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Journal articles on the topic 'D-partition'

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Published: 4 June 2021

Last updated: 1 February 2022

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1

MULZER, WOLFGANG, and YANNIK STEIN. "ALGORITHMS FOR TOLERANT TVERBERG PARTITIONS." International Journal of Computational Geometry & Applications 24, no. 04 (December 2014): 261–73. http://dx.doi.org/10.1142/s0218195914600073.

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Let P be a d-dimensional n-point set. A partition [Formula: see text] of P is called a Tverberg partition if the convex hulls of all sets in [Formula: see text] intersect in at least one point. We say that [Formula: see text] is t-tolerant if it remains a Tverberg partition after deleting any t points from P. Soberón and Strausz proved that there is always a t-tolerant Tverberg partition with ⌈n/(d + 1)(t + 1)⌉ sets. However, no nontrivial algorithms for computing or approximating such partitions have been presented so far. For d ≤ 2, we show that the Soberón-Strausz bound can be improved, and we show how the corresponding partitions can be found in polynomial time. For d ≥ 3, we give the first polynomial-time approximation algorithm by presenting a reduction to the regular Tverberg problem (with no tolerance). Finally, we show that it is coNP-complete to determine whether a given Tverberg partition is t-tolerant.

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2

Hwang, Chyi, Li‐Fong Hwang, and Jyh‐Haur Hwang. "Robust D‐partition." Journal of the Chinese Institute of Engineers 33, no. 6 (September 2010): 811–21. http://dx.doi.org/10.1080/02533839.2010.9671671.

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3

Józefiak, Tadeusz, and Jerzy Weyman. "Representation-theoretic interpretation of a formula of D. E. Littlewood." Mathematical Proceedings of the Cambridge Philosophical Society 103, no. 2 (March 1988): 193–96. http://dx.doi.org/10.1017/s0305004100064768.

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This note is a continuation of our attempts (see [3]) to give a satisfactory representation-theoretic justification of the following formula of D. E. Littlewood:where sI is the Schur symmetric function corresponding to a partition I, |I| is the weight of I, r(I) is the rank of I, and the summation ranges over all self-conjugate partitions (i.e. partitions I such that I = I where I is the partition conjugate to I).

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4

Azhar, Kamran, Sohail Zafar, Agha Kashif, and Zohaib Zahid. "On fault-tolerant partition dimension of graphs." Journal of Intelligent & Fuzzy Systems 40, no. 1 (January 4, 2021): 1129–35. http://dx.doi.org/10.3233/jifs-201390.

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Fault-tolerant resolving partition is natural extension of resolving partitions which have many applications in different areas of computer sciences for example sensor networking, intelligent systems, optimization and robot navigation. For a nontrivial connected graph G (V (G) , E (G)), the partition representation of vertex v with respect to an ordered partition Π = {Si : 1 ≤ i ≤ k} of V (G) is the k-vector r ( v | Π ) = ( d ( v , S i ) ) i = 1 k , where, d (v, Si) = min {d (v, x) |x ∈ Si}, for i ∈ {1, 2, …, k}. A partition Π is said to be fault-tolerant partition resolving set of G if r (u|Π) and r (v|Π) differ by at least two places for all u ≠ v ∈ V (G). A fault-tolerant partition resolving set of minimum cardinality is called the fault-tolerant partition basis of G and its cardinality the fault-tolerant partition dimension of G denoted by P ( G ) . In this article, we will compute fault-tolerant partition dimension of families of tadpole and necklace graphs.

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5

Gramain, Jean-Baptiste, and Jørn B. Olsson. "On bar lengths in partitions." Proceedings of the Edinburgh Mathematical Society 56, no. 2 (March 21, 2013): 535–50. http://dx.doi.org/10.1017/s0013091512000387.

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AbstractWe present, given an odd integer d, a decomposition of the multiset of bar lengths of a bar partition λ as the union of two multisets, one consisting of the bar lengths in its d-core partition cd(λ) and the other consisting of modified bar lengths in its d-quotient partition. In particular, we obtain that the multiset of bar lengths in cd(λ) is a sub-multiset of the multiset of bar lengths in λ. Also, we obtain a relative bar formula for the degrees of spin characters of the Schur extensions of $\mathfrak{S}_n$. The proof involves a recent similar result for partitions, proved by Bessenrodt and the authors.

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6

Last, Günter. "Stationary partitions and Palm probabilities." Advances in Applied Probability 38, no. 03 (September 2006): 602–20. http://dx.doi.org/10.1017/s0001867800001191.

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A stationary partition based on a stationary point process N in ℝ d is an ℝ d -valued random field π={π(x): x∈ℝ d } such that both π(y)∈N for each y∈ℝ d and the random partition {{y∈ℝ d : π(y)=x}: x∈N} is stationary jointly with N. Stationary partitions may be considered as general versions of the stationary random tessellations studied in stochastic geometry. As in the special case of the Voronoi tessellation, a stationary partition can be used to relate the underlying stationary probability measure to the associated Palm probability measure of N. In doing so, we will develop some basic theory for stationary partitions and extend properties of stationary tessellations to our more general case. One basic idea is that the stationary measure is (up to a shift) a weighted version of the Palm measure, where the weight is the volume of the typical cell. We will make systematic use of a known modified probability measure. Finally, we use our approach to extend some recent results on the shift coupling of the stationary distribution and the Palm distribution.

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7

Bogaevskaya, V. G. "Automatic analysis of D-partition." Journal of Physics: Conference Series 788 (January 2017): 012006. http://dx.doi.org/10.1088/1742-6596/788/1/012006.

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8

Dorey, Nick, Timothy J. Hollowood, and Valentin V. Khoze. "The D-instanton partition function." Journal of High Energy Physics 2001, no. 03 (March 26, 2001): 040. http://dx.doi.org/10.1088/1126-6708/2001/03/040.

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9

Neimark, Yu I. "D-partition and robust stability." Computational Mathematics and Modeling 9, no. 2 (April 1998): 160–66. http://dx.doi.org/10.1007/bf02404127.

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10

Hansen, Jennie C. "A functional central limit theorem for the Ewens sampling formula." Journal of Applied Probability 27, no. 1 (March 1990): 28–43. http://dx.doi.org/10.2307/3214593.

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For each n > 0, the Ewens sampling formula from population genetics is a measure on the set of all partitions of the integer n. To determine the limiting distributions for the part sizes of a partition with respect to the measures given by this formula, we associate to each partition a step function on [0, 1]. Each jump in the function equals the number of parts in the partition of a certain size. We normalize these functions and show that the induced measures on D[0, 1] converge to Wiener measure. This result complements Kingman's frequency limit theorem [10] for the Ewens partition structure.

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11

Hansen, Jennie C. "A functional central limit theorem for the Ewens sampling formula." Journal of Applied Probability 27, no. 01 (March 1990): 28–43. http://dx.doi.org/10.1017/s0021900200038407.

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For each n > 0, the Ewens sampling formula from population genetics is a measure on the set of all partitions of the integer n. To determine the limiting distributions for the part sizes of a partition with respect to the measures given by this formula, we associate to each partition a step function on [0, 1]. Each jump in the function equals the number of parts in the partition of a certain size. We normalize these functions and show that the induced measures on D[0, 1] converge to Wiener measure. This result complements Kingman's frequency limit theorem [10] for the Ewens partition structure.

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12

Subbarao, M. V. "Product partitions and recursion formulae." International Journal of Mathematics and Mathematical Sciences 2004, no. 33 (2004): 1725–35. http://dx.doi.org/10.1155/s0161171204307258.

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Utilizing a method briefly hinted in the author's paper written in 1991 jointly with V. C. Harris, we derive here a number of unpublished recursion formulae for a variety of product partition functions which we believe have not been considered before in the literature. These include the functionsp*(n;k,h)(which stands for the number of product partitions ofn>1intokparts of whichhare distinct), andp(d)*(n;m)(which stands for the number of product partitions ofninto exactlymparts with at mostdrepetitions of any part). We also derive recursion formulae for certain product partition functions without the use of generating functions.

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13

Pusadan, Mohammad Yazdi, Joko Lianto Buliali, and Raden Venantius Hari Ginardi. "Optimum partition in flight route anomaly detection." Indonesian Journal of Electrical Engineering and Computer Science 14, no. 3 (June 1, 2019): 1315. http://dx.doi.org/10.11591/ijeecs.v14.i3.pp1315-1329.

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Anomaly detection of flight route can be analyzed with the availability of flight data set. Automatic Dependent Surveillance (ADS-B) is the data set used. The parameters used are timestamp, latitude, longitude, and speed. The purpose of the research is to determine the optimum area for anomaly detection through real time approach. The methods used are: a) clustering and cluster validity analysis; and b) False Identification Rate (FIR). The results archieved are four steps, i.e: a) Build segments based on waypoints; b) Partition area based on 3-Dimension features P<sub>1</sub> and P<sub>2</sub>; c) grouping; and d) Measurement of cluster validity. The optimum partition is generated by calculating the minimum percentage of FIR. The results achieved are: i) there are five partitions, i.e: (n/2, n/3, n/4, n/5) and ii) optimal partition of each 3D, that is: for P<sub>1</sub> was five partitions and the P<sub>2</sub> feature was four partitions

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14

Ahmed, Chwas, Paul Martin, and Volodymyr Mazorchuk. "On the number of principal ideals in d-tonal partition monoids." Annals of Combinatorics 25, no. 1 (January 8, 2021): 79–113. http://dx.doi.org/10.1007/s00026-020-00518-z.

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AbstractFor a positive integer d, a non-negative integer n and a non-negative integer $$h\le n$$ h ≤ n , we study the number $$C_{n}^{(d)}$$ C n ( d ) of principal ideals; and the number $$C_{n,h}^{(d)}$$ C n , h ( d ) of principal ideals generated by an element of rank h, in the d-tonal partition monoid on n elements. We compute closed forms for the first family, as partial cumulative sums of known sequences. The second gives an infinite family of new integral sequences. We discuss their connections to certain integral lattices as well as to combinatorics of partitions.

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15

Kuziak, Dorota, and Ismael G. Yero. "Further new results on strong resolving partitions for graphs." Open Mathematics 18, no. 1 (May 14, 2020): 237–48. http://dx.doi.org/10.1515/math-2020-0142.

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Abstract A set W of vertices of a connected graph G strongly resolves two different vertices x, y ∉ W if either d G (x, W) = d G (x, y) + d G (y, W) or d G (y, W) = d G (y, x) + d G (x, W), where d G (x, W) = min{d(x,w): w ∈ W} and d(x,w) represents the length of a shortest x − w path. An ordered vertex partition Π = {U 1, U 2,…,U k } of a graph G is a strong resolving partition for G, if every two different vertices of G belonging to the same set of the partition are strongly resolved by some other set of Π. The minimum cardinality of any strong resolving partition for G is the strong partition dimension of G. In this article, we obtain several bounds and closed formulae for the strong partition dimension of some families of graphs and give some realization results relating the strong partition dimension, the strong metric dimension and the order of graphs.

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16

Wei, Changcheng, Muhammad Faisal Nadeem, Hafiz Muhammad Afzal Siddiqui, Muhammad Azeem, Jia-Bao Liu, and Adnan Khalil. "On Partition Dimension of Some Cycle-Related Graphs." Mathematical Problems in Engineering 2021 (March 5, 2021): 1–8. http://dx.doi.org/10.1155/2021/4046909.

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Let G be a simple connected graph. Suppose Δ = Δ 1 , Δ 2 , … , Δ l an l -partition of V G . A partition representation of a vertex α w . r . t Δ is the l − vector d α , Δ 1 , d α , Δ 2 , … , d α , Δ l , denoted by r α | Δ . Any partition Δ is referred as resolving partition if ∀ α i ≠ α j ∈ V G such that r α i | Δ ≠ r α j | Δ . The smallest integer l is referred as the partition dimension pd G of G if the l -partition Δ is a resolving partition. In this article, we discuss the partition dimension of kayak paddle graph, cycle graph with chord, and a graph generated by chain of cycles. It has been shown that the partition dimension of the said families of graphs is constant.

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17

Wansard, Guy, Patrick De Deckker, and Ramon Julià. "Variability in ostracod partition coefficients D(Sr) and D(Mg)." Chemical Geology 146, no. 1-2 (April 1998): 39–54. http://dx.doi.org/10.1016/s0009-2541(97)00165-4.

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18

BRENNAN, CHARLOTTE, ARNOLD KNOPFMACHER, and STEPHAN WAGNER. "The Distribution of Ascents of Size d or More in Partitions of n." Combinatorics, Probability and Computing 17, no. 4 (July 2008): 495–509. http://dx.doi.org/10.1017/s0963548308009073.

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A partition of a positive integer n is a finite sequence of positive integers a1, a2, . . ., ak such that a1+a2+ċ ċ ċ+ak=n and ai+1 ≥ ai for all i. Let d be a fixed positive integer. We say that we have an ascent of size d or more if ai+1 ≥ ai+d.We determine the mean, the variance and the limiting distribution of the number of ascents of size d or more (equivalently, the number of distinct part sizes of multiplicity d or more) in the partitions of n.

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19

Nadeem, Asim, Agha Kashif, Sohail Zafar, and Zohaib Zahid. "On 2-partition dimension of the circulant graphs." Journal of Intelligent & Fuzzy Systems 40, no. 5 (April 22, 2021): 9493–503. http://dx.doi.org/10.3233/jifs-201982.

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The partition dimension is a variant of metric dimension in graphs. It has arising applications in the fields of network designing, robot navigation, pattern recognition and image processing. Let G (V (G) , E (G)) be a connected graph and Γ = {P1, P2, …, Pm} be an ordered m-partition of V (G). The partition representation of vertex v with respect to Γ is an m-vector r (v|Γ) = (d (v, P1) , d (v, P2) , …, d (v, Pm)), where d (v, P) = min {d (v, x) |x ∈ P} is the distance between v and P. If the m-vectors r (v|Γ) differ in at least 2 positions for all v ∈ V (G), then the m-partition is called a 2-partition generator of G. A 2-partition generator of G with minimum cardinality is called a 2-partition basis of G and its cardinality is known as the 2-partition dimension of G. Circulant graphs outperform other network topologies due to their low message delay, high connectivity and survivability, therefore are widely used in telecommunication networks, computer networks, parallel processing systems and social networks. In this paper, we computed partition dimension of circulant graphs Cn (1, 2) for n ≡ 2 (mod 4), n ≥ 18 and hence corrected the result given by Salman et al. [Acta Math. Sin. Engl. Ser. 2012, 28, 1851-1864]. We further computed the 2-partition dimension of Cn (1, 2) for n ≥ 6.

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20

CAI, Yan-Guang, Yun ZHANG, and Ji-Xin QIAN. "The d-Subtree Partition Problem." Chinese Journal of Computers 33, no. 4 (May 10, 2010): 652–65. http://dx.doi.org/10.3724/sp.j.1016.2010.00652.

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21

Jung, Sung Soo, Seung Il Cho, Yong Bong Lee, and Woo Seop Lee. "Noise Barriers with 2-D Partition Lattices." Journal of the Korean Physical Society 43, no. 5 (November 15, 2003): 722–26. http://dx.doi.org/10.3938/jkps.43.722.

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22

CHEN, DANNY Z., and EWA MISIOŁEK. "FREE-FORM SURFACE PARTITION IN 3-D." International Journal of Computational Geometry & Applications 21, no. 06 (December 2011): 609–34. http://dx.doi.org/10.1142/s0218195911003834.

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We present a theoretical study of a problem related to optimal free-form surface partitioning, which arises in surface machining in manufacturing. In particular, we consider partitioning a free-form surface in 3-D into two subsurfaces subject to a global objective function. As a key subroutine, we develop new algorithms for the geometric problem of processing an off-line sequence of insertions and deletions of convex polygons alternated with point membership/proximity queries on the common intersection of the polygons. We show how this subroutine can be used to solve surface 2-partitioning. Our algorithm for the 2-partitioning problem takes [Formula: see text] time, where m is the number of polygons of size O(n) each. This improves the asymptotic time complexity of the previous best-known O(m2n2)-time algorithm. Our algorithms may be applicable to other accessibility and partition problems involving free-form surfaces in computer graphics and manufacturing. From the computational geometry point of view, our method combines nontrivial data structures, geometric observations, and algorithmic techniques that may be used to solve other geometric problems. For example, our algorithm can process O(n) insertions and deletions of convex polygons (of size O(n) each) and queries on their intersections in O(n2 log log n) time, improving the "standard" O(n2 log n) time solution.

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23

Bellamy, Gwyn. "The Calogero-Moser partition for G(m, d, n)." Nagoya Mathematical Journal 207 (September 2012): 47–77. http://dx.doi.org/10.1017/s0027763000022303.

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AbstractWe show that it is possible to deduce the Calogero-Moser partition of the irreducible representations of the complex reflection groups G(m,d, n) from the corresponding partition for G(m,1,n). This confirms, in the case W = G(m,d,n), a conjecture of Gordon and Martino relating the Calogero-Moser partition to Rouquier families for the corresponding cyclotomic Hecke algebra.

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24

Kiderlen, Markus, and Florian Pausinger. "Discrepancy of stratified samples from partitions of the unit cube." Monatshefte für Mathematik 195, no. 2 (March 7, 2021): 267–306. http://dx.doi.org/10.1007/s00605-021-01538-4.

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AbstractWe extend the notion of jittered sampling to arbitrary partitions and study the discrepancy of the related point sets. Let $${\varvec{\Omega }}=(\Omega _1,\ldots ,\Omega _N)$$ Ω = ( Ω 1 , … , Ω N ) be a partition of $$[0,1]^d$$ [ 0 , 1 ] d and let the ith point in $${{\mathcal {P}}}$$ P be chosen uniformly in the ith set of the partition (and stochastically independent of the other points), $$i=1,\ldots ,N$$ i = 1 , … , N . For the study of such sets we introduce the concept of a uniformly distributed triangular array and compare this notion to related notions in the literature. We prove that the expected $${{{\mathcal {L}}}_p}$$ L p -discrepancy, $${{\mathbb {E}}}{{{\mathcal {L}}}_p}({{\mathcal {P}}}_{\varvec{\Omega }})^p$$ E L p ( P Ω ) p , of a point set $${{\mathcal {P}}}_{\varvec{\Omega }}$$ P Ω generated from any equivolume partition $${\varvec{\Omega }}$$ Ω is always strictly smaller than the expected $${{{\mathcal {L}}}_p}$$ L p -discrepancy of a set of N uniform random samples for $$p>1$$ p > 1 . For fixed N we consider classes of stratified samples based on equivolume partitions of the unit cube into convex sets or into sets with a uniform positive lower bound on their reach. It is shown that these classes contain at least one minimizer of the expected $${{{\mathcal {L}}}_p}$$ L p -discrepancy. We illustrate our results with explicit constructions for small N. In addition, we present a family of partitions that seems to improve the expected discrepancy of Monte Carlo sampling by a factor of 2 for every N.

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25

Song, Jijian, and Bin Xu. "On Rational Functions with More than Three Branch Points." Algebra Colloquium 27, no. 02 (May 7, 2020): 231–46. http://dx.doi.org/10.1142/s100538672000019x.

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Let d be a positive integer and Λ be a collection of partitions of d of the form (a1, …, ap), (b1, …, bq), (m1 + 1, 1, …, 1), …, (ml + 1, 1, …, 1), where (m1, …, ml) is a partition of p + q − 2 > 0. We prove that there exists a rational function on the Riemann sphere with branch data Λ if and only if max(m1, …, ml) < d/GCD(a1, …, ap, b1, …, bq). As an application, we give a new class of branch data which can be realized by Belyi functions on the Riemann sphere.

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26

RAMAN, A., and C. PANDU RANGAN. "A UNIFIED APPROACH TO PARALLEL ALGORITHMS FOR THE DOMATIC PARTITION PROBLEM ON SPECIAL CLASSES OF PERFECT GRAPHS." Parallel Processing Letters 03, no. 03 (September 1993): 233–41. http://dx.doi.org/10.1142/s0129626493000277.

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A set of vertices D is a dominating set of a graph G=(V, E) if every vertex in V−D is adjacent to at least one vertex in D. The domatic partition of G is the partition of the vertex set V into a maximum number of dominating sets. In this paper, we present efficient parallel algorithms for finding the domatic partition of Interval graphs, Block graphs and K-trees.

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27

Huang, H. Y., and F. Y. Wu. "The Infinite-State Potts Model and Solid Partitions of an Integer." International Journal of Modern Physics B 11, no. 01n02 (January 20, 1997): 121–26. http://dx.doi.org/10.1142/s0217979297000150.

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It has been established that the infinite-state Potts model in d dimensions generates restricted partitions of integers in d-1 dimensions, the latter a well-known intractable problem in number theory for d>3. Here we consider the d=4 problem. We consider a Potts model on an L × M × N × P hypercubic lattice whose partition function GLMNP(t) generates restricted solid partitions on an L × M × N lattice with each part no greater than P. Closed-form expressions are obtained for G222P(t) and we evaluated its zeroes in the complex t plane for different values of P. On the basis of our numerical results we conjecture that all zeroes of the enumeration generating function GLMNP(t) lie on the unit circle |t|=1 in the limit that any of the indices L, M, N, P becomes infinite.

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28

AMBJØRN, J., D. BOULATOV, and V. A. KAZAKOV. "THE BOSONIC STRING REPRESENTED AS Φ3 GRAPHS: NEW MONTE-CARLO SIMULATIONS." Modern Physics Letters A 05, no. 10 (April 20, 1990): 771–85. http://dx.doi.org/10.1142/s0217732390000871.

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We discuss a new method for measuring the critical exponent γ for the partition function of the bosonic string. The statistics seems very good and the fit to γconsistent with the assumed asymptotic form for the partition function for dimensions d=1–6. The results are in agreement with analytical results when the target space dimension is d=0, but disagree when d=1. We conjecture that this is due to the appearance of logarithmic corrections to the asymptotic form of the partition function. These corrections might persist for d>1 and might render the determination of γquite difficult.

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29

Yu, Hai Yan. "An Image Adaptive Watermarking Algorithm Based on Ridgelet Transform and Two-Dimensional Fuzzy Partition." Advanced Materials Research 301-303 (July 2011): 1299–304. http://dx.doi.org/10.4028/www.scientific.net/amr.301-303.1299.

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An image adaptive watermarking algorithm based on ridgelet transform and two- dimensional(2-D) fuzzy partition classification is proposed. In order to obtain a sparse representation of straight edge singularity, the image is first partitioned into small pieces and the ridgelet transform is applied for each piece. After analyzing texture distribution in ridgelet coefficients of each piece, two feature vectors are selected to make up for the ‘wrap around’ effect for FRIT on representation of the image texture. Then the image is classifed into frat regions and texture regions by applying 2-D fuzzy partition classification algorithm with the two feature vectors prepocessed. An watermark sequence is embedded into texture regions with the embedding strength adaptively adjusted by ridgelet coefficients based on the feature of luminance masking and texture masking. Experimental results prove robustness and transparency of the proposed watermarking scheme.

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30

Kudaisya, Medha Malik. "G. D. Birla, big business and India's partition." South Asia: Journal of South Asian Studies 18, sup001 (January 1995): 193–212. http://dx.doi.org/10.1080/00856409508723251.

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31

Epstein, M. "Buoyancy-Driven Exchange Flow Through Small Openings in Horizontal Partitions." Journal of Heat Transfer 110, no. 4a (November 1, 1988): 885–93. http://dx.doi.org/10.1115/1.3250589.

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This paper describes an experimental study of the phenomenon of buoyancy-driven exchange (countercurrent) flow through openings in a horizontal partition. A density-driven exchange flow was obtained by using brine above the partition and fresh water below the partition. In the first part of the study, flow measurements were made with a single opening, for opening ratios L/D in the range 0.01 to 10.0, where L and D are the length of the opening (in the direction normal to the partition) and the diameter of the opening, respectively. Four different flow regimes are identified as L/D is increased through this range. As a result of the competition between two of these regimes, the exchange flow rate versus L/D relation exhibits a peak. The exchange flow rate was found, for all practical purposes, to be independent of viscosity, enabling a universal correlation between Froude number (dimensionless exchange flow rate) and L/D. The second part of the study was an experimental investigation of the same phenomenon, but with two openings in the horizontal partition. Two openings were observed to give rise to three different flow configurations involving both one-way and countercurrent flows within the openings.

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32

González, Yero, and Magdalena Lemńska. "Convex dominating-geodetic partitions in graphs." Filomat 30, no. 11 (2016): 3075–82. http://dx.doi.org/10.2298/fil1611075g.

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The distance d(u,v) between two vertices u and v in a connected graph G is the length of a shortest u-v path in G. A u-v path of length d(u,v) is called u-v geodesic. A set X is convex in G if vertices from all a -b geodesics belong to X for every two vertices a,b?X. A set of vertices D is dominating in G if every vertex of V-D has at least one neighbor in D. The convex domination number con(G) of a graph G equals the minimum cardinality of a convex dominating set in G. A set of vertices S of a graph G is a geodetic set of G if every vertex v ? S lies on a x-y geodesic between two vertices x,y of S. The minimum cardinality of a geodetic set of G is the geodetic number of G and it is denoted by g(G). Let D,S be a convex dominating set and a geodetic set in G, respectively. The two sets D and S form a convex dominating-geodetic partition of G if |D| + |S| = |V(G)|. Moreover, a convex dominating-geodetic partition of G is called optimal if D is a ?con(G)-set and S is a g(G)-set. In the present article we study the (optimal) convex dominating-geodetic partitions of graphs.

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Subbarao, M. V., and A. K. Agarwal. "Further Theorems of the Rogers-Ramanujan Type Theorems*." Canadian Mathematical Bulletin 31, no. 2 (June 1, 1988): 210–14. http://dx.doi.org/10.4153/cmb-1988-032-3.

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AbstractWe give three new partition theorems of the classical Rogers-Ramanujan type which are very much in the style of MacMahon. These are a continuation of four theorems of the same kind given recently by the second author. One of these new theorems, very similar to one of the original Rogers-Ramanuj an - MacMahon type theorems is as follows: Let C(n) denote the number of partitions of n into parts congruent to ±2, ± 3, ±4, ± 5, ±6, ±7 (mod 20). Let D(n) denote the number of partitions of n of the form n = b1 + b2 + … + bt, where bt ≧ 2, bt ≧ bi + 1 and if 1 ≦ i ≦ [(t - 2)/2], bi - bi + 1 ≧ 2. Then C(n) = D(n).

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Koponen, Vera. "A Limit Law of Almost l-partite Graphs." Journal of Symbolic Logic 78, no. 3 (September 2013): 911–36. http://dx.doi.org/10.2178/jsl.7803110.

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AbstractFor integers l ≥ 1, d ≥ 0 we study (undirected) graphs with vertices 1, …, n such that the vertices can be partitioned into l parts such that every vertex has at most d neighbours in its own part. The set of all such graphs is denoted Pn (l, d). We prove a labelled first-order limit law, i.e., for every first-order sentence φ, the proportion of graphs in Pn (l, d) that satisfy φ converges as n → ∞. By combining this result with a result of Hundack, Prömel and Steger [12] we also prove that if 1 ≤ s1 ≤ … ≤ sl are integers, then Forb() has a labelled first-order limit law, where Forb() denotes the set of all graphs with vertices 1, …, n, for some n, in which there is no subgraph isomorphic to the complete (l + 1 )-partite graph with parts of sizes 1, s1, …, sl. In the course of doing this we also prove that there exists a first-order formula ξ, depending only on l and d, such that the proportion of ∈ Pn (l, d) with the following property approaches 1 as n → ∞: there is a unique partition of {1, …, n} into l parts such that every vertex has at most d neighbours in its own part, and this partition, viewed as an equivalence relation, is defined by ξ.

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GROTHAUS, MARTIN, LUDWIG STREIT, and IGOR V. VOLOVICH. "KNOTS, FEYNMAN DIAGRAMS AND MATRIX MODELS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 02, no. 03 (September 1999): 359–80. http://dx.doi.org/10.1142/s0219025799000217.

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A U (N)-invariant matrix model with d matrix variables is studied. It was shown that in the limit N → ∞ and d→0 the model describes the knot diagrams. We realize the free partition function of the matrix model as the generalized expectation of a Hida distribution ΦN,d. This enables us to give a mathematically rigorous meaning to the partition function with interaction. For the generalized function ΦN,d, we prove a Wick theorem and we derive explicit formulas for the propagators.

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Muhana, Muthia, Des Welyyanti, and Narwen Narwen. "Dimensi Partisi Graf Lobster." Jurnal Matematika UNAND 8, no. 1 (July 5, 2019): 215. http://dx.doi.org/10.25077/jmu.8.1.215-218.2019.

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Misalkan terdapat k partisi dengan himpunan terurut S = {S1, S2, ..., Sk} dari himpunan titik V (G) pada graf terhubung G = (V, E), representasi partisi v ∈ V terhadap S adalah koordinat r(v | S) dengan:r(v | S) = (d(v, S1), d(v, S2), ..., d(v, Sk))untuk d(v, Si) menyatakan jarak antara titik v dengan himpunan Si dimana i = [1, k]. Partisi S dari V (G) disebut resolving partition dari G jika ∀v ∈ V (G) memiliki representasi partisi yang berbeda untuk setiap pasangan terurut dari u, v ∈ V maka r(u | S) 6= r(v | S). Resolving partition dengan kardinalitas minimum dari V (G) disebut dimensi partisi dari G, dinotasikan dengan pd(G). Pada penulisan ini akan dibahas tentang penentuan dimensi partisi untuk Graf Lobster.Kata Kunci: Partisi, Resolving Partition, Dimensi Partisi, Graf Lobster

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CHRISTENSEN, OLE, HONG OH KIM, and RAE YOUNG KIM. "CHARACTERISATIONS OF PARTITION OF UNITIES GENERATED BY ENTIRE FUNCTIONS IN." Bulletin of the Australian Mathematical Society 95, no. 2 (January 5, 2017): 281–90. http://dx.doi.org/10.1017/s0004972716001052.

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Collections of functions forming a partition of unity play an important role in analysis. In this paper we characterise for any $N\in \mathbb{N}$ the entire functions $P$ for which the partition of unity condition $\sum _{\mathbf{n}\in \mathbb{Z}^{d}}P(\mathbf{x}+\mathbf{n})\unicode[STIX]{x1D712}_{[0,N]^{d}}(\mathbf{x}+\mathbf{n})=1$ holds for all $\mathbf{x}\in \mathbb{R}^{d}.$ The general characterisation leads to various easy ways of constructing such entire functions as well. We demonstrate the flexibility of the approach by showing that additional properties like continuity or differentiability of the functions $(P\unicode[STIX]{x1D712}_{[0,N]^{d}})(\cdot +\mathbf{n})$ can be controlled. In particular, this leads to easy ways of constructing entire functions $P$ such that the functions in the partition of unity belong to the Feichtinger algebra.

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Khanin, S., R. Zybin, and O. Mordovskaya. "INCREASING THE EFFICIENCY OF THE MATERIAL CLASSIFICATION PROCESS IN CLASSIFYING BALL MILLS PARTITION." Bulletin of Belgorod State Technological University named after. V. G. Shukhov 5, no. 9 (October 3, 2020): 97–107. http://dx.doi.org/10.34031/2071-7318-2020-5-9-97-107.

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Ball mills are widely used for dispersion of materials in various industries, as they are characterized by ease of maintenance, operation and sufficiently high performance. To improve the process of inner-chamber classification of coarse-ground material, a new construction of the classifying partition with blades having cylindrical screening surfaces is proposed. The aim of the research is to substantiate the effectiveness of the application of the developed construction of the classifying partition in an industrial ball mill. The tasks of the research are to study the effectiveness of the developed construction of the classifying partitions on industrial ball mill. The tasks are solved of constructing and analyzing a regression equation that adequately describes the efficiency of the process of classifying coarse material with a cylindrical sieving surface depending on variable factors, determining rational areas of their values; comparing the efficiency of using blades with cylindrical and flat screening surfaces for inner-chamber classification of coarse material; confirmation of the possibility of providing the developed classifying partition with the mass productivity necessary for the operation of the mill. In the course of the work, the methods of simulation and mathematical modeling were used. As a result of the study, the efficiency of using a classifying partition with blades with cylindrical screening surfaces of a ball mill D×L=2×10,5 is substantiated.

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HIRSCHHORN, MICHAEL D., and JAMES A. SELLERS. "ELEMENTARY PROOFS OF PARITY RESULTS FOR 5-REGULAR PARTITIONS." Bulletin of the Australian Mathematical Society 81, no. 1 (July 2, 2009): 58–63. http://dx.doi.org/10.1017/s0004972709000525.

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AbstractIn a recent paper, Calkin et al. [N. Calkin, N. Drake, K. James, S. Law, P. Lee, D. Penniston and J. Radder, ‘Divisibility properties of the 5-regular and 13-regular partition functions’, Integers8 (2008), #A60] used the theory of modular forms to examine 5-regular partitions modulo 2 and 13-regular partitions modulo 2 and 3; they obtained and conjectured various results. In this note, we use nothing more than Jacobi’s triple product identity to obtain results for 5-regular partitions that are stronger than those obtained by Calkin and his collaborators. We find infinitely many Ramanujan-type congruences for b5(n), and we prove the striking result that the number of 5-regular partitions of the number n is even for at least 75% of the positive integers n.

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Arone, Gregory Z., and D. Lukas B. Brantner. "The action of Young subgroups on the partition complex." Publications mathématiques de l'IHÉS 133, no. 1 (March 25, 2021): 47–156. http://dx.doi.org/10.1007/s10240-021-00123-7.

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AbstractWe study the restrictions, the strict fixed points, and the strict quotients of the partition complex $|\Pi_{n}|$ | Π n | , which is the $\Sigma_{n}$ Σ n -space attached to the poset of proper nontrivial partitions of the set $\{1,\ldots,n\}$ { 1 , … , n } .We express the space of fixed points $|\Pi_{n}|^{G}$ | Π n | G in terms of subgroup posets for general $G\subset \Sigma_{n}$ G ⊂ Σ n and prove a formula for the restriction of $|\Pi_{n}|$ | Π n | to Young subgroups $\Sigma_{n_{1}}\times \cdots\times \Sigma_{n_{k}}$ Σ n 1 × ⋯ × Σ n k . Both results follow by applying a general method, proven with discrete Morse theory, for producing equivariant branching rules on lattices with group actions.We uncover surprising links between strict Young quotients of $|\Pi_{n}|$ | Π n | , commutative monoid spaces, and the cotangent fibre in derived algebraic geometry. These connections allow us to construct a cofibre sequence relating various strict quotients $|\Pi_{n}|^{\diamond} \mathbin {\operatorname* {\wedge }_{\Sigma_{n}}^{}} (S^{\ell})^{\wedge n}$ | Π n | ⋄ ∧ Σ n ( S ℓ ) ∧ n and give a combinatorial proof of a splitting in derived algebraic geometry.Combining all our results, we decompose strict Young quotients of $|\Pi_{n}|$ | Π n | in terms of “atoms” $|\Pi_{d}|^{\diamond} \mathbin {\operatorname* {\wedge }_{\Sigma_{d}}^{}} (S^{\ell})^{\wedge d}$ | Π d | ⋄ ∧ Σ d ( S ℓ ) ∧ d for $\ell$ ℓ odd and compute their homology. We thereby also generalise Goerss’ computation of the algebraic André-Quillen homology of trivial square-zero extensions from $\mathbf {F}_{2}$ F 2 to $\mathbf {F}_{p}$ F p for $p$ p an odd prime.

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GONZALEZ, TEOFILO F., MOHAMMADREZA RAZZAZI, and SI-QING ZHENG. "AN EFFICIENT DIVIDE-AND-CONQUER APPROXIMATION ALGORITHM FOR PARTITIONING INTO D-BOXES." International Journal of Computational Geometry & Applications 03, no. 04 (December 1993): 417–28. http://dx.doi.org/10.1142/s0218195993000269.

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Let P be a set of n points located inside a d-box (rectangle if d=2) R. We study the problem of partitioning R into d-boxes by introducing a set of hyperplane (line if d=2) segments of least total (d-1)–volume (length if d=2). Each of the resulting d-boxes in a valid partition cannot contain points from P as interior points. Since this problem is computationally intractable (NP-hard) even when d=2, we present an efficient approximation algorithm for its solution. The partition generated by our approximation algorithm is guaranteed to be within 2d times the optimal solution value. We also present a problem instance for each d≥2 for which the approximation bound is tight for the algorithm. The time complexity for our algorithm is O(dn log n).

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Sutanto, S. J., J. Wenninger, A. M. J. Coenders-Gerrits, and S. Uhlenbrook. "Partitioning of evaporation into transpiration, soil evaporation and interception: a comparison between isotope measurements and a HYDRUS-1D model." Hydrology and Earth System Sciences 16, no. 8 (August 10, 2012): 2605–16. http://dx.doi.org/10.5194/hess-16-2605-2012.

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Abstract. Knowledge of the water fluxes within the soil-vegetation-atmosphere system is crucial to improve water use efficiency in irrigated land. Many studies have tried to quantify these fluxes, but they encountered difficulties in quantifying the relative contribution of evaporation and transpiration. In this study, we compared three different methods to estimate evaporation fluxes during simulated summer conditions in a grass-covered lysimeter in the laboratory. Only two of these methods can be used to partition total evaporation into transpiration, soil evaporation and interception. A water balance calculation (whereby rainfall, soil moisture and percolation were measured) was used for comparison as a benchmark. A HYDRUS-1D model and isotope measurements were used for the partitioning of total evaporation. The isotope mass balance method partitions total evaporation of 3.4 mm d−1 into 0.4 mm d−1 for soil evaporation, 0.3 mm d−1 for interception and 2.6 mm d−1 for transpiration, while the HYDRUS-1D partitions total evaporation of 3.7 mm d−1 into 1 mm d−1 for soil evaporation, 0.3 mm d−1 for interception and 2.3 mm d−1 for transpiration. From the comparison, we concluded that the isotope mass balance is better for low temporal resolution analysis than the HYDRUS-1D. On the other hand, HYDRUS-1D is better for high temporal resolution analysis than the isotope mass balance.

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Thilmany, Jean, and Zina Saunders. "Printed Parts for a Lighter Flight." Mechanical Engineering 140, no. 12 (December 1, 2018): 37. http://dx.doi.org/10.1115/1.2018-dec-5.

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Bastian Schäfer, an Airbus engineer, believes the ability to 3-D print airplane parts—including the plane’s very skeleton structure—will revolutionize air travel. Schäfer is project manager on what his group calls the Bionic Partition Project. The project itself is under the purview of the Airbus Emerging Technologies and Concepts Group, led by Peter Sander. Working under Schäfer, the group has created a 3-D printed partition to separate the seating area on the A320 from the galley.

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Saenpholphat, Varaporn, and Ping Zhang. "Conditional resolvability in graphs: a survey." International Journal of Mathematics and Mathematical Sciences 2004, no. 38 (2004): 1997–2017. http://dx.doi.org/10.1155/s0161171204311403.

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For an ordered setW={w1,w2,…,wk}of vertices and a vertexvin a connected graphG, the code ofvwith respect toWis thek-vectorcW(v)=(d(v,w1),d(v,w2),…,d(v,wk)), whered(x,y)represents the distance between the verticesxandy. The setWis a resolving set forGif distinct vertices ofGhave distinct codes with respect toW. The minimum cardinality of a resolving set forGis its dimensiondim(G). Many resolving parameters are formed by extending resolving sets to different subjects in graph theory, such as the partition of the vertex set, decomposition and coloring in graphs, or by combining resolving property with another graph-theoretic property such as being connected, independent, or acyclic. In this paper, we survey results and open questions on the resolving parameters defined by imposing an additional constraint on resolving sets, resolving partitions, or resolving decompositions in graphs.

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45

Levi, Inessa, and Steve Seif. "COMBINATORIAL TECHNIQUES FOR DETERMINING RANK AND IDEMPOTENT RANK OF CERTAIN FINITE SEMIGROUPS." Proceedings of the Edinburgh Mathematical Society 45, no. 3 (October 2002): 617–30. http://dx.doi.org/10.1017/s0013091501000530.

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AbstractLet $\tau$ be a partition of the positive integer $n$. A partition of the set $\{1,2,\dots,n\}$ is said to be of type $\tau$ if the sizes of its classes form the partition $\tau$ of $n$. It is known that the semigroup $S(\tau)$, generated by all the transformations with kernels of type $\tau$, is idempotent generated. When $\tau$ has a unique non-singleton class of size $d$, the difficult Middle Levels Conjecture of combinatorics obstructs the application of known techniques for determining the rank and idempotent rank of $S(\tau)$. We further develop existing techniques, associating with a subset $U$ of the set of all idempotents of $S(\tau)$ with kernels of type $\tau$ a directed graph $D(U)$; the directed graph $D(U)$ is strongly connected if and only if $U$ is a generating set for $S(\tau)$, a result which leads to a proof if the fact that the rank and the idempotent rank of $S(\tau)$ are both equal to$$ \max\biggl\{\binom{n}{d},\binom{n}{d+1}\biggr\}. $$AMS 2000 Mathematics subject classification: Primary 20M20; 05A18; 05A17; 05C20

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Jones, Albin L. "A polarized partition relation for weakly compact cardinals using elementary substructures." Journal of Symbolic Logic 71, no. 4 (December 2006): 1342–52. http://dx.doi.org/10.2178/jsl/1164060459.

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AbstractWe show that if κ is a weakly compact cardinal, thenfor any ordinals α < κ+ and μ < κ, and any finite ordinals m and n. This polarized partition relation represents the statement that for any partitionof κ × κ+ into m + μ pieces either there are A ∈ [κ]κ, B ∈ [κ]+]α and i < m with A × B ⊆ Ki or there are C ∈ [κ]κ, , and j < μ with C × D ⊆ Lj. Related results for measurable and almost measurable κ are also investigated. Our proofs of these relations involve the use of elementary substructures of set models of large fragments of ZFC.

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ROBERTS, PATRICK, and HARUHIKO TERAO. "MODULAR INVARIANTS OF KAČ-MOODY ALGEBRAS FROM SELF-DUAL LATTICES." International Journal of Modern Physics A 07, no. 10 (April 20, 1992): 2207–18. http://dx.doi.org/10.1142/s0217751x92000983.

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A general method for the construction of modular invariant partition functions of Kač-Moody algebras using a covariant lattice approach is presented. The A-D-E series for [Formula: see text] conformal invariant theories and partition functions of [Formula: see text] are constructed explicitly to demonstrate the technique.

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SINANOGLOU, PENNY. "BRITISH PLANS FOR THE PARTITION OF PALESTINE, 1929–1938." Historical Journal 52, no. 1 (February 27, 2009): 131–52. http://dx.doi.org/10.1017/s0018246x08007346.

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ABSTRACTThe 1937 Peel Commission proposal for the partition of British mandatory Palestine has generally been framed as the precursor to the United Nations partition plan of 1947. This article demonstrates the importance of tracing the roots of the 1937 Peel Commission plan back to conversations taking place in the Colonial Office and government of Palestine as early as 1929. A close analysis of dialogues over territorial division and of preliminary partition plans, particularly those drawn up by L. G. Archer Cust and D. G. Harris, leads to the conclusion that Britain's focus on the ideal of representative government played a primary role in the development of partition proposals. This article argues that inter-ethnic violence played a much smaller role in the development of partition proposals than has previously been thought. Instead, partition was proposed as a solution to the political implications of non-representative government in Palestine, a topic constantly in the spotlight thanks to the League of Nations.

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BAL, SUBRATA, and B. SATHIAPALAN. "HIGH TEMPERATURE LIMIT OF THE N=2 MATRIX MODEL." Modern Physics Letters A 14, no. 40 (December 28, 1999): 2753–65. http://dx.doi.org/10.1142/s0217732399002893.

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The high temperature limit of a system of two D-0 branes is investigated. The partition function can be expressed as a power series in β (inverse temperature). The leading term in the high temperature expression of the partition function and effective potential is calculated exactly. Physical quantities like the mean square separation can also be exactly determined in the high temperature limit.

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Sun, Jiachang, and Xiquan Shi. "B-splines on 3-D tetrahedron partition in four-directional mesh." Science in China Series A: Mathematics 44, no. 4 (April 2001): 491–96. http://dx.doi.org/10.1007/bf02881886.

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