CONTENTS 1. Introduction 2. Symbolic substitution (SS) rules for modified-signed-digit (MSD) addition and subtraction 3. Optical implementation of SS rules 4. Parallel MSD multiplication 4.1. MSD multiplication algorithm 4.2. Optical implementation of the MSD multiplication 5. MSD convergence division 6. Conversion between binary and MSD representations 6.1. From binary code to MSD code 6.2. Fr...

Heaviness refers to a sequence of partial sums maintaining a certain lower bound and was recently introduced and studied in [11]. After a review of basic properties to familiarize the reader with the ideas of heaviness, general principles of heaviness in symbolic dynamics are introduced. The classical Morse sequence is used to study a specific example of heaviness in a system with nontrivial ra...

We prove that every free group of finite rank can be realized as the fundamental group of a planar Rauzy fractal associated with a 4-letter unimodular cubic Pisot substitution. This characterizes all countable fundamental groups for planar Rauzy fractals. We give an explicit construction relying on two operations on substitutions: symbolic splittings and conjugations by free group automorphisms.

In this paper, we present some formulas of symbolic operator summation, which involving Generalization well-know number sequences or polynomial sequences, and mean while we obtain some identities about the sequences by employing M-R‘s substitution rule. Keywords—Generating functions; Operators sequence group; Riordan arrays; R.G Operator group; Combinatorial identities.

We prove that every free group of finite rank can be realized as the fundamental group of a planar Rauzy fractal associated with a 4-letter unimodular cubic Pisot substitution. This characterizes all countable fundamental groups for planar Rauzy fractals. We give an explicit construction relying on two operations on substitutions: symbolic splittings and conjugations by free group automorphisms.

The study is focused with the structure of social representations, common sense theories about relevant themes. It aims at investigating how central the ‘wisdom’ element is for young people’s representations on aging and also if the results provided by social representation questionnaires could be under the influence of normative pressures favoring politically correct responses, the so-called m...

We have designed and implemented an interprocedural algorithm to analyze symbolic value ranges that can be assumed by variables at any given point in a program. Our algorithm contrasts with related work on interprocedural value range analysis in that it extends the ability to handle symbolic range expressions. It builds on our previous work of intraprocedural symbolic range analysis. We have ev...

Higher-order representations of objects such as programs, proofs, formulas and types have become important to many symbolic computation tasks. Systems that support such representations usually depend on the implementation of an intensional view of the terms of some variant of the typed λ-calculus. Various notations have been proposed for λ-terms to explicitly treat substitutions as basis for re...