Competitive Agents in Certain and Uncertain Markets

2020 ◽  
Author(s):  
Robert G. Chambers
Keyword(s):  
Convex Analysis ◽  
Duality Theorem ◽  
Optimization Theory ◽  
Economic Behavior ◽  
Important Special Case ◽  
Special Cases ◽  
Equilibrium Analyses ◽  
Fenchel's Duality Theorem ◽  
Special Case ◽  
Decision Settings

This book uses concepts from optimization theory to develop an integrated analytic framework for treating consumer, producer, and market equilibrium analyses as special cases of a generic optimization problem. The same framework applies to both stochastic and non-stochastic decision settings, so that the latter is recognized as an (important) special case of the former. The analytic techniques are borrowed from convex analysis and variational analysis. Special emphasis is given to generalized notions of differentiability, conjugacy theory, and Fenchel's Duality Theorem. The book shows how virtually identical conjugate analyses form the basis for modeling economic behavior in each of the areas studied. The basic analytic concepts are borrowed from convex analysis. Special emphasis is given to generalized notions of differentiability, conjugacy theory, and Fenchel's Duality Theorem. It is demonstrated how virtually identical conjugate analyses form the basis for modelling economic behaviour in each of the areas studied.

scholarly journals Aperture referral in dioptric systems with stigmatic elements

2012 ◽  
Vol 71 (1) ◽  
Author(s):  
W. F. Harris
Keyword(s):  
General Theory ◽  
Focal Point ◽  
Positive Zero ◽  
Mathematical Form ◽  
Linear Optics ◽  
Important Special Case ◽  
Vector Algebra ◽  
Special Cases ◽  
Special Case ◽  
Projective Field

A previous paper develops the general theory of aperture referral in linear optics and shows how several ostensibly distinct concepts, including the blur patch on the retina, the effective cornealpatch, the projective field and the field of view, are now unified as particular applications of the general theory.  The theory allows for astigmatism and heterocentricity.  Symplecticity and the generality of the approach, however, make it difficult to gain insight and mean that the material is not accessible to readers unfamiliar with matrices and linear algebra. The purpose of this paper is to examine whatis, perhaps, the most important special case, that in which astigmatism is ignored.  Symplecticity and, hence, the mathematics become greatly simplified. The mathematics reduces largely to elementary vector algebra and, in some places, simple scalar algebra and yet retains the mathematical form of the general approach.  As a result the paper allows insight into and provides a stepping stone to the general theory.  Under referral an aperture under-goes simple scalar magnification and transverse translation.  The paper pays particular attention to referral to transverse planes in the neighbourhood of a focal point where the magnification may be positive, zero or negative.  Circular apertures are treated as special cases of elliptical apertures and the meaning of referred apertures of negative radius is explained briefly. (S Afr Optom 2012 71(1) 3-11)

scholarly journals Parameterized Approximation Algorithms for Bidirected Steiner Network Problems

2021 ◽  
Vol 17 (2) ◽  
pp. 1-68
Author(s):  
Rajesh Chitnis ◽  
Andreas Emil Feldmann ◽  
Pasin Manurangsi
Keyword(s):  
Approximation Algorithm ◽  
Approximation Scheme ◽  
Upper And Lower Bounds ◽  
Important Special Case ◽  
Bidirected Graph ◽  
Negative Edge ◽  
Network Problems ◽  
Special Cases ◽  
Bidirected Graphs ◽  
Special Case

The D irected S teiner N etwork (DSN) problem takes as input a directed graph G =( V , E ) with non-negative edge-weights and a set D ⊆ V × V of k demand pairs. The aim is to compute the cheapest network N⊆ G for which there is an s\rightarrow t path for each ( s , t )∈ D. It is known that this problem is notoriously hard, as there is no k 1/4− o (1) -approximation algorithm under Gap-ETH, even when parametrizing the runtime by k [Dinur & Manurangsi, ITCS 2018]. In light of this, we systematically study several special cases of DSN and determine their parameterized approximability for the parameter k . For the bi -DSNP lanar problem, the aim is to compute a solution N⊆ G whose cost is at most that of an optimum planar solution in a bidirected graph G , i.e., for every edge uv of G the reverse edge vu exists and has the same weight. This problem is a generalization of several well-studied special cases. Our main result is that this problem admits a parameterized approximation scheme (PAS) for  k . We also prove that our result is tight in the sense that (a) the runtime of our PAS cannot be significantly improved, and (b) no PAS exists for any generalization of bi-DSNP lanar , under standard complexity assumptions. The techniques we use also imply a polynomial-sized approximate kernelization scheme (PSAKS). Additionally, we study several generalizations of bi -DSNP lanar and obtain upper and lower bounds on obtainable runtimes parameterized by  k . One important special case of DSN is the S trongly C onnected S teiner S ubgraph (SCSS) problem, for which the solution network N⊆ G needs to strongly connect a given set of k terminals. It has been observed before that for SCSS a parameterized 2-approximation exists for parameter  k [Chitnis et al., IPEC 2013]. We give a tight inapproximability result by showing that for k no parameterized (2 − ε)-approximation algorithm exists under Gap-ETH. Additionally, we show that when restricting the input of SCSS to bidirected graphs, the problem remains NP-hard but becomes FPT for k .

ON THE DENSITY OF HAUSDORFF DIMENSIONS OF BOUNDED TYPE CONTINUED FRACTION SETS: THE TEXAN CONJECTURE

2004 ◽  
Vol 04 (01) ◽  
pp. 63-76 ◽  
Author(s):  
OLIVER JENKINSON
Keyword(s):  
Hausdorff Dimension ◽  
Finite Subset ◽  
Continued Fraction ◽  
Cantor Set ◽  
Computational Method ◽  
Accurate Approximation ◽  
Natural Numbers ◽  
Important Special Case ◽  
The One ◽  
Special Case

Given a non-empty finite subset A of the natural numbers, let EA denote the set of irrationals x∈[0,1] whose continued fraction digits lie in A. In general, EA is a Cantor set whose Hausdorff dimension dim (EA) is between 0 and 1. It is shown that the set [Formula: see text] intersects [0,1/2] densely. We then describe a method for accurately computing dimensions dim (EA), and employ it to investigate numerically the way in which [Formula: see text] intersects [1/2,1]. These computations tend to support the conjecture, first formulated independently by Hensley, and by Mauldin & Urbański, that [Formula: see text] is dense in [0,1]. In the important special case A={1,2}, we use our computational method to give an accurate approximation of dim (E{1,2}), improving on the one given in [18].

An Elementary Proof of Suslin Reciprocity

2005 ◽  
Vol 48 (2) ◽  
pp. 221-236 ◽  
Author(s):  
Matt Kerr
Keyword(s):  
Elementary Proof ◽  
Function Fields ◽  
Algebraic Cycles ◽  
Important Special Case ◽  
Special Case

AbstractWe state and prove an important special case of Suslin reciprocity that has found significant use in the study of algebraic cycles. An introductory account is provided of the regulator and norm maps on Milnor K2-groups (for function fields) employed in the proof.

Analytical Techniques for the Optimisation of Rocket Trajectories

1963 ◽  
Vol 14 (2) ◽  
pp. 105-124 ◽  
Author(s):  
Derek F. Lawden
Keyword(s):  
Gravitational Field ◽  
Artificial Satellite ◽  
Analytical Techniques ◽  
Present Position ◽  
Mayer Problem ◽  
Special Cases ◽  
Minimum Expenditure ◽  
Law Of Attraction ◽  
Special Case

SummaryThe development during the last two decades of analytical techniques for the solution of problems relating to the optimisation of rocket trajectories is outlined and the present position in this field of research is summarised. It is shown that the determination of optimal trajectories in a general gravitational field can be expressed as a Mayer problem from the calculus of variations. The known solution to such a problem is stated and applied, first to the special case of the launching of an artificial satellite into a circular orbit with minimum expenditure of propellant and, secondly, to the general astronautical problem of the economical transfer of a rocket between two terminals in a gravitational field. The special cases when the field is uniform and when it obeys an inverse square law of attraction to a point are then considered, and the paper concludes with some remarks concerning areas in which further investigations are necessary.

scholarly journals Water entry of an expanding wedge/plate with flow detachment

2016 ◽  
Vol 797 ◽  
pp. 322-344 ◽  
Author(s):  
Yuriy A. Semenov ◽  
Guo Xiong Wu
Keyword(s):  
Free Surface ◽  
General Solution ◽  
Water Entry ◽  
Hodograph Method ◽  
Fixed Length ◽  
Pressure Distributions ◽  
Initial Stage ◽  
Special Cases ◽  
Wedge Plate ◽  
Special Case

A general similarity solution for water-entry problems of a wedge with its inner angle fixed and its sides in expansion is obtained with flow detachment, in which the speed of expansion is a free parameter. The known solutions for a wedge of a fixed length at the initial stage of water entry without flow detachment and at the final stage corresponding to Helmholtz flow are obtained as two special cases, at some finite and zero expansion speeds, respectively. An expanding horizontal plate impacting a flat free surface is considered as the special case of the general solution for a wedge inner angle equal to ${\rm\pi}$. An initial impulse solution for a plate of a fixed length is obtained as the special case of the present formulation. The general solution is obtained in the form of integral equations using the integral hodograph method. The results are presented in terms of free-surface shapes, streamlines and pressure distributions.

scholarly journals On Reay's Relaxed Tverberg Conjecture and Generalizations of Conway's Thrackle Conjecture

10.37236/6516 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Megumi Asada ◽  
Ryan Chen ◽  
Florian Frick ◽  
Frederick Huang ◽  
Maxwell Polevy ◽  
...  
Keyword(s):  
Geometric Property ◽  
Convex Sets ◽  
Convex Hulls ◽  
Open Problems ◽  
Higher Dimensional ◽  
Special Cases ◽  
Minimum Number ◽  
Point Set ◽  
Colored Version ◽  
Special Case

Reay's relaxed Tverberg conjecture and Conway's thrackle conjecture are open problems about the geometry of pairwise intersections. Reay asked for the minimum number of points in Euclidean $d$-space that guarantees any such point set admits a partition into $r$ parts, any $k$ of whose convex hulls intersect. Here we give new and improved lower bounds for this number, which Reay conjectured to be independent of $k$. We prove a colored version of Reay's conjecture for $k$ sufficiently large, but nevertheless $k$ independent of dimension $d$. Pairwise intersecting convex hulls have severely restricted combinatorics. This is a higher-dimensional analogue of Conway's thrackle conjecture or its linear special case. We thus study convex-geometric and higher-dimensional analogues of the thrackle conjecture alongside Reay's problem and conjecture (and prove in two special cases) that the number of convex sets in the plane is bounded by the total number of vertices they involve whenever there exists a transversal set for their pairwise intersections. We thus isolate a geometric property that leads to bounds as in the thrackle conjecture. We also establish tight bounds for the number of facets of higher-dimensional analogues of linear thrackles and conjecture their continuous generalizations.

Introduction

2020 ◽  
pp. 1-6
Author(s):  
Peter Scholze ◽  
Jared Weinstein
Keyword(s):  
Moduli Spaces ◽  
Function Fields ◽  
Number Fields ◽  
Important Special Case ◽  
Langlands Correspondence ◽  
Key Innovation ◽  
Perfectoid Spaces ◽  
Definition Of ◽  
Special Case

This introductory chapter provides an overview of Drinfeld's work on the global Langlands correspondence over function fields. Whereas the global Langlands correspondence is largely open in the case of number fields K, it is a theorem for function fields, due to Drinfeld and L. Lafforgue. The key innovation in this case is Drinfeld's notion of an X-shtuka (or simply shtuka). The Langlands correspondence for X is obtained by studying moduli spaces of shtukas. A large part of this course is about the definition of perfectoid spaces and diamonds. There is an important special case where the moduli spaces of shtukas are classical rigid-analytic spaces. This is the case of local Shimura varieties. Some examples of these are the Rapoport-Zink spaces.

On the Hyperplane Sections Through two Given Points of an Algebraic Variety

1970 ◽  
Vol 22 (1) ◽  
pp. 128-133 ◽  
Author(s):  
Wei-Eihn Kuan
Keyword(s):  
Simple Proof ◽  
Hyperplane Section ◽  
Rational Points ◽  
Infinite Field ◽  
Important Special Case ◽  
Irreducible Curve ◽  
Irreducible Variety ◽  
Dimension 2 ◽  
Hyperplane Sections ◽  
Special Case

1. Let k be an infinite field and let V/k be an irreducible variety of dimension ≧ 2 in a projective n-space Pn over k. Let P and Q be two k-rational points on V In this paper, we describe ideal-theoretically the generic hyperplane section of V through P and Q (Theorem 1) and prove that the section is almost always an absolutely irreducible variety over k1/pe if V/k is absolutely irreducible (Theorem 3). As an application (Theorem 4), we give a new simple proof of an important special case of the existence of a curve connecting two rational points of an absolutely irreducible variety [4], namely any two k-rational points on V/k can be connected by an irreducible curve.I wish to thank Professor A. Seidenberg for his continued advice and encouragement on my thesis research.

scholarly journals Deception through Half-Truths

2020 ◽  
Vol 34 (06) ◽  
pp. 10110-10117
Author(s):  
Andrew Estornell ◽  
Sanmay Das ◽  
Yevgeniy Vorobeychik
Keyword(s):  
Polynomial Time ◽  
Transition Function ◽  
Np Hard ◽  
Fundamental Issue ◽  
Fake News ◽  
False Information ◽  
Important Special Case ◽  
Bayes Network ◽  
Special Case

Deception is a fundamental issue across a diverse array of settings, from cybersecurity, where decoys (e.g., honeypots) are an important tool, to politics that can feature politically motivated “leaks” and fake news about candidates. Typical considerations of deception view it as providing false information. However, just as important but less frequently studied is a more tacit form where information is strategically hidden or leaked. We consider the problem of how much an adversary can affect a principal's decision by “half-truths”, that is, by masking or hiding bits of information, when the principal is oblivious to the presence of the adversary. The principal's problem can be modeled as one of predicting future states of variables in a dynamic Bayes network, and we show that, while theoretically the principal's decisions can be made arbitrarily bad, the optimal attack is NP-hard to approximate, even under strong assumptions favoring the attacker. However, we also describe an important special case where the dependency of future states on past states is additive, in which we can efficiently compute an approximately optimal attack. Moreover, in networks with a linear transition function we can solve the problem optimally in polynomial time.

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