Search results for "singular"

Glass transitions and scaling laws within an alternative mode-coupling theory

2015

Idealized glass transitions are discussed within an alternative mode-coupling theory (TMCT) proposed by Tokuyama [Physica A 395, 31 (2014)]. This is done in order to identify common ground with and differences from the conventional mode-coupling theory (MCT). It is proven that both theories imply the same scaling laws for the transition dynamics, which are characterized by two power-law decay functions and two diverging power-law time scales. However, the values for the corresponding anomalous exponents calculated within both theories differ from each other. It is proven that the TMCT, contrary to the MCT, does not describe transitions with continuously vanishing arrested parts of the corre…

SPATIAL MULTIFRACTALITY OF ELECTRONIC STATES AND THE METAL-INSULATOR TRANSITION IN DISORDERED SYSTEMS

1993

For the investigation of the spatial behavior of electronic wave functions in disordered systems, we employ the Anderson model of localization. The eigenstates of the corresponding Hamiltonian are calculated numerically by means of the Lanczos algorithm and are analyzed with respect to their spatial multifractal properties. We find that the wave functions show spatial multifractality for all parameter cases not too far away from the metal-insulator transition (MIT) which separates localized from extended states in this model. Exactly at the MIT, multifractality is expected to exist on all length scales larger than the lattice spacing. It is found that the corresponding singularity spectrum…

Mond's conjecture for maps between curves

2017

A theorem by D. Mond shows that if f:(C,0)→C2,0 is finite and has has degree one onto its image (Y, 0), then the Ae-codimension is less than or equal to the image Milnor number μI(f), with equality if and only if (Y, 0) is weighted homogeneous. Here we generalize this result to the case of a map germ f:(X,0)→C2,0, where (X, 0) is a plane curve singularity.

On Severi Type Inequalities for Irregular Surfaces

2017

Let X be a minimal surface of general type and maximal Albanese dimension with irregularity q ≥ 2. We show that K2 X ≥ 4χ(OX) + 4(q − 2) if K2 X < 9 2 χ(OX), and also obtain the characterization of the equality. As a consequence, we prove a conjecture of Manetti on the geography of irregular surfaces if K2 X ≥ 36(q−2) or χ(OX) ≥ 8(q−2), and we also prove a conjecture that the surfaces of general type and maximal Albanese dimension with K2 X = 4χ(OX) are exactly the resolution of double covers of abelian surfaces branched over ample divisors with at worst simple singularities.

The Topology of the Milnor Fibration

2020

The fibration theorem for analytic maps near a critical point published by John Milnor in 1968 is a cornerstone in singularity theory. It has opened several research fields and given rise to a vast literature. We review in this work some of the foundational results about this subject, and give proofs of several basic “folklore theorems” which either are not in the literature, or are difficult to find. Examples of these are that if two holomorphic map-germs are isomorphic, then their Milnor fibrations are equivalent, or that the Milnor number of a complex isolated hypersurface or complete intersection singularity \((X, \underline {0})\) does not depend on the choice of functions that define …

Large time behavior for a porous medium equation in a nonhomogeneous medium with critical density

2014

Abstract We study the large time behavior of solutions to the Cauchy problem for the porous medium equation in nonhomogeneous media with critical singular density | x | − 2 ∂ t u = Δ u m , in R N × ( 0 , ∞ ) , where m > 1 and N ≥ 3 , with nonnegative initial condition u ( x , 0 ) = u 0 ( x ) ≥ 0 . The asymptotic behavior proves to have some interesting and striking properties. We show that there are different asymptotic profiles for the solutions, depending on whether the continuous initial data u 0 vanishes at x = 0 or not. Moreover, when u 0 ( 0 ) = 0 , we show the convergence towards a peak-type profile presenting a jump discontinuity, coming from an interesting asymptotic simplification…

On the influence of lower order terms for propagation of analytic singularities for operators with constant coefficients

1988

Regular and singular pulse and front solutions and possible isochronous behavior in the Extended-Reduced Ostrovsky Equation: Phase-plane, multi-infin…

2016

In this paper we employ three recent analytical approaches to investigate several classes of traveling wave solutions of the so-called extended-reduced Ostrovsky Equation (exROE). A recent extension of phase-plane analysis is first employed to show the existence of breaking kink wave solutions and smooth periodic wave (compacton) solutions. Next, smooth traveling waves are derived using a recent technique to derive convergent multi-infinite series solutions for the homoclinic orbits of the traveling-wave equations for the exROE equation. These correspond to pulse solutions respectively of the original PDEs. We perform many numerical tests in different parameter regime to pinpoint real saddl…

Singular Double Phase Problems with Convection

2020

We consider a nonlinear Dirichlet problem driven by the sum of a $p$ -Laplacian and of a $q$ -Laplacian (double phase equation). In the reaction we have the combined effects of a singular term and of a gradient dependent term (convection) which is locally defined. Using a mixture of variational and topological methods, together with suitable truncation and comparison techniques, we prove the existence of a positive smooth solution.

Energy landscape properties studied using symbolic sequences

2006

We investigate a classical lattice system with $N$ particles. The potential energy $V$ of the scalar displacements is chosen as a $\phi ^4$ on-site potential plus interactions. Its stationary points are solutions of a coupled set of nonlinear equations. Starting with Aubry's anti-continuum limit it is easy to establish a one-to-one correspondence between the stationary points of $V$ and symbolic sequences $\bm{\sigma} = (\sigma_1,...,\sigma_N)$ with $\sigma_n=+,0,-$. We prove that this correspondence remains valid for interactions with a coupling constant $\epsilon$ below a critical value $\epsilon_c$ and that it allows the use of a ''thermodynamic'' formalism to calculate statistical prope…