On generalized algebraic theories and categories with families

We give a syntax independent formulation of finitely presented generalized algebraic theories as initial objects in categories of categories with families (cwfs) with extra structure. To this end, we simultaneously define the notion of a presentation Σ of a generalized algebraic theory and the associated category CwFΣ of small cwfs with a Σ-structure and cwf-morphisms that preserve Σ-structure on the nose. Our definition refers to the purely semantic notion of uniform family of contexts, types, and terms in CwFΣ. Furthermore, we show how to syntactically construct an initial cwf with a Σ-structure. This result can be viewed as a generalization of Birkhoff’s completeness theorem for equational logic. It is obtained by extending Castellan, Clairambault, and Dybjer’s construction of an initial cwf. We provide examples of generalized algebraic theories for monoids, categories, categories with families, and categories with families with extra structure for some type formers of Martin-Löf type theory. The models of these are internal monoids, internal categories, and internal categories with families (with extra structure) in a small category with families. Finally, we show how to extend our definition to some generalized algebraic theories that are not finitely presented, such as the theory of contextual cwfs.

Ergodic equilibration of Rényi entropies and replica wormholes

We study the behavior of Rényi entropies for pure states from standard assumptions about chaos in the high-energy spectrum of the Hamiltonian of a many-body quantum system. We compute the exact long-time averages of Rényi entropies and show that the quantum noise around these values is exponentially suppressed in the microcanonical entropy. For delocalized states over the microcanonical band, the long-time average approximately reproduces the equilibration proposal of H. Liu and S. Vardhan, with extra structure arising at the order of non-planar permutations. We analyze the equilibrium approximation for AdS/CFT systems describing black holes in equilibrium in a box. We extend our analysis to the situation of an evaporating black hole, and comment on the possible gravitational description of the new terms in our approximation.

Khovanov homotopy type, periodic links and localizations

Given an m-periodic link $$L\subset S^3$$ L ⊂ S 3 , we show that the Khovanov spectrum $$\mathcal {X}_L$$ X L constructed by Lipshitz and Sarkar admits a group action. We relate the Borel cohomology of $$\mathcal {X}_L$$ X L to the equivariant Khovanov homology of L constructed by the second author. The action of Steenrod algebra on the cohomology of $$\mathcal {X}_L$$ X L gives an extra structure of the periodic link. Another consequence of our construction is an alternative proof of the localization formula for Khovanov homology, obtained first by Stoffregen and Zhang. By applying the Dwyer–Wilkerson theorem we express Khovanov homology of the quotient link in terms of equivariant Khovanov homology of the original link.

Quantificational Attitudes

The literature contains a popular argument in favor of the position that conditional attitudes (especially intentions and desires) are not simple attitudes with conditional contents but, rather, have a more complex structure. In this paper I show that an analogous argument applies to what we might call quantificational attitudes—like an intention to follow every bit of good advice I receive or a desire to get rabies shots for each bite I incur from an infected bat. The conditions under which these attitudes are satisfied and thwarted are not captured by claiming that they are simple attitudes with quantificational contents. So, the argument supports a novel position—that quantificational attitudes have a more complex structure. After sketching the form of this extra structure, I show how similar considerations count in favor of the existence of genuinely quantificational speech acts.

Causal hierarchy in modified gravity

We investigate the causal hierarchy in various modified theories of gravity. In general relativity the standard causal hierarchy, (key elements of which are chronology, causality, strong causality, stable causality, and global hyperbolicity), is well-established. In modified theories of gravity there is typically considerable extra structure, (such as: multiple metrics, aether fields, modified dispersion relations, Hořava-like gravity, parabolic propagation, etcetera), requiring a reassessment and rephrasing of the usual causal hierarchy. We shall show that in this extended framework suitable causal hierarchies can indeed be established, and discuss the implications for the interplay between “superluminal” propagation and causality. The key distinguishing feature is whether the signal velocity is finite or infinite. Preserving even minimal notions of causality in the presence of infinite signal velocity requires the aether field to be both unique and hypersurface orthogonal, leading us to introduce the notion of global parabolicity.

A simple quad-band printed diversity antenna with high isolation without extra structure

In this paper a Quad-band diversity antenna with a small size of 48 × 28 × 0.8 mm3 printed on FR4 substrate is presented that consists of two same pentagonal-shaped patch elements. Four arc-shaped slots in the radiating elements are introduced to obtain the quad-band operation with good return loss. The antenna is designed for covering 5.91 GHz in IEEE 802.11 b/g/n standards, 7.40 GHz in C-Band, 9.18 and 10.72 GHz in X-Band. To achieve the lowest coupling between two elements, three structures for embedding elements are investigated. The prototype is fabricated, and measured results are in good agreement with simulated results. The final antenna accomplishes a weak mutual coupling below −29.2 dB in the all operational bands. Radiation characteristics, radiation efficiency and diversity performance such as diversity gain and envelope correlation coefficient of the final structure, are presented.

Well-Graded Families and the Union-Closed Sets Conjecture

The union-closed sets conjecture states that if a finite family of sets $\mathcal{F}$ is union-closed, then there must be some element contained in at least half of the sets of $\mathcal{F}$. In this work we study the relationship between the union-closed sets conjecture and union-closed families that have the property of being well-graded. In doing so, we show how the density and other properties are affected by the extra structure contained in well-graded families, and we also give several conditions under which well-graded families satisfy the union-closed sets conjecture.

Visualizing a Supercomputer: a Case of Objects Regrouping

In the paper we present the situation which had required visualization of a large amount of non-trivial objects, such as supercomputer’s tasks. The method of visualization of these objects was hard to find. Then we used additional information about an extra structure on those objects. This knowledge led us to an idea of grouping the objects into new generalized ones. Those new artificial objects were easy to visualize due to their small quantity. And they happened to be enough for the cognition of the original problem. That was a successful change of point of view. As a whole, our work belongs to a high-performance computing performance visualization area. It gains valuable attention from scientists over the whole world, for example [1-2].

Definiteness, Structure and Agreement in Turkish Possessives

The present study argues for different levels of definiteness in the nominal domain based on an analysis of two different kinds of possession phrases in Turkish. In line with related research (e.g. Campbell 1996, Haegeman & Ürögdi 2010 and Jimenez-Fernandez 2012, Zamparelli 2014), I argue that the more definite a possession phrase is in Turkish, the more structure it has. This extra structure also hosts a definiteness operator, which accounts for island-like properties of the more definite possession structures. The proposal can therefore account for interpretational and syntactic properties of different possession structures in Turkish. I also argue that agreement properties are also accounted for with this proposal, when coupled with the island-forming properties of agreement in Turkish (George & Kornfilt 1981). The current proposal then contributes to previous research on levels of definiteness and it does so by focusing on a type of determiner phrases that previous research on levels of definiteness did not focus on.

A partial refining of the Erdős-Kelly regulation

The aim of this note is to advance the refining of the Erdős-Kelly result on graphical inducing regularization. The operation of inducing regulation (on graphs or multigraphs) with prescribed maximum vertex degree is originated by D. König in 1916. As is shown by Chartrand and Lesniak in their textbook Graphs & Digraphs (1996), an iterated construction for graphs can result in a regularization with many new vertices. Erdős and Kelly have presented (1963, 1967) a simple and elegant numerical method of determining for any simple \(n\)-vertex graph \(G\) with maximum vertex degree \(\Delta\), the exact minimum number, say \(\theta =\theta(G)\), of new vertices in a \(\Delta\)-regular graph \(H\) which includes \(G\) as an induced subgraph. The number \(\theta(G)\), which we call the cost of regulation of \(G\), has been upper-bounded by the order of \(G\), the bound being attained for each \(n\ge4\), e.g. then the edge-deleted complete graph \(K_n-e\) has \(\theta=n\). For \(n\ge 4\), we present all factors of \(K_n\) with \(\theta=n\) and next \(\theta=n-1\). Therein in case \(\theta=n-1\) and \(n\) odd only, we show that a specific extra structure, non-matching, is required.