Glider automata on all transitive sofic shifts

For any infinite transitive sofic shift X we construct a reversible cellular automaton (that is, an automorphism of the shift X) which breaks any given finite point of the subshift into a finite collection of gliders traveling into opposing directions. This shows in addition that every infinite transitive sofic shift has a reversible cellular automaton which is sensitive with respect to all directions. As another application we prove a finitary version of Ryan’s theorem: the automorphism group $\operatorname {\mathrm {Aut}}(X)$ contains a two-element subset whose centralizer consists only of shift maps. We also show that in the class of S-gap shifts these results do not extend beyond the sofic case.

4-output Programmable Spin Wave Logic Gate

To bring Spin Wave (SW) based computing paradigm into practice and develop ultra low power Magnonic circuits and computation platforms, one needs basic logic gates that operate and can be cascaded within the SW domain without requiring back and forth conversion between the SW and voltage domains. To achieve this, SW gates have to possess intrinsic fanout capabilities, be input-output data representation coherent, and reconfigurable. In this paper, we address the first and the last requirements and propose a novel 4-output programmable SW logic. First, we introduce the gate structure and demonstrate that, by adjusting the gate output detection method, it can parallelly evaluate any 4-element subset of the 2-input Boolean function set AND, NAND, OR, NOR, XOR, and XNOR. Furthermore, we adjust the structure such that all its 4 outputs produce SWs with the same energy and demonstrate that it can evaluate Boolean function sets while providing fanout capabilities ranging from 1 to 4. We validate our approach by instantiating and simulating different gate configurations such as 4-output AND/OR, 4-output XOR/XNOR, output energy balanced 4-output AND/OR, and output energy balanced 4-output XOR/XNOR by means of Object Oriented Micromagnetic Framework (OOMMF) simulations. Finally, we evaluate the performance of our proposal in terms of delay and energy consumption and compare it against existing state-of-the-art SW and 16nm CMOS counterparts. The results indicate that for the same functionality, our approach provides 3x and 16x energy reduction, when compared with conventional SW and 16nm CMOS implementations, respectively.

4-output Programmable Spin Wave Logic Gate

To bring Spin Wave (SW) based computing paradigm into practice and develop ultra low power Magnonic circuits and computation platforms, one needs basic logic gates that operate and can be cascaded within the SW domain without requiring back and forth conversion between the SW and voltage domains. To achieve this, SW gates have to possess intrinsic fanout capabilities, be input-output data representation coherent, and reconfigurable. In this paper, we address the first and the last requirements and propose a novel 4-output programmable SW logic. First, we introduce the gate structure and demonstrate that, by adjusting the gate output detection method, it can parallelly evaluate any 4-element subset of the 2-input Boolean function set AND, NAND, OR, NOR, XOR, and XNOR. Furthermore, we adjust the structure such that all its 4 outputs produce SWs with the same energy and demonstrate that it can evaluate Boolean function sets while providing fanout capabilities ranging from 1 to 4. We validate our approach by instantiating and simulating different gate configurations such as 4-output AND/OR, 4-output XOR/XNOR, output energy balanced 4-output AND/OR, and output energy balanced 4-output XOR/XNOR by means of Object Oriented Micromagnetic Framework (OOMMF) simulations. Finally, we evaluate the performance of our proposal in terms of delay and energy consumption and compare it against existing state-of-the-art SW and 16nm CMOS counterparts. The results indicate that for the same functionality, our approach provides 3x and 16x energy reduction, when compared with conventional SW and 16nm CMOS implementations, respectively.

On the Spectral-Equipartite Graphs and Eccentricity-Equipartite Graphs

Let G = (V, E) be a graph of order 2n. If A ⊆ V and hAi ∼= hV \Ai, then A is said to be isospectral. If for every n-element subset A of V we have hAi ∼= hV \Ai, then we say that G is spectral-equipartite. In [1], Igor Shparlinski communicated with Bibak et al., proposing a full characterization of spectral-equipartite graphs. In this paper, we gave a characterization of disconnected spectral-equipartite graphs. Moreover, we introduced the concept eccentricity-equipartite graphs.

Hypercycle Systems of 5-Cycles in Complete 3-Uniform Hypergraphs

In this paper, we consider the problem of constructing hypercycle systems of 5-cycles in complete 3-uniform hypergraphs. A hypercycle system C(r,k,v) of order v is a collection of r-uniform k-cycles on a v-element vertex set, such that each r-element subset is an edge in precisely one of those k-cycles. We present cyclic hypercycle systems C(3,5,v) of orders v=25,26,31,35,37,41,46,47,55,56, a highly symmetric construction for v=40, and cyclic 2-split constructions of orders 32,40,50,52. As a consequence, all orders v≤60 permitted by the divisibility conditions admit a C(3,5,v) system. New recursive constructions are also introduced.

Glider automorphisms and a finitary Ryan’s theorem for transitive subshifts of finite type

For any mixing SFT X we construct a reversible shift-commuting continuous map (automorphism) which breaks any given finite point of the subshift into a finite collection of gliders traveling into opposing directions. As an application we prove a finitary Ryan’s theorem: the automorphism group $${{\,\mathrm{Aut}\,}}(X)$$ Aut ( X ) contains a two-element subset S whose centralizer consists only of shift maps. We also give an example which shows that a stronger finitary variant of Ryan’s theorem does not hold even for the binary full shift.

A New Approach Model

The development of a new type of approach model of language learning is assumed to be based on the reference points establishing it conceptually. The general concept of approach is verified in terms of the conventional categories of domain, set, set element (subset), set extension, set intension, and hierarchy. A new type of approach to language learning proposed in the chapter is assumed to be conceptually established in the framework of a theoretical model construed as a functional domain comprising a hierarchical set of elements, approach being the top member but method and technique forming subset elements. The hierarchical approach set has both extensional and intensional projections, which are mapped onto the actual learning and teaching procedure and transformed into external and internal language acquisition segments, correspondingly. Thought-outward target language speech interrelations are thus represented. This is the way the functional correlation between theoretical grounds of the approach conception and curriculum procedure is established. The function characterizing approach dimensions is a multifold dichotomized formation whose duality is being made up of 1) the internal-external opposition, 2) source-target language opposition, 3) verbal dissimilarity opposition. The function in question gets delineated at the intersection of the dichotomies and designated as predication. Predication is an invariant unit of instruction surfaced in the form of the approach radix. The approach under consideration acquires the same contingent name. The main issues considered include the justification of the descriptive approach terminology, the functional modeling of approach, source-target language relationship in approach modeling, types of speech in approach modeling, and the invariant approach to language learning: introductory statements.

New Hope for Relative Overlap Measures of Coherence

Relative overlap measures of coherence have recently been shown to have two devastating properties: (i) according to the plain relative overlap measure, the degree of coherence of any set of propositions cannot be increased by adding further propositions, and (ii) according to the refined relative overlap measure, no set can be more coherent than its most coherent two-element subset. This result has been taken to rule out relative overlap as a foundation for a probabilistic explication of coherence. The present paper shows that this view is premature: we propose a relative overlap measure that does not fall victim to the two properties. The guiding idea is to employ a well-established recipe for the construction of coherence measures and to adapt it to the idea of relative overlap. We show that this new measure keeps up with, or even outperforms, former overlap measures in a set of desiderata for coherence measures and a collection of popular test cases. This result re-establishes relative overlap as a candidate for a proper formalization of coherence.

Harmonious coloring of uniform hypergraphs

A harmonious coloring of a k-uniform hypergraph H is a vertex coloring such that no two vertices in the same edge share the same color, and each k-element subset of colors appears on at most one edge. The harmonious number h(H) is the least number of colors needed for such a coloring. We prove that k-uniform hypergraphs of bounded maximum degree ? satisfy h(H) = O(k?k!m), where m is the number of edges in H which is best possible up to a multiplicative constant. Moreover, for every fixed ?, this constant tends to 1 with k ? ?. We use a novel method, called entropy compression, that emerged from the algorithmic version of the Lov?sz Local Lemma due to Moser and Tardos.

On B(4, 14) non-2-groups

We consider a combinatorial problem in group theory. A group G is said to be a B(n, k) group if for any n-element subset A of G, ∣A2∣ ≤ k. In this paper, a complete characterization of B(4, 14) non-2-groups is given.