scholarly journals The cd-index of Bruhat Intervals

10.37236/1827 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Nathan Reading
Keyword(s):  
Upper Bound ◽  
Recursive Formula ◽  
Bruhat Order ◽  
Boolean Algebras ◽  
Structural Recursion ◽  
Simple Effect ◽  
The Face ◽  
Stacked Polytope ◽  
Special Case ◽  
Bruhat Intervals

We study flag enumeration in intervals in the Bruhat order on a Coxeter group by means of a structural recursion on intervals in the Bruhat order. The recursion gives the isomorphism type of a Bruhat interval in terms of smaller intervals, using basic geometric operations which preserve PL sphericity and have a simple effect on the cd-index. This leads to a new proof that Bruhat intervals are PL spheres as well a recursive formula for the cd-index of a Bruhat interval. This recursive formula is used to prove that the cd-indices of Bruhat intervals span the space of cd-polynomials. The structural recursion leads to a conjecture that Bruhat spheres are "smaller" than polytopes. More precisely, we conjecture that if one fixes the lengths of $x$ and $y$, then the cd-index of a certain dual stacked polytope is a coefficientwise upper bound on the cd-indices of Bruhat intervals $[x,y]$. We show that this upper bound would be tight by constructing Bruhat intervals which are the face lattices of these dual stacked polytopes. As a weakening of a special case of the conjecture, we show that the flag h-vectors of lower Bruhat intervals are bounded above by the flag h-vectors of Boolean algebras (i. e. simplices).

Studying Infection Worries by Ecological Momentary Assessment: Development and Causal Factors in Sweden during the Early Phase of the Covid-19 Pandemic (Preprint)

2020 ◽  
Author(s):  
Peter Schulz ◽  
Elin Andersson ◽  
Nicole Bizzotto ◽  
Margareta Norberg
Keyword(s):  
Study Design ◽  
Preventive Action ◽  
Causal Factors ◽  
Perceived Severity ◽  
Assessment Development ◽  
Infection Susceptibility ◽  
The Face ◽  
New Type ◽  
Ecological Momentary ◽  
Special Case

BACKGROUND The foray of Covid-19 around the globe is sure to have instigated worries in many humans, and lockdown measures may well have created their own worries. Sweden, in contrast to most other countries, had first relied on voluntary measures, but had to change its policy in the face of an increasing number of infections. OBJECTIVE The aim was to better understand the worried reactions to the virus and the lockdown measures. To grasp the reactions, their development over time was studied. METHODS Results were based on an unbalanced panel sample of 261 Swedish participants filling in 3218 interview questionnaires by smartphone in a 7-week period in 2020. Causal factors considered in this study include the perceived severity of an infection, the susceptibility of a person to the threat posed by the virus, the perceived efficacy of safeguarding measures and the assessment of government action against the spread of Covid-19. The effect of these factors on worries was traced in two analytical steps: the effects at the beginning of the study, and the effect on the trend during the study. RESULTS Findings confirmed that the hypothesized causal factors (severity of infection, susceptibility to the threat of the virus, efficacy of safeguarding and the assessment of government preventive action did indeed affect worries. CONCLUSIONS The results confirmed earlier research in a very special case and demonstrated the usefulness of a different study design, which takes a longitudinal perspective, and a new type of data analysis borrowed from multi-level study design.

scholarly journals A Study on f Number of Points for Number of Points with Inter-Distance of Less Than, or for Number of Points with Inter-Distance of or More to Necessarily Exist on Plane

2018 ◽  
Author(s):  
Benjamin Smith
Keyword(s):  
Upper Bound ◽  
Ramsey Number ◽  
Minimum Value ◽  
Special Case

We defined number of points with an inter-distance of β or more to necessarily exist on a plane. Furthermore, we aimed to reduce the range of this minimum value. We first showed that the upper bound of this value could be scaled by , and further reduced the constant that was multiplied. We compared the upper bound of and the Ramsey number in a special case and confirmed that was a better upper bound than except when were both small or trivial.

Poisson approximation for (k1, k2)-events via the Stein-Chen method

2004 ◽  
Vol 41 (4) ◽  
pp. 1081-1092 ◽  
Author(s):  
P. Vellaisamy
Keyword(s):  
Limit Theorem ◽  
Rate Of Convergence ◽  
Upper Bound ◽  
Success Probability ◽  
Random Variable ◽  
Poisson Approximation ◽  
Bernoulli Trials ◽  
Poisson Random Variable ◽  
Markov Dependent Trials ◽  
Special Case

Consider a sequence of independent Bernoulli trials with success probability p. Let N(n; k1, k2) denote the number of times that k1 failures are followed by k2 successes among the first n Bernoulli trials. We employ the Stein-Chen method to obtain a total variation upper bound for the rate of convergence of N(n; k1, k2) to a suitable Poisson random variable. As a special case, the corresponding limit theorem is established. Similar results are obtained for Nk3(n; k1, k2), the number of times that k1 failures followed by k2 successes occur k3 times successively in n Bernoulli trials. The bounds obtained are generally sharper than, and improve upon, some of the already known results. Finally, the technique is adapted to obtain Poisson approximation results for the occurrences of the above-mentioned events under Markov-dependent trials.

On norm of single layer potentials on segments

2016 ◽  
Vol 22 (1) ◽  
Author(s):  
Seyed M. Zoalroshd
Keyword(s):  
Upper Bound ◽  
Single Layer ◽  
Exact Expression ◽  
Operator Norm ◽  
Equal Length ◽  
Layer Potentials ◽  
Schmidt Norm ◽  
Special Case

AbstractWe show that, for a special case, equality of the spectra of single layer potentials defined on two segments implies that these segments must have equal length. We also provide an upper bound for the operator norm and exact expression for the Hilbert–Schmidt norm of single layer potentials on segments.

The Radical Right in Germany, Austria, and Switzerland

2018 ◽  
Author(s):  
Uwe Backes
Keyword(s):  
Party System ◽  
Common Ground ◽  
Civic Culture ◽  
Right Wing ◽  
Constitutional State ◽  
German Speaking ◽  
The Face ◽  
The Common ◽  
The Right ◽  
Special Case

This chapter analyzes and compares political developments in Germany, Austria, and Switzerland. It highlights the common ground between groups on the right-wing fringe of each country’s party system. To an extent the differences between the way right-wing groups developed in each of these countries is due to the different histories of the respective states. Recently however, they have moved closer to each other in the face of very similar problems. To a degree Switzerland is a special case because of its multilingual cantons and the early development of a pluralist civic culture that sustains an extraordinarily dynamic democratic constitutional state. This is particularly true given the autocratic relapses toward right-wing politics in neighboring German-speaking countries.

scholarly journals Analysis of Ring Compression Test by Upper Bound Theorem as Special Case of Non-symmetric Part

2015 ◽  
Vol 132 ◽  
pp. 334-341 ◽  
Author(s):  
A. Moncada ◽  
F. Martín ◽  
L. Sevilla ◽  
A.M. Camacho ◽  
M.A. Sebastián
Keyword(s):  
Compression Test ◽  
Upper Bound ◽  
Symmetric Part ◽  
Ring Compression Test ◽  
Upper Bound Theorem ◽  
Ring Compression ◽  
Bound Theorem ◽  
Special Case

scholarly journals TESTING EMBEDDABILITY BETWEEN METRIC SPACES

2009 ◽  
Vol 20 (02) ◽  
pp. 313-329
Author(s):  
CHING-LUEH CHANG ◽  
YUH-DAUH LYUU ◽  
YEN-WU TI
Keyword(s):  
Metric Space ◽  
Upper Bound ◽  
Metric Spaces ◽  
Graph Isomorphism ◽  
Upper And Lower Bounds ◽  
Wrong Answer ◽  
Probability Of Error ◽  
Finite Metric Space ◽  
Finite Metric Spaces ◽  
Special Case

Let L ≥ 1, ε > 0 be real numbers, (M, d) be a finite metric space and (N, ρ) be a metric space. A query to a metric space consists of a pair of points and asks for the distance between these points. We study the number of queries to metric spaces (M, d) and (N, ρ) needed to decide whether (M, d) is L-bilipschitz embeddable into (N, ρ) or ∊-far from being L-bilipschitz embeddable into N, ρ). When (M, d) is ∊-far from being L-bilipschitz embeddable into (N, ρ), we allow an o(1) probability of error (i.e., returning the wrong answer "L-bilipschitz embeddable"). However, no error is allowed when (M, d) is L-bilipschitz embeddable into (N, ρ). That is, algorithms with only one-sided errors are studied in this paper. When |M| ≤ |N| are both finite, we give an upper bound of [Formula: see text] on the number of queries for determining with one-sided error whether (M, d) is L-bilipschitz embeddable into (N, ρ) or ∊-far from being L-bilipschitz embeddable into (N, ρ). For the special case of finite |M| = |N|, the above upper bound evaluates to [Formula: see text]. We also prove a lower bound of Ω(|N|3/2) for the special case when |M| = |N| are finite and L = 1, which coincides with testing isometry between finite metric spaces. For finite |M| = |N|, the upper and lower bounds thus match up to a multiplicative factor of at most [Formula: see text], which depends only sublogarithmically in |N|. We also investigate the case when (N, ρ) is not necessarily finite. Our results are based on techniques developed in an earlier work on testing graph isomorphism.

scholarly journals On Partitioning and Packing Products with Rectangles

1994 ◽  
Vol 3 (4) ◽  
pp. 429-434 ◽  
Author(s):  
Rudolf Ahlswede ◽  
Ning Cai
Keyword(s):  
Upper Bound ◽  
Complete Graphs ◽  
Minimal Size ◽  
Uniform Hypergraphs ◽  
Image Position ◽  
Special Case

In [1] we introduced and studied for product hypergraphs where ℋi = (i,ℰi), the minimal size π(ℋn) of a partition of into sets that are elements of . The main result was thatif the ℋis are graphs with all loops included. A key step in the proof concerns the special case of complete graphs. Here we show that (1) also holds when the ℋi are complete d-uniform hypergraphs with all loops included, subject to a condition on the sizes of the i. We also present an upper bound on packing numbers.

scholarly journals Collision-avoidance and flocking in the Cucker–Smale-type model with a discontinuous controller

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jianfei Cheng ◽  
Xiao Wang ◽  
Yicheng Liu
Keyword(s):  
Weight Function ◽  
Collision Avoidance ◽  
Upper Bound ◽  
Finite Time ◽  
Relative Velocity ◽  
Sufficient Conditions ◽  
Repulsive Force ◽  
Type Model ◽  
Movement Tendency ◽  
Special Case

<p style='text-indent:20px;'>The collision-avoidance and flocking of the Cucker–Smale-type model with a discontinuous controller are studied. The controller considered in this paper provides a force between agents that switches between the attractive force and the repulsive force according to the movement tendency between agents. The results of collision-avoidance are closely related to the weight function <inline-formula><tex-math id="M1">\begin{document}$ f(r) = (r-d_0)^{-\theta } $\end{document}</tex-math></inline-formula>. For <inline-formula><tex-math id="M2">\begin{document}$ \theta \ge 1 $\end{document}</tex-math></inline-formula>, collision will not appear in the system if agents' initial positions are different. For the case <inline-formula><tex-math id="M3">\begin{document}$ \theta \in [0,1) $\end{document}</tex-math></inline-formula> that not considered in previous work, the limits of initial configurations to guarantee collision-avoidance are given. Moreover, on the basis of collision-avoidance, we point out the impacts of <inline-formula><tex-math id="M4">\begin{document}$ \psi (r) = (1+r^2)^{-\beta } $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ f(r) $\end{document}</tex-math></inline-formula> on the flocking behaviour and give the decay rate of relative velocity. We also estimate the lower and upper bound of distance between agents. Finally, for the special case that agents moving on the 1-D space, we give sufficient conditions for the finite-time flocking.</p>

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