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As early as the 16th century, Simon Jacob, a German reckoning master, noticed that the worst case in computing the greatest common divisor of two numbers by the Euclidean algorithm occurs if these numbers are equimultiples of two consecutive members of the Fibonacci sequence.

In the paper (Hill, 1973) from 1973 R.D. Hill studied linear matrix maps L : ℂ q × q → ℂ n × n which map Hermitian matrices to Hermitian matrices, or equivalently, preserve adjoints, i.e., L ( V ∗ ) = L ( V ) ∗ , via representations of the form L ( V ) = ∑ k , l = 1 m H k l A l V A k ∗ , V ∈ ℂ q × q , for matrices A 1 , … , A m ∈ ℂ n × q and continued his study of such representations in later work, sometimes with co-authors, to completely positive matrix maps and associated matrix reorderings. In this paper we expand the study of such representations, referred to as Hill representations here, in various directions. In particular, we describe which matrices A 1 , … , A m can appear in Hill representations (provided the number m is minimal) and determine the associated Hill matrix H = H k l explicitly. Also, we describe how different Hill representations of L (again with m minimal) are related and investigate further the implication of ∗ -linearity on the linear map L .

In Sarkozy (2001) the third author of this paper presented two conjectures on the additive decomposability of the sequence of ”smooth” (or ”friable”) numbers. Elsholtz and Harper (2015) proved (by using sieve methods) the second (less demanding) conjecture. The goal of this paper is to extend and sharpen their result in three directions by using a different approach (based on the theory of S -unit equations).

The classical Morse–Sard theorem claims that for a mapping v : R n → R m + 1 of class C k the measure of critical values v ( Z v , m ) is zero under condition k ≥ n − m . Here the critical set, or m-critical set is defined as Z v , m = { x ∈ R n : rank ∇ v ( x ) ≤ m } . Further Dubovitskiĭ in 1957 and independently Federer and Dubovitskiĭ in 1967 found some elegant extensions of this theorem to the case of other (e.g., lower) smoothness assumptions. They also established the sharpness of their results within the C k category. Here we formulate and prove a bridge theorem that includes all the above results as particular cases: namely, if a function v : R n → R d belongs to the Holder class C k , α , 0 ≤ α ≤ 1 , then for every q > m the identity H μ ( Z v , m ∩ v − 1 ( y ) ) = 0 holds for H q -almost all y ∈ R d , where μ = n − m − ( k + α ) ( q − m ) . Intuitively, the sense of this bridge theorem is very close to Heisenberg's uncertainty principle in theoretical physics: the more precise is the information we receive on measure of the image of the critical set, the less precisely the preimages are described, and vice versa. The result is new even for the classical C k -case (when α = 0 ); similar result is established for the Sobolev classes of mappings W p k ( R n , R d ) with minimal integrability assumptions p = max ⁡ ( 1 , n / k ) , i.e., it guarantees in general only the continuity (not everywhere differentiability) of a mapping. However, using some N-properties for Sobolev mappings, established in our previous paper, we obtained that the sets of nondifferentiability points of Sobolev mappings are fortunately negligible in the above bridge theorem. We cover also the case of fractional Sobolev spaces. The proofs of the most results are based on our previous joint papers with J. Bourgain and J. Kristensen (2013, 2015). We also crucially use very deep Y. Yomdin's entropy estimates of near critical values for polynomials (based on algebraic geometry tools).

Diaconis and Griffiths (2014) study the multivariate Krawtchouk polynomials orthogonal on the multinomial distribution. In this paper we derive the reproducing kernel orthogonal polynomials Q n ( x , y ; N , p ) on the multinomial distribution which are sums of products of orthonormal polynomials in x and y of fixed total degree n = 0 , 1 , … , N . The Poisson kernel ∑ n = 0 N ρ n Q n ( x , y ; N , p ) arises naturally from a probabilistic argument. An application to a multinomial goodness of fit test is developed, where the chi-squared test statistic is decomposed into orthogonal components which test the order of fit. A new duplication formula for the reproducing kernel polynomials in terms of the 1-dimensional Krawtchouk polynomials is derived. The duplication formula allows a Lancaster characterization of all reversible Markov chains with a multinomial stationary distribution whose eigenvectors are multivariate Krawtchouk polynomials and where eigenvalues are repeated within the same total degree. The χ 2 cutoff time, and total variation cutoff time is investigated in such chains. Emphasis throughout the paper is on a probabilistic understanding of the polynomials and their applications, particularly to Markov chains.

In this paper we make a comparison between certain probabilistic and deterministic point sets and show that some deterministic constructions (especially spherical t -designs) are better or as good as probabilistic ones like the jittered sampling model. We find asymptotic equalities for the discrete Riesz s -energy of sequences of well separated t -designs on the unit sphere S d ⊂ R d + 1 , d ≥ 2 . The case d = 2 was studied in Hesse (2009) and Hesse and Leopardi (2008). In Bondarenko et al., (2015) it was established that for d ≥ 2 , there exists a constant c d , such that for every N > c d t d there exists a well-separated spherical t -design on S d with N points. This paper gives results, based on recent developments that there exists a sequence of well separated spherical t -designs such that t and N are related by N ≍ t d .

In this paper we study the so called optimal polynomial meshes for domains in K ⊂ R d , d ≥ 2 . These meshes are discrete point sets Y n of cardinality c n d which have the property that ‖ p ‖ K ≤ A ‖ p ‖ Y n for every polynomial p of degree at most n with a constant A > 1 independent of n . It was conjectured earlier that optimal polynomial meshes exist in every convex domain. This statement was previously shown to hold for polytopes and C 2 like domains. In this paper we give a complete affirmative answer to the above conjecture when d = 2 .

In this paper we extend the idea of weighted density of Balcerzak et al. (2015) by using a modulus function and introduce the idea f -density of weight g of subsets of ω ≔ { 0 , 1 , … } (at the same time extending the notion of f -density (Aizpuru et al., 2014)), which we name d g f where g : ω → [ 0 , ∞ ) satisfies g ( n ) → ∞ and n ∕ g ( n ) ↛ 0 and f is a modulus function. The aim of this paper is to show that we can get new ideals Z g ( f ) consisting of sets A ⊂ ω for which d g f ( A ) = 0 different from all the previously constructed ideals Z g of Balcerzak et al. (2015) and moreover they retain all the nice properties of the ideals Z g .

Decomposition spaces are simplicial $\infty$-groupoids subject to a certain exactness condition, needed to induce a coalgebra structure on the space of arrows. Conservative ULF functors (CULF) between decomposition spaces induce coalgebra homomorphisms. Suitable added finiteness conditions define the notion of M\"obius decomposition space, a far-reaching generalisation of the notion of M\"obius category of Leroux. In this paper, we show that the Lawvere-Menni Hopf algebra of M\"obius intervals, which contains the universal M\"obius function (but is not induced by a M\"obius category), can be realised as the homotopy cardinality of a M\"obius decomposition space $U$ of all M\"obius intervals, and that in a certain sense $U$ is universal for M\"obius decomposition spaces and CULF functors., Comment: 35 pages. This paper is one of six papers that formerly constituted the long manuscript arXiv:1404.3202. v3: minor expository improvements. Final version to appear in Adv. Math

Let M m be a closed smooth manifold equipped with a smooth involution having fixed point set of the form F n ∪ F 3 , where F n and F 3 are submanifolds with dimensions n and 3, respectively, where 3 n m and with the normal bundles over F n and F 3 being nonbounding. The authors of this paper, together with Patricia E. Desideri, previously showed that, when n is even, then m ≤ n + 4 , which we call a small codimension phenomenon. Further, they showed that this small bound is best possible. In this paper we study this problem for n odd, which is much more complicated, requiring more sophisticated techniques involving characteristic numbers. We show in this case that m ≤ M ( n − 3 ) + 6 , where M ( n ) is the Stong–Pergher number (see the definition of M ( n ) in Section 1). Further, we show that this bound is almost best possible, in the sense that there exists an example with m = M ( n − 3 ) + 5 , which means that for n odd the small codimension phenomenon does not occur and the bound in question is meaningful. The existence of these bounds is guaranteed by the famous Five Halves Theorem of J. Boardman, which establishes that, under the above hypotheses, m ≤ 5 2 n .

Completely multiplicative functions whose sum is zero ($CMO$). The paper deals with $CMO$, meaning completely multiplicative ($CM$) functions $f$ such that $f(1)=1$ and $\sum\limits_1^\infty f(n)=0$. $CM$ means $f(ab)=f(a)f(b)$ for all $(a,b)\in \N^{*2}$, therefore $f$ is well defined by the $f(p)$, $p$ prime. Assuming that $f$ is $CM$, give conditions on the $f(p)$, either necessary or sufficient, both is possible, for $f$ being $CMO$ : that is the general purpose of the authors. The $CMO$ character of $f$ is invariant under slight modifications of the sequence $(f(p))$ (theorem~3). The same idea applies also in a more general context (theorem~4). After general statements of that sort, including examples of $CMO$ (theorem~5), the paper is devoted to ``small'' functions, that is, functions of the form $\frac{f(n)}{n}$, where the $f(n)$ are bounded. Here is a typical result : if $|f(p)|\le 1$ and $Re\, f(p)\le0$ for all $p$, a necessary and sufficient condition for $\big(\frac{f(n)}{n}\big)$ to be $CMO$ is $\sum \, Re\, f(p)/p=-\infty$ (theorem~8). Another necessary and sufficient condition is given under the assumption that $|1+f(p)|\le 1$ and $f(2)\not=-2$ (theorem~7). A third result gives only a sufficient condition (theorem~9). The three results apply to the particular case $f(p)=-1$, the historical example of Euler. Theorems 7 and 8 need auxiliary results, coming either from the existing literature (Hal\'asz, Montgomery--Vaughan), or from improved versions of classical results (Ingham, Ska\l ba) about $f(n)$ under assumptions on the $f*1(n)$, * denoting the multiplicative convolution (theorems~10~and~11).

In this paper, the Taylor sequence space t p θ of non-absolute type is introduced which consists of all sequences whose Taylor transforms of order θ , ( 0 θ 1 ) , are in the space l p . It is shown that the space t p θ is a normed space which includes the space l p where 1 ≤ p ≤ ∞ . Moreover, in this paper a general upper estimate is obtained for the norm of Hausdorff matrices and Norlund matrices as operators from l p into the Taylor sequence space t p θ . Finally, the results are extended to the matrix domain of an arbitrary invertible matrix E in the sequence space l p .

In a paper in 1962, Guido Zappa asked whether a non-trivial coset of a Sylow p-subgroup of a finite group could contain only elements whose orders are powers of p, and also in that case, at least one element of order p. The first question was raised again recently in a 2014 paper by Daniel Goldstein and Robert Guralnick, when generalising an answer by John Thompson in 1967 to a similar question by L.J. Paige. In this paper we give a positive answer to both questions of Zappa, showing somewhat surprisingly that in a number of non-abelian finite simple groups (including PSL ( 3 , 4 ) , PSU ( 5 , 2 ) and the Janko group J 3 ), some non-trivial coset of a Sylow 5-subgroup (of order 5) contains only elements of order 5. Also Zappa's first question is studied in more detail. Various possibilities for the group and its Sylow p-subgroup P are eliminated, and it then follows that | P | ≥ 5 and | P | ≠ 8 . It is an open question as to whether the order of the Sylow p-subgroup can be 7 or 9 or more.

Let E be an elliptic curve defined over an algebraically closed field k whose characteristic is not 2 or 3. Let τ be a translation automorphism of E that is not of order 2. In a previous paper we studied an algebra A = A ( E , τ ) that depends on this data: A ( E , τ ) = ( S ( E , τ ) ⊗ M 2 ( k ) ) Γ where S ( E , τ ) is the 4-dimensional Sklyanin algebra associated to ( E , τ ) , M 2 ( k ) is the ring of 2 × 2 matrices over k, and Γ is ( Z / 2 ) × ( Z / 2 ) acting in a particular way as automorphisms of S and M 2 ( k ) . The action of Γ on S is compatible with the translation action of the 2-torsion subgroup E [ 2 ] on E. Following the ideas and results in papers of Artin–Tate–Van den Bergh, Smith–Stafford, and Levasseur–Smith, this paper examines the line modules, point modules, and fat point modules, over A, and their incidence relations. The right context for the results is non-commutative algebraic geometry: we view A as a homogeneous coordinate ring of a non-commutative analogue of P 3 that we denote by Proj n c ( A ) . Point modules and fat point modules determine “points” in Proj n c ( A ) . Line modules determine “lines” in Proj n c ( A ) . Line modules for A are in bijection with certain lines in P ( A 1 ⁎ ) ≅ P 3 and therefore correspond to the closed points of a certain subscheme L of the Grassmannian G ( 1 , 3 ) . Shelton–Vancliff call L the line scheme for A. We show that L is the union of 7 reduced and irreducible components, 3 quartic elliptic space curves and 4 plane conics in the ambient Plucker P 5 , and that deg ⁡ ( L ) = 20 . The union of the lines corresponding to the points on each elliptic curve is an elliptic scroll in P ( A 1 ⁎ ) . Thus, the lines on that elliptic scroll are in natural bijection with a corresponding family of line modules for A.

In this paper we introduce a counterpart structure to the shamrocks studied in the paper A dual of Macmahon's theorem on plane partitions by M. Ciucu and C. Krattenthaler (2013) [5] , which, just like the latter, can be included at the center of a lattice hexagon on the triangular lattice so that the region obtained from the hexagon by removing it has its number of lozenge tilings given by a simple product formula. The new structure, called a fern, consists of an arbitrary number of equilateral triangles of alternating orientations lined up along a lattice line. The shamrock and the fern seem to be the only such connected structures with this property. It would be interesting to understand the reason for this.

Let q be a prime, P ⩾ 1 and let N q ( P ) denote the number of rational primes p ⩽ P that split in the imaginary quadratic field Q ( − q ) . The first part of this paper establishes various unconditional and conditional (under existence of a Siegel zero) lower bounds for N q ( P ) in the range q 1 / 4 + e ⩽ P ⩽ q , for any fixed e > 0 . This improves upon what is implied by work of Pollack and Benli–Pollack. The second part of this paper is dedicated to proving an estimate for a bilinear form involving Weyl sums for modular square roots (equivalently Salie sums). Our estimate has a power saving in the so-called Polya–Vinogradov range, and our methods involve studying an additive energy coming from quadratic residues in F q . This bilinear form is inspired by the recent automorphic motivation: the second moment for twisted L-functions attached to Kohnen newforms has recently been computed by the first and fourth authors. So the third part of this paper links the above two directions together and outlines the arithmetic applications of this bilinear form. These include the equidistribution of quadratic roots of primes, products of primes, and relaxations of a conjecture of Erdős–Odlyzko–Sarkozy.

This paper studies the approximation of the Hilbert transform f ? = H f of continuous functions f with continuous conjugate f ? based on a finite number of samples. It is known that every sequence { H N f } N ? N which approximates f ? from samples of f diverges (weakly) with respect to the uniform norm. This paper conjectures that all of these approximation sequences even contain no convergent subsequence. A property which is termed strong divergence.The conjecture is supported by two results. First it is proven that the sequence of the sampled conjugate Fejer means diverges strongly. Second, it is shown that for every sample based approximation method { H N } N ? N there are functions f such that ? H N f ? ∞ exceeds any given bound for any given number of consecutive indices N .As an application, the later result is used to investigate a problem associated with a question of Ul'yanov on Fourier series which is related to the possibility to construct adaptive approximation methods to determine the Hilbert transform from sampled data. This paper shows that no such approximation method with a finite search horizon exists.

Markov numbers are integers that appear in the solution triples of the Diophantine equation, $x^2+y^2+z^2=3xyz$, called the Markov equation. A classical topic in number theory, these numbers are related to many areas of mathematics such as combinatorics, hyperbolic geometry, approximation theory and cluster algebras. There is a natural map from the rational numbers between zero and one to the Markov numbers. In this paper, we prove two conjectures seen in Martin Aigner's book, Markov's theorem and 100 years of the uniqueness conjecture, that determine an ordering on subsets of the Markov numbers based on their corresponding rational. The proof relies on a relationship between Markov numbers and continuant polynomials which originates in Frobenius' 1913 paper., 16 pages

Let $X \subseteq \mathbb{P}^r$ be a non-degenerate smooth projective variety of dimension $n$, codimension $e$, and degree $d$ defined over an algebraically closed field of characteristic zero. In this paper, we first show that $\text{reg} (\mathcal{O}_X) \leq d-e$, and classify the extremal and the next to extremal cases. Our result reduces the Eisenbud-Goto regularity conjecture for the smooth case to the problem finding a Castelnuovo-type bound for normality. It is worth noting that McCullough-Peeva recently constructed counterexamples to the regularity conjecture by showing that $\text{reg} (\mathcal{O}_X)$ is not even bounded above by any polynomial function of $d$ when $X$ is not smooth. For a normality bound in the smooth case, we establish that $\text{reg}(X) \leq n(d-2)+1$, which improves previous results obtained by Mumford, Bertram-Ein-Lazarsfeld, and Noma. Finally, by generalizing Mumford's method on double point divisors, we prove that $\text{reg}(X) \leq d-1+m$, where $m$ is an invariant arising from double point divisors associated to outer general projections. Using double point divisors associated to inner projection, we also obtain a slightly better bound for $\text{reg}(X)$ under suitable assumptions., Comment: 23 pages. This paper has been largely rewritten after McCullough-Peeva's counterexamples to the Eisenbud-Goto regularity conjecture, which appeared in J. Amer. Math. Soc. in 2018. We also added new results on the regularity of smooth projective varieties of arbitrary dimension

In this paper, given an integral domain U, we investigate the main properties of a relation ← m o d which is based on the interrelation between subdomains of U and finitely generated unitary submodules of U. We shall characterize it in terms of a second relation ≺ ⋄ between n-tuples ( u 1 , … , u n ) of elements of U and subdomains D of U defined by the vanishing in ( u 1 , … , u n ) of some polynomial p ( Z 1 , … , Z n ) belonging to a specific subset of the polynomial ring in several variables D [ Z 1 , … , Z n ] . Such an equivalence shall be used in order to introduce three specific collections of subdomains X U , B U and P U , whose algebraic properties present a close connection with geometrical and combinatorial properties induced by ← m o d . On the other hand, the characterization of the subdomains of X U leads to the more general problem of finding a map Ψ associating with a subdomain D of U a collection Ψ ( D ) of subdomains of K U such that the intersection of some or of any member of Ψ ( D ) gives D. In this perspective, in the present paper we shall study two further collections of subdomains of U, denoted respectively by E U and L U , whose main properties are related to those of the families P U and B U . Finally, our investigation of all the aforementioned subdomain families shall be also related to the study of pairs ( e , ξ ) , where e ∈ U ∖ { 0 } and ξ is an idempotent ring endomorphism of U whose kernel agrees with the ideal of U generated by e. We shall exhibit several results concerning the membership of ξ ( U ) and of ξ ( U ) [ e ] to the above subdomain families.

In this paper we study the moduli spaces Simp n m of degree n + 1 morphisms A K 1 → A K 1 with “ramification length Simp n m is a Zariski open subset of the space of degree n + 1 polynomials over K up to A u t ( A K 1 ) . It is, in a way, orthogonal to the many papers about polynomials with prescribed zeroes- here we are prescribing, instead, the ramification data. Exploiting the topological properties of the poset that encodes the ramification behavior, we use a sheaf-theoretic argument to compute H ⁎ ( Simp n m ( C ) ; Q ) as well as the etale cohomology H e ´ t ⁎ ( Simp n m / K ; Q l ) for c h a r K = 0 or c h a r K > n + 1 , when n and m are such that n ≥ 3 m . As a by-product we obtain that H ⁎ ( Simp n m ( C ) ; Q ) is independent of n, thus implying rational cohomological stability. When c h a r K > 0 our methods compute H e ´ t ⁎ ( Simp n m ; Q l ) provided c h a r K > n + 1 and show that the etale cohomology groups in positive characteristics do not stabilize.

We show that there is the same number of ( n , l ) -alternating sign trapezoids as there is of column strict shifted plane partitions of class l − 1 with at most n parts in the top row, thereby proving a result that was conjectured independently by Behrend and Aigner. The first objects generalize alternating sign triangles, which have recently been introduced by Ayyer, Behrend and the author who showed that they are counted by the same formula as alternating sign matrices. Column strict shifted plane partitions of a fixed class were introduced in a slightly different form by Andrews, and they essentially generalize descending plane partitions. They also correspond to cyclically symmetric lozenge tilings of a hexagon with a triangular hole in the center. In addition, we also provide three statistics on each class of objects and show that their joint distribution is the same. We prove our result by employing a constant term approach that is based on the author's operator formula for monotone triangles. This paper complements a forthcoming paper of Behrend and the author, where the six-vertex model approach is used to show equinumeracy as well as generalizations involving statistics that are different from those considered in the present paper.

We study the hit problem, set up by F. Peterson, of finding a minimal set of generators for the polynomial algebra $P_k := \mathbb F_2[x_1,x_2,...,x_k]$ as a module over the mod-2 Steenrod algebra, $\mathcal{A}$. In this paper, we study a minimal set of generators for $\mathcal A$-module $P_k$ in some so-call generic degrees and apply these results to explicitly determine the hit problem for $k=4$., Comment: 68 pages, Quy Nhon University preprint, Viet Nam, 2011. A shorter version of this paper was published in Advances in Mathematics

We introduce a covering conjecture and show that it holds below A D R + “ Θ is regular” . We then use it to show that in the presence of mild large cardinal axioms, PFA implies that there is a transitive model containing the reals and ordinals and satisfying A D R + “ Θ is regular” . The method used to prove the Main Theorem of this paper is the core model induction. The paper contains the first application of the core model induction that goes significantly beyond the region of A D + + θ 0 Θ .

This paper provides the theoretical foundation for the component-by-component (CBC) construction of randomly shifted lattice rules that are tailored to integrals over R s arising from practical applications. For an integral of the form ∫ R s f ( y ) ∏ j = 1 s ϕ ( y j ) d y with a univariate probability density ϕ , our general strategy is to first map the integral into the unit cube [ 0 , 1 ] s using the inverse of the cumulative distribution function of ϕ , and then apply quasi-Monte Carlo (QMC) methods. However, the transformed integrand in the unit cube rarely falls within the standard QMC setting of Sobolev spaces of functions with mixed first derivatives. Therefore, a non-standard function space setting for integrands over R s , previously considered by Kuo, Sloan, Wasilkowski and Waterhouse (2010), is required for the analysis. Motivated by the needs of three applications, the present paper extends the theory of the aforementioned paper in several non-trivial directions, including a new error analysis for the CBC construction of lattice rules with general non-product weights, the introduction of an unanchored variant for the setting, the use of coordinate-dependent weight functions in the norm, and the strategy for fast CBC construction with POD (“product and order dependent”) weights.

In this paper we establish a new connection between central sets and the strong coincidence conjecture for fixed points of irreducible primitive substitutions of Pisot type. Central sets, first introduced by Furstenberg using notions from topological dynamics, constitute a special class of subsets of N possessing strong combinatorial properties: Each central set contains arbitrarily long arithmetic progressions, and solutions to all partition regular systems of homogeneous linear equations. We give an equivalent reformulation of the strong coincidence condition in terms of central sets and minimal idempotent ultrafilters in the Stone–Cech compactification β N . This provides a new arithmetical approach to an outstanding conjecture in tiling theory, the Pisot substitution conjecture. The results in this paper rely on interactions between different areas of mathematics, some of which had not previously been directly linked: They include the general theory of combinatorics on words, abstract numeration systems, tilings, topological dynamics and the algebraic/topological properties of Stone–Cech compactification of N .

An m-ballot path of size n is a path on the square grid consisting of north and east unit steps, starting at (0,0), ending at (mn,n), and never going below the line {x=my}. The set of these paths can be equipped with a lattice structure, called the m-Tamari lattice and denoted by T_n^{m}, which generalizes the usual Tamari lattice T_n obtained when m=1. This lattice was introduced by F. Bergeron in connection with the study of diagonal coinvariant spaces in three sets of n variables. The representation of the symmetric group S_n on these spaces is conjectured to be closely related to the natural representation of S_n on (labelled) intervals of the m-Tamari lattice, which we study in this paper. An interval [P,Q] of T_n^{m} is labelled if the north steps of Q are labelled from 1 to n in such a way the labels increase along any sequence of consecutive north steps. The symmetric group S_n acts on labelled intervals of T_n^{m} by permutation of the labels. We prove an explicit formula, conjectured by F. Bergeron and the third author, for the character of the associated representation of S_n. In particular, the dimension of the representation, that is, the number of labelled m-Tamari intervals of size n, is found to be (m+1)^n(mn+1)^{n-2}. These results are new, even when m=1. The form of these numbers suggests a connection with parking functions, but our proof is not bijective. The starting point is a recursive description of m-Tamari intervals. It yields an equation for an associated generating function, which is a refined version of the Frobenius series of the representation. This equation involves two additional variables x and y, a derivative with respect to y and iterated divided differences with respect to x. The hardest part of the proof consists in solving it, and we develop original techniques to do so, partly inspired by previous work on polynomial equations with "catalytic" variables., Comment: 29 pages --- This paper subsumes the research report arXiv:1109.2398, which will not be submitted to any journal

Let D ▵ ( n ) be the double Ringel–Hall algebra of the cyclic quiver △ ( n ) and let D ▵ ( n ) be the modified quantum affine algebra of D ▵ ( n ) . We will construct an integral form D ▵ ( n ) Z for D ▵ ( n ) such that the natural algebra homomorphism from D ▵ ( n ) Z to the integral affine quantum Schur algebra is surjective. Furthermore, we will use Hall algebras to construct the integral form U Z ( g l n ) of the universal enveloping algebra U ( g l n ) of the loop algebra g l n = g l n ( Q ) ⊗ Q [ t , t − 1 ] , and prove that the natural algebra homomorphism from U Z ( g l n ) to the affine Schur algebra over Z is surjective. In a subsequent paper (Fu [10] ), we will use affine Schur algebras to give BLM realization of U Z ( g l n ) , and this enables us to give a new proof of the statements about U Z ( g l n ) given in this paper.

The main purpose of this paper is to study the generalized Hilbert operator {equation*} \mathcal{H}_g(f)(z)=\int_0^1f(t)g'(tz)\,dt {equation*} acting on the weighted Bergman space $A^p_\om$, where the weight function $\om$ belongs to the class $\R$ of regular radial weights and satisfies the Muckenhoupt type condition {equation}\label{Mpconditionaabstract} \sup_{0\le r, Comment: This paper has been accepted for publication in Advances in Mathematics

In this paper, we study degenerate CR embeddings f of a strictly pseudoconvex hypersurface M ⊂ C n + 1 into a sphere S in a higher dimensional complex space C N + 1 . The degeneracy of the mapping f will be characterized in terms of the ranks of the CR second fundamental form and its covariant derivatives. In 2004, the author, together with X. Huang and D. Zaitsev, established a rigidity result for CR embeddings f into spheres in low codimensions. A key step in the proof of this result was to show that degenerate mappings are necessarily contained in a complex plane section of the target sphere (partial rigidity). In the 2004 paper, it was shown that if the total rank d of the second fundamental form and all of its covariant derivatives is n (here, n is the CR dimension of M ), then f ( M ) is contained in a complex plane of dimension n + d + 1 . The converse of this statement is also true, as is easy to see. When the total rank d exceeds n , it is no longer true, in general, that f ( M ) is contained in a complex plane of dimension n + d + 1 , as can be seen by examples. In this paper, we carry out a systematic study of degenerate CR mappings into spheres. We show that when the ranks of the second fundamental form and its covariant derivatives exceed the CR dimension n , then partial rigidity may still persist, but there is a “defect” k that arises from the ranks exceeding n such that f ( M ) is only contained in a complex plane of dimension n + d + k + 1 . Moreover, this defect occurs in general, as is illustrated by examples.

In an earlier paper (arxiv:1108.4292) we introduced a new concept of dimension for metric spaces, the so called topological Hausdorff dimension. For a compact metric space $K$ let $\dim_{H}K$ and $\dim_{tH} K$ denote its Hausdorff and topological Hausdorff dimension, respectively. We proved that this new dimension describes the Hausdorff dimension of the level sets of the generic continuous function on $K$, namely $\sup{\dim_{H}f^{-1}(y) : y \in \mathbb{R}} = \dim_{tH} K - 1$ for the generic $f \in C(K)$, provided that $K$ is not totally disconnected, otherwise every non-empty level set is a singleton. We also proved that if $K$ is not totally disconnected and sufficiently homogeneous then $\dim_{H}f^{-1}(y) = \dim_{tH} K - 1$ for the generic $f \in C(K)$ and the generic $y \in f(K)$. The most important goal of this paper is to make these theorems more precise. As for the first result, we prove that the supremum is actually attained on the left hand side of the first equation above, and also show that there may only be a unique level set of maximal Hausdorff dimension. As for the second result, we characterize those compact metric spaces for which for the generic $f\in C(K)$ and the generic $y\in f(K)$ we have $\dim_{H} f^{-1}(y)=\dim_{tH}K-1$. We also generalize a result of B. Kirchheim by showing that if $K$ is self-similar then for the generic $f\in C(K)$ for every $y\in \inter f(K)$ we have $\dim_{H} f^{-1}(y)=\dim_{tH}K-1$. Finally, we prove that the graph of the generic $f\in C(K)$ has the same Hausdorff and topological Hausdorff dimension as $K$., 20 pages

This paper is mainly concerned with the structure of the centre of a vector lattice. A special attention is paid in the case when E = L p , p ≥ 1 . In this paper we give some characterizations of dense vector sublattices of the centre. Those characterizations will be applied in several directions. At the end of this work we compare various fullness and richness properties of the centre of a vector lattice.

Let a and b be two even integers with 2 ⩽ a < b, and let k be a nonnegative integer. Let G be a graph of order n. Its binding number bind(G) is defined as follows, bind(G)=min|NG(X)||X|:∅≠X⊆V(G),NG(X)≠V(G). In this paper, it is proved that G is an (a, b, k)-critical graph if bind(G)>(a+b-1)(n-1)bn-(a+b)-bk+3 and n⩾(a+b)(a+b-3)b+bkb-1. Furthermore, it is shown that the result in this paper is best possible in some sense.

In the present paper, we study conditions under which the metric projection of a polyhedral Banach space X onto a closed subspace is Hausdorff lower or upper semicontinuous. For example, we prove that if X satisfies (*) (a geometric property stronger than polyhedrality) and Y@?X is any proximinal subspace, then the metric projection P"Y is Hausdorff continuous and Y is strongly proximinal (i.e., if {y"n}@?Y, x@?X and @?y"n-x@?->dist(x,Y), then dist(y"n,P"Y(x))->0). One of the main results of a different nature is the following: if X satisfies (*) and Y@?X is a closed subspace of finite codimension, then the following conditions are equivalent: (a) Y is strongly proximinal; (b) Y is proximinal; (c) each element of Y^@? attains its norm. Moreover, in this case the quotient X/Y is polyhedral. The final part of the paper contains examples illustrating the importance of some hypotheses in our main results.

Berrizbeitia and Olivieri showed in a recent paper that, for any integer r, the notion of @w-prime to base a leads to a primality test for numbers n=1 mod r, that under the Extended Riemann Hypothesis (ERH) runs in polynomial time. They showed that the complexity of their test is at most the complexity of the Miller primality test (MPT), which is O((logn)^4^+^o^(^1^)). They conjectured that their test is more effective than the MPT if r is large. In this paper, we show that their conjecture is not true by showing that the Berrizbeitia-Olivieri primality test (BOPT) has no advantage over the MPT, either for proving primality of a prime under the ERH, or for detecting compositeness of a composite. In particular, we point out that the complexity of the BOPT depends not only on n but also on r and that in the worst cases (usually when n is prime) for both tests, the BOPT is in general at least twice slower than the MPT, and in some cases (usually when n is composite) the BOPT may be much slower. Moreover, the BOPT needs O(rlogn) bit memories. We also give facts and numerical examples to show that, for some composites n and for some r, the rth roots of unity @w do not exist, and thus outputs of the BOPT are ERH conditional, whereas the MPT always quickly and definitely (without ERH) detects compositeness for all odd composites.

In a paper Cheung, Cucker and Peña (in press) [5] that can be seen as the first part of this one, we extended the well-known condition numbers for polyhedral conic systems C(A) Renegar (1994, 1995) [7–9] and C(A) Cheung and Cucker (2001) [3] to versions C¯(A) and C¯(A) that are finite for all input matrices A∈Rn×m. In this paper we compare C¯(A) and C¯(A) with other condition measures for the same problem that are also always finite.

In this paper we consider a family of random Cantor sets on the line. We give some sufficient conditions when the Lebesgue measure of the arithmetic difference is positive. Combining this with the main result of a recent joint paper of the second author with M. Dekking we construct random Cantor sets F1, F2 such that the arithmetic difference set F2 − F1 does not contain any intervals but ℒeb(F2 − F1)> 0 almost surely, conditioned on non-extinction.

The cyclic projections algorithm is an important method for determining a point in the intersection of a finite number of closed convex sets in a Hilbert space. That is, for determining a solution to the ''convex feasibility'' problem. This is the third paper in a series on a study of the rate of convergence for the cyclic projections algorithm. In the first of these papers, we showed that the rate could be described in terms of the ''angles'' between the convex sets involved. In the second, we showed that these angles often had a more tractable formulation in terms of the ''norm'' of the product of the (nonlinear) metric projections onto related convex sets. In this paper, we show that the rate of convergence of the cyclic projections algorithm is also intimately related to the ''linear regularity property'' of Bauschke and Borwein, the ''normal property'' of Jameson (as well as Bakan, Deutsch, and Li's generalization of Jameson's normal property), the ''strong conical hull intersection property'' of Deutsch, Li, and Ward, and the rate of convergence of iterated parallel projections. Such properties have already been shown to be important in various other contexts as well.

For r = ( r 1 ,…, r d ) ∈ ℝ d the mapping τ r :ℤ d →ℤ d given by τ r ( a 1 ,…, a d ) = ( a 2 , …, a d ,−⌊ r 1 a 1 +…+ r d a d ⌋) where ⌊·⌋ denotes the floor function, is called a shift radix system if for each a ∈ ℤ d there exists an integer k > 0 with τ r k ( a ) = 0. As shown in Part I of this series of papers, shift radix systems are intimately related to certain well-known notions of number systems like β -expansibns and canonical number systems. After characterization results on shift radix systems in Part II of this series of papers and the thorough investigation of the relations between shift radix systems and canonical number systems in Part III, the present part is devoted to further structural relationships between shift radix systems and β -expansions. In particular we establish the distribution of Pisot polynomials with and without the finiteness property (F).

This paper is concerned with the algebraic K-theory of locally convex algebras stabilized by operator ideals, and its comparison with topological K-theory. We show that the obstruction for the comparison map between algebraic and topological K-theory to be an isomorphism is (absolute) algebraic cyclic homology and prove the existence of an 6-term exact sequence. We show that cyclic homology vanishes in the case when J is the ideal of compact operators and L is a Frechet algebra with bounded app. unit. This proves the generalized version of Karoubi's conjecture due to Mariusz Wodzicki and announced in his paper "Algebraic K-theory and functional analysis", First European Congress of Mathematics, Vol. II (Paris, 1992), 485--496, Progr. Math., 120, Birkh\"auser, Basel, 1994. We also consider stabilization with respect to a wider class of operator ideals, called sub-harmonic. We study the algebraic K-theory of the tensor product of a sub-harmonic ideal with an arbitrary complex algebra and prove that the obstruction for the periodicity of algebraic K-theory is again cyclic homology. The main technical tools we use are the diffeotopy invariance theorem of Cuntz and the second author (which we generalize), and the excision theorem for infinitesimal K-theory, due to the first author., Comment: Final version, to appear in Advances in Mathematics

This paper is intended to provide an introduction to the theory of substitution tilings. For our purposes, tiling substitution rules are divided into two broad classes: geometric and combinatorial. Geometric substitution tilings include self-similar tilings such as the well-known Penrose tilings; for this class there is a substantial body of research in the literature. Combinatorial substitutions are just beginning to be examined, and some of what we present here is new. We give numerous examples, mention selected major results, discuss connections between the two classes of substitutions, include current research perspectives and questions, and provide an extensive bibliography. Although the author attempts to represent the field as a whole, the paper is not an exhaustive survey, and she apologizes for any important omissions.

Let N be the set of all nonnegative integers, A ⊆ N be a finite set, and 2A be the set of all numbers of the form a + b for all a and b in A. In [G.A. Freiman, Foundations of a Structural Theory of Set Addition, Transl. Math. Monogr., vol. 37, American Mathematical Society, Providence, RI, 1973 (translated from the Russian)] the arithmetic structure of A was optimally characterized when | 2 A | ⩽ 3 | A | − 3 . 2 In [G.A. Freiman, Foundations of a Structural Theory of Set Addition, Transl. Math. Monogr., vol. 37, American Mathematical Society, Providence, RI, 1973 (translated from the Russian)] the structure of A was also characterized without proof when | 2 A | = 3 | A | − 2 . Since then the efforts have been made to generalize these results, see [V. Lev, P.Y. Smeliansky, On addition of two distinct sets of integers, Acta Arith. 70 (1) (1995) 85–91; V. Lev, On the structure of sets of integers with small doubling property, unpublished manuscripts, 1995; Y.O. Hamidoune, A. Plagne, A generalization of Freiman's 3 k − 3 theorem, Acta Arith. 103 (2) (2002) 147–156] for example. However, no optimal characterization of the structure of A has been obtained without imposing extra conditions, until now, when | 2 A | > 3 | A | − 2 . In this paper we optimally characterize, with the help of nonstandard analysis, the arithmetic structure of A when | 2 A | = 3 | A | − 3 + b , where b is positive but not too large. Precisely, we prove that there is a real number ϵ > 0 and there is K ∈ N such that if | A | > K and | 2 A | = 3 | A | − 3 + b for 0 ⩽ b ⩽ ϵ | A | , then A is either a subset of an arithmetic progression of length at most 2 | A | − 1 + 2 b or a subset of a bi-arithmetic progression 3 of length at most | A | + b . An application of this result to the inverse problem for upper asymptotic density is presented near the end of the paper. In the application we improve the most important part of the main theorem in [R. Jin, Solution to the inverse problem for upper asymptotic density, J. Reine Angew. Math. (Crelle's J.) 595 (2006) 121–166].

In 1957, N. G. de Bruijn showed that the symmetric group Sym(\Omega) on an infinite set \Omega contains a free subgroup on 2^{card(\Omega)} generators, and proved a more general statement, a sample consequence of which is that for any group A of cardinality \leq card(\Omega), Sym(\Omega) contains a coproduct of 2^{card(\Omega)} copies of A, not only in the variety of all groups, but in any variety of groups to which A belongs. His key lemma is here generalized to an arbitrary variety of algebras \bf{V}, and formulated as a statement about functors Set --> \bf{V}. From this one easily obtains analogs of the results stated above with "group" and Sym(\Omega) replaced by "monoid" and the monoid Self(\Omega) of endomaps of \Omega, by "associative K-algebra" and the K-algebra End_K(V) of endomorphisms of a K-vector-space V with basis \Omega, and by "lattice" and the lattice Equiv(\Omega) of equivalence relations on \Omega. It is also shown, extending another result from de Bruijn's 1957 paper, that each of Sym(\Omega), Self(\Omega) and End_K (V) contains a coproduct of 2^{card(\Omega)} copies of itself. That paper also gave an example of a group of cardinality 2^{card(\Omega)} that was {\em not} embeddable in Sym(\Omega), and R. McKenzie subsequently established a large class of such examples. Those results are shown to be instances of a general property of the lattice of solution sets in Sym(\Omega) of sets of equations with constants in Sym(\Omega). Again, similar results -- this time of varying strengths -- are obtained for Self(\Omega), End_K (V), and Equiv(\Omega), and also for the monoid \Rel of binary relations on \Omega. Many open questions and areas for further investigation are noted., Comment: 37 pages. Copy at http://math.berkeley.edu/~gbergman/papers is likely to be updated more often than arXiv copy Revised version includes answers to some questions left open in first version, references to results of Wehrung answering some other questions, and some additional new results

Normalized irreducible characters of the symmetric group S(n) can be understood as zonal spherical functions of the Gelfand pair $(S(n)\times S(n),\Diag S(n))$. They form an orthogonal basis in the space of the functions on the group S(n) invariant with respect to conjugations by S(n). In this paper we consider a different Gelfand pair connected with the symmetric group, that is an ``unbalanced'' Gelfand pair $(S(n)\times S(n-1),\Diag S(n-1))$. Zonal spherical functions of this Gelfand pair form an orthogonal basis in a larger space of functions on S(n), namely in the space of functions invariant with respect to conjugations by $S(n-1)$. We refer to these zonal spherical functions as normalized \textit{generalized} characters of S(n). The main discovery of the present paper is that these generalized characters can be computed on the same level as the irreducible characters of the symmetric group. The paper gives a Murnaghan-Nakayama type rule, a Frobenius type formula, and an analogue of the determinantal formula for the generalized characters of S(n).

Many papers study polynomial tractability for multivariate problems. Let n(@?,d) be the minimal number of information evaluations needed to reduce the initial error by a factor of @? for a multivariate problem defined on a space of d-variate functions. Here, the initial error is the minimal error that can be achieved without sampling the function. Polynomial tractability means that n(@?,d) is bounded by a polynomial in @?^-^1 and d and this holds for all (@?^-^1,d)@?[1,~)xN. In this paper we study generalized tractability by verifying when n(@?,d) can be bounded by a power of T(@?^-^1,d) for all (@?^-^1,d)@[email protected], where @W can be a proper subset of [1,~)xN. Here T is a tractability function, which is non-decreasing in both variables and grows slower than exponentially to infinity. In this article we consider the set @W=[1,~)x{1,2,...,d^*}@?[1,@?"0^-^1)xN for some d^*>=1 and @?"[email protected]?(0,1). We study linear tensor product problems for which we can compute arbitrary linear functionals as information evaluations. We present necessary and sufficient conditions on T such that generalized tractability holds for linear tensor product problems. We show a number of examples for which polynomial tractability does not hold but generalized tractability does.

In the present paper we are investigating a certain point measure of a distribution function arising in a paper by Grabner et al. [Combinatorica 22 (2002) 245–267]. This distribution function is defined by means of the subtractive Euclidean algorithm and bears a striking resemblance to the singular?(x)-function of H. Minkowski. Beyond it, we will also consider a whole family of distribution functions arising in a natural way from the above ones. Nevertheless we will prove that all of the corresponding measures of the mentioned functions are mutually singular by using dynamical systems and the ergodic theorem.

Double Hurwitz numbers count branched covers of CP 1 with fixed branch points, with simple branching required over all but two points 0 and ∞ , and the branching over 0 and ∞ specified by partitions of the degree (with m and n parts, respectively). Single Hurwitz numbers (or more usually, Hurwitz numbers) have a rich structure, explored by many authors in fields as diverse as algebraic geometry, symplectic geometry, combinatorics, representation theory, and mathematical physics. The remarkable ELSV formula relates single Hurwitz numbers to intersection theory on the moduli space of curves. This connection has led to many consequences, including Okounkov and Pandharipande's proof of Witten's conjecture. In this paper, we determine the structure of double Hurwitz numbers using techniques from geometry, algebra, and representation theory. Our motivation is geometric: we give evidence that double Hurwitz numbers are top intersections on a moduli space of curves with a line bundle (a universal Picard variety). In particular, we prove a piecewise-polynomiality result analogous to that implied by the ELSV formula. In the case m = 1 (complete branching over one point) and n is arbitrary, we conjecture an ELSV-type formula, and show it to be true in genus 0 and 1. The corresponding Witten-type correlation function has a richer structure than that for single Hurwitz numbers, and we show that it satisfies many geometric properties, such as the string and dilaton equations, and an Itzykson–Zuber-style genus expansion ansatz. We give a symmetric function description of the double Hurwitz generating series, which leads to explicit formulae for double Hurwitz numbers with given m and n, as a function of genus. In the case where m is fixed but not necessarily 1, we prove a topological recursion on the corresponding generating series, which leads to closed-form expressions for double Hurwitz numbers and an analogue of the Goulden–Jackson polynomiality conjecture (an early conjectural variant of the ELSV formula). In a later paper (Faber's intersection number conjecture and genus 0 double Hurwitz numbers, 2005, in preparation), the formulae in genus 0 will be shown to be equivalent to the formulae for “top intersections” on the moduli space of smooth curves M g . For example, three formulae we give there will imply Faber's intersection number conjecture (in: Moduli of Curves and Abelian Varieties, Aspects of Mathematics, vol. E33, Vieweg, Braunschweig, 1999, pp. 109–129) in arbitrary genus with up to three points.

In this paper the authors investigate the representation type of the blocks of the relative (parabolic) category O S for complex semisimple Lie algebras. A complete classification of the blocks corresponding to regular weights is given. The main results of the paper provide a classification of the blocks in the “mixed” case when the simple roots corresponding to the singular set and S do not meet.

Suppose that X- = (X 0 , X 1 ) and Y-=(Y 0 ,Y 1 ) are Banach couples and suppose that T 0 : X 0 → Y 0 and T 1 : X 1 → Y 1 are bounded and linear. Also assume that Γ ∈ (Δ(X-))' and that T 0 and T 1 agree as maps from Δ(X-) ∩ ker Γ to Σ(Y-). If the maps do not agree as maps from all of Δ(X-) we cannot interpolate T 0 and T 1 to a map T : J 0,p (X- → J 0,p (Y-), where J 0,p denotes the classical J-method. This situation can for example be found in an article on interpolation of Hardy-type inequalities by Krugljak, Maligranda and Persson. We will in this paper define functors J 0,p;Γ such that T 0 and T 1 interpolate to a map T: J 0,p:Γ (X-) → J 0,p (Y-). The main purpose of this paper is to make the definition of the J 0,p;Γ (X-) spaces and build a theory for them. We will also do this for more general real parameters. If Γ is bounded on X 0 it holds that J 0,p:Γ (X-)=J 0,p (X 0 ∩ ker Γ, X 1 ). These spaces have been studied by Kalton, Ivanov and Lofstrom. Their results will follow as corollaries to the more general results of this article and our new theory can be thought of as a theory for generalized subcouples of codimension one. In the last section, we apply our theory to a situation considered by Krugljak, Maligranda and Persson in connection with Hardy-type inequalities. We prove new results and provide a new way of understanding that kind of problems.