scholarly journals Short Communication: Optimally Solving the Unit-Demand Envy-Free Pricing Problem with Metric Substitutability in Cubic Time

Algorithms ◽  
2021 ◽  
Vol 14 (10) ◽  
pp. 279
Author(s):  
Marcos M. Salvatierra ◽  
Mario Salvatierra ◽  
Juan G. Colonna
Keyword(s):  
Short Communication ◽  
Competitive Equilibrium ◽  
Shortest Paths ◽  
Single Copy ◽  
Time Algorithm ◽  
Optimal Solutions ◽  
First Case ◽  
Special Cases ◽  
Unit Demand ◽  
Theoretical Results

In general, the unit-demand envy-free pricing problem has proven to be APX-hard, but some special cases can be optimally solved in polynomial time. When substitution costs that form a metric space are included, the problem can be solved in O(n4) time, and when the number of consumers is equal to the number of items—all with a single copy so that each consumer buys an item—a O(n3) time method is presented to solve it. This work shows that the first case has similarities with the second, and, by exploiting the structural properties of the costs set, it presents a O(n2) time algorithm for solving it when a competitive equilibrium is considered or a O(n3) time algorithm for more general scenarios. The methods are based on a dynamic programming strategy, which simplifies the calculations of the shortest paths in a network; this simplification is usually adopted in the second case. The theoretical results obtained provide efficiency in the search for optimal solutions to specific revenue management problems.

scholarly journals A Convergent Collocation Approach for Generalized Fractional Integro-Differential Equations Using Jacobi Poly-Fractonomials

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 979
Author(s):  
Sandeep Kumar ◽  
Rajesh K. Pandey ◽  
H. M. Srivastava ◽  
G. N. Singh
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Collocation Method ◽  
Caputo Fractional Derivative ◽  
Fractional Derivatives ◽  
Special Cases ◽  
Collocation Approach ◽  
Linear And Nonlinear ◽  
Simulation Results ◽  
Theoretical Results

In this paper, we present a convergent collocation method with which to find the numerical solution of a generalized fractional integro-differential equation (GFIDE). The presented approach is based on the collocation method using Jacobi poly-fractonomials. The GFIDE is defined in terms of the B-operator introduced recently, and it reduces to Caputo fractional derivative and other fractional derivatives in special cases. The convergence and error analysis of the proposed method are also established. Linear and nonlinear cases of the considered GFIDEs are numerically solved and simulation results are presented to validate the theoretical results.

scholarly journals A Fast and Exact Greedy Algorithm for the Core–Periphery Problem

Symmetry ◽  
2020 ◽  
Vol 12 (1) ◽  
pp. 94 ◽  
Author(s):  
Dario Fasino ◽  
Franca Rinaldi
Keyword(s):  
Structural Analysis ◽  
Optimization Problem ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Combinatorial Optimization Problem ◽  
Connected Subgraph ◽  
Optimal Solutions ◽  
The Core ◽  
Core Set ◽  
Key Concepts

The core–periphery structure is one of the key concepts in the structural analysis of complex networks. It consists of a partitioning of the node set of a given graph or network into two groups, called core and periphery, where the core nodes induce a well-connected subgraph and share connections with peripheral nodes, while the peripheral nodes are loosely connected to the core nodes and other peripheral nodes. We propose a polynomial-time algorithm to detect core–periphery structures in networks having a symmetric adjacency matrix. The core set is defined as the solution of a combinatorial optimization problem, which has a pleasant symmetry with respect to graph complementation. We provide a complete description of the optimal solutions to that problem and an exact and efficient algorithm to compute them. The proposed approach is extended to networks with loops and oriented edges. Numerical simulations are carried out on both synthetic and real-world networks to demonstrate the effectiveness and practicability of the proposed algorithm.

scholarly journals Edge detection with trigonometric polynomial shearlets

2021 ◽  
Vol 47 (1) ◽  
Author(s):  
Kevin Schober ◽  
Jürgen Prestin ◽  
Serhii A. Stasyuk
Keyword(s):  
Edge Detection ◽  
Trigonometric Polynomial ◽  
Two Dimensions ◽  
Characteristic Functions ◽  
Numerical Examples ◽  
Special Cases ◽  
Periodic Characteristic ◽  
Boundary Curves ◽  
Theoretical Results ◽  
De La Vallée Poussin

AbstractIn this paper, we show that certain trigonometric polynomial shearlets which are special cases of directional de la Vallée Poussin-type wavelets are able to detect step discontinuities along boundary curves of periodic characteristic functions. Motivated by recent results for discrete shearlets in two dimensions, we provide lower and upper estimates for the magnitude of the corresponding inner products. In the proof, we use localization properties of trigonometric polynomial shearlets in the time and frequency domain and, among other things, bounds for certain Fresnel integrals. Moreover, we give numerical examples which underline the theoretical results.

scholarly journals The Computational Complexity of Understanding Binary Classifier Decisions

2021 ◽  
Vol 70 ◽  
Author(s):  
Stephan Waeldchen ◽  
Jan Macdonald ◽  
Sascha Hauch ◽  
Gitta Kutyniok
Keyword(s):  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Boolean Circuits ◽  
Binary Classifier ◽  
Minimal Set ◽  
Limited Size ◽  
Approximation Factor ◽  
Special Cases ◽  
Relevant Variables ◽  
Binary Classifiers

For a d-ary Boolean function Φ: {0, 1}d → {0, 1} and an assignment to its variables x = (x1, x2, . . . , xd) we consider the problem of finding those subsets of the variables that are sufficient to determine the function value with a given probability δ. This is motivated by the task of interpreting predictions of binary classifiers described as Boolean circuits, which can be seen as special cases of neural networks. We show that the problem of deciding whether such subsets of relevant variables of limited size k ≤ d exist is complete for the complexity class NPPP and thus, generally, unfeasible to solve. We then introduce a variant, in which it suffices to check whether a subset determines the function value with probability at least δ or at most δ − γ for 0 < γ < δ. This promise of a probability gap reduces the complexity to the class NPBPP. Finally, we show that finding the minimal set of relevant variables cannot be reasonably approximated, i.e. with an approximation factor d1−α for α > 0, by a polynomial time algorithm unless P = NP. This holds even with the promise of a probability gap.

k-PAIRS NON-CROSSING SHORTEST PATHS IN A SIMPLE POLYGON

1999 ◽  
Vol 09 (06) ◽  
pp. 533-552 ◽  
Author(s):  
EVANTHIA PAPADOPOULOU
Keyword(s):  
Outer Boundary ◽  
Shortest Paths ◽  
Simple Polygon ◽  
Time Algorithm ◽  
Polygonal Domain ◽  
Total Cost

This paper presents a simple O(n+k) time algorithm to compute the set of knon-crossing shortest paths between k source-destination pairs of points on the boundary of a simple polygon of n vertices. Paths are allowed to overlap but are not allowed to cross in the plane. A byproduct of this result is an O(n) time algorithm to compute a balanced geodesic triangulation which is easy to implement. The algorithm extends to a simple polygon with one hole where source-destination pairs may appear on both the inner and outer boundary of the polygon. In the latter case, the goal is to compute a collection of non-crossing paths of minimum total cost. The case of a rectangular polygonal domain where source-destination pairs appear on the outer and one inner boundary12 is briefly discussed.

Vibration Characteristics of Straight Blades of Asymmetrical Aerofoil Cross-Section

1969 ◽  
Vol 20 (2) ◽  
pp. 178-190 ◽  
Author(s):  
W. Carnegie ◽  
B. Dawson
Keyword(s):  
Analytical Solution ◽  
Cross Section ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Limited Range ◽  
First Order ◽  
Special Cases ◽  
Theoretical Results ◽  
Theoretical Procedure ◽  
Good Agreement

SummaryTheoretical and experimental natural frequencies and modal shapes up to the fifth mode of vibration are given for a straight blade of asymmetrical aerofoil cross-section. The theoretical procedure consists essentially of transforming the differential equations of motion into a set of simultaneous first-order equations and solving them by a step-by-step finite difference procedure. The natural frequency values are compared with results obtained by an analytical solution and with standard solutions for certain special cases. Good agreement is shown to exist between the theoretical results for the various methods presented. The equations of motion are dependent upon the coordinates of the axis of the centre of flexure of the beam relative to the centroidal axis. The effect of variations of the centre of flexure coordinates upon the frequencies and modal shapes is shown for a limited range of coordinate values. Comparison is made between the theoretical natural frequencies and modal shapes and corresponding results obtained by experiment.

Helices, Hasimoto Surfaces and Bäcklund Transformations

2000 ◽  
Vol 43 (4) ◽  
pp. 427-439 ◽  
Author(s):  
Thomas A. Ivey
Keyword(s):  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Similar Investigation ◽  
Screw Motion ◽  
Single Screw ◽  
Wave Solutions ◽  
First Case ◽  
Pseudospherical Surfaces ◽  
Special Cases ◽  
Constant Torsion

AbstractTravelling wave solutions to the vortex filament flow generated by elastica produce surfaces in ℝ3 that carry mutually orthogonal foliations by geodesics and by helices. These surfaces are classified in the special cases where the helices are all congruent or are all generated by a single screw motion. The first case yields a new characterization for the Bäcklund transformation for constant torsion curves in R3, previously derived fromthe well-known transformation for pseudospherical surfaces. A similar investigation for surfaces in H3 or S3 leads to a new transformation for constant torsion curves in those spaces that is also derived from pseudospherical surfaces.

scholarly journals Every Local Minimum Value Is the Global Minimum Value of Induced Model in Nonconvex Machine Learning

2019 ◽  
Vol 31 (12) ◽  
pp. 2293-2323 ◽  
Author(s):  
Kenji Kawaguchi ◽  
Jiaoyang Huang ◽  
Leslie Pack Kaelbling
Keyword(s):  
Machine Learning ◽  
Neural Networks ◽  
Local Minimum ◽  
Deep Neural Networks ◽  
Representation Learning ◽  
Minimum Value ◽  
Special Cases ◽  
Optimal Value ◽  
Special Case ◽  
Theoretical Results

For nonconvex optimization in machine learning, this article proves that every local minimum achieves the globally optimal value of the perturbable gradient basis model at any differentiable point. As a result, nonconvex machine learning is theoretically as supported as convex machine learning with a handcrafted basis in terms of the loss at differentiable local minima, except in the case when a preference is given to the handcrafted basis over the perturbable gradient basis. The proofs of these results are derived under mild assumptions. Accordingly, the proven results are directly applicable to many machine learning models, including practical deep neural networks, without any modification of practical methods. Furthermore, as special cases of our general results, this article improves or complements several state-of-the-art theoretical results on deep neural networks, deep residual networks, and overparameterized deep neural networks with a unified proof technique and novel geometric insights. A special case of our results also contributes to the theoretical foundation of representation learning.

scholarly journals Progress of COVID-19 Epidemic in Pakistan

2020 ◽  
Vol 32 (4) ◽  
pp. 154-156 ◽  
Author(s):  
Khadijah Abid ◽  
Yashfika Abdul Bari ◽  
Maryam Younas ◽  
Sehar Tahir Javaid ◽  
Abira Imran
Keyword(s):  
Public Health ◽  
World Health Organization ◽  
Short Communication ◽  
Health Sector ◽  
Public Health Emergency ◽  
World Health ◽  
First Case ◽  
The World ◽  
Health Organization ◽  
Shed Light

The outbreak of corona virus initiated as pneumonia of unknown cause in December 2019 in Wuhan, China, which has been now spreading rapidly out of Wuhan to other countries. On January 30, 2020, the World Health Organization (WHO) declared coronavirus outbreak as the sixth public health emergency of international concern (PHEIC), and on March 11, 2020, the WHO announced coronavirus as pandemic. Coronavirus is thought to be increasing in Pakistan. The first case of coronavirus was reported from Karachi on February 26, 2020, with estimated populace of Pakistan as 204.65 million. Successively, the virus spreads into various regions nationwide and has currently become an epidemic. The WHO has warned Pakistan that the country could encounter great challenge against the outbreak of coronavirus in the coming days. This short communication is conducted to shed light on the epidemic of coronavirus in the country. It would aid in emphasizing the up-to-date situation in a nutshell and the measures taken by the health sector of Pakistan to abate the risk of communication.

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