65,770 research outputs found
Classification of singularities of cluster algebras of finite type
We provide a complete classification of the singularities of cluster algebras
of finite type. Alongside, we develop a constructive desingularization of these
singularities via blowups in regular centers over fields of arbitrary
characteristic. Furthermore, from the same perspective, we study a family of
cluster algebras, which are not of finite type and which arise from a star
shaped quiver.Comment: 36 page
On the numerical classification of the singularities of robot manipulators
This paper is concerned with the task to obtain a complete description of the singularity set of any given non-redundant manipulator,
including the identification and the precise computation of each constituent singularity class. Configurations belonging to the same class are equivalent in terms of the various
types of kinematic and static degeneracy that characterize mechanism singularity. The proposed approach is an extension of recent
work on computing singularities using a numerical method based on linear relaxations. Classification is sought by means of a hierarchy of singularity tests, each formulated as a system of quadratic or linear equations, which yields sets of classes to which an identified singularity cannot belong. A planar manipulator exemplifies the process of classification, and illustrates how, while most singularities get completely classified, for some
lower-dimensional subsets one can only identify a restricted list of possible singularity classes.Postprint (published version
Polar Cremona Transformations and Monodromy of Polynomials
Consider the gradient map associated to any non-constant homogeneous
polynomial f\in \C[x_0,...,x_n] of degree , defined by \phi_f=grad(f):
D(f)\to \CP^n, (x_0:...:x_n)\to (f_0(x):...:f_n(x)) where D(f)=\{x\in \CP^n;
f(x)\neq 0\} is the principal open set associated to and
. This map corresponds to polar Cremona
transformations. In Proposition \ref{p1} we give a new lower bound for the
degree of under the assumption that the projective hypersurface
has only isolated singularities. When , Theorem \ref{t4}
yields very strong conditions on the singularities of .Comment: 8 page
The Łojasiewicz exponent over a field of arbitrary characteristic
Let K be an algebraically closed field and let K((XQ)) denote the field
of generalized series with coefficients in K. We propose definitions of the local
Łojasiewicz exponent of F = ( f1, . . . , fm) ∈ K[[X, Y ]]m as well as of the
Łojasiewicz exponent at infinity of F = ( f1, . . . , fm) ∈ K[X, Y ]m, which generalize
the familiar case of K = C and F ∈ C{X, Y }m (resp. F ∈ C[X, Y ]m), see
Cha˛dzy´nski and Krasi´nski (In: Singularities, 1988; In: Singularities, 1988; Ann Polon
Math 67(3):297–301, 1997; Ann Polon Math 67(2):191–197, 1997), and prove some
basic properties of such numbers. Namely, we show that in both cases the exponent
is attained on a parametrization of a component of F (Theorems 6 and 7), thus being
a rational number. To this end, we define the notion of the Łojasiewicz pseudoexponent
of F ∈ (K((XQ))[Y ])m for which we give a description of all the generalized
series that extract the pseudoexponent, in terms of their jets. In particular, we show
that there exist only finitely many jets of generalized series giving the pseudoexponent
of F (Theorem 5). The main tool in the proofs is the algebraic version of Newton’s
Polygon Method. The results are illustrated with some explicit examples
On maps with unstable singularities
If a continuous map f: X->Q is approximable arbitrary closely by embeddings
X->Q, can some embedding be taken onto f by a pseudo-isotopy? This question,
called Isotopic Realization Problem, was raised by Shchepin and Akhmet'ev. We
consider the case where X is a compact n-polyhedron, Q a PL m-manifold and show
that the answer is 'generally no' for (n,m)=(3,6); (1,3), and 'yes' when:
1) m>2n, (n,m)\neq (1,3);
2) 2m>3(n+1) and the set {(x,y)|f(x)=f(y)} has an equivariant (with respect
to the factor exchanging involution) mapping cylinder neighborhood in X\times
X;
3) m>n+2 and f is the composition of a PL map and a TOP embedding.
In doing this, we answer affirmatively (with a minor preservation) a question
of Kirby: does small smooth isotopy imply small smooth ambient isotopy in the
metastable range, verify a conjecture of Kearton-Lickorish: small PL
concordance implies small PL ambient isotopy in codimension \ge 3, and a
conjecture set of Repovs-Skopenkov.Comment: 46 pages, 5 figures, to appear in Topol Appl; some important
footnotes added in version
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