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text:
\section{Introduction}
	This work is inspired by the recent work of the first author with Ma and Smirnov \cite{MQS} about the preservation of Hilbert function under sufficiently small perturbations which was also inspired by the previous work of Srinivas and Trivedi \cite{ST1}. Taking a small perturbation arises naturally in studying deformations when we change the defining equations by adding terms of high order. In this way we can transform a singularity defined analytically, e.g., as a quotient of a (convergent) power series ring, into an algebraic singularity by truncating the defining equations.

	This problem was first considered  by Samuel in 1956. Let $f \in S = k[[x_1, \ldots, x_d]]$ be a hypersurface with an isolated singularity, i.e. the Jacobian ideal $J(f) = (\frac{\partial f}{\partial x_1}, \ldots, \frac{\partial f}{\partial x_d})$ is $(x_1,\ldots, x_d)$-primary. Then Samuel proved that for every $\varepsilon \in (x_1,\ldots, x_d) J(f)^2$ we have an automorphism of $S$ that maps $f \mapsto f + \varepsilon$. In particular, Samuel's result asserts if $f$ has an isolated singularity and $\varepsilon$ is in a sufficiently large power of $(x_1, \ldots, x_d)$, then the rings $S/(f)$ and $S/(f + \varepsilon)$ are isomorphic. Samuel's result was extended by Hironaka in 1965, who showed that if $S/I$ is an equidimensional reduced isolated singularity, then $S/I \cong S/I'$ for every ideal $I'$ obtained by changing the generators of $I$ by elements of sufficiently large order such that $S/I'$ is still reduced, equidimensional, and same height as $I$.

	The isolated singularity is essential in the both theorems of Samuel and Hironaka. For a local ring $(R, \frak m)$ and a sequence of elements $f_1, \ldots, f_r$, instead of requiring the deformation to give isomorphic rings $R/(f_1, \ldots, f_r) \cong R/(f_1 + \varepsilon_1, \ldots, f_r + \varepsilon_r)$, we consider a weaker question: what properties and invariants are preserved by a sufficiently fine perturbation? For example, Eisenbud \cite{E} showed how to control the homology of a complex under a perturbation and thus showed that Euler characteristic and depth can be preserved. As an application, if $f_1, \ldots, f_r$ is a regular sequence, then so is the sequence $f_1 + \varepsilon_1, \ldots, f_r + \varepsilon_r$ as long as we take a sufficiently small perturbation. Huneke and Trivedi \cite{HT} extended this result for filter regular sequences, a generalization of the notion of regular sequence.

	For numerical invariants, perhaps the most natural direction is to study the behavior of Hilbert function. Srinivas and Trivedi \cite{ST1} showed that the Hilbert function of a sufficiently fine perturbation is at most the original Hilbert function. Furthermore they proved that the Hilbert functions of $R/(f_1, \ldots, f_r)$ and $R/(f_1 + \varepsilon_1, \ldots, f_r + \varepsilon_r)$ coincide under small perturbations provided two conditions: (a) $f_1, \ldots, f_r$ a filter regular sequence; (b) $R/(f_1, \ldots, f_r)$ is generalized Cohen-Macaulay. Recalling that $(R, \frak m)$ is generalized Cohen-Macaulay if all lower local cohomology $H^i_{\frak m}(R), i < \operatorname{dim} R$, have finite length. Moreover, a generalized Cohen-Macaulay ring is Cohen-Macaulay on the punctured spectrum. Srinivas and Trivedi gave examples to show that the condition (a) is essential even if $f_1, \ldots, f_r$ is a part of system of parameters. However they asked whether the condition (b) is superfluous.

	\begin{Notation} Let $(R,{\frak m})$ be a Noetherian local ring and $I = (f_1,\ldots,f_r)$ an ideal of $R$. For each $N > 0$ we denote
		$$C_N(I) = \{ (f_1+\varepsilon_1, \ldots, f_r+\varepsilon_r)\ | \ \varepsilon_i \in {\frak m}^N \}.$$
	\end{Notation}

	Recently, Ma, Smirnov and the first author \cite{MQS} answered affirmatively the above question of Srinivas and Trivedi and proved the following.
	\begin{Theorem}
		Let $(R, \frak m)$ be a Noetherian local ring of dimension $d$, and $I \subseteq R$ is generated by  a filter regular sequence $f_1,\ldots,f_r$. Then there exists $N>0$ such that for all $I_N \in C_N(I)$, the Hilbert functions of
		$R/I$ and $R/I_N$ are equal, i.e.
		$$\ell(R/(I + {\frak m}^n)) = \ell(R/(I_N + \frak m^n))$$
		for all $n \ge 1$.\footnote{Actually, we proved the result for any ideal $J$ such that $(f_1, \ldots, f_r)+J$ is $\frak m$-primary. Although the main result of this paper can be extended for such ideals, we will keep our interest for the maximal ideal for simplicity.}
	\end{Theorem}
	We also asked the question.
	\begin{Question}\label{question}
		Can one obtain explicit bounds on $N$?
	\end{Question}
	A certainly positive answer for the case $r=1$ was given in \cite[Theorem 3.3]{MQS}. If $R$ is a Cohen-Macaulay local ring of dimension $d$, Srinivas and Trivedi \cite[Proposition 1.1]{ST2} provided a formula for $N$ in terms of the multiplicity for any $r \ge 1$. Namely, we can choose
	$$N = (d-r)! \, e(R/I) + 2.$$
	Inspired by the above formula, one can hope to give a bound for $N$ in any local ring by using the extended degree instead of the multiplicity. See the next section for more details about the notion of extended degree. The aim on the present paper is to give an evident for this belief. We will extend the above result of Srinivas and Trivedi for the class of generalized Cohen-Macaulay rings by using the multiplicity and the length of local cohomology $H^i_{\frak m}(R)$.

	Let $(R, \frak m)$ be a local ring and $M$ a generalized Cohen-Macaulay module of dimension $d$. The Buchsbaum invariant of $M$ is defined as follows
	$$I(M) = \sum_{i = 0}^{d-1} \binom{d-1}{i} \ell(H^i_{\frak m}(M)).$$

	We now present the first main result of this paper.
	\begin{Theorem}\label{result 1}
		Let $(R, {\frak m})$ be a generalized Cohen-Macaulay local ring of dimension $d$ and $I \subseteq R$ is generated by a part of system of parameters $f_1,\ldots,f_r$ of $R$. Let $s = d-r $, and
		$$N = s!\, \big(e(R/I) + I(R/I)\big) + (s+1)I(R) + 1.$$
		Then for all $I_N \in C_N(I)$ we have
		the Hilbert functions of
		$R/I$ and $R/I_N$ are equal.
	\end{Theorem}
	The method of our proof of the above result is inspired by the Srinivas and Trivedi one in the Cohen-Macaulay case. Let us mention the most important step in our proof. If $R$ is Cohen-Macaulay and $J = (x_1, \ldots, x_s)$ a minimal reduction of $\frak m$ with respect to $R/I$, then we can choose $N$ such that $I + J = I_N + J$ for all $I_N \in C_N(I)$. The strategy of Srinivas and Trivedi was to transform the Hilbert functions of $R/I$ and $R/I_N$ (with respect to $\frak m$) to the Hilbert functions of $R/I$ and $R/I_N$ with respect to the parameter ideal $J$, and using the following well-known fact for Cohen-Macaulay rings
	$$\ell (R/(I+J^{n+1})) = \binom{n+s}{s} \ell(R/(I+J)) = \binom{n+s}{s} \ell(R/(I_N+J)) = \ell (R/(I_N+J^{n+1})).$$
	For generalized Cohen-Macaulay rings, we also have an explicit formula for the Hilbert function with respect to special parameter ideals, say standard parameter ideals, in terms of the length of lower local cohomology modules (see Theorem \ref{HF of standard}). Therefore we need to control $\ell (H^i_{\frak m}(R/I))$ under sufficiently small perturbations. This is the second main result of this paper.
	\begin{Theorem}\label{result 2}
		Let $(R,{\frak m})$ be a generalized Cohen-Macaulay ring of dimension $d$ and $I \subseteq R$ is generated by a part of system of parameters $f_1,\ldots,f_r$. Let $N = e(R/I) + I(R) + 1$, then for all $I_N \in C_N(I)$ we have
		$$\ell(H^i_{{\frak m}}(R/I)) = \ell(H^i_{{\frak m}}(R/I_N))$$
		for every $i < d-r$.
	\end{Theorem}
	The paper is organized as follows: In the next section, we recall some notations used in this paper. We will prove Theorem \ref{result 2} in Section 3. Section 4 is devoted to prove Theorem \ref{result 1}.

	\begin{acknowledgement} This project resulted from a trip the first author took to the University of Genoa,
		we would like to thank Matteo Varbaro for making that trip possible. This paper was written while the first author visited the Vietnam Institute for Advanced Study in Mathematics (VIASM), he would like to thank the VIASM for the very kind support and hospitality. The authors are grateful to Professor Maria Evelina Rossi for her useful discussions on the first step of the project.
	\end{acknowledgement}

	\section{Preliminaries}

	Throughout this paper, $(R, \frak m)$ denotes a Noetherian local ring. Let $M$ be a finitely generated $R$-module of dimension $d \geq 1$, and $J$ an ${\frak m}$-primary ideal of $R$. The {\it Hilbert function} of $M$ with respect to $J$ is defined by
	$$HF_J(M)(n) := \ell(M/J^{n+1}M)$$
	for all $n\geq 0$.

	The Hilbert function of $M$, denoted by $HF(M)$, is the Hilbert function of $M$ with respect to the maximal ideal ${\frak m}$. It is well-known that for $n$ sufficiently large the Hilbert function $HF_J(M)$ becomes a polynomial in $n$ of degree $d$, and can be written as the following form
	$$HF_J(M)(n) = e_0(J,M)\binom{n+d}{d} - e_1(J,M)\binom{n+d-1}{d-1} + \cdots + (-1)^de_d(J,M)$$
	for all $n \gg 0$, where $e_i(J,M)$ are the integers and they are called the Hilbert coefficients of $M$ with respect to $J$. In particular, $e(J,M) = e_0(J, M)$ is called the {\it multiplicity} of $M$ with respect to $J$ and $e(M) = e({\frak m},M)$ is called the multiplicity of $M$.

	If $M$ is Cohen-Macaulay and $J$ is a parameter ideal we have
	$$HF_J(M)(n) = \ell (M/JM) \, \binom{n+d}{d}$$
	for all $n \ge 0$. In particular $e(J,M) = \ell (M/JM)$. In general we always have the inequality $e(J,M) \le \ell (M/JM)$ for all parameter ideals $J$ of $M$.
	\begin{Definition}
		An $R$-module $M$ is called {\it generalized Cohen-Macaulay} if the difference $\ell (M/JM) - e(J, M)$ is bounded above for every parameter ideal $J$.
	\end{Definition}
	We next recall some well-known facts in the theory of generalized Cohen-Macaulay modules (see \cite{T}).
	\begin{Remark} Let $M$ be an $R$-module of dimension $d$. Then
		\begin{enumerate}
			\item $M$ is generalized Cohen-Macaulay if and only if $H^i_{\frak m}(M)$ has finite length for every $i<d$. Moreover, we have
			$$\ell (M/JM) - e(J, M) \le \sum_{i=0}^{d-1}\binom{d-1}{i} \ell(H^i_{\frak m}(M))$$
			for all parameter ideals $J$. The left hand side, denoted by $I(M)$, and is called the {\it Buchsbaum invariant} of $M$.
			\item If $M$ is generalized Cohen-Macaulay, then for every part of system of parameters $x_1, \ldots, x_r$ we have $I(M) \ge I(M/(x_1, \ldots, x_r)M)$.
			\item If $M$ is generalized Cohen-Macaulay, then every system of parameter is a filter regular sequence of $M$. Recalling that $x_1, \ldots, x_t \in \frak m$ is called a {\it filter regular sequence} of $M$ if
			$$\mathrm{Supp} \big(\frac{(x_1, \ldots, x_{i-1})M: x_i}{(x_1, \ldots, x_{i-1})M} \big) \subseteq \{\frak m\}$$
			for all $i = 1, \ldots, t$.
		\end{enumerate}
	\end{Remark}
	\begin{Definition} Let $M$ be a generalized Cohen-Macalay module of dimension $d$. A parameter ideal $J$ of $M$ is called {\it standard} if
		$$\ell (M/JM) - e(J, M) = \sum_{i=0}^{d-1} \binom{d-1}{i} \ell(H^i_{\frak m}(M)).$$
	\end{Definition}
	\begin{Remark} Let $M$ be a generalized Cohen-Macalay module of dimension $d$. Then there exists a positive integer $N$ such that $J$ is standard for every parameter ideal $J \subseteq \frak m^N$. In fact we can choose $N = I(M)$.
	\end{Remark}
	The Hilbert function of a standard parameter ideal $J$ can be expressed explicitly as follows (see \cite[Corollary 4.2]{T}).
	\begin{Theorem} \label{HF of standard} $J$ is a standard parameter ideal of $M$ if and only if
		$$HF_J(M)(n) =  \binom{n+d}{d}e(J,M) + \sum_{i=1}^d \sum_{j=0}^{d-i}\binom{n+d-i}{d-i}\binom{d-i-1}{j-1}\ell(H_{{\frak m}}^j(M)),$$
		for all $n \geq 0$.
	\end{Theorem}

	In order to capture the complexity of non (generalized) Cohen-Macaulay modules, Vasconcelos et al. \cite{V1, V2} introduced the
	notion of extended degree which is a generalization of the notion of multiplicity. Let $\mathcal{M}(R)$ be the category of finitely generated $R$-modules. An {\it extended degree} on $\mathcal{M}(R)$ is a numerical function $D(\bullet)$ on $\mathcal{M}(R)$ such that the following properties hold for every $R$-module $M \in \mathcal{M}(R)$:
	\begin{enumerate}
		\item $D(M) = D(M/L) + \ell(L)$, where $L = H^0_{\frak m}(M)$,

		\item $D(M) \geq D(M/xM)$ for a generic element $x$ of ${\frak m}$,

		\item $D(M) = e(M)$ if $M$ is a Cohen-Macaulay module.
	\end{enumerate}
	The prototype of an extended degree is the homological degree defined by Vasconcelos in \cite{V1}. If $R$ is a homomorphic image of a Gorenstein ring $S$ with $\operatorname{dim} S = n$ then the {\it homological degree} of $R$-module $M$ is defined by
	$$\operatorname{hdeg}(M):= e(M) + \sum_{i=0}^{d-1}\binom{d-1}{i}\operatorname{hdeg}(\operatorname{Ext}_S^{n-i}(M,S)).$$
	Recently, Cuong and the first author \cite{CQ} introduced a new extended degree, say the {\it unmixed degree}, and denoted by $\mathrm{udeg}(M)$. The readers are encouraged to \cite{CQ} for more details about the construction. If $M$ is generalized Cohen-Macaulay we have
	$$\operatorname{hdeg} (M) = \mathrm{udeg} (M) = e(M) + I(M).$$
	We close this section with some lemmas that will be useful for the proof of the main results.

	\begin{Lemma}\label{presever sop} Let $(R, \frak m)$ be a local ring of dimension $d$ and $I \subseteq R$ is generated by  a part of system of parameters $f_1, \ldots, f_r$. Let $D(\bullet)$ be an any extended degree and set $N = D(R/I) + 1$. Then for every $\varepsilon_1, \ldots, \varepsilon_r \in \frak m^{N}$ we have $f_1 + \varepsilon_1, \ldots, f_r + \varepsilon_r$ is a part of system of parameters of $R$.
	\end{Lemma}

	\begin{proof} Let $x_1, \ldots, x_{d-r}$ be a general sequence of elements of $R/I$. Then
		$$\ell (R/(I + (x_1, \ldots, x_{d-r})) \le D(R/I) = N-1.$$
		This implies that $\frak m^{N-1} \subseteq I + (x_1, \ldots, x_{d-r})$. Therefore for every $\varepsilon_1, \ldots, \varepsilon_r \in \frak m^{N}$ we have
		$$(f_1, \ldots, f_r,x_1, \ldots, x_{d-r}) = (f_1 + \varepsilon_1, \ldots, f_r + \varepsilon_r, x_1, \ldots, x_{d-r}).$$
		Hence $f_1 + \varepsilon_1, \ldots, f_r + \varepsilon_r$ is a part of system of parameters of $R$.
	\end{proof}
	It would be nice if we obtain similar results for (filter) regular sequences instead of system of parameters. These results, if have, will play an important role for an answer for Question \ref{question} in the general case. In the main context of this paper $R$ is generalized Cohen-Macaulay, so these three notions coincide. We will need the following regular \cite[Corollary 3.6]{V1}.
	\begin{Lemma}\label{reduction}
		Let $(R, \frak m)$ be a local ring of dimension $d$ with the infinite residue field. Let $\operatorname{hdeg}(\bullet)$ be the homological degree. Then there exists a minimal reduction $J$ of $\frak m$ with reduction number $\mathrm{r}_J(\frak m) \le (d-r)!\, \operatorname{hdeg}(R) - 1$.
	\end{Lemma}
	\begin{Lemma} \label{prevever reduction} Let $(R, \frak m)$ be a local ring of dimension $d$ and $I \subseteq R$ is generated by  a part of system of parameters $f_1, \ldots, f_r$. Let $J$ be a minimal reduction of $\frak m$ in $R/I$, and $k$ a non-negative integer such that $\mathrm{r}_J(\frak m , R/I) \le k$. Then for all $I_{k+2} \in C_{k+2}(I)$, $J$ is a minimal reduction of $\frak m$ in $R/I_{k+2}$ and $\mathrm{r}_J(\frak m, R/I_{k+2}) \le k+1$.
	\end{Lemma}
	\begin{proof}
		By the assumption we have $\frak m^{k+1} + I =  J\frak m^k + I$. Therefore $\frak m^{k+1} \subseteq J + I$. Hence for every $I_{k+2} \in C_{k+2}(I)$  we have $J + I = J + I_{k+2}$. We are going to prove that $\frak m^{k+2} + I_{k+2} =  J\frak m^{k+1} + I_{k+2}$. We have
		\begin{align*}
			J\frak m^{k+1} + I_{k+2} + \frak m (\frak m^{k+2} + I_{k+2}) &= J\frak m^{k+1} + I_{k+2} + \frak m (\frak m^{k+2} + I)\\
			&= J\frak m^{k+1} + I_{k+2} + \frak m (J\frak m^{k+1} + I)\\
			&= J\frak m^{k+1} + I_{k+2} + \frak m I\\
			&= \frak m( J\frak m^{k}  + I)+ I_{k+2}\\
			&= \frak m( \frak m^{k+1}  + I)+ I_{k+2}\\
			&= \frak m^{k+2}  + I_{k+2} + \frak m I\\
			&= \frak m^{k+2}  + I_{k+2}.
		\end{align*}
		The last equality follows from the fact that $\frak m I \subseteq \frak m^{k+2}  + I = \frak m^{k+2}  + I_{k+2}$. By NAK we have $\frak m^{k+2} + I_{k+2} =  J\frak m^{k+1} + I_{k+2}$. The proof is complete.
	\end{proof}

	\section{Local cohomology under small perturbations}
	Let $(R,{\frak m})$ be a generalized Cohen-Macaulay ring of dimension $d$ and $I \subseteq R$ is generated by  a part of system of parameters $f_1, \ldots, f_r$. In this section we provide a positive integer $N$ depends on $e(R/I)$ and $I(R)$ such that for all $I_N \in C_N(I)$ we have the lengths of $H^i_{{\frak m}}(R/I)$ and $H^i_{{\frak m}}(R/I_N)$ coincide for every $0 \leq i < d-r$.

	The proof of the main result is based on the induction on $r$, where $r$ is the length of the sequence $f_1, \ldots, f_r$. First, for the case $r=1$ we have the following proposition.
	\begin{Proposition} \label{one form} Let $(R,{\frak m})$ be a generalized Cohen-Macaulay ring of dimension $d$, and $f$ a parameter element of $R$. Then for every $\varepsilon \in {\frak m}^{I(R)}$ such that $f + \varepsilon$ is a parameter element of $R$, we have
		$$\ell(H^i_{{\frak m}}(R/(f))) = \ell(H^i_{{\frak m}}(R/(f+\varepsilon)))$$
		for every $i < d-1$.
	\end{Proposition}
	\begin{proof} Following from the short exact sequence
		$$0 \longrightarrow R/(0:f) \overset{f}{\longrightarrow} R \longrightarrow R/(f) \longrightarrow 0$$
		we obtain the following short exact sequence
		$$0 \longrightarrow \frac{H^i_{{\frak m}}(R)}{fH^i_{{\frak m}}(R)} \longrightarrow H^i_{{\frak m}}(R/(f)) \longrightarrow (0:_{H^{i+1}_{{\frak m}}(R)} f)\longrightarrow  0$$
		for every $i < d-1$. Similarly we have the following short exact sequence
		$$0 \longrightarrow \frac{H^i_{{\frak m}}(R)}{(f+\varepsilon)H^i_{{\frak m}}(R)} \longrightarrow H^i_{{\frak m}}(R/(f+\varepsilon)) \longrightarrow (0:_{H^{i+1}_{{\frak m}}(R)} (f+\varepsilon))\longrightarrow  0$$
		for every $i < d-1$. Since $\varepsilon \in {\frak m}^{I(R)}$ we have $\varepsilon\,  H^i_{{\frak m}}(R) = 0$ for all $i <d$. It follows that $(0:_{H^{i+1}_{{\frak m}}(R)} f )\cong (0:_{H^{i+1}_{{\frak m}}(R)} (f+\varepsilon))$ and $\frac{H^i_{{\frak m}}(R)}{fH^i_{{\frak m}}(R)} \cong \frac{H^i_{{\frak m}}(R)}{(f+\varepsilon)H^i_{{\frak m}}(R)}$ for all $i < d-1$. Hence the above two short exact sequences imply
		$$\ell(H^i_{{\frak m}}(R/(f))) = \ell(H^i_{{\frak m}}(R/(f+\varepsilon)))$$
		for all $i < d-1$.
	\end{proof}
	We now present the main result of this section.
	\begin{Theorem} \label{lc}
		Let $(R,{\frak m})$ be a generalized Cohen-Macaulay ring of dimension $d$ and $I \subseteq R$ is generated by  a part of system of parameters $f_1, \ldots, f_r$. Let $N = e(R/I) + I(R) + 1$, then for all $I_N \in C_N(I)$ we have
		$$\ell(H^i_{{\frak m}}(R/I)) = \ell(H^i_{{\frak m}}(R/I_N))$$
		for every $i < d-r$.
	\end{Theorem}
	\begin{proof} Without loss of generality we will always assume that the residue field is infinite. We proceed by induction on $r$. For $r=1$, we have
		$$N \geq e(R/(f_1)) + I(R/(f_1)) + 1 = \operatorname{hdeg}(R/(f_1)) + 1.$$
		So, by Lemma \ref{presever sop}, $f_1+\varepsilon_1$ is a parameter element for every $\varepsilon_1 \in {\frak m}^N$. Hence we are done by Proposition \ref{one form}.

		For $r > 1$ and $I_N = (f_1+\varepsilon_1,\ldots,f_r+\varepsilon_r)$, where $\varepsilon_1,\ldots,\varepsilon_r \in {\frak m}^N$. Let $R_1 = R/(f_1)$. For simplicity, we will identify $f_i$ with its image in $R_1$. Since $N \geq e(R_1/({f_2},\ldots,{f_r})R_1) + I(R_1) + 1$ and $\varepsilon_2,\ldots,\varepsilon_r \in {\frak m}^N$, by induction we get
		$$\ell(H^i_{{\frak m}}(R/(f_1,f_2,\ldots,f_r))) = \ell(H^i_{{\frak m}}(R/(f_1,f_2+\varepsilon_2,\ldots,f_r+\varepsilon_r)))$$
		for every $i < d-r$.  Since $N \geq \operatorname{hdeg}(R/I)+1$ and $\varepsilon_1,\ldots,\varepsilon_r \in {\frak m}^N$, by Lemma \ref{presever sop} we have $f_1,f_2+\varepsilon_2,\ldots,f_r+\varepsilon_r$ and $f_1+\varepsilon_1,f_2+\varepsilon_2,\ldots,f_r+\varepsilon_r$ are the parts of system of parameters of $R$. Let $R_2 = R/(f_2+\varepsilon_2,\ldots,f_r+\varepsilon_r)$, we have $f_1$ and $f_1+\varepsilon_1$ are parameter elements of $R_2$. Moreover, $N \geq I(R_2)$ and $\varepsilon_1 \in {\frak m}^N$, by Proposition \ref{one form} we get
		$$\ell(H^i_{{\frak m}}(R_2/f_1R_2)) = \ell(H^i_{{\frak m}}(R_2/(f_1+\varepsilon_1)R_2))$$
		for every $i < d-r$. That is
		$$\ell(H^i_{{\frak m}}(R/(f_1,f_2+\varepsilon_2,\ldots,f_r+\varepsilon_r))) = \ell(H^i_{{\frak m}}(R/(f_1+\varepsilon_1,f_2+\varepsilon_2,\ldots,f_r+\varepsilon_r)))$$
		for every $i < d-r$. Hence we obtain the desired assertion. The proof is complete.
	\end{proof}
	It is natural to ask the following question in general case.
	\begin{Question} Let $(R, \frak m)$ be a local ring and $I \subseteq R$ is generated by a filter regular sequence  $f_1, \ldots, f_t$ . Does there exist a positive integer $N$ such that for all $I_N \in C_N(I)$ we have
		$$\ell (H^0_{\frak m}(R/I)) = \ell (H^0_{\frak m}(R/I_N))?$$
	\end{Question}

	\section{Hilbert funcion under small perturbations}
	In this section, let $(R,{\frak m})$ be a generalized Cohen-Macaulay ring of dimension $d$, and $I \subseteq R$ is generated  by a part of system of parameters $f_1,\ldots,f_r$. We will find an explicitly positive integer $N$ depends on $\operatorname{hdeg}(R/I)$ and $I(R)$ such that for all $I_N \in C_N(I)$ the Hilbert functions of $R/I$ and $R/I_N$ coincide. The following lemma is a special case of Lemma \ref{reduction} and Lemma \ref{prevever reduction}.
	\begin{Lemma}\label{reduction gCM}
		Let $(R,{\frak m})$ be a generalized Cohen-Macaulay ring of dimension $d$ with the infinite residue field, and $I \subseteq R$ is generated  by a part of system of parameters $f_1,\ldots,f_r$. Let $s = d-r$, and $k = s!\, \operatorname{hdeg}(R/I)+1$. Then there exists a minimal reduction $J$ of ${\frak m}$ in $R/I$ such that
		\begin{enumerate}
			\item ${\frak m}^{k+m} + I = J^{m+1}{\frak m}^{k-1}+ I$ for all $m \geq 0$.
			\item For every $I_k \in C_k(I)$ one has ${\frak m}^{k+m} + I_k = J^{m+1}{\frak m}^{k-1}+ I_k$ for all $m \geq 0$.
		\end{enumerate}
	\end{Lemma}
	\begin{proof}
		By Lemma \ref{reduction} there exists a minimal reduction $J=(x_1,\ldots,x_{d-r})$ of $\frak m$ in $R/I$ such that $\mathrm{r}_J(\frak m , R/I) \le (d-r)!\, \operatorname{hdeg}( R/I) - 1 = k - 2$. Hence ${\frak m}^k + I = J{\frak m}^{k-1}+I$. By Lemma \ref{prevever reduction} one has $J$ is a minimal reduction of ${\frak m}$ in $R/I_k$ and $\mathrm{r}_J(\frak m,R/I_k) \le k-1$.  Hence ${\frak m}^k + I_k = J{\frak m}^{k-1} + I_k$. Therefore, for all $m \ge 0$ we have $\frak m^{k+m} R/I = J^{m+1} \frak m^{k-1} R/I$ and $\frak m^{k+m} R/I_k = J^{m+1} \frak m^{k-1} R/I_k$. The claims are now clear.
	\end{proof}

	The following theorem is the main result of this section. It extends the result of Srinivas and Trivedi \cite[Proposition 1]{ST2} for generalized Cohen-Macaulay rings.
	\begin{Theorem}
		Let $(R,{\frak m})$ be a generalized Cohen-Macaulay ring of dimension $d$, and $I \subseteq R$ is generated  by a part of system of parameters $f_1,\ldots,f_r$. Let $s = d-r $, and $$N = s!\, \operatorname{hdeg}(R/I) + (s+1)I(R) + 1.$$ Then for all $I_N \in C_N(I)$ we have
		$$HF(R/I) = HF(R/I_N).$$
	\end{Theorem}
	\begin{proof} Without loss of generality we may assume that the residue field is infinite.
		Let $k = s!\, \operatorname{hdeg}(R/I) + 1$, by Lemma \ref{reduction gCM}  there exists ideal $J = (x_1,\ldots,x_s) \subseteq {\frak m}$ such that
		$${\frak m}^{k+m} + I = J^{m+1}{\frak m}^{k-1}+I$$
		for all $m \geq 0$. Moreover, since $I_N \in C_N(I) \subseteq C_k(I)$ we also have
		$${\frak m}^{k+m} + I_N = J^{m+1}{\frak m}^{k-1}+I_N$$
		for all $m \geq 0$. Let $t= \max\{I(R),1\}$. For all $0 \le i \le t-1$ set $N_i= k + s(t-1) + i$. We have $N_i \le N$ for all $i \le t-1$. Since $I_N \in C_N(I) \subseteq C_{N_0}(I)$ we have
		$${\frak m}^i + I = {\frak m}^i + I_N$$
		for all  $ i \leq N_0$. Therefore, it is enough to prove that
		$$\ell \left(\frac{R}{{\frak m}^n+I}\right) = \ell \left(\frac{R}{{\frak m}^n+I_N}\right) \quad \quad (1)$$
		for all $n \geq N_0.$ We will prove it in the following equivalent form
		$$\ell \left(\frac{R}{{\frak m}^{N_i + mt}+I}\right) = \ell \left(\frac{R}{{\frak m}^{N_i + mt}+I_N}\right) \quad \quad (2)$$
		for all $0 \le i \le t-1$ and all $m \ge 0$.
		Set $J' = (x_1^t,\ldots,x_s^t)$, one has $J' \subseteq \frak m^t$ is a standard parameter ideal of $R/I$ and $J' J^{(s-1)(t-1)} = J^{s(t-1)+1}$.
		\vskip 0.2cm
		\noindent {\bf Claim 1.} For all $0 \le i \le t-1$ and all $m \ge 0$ we have
		$${\frak m}^{N_i+mt} + I = J'^{m+1}{\frak m}^{N_i-t}+I, $$
		and
		$${\frak m}^{N_i+mt} + I_N = J'^{m+1}{\frak m}^{N_i-t} + I_N$$
		for all $I_N \in C_N(I)$
		\begin{proof}[Proof of Claim 1] We have
			\begin{eqnarray*}
				\frak m^{N_i + mt}R/I &=& J^{s(t-1)+mt+i+1}\frak m^{k-1}R/I\\
				&=& J'^{m+1} J^{(s-1)(t-1) + i }\frak m^{k-1}R/I\\
				& =& J'^{m+1} \frak m^{N_i-t}R/I
			\end{eqnarray*}
			for all $m \ge 0$. Therefore
			$${\frak m}^{N_i+mt} + I = J'^{m+1}{\frak m}^{N_i-t}+I$$
			for all $0 \le i \le t-1$ and all $m \ge 0$. The second assertion can be proved similarly. The Claim is proved.
		\end{proof}
		By Claim 1, in order to prove the equality (2) it is enough to show
		$$\ell \left(\frac{R}{J'^{m+1}{\frak m}^{N_i-t}+I}\right) = \ell \left(\frac{R}{J'^{m+1}{\frak m}^{N_i-t}+I_N}\right) \quad \quad (3)$$
		for all $0 \le i \le t-1$ and all $m \ge 0$. On the other hand, since $N \ge e(R/I) + I(R) + 1$ we have
		$$\ell(H^i_{{\frak m}}(R/I)) = \ell(H^i_{{\frak m}}(R/I_N))$$
		for all $i<s$ by Theorem \ref{lc}. We also have $I + J' = I_N + J'$ since $I + J' \supseteq {\frak m}^{N-1}$. Hence $J'$ is a standard parameter ideal of $R/I_N$ and $e(J',R/I_N)= e(J',R/I)$. By Theorem \ref{HF of standard} we have
		$$\ell \left(\frac{R}{J'^{m+1}+I}\right) = \ell \left(\frac{R}{J'^{m+1} +I_N}\right) \quad \quad (4)$$
		for all $m \ge 0$. Therefore in order to prove the equality (3) it is sufficient to prove that
		$$\ell \left(\frac{J'^{m+1}+I}{J'^{m+1}{\frak m}^{N_i-t}+I}\right) = \ell \left(\frac{J'^{m+1} +I_N}{J'^{m+1}{\frak m}^{N_i-t}+I_N}\right) \quad \quad (5)$$
		for all $0 \le i \le t-1$ and all $m \ge 0$. Let $K' = I + J' = I_N + J'$. We will prove (5) in the following equivalent form
		$$\ell \left(\frac{K'^{m+1}+I}{K'^{m+1}{\frak m}^{N_i-t}+I}\right) = \ell \left(\frac{K'^{m+1} +I_N}{K'^{m+1}{\frak m}^{N_i-t}+I_N}\right) \quad \quad (6)$$
		for all $0 \le i \le t-1$ and all $m \ge 0$.
		\vskip 0.2cm
		\noindent {\bf Claim 2.} For all $m \ge 0$ we have $K'^{m+1} \cap I = IK'^m$ and $K'^{m+1} \cap I_N = I_NK'^m$.
		\begin{proof}[Proof of Claim 2]
			Notice that $x_1^t,\ldots,x_s^t$ forms a d-sequence of $R/I$, by \cite[Theorem 2.1]{H} we have
			$$J'^{m+1} \cap I \subseteq J'^m I.$$
			Hence, for all $m\geq0$ we have
			\begin{align*}
				K'^{m+1} \cap I &= (I+J')^{m+1} \cap I\\
				&= (IK'^m + J'^{m+1}) \cap I\\
				&= IK'^m + J'^{m+1} \cap I\\
				&= IK'^m.
			\end{align*}
			The second assertion can be proved similarly.
		\end{proof}
		We continue the proof of our theorem. Follows from Claim 2 we have
		\begin{align*}
			K'^{m+1}{\frak m}^{N_i-t}+K'^{m+1} \cap I &= K'^{m+1}{\frak m}^{N_i-t} + K'^mI\\
			&= K'^m(K'{\frak m}^{N_i-t} + I)\\
			&= K'^m(J'{\frak m}^{N_i-t}+ I)\\
			&= K'^m({\frak m}^{N_i}+I)\\
			&= K'^m({\frak m}^{N_i}+I_N)\\
			&= K'^m(J'{\frak m}^{N_i-t} + I_N)\\
			&= K'^m(K'{\frak m}^{N_i-t} + I_N)\\
			&= K'^{m+1}{\frak m}^{N_i-t} + K'^{m+1} \cap I_N
		\end{align*}
		for all $0 \le i \le t-1$ and all $m \ge 0$. Therefore
		\begin{eqnarray*}
			\frac{K'^{m+1}+I}{K'^{m+1}{\frak m}^{N_i-t}+I} & \cong& \frac{K'^{m+1}}{K'^{m+1}{\frak m}^{N_i-t}+K'^{m+1}\cap I}\\
			&\cong & \frac{K'^{m+1}}{K'^{m+1}{\frak m}^{N_i-t}+K'^{m+1}\cap I_N}\\
			&\cong & \frac{K'^{m+1} +I_N}{K'^{m+1}{\frak m}^{N_i-t}+I_N}
		\end{eqnarray*}
		for all $0 \le i \le t-1$ and all $m \ge 0$. The equality (6) is now clear. The proof is complete.
	\end{proof}
	We close the paper with the following.
	\begin{Remark} If $R$ is Cohen-Macaulay, our formula $N = s!\, e(R/I) + 1$ slightly improves the formula of Srinivas and Trivedi. If $R$ is generalized Cohen-Macaulay but not Cohen-Macaulay, according to the proof we can choose
		$$N = N_{t-1} = s!\, \operatorname{hdeg}(R/I) + (s+1)I(R)-s$$
	\end{Remark}

redpajama_set_name: RedPajamaArXiv


Cartel improves capacity and coverage for communications networks in buildings, parkades, stadium arenas, tunnels, airports, utility plants, hospitals, and mining areas. We engineer solutions that take wireless signals where they don't go naturally.
We seek innovative and cost effective methods to address intricate communication problems, designing and building systems that support daily operations, critical requirements, and public safety specifications. Whether in multiple buildings, underground, outside yards, stairwells, elevators, or common areas, we will design a solution for any environment where predictable coverage is necessary.
Designed and built a robust radio system to provide operational signal first responders public safety coverage on the Evergreen LRT extension with particular attention to an underground tunnel.
From architecture to move-in, providing project management, logistic management, equipment supplies, supervising install, commissioning, and on-going support services, we work closely with you carefully planning and meeting your schedule.
Wireless signals, and the requirements they cover, continue to evolve. New radio spectrum, technology, and demands on the systems make, enhancement possibilities almost endless. Cartel makes sure you will be able to transmit well into the future.


text:
Samsung Fascinate - A Great Phone Made Less So by Verizon
in Phones and Mobile, News, Phone Accessories, Cell Phones :: 3 comments
The Samsung Fascinate, the latest Galaxy S phone, is now available on Verizon. Similar to the Samsung Captivate (AT&T) and the Samsung Vibrant (T-Mobile), the Fascinate is a slim, light Android phone with a beautiful 4-inch display.
But the Samsung Fascinate does offer some great features which the Vibrant and Captivate do not. Notably, the ability to act as a Wi-Fi hotspot for up to five devices and an LED flash to help with photos in low-light conditions. That hotspot will require Verizon's optional Mobile Broadband plan, though, which costs an additional $20 per month and has a 2GB data cap. Go over the cap and you will be charged 5 cents per MB.
The downside of the Fascinate versus the other Samsung Galaxy S phones is a deal Verizon struck with Microsoft to make Bing the default search engine for the device and browser, rather than Google. Now, this wouldn't be so bad if you could choose to replace Bing with Google, but you can't—Verizon has it locked down.
I also think it is unfortunate that Verizon replaced Google Navigation with Verizon Navigator. In case you're wondering why that's a bad thing, here's all you need to know—Google's excellent map and navigation services are free, Verizon Navigator requires a $9.99 a month service plan. You can download Google Navigation through the Android Marketplace, though we have not been able to confirm whether that will replace Verizon Navigator across all GPS functions on the phone. In any case, it certainly would not be obvious to the average purchaser of the phone to do this, and I assume this is what Verizon is counting on.
The Fascinate also offers less on-board memory than the Vibrant 2 GB versus 16 GB. Whether this is a fair trade-off for the hotspot and LED Flash will depend on the user.
Beyond these differences, the Fascinate is largely identical to the Vibrant, which we wrote a full review on back in July. If it weren't for the Bing and Verizon Navigator issues, I would rank the Fascinate as the best of the Galaxy S devices, and a top choice for Verizon customers.
The Fascinate is available at Verizon Wireless for $199 with a two-year agreement.
Bing and Verizon Nav?
From RF on September 08, 2010 :: 1:59 pm
Bing and Verizon Nav? Absolute deal breakers. I wanted this phone so bad. Now I would never consider it, if on principle alone (never mind that Bing sucks and Verizon Nav costs extra!)
Bing is actually better than Google search
From Buckhoe on June 25, 2011 :: 12:58 pm
You, like many stay with Google because in the past it was better than all other search engines. You need to spend some time searching on Bing. You won't go back. It is the equal to Google in search and does so much more. Try shopping or image searches on Bing. You will love it.
Google navigation can be installed
From Dave on September 09, 2010 :: 12:05 am
Google navigation can be installed from the market.

redpajama_set_name: RedPajamaCommonCrawl


Joe Rogers, a Denver lawyer and politician, believes the city's schools are so bad that they violate students' civil rights.
On November 3, 1996, Joe Rogers, a Republican, lost in his bid to become U.S. representative from Colorado's 1st Congressional District, which includes the city of Denver, a Democratic stronghold. His opponent, Diana DeGette, won the race easily, taking 57 percent of the votes. Rogers, a 33-year-old African American lawyer who had spent the better part of two years campaigning for the retiring Patricia Schroeder's seat in a race that attracted national attention, suddenly was out of the spotlight. During the campaign, Newt Gingrich, Colin Powell, and Oklahoma Representative J.C. Watts had all journeyed to Colorado to stump for the candidate. The New York Times had called him "young, articulate, energetic, and well-prepared." But now he was just another losing candidate with a lot of time on his hands.
"After the election," he told a Denver Post reporter in January 1997, "people think you're dead. You feel like you've been to your own funeral."
These days, however, Joe Rogers is very much alive.
Not only is he running for lieutenant governor, but he is also leading a controversial class action lawsuit against the Denver Public Schools and the Colorado State Board of Education. Filed last May in Denver District Court on behalf of about 40 African American parents and their children, the suit charges the school system with failing to teach basic skills to poor and minority students, and it calls for the district to grant publicly funded vouchers to low- and middle-income parents who want to send their children to private or parochial schools. This past fall, Rogers amended the suit to include the names of more than 3,000 additional plaintiffs—equally divided, he says, among African Americans, Hispanics, and whites. Rogers and his supporters collected the signatures at grocery stores, churches, and schools throughout Denver. "I think if we had kept going," he says, "we would have had double the number of people we have now. I really do. You can't underestimate how disappointed and frustrated many parents are with the Denver Public Schools."
Both the school district and the state board have filed motions seeking to have the suit dismissed. "We don't think there are grounds for this kind of lawsuit," says Denver school board member Laura Lefkowits, "or that it's a constructive way to deal with the problem." Others—including some parents who have signed on to the suit—accuse Rogers of political opportunism. They say the lawyer initiated the class action in an attempt to gain visibility in his bid for the lieutenant governorship, and they question his commitment to solving the problems spelled out in the lawsuit.
"Quality education is a civil right," Rogers likes to say. And the court system, he believes, is just the place to make certain that right is upheld. "It's really in the classic American tradition," he says, "to ask the courts to resolve a problem that a school district refuses to solve on its own, and an elected school board has no power to solve."
Legal experts say Rogers will have a hard time winning his case. Nationwide, only a handful of similar suits have been filed, none of them successful. But whatever the outcome, the suit underscores a trend: the increasing popularity of vouchers—or rather, the promise of vouchers—among blacks and Hispanics, many of whom are fed up with the public schools. Last year, a poll taken by the Joint Center for Political and Economic Studies, a Washington, D.C., research group that focuses on black issues, found that 57.3 percent of African Americans and 65.4 percent of Hispanics support school vouchers, while whites are evenly divided on the issue. The strongest support—86.5 percent—comes from blacks between the ages of 26 and 35.
One of the most outspoken proponents of vouchers these days is the Rev. Floyd Flake—a liberal, black, six-term Democratic congressman from New York who resigned from the House last year to devote more time to his church in the Jamaica section of Queens. Like Rogers, he too calls vouchers "a moral issue, a civil rights issue."
"Many of my friends are saying we don't need any alternatives, we can do it with public education," he said in a speech last year at the conservative Manhattan Institute. "If you can do it with public education, why have you not done it?"
"Quality education is a civil right."
The National Education Association and the American Federation of Teachers continue to oppose vouchers, as does the NAACP. And the publicly funded voucher programs that have been established in the United States—in Milwaukee, Cleveland, and rural Vermont—face challenges in the courts. But conservatives hope that the growing interest in vouchers among minorities will turn the tide in their favor. The traditional liberal-conservative lines on the issue, they say, are blurring, and strange political alliances are forming.
"Democrat, Republican—none of that stuff really matters," Rogers says. "People, many of them Democrats, have said to me, 'Listen, I don't really care if you're a Republican. The solution makes sense for our kids.' "
Sharlot Smith, one of the plaintiffs in the lawsuit, is a case in point. "I'm a registered Democrat," she says, "but that doesn't play a part in how I feel about vouchers. My decision is one from the heart, from being a parent. It's not a political thing."
A few days before the 1996 election, one Denver reporter referred to Rogers as "at once charismatic and enigmatic." It's an apt description. A passionate orator, he can easily bring a crowd to its feet. In conversation, he's affable and engaging, but his voice fills with emotion when he talks about the public schools. During a nearly two-hour lunch at the Airport Broker Restaurant, not far from his Northeast Denver home, Rogers, wearing a dark-blue business suit, often pounds his fist on the table to emphasize a point. Then he apologizes for getting so worked up.
Married and the father of three young children, Rogers grew up poor in Commerce City, adjacent to Denver. "Joe Rogers knows what it's like to use food stamps," he would say in campaign speeches, using the third-person voice favored by many politicians. But he's no fan of federal assistance programs. "Success," he has said, "should be measured by how few people we have on welfare." He attended public schools, graduating from Adams City High School, where he was a top student. "I believe in public schools," he says. "I always attended public schools. But I'll be doggone if I'm going to tell you or anyone else that you have to send your child there." He says his own experience merely proves that "all cream rises to the top, regardless of the school."
Rogers had hoped that his personality and message would appeal to members of both political parties, but when the votes were counted, it was clear that few Democrats had switched ranks. "I met so many people who said they agreed with 95 percent of what I said, but they couldn't vote for me because I'm a Republican," he said a few months after the election. "I thought people would look at me as Joe Rogers, but the Democrats put out this message that a vote for Joe Rogers was a vote for Newt Gingrich. They pushed Denver to vote Democratic."
Not long after the election, Rogers was at home with his family one night watching a Disney Channel miniseries called Separate But Equal about Thurgood Marshall and the Brown v. Board decision. "I was so fascinated by it," he says. "And what struck me about it was that in 1954, they were dealing with many of the same issues we're dealing with now." Rogers was surprised to learn that the landmark case was more than just an effort to end segregation in the public schools. "The issue in '54 was two-fold," he says. "One part dealt with race, but the other tier that nobody pays attention to was quality of education." That is, Marshall and the NAACP argued that the so-called "separate but equal" doctrine was inherently unfair because black schools were, in fact, inferior to white schools.
Our special report "Quality Counts '98" contains a comprehensive overview of the challenges facing urban school districts in Colorado, with a particular focus on the problems of Denver's public schools. For background, previous stories, and Web links, see our Issues Page on Choice and Vouchers.


text:
10 Great Board Games Like Descent
Published on August 6, 2021 February 11, 2021 by Khadija Osman
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Descent is a dungeon-crawler epic that has fit well into the collection of all tabletop adventurers for many years now. So, it makes absolute sense that you're looking for board games like Descent. You must have played it a thousand times now.
With a light sprinkling of role-play gaming and strategizing and a liberal scoop of sadism on the part of the Overlord it's hard to get tired of a classic but here are some titles that will spice up your table without changing the script completely.
You won't be disappointed, all of the games we will be suggesting in this article will contain the classic mix of puzzles, traps, monsters and heroes that make this kind of board game so enticing.
10 Games like Descent
One Deck Dungeon: Forest of Shadows
Mice and Mystics
Mansions of Madness: Second Edition
Escape The Dark Castle
Welcome to the Dungeon
Doom: Board Game
One game similar to Descent but with a fresh take is One Deck Dungeon: Forest of Shadows. It can be run as a solo game or done with two players and is an excellent dungeon-crawl experience that occurs in a randomised forest.
Players character cards will determine the dice pool they're allowed to work with throughout the game and with it, they will work their way through an assortment of different foes, poisons and perils to gain rewards. As they continue through it they will see harder challenges — and a dungeon boss! — until they make their way through the entire deck.
This game is different every time it's played because of the random order of the deck and different characters you can play so it makes a nice little investment that will keep giving new experiences.
In Mice and Mystics, one to four players will work together to get out of a castle.
However, the players will be surprised to take on the role of heroes turned to mice who need to overcome the countless perils that might befall the very small, not to mention their ferocious furry foes the rat guards, the cockroaches and a housecat who lives in the castle.
The mice are able to become more powerful by collecting crumbs of cheese to gain new abilities and fight more effectively.
This is also a game that can be played over multiple sessions, players continue with the same character items and development so this can offer some long-term enjoyment with a prolonged series of game-night.
An app-driven board game is still a somewhat futuristic idea despite us being in the 21st Century. However, the modernity of Mansions of Madness is what makes it such a great suggestion for those looking for board games similar to Descent.
This can have up to five players on the board at once and the app will manage monster damage, effects and narrate the game for the players in the role of what was a keeper in the first edition.
This means that all the players need to do is follow the mystery and try not to get too scared. Which might prove difficult when they find themselves wandering the monster-infested halls of this cursed mansion.
Already know and love this game? Check out our list of more games like Mansions of Madness!
This stylish, bleak black-and-white horror-inspired dungeon crawler is quite the looker, with beautiful illustrations, exotic-looking D6s and two different sized decks. For anyone who's excited by uniquely styled titles, Escape the Dark Castle should be your choice for a board game like Descent.
In this game, one to four different players will take on the role of prisoners trying to break out of the titular castle and must agree between themselves who should take the next turn overturning Chapter Cards and moving through the castle, how the group should react to each situation and who should receive which items.
Once again, this game has a number of different ways it can go so it keeps it's novelty far past the 10th or 30th play.
Arcadia Quest veers to a slightly younger audience with its game board and miniatures done in an iconic cartoonish style that draws the eye in. But it still has just as much violence as you'd expect from a board game similar to Descent.
Two to four players will make competing guilds using three heroes each to try and clear the land of all the monsters in the area so that they can take control. They also want to clear out the opposing guilds, they're just more people to wrestle for domination.
Players will be trying to complete a series of quests in order to win a scenario by games end but, unlike most younger-aimed games, they can be picked up again later with winners choosing where to move the campaign next.
Love complex miniature games? Check out our list of games like Warhammer 40k!
This game that's not stopped being talked about since it first launched and there's a reason for that. Just like Descent, this classic is a peak example of optimal cooperative gaming, taking one to four players on an epic adventure that will last many game nights.
I would suggest this game for someone who has played a lot of other games like Descent and now that they're familiar with the genre is looking for something innovative and dense. Because that is Gloomhaven all over.
With a lot to learn from level-dependent decks, health and exhaustion to keep an eye on, alternating card use; this isn't a game for the player who faints at the sight of a rules booklet. But if you want to meet the daddy of dungeon-crawlers then look no further than this mammoth game.
Already know this game? Check out our list of games like Gloomhaven!
Mage Knight, just like Gloomhaven, is very complex and not something I would say the board game novice needs to pick up. But for those who have been around the block, this is a very fun take on the roleplay-inspired board game like Descent.
In this game, one to four players will each be working as an extraordinary Mage Knight that will be bolstering armies to fight monsters, collect artefacts and conquer lands as they travel.
Players will have decks filled with their abilities that they can try to grow throughout the run of the game. They will be able to buy their characters spells and influence and level up as they go along the way as well.
Players can choose from many different scenarios to play and different ones are either cooperative or combative so there's room for tricksy behaviours or group tactics.
Almost the opposite of the last suggestion now, we have Welcome to the Dungeon, a very simple game that allows two to four players to gamble with their lives.
Instead of making their way through the dungeon step by step in this push-your-luck game players will build up the difficulty of the dungeon by each placing different monsters down to make a dungeon pile and will lower it along with their own strength by giving up their own weapons as sacrifices.
The bravest player, or the unluckiest, will be the last to not pass on the betting pool and they will then have to take on the entire dungeon pile with whatever equipment they have left and their character card.
This can be one nail-biting and hilarious game like Descent, but it's speed and easy ruleset make it a great game to pull out at parties.
You probably thought a list of board games like Descent was already nerdy enough, huh? For fans of both science-fiction and strategy board games, there could not be a better choice than Star Wars: Imperial Assault.
In this one, players will take on the commanding role of the Galactic Empire ready to destroy and the one to four remaining players who are heroes of the Rebel Alliance trying to make their way through a campaign of separate storylines.
There is an option to play the Skirmish Mode which is a one-shot and a smaller number of players but either way, it's an exciting adventure for anyone who's especially into either franchise or its genre.
Finally, I couldn't finish this list without suggesting the game that started it all. First coming out in 2005, Descent was actually based on the Doom: Board Game based on the video game and very well-received.
It has continued to be improved on from that starting point to Descent and then to a number of new editions of Doom that have become well-loved today.
In this game, much like in the board game Descent, one player will be in charge of the demons, working to create a more dangerous board for the other players who will be taking control of marines. This game can have two to five players take part in this epic battle between the legions of hell and this small battalion of marines.
The game also has some amazing design choices and very high-quality product as you'd expect from such a famous IP.
All of these suggestions for board games like Descent bring together the game's greatest features. From the cooperative nature, or semi-cooperative in the case of a Dungeon master, to the levelling, to the quickfire thinking needed to finish these dungeons alive. It really is an amazing time for people looking to pick up board games as excellent dungeon-crawlers just keep being released.
Not to mention all the other Descent titles that are out there for you to discover like Descent: Tomb of Ice, Descent: Sea of Blood, Descent: The Road to Legend and Descent: Altar of Despair. I know that you'll be pleased with any one of these choices so get to picking!
Are you looking for more games like Descent? Have any recommendations that didn't make the list? Let us know in the comments!
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About Khadija Osman
Khadija is a writer for Games Like This One. She is a mid-twenties, mid-crisis, middle-of-the-pack writer from London who loves board games and writing about them. She made her start in copywriting and is excited to continue sharing her tabletop opinions with you for as long as you'll let her.
Welcome to Games Like This One! We're passionate about helping you find your next game to play – just like the one you recently finished!
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The temple is located on Fuang Nakhon Road near Wat Pho. Built by King Rama V in 1869, it was in keeping with tradition that each monarch constructed a temple to mark his reign. The temple is a mixture of local and western styles, showing an awakening interest in new ideas and a desire to experiment with them.
Bangkok Siam Exchange BTS Station to Wat Chana Songkhram Rachawora Mahawiharn.


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Home > Missing Children > Child Rescue Alert
By: Beth Morrisey MLIS - Updated: 11 Aug 2010 | *Discuss
The Child Rescue Alert initiative was launched in 2006 by the UK Police National Missing Persons Bureau. The Alert system works by posting immediate newsflashes – sometimes breaking into television or radio programmes if need be – to notify the public that a young person has been abducted and is believed to be at risk of physical harm or even death.
The Alert system is not used often, and the criteria for issuing such a newsflash is quite stringent, as the Bureau does not want the public to become numb to these warnings. Instead, the alerts are saved for cases in which the welfare of the child, and not simply the location of the child, is truly in question.
Criteria for Issuing a Child Rescue Alert
There are four key criteria which must be met before a Child Rescue Alert can be issued. Though the criteria are subjective, a police officer of at least the grade of a superintendent will make the final decision of whether or not information will be released for a newsflash. These key criteria include:
That the child is believed to be under the age of 18.
That the child is believed to have been kidnapped or Otherwise Abducted.
That the child is believed to be at great risk of serious harm or death.
That there is enough information available that the public may be able to help.
A Child Rescue Alert News Flash
When a Child Rescue Alert is issued, the newsflash will be carried by any radio or television station with which the individual police force has an agreement. The alerts will be transmitted every 15 minutes for a 4 hour period in the hope that a member of the public will recognise something about the case and contact the police. Common information transmitted during an alert includes:
A description of the child.
A photograph of the child.
The location of the believed abduction.
The nature of the believed abduction.
A description of the believed abductor.
A photo or CCTV footage of the believed abductor.
A description or other details of the vehicle used in the believed abduction.
Child Rescue Alert Success
The Child Rescue Alert is a relatively new initiative in the UK, however, it is based on a system known as the Amber Alert in the United States. The Amber Alert, which has successfully recovered over 300 children since 1996, was launched after the abduction and murder of a young girl named Amber in Texas. While there were no alerts to help young Amber, the Amber Alert has helped bring hundreds of other young children home since. Police forces across the UK are hopeful that similar successes will result in the UK.
Further Information about the Child Rescue Alert
For those interested in obtaining further information about the Child Rescue Alert in the UK, the Police National Missing Persons Bureau has set up a website at www.childrescuealert.com, or the Bureau can be contacted at 01256 602 979. For further information about a particular alert, however, please pay attention to the newsflashes in your area and follow their particular instructions for reporting information on a specific missing child.
The Child Rescue Alert has been hailed as a groundbreaking resource for keeping children safe even after they have been kidnapped or abducted. If an alert is broadcast in your area, please pay attention to the information and report any details that you may have. A child's life may depend upon it.


A meeting of the Consumer Committee in Brussels last Friday (September 21) has resulted in the EU expressing support for a consumer voice in the reform of the Common Agricultural Policy (CAP). Franz Fischler, Commissioner for Agriculture, Rural Development and Fisheries, stated: "Everybody has strong views on where we have to go in food production and agriculture in the future and we cannot ignore that the discussion on food safety and food quality touches on the basic issues of our lives. It raises the question of what we eat every day.


The Cedar Rapids Area Chamber of Commerce will host its Hiawatha Business Summit at Kirkwood Linn County Regional Center. You're invited to a morning of networking with the Hiawatha business community! This morning event includes networking and a short program. The Hiawatha Economic Development Corporation, the City of Hiawatha and the Cedar Rapids Metro Economic Alliance collaborate to bring you this event. For more information, visit the Cedar Rapids Area Chamber of Commerce's website at https://www.cedarrapids.org.


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First 831 model with a deep blue dial from CITIZEN Series 8 collection of modern and sporty mechanical watches
CITIZEN is releasing a new model under its CITIZEN 8 brand of mechanical watches that combine practicality and functionality with modern and sporty designs. This new 831 mechanical model has an eye-catching deep blue dial – a first for the brand - and it also features CITIZEN's Caliber 9051 movement for enhanced magnetic resistance.
CITIZEN Series 8 831 Mechanical Model NB6010-81L
This new model combines a stylish octagonal case with a simple but eye-catching blue dial as the latest addition to the line-up of 831 Mechanical watches that deliver practicality, functionality, and modern designs. The sleek lines of the elegant octagonal case accent the deep blue dial with a starburst pattern that produces subtle changes in light and shade depending on the viewing angle. The result is simple yet bold, with a refreshingly modern look.
The watch is designed to be both practical and functional, and it is equipped with CITIZEN's Caliber 9051 movement for enhanced magnetic resistance. This helps to maintain the accuracy of the watch by reducing the effects of magnetic fields produced by smartphones and tablets in our digital society. The movement is just 4.1 mm thick, resulting in a thinner case and an extremely comfortable fit.
With their modern, dynamic styling, Series 8 watches suit any scene, from business occasions to casual days off. The brand's mechanical watches provide a range of practical features, including magnetic resistance to protect them against magnetic fields in everyday life, as well as a supremely comfortable fit.
■ The first Series 8 831 Mechanical watch with a blue dial
This new watch combines an octagonal case with a simple but eye-catching deep blue dial as the latest addition to the line-up of 831 Mechanical watches that deliver practicality, functionality, and modern designs. The sleek, streamlined hands extend toward the outer edge of the dial, and the detailed indices create a dynamic look.
■ Caliber 9051 movement with enhanced magnetic resistance
Enhanced magnetic resistance helps to maintain the accuracy of the watch by reducing the effects of magnetic fields produced by smartphones and tablets.
Series 8 / 831 Mechanical
Case / Band
Sapphire glass with anti-reflective coating
Case diameter / thickness
40.0mm / 10.1mm( design specification only )
Cal.9051 / Automatic & Manual winding / Accuracy of average -10~+20sec/day */ Running time of approx. 42 hours (when fully wound)/ Frequency : 28,800 vph / 24 jewels / Water resistant to 10 BAR
*Accuracy measured in a stationary position. Due to the characteristics of mechanical watches, daily accuracy may be outside stated figures due to variations in conditions of use (including length of time worn, the position of the watch, wrist movement, and the winding of the spring
<About CITIZEN WATCH>
CITIZEN WATCH is a true manufacture d'horlogerie with a comprehensive manufacturing process that extends from creating a watch's individual components to its final assembly. The company operates in more than 140 countries and regions around the world. Since its founding in 1918, CITIZEN has held the belief of "Better Starts Now" — that is, no matter who you are and what you do, it is always possible to make something better, and now is the time to start doing it. Based this belief, we have created watches, invented and improved technologies and explored the future of watches such as our proprietary light-powered Eco-Drive technology and state-of-the-art satellite-synchronized timekeeping.
*The final design and specifications are subject to change without notice.
*The availability, specifications, and related information, such as the product code for the models featured in this site, may vary by markets. If you would like to know more details, including the availability of these products in your area, please select your local website from our global site.


The Ethereum price in the back end of trading on Friday is seen trading with minor gains of some 1.33%.
ETH/USD price action remains within the confinements of a bearish pennant structure via the daily frame.
The Ethereum price in the latter part of trading on Friday was holding very minor gains of some 1.40%. Bulls bounced back across the cryptocurrency market today, after their absence during Thursday, where much cooling was observed.
ETH/USD price action continues to move within a bearish pennant pattern structure, subject to potential downside risks. The formation of this has been taking shape since the back end of February, after the big drop on 24th February.
Technically, given the above-mentioned pattern formation, risks do appear to be pointing to the downside. Key near-term underlying support is seen at $135, the lower tracking trend line of the bearish pennant. Resistance to the upside is observed just above at $141, a daily break and closure above would rule out this bearish structure.


Flavia Liz – Personal Guide – Flavia Liz é considerada uma lenda urbana pela Condé Nast Traveller U.K.'s 'The Experts' series.
Condé Nast Traveller U.K.'s 'The Experts' series."
Surprise yourself with remarkable experiences!
Connect with unique people and places!
Get enchanted with the financial and cultural capital of Brazil!
The Flavia Liz tours provide special moments and unique interactive experiences with the culture, history and with anything that is interesting and exclusive in the city of São Paulo.
This is the perfect tour for those who do not have a lot of time in the city. It will take the visitor to see the most important attractions of São Paulo.
A look at the urban interventions of world-renowned Brazilian artists that re-draw the personality of the metropolis. The tour may include a chat with an artist, a visit to a street art gallery or to an important social project.
Have access to the exclusive designer ateliers of important designers that are part of the avant-garde fashion world and full of Brazilian creativity.
Go to the best bird watching spots, from swamps to the Atlantic Forest, all inside the urban perimeter.
Conheça a diversidade de cenários dos Parques da cidade – dos naturais com áreas verdes intensas de Mata Atlântica aos urbanos e planejados por paisagistas e arquitetos de renome internacional.
See for yourself the reality of the favelas and their cultural and artistic expressions which paint an authentic portrait of the most humble population of the city of São Paulo.
Na favela de Paraisópolis mora Estevão na sua Casa de Pedra. Conhecido como o "Gaudi brasileiro", ele é um artista singular que tem o poder de expandir a visão do visitante sobre o uso de objetos do dia a dia. Outro artista de Paraisópolis é Seu Berbela que transforma magicamente elementos da sua oficina de carros em esculturas de surpreendente delicadeza e movimento.
Integrate yourself to the backstreets and alleys of Monte Azul, a model favela where the community adopted Rudolph Steiner?s philosophy of anthroposophy, Steiner also created the Waldorf pedagogy.
Dive into the exciting universe of Brazilian soccer, where you will discover the sport's history, its curiosities, the trophy cases and, even better, get access to the dressing rooms and fields that were immortalized by the soccer pros who made this the most popular sport on Earth.
A tour around the city that shows the utmost expression of modern architecture and designs, preserved throughout the streets and avenues of São Paulo, on the facades and interiors of private residences and public buildings.
Considered one of the greatest world capitals of gastronomy, São Paulo offers flavors from the most exquisite and luxurious modern culinary to our own regional cuisine with some very peculiar spices.
Clothing, accessories, jewelry, arts and crafts, design and decoration objects, furniture, collector?s items, etc. Depending on the profile, the client will be taken to places with the amazing shopping opportunities that São Paulo offers.
Perfect for collectors and lovers of art, this tour opens the doors to private collections, contemporary art galleries, museums and ateliers. It also includes bits of academic and modern arts.
Visits to Buddhist temples, mosques, synagogues, umbanda squares and specially, baroque churches: a tour to understand the religious syncretism in the most catholic country in the world.
From houses of samba and chorinho shows to the most modern dance clubs – the metropolis has nocturnal attractions that inspire and motivate all kinds of people.
An encounter with Brazil?s history in the beautiful coffee farms in the city?s outskirts, only one hour from the capital, where the city and country?s economic wealth was built in the 19th century.
To head down to the beaches from the city, one will experience the beautiful forest-filled mountains and fantastic views of the Atlantic Ocean, until reaching the beautiful tropical beaches with white sand, just one hour away from the capital.
The Zapping SP tour can focus on interesting activities for kids and be a fun way for the little ones to get to know the history of the city and of the country. Another option can include visits to natural parks and theme parks.
Try a portion of each of the Flavia Liz tours and gain a comprehensive vision of São Paulo culture.
Created for those who plan to move to São Paulo, this presentation tour of the city identifies the advantages and disadvantages of each area of the city to the expat.
Would you like to do a tour that is not on the menu? We will create a tour that is 100% personalized for you.
A) Touristic presentantion of the city with focus on potential areas for business development.
B) In partnership with experienced market researchers, she is a facilitator between foreigners that want to invest in the city or in the country, opening businesses, developing social projects or partnerships along with other possibilities, and the governmental entities, institutions, commerce chambers, and Brazilian companies.
Customized cultural itineraries, guided by Flavia Liz Di Paolo, a specialist in experiential tourism in the city of São Paulo. They offer the participants contact with the less publicized facet of the city, prompting a surprised and delighted reaction to the financial capital of Latin America. They can be made on foot, by subway train, bicycle, executive car, armored car, helicopter, van, microbus or bus.
Recommended for visitors and local inhabitants who wish to have an unforgettable experience and to sharpen their awareness of cultural and historical aspects of São Paulo.
The participants have access to significant contents, which bring about an understanding of tourism equipment, collections of the most important museums in São Paulo, comprehension of the different profiles of visitors and understanding of the concept of luxury tourism. The learning allows a fresh understanding of objectives and desires, enabling an assertive and remarkable relationship with their clients.
Recommended for concierges, receptionists, marketing and sales employees of hotels, luxury tourism agencies and operators and transport companies working with executive cars, vans, microbuses and buses.
Minimum hours of lessons are 5h of practice and 3h of theoretical content.
Lectures focused on the essence of luxury tourism in São Paulo, addressing topics such as architecture, design, fashion, graffiti, favela, gastronomy and contemporary art, among others. They allow the participants to gain a unique perspective through values, feelings and knowledge.
Recommended for audiences interested in culture and the luxury market.
Exclusive consulting made to measure for each case; specialized in luxury tourism and the tourism potential of the city of São Paulo. Offers specific knowledge that generates and enhances the understanding of the luxury market and its audience.
Recommended for organizational structures of all sizes related to tourism in the city of São Paulo, which seek to optimize the understanding and delivery of services to the luxury market.
Private or group lessons in professional education and development, designed to bring about an understanding of the beauty of the history of São Paulo and general culture with topics that include architecture, design, fashion, graffiti, favela and contemporary art, among others.
Recommended for audiences interested in acquiring and enhancing knowledge of the city of São Paulo.
Development of customized content, focused on the luxury market, for printed, digital and audiovisual publications, such as newspapers, magazines, websites, blogs, TVs and special events.
Recommended for organizations with a demand for differentiated material, which wish to build a closer relationship with specific audiences.
Flavia Liz is simply one of the best professionals in her sector in the entire world!
Wow! I have never experienced a city this magnanimous. Your city is breathtaking, very beautiful an warm people.
Te agradeço por ter me proporcionado tamanho emoção, não imaginava que após ter vistos os melhores museus do mundo pudesse me emocionar tanto conhecendo uma obra dentro de uma favela. Não foi só emocionante, foi uma grande lição de vida! Muito obrigada cara!
Your perfect English, excellent driving skills, and bottomless local knowledge was amazing. It was so nice to have you adapt the tour to our particular interests. While before our trip, we had some initial hesitance to include Sao Paulo on our Brazil itinerary, after our day with you, we now hope to return again and give Sao Paulo the extra time it deserves.
As we visited her city, Flavia Liz added significant historical details, which allowed us not to simply 'see' a building, but know its importance to both the city but country. Flavia had so many contacts! Everywhere we went, people came up and greeted her by name and a smile! She is such a nice person, generous with her time. I can't wait to return and have Flavia Liz show us more! Thanks again for a great day in Sao Paulo!


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Beltiug (în maghiară: Krasznabéltek, în germană: Bildegg) este o comună în județul Satu Mare, Transilvania, România, formată din satele Beltiug (reședința), Bolda, Ghirișa, Giungi, Rătești și Șandra.

Comuna Beltiug este situată în sudul județului Satu Mare în zona de contact a Câmpiei Crasnei cu versanții vestici ai dealurilor Codrului, fiind străbătută de râul Crasna , la o distanță de 35 km de municipiul Satu Mare pe drumul E81, care leagă municipiile Satu Mare, trecând prin municipiul Zalău cu municipiul Cluj-Napoca. Comuna Beltiug se învecinează: la nord cu orașul Ardud, la est și sud-est cu comuna Socond, la sud cu comuna Supur, la vest cu Comuna Craidorolț și cu Comuna Acâș.
Suprafața totală a comunei este de 11703 ha, intravilan 897 ha iar în extravilan de 10806 ha.

Relief
Din punct de vedere geologic comuna Beltiug face parte din unitatea tectonică de bordure a Depresiunii Panonice, unitate ce s-a individualizat în urma evenimentelor geologice din teritoriul superior în cuaternar. Relieful comunei se caracterizează prin peisajul dat de contactul dintre Piemontul Codrului și zona de câmpie a Crasnei.

Câmpia Crasnei face parte dintr-o unitate geografică mai mare, care este Câmpia Someșului. Este brăzdată de râul cu același nume, Crasna, și ocupă partea centrală și nordică din regiunea hotarului comunei. Relieful actual al șesului s-a conturat în urma retragerii apelor Mării Panonice, care intră asemenea unui golf până la poalele Culmii Codrului și la marginea dealurilor Silvaniei. În urma retragerii apelor mării a rămas un mâl la suprafața scoarței terestre, care a fost ridicat de vânturi și depus sub formă de loess (lut gălbui). Acest strat poate fi găsit și în zilele noastre și observat la forarea fântânilor, la adâncime de 0,5-1 metri, suprapus peste straturi de nisip și pietriș, ceea ce determină dificultățile de cultivare a acestor soluri. Câmpia ocupă partea vestică și sudică a comunei.

Culmea Codrului se întinde până la marginea estică a hotarului comunei Acâș și este cunoscută sub numele de Dealurile Beltiugului și Dealurile Dobrei. Suprafața deluroasă este o zonă în care munții s-au prăbușit și stau înecați sub sedimente groase, cu excepția spinării puțin înalte de șisturi cristaline - Culmea Codrului , cu partea vestică care trece destul de brusc în Câmpia Crasnei .
La poalele masivului cristalin al Culmii Codrului se așterne o zonă de dealuri piemontane alcătuite dintr-un material foarte complex de nisipuri intercalate cu pietrișuri, argile și marne. O parte din aceste dealuri sunt împădurite, o parte ocupate cu pajiști naturale, culturi agricole și plantații de pomi fructiferi și viță de vie. În zona colinară s-au semnalat slabe alunecări de teren fiind luate măsuri de combatere a eroziunii prin plantări de pomi.

Clima

Teritoriul comunei Beltiug se încadrează în sectorul climatic al Câmpiei de Vest. Aceasta se caracterizează printr-o relativă uniformitate teritorială. Climatul corespunde tipului de climă temperat – continentală moderată, cu un regim termic cald ( veri călduroase și ierni ceva mai blânde decât în restul țării ). Verile sunt moderate 19 -20̊ C iar iernile lungi și reci ( -1̊… -2̊ C), temperatura medie anuală fiind cu 1,2̊ până la 1,3̊ mai mică decât în sud. Numărul zilelor de îngheț de iarnă este de numai 50-60 pe an, dar se mai ivesc înghețuri târzii de primăvară, de obicei la sfârșitul lunii aprilie. Temperaturile maxime absolute nu depășesc 40̊ C iar cele minime nu scad sub – 30̊ C. Climatul oferă condiții convenabile pentru culturile agricole.

Precipitații
Nebulozitatea care este relativ redusă 5, 5 condiționează un număr însemnat de zile însorite (70-75) însumând circa 2000 de ore pe an. Cantitatea anuală de precipitații variază în jur de 600 mm, din care aproape jumătate ( 45,6 % ) cad la sfârșitul primăverii și vara. Perioada maximă apare în luna mai și iunie ( circa 25 % ). Numărul anual de zile cu precipitații este de 120 – 130 pe an, din care 20 sânt cu ninsoare. De obicei prima zăpadă cade în prima decadă a lunii decembrie, dar nu se menține.

Presiunea atmosferică și regimul vânturilor
Dată fiind altitudinea mică ( aprox. 200 m ), presiunea atmosferică înregistrează valori ridicate în jur de 1000 mb, cu mici oscilații de la un anotimp la altul. Regimul eolian indică o predominanță din sectorul nord-vest ( 75 – 80 % ), primăvara și vara fiind mai frecvente vânturile vestice, cu maxima în iulie, iar toamna și iarna cele estice și nord-estice. Vânturile afectează în mare măsură regimul precipitațiilor, în special cele de vest care generează ploile de primăvară și vară. Predomină de obicei vânturile din sectorul nord-vestic, care favorizează frecventele invazii de aer rece și umed, atras de cicloni proveniți din zona Atlanticului.
Zona Beltiug dispune de un mediu climatic relativ omogen, cu o mică, nuanță între climatul de câmpie și ce le deal.

Clima localiății Beltiug se caracterizează ca fiind una temperat continentală cu un regim termic mai cald, cu desprimăvăriri timpurii și cu precipitații relativ reduse, fiind o climă favorabilă majorității culturilor agricole precum și pentru pomi fructiferi și viticultură.

Hidrografia

La sud- vest de localitatea Beltiug curge râul Crasna, care izvorăște în Munții Apuseni și se varsă în Tisa, iar la nord-est de aceasta localitate își are cursul râul Bolda. La 2,8 km de localitatea Beltiug, se află forajul de apă termală ( sonda FB 1 ) cu apă minerală iodurată, bicarbonatată, clorurată, sodică, hipermetală, hiptonă, cu o temperatură de 65̊ C.
Alimentarea cu apă a comunei Beltiug se face din fântâni proprii ale populației.

Flora și fauna

Răspândirea vegetației și faunei este strâns legată de climă și relief.
Vegetația este arborescentă și pădurile sunt reprezentate prin stejar ( Quercus robur), carpen (Carpinus betulus), frasin (Fraxinus), plop ( Populus), jugastru (Acer campestre) și altele. Pajiștile întâlnite la pășuni și fânețe au plante caracteristice precum păiușul (Festuca porcii), obsiga (Bromus tectorium), firuța bulboasă (Poa bulbosa), ghizdeiul ( Lotus corniculatus), iar în locurile mlăștinoase, specii de rogoz, pipirig și stuf. Este evident impactul omului în acest areal, deoarece a desființat aproape total arealul silvostepei, cat și cel al pădurii, înlocuindul cu culturi. Au rămas doar câteva petice de stepă secundară sau pâlcuri de pădure.

Fauna nu este pre bogată dar merită să fie amintite insecte ce se găsesc în număr mare ca libelula, greierii, cosașul verde (Tettigonia viridissima), lăcusta călătoare, gândacul de colorado (Leptinotarsa decemlineata) ( care provoacă pagube culturilor de cartofi) , cărăbușul de mai(Melolontha melolontha), rădașca (Lucanus cervus), cărăbușul negru care este prezent în păduri, precum și o mare varietate de fluturi. Păsările au devenit mai puține datorită asanării mlaștinilor și tăierii arbuștilor de pe câmp. Se întâlnesc mierla, pițigoiul de grădină, cioara de semănături, vrabia, guguștiucul (Streptopelia decaocto) și rar rața și gâsca sălbatică precum și fazan (Phasianus colchicus), prepelița (Coturnix coturnix) și potârnichea (Perdix perdix). Avifauna este reprezentată de o serie de păsări răpitoare, mai ales care se hrănesc cu rozătoare precum: vânturelul roșu ( Falco tinnunculus), șoimul rândunelelor ( Falco subbuteo), uliul porumbar și uliul păsărar. Dintre răpitoare de noapte: cucuveaua (Athene noctua), ciuful de pădure ( Asio otus) etc.
Dintre mamifere putem aminti căprioara ( Capreolus capreolus), mistrețul (Sus scrofa), vulpea ( Canis vulpes), iepurele de câmp ( Lepus europaeus), dihorul (Mustela putorius) - în păduri. Pe terenurile arabile și pășuni sunt caracteristice o serie de rozătoare ca: popândăul (Citellus citellus), hârciogul ( Cricetus cricetus), șoarecele de câmp ( Microtus arvalis), ariciul, șoarecele comun ( Buteo buteo) schimbat iarna de șorecarul încălțat ( Buteo Lagopus).
Dintre amfibieni putem aminti broasca râioasă brună ( Bufo bufo) și broasca râioasă verde ( Bufo viridis).

Demografie

Conform recensământului efectuat în 2011, populația comunei Beltiug se ridică la  de locuitori, în scădere față de recensământul anterior din 2002, când se înregistraseră  de locuitori. Nu există o etnie majoritară, locuitorii fiind români (35,1%), maghiari (31,54%), romi (18%) și germani (11,4%). Pentru 3,53% din populație, apartenența etnică nu este cunoscută.
Din punct de vedere confesional, nu există o religie majoritară, locuitorii fiind romano-catolici (37,61%), ortodocși (32,06%), reformați (18,25%), penticostali (5,27%) și creștini după evanghelie (1,46%). Pentru 3,59% din populație, nu este cunoscută apartenența confesională.

Politică și administrație
Comuna Beltiug este administrată de un primar și un consiliu local compus din 13 consilieri. Primarul, , de la , este în funcție din . Începând cu alegerile locale din 2020, consiliul local are următoarea componență pe partide politice:

Note

Bibliografie
• Ghițescu, L. (martie 1972) Județul Satu Mare - condiții naturale, Pagini Sătmărene

• Memoriu tehnic pentru Avizul Natura 2000- "Parcul Balnear în Regiunea de Nord-Vest – localitățiile Tășnad, Beltiug și Marghita", locația Beltiug

Beltiug

redpajama_set_name: RedPajamaWikipedia