On Gaussian Correlation Inequalities for Rectangles and Symmetric Sets

2002 ◽  
Vol 53 (3-4) ◽  
pp. 245-248
Author(s):  
Subir K. Bhandari ◽  
Ayanendranath Basu
Keyword(s):  
Recent Result ◽  
Convex Sets ◽  
Convex Set ◽  
Correlation Inequalities ◽  
The Other ◽  
Symmetric Convex ◽  
Complete Proof ◽  
Other Hand

Pitt's conjecture (1977) that P( A ∩ B) ≥ P( A) P( B) under the Nn (0, In) distribution of X, where A, B are symmetric convex sets in IRn still lacks a complete proof. This note establishes that the above result is true when A is a symmetric rectangle while B is any symmetric convex set, where A, B ∈ IRn. We give two different proofs of the result, the key component in the first one being a recent result by Hargé (1999). The second proof, on the other hand, is based on a rather old result of Šidák (1968), dating back a period before Pitt's conjecture.

scholarly journals New Generalized Projection Speeds Up Audio Declipping

2019 ◽  
Author(s):  
Pavel Rajmic ◽  
Pavel Záviška ◽  
Vítězslav Veselý ◽  
Ondřej Mokrý
Keyword(s):  
Signal Processing ◽  
Explicit Formula ◽  
Convex Sets ◽  
The Other ◽  
Generalized Projection ◽  
Other Hand ◽  
Linear Transforms ◽  
Speed Up ◽  
Linear Transform

In theory and applications, it is often inevitable to work with projectors onto convex sets, where a linear transform is involved. In this article, a novel projector is presented, which generalizes previous results in that it admits a broader family of linear transforms, but on the other hand it is limited to box-type convex sets in the transformed domain. The new projector has an explicit formula and it can be interpreted within the framework of proximal optimization. The benefit of the new projector is demonstrated on an example from signal processing, where it was possible to speed up the convergence of a signal declipping algorithm by a factor of more than two.

Separating Translates in the Plane: Combinatorial Bounds and an Algorithm

1997 ◽  
Vol 07 (06) ◽  
pp. 551-562
Author(s):  
Jurek Czyzowicz ◽  
Hazel Everett ◽  
Jean-Marc Robert
Keyword(s):  
Pairwise Disjoint ◽  
Convex Sets ◽  
Convex Set ◽  
Time Algorithm ◽  
The Other ◽  
Separation Problem ◽  
Disjoint Convex

Given a family of pairwise disjoint convex sets S in the plane, a set [Formula: see text] is separated from a subset X ⊆ S if there exists a line l such that [Formula: see text] lies on one side of l and the sets in X lie on the other side. In this paper, we establish two combinatorial bounds related to the separation problem for families of n pairwise disjoint translates of a convex set in the plane: 1) there exists a line which separates one translate from at least [Formula: see text] translates, for some constant [Formula: see text] that depends only on the shape of the translates and 2) there is a function [Formula: see text], defined only by the shape of [Formula: see text], such that there exists a line with orientation Θ or [Formula: see text] which separates one translate from at least [Formula: see text] translates, for any orientation Θ. We also present an O(n log (n + k) + k) time algorithm for finding a translate which can be separated from the maximum number of translates amongst families of n pairwise disjoint translates of convex k-gons in the plane.

scholarly journals On the area of planar convex sets containing many lattice points

1987 ◽  
Vol 35 (3) ◽  
pp. 441-454
Author(s):  
P. R. Scott
Keyword(s):  
Large Class ◽  
Convex Sets ◽  
Convex Set ◽  
Lattice Points ◽  
Symmetric Convex ◽  
Classical Theorem ◽  
Planar Convex

A classical theorem of van der Corput gives a bound for the volume of a symmetric convex set in terms of the number of lattice points it contains. This theorem is here generalized and extended for a large class of non-symmetric sets in the plane.

scholarly journals Triforce and corners

2019 ◽  
Vol 169 (1) ◽  
pp. 209-223
Author(s):  
JACOB FOX ◽  
ASHWIN SAH ◽  
MEHTAAB SAWHNEY ◽  
DAVID STONER ◽  
YUFEI ZHAO
Keyword(s):  
Recent Result ◽  
Higher Dimensions ◽  
The Other ◽  
Edge Density ◽  
Uniform Hypergraph ◽  
Other Hand

AbstractMay the triforce be the 3-uniform hypergraph on six vertices with edges {123′, 12′3, 1′23}. We show that the minimum triforce density in a 3-uniform hypergraph of edge density δ is δ4–o(1) but not O(δ4).Let M(δ) be the maximum number such that the following holds: for every ∊ > 0 and $G = {\mathbb{F}}_2^n$ with n sufficiently large, if A ⊆ G × G with A ≥ δ|G|2, then there exists a nonzero “popular difference” d ∈ G such that the number of “corners” (x, y), (x + d, y), (x, y + d) ∈ A is at least (M(δ)–∊)|G|2. As a corollary via a recent result of Mandache, we conclude that M(δ) = δ4–o(1) and M(δ) = ω(δ4).On the other hand, for 0 < δ < 1/2 and sufficiently large N, there exists A ⊆ [N]3 with |A| ≥ δN3 such that for every d ≠ 0, the number of corners (x, y, z), (x + d, y, z), (x, y + d, z), (x, y, z + d) ∈ A is at most δc log(1/δ)N3. A similar bound holds in higher dimensions, or for any configuration with at least 5 points or affine dimension at least 3.

scholarly journals A stochastic Prékopa–Leindler inequality for log-concave functions

2020 ◽  
pp. 2050019
Author(s):  
Peter Pivovarov ◽  
Jesus Rebollo Bueno
Keyword(s):  
Stochastic Dominance ◽  
Convex Sets ◽  
Minkowski Inequality ◽  
The Other ◽  
Concave Functions ◽  
Compact Sets ◽  
Stochastic Approximations ◽  
Random Polytopes ◽  
Other Hand ◽  
Integrable Functions

The Brunn–Minkowski and Prékopa–Leindler inequalities admit a variety of proofs that are inspired by convexity. Nevertheless, the former holds for compact sets and the latter for integrable functions so it seems that convexity has no special signficance. On the other hand, it was recently shown that the Brunn–Minkowski inequality, specialized to convex sets, follows from a local stochastic dominance for naturally associated random polytopes. We show that for the subclass of log-concave functions and associated stochastic approximations, a similar stochastic dominance underlies the Prékopa–Leindler inequality.

A study of cometary eruptions through OH radio-lines

1999 ◽  
Vol 173 ◽  
pp. 249-254
Author(s):  
A.M. Silva ◽  
R.D. Miró
Keyword(s):  
Short Duration ◽  
Growth Curves ◽  
The Other ◽  
Initial Mass ◽  
Radio Lines ◽  
Other Hand ◽  
Sudden Rise

AbstractWe have developed a model for theH2OandOHevolution in a comet outburst, assuming that together with the gas, a distribution of icy grains is ejected. With an initial mass of icy grains of 108kg released, theH2OandOHproductions are increased up to a factor two, and the growth curves change drastically in the first two days. The model is applied to eruptions detected in theOHradio monitorings and fits well with the slow variations in the flux. On the other hand, several events of short duration appear, consisting of a sudden rise ofOHflux, followed by a sudden decay on the second day. These apparent short bursts are frequently found as precursors of a more durable eruption. We suggest that both of them are part of a unique eruption, and that the sudden decay is due to collisions that de-excite theOHmaser, when it reaches the Cometopause region located at 1.35 × 105kmfrom the nucleus.

Closing the GAP—A 5Å Scanning Microscope

1969 ◽  
Vol 27 ◽  
pp. 6-7
Author(s):  
A. V. Crewe
Keyword(s):  
Normal Form ◽  
The Other ◽  
Resolving Power ◽  
Scanning Microscope ◽  
Conventional Microscope ◽  
Other Hand ◽  
Closing The Gap ◽  
Thermal Sources ◽  
Better Than ◽  
Transmission Microscope

We have become accustomed to differentiating between the scanning microscope and the conventional transmission microscope according to the resolving power which the two instruments offer. The conventional microscope is capable of a point resolution of a few angstroms and line resolutions of periodic objects of about 1Å. On the other hand, the scanning microscope, in its normal form, is not ordinarily capable of a point resolution better than 100Å. Upon examining reasons for the 100Å limitation, it becomes clear that this is based more on tradition than reason, and in particular, it is a condition imposed upon the microscope by adherence to thermal sources of electrons.

Life Beyond 1MeV

1980 ◽  
Vol 38 ◽  
pp. 30-33
Author(s):  
K.H. Westmacott
Keyword(s):  
Electron Microscopy ◽  
Past Experience ◽  
The Other ◽  
Good Case ◽  
Other Hand ◽  
The U.S

Life beyond 1MeV – like life after 40 – is not too different unless one takes advantage of past experience and is receptive to new opportunities. At first glance, the returns on performing electron microscopy at voltages greater than 1MeV diminish rather rapidly as the curves which describe the well-known advantages of HVEM often tend towards saturation. However, in a country with a significant HVEM capability, a good case can be made for investing in instruments with a range of maximum accelerating voltages. In this regard, the 1.5MeV KRATOS HVEM being installed in Berkeley will complement the other 650KeV, 1MeV, and 1.2MeV instruments currently operating in the U.S. One other consideration suggests that 1.5MeV is an optimum voltage machine – Its additional advantages may be purchased for not much more than a 1MeV instrument. On the other hand, the 3MeV HVEM's which seem to be operated at 2MeV maximum, are much more expensive.

Can the Academic Self-Concept be Positive and Realistic?

2005 ◽  
Vol 19 (3) ◽  
pp. 129-132 ◽  
Author(s):  
Reimer Kornmann
Keyword(s):  
Academic Performance ◽  
Opposite Effect ◽  
Self Esteem ◽  
Point Of View ◽  
The Other ◽  
Self Concept ◽  
High Ability ◽  
Self Image ◽  
Other Hand

Summary: My comment is basically restricted to the situation in which less-able students find themselves and refers only to literature in German. From this point of view I am basically able to confirm Marsh's results. It must, however, be said that with less-able pupils the opposite effect can be found: Levels of self-esteem in these pupils are raised, at least temporarily, by separate instruction, academic performance however drops; combined instruction, on the other hand, leads to improved academic performance, while levels of self-esteem drop. Apparently, the positive self-image of less-able pupils who receive separate instruction does not bring about the potential enhancement of academic performance one might expect from high-ability pupils receiving separate instruction. To resolve the dilemma, it is proposed that individual progress in learning be accentuated, and that comparisons with others be dispensed with. This fosters a self-image that can in equal measure be realistic and optimistic.

Stories and the Self

2020 ◽  
Vol 32 (2) ◽  
pp. 47-58 ◽  
Author(s):  
Stefan Krause ◽  
Markus Appel
Keyword(s):  
Meta Analysis ◽  
The Self ◽  
The Other ◽  
Lower Degree ◽  
Contrast Effects ◽  
Self Perception ◽  
Self Perceptions ◽  
Other Hand ◽  
The One ◽  
Implicit And Explicit

Abstract. Two experiments examined the influence of stories on recipients’ self-perceptions. Extending prior theory and research, our focus was on assimilation effects (i.e., changes in self-perception in line with a protagonist’s traits) as well as on contrast effects (i.e., changes in self-perception in contrast to a protagonist’s traits). In Experiment 1 ( N = 113), implicit and explicit conscientiousness were assessed after participants read a story about either a diligent or a negligent student. Moderation analyses showed that highly transported participants and participants with lower counterarguing scores assimilate the depicted traits of a story protagonist, as indicated by explicit, self-reported conscientiousness ratings. Participants, who were more critical toward a story (i.e., higher counterarguing) and with a lower degree of transportation, showed contrast effects. In Experiment 2 ( N = 103), we manipulated transportation and counterarguing, but we could not identify an effect on participants’ self-ascribed level of conscientiousness. A mini meta-analysis across both experiments revealed significant positive overall associations between transportation and counterarguing on the one hand and story-consistent self-reported conscientiousness on the other hand.

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