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abstract: 'In this paper we calculate the spectrum of Neumann matrix with zero modes in the presence of the constant B field in Witten’s cubic string field theory. We find both the continuous spectrum inside $[{-1\over3}, 0)$ and the constraint on the existence of the discrete spectrum. For generic $\theta$, $-1/3$ is not in the discrete spectrum but in the continuous spectrum. For each eigenvalue in the continuous spectrum there are four twist-definite degenerate eigenvectors except for $-1/3$ at which the degeneracy is two. However, for each twist-definite eigenvector the twist parity is opposite among the two spacetime components. Based upon the result at $-1/3$ we prove that the ratio of brane tension to be one as expected. Furthermore, we discuss the factorization of star algebra in the presence of B field under zero-slope limit and comment on the implications of our results to the recent proposed map of Witten’s star to Moyal’s star.'
---

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**[Star Spectroscopy in the Constant B field Background]{}**

Bin Chen$^{*}$[^1] and Feng-Li Lin $^{\dagger}$[^2]

*${}^*$High Energy Section*

*the Abdus Salam ICTP*

*Strada Costiera, 11*

*34014 Trieste, Italy*

*${}^\dagger$Department of Physics*

*Tamkang University*

*Tamsui, Taipei 25137, Taiwan*

Introduction
============

The non-perturbative physics of Witten’s cubic string field theory [@Witten; @GJ1; @GJ2] has been enhanced a lot since the proposal of vacuum string field theory(VSFT) by Rastelli, Sen and Zwiebach in [@RSZ1] to [@RSZ7], see also the related works [@GT1] to [@Okuyama2]. The basic idea is to assume the universality of the BRST operator around the closed string vacuum after tachyon condensation such that the ghost and matter sector decompose and the classical solution of the matter string field is a projector, or the so called sliver state and its tachyonic lump generalization [@RSZ2].

These projector states are understood as the non-perturbative D-brane soliton of the VSFT, so the ratio of the branes’ tension provide a nontrivial test of the VSFT proposal. The ratio is mainly determined by the spectrum of the Neumann matrix with and without the zero modes as following [@RSZ2; @Okuyama2] \[R1\] R=[T\_p 2 T\_[p+1]{}]{}=[3(V\_[00]{}+[b2]{})\^2 ]{} [(1-M\^[’]{})\^[34]{}(1+3M\^[’]{})\^[14]{}(1-M)\^[34]{}(1+3M)\^[14]{}]{}, where $M$ is the Neumann matrix in the oscillator basis without zero modes, and $M^{'}$ is the one with zero modes. The ratio has been first tested numerically to be almost one in [@RSZ1] and then proved to be one analytically by Okuyama in [@Okuyama2]. Since the spectrum of both $M$ and $M^{'}$ contain $-1/3$ so that the ratio blows up and requires properly regularization related to the issue of twist anomaly investigated in [@Hata1].

The spectrum of the Neumann matrix $M$ is constructed in [@RSZ8] which shows that it has an doubly degenerate continuous spectrum except at $-1/3$ which is non-degenerate. Later on, the explicit calculation of the spectrum of the Neumann matrix $M^{'}$ is also carried out in [@Feng1]. The result shows that there are additional discrete spectrum besides the continuous one found in [@RSZ8]. However, it also leads to further puzzle that $M^{'}$ has doubly degenerate eigenvectors at $-1/3$ so that naively the ratio R could be zero instead of one as expected. More detailed study of the density of states is required to resolve the issue.

On the other hand, Witten’s cubic SFT is believed to be background independent although there is no general proof for the background independence except some general arguments based upon BCFT in the context of VSFT [@RSZ3] to [@RSZ5]. It is then an issue to check the background independence case by case. A simple case is to turn on the constant background B field, some discussions for Witten’s SFT on this background have been discussed in [@Kawano1; @Kawano2; @Sugino], there they argue that the string vertexs can be obtained by an unitary transformation from the one without the B-field. Although this is the case, it is still interesting to understand the background independence issue in VSFT where the non-perturbative D-brane physics can be worked out.

The construction of the sliver state and the tachyonic lump state in the constant B field background based upon [@Kawano1; @Kawano2; @Sugino] is done in [@Bonora], and the ratio of the brane tension between $Dp$ and $D(p+2)$ branes is given by \[R2\] [R]{}=[(DetG)\^[14]{}(2)\^2]{}[T\_p T\_[p+2]{}]{}=([\^2+12(V\_[00]{}+[b2]{})\^2 4 (DetG)\^[18]{} ]{})\^2[(1-[M]{}\^[’]{})\^[34]{}(1+3[M]{}\^[’]{})\^[14]{}(1-M)\^[32]{}(1+3M)\^[12]{}]{}, where $G$ stands for the open string metric. ${\cal M}^{'}$ is the Neumann matrix with the zero modes in the oscillator basis, and due to the B field there is mixing between two coordinate directions so that the dimension of $\cMp$ is twice larger than the one of $M$. In order to determine the ratio one needs to have the spectrum of $\cMp$, at least one should make sure if $\cMp$ has a eigenvalue at $-1/3$ in order for the ratio to be finite [^3].

Another interesting issue to which our results will be relevant is the generalization of the recent proposed map of Witten’s star to Moyal’s star [@Douglas] to the case with zero modes in the constant B-field. There the Moyal conjugate pairs are the Fourier transform of the twist-even coordinate modes and twist-odd momentum modes, the Fourier basis are the eigenvectors of $M$ or $\cMp$ in our case, and the twist parity is defined with respect to left and right part of the open string in Witten’s cubic string field theory. So we need to construct the complete orthogonal eigenvector basis of $\cMp$ in order to find out the proper Moyal conjugate pairs such that the proposed map makes sense. If so, then Witten’s cubic string field theory can be reformulated in the language of the noncommutative field theory, for which the soliton is just a projector.

In section 2, we will summarize the basic of the VSFT by setting up our conventions and notations, and warm up by calculating the $1+\cMp$ part of the ratio $\cR$. Section 3 contains the main results of our paper in which we will calculate both the continuous and discrete spectrum of the Neumann matrix. In section 4 we will conclude our paper by discussing the implication of our results to the ratio $\cR$ and to the map of Witten’s star to Moyal star. In the appendix we compute the ratio $\cR$ explicitly and show that it is exactly one.

: Upon finishing our paper we find that there appears two papers [@Feng2] and [@Bonora2] dealing with the same topic, and the first one has substantial overlap with ours.

Neumann matrix in the constant B field background
=================================================

3-string vertex in the constant B-field background
--------------------------------------------------

To construct the sliver state which is D25-brane in VSFT, we need to use the 3-string vertex for the star product in the constant B field background [@Kawano1; @Kawano2; @Sugino], which is given by \[3stringp\] |V\_3> &=& dp\^[(1)]{}dp\^[(2)]{}dp\^[(3)]{} (p\^[(1)]{}+p\^[(2)]{}+p\^[(3)]{}) ( - \_[r,s3]{} \[\_[m,n1]{} V\^[rs]{}\_[mn]{} a\^[(r)]{}\_m a\^[(s)]{}\_n .\
& & . +2 \_[m 1]{} V\^[rs]{}\_[m0]{} a\^[(r)]{}\_m p\^[(s)]{}+ V\^[rs]{}\_[00]{} p\^[(r)]{}p\^[(s)]{}\]-\_[r<s]{}p\^[(r)]{} p\^[(s)]{} )\
& & (|p\^[(1)]{}>|p\^[(2)]{}>|p\^[(3)]{}>) where |p>=(-(pp) + (a\^\_0p) -(a\^\_0 a\^\_0)) |0> satisfy $\hat{p}|k> =k |k>$. The dot products are with respect to the closed string metric along directions without $B$ field and to the open string metric $G_{\a\b}$ along directions with $B$ field. The $\theta$ is the noncommutative parameter and the cross product means the contraction with the anti-symmetric symbol $\epsilon^{\a\b}$.

As shown in [@Kawano1; @Kawano2; @Sugino] the B field will change the Neumann boundary conditions of the free open string to the mixed one and introduce the Moyal-like phase in the string vertex besides replacing the closed string metric by the open string one in the B-field directions. Obviously one can decompose the above vertex into the transverse part $|V_3^{\bot}>$ and the parallel part $|V_3^{\|}>$ with respect to the $B$ directions.

To construct the lower dimensional brane to which the transverse directions is no longer translational invariant, we need to use the 3-string vertex in the oscillator basis as the star product, which can be obtained by integrating out the momentum in (\[3stringp\]), \[Vper\] |V\_3\^>=( )\^[d\^/2]{} ( -\_[r,s3]{} \_[m,n0]{} [V]{}\^[rs]{}\_[, mn]{} a\^[(r)]{}\_m a\^[(s)]{}\_n ) (|>|> |>), where $V_{00}\equiv V^{rr}_{00}\;$.

The relations between $\cV$ and $V$ is obtained in [@Bonora] and we summarize the results in the following \[V1\] [V]{}\^[, rs]{}\_[mn]{}&=& G\^V\^[rs]{}\_[mn]{}- \_[t,v=1]{}\^3 V\^[rv]{}\_[m0]{}U\^[,vt]{}V\^[ts]{}\_[0n]{}\
\[V2\] [V]{}\^[, rs]{}\_[0n]{}&=& \_[t=1]{}\^3 U\^[,rt]{}V\^[ts]{}\_[0n]{}\
\[V3\] [V]{}\^[, rs]{}\_[00]{}&=&G\^\^[rs]{}-[N]{}\_ U\^[,rs]{} where U\^[,rs]{}G\^\^[rs]{}-i a \^\^[rs]{}, with , a , and \^[rs]{}=( 0&1&-1\
-1&0&1\
1&-1&0 ), \^[rs]{}=( 1&-1/2&-1/2\
-1/2&1&-1/2\
-1/2&-1/2&1 )

Instead of using $V^{rs}$ one defines the matrices $M^{rs}$ as following M\^[rs]{}=CV\^[rs]{}, where $C_{mn}=(-1)^n\delta_{m,n}, m,n \ge 1$ is the twist matrix. The matrices $M^{rs}$ are commuting to each other so that it is useful to solve the algebraic equation for the projector states. Similarly, as shown in [@Bonora] the matrices \^[’rs]{}=C\^[’]{}\^[rs]{} are also commuting to each other. The generalized twist matrix $C^{'}_{mn}=(-1)^n\delta_{m,n}, m,n \ge 0$. Although there are nine Neumann matrices, only three of them are independent, in the following we will only focus on one of them, that is, $M\equiv
M^{11}$, and the one with zero modes in the constant B-field background $\cMp\equiv {\cal M}^{'11}$. These two are the only matrices appear in the ratio of brane tension in (\[R2\]), and will be our focus to find out their spectrum.

The form of $\cMp$
------------------

In the following we write down the explicit form of $\cMp$ by adopting the basis introduced firstly in [@RSZ2]. After some calculations, we arrive \^[’]{}\_[00]{}&=&G\^(1-[N]{}\_),\
[M]{}\^[’]{}\_[0n]{}&=&-\_(G\^ <v\_e|-i \^[2a ]{} <v\_o|),\
[M]{}\^[’]{}\_[mn]{}&=&G\^(M\_[mn]{}-[2b]{}[N]{}\_(|v\_e><v\_e|-|v\_o><v\_o|))\
&&+i\^ [2a ]{}[2b]{}[N]{}\_ (|v\_e><v\_o|+|v\_o><v\_e|), m,n1, where the vectors $|v_e>$ and $|v_o>$ are the same as defined in [@Feng1] and [@Okuyama2]. Note that $(|v_{e,o}>)^T=<v_{e,o}|$ and C|v\_e>=|v\_e> ,C|v\_o>=-|v\_o>. Explicitly, =M(k)|k>, where M(k)=-[12+1]{}, and the eigenvector |k>=(v\_1\^k,v\_2\^k,v\_3\^k,…)\^T, is generated by f\_k(z)=\_[n=1]{}\^[v\_n ]{}z\^n=[1k]{}(1-e\^[-k \^[-1]{}z)]{},.

The eigenvectors constitute a set of complete orthogonal basis and \^\_[-]{} [|k><k| (k)]{}=1, where . However, $k$ is not an twist definite eigenstate but C|k>=-|-k>. Since $C$ commutes with $M$[^4] so that each eigenvalue has doubly degenerate twist-definite states except for $|k=0>$ which is twist odd [@RSZ8].

Later on it is also useful to have the following facts [@Hata2; @Okuyama2] about $|v_{e,o}>$, \[vk\] <k|v\_e>=[1k]{}[-1 2+1]{}, =[ k]{} [ 2+1]{}, =[14]{}V\_[00]{},\
<v\_o|[11-M]{}|v\_o>=[34]{} V\_[00]{}.

Since there is an eigenvalue of $M$ at $-1/3$ so that the ratio $\cR$ is not well-defined unless $-1/3$ is also an eigenvalue of $\cMp$ with multiplicity two. In order to calculate $\cR$ we need to study the spectrum of $\cMp$, especially at $-1/3$, this will be done in the next section.

However, it is straightforward to calculate the ratio of the tension. As for $Det(1-\cMp)$, based upon the above expression of $\cMp$ and the facts det ( A &B\
C& D ) &=& detA det(D-CA\^[-1]{}B),\
det(1+|u><v|) &=&1+<v|u>, one obtains Det\^[1/2]{}(1-[M]{}\^) det(1-M) &=& [N]{}\_det( 1-[N]{}\_ (1+)[1 ]{}|\_o><\_o|)\
&=& . when $\theta=0$, this recovers the result in [@Okuyama2].

As for the calculation of $Det(1+3\cMp)$ part, one has to introduce the regulator to overcome the difficulty brought by the existence of $-1/3$ eigenvalue. It has been worked out in [@Bonora2] and at last one find that the ratio of the brane tension is exactly $1$, as expected. For completeness, the detailed calculation is given in the appendix with a slightly different way of calculating the $Det(1+3\cMp)$ from the one used in [@Bonora2].

Finding the spectrum of $\cMp$
==============================

In this section we will find out the spectrum of ${\cal M}^\prime$ by using the method in [@Feng1].

The eigen-equations
-------------------

Let’s assume the form of the eigenstate as v=( g\_1\
g\_2\
h\_1(k)|k>\
h\_2(k)|k> ) where $g_1, g_2$ are some complex numbers corresponding to the zero modes and $h_1(k), h_2(k)$ are the coefficients of the expansion on the $|k>$ basis. Note that due to the complex elements in $\cMp$, we assume that $h_i(k)$ could be complex. Let $\lambda$ be the corresponding eigenvalue, then $\cMp v =\lambda v$ encodes the following relations: g\_1&=&(1-)g\_1-(h\_1(k)<v\_e|k> + h\_2(k)<v\_o|k>),\
g\_2&=&(1-)g\_2-(h\_2(k)<v\_e|k> - h\_1(k)<v\_o|k>),\
h\_1(k)|k> &=& -(g\_1 |v\_e> +g\_2|v\_o>) + h\_1(k)M(k)|k>\
& & -(|v\_e> h\_1(k)<v\_e|k> -|v\_o> h\_1(k)<v\_o|k>)\
& &+((|v\_e> h\_2(k)<v\_o|k> + |v\_o> h\_2(k)<v\_e|k>) \[eigen1\]\
h\_2(k)|k> &=& -(g\_2|v\_e> -g\_1|v\_o>) + h\_2(k)M(k)|k>\
& & -(|v\_e> h\_2(k)<v\_e|k> -|v\_o> h\_2(k)<v\_o|k>)\
& & +((|v\_e> h\_1(k)<v\_o|k> +|v\_o> h\_1(k)<v\_e|k>). \[eigen2\]

Define =h\_1(k)<v\_e|k> &=h\_1(k)<v\_o|k>\
=h\_2(k)<v\_e|k> &=h\_2(k)<v\_o|k> . And from the first two relations, we can rewrite $g_1,g_2$ as g\_1&=&(+)\
g\_2&=&(-) Put the above expressions into the Eqs.(\[eigen1\],\[eigen2\]), and using the expansions |v\_e>&=& |k>\
|v\_o>&=& |k> one have\
&=& {c\_1+ic\_2}|k> +{c\_3+ic\_2}|k>\
\
&=& { c\_1-ic\_2}|k> +{-ic\_2+c\_3}|k> where c\_1&&-[2b]{}\
c\_2&&-\
c\_3&&()(1-)

From the argument in [@Feng1], one could expect -(k-k\_0)r\_1(k)&=& -(-M(k))h\_1(k)+(c\_1 +ic\_2) +(c\_3+ic\_2)\
-(k-k\_0\^)r\_2(k)&=& -(-M(k))h\_2(k)+(c\_1 -ic\_2) +(c\_3-ic\_2) where $r_i(k)$ undetermined, with $r_1(k_0)=0, r_2(k_0^\prime)=0$. Now solve $h_1(k), h_2(k)$ from the above two relations, and applying the operations $\int dk <v_e|k>$ and $\int dk <v_o|k>$, we obtain &=&A\_[ee]{}(c\_1+ic\_2)+B\_[1e]{}\
&=&A\_[oo]{}(c\_3+ic\_2)+B\_[1o]{}\
&=&A\_[ee]{}(c\_1-ic\_2)+B\_[2e]{}\
&=&A\_[oo]{}(c\_3-ic\_2)+B\_[2o]{} where A\_[ee]{}()&&=<v\_e||v\_e>,\
A\_[oo]{}()&&=<v\_o||v\_o>,\
B\_[ie]{}&& \[Be\]\
B\_[io]{}&&\[Bo\]\
It’s easy to find that one can group the above equations into two sets: \[set1\] (1-c\_1A\_[ee]{})-ic\_2A\_[ee]{}&=& B\_[1e]{}\
ic\_2A\_[oo]{}+ (1-c\_3A\_[oo]{})&=& B\_[2o]{} and (1-c\_3A\_[oo]{})-ic\_2A\_[oo]{}&=& B\_[1o]{}\
ic\_2A\_[ee]{}+ (1-c\_1A\_[ee]{})&=& B\_[2e]{} \[set2\]

Continuous spectrum
-------------------

If the determinant factor Det (1-c\_1 A\_[ee]{})(1-c\_3 A\_[oo]{})-c\^2\_2A\_[oo]{}A\_[ee]{} is not zero, then we can invert the above two sets of equation (\[set1\]) to (\[set2\]) and obtain the formal solutions \[sol1\] &=&[B\_[1e]{}(1-c\_3A\_[oo]{})+ic\_2A\_[ee]{}B\_[2o]{} Det]{},\
&=&[B\_[1o]{}(1-c\_1A\_[ee]{})+ic\_2A\_[oo]{}B\_[2e]{} Det]{},\
&=&[B\_[2e]{}(1-c\_3A\_[oo]{})-ic\_2A\_[ee]{}B\_[1o]{} Det]{},\
&=&[B\_[2o]{}(1-c\_1A\_[ee]{})-ic\_2A\_[oo]{}B\_[1e]{} Det]{}. \[sol4\]

In order to have nontrivial solutions, that is, the above $B$ functions are not all zero, we need to have $\lambda -M(k)=0$ at $k=k_0=k^{'}_0\ne 0$ to cancel the zero of $r_{1,2}(k)$ at $k=k_0$. In this case, we will have a continuous spectrum $\lambda=M(k_0)=-{1\over 2\cosh{\pi k_0\over 2}+1} \in [-1/3,0)$. Explicitly, - M(k)&=&M(k\_0)-M(k)=-M\^[(1)]{}(k\_0)(k-k\_0)-[12]{}M\^[(2)]{}(k\_0)(k-k\_0)\^2+…,\
&=& -[ (1+2)\^2]{}(k-k\_0)-[12]{}[\^2+[\^22]{}-\^2\^2[k\_0 2]{}(1+2)\^3]{}(k-k\_0)\^2+…where $M^{(n)}(k_0)\equiv {d^n M \over dk^n}|_{k_0}$. Note that $M^{(1)}(k_0)$ is non-vanishing except at $k_0=0$, however, $M^{(2)}(0)\ne 0$. Therefore, if $k_0= 0$ then $\lam -M(k) \sim
-{1\over 2}M^{(2)}(0)k^2$, otherwise $\lam -M(k) \sim
-M^{(1)}(k_0)(k-k_0)$, to the leading order.

In this subsection we focus on the case of $k_0 \ne 0$ and will discuss the special case $\lam=-1/3$($k_0=0$) in the next subsection. Based upon the behavior of $\lam -M(k)$ near $k=k_0$ we will have three kind of choices for the behavior of $r_i(k)$ around $k=k_0$. The first one is to let $r_i(k)\sim d_i(k-k_0)$ for $i=1,2$, then we have nonzero \[Be0\] B\_[ie]{}(k\_0)&=&[-d\_i(-1) k\_0 (2+1)M\^[(1)]{}(k\_0)]{},\
\[Bo0\] B\_[io]{}(k\_0)&=&[-d\_ik\_0(2+1) M\^[(1)]{}(k\_0)]{}, and the corresponding eigenvector is v(k\_0)=(\
-\
|v\_e> + |v\_o>-[d\_1M\^[(1)]{}(k\_0)]{}|k\_0>\
|v\_e> + |v\_o>-[d\_2M\^[(1)]{}(k\_0)]{}|k\_0>). \[vge\] Note that $v(k_0)$ is degenerate with $v(-k_0)$ which is in general different from $v(k_0)$ because B\_[ie]{}(-k\_0)=B\_[ie]{}(k\_0), B\_[io]{}(-k\_0)=-B\_[io]{}(k\_0) and $A_{ee}$ and $A_{oo}$ are also even functions of $k_0$.

The second choice is $r_1(k)\sim (k-k_0)$ but $r_2(k)\sim
(k-k_0)^2$ or higher power of $(k-k_0)$ so that $B_{1e}$ and $B_{1o}$ are the same as the ones in (\[Be0\]) and (\[Bo0\]) but B\_[2e]{}=B\_[2o]{}=0.

In this case we can form a pair of “twist-definite” states from $v(\pm k_0)$ as following, v\_[+-]{}(k\_0)&=&[12]{}(v(k\_0)+v(-k\_0))|\_[B\_[2e]{}=B\_[2o]{}=0]{}\
&=&[B\_[1e]{}Det]{}( (1+[2(-3+4a\^2)3b]{}A\_[oo]{})\
0\
(c\_1-(c\_1c\_3-c\_2\^2)A\_[oo]{})|v\_e>\
-ic\_2|v\_o> )- ( 0\
0\
[12M\^[(1)]{}(k\_0)]{}(|k\_0>-|-k\_0>)\
0). \[vpm\] and v\_[-+]{}(k\_0)&=&[12]{}(v(k\_0)-v(-k\_0))|\_[B\_[2e]{}=B\_[2o]{}=0]{}\
&=&[B\_[1o]{}Det]{}( 0\
i (1+A\_[ee]{})\
(c\_3-(c\_1c\_3-c\_2\^2)A\_[ee]{}) |v\_o>\
ic\_2|v\_e> )- ( 0\
0\
[12M\^[(1)]{}(k\_0)]{}(|k\_0>+|-k\_0>)\
0). \[vmp\] Here the subscript indices $+$ or $-$ indicate the twist parity of the third and fourth components with respect to the twist matrix $C_{mn}$. Note that both $v_{+-}$ and $v_{-+}$ are the “twist-definite” eigenstates although their third and fourth components have different twist parity, this implies that two spacetime directions cannot have the same twist parity for each “twist-definite” eigenstate.

The third choice is $r_2(k)\sim (k-k_0)$ but $r_1(k)\sim
(k-k_0)^2$ or higher power of $(k-k_0)$ so that $B_{2e}$ and $B_{2o}$ are the same as the ones in (\[Be0\]) and (\[Bo0\]) but B\_[1e]{}=B\_[1o]{}=0. We will then get the corresponding eigenvectors similar to (\[vpm\]) and (\[vmp\]), u\_[-+]{}(k\_0)&=&[12]{}(v(k\_0)+v(-k\_0))|\_[B\_[1e]{}=B\_[1o]{}=0]{}\
&=&[B\_[2e]{}Det]{}( 0\
-(1+[2(-3+4a\^2)3b]{}A\_[oo]{})\
ic\_2|v\_o>\
(c\_1-(c\_1c\_3-c\_2\^2)A\_[oo]{}) |v\_e> )- ( 0\
0\
0\
[12M\^[(1)]{}(k\_0)]{}(|k\_0>-|-k\_0>) ). \[ump\] and u\_[+-]{}(k\_0)&=&[12]{}(v(k\_0)-v(-k\_0))|\_[B\_[1e]{}=B\_[1o]{}=0]{}\
&=&[B\_[2o]{}Det]{}( i (1+A\_[ee]{})\
0\
ic\_2|v\_e>\
(c\_3-(c\_1c\_3-c\_2\^2)A\_[ee]{}) |v\_o> )- ( 0\
0\
0\
[12M\^[(1)]{}(k\_0)]{}(|k\_0>+|-k\_0>) ). \[upm\] and it is easy to see that they are independent set from (\[vpm\]) and (\[vmp\]); moreover, the general state (\[vge\]) is the linear combinations of the four independent “twist-definite” eigenstates. We then conclude that there are four “twist-definite” eigenstates for each eigenvalue except $-1/3$. This is a natural generalization of the results for $M$ [@RSZ8] and $M^{'}$ [@Feng1] since our $\cMp$ involves two spatial directions, also we have four independent parameters $B_{ie}$ and $B_{io}$ in the general eigenstate (\[vge\]).

If we define the inner product of the vectors $u, v$ as $u^\dagger v$, then it is not hard to see that among the four independent eigenvectors, the ones with different twist parities are orthogonal to each other.

The $\lam =-1/3$ case
---------------------

Now, let us take $\lam=-1/3$ as an eigenvalue in the continuous spectrum. Then we have $k_0=0$ such that $M(k_0)=-1/3$. From the requirement that $r_i(k)/(k-M(k_0))$ should have no poles at $k_0$, one can determine the form of the functions $r_i(k)$ up to a scale factor. Taking the freedom of choosing the scale factor and to the leading order, we set r\_1(k)=r\_1 k\^2, r\_2(k)=r\_2 k\^2 where $r_{1,2}$ are arbitrary constants[^5] . And B\_[2e]{}&=&B\_[1e]{}=0\
r\^[-1]{}\_2B\_[2o]{}&=&r\^[-1]{}\_1B\_[1o]{}=- Then we obtain &=&\
&=&\
&=&\
&=&. Note that all of ${\cal C,D}s$ are dependent on $A_{ee}$ explicitly but dependent on $A_{oo}$ only through $Det$.

It is straightforward to get the eigenvector for the $\lam=-1/3$ eigenvalue. The explicit form is v=( (1+A\_[ee]{})\
-(1+A\_[ee]{})\
|v\_e> + |v\_o>-|k=0>\
|v\_e> + |v\_o>-|k=0> ). \[vect\]

In [@Feng1], it has been argued that the $-1/3$ eigenvalue is doubly degenerate, with two independent eigenvectors. At first looking, it seems that we have only one eigenvector. This is not the case. In fact, due to our freedom to choose $r_i$, we can set either $r_1(k)$ or $r_2(k)$ to have higher power expansion in terms of $(k-k_0)$ such that one of $B_{io}, i=1,2$ vanishes while the other finite. This will not spoil the whole story. On the contrary, we will find that the choice will lead to two independent eigenvectors and the above one is just the superposition of them.

First let $B_{1o}=0$, which could be achieved by choosing $r_1(k)\propto k^3$ or higher power of $k$. In this case, the above eigenvector turns to be v\_[+-,[-13]{}]{}=[B\_[2o]{}Det]{}( (1+A\_[ee]{})\
0\
ic\_2|v\_e>\
(c\_3-(c\_1c\_3-c\_2\^2)A\_[ee]{}) |v\_o> )-( 0\
0\
0\
|k=0>). \[vect1\]

On the other hand, one could choose $B_{2o}=0$ by chooseing $r_1(k)\propto k^3$ or higher power of $k$. Then the eigenvector has the form v\_[-+,[-13]{}]{}=[B\_[1o]{} Det]{}( 0\
-(1+A\_[ee]{})\
(c\_3-(c\_1c\_3-c\_2\^2)A\_[ee]{})|v\_o>\
-ic\_2|v\_e> )-( 0\
0\
|k=0>\
0). \[vect2\]

It is not hard to see that the general eigenvector (\[vect\]) is just the superposition of $v_{+-,{-1\over 3}}$ and $v_{-+,{-1\over
3}}$. Moreover $v_{+-,{-1\over3}}$ and $v_{-+,{-1\over3}}$ are kind of twist definite states as mentioned in the $k_0 \ne 0$ case.

One may wonder if (\[vect1\]) (\[vect2\]) are really the eigenvectors of ${\cal M}^\prime$ with eigenvalue $-1/3$. This could be checked directly with the explicit form of ${\cal
M}^\prime$. Unlike the matrix $M$ which has only one twist odd state $|k=0>$ at $-1/3$, our $\cMp$ is doubly degenerate at $-1/3$ in the continuous spectrum. However taking into account the space dimensions involved in our discussion, we find that the degeneracy at $-1/3$ in the case with $B$ field is only half of the one in the case without $B$ field. Nevertheless, the degeneracy of $\cMp$ at $-1/3$ matches the one of $M$.

A subtlety is about the discontinuity of $A_{oo}$ at $-1/3$ discovered in [@Feng1]. As shown there, $A_{oo}(({-1\over
3})^+)={3\over 4}ln27$ but $A_{oo}(({-1\over 3})^+)=-\infty$. Since in our (\[vect1\]) and (\[vect2\]) $A_{oo}$ only appears in $Det$ which blows up if $A_{oo} \rightarrow -\infty$, in this case the vectors in (\[vpm\]) and (\[vmp\])are just reduced to $(0,0,|k=0>,0)^T$ or $(0,0,0,|k=0>)^T$ which obviously cannot satisfy the eigen-equations. So the ambiguity is lifted and we should choose $A_{oo}(({-1\over 3})^+)$ instead of $A_{oo}(({-1\over 3})^-)$; therefore the degeneracy at $-1/3$ is two.

Discrete spectrum
-----------------

When the determinant factor $Det$ vanish, then one could have nontrivial solutions only if $B_{ie}=B_{io}=0, i=1,2$. The solutions of $Det=0$ correspond to the so-called discrete spectrum. Being the function of $\lam$, the equation has an integral form, which make it very hard to solve analytically. However, what we are really interested in is to see if $\lam=-1/3$ is an eigenvalue or not. The reasons are twofold: on the one hand, from the form of the ratio of tension, the $-1/3$ eigenvalue of ${\cal M}^\prime$ is essential to cancel the zero from det(1+3M). In order to have the finite ratio, one should expect the degeneracy at $\lam=-1/3$ of ${\cal M}^\prime$ match the one of $M$. On the other hand, from the experience of ${\cal M}$ without $B$ field, $-1/3$ exists as an eigenvalue in the discrete spectrum and make the degeneracy higher[@Feng1]. Therefore, it is quite important to investigate the possibility of $-1/3$ as an eigenvalue in the discrete spectrum. To this aim, taking $A_{ee}=-(3/4)V_{00}, \lam=-1/3$, we have Det= Obviously, $\theta=0$ is a solution. In fact, this recovers the well-known result that $\lam=-1/3$ belongs to the discrete spectrum, no matter what $b$ is, in the case of $\theta=0$. (see [@Feng1]). Certainly it is possible for specified values of $\theta$, $Det(\lam=-1/3)=0$. In this case, one find \^2=-12(V\_[00]{}+b/2)\^2. Since $\theta$ is real, we have to require 8bA\_[oo]{}V\_[00]{}-3(V\_[00]{}+b/2)\^2 0 The above relation set the range of $b$, in which it is possible to find a $\theta$ such that $\lam=-1/3$ is a discrete eigenvalue. However, one could expect that for generic value of $\theta$, $\lam=-1/3$ does not belong to discrete spectrum, although it exists in the continuous spectrum.

We will not discuss carefully the discrete spectrum here. In [@Feng2], the careful discussion on discrete spectrum and eigenvector has been worked out.

The spectrum of $\cM^{\prime 12}$ and $\cM^{\prime 21}$
-------------------------------------------------------

Due to the fact that the matrices $\cM^{\prime rs}$ enjoy the same property as the usual matrices $M^\prime$, it is straightforward to obtain the spectrum of $\cM^{\prime 12}$ and $\cM^{\prime 21}$, once we know the spectrum of $\cMp$. More precisely, since the matrices $\cM^{\prime rs}$ are commuting with each other and satisfy the relations +\^[12]{}+\^[21]{}=()\^2+(\^[12]{})\^2+(\^[21]{})\^2=1, \^[12]{}\^[21]{}=(-1), they share the same eigenvectors and their eigenvalues satisfy \^[12]{}(k)-\^[21]{}(k)&=&\
\^[12]{}(k)+\^[21]{}(k)&=&1-(k). From the above two relations, it’s easy to read out the spectrum of $\cM^{\prime 12}$ and $\cM^{\prime 21}$.

Discussions and Conclusions
===========================

In this paper we have calculated the spectrum of the Neumann matrix with zero modes in the constant B field background. We find both the continuous and discrete spectrum. The existence of the discrete spectrum will set a constraint on $b$ for a given $\theta$; moreover, for the generic $\theta$ there is no $-1/3$ in the discrete spectrum. We will take this as a good point to the finite and nonzero ratio of tension $\cR$.

The continuous one is similar to the one of $M$ but with four degenerate twist-definite eigenstates at each eigenvalue except at $-1/3$ where it is only doubly degenerate. The doubling of the degeneracy at each eigenvalue compared to the one of $M$ is expected since the B field mixes two spatial directions and doubles the dimensions of the eigen-space; moreover, the double degeneracy at $-1/3$ is required for the the ratio of the tension $\cR$ to be well-defined. However, to determine precisely the ratio from the spectrum requires the details of the density of states which is beyond the scope of this paper. On the other hand there is an analytic way to determine the ratio $\cR$ to be one a la the regularization method of Okuyama [@Okuyama2] as shown in [@Bonora2]. The agreement adds more weights to VSFT and its background independence.

Another issue on which our results are concerned is the new proposed map of Witten star to Moyal star [@Douglas] where the Moyal pair is the Fourier transform of a twist-even coordinate and the twist odd momentum. The Fourier basis are the twist-definite eigenstates of $M$. In [@MT] and [@Douglas] it is shown that at the zero slope limit, the star algebra factorizes into two subalgebras which was first proposed in [@Witten2]: ${\cal A}
\longrightarrow {\cal A}_0 \otimes {\cal A}_1$. The ${\cal A}_0$ corresponds to the subalgebra of zero-momentum sector and the ${\cal A}_1$ corresponds to the $C^*$-algebra of spacetime functions. The 3-string vertex factorizes accordingly |V\_3> |V\_3\^[(0)]{}> |V\_3\^[(1)]{}>, where $<x_1|\otimes <x_2|\otimes <x_3||V_3^{(1)}>$ can be identified as the the kernel for the usual Moyal product with zero noncommutativity [@Douglas].

In the presence of $B$ field, the directions without $B$ field could be treated along the above way, while the directions with $B$ field are a little different. It is easy to see that the factorization of the star algebra still works. However, one should replace commutative $C^*$-algebra ${\cal A}_1$ with the noncommutative one. The zero slope limit is the equivalent to (set $b=2$) \^[,rs]{}\_[mn]{} &&G\^V\^[rs]{}\_[mn]{} +[O]{}(\^2)\
[V]{}\^[,rs]{}\_[0n]{} &&0+ [O]{}()\
[V]{}\^[,rs]{}\_[00]{} &&G\^ (\^[rs]{}-\^[rs]{})+i\^[412+\^2]{}\^[rs]{}, and we can obtain the ${\cal A}_1$ part of the longitudinal 3-string vertex $|V_{3}^{\|}>$, it is (([-4+\^2 12+\^2]{})\_[r]{}a\_0\^[(r)]{}a\_0\^[(r)]{} -([812+\^2]{})\_[r<s]{}a\_0\^[(r)]{}a\_0\^[(s)]{} -([4i12+\^2]{})\_[r<s]{}a\_0\^[(r)]{}a\_0\^[(s)]{} )|0> which is exactly the same[^6] as the 3-string vertex for the canonical Moyal product(see (2.26) constructed in [@Douglas]). This confirms that the spacetime algebra is a noncommutative one, with Moyal product replacing the usual pointwise product.

Moreover, it deserves to mention that: in the recent paper [@Bonora2] they show that the zero mode of the tachyonic lump state in the constant B field background is nothing but the usual noncommutative soliton as a projector discovered in [@GMS]. This is complimentary to our above observation that the low energy zero mode of the SFT in constant B field background is governed by the noncommutative $C^*$-algebra.

It is interesting to generalize the proposed map of Moyal star to Witten star in [@Douglas] to the 3-string vertex with zero modes and with B field. This requires the eigenvectors of $\cMp$ to form a complete orthogonal set in order to obtain the proper Moyal conjugate pairs which are the twist-definite eigenvector by the Fourier transform based on these eigenvectors. Future work are required to carry out the program.

In this paper we did find the twist-definite eigenvectors of $\cMp$ , however, for each twist-definite eigenvector the twist parity is opposite among the two spatial components. This implies that we need to choose opposite twist parity assignments for the Moyal conjugate pairs between the two spatial directions parallel to the B field, that is, if we choose the twist-even coordinate modes and the twist-odd momentum modes as the Moyal conjugate pairs for one parallel direction to the B field, we should choose the twist-odd coordinate modes and the twist-even momentum modes as the conjugate pairs for another parallel direction. This kind of the flip of the twist parity for the Moyal pairs in two spatial directions is unexpected, and it deserves more study to understand its physical implication to the equivalence between Witten’s string field theory and the generalized noncommutative field theory.

Acknowledgment {#acknowledgment .unnumbered}
==============

BC would like to thank ITP, Chinese Academy of Science for the hospitality during his visit. FLL would like to thank Pei-Ming Ho and Sanefumi Moriyama for helpful discussions, Hsien-Chung Kao for his kind support, and the courtesy of the CTP at NTU, the NCTS, and the CosPA project, Taiwan, he was supported by the NSC grant No. NSC89-2112-M-032-002.

Appendix: The calculation of the ratio $\cR$
============================================

In this appendix, we show how to get the ratio of tension ${\cal
R}=1$, even in the case with $B$ field.

The formula for the ratio $\cR$ has been given in (\[R2\]), and in section 2.3, we have found = [4b\^2 \^2+ 12(V\_[00]{}+[b2]{})\^2]{}. \[-M\] As for the part involving $(1+3\cMp)$, we need to use regulator as in [@Okuyama2] since there are double degenerate eigenvectors at $-1/3$ as shown section 3.3. First we can rewrite \[3M1\] Det(1+3)=(4-3)\^2Det(\^(1+3M+Q) + i\^P) where Q& =& -|v\_e><v\_e| +|v\_o><v\_o|\
P&=& (|v\_e><v\_e|+|v\_o><v\_o|) with &=& 48(V\_[00]{}+[b2]{}) \^2+12V\_[00]{}(V\_[00]{}+[b2]{})\
&=& 48 V\_[00]{} \^2+12V\_[00]{}(V\_[00]{}+[b2]{})\
&=& 8 \^2+12V\_[00]{}(V\_[00]{}+[b2]{}) It is easy to see that \[abc\] +\^2=[4V\_[00]{}]{}, which will be useful later.

Further simplifying (\[3M1\]) we get && [Det(1+3)det(1+3M)\^2]{}\
&=& (4-3)\^2det(1+[1 1+3M]{}Q)det(1+[1 1+3M]{}(Q- P[1 1+3M+Q]{})P), \[Det\]\
&=& (4-3)\^2(1-H\_[ee]{})(1+H\_[oo]{})(1-(+q\^2)H\_[ee]{})(1+(-p\^2)H\_[oo]{}) \[3M2\], where p&=&<v\_e|[1 1+3M+Q]{}|v\_e> = [H\_[ee]{}1-H\_[ee]{}]{}\
q&=&<v\_o|[1 1+3M+Q]{}|v\_o> = [H\_[oo]{}1+H\_[oo]{}]{} and H\_[ee]{}&=&<v\_e|[11+3M]{}|v\_e> =V\_[00]{}\
H\_[oo]{}&=&<v\_o|[11+3M]{}|v\_o>. Note that $H_{oo}$ is not well-defined as shown in [@Feng1] and it requires proper regularization. Here we just keep it formally and will regularize it later.

We then combine the 2st(3rd) and the 4th(5th) factors in the (\[3M2\]), and use (\[abc\]) we get =(4-3)\^2 (1-H\_[ee]{}+(1-[4V\_[00]{}]{}H\_[ee]{})H\_[oo]{})\^2. The factor before $H_{oo}$ is formally zero, however, we know $H_{oo}$ is divergent. Using the proper regularization as done in [@Okuyama2], one has the final result as following (1-[4V\_[00]{}]{}H\_[ee]{})H\_[oo]{}=

In the end we get = ( \^2 + 4\^2 \^2+ 12(V\_[00]{}+[b2]{})\^2 )\^2 \[3M\]

Combing with (\[3M\]) and using the fact that $DetG=(1+({\theta
\over 2\pi})^2)^2$ which appears in the ratio formula (\[R2\]), we obtain =1 as expected.

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L. Rastelli, A. Sen and B. Zwiebach, “String field theory around the tachyon vacuum”, hep-th/0012251.

L. Rastelli, A. Sen, B. Zwiebach, “Classical solutions in string field theory around the tachyon vacuum”, hep-th/0102112.

L. Rastelli, A. Sen, B. Zwiebach, “Half Strings, projectors, and multiple D-branes in vacuum string field theory”, JHEP [**0111**]{}, 035 (2001), hep-th/0105058.

L. Rastelli, A. Sen, B. Zwiebach, “Boundary CFT construction of D-branes in vacuum string field theory”, JHEP [**0111**]{}, 045 (2001), hep-th/0105168.

L. Rastelli, A. Sen, B. Zwiebach, “Vacuum string field theory”, hep-th/0106010.

D. Gaiotto, L. Rastelli, A. Sen, B. Zwiebach, “Ghost structure and closed string in in vacuum string field theory”, hep-th/0111129.

L. Rastelli, A. Sen, B. Zwiebach, “A note on the proposal for the tachyon state in vacuum string field theory”, JHEP [**0111**]{}, 035 (2001), hep-th/0111153.

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D. Gross and W. Taylor, “Split string field theory, II.”, JHEP [**0108**]{}, 010 (2001), hep-th/0106036.

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H. Hata and S. Moriyama, “Exact results on twist anomaly”, JHEP [**0202**]{}, 036 (2002), hep-th/0201177.

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K. Okuyama, “Ratio of tensions from vacuum string field theory”, hep-th/0201136.

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L. Rastelli, A. Sen, B. Zwiebach, “Star algebra spectroscopy”, hep-th/0111281.

B. Feng, Y.-H. He and N. Moller, “The spectrum of the Neumann matrix with zero modes”, hep-th/0202176.

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[^1]: chenb@ictp.trieste.it

[^2]: linfl@mail.tku.edu.tw

[^3]: The finiteness of the ratio is more subtle since some regularization is required, moreover, in order to precisely determine the density states of the continuous spectrum. However, $-1/3$ is at the end of the spectrum so that we believe the match of the degeneracy of $-1/3$ for $M$ and $\cMp$ is essential for the ratio to be well-defined.

[^4]: However, C does not commute with $M^{12}$ and $M^{21}$.

[^5]: There is a freedom to choose the value of $r_{1,2}$. In this paper, if we don’t let the $r_i$ vanishing, then we just choose $r_i=1$. Choosing one of $r_i$ vanishing equivalent to choosing higher order of $k$.

[^6]: It is up to an overall constant factor $3\pi/4$.


Meta-analytic comparison of robotic and transoral laser surgical procedures in supraglottic carcinoma.
To assess published reports of oncological surgical success rates in patients who underwent transoral laser supraglottic surgery and robotic surgery for supraglottic cancer. A systematic review of the literature was conducted and a meta-analysis of published data was performed. PubMed, Sage, Medline and Cochrane data sources were investigated. Overall survival rates, disease-specific survival rates, additional treatments and recurrence rates were investigated to determine the success of the surgical procedures. The meta-analysis included 24 studies; 1617 studies were excluded. There were no statistically significant differences between the transoral laser supraglottic surgery and transoral robotic supraglottic surgery groups in terms of overall survival (77.0 per cent and 82.4 per cent respectively) and disease-specific survival (75.8 per cent and 87.0 per cent respectively). There was recurrence in 164 out of 832 patients (19.7 per cent) in the transoral laser supraglottic surgery group and in only 6 out of 66 patients (9 per cent) in the transoral robotic supraglottic surgery group. Transoral laser surgery and robotic surgery appear to have comparable and acceptable oncological success rates.


/**
 * @file
 * Test code for mutt_extract_message_id()
 *
 * @authors
 * Copyright (C) 2019 Richard Russon <rich@flatcap.org>
 *
 * @copyright
 * This program is free software: you can redistribute it and/or modify it under
 * the terms of the GNU General Public License as published by the Free Software
 * Foundation, either version 2 of the License, or (at your option) any later
 * version.
 *
 * This program is distributed in the hope that it will be useful, but WITHOUT
 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
 * FOR A PARTICULAR PURPOSE.  See the GNU General Public License for more
 * details.
 *
 * You should have received a copy of the GNU General Public License along with
 * this program.  If not, see <http://www.gnu.org/licenses/>.
 */

#define TEST_NO_MAIN
#include "config.h"
#include "acutest.h"
#include <search.h>
#include "mutt/lib.h"
#include "address/lib.h"
#include "email/lib.h"
#include "test_common.h"

int cmp(const void *a, const void *b)
{
  return strcmp(a, *(char **) b);
}

struct TestData
{
  const char *in; // string to parse
  char *out;      // expected result
  size_t len;     // expected parsed len
} test[] = {
  { NULL, NULL, 0 },
  { "apple", NULL, 0 },
  { "foo bar <baz", NULL, 0 },
  { "foo bar > baz", NULL, 0 },
  { "foo bar <foo.bar> boo bar", "<foo.bar>", 17 },
};

void test_mutt_extract_message_id(void)
{
  for (size_t i = 0; i < mutt_array_size(test); i++)
  {
    size_t len = 0;
    char *out = mutt_extract_message_id(test[i].in, &len);
    TEST_CHECK_STR_EQ(test[i].out, out);
    if (!TEST_CHECK(test[i].len == len))
    {
      TEST_MSG("Expected: %zu", test[i].len);
      TEST_MSG("Actual  : %zu", len);
    }
    FREE(&out);
  }

  {
    const char *tokens[] = { "foo bar ", "<foo@bar.baz>", " moo mar", "<moo@mar.maz>" };
    char buf[1024];
    size_t off = 0;
    for (size_t i = 0; i < mutt_array_size(tokens); i++)
    {
      off += mutt_str_copy(&buf[0] + off, tokens[i], sizeof(buf) - off);
    }

    char *tmp = NULL;
    off = 0;
    size_t elems = mutt_array_size(tokens);
    for (const char *it = &buf[0]; (tmp = mutt_extract_message_id(it, &off)); it += off)
    {
      TEST_CHECK(tmp[0] == '<');
      char *found = lfind(tmp, &tokens[0], &elems, sizeof(char *), cmp);
      TEST_CHECK(found != NULL);
      FREE(&tmp);
    }
  }
}


Internally Displaced Persons, 2005

Table includes countries in which persecution, armed conflict, or
widespread violence has internally displaced the largest numbers of
civilians. Although internally displaced persons share many
characteristics with refugees, they are not protected by international
refugee law because they remain inside their own countries.


| file://:0:0:0:0 | Module sys | isUsedAsModule |
| imported.py:0:0:0:0 | Module imported | isUsedAsModule |
| main.py:0:0:0:0 | Module main | isUsedAsScript |
| myscript.py:0:0:0:0 | Script myscript | isUsedAsScript |
| script:0:0:0:0 | Script script | isUsedAsScript |


Economic pressure requires agricultural processes, alike industrial processes, to operate mechanically, thus with a minimum of manpower, and to yield superior results in the shortest possible time. Consequently, for the sake of efficient tree harvesting and fruit collection, fruit-tree grooves are planted in successive rows between which dedicated fruit harvesting and fruit collection equipment operates. Nowadays for example, mechanical harvesting is achieved by running a motorized harvester along a first row-side of a row of trees, stopping to clamp a vibrator on each successive tree-trunk, and using a deflector to guide the harvested fruit to the second row-side, for collection. In association with the deflector supported by the harvester on the first row-side, the fruit collection equipment being operated on the second row-side receives the fruit and forwards it for further transportation.
In U.S. Pat. Nos. 5,469,695 and 5,816,037, the present applicants disclose equipment for mechanical harvesting and for the collection of fruit from harvested trees, but those means are rather bulky, and expensive to acquire and maintain.
The terms “mechanical” and “mechanically” are used hereinbelow to define operations not requiring manual labor or manual intervention, besides the operation of controls by an operator or a driver.
Fruit collection equipment is required to feature simple construction, operate fast and reliably at minimal maintenance costs, and be offered to the market at low acquisition price. This contrasts with the hereinabove-mentioned fruit collection means, which are expensive to acquire and to maintain.


Q:

Instanciando o contexto da base de dados no controller

Olá, tenho uma dúvida relacionado ao modo de instanciamento do DbContext no controlador. Qual a diferença entre os dois métodos abaixo de instanciamento ?
1.
    private ApplicationDbContext _db;
    public ApplicationDbContext db
    {
        get { return _db ?? new ApplicationDbContext(); }
        set { _db = value; }
    }

2.
private readonly ApplicationDbContext db = new ApplicationDbContext();

Pois atualmente eu utilizo a segunda, porem em vários modelos de ASP.NET MVC 5 vejo que vem como padrão a primeira opção...

A:

Bem, a única diferença que vejo é que na primeira opção caso o valor de _db estiver nulo ele ira gerar uma nova instância, adiciona uma certa segurança a sua implementação, mas, como foi já foi comentado, a não ser que este seja apenas um projeto de testes, vejo dois problemas nisso:
1 - Ao utilizar o "new" é criado uma dependencia entre o seu controller e a classe "ApplicationDbContext", o ideal seria utilizar Injeção de dependencia para reduzir o acoplamento entre as camadas de sua aplicação.
2 - Ao realizar consultas no banco diretamente do controller, você está impedindo que ela possa ser reutilizada em outras partes do projeto, pois só estará acessivel dentro desse controller
Mas da pra citar vários outros problemas nessa abordagem....


Generally, in this kind of fishing reel the spool is freely rotatable for casting, with the spool rotating at a higher speed than a fishing line drawn out of the spool causing backlash, whereby the line wound on the spool is entangled or caught in a gap between the spool and a reel frame. In order to solve this problem, a centrifugal brake usually is used to impart a predetermined rotational resistance to the spool. However, a sufficient braking effect has not been achieved with this type of brake and the centrifugal brake, which exerts the braking action by slidable contact of a brake shoe with a brake drum, may have its braking force changed due to wearing of the brake shoe.
There has been proposed a spool braking device using a magnet and a metallic conductor, one of them being rotatable together with the spool and the other being fixed to the reel body so that the rotating field of the rotating member generates an eddy current. Hence, the rotating member, due to Fleming's left-hand rule, is given a magnetic force against its rotation, thereby exerting a braking action on the spool.
This method can increase the braking effect and since there is no contact-portion at the brake member variations in the braking force can be eliminated. However, with this type of mechanism a bearing for the spool shaft becomes magnetized so that the magnetic flux links the bearing and the spool shaft which is made of a magnetic material of high strength, such as stainless steel, thereby increasing the rotational resistance against the rotation of the spool which is integral with the spool shaft.


28 Wis.2d 431 (1965)
ESTATE OF KING: STATE, Appellant,
v.
MARINE NATIONAL EXCHANGE BANK OF MILWAUKEE, Executor, Respondent.
Supreme Court of Wisconsin.
September 13, 1965.
October 5, 1965.
*433 For the appellant the cause was argued by E. Weston Wood, assistant attorney general, with whom on the brief were Bronson C. La Follette, attorney general, Harold H. Persons, assistant attorney general, attorneys, and Richard E. Williams, inheritance tax counsel, of counsel.
For the respondent there was a brief by Shea, Hoyt & Greene, and oral argument by Paul F. Meissner, all of Milwaukee.
HEFFERNAN, J.
The sole question before the court is the taxability of the pension payments to be made to Mrs. King. The state contends that a tax is imposed by sec. 72.01 (3) (b), Stats.:
"When a transfer is of property, made without an adequate and full consideration in money or money's worth . . . by . . . gift, intended to take effect in possession or enjoyment at or after the death of the . . . donor, . . ."
It is our conclusion that this case is controlled by the Estate of Sweet (1955), 270 Wis. 256, 70 N. W. (2d) 645. We there held that no tax was assessable against a widow's pension provided for by the Federal Employees' Pension Act, where the pension, by the terms of the statute, was payable after the death of the employee only to the wife. Our court concluded that there would be no tax when "the federal *434 retirement system prescribes the beneficiary and the employee cannot change it." Estate of Sweet, supra, at page 262.
The principles upon which the Estate of Sweet was decided were amplified in Estate of Stone (1960), 10 Wis. (2d) 467, 103 N. W. (2d) 663.
In Stone the decedent elected to come under a "joint and survivor option" of the trust agreement. Under the terms of this option, Stone elected to receive a reduced pension during his lifetime, and by so doing he triggered the provision of the agreement that extended the payment of the pension to his widow for her lifetime following his death. In the Stone Case the decedent chose to forego some of the pension payments that might have been payable in his lifetime that the widow might receive benefits for her life. In the Stone Case this court said:
"We have no hesitancy in deciding that Mr. Stone's election of a joint and survivor option was a `transfer' of whatever interest he had in the trust fund, . . ." Estate of Stone, supra, at page 470.
The facts in Stone were distinguished from those in Sweet. It was pointed out that Mr. Sweet had no option to designate his beneficiary or affect the amount to be paid after his death. The entire devolution of the pension was controlled by statute and was beyond Sweet's control. Stone, "however, was free to retain all the benefits . . . for himself, or he could, by reducing his own benefits, provide benefits to his wife if she survived." Estate of Stone, supra, at page 471.
The court summarized the Sweet holding by saying:
". . . we consider that the Sweet decision is authority only as to benefits which, under a particular retirement plan, are payable to another upon the death of the employee without any option having been given to the employee as to the distribution of such benefits." Estate of Stone, supra, at page 474.
*435 It appears to us that, in this controlling particular, the facts in the case before us are substantially the same as in Sweet. The terms of the Journal pension trust agreement determine how the benefits of the trust are to be distributed to the retired employee and to his widow. No power or discretion is vested in the decedent to alter this distribution. This situation is different from that in Stone, and as in Sweet we deem that no transfer has been made by the decedent.
The attorney general, appearing for the state of Wisconsin, urges that we overrule the Sweet decision on the basis of the reasoning expressed by Mr. Justice CURRIE for the dissenters in Sweet. The dissenters therein took the position that "by voluntarily rendering services for the federal government under the terms of the employment . . . performed the acts which effected the transfer to his widow." Estate of Sweet, supra, at page 264. The dissent also pointed out that the pension payments were not in fact a gratuity, but were in fact additional compensation, the payment of which was deferred until retirement, and hence the fund to the extent of his interest was the property of the decedent.
The logic of the dissent has much to commend it, but we are not in a position to consider this question as an original proposition. Since the decision in Sweet in 1955, our position was clarified and re-enunciated in Stone in 1960. One commentator stated:
"When the Stone and Sweet cases are considered together, it is clear that there will not be a tax on the amount which the wife receives if the plan does not give the husband an option." 1961 Wisconsin Law Review, 153, 156.
The Stone opinion pointed out that the refusal of the legislature to affirm taxability is not the equivalent of an express tax exemption or an affirmance of the court's decision. Nevertheless, as set forth in Stone, the legislature has declined to alter the statute in order to eliminate the effect of *436 Sweet, neither has there been any legislative change since the Stone Case that would cast doubt upon the distinctions that were drawn therein.
It appears, therefore, that the legislative history of sec. 72.01 (3) (b), Stats., together with the interpretation that it has been given in Sweet, Stone, and the cases that preceded, would justify the analysis of the commentator quoted above.
This court has previously stated that it would depart from stare decisis in unintentional tort actions or in situations involving conduct that would not in the usual course of things be planned in advance and when it appeared that there were compelling policy reasons that the established court-made rule should be altered. Kojis v. Doctors Hospital (1961), 12 Wis. (2d) 367, 107 N. W. (2d) 131; Holytz v. Milwaukee (1962), 17 Wis. (2d) 26, 115 N. W. (2d) 618. As we said in Wilcox v. Wilcox (1965), 26 Wis. (2d) 617, 623, 133 N. W. (2d) 408:
"The rules of stare decisis attempt to give certainty to our law, so conduct can be planned in light of foreseeable legal consequences. Certainty, however, is less relevant in the law of unintentional torts, where conduct is not planned, than in the law of contracts or, more particularly, in the law of real property."
Certainly we should apply a similar rationale when considering a reinterpretation of a statute involving inheritance taxes. While no precedent of this court is under all circumstances immune from re-examination, nevertheless, where a course of action is likely to be determined only after a considered examination of the existing case law, we are obviously more reluctant to disturb the posture of the law than where the conduct is of the spur of the moment. The entire field of estate planning presupposes a stability and continuity to the law that will enable a well-informed lawyer to predict the tax consequences of his client's conduct in *437 regard to various arrangements that may be devised for the disposition or transfer of his property.
We, therefore, cannot properly consider this problem ab initio. In the absence of a legislative change which would clearly make the transaction herein taxable, or in the absence of a showing of compelling policy reasons (and there has been no such showing) that our previous rulings have resulted in the inequitable or unfair administration of the inheritance-tax laws, we consider that this case is controlled by our previous decisions.
By the Court.Order affirmed.


#include "elf.h"
	.arch sledgehammer
	.code32
	.equ	FLAT_CODE_SEG,_pmcs-_gdt
	.equ	FLAT_DATA_SEG,_pmds-_gdt
	.equ	MSR_K6_EFER,   0xC0000080
	.equ	EFER_LME,      0x00000100
	.equ	X86_CR4_PAE,   0x00000020
	.equ	CR0_PG,        0x80000000

	.section ".prefix", "ax", @progbits

#define LOAD_ADDR 0x10000

	/* ELF Header */
	.globl elf_header
elf_header:
e_ident:	.byte	0x7f, 'E', 'L', 'F', 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0
e_type:		.short	ET_EXEC
e_machine:	.short	EM_X86_64
e_version:	.long	1
e_entry:	.long	LOAD_ADDR + elf_start - elf_header
e_phoff:	.long	elf_program_header - elf_header
e_shoff:	.long	0
e_flags:	.long	0
e_ehsize:	.short	elf_header_end - elf_header
e_phentsize:	.short	ELF32_PHDR_SIZE
e_phnum:	.short	(elf_program_header_end - elf_program_header)/ELF32_PHDR_SIZE
e_shentsize:	.short	0
e_shnum:	.short	0
e_shstrndx:	.short	0
elf_header_end:

elf_program_header:
phdr1_p_type:	.long	PT_NOTE
phdr1_p_offset:	.long	elf_note - elf_header
phdr1_p_vaddr:	.long	elf_note
phdr1_p_paddr:	.long	elf_note
phdr1_p_filesz:	.long	elf_note_end - elf_note
phdr1_p_memsz:	.long	elf_note_end - elf_note
phdr1_p_flags:	.long	PF_R | PF_W | PF_X
phdr1_p_align:	.long	0

/* The decompressor */
phdr2_p_type:	.long	PT_LOAD
phdr2_p_offset:	.long	0
phdr2_p_vaddr:	.long	elf_header
phdr2_p_paddr:	.long	LOAD_ADDR
phdr2_p_filesz:	.long	_verbatim_size
phdr2_p_memsz:	.long	_image_size
phdr2_p_flags:	.long	PF_R | PF_W | PF_X
phdr2_p_align:	.long	16

elf_program_header_end:

	.globl elf_note
elf_note:
	.balign 4
	.int 2f - 1f
	.int 4f - 3f
	.int EIN_PROGRAM_NAME
1:	.asciz "ELFBoot"
2:
	.balign 4
3:
	.asciz "Etherboot"
4:

	.balign 4
	.int 2f - 1f
	.int 4f - 3f
	.int EIN_PROGRAM_VERSION
1:	.asciz "ELFBoot"
2:
	.balign 4
3:
	.asciz VERSION
4:

#if 0
	.balign 4
	.int 2f - 1f
	.int 4f - 3f
	.int EIN_PROGRAM_CHECKSUM
1:	.asciz "ELFBoot"
2:
	.balign 4
3:
	.word 0
4:
#endif
	.balign 4
elf_note_end:

elf_start:
	.code64
	/* Reload the gdt to something I know */
	leaq	_gdt(%rip), %rax
	movq	%rax, 0x02 + gdtptr(%rip)
	lgdt	gdtptr(%rip)

	/* Enter 32bit compatibility mode */
	leaq	elf_start32(%rip), %rax
	movl	%eax, 0x00 + elf_start32_addr(%rip)
	ljmp	*elf_start32_addr(%rip)

elf_start32:
	.code32
	/* Reload the data segments */
	movl	$FLAT_DATA_SEG, %eax
	movl	%eax, %ds
	movl	%eax, %es
	movl	%eax, %ss

	/* Disable paging */
	movl	%cr0, %eax
	andl	$~CR0_PG, %eax
	movl	%eax, %cr0

	/* Disable long mode */
	movl	$MSR_K6_EFER, %ecx
	rdmsr
	andl	$~EFER_LME, %eax
	wrmsr

	/* Disable PAE */
	movl	%cr4, %eax
	andl	$~X86_CR4_PAE, %eax
	movl	%eax, %cr4

	/* Save the first argument */
	pushl	%ebx
	jmp	_start

gdtptr:
	.word	_gdt_end - _gdt -1
	.long	_gdt
	.long	0
_gdt:
elf_start32_addr:
	.long	elf_start32
	.long	FLAT_CODE_SEG
_pmcs:
	/* 32 bit protected mode code segment, base 0 */
	.word	0xffff,0
	.byte	0,0x9f,0xcf,0

_pmds:
	/* 32 bit protected mode data segment, base 0 */
	.word	0xffff,0
	.byte	0,0x93,0xcf,0
_gdt_end:

	/* Dummy routines to satisfy the build */
	.section ".text16", "ax", @progbits
	.globl	prefix_exit
prefix_exit:

	.globl	prefix_exit_end
prefix_exit_end:
	.previous


tedivm/fetch

description: A PHP IMAP Library

keywords: email imap pop3

type: library

satooshi/php-coveralls: dev-master

Fetch: src/


“Today the Federal Energy Regulatory Commission is announcing two major enforcement actions against Amaranth and Energy Transfer Partners for manipulating natural gas markets. This morning we issued show cause orders that make preliminary findings these companies have manipulated natural gas markets.

These enforcement actions are very significant. They represent the first prosecution of market manipulation by the Commission with the new enforcement powers Congress gave us two years ago. These actions also represent the first time the Commission has proposed maximum civil penalties.

As detailed in our orders, the evidence against both companies is significant. In both instances, there are tapes or instant messages that reveal an intent to manipulate prices. Before further describing our actions today, however, I want to emphasize two things we are not doing. First, the Commission is not seeking to punish risk taking in energy markets. Well functioning markets are critical to serving the Nation's energy needs and investors are expected to take risks in these markets. These cases do not involve honest risk taking, but rather well designed plans to manipulate markets.

Second, the Commission is not making final conclusions in these show cause orders. Rather, they represent our belief, based on the existing record, that these companies may have manipulated markets and therefore violated the law. Both companies will have the opportunity to rebut the preliminary conclusions set forth in the show cause orders. We will give full and fair consideration to their response.

The Commission proposes to impose maximum penalties and penalties that approach the maximum for certain violations by Amaranth and Energy Transfer Partners. In the case of Amaranth we propose civil penalties and remedies in the amount of $291 million. We propose civil penalties and remedies in the amount of $167 million on Energy Transfer Partners.

The Commission has been exercising its new civil penalty authority since January. This is the first instance in which we propose maximum penalties for certain violations. We do so here for a number of reasons. First, the violations in question involve market manipulation. Manipulation is one of the most serious violations of Commission requirements.

Second, the harm to the market caused by Amaranth and Energy Transfer Partners was significant.

Third, in both cases, we find the violations were intentional. In both cases, we find there was involvement of senior management. Further, in both cases, we find the level of cooperation does not merit any reduction in civil penalties.

Market manipulation may tend to be subject to maximum civil penalties, because of the seriousness of the violation, especially when combined with clear intent, involvement of senior management, and a lack of cooperation with our investigation.

The manipulative schemes in question were designed to lower prices in one market in order to benefit positions held in a related market. Manipulation that lowers prices is as offensive as manipulation that raises prices. Market manipulation undermines the integrity of markets, regardless of whether prices are manipulated upward or downward. Manipulation that raises prices harms consumers immediately and in ways that are easy to understand. Manipulation that lowers prices also hurts consumers, over a longer period and more indirectly.

It is also important to recognize that a company that manipulates prices downward today may try to manipulate prices upward tomorrow.

We do not allege Amaranth and Energy Transfer Partners colluded in their manipulative schemes - they pursued different schemes designed to manipulative prices in different markets. These are separate investigations.

These investigations were extensive and span a period of months. I want to congratulate the Commission's Office of Enforcement staff for their fine work on these investigations. Manipulative schemes can be highly complicated, and market manipulation can be difficult to detect and difficult to prosecute. The Amaranth investigation began last year, when Commission oversight staff observed suspicious movements in natural gas prices. That proves the quality and effectiveness of the Commission's market oversight function.

I also want to praise the Commodity Futures Trading Commission (CFTC) for being a strong partner in coordinating investigations regarding market manipulation. In particular, I want to commend the leadership of Acting Chairman Lukken and former Chairman Jeffrey. FERC and the CFTC have collaborated closely in these and other investigations. It is fair to say that the coordination between FERC and CFTC enforcement has never been as strong as it is now. That close coordination has improved our respective ability to detect and sanction market manipulation.

To conclude, these show cause orders represent the conclusion of extensive investigations. We have made preliminary findings that the practices in question constitute market manipulation, and are taking the first step to initiate administrative litigation. Amaranth and Energy Transfer Partners now have the chance to dispute our findings on the record. We are not taking final action today and are not rendering final judgment.
"


---
abstract: 'An impurity coupling to both spin and charge degrees of freedom is added to a periodic $t$-$J$ chain such that its interaction with the bulk can be varied continuously without losing integrability. Ground state properties, impurity contributions to the susceptibilities and low temperature specific heat are studied as well as transport properties. The impurity phase–shifts are calculated to establish the existence of an impurity bound state in the holon sector.'
address:
- '$^1$Institut für Theoretische Physik, Universität Hannover, D-30167 Hannover, Germany'
- |
    $^2$Department of Physics, Theoretical Physics, Oxford University\
    1 Keble Road, Oxford OX1 3NP, Great Britain
author:
- 'Gerald Bedürftig$^1$, Fabian H.L.Eßler$^2$, and Holger Frahm$^1$'
date: September 1996
title: 'Integrable impurity in the supersymmetric $t$-$J$ model'
---

\#1[preprint]{}

[2]{} Quantum fluctuations are known to play an important role in the physics of low dimensional strongly correlated electron systems: the low temperature properties of such systems in one spatial dimension have to be described in terms of a Luttinger liquid rather than a Fermi liquid. From an experimental point of view the transport properties of these systems in the presence of boundaries and potential scatterers are of particular interest. Recently several attempts have been made to describe such a situation: Using renormalization group techniques the transport properties of a 1D interacting electron gas in the presence of a potential barrier have been studied by Kane and Fisher [@kafi:92]. Their surprising findings triggered further work using different techniques like boundary conformal field theory [@card:89] and an exact solution by means of a mapping to the boundary sine-Gordon model [@Fendley; @tsve:95a]. In particular the low temperature properties of magnetic (Kondo) impurities in a Luttinger liquid [@Affleck; @FrojdhJ] have been investigated in great detail. In the present work we will investigate the effects of a particular type of potential impurity in a Luttinger liquid (where both spin-and charge degrees of freedom are gapless) by means of an exact solution through the Quantum Inverse Scattering Method (QISM) [@vladb].

Attempts to study effects due to the presence of impurities in many-body quantum system in the framework of integrable models have a long successful history [@anjo:84; @LeeSchl; @Vega; @bares:95; @KondoBA]. As far as lattice models are concerned the underlying principle in these exact solutions is the fact that the QISM allows for the introduction of certain “inhomogeneities” into vertex models without spoiling integrability. The [*local*]{} vertices—so called ${\cal L}$-operators—are objects depending on a complex valued spectral parameter acting on an auxiliary matrix space in addition to the quantum space of the model. They are solutions of a Yang-Baxter equation with an ${\cal R}$ matrix which itself acts on two copies of the matrix space and depends on the difference of the corresponding spectral parameters only. This allows to to build families of vertex models with site-dependent shifts of the spectral parameters and even different quantum spaces on different sites. The first fact has been widely used in solving models for particles with an internal degree of freedom by means of the nested Bethe Ansatz [@yang:67]. The second approach has been first applied by Andrei and Johannesson to study the properties of an $S={1\over2}$ Heisenberg chain with an additional site carrying spin $S$ [@anjo:84] (see also [@LeeSchl]).

In this letter we study the properties of the supersymmetric $t$–$J$ model with one vertex replaced by an ${\cal L}$ operator acting on a four–dimensional quantum space. This preserves the $gl(2|1)$ supersymmetry of the model but at the same time lifts the restriction of no double occupancy present in the $t$–$J$ model at the impurity site. The existence of a free parameter in the four–dimensional representations of the superalgebra [@brax:94] allows to tune the coupling of the impurity to the host chain. As will be shown below, the present model allows to study some aspects of a more general situation than the ones mentioned above: the impurity introduced here couples to [*both*]{} spin- and charge degrees of freedom of the bulk Luttinger liquid. The extension of our calculation to the case of many impurities is straightforward.

The solution of the model is completely analogous to that of the pure $t$–$J$ model [@QISM_tJ]: The transfer matrix generating the hamiltonian and the other conserved quantities is the trace of a product of the local ${\cal L}$ operators chosen as ${\cal L}_{tJ}= (\lambda+i\Pi)/
(\lambda+i)$ for the regular sites ($\Pi$ is a graded permutation operator acting on the tensor product of the auxiliary and the quantum space) and ${\cal L}_{34}\propto\lambda-i({\alpha \over 2}+1)+i\tilde{\cal L}$ on the site associated to the impurity. Written as a matrix in the three dimensional auxiliary space $\tilde{\cal L}$ reads $$\tilde{\cal L}\!\!=\!\!
  \left( \begin{array}{ccc}
  X_2^{\downarrow \downarrow}+X_2^{00} &
  - X_2^{\downarrow \uparrow} &
   Q_{\uparrow}
\\
  - X_2^{\uparrow \downarrow} &
  X_2^{\uparrow \uparrow}+X_2^{00} &
  Q_{\downarrow}
\\
  Q^\dagger_{\uparrow} &
  Q^\dagger_{\downarrow} &
 \!\alpha+  X_2^{\uparrow \uparrow}+X_2^{\downarrow \downarrow}+2 X_2^{00}\!\!
\end{array} \right)\!\!.$$ Here $Q_{\sigma}=\sqrt{\alpha+1}X_2^{0
\sigma}-\sigma\sqrt{\alpha}X_2^{-\sigma 2}$ with $\sigma=\pm$, where $+$ ($-$) corresponds to $\uparrow$ ($\downarrow$) and where the Hubbard projection operators are given by $X^{ab}=|a\rangle\langle b|$ with $a,b=\uparrow,\downarrow,2,0$. The hamiltonian is then given by the logarithmic derivative of the transfer matrix at spectral parameter $\lambda=0$. In general the form of this hamiltonian is rather complicated and will be given elsewhere [@bef:up]. In any case the precise form of the lattice (impurity) interactions is not essential as far as low-energy properties are concerned: in the continuum limit only a small number of terms with scaling dimensions smaller than two will survive (taking the continuum limit and identifying the scaling dimensions of the composite operators at the impurity site is rather nontrivial though). Physically the model describes an impurity with four allowed states (spin-up/down electrons, empty/doubly occupied site) that couples to two neighbouring $t$-$J$ sites and also modifies the interaction between the $t$-$J$ sites (see Fig. \[fig:im.ps\]). In the limiting cases $\alpha\to0(\infty)$ the hamiltonian simplifies essentially: In the first case the impurity acts as a ordinary $t$–$J$ site in the ground state below half filling, for $\alpha\to\infty$ the impurity site is doubly occupied and induces a twist in the boundary conditions of the host chain.

The eigenstates of the hamiltonian for $N_\uparrow$ ($N_\downarrow$) electrons with spin $\uparrow$ ($\downarrow$) are constructed by means of the nested algebraic Bethe Ansatz (NABA) leading to a system of algebraic equations for the spectral parameters $\lambda_j$ ($j=1,\ldots,N_e=
N_\uparrow + N_\downarrow$) and $\lambda_\alpha^{(1)}$ ($\alpha=1,\ldots,N_\downarrow$) $$\begin{aligned}
 \left( \frac{\lambda_j-{i\over 2}}{\lambda_j+{i\over 2}} \right)^{L}
 \left( \frac{\lambda_j-{\alpha+1 \over 2}i}{\lambda_j+{\alpha+1 \over 2}i}
 \right) &=& \prod^{N_\downarrow}_{\alpha=1} \frac
 {\lambda_j-\lambda^{(1)}_\alpha-{i\over 2}}
 {\lambda_j-\lambda^{(1)}_\alpha+{i\over 2}}\ , \nonumber \\
 \prod^{N_e}_{j=1} \frac{\lambda^{(1)}_\alpha-\lambda_j+{i\over 2}}
 {\lambda^{(1)}_\alpha-\lambda_j -{i\over 2}} &=& -
 \prod^{N_\downarrow}_{\beta=1}
 \frac{\lambda^{(1)}_\alpha-\lambda^{(1)}_\beta+i}
 {\lambda^{(1)}_\alpha-\lambda^{(1)}_\beta-i}\ .
\nonumber\end{aligned}$$ The corresponding eigenvalues of the hamiltonian in the grand canonical ensemble are $E= -\mu N_e -(H/2)(N_\uparrow-N_\downarrow)
+\sum_{j=1}^{N_e}{1/( \lambda_j^2+{1\over 4})}$.

The configuration of spectral parameters leading to the lowest energy state for given chemical potential and magnetic field are found in complete analogy to the pure $t$–$J$ chain: the ground state for finite $H$ is described by two filled Fermi seas of $\lambda$-$\lambda^{(1)}$-“strings” $\lambda_\pm=\lambda^{(1)} \pm \frac{i}{2}$ with real $\lambda^{(1)}$ associated with the holon excitations and real solutions $\lambda_j$ describing spin degrees of freedom. In the thermodynamic limit dressed energies $\epsilon_c$ and $\epsilon_s$ can be associated with the excitations of these objects. They are given in terms of coupled integral equations which are identical to those found for the chain without impurities [@schl:87]. In the resulting ground state energy the impurity contribution can be identified from its $L$-dependence which allows to compute the occupation, magnetization and susceptibilities of the impurity site. Analytical results for these expectation values are available only in limiting cases close to half filling and for densities near or below the critical density $n_c$ related to the magnetic field by $H=4\sin^2(\pi n_c/2)$ where the ground state is ferromagnetic and only real $\lambda_j$ are present [@bef:up]. For general values of band filling and magnetic field the magnetization and particle number on the impurity site can be determined by numerically solving a set of two coupled integral equations (see Fig. \[fig:gs\_prop\]): For $\alpha\to0$ the impurity mimics the bulk behaviour as discussed above. For large $\alpha$ the impurity occupation number is close to 1 (2) for $n_e<n_c$ ($>n_c$) leading to an enhanced (reduced) magnetization at small (large) electron densities. The magnetic susceptibility of the impurity near half filling is found to be $\sim \chi_{bulk}/\alpha$. We note that the magnetization curves for sufficiently large $\alpha$ intersect the ones for small $\alpha$.

In addition to the ground state properties the Bethe Ansatz allows to study the finite temperature behaviour of the system. The thermodynamic Bethe Ansatz equations for the system with impurity are the same as in the pure case [@schl:87]. Again the impurity contribution can be isolated in the free energy which can evaluated explicitly in the limit $T\to\infty$ and for $H\gg T$ using Takahashi’s method [@taka:74]. In the high temperature limit we find $$\begin{aligned}
  F_{\rm bulk}&=&-LT \ln\left(1+e^{\frac{\mu}{T}}2\cosh\frac{H}{2T}\right)
\nonumber\\
F_{\rm imp}&=& -\frac{2\alpha}{\alpha+2}-T
\ln\left(1+e^{\frac{2\mu}{T}}+2e^{\frac{\mu}{T}}\cosh\frac{H}{2T}\right)
\nonumber\end{aligned}$$ giving the correct entropy in this limit. Note that the parameter $\alpha$ enters the leading term in this expansion in a trivial way only.

For low temperatures $T\ll H$ we can determine the phase diagram of the system. Most interesting is the behaviour at half filling where the impurity contribution to the specific heat is found to show a different temperature dependence than the one from the bulk: for $2\mu>H>4$ the system is ferromagnetic and the low T free energy is given by (we suppress the contribution from the ground state energy) $$F_{\rm bulk} \approx -{L \over 2 \sqrt{\pi}}T^{3/2} e^{4-H \over T}, \qquad
F_{\rm imp}\approx  -{T} e^{H/2-\mu \over T}\ .$$ For smaller magnetic fields the thermodynamic equilibrium state is not ferromagnetically ordered, the bulk free energy is $F_{\rm bulk}
\approx-\pi L T^2/(6v_s)$ with the spinon velocity $v_s$ and the impurity contribution is $F_{\rm imp}\approx -{2 \over \alpha}T^{3/2}$ up to factors that cannot be caculated in closed form in general. Near $H=H_c$ this factor becomes $\sqrt{3 \over 8}(4-H)^{-{3 \over 4}} \exp\left({1\over
T}(H/2-\mu+{2\over 3 \pi} (4-H)^{3/2})\right)$.

The effect of the impurities on the transport properties of the system can be studied by calculating the spin- and charge stiffnesses from the finite size corrections to the ground state energy of the model subject to twisted boundary conditions [@Twist]. Following the analysis in [@bcfh:92] we introduce twist angles $\phi_c$ and $\phi_s$ affecting charge and spin degrees of freedom, respectively. The leading term in the resulting shift of the ground state energy can then be written as $\Delta
E(\phi_c,\phi_s)=L^{-1} \phi_\alpha D_{\alpha\beta}(\alpha) \phi_\beta$ where charge (spin) stiffness are defined as $D^{(\rho)}=(L/2)\partial^2_\phi \Delta E(\phi,0)=D_{cc}$ ($D^{(\sigma)}=(L/2)\partial^2_\phi \Delta E(\phi,-2\phi)$). The analysis of the finite-size corrections to the ground state energy yields the result that for the case of a single impurity the stiffnesses are not modified to leading order in $L^{-1}$. Hence, in spite of the presence of the impurity we find an infinite dc-conductivity. This is completely different from the situation in the “weak-link” type potential impurity discussed in [@kafi:92; @tsve:95a]: such a weak link drives the Luttinger liquid to a strong coupling fixed point characterized by a vanishing conductivity. We believe that the behaviour of the system considered here is related to its integrability and the absence of backscattering at the impurity.

The transport properties of the system do change if one considers a finite density $n_i$ of impurity sites. In this situation the band filling can take values larger than 1 as the impurity sites allow for double occupancies. Comparing the charge stiffness to that of the pure $t$–$J$ case one observes a reduction for densities just above the critical one. For larger band fillings the presence of the impurities leads to an enhancement of the stiffness (see Fig. \[fig:stiff\]). This is easily understood: $D^{(\rho)}$ vanishes at half filling in the $t$–$J$ chain. The impurities do allow double occupancies thereby enlarging the phase space for the electrons which leads to an increase in the stiffness. At large fillings the stiffness increases as a function of $\alpha$ as the average occupation number of the impurity sites are close to doubly occupied which makes the movement of electrons between the $t$–$J$ sites easier. In particular, the “absorption” of particles by the impurity sites for $\alpha=\infty$ leads to plateaus in the stiffness for $n_e<n_i$ (where it vanishes) and for $n_i+(1-n_i)n_c<n_e<2n_i+(1-n_i)n_c$. For other fillings the stiffness is simply given by that of the $t$–$J$ model (up to a rescaling). Similarly the reduction of the spin stiffness due to the addition of impurities can be understood.

Finally, to study the effects of the impurity on the excitations in the model we have computed the phase shifts acquired by holons and spinons due to scattering off the impurity in the case of a microscopic number of holes in the half filled ground state at vanishing magnetic field. The basic ingredient for this calculation is the quantization condition for factorized scattering of two particles with rapidities $\lambda_1$ and $\lambda_2$ on a ring of length $N$, namely $\exp(iNk(\lambda_1))
S(\lambda_1-\lambda_2) \exp(\psi(\lambda_1))=1$ where $k(\lambda)$ is the physical momentum in the infinite periodic system, $S(\lambda)$ is the bulk scattering matrix for scattering of particles 1 and 2 and $\psi(\lambda_1)$ is the phase shift acquired when scattering off the impurity (note that this incorporates the fact that there is no backscattering at the impurity). Using the known result for $S$ [@babo:91] one extracts the impurity phase shifts using the method of [@kore:79; @al]: both the spinon and holon impurity phase shifts are proportional to $\exp(-ik)$, where $k$ denotes the physical momentum of spinons/holons in the $t$-$J$ model [*without impurity*]{}. This reflects the fact that the impurity essentially decouples from the chain at half filling leading to a chain of $N-1$ sites. In addition the holons pick up a phase shift $(2i\lambda-\alpha)/(2i\lambda+i\alpha)$ due to the fact that the impurity site is charged. The pole at $\lambda=i\alpha/2$ corresponds to an impurity bound state for $\alpha<2$.

To summarize, we have studied the effects of the addition of integrable impurities to the supersymmetric $t$–$J$ model on certain zero and finite temperature properties of the system. The properties of the impurity, which couples to both spin and charge degrees of freedom, can be tuned by adjusting a continous parameter $\alpha$. Compared to the “weak-link” type impurities investigated by Kane and Fisher [@kafi:92] it appears to be very special in that its dc-conductivity is unchanged by the addition of a single impurity. We have argued that this is due to the absence of backscattering terms on the level of the dressed excitations (holons and spinons). Although the verification by explicit construction of the continuum limit for this system appears difficult one may speculate that similar to the case of a Kondo impurity in a Luttinger liquid [@FrojdhJ] a backscattering term would drive the system to a new fixed point. Hence the present model can be interpreted as an unstable fixed point from a renormalization group point of view.

This work has been supported in parts by the Deutsche Forschungsgemeinschaft under Grant No. Fr 737/2–2.

[10]{}

C. L. Kane and M. P. A. Fisher, Phys. Rev. B [**46**]{}, 15233 (1992).

J. L. Cardy, Nucl. Phys. B [**324**]{}, 581 (1989).

P. Fendley and H. Saleur, Phys. Rev. Lett. [**75**]{}, 4492 (1995); P. Fendley, A. W. W. Ludwig, and H. Saleur, Phys. Rev. B [**52**]{}, 8934 (1995).

A. M. Tsvelik, J. Phys. A [**28**]{}, L625 (1995).

I. Affleck, Nucl. Phys. B [**336**]{}, 517 (1990); I. Affleck and A. W. W. Ludwig, Nucl. Phys. B [**352**]{}, 849 (1991).

P. Fr[ö]{}jdh and H. Johannesson, Phys. Rev. Lett. [**75**]{}, 300 (1995); Phys. Rev. B [**53**]{}, 3211 (1996).

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=0.45

=0.4\
=0.4

=0.4


Determination of hydrogen peroxide generated by reduced nicotinamide adenine dinucleotide oxidase.
Hydrogen peroxide can be conveniently determined using horseradish peroxidase (HRP) and 2,2'-azino-bis(3-ethylbenzthiazoline-6-sulfonic acid). However, interference occurs among assay components in the presence of reduced nicotinamide adenine dinucleotide (NADH) that is also a substrate of NADH oxidase. So, depletion of NADH is required before using the HRP method. Here, we report simple and rapid procedures to accurately determine hydrogen peroxide generated by NADH oxidase. All procedures developed were based on the extreme acid lability of NADH and the stability of hydrogen peroxide, because NADH was decomposed at pH 2.0 or 3.0 for 10 min, while hydrogen peroxide was stable at pH 2.0 or 3.0 for at least 60 min. Acidification and neutralization were carried out by adjusting sample containing NADH up to 30 microM to pH 2.0 for 10 min before neutralizing it back to pH 7.0. Then, hydrogen peroxide in the sample was measured using the HRP method and its determination limit was found to be about 0.3 microM. Alternatively, hydrogen peroxide in samples containing NADH up to 100 microM could be quantitated using a modified HRP method that required an acidification step only, which was found to have a determination limit of about 3 microM hydrogen peroxide in original samples.