scholarly journals A Partial Solution To An Open Problem

2015 ◽  
Vol 5 (2) ◽  
Author(s):  
Şükran KONCA
Keyword(s):  
Open Problem ◽  
Partial Solution

A partial solution to an open problem of Amadio and Curien

2018 ◽  
Vol 260 ◽  
pp. 126-134
Author(s):  
Xiaoyong Xi ◽  
Jinbo Yang ◽  
Hui Kou
Keyword(s):  
Open Problem ◽  
Partial Solution

Extensional quotients for type theory and the consistency problem for NF

1998 ◽  
Vol 63 (1) ◽  
pp. 247-261
Author(s):  
Gian Aldo Antonelli
Keyword(s):  
Equivalence Relation ◽  
Open Problem ◽  
Type Theory ◽  
Partial Solution ◽  
Weak Version ◽  
New Foundations ◽  
Multiple Copies ◽  
Consistency Problem ◽  
Membership Relation ◽  
Main Ideas

Quine's “New Foundations” (NF) was first presented in Quine [10] and later on in Quine [11]. Ernst Specker [15, 13], building upon a previous result of Ehrenfeucht and Mostowski [5], showed that NF is consistent if and only if there is a model of the Theory of Negative (and positive) Types (TNT) with full extensionality that admits of a “shifting automorphism,” but the existence of such a model remains an open problem.In his [8], Ronald Jensen gave a partial solution to the problem of the consistency of Quine's NF. Jensen considered a version of NF—referred to as NFU—in which the axiom of extensionality is weakened to allow for Urelemente or “atoms.” He showed, modifying Specker's theorem, that the existence of a model of TNT with atoms admitting of a “shifting automorphism” implies the consistency of NFU, proceeding then to exhibit such a model.This paper presents a reduction of the consistency problem for NF to the existence of a model of TNT with atoms containing certain “large” (unstratified) sets and admitting a shifting automorphism. In particular we show that such a model can be “collapsed” to a model of pure TNT in such a way as to preserve the shifting automorphism. By the above-mentioned result of Specker's, this implies the consistency of NF.Let us take the time to explain the main ideas behind the construction. Suppose we have a certain universe U of sets, built up from certain individuals or “atoms.” In such a universe we have only a weak version of the axiom of extensionality: two objects are the same if and only if they are both sets having the same members. We would like to obtain a universe U′ that is as close to U as possible, but in which there are no atoms (i.e., the only memberless object is the empty set). One way of doing this is to assign to each atom ξ, a set a (perhaps the empty set), inductively identifying sets that have members that we are already committed to considering “the same.” In doing this we obtain an equivalence relation ≃ over U that interacts nicely with the membership relation (provided we have accounted for multiplicity of members, i.e., we have allowed sets to contain “multiple copies” of the same object). Then we can take U′ = U/≃, the quotient of U with respect to ≃. It is then possible to define a “membership” relation over U′ in such a way as to have full extensionality. Relations such as ≃ are referred to as “contractions” by Hinnion and “bisimulations” by Aczel.

The Yukawa Potential Function and Reciprocal Univalent Function

2013 ◽  
Vol 3 (2) ◽  
pp. 197-202
Author(s):  
Amir Pishkoo ◽  
Maslina Darus
Keyword(s):  
Mathematical Model ◽  
Unit Disk ◽  
Potential Function ◽  
Univalent Function ◽  
Open Problem ◽  
Yukawa Potential ◽  
Reciprocal Theorem ◽  
Problem Statement ◽  
Fundamental Forces

This paper presents a mathematical model that provides analytic connection between four fundamental forces (interactions), by using modified reciprocal theorem,derived in the paper, as a convenient template. The essential premise of this work is to demonstrate that if we obtain with a form of the Yukawa potential function [as a meromorphic univalent function], we may eventually obtain the Coloumb Potential as a univalent function outside of the unit disk. Finally, we introduce the new problem statement about assigning Meijer's G-functions to Yukawa and Coloumb potentials as an open problem.

Big Questions in Ecology and Evolution

2009 ◽  
Author(s):  
Thomas N. Sherratt ◽  
David M. Wilkinson
Keyword(s):  
Partial Solution ◽  
Fundamental Importance ◽  
Evolutionary Perspective ◽  
Text Book ◽  
Broad Audience ◽  
The Future ◽  
The Tropics ◽  
Made In ◽  
Do So

Why do we age? Why cooperate? Why do so many species engage in sex? Why do the tropics have so many species? When did humans start to affect world climate? This book provides an introduction to a range of fundamental questions that have taxed evolutionary biologists and ecologists for decades. Some of the phenomena discussed are, on first reflection, simply puzzling to understand from an evolutionary perspective, whilst others have direct implications for the future of the planet. All of the questions posed have at least a partial solution, all have seen exciting breakthroughs in recent years, yet many of the explanations continue to be hotly debated. Big Questions in Ecology and Evolution is a curiosity-driven book, written in an accessible way so as to appeal to a broad audience. It is very deliberately not a formal text book, but something designed to transmit the excitement and breadth of the field by discussing a number of major questions in ecology and evolution and how they have been answered. This is a book aimed at informing and inspiring anybody with an interest in ecology and evolution. It reveals to the reader the immense scope of the field, its fundamental importance, and the exciting breakthroughs that have been made in recent years.

scholarly journals Compact Differences of Composition Operators on Bergman Spaces Induced by Doubling Weights

2021 ◽  
Author(s):  
Bin Liu ◽  
Jouni Rättyä ◽  
Fanglei Wu
Keyword(s):  
Bergman Space ◽  
Open Problem ◽  
Lebesgue Space ◽  
Composition Operators ◽  
Bergman Spaces ◽  
Weighted Bergman Space ◽  
Carleson Measures ◽  
Doubling Weights ◽  
The Way

AbstractBounded and compact differences of two composition operators acting from the weighted Bergman space $$A^p_\omega $$ A ω p to the Lebesgue space $$L^q_\nu $$ L ν q , where $$0<q<p<\infty $$ 0 < q < p < ∞ and $$\omega $$ ω belongs to the class "Equation missing" of radial weights satisfying two-sided doubling conditions, are characterized. On the way to the proofs a new description of q-Carleson measures for $$A^p_\omega $$ A ω p , with $$p>q$$ p > q and "Equation missing", involving pseudohyperbolic discs is established. This last-mentioned result generalizes the well-known characterization of q-Carleson measures for the classical weighted Bergman space $$A^p_\alpha $$ A α p with $$-1<\alpha <\infty $$ - 1 < α < ∞ to the setting of doubling weights. The case "Equation missing" is also briefly discussed and an open problem concerning this case is posed.

scholarly journals Graph Learning for Combinatorial Optimization: A Survey of State-of-the-Art

2021 ◽  
Author(s):  
Yun Peng ◽  
Byron Choi ◽  
Jianliang Xu
Keyword(s):  
Machine Learning ◽  
Combinatorial Optimization ◽  
Graph Embedding ◽  
Partial Solution ◽  
Complex Data ◽  
Learning Methods ◽  
Graph Learning ◽  
Second Stage ◽  
End To End ◽  
Embedding Methods

AbstractGraphs have been widely used to represent complex data in many applications, such as e-commerce, social networks, and bioinformatics. Efficient and effective analysis of graph data is important for graph-based applications. However, most graph analysis tasks are combinatorial optimization (CO) problems, which are NP-hard. Recent studies have focused a lot on the potential of using machine learning (ML) to solve graph-based CO problems. Most recent methods follow the two-stage framework. The first stage is graph representation learning, which embeds the graphs into low-dimension vectors. The second stage uses machine learning to solve the CO problems using the embeddings of the graphs learned in the first stage. The works for the first stage can be classified into two categories, graph embedding methods and end-to-end learning methods. For graph embedding methods, the learning of the the embeddings of the graphs has its own objective, which may not rely on the CO problems to be solved. The CO problems are solved by independent downstream tasks. For end-to-end learning methods, the learning of the embeddings of the graphs does not have its own objective and is an intermediate step of the learning procedure of solving the CO problems. The works for the second stage can also be classified into two categories, non-autoregressive methods and autoregressive methods. Non-autoregressive methods predict a solution for a CO problem in one shot. A non-autoregressive method predicts a matrix that denotes the probability of each node/edge being a part of a solution of the CO problem. The solution can be computed from the matrix using search heuristics such as beam search. Autoregressive methods iteratively extend a partial solution step by step. At each step, an autoregressive method predicts a node/edge conditioned to current partial solution, which is used to its extension. In this survey, we provide a thorough overview of recent studies of the graph learning-based CO methods. The survey ends with several remarks on future research directions.

Hamiltonicity of a class of toroidal graphs

2020 ◽  
Vol 70 (2) ◽  
pp. 497-503
Author(s):  
Dipendu Maity ◽  
Ashish Kumar Upadhyay
Keyword(s):  
Connected Graph ◽  
Hamiltonian Cycle ◽  
Partial Solution ◽  
The Other ◽  
Hamiltonian Cycles ◽  
The Face ◽  
Toroidal Graphs

Abstract If the face-cycles at all the vertices in a map are of same type then the map is said to be a semi-equivelar map. There are eleven types of semi-equivelar maps on the torus. In 1972 Altshuler has presented a study of Hamiltonian cycles in semi-equivelar maps of three types {36}, {44} and {63} on the torus. In this article we study Hamiltonicity of semi-equivelar maps of the other eight types {33, 42}, {32, 41, 31, 41}, {31, 61, 31, 61}, {34, 61}, {41, 82}, {31, 122}, {41, 61, 121} and {31, 41, 61, 41} on the torus. This gives a partial solution to the well known Conjecture that every 4-connected graph on the torus has a Hamiltonian cycle.

scholarly journals Optimal regularity of stable solutions to nonlinear equations involving the p-Laplacian

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Xavier Cabré ◽  
Pietro Miraglio ◽  
Manel Sanchón
Keyword(s):  
Dirichlet Problem ◽  
Optimal Condition ◽  
Bounded Domain ◽  
Laplace Operator ◽  
Nonlinear Equations ◽  
Open Problem ◽  
Optimal Range ◽  
Optimal Regularity ◽  
Smooth Bounded Domain ◽  
Stable Solutions

AbstractWe consider the equation {-\Delta_{p}u=f(u)} in a smooth bounded domain of {\mathbb{R}^{n}}, where {\Delta_{p}} is the p-Laplace operator. Explicit examples of unbounded stable energy solutions are known if {n\geq p+\frac{4p}{p-1}}. Instead, when {n<p+\frac{4p}{p-1}}, stable solutions have been proved to be bounded only in the radial case or under strong assumptions on f. In this article we solve a long-standing open problem: we prove an interior {C^{\alpha}} bound for stable solutions which holds for every nonnegative {f\in C^{1}} whenever {p\geq 2} and the optimal condition {n<p+\frac{4p}{p-1}} holds. When {p\in(1,2)}, we obtain the same result under the nonsharp assumption {n<5p}. These interior estimates lead to the boundedness of stable and extremal solutions to the associated Dirichlet problem when the domain is strictly convex. Our work extends to the p-Laplacian some of the recent results of Figalli, Ros-Oton, Serra, and the first author for the classical Laplacian, which have established the regularity of stable solutions when {p=2} in the optimal range {n<10}.

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