Tuples Of Variables Research Articles

Identification of Relevant Features based on the Variation of Chrono-Valued Descriptors

In this paper the problem of feature selection in chronological information type is dealt. Objects are described by several tuples chained chrono-valued variables. We work with a universe of data consisting of sets of objects separated into classes wherein each object is described by t tuples of variables and also each tuple contains n variables. In general we can speak of two types of problems where data points are used and where no specific data are used. In the first case the data obtained from static objects, such as images; in the second case the data are obtained from dynamic objects, such as videos. Found research in which they treat information that care about the order of the data but with objects that are represented by a single tuple of variables, that is, are objects that represent the same phenomenon. This research is of great importance because it is intended to work with information that has two main features. The first feature is that objects are described by several tuples chained chrono-valued variables, that is, each object is represented in matrix form. The second feature is that a phenomenon described by the same object but at different points in time.

Almost everywhere elimination of probability quantifiers

AbstractWe obtain an almost everywhere quantifier elimination for (the noncritical fragment of) the logic with probability quantifiers, introduced by the first author in [10]. This logic has quantifiers like ∃≥3/4y which says that “for at least 3/4 of all y”. These results improve upon the 0-1 law for a fragment of this logic obtained by Knyazev [11]. Our improvements are:1. We deal with the quantifier ∃≥ry, where y is a tuple of variables.2. We remove the closedness restriction, which requires that the variables in y occur in all atomic subformulas of the quantifier scope.3. Instead of the unbiased measure where each model with universe n has the same probability, we work with any measure generated by independent atomic probabilities PR for each predicate symbol R.4. We extend the results to parametric classes of finite models (for example, the classes of bipartite graphs, undirected graphs, and oriented graphs).5. We extend the results to a natural (noncritical) fragment of the infinitary logic with probability quantifiers.6. We allow each PR, as well as each r in the probability quantifier (∃≥ry), to depend on the size of the universe.

On the complexity of Boolean functions with small number of ones

We consider the class of Boolean functions Fn,k consisting of all functions in n variables such that each of them takes value one exactly for k tuples of variables. We obtain linear in n estimates of the complexity of realisation of functions in Fn,k by circuits of functional elements over the basis containing all Boolean functions in two variables except the linear functions x ⊕ y and x ⊕ y ⊕ 1. It follows from these estimates that for small k, for example, for k < ln n, the well-known Finikov method provides asymptotically minimal circuits for all functions of Fn,k. In some cases, the known lower bounds for complexity of circuits give a possibility to prove the minimality of the corresponding circuits.

General Properties and Termination Conditions for Soft Constraint Propagation

Soft constraints based on semirings are a generalization of classical constraints, where tuples of variables' values in each soft constraint are associated to elements from an algebraic structure called semiring. This framework is able to express, for example, fuzzy, classical, weighted, valued and over-constrained constraint problems. Classical constraint propagation has been extended and adapted to soft constraints by defining a schema for soft constraint propagation l8r. On the other hand, in l1–3r it has been proven that most of the well known constraint propagation algorithms for classical constraints can be cast within a single schema. In this paper we combine these two schemas and we provide a more general framework where the schema of l3r can be used for soft constraints. In doing so, we generalize the concept of soft constraint propagation, and we provide new sufficient and independent conditions for its termination.

Approximate Solutions of Polynomial Equations

In this paper, we introduce “approximate solutions" to solve the following problem: given a polynomial F(x, y) over Q, where x represents an n -tuple of variables, can we find all the polynomials G(x) such that F(x, G(x)) is identically equal to a constant c in Q ? We have the following: let F(x, y) be a polynomial over Q and the degree of y in F(x, y) be n. Either there is a unique polynomial g(x) ∈Q [ x ], with its constant term equal to 0, such that F(x, y) =∑j=0ncj(y−g(x))jfor some rational numbers cj, hence, F(x, g(x) +a) ∈Q for all a∈Q, or there are at most t distinct polynomials g1(x),⋯ , gt(x), t≤n, such that F(x, gi(x)) ∈Q for 1 ≤i≤t. Suppose that F(x, y) is a polynomial of two variables. The polynomial g(x) for the first case, or g1(x),⋯ , gt(x) for the second case, are approximate solutions of F(x, y), respectively. There is also a polynomial time algorithm to find all of these approximate solutions. We then use Kronecker’s substitution to solve the case of F(x, y).

Avoidable algebraic subsets of Euclidean space

Fix an integer $n\ge 1$ and consider real $n$-dimensional $\mathbb R^n$. A partition of $\mathbb R^n$ avoids the polynomial $p(x_0,x_1,\dotsc ,x_{k-1})\in \mathbb R[x_0,x_1,\dotsc ,x_{k-1}]$, where each $x_i$ is an $n$-tuple of variables, if there is no set of the partition which contains distinct $a_0,a_1,\dotsc ,a_{k-1}$ such that $p(a_0,a_1,\dotsc ,a_{k-1})=0$. The polynomial is avoidable if some countable partition avoids it. The avoidable polynomials are studied here. The polynomial $\|x-y\|^2-\|y-z\|^2$ is an especially interesting example of an avoidable one. We find (1) a countable partition which avoids every avoidable polynomial over $Q$, and (2) a characterization of the avoidable polynomials. An important feature is that both the “master” partition in (1) and the characterization in (2) depend on the cardinality of $\mathbb R$.