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We bound the third Chern number of a minimal smooth threefold with ample canonical bundle by quadratic functions in the first two Chern numbers.

We give a lower bound on the number of distinct dimensions of irreducible components of the Hilbert scheme of codimension 2 subvarieties in P1W, for n < 5 (respectively, the moduli space of surfaces or 3-folds) in terms of the Hilbert polynomial (resp. Chern numbers). Let Hilbp be the Hilbert scheme of subvarieties in the projective space with fixed Hilbert polynomial P (respectively, let M be a moduli space of varieties with fixed Chern numbers). It is known that Hilbp (resp. M) has finitely many irreducible components and that the number of these components is bounded by some function of the Hilbert polynomial (resp. the Chern numbers). For work on the number of components of the Hilbert scheme (resp. the moduli space), see [EHM] for curves in P3 and [Chl] for codimension 2 subvarieties in Ipn with n ?5 (resp. [Cal], [Ca2], [Ca3], [M] for surfaces and [Chl] for surfaces and 3-folds). The next question to ask is whether the Hilbert scheme (resp. moduli space) is equidimensional if it is reducible. Catanese [Ca3] has shown that for M, the moduli space of surfaces, the number of distinct dimensions can be arbitrarily large. In this note we study the number of distinct dimensions of the components of the Hilbert scheme Hilbp (resp. moduli space M) parametrizing subschemes with intersection numbers H'Kn-2-i (resp. Chern numbers), where H is the hyperplane class and K is the canonical class. We define n(d, HK, K2) = #{dimHIH is a component of the Hilbert scheme of surfaces in JR'4 with intersection numbers d, HK, K2 }, n(d, H2K, HK2, K3) = #{dim HIH is a component of the Hilbert schemes of 3-folds in I5 with intersection numbers d, H2K, HK2, K3}, n(K2, C2) = #{dim HIH is a component of the moduli space of surfaces with Chern numbers K2, c21, n(K3,c1c2,c3) = #{dimHIH is a component of the moduli space of 3-folds with Chern numbers K3, C1 C2, C3 }. Received by the editors October 5, 1995 and, in revised form, March 14, 1996. 1991 Mathematics Subject Classification. Primary 14J29; Secondary 14M07, 14M12.

Let $p$ be the characteristic of $\mathbb{F}_q$ and let $q$ be a primitive root modulo a prime $r = 2n + 1$. Let $\beta \in \mathbb{F}_{q^{2n}}$ be a primitive $r$th root of unity. We prove that the multiplicative order of the Gauss period $\beta + \beta^{-1}$ is at least $(\log p)^{c\log n}$ for some $c \gt 0$. This improves the bound obtained by Ahmadi, Shparlinski and Voloch when $p$ is very large compared with $n$. We also obtain bounds for "most" $p$.

We experimentally investigate the Lagrangian-Eulerian dynamics of fluid motion and wave-form evolution for a breaking shallow water wave approaching a slope by tracking tracer motions. It is found that, before breaking, the surface element can climb over the crest and exhibits cyclic oscillation with small forward drift. The increasing asymmetric tangential compression (accumulation) and rarefaction (depletion) in the crest front and the crest are the keys for the crest front steepening with the increasing particle cyclic excursion and forward Stoke drift. Eventually, the surface layer cannot climb over the crest with the vertical front. It curls up and forms an overhanging plunging jet leading the crest, while the lower flow still can reach the crest rear. This process leads to wave breaking with the rapid drop of crest height and the transition from slow divergence to rapid divergence of the adjacent fluid trajectories.

Intermittent dust acoustic wave turbulence self-excited by downward ion flow in dissipative dusty plasma is experimentally observed and investigated. The power spectra of the temporal dust density fluctuation show distinct bumps in the low-frequency regime and power-law scaling in the high-frequency regime. The structure-function analysis demonstrates the multifractal dynamics of the wave turbulence. Decreasing dissipation by decreasing neutral pressure leads to a more turbulent state with a less distinct low-frequency bump in the power spectrum, more stretched non-Gaussian tails in the histogram of the wave-height increment at a small time interval $\ensuremath{\tau}$, and a higher degree of multifractality. The loss of long time memory with increasing $\ensuremath{\tau}$ for a more turbulent state causes a change from the distribution with stretched non-Gaussian tails to Gaussian with increasing $\ensuremath{\tau}$.

We establish new estimates on short character sums for arbitrary composite moduli with small prime factors. Our main result improves on the Graham-Ringrose bound for square-free moduli and also on the result due to Gallagher and Iwaniec when the core q′ = Π p|q p of the modulus q satisfies log q′ ∼ log q. Some applications to zero free regions of Dirichlet L-functions and the Polya and Vinogradov inequalities are indicated.

We experimentally investigate the micro-origin of the absence of trough trapping in nonlinear traveling dust acoustic waves self-excited by the downward ion flow in the dissipative dusty plasma. The wave forms of dust density, the drag force from the background neutrals, ions, and dusts, and the effective potential energy for dusts are constructed by tracking dust motion and measuring the velocity and the position-dependent forces. The tilted washboard type potential wave form with a slight phase lead to the dust density wave form is obtained. It provides sufficient kinetic energy to compensate drag dissipation and move dusts from the dust density trough to the crest front. The dusts with sufficient energy overcome the downward pushing by the crest front, climb over the crest, and sustain the oscillatory motion with upward drift. Those dusts with insufficient energy to climb over the potential barrier of the crest are trapped in and move downward with the crest front, until kicked upward by fluctuation. The upward neutral dominated drag force prevents them from sliding down the potential energy hill at the crest front and further oscillating in the trough. It leads to the absence of trough trapping.

We investigate experimentally the dynamics of the wake field and the high speed projectile travelling along the channels formed by the surrounding long vertical chains of the chain‐bundle dusty plasma liquids (CBDPLs) consisting of negatively charged dusts suspended in weakly ionized low pressure discharges, in which the downward ion flow not only causes the formation of long vertical chains but also can temporarily sustain the downward dust acoustic waves trigged by external perturbations. For the supersonic projectile channeling in the quiescent CBDPL, the excitation of the downward travelling narrow wake field associated with the elliptical motions of the background dusts is observed. The continuous increase and the subsequent oscillation of the drag force on the projectile as the projectile speed goes below the resonant velocity for wave excitation manifest the strong projectile‐wake field interaction, especially after the wake field catches up the subsonic projectile. The acceleration and surfing of ...

Large amplitude dust acoustic wave can be self‐excited by the strong downward ion flow in a dusty plasma liquid formed by negatively charged dusts suspended in a weakly ionized low pressure discharge. In this work, we investigate experimentally the wave‐particle phase space dynamics of the large amplitude dust acoustic wave by connecting the Lagrangian and Eulerian views, through directly tracking particle motion and measuring local dust density fluctuations. The microscopic pictures of wave steepening and breaking, resonant particle‐wave crest trapping, and the absence of trough trapping observed in our experiment are constructed.

The wave-particle microdynamics in the breaking of the self-excited dust acoustic wave growing in a dusty plasma liquid is investigated through directly tracking dust micromotion. It is found that the nonlinear wave growth and steepening stop as the mean oscillating amplitude of dust displacement reaches about 1/k (k is the wave number), where the vertical neighboring dust trajectories start to crossover and the resonant wave heating with uncertain crest trapping onsets. The dephased dust oscillations cause the abrupt dropping and broadening of the wave crest after breaking, accompanied by the transition from the liquid phase with coherent dust oscillation to the gas phase with chaotic dust oscillation. Corkscrew-shaped phase-space distributions measured at the fixed phases of the wave oscillation cycle clearly indicate how dusts move in and constitute the evolving waveform through dust-wave interaction.

The purpose of the paper is to present new estimates on incomplete character sums in finite fields that are of the strength of Burgess’ celebrated theorem for prime fields. More precisely, an inequality of this type is obtained in $${F_{p^2}}$$ and also for binary quadratic forms, improving on the work of Davenport–Lewis and on several results due to Burgess. The arguments are based on new estimates for the multiplicative energy of certain sets that allow us to improve the amplification step in Burgess’ method.

Let $\chi$ be a nontrivial multiplicative character of $\Bbb F_{p^n}$ . We obtain the following results. ¶ (1) Let $\varepsilon>0$ be given. If $B=\big\{ \sum _{j=1}^n x_j \omega_j \;: x_j \in [N_j+1, N_j+H_j]\cap \Bbb Z , j=1,\ldots, n %\text{ for all } j \big\}$ is a box satisfying ${\mathop\Pi\limits}_{j=1}^{n}H_j>p^{({2}/{5}+\varepsilon)n},$ then for $p>p(\varepsilon)$ we have, denoting $\chi$ a nontrivial multiplicative character, \[ \Big| \sum_{x \in B} \chi(x) \Big| \ll_np^{-{\varepsilon^{2}}/4} |B| \] unless $n$ is even, $\chi$ is principal on a subfield $F_2$ of size $p^{n/2}$ , and $\max_\xi\!\!|B\cap \xi F_2| >p^{-\varepsilon}|B|$ . ¶ (2) Assume that $A, B \subset \Bbb F_p$ so that \[|A|> p^{(4/9)+\varepsilon},\qquad |B|> p^{(4/9)+\varepsilon},\qquad |B+B| \lt K|B|.\] Then \[\Big|\sum_{x\in A, y\in B} \chi(x+y)\Big| \lt p^{-\tau}|A|\;|B|.\] ¶ (3) Let $I\subset \Bbb F_p$ be an interval with $|I|=p^{\beta}$ , and let $\mathcal D\subset \Bbb F_p$ be a $p^\beta$ -spaced set with $|\mathcal D|=p^\sigma$ . Assume that $2\beta+\sigma-{\beta\sigma}/{(1-\beta)}> 1/2+\delta$ . Then for a nonprincipal multiplicative character $\chi$ , \[\Big|\sum_{x\in I, y\in \mathcal D}\chi(x+y)\Big| \lt p^{-{\delta^2}/{12}}|I|\;\;|\mathcal D|.\] We also slightly improve a result of Karacuba [K3]

In this article, we consider the problem of finding upper bounds on the minimum norm of representatives in residue classes in quotient $O/I$ , where $I$ is an integral ideal in the maximal order $O$ of a number field $K$ . In particular, we answer affirmatively a question of Konyagin and Shparlinski [KS], stating that an upper bound $o(N(I))$ holds for most ideals $I$ , denoting $N(I)$ the norm of $I$ . More precise statements are obtained, especially when $I$ is prime. We use the method of exponential sums over multiplicative groups, essentially exploiting some new bounds obtained by the authors

Earlier bounds involved exponential dependence in αin the second estimate. Ourargument combines I. Ruzsa’s method, which we improve in several places, as well asY. Bilu’s proof of Freiman’s theorem.A fundamental result in the theory of set addition is Freiman’s theorem. Let A ⊂Z be a finite set of integers with small sumset; thus assume|A + A

We experimentally demonstrate the direct observation of defect mediated wave turbulence with fluctuating defects and low amplitude hole filaments, from a 3D self-excited plane dust acoustic wave in a dusty plasma by reducing dissipation. The waveform undulation is found to be the origin for the amplitude and the phase modulations of the local dust density oscillation, the broadening of the sharp peaks in the frequency spectrum, and the fluctuating defects. The corrugated wave crest surface also causes the observed high and low density patches in the transverse (xy) plane. Low oscillation amplitude spots (holes) share the same positions with the defects. Their trajectories in the xyt space appear in the form of chaotic filaments without long term predictability, through uncertain pair generation, propagation, and pair annihilation.

The microscopic channeling dynamics of projectiles in subexcitable chain bundle dusty plasma liquids consisting of long chains of negatively charged dusts suspended in low pressure glow discharges is investigated experimentally using fast video-microscopy. The long distance channeling of the projectile in the channel formed by the surrounding dust chain bundles and the excitation of a narrow wake associated with the elliptical motions of the background dusts are demonstrated. In the high projectile speed regime, the drag force due to wake wave excitation increases with the decreasing projectile speed. The excited wave then leads the slowed down projectile after the projectile speed is decreased below the resonant speed of wave excitation. The wave-projectile interaction causes the increasing projectile drag below the resonant speed and the subsequent oscillation around a descending average level, until the projectile settles down to the equilibrium point. Long distance projectile surfing through the resonant crest trapping by the externally excited large amplitude solitary wave is also demonstrated.

The Eulerian/Lagrangian dynamics in the narrow wake field of the dusty plasma bubble is explored by directly tracking dust motion at the microscopic level. The bubble is induced by the focused laser pulse ablation in three-dimensional quiescent dusty plasma liquids operated in the pressure higher than the critical pressure for the self-excitation of dust acoustic wave by the downward ion wind. It is found that, after bubble expansion ceases, the collective excitation maintains its width and travels downward as a solitary wave, led by an ultrasonic rarefaction front contributed by the dust motion below the lower boundary, and trailed by the few compressional crests with descending crest heights and speeds in the narrow wake, under the symmetry breaking by the downward ion flow. The quick damping of the waves propagating along other directions leads to a narrow wake. Increasing the background pressure causes the more isotropic collapsing of the bubble without wake field oscillation. The role played by dust motion on interacting with and sustaining the wake field evolution is identified and discussed.

We experimentally demonstrate the observation of several pathways with a few tens of ms inter-cluster delay but negligible intra-cluster delay, in the spontaneous synchronous burstings (SSBs) of clusterized cortical neuronal networks in vitro through epifluorescence microscopy of Ca 2+ indicator. Nearby clusters sharing the similar relative firing timings in different SSB events form a group. Switching among different SSB pathways initiated by different master groups is found. Under chemical tuning such as decreasing the mutual coupling by increasing [Mg 2+ ] or lowering the effective firing threshold by increasing [K + ], the firing sequence of each pathway remains robust, but the occurrence probability and the temporal persistence of each pathway, and the inter-cluster delay times in the firing sequences can be varied.

Let $A$ be a set of $N$ matrices. Let $g(A)\ccolon=|A+A|+|A\cdot A|$, where $A+A=\{a_1 + a_2 \mid a_i \in A\}$ and $A\cdot A=\{a_1a_2 \mid a_i \in A\}$ are the sum set and product set. We prove that if the determinant of the difference of any two distinct matrices in $A$ is nonzero, then $g(A)$ cannot be bounded below by $cN$ for any constant $c$. We also prove that if $A$ is a set of $d\times d$ symmetric matrices, then there exists $\varepsilon=\varepsilon (d)>0$ such that $g(A)>N^{1+\varepsilon}.$ For the first result, we use the bound on the number of factorizations in a generalized progression. For the symmetric case, we use a technical proposition which provides an affine space $V$ containing a large subset $E$ of $A$, with the property that if an algebraic property holds for a large subset of $E$, then it holds for $V$. Then we show that the system $\{a^2\ccolon a\in V\}$ is commutative, allowing us to decompose ${\mathbb R}^d$ as eigenspaces simultaneously, so we can finish the proof with induction and a variant of the Erdos–Szemeredi argument.

Let E be a stable rank 2 vector bundle on n )3. Assume E is normalized, i.e., c~(E)=0 or 1 . By Grothedieck's theorem the restriction of E on a line L is isomorphic to (~L(bL)O(gL(--bL) if Cl=O[(gL(bL--1)t~OL(--b L) if C1=-1 ] , for some nonnegative integer b L. This b L is an upper semicontinuous function of L, which in char. 0, according to the Grauert-Miilich theorem [G-M], assumes the value 0 on an open set. If bE>0, then L is said to be a jumping line of order b L. For c 1 =0, Barth [B] has shown that the set of jumping lines can be given a scheme structure, making it a divisor S E on the Grassmannian G=G(1, 3) which via G ~IP 5, has degree 2c2(E), and for any line L in IP 3, b L < PL(SE), where #~.(S~) is the multiplicity of S t at L. Since any point on a projective variety of degree k has multiplicity < k, it follows that b L < 2c 2. However, this bound is not sharp. In this paper, we will give a sharp bound for b L, valid for arbitrary c r Namely, if c~ =0

We investigate the moduli spaces of stable rank two reflexive sheaves on P3 with small Chern classes. As an application to curves of low degree in P3, we prove the curve has maximal rank and that the corresponding Hilbert scheme is irreducible and unirational. Introduction. In the past few years, the subject of vector bundles on projective spaces and, in particular, the case of rank 2 on P3, has received much attention. Several basic theorems have been proved, e.g., the existence of the moduli space of stable vector bundles [M], though so far very few of these moduli spaces have been studied in detail. The interest of vector bundles partly stems from their connection with curves. However, the class of curves obtained in this way is rather restricted. Recently, Hartshorne [SRS] has focused attention on reflexive sheaves of rank 2 on P3. On the one hand, most results proved for vector bundles also turn out to be true for reflexive sheaves. On the other hand, reflexive sheaves have two significant advantages. First, they correspond to quite general curves; second, and most importantly, one has the "reduction step" introduced by Hartshorne, which is an effective tool in studying vector bundles, by relating them to simpler reflexive sheaves. In this paper, we investigate the moduli spaces of stable rank 2 reflexive sheaves on P3 with c2 3. For c2 < 2, we prove the moduli spaces are irreducible, nonsingular and rational; we also classify the related unstable planes. For c2= 3, in most cases we show that the moduli space is irreducible and unirational. As one of the applications to curves of low degree in P3 (cf. [SRS, 4.1]), we prove that the curve has maximal rank and that the corresponding Hilbert scheme is irreducible and unirational. Hence, as a corollary, we conclude that the variety of moduli '1g of curves of genus g is unirational, for g = 5, 6, 7, 8. In ?1, we give facts on multiple lines and deduce a criterion for unstable planes. ?2 contains our classification of semistable sheaves with c2 < 2. ?3 is about sheaves with c2 = 3. ?4 gives the applications to curves. ACKNOWLEDGMENT. I would like to thank my thesis advisor Robin Hartshorne. He has been very generous in his advice and time. In addition, he attracted to Received by the editors September 1, 1982 and, in revised form, March 18, 1983. 1980 Mathematics Subject Classification. Primary 14F05; Secondary 14D22. I1984 American Mathematical Society 0002-9947/84 $1.00 + $.25 per page

Let E be a vector bundle on Pn . We say E is Buchsbaum if Hp(EM(∗)) has trivial module structure for each linear space M and for all 1≤p≤dimM−1. In this note, we show that a Buchsbaum bundle is either an Ω-bundle (i.e. a direct sum of Ωp(h)'s, where Ωp is the pth exterior power of the contangent bundle of Pn) or the quotient of an Ω-bundle by a direct sum of line bundles. We also give a strategy to characterize/construct Buchsbaum bundles, give conditions for an Ω-bundle to be extendable.