scholarly journals The relative Bruce–Roberts number of a function on a hypersurface

2021 ◽  
pp. 1-13
Author(s):  
B. K. Lima-Pereira ◽  
J. J. Nuño-Ballesteros ◽  
B. Oréfice-Okamoto ◽  
J. N. Tomazella
Keyword(s):  
Ambient Space ◽  
Milnor Number ◽  
Hypersurface Singularity ◽  
Characteristic Variety ◽  
The Difference

Abstract We consider the relative Bruce–Roberts number $\mu _{\textrm {BR}}^{-}(f,\,X)$ of a function on an isolated hypersurface singularity $(X,\,0)$ . We show that $\mu _{\textrm {BR}}^{-}(f,\,X)$ is equal to the sum of the Milnor number of the fibre $\mu (f^{-1}(0)\cap X,\,0)$ plus the difference $\mu (X,\,0)-\tau (X,\,0)$ between the Milnor and the Tjurina numbers of $(X,\,0)$ . As an application, we show that the usual Bruce–Roberts number $\mu _{\textrm {BR}}(f,\,X)$ is equal to $\mu (f)+\mu _{\textrm {BR}}^{-}(f,\,X)$ . We also deduce that the relative logarithmic characteristic variety $LC(X)^{-}$ , obtained from the logarithmic characteristic variety $LC(X)$ by eliminating the component corresponding to the complement of $X$ in the ambient space, is Cohen–Macaulay.

NON-NEGATIVE DEFORMATIONS OF WEIGHTED HOMOGENEOUS SINGULARITIES

2017 ◽  
Vol 60 (1) ◽  
pp. 175-185 ◽  
Author(s):  
J. J. NUÑO-BALLESTEROS ◽  
B. ORÉFICE-OKAMOTO ◽  
J. N. TOMAZELLA
Keyword(s):  
Sufficient Conditions ◽  
Additional Term ◽  
Milnor Number ◽  
Curve Singularity ◽  
Hypersurface Singularity ◽  
Weighted Degree ◽  
Function Germ ◽  
Analytic Variety ◽  
Complex Analytic Variety ◽  
Necessary And Sufficient

AbstractWe consider a weighted homogeneous germ of complex analytic variety (X, 0) ⊂ (ℂn, 0) and a function germ f : (ℂn, 0) → (ℂ, 0). We derive necessary and sufficient conditions for some deformations to have non-negative degree (i.e., for any additional term in the deformation, the weighted degree is not smaller) in terms of an adapted version of the relative Milnor number. We study the cases where (X, 0) is an isolated hypersurface singularity and the invariant is the Bruce-Roberts number of f with respect to (X, 0), and where (X, 0) is an isolated complete intersection or a curve singularity and the invariant is the Milnor number of the germ f: (X, 0) → ℂ. In the last part, we give some formulas for the invariants in terms of the weights and the degrees of the polynomials.

On the Pole Order and Hodge Filtrations of Isolated Hypersurface Singularities

1993 ◽  
Vol 36 (3) ◽  
pp. 368-372
Author(s):  
John Scherk
Keyword(s):  
Hypersurface Singularity ◽  
Projective Hypersurface ◽  
Pole Order ◽  
Hypersurface Singularities ◽  
The Difference

AbstractUnlike for a smooth projective hypersurface, for an isolated hypersurface singularity, the pole order and Hodge filtrations do not in general coincide. This note studies the difference between the two.

scholarly journals The Bruce–Roberts Number of A Function on A Hypersurface with Isolated Singularity

2020 ◽  
Vol 71 (3) ◽  
pp. 1049-1063
Author(s):  
J J Nuño-Ballesteros ◽  
B Oréfice-Okamoto ◽  
B K Lima-Pereira ◽  
J N Tomazella
Keyword(s):  
Complete Intersection ◽  
Hypersurface Singularity ◽  
Isolated Singularity ◽  
Characteristic Variety ◽  
Intersection Singularity ◽  
Complete Intersection Singularity ◽  
Isolated Complete Intersection Singularity

Abstract Let $(X,0)$ be an isolated hypersurface singularity defined by $\phi \colon ({\mathbb{C}}^n,0)\to ({\mathbb{C}},0)$ and $f\colon ({\mathbb{C}}^n,0)\to{\mathbb{C}}$ such that the Bruce–Roberts number $\mu _{BR}(f,X)$ is finite. We first prove that $\mu _{BR}(f,X)=\mu (f)+\mu (\phi ,f)+\mu (X,0)-\tau (X,0)$, where $\mu $ and $\tau $ are the Milnor and Tjurina numbers respectively of a function or an isolated complete intersection singularity. Second, we show that the logarithmic characteristic variety $LC(X,0)$ is Cohen–Macaulay. Both theorems generalize the results of a previous paper by some of the authors, in which the hypersurface $(X,0)$ was assumed to be weighted homogeneous.

scholarly journals Eggplant culture (Solanum melongena L.) in Siberia

2020 ◽  
Author(s):  
S.V. Ugarova ◽  
Keyword(s):  
Species Diversity ◽  
Plant Species ◽  
Solanum Melongena ◽  
Morphological Features ◽  
Biological Characteristic ◽  
Plant Selection ◽  
Climatic Parameter ◽  
Characteristic Variety ◽  
Botanical Species ◽  
The Difference

Under Siberian conditions, aubergine (eggplant) is stressed by the difference between region climatic parameter and the thermophilic plant species requirements. Plant selection with reference to the crop botanical species diversity and the full use of worldwide biological characteristic variety and morphological features of plants provides the adaptation of species.

scholarly journals Topological type of discriminants of some special families

2021 ◽  
Author(s):  
Evelia R. García Barroso ◽  
M. Fernando Hernández Iglesias
Keyword(s):  
Topological Type ◽  
Smooth Curve ◽  
Milnor Number ◽  
Irreducible Curve ◽  
Tjurina Number ◽  
Discriminant Curve ◽  
The Difference

AbstractWe will describe the topological type of the discriminant curve of the morphism $$(\ell , f)$$ ( ℓ , f ) , where $$\ell $$ ℓ is a smooth curve and f is an irreducible curve (branch) of multiplicity less than five or a branch such that the difference between its Milnor number and Tjurina number is less than 3. We prove that for a branch of these families, the topological type of the discriminant curve is determined by the semigroup, the Zariski invariant and at most two other analytical invariants of the branch.

The Origin of the Moon

1962 ◽  
Vol 14 ◽  
pp. 149-155 ◽  
Author(s):  
E. L. Ruskol
Keyword(s):  
Chemical Composition ◽  
Solar System ◽  
The Moon ◽  
The Earth ◽  
The Difference ◽  
Origin Of The Moon

The difference between average densities of the Moon and Earth was interpreted in the preceding report by Professor H. Urey as indicating a difference in their chemical composition. Therefore, Urey assumes the Moon's formation to have taken place far away from the Earth, under conditions differing substantially from the conditions of Earth's formation. In such a case, the Earth should have captured the Moon. As is admitted by Professor Urey himself, such a capture is a very improbable event. In addition, an assumption that the “lunar” dimensions were representative of protoplanetary bodies in the entire solar system encounters great difficulties.

Influence of Cell Wall Composition on the Fossilisation of Bacteria and the Implications for the Search for Early Life Forms

1997 ◽  
Vol 161 ◽  
pp. 491-504 ◽  
Author(s):  
Frances Westall
Keyword(s):  
Cell Wall ◽  
Life Forms ◽  
Cation Binding ◽  
Positive Bacterium ◽  
Gram Positive ◽  
Gram Positive Bacteria ◽  
Transmission Electron ◽  
Hydrothermal Sediments ◽  
Identification Of Bacteria ◽  
The Difference

AbstractThe oldest cell-like structures on Earth are preserved in silicified lagoonal, shallow sea or hydrothermal sediments, such as some Archean formations in Western Australia and South Africa. Previous studies concentrated on the search for organic fossils in Archean rocks. Observations of silicified bacteria (as silica minerals) are scarce for both the Precambrian and the Phanerozoic, but reports of mineral bacteria finds, in general, are increasing. The problems associated with the identification of authentic fossil bacteria and, if possible, closer identification of bacteria type can, in part, be overcome by experimental fossilisation studies. These have shown that not all bacteria fossilise in the same way and, indeed, some seem to be very resistent to fossilisation. This paper deals with a transmission electron microscope investigation of the silicification of four species of bacteria commonly found in the environment. The Gram positiveBacillus laterosporusand its spore produced a robust, durable crust upon silicification, whereas the Gram negativePseudomonas fluorescens, Ps. vesicularis, andPs. acidovoranspresented delicately preserved walls. The greater amount of peptidoglycan, containing abundant metal cation binding sites, in the cell wall of the Gram positive bacterium, probably accounts for the difference in the mode of fossilisation. The Gram positive bacteria are, therefore, probably most likely to be preserved in the terrestrial and extraterrestrial rock record.

Structuring, Motions and Shapes of Corona Emission Line Profiles

1994 ◽  
Vol 144 ◽  
pp. 421-426
Author(s):  
N. F. Tyagun
Keyword(s):  
Emission Line ◽  
Filling Factor ◽  
Direct Relationship ◽  
Coronal Plasma ◽  
Line Of Sight ◽  
Line Profiles ◽  
The Difference ◽  
Yellow Line ◽  
Red Line

AbstractThe interrelationship of half-widths and intensities for the red, green and yellow lines is considered. This is a direct relationship for the green and yellow line and an inverse one for the red line. The difference in the relationships of half-widths and intensities for different lines appears to be due to substantially dissimilar structuring and to a set of line-of-sight motions in ”hot“ and ”cold“ corona regions.When diagnosing the coronal plasma, one cannot neglect the filling factor - each line has such a factor of its own.

Difference Fourier Analysis of Glucose Embedded and Frozen Hydrated Purple Membrane

1982 ◽  
Vol 40 ◽  
pp. 74-75
Author(s):  
Jules S. Jaffe ◽  
Robert M. Glaeser
Keyword(s):  
Purple Membrane ◽  
Data Set ◽  
High Resolution Data ◽  
X Ray ◽  
X Ray Crystallography ◽  
Fourier Techniques ◽  
Versus Protein ◽  
The Difference ◽  
Difference Fourier ◽  
Ideal Method

Although difference Fourier techniques are standard in X-ray crystallography it has only been very recently that electron crystallographers have been able to take advantage of this method. We have combined a high resolution data set for frozen glucose embedded Purple Membrane (PM) with a data set collected from PM prepared in the frozen hydrated state in order to visualize any differences in structure due to the different methods of preparation. The increased contrast between protein-ice versus protein-glucose may prove to be an advantage of the frozen hydrated technique for visualizing those parts of bacteriorhodopsin that are embedded in glucose. In addition, surface groups of the protein may be disordered in glucose and ordered in the frozen state. The sensitivity of the difference Fourier technique to small changes in structure provides an ideal method for testing this hypothesis.

Destruction of Actin Filaments by Osmium Tetroxide

1977 ◽  
Vol 35 ◽  
pp. 466-467
Author(s):  
P. Maupin-Szamier ◽  
T. D. Pollard
Keyword(s):  
Actin Filaments ◽  
Time Course ◽  
Osmium Tetroxide ◽  
Rabbit Muscle ◽  
Muscle Actin ◽  
Sodium Phosphate Buffer ◽  
Transmission Electron ◽  
The Difference ◽  
Rates Of Decay ◽  
Short Time

We have studied the destruction of rabbit muscle actin filaments by osmium tetroxide (OSO4) to develop methods which will preserve the structure of actin filaments during preparation for transmission electron microscopy.Negatively stained F-actin, which appears as smooth, gently curved filaments in control samples (Fig. 1a), acquire an angular, distorted profile and break into progressively shorter pieces after exposure to OSO4 (Fig. 1b,c). We followed the time course of the reaction with viscometry since it is a simple, quantitative method to assess filament integrity. The difference in rates of decay in viscosity of polymerized actin solutions after the addition of four concentrations of OSO4 is illustrated in Fig. 2. Viscometry indicated that the rate of actin filament destruction is also dependent upon temperature, buffer type, buffer concentration, and pH, and requires the continued presence of OSO4. The conditions most favorable to filament preservation are fixation in a low concentration of OSO4 for a short time at 0°C in 100mM sodium phosphate buffer, pH 6.0.

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