What are the rules for rebinding the suddenly freed arm of a bound prisoner to prevent them from attacking?

We are working on a scenario where a lich is bound and gagged but one hand gets free.

We are working on how to play out the scenario where relatively lower level PCs are trying to restrain the freed arm of the lich while it is attacking them with its Paralyzing Touch.

Are there any prior examples in adventures, guidance or rules around how the party would tie up the freed arm of a prisoner?

We looked at grappling but it appears the lich could still use its Paralyzing Touch because a grapple only grabs an opponent and doesn’t stop attacks.

We believe this question is different from this prior question as we are not asking about all the different ways to invoke the restraining condition, but rather specifically the prior art, guidance or rules around a specific scenario seen with tied prisoners.

What’s an upper bound for this recurrence so I can take advantage of the Master Theorem?

Let

$ $ T(N) = \begin{cases}1 & \text{if } N = 1\ T(\varphi(N)) + 2T(\sqrt{N}) + \lg(\varphi(N))^3 & \text{otherwise} \end{cases}$ $

where $ \varphi(N)$ is Euler’s totient function. My objective is to find an upper bound so that I can apply the Master Theorem and find a closed-form formula.

Question about the lower bound for $k \times k$-clique (under ETH) shown in “Slightly Superexponential Parameterized Problems”

I am reading the paper Slightly Superexponential Parameterized Problems at the moment and have two questions about it:

First question: The paper gives a proof of the following statement

Theorem 2.1: Assuming ETH, there is no $ 2^{o(klogk)}$ time algorithm for $ k \times k$ -Clique.

They prove this statement by sophisticated reduction from $ 3$ -coloring. They then state that this construction runs in time polynomial in $ k$ . They state:

The graph $ G$ has $ k^2$ vertices and the time required to construct G is polynomial in $ k$ . […] Therefore, the total running time is $ 2^{o(k \log(k)} \cdot k^{O(1)}$

Why is this not $ 2^{o(k \log(k)}) + k^{O(1)}$ instead? As far as I can see, we have only have to construct the graph once, and then we can run the presumed $ 2^{o(k \log(k)}$ algorithm.

Second question: From this theorem, it follows that $ k \times k$ -Clique can not have a $ k^{o(k)}$ algorithm under the Exponential Time Hypothesis. This follows from the abstract, which states that $ k^{O(k)} = 2^{O(k \log(k))}$ . What is a good way to proof this statement?

How to prove an implication about an upper bound mentioned in the proof of master theorem?

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How can we prove rigorously the proposition “Suppose the if in case 1 is true, the equation 4.23 is true”? For given constant b and j, the implication in green makes sense. If the upper bound of j was fixed, the equation 4.23 follows directly. However, when n increases, the upper bound of j also increases, though is slower. It is where I find difficult to prove there always exists a value m > 0 such that for all n >= m, equation 4.23 is true.

PDOStatement::execute(): SQLSTATE[HY093]: Invalid parameter number: number of bound variables does not match number of tokens in

Buenas tardes necesito ayuda, estoy intentando hacer un insert uso mysql y la conexion tipo pdo y me aparece el siguiente error:

Warning: PDOStatement::execute(): SQLSTATE[HY093]: Invalid parameter number: number of bound variables does not match number of tokens in C:\xampp\htdocs\crud2\datos\CampusDao.php on line 121 

este es la pagina donde aparece el error

    /**  * Metodo que sirve para crear y editar campus  *  * @param      object         $  campus  * @return     boolean  */ public static function crearCampus($  campus) {     if (is_null($  campus->getId_campus())) {         $  query = "INSERT INTO campus (nombre,direccion,estado) VALUES (:nombre,:direccion,:estado)";     } else {         $  query = "UPDATE campus SET nombre=:nombre,direccion=:direccion,estado=:estado WHERE id_campus=:id_campus";     }      self::getConexion();      $  resultado = self::$  cnx->prepare($  query);      $  nombre     = $  campus->getNombre();     $  direccion  = $  campus->getDireccion();     $  estado     = $  campus->getEstado();      $  resultado->bindParam(":nombre", $  nombre);     $  resultado->bindParam(":direccion", $  direccion);     $  resultado->bindParam(":estado", $  estado);    --------esta es la linea del error------- if ($  resultado->execute()) {         return true;     }      return false; } 

}

la tabla campus solo tiene 4 campos que son: id_campus, nombre, direccion y estado. no se cual podria ser el error

Are Cellular Automata models related to the Bekenstein bound?

Cellular Automata are discrete models which have a regular finite dimensional grid of cells, each in one of a finite number of states, such as on and off.

There are various scientists that have combined Cellular Automata with the Holographic Principle (like Gerard ‘t Hooft, Seth Lloyd, Paola Zizzi…etc) to describe the universe.

This makes me think that all Cellular Automata models are directly with the Bekenstein bound (https://en.wikipedia.org/wiki/Bekenstein_bound)and thus with holography, but I would need confirmation from an expert.

So, are literally all cellular automata models related with the Bekenstein bound or holography in general?

Bound on difference of log of unitary matrices

Suppose I have two unitary matrices $ u, v$ such that $ \|u-v\|<\epsilon$ in the operator norm. Is there a way to bound the quantity $ \|\log u-\log v\|$ ? We can assume that $ \epsilon$ is sufficiently small and we choose a branch cut for the logarithm such that eigenvalues of $ u$ and $ v$ will not be split up. Ideally I would like to get a bound in the form $ \|\log u-\log v\|<C\epsilon$ where $ C$ is a constant independent of the dimension of the matrix. Thanks!

Understanding Gillman’s proof of the Chernoff bound for expander graphs

My question is about Claim 1 in the proof here: Gillman (1993). At the end of the proof, the author says:

The matrix product $ U^\top\sqrt{D^{-1}}(P+(\mathrm{e}^x-1)B(0)-\mu I)\sqrt{D}U$ , which is equal to $ (D’-\mu I)(I+(D’-\mu I)^{-1}(\mathrm{e}^x-1)D’U^\top D_A U)$ , is singular. Therefore,

\begin{align*} 1&\leq \lVert (D’-\mu I)^{-1}(\mathrm{e}^x-1)D’U^\top D_A U \rVert_2 \ &\leq \frac{1}{\mu-\lambda_2}(\mathrm{e}^x -1). \end{align*}

(The first inequality uses the continuity of the function $ \lambda_2(y)$ .)

I understand why the two expressions at the beginning are equal and I understand the second inequality, but I do not understand the first inequality and also why the matrix product is singular.

Let me provide the definitions so you can avoid reading the whole paper. There is a weighted undirected graph $ G=(V, E, w)$ where $ w_{ij}=0$ if $ \{i,j\}\notin E$ . Denote $ w_i:=\sum_j w_{ij}$ . Let $ P$ denote the transition matrix, so $ P_{ij}:=\frac{w_{ij}}{w_i}$ . Denote by $ \lambda_2$ the second largest eigenvalue of $ P$ and by $ \epsilon:=1-\lambda_2$ the spectral gap. Next, let $ M$ be the weighted adjacency matrix $ M_{ij}:=w_{ij}$ . Let $ A$ be a set of vertices and $ \chi_A$ be an indicator function. Some more definitions are:

\begin{align*} &E_r:=\operatorname{diag}(\mathrm{e}^{r\chi_A}) & &P(r):=PE_r \ &D:=\operatorname{diag}(\frac{1}{w_i}) & &S:=\sqrt{D}M\sqrt{D} \ &S_r:=\sqrt{DE_r}M\sqrt{DE_r} & & B(r):=\frac{1}{\mathrm{e}-1}(P(r+1)-P(r)) \end{align*}

Moreover, since $ S$ is unitarily diagonalizable, there exist a unitary matrix $ U$ and a diagonal matrix $ D’$ such that $ D’=U^\top SU$ . Furthermore, there exists a diagonal matrix $ D_A$ such that $ B(0)=PD_A$ .

Define $ \lambda(r)$ to be the largest eigenvalue of $ P(r)$ and $ \lambda_2(r)$ to be its second largest eigenvalue. As before, $ \epsilon_r := \lambda(r)-\lambda_2(r)$ is the spectral gap.

In Claim 1 the author lets $ 0\leq x\leq r$ be some number. He also defines $ \mu<\lambda(x)$ to be any other eigenvalue of $ P(x)$ . At the end of the proof, we are only interested in $ \mu>\lambda_2$ .

Some other facts are:

\begin{align*} &\lVert D’ \rVert_2 = \lVert D_A \rVert_2 = 1 & &D’=U^\top\sqrt{D^{-1}}P\sqrt{D}U \ &P(0)=P & &\lambda(0)=1 & &\lambda_2(0)=\lambda_2 \ &P=\sqrt{D}S\sqrt{D^{-1}} & &P(r)=\sqrt{DE_r^{-1}}S_r\sqrt{E_rD^{-1}} \end{align*}

I hope I didn’t miss anything relevant. Thank you for your help.

A uniform upper bound for Fredholm index of quasi Laplace operators on a compact parallelizable manifold

Assume that $ M$ is a compact parallelizable manifold. Is there an upper bound for the absolute value of Fredholm index of all operators in the form $ D=\sum_{i=1}^n \partial^2/\partial{X_i^2}$ where $ \{X_1,X_2,\ldots,X_n\}$ is a global smooth frame?