthetic if it includes at least one synthetic inferential step. Finally, it will be shown that the proposed definition is not sensitive to different ways of driving the same conclusion from the same assumptions.

The finite coarse shape groups

In this paper we define a new pointed finite coarse shape invariant - k-th finite coarse shape group of a pointed topological space and compare it with the corresponding shape and coarse shape group. We also define a functor πˇk⊛ from the pointed finite coarse shape category Sh⋆⊛ to the category Grp of all groups (for k=0 to the category Set⋆ of all pointed sets) associating with every pointed topological space (X,x0) a k-th finite coarse shape group πˇk⊛(X,x0) and with every finite coarse shape morphism F⊛:(X,x0)→(Y,y0) a homomorphism (a base point preserving function) πˇk⊛(F⊛):πˇk⊛(X,x0)→πˇk⊛(Y,y0). We show that, in a case of a pointed compact polyhedral system, functor πˇk⊛ is continuous, i.e., commutes with the inverse limit. Finally, finite coarse shape groups of some interesting spaces (dyadic solenoid and arbitrary stable pointed space) are calculated.

Tableless Calculation of Circular Functions on Dyadic Rationals

I would like to tell a story. A story about a beautiful mathematical relationship that elucidates the computational view on the classic subject of trigonometry. All stories need a language, and for this particular story an algorithmic language ought to do well. What makes a language algorithmic? From our perspective as the functional programming community, an algorithmic language provides means to express computation in terms of functions, with no implementation-imposed limitations. We develop a new algorithm for the computation of trigonometric functions on dyadic rationals, together with the language used to express it, in Scheme. We provide a mechanically-derived algorithm for the computation of the inverses of our target functions. We address efficiency and accuracy concerns that pertain to the implementation of the proposed algorithm either in hardware or software.

Conditions of A-completeness for linear automata over dyadic rationals

Abstract We consider the problem of A-completeness in the class of linear automata such that the sets of inputs, outputs and states are Cartesian products of dyadic rationals; systems checked for completeness are comprised of a variable finite set and a fixed additional set. We obtain conditions of A-completeness in terms of maximal subclasses in the cases when the additional set is the set of all unary automata and when the additional set consists of the adder.

Hausdorff measures, dyadic approximations, and the Dobiński set

The Dobiński set D is an exceptional set for a certain infinite product identity, whose points are characterized as having exceedingly good approximations by dyadic rationals. We study the Hausdorff dimension and logarithmic measure of D by means of the mass transference principle and by the construction of certain appropriate Cantor-like sets, termed willow sets, contained in D.

Conditions of A-completeness for linear automata over dyadic rationals

Рассматривается задача об $A$-полноте конечных систем автоматов с добавками в классе линейных автоматов, у которых в качестве алфавита состояний, входного и выходного алфавитов выступают декартовы произведения множества двоично-рациональных чисел. В терминах предполных классов описываются условия $A$-полноты для систем, содержащих все одноместные автоматы и конечную добавку, а также конечных систем, содержащих сумматор.

Operator-Algebraic Construction of Gauge Theories and Jones’ Actions of Thompson’s Groups

Using ideas from Jones, lattice gauge theory and loop quantum gravity, we construct 1+1-dimensional gauge theories on a spacetime cylinder. Given a separable compact group $G$, we construct localized time-zero fields on the spatial torus as a net of C*-algebras together with an action of the gauge group that is an infinite product of $G$ over the dyadic rationals and, using a recent machinery of Jones, an action of Thompson's group $T$ as a replacement of the spatial diffeomorphism group. Adding a family of probability measures on the unitary dual of $G$ we construct a state and obtain a net of von Neumann algebras carrying a state-preserving gauge group action. For abelian $G$, we provide a very explicit description of our algebras. For a single measure on the dual of $G$, we have a state-preserving action of Thompson's group and semi-finite von Neumann algebras. For $G=\mathbf{S}$ the circle group together with a certain family of heat-kernel states providing the measures, we obtain hyperfinite type III factors with a normal faithful state providing a nontrivial time evolution via Tomita-Takesaki theory (KMS condition). In the latter case, we additionally have a non-singular action of the group of rotations with dyadic angles, as a subgroup of Thompson's group $T$, for geometrically motivated choices of families of heat-kernel states.

Collatz on the Dyadic Rationals in [0.5, 1) with Fractals: How Bit Strings Change Their Length Under 3x + 1

The reduced Collatz map (Syracuse function) can be stated as “for any odd positive integer x, calculate and then divide by 2 until the result is odd.” We calculate the change in bit string length caused by this map. The result arises from a novel reformulation of the Collatz process “for any fraction in [0.5, 1) with a binary representation of finite length, append If the resulting number is smaller than 2/3 then multiply by 3/2 otherwise multiply by 3/4.” The domain is where strings may shrink versus for may grow. If the Collatz map has non-trivial periodic orbits then they will arise from a fractal that has been added to a map that lacks periodic orbits. If there are an infinite number of such orbits then as their length increases, they will make roughly 0.71 visits to the may-shrink domain for each visit to the may-grow domain.

WHT and Matrix Decomposition-Based Approximated IDCT Architecture for HEVC

This brief proposes a new model of inverse discrete cosine transform (IDCT) kernel for high efficiency video coding (HEVC). The kernel supports IDCT of order 4, 8, 16, and 32. The proposed architecture uses the Chen’s algorithm as well as Walsh–Hadamard transform-based matrix decomposition method for complexity reduction. Additionally, an approximation scheme is proposed to replace all the coefficients by dyadic fractions on the basis of orthogonality and normality errors. As a consequence, the entire transform can be realized by using add and shift operations and hence, rotation unit is inessential. This reduces hardware cost significantly and the speed of the architecture also gets enhanced. The proposed approximated model complies with the requirements of HEVC and during video coding, less than 0.5 dB PSNR variation is observed as compared to that of the HEVC reference software. The proposed integer IDCT architecture is implemented on CMOS 90-nm ASIC platform. It is observed that it requires 36.6K logic gates at 200 MHz operating frequency.

Duality for dyadic triangles

This paper is a direct continuation of the paper “Duality for dyadic intervals” by the same authors, and can be considered as its second part. Dyadic rationals are rationals whose denominator is a power of 2. Dyadic triangles and dyadic polygons are, respectively, defined as the intersections with the dyadic plane of a triangle or polygon in the real plane whose vertices lie in the dyadic plane. The one-dimensional analogues are dyadic intervals. Algebraically, dyadic polygons carry the structure of a commutative, entropic and idempotent groupoid under the binary operation of arithmetic mean. The first paper dealt with the structure of finitely generated subgroupoids of the dyadic line, which were shown to be isomorphic to dyadic intervals. Then a duality between the class of dyadic intervals and the class of certain subgroupoids of the dyadic unit square was described. The present paper extends the results of the first paper, provides some characterizations of dyadic triangles, and describes a duality for the class of dyadic triangles. As in the case of intervals, the duality is given by an infinite dualizing (schizophrenic) object, the dyadic unit interval. The dual spaces are certain subgroupoids of the dyadic unit cube, considered as (commutative, idempotent and entropic) groupoids with additional constant operations.

Duality for dyadic intervals

In an earlier paper, Romanowska, Ślusarski and Smith described a duality between the category of (real) polytopes (finitely generated real convex sets considered as barycentric algebras) and a certain category of intersections of hypercubes, considered as barycentric algebras with additional constant operations. This paper is a first step in finding a duality for dyadic polytopes, analogues of real convex polytopes, but defined over the ring [Formula: see text] of dyadic rational numbers instead of the ring of reals. A dyadic [Formula: see text]-dimensional polytope is the intersection with the dyadic space [Formula: see text] of an [Formula: see text]-dimensional real polytope whose vertices lie in the dyadic space. The one-dimensional analogues are dyadic intervals. Algebraically, dyadic polytopes carry the structure of a commutative, entropic and idempotent groupoid under the operation of arithmetic mean. Such dyadic polytopes do not preserve all properties of real polytopes. In particular, there are infinitely many (pairwise non-isomorphic) dyadic intervals. We first show that finitely generated subgroupoids of the groupoid [Formula: see text] are all isomorphic to dyadic intervals. Then, we describe a duality for the class of dyadic intervals. The duality is given by an infinite dualizing (schizophrenic) object, the dyadic unit interval. The dual spaces are certain subgroupoids of the square of the dyadic unit interval with additional constant operations. A second paper deals with a duality for dyadic triangles.

Construction of an exotic measure: Dyadic doubling and triadic doubling does not imply doubling

What kinds of properties of measures, and of functions, must necessarily hold in a continuous setting if they hold in certain martingale settings? And for which martingale settings? Consider the doubling property enjoyed by some measures, of importance in both harmonic analysis and complex analysis. It is known that if a measure is (dyadic) doubling with respect to the standard lattice of dyadic intervals in the real line, and also with respect to certain translations of that dyadic lattice, then the measure has the doubling property on all intervals. Analogous results hold for functions of bounded mean oscillation (BMO), and also for Muckenhoupt's Ap weights, for reverse Hölder weights, and for functions of vanishing mean oscillation (VMO), both on Euclidean spaces and on one-parameter or product spaces of homogeneous type in the sense of Coifman and Weiss. By contrast, as we show here, a measure that is doubling both with respect to the standard lattice of dyadic intervals and also with respect to a lattice of triadic intervals need not be doubling on all intervals. In other words, the doubling property is not necessarily inherited from the dyadic and triadic martingale settings to the continuous setting. The proof is by means of an intricate construction, yielding measures that are both dyadic and triadic doubling but not doubling, and making use of some number-theoretic properties of dyadic and triadic rational numbers.

Low-Complexity Approximate Convolutional Neural Networks.

In this paper, we present an approach for minimizing the computational complexity of the trained convolutional neural networks (ConvNets). The idea is to approximate all elements of a given ConvNet and replace the original convolutional filters and parameters (pooling and bias coefficients; and activation function) with an efficient approximations capable of extreme reductions in computational complexity. Low-complexity convolution filters are obtained through a binary (zero and one) linear programming scheme based on the Frobenius norm over sets of dyadic rationals. The resulting matrices allow for multiplication-free computations requiring only addition and bit-shifting operations. Such low-complexity structures pave the way for low power, efficient hardware designs. We applied our approach on three use cases of different complexities: 1) a "light" but efficient ConvNet for face detection (with around 1000 parameters); 2) another one for hand-written digit classification (with more than 180 000 parameters); and 3) a significantly larger ConvNet: AlexNet with million matrices. We evaluated the overall performance on the respective tasks for different levels of approximations. In all considered applications, very low-complexity approximations have been derived maintaining an almost equal classification performance.

Infinitely supported Liouville measures of Schreier graphs

We provide equivalent conditions for Liouville property of actions of groups. As an application, we show that there is a Liouville measure for the action of the Thompson group F on dyadic rationals. This result should be compared with a recent result of Kaimanovich, where he shows that the action of the Thompson group F on dyadic rationals is not Liouville for all finitely supported measures. As another application we show that there is a Liouville measure for lamplighter actions. This gives more examples of non-amenable Liouville actions.