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We describe a graph parametrization of rational quadratic differentials with presence of a simple pole, whose critical trajectories form a network depending on parameters focusing on the network topological jumps. Obtained bifurcation diagrams are associated with the Stasheff polytopes.

Let X X be the solution of a stochastic differential equation in Euclidean space driven by standard Brownian motion, with measurable drift and Sobolev diffusion coefficient. In our main result we show that when the drift is measurable and the diffusion coefficient belongs to an appropriate Sobolev space, the law of X X satisfies Talagrand’s inequality with respect to the uniform distance.

Watkins conjectured that for an elliptic curve $E$ over $\mathbb{Q}$ of Mordell-Weil rank $r$, the modular degree of $E$ is divisible by $2^r$. If $E$ has non-trivial rational $2$-torsion, we prove the conjecture for all the quadratic twists of $E$ by squarefree integers with sufficiently many prime factors.

When ordered by discriminant, it is known that about 83% of quartic fields over Q have associated Galois group S_4, while the remaining 17% have Galois group D_4. We study these proportions over a general number field F. We find that asymptotically 100% of quadratic number fields have more D_4 extensions than S_4 and that the ratio between the number of D_4 and S_4 quartic extensions is biased arbitrarily in favor of D_4 extensions. Under GRH, we give a lower bound that holds for general number fields., Fixed a typo with Theorem 1.3 that is present in the published version of the paper. The main results remain unchanged

We prove central limit theorems (under suitable growth conditions) for sums of quadratic characters, families of Hecke eigenforms of level 1 1 and weight k k , and families of elliptic curves, twisted by an L L -function satisfying certain properties. As a corollary, we obtain a central limit theorem for products χ ( p ) a f ( p ) \chi (p)a_f(p) where χ \chi is a quadratic Dirichlet character and f f is a normalized Hecke eigenform.

Let $M$ and $N$ be two monomials of the same degree, and let $I$ be the smallest Borel ideal containing $M$ and $N$. We show that the toric ring of $I$ is Koszul by constructing a quadratic Grobner basis for the associated toric ideal. Our proofs use the construction of graphs corresponding to fibers of the toric map. As a consequence, we conclude that the Rees algebra is also Koszul.

In 1984, the second author conjectured a quadratic transformation formula which relates two hypergeometric 2 F 1 {}_2\hspace {-1pt}F_1 functions over a finite field F q \mathbb {F}_q . We prove this conjecture in Theorem 2. The proof depends on a new linear transformation formula for pseudo-hypergeometric functions over F q \mathbb {F}_q . Theorem 2 is then applied to give an elegant new transformation formula (Theorem 3) for 2 F 1 {}_2\hspace {-1pt}F_1 functions over finite fields.

The algebraic underpinning of the tridiagonalization procedure is investigated. The focus is put on the tridiagonalization of the hypergeometric operator and its associated quadratic Jacobi algebra. It is shown that under tridiagonalization, the quadratic Jacobi algebra becomes the quadratic Racah–Wilson algebra associated to the generic Racah–Wilson polynomials. A degenerate case leading to the Hahn algebra is also discussed.

We show that convex entire solutions to Donaldson’s equation are quadratic, using a result of Weiyong He. We also exhibit entire solutions to the Donaldson equation that are not of the form discussed by He. In the process we discover some nontrivial entire solutions to complex Monge-Ampère equations.

We use the Burgess bound and combinatorial sieve to obtain an upper bound on the number of primes $p$ in a dyadic interval $[Q,2Q]$ for which a given interval $[u+1,u+\psi(Q)]$ does not contain a quadratic non-residue modulo $p$. The bound is nontrivial for any function $\psi(Q)\to\infty$ as $Q\to\infty$. This is an analogue of the well known estimates on the smallest quadratic non-residue modulo $p$ on average over primes $p$, which corresponds to the choice $u=0$.

Given a number field $K$ and a finite set $S$ of places of $K$, the first main result of this paper shows that the quadratic rational maps $\phi:{\mathbb P}^1\to{\mathbb P}^1$ defined over $K$ which have good reduction at all places outside $S$ comprise a Zariski-dense subset of the moduli space ${\mathcal M}_2$ parametrizing all isomorphism classes of quadratic rational maps. We then consider quadratic rational maps with double unramified fixed-point structure, and our second main result establishes a geometric Shafarevich-type non-Zariski-density result for the set of such maps with good reduction outside $S$. We also prove a variation of this result for quadratic rational maps with unramified 2-cycle structure., Comment: This version fixes a few LaTeX errors

We refine a result of R. Benedetto in p-adic analysis in order to exhibit infinitely many quadratic polynomials with rational coefficients having a specified graph of rational preperiodic points.

Let ϕ \phi be a morphism of P N \mathbb {P}^N defined over a number field K . K. We prove that there is a bound B B depending only on ϕ \phi such that every twist of ϕ \phi has no more than B B K K -rational preperiodic points. (This result is analogous to a result of Silverman for abelian varieties.) For two specific families of quadratic rational maps over Q \mathbb {Q} , we find the bound B B explicitly.

A classical theorem of Paley asserts the existence of an infinite family of quadratic characters whose character sums become exceptionally large. In this paper, we establish an analogous result for characters of any fixed even order. Previously our bounds were only known under the assumption of the Generalized Riemann Hypothesis., 6 pages; minor correction in Proposition 2.4

We prove cancellation in a sum of Fourier coefficents of a G L ( 3 ) GL(3) form F F twisted by additive characters, uniformly in the form F F . Previously, this type of result was available only when F F is a symmetric square lift.

The spider principle is used for establishing a formula for a finite quadratic sequence which determines the multiplicity sequences of all the sprouts which are founded upon the given finite quadratic sequence. This formula is basic for the theories of curvettes and dicriticals.

The generic incarnation of a quadratic transform is obtained by adjoining an indeterminate. This is applied to proving some useful facts about intersection, such as the commutativity of contact numbers and the mixed characteristic generalization of Max Noether’s Theorem on Refined Bezout.

We construct a compact Kähler manifold of nonnegative quadratic bisectional curvature which does not admit any Kähler metric of nonnegative orthogonal bisectional curvature. The manifold is a 7-dimensional Kähler C C -space with second Betti number equal to 1, and its canonical metric is a Kähler-Einstein metric of positive scalar curvature.

Let q q be a prime and γ = ( γ 0 , γ 1 , γ 2 , … ) \gamma =(\gamma _0,\,\gamma _1,\,\gamma _2,\,\dots ) with γ i ∈ { 0 , … , q − 1 } \gamma _i\in \{0,\,\dots ,\,q-1\} and γ i = 0 \gamma _i=0 for i ≥ i 0 i\geq i_0 be a weight sequence. We study lim N → ∞ # { 0 ≤ n > N | s q , γ ( n 2 ) ≡ d ( mod q ) } / N \lim _{N\rightarrow \infty }{\#\left \{0\leq n> N|s_{q,\gamma }(n^2)\equiv d\,(\text {mod }q)\right \}}/N with s q , γ ( n 2 ) s_{q,\gamma }(n^2) the weighted sum of digits of n 2 n^2 in base q q and we use the results to classify the digital sequences ( x n ) n ≥ 0 \left (\boldsymbol {x}_n\right )_{n\geq 0} in the sense of Niederreiter, generated by matrices with finite rows, for which ( x n 2 ) n ≥ 0 \left (\boldsymbol {x}_{n^2}\right )_{n\geq 0} is uniformly distributed. Finally we derive an upper bound for the star discrepancy of these uniformly distributed subsequences.

We study the factorization into irreducibles of iterates of a quadratic polynomial f f over a finite field. We call f f settled when the factorization of its n n th iterate for large n n is dominated by “stable” polynomials, namely those that are irreducible under post-composition by any iterate of f f . We prove that stable polynomials may be detected by their action on the critical orbit of f f and that the critical orbit also gives information about the splitting of non-stable polynomials under post-composition by iterates of f f . We then define a Markov process based on the critical orbit of f f and conjecture that its limiting distribution describes the full factorization of large iterates of f f . This conjecture implies that almost all quadratic f f defined over a finite field are settled. We give several types of evidence for our conjecture.

A. Mukhopadhyay, M. R. Murty and K. Srinivas have recently studied various arithmetic properties of the discriminant Δ n (a, b) of the trinomial f n,a,b (t) = t n + at + b, where n > 5 is a fixed integer. In particular, it is shown that, under the abc-conjecture, for every n ≡ 1 (mod 4), the quadratic fields ℚ (√Δn(a,b))are pairwise distinct for a positive proportion of such discriminants with integers a and b such that f n,a,b is irreducible over ℚ and |Δ n (a, b)| ≤ X, as X → ∞. We use the square-sieve and bounds of character sums to obtain a weaker but unconditional version of this result.

We give a complete list of all quadratic modules and inclusions between them in the ring R[[X]] of formal power series in one variable X over a euclidean field R.

Recently P. Hubert, C. Mauduit and A. Sarkozy introduced and studied the notion of pseudorandomness of binary lattices and gave a pseudorandom binary lattice. Later in other papers C. Mauduit and A. Sarkozy constructed some large families of "good" binary lattices. In this paper a large family of pseudorandom binary lattices is presented by using the multiplicative inverse and the quadratic character of finite fields.

For real C(K) spaces, we show that being injected in a Hilbert space is a 3-space property. As a consequence, we obtain that, when K does not carry a strictly positive Radon measure, every quadratic continuous homogeneous real-valued polynomial on C(K) admits a linear zero subspace enjoying a property which implies non-separability.

It is shown that over infinitely many imaginary quadratic fields there exists a Hermitian lattice in all even ranks n ≥ 2 n \ge 2 which is generated by its 4 n 4n minimal vectors but which is not generated by 2 n − 1 2n-1 minimal vectors.

This paper is concerned with the behaviour of the period function of the quadratic reversible centers. In this context the interesting stratum is the family of the so-called Loud's dehomogenized systems, namely { x = -y + xy, {y = x+Dx 2 +Fy 2 . In this paper we show that the bifurcation set of the period function of these centers is contained in the rectangle K = (-7, 2) x (0,4). More concretely, we prove that if (D, F) ∉ K, then the period function of the center is monotonically increasing.

A characterization is given for the integral binary quadratic forms for which the set of represented values is closed under products. It is also proved that for an integral binary quadratic lattice over a Dedekind domain, the product of three values represented by the form is again a value represented by the form. This generalizes the trigroup property observed by V. Arnold in the case of integral binary quadratic forms.

This paper is devoted to studying the period function of the quadratic reversible centers. In this context the interesting stratum is the family of the so-called Loud's dehomogenized systems, namely { x = -y + xy, { y = x + Dx 2 + Fy 2 . We determine several regions in the parameter plane for which the corresponding center has a monotonic period function. To this end we first show that any of these systems can be brought by means of a coordinate transformation to a potential system. Then we apply a monotonicity criterium of R. Schaaf.

This paper provides a description for the quadratic polynomials on the boundary of the Mandelbrot set M which are typical in the sense of harmonic measure. In particular, it is shown that a typical point on the boundary of M has a unique parameter ray landing on it. Applications of this result in the study of embedded arcs in M and the lamination associated with M are given.

We prove that the set of Lévy constants for quadratic irrationals is dense in [ log ⁡ 5 + 1 2 , ∞ ) [\log \frac {\sqrt {5}+1}{2}, \infty ) .

Let b 1 , … , b 5 b_1, \ldots , b_5 be non-zero integers and n n any integer. Suppose that b 1 + ⋯ + b 5 ≡ n ( mod 24 ) b_1+\cdots +b_5 \equiv n \pmod {24} and ( b i , b j ) = 1 (b_i,b_j)=1 for 1 ≤ i > j ≤ 5 1 \leq i > j \leq 5 . In this paper we prove that (i) if the b j b_j are not all of the same sign, then the above quadratic equation has prime solutions satisfying p j ≪ | n | + max { | b j | } 25 / 2 + ε ; p_j\ll \sqrt {|n|}+ \max \{|b_j|\}^{25/2+\varepsilon }; and (ii) if all the b j b_j are positive and n ≫ max { | b j | } 26 + ε n \gg \max \{|b_j|\}^{26+\varepsilon } , then the quadratic equation b 1 p 1 2 + ⋯ + b 5 p 5 2 = n b_1p_1^2+\cdots +b_5p_5^2=n is soluble in primes p j . p_j. Our previous results are max { | b j | } 20 + ε \max \{|b_j|\}^{20+\varepsilon } and max { | b j | } 41 + ε \max \{|b_j|\}^{41+\varepsilon } in place of max { | b j | } 25 / 2 + ε \max \{|b_j|\}^{25/2+\varepsilon } and max { | b j | } 26 + ε \max \{|b_j|\}^{26+\varepsilon } above, respectively.

We show that there are ≫ q l / ( 2 g ) \gg q^{l/(2g)} polynomials D ∈ F q [ t ] D\in \mathbb {F}_q[t] with deg ⁡ ( D ) ≤ l \deg (D)\le l such that the ideal class group of the real quadratic extensions F q ( t , D ) \mathbb {F}_q(t,\sqrt D) has an element of order g g .

We consider associative PI-algebras over a field of characteristic zero. We study the asymptotic behavior of the sequence of multiplicities of the cocharacters for some significant classes of algebras. We also give a characterization of finitely generated algebras for which this behavior is linear or quadratic.

For m ≥ 1, let α: α 0 < …< α m be a collection of (m+1) positive weights. The Quadratically Hyponormal Completion Problem seeks necessary and sufficient conditions on a to guarantee the existence of a quadratically hyponormal unilateral weighted shift W with a as the initial segment of weights. We prove that a admits a quadratically hyponormal completion if and only if the self-adjoint m x m matrix formula math. is positive and invertible, where q k := u k + |s| 2 v k , r k := s√w k , u k := α 2 k-α 2 k-1, v k := α 2 kα 2 k+1-α 2 k-1α 2 k-2, w k := α 2 k(α 2 k+1-α 2 k-1) 2 , and, for notational convenience, α -2 = α -1 = 0. As a particular case, this result shows that a collection of four positive numbers α 0 < α 1 < α 2 < α 3 always admits a quadratically hyponormal completion. This provides a new qualitative criterion to distinguish quadratic hyponormality from 2-hyponormality.

For bounded linear operators on Hilbert space, positive quadratic hyponormality is a property strictly between subnormality and hyponormality and which is of use in exploring the gap between these more familiar properties. Recently several related positively quadratically hyponormal weighted shifts have been constructed. In this note we establish general criteria for the positive quadratic hyponormality of weighted shifts which easily yield the results for these examples and other such weighted shifts.

We give a class of quadratic systems without rational first integral which contains irreducible algebraic solutions of arbitrarily high degree. The construction gives a negative answer to a conjecture of Lins Neto and others.