Even-odd Harmonious Labeling of Some Graphs

Let G = be a graph, with and . An injective mapping is called an even-odd harmonious labeling of the graph G, if an induced edge mapping such that (i) is bijective mapping (ii) The graph acquired from this labeling is called even-odd harmonious graph. In this paper, we discovered some interesting results like H-graph, comb graph, bistar graph and graph for even-odd harmonious labeling.

Ripplet transform and its extension to Boehmians

First, we correct the mistake in the inversion theorem of the ripplet transform in the literature. Next, we prove a convolution theorem for the ripplet transform and extend the ripplet transform as a continuous, linear, injective mapping from a suitable Boehmian space into another Boehmian space.

Disjoint hypercyclic weighted translations on locally compact hausdorff spaces

Let G be a locally compact second countable Hausdorff space with a positive regular Borel measure λ, where λ is invariant under a continuous injective mapping φ : G → G. We characterize the disjoint hypercyclicity of finite weighted translations generated by φ acting on the weighted space Lp(G, ω) (1 ≤ p < ∞).

Boolean functions as points on the hypersphere in the Euclidean space

A new approach to the study of algebraic, combinatorial, and cryptographic properties of Boolean functions is proposed. New relations between functions have been revealed by consideration of an injective mapping of the set of Boolean functions onto the sphere in a Euclidean space. Moreover, under this mapping some classes of functions have extremely regular localizations on the sphere. We introduce the concept of curvature of a Boolean function, which characterizes its proximity (in some sense) to maximally nonlinear functions.

Minimal Reversible Deterministic Finite Automata

We study reversible deterministic finite automata (REV-DFAs), that are partial deterministic finite automata whose transition function induces an injective mapping on the state set for every letter of the input alphabet. We give a structural characterization of regular languages that can be accepted by REV-DFAs. This characterization is based on the absence of a forbidden pattern in the (minimal) deterministic state graph. Again with a forbidden pattern approach, we also show that the minimality of REV-DFAs among all equivalent REV-DFAs can be decided. Both forbidden pattern characterizations give rise to [Formula: see text]-complete decision algorithms. In fact, our techniques allow us to construct the minimal REV-DFA for a given minimal DFA. These considerations lead to asymptotic upper and lower bounds on the conversion from DFAs to REV-DFAs. Thus, almost all problems that concern uniqueness and the size of minimal REV-DFAs are solved.

Orthogonal labeling

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>Let ∆</span><span>G </span><span>be the maximum degree of a simple connected graph </span><span>G</span><span>(</span><span>V,E</span><span>). An injective mapping </span><span>P </span><span>: </span><span>V </span><span>→ </span><span>R</span><span>∆</span><span>G </span><span>is said to be an orthogonal labeling of </span><span>G </span><span>if </span><span>uv,uw </span><span>∈ </span><span>E </span><span>implying (</span><span>P</span><span>(</span><span>v</span><span>) </span><span>− </span><span>P</span><span>(</span><span>u</span><span>)) </span><span>· </span><span>(</span><span>P</span><span>(</span><span>w</span><span>) </span><span>− </span><span>P</span><span>(</span><span>u</span><span>)) = 0, where </span><span>· </span><span>is the usual dot product defined in Euclidean space. A graph </span><span>G </span><span>which has an orthogonal labeling is called an orthogonal graph. This labeling is motivated by the existence of several labelings defined by some algebraic structure, i.e. harmonious labeling and group distance magic labeling. In this paper we study some preliminary results on orthogonal labeling. One of the early result is the fact that cycle graph with even vertices are orthogonal, while ones with odd vertices are not. The main results in this paper state that any graph containing </span><span>K</span><span>3 </span><span>as its subgraph is non-orthogonal and that a graph </span><span>G</span><span>′ </span><span>obtained from adding a pendant to a vertex in orthogonal graph </span><span>G </span><span>is orthogonal. In the end of the paper we state the corollary that any tree is orthogonal.<br /> </span></p></div></div></div>

The fundamental theorems of affine and projective geometry revisited

The fundamental theorem of affine geometry is a classical and useful result. For finite-dimensional real vector spaces, the theorem roughly states that a bijective self-mapping which maps lines to lines is affine-linear. In this paper, we prove several generalizations of this result and of its classical projective counterpart. We show that under a significant geometric relaxation of the hypotheses, namely that only lines parallel to one of a fixed set of finitely many directions are mapped to lines, an injective mapping of the space must be of a very restricted polynomial form. We also prove that under mild additional conditions the mapping is forced to be affine-additive or affine-linear. For example, we show that five directions in three-dimensional real space suffice to conclude affine-additivity. In the projective setting, we show that [Formula: see text] fixed projective points in real [Formula: see text]-dimensional projective space, through which all projective lines that pass are mapped to projective lines, suffice to conclude projective-linearity.

Systems of knowledge representation based on stratified graphs. Application in Natural Language Generation

The concept of stratified graph introduces some method of knowledge representation (see [T¸ and ˘ areanu, N., ˘ Knowledge representation by labeled stratified graphs, Proc. 8th World Multi-Conference on Systemics, Cybernetics and Informatics, 5 (2004), 345–350; T¸ and ˘ areanu, N., ˘ Proving the Existence of Labelled Stratified Graphs, An. Univ. Craiova Ser. Mat. Inform., 27 (2000), 81–92]) The inference process developed for this method uses the paths of the stratified graphs, an order between the elementary arcs of a path and some results of universal algebras. The order is defined by considering a structured path instead of a regular path. In this paper we define the concept of system of knowledge representation as a tuple of the following components: a stratified graph G, a partial algebra Y of real objects, an embedding mapping (an injective mapping that embeds the nodes of G into objects of Y ) and a set of algorithms such that each of them can combine two objects of Y to get some other object of Y . We define also the concept of inference process performed by a system of knowledge processing in which the interpretation of the symbolic elements is defined by means of natural language constructions. In this manner we obtained a mechanism for texts generation in a natural language (for this approach, Romanian).

PARIKH MATRICES, AMIABILITY AND ISTRAIL MORPHISM

Using the fact that the Parikh matrix mapping is not an injective mapping, the paper investigates some properties of the set of words having the same Parikh matrix; these words are called "amiable" or "M - equivalent". The presented paper uses the results obtained in [3] for the binary case. The aim is to distinguish the amiable words by using a morphism that provides additional information about them. The morphism proposed here is the Istrail morphism.