A New Deterministic Method for Computing Milnor Number of an ICIS

2021 ◽  
pp. 391-408
Author(s):  
Shinichi Tajima ◽  
Katsusuke Nabeshima
Keyword(s):  
Milnor Number ◽  
Deterministic Method

scholarly journals Conjectures on Stably Newton Degenerate Singularities

2021 ◽  
Author(s):  
Jan Stevens
Keyword(s):  
Characteristic Zero ◽  
Arbitrary Number ◽  
Milnor Number ◽  
Plane Curves ◽  
Newton Diagram ◽  
Finite Characteristic ◽  
Vanishing Cycles

AbstractWe discuss a problem of Arnold, whether every function is stably equivalent to one which is non-degenerate for its Newton diagram. We argue that the answer is negative. We describe a method to make functions non-degenerate after stabilisation and give examples of singularities where this method does not work. We conjecture that they are in fact stably degenerate, that is not stably equivalent to non-degenerate functions.We review the various non-degeneracy concepts in the literature. For finite characteristic, we conjecture that there are no wild vanishing cycles for non-degenerate singularities. This implies that the simplest example of singularities with finite Milnor number, $$x^p+x^q$$ x p + x q in characteristic p, is not stably equivalent to a non-degenerate function. We argue that irreducible plane curves with an arbitrary number of Puiseux pairs (in characteristic zero) are stably non-degenerate. As the stabilisation involves many variables, it becomes very difficult to determine the Newton diagram in general, but the form of the equations indicates that the defining functions are non-degenerate.

scholarly journals The Image Milnor Number and Excellent Unfoldings

2021 ◽  
Author(s):  
R Giménez Conejero ◽  
J J Nuño-Ballesteros
Keyword(s):  
Milnor Number ◽  
Basic Properties ◽  
Topological Invariance

Abstract We show three basic properties of the image Milnor number µI(f) of a germ $f\colon(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{n+1},0)$ with isolated instability. First, we show the conservation of the image Milnor number, from which one can deduce the upper semi-continuity and the topological invariance for families. Second, we prove the weak Mond’s conjecture, which states that µI(f) = 0 if and only if f is stable. Finally, we show a conjecture by Houston that any family $f_t\colon(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{n+1},0)$ with $\mu_I(\,f_t)$ constant is excellent in Gaffney’s sense. For technical reasons, in the last two properties, we consider only the corank 1 case.

scholarly journals The jump of the Milnor number in the X 9 singularity class

2014 ◽  
Vol 12 (3) ◽  
Author(s):  
Szymon Brzostowski ◽  
Tadeusz Krasiński
Keyword(s):  
Milnor Number ◽  
Isolated Singularity ◽  
Milnor Numbers

AbstractThe jump of the Milnor number of an isolated singularity f 0 is the minimal non-zero difference between the Milnor numbers of f 0 and one of its deformations (f s). We prove that for the singularities in the X 9 singularity class their jumps are equal to 2.

Deterministic Method to Evaluate the Threshold Voltage Variability Induced by Discrete Trap Charges in Si-Nanowire FETs

2012 ◽  
Vol 59 (5) ◽  
pp. 1462-1467 ◽  
Author(s):  
Abderrezak Bekaddour ◽  
Marco G. Pala ◽  
Nasr-Eddine Chabane-Sari ◽  
Gérard Ghibaudo
Keyword(s):  
Threshold Voltage ◽  
Deterministic Method ◽  
Si Nanowire ◽  
Trap Charges

scholarly journals Coherently controlled quantum features in a coupled interferometric scheme

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Byoung S. Ham
Keyword(s):  
Optical Materials ◽  
Bell Inequality ◽  
Third Order ◽  
Deterministic Method ◽  
Photon Pairs ◽  
Wave Nature ◽  
Parametric Down Conversion ◽  
Pair Generation ◽  
Entangled Photon Pairs ◽  
Down Conversion

AbstractOver the last several decades, entangled photon pairs generated by spontaneous parametric down conversion processes in both second-order and third-order nonlinear optical materials have been intensively studied for various quantum features such as Bell inequality violation and anticorrelation. In an interferometric scheme, anticorrelation results from photon bunching based on randomness when entangled photon pairs coincidently impinge on a beam splitter. Compared with post-measurement-based probabilistic confirmation, a coherence version has been recently proposed using the wave nature of photons. Here, the origin of quantum features in a coupled interferometric scheme is investigated using pure coherence optics. In addition, a deterministic method of entangled photon-pair generation is proposed for on-demand coherence control of quantum processing.

scholarly journals Scaling High-Quality Pairwise Link-Based Similarity Retrieval on Billion-Edge Graphs

2022 ◽  
Vol 40 (4) ◽  
pp. 1-45
Author(s):  
Weiren Yu ◽  
Julie McCann ◽  
Chengyuan Zhang ◽  
Hakan Ferhatosmanoglu
Keyword(s):  
Web Search ◽  
Similarity Score ◽  
High Quality ◽  
Deterministic Method ◽  
Large Graphs ◽  
Guaranteed Accuracy ◽  
Semantic Difference ◽  
Speed Up ◽  
Novel Method ◽  
Search Quality

SimRank is an attractive link-based similarity measure used in fertile fields of Web search and sociometry. However, the existing deterministic method by Kusumoto et al. [ 24 ] for retrieving SimRank does not always produce high-quality similarity results, as it fails to accurately obtain diagonal correction matrix  D . Moreover, SimRank has a “connectivity trait” problem: increasing the number of paths between a pair of nodes would decrease its similarity score. The best-known remedy, SimRank++ [ 1 ], cannot completely fix this problem, since its score would still be zero if there are no common in-neighbors between two nodes. In this article, we study fast high-quality link-based similarity search on billion-scale graphs. (1) We first devise a “varied- D ” method to accurately compute SimRank in linear memory. We also aggregate duplicate computations, which reduces the time of [ 24 ] from quadratic to linear in the number of iterations. (2) We propose a novel “cosine-based” SimRank model to circumvent the “connectivity trait” problem. (3) To substantially speed up the partial-pairs “cosine-based” SimRank search on large graphs, we devise an efficient dimensionality reduction algorithm, PSR # , with guaranteed accuracy. (4) We give mathematical insights to the semantic difference between SimRank and its variant, and correct an argument in [ 24 ] that “if D is replaced by a scaled identity matrix (1-Ɣ)I, their top-K rankings will not be affected much”. (5) We propose a novel method that can accurately convert from Li et al.  SimRank ~{S} to Jeh and Widom’s SimRank S . (6) We propose GSR # , a generalisation of our “cosine-based” SimRank model, to quantify pairwise similarities across two distinct graphs, unlike SimRank that would assess nodes across two graphs as completely dissimilar. Extensive experiments on various datasets demonstrate the superiority of our proposed approaches in terms of high search quality, computational efficiency, accuracy, and scalability on billion-edge graphs.

Convergence Study on Application of the Over-Deterministic Method for Determination of Near-Tip Fields in a Cracked Plate Loaded in Mixed-Mode

2012 ◽  
Vol 249-250 ◽  
pp. 76-81 ◽  
Author(s):  
Lucie Šestáková ◽  
Václav Veselý
Keyword(s):  
Mixed Mode ◽  
Brittle Materials ◽  
Higher Order ◽  
Displacement Fields ◽  
Deterministic Method ◽  
Crack Behavior ◽  
Stress And Displacement Fields ◽  
Higher Order Terms ◽  
Convergence Study

Multi-parameter description of crack behavior in quasi-brittle materials offers still enough space for investigations. Several studies have been carried out by the authors in this field [1-3]. One part of the publications by the authors (this work included) contain analyses of the accuracy, convergence and/or tuning of the over-deterministic method that enables determination of the coefficients of the higher-order terms in Williams expansion approximating the stress and displacement fields in a cracked body without any complicated FE formulations. These intermediate studies should bring together a list of recommendations how to use the ODM as effectively as possible and obtain reliable enough values of coefficients of the higher-order terms. Thus, the stress/displacement field can be determined precisely even in a larger distance from the crack tip, which is crucial for assessment of the fracture occurring in quasi-brittle materials.

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