pinecone/core-2020-05-10-deduplication huggingface dataset:
AIC, Overfitting Principles, and the Boundedness of Moments of Inverse Matrices for Vector Autotregressions and Related Models

AbstractIn his somewhat informal derivation, Akaike (in “Proceedings of the 2nd International Symposium Information Theory” (C. B. Petrov and F. Csaki, Eds.), pp. 610–624, Academici Kiado, Budapest, 1973) obtained AIC's parameter-count adjustment to the log-likelihood as a bias correction: it yields an asymptotically unbiased estimate of the quantity that measures the average fit of the estimated model to an independent replicate of the data used for estimation. We present the first mathematically complete derivation of an analogous property of AIC for comparing vector autoregressions fit to weakly stationary series. As a preparatory result, we derive a very general “overfitting principle,” first formulated in a more limited context in Findley (Ann. Inst. Statist. Math.43, 509–514, 1991), asserting that a natural measure of an estimated model's overfit due to parameter estimation is equal, asymptotically, to a measure of its accuracy loss with independent replicates. A formal principle of parsimony for fitted models is obtained from this, which for nested models, covers the situation in which all models considered are misspecified. To prove these results, we establish a set of general conditions under which, for each τ⩾1, the absolute τth moments of the entries of the inverse matrices associated with least squares estimation are bounded for sufficiently large sample sizes
The Density ofBh[g] Sequences and the Minimum of Dense Cosine Sums

AbstractA setEof integers is called aBh[g] set if every integer can be written in at mostgdifferent ways as a sum ofhelements ofE. We give an upper bound for the size of aBh[1] subset {n1, …,nk} of {1, …,n} wheneverh=2mis an even integer:[formula]For the caseh=2 (h=4) this has already been proved by Erdo&#x030B;s and Turán (by Lindström). It has been independently proved for all evenhby Jia [9] who used an elementary combinatorial argument. Our method uses a result, which we prove, related to the minimum of dense cosine sums which roughly states that if 1⩽λ1<…<λN⩽(2−ε)NareNdifferent integers then[formula]Finally we exhibit some dense finite and infiniteB2[2] sequences
On Prielipp's Problem on Signed Sums ofkth Powers

AbstractIn this paper we consider the representation of and integer,n, in the form[formula]wheremis a positive integer, depending onnand theεiare all either ± the particular choices of theεidepending onnandm. Among other things we prove such a representation always exists; in fact, infinitely many exist. The proof is algorithmic, so that a polynomial time method of finding the representation is presented. We get asymptotic estimates for the minimal value ofm, in each of the cases (a) kfixed andngrows to infinity and (b) nfixed andkgrows to infinity. A number of related problems and conjectures are also presented
Class Groups and Selmer Groups

AbstractIt is often the case that a Selmer group of an abelian variety and a group related to an ideal class group can both be naturally embedded into the same cohomology group. One hopes to compute one from the other by finding how close each is to their intersection. In this paper we compute the two groups and their intersection explicitly in the local case and put together the local information to get sharp upper bounds in the global case. The techniques in this paper can be used for arbitrary abelian varieties, isogenies and number fields assuming a frequently occurring condition. Several examples are worked out for the Jacobians of elliptic and hyperelliptic curves
Complex Multiplication Structure of Elliptic Curves

AbstractLetkbe a finite field and letEbe an elliptic curve overk. In this paper we describe, for each finite extensionlofk, the structure of the groupE(l) of points ofEoverlas a module over the ringRof endomorphisms ofEthat are defined overk. If the Frobenius endomorphismπofEoverkdoes not belong to the subringZofR, then we find thatE(l)≅R/R(πn−1), wherenis the degree ofloverk; and ifπdoes belong toZthenE(l) is, as anR-module, characterized byE(l)⊕E(l)≅R/R(πn−1). The arguments used in the proof of these statements generalize to yield a description of the group of points of an elliptic curve over an algebraically closed field as a module over suitable subrings of the endomorphism ring of the curve. It is shown that straightforward generalizations of the results of this paper to abelian varieties of dimension greater than 1 cannot be expected to exist
Jacobi Forms of Several Variables and the Maaß Space

AbstractIn this paper we describe the space of Jacobi forms on H×Cn. This type of Jacobi forms appears in the Fourier–Jacobi expansions of all kinds of modular forms of several variables. We can characterize the space of Jacobi forms of weightkby vector valued elliptic modular forms of weightk−n/2. Then we use the Jordan theoretic language in order to describe modular forms on the orthogonal groupO(2, n+2), whose Fourier–Jacobi expansions also yield Jacobi forms of our type. Finally we determine a Maaß space as the isomorphic image of a particular space of Jacobi forms. Especially our procedure guarantees the existence of certain non-trivial singular modular forms
Bh[g]-Sequences with Large Upper Density

AbstractFor any positive integersh, g, letm=m(h, g) be the largest possible integer such that for everya∈Zmthe equationa≡x1+x2+…+xh(modm) has at mostgsolutions (up to rearrangement of thexi's). It is proved in this paper that there exists aBh[g]-sequenceB⊆[1, n] with[formula]In particular, there exists aB2[2]-sequenceB⊆[1, n] such that[formula]which improves an earlier result of Kolountzakis. It is also proved that for everyg⩾2 there exists an infiniteB2[g]-sequenceB={n1<n2<…} with[formula]Some open problems are also discussed in this paper
Average Distribution of Supersingular Drinfeld Modules

AbstractLetφbe a rank 2 Drinfeld module defined over Fq(T). For each monic prime polynomialp∈Fq(T) which is a regular prime ofφ, the reduction ofφatpis a rank 2 Drinfeld moduleφpover the finite field Fq(T)/(p); depending on the structure of the ring End(φp), the regular primepis either a supersingular or an ordinary prime ofφ. We prove in this paper that,on average, supersingular primes are distributed according to the Lang-Trotter conjecture (for Drinfeld modules). We first show this result averaging over all Drinfeld modules, and then over all isomorphism classes of Drinfeld modules
Substitution and Closure of Sets under Integer-Valued Polynomials

AbstractLetRbe a domain andKits quotient-field. For a subsetSofK, let FR(S) be the set of polynomialsf∈K[x] withf(S)⊆Rand define theR-closure ofSas the set of thoset∈Kfor whichf(t)∈Rfor allf∈FR(S). The concept ofR-closure was introduced by McQuillan (J. Number Theory39(1991), 245–250), who gave a description in terms of closure inP-adic topology, whenRis a Dedekind ring with finite residue fields. We introduce a toplogy related to, but weaker thanP-adic topology, which allows us to treat ideals of infinite index, and derive a characterization ofR-closure whenRis a Krull ring. This gives us a criterion for FR(S)=FR(T), whereSandTare subsets ofK. As a corollary we get a generalization to Krull rings ofR. Gilmer's result (J. Number Theory33(1989), 95–100) characterizing those subsetsSof a Dedekind ring with finite residue fields for which FR(S)=FR(R)
Rational Period Functions and Parabolic Cohomology

AbstractIn [K], M. Knopp defined a generalization of Eichler cohomology by considering rational functions as possible periods for the action, by way of the usual slash operator, of a Fuchsian group upon functions defined on the upper half-plane. This theory of rational period functions already enjoys a rich history. A significant step was made by A. Ash [A], who applied cohomological techniques in the setting of the finite index subgroups of the modular group to provide a classification of their rational period functions. Here we show that Ash's theorem, with appropriate adjustments, is valid for all finitely generated Fuchsian groups of the first kind with parabolic elements. Ash's proof relies heavily upon the Borel–Serre compactification forarithmeticgroups. We show that this compactification is valid in our wider setting and proceed to give a simplified version of the Ash proof. Applications to the classification of rational period functions of the Hecke groups are provided
Simultaneous Representation of Integers by a Pair of Ternary Quadratic Forms—With an Application to Index Form Equations in Quartic Number Fields

AbstractLetQ1, Q2∈Z[X, Y, Z] be two ternary quadratic forms andu1, u2∈Z. In this paper we consider the problem of solving the system of equations[formula]According to Mordell [12] the coprime solutions of[formula]can be presented by finitely many expressions of the formx=fx(p, q),y=fy(p, q),z=fz(p, q), wherefx, fy, fz∈Z[P, Q] are binary quadratic forms andp, qare coprime integers. Substituting these expressions into one of the equations of (1), we obtain a quartic homogeneous equation in two variables. If it is irreducible it is a quartic Thue equation, otherwise it can be solved even easier. The finitely many solutionsp, qof that equation then yield all solutionsx, y, zof (1). We also discuss two applications. In [8] we showed that the problem of solving index form equations in quartic numbers fieldsKcan be reduced to the resolution of a cubic equationF(u, v)=iand a corresponding system of quadratic equationsQ1(x, y, z)=u,Q2(x, y, z)=v, whereFis a binary cubic form andQ1, Q2are ternary quadratic forms. In this case the application of the new ideas for the resolution of (1) leads to quartic Thue equations which split over the same quartic fieldK. The second application is to the calculation of all integral points of an elliptic curve
Recurrent Sequences and Affine Functional Equations

AbstractSárközy and other authors have characterized the multiplicative and the additive sequences among the solutionsg:N→C of homogeneous linear recurrence equations with complex coefficients. Their results are special cases of a much more general theorem concerning recurrent sequencesgsatisfying certain functional equations of the typeg(nq+a)=q*g(n)+a* (n=1, …, N) for sufficiently largeq, N∈N with some coefficientsa∈N,a*, q*∈C,q*≠0. The results and methods of proof given in the present paper are new
Haar Bases forL2(Rn) and Algebraic Number Theory

AbstractGröchenig and Madych showed that a Haar-type orthonormal wavelet basis ofL2(Rn) can be constructed from the characteristic functionχQof a setQif and only ifQis an affine image of an integral self-affine tileTwhich tiles Rnusing the integer lattice Zn. An integral self-affine tileT=T(A, D) is the attractor of an iterated function systemT=∪mi=1A−1(T+di) where A∈Mn(Z) is an expandingn×ninteger matrix and the digit set D={d1, d2, …, dm}⊆Znhasm=|det(A)|, provided that the Lebesgue measureμ(T)>0. Two necessary conditions forT(A, D) to tile Rnwith the integer lattice Znare that D be a complete set of coset representatives of Zn/A(Zn) and that Z[A, D]=Zn, where Z[A, D] is the smallest A-invariant lattice containing all {di−dj:i≠j}. These two conditions are necessary and sufficient in the special case that |det(A)|=2. We study these two conditions for an arbitrary matrix A∈Mn(Z). We prove that a digit set D satisfying the two conditions exists whenever |det(A)|⩾n+1. When |det(A)|=2 there are number-theoretic obstructions to the existence of such D. Using these we exhibit a (non-expanding) A∈M2(Z) for which no digit set has Z[A, D]=Z2. However we show that for all expanding integer matrices A in dimensions 2 and 3, there exists some digit set D that satisfies the two conditions. Could this be true for all expanding integer matrices in dimensionsn⩾4? A necessary condition is that the (non-Galois) field[formula]have class number one for alln⩾4
On the Order of Finitely Generated Subgroups of Q*(modp) and Divisors ofp−1

AbstractLetΓbe a finitely generated subgroup of Q* with rankr. We study the size of the order |Γp| ofΓmodpfor density-one sets of primes. Using a result on the scarcity of primesp⩽xfor whichp−1 has a divisor in an interval of the type [y,  yexplogτy] (τ∼0.15), we deduce that |Γp|⩾pr/(r+1)explogτpfor almost allpand, assuming the Generalized Riemann Hypothesis, we show that |Γp|⩾p/ψ(p) (ψ→∞) for almost allp. We also apply this to the Brown–Zassenhaus Conjecture concerned with minimal sets of generators for primitive roots
Quintic Forms overp-adic Fields

AbstractWe prove that a quintic form in 26 variables defined over ap-adic fieldKalways has a nontrivial zero overKif the residue class field ofKhas at least 47 elements. This is in agreement with the theorem of Ax–Kochen which states that a homogeneous form of degreedind2+1 variables defined overQphas a nontrivialQp-rational zero ifpis sufficiently large. The Ax–Kochen theorem gives no results on the bound forp. Ford=1, 2, 3 it has been known for a long time that there is a nontrivialQp-rational zero for all values ofp. Ford=4, Terjanian gave an example of a form in 18 variables overQ2having no nontrivialQ2-rational zero. This is the first result which gives an effective bound for the cased=5
Computation of Partial Zeta Values ats=0 over a Totally Real Cubic Field

AbstractIn theory the problem of computing the special values of partial zeta functions of a totally real field was solved by Shintani. In practice it rarely works as the fundamental domain of the action of the relevant unit group is too cumbersome to apply in a computational setting. The object of this article is to give a computationally efficient version of Shintani's approach in the cubic case. The basic idea is to take the convex closure of the lattice points, in the first octant, and use the boundary points, of this convex set, to construct a more amenable fundamental domain. Thus the basic thrust is to arrive at an algorithm to find points on this boundary. For us the case of interest is whens=0. In this case we find a cubic analogue of the CF-formula for real quadratic number fields
On the Algebraic Structure of Gaussian Periods

AbstractWe show that the classical cyclotomic constants associated to a primepand a positive integerndividingp−1 can be used to define an algebraic object which we call an assembly and which is close to being a finite commutative hypergroup. We study these objects abstractly and show that subject to a condition of cyclical symmetry they are determined uniquely by two quantities which we call total mass and order. This shows that the theory of cyclotomy can in some sense be viewed as a chapter of abstract algebra
On Denominators of Algebraic Numbers and Integer Polynomials

AbstractLetA(x)=adxd+…+a0be the minimal polynomial ofαover Z. Recall that the denominator ofα, denoted den(α), is defined as the least positive integernfor whichnαis an algebraic integer. It is well known that den(α)∣ad. In this paper we study the density of algebraic numbersαof fixed degreedsuch that den(α)=ad. We show that this density is given by[formula]Note that the above density approaches 1/ζ(3) asd→∞. As a result, we show, loosely speaking, that the chance that an algebraic numberαsatisfies den(α)=adis 1/ζ(3). In order to prove these results we introduce the concept of the denominator of an integer polynomialA. Several formulas for computing denominators of integer polynomials are derived
A Polynomial Time Algorithm for the Hausdorff Dimension of Continued Fraction Cantor Sets

AbstractFor any finite setAof positive integers, letEA:={α∈(0, 1):αis irrational, and every partial quotient in the (infinite) simple continued fraction expansion ofαis an element ofA}. For setsAwith fewer than two elements,EAis uninteresting. For |A|⩾2,EAis a kind of Cantor fractal dust, with a Hausdorff dimension (dimEA) between 0 and 1. This work presents an algorithm which, given a finite setAof between 2 andNpositive integers 2N, determines dimEAto within ±2−NusingO(N7) elementary bit operations. There is also a convenient implementation of the algorithm in Mathematica® code, together with a small table and some conjectures
On a Question of Davenport

AbstractLetkbe a number field and denote by okits ring of integers. Let p be a non-zero prime ideal of ok. Denote byfthe polynomial derived fromfby reducing the coefficients modulo p. SetVp(f)={f(u)∣u∈ok/p}. Davenport raised the following question (withkbeing the rationals). Supposefandgare polynomials in ok[X] such thatVp(f)=Vp(g) for all but finitely many non-zero prime ideals of ok. Does this implyf(X)=g(aX+b) for somea, b∈k? Extending work of M. Fried, we give an affirmative answer under rather general conditions, and also new types of counter-examples
Structure Theorem for Multiple Addition and the Frobenius Problem

AbstractLetA⊆[0; l] be a set ofnintegers, and leth⩾2. By how much does |hA| exceed |(h−1)A| ? How can one estimate |hA| in terms ofn, l? We give sharp lower bounds extending and generalizing the well-known theorem of Freiman for |2A|. A number of applications are provided as well. In particular, we give a solution for the old extremal problem of Frobenius–Erdo&#x030B;s–Graham concerning estimating of the largest integer, non-representable by a linear form. In a sense, our solution can not be improved
Symmetric and Asymmetric Primes

AbstractIn a well-known proof of the quadratic reciprocity law, one counts the lattice points inside the rectangle with sides parallel to the axes and opposite vertices at the origin and (p/2, q/2), wherepandqare distinct odd primes. In particular, the Legendre symbols (p/q) and (q/p) depend, respectively, on the number of lattice points in the rectangle above and below the main diagonal. Sayp,nbsp;qform asymmetric pairif the number of lattice points above the main diagonal is equal to the number of lattice points below. Say a primepissymmetricif it belongs to some symmetric pair, and otherwise call itasymmetric. We first characterize symmetric pairsp, qwith the condition (p−1, q−1)=|p−q|. In particular, twin primes form a symmetric pair. Of the first 100,000 odd primes, about 5/6 of them are symmetric. However, we are able to prove that, asymptotically, almost all primes are asymmetric
The Truncated Second Main Theorem of Function Fields

AbstractWe formulate some results of function fields which are analogous to Cartan's Second Main Theorem with truncated counting functions, Cartan's conjecture, and the generalized Picard Theorem. We also give a generalization of function fields version ofabc-conjecture due to Mason, Voloch, Brownawell, and Masser. A result of the finiteness of theS-integral points of function fields of characteristic 0 in the complement of 2n+1 hyperplanes with constant coefficients and in general position is obtained in this paper
Simultaneous Diophantine Approximation and Asymptotic Formulae on Manifolds

AbstractLetψ(r),r=1, 2, … be a positive decreasing sequence such that ∑r=1∞ψ(r)kdiverges. Using a powerful variance argument due to Schmidt, an asymptotic formula is obtained for the number of integer solutions q of the system of Diophantine inequalities[formula]which holds for almost all points (x1, …, xk) on a smoothm-dimensional submanifoldMof Rk. The manifold satisfies certain curvature conditions which entail restrictions on the codimension. This result extends the known result when the 0?points are not constrained to lie in a submanifold, (i.e., whenM=Rk) to a reasonably general class of manifolds
Diophantine Undecidability over Algebraic Function Fields over Finite Fields of Constants

AbstractWe show that Diophantine problem (otherwise known as Hilbert's Tenth Problem) is undecidable over the fields of algebraic functions over the finite fields of constants of characteristic greater than two. This is the first example of Diophantine undecidability over any algebraic field. We also show that the Diophantine class of a holomorphy ring of an above mentioned algebraic function field does not change if the set of primes at which the functions of the ring are allowed to have poles is changed by adding or removing of finitely many primes
On Mordell's Inverse Problem in Dimension Three

AbstractGiven a latticeL=AZ3with determinantd(L)=|det(A)|>0, letκ(L)= sup{vol(P)/(8d(L)} where the supremum is taken over allo-symmetric parallelepipedsPwith faces parallel to the coordinate axes such thatP∩L={o}. We prove the following conjecture by Gruber: The absolute minimum of the functionκ(L) has value 8/7cos2(π/7) cos(2π/7)=0.578416…=κ(L*) and is uniquely attained at the critical latticeL* of the star body |x1x2x3|⩽1. Moreover, this minimum is isolated: There exists a positiveηsuch thatκ(L)>κ(L*)+ηholds for any lattice which is not equivalent toL*. We also state a conjecture concerning higher minima ofκ(L)
Arithmetic of Weighted Diagonal Surfaces over Finite Fields

AbstractIn this paper, we discuss several fundamental properties of weighted diagonal surfaces over finite fields. Weighted diagonal surfaces are defined to be diagonal surfaces in weighted projective 3-spaces. In general, they have cyclic quotient singularities. For our arithmetic investigations, we choose their minimal resolutions. We are particularly interested in their Picard numbers and the orders of their Brauer groups. First we prove that the (congruence) zeta-functions of minimal resolutions of weighted diagonal surfaces can be described in terms of twisted Jacobi sums. The proof is based on an explicit description of the set of rational points in weighted projective spaces. Secondly, we give a formula for the Picard numbers of minimal resolutions of weighted diagonal surfaces. The Tate conjecture asserts that their Picard numbers are equal to the orders of certain poles of their zeta-functions. Thus, we first prove the Tate conjecture for the minimal resolutions. Then we calculate the orders of poles of their zeta-functions. Here the validity of the conjecture will be established, in fact, for more general surfaces. Thirdly, we look into quantities associated to special values of zeta-functions, especially the orders of Brauer groups, of certain minimal resolutions of weighted diagonal surfaces. Specifically, we confine ourselves to the minimal resolutions whose Néron–Severi groups have Z-bases consisting only of canonical divisors and exceptional divisors. For such surfaces, we compute the orders of Brauer groups in terms of twisted Jacobi sums
The Distribution of Zeros of an Irreducible Curve over a Finite Field

AbstractLetf(x, y) be a polynomial defined overZin two variables of total degreed⩾2, and letVp={(x, y)∈Cp: f(x, y)≡0(modp)} for each primep, whereCp={(x, y)∈Z2: 0⩽x<pand 0⩽y<p}. In this paper, we show that iff(x, y) is absolutely irreducible modulopfor all sufficiently largep, then we have the following distribution formula for the zeros off(x, y) modulop,[formula]whereδis any given real number with 0⩽δ⩽1, and[formula]Here ‖θ‖ denotes the nearest distance fromθto integer, andβis a real number with 0⩽β⩽12
p-Adic Limits of Shintani Generating Functions for a Real Quadratic Field

AbstractFor a totally real number fieldF, the values of partial ray-class zeta-functions at −k,k∈N, may be calculated using generating functions due to Shintani. These are associated to “cone decompositions” inF⊗R which, however, seem rather non-canonical forF≠Q. Motivated by Iwasawa Theory andp-adicL-functions, we obtain new generating functions by passing to a certainp-adic limit in the case [F: Q]=2,h+(F)=1. We investigate these limiting functions, showing that they possess several pleasing properties not shared by Shintani's
The Diophantine Equation 3u2−2=v6

AbstractThe theorem of Delaunay–Nagell states that:If d is a cube-free integer>1,then the equation x3+dy3=1has at most one solution in non-zero integers x, y, and if such a solution exists then[formula]is either the fundamental unit of the field[formula]or its square, the latter occurring for only finitely many values of d. Investigation of these exceptionaldvalues has led to the equation of the title [5, 3.9], which has only finitely many solutions. We prove that the title equation has no integer solution other than |u|=|v|=1, which give the known valuesd=19, 20, 28, therefore there are no otherdvalues