Preimage Sets Research Articles

Sorting with a popqueue

We introduce a new sorting device for permutations, which we call popqueue. It consists of a special queue, having the property that any time one wants to extract elements from the queue, actually all the elements currently in the queue are poured into the output. We illustrate two distinct optimal algorithms, called Min and Cons, to sort a permutation using such a device, which allow us also to characterize sortable permutations in terms of pattern avoidance. We next investigate what happens by making two passes through a popqueue, showing that the set of sortable permutations is not a class for Min, whereas it is for Cons. In the latter case we also explicitly find the basis of the class of sortable permutations. Finally, we study preimages under Cons (by means of an equivalent version of the algorithm), and find a characterization of the set of preimages of a given permutation. We also give some enumerative results concerning the number of permutations having k preimages, for k = 1, 2, 3, and we conclude by observing that there exist permutations having k preimages for any value of k ≥ 0.

Uniqueness of L functions with special class of meromorphic functions via sharing of pre-image sets

Uniqueness of L functions with special class of meromorphic functions via sharing of pre-image sets

Value Distributions of Perfect Nonlinear Functions

In this paper, we study the value distributions of perfect nonlinear functions, i.e., we investigate the sizes of image and preimage sets. Using purely combinatorial tools, we develop a framework that deals with perfect nonlinear functions in the most general setting, generalizing several results that were achieved under specific constraints. For the particularly interesting elementary abelian case, we derive several new strong conditions and classification results on the value distributions. Moreover, we show that most of the classical constructions of perfect nonlinear functions have very specific value distributions, in the sense that they are almost balanced. Consequently, we completely determine the possible value distributions of vectorial Boolean bent functions with output dimension at most 4. Finally, using the discrete Fourier transform, we show that in some cases value distributions can be used to determine whether a given function is perfect nonlinear, or to decide whether given perfect nonlinear functions are equivalent.

On cardinalities of Green’s equivalence classes in the semigroup of difunctional binary relations

In this paper, we considered the set DX, consisting of all binary relations α ⊆ X × X satisfying (∀x,y,u,v ∈ X) (x,u),(x,v),(y,u)∈α ⇒(y,v) ∈ α. This set is an inverse semigroup under a binary operation defined by xα = yβ−1 ≠ ∅, where xα denotes the set of images of x under α, and yβ−1 denotes the set of pre-images of y under β. Combinatorial results relating to Green’s relations in semigroup are obtained. In particular, we obtained cardinalities of Green’s equivalence classes in the semigroup for the case where X is finite. Also, we obtained the number of idempotent elements in to be equal to , where n = |X| and B(k) is the Bell number defined as the number of partitions of a set of k elements.

Preimages under the Bubblesort Operator

We study preimages of permutations under the bubblesort operator $\mathbf{B}$. We achieve a description of these preimages much more complete than what is known for the more complicated sorting operators $\mathbf{S}$ (stacksort) and $\mathbf{Q}$ (queuesort). We describe explicitly the set of preimages under $\mathbf{B}$ of any permutation $\pi$ from the left-to-right maxima of $\pi$, showing that there are $2^{k-1}$ such preimages if $k$ is the number of these left-to-right maxima. We further consider, for each $n$, the tree $T_n$ recording all permutations of size $n$ in its nodes, in which an edge from child to parent corresponds to an application of $\mathbf{B}$ (the root being the identity permutation), and we present several properties of these trees. In particular, for each permutation $\pi$, we show how the subtree of $T_n$ rooted at $\pi$ is determined by the number of left-to-right maxima of $\pi$ and the length of the longest suffix of left-to-right maxima of $\pi$.
 Building on this result, we determine the number of nodes and leaves at every height in such trees, and we recover (resp. obtain) the average height of nodes (resp. leaves) in $T_n$.

New results on vectorial dual-bent functions and partial difference sets

Bent functions \(f: V_{n}\rightarrow \mathbb {F}_{p}\) play an important role in constructing partial difference sets, where \(V_{n}\) denotes an n-dimensional vector space over \(\mathbb {F}_{p}\), p is an odd prime. In [2, 3], the so-called vectorial dual-bent functions are considered to construct partial difference sets. In [2], Çeşmelioğlu et al. showed that for certain vectorial dual-bent functions \(F: V_{n}\rightarrow V_{s}\), the preimage set of 0 for F forms a partial difference set. In [3], Çeşmelioğlu et al. showed that for a class of Maiorana-McFarland vectorial dual-bent functions \(F: V_{n}\rightarrow \mathbb {F}_{p^s}\), the preimage set of the squares (non-squares) in \(\mathbb {F}_{p^s}^{*}\) for F forms a partial difference set. In this paper, we further study vectorial dual-bent functions and partial difference sets. We prove that for certain vectorial dual-bent functions \(F: V_{n}\rightarrow \mathbb {F}_{p^s}\), the preimage set of the squares (non-squares) in \(\mathbb {F}_{p^s}^{*}\) for F and the preimage set of any coset of some subgroup of \(\mathbb {F}_{p^s}^{*}\) for F form partial difference sets. Furthermore, explicit constructions of partial difference sets are yielded from some (non-)quadratic vectorial dual-bent functions. In this paper, we illustrate that many results of using weakly regular p-ary bent functions to construct partial difference sets are special cases of our results. In [2], the authors considered weakly regular p-ary bent functions f with \(f(0)=0\). They showed that if such a function f is an l-form with \(gcd(l-1, p-1)=1\) for some integer \(1\le l\le p-1\), then f is vectorial dual-bent. We prove that the converse also holds, which answers one open problem proposed in [3].

ON THE SEMIGROUP OF DIFUNCTIONAL BINARY RELATIONS

In this paper, we have examine some properties of elements of the semigroup , where DX, is the set of all binary relations α ⊆ X × X satisfying , (), and is a binary operation on DX defined by () , with xα denoting set of images of x under α, and yβ−1 denoting set of pre-images of y under β. In particular, we showed that in the semigroup there is no distinction between the concepts of reflexive and symmetric relations. We also presented a characterization of idempotent elements in in term of equivalence relations.

Collisions and incidence of vertices and components in the graph ofk-fold iteration of the uniform random mapping

AbstractThe probabilistic characteristics of the graph ofk-fold iteration of uniform random mapping are studied. Formulas for the distribution of the length of the aperiodicity segment of an arbitrary vertex with some restrictions are calculated. We obtain exact expressions for the probabilities that two arbitrary vertices belong to the same connected component, that an arbitrary vertex belongs to the preimage set of another vertex and that there exists a collision in the considered graph.

Preimages under the Queuesort algorithm

Following the footprints of what has been done with the algorithm Stacksort, we investigate the preimages of the map associated with a slightly less well known algorithm, called Queuesort. After having described an equivalent version of Queuesort, we provide a recursive description of the set of all preimages of a given permutation, which can be also translated into a recursive procedure to effectively find such preimages. We then deal with some enumerative issues. More specifically, we investigate the cardinality of the set of preimages of a given permutation, showing that all cardinalities are possible, except for 3. We also give exact enumeration results for the number of permutations having 0,1 and 2 preimages. Finally, we consider the special case of those permutations π whose set of left-to-right maxima is the disjoint union of a prefix and a suffix of π: we determine a closed formula for the number of preimages of such permutations, which involves two different incarnations of ballot numbers, and we show that our formula can be expressed as a linear combination of Catalan numbers.

Bidirectional Backpropagation

We extend backpropagation (BP) learning from ordinary unidirectional training to bidirectional training of deep multilayer neural networks. This gives a form of backward chaining or inverse inference from an observed network output to a candidate input that produced the output. The trained network learns a bidirectional mapping and can apply to some inverse problems. A bidirectional multilayer neural network can exactly represent some invertible functions. We prove that a fixed three-layer network can always exactly represent any finite permutation function and its inverse. The forward pass computes the permutation function value. The backward pass computes the inverse permutation with the same weights and hidden neurons. A joint forward–backward error function allows BP learning in both directions without overwriting learning in either direction. The learning applies to classification and regression. The algorithms do not require that the underlying sampled function has an inverse. A trained regression network tends to map an output back to the centroid of its preimage set.

On Univalent Mappings Performed by the Generalized Christoffel–Schwarz Formula

We examine the existence of an univalent mapping of the upper half-plane with a fixed set of preimages of vertices to a polygonal domain.

On the density of pre-orbits under linear toral endomorphisms

It is well known for non-injective endomorphisms that if for every point the set of preimages is dense in the manifold then the endomorphism is transitive (i. e. there exists a point that its orbit is dense in the manifold). But it has not yet been completely investigated that if the pre-orbit of points are dense under Anosov endomorphisms or what are the necessary conditions that make the pre-orbits of each point dense. By making a great use of the integral lattice properties, we construct our proof on the pre-image sets of points under the iterations of the linear dynamical system. We introduce a class of hyperbolic linear endomorphism that is called absolutely hyperbolic and show that if A : Tm → Tm is an absolutely hyperbolic linear endomorphism of degree more than 1 then the pre-orbit of each point is dense in Tm.

Tree-Structured Polyhedral Invariant Set Calculations

This letter provides a description of how hierarchical dependencies between inequalities can be exploited in order to efficiently calculate polyhedral approximations of maximal robust positive invariant sets using geometrically motivated methods. Due to the hierarchical dependencies, the calculations of preimage sets and minimal representations can be alleviated. It is also shown that as a byproduct from the calculations of minimal representations, a stopping criterion is obtained, which means that the commonly used subset test is superfluous.

Large arboreal Galois representations

Given a field K, a polynomial f∈K[x] of degree d, and a suitable element t∈K, the set of preimages of t under the iterates f∘n carries a natural structure of a d-ary tree. We study conditions under which the absolute Galois group of K acts on the tree by the full group of automorphisms. When d≥20 is even and K=Q we exhibit examples of polynomials with maximal Galois action on the preimage tree, partially affirming a conjecture of Odoni. We also study the case of K=F(t) and f∈F[x] in which the corresponding Galois groups are the monodromy groups of the ramified covers f∘n:PF1→PF1.

DIVISOR‐SUM FIBERS

Let $s(\cdot)$ denote the sum-of-proper-divisors function, that is, $s(n) = \sum_{d\mid n,~d<n}d$. Erd\H{o}s-Granville-Pomerance-Spiro conjectured that for any set $\mathcal{A}$ of asymptotic density zero, the preimage set $s^{-1}(\mathcal{A})$ also has density zero. We prove a weak form of this conjecture: If $\epsilon(x)$ is any function tending to $0$ as $x\to\infty$, and $\mathcal{A}$ is a set of integers of cardinality at most $x^{\frac12+\epsilon(x)}$, then the number of integers $n\le x$ with $s(n) \in \mathcal{A}$ is $o(x)$, as $x\to\infty$. In particular, the EGPS conjecture holds for infinite sets with counting function $O(x^{\frac12 + \epsilon(x)})$. We also disprove a hypothesis from the same paper of EGPS by showing that for any positive numbers $\alpha$ and $\epsilon$, there are integers $n$ with arbitrarily many $s$-preimages lying between $\alpha(1-\epsilon)n$ and $\alpha(1+\epsilon)n$. Finally, we make some remarks on solutions $n$ to congruences of the form $\sigma(n) \equiv a\pmod{n}$, proposing a modification of a conjecture appearing in recent work of the first two authors. We also improve a previous upper bound for the number of solutions $n \leq x$, making it uniform in $a$.

An Example of a Deterministic Cellular Automaton Exhibiting Linear-exponential Convergence to the Steady State

In a recent paper [arXiv:1506.06649 [nlin.CG]], we presented an example of a 3-state cellular automaton which exhibits behaviour analogous to degenerate hyperbolicity often observed in finite-dimensional dynamical systems. We also calculated densities of 0, 1 and 2 after n iterations of this rule, using finite state machines to conjecture patterns present in preimage sets. Here, we re-derive the same formulae in a rigorous way, without resorting to any semi-empirical methods. This is done by analysing the behaviour of continuous clusters of symbols and by considering their interactions.

Measure-theoretic degrees and topological pressure for non-expanding transformations

We consider invariant sets Λ of saddle type, for non-invertible smooth maps f, and equilibrium measures μϕ associated to Hölder potentials ϕ on Λ. We define a notion of measure-theoretic asymptotic degree off|Λ:Λ→Λ, with respect to the measure μϕ on the fractal set Λ. In our case, the equilibrium measure μϕ is the unique linear functional in C(Λ)⁎ tangent to the pressure function P(⋅):C(Λ)→R at ϕ. In particular, for the measure of maximal entropy μ0 of f|Λ, we obtain the asymptotic degree of f|Λ, which represents the average rate of growth of the number of n-preimages of x that remain in Λ when n→∞; notice that, in general, Λ is not totally invariant for f. To this end, we will obtain first a formula for the Jacobians of the probability μϕ, with respect to arbitrary iterates fm, m≥2. We then show that a formula for the topological pressure P(ϕ) that holds in the expanding case, is no longer true on saddle sets. In the saddle case we find a new formula for the pressure, involving weighted sums on preimage sets. We also apply the asymptotic degrees, together with various pressure functionals, in order to obtain estimates for the Hausdorff dimension of stable slices through certain sets of full μϕ-measure in the fractal Λ. In the end, we give also some concrete examples on saddle folded sets.

On the exceptional set in a conditional theorem of Littlewood

In 1952, Littlewood stated a conjecture about the average growth of spherical derivatives of polynomials, and showed that it would imply that for entire function of finite order, “most” preimages of almost all points are concentrated in a small subset of the plane. In 1988, Lewis and Wu proved Littlewood’s conjecture. Using techniques from complex dynamics, we construct entire functions of finite order with a bounded set of singular values for which the set of exceptional preimages is infinite, with logarithmically growing cardinality.

Sets of zero-difference balanced functions and their applications

Zero-difference balanced (ZDB) functions can be employed in many applications, e.g., optimal constant composition codes, optimal and perfect difference systems of sets, optimal frequency hopping sequences, etc. In this paper, two results are summarized to characterize ZDB functions, among which a lower bound is used to achieve optimality in applications and determine the size of preimage sets of ZDB functions. As the main contribution, a generic construction of ZDB functions is presented, and many new classes of ZDB functions can be generated. This construction is then extended to construct a set of ZDB functions, in which any two ZDB functions are related uniformly. Furthermore, some applications of such sets of ZDB functions are also introduced.

Inverse continuity on the boundary of the numerical range

Let . We consider the mapping , defined on the unit sphere in . The map has a multi-valued inverse , and the continuity properties of are considered in terms of the structure of the set of pre-images for points in the numerical range. It is shown that there may be only finitely many failures of continuity of , and conditions for where these failure occur are given. Additionally, we give a necessary and sufficient condition for weak inverse continuity to hold for and a sufficient condition for .