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Register a new userUniqueness of nonnegative weak solution to $u^ple(-Delta)^frac{alpha}{2}u$ on $mathbb R^N$
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This note shows that under $(p,alpha, N)in (1,infty)times(0,2)timesmathbb Z_+$ the fractional order differential inequality $$ (dagger)quad u^p le (-Delta)^{frac{alpha}{2}} uquadhbox{in}quadmathbb R^{N} $$ has the property that if $Nlealpha$ then a nonnegative solution to $(dagger)$ is unique, and if $N>alpha$ then the uniqueness of a nonnegative weak solution to $(dagger)$ occurs when and only when $ple N/(N-alpha)$, thereby innovatively generalizing Gidas-Sprucks result for $u^p+Delta ule 0$ in $R^N$ discovered in cite{GS}.
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We consider critical exponent semi-linear elliptic equation with coefficient K(x) periodic in its first k variables, with 2k smaller than n-2. Under some natural conditions on K near a critical point, we prove the existence of multi-bump solutions where the centers of bumps can be placed in some lattices in Rk, including infinite lattices. We also show that for 2k greater than or equal to n-2, no such solutions exist.
In this paper we prove the equivalence among (i) the weakly coupled worldsheet string theory described by the coset sigma model $frac{SL(2,mathbb{R})_ktimes U(1)}{U(1)}times S^3 times T^4$ with $SL(2,mathbb{R})$ WZW level $kgeq 2$, (ii) the full near horizon theory of the NS5 branes with $k$ NS5 branes wrapping $T^4times S^1$, $pgg1$ F1 strings wrapping $S^1$ and $n$ units of momentum along the $S^1$ and (iii) the single trace $Tbar{T}$ deformation of string theory in $AdS_3times S^3times T^4$. As a check we compute the spectrum of the spacetime theory by performing BRST quantization of the coset description of the worldsheet theory and show that it matches exactly with the one derived in the case of single trace $Tbar{T}$ deformed string theory in $AdS_3$. Secondly, we compute the two-point correlation function of local operators of the spacetime theory using the worldsheet coset approach and reproduce the same two-point function from the supergravity approach.
In this paper, we prove two improv
Let $mathbb{F}_{2^m}$ be a finite field of $2^m$ elements, and $R=mathbb{F}_{2^m}[u]/langle u^krangle=mathbb{F}_{2^m}+umathbb{F}_{2^m}+ldots+u^{k-1}mathbb{F}_{2^m}$ ($u^k=0$) where $k$ is an integer satisfying $kgeq 2$. For any odd positive integer $n$, an explicit representation for every self-dual cyclic code over $R$ of length $2n$ and a mass formula to count the number of these codes are given first. Then a generator matrix is provided for the self-dual and $2$-quasi-cyclic code of length $4n$ over $mathbb{F}_{2^m}$ derived by every self-dual cyclic code of length $2n$ over $mathbb{F}_{2^m}+umathbb{F}_{2^m}$ and a Gray map from $mathbb{F}_{2^m}+umathbb{F}_{2^m}$ onto $mathbb{F}_{2^m}^2$. Finally, the hull of each cyclic code with length $2n$ over $mathbb{F}_{2^m}+umathbb{F}_{2^m}$ is determined and all distinct self-orthogonal cyclic codes of length $2n$ over $mathbb{F}_{2^m}+umathbb{F}_{2^m}$ are listed.
Let $ E subset mathbb{R}^2 $ be a finite set, and let $ f : E to [0,infty) $. In this paper, we address the algorithmic aspects of nonnegative $C^2$ interpolation in the plane. Specifically, we provide an efficient algorithm to compute a nonnegative $C^2(mathbb{R}^2)$ extension of $ f $ with norm within a universal constant factor of the least possible. We also provide an efficient algorithm to approximate the trace norm.
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