a class of spectral moran measures abstract let d k k 1 be a sequence of digit sets in n and let b k k 1 be a sequence of integer numbers bigger than 1 we call the family f k d k x b k 1 x d d d k k 1 a moran iterated function system ifs which is a natural generalization of an ifs we prove under certain conditions in terms of b k d k that the associated moran measure is a spectral measure i e there exists a countable set n such that e 2 i x is an orthonormal basis for l 2

spectrality of a class of infinite convolutions for a finite set d z and an integer b 2 we say that b d is compatible with c z if e 2 idc b d d c c is a hadamard matrix let e 1 e a e a denote the uniformly discrete probability measure on e we prove that the class of infinite convolution moran measure b dk b 1d1 b 2d2 is a spectral measure provided that there is a common c z compatible to all the b dk and c c 0 1 b 1 we also give some examples to illustrate the hypotheses and results in particular the last condition on c is essential

spectral property of cantor measures with consecutive digits abstract we consider equally weighted cantor measures q b arising from iterated function systems of the form b 1 x i i 0 1 q 1 where q b we classify the q b so that they have infinitely many mutually orthogonal exponentials in l 2 q b in particular if q divides b the measures have a complete orthogonal exponential system and hence spectral measures improving the construction by dutkay et al 2009 3 we characterize all the maximal orthogonal sets when q divides b via a maximal mapping on the q adic tree in which all elements in are represented uniquely in finite b adic expansions and we can separate the maximal orthogonal sets into two types regular and irregular sets for a regular maximal orthogonal set we show that its completeness in l 2 q b is crucially determined by the certain growth rate of non zero digits in the tail of the b adic expansions of the elements furthermore we exhibit complete orthogonal exponentials with zero beurling dimensions these examples show that the technical condition in theorem 3 5 of dutkay et al 2011 4 cannot be removed for an irregular maximal orthogonal set we show that under some conditions its completeness is equivalent to that of the corresponding regularized mapping

mib galerkin method for elliptic interface problems material interfaces are omnipresent in the real world structures and devices mathematical modeling of material interfaces often leads to elliptic partial differential equations pdes with discontinuous coefficients and singular sources which are commonly called elliptic interface problems the development of high order numerical schemes for elliptic interface problems has become a well defined field in applied and computational mathematics and attracted much attention in the past decades despite of significant advances challenges remain in the construction of high order schemes for nonsmooth interfaces i e interfaces with geometric singularities such as tips cusps and sharp edges the challenge of geometric singularities is amplified when they are associated with low solution regularities e g tip geometry effects in many fields the present work introduces a matched interface and boundary mib galerkin method for solving two dimensional 2d elliptic pdes with complex interfaces geometric singularities and low solution regularities the cartesian grid based triangular elements are employed to avoid the time consuming mesh generation procedure consequently the interface cuts through elements to ensure the continuity of classic basis functions across the interface two sets of overlapping elements called mib elements are defined near the interface as a result differentiation can be computed near the interface as if there is no interface interpolation functions are constructed on mib element spaces to smoothly extend function values across the interface a set of lowest order interface jump conditions is enforced on the interface which in turn determines the interpolation functions the performance of the proposed mib galerkin finite element method is validated by numerical experiments with a wide range of interface geometries geometric singularities low regularity solutions and grid resolutions extensive numerical studies confirm the designed second order convergence of the mib galerkin method in the l and l 2 errors some of the best results are obtained in the present work when the interface is c 1 or lipschitz continuous and the solution is c 2 continuous

spectral property of self affine measures on mathbb r n abstract we study spectral properties of the self affine measure m d generated by an expanding integer matrix m m n z and a consecutive collinear digit set d 0 1 q 1 v where v z n 0 and q 2 is an integer some sufficient conditions for m d to be a spectral measure or to have infinitely many orthogonal exponentials are given moreover for some special cases we can obtain a necessary and sufficient condition on the spectrality of m d our study generalizes the one dimensional results proved by dai et al 4 5

on spectral n bernoulli measures abstract for 0 1 and n 1 an integer let be the self similar measure defined by i 0 n 1 1 n 1 i we prove that l 2 has an exponential orthonormal basis if and only if 1 q for some q 0 and n divides q the special case is the cantor measure with 1 2 k and n 2 16 which was proved recently to be the only spectral measure among the bernoulli convolutions with 0 1 4

the cardinality of orthogonal exponentials of planar self affine measures with three element digit sets abstract in this paper we consider the planar self affine measures m d generated by an expanding matrix m m 2 z and an integer digit set d 0 0 1 2 1 2 with 1 2 2 1 0 we show that if det m 3 z then the mutually orthogonal exponential functions in l 2 m d is finite and the exact maximal cardinality is given

self affine tiles in ℝn abstract a self affine tile in r n is a set t of positive measure with a t d d t d where a is an expanding n n real matrix with det a m an integer and d d d 2 d m r n is a set of m digits it is known that self affine tiles always give tilings of r n by translation this paper extends known characterizations of digit sets d yielding self affine tiles it proves several results about the structure of tilings of r n possible using such tiles and gives examples showing the possible relations between self replicating tilings and general tilings which clarify results of kenyon on self replicating tilings

spectra of a class of self affine measures abstract in the present paper we will study the spectral property of a class of self affine measures under the condition of compatible pair we first answer a question of dutkay and jorgensen concerning the relationship between spectral self affine measure and compatible pair we then consider the spectra of bernoulli convolutions and obtain a sharp result which extends the corresponding result of jorgensen kornelson and shuman finally we provide a structural property for the integer spectrum of a spectral self affine measure

on spectral cantor measures a probability measure in rd is called a spectral measure if it has an orthonormal basis consisting of exponentials in this paper we study spectral cantor measures we establish a large class of such measures and give a necessary and sufficient condition on the spectrum of a spectral cantor measure these results extend the studies by jorgensen and pedersen j anal math 75 1998 185 228 and strichartz j d analyse math 81 2000 209 238

on the orthogonal exponential functions of a class of planar self affine measures abstract for an expanding real matrix m 1 c 0 1 m 2 r and a digit set d 0 0 t 1 0 t 0 1 t let m d be the self affine measure generated by m and d in this paper we show that l 2 m d admits an infinite orthogonal set of exponential functions if and only if q p 1 r and c 1 for some positive integers p q r with p 3 z gcd p q 1 and q moreover if l 2 m d does not admit any infinite orthogonal set of exponential functions we estimate the number of orthogonal exponential functions in l 2 m d and give the exact maximal cardinality

orthogonal exponentials translations and bohr completions abstract we are concerned with an harmonic analysis in hilbert spaces l 2 where is a probability measure on r n the unifying question is the presence of families of orthogonal complex exponentials e x exp 2 i x in l 2 this question in turn is connected to the existence of a natural embedding of l 2 into an l 2 space of bohr almost periodic functions on r n in particular we explore when l 2 contains an orthogonal basis of e functions for in a suitable discrete subset in r n i e when the measure is spectral we give a new characterization of finite spectral sets in terms of the existence of a group of local translation we also consider measures that arise as fixed points in the sense of hutchinson of iterated function systems ifss and we specialize to the case when the function system in the ifs consists of affine and contractive mappings in r n we show in this case that if is then assumed spectral then its partitions induced by the ifs at hand have zero overlap measured in this solves part of the aba wang conjecture as an application of the new non overlap result we solve the spectral pair problem for bernoulli convolutions advancing in this way a theorem of ka sing lau in addition we present a new perspective on spectral measures and orthogonal fourier exponentials via the bohr compactification

on spectral eigenvalue problem of a class of generalized cantor measures abstract let q 4 r with integer r 1 let a a 1 a 2 b b 1 b 2 0 1 2 3 with a b i e a n b n for all n 1 we consider the following generalized cantor measure q a b 1 q a 1 b 1 1 q 2 a 2 b 2 where e 1 e a e a denotes the uniformly discrete probability on e in this paper we investigate the spectral eigenvalues of the second type of the measure q a b the spectrality of q a b is settled by he et al 2017 17 more precisely we characterize the possible real numbers t satisfying that there exists a countable set r such that and t are spectra of q a b simultaneously i e the sets e e 2 i x and e t e 2 i t x are both orthonormal basis for l 2 q a b

fourier frequencies in affine iterated function systems abstract we examine two questions regarding fourier frequencies for a class of iterated function systems ifs these are iteration limits arising from a fixed finite families of affine and contractive mappings in r d and the ifs refers to such a finite system of transformations or functions the iteration limits are pairs x where x is a compact subset of r d the support of and the measure is a probability measure determined uniquely by the initial ifs mappings and a certain strong invariance axiom the two questions we study are 1 existence of an orthogonal fourier basis in the hilbert space l 2 x and 2 explicit constructions of fourier bases from the given data defining the ifs

scaling of spectra of self similar measures with consecutive digits abstract we consider the equally weighted cantor measures p q generated by the iterated function system ifs f i x x p i q i 0 q 1 where 2 q z and q p r it is known that if q divides p then p q is a spectral measure with a spectrum p q 0 1 q 1 p 0 1 q 1 p 2 0 1 q 1 finite sum dai he and lai 2013 6 in this paper the authors study some positive integers b such that b p q is also a spectrum of p q

enhancing the security of quantum communication by hiding the message in a superposition a new method to hide specific quantum states in a superposition will be proposed the proposed method is used to hide an encrypted message in a superposition the process of hiding and unhiding the target quantum states is fast by using a single iteration of the amplitude amplification operators this makes the message 100 safe by direct measurement from an eavesdropper no entanglement sharing is required among the communicating parties

spectral property of the bernoulli convolutions for 0 1 let be the bernoulli convolution associated with jorgensen and pedersen p jorgensen s pedersen dense analytic subspaces in fractal l2 spaces j anal math 75 1998 185 228 proved that if 1 q where q is an even integer then l2 has an exponential orthonormal basis we show that for any 0 1 l2 contains an infinite orthonormal set of exponential functions if and only if is the nth root of a fraction p q where p is an odd integer and q is an even integer

non spectrality of planar self affine measures with three elements digit set abstract the self affine measure m d associated with an affine iterated function system d x m 1 x d d d is uniquely determined the problems of determining the spectrality or non spectrality of a measure m d have been received much attention in recent years one of the non spectral problem on m d is to estimate the number of orthogonal exponentials in l 2 m d and to find them in the present paper we show that for an expanding integer matrix m m 2 z and the three elements digit set d given by m a b d c and d 0 0 1 0 0 1 if a c b d 3 z then there exist at most 3 mutually orthogonal exponentials in l 2 m d and the number 3 is the best this confirms the three elements digit set conjecture on the non spectrality of self affine measures in the plane

when does a bernoulli convolution admit a spectrum abstract in this paper we solve a long standing problem on bernoulli convolutions in particular we show that the bernoulli convolution with contraction rate 0 1 admits a spectrum if and only if is the reciprocal of an even integer

non spectral problem for a class of planar self affine measures abstract the self affine measure m d corresponding to an expanding matrix m m n r and a finite subset d r n is supported on the attractor or invariant set of the iterated function system d x m 1 x d d d the spectral and non spectral problems on m d including the spectrum tiling problem implied in them have received much attention in recent years one of the non spectral problem on m d is to estimate the number of orthogonal exponentials in l 2 m d and to find them in the present paper we show that if a b c z a 1 c 1 and a c z 3 z m a b 0 c and d 0 0 1 0 0 1 then there exist at most 3 mutually orthogonal exponentials in l 2 m d and the number 3 is the best this extends several known conclusions the proof of such result depends on the characterization of the zero set of the fourier transform m d and provides a way of dealing with the non spectral problem

on the μm d orthogonal exponentials abstract let m d be a self affine measure corresponding to a given affine iterated function system d x m 1 x d d d in the present paper we will study the problem of how to determine the l 2 m d space has finite or infinite orthogonal exponentials such research is motivated by a conjecture on the non spectrality of m d we first obtain a non spectral criterion which extends the result of dutkay and jorgensen in opposition to the condition of this criterion we then obtain some conditions which imply the infinite orthogonal systems in l 2 m d these are necessary for further investigation on the spectrality of m d as an application we completely settle the corresponding problem for the generalized planar sierpinski family