On Packings of Unequal Spheres in Rn

1968 ◽  
Vol 20 ◽  
pp. 967-969 ◽  
Author(s):  
D. G. Larman
Keyword(s):  
Lower Bound ◽  
Recent Work ◽  
Higher Dimensions ◽  
Unit Cube ◽  
Two Dimensions

Suppose that a sequence of spheres is packed in order of decreasing diameters into the unit cube In of Rn. In recent work (2), I have shown that for n = 2, there exist positive constants K2, s ( = 0.97) such that the area of has an asymptotic lower bound K2(d(Sm))s. Although the methods used were complicated and possibly only viable in two dimensions, it is intuitively clear that such a result should also be true in higher dimensions.

scholarly journals Semi-Lagrangian Subgrid Reconstruction for Advection-Dominant Multiscale Problems with Rough Data

2021 ◽  
Vol 87 (2) ◽  
Author(s):  
Konrad Simon ◽  
Jörn Behrens
Keyword(s):  
Nonlinear Problems ◽  
Higher Dimensions ◽  
Two Dimensions ◽  
Multiscale Methods ◽  
Galerkin Methods ◽  
Standard Methods ◽  
Multiscale Problems ◽  
Eulerian Method ◽  
Lower Order Terms ◽  
New Framework

AbstractWe introduce a new framework of numerical multiscale methods for advection-dominated problems motivated by climate sciences. Current numerical multiscale methods (MsFEM) work well on stationary elliptic problems but have difficulties when the model involves dominant lower order terms. Our idea to overcome the associated difficulties is a semi-Lagrangian based reconstruction of subgrid variability into a multiscale basis by solving many local inverse problems. Globally the method looks like a Eulerian method with multiscale stabilized basis. We show example runs in one and two dimensions and a comparison to standard methods to support our ideas and discuss possible extensions to other types of Galerkin methods, higher dimensions and nonlinear problems.

scholarly journals NEW DISCRETE STATES IN TWO-DIMENSIONAL SUPERGRAVITY

2007 ◽  
Vol 22 (07) ◽  
pp. 1375-1394 ◽  
Author(s):  
DIMITRI POLYAKOV
Keyword(s):  
Operator Algebra ◽  
Higher Dimensions ◽  
Two Dimensions ◽  
Field Theories ◽  
Two Dimensional ◽  
Discrete States ◽  
Structure Constants ◽  
Area Preserving ◽  
Volume Preserving ◽  
Lowering Operator

Two-dimensional string theory is known to contain the set of discrete states that are the SU (2) multiplets generated by the lowering operator of the SU (2) current algebra. Their structure constants are defined by the area preserving diffeomorphisms in two dimensions. In this paper we show that the interaction of d = 2 superstrings with the superconformal β - γ ghosts enlarges the actual algebra of the dimension 1 currents and hence the new ghost-dependent discrete states appear. Generally, these states are the SU (N) multiplets if the algebra includes the currents of ghost numbers n : -N ≤ n ≤ N - 2, not related by picture changing. We compute the structure constants of these ghost-dependent discrete states for N = 3 and express them in terms of SU (3) Clebsch–Gordan coefficients, relating this operator algebra to the volume preserving diffeomorphisms in d = 3. For general N, the operator algebra is conjectured to be isomorphic to SDiff (N). This points at possible holographic relations between two-dimensional superstrings and field theories in higher dimensions.

Critical intensities of Boolean models with different underlying convex shapes

2002 ◽  
Vol 34 (01) ◽  
pp. 48-57
Author(s):  
Rahul Roy ◽  
Hideki Tanemura
Keyword(s):  
Unit Area ◽  
Higher Dimensions ◽  
Boolean Model ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Boolean Models ◽  
Regular Polytope ◽  
Minimum Value ◽  
Critical Intensity ◽  
Poisson Boolean Model

We consider the Poisson Boolean model of percolation where the percolating shapes are convex regions. By an enhancement argument we strengthen a result of Jonasson (2000) to show that the critical intensity of percolation in two dimensions is minimized among the class of convex shapes of unit area when the percolating shapes are triangles, and, for any other shape, the critical intensity is strictly larger than this minimum value. We also obtain a partial generalization to higher dimensions. In particular, for three dimensions, the critical intensity of percolation is minimized among the class of regular polytopes of unit volume when the percolating shapes are tetrahedrons. Moreover, for any other regular polytope, the critical intensity is strictly larger than this minimum value.

On a generalization of Jensen's □κ, and strategic closure of partial orders

1983 ◽  
Vol 48 (4) ◽  
pp. 1046-1052 ◽  
Author(s):  
Dan Velleman
Keyword(s):  
Lower Bound ◽  
Partial Order ◽  
Recent Work ◽  
Partial Orders ◽  
Closure Condition ◽  
Forcing Axioms ◽  
Combinatorial Principles ◽  
The Relationship ◽  
Mahlo Cardinals ◽  
Forcing Axiom

It is well known that many statements provable from combinatorial principles true in the constructible universe L can also be shown to be consistent with ZFC by forcing. Recent work by Shelah and Stanley [4] and the author [5] has clarified the relationship between the axiom of constructibility and forcing by providing Martin's Axiom-type forcing axioms equivalent to ◊ and the existence of morasses. In this paper we continue this line of research by providing a forcing axiom equivalent to □κ. The forcing axiom generalizes easily to inaccessible, non-Mahlo cardinals, and provides the motivation for a corresponding generalization of □κ.In order to state our forcing axiom, we will need to define a strategic closure condition for partial orders. Suppose P = 〈P, ≤〉 is a partial order. For each ordinal α we will consider a game played by two players, Good and Bad. The players choose, in order, the terms in a descending sequence of conditions 〈pβ∣β < α〉 Good chooses all terms pβ for limit β, and Bad chooses all the others. Bad wins if for some limit β<α, Good is unable to move at stage β because 〈pγ∣γ < β〉 has no lower bound. Otherwise, Good wins. Of course, we will be rooting for Good.

Mass transference principle for limsup sets generated by rectangles

2015 ◽  
Vol 158 (3) ◽  
pp. 419-437 ◽  
Author(s):  
BAO-WEI WANG ◽  
JUN WU ◽  
JIAN XU
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Hausdorff Measure ◽  
Unit Cube ◽  
Transference Principle ◽  
Theoretical Statement ◽  
Formula Group ◽  
Image Position ◽  
Full Lebesgue Measure

AbstractWe generalise the mass transference principle established by Beresnevich and Velani to limsup sets generated by rectangles. More precisely, let {xn}n⩾1 be a sequence of points in the unit cube [0, 1]d with d ⩾ 1 and {rn}n⩾1 a sequence of positive numbers tending to zero. Under the assumption of full Lebesgue measure theoretical statement of the set \begin{equation*}\big\{x\in [0,1]^d: x\in B(x_n,r_n), \ {{\rm for}\, {\rm infinitely}\, {\rm many}}\ n\in \mathbb{N}\big\},\end{equation*} we determine the lower bound of the Hausdorff dimension and Hausdorff measure of the set \begin{equation*}\big\{x\in [0,1]^d: x\in B^{a}(x_n,r_n), \ {{\rm for}\, {\rm infinitely}\, {\rm many}}\ n\in \mathbb{N}\big\},\end{equation*} where a = (a1, . . ., ad) with 1 ⩽ a1 ⩽ a2 ⩽ . . . ⩽ ad and Ba(x, r) denotes a rectangle with center x and side-length (ra1, ra2,. . .,rad). When a1 = a2 = . . . = ad, the result is included in the setting considered by Beresnevich and Velani.

Asymptotic properties of the equilibrium probability of identity in a geographically structured population

1977 ◽  
Vol 9 (2) ◽  
pp. 268-282 ◽  
Author(s):  
Stanley Sawyer
Keyword(s):  
Nearest Neighbor ◽  
Asymptotic Properties ◽  
Higher Dimensions ◽  
Two Dimensions ◽  
Three Dimensions ◽  
One Dimension ◽  
Order Zero ◽  
Structured Population ◽  
Equilibrium Probability ◽  
Migration Law

Let I(x, u) be the probability that two genes found a vector distance x apart are the same type in an infinite-allele selectively-neutral migration model with mutation rate u. The creatures involved inhabit an infinite of colonies, are diploid and are held at N per colony. Set in one dimension and in higher dimensions, where σ2 is the covariance matrix of the migration law (which is assumed to have finite fifth moments). Then in one dimension, in two dimensions, and in three dimensions uniformly for Here C0 is a constant depending on the migration law, K0(y) is the Bessel function of the second kind of order zero, and are the eigenvalues of σ2. For symmetric nearest-neighbor migrations, in one dimension and log mi in two. For is known in one dimension and C0 does not appear. In two dimensions, These results extend and make more precise earlier work of Malécot, Weiss and Kimura and Nagylaki.

scholarly journals AdS3/AdS2 degression of massless particles

2021 ◽  
Vol 2021 (9) ◽  
Author(s):  
Konstantin Alkalaev ◽  
Alexander Yan
Keyword(s):  
Massless Particle ◽  
Higher Dimensions ◽  
Two Dimensions ◽  
Spin Representation ◽  
Theoretical Terms ◽  
Finite Spectrum ◽  
Branching Rules ◽  
Higher Spin ◽  
Massless Fields ◽  
Kaluza Klein

Abstract We study a 3d/2d dimensional degression which is a Kaluza-Klein type mechanism in AdS3 space foliated into AdS2 hypersurfaces. It is shown that an AdS3 massless particle of spin s = 1, 2, …, ∞ degresses into a couple of AdS2 particles of equal energies E = s. Note that the Kaluza-Klein spectra in higher dimensions are always infinite. To formulate the AdS3/AdS2 degression we consider branching rules for AdS3 isometry algebra o(2,2) representations decomposed with respect to AdS2 isometry algebra o(1,2). We find that a given o(2,2) higher-spin representation lying on the unitary bound (i.e. massless) decomposes into two equal o(1,2) modules. In the field-theoretical terms, this phenomenon is demonstrated for spin-2 and spin-3 free massless fields. The truncation to a finite spectrum can be seen by using particular mode expansions, (partial) diagonalizations, and identities specific to two dimensions.

scholarly journals Splitting Algorithms for Rare Events of Semimartingale Reflecting Brownian Motions

2021 ◽  
Author(s):  
Kevin Leder ◽  
Xin Liu ◽  
Zicheng Wang
Keyword(s):  
Growth Rate ◽  
Monte Carlo ◽  
Rare Event ◽  
Small Neighborhood ◽  
Higher Dimensions ◽  
Two Dimensions ◽  
Superior Performance ◽  
Regularity Conditions ◽  
Brownian Motions ◽  
Starting Point

We study rare event simulations of semimartingale reflecting Brownian motions (SRBMs) in an orthant. The rare event of interest is that a d-dimensional positive recurrent SRBM enters the set [Formula: see text] before hitting a small neighborhood of the origin [Formula: see text] as [Formula: see text] with a starting point outside the two sets and of order o(n). We show that, under two regularity conditions (the Dupuis–Williams stability condition of the SRBM and the Lipschitz continuity assumption of the associated Skorokhod problem), the probability of the rare event satisfies a large deviation principle. To study the variational problem (VP) for the rare event in two dimensions, we adapt its exact solution from developed by Avram, Dai, and Hasenbein in 2001. In three and higher dimensions, we construct a novel subsolution to the VP under a further assumption that the reflection matrix of the SRBM is a nonsingular [Formula: see text]-matrix. Based on the solution/subsolution, particle-based simulation algorithms are constructed to estimate the probability of the rare event. Our estimator is asymptotically optimal for the discretized problem in two dimensions and has exponentially superior performance over standard Monte Carlo in three and higher dimensions. In addition, we establish that the growth rate of the relative bias term arising from discretization is subexponential in all dimensions. Therefore, we can estimate the probability of interest with subexponential complexity growth in two dimensions. In three and higher dimensions, the computational complexity of our estimators has a strictly smaller exponential growth rate than the standard Monte Carlo estimators.

scholarly journals On Proinov’s Lower Bound for the Diaphony

2020 ◽  
Vol 15 (2) ◽  
pp. 39-72
Author(s):  
Nathan Kirk
Keyword(s):  
Lower Bound ◽  
Uniform Distribution ◽  
Lower Bounds ◽  
Unit Cube ◽  
Integral Approximation ◽  
Quantitative Theory ◽  
Dimensional Unit ◽  
Explicit Lower Bound ◽  
Irregularities Of Distribution ◽  
Infinite Sequences

AbstractIn 1986, Proinov published an explicit lower bound for the diaphony of finite and infinite sequences of points contained in the d−dimensional unit cube [Proinov, P. D.:On irregularities of distribution, C. R. Acad. Bulgare Sci. 39 (1986), no. 9, 31–34]. However, his widely cited paper does not contain the proof of this result but simply states that this will appear elsewhere. To the best of our knowledge, this proof was so far only available in a monograph of Proinov written in Bulgarian [Proinov, P. D.: Quantitative Theory of Uniform Distribution and Integral Approximation, University of Plovdiv, Bulgaria (2000)]. The first contribution of our paper is to give a self contained version of Proinov’s proof in English. Along the way, we improve the explicit asymptotic constants implementing recent, and corrected results of [Hinrichs, A.—Markhasin, L.: On lower bounds for the ℒ2-discrepancy, J. Complexity 27 (2011), 127–132.] and [Hinrichs, A.—Larcher, G.: An improved lower bound for the ℒ2-discrepancy, J. Complexity 34 (2016), 68–77]. (The corrections are due to a note in [Hinrichs, A.—Larcher, G. An improved lower bound for the ℒ2-discrepancy, J. Complexity 34 (2016), 68–77].) Finally, as a main result, we use the method of Proinov to derive an explicit lower bound for the dyadic diaphony of finite and infinite sequences in a similar fashion.

Reorganizing topologies of Steiner trees to accelerate their eliminations

2019 ◽  
Vol 12 (01) ◽  
pp. 2050003
Author(s):  
Aymeric Grodet ◽  
Takuya Tsuchiya
Keyword(s):  
Lower Bound ◽  
Higher Dimensions ◽  
Exact Algorithms ◽  
Steiner Trees ◽  
Steiner Tree Problem ◽  
Original Algorithm ◽  
Steiner Points ◽  
Euclidean Steiner Tree Problem ◽  
Correct Topology ◽  
Enumeration Scheme

We describe a technique to reorganize topologies of Steiner trees by exchanging neighbors of adjacent Steiner points. We explain how to use the systematic way of building trees, and therefore topologies, to find the correct topology after nodes have been exchanged. Topology reorganizations can be inserted into the enumeration scheme commonly used by exact algorithms for the Euclidean Steiner tree problem in [Formula: see text]-space, providing a method of improvement different than the usual approaches. As an example, we show how topology reorganizations can be used to dynamically change the exploration of the usual branch-and-bound tree when two Steiner points collide during the optimization process. We also turn our attention to the erroneous use of a pre-optimization lower bound in the original algorithm and give an example to confirm its usage is incorrect. In order to provide numerical results on correct solutions, we use planar equilateral points to quickly compute this lower bound, even in dimensions higher than two. Finally, we describe planar twin trees, identical trees yielded by different topologies, whose generalization to higher dimensions could open a new way of building Steiner trees.

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