sharp estimates for solutions of multi bubbles in compact riemann surfaces in this paper we consider a sequence of multibubble solutions u k of the equation 0 1 0 u k k he uk f m he u k d 1 0 in m where h is a c 2 positive function in a compact riemann surface m and k is a constant satisfying lim k k 8m for some positive integer m 1 we prove among other things that k 8m 2 m m j 1h 1 p k j 0 log h p k j 8m 2k p k j k j e k j o e k j where p k j are centers of the bubbles of u k and k j are the local maxima of u k after adding a constant this yields a uniform bound of solutions as k converges to 8m from below provided that 0 log h p k j 8m 2k p k j 0 it generalizes a previous result due to ding jost li and wang 18 and nolasco and tarantello 31 which says that any sequence of minimizers u k is uniformly bounded if k 0 for any maximum point p of the sum of 2 log h and the regular part of the green function where k is the gaussian curvature of m the analytic work of this paper is the first step toward computing the topological degree of 0 1 which was initiated by li 24

optimal detection of digital data over the nonselective rayleigh fading channel with diversity reception based on the criterion of minimum symbol error probability an analysis is made of symbol by symbol detection of a sequence of digital data transmitted using linear suppressed carrier modulation over l independent diversity channels with awgn additive white gaussian noise and slow nonselective rayleigh fading the optimal receiver is derived but is found to be difficult to implement in practice because of its exponential growth in complexity as a function of sequence length suboptimal decision feedback approximations are then suggested which are linear and readily implementable and can be integrated as generalized differentially coherent receivers the exact bit error probabilities of these suboptimal receivers are obtained tight upper bounds on these error probabilities are also obtained which show simply how they behave as a function of signal to noise ratio and order of diversity a main conclusion of this work is that optimal data detection on a fading channel should be performed using mmse minimum mean squared error estimates of the quadrature amplitudes of the channel fading processes as a coherent reference

on the relation between the standard phase field model and a thermodynamically consistent phase field model abstract a further comparison is made between the standard phase field equations t 2 1 2 g u u t 2 u 1 2 l t and the relevant thermodynamically consistent model of phase transitions proposed by the authors physica d 43 1990 44 62 here we concentrate on the usual case where g 3 and for comparison purposes retain this expression for the analogous nonlinear functions in the latter model it is brought out among other things that the standard model is thermodynamically consistent in the sense of being derivable from a free energy functional however this free energy functional is of a somewhat unusual kind it implies that at constant temperature the energy density varies linearly with the order parameter and the entropy density is a non concave function of the example of a hard sphere system indicates that such behaviour is not impossible but in most other models the energy density and the entropy are both strictly concave in

a singular sphere covering inequality uniqueness and symmetry of solutions to singular liouville type equations we derive a singular version of the sphere covering inequality which was recently introduced in gui and moradifam invent math https doi org 10 1007 s00222 018 0820 2 2018 suitable for treating singular liouville type problems with superharmonic weights as an application we deduce new uniqueness results for solutions of the singular mean field equation both on spheres and on bounded domains as well as new self contained proofs of previously known results such as the uniqueness of spherical convex polytopes first established in luo and tian proc am math soc 116 4 1119 1129 1992 furthermore we derive new symmetry results for the spherical onsager vortex equation

mean field equation of liouville type with singular data topological degree we consider the following mean field equation gv h ev mh ev 1 4 j 1n j qj 1 on m where m is a compact riemann surface with volume 1 h is a positive c1 function on m and and j are constants satisfying j 1 in this paper we derive the topological degree counting formula for noncritical values of we also give several applications of this formula including existence of the curvature 1 metric with conic singularities doubly periodic solutions of electroweak theory and a special case of self gravitating strings 2015 wiley periodicals inc

solutions to a penrose fife model of phase field type abstract in a recent paper sprekels and zheng study global existence of smooth solutions to a model of phase field type for the kinetics of phase transitions which was proposed by penrose and fife in this paper we study this model with constraint and show existence of solutions to that problem

an improved geometric inequality via vanishing moments with applications to singular liouville equations we consider a class of singular liouville equations on compact surfaces motivated by the study of electroweak and self dual chern simons theories the gaussian curvature prescription with conical singularities and onsager s description of turbulence we analyse the problem of existence variationally and show how the angular distribution of the conformal volume near the singularities may lead to improvements in the moser trudinger inequality and in turn to lower bounds on the euler lagrange functional we then discuss existence and non existence results

the conserved penrose fife system with fourier heat flux law abstract a system of phase field equations of penrose fife type governing the dynamics of phase transitions with a conserved order parameter is considered as in the original model the heat flux is assumed to be given by the fourier law existence of weak solutions is proved for the related initial neumann boundary value problem

elliptic functions green functions and the mean field equations on tori we show that the green functions on flat tori can have either 3 or 5 critical points only there does not seem to be any direct method to attack this problem instead we have to employ sophisticated non linear partial differential equations to study it we also study the distribution of number of critical points over the moduli space of flat tori through deformations the functional equations of special theta values provide important inequalities which lead to a solution for all rhombus tori the general picture is also emerged though some of the necessary technicality is still to de developed 1 introduction and statement of results the study of geometric or analytic problems on two dimensional tori is the same as the study of problems on r 2 with doubly periodic data such situations occur naturally in sciences and mathematics since early days the mathematical foundation of elliptic functions was subsequently devel oped in the 19th century it turns out that these special functions are rather deep objects by themselves tori of different shape may result in very dif ferent behavior of the elliptic functions and their associated objects arith metic on elliptic curves is perhaps the eldest and the most vivid example in this paper we show that this is also the case for certain non linear par tial differential equations indeed researches on doubly periodic problems in mathematical physics or differential equations often restrict the study to rectangular tori for simplicity this leaves the impression that the theory for general tori may resemble much the same way as for the rectangular case however this turns out to be false we will show that the solvability of themean field equation dependson the shape of the green function which in turn depends on the geometry of the tori in an essential way recall that the green function g z w on a flat torus t c z 1 z 2 is the unique function on t t which satisfies zg z w w z 1 t

the conserved penrose fife system with temperature dependent memory abstract a nonlinear parabolic system of penrose fife type with a singular evolution term arising from modelling dynamic phenomena of the nonisothermal diffusive phase separation is studied here we consider the evolution of a material in which the heat flux is a superposition of two different contributions one part is proportional to the spacial gradient of the inverse of the absolute temperature while the other agrees with the gurtin pipkin law introduced in the theory of materials with thermal memory the phase transition here is described through the evolution of the conserved order parameter which may represent the density or concentration of some substance it is shown that an initial boundary value problem for the resulting state equations has a unique solution

sharp sobolev inequalities on the sphere and the moser trudinger inequality where de denotes normalized surface measure v is the conformal gradient and q 2n n 2 a modern folklore theorem is that by taking the infinitedimensional limit of this inequality one obtains the gross logarithmic sobolev inequality for gaussian measure which determines nelson s hypercontractive estimates for the hermite semigroup see 8 one observes using conformal invariance that the above inequality is equivalent to the sharp sobolev inequality on rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization the roots here go back to hardy and littlewood the advantage of casting the problem on the sphere is that the role of the constants is evident and one is led immediately to the conjecture that this inequality should hold whenever possible for example 2 q 0o if n 2 this is in fact true and will be demonstrated in section 2 a clear question at this point is what is the situation in dimension 2 two important arguments 25 26 27 dealt with this issue both motivated by geometric variational problems because q goes to infinity for dimension 2 the appropriate function space is the exponential class responding in part

thermodynamically consistent models of phase field type for the kinetics of phase transitions abstract a general framework is given for the phenomenological kinetics of phase transitions in which not only the order parameter but also the temperature may vary in time and space instead of a ginzburg landau free energy functional as used in formulating the cahn hilliard equation we use the analogous entropy functional model entropy functionals and the kinetic equations resulting from them are constructed for various cases phase transitions with and without a critical point and in the former case with or without a latent heat the class considered is general enough to include the entropy functionals for the ising model in mean field approximation the van der waals fluid and a simplified version of the density functional theory of freezing a case without critical point for which the energy is conserved but the order parameter is not provides a thermodynamically consistent derivation of the phase field equations studied by caginalp fix and others and also leads in a natural way to the lyapunov functional given by langer for these equations but the treatment also suggests that a modified version of the phase field equations might provide a more realistic model of freezing

generalized quadratic receivers for orthogonal signals over the gaussian channel with unknown phase fading the orthogonal signal structure has been shown to be the superposition of an antipodal signal set and an unmodulated pilot tone component which can be used for channel measurement starting from this point of view the quadratic receiver for orthogonal signals over the gaussian channel with unknown phase fading has been shown to be equivalent to a detector estimator receiver the estimator makes an optimum estimate of the unknown complex channel gain based on the channel measurement provided by the unmodulated component of the received signal this channel estimate then forms a partially coherent reference for the detector in detecting the data carried by the antipodal signaling component of the received signal this paper exploits this detector estimator structure of the quadratic receiver and generalizes it to a receiver in which the estimator makes an estimate of the channel gain in each signaling interval based on the totality of signals received over all the signaling intervals or a subset of these intervals the generalized quadratic receiver is just as simple to implement as the conventional quadratic receiver and theoretical and simulation results show that it can achieve substantial performance gains over the conventional receiver a theory is presented to show that the generalized quadratic receiver is an implementable approximation to the optimum symbol by symbol receiver for uncoded orthogonal signals over the gaussian channel with unknown phase fading the theory shows that the structure provides a unified and systematic approach to the design of coherent symbol by symbol receivers and shows that the conventional carrier loop type receivers are ad hoc

existence and uniqueness for mean field equations on multiply connected domains at the critical parameter we consider the mean field equation begin aligned left begin array l quad l vardelta u rho frac e u int varomega e u 0 hbox in varomega u 0 hbox on partial varomega end array right end aligned 1 where varomega subset mathbb r 2 is an open and bounded domain of class c 1 in his 1992 paper suzuki proved that if varomega is a simply connected domain then eq 1 admits a unique solution for rho in 0 8 pi this result for varomega a simply connected domain has been extended to the case rho 8 pi by chang chen and the second author however the uniqueness result for varomega a multiply connected domain has remained a long standing open problem which we solve positively here for rho in 0 8 pi to obtain this result we need a new version of the classical bol s inequality suitable to be applied on multiply connected domains our second main concern is the existence of solutions for 1 when rho 8 pi we obtain a necessary and sufficient condition for the solvability of the mean field equation at rho 8 pi which is expressed in terms of the robin s function gamma for varomega for example if eq 1 has no solution at rho 8 pi then gamma has a unique nondegenerate maximum point as a by product of our results we solve the long standing open problem of the equivalence of canonical and microcanonical ensembles in the onsager s statistical description of two dimensional turbulence on multiply connected domains

non degeneracy and uniqueness of solutions to singular mean field equations on bounded domains abstract the aim of this paper is to complete the program initiated in 51 23 and then carried out by several authors concerning non degeneracy and uniqueness of solutions to mean field equations in particular we consider mean field equations with general singular data on non smooth domains the argument is based on the alexandrov bol inequality and on the eigenvalues analysis of linearized singular liouville type problems

conformal deformation of metrics on s 2 soit d un domaine lisse de s 2 et k une fonction lisse definie sur d alors pour que le probleme aux limites de neumann u k exp 2u 1 sur d u n 0 sur d ait une solution u on a a quand aire d max x d k x 0

a special class of stationary flows for two dimensional euler equations a statistical mechanics description we consider the canonical gibbs measure associated to an vortex system in a bounded domain at inverse temperature widetilde beta and prove that in the limitn widetilde beta n n 1 where 8 here denotes the vorticity intensity of each vortex the one particle distribution function n nx x converges to a superposition of solutions of the following mean field equation left begin array 20 c varrho beta x frac e beta psi mathop smallint limits lambda e beta psi delta psi varrho beta in lambda psi partial lambda 0 end array right

asymptotic blow up analysis for singular liouville type equations with applications abstract we generalize the pointwise estimates obtained in 2 19 and 34 concerning blow up solutions of the liouville type equation u x 2 w x e u in with r 2 open and bounded 1 and w any lipschitz continuous function which satisfies 0 a w b we focus to the case left open in 2 and 34 where the parameter n whose analysis is much more involved as we need to resolve the difficulty of a genuinely non radial behaviour of blow up solutions in the worst situation there is no chance in general to resolve the profile in the form of a solution of a liouville equation in r 2 instead we need to adopt iterated blow up arguments next we refine our blow up analysis to cover a class of planar liouville type problems see 1 27 1 28 below arising from the study of cosmic strings cfr 28 35 in this context we are able to distinguish between a single blow up radial profile and the case of multiple blow up profiles typical of non radial solutions as a consequence we obtain a radial symmetry result which is interesting in itself but also contributes towards the sharp solvability issue for the planar problem 1 27 1 28

on a class of elliptic problems in r symmetry and uniqueness results in the plane r 2 we classify all solutions for an elliptic problem of liouville type involving a radial weight function as a consequence we clarify the origin of the non radially symmetric solutions for the given problem as established by chanillo and kiessling for a more general class of liouville type problems we show that rather than radial symmetry the solutions always inherit the invariance of the problem under inversion with respect to suitable circles this symmetry result is derived with the help of a shrinking sphere method

generic multiplicity for a scalar field equation on compact surfaces we prove generic multiplicity of solutions for a scalar field equation on compact surfaces via morse inequalities in particular our result improves significantly the multiplicity estimate which can be deduced from the degree counting formula in chen and lin 2003 12 related results are derived for the prescribed q curvature equation

supercritical mean field equations on convex domains and the onsager s statistical description of two dimensional turbulence we are motivated by the study of the microcanonical variational principle within onsager s description of two dimensional turbulence in the range of energies where the equivalence of statistical ensembles fails we obtain sufficient conditions for the existence and multiplicity of solutions for the corresponding mean field equation on convex and thin enough domains in the supercritical with respect to the moser trudinger inequality regime this is a brand new achievement since existence results in the supercritical region were previously known only on multiply connected domains we then study the structure of these solutions by the analysis of their linearized problems and we also obtain a new uniqueness result for solutions of the mean field equation on thin domains whose energy is uniformly bounded from above finally we evaluate the asymptotic expansion of those solutions with respect to the thinning parameter and combining it with all the results obtained so far we solve the microcanonical variational principle in a small range of supercritical energies where the entropy is shown to be concave

uniqueness and symmetry results for solutions of a mean field equation on 2 via a new bubbling phenomenon motivated by the study of gauge field vortices we consider a mean field equation on the standard two sphere involving a dirac distribution supported at a point p of s 2 as needed for the applications we show that solutions concentrate exactly at p for some limiting value of a given parameter we use this fact to obtain symmetry about the axis op and uniqueness property for the solution the presence of the dirac measure makes such a task particularly delicate indeed we need to rule out the possibility that after blow up in a suitable scale the solution sequence may admit a double peak profile described in terms of appropriate limiting problems for instance we need to account for the presence of non axially symmetric solutions in such limiting problems in this process we establish a symmetry result about a maximal circle through p and its antipodal point p that applies to more general situations where the full axial symmetry cannot be expected

non linear system for non isothermal diffusive phase separation abstract an evolution system which extends the cahn hilliard model to non isothermal situations subject to convex constraints for solutions is studied the problem can be considered as representing a simple model of non isothermal phase separation driven by diffusive mechanisms a variational formulation of the problem is introduced using t dependent subdifferential operator techniques in hilbert spaces existence and uniqueness of weak solutions are proved

weak solutions to a penrose fife model with fourier law for the temperature abstract we prove global existence in time of weak solutions to the penrose fife model when the heat flux is given by the fourier law and the homogeneous free energy density by the classical double well potential the main feature of these weak solutions is the almost everywhere positivity of the absolute temperature throughout time evolution

binary orthogonal signaling over the gaussian channel with unknown phase fading new results and interpretations the problem of binary orthogonal signaling over a gaussian noise channel with unknown phase fading is considered by viewing the problem in a rotated coordinate system the orthogonal signal structure is considered as the combination of an antipodal signal set and a pilot tone for channel measurement for data detection the optimum matched filter envelope detector is shown to be identical to a novel detector estimator receiver in which the detector performs partially coherent detection using an absolute coherent reference generated by the estimator from the channel measurement provided by the pilot tone component of the orthogonal signal structure this detector estimator interpretation shows that it is incorrect to refer to the optimum receiver as a noncoherent receiver it also leads to the development of new approaches for analyzing the error probability of the receiver an exponential chernoff upper bound is obtained for the rician channel

on steady distributions of self attracting clusters under friction and fluctuations we consider the vlasov fokker planek equation with a newtonian attracting potential and study its stationary solutions given by the generalized lane emden equation in a two dimensional domain we obtain the existence of a critical mass beyond which the system may admit a gravitational collapse for a one dimensional model we prove some results on existence uniqueness stability and symmetry breaking of stationary solutions

weighted barycentric sets and singular liouville equations on compact surfaces given a closed surface we prove a general existence result for some elliptic pde with exponential nonlinearities and negative dirac deltas extending a theory recently obtained for the regular case this is done by global methods since the associated euler functional might be unbounded from below we define a new model space generalizing the so called space of formal barycenters and characterizing up to homotopy equivalence its low sublevels as a result the analytic problem is reduced to a topological one concerning the contractibility of this model space to this aim we prove a new functional inequality in the spirit of chen and li 1991 11 and then employ a min max scheme based on conical construction jointly with the blow up analysis in bartolucci and montefusco 2007 4 after bartolucci and tarantello 2002 brezis and merle 1991 and this study is motivated by abelian chern simons theory in self dual regime or from the problem of prescribing the gaussian curvature with conical singularities generalizing a problem raised in kazdan and warner 1974 24

on the best constant in the moser onofri aubin inequality let s2 be the 2 dimensional unit sphere and let j denote the nonlinear functional on the sobolev space h1 s2 defined by j alpha u frac alpha 16 pi int s 2 nabla u 2 d mu 0 frac 1 4 pi int s 2 u d mu 0 rm ln int s 2 e u frac d mu 0 4 pi where d mu 0 sin theta d theta wedge d phi onofri had established that j is non negative on h1 s2 provided 1 in this note we show that if j is restricted to those u in h 1 s 2 that satisfies the aubin condition int s 2 e u x j d mu 0 0 quad rm for all 1 leq j leq 3 then the same inequality continues to hold i e j u 0 whenever alpha geq frac 2 3 epsilon 0 for some epsilon 0 0 the question of chang yang on whether this remains true for all alpha geq frac 1 2 remains open

generalized quadratic receivers for unitary space time modulation over rayleigh fading channels we propose the generalized quadratic receivers gqrs for unitary space time modulation over flat rayleigh fading channels the gqrs realize the performance improvement potential known to be approximately 2 4 db between the quadratic receiver qr and the coherent receiver cr by performing channel estimation without the help of additional training signals that consume additional bandwidth they are designed for various unitary space time constellations ustc in which signal matrices may or may not contain explicit inherent training blocks and may be orthogonal or nonorthogonal to one another as the channel memory span exploited for channel estimation increases the error probability of the gqrs reduces from that of the qr to that of the cr the gqrs work well for both slow and fast fading channels and the performance improvement increases as the channel fade rate decreases for a class of ustc with the orthogonal design structure the gqr is simplified to a form whose complexity can be less than the complexity of the qr or even that of the simplified form of the qr

maximum likelihood carrier phase recovery for coherently orthogonal cpfsk signals ml estimation of carrier phase for coherently orthogonal continuous phase frequency shift keying cocpfsk signals is considered although the estimator in general is nonimplementable its high and low signal to noise ratio approximations both lead to linear readily implementable receiver structures the high snr approximation yields a da receiver whereas the low snr approximation yields an nda receiver the performance of both receivers in term of bit error probability is analyzed the existence of an unmodulated component in the sufficient statistical representation of a cocpfsk signal is pointed out and it is shown how this component enters directly into maximum like carrier recovery this leads to interpretation of the nda receiver as a generalization of the conventional matched filter envelope detector receiver the insights gained here are useful to the problem of ml carrier recovery for viterbi decoding of continuous phase modulation signals

on a planar liouville type problem in the study of selfgravitating strings abstract motivated by the construction of selfgravitating strings cf yang 2001 1994 22 23 we analyze a liouville type equation on the plane derived in yang 1994 23 we establish sharp existence and uniqueness properties for the corresponding radial solutions we investigate also when the problem allows for non radial solutions

topological degree for a mean field equation on riemann surfaces we consider the following mean field equations 0 1 where m is a compact riemann surface with volume 1 h is a positive continuous function on m is a real number 0 2 and where is a bounded smooth domain h is a c1 positive function on and based on our previous analytic work 14 we prove among other things that the degree counting formula for 0 1 is given by for 8m 8 m 1 2003 wiley periodicals inc

weak solutions to the penrose fife phase field model for a class of admissible heat flux laws abstract this paper is concerned with a thermodynamically consistent model for diffusive phase transitions proposed by penrose and fife the model has recently received a good deal of attention and the related initial boundary value problems have been investigated from the mathematical point of view in the case where the order parameter is not conserved the common working assumption for the heat flow was that such a flux is proportional to the gradient of the inverse absolute temperature this position seems to be helpful for the analysis due to the coupling term in the phase field equation which depends right on the inverse temperature here we prove the existence of weak solutions in a wide setting of constitutive laws for the heat flux different from the previous ones and considerably more significant owing to their behaviour for high temperatures we also discuss the limiting situation of zero interfacial energy for the order parameter

uniqueness of solutions for a mean field equation on torus abstract we consider on a two dimensional flat torus t the following equation u e u t e u 1 t 0 when the fundamental domain of the torus is 0 a 0 b a b we establish that the constants are the unique solutions whenever 8 if b a 4 32 b a if b a 4 and this result is sharp if b a 4 a similar conclusion is obtained for general two dimensional torus by considering the length of the shortest closed geodesic these results are derived by comparing the isoperimetric profile of the torus t with the one of the two dimensional canonical sphere which has same volume as t

prescribing curvature on compact surfaces with conical singularities we study the berger nirenberg problem on surfaces with conical singularities i e we discuss conditions under which a function on a riemann surface is the gaussian curvature of some conformal metric with a prescribed set of singularities of conical types

a dynamic programming algorithm for simultaneous phase estimation and data decoding on random phase channels the problem of simultaneously estimating phase and decoding data symbols from baseband data is posed the phase sequence is assumed to be a random sequence on the circle and the symbols are assumed to be equally likely symbols transmitted over a perfectly equalized channel a dynamic programming algorithm viterbi algorithm is derived for decoding a maximum em a posteriori map phase symbol sequence on a finite dimensional phase symbol trellis a new and interesting principle of optimality for simultaneously estimating phase and decoding phase amplitude coded symbols leads to an efficient two step decoding procedure for decoding phase symbol sequences simulation results for binary 8 ary phase shift keyed psk and 16 quadrature amplitude shift keyed qask symbol sets transmitted over random walk and sinusoidal jitter channels are presented and compared with results one may obtain with a decision directed algorithm or with the binary viterbi algorithm introduced by ungerboeck when phase fluctuations are severe and when occasional large phase fluctuations exist map phase symbol sequence decoding on circles is superior to ungerboeck s technique which in turn is superior to decision directed techniques