Bijective Map Research Articles

Imprecise probability through f-probability and its statistical physical implications

Abstract In this work, we present an approach to describe imprecise probability through an effective probability theory, called the f-probability. We develop a bijective and monotonous map from the precise probability in order to construct the f-probability theory based on the f-addition, f-subtraction, f-multiplication and f-division. We apply the f-probability to the Bernoulli trial and derive the f-binomial distribution. Finally, we obtain the non-extensive entropy through the f-probability theory, and give its statistical physical implications on several areas of potential applications. PACS number(s): 02.50.Cw, 05.20.-y; 05.90.+m

Spectral Alignment of Graphs

Graph alignment refers to the problem of finding a bijective mapping across vertices of two graphs such that, if two nodes are connected in the first graph, their images are connected in the second graph. This problem arises in many fields, such as computational biology, social sciences, and computer vision and is often cast as a quadratic assignment problem (QAP). Most standard graph alignment methods consider an optimization that maximizes the number of matches between the two graphs, ignoring the effect of mismatches. We propose a generalized graph alignment formulation that considers both matches and mismatches in a standard QAP formulation. This modification can have a major impact in aligning graphs with different sizes and heterogeneous edge densities. Moreover, we propose two methods for solving the generalized graph alignment problem based on spectral decomposition of matrices. We compare the performance of proposed methods with some existing graph alignment algorithms including Natalie2, GHOST, IsoRank, NetAlign, Klau's approach as well as a semidefinite programming-based method over various synthetic and real graph models. Our proposed method based on simultaneous alignment of multiple eigenvectors leads to consistently good performance in different graph models. In particular, in the alignment of regular graph structures, which is one of the most difficult graph alignment cases, our proposed method significantly outperforms other methods.

On Iteration of Bijective Functions with Discontinuities

Abstract We present three different types of bijective functions f : I → I on a compact interval I with finitely many discontinuities where certain iterates of these functions will be continuous. All these examples are strongly related to permutations, in particular to derangements in the first case, and permutations with a certain number of successions (or small ascents) in the second case. All functions of type III form a direct product of a symmetric group with a wreath product. It will be shown that any iterative root F : J → J of the identity of order k on a compact interval J with finitely many discontinuities is conjugate to a function f of type III, i.e., F = φ− 1 ∘ f ∘ φ where φ is a continuous, bijective, and increasing mapping between J and [0, n] for some integer n.

Exponential meanness of some ladder related graphs

For a graph G = (V, E) define an injective function χ from V (G) to the set {1, 2, 3…, q + 1} and an induced bijective function χ* from E(G) to the set {1, 2, 3…, q} as Then χ is called an exponential mean labeling of G. A graph which allows an exponential mean labeling is called an exponential mean graph. In this article, we discuss an exponential mean labeling of some ladder graphs.

COMBINATORIAL POLYNOMIALLY COMPUTABLE CHARACTERISTICS OF SUBSTITUTIONS AND THEIR PROPERTIES

The construction and selection of a suitable bijective function, that is, substitution, is now becoming an important applied task, particularly for building block encryption systems. Many articles have suggested using different approaches to determining the quality of substitution, but most of them are highly computationally complex. The solution of this problem will significantly expand the range of methods for constructing and analyzing scheme in information protection systems. The purpose of research is to find easily measurable characteristics of substitutions, allowing to evaluate their quality, and also measures of the proximity of a particular substitutions to a random one, or its distance from it. For this purpose, several characteristics were proposed in this work: difference and polynomial, and their mathematical expectation was found, as well as variance for the difference characteristic. This allows us to make a conclusion about its quality by comparing the result of calculating the characteristic for a particular substitution with the calculated mathematical expectation. From a computational point of view, the thesises of the article are of exceptional interest due to the simplicity of the algorithm for quantifying the quality of bijective function substitutions. By its nature, the operation of calculating the difference characteristic carries out a simple summation of integer terms in a fixed and small range. Such an operation, both in the modern and in the prospective element base, is embedded in the logic of a wide range of functional elements, especially when implementing computational actions in the optical range, or on other carriers related to the field of nanotechnology.

Automated Detection of Interesting Properties in Regular Polygons

We demonstrate a systematic, automated way of discovery of a large number of new geometry theorems on regular polygons. The applied theory includes a formula by Watkins and Zeitlin on minimal polynomials of cos frac{2pi }{n}, and a method by Recio and Vélez to discover a property in a plane geometry construction. This method exploits Wu’s idea on algebraizing the geometric setup and utilizes the theory of Gröbner bases. Also a bijective function is given that maps the investigated cases to the first natural numbers. Finally, several examples are shown that are all previously unknown results in planar Euclidean geometry.

The multivariate method strikes again: New power functions with low differential uniformity in odd characteristic

Let f(x) = xd be a power mapping over mathbb {F}_{n} and mathcal {U}_{d} the maximum number of solutions xin mathbb {F}_{n} of {Delta }_{f,c}(x):=f(x+c)-f(x)=atext {, where }c,ain mathbb {F}_{n}text { and } cneq 0. f is said to be differentially k-uniform if mathcal {U}_{d} =k. The investigation of power functions with low differential uniformity over finite fields mathbb {F}_{n} of odd characteristic has attracted a lot of research interest since Helleseth, Rong and Sandberg started to conduct extensive computer search to identify such functions. These numerical results are well-known as the Helleseth-Rong-Sandberg tables and are the basis of many infinite families of power mappings x^{d_{n}},n in mathbb {N}, of low uniformity (see e.g. Dobbertin et al. Discret. Math. 267, 95–112 2003; Helleseth et al. IEEE Trans. Inform Theory, 45, 475–485 1999; Helleseth and Sandberg AAECC, 8, 363–370 1997; Leducq Amer. J. Math. 1(3) 115–123 1878; Zha and Wang Sci. China Math. 53(8) 1931–1940 2010). Recently the crypto currency IOTA and Cybercrypt started to build computer chips around base-3 logic to employ their new ternary hash function Troika, which currently increases the cryptogrpahic interest in such families. Especially bijective power mappings are of interest, as they can also be employed in block- and stream ciphers. In this paper we contribute to this development and give a family of power mappings x^{d_{n}} with low uniformity over mathbb {F}_{n}, which is bijective for p ≡ 3 mod 4. For p = 3 this yields a family x^{d_{n}} with 3leq mathcal {U}_{d_{n}}leq 4, where the family of inverses has a very simple description. These results explain “open entries” in the Helleseth-Rong-Sandberg tables. We apply the multivariate method to compute the uniformity and thereby give a self-contained introduction to this method. Moreover we will prove for a related family of low uniformity introduced in Helleseth and Sandberg (AAECC, 8 363–370 1997) that it yields permutations.

Certain Results on Prime and Prime Distance Labeling of Graphs

Let G be a graph on n vertices. A bijective function f : V (G) → {1, 2,…,n} is said to be a prime labeling if for every e = xy, GCD{f (x),f (y)} = 1. A graph which permits a prime labeling is a “prime graph”. On the other hand, a graph G is a prime distance graph if there is an injective function g : V(G) → Z (the set of all integers) so that for any two vertices s & t which are adjacent, the integer |g(s) – g(t)| is a prime number and g is called a prime distance labeling of G. A graph G is a prime distance graph (PDL) iff there exists a “prime distance labeling” (PDL) of G. In this paper, we obtain the prime labeling and prime distance labeling of certain classes of graphs.

The upper bound of vertex local antimagic edge labeling on graph operations

Let G(V(G), E(G)) be a connected simple graph and let u, v be two vertices of graph G. A bijective function f from the edge set of G to the natural number up to the number of edges in G is called a vertex local antimagic edge labeling if for any two adjacent vertices v and v′, w(v) = w(v′), where , and E(v) is the set of vertices which is incident to v. Thus any vertex local antimagic edge labeling induces a proper vertex coloring of G where the vertex v is assigned with the color w(v). The vertex local antimagic chromatic number χla (G) is the minimum number of colors taken over all coloring induced by vertex local antimagic edge labeling of G. In this paper, we study the vertex local antimagic edge labeling of graphs and determine the upper bound of vertex local antimagic chromatic number on graph operation, namely cartesian product of graphs, corona product of graphs, and power graph.

A comprehensive survey on prime cordial and divisor cordial labeling of graphs

This article presents a short and concise survey on prime cordial and divisor cordial labeling of graphs. A prime cordial labeling of a graph G(V,E) is a bijective function f:V(G) → {1,2,…,|V|} such that if each edge xy is assigned the label 1 if gcd(f(x),f(y)) = 1 and 0 if gcd(f(x),f(y)) > 1, then the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. Further, a divisor cordial labeling of G is a bijection g: V(G) → {1,2,…,|V|} such that an edge st is assigned the label 1 if one g(s) or g(t) divides the other and 0 otherwise, then the number of edges labeled with 0 and the number of edges labelled with 1 differ by at most 1. We call G a divisor cordial graph if it admits a divisor cordial labeling. This article stands divided into five sections. The first and fifth sections are reserved respectively for introduction and some important references. The second section deals with the prime cordial labeling of certain classes of graphs wherein some important known results have been recalled. The third section deals with the divisor cordial labeling of graphs in which a few known results of high interest have been outlined. In the fourth section we highlight certain conjectures and open problems in respect of the above mentioned labelling that still remain unsolved.

Numerical roadmap of smooth bounded real algebraic surface

For a smooth bounded real algebraic surface in Rn, a roadmap of it is a one-dimensional semi-algebraic subset of the surface whose intersection with each connected component of the surface is nonempty and semi-algebraically connected. In this paper, we introduce the notion of a numerical roadmap of a surface, which is a set of disjoint polygonal chains such that there is a bijective map between the chains and the connected components of a given roadmap of the surface. Moreover, the chains are ϵ-close to the connected components. We present an algorithm to compute such a numerical roadmap through constructing a topological graph. The topological graph also enables us to compute an approximate graph and a more intrinsic connectivity graph to represent the roadmap and its connectivity property, which is important for applications such as determining if two points on the surface belong to the same connected component and if so, finding a connected path between them.

The consecutively super edge-magic deficiency of graphs and related concepts

A bipartite graph G with partite sets X and Y is called consecutively super edge-magic if there exists a bijective function f : V ( G ) ⋃ E ( G ) → {1,2,...,| V ( G )| + | E ( G )|} with the property that f ( X ) = {1,2,...,| X |}, f ( Y ) = {| X |+1, | X |+2,...,| V ( G )|} and f ( u )+ f ( v ) + f ( uv ) is constant for each uv ∈ E ( G ). The question studied in this paper is for which bipartite graphs it is possible to add a finite number of isolated vertices so that the resulting graph is consecutively super edge-magic. If it is possible for a bipartite graph G , then we say that the minimum such number of isolated vertices is the consecutively super edge-magic deficiency of G ; otherwise, we define it to be +∞. This paper also includes a detailed discussion of other concepts that are closely related to the consecutively super edge-magic deficiency.

A new algorithm design technique for hard problems

Heuristic algorithms have been used for a long time to tackle problems that are known or conjectured intractable. A heuristic algorithm is one that provides a correct decision for most inputs, but may fail on some. We focus on the case when failure means that the algorithm does not return any answer, rather than returning a wrong result. These algorithms are called errorless heuristics.A reasonable quality measure for heuristics is the failure rate over the set of n-bit instances. When no efficient exact algorithm is available for a problem, then, ideally, we would like one with vanishing failure rate. We show, however, that this is hard to achieve: unless a complexity theoretic hypothesis fails, some NP-complete problems cannot have a polynomial-time errorless heuristic algorithm with any vanishing failure rate.On the other hand, we prove that vanishing, even exponentially small, failure rate is achievable, if we use a somewhat different accounting scheme to count the failures. This is based on special sets, that we call α-spheres, which are the images of the n-bit strings under a bijective, polynomial-time computable and polynomial-time invertible encoding function α. Our main result is that polynomial-time errorless heuristic algorithms exist, with exponentially low failure rates on the α-spheres, for a large class of decision problems. This class includes, surprisingly, all known intuitively natural NP-complete problems. We also explore some connections to the theory of average case complexity.

Theory and algorithm of the homeomorphic Fourier transform for optical simulations

The introduction of the fast Fourier transform (FFT) constituted a crucial step towards a faster and more efficient physio-optics modeling and design, since it is a faster version of the Discrete Fourier transform. However, the numerical effort of the operation explodes in the case of field components presenting strong wavefront phases-very typical occurrences in optics- due to the requirement of the FFT that the wrapped phase be well sampled. In this paper, we propose an approximated algorithm to compute the Fourier transform in such a situation. We show that the Fourier transform of fields with strong wavefront phases exhibits a behavior that can be described as a bijective mapping of the amplitude distribution, which is why we name this operation "homeomorphic Fourier transform." We use precisely this characteristic behavior in the mathematical approximation that simplifies the Fourier integral. We present the full theoretical derivation and several numerical applications to demonstrate its advantages in the computing process.

New Constructions for the n-Queens Problem

Let D be a digraph, possibly with loops. A queen labeling of D is a bijective function $$l:V(G)\longrightarrow \{1,2,\ldots ,|V(G)|\}$$ such that, for every pair of arcs in E(D), namely (u, v) and $$(u',v')$$ we have (i) $$l(u)+l(v)\ne l(u')+l(v')$$ and (ii) $$l(v)-l(u)\ne l(v')-l(u')$$. Similarly, if the two conditions are satisfied modulo $$n=|V(G)|$$, we define a modular queen labeling. There is a bijection between (modular) queen labelings of 1-regular digraphs and the solutions of the (modular) n-queens problem. The $$\otimes _h$$-product was introduced in 2008 as a generalization of the Kronecker product and since then, many relations among labelings have been established using the $$\otimes _h$$-product and some particular families of graphs. In this paper, we study some families of 1-regular digraphs that admit (modular) queen labelings and present a new construction concerning to the (modular) n-queens problem in terms of the $$\otimes _h$$-product, which in some sense complements a previous result due to Polya.

On maps preserving Lie products equal to a rank-one nilpotent

Let ϕ be a bijective linear map on the algebra of n×n complex matrices such that ϕ(e12)=e12 and [ϕ(A),ϕ(B)]=e12 whenever [A,B]=e12. The purpose of this paper is to describe ϕ. Surprisingly, ϕ has a different description from maps preserving zero Lie products.

On rainbow antimagic coloring of some special graph

Let G = (V, E) be a connected and simple graphs with vertex set V and edge set E. A coloring of graph G is rainbow connected if there is a rainbow path that connects each two vertices of graph G. The minimum k such that G has a rainbow-connected using k colors of the edges of G is the rainbow connection number rc(G) of G. A graph with a bijective mapping f : E → {1, 2, …, |E|}. The sums of each paired vertex has distinct value, defined as ∑ e ∈E(v)f(e). Thus, the function of G clearly an antimagic labeling if the sums of each paired vertex has distinct value. It is clear that rainbow antimagic connection number is the smallest number of colors which are needed to make G rainbow connected, denoted by rcA (G). A bijection function f : E → {1, 2, …, |E|} is called a rainbow antimagic labeling if there is a rainbow path between every pair of vertices and for each edge e = uv ∈ E(G), the weight w(e) = f(u) + f(v). A graph G is rainbow antimagic if G has a rainbow antimagic labeling. In this paper, we will analyze the rainbow antimagic coloring of related book graph.

Elliptic Curve Elgamal Encryption Scheme using Higher-Order Golden Matrices

In this paper, we proposed ElGamal encryption scheme of elliptic curves based on the golden matrices. This algorithm works with a bijective function identified as characters of ASCII from the elliptic curve points and the matrix produced the additional private key, which was obtained from golden matrices defined by A.P Stakhov.

Edge δ− Graceful Labeling for Some Cyclic-Related Graphs

In this paper, we introduce a new type of labeling of a graph G with p vertices and q edges called edge δ− graceful labeling, for any positive integer δ, as a bijective mapping f of the edge set EG into the set δ,2δ,3δ,⋯,qδ such that the induced mapping f∗:VG→0,δ,2δ,3δ,⋯,qδ−δ, given by f∗u=∑uv∈EGfuvmodδk, where k=maxp,q, is an injective function. We prove the existence of an edge δ− graceful labeling, for any positive integer δ, for some cycle-related graphs like the wheel graph, alternate triangular cycle, double wheel graph Wn,n, the prism graph Πn, the prism of the wheel PWn, the gear graph Gn, the closed helm CHn, the butterfly graph Bn, and the friendship Frn.

Power commuting traces of bilinear maps on invertible elements

Let [Formula: see text] be a unital simple ring. Under a mild technical restriction on [Formula: see text], we characterize bilinear mappings [Formula: see text] satisfying [Formula: see text], and [Formula: see text] for all unit [Formula: see text] and [Formula: see text], where [Formula: see text]. As an application we describe bijective linear maps [Formula: see text] satisfying [Formula: see text] for all invertible [Formula: see text]. Precisely, we will show that [Formula: see text] is an isomorphism.