robust models in probability sampling in the estimation of a population mean or total from a random sample certain methods based on linear models are known to be automatically design consistent regardless of how well the underlying model describes the population a sufficient condition is identified for this type of robustness to model failure the condition which we call internal bias calibration relates to the combination of a model and the method used to fit it included among the internally bias calibrated models in addition to the aforementioned linear models are certain canonical link generalized linear models and nonparametric regressions constructed from them by a particular style of local likelihood fitting other models can often be made robust by using a suboptimal fitting method thus the class of model based but design consistent analyses is enlarged to include more realistic models for certain types of survey variable such as binary indicators and counts particular applications discussed are the estimation of the size of a population subdomain as arises in tax auditing for example and the estimation of a bootstrap tail probability

constructing symmetric boolean functions with maximum algebraic immunity symmetric boolean functions with even variables 2k and maximum algebraic immunity ai f k have been constructed in braeken s thesis 2006 in this paper we show more constructions of such boolean functions including the generalization of a result and prove a conjecture raised in braeken s thesis 2006

equivariant stable homotopy and segal s burnside ring conjecture analogous results were proved later for ko in the generality of compact lie groups by atiyah and segal 8 and for kfq the algebraic k theory spectrum associated to the finite field fq by rector 29 using quillen s 26 computation of it kfq in each case the answer involves an appropriate completed representation ring of g and the cohomology theory in question is constructed from the permutative category of finite dimensional vector spaces over a field see 31 if one considers cohomology theories constructed from other permutative categories one expects to find analogous computations in terms of a completed representation ring of g in the given category appropriately defined in particular stable cohomotopy ts is constructed from the category of finite sets 31 and for this category the analogue of the representation ring is a well known object the burnside ring a g 13 a g is a commutative ring with augmentation so one may speak of a g the completed burnside ring moreover there is

the equivariant conner floyd isomorphism this paper proves two equivariant generalizations of the conner floyd isomorphism relating unitary cobordism and k theory it extends a previous result of okonek for abelian groups to all compact lie groups we also show that the result for finite groups is true using either the geometric or homotopical versions of cobordism 1 introduction in 6 conner and floyd established a relation between cobordism and rf theory they proved that mu x mu k k x where mu is unitary cobordism and k is complex a theory a generalization of this result to the equivariant context was proved by okonek 15 this paper improves upon that generalization in two ways first it expands the class of groups considered from abelian groups to all compact lie groups second it shows that both of the usual generalizations of cobordism to the equivariant context can be used the rf theory that we use is kg the equivariant k theory of segal 17 which is the obvious generalization of k theory to the equivariant world atiyah 2 proved that this theory has bott periodicity it is periodic with respect to spinc representations of the group there are two common ways of generalizing cobordism we will define these carefully in 2 one gives geometric cobordism which we denote by ug and the other homotopical cobordism mug both of these are multiplicative cohomology theories there are multiplicative maps p u g k g and p mug kg writing u g for the ring ug and similarly for the other theories the two main results we will show are theorem a p ug x w kg kg x is an isomorphism when g is a finite group theorem b p mug x mu kg kg x is an isomorphism for every compact lie group g it should be pointed out that these results use integer grading and not ro g grading we will use integer grading throughout this paper except on those occa sions when it is necessary or particularly convenient to use ro g we will make it clear at these points that we are using the larger grading theorem b has already been shown for g abelian in 15 and we will use that result as a starting point theorem a follows from theorem b as shown in 3 by

correlation immunity of nonlinear combining functions for cryptographic applications corresp pseudonoise generators for cryptographic applications consisting of several linear feedback shift registers with a nonlinear combining function have been proposed as running key generators in stream ciphers these running key generators eau sometimes be broken by ciphertext only correlation attacks on individual subsequences a new class of combining functions is presented which provides better security against such attacks the security is quantified by the smallest number m 1 of subsequences that must be simultaneously considered in a correlation attack a necessary condition for such m th order correlation immunity is proved a recursive construction is given that permits the construction of an m th order immune combining function for n subsequences for any m and n with 1 leq m finally the trade off between the length of the linear equivalent of the nonlinear generator and the order m of its immunity against correlation attacks is considered

asymptotically design unbiased predictors in survey sampling summary a probability sampling plan is suggested in this paper for drawing the sample so that the linear least squares predictor of the total of a finite population is made asymptotically design unbiased conditions are given for the sampling plan to reduce to the more familiar sample designs discussed in the literature the linear least squares predictor of the total of a finite population royall 1970 1976 derived under a superpopulation model with explanatory variables omitted is still best under the correct model provided that it is unbiased under the correct model tam 1986 the present paper specifies probability sampling schemes which render the linear least squares predictor asymptoti cally design unbiased such probability schemes enable some general form of balancing in the sample for the explanatory variables omitted from the assumed superpopulation model and provide protection of the predictor against this type of model failure conditions are also presented in this paper for the specified sample design to become the equal probability selection sample design or the optimal sample design considered by godambe 1955 1982

the lower bounds on the second order nonlinearity of three classes of boolean functions with high nonlinearity the rth order nonlinearity of boolean functions is an important cryptographic criterion associated with some attacks on stream and block ciphers it is also very useful in coding theory since it is related to the covering radii of reed muller codes this paper tightens the lower bounds of the second order nonlinearity of three classes of boolean functions in the form f x tr x d in n variables where 1 d 2 m 1 3 and n 2m or 2 d 2 m 2 m 1 2 1 n 2m and m is odd or 3 d 2 2 r 2 r 1 1 and n 4r

survey design under the regression superpopulation model abstract the construction of sample designs and estimators under a linear regression superpopulation model is considered the anticipated variance the variance of the predictor computed with respect to the sampling design and the superpopulation model is used as a criterion for evaluating probability designs and model unbiased predictors regression predictors that are model unbiased and design consistent are constructed

construction of highly nonlinear 1 resilient boolean functions with optimal algebraic immunity and provably high fast algebraic immunity in 2013 tang carlet and tang ieee tit 59 1 653 664 2013 presented two classes of boolean functions the functions in the first class are unbalanced and the functions in the second one are balanced both of those two classes of functions have high nonlinearity high algebraic degree optimal algebraic immunity and high fast algebraic immunity however they are not 1 resilient which represents a drawback for their use as filter functions in stream ciphers in this paper we first propose a large family of 1 resilient boolean functions having high lower bound on nonlinearity optimal algebraic immunity and optimal algebraic degree that is meeting the siegenthaler bound most notably we can mathematically prove that every function in n variables belonging to this family has fast algebraic immunity no less than n 6 which is the first time that an infinite family of 1 resilient functions with provably high fast algebraic immunity has been invented furthermore we exhibit a subclass of the family which has higher lower bound on nonlinearity than all the known 1 resilient functions with potentially optimal algebraic immunity and potentially high fast algebraic immunity

nilpotence and descent in equivariant stable homotopy theory abstract let g be a finite group and let f be a family of subgroups of g we introduce a class of g equivariant spectra that we call f nilpotent this definition fits into the general theory of torsion complete and nilpotent objects in a symmetric monoidal stable category with which we begin we then develop some of the basic properties of f nilpotent g spectra which are explored further in the sequel to this paper in the rest of the paper we prove several general structure theorems for categories of module spectra over objects such as equivariant real and complex k theory and borel equivariant mu using these structure theorems and a technique with the flag variety dating back to quillen we then show that large classes of equivariant cohomology theories for which a type of complex orientability holds are nilpotent for the family of abelian subgroups in particular we prove that equivariant real and complex k theory as well as the borel equivariant versions of complex oriented theories have this property

two classes of symmetric boolean functions with optimum algebraic immunity construction and analysis this paper discusses two classes of symmetric boolean functions for each class a necessary and sufficient condition for having optimum algebraic immunity is proposed the algebraic degree and nonlinearity of the boolean functions are also completely determined and then we prove several of braeken s conjectures about the algebraic degree and nonlinearity of the boolean functions with optimum algebraic immunity in the two classes

results on highly nonlinear boolean functions with provably good immunity to fast algebraic attacks abstract in the last decade algebraic and fast algebraic attacks are regarded as the most successful attacks on lfsr based stream ciphers since the notion of algebraic immunity was introduced the properties and constructions of boolean functions with maximum algebraic immunity have been researched in a large number of papers however there are few results with respect to boolean functions with provably good immunity against fast algebraic attacks in previous literatures only carlet feng function was proven to have good immunity to fast algebraic attacks in this paper we first study a large family of highly nonlinear boolean functions in terms of the immunity to fast algebraic attacks which includes the functions of tu deng the functions of tang et al and the functions of jin et al based on a sufficient and necessary condition for measuring the immunity of boolean functions against fast algebraic attacks using bivariate polynomial representation we propose an efficient method for estimating the immunity of the functions of such family then we prove that a family of 2 k variable boolean functions including the function recently constructed by tang et al are almost perfect algebraic immune for any integer k 3 more exactly they achieve optimal algebraic immunity and almost perfect immunity to fast algebraic attacks the functions of such family are balanced and have optimal algebraic degree besides we prove a lower bound on their nonlinearity based on the work of tang et al which is better than that of carlet feng function it is also checked for 3 k 9 that the exact nonlinearity of such functions is very good which is slightly smaller than that of carlet feng function and some functions of this family even have a slightly larger nonlinearity than tang s et al function to sum up among the known functions with provably good immunity against fast algebraic attacks the functions of this family make a trade off between the exact value and the lower bound of nonlinearity

hybrid classes of balanced boolean functions with good cryptographic properties abstract cryptographically strong boolean functions play an imperative role in the design of almost every modern symmetric cipher in this context the cryptographic properties of boolean functions such as non linearity algebraic degree correlation immunity and propagation criteria are critically considered in the process of designing these ciphers more recently with the emergence of algebraic and fast algebraic attacks algebraic immunity has also been included as an integral property to be considered as a result several constructions of boolean functions with high non linearity maximal algebraic degree and optimal algebraic immunity have been devised since then this paper focuses on some of these constructions and presents two hybrid classes of boolean functions the functions constructed within these classes possess maximal algebraic degree for balanced functions optimal algebraic immunity high non linearity and good resistance to algebraic and fast algebraic attacks a hybrid class of 1 resilient functions has also been proposed that also possesses high algebraic degree optimal algebraic immunity high non linearity and good resistance to algebraic and fast algebraic attacks

on the construction of cryptographically significant boolean functions using objects in projective geometry spaces recently several construction methods of highly nonlinear boolean functions with relatively good algebraic properties were proposed these approaches manage in optimizing most of the relevant cryptographic criteria but not all of them at the same time usually either the nonlinearity bounds are rather loose though the actual nonlinearity is relatively high or the functions do not provide a good resistance to fast algebraic cryptanalysis in this paper we develop a theoretical framework for using objects in suitable projective geometry spaces for construction of highly nonlinear boolean functions this allows us to establish tight bounds on the nonlinearity using simple counting arguments thus avoiding rather complicated estimates of certain trace sums our method generates a class of almost fully optimized functions that is the functions apart from very high nonlinearity also have the maximum algebraic degree and optimal algebraic immunity compared to the classes of functions proposed by carlet and feng wang and zeng our functions achieve a slightly better nonlinearity which is traded off against a little worse resistance against fast algebraic attacks on the other hand compared to the functions by tang and tu and deng our nonlinearity is somewhat lower but the algebraic properties are slightly better

best affine and quadratic approximations of particular classes of boolean functions in this paper we consider the problem of computing best low order approximations of boolean functions we focus on the best quadratic approximations of a subclass of cubic functions with arbitrary number of variables and we provide formulas for their efficient calculation our methodology is developed upon properties of the best affine approximations of quadratic functions for which formulas for their direct computation not by means of the walsh hadamard transform are given we determine the cubic functions in the above subclass that achieve the maximum second order nonlinearity thus yielding a lower bound for the covering radius of the second order reed muller code ssr rm 2 n in ssr rm 3 n simple extensions of these results to some special cases of higher degree functions are seen to hold furthermore a preliminary analysis of well known constructions for bent functions in terms of their second order nonlinearity is performed that indicates potential weaknesses if construction parameters are not properly chosen

algebraic immunity for cryptographically significant boolean functions analysis and construction recently algebraic attacks have received a lot of attention in the cryptographic literature it has been observed that a boolean function f used as a cryptographic primitive and interpreted as a multivariate polynomial over f sub 2 should not have low degree multiples obtained by multiplication with low degree nonzero functions in this paper we show that a boolean function having low nonlinearity is also weak against algebraic attacks and we extend this result to higher order nonlinearities next we present enumeration results on linearly independent annihilators we also study certain classes of highly nonlinear resilient boolean functions for their algebraic immunity we identify that functions having low degree subfunctions are weak in terms of algebraic immunity and we analyze some existing constructions from this viewpoint further we present a construction method to generate boolean functions on n variables with highest possible algebraic immunity spl lceil n 2 spl rceil this construction first presented at the 2005 workshop on fast software encryption fse 2005 has been the first one producing such functions these functions are obtained through a doubly indexed recursive relation we calculate their hamming weights and deduce their nonlinearities we show that they have very high algebraic degrees we express them as the sums of two functions which can be obtained from simple symmetric functions by a transformation which can be implemented with an algorithm whose complexity is linear in the number of variables we deduce a very fast way of computing the output to these functions given their input

homology of perfect complexes abstract it is proved that the sum of the loewy lengths of the homology modules of a finite free complex f over a local ring r is bounded below by a number depending only on r this result uncovers in the structure of modules of finite projective dimension obstructions to realizing r as a closed fiber of some flat local homomorphism other applications include as special cases uniform proofs of known results on free actions of elementary abelian groups and of tori on finite cw complexes the arguments use numerical invariants of objects in general triangulated categories introduced here and called levels they allow one to track through changes of triangulated categories homological invariants like projective dimension as well as structural invariants like loewy length an intermediate result sharpens with a new proof the new intersection theorem for commutative algebras over fields under additional hypotheses on the ring r stronger estimates are proved for loewy lengths of modules of finite projective dimension

some results on robust estimation in finite population sampling abstract the present article examines the robustness of sampling strategies that is estimators and sample designs from the point of view of attaining the minimum value of the godambe joshi lower bound to the expected variance of a strategy it is shown that asymptotic design unbiasedness is necessary but insufficient for robustness defined in the above sense the concepts of weak and strong robustness of estimators for given sampling strategies are introduced the article establishes that a sufficient condition for a sampling strategy to be robust is that the estimator is model unbiased and weakly robust sufficient conditions are also given for an estimator to be weakly robust these conditions are then used to examine the robustness of strategies when the design matrix or the covariance matrix of the superpopulation model is misspecified the robustness of model based predictors is also examined

spectral mackey functors and equivariant algebraic k theory i abstract spectral mackey functors are homotopy coherent versions of ordinary mackey functors as defined by dress we show that they can be described as excisive functors on a suitable category and we use this to show that universal examples of these objects are given by algebraic k theory more importantly we introduce the unfurling of certain families of waldhausen categories bound together with suitable adjoint pairs of functors this construction completely solves the homotopy coherence problem that arises when one wishes to study the algebraic k theory of such objects as spectral mackey functors finally we employ this technology to introduce fully functorial versions of a theory upside down a theory and the algebraic k theory of derived stacks we use this to give what we think is the first general construction of 1 et equivariant algebraic k theory for profinite etale fundamental groups this is key to our approach to the mackey functor case of a sequence of conjectures of gunnar carlsson

optimal sampling and estimation strategies under the linear model in some cases model based and model assisted inferences can lead to very different estimators these two paradigms are not so different if we search for an optimal strategy rather than just an optimal estimator a strategy being a pair composed of a sampling design and an estimator we show that under a linear model the optimal model assisted strategy consists of a balanced sampling design with inclusion probabilities that are proportional to the standard deviations of the errors of the model and the horvitz thompson estimator if the heteroscedasticity of the model is fully explainable by the auxiliary variables then this strategy is also optimal in a model based sense moreover under balanced sampling and with inclusion probabilities that are proportional to the standard deviation of the model the best linear unbiased estimator and the horvitz thompson estimator are equal finally it is possible to construct a single estimator for both the design and model variance the inference can thus be valid under the sampling design and under the model

on the construction of boolean functions with optimal algebraic immunity in this correspondence we introduce a method to construct boolean functions in any number of variables with optimal algebraic immunity remarkably all functions of this type with an odd number of variables can be obtained in this way we study some cryptographic properties such as balancedness algebraic degree of the constructed functions moreover a lower bound of the number of boolean functions with optimal algebraic immunity is given

generalized group characters and complex oriented cohomology theories let bg be the classifying space of a finite group g given a multiplicative cohomology theory e the assignment g 7i e bg is a functor from groups to rings endowed with induction transfer maps in this paper we investigate these functors for complex oriented cohomology theories e using the theory of complex representations of finite groups as a model for what one would like to know an analogue of artin s theorem is proved for all complex oriented e the abelian subgroups of g serve as a detecting family for e bg modulo torsion dividing the order of g when e is a complete local ring with residue field of characteristic p and associated formal group of height n we construct a character ring of class functions that computes 1 e bg the domain of the characters is gn p the set of n tuples of elements in g each of which has order a power of p a formula for induction is also found the ideas we use are related to the lubin tate theory of formal groups the construction applies to many cohomology theories of current interest completed versions of elliptic cohomology e n theory etc the nth morava k theory euler characteristic for bg is computed to be the number of g orbits in gn p for various groups g including all symmetric groups we prove that k n bg concentrated in even degrees our results about e bg extend to theorems about e eg gx where x is a finite g cw complex

constructions of cryptographically significant boolean functions using primitive polynomials it is known that boolean functions used in stream and block ciphers should have good cryptographic properties to resist algebraic attacks up until now there have been several constructions of boolean functions achieving optimum algebraic immunity however most of their nonlinearities are very low carlet and feng studied a class of boolean functions with optimum algebraic immunity and deduced the lower bound of its nonlinearity which is good but not very high moreover the main practical problem with this construction is that it cannot be implemented efficiently in this paper we put forward a new method to construct cryptographically significant boolean functions by using primitive polynomials and construct three infinite classes of boolean functions with good cryptographic properties balancedness optimum algebraic degree optimum algebraic immunity and a high nonlinearity

how nearly can model based prediction and design based estimation be reconciled abstract two general linear model based predictors one of the expectation of a finite population total and one of that total itself are compared with the design based generalized regression estimator gre first the predictors are made to conform to the gre by modifying the regression parameter estimators but retaining the same optimal inclusion probabilities second the gre is made to conform with each of the predictors in turn by modifying the inclusion probabilities but retaining the generalized least squares gls or best linear unbiased form for the estimators of the regression parameters it is shown that the choice of inclusion probabilities is more important asymptotically than the choice of estimator for the regression parameters and hence that predictors obtained by the first method generally have smaller asymptotic expected variances than those obtained by the second method using the first method certain special cases are shown to correspond to familiar estimators if there is only one

complete modules and torsion modules suppose that r is a ring and that a is a chain complex over r inside the derived category of differential graded a modules there are naturally defined subcategories of a torsion objects and of a complete objects under a finiteness condition on a we develop a morita theory for these subcat egories find conceptual interpretations for some associated algebraic functors and in appropriate commutative situations identify the associated functors as local homology or local cohomology some of the results are suprising even in the case r z and a z p 1 introduction let r be a ring and r mod the derived category of chain complexes of left modules see section 1 2 we choose a fixed complex a which is perfect in other words isomorphic in mod to a complex of finite length in which the entries are finitely generated projective modules we de clare another complex n to be a trivial if hom a a0 0 where hom denotes the chain complex of morphisms in r mod going further we say that x is a torsion if hom x at 0 for all a trivial n9 and that x is a complete if homr n9x 0 for all a trivial n we then study the category ators of a torsion complexes and the category acomp of a complete complexes both of these are triangulated full subcategories of r mod it turns out that these categories are equivalent to one another and also equivalent to the derived category of dif ferential graded modules over the endomorphism complex of a we construct approximation functors celu r mod at0rs and a r mod acomp for an object m of r mod9 the complex celu m is a kind of a cellular approxi mation to m in the sense that it is the best approximation to m which can be cobbled together from a and its suspensions the complex ma is the bousfield localization of m with respect to a homology theory on r mod derived from a we provide algebraic formulas for the functors and find that the functors are related in interesting ways one of which involves an arithmetic square we also show that if cciia r has certain finite dimensionality properties then an object m of r mod is a torsion or a complete if and only if the homology groups 77 m individually satisfy appropriate torsion or completeness conditions suppose now that r is a commutative ring that 7 c r is a finitely generated ideal that a r i9 and that k is the associated koszul complex the complex a

algebras and modules in monoidal model categories in recent years the theory of structured ring spectra formerly known as a infty and e infty ring spectra has been simplified by the discovery of categories of spectra with strictly associative and commutative smash products now a ring spectrum can simply be defined as a monoid with respect to the smash product in one of these new categories of spectra in this paper we provide a general method for constructing model category structures for categories of ring algebra and module spectra this provides the necessary input for obtaining model categories of symmetric ring spectra functors with smash product gamma rings and diagram ring spectra algebraic examples to which our methods apply include the stable module category over the group algebra of a finite group and unbounded chain complexes over a differential graded algebra 1991 mathematics subject classification primary 55u35 secondary 18d10

the balmer spectrum of the equivariant homotopy category of a finite abelian group for a finite abelian group a we determine the balmer spectrum of mathrm sp a omega the compact objects in genuine a spectra this generalizes the case a mathbb z p mathbb z due to balmer and sanders invent math 208 1 283 326 2017 by establishing a corrected version of their hbox log p conjecture for abelian groups we also work out the consequences for the chromatic type of fixed points and establish a generalization of kuhn s blue shift theorem for tate constructions kuhn in invent math 157 2 345 370 2004

on symmetric boolean functions with high algebraic immunity on even number of variables in this paper we put forward an efficient method to study the symmetric boolean functions with high algebraic immunity on even number of variables we obtain some powerful necessary conditions for symmetric boolean functions to achieve high algebraic immunity by studying the weight support of some specific types of boolean functions of low degrees with these results we prove that the algebraic immunity of a large class of symmetric correlation immune boolean functions namely the symmetric palindromic functions is not high besides we construct all symmetric boolean functions with maximum algebraic immunity and give a description for those with submaximum algebraic immunity we also determine the hamming weight degrees and nonlinearity of the symmetric boolean functions with maximum algebraic immunity

reed muller codes achieve capacity on erasure channels we introduce a new approach to proving that a sequence of deterministic linear codes achieves capacity on an erasure channel under maximum a posteriori decoding rather than relying on the precise structure of the codes our method exploits code symmetry in particular the technique applies to any sequence of linear codes where the blocklengths are strictly increasing the code rates converge and the permutation group of each code is doubly transitive in other words we show that symmetry alone implies near optimal performance an important consequence of this result is that a sequence of reed muller codes with increasing blocklength and converging rate achieves capacity this possibility has been suggested previously in the literature but it has only been proven for cases where the limiting code rate is 0 or 1 moreover these results extend naturally to all affine invariant codes and thus to extended primitive narrow sense bch codes this also resolves in the affirmative the existence question for capacity achieving sequences of binary cyclic codes the primary tools used in the proof are the sharp threshold property for symmetric monotone boolean functions and the area theorem for extrinsic information transfer functions

motivic tambara functors let k be a field and denote by mathcal sh k the motivic stable homotopy category recall its full subcategory mathcal sh k text eff heartsuit bachmann in j topol 10 4 1124 1144 arxiv 1610 01346 2017 write mathrm nalg mathcal sh k for the category of mathrm s mathrm m normed spectra bachmann hoyois in arxiv 1711 03061 2017 recall that there is a forgetful functor u mathrm nalg mathcal sh k rightarrow mathcal sh k let mathrm nalg mathcal sh k text eff heartsuit subset mathrm nalg mathcal sh k denote the full subcategory on normed spectra e such that ue in mathcal sh k text eff heartsuit in this article we provide an explicit description of mathrm nalg mathcal sh k text eff heartsuit as the category of effective homotopy modules with etale norms at least if char k 0 a weaker statement is available if k is perfect of characteristic 2

ideals in triangulated categories phantoms ghosts and skeleta we begin by showing that in a triangulated category specifying a projective class is equivalent to specifying an ideal i of morphisms with certain properties and that if i has these properties then so does each of its powers we show how a projective class leads to an adams spectral sequence and give some results on the convergence and collapsing of this spectral sequence we use this to study various ideals in the stable homotopy category we examine phantom maps skeletal phantom maps superphantom maps and ghosts a ghost is a map which induces the zero map of homotopy groups we show that ghosts lead to a stable analogue of the lusternik schnirelmann category of a space and we calculate this stable analogue for low dimensional real projective spaces we also give a relation between ghosts and the hopf and kervaire invariant problems in the case ofa modules over ana ring spectrum the ghost spectral sequence is a universal coefficient spectral sequence from the phantom projective class we derive a generalized milnor sequence for filtered diagrams of finite spectra and from this it follows that the group of phantom maps fromxtoycan always be described as alim 1group the last two sections focus on algebraic examples in the derived category of an abelian category we study the ideal of maps inducing the zero map of homology groups and find a natural setting for a result of kelly on the vanishing of composites of such maps we also explain how pure exact sequences relate to phantom maps in the derived category of a ring and give an example showing that phantoms can compose non trivially

finite population sampling with multivariate auxiliary information abstract this article examines strategies that are approximately design unbiased and nearly optimal assuming a large sample survey and a regression superpopulation model a new class of predictors is proposed to link certain features of optimal design unbiased and model unbiased predictors generalized regression predictors are shown to pervade the subclass of asymptotically design unbiased adu predictors generalized regression predictors are combined with model based stratification to construct highly efficient adu strategies

an algebraic model for free rational g spectra for connected compact lie groups g we show that the category of free rational g spectra for a connected compact lie group g is quillen equivalent to the category of torsion differential graded modules over the polynomial ring h bg the ingredients are the enriched morita equivalences of schwede and shipley topology 42 1 103 153 2003 the functors of shipley am j math 129 351 379 2007 making rational spectra algebraic koszul duality and thick subcategory arguments based on the simplicity of the derived category of a polynomial ring

local duality in algebra and topology abstract the first goal of this paper is to provide an abstract framework in which to formulate and study local duality in various algebraic and topological contexts for any stable category c together with a collection of compact objects k c we construct local cohomology and local homology functors satisfying an abstract version of local duality when specialized to the derived category of a commutative ring a and a suitable ideal in a we recover the classical local duality due to grothendieck as well as generalizations by greenlees and may more generally applying our result to the derived category of quasi coherent sheaves on a quasi compact and separated scheme x implies the local duality theorem of alonso tarrio jeremias lopez and lipman as a second objective we establish local duality for quasi coherent sheaves over many algebraic stacks in particular those arising naturally in stable homotopy theory after constructing an appropriate model of the derived category in terms of comodules over a hopf algebroid we show that in familiar cases the resulting local cohomology and local homology theories coincide with functors previously studied by hovey and strickland furthermore our framework applies to global and local stable homotopy theory in a way which is compatible with the algebraic avatars of these theories in order to aid computability we provide spectral sequences relating the algebraic and topological local duality contexts

more balanced boolean functions with optimal algebraic immunity and good nonlinearity and resistance to fast algebraic attacks in this paper three constructions of balanced boolean functions with optimal algebraic immunity are proposed it is checked that at least for small numbers of input variables these functions have good behavior against fast algebraic attacks as well other cryptographic properties such as algebraic degree and nonlinearity of the constructed functions are also analyzed lower bounds on the nonlinearity are proved which are similar to the best bounds obtained for known boolean functions resisting algebraic attacks and fast algebraic attacks moreover it is checked that for the number n of variables with 5 n 19 the proposed n variable boolean functions have in fact very good nonlinearity

fast algebraic attacks and decomposition of symmetric boolean functions in this correspondence first we give a decomposition of symmetric boolean functions then we show that almost all symmetric boolean functions including these functions with good algebraic immunity behave badly against fast algebraic attacks besides we improve the relations between algebraic degree and algebraic immunity of symmetric boolean functions

symmetric boolean functions depending on an odd number of variables with maximum algebraic immunity to resist algebraic attacks boolean functions should possess high algebraic immunity in 2003 courtois and meier showed that the algebraic immunity of an n variable boolean function is upper bounded by spl lceil n 2 spl rceil and then several papers studied how to find symmetric boolean functions with maximum algebraic immunity in this correspondence we prove that for each odd n there is exactly one trivially balanced n variable symmetric boolean function achieving the maximum algebraic immunity

on the resistance of boolean functions against algebraic attacks using univariate polynomial representation in the past few years algebraic attacks against stream ciphers with linear feedback function have been significantly improved as a response to the new attacks the notion of algebraic immunity of a boolean function f was introduced defined as the minimum degree of the annihilators of f and f 1 an annihilator of f is a nonzero boolean function g such that f g 0 while several constructions of boolean functions with optimal algebraic immunity have been proposed there is no significant progress concerning the resistance against the so called fast algebraic attacks in this paper we provide a framework to assess the resistance of boolean functions against the new algebraic attacks including fast algebraic attacks the analysis is based on the univariate polynomial representation of boolean functions and necessary and sufficient conditions are presented for a boolean function to have optimal behavior against all the new algebraic attacks finally we introduce a new infinite family of balanced boolean functions described by their univariate polynomial representation by applying the new framework we prove that all the members of the family have optimal algebraic immunity and we efficiently evaluate their behavior against fast algebraic attacks

the galois group of a stable homotopy theory abstract to a stable homotopy theory a presentable symmetric monoidal stable category we naturally associate a category of finite etale algebra objects and using grothendieck s categorical machine a profinite group that we call the galois group we then calculate the galois groups in several examples for instance we show that the galois group of the periodic e algebra of topological modular forms is trivial and that the galois group of k n local stable homotopy theory is an extended version of the morava stabilizer group we also describe the galois group of the stable module category of a finite group a fundamental idea throughout is the purely categorical notion of a descendable algebra object and an associated analog of faithfully flat descent in this context

on asymptotically design unbiased estimators of a finite population mean summary certain regression type estimators of a finite population mean are examined for their asymptotic design properties a sufficient condition that implies the design consistency or asymptotic design unbiasedness is discussed for a general regression estimator the leading term of its asymptotic bias is explicitly derived when this condition holds as well as when it does not hold

enriched categories via non symmetric operads abstract we set up a general theory of weak or homotopy coherent enrichment in an arbitrary monoidal category our theory of enriched categories has many desirable properties for instance if the enriching category v is presentably symmetric monoidal then cat v is as well these features render the theory useful even when an category of enriched categories comes from a model category as is often the case in examples of interest e g dg categories spectral categories and n categories this is analogous to the advantages of categories over more rigid models such as simplicial categories for example the resulting categories of functors between enriched categories automatically have the correct homotopy type we construct the homotopy theory of v enriched categories as a certain full subcategory of the category of many object associative algebras in v the latter are defined using a non symmetric version of lurie s operads and we develop the basics of this theory closely following lurie s treatment of symmetric operads while we may regard these many object algebras as enriched categories we show that it is precisely the full subcategory of complete objects in the sense of rezk i e those whose spaces of objects are equivalent to their spaces of equivalences that are local with respect to the class of fully faithful and essentially surjective functors we also consider an alternative model of enriched categories as certain presheaves of spaces satisfying analogues of the segal condition for rezk s segal spaces lastly we present some applications of our theory most notably the identification of associative algebras in v as a coreflective subcategory of pointed v enriched categories as well as a proof of a strong version of the baez dolan stabilization hypothesis

highly nonlinear boolean functions with optimal algebraic immunity and good behavior against fast algebraic attacks inspired by the previous work of tu and deng we propose two infinite classes of boolean functions of 2k variables where k 2 the first class contains unbalanced functions having high algebraic degree and nonlinearity the functions in the second one are balanced and have maximal algebraic degree and high nonlinearity as shown by a lower bound that we prove as a byproduct we also prove a better lower bound on the nonlinearity of the carlet feng function thanks to a combinatorial fact first conjectured by the authors and later proved by cohen and flori we are able to show that they both possess optimal algebraic immunity it is also checked that at least for numbers of variables n 16 functions in both classes have a good behavior against fast algebraic attacks compared with the known boolean functions resisting algebraic attacks and fast algebraic attacks both of them possess the highest lower bounds on nonlinearity these bounds are however not enough for ensuring a sufficient nonlinearity for allowing resistance to fast correlation attack nevertheless as for previously found functions with the same features there is a gap between the bound that we can prove and the actual values computed for bounded numbers of variables n 38 moreover these values are very good the infinite class of functions we propose in construction 2 presents among all currently known constructions the best provable tradeoff between all the important cryptographic criteria

improving the upper bounds on the covering radii of binary reed muller codes by deriving bounds on character sums of boolean functions and by using the characterizations due to kasami of those elements of the reed muller codes whose hamming weights are smaller than twice and a half the minimum distance we derive an improved upper bound on the covering radius of the reed muller code of order 2 and we deduce improved upper bounds on the covering radii of the reed muller codes of higher orders

a new upper bound on the block error probability after decoding over the erasure channel motivated by cryptographic applications we derive a new upper bound on the block error probability after decoding over the erasure channel the bound works for all linear codes and is in terms of the generalized hamming weights it turns out to be quite useful for reed muller codes for which all the generalized hamming weights are known whereas the full weight distribution is only partially known for these codes the error probability is related to the cryptographic notion of algebraic immunity we use our bound to show that the algebraic immunity of a random balanced m variable boolean function is of order m 2 1 o 1 with probability tending to 1 as m goes to infinity

counting boolean functions with specified values in their walsh spectrum the problem of counting boolean functions with specified number s of walsh coefficients w in their walsh spectrum is discussed in this paper strategies to solve this problem shall help solving many more problems related to desired cryptographic features of boolean functions such as nonlinearity resiliency algebraic immunity etc in an attempt to study this problem we present a new framework of solutions we give results for w 2 n 1 and for all s in line with a previous work of wu 1998 12 we also provide various results such as existence and construction for some s when w 0 multiplicities for all w and naive bounds on s for w 2 n 2