propagation of singularities and maximal functions in the plane in this work we mainly generalize bourgain s circular maximal function to include variable coefficient averages our techniques involve a combination of bourgain s basic ideas plus microlocal analysis in particular to see the role of curvature we rely heavily on methods used in studying propogation of singularities for hyperbolic differential equations we also show that forp 2 there is local smoothing inl p for solutions to the wave equation

from harmonic analysis to arithmetic combinatorics arithmetic combinatorics or additive combinatorics is a fast developing area of research combining elements of number theory combinatorics harmonic analysis and ergodic theory its arguably best known result and the one that brought it to global prominence is the proof by ben green and terence tao of the long standing conjecture that primes contain arbitrarily long arithmetic progressions there are many accounts and expositions of the green tao theorem including the articles by kra 119 and tao 182 in the bulletin the purpose of the present article is to survey a broader highly interconnected network of questions and results built over the decades and spanning several areas of mathematics of which the green tao theorem is a famous descendant an old geometric problem lies at the heart of key conjectures in harmonic analysis a major result in partial differential equations invokes combinatorial theorems on intersecting lines and circles an unexpected argument points harmonic analysts towards additive number theory with consequences that could have hardly been anticipated we will not try to give a comprehensive survey of harmonic analysis combinatorics or additive number theory we will not even be able to do full justice to our specific areas of focus instead referring the reader to the more complete expositions and surveys listed in section 7 our goal here is to emphasize the connections between these areas we will thus concentrate on relatively few problems chosen as much for their importance to their fields as for their links to each other the article is written from the point of view of an analyst who in the course of her work was gradually introduced to the questions discussed here and found them fascinating we hope that the reader will enjoy a taste of this experience

a capacity achieving simple decoder for bias based traitor tracing schemes we investigate alternative suspicion functions for bias based traitor tracing schemes and present a practical construction of a simple decoder that attains capacity in the limit of large coalition size c we derive optimal suspicion functions in both the restricted digit model and the combined digit model these functions depend on information that is usually not available to the tracer the attack strategy or the tallies of the symbols received by the colluders we discuss how such results can be used in realistic contexts we study several combinations of coalition attack strategy versus suspicion function optimized against some attack another attack or the same in many of these combinations the usual codelength scaling ell propto c 2 changes to a lower power of c e g c 3 2 we find that the interleaving strategy is an especially powerful attack the suspicion function tailored against interleaving is the key ingredient of the capacity achieving construction

tardos fingerprinting is better than we thought tardos has proposed a randomized fingerprinting code that is provably secure against collusion attacks we revisit his scheme and show that it has significantly better performance than suggested in the original paper first we introduce variables in place of tardos hard coded constants and we allow for an independent choice of the desired false positive fp and false negative fn error rates following through tardos proofs with these modifications we show that the code length can be reduced by more than a factor of two in typical content distribution applications where high fn rates can be tolerated second we study the statistical properties of the code under some reasonable assumptions the accusation sums can be regarded as gaussian distributed stochastic variables in this approximation the desired error rates are achieved by a code length twice shorter than in the first approach overall typical fp and fn error rates may be achieved with a code length approximately five times shorter than in the original construction

achieve higher efficiency at maximum power with finite time quantum otto cycle the optimization of heat engines was intensively explored to achieve higher efficiency while maintaining the output power however most investigations were limited to a few finite time cycles e g the carnot like cycle due to the complexity of the finite time thermodynamics in this paper we propose a class of finite time engine with quantum otto cycle and demonstrate a higher achievable efficiency at maximum power the current model can be widely utilized benefitting from the general c 2 scaling of extra work for a finite time adiabatic process with long control time we apply the adiabatic perturbation method to the quantum piston model and calculate the efficiency at maximum power which is validated with an exact solution

a kakeya type problem for circles we prove full hausdorff dimension in a variant of the kakeya problem involving circles in the plane and also sharp estimates for the relevant maximal function these results can also be formulated in terms of the wave equation in two space variables a novelty in our approach is the use of ideas from computational geometry

low dissipation carnot like heat engines at maximum efficient power we study the optimal performance of carnot like heat engines working in the low dissipation regime using the product of the efficiency and the power output also known as the efficient power as our objective function efficient power function represents a trade off between power and efficiency of a heat engine we find lower and upper bounds on the efficiency in the case of extreme asymmetric dissipation when the ratio of dissipation coefficients at the cold and the hot contacts approaches respectively zero or infinity in addition we obtain the form of efficiency for the case of symmetric dissipation we also discuss the universal features of efficiency at maximum efficient power and derive the bounds on the efficiency using a global linear irreversible framework introduced recently by one of the authors

efficiency at maximum power output of linear irreversible carnot like heat engines the efficiency at maximum power output of linear irreversible carnot like heat engines is investigated based on the assumption that the rate of irreversible entropy production of the working substance in each isothermal process is a quadratic form of the heat exchange rate between the working substance and the reservoir it is found that the maximum power output corresponds to minimizing the irreversible entropy production in two isothermal processes of the carnot like cycle and that the efficiency at maximum power output has the form ensuremath eta mp ensuremath eta c 2 ensuremath ensuremath gamma ensuremath eta c where ensuremath eta c is the carnot efficiency while ensuremath gamma depends on the heat transfer coefficients between the working substance and two reservoirs the value of ensuremath eta mp is bounded between ensuremath eta ensuremath ensuremath equiv ensuremath eta c 2 and ensuremath eta ensuremath equiv ensuremath eta c 2 ensuremath ensuremath eta c these results are consistent with those obtained by chen and yan j chem phys 90 3740 1989 based on the endoreversible assumption those obtained by esposito et al phys rev lett 105 150603 2010 based on the low dissipation assumption and those obtained by schmiedl and seifert europhys lett 81 20003 2008 for stochastic heat engines which in fact also satisfy the low dissipation assumption additionally we find that the endoreversible assumption happens to hold for carnot like heat engines operating at the maximum power output based on our fundamental assumption and that the carnot like heat engines that we focused on do not strictly satisfy the low dissipation assumption which implies that the low dissipation assumption or our fundamental assumption is a sufficient but non necessary condition for the validity of ensuremath eta mp ensuremath eta c 2 ensuremath ensuremath gamma ensuremath eta c as well as the existence of two bounds ensuremath eta ensuremath ensuremath equiv ensuremath eta c 2 and ensuremath eta ensuremath equiv ensuremath eta c 2 ensuremath ensuremath eta c

connection between maximum work and maximum power thermal cycles a new connection between maximum power curzon ahlborn thermal cycles and maximum work reversible cycles is proposed this linkage is built through a mapping between the exponents of a class of heat transfer laws and the exponents of a family of heat capacities depending on temperature this connection leads to the recovery of known results and to a wide and interesting set of results for a class of thermal cycles among other results it was found that it is possible to use analytically closed expressions for maximum work efficiencies to calculate good approaches to maximum power efficiencies behind the proposed connection is an interpretation of endoreversibility hypothesis additionally we suggest that certain reversible maximum work cycles depending on working substance can be used as reversible landmarks for ftt maximum power cycles which also depend on working substance properties

ecological optimization and coefficient of performance bounds of general refrigerators abstract an analysis of cop and its bounds at maximum ecological criterion for general refrigerators is conducted for generality both the non isothermal heat transfer processes and the internal dissipations are considered under different situations the cop under the maximum ecological criterion have been studied systematically and the general upper and lower bounds of the optimal cop have been obtained furthermore under maximum ecological criterion the cop of general endoreversible refrigerators have also been studied and the cop bounds of different kinds of refrigerators have been analyzed as actual refrigerators may not operate under the condition of maximum cop or maximum cooling load but operate under the maximum ecological condition which indicates the best compromise between the refrigeration rate and the loss of refrigeration rate this paper could provide a practical insight for designing and operating actual refrigerators

wave front sets local smoothing and bourgain s circular maximal theorem the purpose of this paper is to improve certain known regularity results for the wave equation and to give a simple proof of bourgain s circular maximal theorem 1 we use easy wave front analysis along with techniques previously used in proofs of the carleson sj6lin theorem see 3 5 7 and in the proof of sharp regularity properties of fourier integral operators 13 the circular means operators are defined by

ecological optimization for general heat engines abstract we conducted an analysis of efficiency and its bounds for general heat engines under the maximum ecological criterion for generality both nonisothermal heat exchanging processes and internal dissipation were taken into consideration when the product of the internal dissipation and the heat capacity ratio is one the efficiency under the maximum ecological criterion is the same as that of the irreversible carnot model however the efficiencies have different physical meanings and optimization spaces furthermore the efficiency is independent of the time it takes to complete each process and the heat conductance for other situations numerical calculations were conducted to investigate the parameters effects on optimal efficiency when the dimensionless contact times approach zero the irreversible carnot model is recovered the general upper and lower bounds of optimal efficiency are obtained by applying the asymmetric heat capacity ratio limits when the dimensionless contact times approach infinity in addition the efficiency of general endoreversible heat engines was investigated the efficiency bounds of different real life heat engines under the maximum ecological criterion are proposed

a generalization of bourgain s circular maximal theorem where d r is the normalized surface measure on r s1 it is easy to see that m is not bounded on l2 see example 1 1 below a well known result of bourgain 1 asserts that m is bounded on lp for 2 p we will consider the question of boundedness of m and m from lp to lq unless stated to the contrary we will be dealing only with functions defined on r2 absolute constants will be denoted by c and the notation will mean up to a constant

maximal operators and differentiation theorems for sparse sets we study maximal averages associated with singular measures on r our main result is a construction of singular cantor type measures supported on sets of hausdorff dimension 1 e with 0 e 1 e 1 e as a consequence we are able to answer a question of aversa and preiss on density and differentiation theorems for singular measures in one dimension our proof combines probabilistic techniques with the methods developed in multidimensional euclidean harmonic analysis in particular there are strong similarities to bourgain s proof of the circular maximal theorem in two dimensions

optimization induced by stability and the role of limited control near a steady state a relationship between stability and self optimization is found for weakly dissipative heat devices the effect of limited control on operation variables around an steady state is such that after instabilities the paths toward relaxation are given by trajectories stemming from restitution forces which improve the system thermodynamic performance power output efficiency and entropy generation statistics over random trajectories for many cycles shows this behavior as well two types of dynamics are analyzed one where an stability basin appears and another one where the system is globally stable under both dynamics there is an induced trend in the control variables space due to stability in the energetic space this behavior translates into a preference for better thermodynamic states and thus stability could favor self optimization under limited control this is analyzed from the multiobjective optimization perspective as a result the statistical behavior of the system is strongly influenced by the pareto front the set of points with the best compromise between several objective functions and the stability basin additionally endoreversible and irreversible behaviors appear as very relevant limits the first one is an upper bound in energetic performance connected with the pareto front and the second one represents an attractor for the stochastic trajectories

a bilinear approach to the restriction and kakeya conjectures the purpose of this paper is to investigate bilinear variants of the restriction and kakeya conjectures to relate them to the standard formulations of these conjectures and to give applications of this bilinear approach to existing conjectures the methods used are based on several observations and results of bourgain see 2 6 together with some refinements by moyua vargas and vega 17 18 this paper is organized as follows in the first section we discuss bilinear restriction estimates and show how one can pass back and forth between these estimates and the standard restriction estimates we also generalize the 12 7 bilinear restriction estimate of 18 to higher dimensions in the second section we give analogues of the above results for the kakeya operator in particular we give a bilinear improvement to wolff s kakeya theorem in arbitrary dimension in the third section we give applications of these bilinear estimates in three dimensions for example we are able to improve the 42 11 exponent in wolff s restriction theorem to 34 9 we are also able to prove a sharp lp lq restriction theorem which improves on the classical l2 l4 tomas stein theorem and also give some concrete progress on a bilinear restriction conjecture of klainerman and machedon we also give a non bilinear approach to these estimates which gives weaker results but is more direct and probably has a wider range of application

on continuum incidence problems related to harmonic analysis abstract we consider averaging operators over curves and surfaces satisfying the rotational curvature condition of phong and stein using combinatorial arguments restricted weak type versions of the standard estimates are obtained under weaker assumptions on the regularity of these surfaces we also consider maximal functions over circles in the plane and obtain a simpler proof of wolff s endpoint result

a geometric proof of the circular maximal theorem is bounded on l r for p 2 simple examples show that this fails for p 2 in this note we derive bourgain s result by geometric and combinatorial methods in particular we do not use the fourier transform in any way our proof is based on a combinatorial argument from 6 which in turn uses marstrand s three circle lemma 7 and a lemma involving two circles that seems to originate in 10 the three circle lemma was used in 7 to prove the following result which is a simple consequence of bourgain s theorem suppose a planar set e has the property for every point in the plane e contains some circle with that point as center then e has positive measure thus we show here that marstrand s lemma in combination with other ideas does indeed allow one to establish the stronger maximal function estimate furthermore we demonstrate in section 4 how to obtain the entire known range of l l estimates for the circular maximal function which is optimal possibly up to endpoints see 9 and 11 by using the methods from sections 2 and 3 one perhaps significant distinction from the techniques developed in 1 8 11 and 12 which involve the fourier transform is the fact that the methods presented here do not seem to yield estimates for the global maximal function mf x sup 0 t

efficiency at maximum power output of an irreversible carnot like cycle with internally dissipative friction we investigate the efficiency at the maximum power output emp of an irreversible carnot engine performing finite time cycles between two reservoirs at constant temperatures t h and t c t h taking into account the internally dissipative friction in two adiabatic processes the emp is retrieved to be situated between c 2 and c 2 c with c 1 t c t h being the carnot efficiency whether the internally dissipative friction is considered or not when dissipations of two isothermal and two adiabatic processes are symmetric respectively and the time allocation between the adiabats and the contact time with the reservoir satisfy a certain relation the curzon ahlborn ca efficiency ca 1 sqrt t c t h is derived

optimal probabilistic fingerprint codes we construct binary codes for fingerprinting digital documents our codes for n users that are e secure against c pirates have length o c2log n e this improves the codes proposed by boneh and shaw l1998r whose length is approximately the square of this length the improvement carries over to works using the boneh shaw code as a primitive for example to the dynamic traitor tracing scheme of tassa l2005r by proving matching lower bounds we establish that the length of our codes is best within a constant factor for reasonable error probabilities this lower bound generalizes the bound found independently by peikert et al l2003r that applies to a limited class of codes our results also imply that randomized fingerprint codes over a binary alphabet are as powerful as over an arbitrary alphabet and the equal strength of two distinct models for fingerprinting