Related papers: Tree Codes and a Conjecture on Exponential Sums
In this paper we give an elementary proof of the local sum conjecture in two dimensions. In a remarkable paper [CMN, arXiv:1810.11340], this conjecture has been established in all dimensions using sophisticated, powerful techniques from a…
New families of unit memory as well as multi-memory convolutional codes are constructed algebraically in this paper. These convolutional codes are derived from the class of group character codes. The proposed codes have basic generator…
Using well-known results from statistical physics, concerning the almost-sure behavior of the free energy of directed polymers in a random medium, we prove that random tree codes achieve the distortion-rate function almost surely under a…
We study the complexity and expressive power of conjunctive queries over unranked labeled trees represented using a variety of structure relations such as ``child'', ``descendant'', and ``following'' as well as unary relations for node…
Some class of sums which naturally include the sums of powers of integers is considered. A number of conjectures concerning a representation of these sums is made.
An alphabetic binary tree formulation applies to problems in which an outcome needs to be determined via alphabetically ordered search prior to the termination of some window of opportunity. Rather than finding a decision tree minimizing…
We give new general formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay (1983) for regular graphs. The general answer involves a quantity for infinite graphs that we call…
Tree-structured neural networks encode a particular tree geometry for a sentence in the network design. However, these models have at best only slightly outperformed simpler sequence-based models. We hypothesize that neural sequence models…
We investigate the structural relationship between prefix-free codes over the binary alphabet and a class of unlabeled rooted trees, which we call \emph{symmetric} trees. We establish a canonical correspondence between prefix-free codes and…
A Direct Sum Theorem holds in a model of computation, when solving some k input instances together is k times as expensive as solving one. We show that Direct Sum Theorems hold in the models of deterministic and randomized decision trees…
We consider the problem of computing the measure of a regular language of infinite binary trees. While the general case remains unsolved, we show that the measure of a language defined by a first-order formula with no descendant relation or…
Pairwise ordered tree alignment are combinatorial objects that appear in RNA secondary structure comparison. However, the usual representation of tree alignments as supertrees is ambiguous, i.e. two distinct supertrees may induce identical…
In this paper, the formulas of some exponential sums over finite field, related to the Coulter's polynomial, are settled based on the Coulter's theorems on Weil sums, which may have potential application in the construction of linear codes…
We formalize an existing computability-theoretic method of presenting first-order structures whose domains have the cardinality of the continuum. Work using these methods until now has emphasized their topological properties. We shift the…
We reduce the problem of constructing asymptotically good tree codes to the construction of triangular totally nonsingular matrices over fields with polynomially many elements. We show a connection of this problem to Birkhoff interpolation…
The tree code for the approximate evaluation of gravitational forces is extended and substantially accelerated by including mutual cell-cell interactions. These are computed by a Taylor series in Cartesian coordinates and in a completely…
We propose a new arithmetic for non-empty rooted unordered trees simply called trees. After discussing tree representation and enumeration, we define the operations of tree addition, multiplication and stretch, prove their properties, and…
Regular tree grammars and regular path expressions constitute core constructs widely used in programming languages and type systems. Nevertheless, there has been little research so far on frameworks for reasoning about path expressions…
In this letter we consider the ensemble of codes formed by the serial concatenation of a Hamming code and two accumulate codes. We show that this ensemble is asymptotically good, in the sense that most codes in the ensemble have minimum…
We consider the rooted trees which not have isomorphic representation and introduce a conception of complexity a natural number also. The connection between quantity such trees with $n$ edges and a complexity of natural number $n$ is…