Two-dimensional partial orderings: Undecidability

1980 ◽  
Vol 45 (1) ◽  
pp. 133-143 ◽  
Author(s):  
Alfred B. Manaster ◽  
Joseph G. Rosenstein
Keyword(s):  
Bipartite Graph ◽  
Bipartite Graphs ◽  
The Other ◽  
Two Dimensional ◽  
One Dimensional ◽  
Recursive Model ◽  
Linear Orderings ◽  
Other Hand ◽  
Partial Orderings ◽  
Planar Lattices

In this paper we examine the class of two-dimensional partial orderings from the perspective of undecidability. We shall see that from this perspective the class of 2dpo's is more similar to the class of all partial orderings than to its one-dimensional subclass, the class of all linear orderings. More specifically, we shall describe an argument which lends itself to proofs of the following four results:(A) the theory of 2dpo's is undecidable:(B) the theory of 2dpo's is recursively inseparable from the set of sentences refutable in some finite 2dpo;(C) there is a sentence which is true in some 2dpo but which has no recursive model;(D) the theory of planar lattices is undecidable.It is known that the theory of linear orderings is decidable (Lauchli and Leonard [4]). On the other hand, the theories of partial orderings and lattices were shown to be undecidable by Tarski [14], and that each of these theories is recursively inseparable from its finitely refutable statements was shown by Taitslin [13]. Thus, the complexity of the theories of partial orderings and lattices is, by (A), (B) and (D), already reflected in the 2dpo's and planar lattices.As pointed out by J. Schmerl, bipartite graphs can be coded into 2dpo's, so that (A) and (B) could also be obtained by applying a Rabin-Scott style argument [9] to Rogers' result [11] that the theory of bipartite graphs is undecidable and to Lavrov's result [5] that the theory of bipartite graphs is recursively inseparable from the set of sentences refutable in some finite bipartite graph. (However, (C) and (D) do not seem to follow from this type of argument.)

Two-dimensional partial orderings: Recursive model theory

1980 ◽  
Vol 45 (1) ◽  
pp. 121-132 ◽  
Author(s):  
Alfred B. Manaster ◽  
Joseph G. Rosenstein
Keyword(s):  
Model Theory ◽  
Companion Paper ◽  
Partial Ordering ◽  
Linear Ordering ◽  
Two Dimensional ◽  
Recursive Model ◽  
Linear Orderings ◽  
Rational Plane ◽  
Partial Orderings

In this paper and the companion paper [9] we describe a number of contrasts between the theory of linear orderings and the theory of two-dimensional partial orderings.The notion of dimensionality for partial orderings was introduced by Dushnik and Miller [3], who defined a partial ordering 〈A, R〉 to be n-dimensional if there are n linear orderings of A, 〈A, L1〉, 〈A, L2〉 …, 〈A, Ln〉 such that R = L1 ∩ L2 ∩ … ∩ Ln. Thus, for example, if Q is the linear ordering of the rationals, then the (rational) plane Q × Q with the product ordering (〈x1, y1〉 ≤Q×Q 〈x2, y2, if and only if x1 ≤ x2 and y1 ≤ y2) is 2-dimensional, since ≤Q×Q is the intersection of the two lexicographic orderings of Q × Q. In fact, as shown by Dushnik and Miller, a countable partial ordering is n-dimensional if and only if it can be embedded as a subordering of Qn.Two-dimensional partial orderings have attracted the attention of a number of combinatorialists in recent years. A basis result recently obtained, independently, by Kelly [7] and Trotter and Moore [10], describes explicitly a collection of finite partial orderings such that a partial ordering is a 2dpo if and only if it contains no element of as a subordering.

scholarly journals The influence of resistivity gradients on shock conditions for a Petschek reconnection geometry

2016 ◽  
Vol 34 (4) ◽  
pp. 421-425
Author(s):  
Christian Nabert ◽  
Karl-Heinz Glassmeier
Keyword(s):  
Magnetic Field ◽  
Necessary Conditions ◽  
The Other ◽  
Diffusion Region ◽  
Two Dimensional ◽  
Characteristic Velocity ◽  
Magnetohydrodynamic Equations ◽  
One Dimensional ◽  
The Magnetic Field ◽  
Alfven Velocity

Abstract. Shock waves can strongly influence magnetic reconnection as seen by the slow shocks attached to the diffusion region in Petschek reconnection. We derive necessary conditions for such shocks in a nonuniform resistive magnetohydrodynamic plasma and discuss them with respect to the slow shocks in Petschek reconnection. Expressions for the spatial variation of the velocity and the magnetic field are derived by rearranging terms of the resistive magnetohydrodynamic equations without solving them. These expressions contain removable singularities if the flow velocity of the plasma equals a certain characteristic velocity depending on the other flow quantities. Such a singularity can be related to the strong spatial variations across a shock. In contrast to the analysis of Rankine–Hugoniot relations, the investigation of these singularities allows us to take the finite resistivity into account. Starting from considering perpendicular shocks in a simplified one-dimensional geometry to introduce the approach, shock conditions for a more general two-dimensional situation are derived. Then the latter relations are limited to an incompressible plasma to consider the subcritical slow shocks of Petschek reconnection. A gradient of the resistivity significantly modifies the characteristic velocity of wave propagation. The corresponding relations show that a gradient of the resistivity can lower the characteristic Alfvén velocity to an effective Alfvén velocity. This can strongly impact the conditions for shocks in a Petschek reconnection geometry.

Polymorphs of phenazine hexacyanoferrate(II) hydrate: supramolecular isomerism in a 2D hydrogen-bonded network

2021 ◽  
Vol 77 (2) ◽  
pp. 211-218
Author(s):  
Ivica Cvrtila ◽  
Vladimir Stilinović
Keyword(s):  
Hydrogen Bonding ◽  
Hydrogen Bonds ◽  
Crystal Structures ◽  
The Other ◽  
Two Dimensional ◽  
Structural Motifs ◽  
Supramolecular Isomerism ◽  
Other Hand ◽  
Π Stacking ◽  
Hydrogen Bonded

The crystal structures of two polymorphs of a phenazine hexacyanoferrate(II) salt/cocrystal, with the formula (Hphen)3[H2Fe(CN)6][H3Fe(CN)6]·2(phen)·2H2O, are reported. The polymorphs are comprised of (Hphen)2[H2Fe(CN)6] trimers and (Hphen)[(phen)2(H2O)2][H3Fe(CN)6] hexamers connected into two-dimensional (2D) hydrogen-bonded networks through strong hydrogen bonds between the [H2Fe(CN)6]2− and [H3Fe(CN)6]− anions. The layers are further connected by hydrogen bonds, as well as through π–π stacking of phenazine moieties. Aside from the identical 2D hydrogen-bonded networks, the two polymorphs share phenazine stacks comprising both protonated and neutral phenazine molecules. On the other hand, the polymorphs differ in the conformation, placement and orientation of the hydrogen-bonded trimers and hexamers within the hydrogen-bonded networks, which leads to different packing of the hydrogen-bonded layers, as well as to different hydrogen bonding between the layers. Thus, aside from an exceptional number of symmetry-independent units (nine in total), these two polymorphs show how robust structural motifs, such as charge-assisted hydrogen bonding or π-stacking, allow for different arrangements of the supramolecular units, resulting in polymorphism.

On Wake-Induced Flutter of a Circular Conductor in the Wake of Another

1980 ◽  
Vol 102 (2) ◽  
pp. 125-137 ◽  
Author(s):  
Y. T. Tsui ◽  
C. C. Tsui
Keyword(s):  
Transmission Line ◽  
Coupling Coefficient ◽  
Frequency Ratio ◽  
Horizontal Axis ◽  
The Other ◽  
Two Dimensional ◽  
Coupling Ratio ◽  
Aeroelastic Stability ◽  
Other Hand ◽  
The Stability

This paper, which is an extension of [1], treats two-dimensional aeroelastic stability of two coupled conductors. It is found that the wake-induced flutter is symmetric with respect to the horizontal axis of the wake for all cases provided that the sign of the static coupling coefficient, ε = kxy/kxx, is changed. It appears that the spacer coupling ratio, K/kxx = Ω/ωx, is the most important factor in determining stability. For practical purposes, the system is almost always stable for K/kxx = Ω/ωx = 0.8, because the frequency ratio, κ = ωy/ωx, deviates less than ten percent from unity for a typical transmission line. On the other hand, within our range of interest, damping has little or no effect on the stability of coupled conductors. When the windward conductor is fixed, i.e., K = 0, then damping does influence the stability of the leeward conductor.

CONTEXT-SENSITIVITY OF TWO-DIMENSIONAL REGULAR ARRAY GRAMMARS

1989 ◽  
Vol 03 (03n04) ◽  
pp. 295-319 ◽  
Author(s):  
YASUNORI YAMAMOTO ◽  
KENICHI MORITA ◽  
KAZUHIRO SUGATA
Keyword(s):  
Context Sensitivity ◽  
The Other ◽  
Two Dimensional ◽  
Regular Array ◽  
Nonterminal Symbol ◽  
Left Hand ◽  
Other Hand ◽  
Context Free ◽  
Rewriting Rules

Regular array grammars (RAGs) are the lowest subclass in the Chomsky-like hierarchy of isometric array grammars. The left-hand side of each rewriting rule of RAGs has one nonterminal symbol and at most one "#" (a blank symbol). Therefore, the rewriting rules cannot sense contexts of non-# symbols. However, they can sense # as a kind of context. In this paper, we investigate this #-sensing ability. and study the language generating power of RAGs. Making good use of this ability, We show a method for RAGs to sense the contexts of local shapes of a host array in a derivation. Using this method, we give RAGs which generate the sets of all solid upright rectangles and all solid squares. On the other hand. it is proved that there is no context-free array grammar (and thus no RAG) which generates the set of all hollow upright rectangles.

Degree spectra of relations on computable structures in the presence of Δ20 isomorphisms

2002 ◽  
Vol 67 (2) ◽  
pp. 697-720 ◽  
Author(s):  
Denis R. Hirschfeldt
Keyword(s):  
Point Of View ◽  
The Other ◽  
Computable Structure ◽  
Sufficient Condition ◽  
Computable Structures ◽  
Linear Orderings ◽  
Degree Spectra ◽  
Other Hand ◽  
Infinite Degree ◽  
Degree Spectrum

AbstractWe give some new examples of possible degree spectra of invariant relations on Δ20-categorical computable structures, which demonstrate that such spectra can be fairly complicated. On the other hand, we show that there are nontrivial restrictions on the sets of degrees that can be realized as degree spectra of such relations. In particular, we give a sufficient condition for a relation to have infinite degree spectrum that implies that every invariant computable relation on a Δ20-categorical computable structure is either intrinsically computable or has infinite degree spectrum. This condition also allows us to use the proof of a result of Moses [23] to establish the same result for computable relations on computable linear orderings.We also place our results in the context of the study of what types of degree-theoretic constructions can be carried out within the degree spectrum of a relation on a computable structure, given some restrictions on the relation or the structure. From this point of view we consider the cases of Δ20-categorical structures, linear orderings, and 1-decidable structures, in the last case using the proof of a result of Ash and Nerode [3] to extend results of Harizanov [14].

scholarly journals Feller chains and random functions

2015 ◽  
Vol 56 ◽  
Author(s):  
Vytautas Kazakevičius
Keyword(s):  
Random Function ◽  
Transition Probability ◽  
The Other ◽  
One Dimensional ◽  
Random Functions ◽  
Other Hand ◽  
Dimensional Distribution ◽  
The One

We prove that each Feller transition probability is the one-dimensional distribution of some stochastically continuous random function. We also introduce the notion of a regular random function and show, on one hand, that every random  function has a regular modification, and on the other hand, that the composition of independent regular stochastically continuous random functions is stochastically continuous as well.

scholarly journals A Note on Path Factors of $(3,4)$-Biregular Bipartite Graphs

10.37236/705 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Carl Johan Casselgren
Keyword(s):  
Graph Theory ◽  
Bipartite Graph ◽  
Edge Coloring ◽  
Bipartite Graphs ◽  
The Other ◽  
Spanning Subgraph ◽  
Interval Coloring ◽  
Proper Edge Coloring

A proper edge coloring of a graph $G$ with colors $1,2,3,\dots$ is called an interval coloring if the colors on the edges incident with any vertex are consecutive. A bipartite graph is $(3,4)$-biregular if all vertices in one part have degree $3$ and all vertices in the other part have degree $4$. Recently it was proved [J. Graph Theory 61 (2009), 88-97] that if such a graph $G$ has a spanning subgraph whose components are paths with endpoints at 3-valent vertices and lengths in $\{2, 4, 6, 8\}$, then $G$ has an interval coloring. It was also conjectured that every simple $(3,4)$-biregular bipartite graph has such a subgraph. We provide some evidence for this conjecture by proving that a simple $(3,4)$-biregular bipartite graph has a spanning subgraph whose components are nontrivial paths with endpoints at $3$-valent vertices and lengths not exceeding $22$.

scholarly journals Investigation of the Relationship Between the Two-Dimensional Self-Esteem Perceptions and Leadership Orientations of the Faculty of Sports Sciences Students

2021 ◽  
Vol 4 (4) ◽  
Author(s):  
Ismail Karatas ◽  
◽  
Hayri Akyuz
Keyword(s):  
Charismatic Leadership ◽  
Self Esteem ◽  
The Self ◽  
The Other ◽  
Two Dimensional ◽  
Leadership Orientations ◽  
Other Hand ◽  
Questionnaire Form ◽  
Sports Sciences ◽  
The Relationship

This research was carried out to investigate of the relationship between the two-dimensional self-esteem perceptions and leadership orientations of the students of the faculty of sports sciences. In this context, the relational survey model, which is consistent with the main purpose of the study, was used in this quantitative study. A total of 323 students, 125 females and 198 males at the Faculty of Sports Sciences of Bartın University constitute the sample of the research. Convenience sampling method, one of the non-probabilistic sampling approaches, was used in the selection of the research group. Questionnaire form was used as data collection tool and this form consisted of three parts. The first part includes the “Personal Information Form,” the second part includes the “Two-Dimensional Self-Esteem: Self-Liking/Self-Competence Scale” and the third part includes the “Multidimensional Leadership Orientations Scale.” The descriptive statistics of the raw data obtained through the questionnaire form were first calculated by considering the data type. Then, the reliability of the scales related to the obtained data were investigated, and the difference and correlation tests were used in the statistical evaluation. In this direction, it has been determined that there are significant correlations within the scope of age and family income level variables. However, there was no significant relationship within the scope of personal income level variable. On the other hand, it was found that there are significant differences in the scope of department and actively doing sports variables. However, it was observed that there were no significant differences in the scope of gender, grade, and place of residence variables. On the other hand, it was determined that there were positive and moderately significant correlations between the participants’ scores of self-liking and political leadership, human resources leadership, charismatic leadership and structural leadership. In addition, it was found that there were positive and moderately significant correlations between the self-competence scores of the participants and the scores of political leadership, charismatic leadership and structural leadership. On the other hand, it was understood that there was a statistically significant positive and low-level correlation between the participants' self-competence scores and their human resources leadership scores. As a result, it can be said that as the self-esteem of the participants increases, their leadership orientation also increases. In this context, it can be said that increasing the self-esteem of the participants is an important concept in the context of leadership orientations.

scholarly journals A stronger recognizability condition for two-dimensional languages

2013 ◽  
Vol Vol. 15 no. 2 (Automata, Logic and Semantics) ◽  
Author(s):  
Marcella Anselmo ◽  
Maria Madonia
Keyword(s):  
Finite Automata ◽  
The Other ◽  
Two Dimensional ◽  
Minimal Size ◽  
One Dimensional ◽  
Tiling System ◽  
International Audience ◽  
The One

Automata, Logic and Semantics International audience The paper presents a condition necessarily satisfied by (tiling system) recognizable two-dimensional languages. The new recognizability condition is compared with all the other ones known in the literature (namely three conditions), once they are put in a uniform setting: they are stated as bounds on the growth of some complexity functions defined for two-dimensional languages. The gaps between such functions are analyzed and examples are shown that asymptotically separate them. Finally the new recognizability condition results to be the strongest one, while the remaining ones are its particular cases. The problem of deciding whether a two-dimensional language is recognizable is here related to the one of estimating the minimal size of finite automata recognizing a sequence of (one-dimensional) string languages.

Sign in / Sign up

Export Citation Format

Share Document