elliptic equations strongly degenerate at a point abstract in this paper we will mainly propose some problems for a class of degenerate elliptic equations either linear or nonlinear we will study some special cases of these problems and reveal some phenomena which may not have been noticed previously our problems originated from the self similar solutions of the heat flow of harmonic maps we will prove that the self similar solutions or the so called quasi harmonic spheres are discontinuous at infinity for the equivariant case in other words the equivariant quasi harmonic spheres are not continuous images of topological spheres

subjective diffuseness of music signals convolved with binaural impulse responses abstract the spatial impression of sound in a hall can be quantified using sound field factors such as the interaural cross correlation coefficient iacc calculated from binaural impulse response bir henceforth denoted by iaccir the subjective diffuseness for the listener is a spatial attribute which depends on factors associated both with the source signal and with the actual sound field and is quantified using the iacc of the signal received by the listener henceforth denoted by iaccsr therefore the subjective diffuseness in a given hall may change with the music the aims of this study are to estimate the iaccsr from the iaccir and the factors which is obtained from autocorrelation function acf of music signal and to evaluate the subjective diffuseness by these factors first the relationship between the iaccir and iaccsr was investigated second subjective diffuseness was measured by a psycho acoustical experiment as a result the iaccsr could be estimated from the iaccir of the bir and the effective duration e from the acf of music signal it was found that the effects of birs on subjective diffuseness could be evaluated by iaccir for almost all subjects while the effects of music signals could be evaluated by the e and the width of the peak at 0 w 0 of the acf

line transect sampling with unknown probability of detection along the transect summary a model is given for line transect sampling for the purpose of estimating animal population density which makes no assumption about the value of the detection probability along the transect or about the form of the detection probability function other than continuity in addition animals are permitted to be available or not according to a process that is independent of the sampling process the model requires sampling at two effort levels either by varying the speed of the observers and or their number the unconditional maximum likelihood estimate does not exist whereas the maximum likelihood estimate given the total numbers detected at each effort level exists but does very poorly a generalization of a moment like estimator given by cooke 1985 report of the international whaling commission 35 319 323 is proposed instead asymptotic and small sample behavior is examined the estimator is also applied to a field experiment

kernel methods in line and point transect sampling we consider kernel estimation of population density based on size and distance data in line and point transect sampling asymptotic normality of the estimators is established in each setting with very weak assumptions imposed on the random sample size it is pointed out that the kernel approach does not require specifying a horizon a severe shortcoming in the fourier series method some numerical examples based on simulated and real data are presented the results suggest that the kernel method is a viable alternative to other existing estimators

duration sensation when listening to bandpass noises abstract the duration sensation of bandpass noise was examined while changing the factors 1 1and e extracted from the autocorrelation function the white noise and bandpass noises with six center frequencies 125 250 500 1000 2000 and 4000 hz with different bandwidths were used as stimuli a paired comparison test comparing the white noise duration with bandpass noise duration was conducted under the conditions of constant sound pressure level spl 80 db a and rise and fall times 1 ms results indicate that the duration of bandpass noise is judged to be longer than that of the white noise the duration sensation of the bandpass stimuli with longer 1is significantly longer than that of the stimuli with shorter 1 p

singularity formation for the two dimensional harmonic map flow into s 2 we construct finite time blow up solutions to the 2 dimensional harmonic map flow into the sphere s 2 begin aligned u t delta u nabla u 2 u quad text in omega times 0 t u varphi quad text on partial omega times 0 t u cdot 0 u 0 quad text in omega end aligned where omega is a bounded smooth domain in mathbb r 2 u omega times 0 t rightarrow s 2 u 0 bar omega rightarrow s 2 is smooth and varphi u 0 big partial omega given any k points q 1 ldots q k in the domain we find initial and boundary data so that the solution blows up precisely at those points the profile around each point is close to an asymptotically singular scaling of a 1 corotational harmonic map we build a continuation after blow up as a h 1 weak solution with a finite number of discontinuities in space time by reverse bubbling which preserves the homotopy class of the solution after blow up furthermore we prove the codimension one stability of the one point blow up phenomenon

brownian motion and the formation of singularities in the heat flow for harmonic maps we develop a general framework for a stochastic interpretation of certain nonlinear pdes on manifolds the linear operation of takin expectations is replaced by the concept of martingale means namely the notion of deterministic starting points of martingales with respect to the levi civita connection ending up at a prescribed state we formulate a monotonicity condition for the riemannian quadratic variation of such martingales that allows us to turn smallness of the quadratic variation into a priori gradient bounds for solutions of the nonlinear heat equation such estimates lead to simple criteria for blow ups in the nonlinear heat flow for harmonic maps with small initial energy

the landis oleinik conjecture in the exterior domain abstract in 1974 landis and oleinik conjectured that if a bounded solution of a parabolic equation decays fast at a time then the solution must vanish identically before that time provided the coefficients of the equation satisfy appropriate conditions at infinity we prove this conjecture under some reasonable assumptions of the coefficients which improved the earlier results

partial regularity for weak heat flows into spheres in this paper we show that a weak heat flow of harmonic maps from a compact riemannian manifold possibly with boundary into a sphere satisfying the monotonicity inequality and the energy inequality is regular off a closed set of m dimensional hausdorff measure zero

asymptotic stability of harmonic maps under the schrodinger flow for schr odinger maps from r 2 r to the 2 sphere s 2 it is not known if finite energy solutions can form singularities blow up in finite time we consider equivariant solutions with energy near the energy of the two parameter family of equivariant harmonic maps we prove that if the topological degree of the map is at least four blowup does not occur and global solutions converge in a dispersive sense i e scatter to a fixed harmonic map as time tends to infinity the proof uses among other things a time dependent splitting of the solution the generalized hasimoto transform and strichartz dispersive estimates for a certain two space dimensional linear schr odinger equation whose potential has critical power spatial singularity and decay along the way we establish an energy space local well posedness result for which the existence time is determined by the length scale of a nearby harmonic map

long time existence for yang mills flow we establish that finite time singularities do not occur in four dimensional yang mills flow confirming the conjecture of schlatter et al am j math 120 117 128 1998 the proof relies on a weighted energy identity and sharp decay estimates in the neck region

stable blowup dynamics for the 1 corotational energy critical harmonic heat flow we exhibit a stable finite time blowup regime for the 1 corotational energy critical harmonic heat flow from 2 into a smooth compact revolution surface of 3 that reduces to the semilinear parabolic problem for a suitable class of functions f the corresponding initial data can be chosen smooth well localized and arbitrarily close to the ground state harmonic map in the energy critical topology we give sharp asymptotics on the corresponding singularity formation that occurs through the concentration of a universal bubble of energy at the speed predicted by van den berg hulshof and king our approach lies in the continuation of the study of the 1 equivariant energy critical wave map and schrodinger map with 2 target by merle raphael and rodnianski 2012 wiley periodicals inc

perception of coloration in sound fields in relation to the autocorrelation function the perception of coloration was investigated in a sound field composed of a strong primary sound and a time shifted t1 weak sound as source signal a continuous bandpass noise was used subjects judged the sensation level of the weak sound in comparison to the situation in which only the primary sound was presented the results are interpreted by means of the autocorrelation function of the bandpass noise signal

backward uniqueness for parabolic equations it is shown that a function u satisfying t u m u u u x t mem x 2 in n br 0 t and u x 0 0 for x n br must vanish identically in n br 0 t

well posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data we investigate the well posedness of 1 the heat flow of harmonic maps from mathbb r n to a compact riemannian manifold n without boundary for initial data in bmo and 2 the hydrodynamic flow u d of nematic liquid crystals on mathbb r n for initial data in bmo 1 bmo

remarks on the regularity criterion for the nematic liquid crystal flows in r 3 in this paper we investigate the regularity criteria for the 3d nematic liquid crystal flows in terms of the components of the velocity u or the gradient of the velocity field u more precisely the anisotropic integrability regularity criterion to the 3d nematic liquid crystal flows involving i u i 1 2 3 is obtained firstly and then it is proved that the solution is regular if u3 3 or 3u3 3 satisfies the prodi serrin type conditions

radial distance models for the line transect method line transect estimation of wildlife populations can be based on either right angle distances or radial distances estimation based on radial distances requires more restrictive assumptions about the mechanism by which observations are generated we examine this mechanism and show that the mean sighting angle may frequently be larger than predicted by hayne s fixed distance model or by eberhardt s variable distance model the modified and generalized hayne s models are also investigated an alternative approach based on hazard rate analysis is suggested but we conclude that a nonparametric right angle distance model will generally result in more reliable estimation than any radial distance model

heat flow of harmonic maps whose gradients belong to l n _ x l infty _ t for any compact n dimensional riemannian manifold m g without boundary a compact riemannian manifold n subset mathbb r k without boundary and 0 t we prove that for n 4 if u m 0 t n is a weak solution to the heat flow of harmonic maps such that nabla u in l n x l infty t m times 0 t then u c m 0 t n as a consequence we show that for n 3 if 0 t is the maximal time interval for the unique smooth solution u c m 0 t n of 1 1 then nabla u t l n m blows up as t t

on the evolution of harmonic maps in higher dimensions on etablit des resultats de regularite partielle et l existence de solutions regulieres globales pour le probleme d evolution pour des applications harmoniques avec des donnees petites

a regularity criterion for the solution of nematic liquid crystal flows in terms of the b 1 norm abstract in this paper we investigate the regularity criterion for the solution of nematic liquid crystal flows in three dimensions n 3 and in two dimensions n 2 we prove that the solution u d is smooth up to time t provided that there exists a positive constant e 0 0 such that i for n 3 u d l 0 t b 1 e 0 and ii for n 2 d l 0 t b 1 e 0

strong solutions of the compressible nematic liquid crystal flow abstract we study strong solutions of the simplified ericksen leslie system modeling compressible nematic liquid crystal flows in a domain r 3 we first prove the local existence of a unique strong solution provided that the initial data 0 u 0 d 0 are sufficiently regular and satisfy a natural compatibility condition the initial density function 0 may vanish on an open subset i e an initial vacuum may exist we then prove a criterion for possible breakdown of such a local strong solution at finite time in terms of blow up of the quantities l t l x and d l t 3 l x

singularities of first kind in the harmonic map and yang mills heat flows by constructing examples which are explicit up to solving an ode we prove that singularities of first kind exist for the harmonic map and yang mills heat flows as a by product we also get a simplified proof of ratto s ding s theorem about the existence of harmonic hopf constructions

regularity of solutions to the liquid crystal flows with rough initial data abstract in this paper we are concerned with the regularity of solutions to the liquid crystal flows with rough initial data in r n we prove that the solution constructed by wang 2011 in 23 has higher regularity moreover we obtain a decay estimate in time for any space derivative