Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation

We consider the critical nonlinear Schrödinger equation $iu_{t} = -\Delta u-|u|^{4/N}$ with initial condition u(0, x) = u₀.For u₀$\in$H¹, local existence in time of solutions on an interval [0, T) is known, and there exist finite time blow-up solutions, that is u₀ such that $\textrm{lim} _{t\uparrow T <+\infty}|\nabla u(t)|_{L^{2}}=+\infty$. This is the smallest power in the nonlinearity for which blow-up occurs, and is critical in this sense.The question we address is to control the blow-up rate from above for small (in a certain sense) blow-up solutions with negative energy. In a previous paper [MeR], we established some blow-up properties of (NLS) in the energy space which implied a control $|\nabla u(t)|_{L^{2}} \leq C \frac{|\ln(T-t)|^{N/4}}{\sqrt{T-t}}$ and removed the rate of the known explicit blow-up solutions which is $\frac{C}{T-t}$.In this paper, we prove the sharp upper bound expected from numerics as$|\nabla u(t)|_{L^{2}} \leq C \left(\frac{\ln|\ln(T-t)|}{T-t} \right)^{1/2}$by exhibiting the exact geometrical structure of dispersion for the problem.     

On an upper bound for the global dimension of Auslander–Dlab–Ringel algebras

Archiv der Mathematik - Lin and Xi introduced Auslander–Dlab–Ringel (ADR) algebras of semilocal modules as a generalization of original ADR algebras and showed that they are...     

Sharp estimates for Hübner’s upper bound function with applications

New better estimates, which are given in terms of elementary functions, for the function r → (2/π)(1 - r2)K(r)K (r) + log r appearing in Hübner's sharp upper bound for the Hersch-Pfluger distortion function are obtained. With these estimates, some known bounds for the Hersch-Pfluger distortion function in quasiconformal theory are improved, thus improving the explicit quasiconformal Schwarz lemma and some known estimates for the solutions to the Ramanujan modular equations.     

An upper bound for a Hilbert polynomial on quaternionic Kähler manifolds

In this article we prove an upper bound for a Hilbert polynomial on quaternionic Kähler manifolds of positive scalar curvature. As corollaries we obtain bounds on the quaternionic volume and the degree of the associated twistor space. Moreover, the article contains some details on differential equations of finite type. Part of this article is used in the proof of the main theorem.     

Are there elimination algorithms for the permanent?

We define the class of elimination algorithms. There are algebraic algorithms for evaluating multivariate polynomials, and include as a special case Gaussian elimination for evaluating the determinant. We show how to find the linear symmetries of a polynomial, defined appropriately, and use these methods to find the linear symmetries of the permanent and determinant. We show that in contrast to the Gaussian elimination algorithm for the determinant, there is no elimination algorithm for the permanent.     

Strengthening the Lovász bound for graph coloring

The Lovász -number is a way to approximate the independence number of a graph, but also its chromatic number. We express the Lovász bound as the continuous relaxation of a discrete Lovász -number which we derive from Karger et al.s formulation, and which is equal to the chromatic number. We also give another relaxation à la Schrijver-McEliece, which is better than the Lovász -number.     

A low bound for the de Bruijn-newman constant Λ

Summary A lower bound is constructively found for the de Bruijn-Newman constant , which is related to the Riemann Hypothesis. This lower bound is determined by explicitly exhibiting an associated jensen polynomial with nonreal zeros.Dedicated to the memory of our good friend Peter Henrici (Sept. 13, 1923-March 13, 1987)Research supported by the Airforce Office of Scientific Research and the National Science Foundation     

Two upper bounds for the Erd?s-Hooley Delta-function

For integer n≥1 and real u,let Δ(n,u):=|{d:d] n,e^u u+1}|.The Erd?s-Hooley Deltafunction is then defined by Δ(n):=Max_u∈R Δ(n,u).We improve the current upper bounds for the average and normal orders of this arithmetic function.     

Improvements of the Weil bound for Artin–Schreier curves

For the Artin–Schreier curve y ^q ? y = f(x) defined over a finite field \({{\mathbb F}_q}\) of q elements, the celebrated Weil bound for the number of \({{\mathbb F}_{q^r}}\)-rational points can be sharp, especially in super-singular cases and when r is divisible. In this paper, we show how the Weil bound can be significantly improved, using ideas from moment L-functions and Katz’s work on ?-adic monodromy calculations. Roughly speaking, we show that in favorable cases (which happens quite often), one can remove an extra \({\sqrt{q}}\) factor in the error term.     

Generalized lower and upper functions for the φ-Laplacian

We give definitions of generalized lower and upper functions and generalized solution, which differs from a solution in the conventional sense in that the derivative of a generalized solution can be equal to +∞ and ?∞. We show how to use generalized solutions for obtaining classical results.     

An improved bound for the de Bruijn–Newman constant

The Riemann hypothesis is equivalent to the conjecture that the de Bruijn–Newman constant satisfies 0. However, so far all the bounds that have been proved for go in the other direction, and provide support for the conjecture of Newman that 0. This paper shows how to improve previous lower bounds and prove that –2.710^–9<. This can be done using a pair of zeros of the Riemann zeta function near zero number 10²⁰ that are unusually close together. The new bound provides yet more evidence that the Riemann hypothesis, if true, is just barely true.     

Köthe’s upper nil radical for modules

Let M be a left R-module. In this paper a generalization of the notion of an s-system of rings to modules is given. Let N be a submodule of M. Define $\mathcal{S}(N):=\{ {m\in M}:\, \mbox{every } s\mbox{-system containing } m \mbox{ meets}~N \}$ . It is shown that $\mathcal{S}(N)$ is equal to the intersection of all s-prime submodules of M containing N. We define $\mathcal{N}({}_{R}M) = \mathcal{S}(0)$ . This is called (Köthe’s) upper nil radical of M. We show that if R is a commutative ring, then $\mathcal{N}({}_{R}M) = {\mathop{\mathrm{rad}}\nolimits}_{R}(M)$ where ${\mathop{\mathrm{rad}}\nolimits}_{R}(M)$ denotes the prime radical of M. We also show that if R is a left Artinian ring, then ${\mathop{\mathrm{rad}}\nolimits}_{R}(M)=\mathcal{N}({}_{R}M)= {\mathop{\mathrm{Rad}}\nolimits}\, (M)= {\mathop{\mathrm{Jac}}\nolimits}\, (R)M$ where ${\mathop{\mathrm{Rad}}\nolimits}\, (M)$ denotes the Jacobson radical of M and ${\mathop{\mathrm{Jac}}\nolimits}\, (R)$ the Jacobson radical of the ring R. Furthermore, we show that the class of all s-prime modules forms a special class of modules.     

Upper bound involving parameter σ 2 for the rainbow connection number

Let G be a connected graph of order n. The rainbow connection number rc（G） of G was introduced by Chartrand et al. Chandran et al. used the minimum degree of G and obtained an upper bound that rc（G） 〈_ 3n/（ δ＋ 1） - 3, which is tight up to additive factors. In this paper, we use the minimum degree-sum a2 6n of G to obtain a better bound rc（G） _〈 - 8, especially when is small （constant） but a2 is large （linear in n）.     

Further improvements on the Feng–Rao bound for dual codes

Salazar, Dunn and Graham in [16] presented an improved Feng–Rao bound for the minimum distance of dual codes. In this work we take the improvement a step further. Both the original bound by Salazar et al., as well as our improvement are lifted so that they deal with generalized Hamming weights. We also demonstrate the advantage of working with one-way well-behaving pairs rather than weakly well-behaving or well-behaving pairs.