We study a localization of gravity through the matching point between non-inertial frames and local inertial frames. This localization of gravity is done through defining relative gravitational red-shift. This lead to an emergence of a timeless state in a mathematically consistent way. In this timeless state of gravity, we find a geometric interpretation of the speed of light and mass. The experimental support of the timeless state is the quantum entanglement and internal symmetries that are independent of time. Therefore gravity would be responsible for measurements independent of time including quantum entanglement. Timeless state in that sense explain why do we have gravity/gauge correspondence. We conjecture that the universe emerged from a black hole with a global $SU(3)\times SU(2)\times U(1)$ symmetry on its event horizon that produces gauge fields Electromagnetism, weak and strong nuclear force through localization of this global symmetry. Through the localization in the gravity field, the timeless measurements such as spin will be correlated with the varying of timeless measurement which is relative gravitational red-shift. We present a gravitational or geometric interpretation of spin-0, spin-1, and spin-1/2 states. We present an interpretation of why do we measure matter rather than anti-matter. We found that the Higgs scalar field is represented by the gravitational red-shift at every point in the space around the black hole. We derive the numerical value of the cosmological constant that agrees with experimental observations.

A sparse language is a formal language such that the number of strings of length $n$ is bounded by a polynomial function of $n$. We create a class with the opposite definition, that is a class of languages that are dense instead of sparse. We define a dense language on $m$ as a formal language (a set of binary strings) where there exists a positive integer $n_{0}$ such that the counting of the number of strings of length $n \geq n_{0}$ in the language is greater than or equal to $2^{n - m}$ where $m$ is a real number and $0 < m \leq 1$. We call the complexity class of all dense languages on $m$ as $DENSE(m)$. We prove that there exists an $\textit{NP--complete}$ problem that belongs to $DENSE(m)$ for every possible value of $0 < m \leq 1$.

The aim of this paper is to discuss the relations between various notions of sequential completeness and the corresponding notions of completeness by nets or by filters in the setting of quasi-metric spaces. We propose a new definition of right $K$-Cauchy net in a quasi-metric space for which the corresponding completeness is equivalent to the sequential completeness. In this way we complete some results of R.~A. Stoltenberg, Proc. London Math. Soc. \textbf{17} (1967), 226--240, and V.~Gregori and J.~Ferrer, Proc. Lond. Math. Soc., III Ser., \textbf{49} (1984), 36. A discussion on nets defined over ordered or pre-ordered directed sets is also included.

Natural hazards threaten millions of people all over the world. To address the risk, exposure and vulnerability models with high resolution data are essential. However, in many areas of the world, exposure models are rather coarse and aggregated over large areas. Although OpenStreetMap (OSM) offers great potential to assess risk at a detailed building-by-building level, completeness of OSM building footprints is still heterogeneous. We present an approach to close this gap by means of crowdsourcing based on the mobile App MapSwipe, where volunteers swipe through satellite images of a region collecting user feedback on classification tasks. For our application, MapSwipe was extended by a completeness feature that allows to classify a tile as “no building”, “complete” or “incomplete”. To assess the quality of the produced data, the completeness feature was applied at four regions. Our results show that the crowdsourced approach yields a reasonable classification performance of the completeness of OSM building footprints. Nevertheless, this study also revealed that volunteers tend to classify nearly completely mapped tiles as “complete”, especially in areas with a high OSM building density. Another factor that influenced the classification performance was the level of alignment of the OSM layer with the satellite imagery.

Two representable substructural logics ${\rm {\bf UL}}_\omega $ and ${\rm {\bf IUL}}_\omega $ are logics for finite UL and IUL-algebras, respectively. In this paper, the standard completeness of ${\rm {\bf UL}}_\omega $ and ${\rm {\bf IUL}}_\omega $ is proved by the method developed by Jenei, Montagna, Esteva, Gispert, Godo and Wang. This shows that ${\rm {\bf UL}}_\omega $ and ${\rm {\bf IUL}}_\omega $ are substructural fuzzy logics.

Our method for density elimination is generalized to the non-commutative substructural logic GpsUL∗. Then the standard completeness of HpsUL∗follows as a lemma by virtue of previous workbyMetcalfeandMontagna. This result shows that HpsUL∗ is the logic of pseudo-uninorms and their residua and answered the question posed by Prof. Metcalfe, Olivetti, Gabbay and Tsinakis.

We present a uniform method of density elimination for several semilinear substructural logics. Especially, the density elimination for the involutive uninorm logic IUL is proved. Then the standard completeness of IUL follows as a lemma by virtue of previous work by Metcalfe and Montagna.

We prove a version of Ekeland Variational Principle (EkVP) in a weighted graph $G$ and its equivalence to Caristi fixed point theorem and to Takahashi minimization principle. The usual completeness and topological notions are replaced with some weaker versions expressed in terms of the graph $G$. The main tool used in the proof is the OSC property for sequences in a graph. Converse results, meaning the completeness of graphs for which one of these principles holds is also considered.

To ensure the security of spatial-temporal Top-k query in two-tiered wireless sensor networks, many schemes have been proposed in the literature in the past decade. However, most of them only consider the scenario where sensor nodes are static, and cannot achieve the security goal for spatial-temporal Top-k query in mobile sensor networks, because the mobility of the sensor nodes will affect the spatial-temporal relationships of the sensory data items generated by the sensor nodes. Although we have proposed some schemes for two-tiered mobile wireless sensor networks (TMWSNs) in our previous work, there is still large room to improve their performances. In this paper, we proposed a novel scheme named STQ-TMWSN for secure fine-grained spatial-temporal Top-k query in TMWSNs based on the virtual-grid construction and the size-order encryption binding. Theoretic analysis shows that STQ-TMWSN can achieve low computation complexity and high security performance. Simulation results indicate that STQ-TMWSN brings much lower communication cost than the state-of-the-art schemes on securing Top-k query in TMWSNs.

The previous two parts of the paper (correspondingly, https://philpapers.org/rec/PENFLT-2 and https://philpapers.org/rec/PENFLT-3) demonstrate that the interpretation of Fermat’s last theorem (FLT) in Hilbert arithmetic meant both in a narrow sense and in a wide sense can suggest a proof by induction in Part I and by means of the Kochen - Specker theorem in Part II. The same interpretation can serve also for a proof FLT based on Gleason’s theorem and partly similar to that in Part II. The concept of (probabilistic) measure of a subspace of Hilbert space and especially its uniqueness can be unambiguously linked to that of partial algebra or incommensurability, or interpreted as a relation of the two dual branches of Hilbert arithmetic in a wide sense. The investigation of the last relation allows for FLT and Gleason’s theorem to be equated in a sense, as two dual counterparts, and the former to be inferred from the latter, as well as vice versa under an additional condition relevant to the Gödel incompleteness of arithmetic to set theory. The qubit Hilbert space itself in turn can be interpreted by the unity of FLT and Gleason’s theorem. The proof of such a fundamental result in number theory as FLT by means of Hilbert arithmetic in a wide sense can be generalized to an idea about “quantum number theory”. It is able to research mathematically the origin of Peano arithmetic from Hilbert arithmetic by mediation of the “nonstandard bijection” and its two dual branches inherently linking it to information theory. Then, infinitesimal analysis and its revolutionary application to physics can be also re-realized in that wider context, for example, as an exploration of the way for physical quantity of time (respectively, for time derivative in any temporal process considered in physics) to appear at all. Finally, the result admits a philosophical reflection of how any hierarchy arises or changes itself only thanks to its dual and idempotent counterpart.