analysis of d2d underlaid cellular networks sir meta distribution and mean local delay we study the performance of device to device d2d communication underlaying cellular wireless network in terms of the meta distribution of the signal to interference ratio sir which is the distribution of the conditional sir distribution given the locations of the wireless nodes modeling d2d transmitters and base stations as poisson point processes ppps moments of the conditional sir distribution are derived in order to calculate analytical expressions for the meta distribution and the mean local delay of the typical d2d receiver and cellular downlink user it turns out that for d2d users the total interference from the d2d interferers and base stations is equal in distribution to that of a single ppp while for downlink users the effect of the interference from the d2d network is more complicated we also derive the region of transmit probabilities for the d2d users and base stations that result in a finite mean local delay and give a simple inner bound on that region finally the impact of increasing the base station density on the mean local delay the meta distribution and the density of users reliably served is investigated with numerical results

abstract volterra integrodifferential equations with applications to parabolic models with memory in this paper we are concerned with local existence regularity and continuous dependence upon the initial data of epsilon regular mild solutions for the abstract integrodifferential equation 1 2 we also present a result on unique continuation and a blow up alternative for an epsilon regular mild solution of 1 2 finally we apply our results to three interesting models navier stokes equations with memory diffusion equations with memory and a strongly damped plate equation with memory

a theorem of real analysis and its application to free boundary problems in this note we show how comparison theorems for solutions to equations with non smooth coefficients yield in a very simple fashion some free boundary regularity theorems in cases where there is a priori geometrical information for instance when one knows that the solution u is monotone in some given direction

gaussian bounds for degenerate parabolic equations abstract let a be a real symmetric degenerate elliptic matrix whose degeneracy is controlled by a weight w in the a 2 or qc class we show that there is a heat kernel w t x y associated to the parabolic equation w u t div a u and w t satisfies classic gaussian bounds w t x y c 1 t n 2 exp c 2 x y 2 t we then use this bound to derive a number of other properties of the kernel

downlink cellular network analysis with multi slope path loss models existing cellular network analyses and even simulations typically use the standard path loss model where received power decays like vert x vert alpha over a distance vert x vert this standard path loss model is quite idealized and in most scenarios the path loss exponent alpha is itself a function of vert x vert typically an increasing one enforcing a single path loss exponent can lead to orders of magnitude differences in average received and interference powers versus the true values in this paper we study multi slope path loss models where different distance ranges are subject to different path loss exponents we focus on the dual slope path loss function which is a piece wise power law and continuous and accurately approximates many practical scenarios we derive the distributions of sir snr and finally sinr before finding the potential throughput scaling which provides insight on the observed cell splitting rate gain the exact mathematical results show that the sir monotonically decreases with network density while the converse is true for snr and thus the network coverage probability in terms of sinr is maximized at some finite density with ultra densification network density goes to infinity there exists a phase transition in the near field path loss exponent alpha 0 if alpha 0 1 unbounded potential throughput can be achieved asymptotically if alpha 0 ultra densification leads in the extreme case to zero throughput

harnack inequality and exponential separation for oblique derivative problems on lipschitz domains abstract we consider the oblique derivative problem for linear nonautonomous second order parabolic equations with bounded measurable coefficients on bounded lipschitz cylinders we derive an optimal elliptic type harnack inequality for positive solutions of this problem and use it to show that each positive solution exponentially dominates any solution which changes sign for all times we show several nontrivial applications of both the exponential estimate and the derived harnack inequality

decay estimates for time fractional and other non local in time subdiffusion equations in mathbb r d we prove optimal estimates for the decay in time of solutions to a rather general class of non local in time subdiffusion equations in mathbb r d an important special case is the time fractional diffusion equation which has seen much interest during the last years mostly due to its applications in the modeling of anomalous diffusion processes we follow three different approaches and techniques to study this particular case a estimates based on the fundamental solution and young s inequality b fourier multiplier methods and c the energy method it turns out that the decay behaviour is markedly different from the heat equation case in particular there occurs a critical dimension phenomenon the general subdiffusion case is treated by method b and relies on a careful estimation of the underlying relaxation function several examples of kernels including the ultraslow diffusion case illustrate our results

a carleson type inequality for fully nonlinear elliptic equations with non lipschitz drift term abstract this paper concerns the boundary behavior of solutions of certain fully nonlinear equations with a general drift term we elaborate on the non homogeneous generalized harnack inequality proved by the second author in 26 to prove a generalized carleson estimate we also prove boundary holder continuity and a boundary harnack type inequality

closed form hop count distributions in random networks with arbitrary routing we contribute a new solution to the problem of establishing an analytical relationship between hop counts under a certain routing policy and euclidean distances in random networks both in the linear and planar cases the contributed solution is unified in that hop count distributions have similar expressions both in the 1d and the 2d cases general in terms of routing policies in that the effect of any given policy is accounted for by means of a single parameter closed form such that hop count probability mass functions pmf s are given in terms of scaled versions of the closed form pmf s of the number of nodes and mathematically tractable since the derived hop count distributions are in the form of a difference of the well known nakagami m cumulative density functions cdf s direct and kullback leibler divergence comparisons against empirical data demonstrate the high accuracy of our solution the simplicity accuracy and generality of the result owes partly to a self imposed confinement to connected networks defined formally in stochastic geometric terms which allows for the elimination of recursions and multivariate marginalization commonly required by existing solutions the contributed results find application in the design and analysis of ad hoc networks cooperative localization algorithms or latency and energy consumption analysis

sobolev interpolation inequalities with weights we study weighted local sobolev interpolation inequalities of the form 1 r lu x i ph ui 2 x dx s 1 r lu x ip ii x dx h 1 1172 b j bl b j b x ibi p n r ly u x ipw l x dx 1 r lu x ipu x dx wl b jb v b jb where i i b is a ball in r n and 11 111 1 and 1112 are weight functions the case p 2 is of special importance in deriv ing regularity results for solutions of degenerate parabolic equations we also study the analogous inequality without the second summand on the right in the case u has compact support in b and we derive global landau inequalities iiy uiilq s cllull ally 2ull p 0 a i i p s q 00 when u has ii compact support

a fine grained analysis of millimeter wave device to device networks enabling device to device d2d communications in millimeter wave mm wave networks is of critical importance for the next generation mobile networks to support very high data rates multi gigabits per second for mobile devices in this paper we provide a fine grained performance analysis of the mm wave d2d communication networks specifically we first establish a general and tractable framework to investigate the performance of mm wave d2d networks using the poisson bipolar model integrated with several features of the mm wave band to show what fraction of users in the network achieve target reliability if the required signal to interference plus noise ratio sinr or qos requirement is given we derive the meta distributions of the sinr and the data rate interestingly in mm wave d2d networks the standard beta approximation for the meta distribution does not work very well when highly directional antenna arrays are used or the node density is small to resolve this issue we provide a modified approximation by using higher moments of the conditional sinr distribution which is shown to be closer to the exact result on this basis we also derive the mean local delay and spatial outage capacity to provide a comprehensive investigation on the impact of mm wave features on the performance of d2d communication

regularity and stability analysis for a class of semilinear nonlocal differential equations in hilbert spaces abstract we deal with a class of semilinear nonlocal differential equations in hilbert spaces which is a general model for some anomalous diffusion equations by using the theory of integral equations with completely positive kernel together with local estimates some existence regularity and stability results are established an application to nonlocal partial differential equations is shown to demonstrate our abstract results

a geometric interpretation of fading in wireless networks theory and applications in wireless networks with random node distribution the underlying point process model and the channel fading process are usually considered separately a unified framework is introduced that permits the geometric characterization of fading by incorporating the fading process into the point process model concretely assuming nodes are distributed in a stationary poisson point process in rd the properties of the point processes that describe the path loss with fading are analyzed the main applications are single hop connectivity and broadcasting

analysis and computation of a discrete costly observation model for growth estimation and management of biological resources abstract early estimation of biological growth of organisms is an indispensable task in ecology and related research areas the biological growth is always time continuous while our observations of the phenomenon are time discrete in practice the formalism of the discrete costly observation dco enables us to mathematically bridge the two qualitatively different processes this formalism is still germinating and its practical applications have not been carried out this paper presents a first application of the dco formalism to a cost effective early estimation problem of the biological growth and its mathematical and numerical analysis growth dynamics of organisms which are fishery resources in this paper is governed by a stochastic differential equation whose solution is observed discretely the optimality equation to be solved for finding the most cost effective observation policy is derived as a fixed point problem based on degenerate parabolic partial differential equations the fixed point problem turns out to be uniquely solvable a recursive approximation of the fixed point problem is presented and its solvability in a viscosity sense is discussed a finite different scheme is then employed to fully discretize the recursive equations the present model is finally applied to a problem of japanese smelt plecoglossus altivelis altivelis ayu an important inland fishery resource in japan

regularity theory for a nonlinear fractional reaction diffusion equation abstract this paper is dedicated to the study of a nonlinear fractional reaction diffusion equation we analyze the behavior of the resolvent family associated with the problem in the scale of fractional power spaces associated with the laplace operator we ensure the existence and uniqueness of regular mild solutions to the problem in the l q setting furthermore we consider global existence or non continuation by a blow up of such solutions

spectral efficiency scaling laws in dense random wireless networks with multiple receive antennas this paper considers large random wireless networks where transmit and receive node pairs communicate within a certain range while sharing a common spectrum by modeling the spatial locations of nodes as a poisson point process analytical expressions for the ergodic spectral efficiency of a typical node pair are derived as a function of the channel state information available at a receiver csir in terms of relevant system parameters the density of communication links the number of receive antennas the path loss exponent and the operating signal to noise ratio one key finding is that when the receiver only exploits csir for the direct link the sum spectral efficiency increases linearly with the density provided the number of receive antennas increases as a certain superlinear function of the density when each receiver exploits csir for a set of dominant interfering links in addition to that of the direct link the sum spectral efficiency increases linearly with both the density and the path loss exponent if the number of antennas is a linear function of the density this observation demonstrates that having csir for dominant interfering links provides an order gain in the scaling law it is also shown that this linear scaling holds for direct csir when incorporating the effect of the receive antenna correlation provided that the rank of the spatial correlation matrix scales superlinearly with the density these scaling laws are derived from integral representations of the distribution of the signal to interference and noise ratio which are of independent interest and which in turn derived from stochastic geometry and more precisely from the theory of shot noise fields simulation results back the scaling laws and the integral representations

design modeling and performance analysis of multi antenna heterogeneous cellular networks this paper presents a stochastic geometry based framework for the design and analysis of downlink multi user multiple input multiple output mimo heterogeneous cellular networks with linear zero forcing transmit precoding and receive combining assuming rayleigh fading channels and perfect channel state information the generalized tiers of base stations may differ in terms of their poisson point process spatial density number of transmit antennas transmit power artificial biasing weight and number of user equipments served per resource block the spectral efficiency of a typical user equipped with multiple receive antennas is characterized using a non direct moment generating function based methodology with closed form expressions of the useful received signal and aggregate network interference statistics systematically derived in addition the area spectral efficiency is formulated under different space division multiple access and single user beamforming transmission schemes we examine the impact of different cellular network deployments propagation conditions antenna configurations and mimo setups on the achievable performance through theoretical and simulation studies based on the state of the art system parameters the results highlight the inherent limitations of baseline single input single output transmission and conventional sparse macro cell deployment as well as the promising potential of multi antenna communications and small cell solution in interference limited cellular environments

generalized labeled multi bernoulli approximation of multi object densities in multiobject inference the multiobject probability density captures the uncertainty in the number and the states of the objects as well as the statistical dependence between the objects exact computation of the multiobject density is generally intractable and tractable implementations usually require statistical independence assumptions between objects in this paper we propose a tractable multiobject density approximation that can capture statistical dependence between objects in particular we derive a tractable generalized labeled multi bernoulli glmb density that matches the cardinality distribution and the first moment of the labeled multiobject distribution of interest it is also shown that the proposed approximation minimizes the kullback leibler divergence over a special tractable class of glmb densities based on the proposed glmb approximation we further demonstrate a tractable multiobject tracking algorithm for generic measurement models simulation results for a multiobject track before detect example using radar measurements in low signal to noise ratio snr scenarios verify the applicability of the proposed approach

behavior near the boundary of positive solutions of second order parabolic equations ii a boundary backward harnack inequality is proved for positive solutions of second order parabolic equations in non divergence form in a bounded cylinder q q x 0 t which vanish on o q oq x 0 t where q is a bounded lipschitz domain in r this inequality is applied to the proof of the holder continuity of the quotient of two positive solutions vanishirlg on a portion of o q

mean internodal distance in regular and random multihop networks the minimum necessary aggregate link capacity in a telecommunication network is directly proportional to the mean distance between nodes the mean internodal distance is therefore an important network characteristic it is shown that most network topologies including those constructed at random display mean internodal distances comparable to those of many carefully designed networks thus careful selection of network topology to minimize the mean internodal distance may be important in only the most sensitive applications furthermore even in such sensitive applications an almost randomly chosen network topology may be the best choice

boundary behavior and the martin boundary problem for p harmonic functions in lipschitz domains boundary behavior and the martin boundary problem for p harmonic functions in lipschitz domains

the asymptotics of the solutions to the anomalous diffusion equations in this paper we study the asymptotics of the solutions to the anomalous diffusion equations the fractional anomalous diffusion equations are obtained from the existing anomalous diffusion and typical diffusion equations by replacing the first order time derivative with fractional derivatives of order email protected 0 1 for the sub diffusion and email protected 1 2 for the super diffusion respectively in most situations fractional derivatives mean riemann liouville derivative or caputo derivative in this paper we use these two kinds of fractional derivatives using laplace transform and fourier transform we obtain the asymptotics estimates of solutions to the anomalous diffusion equations

simple approximations of the sir meta distribution in general cellular networks compared to the standard success coverage probability the meta distribution of the signal to interference ratio sir provides much more fine grained information about the network performance we consider general heterogeneous cellular networks hcns with base station tiers modeled by arbitrary stationary and ergodic non poisson point processes the exact analysis of non poisson network models is notoriously difficult even in terms of the standard success probability let alone the meta distribution hence we propose a simple approach to approximate the sir meta distribution for non poisson networks based on the asappp approximate sir analysis based on the poisson point process method we prove that the asymptotic horizontal gap g 0 between its standard success probability and that for the poisson point process exactly characterizes the gap between the b th moment of the conditional success probability as the sir threshold goes to 0 the gap g 0 allows two simple approximations of the meta distribution for general hcns 1 the per tier approximation by applying the shift g 0 to each tier and 2 the effective gain approximation by directly shifting the meta distribution for the homogeneous independent poisson network given the generality of the model considered and the fine grained nature of the meta distribution these approximations work surprisingly well

on distances in uniformly random networks the distribution of euclidean distances in poisson point processes is determined the main result is the density function of the distance to the n nearest neighbor of a homogeneous process in ropfm which is shown to be governed by a generalized gamma distribution the result has many implications for large wireless networks of randomly distributed nodes

distance statistics of the communication best neighbor in a poisson field of nodes in a wireless network while the signal attenuation due to the distance related path loss is minimum between a node and its nearest neighbor this is not the case for the overall attenuation when the fading effect is also taken into account hence the communication best neighbor cbn of a node is defined as the neighbor of that node with which it has the highest channel power gain recently the statistics of the physical neighborhood index of the cbn have been derived for a general fading environment in which the nodes have two dimensional poisson distribution in this paper the statistics of the distance between a node and its cbn as well as the joint cbn distance index statistics are obtained for the same scenario then the analytical expressions are specialized for the case where the communication channels follow the generalized gamma fading model also the results are verified and illustrated by computer simulations and numerical results further it is shown how the obtained analytical results can be used to identify the cbn more efficiently by limiting the search region and the number of neighbors that are examined

boundedness of weak solutions to evolutionary partial integro differential equations with discontinuous coefficients we investigate linear and quasilinear evolutionary partial integro differential equations of second order which include time fractional evolution equations of time order less than one by means of suitable energy estimates and de giorgi s iteration technique we establish results asserting the global boundedness of appropriately defined weak solutions of these problems we also show that a maximum principle holds for such equations

fractional cauchy problems on bounded domains fractional cauchy problems replace the usual first order time derivative by a fractional derivative this paper develops classical solutions and stochastic analogues for fractional cauchy problems in a bounded domain dr d with dirichlet boundary conditions stochastic solutions are constructed via an inverse stable subordi nator whose scaling index corresponds to the order of the fractional time derivative dirichlet problems corresponding to iterated brow nian motion in a bounded domain are then solved by establishing a correspondence with the case of a half derivative in time 1 introduction in this paper we extend the approach of meerschaert and scheffler 23 and meerschaert et al 24 to fractional cauchy problems on bounded domains our methods involve eigenfunction expansions killed markov processes and inverse stable subordinators in a recent related paper 7 we establish a connection between fractional cauchy problems with index 1 2 on an unbounded domain and iterated brownian motion ibm defined as zt b yt where b is a brownian motion with values in r d and y is an independent one dimensional brownian motion since ibm is also the stochastic solution to a cauchy problem involving a fourth order derivative in space 2 14 that paper also establishes a connection between certain higher order cauchy problems and their time fractional analogues more generally baeumer meerschaert and nane 7 shows a connection between fractional cauchy problems with 1 2 and higher order cauchy problems that involve the square of the generator in the present paper we

on some impulse control problems with long run average cost some particular impulse control problems with infinite horizon and long run average cost are considered for markov processes having nice ergodicity properties the main feature of the method is to start with discounted cost and then let the discount factor go to zero this also gives an opportunity to study the asymptotic behavior of some optimal stopping time problems when the discount factor goes to zero probabilistic and analytical methods are used and examples are given especially for markov jump process and diffusion processes with reflection

cauchy problem for fractional diffusion equations abstract we consider an evolution equation with the regularized fractional derivative of an order 0 1 with respect to the time variable and a uniformly elliptic operator with variable coefficients acting in the spatial variables such equations describe diffusion on inhomogeneous fractals a fundamental solution of the cauchy problem is constructed and investigated

asymptotic deployment gain a simple approach to characterize the sinr distribution in general cellular networks in cellular network models the base stations are usually assumed to form a lattice or a poisson point process ppp in reality however they are deployed neither fully regularly nor completely randomly accordingly in this paper we consider the very general class of motion invariant models and analyze the behavior of the outage probability the probability that the signal to interference plus noise ratio sinr is smaller than a threshold as the threshold goes to zero we show that remarkably the slope of the outage probability in db as a function of the threshold also in db is the same for essentially all motion invariant point processes the slope merely depends on the fading statistics using this result we introduce the notion of the asymptotic deployment gain adg which characterizes the horizontal gap between the success probabilities of the ppp and another point process in the high reliability regime where the success probability is near 1 to demonstrate the usefulness of the adg for the characterization of the sinr distribution we investigate the outage probabilities and the adgs for different point processes and fading statistics by simulations

regularity estimates for the solution and the free boundary of the obstacle problem for the fractional laplacian we use a characterization of the fractional laplacian as a dirichlet to neumann operator for an appropriate differential equation to study its obstacle problem we write an equivalent characterization as a thin obstacle problem in this way we are able to apply local type arguments to obtain sharp regularity estimates for the solution and study the regularity of the free boundary

non divergence parabolic equations of second order with critical drift in lebesgue spaces abstract we consider uniformly parabolic equations and inequalities of second order in the non divergence form with drift u t l u u t i j a i j d i j u b i d i u 0 0 0 in some domain q r n 1 we prove growth theorems and the interior harnack inequality as the main results in this paper we will only assume the drift b is in certain lebesgue spaces which are critical under the parabolic scaling but not necessarily to be bounded in the last section some applications of the interior harnack inequality are presented in particular we show there is a universal spectral gap for the associated elliptic operator the counterpart for uniformly elliptic equations of second order in non divergence form is shown in 19

on an inverse type problem for the heat equation in parabolic regular graph domains in this paper we prove some results concerning inverse free boundary type problems below the continuous threshold for the heat equation in the setting of parabolic regular graph domains

regularity of lipschitz free boundaries in two phase problems for the p laplace operator in this paper we continue the study in ln6 concerning the regularity of the free boundary in a general two phase free boundary problem for the p laplace operator by proving regularity of the free boundary assuming that the free boundary is close to a lipschitz graph

a non local in time telegraph equation abstract in this work we derive a non local in time telegraph equation our model includes as particular cases the classical telegraph equation and the fractional in time telegraph equation among others further we define the fundamental solution of the problem and we prove that it can be interpreted as a probability density function finally using versions of the karamata feller tauberian theorem we study the temporal behavior of the variance of the distribution process associated with the solution of the equation in large times as well as in short times

regularity of solutions to the parabolic fractional obstacle problem in recent years there has been an increasing interest in studying constrained variational problems with a fractional diffusion one of the motivations comes from mathematical finance jumpdiffusion processes where incorporated by merton 14 into the theory of option evaluation to introduce discontinuous paths in the dynamics of the stock s prices in contrast with the classical lognormal diffusion model of black and scholes 2 these models allow to take into account large price changes and they have become increasingly popular for modeling market fluctuations both for risk management and option pricing purposes let us recall that an american option gives its holder the right to buy a stock at a given price prior but not later than a given time t 0 if v x represents the rational price of an american option with a payoff at time t 0 then v will solve in the viscosity sense the following obstacle problem min lv v 0 v t

ergodic impulsive control of feller processes with costly information abstract in this paper we study the problem of ergodic impulsive control of feller processes with costly information we prove continuity of the value functions for optimal stopping and impulsive control with long run average cost we characterize the value functions as generalized solutions of respective quasi variational inequalities and describe optimal policies we study also an equation associated to impulsive control with long run average cost

boundary estimates for solutions to linear degenerate parabolic equations let omega subset mathbb r n be a bounded nta domain and let omega t omega times 0 t for some t 0 we study the boundary behaviour of non negativesolutions to the equation hu parti

distributed order calculus and equations of ultraslow diffusion abstract we consider equations of the form d u t x u t x f t x t 0 x r n where d is a distributed order derivative that is d t 0 1 d t d d is the caputo dzhrbashyan fractional derivative of order is a positive weight function the above equation is used in physical literature for modeling diffusion with a logarithmic growth of the mean square displacement in this work we develop a mathematical theory of such equations study the derivatives and integrals of distributed order

quantitative a priori estimates for fast diffusion equations with caffarelli kohn nirenberg weights harnack inequalities and holder continuity abstract we study a priori estimates for a class of non negative local weak solution to the weighted fast diffusion equation u t x x u m with 0 m 1 posed on cylinders of 0 t r n the weights x and x with n and 2 n 2 n can be both degenerate and singular and need not belong to the class a 2 a typical assumption for this kind of problems this range of parameters is optimal for the validity of a class of caffarelli kohn nirenberg inequalities which play the role of the standard sobolev inequalities in this more complicated weighted setting the weights that we consider are not translation invariant and this causes a number of extra difficulties and a variety of scenarios for instance the scaling properties of the equation change when considering the problem around the origin or far from it we therefore prove quantitative with computable constants upper and lower estimates for local weak solutions focussing our attention where a change of geometry appears such estimates fairly combine into forms of harnack inequalities of forward backward and elliptic type as a consequence we obtain holder continuity of the solutions with a quantitative even if non optimal exponent our results apply to a quite large variety of solutions and problems the proof of the positivity estimates requires a new method and represents the main technical novelty of this paper our techniques are flexible and can be adapted to more general settings for instance to a wider class of weights or to similar problems posed on riemannian manifolds possibly with unbounded curvature in the linear case m 1 we also prove quantitative estimates recovering known results in some cases and extending such results to a wider class of weights

a parabolic analogue of the higher order comparison theorem of de silva and savin abstract we show that the quotient of two caloric functions which vanish on a portion of the lateral boundary of a h k domain is h k up to the boundary for k 2 in the case k 1 we show that the quotient is in h 1 if the domain is assumed to be space time c 1 regular this can be thought of as a parabolic analogue of a recent important result in 8 and we closely follow the ideas in that paper we also give counterexamples to the fact that analogous results are not true at points on the parabolic boundary which are not on the lateral boundary i e points which are at the corner and base of the parabolic boundary

decay estimates for evolution equations with classical and fractional time derivatives abstract using energy methods we prove some power law and exponential decay estimates for classical and nonlocal evolutionary equations the results obtained are framed into a general setting which comprise among the others equations involving both standard and caputo time derivative complex valued magnetic operators fractional porous media equations and nonlocal kirchhoff operators both local and fractional space diffusion are taken into account possibly in a nonlinear setting the different quantitative behaviours which distinguish polynomial decays from exponential ones depend heavily on the structure of the time derivative involved in the equation

on nonlocal backward problems for fractional stochastic diffusion equations abstract this paper is devoted to solving a nonlocal backward problem for fractional stochastic diffusion equations based on the eigenfunction expansion of the solution the nonlocal backward problem for searching the previous values is reduced to an integral equation we obtain the boundedness strong continuity and compactness of the new operators generated by the nonlocally terminal value term and then we proceed to obtain the existence of mild solutions for the nonlocal backward problem

pointwise bounds and blow up for systems of nonlinear fractional parabolic inequalities abstract we investigate nonnegative nonlocal solutions u x t and v x t of the nonlinear system of inequalities 0 t u v 0 t v u in r n r n 1 satisfying the initial conditions u v 0 in r n 0 where and are positive constants specifically using the definition of the fractional heat operator t given in taliaferro 2020 we obtain when they exist optimal pointwise upper bounds on r n 0 for nonnegative nonlocal solutions u and v of this initial value problem with particular emphasis on these bounds as t 0 and as t when 1 we compare our results for nonlocal solutions to those of escobedo and herrero 1991 for classical pointwise solutions

caloric measure in parabolic flat domains in this paper we define parabolic chord arc domains and show that in a parabolic chord arc domain with vanishing constant the logarithm of the density of caloric measure with respect to a certain projective lebesgue measure is of vanishing mean oscillation we also obtain a partial converse to this result our work generalizes to the heat equation recent work of kenig and toro for the laplacian

weighted w1 p estimates for weak solutions of degenerate elliptic equations with coefficients degenerate in one variable abstract this paper studies sobolev regularity of weak solution of degenerate elliptic equations in divergence form div a x u div f x where x x y r n r the coefficient matrix a x is a symmetric measurable n 1 n 1 matrix and it could be degenerate or singular in the one dimensional y variable as a weight function in the a 2 muckenhoupt class our results give weighted sobolev regularity estimates of calderon zygmund type for weak solutions of this class of degenerate singular equations as an application of these estimates we establish global fractional sobolev regularity estimates for solutions of the spectral fractional elliptic equation with measurable coefficients this result can be considered as the sobolev counterpart of the recently established schauder regularity theory of fractional elliptic equations

event detection in wireless sensor networks in random spatial sensors deployments we develop a new class of event detection algorithms in wireless sensor networks where the sensors are randomly deployed spatially we formulate the detection problem as a binary hypothesis testing problem and design the optimal decision rules for two scenarios namely the poisson point process and binomial point process random deployments to calculate the intractable marginal likelihood density we develop three types of series expansion methods which are based on an askey orthogonal polynomials in addition we develop a novel framework to provide guidance on which series expansion is most suitable i e most accurate to use for different system parameters extensive monte carlo simulations are carried out to illustrate the benefits of this framework as well as the quality of the series expansion methods and the impacts that different parameters have on detection performance via the receiver operating curves roc

a de giorgi nash type theorem for time fractional diffusion equations we study the regularity of weak solutions to linear time fractional diffusion equations in divergence form of arbitrary time order alpha in 0 1 the coefficients are merely assumed to be bounded and measurable and they satisfy a uniform parabolicity condition our main result is a de giorgi nash type theorem which gives an interior holder estimate for bounded weak solutions in terms of the data and the l infty bound of the solution the proof relies on new a priori estimates for time fractional problems and uses de giorgi s technique and the method of non local growth lemmas which has been introduced recently in the context of nonlocal elliptic equations involving operators like the fractional laplacian

caloric functions in lipschitz domains and the regularity of solutions to phase transition problems in some domain d c rn l that vanishes locally on some distinguished part of od the term refers to the fact that in some sense along that part of od u resembles the fundamental solution of the heat equation in d this is particularly the case when the domain under consideration d is the intersection of some n 1 dimensional cube q with q one side of a lipschitz graph i e q fx f x t that is d q n q and the distinguished part of od is precisely q n q this is the main first area of study of this paper relying on work of fabes garofalo salsa fgs on backward harnack type inequalities for such domains lipschitz regularity in time versus lipschitz regularity in space is not of course the natural homogeneity balance for the study of parabolic equations but it is so for the study of phase transition relations of the form

wiener s criterion for the heat equation this paper establishes a necessary and sufficient condition for a boundary point of an arbitrary bounded open subset of r 1 to be regular for the heat equation our criterion is in the sense described below the exact analogue of winner s test w for boundary regularity of harmonic functions a careful statement of our result requires that we first of all introduce a great deal of terminology let 2 q r 1 n 1 denote any bounded open subset and 8 2 its topological boundary we will often write a typical point z e r 1 as z x t xer te r if ue c2 q r we may define the heat operator

analysis and computation of an optimality equation arising in an impulse control problem with discrete and costly observations abstract the framework of discrete and costly observations provides an attractive mathematical tool for stochastic control of biological and ecological systems in natural environment however this framework is still germinating and its applications in these research areas are still rare in this framework the system dynamics to be controlled is continuous time while the observations by the decision maker are discrete time the control problem of the system dynamics has a time hybridized type nature by which the optimality equation has a different form from the conventional ones the objectives as well as contributions of this paper are formulation of the optimality equation focusing on population management problems its detailed mathematical analysis numerical discretization and a demonstrative application we show that the optimality equation is solvable uniquely and its solution can be approximated recursively a numerical method to discretize the optimality equation is established with a simple finite difference scheme and the recursive formula mathematical analysis results such as monotonicity stability and convergence of the numerical method are discussed as well the present model is then applied to a population management problem finally an advanced problem subject to model ambiguity is also analyzed

the cauchy schwarz divergence for poisson point processes in this paper we extend the notion of cauchy schwarz divergence to point processes and establish that the cauchy schwarz divergence between the probability densities of two poisson point processes is half the squared l 2 distance between their intensity functions extension of this result to mixtures of poisson point processes and in the case where the intensity functions are gaussian mixtures closed form expressions for the cauchy schwarz divergence are presented our result also implies that the bhattacharyya distance between the probability distributions of two poisson point processes is equal to the square of the hellinger distance between their intensity measures we illustrate the result via a sensor management application where the system states are modeled as point processes

monotonicity of generalized frequencies and the strong unique continuation property for fractional parabolic equations abstract we study the strong unique continuation property backwards in time for the nonlocal equation in r n 1 0 1 t s u v x t u s 0 1 our main result theorem 1 2 can be thought of as the nonlocal counterpart of the result obtained in 30 for the case when s 1 in order to prove theorem 1 2 we develop the regularity theory of the extension problem for the equation 0 1 with such theory in hands we establish i a basic monotonicity result for an adjusted frequency function which plays a central role in this paper see theorem 1 3 below ii an extensive blowup analysis of the so called almgren rescalings associated with the extension problem we feel that our work will also be of interest e g in the study of certain basic open questions in free boundary problems as well as in nonlocal segregation problems

regularity and free boundary regularity for the p laplace operator in reifenberg flat and ahlfors regular domains regularity and free boundary regularity for the p laplace operator in reifenberg flat and ahlfors regular domains

existence and uniqueness results for a time fractional nonlinear diffusion equation abstract in this work we consider a nonlinear ordinary integro differential equation which arises in the studies of time fractional porous medium equation the nonlocality of the resulting free boundary problem is governed by the erdelyi kober operator which requires using other than classical proof techniques to prove the existence and uniqueness of a compactly supported solution we reduce the free boundary case to the initial value problem next we use the sub and supersolution technique to show that there exists a globally defined unique solution as a side product some estimates on the exact solution are found

a new proof of moser s parabolic harnack inequality using the old ideas of nash in 1958 nash published his fundamental work on the local holder continuity of solutions of second order parabolic equations with coefficients which need not be smooth 8 the primary purpose of that work was to study the properties of the fundamental solution corresponding to the parabolic operator and from these properties to derive regularity for a general solution though the work is often cited in the literature about weak solutions of elliptic and parabolic equations one feels that nash s ideas were never fully understood and maybe still are not and that because of this the more understandable and seemingly more fruitful ideas of de giorgi 4 and moser 6 7 were subsequently adopted

square functions nontangential limits and harmonic measure in codimension larger than 1 we characterize the rectifiability both uniform and not of an ahlfors regular set e of arbitrary codimension by the behavior of a regularized distance function in the complement of that set in particular we establish a certain version of the riesz transform characterization of rectifiability for lower dimensional sets we also uncover a special situation in which the regularized distance is itself a solution to a degenerate elliptic operator in the complement of e this allows us to precisely compute the harmonic measure of those sets associated to this degenerate operator and prove that in sharp contrast with the usual setting of codimension 1 a converse to dahlberg s theorem must be false on lower dimensional boundaries without additional assumptions

global existence and asymptotic behavior of solutions for nonlinear parabolic equations on unbounded domains we study the existence and the asymptotic behavior of positive solutions for the parabolic equation a u tu vup 0 on d 0 where a 0 d is a some unbounded domain in rn n 3 and v belongs to a new parabolic class j of singular potentials generalizing the well known parabolic kato class at infinity p introduced recently by zhang we also show that the choice of this class is essentially optimal

regularity of the obstacle problem for a fractional power of the laplace operator given a function and s 0 1 we will study the solutions of the following obstacle problem u in r n s u 0i nr n s u x 0 for those x such that u x x lim x u x 0 we show that when is c 1 s or smoother the solution u is in the space c 1 for every s in the case where the contact set u is convex we prove the optimal regularity result u c 1 s when is only c 1 for a s we prove that our solution u is c 1 for every c 2006 wiley periodicals inc

doubling properties of second order parabolic equations we prove the doubling property of l caloric measure corresponding to the second order parabolic equation in the whole space and in lipschitz domains for parabolic equations in the divergence form a weaker form of the doubling property follows easily from a recent result the backward harnack inequality and known estimates of green s function our method works for both the divergence and nondivergence cases moreover the backward harnack inequality and estimates of green s function are not needed in the course of proof

decay of solutions to parabolic type problem with distributed order caputo derivative abstract we consider the decay of solution to a fractional diffusion equation with distributed order caputo derivative we assume that the elliptic operator is time dependent and that the weight function contained in the definition of the distributed order caputo derivative is just integrable we establish the relation between behavior of the weight function near zero and the decay rate of solution

a parabolic problem with a fractional time derivative we study regularity for a parabolic problem with fractional diffusion in space and a fractional time derivative our main result is a de giorgi nash moser holder regularity theorem for solutions in a divergence form equation we also prove results regarding existence uniqueness and higher regularity in time

extension properties and boundary estimates for a fractional heat operator the square root of the heat operator sqrt partial t delta can be realized as the dirichlet to neumann map of the heat extension of data on mathbb r n 1 to mathbb r n 2 in this n

studying the sinr process of the typical user in poisson networks using its factorial moment measures based on a stationary poisson point process a wireless network model with random propagation effects shadowing and or fading is considered in order to examine the process formed by the signal to interference plus noise ratio sinr values experienced by a typical user with respect to all the base stations in the down link channel this sinr process is completely characterized by deriving its factorial moment measures which involve numerically tractable explicit integral expressions this novel framework naturally leads to expressions for the k coverage probability including the case of random sinr threshold values considered in multi tier network models while the k coverage probabilities correspond to the marginal distributions of the order statistics of the sinr process a more general relation is presented connecting the factorial moment measures of the sinr process to the joint densities of these order statistics this gives a way for calculating the exact values of the coverage probabilities arising in a general scenario of signal combination and interference cancellation between base stations the presented framework consisting of the mathematical representations of sinr characteristics with respect to the factorial moment measures holds for the whole domain of sinr and is amenable to considerable model extension

lyapunov functions and convergence to steady state for differential equations of fractional order we study the asymptotic behaviour as t of bounded solutions to certain integro differential equations in finite dimensions which include differential equations of fractional order between 0 and 2 we derive appropriate lyapunov functions for these equations and prove that any global bounded solution converges to a steady state of a related equation if the nonlinear potential mathcal e occurring in the equation satisfies the ojasiewicz inequality

full duplex radio for uplink downlink wireless access with spatially random nodes a full duplex fd multiple antenna access point ap communicating with single antenna half duplex hd spatially random users to support simultaneous uplink ul downlink dl transmissions is investigated since fd nodes are inherently constrained by the loopback interference li we study precoding schemes for the ap based on maximum ratio combining mrc maximal ratio transmission mrt zero forcing and the optimal scheme for ul and dl sum rate maximization using tools from stochastic geometry in order to shed insights into the systems performance simple expressions for single antenna perfect li cancellation negligible internode interference cases are also presented we show that fd precoding at ap improves the ul dl sum rate and hence a doubling of the performance of the hd mode is achievable in particular our results show that these impressive performance gains remain substantially intact even if the li cancellation is imperfect furthermore relative performance gap between fd and hd modes increases as the number of transmit receive antennas becomes large while with the mrc mrt scheme increasing the receive antenna number at fd ap is more beneficial in terms of sum rate than increasing the transmit antenna number

on time dependent domains for the degenerate p parabolic equation carleson estimate and holder continuity in this paper we propose a definition of parabolic nta for solutions to the degenerate p parabolic equation given this definition we prove the carleson estimate originally proved for this equation in avelin et al j eur math soc 2015 for cylindrical domains moreover we study a non optimal stronger outer corkscrew condition such that we obtain holder continuity up to the boundary for non negative solutions vanishing on a part of the boundary

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

sir meta distribution of k tier downlink heterogeneous cellular networks with cell range expansion heterogeneous cellular networks hcns constitute a necessary step in the evolution of cellular networks in this paper we apply the signal to interference ratio sir meta distribution framework for a refined sir performance analysis of hcns focusing on k tier heterogeneous cellular networks based on the homogeneous independent poisson point process ppp model with range expansion bias offloading bias in each tier expressions for the b th moment of the conditional success probability for both the entire network and each tier are derived based on which the exact meta distributions and the beta approximations are evaluated and compared key performance metrics including the mean success probability the variance of the conditional success probability the mean local delay and the asymptotic sir gains of each tier are obtained the results show that the biases are detrimental to the overall mean success probability of the whole network and that the b th moment curve of the conditional success probability of each tier can be tightly approximated by the horizontal shifted versions of the first moment curve of the single tier ppp network we also provide lower bounds for the region of the active probabilities of the base stations to keep the mean local delay of each tier finite

differentiability at lateral boundary for fully nonlinear parabolic equations abstract for fully nonlinear uniformly parabolic equations the first derivatives regularity of viscosity solutions at lateral boundary is studied under new dini type conditions for the boundary which is called reifenberg dini conditions and is weaker than usual dini conditions

wiener s criterion for parabolic equations with variable coefficients and its consequences dans un ensemble borne de r n 1 on etudie la regularite des points limites pour le probleme de dirichlet pour un operateur parabolique a coefficients lisses on donne une caracterisation geometrique de ces points limites qui sont reguliers

on the regularity of the non dynamic parabolic fractional obstacle problem abstract in the class of the non dynamic fractional obstacle problems of parabolic type it is shown existence uniqueness apriori bounds of the solution and optimal regularity of the space derivatives of the solution furthermore at free boundary points of positive parabolic density it is proven that the time derivative of the solution is holder continuous at such free boundary points space time regularity of the corresponding free boundary is obtained for any fraction s 0 1